design & analysis of a tensioner for a belt-driven integrated starter

DESIGN amp ANALYSIS OF A TENSIONER FOR A

BELT-DRIVEN INTEGRATED STARTER-

GENERATOR SYSTEM

OF MICRO-HYBRID VEHICLES

by

Adebukola O Olatunde

A Thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

copy Copyright by Adebukola O Olatunde 2008

ii

ABSTRACT

DESIGN AND ANALYSIS OF A TENSIONER FOR A BELT-DRIVEN INTEGRATED

STARTER-GENERATOR SYSTEM OF MICRO-HYBRID VEHICLES


Master of Applied Science


University of Toronto 2008

The thesis presents the design and analysis of a Twin Tensioner for a Belt-driven Integrated

Starter-generator (B-ISG) system The B-ISG is an emerging hybrid transmission closely

resembling conventional serpentine belt drives Models of the B-ISG system‟s geometric

properties and dynamic and static states are derived and simulated The objective is to reduce

the magnitudes of static tension in the belt for the ISG-driving phase A literature review of

hybrid systems serpentine belt drive modeling and automotive tensioners is included A

parametric study evaluates tensioner parameters with respect to their impact on static tensions

Design variables are selected from these for an optimization study The optimization uses a

genetic algorithm (GA) and a hybrid GA Results of the optimization indicate the optimal

system contains spans with static tensions that are significantly lower in magnitude than that of

the original design Implications of the research on future work are discussed in closing

iii

A testament unto the LORD God lsquowho answered me in the day of my distress and was with me in

the way which I wentrsquo

To my parents Joseph and Beatrice for your strength and persistent prayers

To my siblings Shade Charlene and Kevin for being a listener an editor and a relief when I

needed it

To my friends Samantha Esther and Yasmin who kept me motivated

amp

With love to my sweetheart Nana whose patience support and companionship has made life

sweeter

iv

ACKNOLOWEDGEMENTS

I would like to express deep gratitude to Dr Jean Zu for her guidance throughout the duration of

my studies and for providing me with the opportunity to conduct this thesis

I wish to thank the individuals of Litens Automotive who have provided guidance and data for

the research work Special thanks to Mike Clark Seeva Karuendiran and Dr Qiu for their time

and help

I thank my committee members Dr Naguib and Dr Sun for contributing their time to my

research work

My sincerest thanks to my research colleague David for his knowledge and support Many

thanks to my lab mates Qiming Hansong Ali Ming Andrew and Peyman for their guidance

I want to especially thank Dr Cleghorn Leslie Sinclair and Dr Zu for the opportunities to

teach These experiences have served to enrich my graduate studies As well thank you to Dr

Cleghorn for guidance in my research work

I am also in debt to my classmates and teaching colleagues throughout my time at the University

of Toronto especially Aaron and Mohammed for their support in my development as a graduate

researcher and teacher

v

CONTENTS

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES xi

LIST OF SYMBOLS xvi

Chapter 1 INTRODUCTION 1

11 Background 1

12 Motivation 3

13 Thesis Objectives and Scope of Research 4

14 Organization and Content of Thesis 5

Chapter 2 LITERATURE REVIEW 7

21 Introduction 7

22 B-ISG System 8

221 ISG in Hybrids 8

2211 Full Hybrids 9

2212 Power Hybrids 10

2213 Mild Hybrids 11

2214 Micro Hybrids 11

222 B-ISG Structure Location and Function 13

2221 Structure and Location 13

2222 Functionalities 14

23 Belt Drive Modeling 15

24 Tensioners for B-ISG System 18

241 Tensioners Structures Function and Location 18

242 Systematic Review of Tensioner Designs for a B-ISG System 20

25 Summary 24

vi

Chapter 3 MODELING OF B-ISG SYSTEM 25

31 Overview 25

32 B-ISG Tensioner Design 25

33 Geometric Model of a B-ISG System with a Twin Tensioner 27

34 Equations of Motion for a B-ISG System with a Twin Tensioner 32

341 Dynamic Model of the B-ISG System 32

3411 Derivation of Equations of Motion 32

3412 Modeling of Phase Change 41

3413 Natural Frequencies Mode Shapes and Dynamic Responses 42

3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44

3415 ISG Pulley Driving Torque Acceleration and Displacement 46

3416 Tensioner Arms Dynamic Torques 48

3417 Dynamic Belt Span Tensions 49

342 Static Model of the B-ISG System 49

35 Simulations 50

351 Geometric Analysis 51

352 Dynamic Analysis 52

3521 Natural Frequency and Mode Shape 54

3522 Dynamic Response 58

3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61

3524 Tensioner Arm Torque Requirement 62

3525 Dynamic Belt Span Tension 63

353 Static Analysis 66

36 Summary 69

Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71

41 Introduction 71

42 Methodology 71

43 Results and Discussion 74

431 Influence of Tensioner Arm Stiffness on Static Tension 74

432 Influence of Tensioner Pulley Diameter on Static Tension 78

433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80


vii

44 Conclusion 92

Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95

51 Optimization Problem 95

511 Selection of Design Variables 95

512 Objective Function amp Constraints 97

52 Optimization Method 100

521 Genetic Algorithm 100

522 Hybrid Optimization Algorithm 101


531 Parameter Settings amp Stopping Criteria for Simulations 101

532 Optimization Simulations 102

533 Discussion 106

54 Conclusion 109

Chapter 6 CONCLUSION AND RECOMMENDATIONS111

61 Summary 111

62 Conclusion 112

63 Recommendations for Future Work 113

REFERENCES 116

APPENDICIES 123

A Passive Dual Tensioner Designs from Patent Literature 123

B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138

C MathCAD Scripts 145

C1 Geometric Analysis 145

C2 Dynamic Analysis 152

C3 Static Analysis 161

D MATLAB Functions amp Scripts 162

D1 Parametric Analysis 162

D11 TwinMainm 162

D12 TwinTenStaticTensionm 168

D2 Optimization 168

D21 OptimizationTwinm - Optimization Function 168

viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

21 Passive Dual Tensioner Designs from Patent Literature

31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span

32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot

33 Geometric Results of B-ISG System with Twin Tensioner

34 Data for Input Parameters used in Dynamic and Static Computations

35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG

Serpentine Belt Drive with a Single Tensioner


Serpentine Belt Drive with a Twin Tensioner

41 Initial Values Increments and Ranges for Parameters of Twin Tensioner

51 Summary of Parametric Analysis Data for Twin Tensioner Properties

52a GA Optimization Results for Twin Tensioner Parameters and Objective Function

52b Computations for Tensions and Angles from GA Optimization Results

53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function

53b Computations for Tensions and Angles from Hybrid Optimization Results

54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective

Function

54b Computations for Tensions and Angles from Non-Weighted Optimizations

x

55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a

Twin Tensioner

56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a

Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions

31 Schematic of the Twin Tensioner

32 B-ISG Serpentine Belt Drive with Twin Tensioner

33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys

34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System

35 Free Body Diagram for Non-Tensioner Pulleys

36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid

Bodies

36b ISG Driving Case First Mode Responses

36c ISG Driving Case Second Mode Responses

37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive

Rigid Bodies

37b Crankshaft Driving Case First Mode Responses

37c Crankshaft Driving Case Second Mode Responses

38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)

39 ISG Pulley Dynamic Response (for ISG driven case)

310 Air Conditioner Pulley Dynamic Response

311 Tensioner Pulley 1 Dynamic Response

xii


313 Tensioner Arm 1 Dynamic Response


315 Required Driving Torque for the ISG Pulley

316 Required Driving Torque for the Crankshaft Pulley

317 Dynamic Torque for Tensioner Arm 1


319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension

320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension

321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension

322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension

323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension

324 B-ISG Serpentine Belt Drive with Single Tensioner

41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1

amp 2

41b Regions 1 and 2 in Cartesian Space

42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)

43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)


xiii

45 Parametric Analysis for Pulley 1 Diameter D3 (m)


47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span

Tension in Crankshaft Driving Case

48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span



Tension in ISG Driving Case











51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of

Tensioner Arms 1 and 2

A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos

EP1420192-A2 and DE10253450-A1

A2a First of four proposed designs (and its various configurations) by Bosch GMBH

corresponding to patent no WO0026532-A1

A2b Second of four proposed designs by Bosch GMBH corresponding to patent no

WO0026532-A1

A2c Third of four proposed designs (and its various configurations) by Bosch GMBH


A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH


A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1

A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2

xiv

A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A

A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-

A1

A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1

A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1

A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1

A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1


A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1


A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-

A1

A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-

A1and WO2006108461-A1

A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no

US20010007839-A1

A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp

corresponding to patent no JP2005083514-A

A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2

A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A

A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos

EP1658432 and WO2005015007

B1 Single Tensioner B-ISG System

B2 Free-body Diagram of ith Pulley

xv

B3 Free-body Diagram of Single Tensioner

C1 Schematic of B-ISG System with Twin Tensioner

C2 Possible Contact Points

xvi

LIST OF SYMBOLS

Latin Letters

A Belt cord cross-sectional area

C Damping matrix of the system

cb Belt damping

119888119894119887 Belt damping constant of the ith belt span

119914119946119946 Damping matrix element in the ith row and ith column

ct Damping acting between tensioner arms 1 and 2

cti Damping of the ith tensioner arm

DCS Diameter of crankshaft pulley

DISG Diameter of ISG pulley

ft Belt transition frequency

H(n) Phase change function

I Inertial matrix of the system

119920119938 Inertial matrix under ISG driving phase

119920119940 Inertial matrix under crankshaft driving phase

Ii Inertia of the ith pulley

Iti Inertia of the ith tensioner arm

119920120784120784 Submatrix of inertial matrix I

j Imaginary coordinate (ie (-1)12

)

K Stiffness matrix of the system

xvii

119896119887 Belt factor

119870119887 Belt cord stiffness

119896119894119887 Belt stiffness constant of the ith belt span

kt Spring stiffness acting between tensioner arms 1 and 2

kti Coil spring of the ith tensioner arm

119922120784120784 Submatrix of stiffness matrix K

Lfi Lbi Lengths of possible belt span connections from the ith pulley

Lti Length of the ith tensioner arm

Modeia Mode shape of the ith rigid body in the ISG driving phase

Modeic Mode shape of the ith rigid body in the crankshaft driving phase

n Engine speed

N Motor speed

nCS rpm of crankshaft pulley

NF Motor speed without load

nISG rpm of ISG pulley

Q Required torque matrix

qc Amplitude of the required crankshaft torque

QcsISG Required torque of the driving pulley (crankshaft or ISG)

Qm Required torque matrix of driven rigid bodies

Qti Dynamic torque of the ith tensioner arm

Ri Radius of the ith pulley

T Matrix of belt span static tensions

xviii

Trsquo Dynamic belt tension matrix

119931119940 Damping matrix due to the belt

119931119948 Stiffness matrix due to the belt

Ti Tension of the ith belt span

To Initial belt tension for the system

Ts Stall torque

Tti Tension for the neighbouring belt spans of the ith tensioner pulley

(XiYi) Coordinates of the ith pulley centre

XYfi XYbi XYfbi

XYbfi Possible connection points on the ith pulley leading to the ith belt span

XYf2i XYb2i

XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span

Greek Letters

αi Angle between the datum and the line connecting the ith and (i+1)th pulley

centres

βji Angle of orientation for the ith belt span

120597θti(t) 120579 ti(t)

120579 ti(t)

Angular displacement velocity and acceleration (rotational coordinate) of the

ith tensioner arm

120637119938 General coordinate matrix under ISG driving phase

120637119940 General coordinate matrix under crankshaft driving phase

θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and

Lbi respectively

Θi Amplitude of displacement of the ith pulley

xix

θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith

pulley

θti Angle of the ith tensioner arm

θtoi Initial pivot angle of the ith tensioner arm

θm Angular displacement matrix of driven rigid bodies

Θm Amplitude of displacement of driven rigid bodies

ρ Belt cord density

120601119894 Belt wrap angle on the ith pulley

φmax Belt maximum phase angle

φ0deg Belt phase angle at zero frequency

ω Frequency of the system

ωcs Angular frequency of crankshaft pulley

ωISG Angular frequency of the ISG pulley

120654119951 Natural frequency of system

1

CHAPTER 1 INTRODUCTION

11 Background

Belt drive systems are the means of power transmission in conventional automobiles The

emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-

ISG) has placed higher demands on belt drives than ever before The presence of an integrated

starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to

premature belt failure This phenomenon has motivated automotive makers to design a tensioner

that is suitable for the B-ISG system

The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the

belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the

role of this system is to transmit torque generated by an internal combustion engine (ICE) in

order to reliably drive multiple peripheral devices mounted on the engine block The high speed

torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a

single continuous member that winds around the driving and driven accessory pulleys of the

drive system Serpentine belts used in automotive applications consist of several layers The

load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a

protective layer to guard against mechanical damage and is bound below by a visco-elastic layer

that provides the required shock absorption and grip against the rigid pulleys [1] The accessory

devices may include an alternator power steering pump water pump and air conditioner

compressor among others

Introduction 2

The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin

to traditional belt drives differing in the fact that an electric motor called an integrated starter-

generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides

braking regeneration [2] The re-start function of the micro-hybrid transmission is known as

stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed

of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and

upwards to produce an instantaneous start in the start-stop process [3] The high rotations per

minute (rpm) produced by the ISG consistently places much higher tension requirements on the

belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to

800N of tension on the belt since this exceeds the safe operating conditions of belts used in most

traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm

tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack

belts spans occurring in the ISG phase of operation for the B-ISG system

In order for the belt to transfer torque in a drive system its initial tension must be set to a value

that is sufficient to keep all spans rigid This value must not be too low as to allow any one span

to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled

with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the

drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in

an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning

of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for

the system to cope with moderate to severe changes in belt span tensions

Introduction 3

Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded

arm This type of tensioner is normally designed to provide a passive response to changes in belt

span tension The introduction of the ISG electric motor into the traditional belt drive with a

single-arm tensioner results in the presence of excessively slack spans and excessively tight

spans in the belt The tension requirements in the ISG-driving phase which differ from the

crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner

Tensioners can be divided into two general classes passive and active In both classes the

single-arm tensioner design approach is the norm The passive class of tensioners employ purely

mechanical power to achieve tensioning of the belt while the active class also known as

automatic tensioners typically use some sort of electronic actuation Automatic tensioners have

been employed by various automotive manufacturers however ldquosuch devices add mass

complication and cost to each enginerdquo [5]

12 Motivation

The motivation for the research undertaken arises from the undesirable presence of high belt

tension in automotive belt drives Manufacturers of automotive belt drives have presented

numerous approaches for tension mechanism designs As mentioned in the preceding section

the automation of the traditional single-arm tensioner has disadvantages for manufacturers A

survey of the literature reveals that few quantitative investigations in comparison to the

qualitative investigations provided through patent literature have been conducted in the area of

passive and dual tensioner configurations As such the author of the research project has selected

to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner

Introduction 4

configuration is motivated by research and developments of industry partner Litens

Automotivendash a manufacturer of automotive belt drive systems and components Litens‟

specialty in automotive tensioners has provided a basis for the research work conducted

13 Thesis Objectives and Scope of Research

The objective of this project is to model and investigate a system containing a passive twin-arm

tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a

crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in

continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority

software known as FEAD at the University of Toronto Firstly for the B-ISG system with a

twin-arm tensioner the geometric state and its equations of motion (EOM) describing the

dynamic and static states are derived The modeling approach was verified by deriving the

geometric properties and the EOM of the system with a single tensioner arm and comparing its

crankshaft-phase‟s simulation results with FEAD software simulations This also provides

comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner

equipped belt drive model Secondly the model for the static system is investigated through

analysis of the tensioner parameters Thirdly the design variables selected from the parametric

analysis are used for optimization of the new system with respect to its criteria for desired

performance

Introduction 5

14 Organization and Content of the Thesis

This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG

serpentine belt drive system which is distinguished by having its driving pulley alternate

between a crankshaft pulley and an ISG pulley

Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of

the research discusses the function and location of the ISG in hybrid technologies in order to

provide a background for the B-ISG system The attributes of the B-ISG are then discussed

Subsequently a description is given of the developments made in modeling belt drive systems

At the close of the chapter the prior art in tensioner designs and investigations are discussed

The third chapter describes the system models and theory for the B-ISG system with a twin-arm

tensioner Models for the geometric properties and the static and dynamic cases are derived The

simulation results of the system model are presented

Then the fourth chapter contains the parametric analysis The methodologies employed results

and a discussion are provided The design variables of the system to be considered in the

optimization are also discussed

The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5

The evaluation of optimization methods results of optimization and discussion of the results are

included Chapter 6 concludes the thesis work in summarizing the response to the thesis

Introduction 6

objectives and concluding the results of the investigation of the objectives Recommendations for

future work in the design and analysis of a B-ISG tensioner design are also described

7

CHAPTER 2 LITERATURE REVIEW

21 Introduction

This literature review justifies the study of the thesis research the significance of the topic and

provides the overall framework for the project The design of a tensioner for a Belt-driven

Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission

developments in hybrid automobiles This chapter will begin with the context of the B-ISG

followed by a review of the hybrid classifications and the critical role of the ISG for each type

The function location and structure of the B-ISG system are then discussed Then a discussion

of the modeling of automotive belt transmissions is presented A systematic review of the prior

art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and

research evidence relevant to the thesis topic which is the design of a B-ISG tensioner

The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is

distinguished from other hybrid classes by the structure functions and location of its ISG The

B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-

ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption

in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]

Environmental and legislative standards for reducing CO2 emissions in vehicles have called for

carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The

transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional

automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG

Literature Review 8

transmission to that of conventional cars provide the motivation for the thesis research

Consequently the micro-hybrid class of cars is especially discussed in the literature review since

it contains the B-ISG type of transmission system The micro-hybrid class is one of several

hybrid classes

A look at the performance of a belt-drive under the influence of an ISG is rooted in the

developments of hybrid technology The distinction of the ISG function and its location in each

hybrid class is discussed in the following section

22 B-ISG System

221 ISG in Hybrids

This section of the review discusses the standard classes of hybrid cars which are full power

mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving

internal combustion engines (ICEs) as the main power source This is done for the sake of

comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-

hybrids which is the focus of the research The term conventional car vehicle or automobile

henceforth refers to a vehicle powered solely by a gas or diesel ICE

A hybrid vehicle has a drive system that uses a combination of energy devices This may include

an ICE a battery and an electric motor typically an ISG Two systems exist in the classification

of hybrid vehicles The older system of classification separates hybrids into two classes series

hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of

operation matching both categories classifying them under either of the two classes [9] The

Literature Review 9

new system of classification has four classes full power mild and micro Under these classes

vehicles are more often under a sole category [9] In both systems an ICE may act as the primary

source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)

diesel or an alternative fuel such as ethanol bio-diesel or natural gas

2211 Full Hybrids

In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores

electrical energy in the batteries to be used to power an electric traction motor [8] The electric

traction motor is akin to a second ISG as it generates power and provides torque output It also

supplies an extra boost to the wheels during acceleration and drives up steep inclines A full

hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering

the electric traction motor without the engine running This silent acceleration known as electric

launch is normally employed when accelerating from standstill [9] Full hybrids can generate

and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]

The ISG allows this by converting from an electric traction motor to a generator when braking or

decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored

in the batteries For full hybrids to achieve this they often use break-by-wire a form of

electronically controlled braking technology

A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the

engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at

approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In

full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled

Literature Review 10

from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called

an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a

vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling

[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of

the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at

standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is

started only when needed which is often several seconds after acceleration has begun The

engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE

when it is not needed to assist in driving the wheels or to produce electricity for recharging the

batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids

over conventional vehicles particularly in urban driving conditions [9] Since the ICE is

required to provide only the average horsepower used by the vehicle the engine is downsized in

comparison to a conventional automobile that obtains all its power from an ICE Frequently in

full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has

a higher efficiency while having a lower power output Examples of full hybrids include the

Ford Escape and the Toyota Prius [9]

2212 Power Hybrids

Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch

regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is

the ICE is not downsized to meet only the average power demand [9] Thus the engine of a

power hybrid is large and produces a high amount of horsepower compared to the former

Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a


greater acceleration performance than a full hybrid or a conventional vehicle with the same size

ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]

2213 Mild Hybrids

In the hybrid types discussed thus far the ISG is positioned between the engine and transmission

to provide traction for the wheels and for regenerative braking Often times the armature or rotor

of the electric motor-generator which is the ISG replaces the engine flywheel in full and power

hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is

not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft

with the ICE In this environment the electric launch feature does not exist since the ISG does

not turn the wheels independently of the engine and energy cannot be generated and consumed

at the same time However the ISG of the mild hybrid allows for the remaining features of the

full hybrid regenerative braking and engine idle-stop including the fact that the engine is

downsized to meet only the average demand for horsepower Mild hybrid vehicles include the

GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]

2214 Micro Hybrids

Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest

to modern conventional vehicles This class normally features a gas or diesel ICE [9] The

conventional automobile is modified by installing an ISG unit on the mechanical drive in place

of or in addition to the starter motor The starter motor typically 12-volts is removed only in

the case that the ISG device passes cold start testing which is also dependent on the engine size

[10] Various mechanical drives that may be employed include chain gear or belt drives or a


clutchgear arrangement The majority of literature pertaining to mechanical driven ISG

applications does not pursue clutchgear arrangements since it is associated with greater costs

and increased speed issues Findings by Henry et al [11] show that the belt drive in

comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted

directly to the accessory drive) has no need for lubrication has less restriction in the packaging

environment and produces very low noise Also mounting the ISG unit on a separate belt from

that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory

belt drive requires less engine transmission or vehicle modifications

As with full power and mild hybrids the presence of the ISG allows for the start-stop feature

The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a

separate ECU) is added to the conventional car in order to shut down the engine when the

vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This

entails that it dictates the field current by way of a microprocessor to allow the system to defer

battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge

the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid

and its various components are discussed in the subsequent section Examples of micro hybrid

vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini

Cooper D and various others of BMW‟s European models [15]


Figure 21 Hybrid Functions

Source Dr Daniel Kok FFA July 2004 modified [16]

Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions

it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may

also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed

in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for

Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has

characteristics of a mild hybrid such as cold cranks and engine assists [17]

222 B-ISG Structure Location and Function

2221 Structure and Location

The ISG is composed of an electrical machine normally of the inductive type which includes a

stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter

comprising of a regulator a modulator switches and filters There are various configurations to

integrate the ISG unit into an automobile power train One configuration situates the ISG

directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact

however it results in a longer power train which becomes a potential concern for transverse-


mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to

describe the configuration of mounting the electrical device on the side of the mechanical drive

[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis

research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system

In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may

be removed This design allows for the functions of the ISG system mentioned in the description

of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG

is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood

layoutrdquo [11]

2222 Functionalities

The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the

engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]

and in some high voltage settings to provide cold starting

The cold starting operation of the ISG refers to starting the engine from its off mode rather than

idle mode andor when the engine is at a low temperature for example -29 to -50 degrees

Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque

requirement for the cold starting operation of the target vehicle since it is greater than the

nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]

18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to

[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt

architecture most commonly found in vehicles does not supply sufficient voltage [2] The

introduction of the ISG machine and other electrical units in vehicles encourages a transition


from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt

architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]

At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine

cranking the conventional starter will [remain] in the system and perform only for the initial

cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch

assist torque the torque required to bring the engine from idle speed to the speed at which it can

develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]

Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by

researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck

[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine

re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the

12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]

23 Belt Drive Modeling

The modeling of a serpentine belt drive and tensioning mechanism has typically involved the

application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of

motion for the system There are two modes of motion in a serpentine belt drive transverse

motion and rotational motion The former can be viewed as the motion of the belt directed

normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut

string that is fixed at either end However the study of the rotational motion in a belt drive is the

focus of the thesis research


Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s

models helped to understand belt performance and the influence of driving and driven pulleys on

the tension member The first description of a serpentine belt drive for automotive use was in

1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge

on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design

methodology to improve the dynamic performance of instability mechanisms for belt tensioner

systems The mathematical model developed by Ulsoy et al [24] coupled the equations of

motion that were obtained through a dynamic equilibrium of moments about a pivot point the

equations of motion for the transverse vibration of the belt and the equations of motion for the

belt tension variations appearing in the transverse vibrations This along with the boundary and

initial conditions were used to describe the vibration and stability of the coupled belt-tensioner

system Their system also considered the geometry of the belt drive and tensioner motion

Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a

driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations

of transverse motion for the respective belt spans with the equations of motion for pulleys and a

tensioner This was used to form the free response of the system and evaluate its response

through a closed-form solution of the system‟s natural frequencies and mode shapes

A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]

to determine the effect of damping on rotational vibration mode solutions The equations of

motion developed for a multi-pulley flat belt system with viscous damping and elastic


properties including the presence of a rotary tensioner were manipulated to carry out the modal

analysis

Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-

tensioner system by way of nonlinear numerical methods and an approximated linear closed-

form method The authors used this strategy to develop a single design parameter referred to as

a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its

operating state from a reference state The authors considered the steady state tensions in belt

spans as a result of accessory loads belt drive geometry and tensioner properties

Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive

system A non-iterative approach was used to explicitly form the equations for the system‟s

natural frequencies An exact closed-form expression for the dynamic response of the system

using eigenfunction expansion was derived with the system under steady-state conditions and

subject to harmonic excitation

The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive

(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD

was found analytically by considering the system to be piecewise linear about the equilibrium

angular deflections The effect of their nonlinear terms was considered through numerical

integration of the derived equations of motion by way of the iterative methodndash fourth order

Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of

the belt spans only


The first to carry out the analysis of a serpentine belt drive system containing a two-pulley

tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a

serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He

employed Runge Kutta method as well to solve the equations of motion to find the response of

the system under a general input from the crankshaft The author‟s work also included the

optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft

excitation Furthermore the author applied active control techniques to the tensioner in a belt

drive system

The works discussed have made significant contributions to the research and development into

tensioner systems for serpentine belt drives These lead into the requirements for the structure

function and location of tensioner systems particularly for B-ISG transmissions

24 Tensioners for B-ISG System

241 Tensioners Structure Function and Location

Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the

tensioning mechanism redesign This mechanism as shown by researchers including

Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt

(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt

and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-

drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction

of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an

adjustable tensioner


The two types of tensioners are passive and active tensioners The former permits an applied

initial tension and then acts as an idler and normally employs mechanical power and can include

passive hydraulic actuation This type is cheaper than the latter and easier to package The latter

type is capable of continually adjusting the belt tension since it permits a lower static tension

Active tensioners typically employ electric or magnetic-electric actuation andor a combination

of active and passive actuators such as electrical actuation of a hydraulic force

Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler

pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler

pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by

a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine

housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-

tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the

changes in belt length [31] When the belt undergoes stretch under a load the spring drives the

tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of

operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a

steady-state condition and auxiliary components continue to rotate still in their own inherent

inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt

flutter or skip and slip that may damage other components present in the belt drive [31]

Locating the tensioner on the slack side of the belt is intended to lower the initial static tension

[11] In conventional vehicles the engine always drives the alternator so the tensioner is located

in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span


of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft

[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table

21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is

driving instead of in the slack span when the crankshaft is driving allows for easier packaging

and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the

same span as Henry et al [11] or in both the slack and taut spans if using a double

tensioneridler configuration

242 Systematic Review of Tensioner Designs for a B-ISG System

The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-

ISG settings are largely in patent records as the re-design of a tensioner has been primarily a

concern of automotive makers thus far A systematic review of the patent literature has been

conducted in order to identify evaluate and collate relevant tensioning mechanism designs

applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner

configuration for the thesis study

The predefined search strategy used by the researcher has been to consider patents dating only

post-2000 as many patents dating earlier are referred to in later patents as they are developed on

in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may

refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any

successor were also considered The inclusion and exclusion criteria and rationales that were

used to assess potential patents are as follows

Inclusion of


tensioner designs with two arms andor two pivots andor two pulleys

mechanical tensioners (ie exclusion of magnetic or electrical actuators or any

combination of active actuators) in order to minimize cost

tension devices that are an independent structure apart from the ISG structure in order to

reduce the required modification to the accessory belt drive of a conventional automobile

and

advanced designs that have not been further developed upon in a subsequent patent by the

inventor or an outside party

Table 21 provides a collation of the results for the systematic review based on the selection

criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that

the patent literature pertaining to these designs in most cases provides minimal numerical data

for belt tensions achieved by the tensioning mechanism In most cases only claims concerning

the outcome in belt performance achievable by the said tension device is stated in the patent

Table 21 Passive Dual Tensioner Designs from Patent Literature

Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach

2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation

A torsion spring is used at bearing bushings to mount tension arms at ISG shaft

Each tension arm cooperates with torsion spring mechanism to rotate through a damping

device in order to apply appropriate pressure to taut and slack spans of the belt in

different modes of operation

Bosch GMBH Patent WO0026532 et al [34]

Design Approach

2 tension pulleys each one is mounted on the return and load spans of the driven and

driving pulley respectively

Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point


Idlers‟ motions are independent of each other and correspond to the tautness or

slackness in their respective spans

Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at

its axis of rotation

Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach

2 idlers are given a working force by a self-aligning bearing

Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle

auxiliary unit pulley

Dayco Products

LLC

Patents US6942589-B2 et al [36]

Design Approach

2 tension arms are each rotatably coupled to an idler pulley

One idler pulley is on the tight belt span while the other idler pulley is on the slack belt

span

Tension arms maintain constant angle between one another

One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt

Idler pulleys are on opposite sides of the ISG pulley

Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]

Design Approach

A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)

A tensioner is asymmetrically biased in direction tending to cause power transmission

belt to be under tension

McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach

2 tension pulleys and carrier arms with a central pivot are mounted to the engine

One tension arm and pulley moderately biases one side of belt run to take up slack

during engine start-up while other tension arm and pulley holds appropriate bias against

taut span of belt

A hydraulic strut is connected to one arm to provide moderate bias to belt during normal

engine operation and velocity sensitive resistance to increasing belt forces during engine

start-up

INA Schaeffler

KG et al

Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]

DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]

WO2006108461-A1 et al [45]

Design Approach

2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same

plane of the housing part

o Idler pulley corresponding to each tensioning arm engages with different

sections of belt

o When high tension span alternates with slack span of belt drive one tension

arm will increase pressure on current slack span of belt and the other will

decrease pressure accordingly on taut span

o Or when the span under highest tension changes one tensioner arm moves out

of the belt drive periphery to a dead center due to a resulting force from the taut

span of the ISG starting mode

o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span


o The 2 lever arms are of different lengths and each have an idler pulley of

different diameters and different wrap angles of belt (see DE10045143-A1 et

al)

1 tensioner arm and 2 pulleys approach ndash

o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably

linked to the beam o The ISG machine is supported by a shock absorber

o During ISG start-up one idler pulley is induced to a dead center position while

it pulls the remaining idler pulley into a clamping position until force

equilibrium takes place

o A shock absorber is laid out such that its supporting spring action provides

necessary preloading at the idler pulley in the direction of the taut span during

ISG start-up mode

Litens Automotive

Group Ltd


Design Approach

2 tension pulleys on opposite sides of the ISG pulley engage the belt

They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot

The pivot axis varies relative to the force applied to each tension pulley

Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the

tensioner pulleys are approximately equal

A limited swivel angle for the tensioner arms work cycle is permitted

Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach

2 tensioners are used

1 tensioner is held on the slack span of the driving pulley in a locked condition and a

second tensioner is held on the slack side of the starting (driven) pulley in a free condition

Nissan Patents JP3565040-B2 et al [48]

Design Approach

A single tensioner is on the slack span once ISG pulley is in start-up mode

The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism

(a plunger which performs retreat actuation according to the energizing force of the oil

pressure spring and load received from the ISG)

The tensioner is equipped with a relief valve to keep a predetermined load lower than the

maximum load added by the ISG device

NTN Corp Patent JP2006189073-A [49]

Design Approach

An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a

screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat

in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of

the cylinder nut) a spring support (arranged on varying diameter stepped recessed

sections of the rod seat) and a check valve with an openingclosing passage

The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG

start-up mode of operation

Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach

ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and


forwards at least partially outside a support element (200) and is characterized in that

the free front end (23) of said shaft section (22) is carried by an arm (206) connected to

the support element (200)rdquo

The author notes that published patents and patent applications may retain patent numbers for multiple patent

offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in

the absence of such a number the published patent application number has been specified However published

patent documents in the above cases also served as the document (ie identical) to the published patent if available

Quoted from patent abstract as machine translation is poor

25 Summary

The research on tensioner designs from the patent literature demonstrates a lack of quantifiable

data for the performance of a twin tensioner particularly suited to a B-ISG system The review of

the literature for the modeling theory of serpentine belt drives and design of tensioners shows

few belt drive models that are specific to a B-ISG setting Hence the literature review supports

the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin

tensioner configuration and as well measuring the tensioner‟s performance The survey of

hybrid classes reveals that the micro-hybrid class is the only class employing a closely

conventional belt transmission and hence its B-ISG transmission is applicable for tensioner

investigation The patent designs for tensioners contribute to the development of the tensioner

design to be studied in the following chapter

25

CHAPTER 3 MODELING OF B-ISG SYSTEM

31 Overview

The derivation of a theoretical model for a B-ISG system uses real life data to explore the

conceptual system under realistic conditions The literature and prior art of tensioner designs

leads the researcher to make the following modeling contributions a proposed design for a

passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the

proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic

and static states as well as deriving the EOM for the B-ISG system with only a passive single-

pulley tensioner for comparison The principles of dynamic equilibrium are applied to the

conceptual system to derive the EOM

32 B-ISG Tensioner Design

The proposed design for a passive two pulley tensioner configures two tensioners about a single

fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm

coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of

the pulley The other end of each arm is pivoted about a point so that the arms share the same

axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows

a schematic for the proposed design

Modeling of B-ISG 26

Figure 31 Schematic of the Twin Tensioner

The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti

and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting

between the two arms and spring stiffness kti acting between each arm and the pivot point The

motion of each arm is dampened by dampers and akin to the springs a damper acts between the

two arms ct and a damper cti acts between each arm and the pivot point The result is a

tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations

of the two pulleys and two arms

The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a

Twin Tensioner to their respective motions


33 Geometric Model of a B-ISG System with a Twin Tensioner

The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive

provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces

These variables are crucial to resolve the components of forces and moment arms acting on each

rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s

geometric modeling approach [51] used in the development of the software FEAD was applied

to the Twin Tensioner system to compute the system‟s unique geometric attributes

Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner

It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and

the overall system respectively that for the purpose of the geometric computations the forward

direction follows the convention of the numbering order counterclockwise The numbering

order is in reverse to the actual direction of the belt motion which is in the clockwise direction in

this study The fourth pulley is identified as an ISG unit pulley However the properties used

for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional


alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving

pulley at which point the requirements of the ISG are solved for and its non-inertia attributes

are not needed to be ascribed

Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each

pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and

its possible belt connection points respectively Length Lfi is the length of the span connecting

the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively

Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the

ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line

connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with

lengths Lfi and Lbi respectively Ri is the radius of the ith pulley

Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys

[modified] [51]


The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is

calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine

trigonometric relation

i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)

The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi

The centre point coordinates and the radii of the pulleys are related through the solution of

triangles which they form to define values of the possible belt span lengths

Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)

The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the

line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley

perimeter


(33a)

(33b)

The array of possible belt connection points comes about from the use of the pulley centre

coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle

αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations

(33a) and (33b) The formula to compute the array of points is shown in equations (34) and

(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point

on the ith pulley which is in the span leading forward to the next (i+1)th pulley

(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1


Equation (35) describes the backward belt connection point on the ith pulley This point sits on

the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th

pulley

(35a)

(35b)

(35c)

(35d)

The selection of the coordinates from the array of possible connection points requires a graphic

user interface allowing for the points to be chosen based on observation This was achieved

using the MathCAD software package as demonstrated in the MathCAD scripts found in

Appendix C The belt connection points can be chosen so as to have a pulley on the interior or

exterior space of the serpentine belt drive The method used in the thesis research was to plot the

array of points in the MathCAD environment with distinct symbols used for each pair of points

and to select the belt connection points accordingly By observation of the selected point types

the type of belt span connection is also chosen Selected point and belt span types are shown in

Table 31


Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span

Pulley Forward Contact

Point

Backwards Contact

Point

Belt Span

Connection

1 Crankshaft XYf1 XYbf21 Lf1

2 Air Conditioning XYfb2 XYf22 Lb2

3 Tensioner 1 XYbf3 XYfb23 Lb3

4 AlternatorISG XYfb4 XYbf24 Lb4


The inscribed angles βji between the datum and the forward connection point on the ith pulley

and βji between the datum and its backward connection point are found through solving the

angle of the arc along the pulley circumference between the datum and specified point The

wrap angle ϕi is found as the difference between the two inscribed angles for each connection

point on the pulley The angle between each belt span and the horizontal as well as the initial

angle of the tensioner arms are found using arctangent relations Furthermore the total length of

the belt is determined by the sum of the lengths of the belt spans

34 Equations of Motion for a B-ISG System with a Twin Tensioner

341 Dynamic Model of the B-ISG System

3411 Derivation of Equations of Motion

This section derives the inertia damping stiffness and torque matrices for the entire system

Moment equilibrium equations are applied to each rigid body in the system and net force

equations are applied to each belt span From these two sets of equations the inertia damping


and stiffness terms are grouped as factors against acceleration velocity and displacement

coordinates respectively and the torque matrix is resolved concurrently

A system whose motion can be described by n independent coordinates is called an n-DOF

system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each

pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are

constrained to rotate about their respective central axes and the arms are free to rotate about their

respective pivot points then at any time the position of each pulley can be described by a

rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the

tensioner system comprises of four rigid bodies where each is described by one coordinate and

hence is a four-DOF system It is important to note that each rigid body is treated as a point

mass In addition inertial rotation in the positive direction is consistent with the direction of belt

motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys

Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System


For the serpentine belt system considered in the thesis research there are seven rigid bodies each

having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled

differential equations meaning that each equation includes all of the general coordinates and

includes up to the second-order time derivatives of these coordinates The EOM can be

obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point

including the couples equals to zero Therefore the inertial couple the product of the inertia and

acceleration is equated to the moment sum as shown in equation (35)

I ∙ θ = ΣM (35)

The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive

direction is in the clockwise direction are shown in equations (36) through to (310) The

numbering convention used for each rigid body corresponds to the labeled serpentine belt drive

system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the

damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and

120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith

tensioner arm The initial angle of the ith tensioner arm is described by θtoi

minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)

minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)


It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙

partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)

It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus

ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)

partθt1 = θt1 minus θto 1 (310a)

partθt2 = θt2 minus θto 2 (310b)

The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley

and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of

rotation are taken for these structures in equations (311) through to (313)

Figure 35 Free Body Diagram for Non-Tensioner Pulleys


I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)



The relationship between belt tensions and rigid body displacements is in the general form of

equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively

with each factorized by a radial arm length This relationship is described for each span in

equations (315) through to (320) The belt damping constant for the ith belt span is cib

119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)

T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1

b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)

T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2

b ∙ [R2 ∙ θ 2 minus R3 ∙

θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)


b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙


Tprime = Ti minus To (320)

Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium

equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each

other Hence equations (321) and (322) are denoted as follows

Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)

Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described

in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged

and written in matrix form as shown in equations (323) through to (328) The general

coordinate matrix 120521 and its first and second derivatives are shown in the EOM below

119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)


The inertia matrix I includes the inertia of each rigid body in its diagonal elements The

damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its

angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith

tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span

stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping

matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation

(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from

experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and

ϕi is the belt wrap angle on the ith pulley


119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5

b ∙ R12 + c1 minusc1

b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5

b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5

minusc1b ∙ R1 ∙ R2 c2

b ∙ R22 + c1


b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0

0 minusc2b ∙ R2 ∙ R3 c3

b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0

0 0 minusc3b ∙ R3 ∙ R4 c4

b ∙ R42 + c3


b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4


minusc5b ∙ R1 ∙ R5 0 0 minusc4

b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57

0 0 0 0 0 ct +ct1 minusct

0 0 0 0 0 minusct ct +ct1

(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5


minusk1b ∙ R1 ∙ R2 k2

b ∙ R22 + k1

b ∙ R22 minusk2

b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0

0 minusk2b ∙ R2 ∙ R3 k3

b ∙ R32 + k2

b ∙ R32 minusk3

b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2

b ∙ sin θto 1 minus βj2 ] 0

0 0 minusk3b ∙ R3 ∙ R4 k4

b ∙ R42 + k3

b ∙ R42 minusk4

b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4


minusk5b ∙ R1 ∙ R5 0 0 minusk4

b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5

2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4

b ∙ sin θto 2 minus βj4 ]

0 0 0 0 0 kt +kt1 minuskt

0 0 0 0 0 minuskt kt +kt1

(326a)

k119894b =

Kb

Li + kb ∙ Ri ∙ϕi+1

2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)


3412 Modeling of Phase Change

The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the

driving pulley is described through a conditional equality based on a set of Boolean conditions

When the crankshaft is driving the rows and the columns of the EOM are swapped such that the

new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner

pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)

When the ISG is driving the order is the same except that the second row and second column

terms relating to the ISG pulley become the first row and first column while the crankshaft

pulley terms (previously in the first row and first column) become the second row and second

column Hence the order for all rows and columns of the matrices making up the EOM in

equation (322) switches between 1423567 (when the crankshaft pulley is driving) and

4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG

driving phases the general coordinate matrix and the inertia matrix become the following

120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)


where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving

phase respectively

The condition for phase change is based on the engine speed n in units of rpm Equation (330)

demonstrates the phase change

H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)

(330)

When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and

following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the

driven pulley These modes of operation mean that the system will predict two different sets of

natural frequencies and mode shapes Using a Boolean condition to allow for a swap between

the first and second rows as well as between the first and second columns of the EOM matrices

I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley

throughout its driving and driven phases as well as for that of the crankshaft pulley

3413 Natural Frequencies Mode Shapes and Dynamic Responses

Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899

and modeshapes are found by solving the eigenvalue problem shown in equation (331a)

ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)

The displacement amplitude Θm is denoted implicitly in equation (331d)


120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)

120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)

θm = 120495119846 ∙ sin(ω ∙ t) (331d)

I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of

the original matrices are removed The eigenvalue problem is reached by considering the

undamped and unforced motion of the system Furthermore the dynamic responses are found by

knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner

arms are zero in the dynamic case which signifies a response of the system to an input solely

from the driving pulley

I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1

In the case of equation (331) θm is the submatrix identified in equations (331b) through to

(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the

phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and

columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in

section 3412 For simple harmonic motion the motion of the driven pulleys are described as

1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG

torque 119876119868119878119866 when the ISG pulley is in its driving function


θm = 120495119846 ∙ sin(ω ∙ t) (333)

The dynamic response of the system to an input from the driving pulley under the assumption of

sinusoidal motion is expressed in equation (334)

120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)

3414 Crankshaft Pulley Driving Torque Acceleration and Displacement

Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly

discussed It is assumed in the thesis research for the purpose of modeling that the engine

serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder

engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which

is represented by the factor of 2 on the steady component of the angular velocity in equation

(335) The torque requirement of the crankshaft pulley when it is the driving pulley is

Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)

The amplitude of the required crankshaft torque qc is expressed in equation (336) and is

derived from equation (332)

qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)


The angular frequency for the system in radians per second (rads) ω when the crankshaft

pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n

and by taking into account the double pulse per crankshaft revolution

ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)

The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed

to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude

of the excitation angular input from the engine is shown in equation (339b) and is found as a

result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


3415 ISG Pulley Driving Torque Acceleration and Displacement

Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is

discussed The torque for the ISG when it is in its driving function is assumed constant Ratings

for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an

Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which

can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]

As well it was noted that the ISG pulley was sized appropriately and the engine speed was

limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall

torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]

The formula for the torque of a permanent magnet DC motor for any given speed (equation

(340)) is used to approximate the torque of the ISG in its driving mode[52]

QISG = Ts minus (N ∙ Ts divide NF) (340)2

Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is

not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N

Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG

re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The

pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm

(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the

2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in

(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for

the required driving torque See Figure 315 for comparison of these results


crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG

speed in equation (341)

nISG = nCS ∙DCS

DISG

(341)

For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to

be approximately 220656 rpm Thus the ISG torque during start-up is found from equation

(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall

torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm

for the ISG The acceleration of the ISG pulley is found by taking into account the torque

developed by the rotor and the polar moment of inertia of the pulley [54]

A1ISG = θ 1ISG = QISG IISG (342)

In torsional motion the function for angular displacement of input excitation is sinusoidal since

the electric motor is assumed to be resonating As a result of constant angular acceleration the

angular displacement of the ISG pulley in its driving mode is found in equation 343

θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)

Knowing that acceleration is the second derivative of the displacement the amplitude of

displacement is solved subsequently [55]


θ 1ISG = minusωISG2 ∙ Θ

1ISG ∙ sin(ωISG ∙ t) (344)


1ISG

(345a)

Θ1ISG =minusQISG IISG

ωISG2

(345b)

In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular

frequency of the ISG pulley which can be expressed as a function of its speed in rpm

ω = ωISG =2 ∙ π ∙ nISG

60

(346a)

or in terms of the crankshaft rpm by substituting equation (341) into (346a)

ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)

3416 Tensioner Arms Dynamic Torques

The dynamic torque for the tensioner arms are shown in equations (347) and (348)

Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)

3417 Dynamic Belt Span Tensions

Furthermore the dynamic belt span tensions are derived from equation (314) and described in

matrix form in equations (349) and (350)

119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)

342 Static Model of the B-ISG System

It is fitting to pursue the derivation of the static model from the system using the dynamic EOM

For the system under static conditions equations (314) and (323) simplify to equations (351)

and (352) respectively

119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)


As noted in other chapters the focus of the B-ISG tensioner investigation especially for the

parametric and optimization studies in the subsequent chapters is to determine its effect on the

static belt span tensions Therein equations (351) and (352) are used to derive the expressions

for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span

Equation (320) demonstrates the relationship between the tension due to belt response and the

initial tension also known as pre-tension The static tension 119931 is found by summing the initial

tension 1198790 with the expression for the dynamic tension shown in equations (315) through to

(319) and by substituting the expressions for the rigid bodies‟ displacements from equation

(352) and the relationship shown in equation (320) into equation (351)

119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations

The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-

ISG system in the previous sections of this chapter were verified using the software FEAD The

input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the

crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software

FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s

geometric model and EOMs in MathCAD software Furthermore the geometric simulation

3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the

torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its

multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft

driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for

the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in

the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix

C3 of MathCAD scripts


results for both of the twin and single tensioner B-ISG systems were found to be in agreement as

well

351 Geometric Analysis

The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner

pivot point are described as Cartesian coordinates and shown in Table 32 which also includes

the diameters for the pulleys

Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]

Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]

1Crankshaft Pulley 20065 [00]

2 Air Conditioner Pulley 10349 [224 -6395]

3 Tensioner Pulley 1 7240 [292761 87]

4 ISG Pulley 6820 [24759 16664]


6 Tensioner Arm Pivot --- [201384 62516]

The geometric results for the B-ISG system are shown in Table 33

Table 33 Geometric Results of B-ISG System with Twin Tensioner

Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949

Total Length of Belt (mm) 1243


352 Dynamic Analysis

The dynamic results for the system include the natural frequencies mode shapes driven pulley

and tensioner arm responses the required torque for each driving pulley the dynamic torque for

each tensioner arm and the dynamic tension for each belt span These results for the model were

computed in equations (331a) through to (331d) for natural frequencies and mode shapes in

equation (334) for the driven pulley and tensioner arm responses in equation (336) for the

crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in

equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the

dynamic tension of each belt span Figures 36 through to 323 respectively display these

results The EOM simulations can also be contrasted with those of a similar system being a B-

ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley

which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG

with a single tensioner is presented in Appendix B

It is assumed for the sake of the dynamic and static computations that the system

does not have an isolator present on any pulley

has negligible rotational damping of the pulley shafts

has negligible belt span damping and that this damping does not differ amongst

spans (ie c1b = ∙∙∙ = ci

b = 0)

has quasi-static belt stretch where its belt experiences purely elastic deformation

has fixed axes for the pulley centres and tensioner pivot

has only one accessory pulley being modeled as an air conditioner pulley and


has a rotational belt response that is decoupled from the transverse response of the

belt

The input parameter values of the dynamic (and static) computations as influenced by the above

assumptions for the present system equipped with a Twin Tensioner are shown in Table 34

Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]

Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]

Crankshaft 10 000 0 0 4

Air Conditioner 2 230 0 0 2

Tensioner 1 300 1x10-4

0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data

Initial belt tension [N] To 300

Belt cord stiffness [Nmmmm] Kb 120 00000

Belt phase angle at zero frequency [deg] φ0deg 000

Belt transition frequency [Hz] ft 000

Belt maximum phase angle [deg] φmax 000

Belt factor [magnitude] kb 0500

Belt cord density [kgm3] ρ 1000

Belt cord cross-sectional area [mm2] A 693


These values are for the driven cases for the ISG and crankshaft pulleys respectively In the

driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving

torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys

respectively

It is noted that because of the belt data for the phase angle at zero frequency the transition

frequency and the maximum phase angle are all zero and hence the belt damping is assumed to

be constant between frequencies These three values are typically used to generate a phase angle

versus frequency curve for the belt where the phase angle is dependent on the frequency The

curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at

0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt

damping would then be found by solving for cb in the following equation

tanφ = cb ∙ 2 ∙ π ∙ f (354)

Nevertheless the assumption for constant damping between frequencies is also in harmony with

the remaining assumptions which assume damping of the belt spans to be negligible and

constant between belt spans

3521 Natural Frequency and Mode Shape

The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37

under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing

frequency for the system differs for each case due to the change in driving pulley Modeic and

Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in

the crankshaft and ISG driving cases respectively


Natural Frequency ωn [Hz]

Crankshaft Pulley ΔΘ4

Air Conditioner Pulley ΔΘ

Tensioner Pulley 1 ΔΘ


Tensioner Arm 1 ΔΘ


Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive

Rigid Bodies

Figure 36b ISG Driving Case First Mode Responses

4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body


Figure 36c ISG Driving Case Second Mode Responses


ISG Pulley ΔΘ5






Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for

Responsive Rigid Bodies



Figure 37b Crankshaft Driving Case First Mode Responses

Figure 37c Crankshaft Driving Case Second Mode Responses


3522 Dynamic Response

The dynamic response specifically the magnitude of angular displacement for each rigid body is

plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is

fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the

system in particular it determines whether the crankshaft pulley or ISG pulley is the driving

pulley

Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)

Figure 39 ISG Pulley Dynamic Response (for ISG driven case)


Figure 310 Air Conditioner Pulley Dynamic Response

Figure 311 Tensioner Pulley 1 Dynamic Response

Crankshaft Driving Phase ISG

Driving

Phase


Driving

Phase



Figure 313 Tensioner Arm 1 Dynamic Response


Driving

Phase


Driving Phase



3523 ISG Pulley and Crankshaft Pulley Torque Requirement

Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving

mode and the crankshaft pulley in its driving mode

Figure 315 Required Driving Torque for the ISG Pulley


Driving Phase


Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line

labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a

permanent magnet DC motor The additional solid line labeled as qamod uses the formula in

equation (336) which determines the load torque of the driving pulley based on the pulley

responses Figure 315 provides a comparison of the results

Figure 316 Required Driving Torque for the Crankshaft Pulley

3524 Tensioner Arms Torque Requirements

The torque for the tensioner arms are shown in Figures 317 and 318


Figure 317 Dynamic Torque for Tensioner Arm 1


3525 Dynamic Belt Span Tension

The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values

plotted represent the magnitude of the dynamic tension


Driving Phase


Driving

Phase


Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension

Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension


Driving Phase


Driving

Phase


Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension

Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension


Driving

Phase


Driving Phase


Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension

The dynamic results for the system serve to show the conditions of the system for a set of input

parameters The following chapter targets the focus of the thesis research by analyzing the affect

of changing the input parameters on the static conditions of the system It is the static results that

are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and

optimization studies respectively The dynamic analysis has been used to complete the picture of

the system‟s state under set values for input parameters

353 Static Analysis

Before looking at the static results for the system under study in brevity the static results for a

B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the

tensioner arm and tensioner pulley that interacts with the span between the air conditioner and

ISG pulleys of the original system are removed as shown in Figure 324


Driving

Phase


Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner

The complete static model as well as the dynamic model for the system in Figure 324 is found

in Appendix B The results of the static tension for each belt span of the single tensioner system

when the crankshaft is driving and the ISG is driving are shown in Table 35

Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-

ISG Serpentine Belt Drive with a Single Tensioner

Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]

Crankshaft ndash Air Conditioner 481239 -361076

Air Conditioner ndash ISG 442588 -399727

ISG ndash Tensioner 29596 316721

Tensioner ndash Crankshaft 29596 316721

The tensions in Table 35 are computed with an initial tension of 300N This value for pre-

tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned


Whereas in the case of the ISG pulley driving the first and second spans are excessively slack

Therein an additional pretension of approximately 400N would be required which would raise

the highest tension span to over 700N This leads to the motivation of the thesis researchndash to

reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these

tensions should be minimized to prolong belt life preferably within the range of 600 to 800N

As well it is desirable to minimize the amount of pretension exerted on the belt The current

design uses a pre-tension of 300N The above results would lead to a required pre-tension of

more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow

the belt to exhibit high performance in power transmission and come near to the safe threshold

This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for

the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner

the following static results in Table 36 are achieved


ISG Serpentine Belt Drive with a Twin Tensioner



Air Conditioner ndash Tensioner 1 427197 -322803

Tensioner 1 ndash ISG 427197 -322803

ISG ndash Tensioner 2 28057 393645

Tensioner 2 ndash Crankshaft 28057 393645

The results in Table 36 show that the span following the ISG in the case between the Tensioner

1 and ISG pulleys is less slack than in the former single tensioner set-up However there

remains an excessive amount of pre-tension required to keep all spans suitably tensioned


36 Summary

The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode

shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when

the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of

the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are

negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving

phase there is greater dynamic response for the driven rigid bodies of the system including for

that of the ISG pulley

As the engine speed increases the torque requirement for the ISG was found to vary between

approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine

speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm

and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks

at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm

respectively The dynamic torque of the first tensioner arm was shown to peak at approximately

15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an

approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an

engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of

approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was

seen at an engine speed of 5000rpm

The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the

B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that


of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span

peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm

5000rpm respectively For the dynamic tension of the second belt span peaks of approximately

1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of

the former span At these same engine speeds the third belt span exhibited tension peaks at

approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were

approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm

The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same

respective engine speeds of the fourth span

The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal

that taut spans of the crankshaft driving case are lower in the ISG driving case The largest

change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase

by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the

crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller

change between the tension values for the spans in the least tension in the crankshaft driving

phase and their corresponding span in the ISG driving phase

The summary of the simulation results are used as a benchmark for the optimized system shown

in Chapter 5 The static tension simulation results are investigated through a parametric study of

the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the

selected design variables from the outcome of Chapter 4

71

CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG

TWIN TENSIONER

41 Introduction

The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator

(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand

how changing one parameter influences the static belt span tensions for the system Since the

thesis research focuses on the design of a tensioning mechanism to support static tension only

the parameters specific to the actual Twin Tensioner and applicable to the static case were

considered The parameters pertaining to accessory pulley properties such as radii or various

belt properties such as belt span stiffness are not considered In the analyses a single parameter

is varied over a prescribed range while all other parameters are held constant The pivot point

described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases

42 Methodology

The parametric study method applies to the general case of a function evaluated over changes in

one of its dependent variables The methodology is illustrated for the B-ISG system‟s function

for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters

The original data used for the system is based on sample vehicle data provided by Litens [56]

Table 41 provides the initial data for the parameters as well as the incremental change and

maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits

for each parameter have been chosen to be plus or minus sixty percent of its initial value

Parametric Analysis 72

Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner

Parameter Name Initial Value Increment (+- Δi) Minimum

value Maximum value

Coupled Spring

Stiffness kt

20626

N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad

Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761

0087] m See Figure 41 for region of possible tensioner pulley

coordinates Tensioner Pulley 2

Initial Coordinates

[012057

009193] m

The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley

2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b

Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner

Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I


Figure 41b Regions 1 and 2 in Cartesian Space

The selection for the minimum and maximum tensioner pulley centre coordinates and their

increments are not selected arbitrarily or without derivation as the other tensioner parameters

The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a

graphing suite in MathCAD to model the regions of space The following descriptions are used

to describe the possible positions for the tensioner pulleys

Tensioner pulleys are situated such that they are exterior to the interior space created by

the serpentine belt thus they sit bdquooutside‟ the belt loop

The highest point on the tensioner pulley does not exceed the tangent line connecting the

upper hemispheres of the pulleys on either side of it

The tensioner pulleys may not overlap any other pulley


Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4

and 5 is selected based on the above criteria and their lower boundaries are selected

arbitrarily

These criteria were used to define the equation for each boundary line and leads to a set of

Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The

density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-

points in each space for the purposes of the parametric analysis The outline of this method is

described in the MATLAB scripts contained in Appendix D

The results of the parametric analysis are shown for the slackest and tautest spans in each driving

case As was demonstrated in the literature review the tautest span immediately precedes the

driving pulley and the slackest span immediately follows the driving pulley in the direction of

the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span

and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in

the fourth span and the slackest span is in the third span Hence the parametric figures in this

chapter display only the tautest and slackest span values for both driving cases so as to describe

the maximum and minimum values for tension present in the given belt

43 Results amp Discussion

431 Influence of Tensioner Arm Stiffness on Static Tension

The parametric analysis begins with changing the stiffness value for the coil spring coupled

between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty

percent less than its initial value kt0 to sixty percent more than its original value as shown in


Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest

spans for both the crankshaft and ISG driving cases

Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)

As kt increases in the crankshaft driving phase for the B-ISG system the highest tension

decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N

In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest

tension increases from -3384N to -3167N Thus a change of approximately -45N is found in

the crankshaft driving case and approximately +22N is found in the ISG driving case for both the

tautest and slackest spans


The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are

shown in Figure 43

Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)

In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of

the tautest span when the crankshaft is driving which is a change of +53N The same value for

net change is found in the slackest span for the same driving condition whose tension increases

from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the

change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and

for the slackest span that changes from -3077N to -3338N


The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in

Figure 44


In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the

crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which

decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in

the ISG driving case a more noticeable change is once again found a difference of +144N

This is a result of the tautest span increasing from 3863N to 4007N and the slackest span

increasing from -3301N to -3157N


432 Influence of Tensioner Pulley Diameter on Static Tension

The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in

Figure 45

Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)

The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is

from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and

2946N for the respective spans This is a change of approximately +100N and a maximum

change of 1551N for both spans For the ISG driving case the tautest and slackest spans

decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N

for the respective spans are seen This nets a change of approximately -467N and a maximum

change of approximately -759N


The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in

Figure 46


The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from

4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is

that for both spans there is an increase in tension of approximately 14N Contrastingly the spans

experience a decrease in the ISG driving case as D5 increases The tension of the tautest span

goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This

equals a decrease of approximately 66N for both spans


433 Influence of Tensioner Pulley 1 Coordinates on Static Tension

The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span

for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47

Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span


The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for

tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue

area reaching to the perimeter of the plot It is evident in the darkest red region of the plot

where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is

(N)


between approximately 026m and 031m that the highest value of tension is experienced in the

tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span

when the crankshaft is driving is between a maximum of approximately 500N and a minimum of

approximately 300N This equals an overall difference of 200N in tension for the tautest span by

moving the position of pulley 1 The lowest values for tension are obtained when the pulley

coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m

to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is

also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is

approximately between 026m and 027m

The changes in tension for the slackest span under the condition of the crankshaft pulley being

the driving pulley are shown in Figure 48


Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span


Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are

represented by the non-blue region For the slackest span in the crankshaft driving case it is seen

that the lowest tension is approximately 125N while the highest tension is approximately 325N

This is an overall change of 200N that is achieved in the region The highest values are achieved

in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate

ranges from 026m to 031m which corresponds to the deep red region The lowest tension

values are achieved in the space where the y-coordinate ranges from approximately -0025m to

015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-

green region of the plot The area containing a y-coordinate of approximately 0m and x-

(N)


coordinates that are approximately between 026m and 027m also show minimum tension for

the slack span The regions of the x-y coordinates for the maximum and minimum tensions are

alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure

47

The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found

in Figure 49



(N)


Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in

Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values

in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension

values are achieved in the medium to light blue region This includes y-coordinates of

approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The

maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m

and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-

coordinates of approximately 026m to 027m It can be observed that aforementioned regions

for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and

48 for the crankshaft driving case

The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is

shown in Figure 410




Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG

driving in the non-yellow-green area The minimum tension values are achieved in the darkest

blue area where the minimum tension is approximately -500N This area corresponds to y-

coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to

03m The area of a maximum tension is approximately 400N and corresponds to the darkest red

area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as

the coordinates for y equaling approximately 0m and for x equaling approximately 026m to

027m The difference between maximum and minimum tensions in this case is approximately

900N It is noticed once again that the space of x- and y-coordinates containing the maximum

(N)


tension is in the similar location to that of the described space for minimum tension in the

crankshaft driving case in Figure 47 and 48


The influence of pulley 2 coordinates on the tension value for the tautest span when the

crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values

corresponding to the non-blue area



In Figure 411 the possible coordinates are contained within Region 2 The maximum tension

value is approximately 500N and is found in the darkest red space including approximately y-

(N)


coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m

to 02m corresponding to the x-coordinate at 0175m A minimum tension value of

approximately 350N is found in the yellow space and includes approximately y-coordinates

008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N

The analysis of the change in coordinates for tension pulley 2 on the value for tension in the

slackest span is shown in Figure 412 in the non-blue region



The value of 325N is the highest tension for the slack span in the crankshaft driving case of the

B-ISG system and is found in the deep-red region where the y-coordinates are between

(N)


approximately 004m and 013m and the x-coordinates are approximately between 0025m and

016m as well as where y is between 013m and 02m and x is approximately 0175m The

lowest tension value for the slack span is approximately 150N and is found in the green-blue

space where y-coordinates are between approximately 01m and 022m and the x-coordinates

are between approximately 016m and 021m The overall difference in minimum and maximum

tension values is 175N The spaces for the maximum and minimum tension values are similar in

location to that found in Figure 411 for the tautest span in the crankshaft driving case

Figure 413 provides the theoretical data for the tension values of the tautest span as the position

of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in

the space of values which correspond to the non-dark-blue region in Figure 413




In Figure 413 the region for high tension reaches a value of approximately 950N and the region

for low tension reaches approximately 250N This equals a difference of 700N between

maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving

case The coordinate points within the space that maximum tension is reached is in the dark red

region and includes y-coordinates from approximately 008m to 022m and x-coordinates from

approximately 016m to 021m The coordinate points within the space that minimum tension is

reached is in the blue-green region and includes y-coordinates from approximately 004m to

013m and the corresponding x-coordinates from approximately 0025m to 015m An additional

small region of minimum tension is seen in the area where the x-coordinate is approximately

(N)


0175m and the y-coordinates are approximately between 013m and 02m The location for the

area of pulley centre points that achieve maximum and minimum tension values is approximately

located in the reverse positions on the plot when compared to that of the case for the crankshaft

driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second

tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points

for maximum tension in the crankshaft driving case becomes the approximate area of points for

minimum tension in the ISG driving case and vice versa

In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner

pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to

earlier figures the non-dark yellow region represents Region 2 that contains the possible points

for the pulley‟s Cartesian coordinates


Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest

Span Tension in ISG Driving Case

Figure 414 demonstrates a difference of approximately 725N between the highest and lowest

tension values for the slackest span of the B-ISG system in the ISG driving case The highest

tension values are approximately 225N The area of points that allow the second tension pulley

to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to

022m and the corresponding x-coordinates from approximately 016m to 021m This

corresponds to the darkest red region in Figure 414 The coordinate values where the lowest

tension being approximately -500N is achieved include y-coordinate values from

approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m

to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also

(N)


seen in the area where y is approximately between 013m and 02m and the x-coordinate is

approximately 0175m The regions for maximum and minimum tension values are observed to

be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found

in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for

the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the

ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft

driving case and vice versa for the maximum tension of the ISG driving case and the minimum

tension for the crankshaft driving case in Region 2

44 Conclusion

Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving

portion that the B-ISG system‟s belt span tensions experience the following effect

Minimum tension for the tautest span is achieved when tensioner pulley 1 centre

coordinates are located closer to the right side boundary and bottom left boundary of

Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space

(near to the ISG pulley) and the space closer to the top boundary of Region 2

Maximum tension for the slackest span is achieved when the first tensioner pulley‟s

coordinates are located in the mid space and near to the bottom boundary of Region 1

and when the second tensioner pulley‟s coordinates are located near to the bottom left

boundary of Region 2 which is the boundary nearest to the crankshaft pulley


The trend for minimizing the tautest span signifies that the tension for the slackest span is also

minimized at the same time As well maximizing the slackest span signifies that the tension for

the tautest span is also maximized at the same time too

The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and

414 is approximately in reverse to that of the crankshaft driving case for the system Wherein

points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately

the same as points corresponding to maximum tension in the Regions for the crankshaft case and

vice versa for the ISG cases‟ areas of maximum tension

Minimum tension for the tautest span is present when the first tensioner pulley‟s

coordinates are near to mid to lower boundary of Region 1 and when the second

tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which

is the furthest boundary from the ISG pulley and closest to the crankshaft pulley

Maximum tension for the slackest span is achieved when the first tensioner pulley is

located close to the right boundary of Region 1 and when the second tensioner pulley is

located near the right boundary and towards the top right boundary of Region 2

It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest

spans decrease or increase together Thus it can be assumed that the tension values in these

spans and likely the remaining spans outside of the tautest and slackest spans follow suit

Therein when parameters are changed to minimize one belt span‟s tension the remaining spans

will also have their tension values reduced Figures 42 through to 413 showed this clearly

where the overall change in the tension of the tautest and slackest spans changed by


approximately the same values for each separate case of the crankshaft driving and the ISG

driving in the B-ISG system

Design variables are selected in the following chapter from the parameters that have been

analyzed in the present chapter The influence of changing parameters on the static tension

values for the various spans is further explored through an optimization study of the static belt

tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter

5

95

CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER

The objective of the optimization analysis is to minimize the absolute magnitude of the static

tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to

optimize the performance of the proposed Twin Tensioner design by using its properties as the

design variables for the objective function The optimization task begins with the selection of

these design variables for the objective function and then the selection of an optimization

method The results of the optimization will be compared with the results of the analytical

model for the static system and with the parametric analysis‟ data

51 Optimization Problem

511 Selection of Design Variables

The optimal system corresponds to the properties of the Twin Tensioner that result in minimized

magnitudes of static tension for the various belt spans Therein the design variables for the

optimization procedure are selected from amongst the Twin Tensioner‟s properties In the

parametric analysis of Chapter 4 the tensioner properties presented included

coupled stiffness kt

tensioner arm 1 stiffness kt1


tensioner pulley 1 diameter D3


tensioner pulley 1 initial coordinates [X3Y3] and

Optimization 96

tensioner pulley 2 initial coordinates [X5Y5]

It was observed in the former chapter that perturbations of the stiffness and geometric parameters

caused a change between the lowest and highest values for the static tension especially in the

case of perturbations in the geometric parameters diameter and coordinates Table 51

summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner

parameters‟ maximum and minimum values

Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties

Parameter Symbol

Original Tensions in TautSlack Span (Crankshaft

Mode) [N]

Tension at

Min | Max Parameter6 for

Crankshaft Mode [N]

Percent Change from Original for

Min | Max Tensions []

Original Tension in TautSlack Span (ISG Mode)

[N]

Tension at

Min | Max Parameter Value in ISG Mode [N]

Percent Change from Original Tension for


kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239

6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum

values of the parameter listed in each case such that the columns of identical colour correspond to each other For

the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these

cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed

Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697

The results of the parametric analyses for the Twin Tensioner parameters show that there is a

noticeable percent change between the initial tensions and the tensions corresponding to each of

the minima and maxima parameter values or in the case of the coordinates between the

minimum and maximum tensions for the spans Thus the parametric data does not encourage

exclusion of any of the tensioner parameters from being selected as a design variable As a

theoretical experiment the optimization procedure seeks to find feasible physical solutions

Hence economic criteria are considered in the selection of the design variables from among the

Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the

tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of

increasing the diameter and consequently its inertia increases the cost of material Material cost

is most significant in the manufacture process of pulleys as their manufacturing is largely

automated [4] Furthermore varying the structure of a pulley requires retooling which also

increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are

excluded from being selected as design variables The remaining tensioner properties the

stiffness parameters and the initial coordinates of the pulley centres are selected as the design

variables for the objective function of the optimization process

512 Objective Function amp Constraints

In order to deal with two objective functions for a taut span and a slack span a weighted

approach was employed This emerges from the results of Chapter 3 for the static model and

Chapter 4 for the parametric study for the static system which show that a high tension span and

Optimization 98

a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first

objective function of equation (51a) is described as equaling fifty percent of the absolute tension

value of the tautest span and fifty percent of the absolute tension value of the slackest span for

the case of the ISG driving only The second objective function uses a non-weighted approach

and is described as the absolute tension of the slackest span when the ISG is driving A non-

weighted approach is motivated by the phenomenon of a fixed difference that is seen between

the slackest and tautest spans of the optimal designs found in the weighted optimization

simulations Equations (51a) through to (51c) display the objective functions

The limits for the design variables are expanded from those used in the parametric analysis for

the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from

approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20

respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-

coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which

were prescribed in Chapter 4 Figure 41a and 41b

Aside from the design variables design constraints on the system include the requirement for

static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve

the goal of the objective function which is to minimize the absolute tensions in the system

However for an optimal solution to be feasible the movement of the tensioner arm must remain

within an appropriate threshold In practice an automotive tensioner arm for the belt

transmission may be considered stable if its movement remains within a 10 degree range of

Optimization 99

motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1

and θt2 respectively in the listed constraints

The optimization task is described in equations 51a to 52 Variables a through to g represent

scalar limits for the x-coordinate for corresponding ranges of the y-coordinate

Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to

(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)

119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892

1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891

1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg

(52)

The functions for the taut and slack spans represent the fourth and third span respectively in the

case of the ISG driving The equations for the tensions of the aforementioned spans are shown

in equation 51a to 51c and are derived from equation 353 The constraints for the

optimization are described in equation 52

Optimization 100

52 Optimization Method

A twofold approach was used in the optimization method A global search alone and then a

hybrid search comprising of a global search and a local search The Genetic Algorithm is used

as the global search method and a Quadratic Sequential Programming algorithm is used for the

local search method

521 Genetic Algorithm

Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The

optimization approach differs from classical search approaches by its ease of use and global

perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable

information random mutation and selection on the basis of fitness between generationsrdquo [58] to

form a robust search algorithm that requires minimal problem information [57] The parameter

sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are

measured against how well they allow the model to achieve the optimization task [58] GA is

stochastic which means that its algorithm uses random choices to generate subsequent sampling

points rather than using a set rule to generate the following sample This avoids the pitfall of

gradient-based techniques that may focus on local maxima or minima and end-up neglecting

regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of

the GA‟s search strategy it is able to search a population (a region of possible parameter sets)

faster than other optimization techniques The GA approach is viewed as a universal

optimization approach while many classical methods viewed to be efficient for one optimization

problem may be seen as inefficient for others However because GA is a probabilistic algorithm

its solution for the objective function may only be near to a global optimum As such the current

Optimization 101

state of stochastic or global optimization methods has been to refine results of the GA with a

local search and optimization procedure

522 Hybrid Optimization Algorithm

In order to enhance the result of the objective function found by the GA a Hybrid optimization

function is implemented in MATLAB software The Hybrid optimization function combines a

global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid

process refines the value of the objective function found through GA by using the final set of

points found by the algorithm as the initial point of the SQP algorithm The GA function

determines the region containing a global optimum and then the SQP algorithm uses a gradient

based technique to find a solution closer to the global optimum The MATLAB algorithm a

constrained minimization function known as fmincon uses an SQP method that approximates the

Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a

quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution

for the QP provides the search direction of the line search procedure used when each iteration is

performed [59]

53 Results and Discussion

531 Parameter Settings amp Stopping Criteria for Simulations

The parameter settings for the optimization procedure included setting the stall time limit to

200s This is the interval of time the GA is given to find an improvement in the value of the

objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time

limit allows for the optimization search to consistently converge without being limited by time

Optimization 102

The population size used in finding the optimal solution is set at 100 This value was chosen

after varying the population size between 50 and 2000 showed no change in the value of the

objective function The max number of generations is set at 100 The time limit for the

algorithm is set at infinite The limiting factor serving as the stopping condition for the

optimization search was the function tolerance which is set at 1x10-6

This allows the program

to run until the ratio of the change in the objective function over the stall generations is less than

the value for function tolerance The stall generation setting is defined as the number of

generations since the last improvement of the objective function value and is limited to 50

532 Optimization Simulations

The results of the genetic algorithm optimization simulations performed in MATLAB are shown

in the following tables Table 52a and Table 52b

Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function

Trial

No

Genetic Algorithm Optimization Method

Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103

Table 52b Computations for Tensions and Angles from GA Optimization Results

Trial No


Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571

For each trial above the GA function required 4 generations each consisting of 20 900 function

evaluations before finding no change in the optimal objective function value according to

stopping conditions

The results of the Hybrid function optimization are provided in Tables 53a and 53b below

Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function

Trial

No

Hybrid Optimization Method

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Table 53b Computations for Tensions and Angles from Hybrid Optimization Results

Trial No

Hybrid Algorithm Optimization Method


Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571

In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search

algorithm following the GA to find an optimal solution Once again the GA function

computed 4 generations which consisted of approximately 20 900 function evaluations before

securing an optimum solution

The simulation results of the non-weighted hybrid optimization approach are shown in tables

54a and 54b below

Optimization 105

Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective

Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations

Trial No Slackest Tension [N] Tautest Tension [N]

Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571

The weighted optimization data of Table 54a shows that the GA simulation again used 4

generations containing 20 900 function evaluations to conduct a global search for the optimal

system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function

evaluations) after its GA run which used the same number of generations and function

evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data

Optimization 106

for only one trial for each of the non-weighted GA and hybrid methods since only a single

optimal point exists in this case

533 Discussion

The optimal design from each search method can be selected based on the least amount of

additional pre-tension (corresponding to the largest magnitude of negative tension) that would

need to be added to the system This is in harmony with the goal of the optimization of the B-

ISG system as stated earlier to minimize the static tension for the tautest span and at the same

time minimize the absolute static tension of the slackest span for the ISG driving case As well

the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is

under static stability Therein the optimal solution may be selected as the design parameters

corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either

of the trials for the non-weighted optimization simulations

Given the ability of the Hybrid optimization to refine the results obtained in the GA

optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal

design from the weighted objective function approaches It is interesting to note that the Hybrid

case for the least slackness in belt span tension corresponds to a significantly larger number of

function evaluations than that of the remaining Hybrid cases This anomaly however does not

invalidate the other Hybrid cases since each still satisfy the design constraints Using the data

for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt

spans in both of the B-ISG‟s phases of operation are as follows

Optimization 107

Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a

Twin Tensioner


Optimized Original Optimized Original

Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152

Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803

Tensioner 1 ndash ISG 3540088 427197 -269178 -322803

ISG ndash Tensioner 2 2073813 28057 6895304 393645

Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645

Additional Pretension

Required (approximate) + 27000 +322803 + 27000 +322803

In Table 54b it is evident that the non-weighted class of optimization simulations achieves the

least amount of required pre-tension to be added to the system The computed tension results

corresponding to both of the non-weighted GA and Hybrid approaches are approximately

equivalent Therein either of their solution parameters may also be called the most optimal

design The Hybrid solution parameters are selected as the optimal design once again due to the

refinement of the GA output contained in the Hybrid approach and its corresponding belt

tensions are listed in Table 56 below

Optimization 108

Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System

with a Twin Tensioner










The results of the simulation experiments are limited by the following considerations

System equations are coupled so that a fixed difference remains between tautest and

slackest spans

A limited number of simulation trials have been performed

There are multiple optimal design points for the weighted optimization search methods

Remaining tensioner parameters tensioner pulley diameters and their stiffness have not

been included in the design variables for the experiments

The belt factor kb used in the modeling of the system‟s belt has been obtained

experimentally and may be open to further sources of error

Therein the conclusions obtained and interpretations of the simulation data can be limited by the

above noted comments on the optimization experiments

Optimization 109

54 Conclusion

The outcomes the trends in the experimental data and the optimal designs can be concluded

from the optimization simulations The simulation outcomes demonstrate that in all cases the

weighted optimization functions reached an identical value for the objective function whereas

the values reached for the parameters varied widely

The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA

optimization approach In reiteration in the presence of slack spans the tension value of the

slackest span must be added to the initial static tension for the belt Therein for the former case

an amount of at least 409N would need to be added to the 300N of pre-tension already applied to

the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of

the weighted Hybrid optimization approach and in both trials of the non-weighted optimization

approaches In the former the weighted Hybrid trial the tension value achieved in the slackest

span was approximately -27N signifying that only at least 27N would need to be added to the

present pre-tension value for the system The tension of the slackest span in the non-weighted

approach was approximately 0N signifying that the minimum required additional tension is 0N

for the system

The optimized solutions for the tension values in each span show that there is consistently a fixed

difference of 716448N between the tautest and slackest span tension values as seen in Tables

52b 53b and 54b This difference is identical to the difference between the tautest and slackest

spans of the B-ISG system for the original values of the design parameters while in its ISG

mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are

Optimization 110

greater than their original values except for that of the stiffness factor of tensioner arm 1

Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their

original values without exceptions The coordinates of the optimal solutions are within close

approximation to each other and also both match the regions for moderately low tension in

Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414

The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid

optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner

In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and

to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal

designs allow for one of the lowest amounts of additional pre-tension to be added to the system

out of all the findings from the simulations which were conducted

111

CHAPTER 6 CONCLUSION

61 Summary

The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a

Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a

Twin Tensioner It is established that the operating phases of the B-ISG system produced two

cases for static tension outcomes an ISG driving case and a crankshaft driving case The

approach taken in this thesis study includes the derivation of a system model for the geometric

properties as well as for the dynamic and static states of the B-ISG system The static state of a

B-ISG system with a single tensioner mechanism is highlighted for comparison with the static

state of the Twin Tensioner-equipped B-ISG system

It is observed that there is an overall reduction in the magnitudes of the static belt tensions with

the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric

and stiffness properties of the Twin Tensioner affecting the static tensions in the system are

analyzed in a parametric study It is found that there is a notable change in the static tensions

produced as result of perturbations in each respective tensioner property This demonstrates

there are no reasons to not further consider a tensioner property based solely on its influence on

the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in

the ISG mode of operation over that of the crankshaft mode of operation particularly in

excessively slack spans provides the motivation for optimizing the ISG case alone for static

tension The optimization method uses weighted and non-weighted approaches with genetic

algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in

Conclusion 112

which the magnitude of tension in the excessively slack spans in the ISG driving case are

significantly lower than in that of the original B-ISG Twin Tensioner design

62 Conclusion

The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system

include the following

1 The simulations of the dynamic model demonstrate that the mode shapes for the system

are greater in the ISG-phase of operation

2 It was observed in the output of the dynamic responses that the system‟s rigid bodies

experienced larger displacements when the crankshaft was driving over that of the ISG-

driving phase It was also noted that the transition speed marking the phase change from

the ISG driving to the crankshaft driving occurred before the system reached either of its

first natural frequencies

3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system

are consistently greater in its ISG operating phase than in its crankshaft operating phase

4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans

of a B-ISG system in comparison to when only a single tensioner mechanism is present

5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt

spans decrease or increase together for either phase of operation

6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a

significant influence on the magnitudes of the static tension of the slackest and tautest

belt spans The coordinate position of each pulley in the Twin Tensioner configuration

Conclusion 113

has the greatest influence on the belt span static tensions out of all the tensioner

properties considered

7 Optimization of the B-ISG system shows a fixed difference trend between the static

tension of the slackest and tautest belt spans for the B-ISG system

8 The values of the design variables for the most optimal system are found using a hybrid

algorithm approach The slackest span for the optimal system is significantly less slack

than that of the original design Therein less additional pretension is required to be added

to the system to compensate for slack spans in the ISG-driving phase of operation

63 Recommendation for Future Work

The investigation of the B-ISG Twin Tensioner encourages the following future work

1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner

properties would provide a complete picture as to the highest possible performance

outcome that the Twin Tensioner is able to have with respect to the static tensions

achieved in the belt spans

2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA

under weighted and other approaches would investigate the scope of optimal designs

available in the Twin Tensioner for the B-ISG system

3 A model of the system without the simplification of constant damping may produce

results that are more representative of realistic operating conditions of the serpentine belt

drive A finite element analysis of the Twin Tensioner B-ISG system may provide more

applicable findings

Conclusion 114

4 Investigation of the transverse motion coupled with the rotational belt motion in an

optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look

at the system under realistic conditions In addition the affect of the Twin Tensioner on

transverse motion can determine whether significant improvements in the magnitudes of

static belt span tensions are still being achieved

5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is

motivated by the fixed difference trend between the slackest and tautest belt span

tensions shown in Chapter 5 The modal decoupling of the system would allow for its

matrices comprising the equations of motion to be diagonalized and therein to decouple

the system equations Modal analysis would transform the system from physical

coordinates into natural coordinates or modal coordinates which would lead to the

decoupling of system responses

6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system

with a Twin Tensioner would increase understanding of the full impact of a Twin

Tensioner mechanism on the state of the B-ISG system It would be informative to

analyze the mode shapes of the first and second modes as well as the required torques of

the driving pulleys and the resulting torque of each of the tensioner arms The

observation of the dynamic belt span tensions would also give direction as to how

damping of the system may or may not be changed

7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states

including the Twin Tensioner‟s stability in each versus a B-ISG system with a single

tensioner would further demonstrate the improvements or dis-improvements in the Twin

Tensioner‟s performance on a B-ISG system

Conclusion 115

8 The influence of the belt properties on the dynamic and static tensions for a B-ISG

system with a Twin Tensioner can also be investigated This again will show the

evidence of improvements or dis-improvement in the Twin Tensioner‟s performance

within a B-ISG setting

9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be

designed and constructed Suitable instrumentation can be set-up to measure belt span

tensions (both static and dynamic) belt motion and numerous other system qualities

This would provide extensive guidance as to finding the most appropriate theoretical

model for the system Experimental data would provide a bench mark for evaluating the

theoretical simulation results of the Twin Tensioner-equipped B-ISG system

116

REFERENCES

[1] A Tonoli N Amati and E Zenerino Dynamic Modeling of Belt Drive Systems Effects of

the Shear Deformations Journal of Vibration and Acoustics vol 128 pp 555 2006

[2] W Cai Comparison and Review of Electric Machines for Integrated Starter Alternator

Applications IEEE Industry Applications Conference vol 39th IAS Annual Meeting pp

386-393 Oct 3 2004

[3] J E Walters R J Krefta G Gallegos-Lopez and G T Fattic Technology Considerations

for Belt Alternator Starter Systems SAE Technical Papers 2004 Document no 2004-01-

0566

[4] Litens Automotive Group Ltd Project Meeting Apr 29 2008

[5] J N Fawcett Improvements in belt tension setting procedures on internal combustion

engines Proceedings of the Institution of Mechanical Engineers Part D Journal of

Automobile Engineering vol 213 pp 119 1999

[6] J W Zu NSERC Proposal 2006

[7] G Tamai Saturn Engine Stop-start System with an Automatic Transmission SAE

Technical Papers 2001 Document no 2001-01-0326

[8] D Naunin Hybrid Electric Vehicles ndash the Technology of the Near Future Business

Briefing Global Automotive Manufacturing amp Technology 2003

[9] A Ebron and R Cregar Introducing Hybrid Technology NAFTC Earth Toys Inc

[Electronic] Jun 2005 [2007 Feb 1] Available at

httpwwwearthtoyscomemagazinephpissue_number=050601amparticle=naftcamp

[10] G T Fattic J E Walters and F S Gunawan Cold Starting Performance of a 42-Volt

Integrated Starter Generator System SAE Technical Papers 2002 Document no 2002-

01-0523

References 117

[11] R R Henry B Lequesne S Chen J J Ronning and Y Xue (Delphi Automotive Systems)

Belt-driven Starter-Generator for Future 42-Volt Systems SAE Technical Paper

Document no 2001-01-0728

[12] FEV Optimization of Hybrid Concepts through Simulation Engineering Services Hybrid

Technology [Electronic] 2006 [2007 Jun 11] Available at

httpwwwfevcomcontentpublicdefaultaspxid=557

[13] K Itagaki T Teratani K Kuramochi S Nakamura T Tachibana H Nakai and Y Kamijo

(Toyota Motor Corp) Development of the Toyota Mild-Hybrid System (THS-M) SAE


[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff

NAFTC Instructors Digest 1(1) pp Nov 10 2006 Available

httpwwwnaftcwvuedutechnicalNAFTCdigestvol1vol1htm

[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60

MPG US [Electronic] May 25 2007 [2007 Jun 4] Available at

httpwwwgreencarcongresscom200705bmw_to_apply_sthtml

[16] H Jeneacute E Scheid and H Kemper Hybrid electric vehicle (HEV) concepts - fuel savings

and costs ICAT 2004 Conference on Automotive Technology Future Automotive

Technologies on Powertrain and Vehicle IstanbulTurkey vol19 Feb 1 2007 Available

httpwwwosdorgtricat2004htm

[17] L Gaedt D Kok (Ford Forschungszentrum Aachen GmbH) C Goodfellow D Tonkin

(Ricardo UK Ltd) C Picod (Valeo Electrical Systems) and M Neu (Gates Corportation)

HyTrans ndash A micro-hybrid transit from ford (Year unknown) Available

wwwosdorgtr18pdf

[18] S Chen B Lequesne R R Henry Y Xue and J J Ronning Design and Testing of a

Belt-driven Induction Starter-generator IEEE Transactions on Industry Applications vol

38 pp 1525-1533 NovDec 2002

References 118

[19] J L Broge (2001 Focus on electronics ndash Siemens startergenerator for 2002 SAE

International Automotive Engineering International Online Mar 6 2005 Available

wwwsaeorgautomagelectronics01-2001

[20] J A MacBain (Delphi Automotive Systems Energenix Center) Simulation Influence in

the Design Process of Mild Hybrid Vehicles SAE Technical Papers 2002 Document no

2002-01-1196

[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P

Sczomak (General Motors Powertrain) J P Absmeier (Delphi Corp) and G T Fattic

(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE

Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569

[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical

Sciences vol 12 pp 1053-1063 1970

[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life

Accessory Drive SAE Papers 1979 Document no 760699

[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for

Dynamic Stability Journal of Vibration Acoustics Stress and Reliability in Design

Transactions of the ASME vol 107 pp 282-290 July 1985

[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive

Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp

406-413 1996

[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley

System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical

Design Transactions of the ASME vol 118 pp 306-311 Jun 1996

[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive

Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical


References 119

[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of

Sound and Vibration vol 222 pp 259-279 1999

[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a

decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005

[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with

Two-pulley Tensioners University of Toronto 2005

[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has

electronic actuator associated with clutch spring for engaging International

WO2005119089-A1 Jun 6 2005 2005

[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter

system Feb 27 2002

[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an

envelope drive of a device especially a belt drive of a starter generator of an internal

combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003

[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal

combustion engines to drive accessories in an automobile Germany DE19849886-A1

May 11 2000 1998

[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)

Belt tensioning device for belt drive between engine and starter generator of motor

vehicle has self-aligning bearing that supports auxiliary unit and provides working force to

tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003

[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle

engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002

[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with

tensioner asymmetrically biased in direction tending to cause power transmission belt to be

under tension International WO2003038309-A1 May 8 2003 2002

References 120

[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory

drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005

[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving

internal combustion engine units has arrangement for demand regulated setting of tension

consisting of unit with housing with limited rotation and pulley German DE10044645-

A1 Mar 21 2002 2000

[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC

engine has locking system for locking tensioning element in an engine operating mode

locking system is directly connected to pivot arm follows arm control movements

German DE10159073-A1 Jun 12 2003 2001

[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp

belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via

freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005

[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter

generator and a belt drive has tensioner with spring and counter hydraulic force Germany

DE10359641-A1 Jul 28 2005 2003

[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction

drive especially belt drive for short-duration driving of starter generator has tensioning

device with lever arm deflectable against restoring force and with end stop limiting

deflection travel Europe EP1738093-A1 Jan 3 2007 2005

[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as

combustion engine has belt or chain drive with wheels turning and connected with starter

generator and unit has two idlers arranged at clamping arm with machine stored by shock

absorber Germany DE102004012395-A1 Sep 29 2005 2004

[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly

has limited movement at the starter belt section tensioner roller bringing it into a dead point

position on starting the motor International WO2006108461-A1 Oct 19 2006 2006

References 121

[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes

tension device positioned within belt drive system having combination starter United

States US2001007839-A1 Jul 12 2001 2001

[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng

KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator

in vehicle involves shifting auto-tensioners between lock state and free state to adjust

tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005

2003

[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has

tensioner tightening chain while cranking engine and slackens chain after start of engine

provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998

[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has

cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder

acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005

[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a

thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005

[51] M Zhen University of Toronto and Litens Automotive Group Ltd FEAD vol 50

2005

[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at

httpwwwgizmologynetmotorshtm

[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov

23 2007

[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7

[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf

[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer

Science [Electronic] Feb 14 2008 [2008 June] Available at

httpwwwcsuiowaedu~jonesstepphysicshtml

References 122

[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project

[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley

amp Sons Ltd 2001 pp 81-85

[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford

[httpwwwengoxacuksamppubsgawbreppdf]

[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007

123

APPENDIX A

Passive Dual Tensioner Designs from Patent Literature

Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1

Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]

Figure A1 label identification 1 ndash tightner 2 ndash belt drive

3 ndash starter generator

4 ndash internal-combustion engine

4‟ ndash crankshaft-lateral drive disk

5 ndash generator housing

6 ndash common axis of rotation

7 ndash featherspring of tiltable clamping arms

8 ndash clamping arm


10 11 ndash idlers

12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum

14 ndash carry-hurries 15 ndash generator wave

16 ndash bush

17 ndash absorption mechanism

18 18‟ ndash support arms

19 19‟ ndash auxiliary straight lines

20 ndash pipe

21 ndash torsion bar

22 ndash breaking through

23 ndash featherspring

24 ndash friction disk

25 ndash screw connection 26 ndash Wellscheibe

(European Patent Office May 2007) [33]

Appendix A 124

Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH


Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)

epespacenetcom [34]

Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no

WO0026532-A1

Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]

Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH



epespacenetcom [34]

Appendix A 125

Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH



epespacenetcom [34]

Figure A2a through to A2d label identification 10 ndash engine wheel

11 ndash [generator] 13 ndash spring

14 ndash belt

16 17 ndash tensioning pulleys

18 19 ndash springs

20 21 ndash fixed points

25ab ndash carriers of idlers

25c ndash gang bolt

(European Patent Office June 2007) [34]

Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1

Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)

epespacenetcom [35]

Figure A3 label identification

Appendix A 126

10 12 ndash belt pulleys

14 ndash auxiliary unit

16 ndash belt

22-1 22-2 ndash belt tensioners


Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2

Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)

epespacenetcom [36]

Figure A4 label identification 12 ndash belt

14 ndash tensioner

16 ndash generator pulley

18 ndash crankshaft pulley

22 ndash slack span 24 ndash tight span

32 34 ndash arms

33 35 ndash pulley


Appendix A 127

Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A

Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)

epespacenetcom [37]

Figure A5 label identification 12 ndash motorgenerator

14 ndash motorgenerator pulley 26 ndash belt tensioner

28 ndash belt tensioner pulley

30 ndash transmission belt


Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1

Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)

epespacenetcom [38]

Appendix A 128

Figure A6 label identification 28 ndash tensioner

32 ndash carrier arm

34 ndash secondary carrier arm

46 ndash tensioner pulley

58 ndash secondary tensioner pulley


Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1

Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)

epespacenetcom [39]

Figure A7 label identification 2 ndash internal combustion engine

3 ndash traction element

11 ndash housing with limited rotation 12 13 ndash direction changing pulleys


Appendix A 129

Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1


epespacenetcom [40]

Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1


epespacenetcom [40]

Appendix A 130

Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no

DE10159073-A1


epespacenetcom [40]

Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]

2 ndash idler

3 ndash drawing means

4 ndash swivel arm

5 ndash axis of rotation

6 ndash drawing means impulse [belt]

7 ndash crankshaft


9 ndash bolting volume 10a ndash bolting device system

10b ndash bolting device system

10c ndash bolting device system

11 ndash friction body

12 ndash lateral surface

13 ndash bolting tape end


15 ndash control member


17 ndash base

18 ndash pylon

19 ndash hub

20 ndash annular gap

21 ndash Gleitlagerbuchse

23 ndash [nil]


24 ndash turning camps 25 ndash teeth

26 ndash elbow levers

27 ndash clamping wedge

28 ndash internal contour

29 ndash longitudinal guidance

30 ndash system

31 ndash sensor

32 ndash clamping gap


Appendix A 131



epespacenetcom [42]

Figure A9 label identification 8 ndash cylinder

10 ndash rod

12 ndash spring plate

13 ndash spring

14 ndash pressure lead


Appendix A 132

Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1

Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]

Figure A10 label identification 4 ndash pulley

5 ndash hydraulic element 11 ndash freewheel

12 ndash shaft



Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)

epespacenetcom [43]

Figure A11 label identification 1 ndash traction drive

2 ndash traction belt


Appendix A 133

7 ndash tension device

9 ndash lever arm

10 ndash guide roller

16 ndash end stop



Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]

Figure A12 label identification 1 ndash belt drive

2 ndash belts

3 ndash wheel of the internal-combustion engine

4 ndash wheel of a Nebenaggregats

5 ndash wheel of the starter generator

6 ndash clamping unit

7 ndash idler

8 ndash idler

9 ndash scale beams

10 ndash drive place


12 ndash camps

13 ndash coupling point

14 ndash shock absorber element

15 ndash arrow


Appendix A 134

Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1

Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]


2 ndash wheel of the crankshaft KW

3 ndash wheel of a climatic compressor AC

4 ndash wheel of a starter generator SG

5 ndash wheel of a water pump WP

6 ndash first clamping system

7 ndash first tension adjuster lever arm

8 ndash first tension adjuster role

9 ndash second clamping system

10 ndash second tension adjuster lever arm

11 ndash second tension adjuster role 12 ndash guide roller

13 ndash drive-conditioned Zugtrum

(generatorischer enterprise (GE))

13 ndash starter-conditioned Leertrum

(starter enterprise (SE))

14 ndash drive-conditioned Leertrum (GE)

14 ndash starter-conditioned Zugtrum (SE)

14a ndash drive-conditioned Leertrumast (GE)

14a ndash starter-conditioned Zugtrumast (SE)

14b ndash drive-conditioned Leertrumast (GE)

14b ndash starter-conditioned Zugtrumast (SE)


Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no

US20010007839-A1

Appendix A 135


epespacenetcom [46]

Figure A14 label identification E - belt

K - crankshaft

R1 ndash first tension pulley

R2 ndash second tension pulley

S ndash tension device

T ndash drive system

1 ndash belt pulley

4 ndash belt pulley


Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding

to patent no JP2005083514-A

Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]

Figure A15 label identification 1 ndash Pulley for Starting

2 ndash Crank Pulley

3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller

5 ndash Idler Pulley

6 ndash Belt

7c ndash Starter generator control section

7d ndash Idle stop control means

8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine

11 ndash Starter Generator

12 ndash Driving Shaft

Appendix A 136

7 ndash ECU

7a ndash 1st auto tensioner control section (the 1st auto

tensioner control means)

7b ndash 2nd auto tensioner control section (the 2nd auto


13 ndash Air-conditioner Compressor

14a ndash 1st auto tensioner

14b ndash 2nd auto tensioner

18 ndash Water Pump

(Industrial Property Digital Library May 2007) [47]

Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2

Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]

Figure A16 label identification 3 ndash chain [or belt]

5 ndash tensioner

4 ndash belt pulley


Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A

Appendix A 137

Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)

epespacenetcom [49]

Figure A17 label identification 5d - flange

6 ndash tensile strength spring

10 ndash actuator

10c ndash cylinder

12 ndash rod

20 ndash hydraulic damper mechanism

21 ndash cylinder nut

22 ndash screw bolt


Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos

EP1658432 and WO2005015007

Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007

accessed May 2007) epespacenetcom [50]

Figure A18 abbreviated list of label identifications

10 ndash starter

22 ndash shaft section

23 ndash free front end

80 ndash pulley

200 ndash support element

206 - arm


The author notes that the list of labels corresponding to Figures A1a through to A7 are generated

from machine translations translated from the documentrsquos original language (ie German)

Consequently words may be translated inaccurately or not at all

138

APPENDIX B

B-ISG Serpentine Belt Drive with Single Tensioner

Equation of Motion

The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are

shown The EOM has been derived similarly to that of the same system with a twin tensioner

that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are

applicable for the single tensioner B-ISG system as well

Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are

numbered 1 through 4 and their associated belt spans are numbered accordingly

Figure B1 Single Tensioner B-ISG System

Appendix B 139

The free-body diagram for the ith non-tensioner pulley in the system shown above is found in

Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular

displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The

required torque is Qi the angular damping is Ci and the tension of the ith span is Ti

Figure B2 Free-body Diagram of ith Pulley

The positive motion designated is assumed to be in the clockwise direction The radius for the

ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows

I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)



Appendix B 140

A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational

stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular

rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively

Figure B3 Free-body Diagram of Single Tensioner

From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of

orientation for the ith belt span is designated by 120573119895119894

minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)

It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t

(B5)

Appendix B 141

partθt = θt minus θto (B6)

The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices

that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894

119887 ) respectively

119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)

The initial tension is represented by To and the initial angle of the tensioner arm is represented

by 120579119905119900 The equation for the tension of the ith span is shown in the following equations


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)

T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3

b ∙ [R3 ∙ θ 3 minus R4 ∙

θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)


b ∙ [R4 ∙ θ 4 minus R1 ∙


Tprime = Ti minus To (B12)

Tt = T3 = T4 (B13)

Appendix B 142

The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to

B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C

stiffness matrix K and the required torque matrix Q as well as the angular displacement

velocity and acceleration matrices 120521 120521 and 120521 respectively

119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)

The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of

the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt

damping for each span and βji is the angle of orientation for the span between the jth and ith

pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical

subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque

matrix and represents the required torque for the tensioner arm As well the term It1 represents

the moment of inertia for the tensioner arm

Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)

Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD

[51] software for dynamic and static cases This allowed for the methodology for deriving the

EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural

frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft

required torque only and the dynamic tensions were solved from the EOM as described in (331)

to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3

This permitted verification of the complete derivation methodology and allowed for comparison

of the static tension of the B-ISG system belt spans in the case that a single tensioner is present

and in the case that a Twin Tensioner is present [51]

145

APPENDIX C

MathCAD Scripts

C1 Geometric Analysis

1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5

Figure C1 Schematic of B-ISG

System with Twin Tensioner

Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )

XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146

i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center

Adjust last number in range variable p to correspond to number of pulleys

p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise

condition1 p( ) acos

XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

condition2 p( ) 2 acos

XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1

condition1 p( ) condition2 p( )

Lfi Lbi - connection belt span lengths

D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm

D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi

(or Pfi) respecti vely l

f p( ) atanLf p( ) mm

Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147

XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley

XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley

XYf2 p( ) XYk p( ) 0

Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )

XYb2 p( ) XYk p( ) 0

Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )

XYfb2 p( ) XYk p( ) 0

Dk p( )

2 mmcos p( ) b p( ) XY

k p( ) 1

Dk p( )

2 mmsin p( ) b p( )

XYbf2 p( ) XYk p( ) 0

Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )

Row 1 --gt Pulley 1 Row i --gt Pulley i

XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1

Row 1 --gt Pulley 2 Row i --gt Pulley i+1

XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210

250Figure C2 Possible Contact Points

250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y

500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x

Appendix C 150

Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley

XY15 XYbf2iT 4

XY12 XYfiT 0

Pulley 1 contact pts

XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4


By observation the lengths of span i is the following

L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li

L1

L2

L3

L4

L5

mm

i Angle between horizontal and span of ith pulley

i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151

Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5

12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51

Wrap angle i for the ith pulley

1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp

Input Data for B-ISG System

Kt 20626Nm

rad (spring stiffness between tensioner arms 1

and 2)

Kt1 10314Nm

rad (stiffness for spring attached at arm 1 only)

Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis

I C K moment of inertia damping and stiffness matrices respectively

u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)

RaD

2

Appendix C 153

RaD

2

Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms

Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2

Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms

1000kg

m3

CrossArea 693mm2

0 M CrossArea Lbelt M 0086kg

cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0

Kbi 1( ) Lt1 mm sin t1 32


0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk

New Equations of Motion for Dual Drive System

I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column

IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0

Ia stack I1kgmm2 IAT 0

T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T

Ic stack I1kgmm2 ICT 3

T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc

CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T

Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley

Modified Submatricies for ISG Driving Phase --gt CS Driving Phase

Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN

case (ie ISG driving case)

H n( ) 1 n 750if

0 n 750if

lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n

reaches start speed)

I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if

I22mod n( )submatrix Ic 1 6 1 6( )

UnitsOf I( )H n( ) 1if

submatrix Ia 1 6 1 6( )


K11mod n( )

Kc0 0

UnitsOf K( )H n( ) 1if

Ka0 0


C11modn( )

Cc0 0

UnitsOf C( )H n( ) 1if

Ca0 0


K22mod n( )submatrix Kc 1 6 1 6( )


submatrix Ka 1 6 1 6( )


C22modn( )submatrix Cc 1 6 1 6( )


submatrix Ca 1 6 1 6( )


















Appendix C 157

2mod n( ) I22mod n( )1

K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )

2

EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0

max eigenvec 2mod n( ) mod n( )0

eigenvec 2mod n( ) mod n( )1










modeshapesmod n( ) stack nmod n( )T

EVmodn( )

t 0 0001 1

mode1a t( ) EVmod100( )0

sin nmod 100( )0 t mode2a t( ) EVmod100( )1

sin nmod 100( )1 t

mode1c t( ) EVmod800( )0

sin nmod 800( )0 t mode2c t( ) EVmod800( )1

sin nmod 800( )1 t

Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively

The system equation becomes

I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc

I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0

Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1

Torque requirement for crank shaft Pulley 1

qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm

Torque requirement for alternator shaft Pulley 4

qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158

Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on

engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if

Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )

j mod n( ) C22modn( )

mmod n( ) Ymod n( )( )1

K21mod n( ) j mod n( ) C21modn( )

Crankshaft amp ISG required torques

Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration

Ac n( ) 650 1 n( )Ac n( )

n( ) 2

Let Qm = QmQ1(n) for n lt 750

and Qm = QmQ4(n) for n gt 750

Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4

qcmod n( ) K11mod n( ) mod n( ) 2

I11mod n( )


1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )

H n( ) 1if

Qc0 H n( ) 0if

qamod n( ) K11mod n( ) mod n( ) 2

I11mod n( )


1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0

H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000

(ISG torque requirement alternate equation)

Appendix C 159

Dynamic tensioner arm torques

Qtt1mod n( )Kt Kt1

UnitsOf Kt( )j mod n( )

Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160

Dynamic belt span tensions

d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

(tensions for driving pulley belt spans)

Appendix C 161

C3 Static Analysis

Static Analysis using K Tk amp Q matricies amp Ts

For static case K = Q

Tension T = T0 + Tks

Thus T = K-1QTks + T0

Q1 68N m Qt1 0N m Qt2 0N m Ts 300N

Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D

MATLAB Functions amp Scripts

D1 Parametric Analysis

D11 TwinMainm

The following function script performs the parametric analysis for the B-ISG system with a Twin

Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a

single input parameter for the called function TwinTenStaticTensionm The main function takes

an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and

geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to

increment by six percent over a range from sixty percent below its initial value to sixty percent

above its initial value The coordinate parameters are incremented through a mesh of Cartesian

points with prescribed boundaries The TwinMainm function plots the parametric results

______________________________________________________________________________

clc

clear all

Static tension for single tensioner system for CS and Alt driving

Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193

Pertubations of initial parameters

Kt = (Kto-060Kto)006Kto(Kto+060Kto)

Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)


D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163

[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)

end

figure

[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on

H3 = line(Kt()T(5)ParentAX(1)) hold on

H4 = line(Kt()Ta(3)Parent AX(2)) hold off

set(AXYLimModeautoYTickModeauto)

title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)

xlabel(Kt (Nmrad))

set(get(AX(1)Ylabel)StringCS Span Tension (N))

set(get(AX(2)Ylabel)StringISG Span Tension (N))

set(H1LineStyle-Colorb)

set(H2LineStyle-Colorg)

set(H3LineStyleColorget(H1Color))


legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG

Slackest SpanLocationBest)

for i = 1s

[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)

end

figure

[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on

H3 = line(Kt1()T(5)ParentAX(1)) hold on

H4 = line(Kt1()Ta(3)Parent AX(2)) hold off


title(Static Tension vs Tensioner Arm 1 Stiffness)

xlabel(Kt1 (Nmrad))









for i = 1s

[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)

end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s

[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)

end

figure

[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on

H3 = line(D3()T(5)ParentAX(1)) hold on

H4 = line(D3()Ta(3)Parent AX(2)) hold off


title(Static Tension vs Tensioner Pulley 1 Diameter)

xlabel(D3 (m))









for i = 1s

[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)

end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)

[ixxiyy] = meshgrid(1m1n)

minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641

X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)

Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)

for i = 1n

for j = 1m

if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-

006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-

024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-

11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b

lt= y lt= circle4

[T(ij)Ta(ij)] =

TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)

else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar

title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)

xlabel(X-coordinate (m))

ylabel(Y-coordinate (m))

zlabel(CS Span Tension (N))

figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar

title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)



Appendix D 166


figure

Za4 = squeeze(Ta(4))

surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar

title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)



zlabel(ISG Span Tension (N))

figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar

title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)




minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if

(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-

(X5(ij)^2))))

[T(ij)Ta(ij)] =

TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))

elseif

((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0

146468))

[T(ij)Ta(ij)] =


elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-

024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar





D12 TwinTenStaticTensionm

The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a

Twin Tensioner This function returns 3 vectors the static tension of each belt span in the

crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid

body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm

stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner

arm pulley coordinates The function is called in the parametric analysis solution script

TwinMainm and in the optimization solution script OptimizationTwinm

D2 Optimization

D21 OptimizationTwinm

The following script is for the main function OptimizationTwinm It performs an optimization

search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector

containing values for the design variables The program calls the objective function

objfunTwinm and the constraint function confunTwinm The program can perform a genetic

algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search

The optimal solution vector corresponding to the design variables and the optimal objective

function value is returned The program inputs the optimized values for the design variables into

TwinTenStaticTensionm This called function returns the optimized static state of tensions for

the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in

the ISG driving phase

______________________________________________________________________________

clc

clear all

Initial values for variables

Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193

w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)

Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto

minKt1 = Kt1o - 1Kt1o

maxKt1 = Kt1o + 1Kt1o



minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

ObjectiveFunction = objfunTwin

nvars = 7 Number of variables

ConstraintFunction = confunTwin

Uncomment next two lines (and comment the two functions after them) to use GA algorithm

for optimization

options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3

minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5

maxY5]PopulationSize100Displayfinal)

[wfvalexitflagoutput] =

ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)

fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not

provided



maxY5]PopulationSize100HybridFcnfmincon fminconOptions)

[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)

[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))

D22 confunTwinm

The constraint function confunTwinm is used by the main optimization program to ensure

input values are constrained within the prescribed regions The function makes use of inequality

constraints for seven constrained variables corresponding to the design variables It takes an

input vector corresponding to the design variables and returns a data set of the vector values that

satisfy the prescribed constraints

Appendix D 170

D23 objfunTwinm

This function is the objective function for the main optimization program It outputs a value for

a weighted objective function or a non-weighted objective function relating the optimization of

the static tension The program takes an input vector containing a set of values for the design

variables that are within prescribed constraints The description of the function is similar to

TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the

value of the objective function

171

VITA

ADEBUKOLA OLATUNDE

Email adebukolaolatundegmailcom

Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto

Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster

University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate

degree in mechanical engineering at the University of Toronto with a specialization in

mechanical systems dynamics and vibrations and environmental engineering In September

2008 she completed the requirements for the Master of Applied Science degree in Mechanical

Engineering She has held the position of teaching assistant for undergraduate courses in

dynamics and vibrations Adebukola has completed course work in professional education She

is a registered member of professional engineering organizations including the Professional

Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical

Engineers and the National Society of Black Engineers She intends to practice as a professional

engineering consultant in mechanical design

ii

ABSTRACT

DESIGN AND ANALYSIS OF A TENSIONER FOR A BELT-DRIVEN INTEGRATED

STARTER-GENERATOR SYSTEM OF MICRO-HYBRID VEHICLES


Master of Applied Science


University of Toronto 2008

The thesis presents the design and analysis of a Twin Tensioner for a Belt-driven Integrated

Starter-generator (B-ISG) system The B-ISG is an emerging hybrid transmission closely

resembling conventional serpentine belt drives Models of the B-ISG system‟s geometric

properties and dynamic and static states are derived and simulated The objective is to reduce

the magnitudes of static tension in the belt for the ISG-driving phase A literature review of

hybrid systems serpentine belt drive modeling and automotive tensioners is included A

parametric study evaluates tensioner parameters with respect to their impact on static tensions

Design variables are selected from these for an optimization study The optimization uses a

genetic algorithm (GA) and a hybrid GA Results of the optimization indicate the optimal

system contains spans with static tensions that are significantly lower in magnitude than that of

the original design Implications of the research on future work are discussed in closing

iii





needed it


amp


sweeter

iv

ACKNOLOWEDGEMENTS





and help


research work









v

CONTENTS

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES xi

LIST OF SYMBOLS xvi


11 Background 1

12 Motivation 3




21 Introduction 7

22 B-ISG System 8


2211 Full Hybrids 9











25 Summary 24

vi


31 Overview 25













35 Simulations 50









36 Summary 69


41 Introduction 71

42 Methodology 71






vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














iii





needed it


amp


sweeter

iv

ACKNOLOWEDGEMENTS





and help


research work









v

CONTENTS

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES xi

LIST OF SYMBOLS xvi


11 Background 1

12 Motivation 3




21 Introduction 7

22 B-ISG System 8


2211 Full Hybrids 9











25 Summary 24

vi


31 Overview 25













35 Simulations 50









36 Summary 69


41 Introduction 71

42 Methodology 71






vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














iv

ACKNOLOWEDGEMENTS





and help


research work









v

CONTENTS

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES xi

LIST OF SYMBOLS xvi


11 Background 1

12 Motivation 3




21 Introduction 7

22 B-ISG System 8


2211 Full Hybrids 9











25 Summary 24

vi


31 Overview 25













35 Simulations 50









36 Summary 69


41 Introduction 71

42 Methodology 71






vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














v

CONTENTS

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES xi

LIST OF SYMBOLS xvi


11 Background 1

12 Motivation 3




21 Introduction 7

22 B-ISG System 8


2211 Full Hybrids 9











25 Summary 24

vi


31 Overview 25













35 Simulations 50









36 Summary 69


41 Introduction 71

42 Methodology 71






vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














vi


31 Overview 25













35 Simulations 50









36 Summary 69


41 Introduction 71

42 Methodology 71






vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














vii

44 Conclusion 92











533 Discussion 106

54 Conclusion 109


61 Summary 111

62 Conclusion 112


REFERENCES 116

APPENDICIES 123









D11 TwinMainm 162


D2 Optimization 168


viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














viii

D22 confunTwinm 169

D23 objfunTwinm 170

VITA 171

ix

LIST OF TABLES

















Function


x


Twin Tensioner


Twin Tensioner

xi

LIST OF FIGURES

21 Hybrid Functions







Bodies




Rigid Bodies







xii















amp 2





xiii






















EP1420192-A2 and DE10253450-A1




WO0026532-A1







xiv



A1



DE10159073-A1


DE10159073-A1


DE10159073-A1





A1


A1and WO2006108461-A1


US20010007839-A1






EP1658432 and WO2005015007



xv




xvi

LIST OF SYMBOLS

Latin Letters



cb Belt damping
















)


xvii











n Engine speed

N Motor speed











xviii






Ts Stall torque



XYfi XYbi XYfbi


XYf2i XYb2i


Greek Letters


centres


120597θti(t) 120579 ti(t)

120579 ti(t)


ith tensioner arm




Lbi respectively


xix


pulley













1


11 Background



















Introduction 2






















Introduction 3













12 Motivation









Introduction 4


















performance

Introduction 5



















Introduction 6



7


21 Introduction



















Literature Review 8




hybrid classes




22 B-ISG System

221 ISG in Hybrids












Literature Review 9





2211 Full Hybrids


































2212 Power Hybrids









2213 Mild Hybrids












2214 Micro Hybrids




















































layoutrdquo [11]


























































analysis


















the belt spans only









drive system

























































Inclusion of







and









Bayerische

Motoren Werke

AG

Patents EP1420192-A2 DE10253450-A1 [33]

Design Approach







Design Approach









Daimler Chrysler

AG

Patents DE10324268-A1 [35]

Design Approach




Dayco Products

LLC


Design Approach



span





Design Approach




McVicar et al

(Firm General

Motors Corp)

Patent US20060287146-A1 [38]

Design Approach




taut span of belt



start-up

INA Schaeffler

KG et al



WO2006108461-A1 et al [45]

Design Approach




sections of belt











al)









ISG start-up mode

Litens Automotive

Group Ltd


Design Approach







Mitsubishi Jidosha

Eng KK

Mitsubishi Motor

Corp

Patents JP2005083514-A [47]

Design Approach





Design Approach








Design Approach








Valeo Equipment

Electriques

Moteur

Patents EP1658432 WO2005015007 [50]

Design Approach











25 Summary











25


31 Overview























































[modified] [51]





i acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31a)

i 2 acos

Xi 1

Xi

Xi 1

Xi

2

Yi 1

Yi

2

Yi 1

Yi

if

(31b)




Lfi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32a)

Lbi

Xi 1

Xi

2

Yi 1

Yi

2

Ri 1

Ri

2

(32b)



perimeter


(33a)

(33b)







(34a)

(34b)

(34c)

(34d)

bi atan

Lbi

Ri

Ri 1




pulley

(35a)

(35b)

(35c)

(35d)









Table 31




Point

Backwards Contact

Point

Belt Span

Connection









































I ∙ θ = ΣM (35)





























119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)


b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)


b ∙ [R2 ∙ θ 2 minus R3 ∙



b ∙ [R3 ∙ θ 3 minus R4 ∙




b ∙ [R4 ∙ θ 4 minus R5 ∙



b ∙ [R5 ∙ θ 5 minus R1 ∙






Tt1 = T2 = T3 (321)

Tt2 = T4 = T5 (322)





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)












119816 =

I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(324)

119810 =

c1

b ∙ R12 + c5


b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5



b ∙ R22 + c1




b ∙ R32 + c2


b ∙ R3 ∙ R4 0 C36 0


b ∙ R42 + c3





b ∙ R4 ∙ R5 c5b ∙ R5

2 + c4b ∙ R5

2 + c5 0 C57



(325a)

C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882

119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)

C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884

119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)


119818 =

k1

b ∙ R12 + k5

b ∙ R12 minusk1

b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5



b ∙ R22 + k1

b ∙ R22 minusk2



b ∙ R32 + k2

b ∙ R32 minusk3




b ∙ R42 + k3

b ∙ R42 minusk4




b ∙ R4 ∙ R5 k5b ∙ R5

2 + k4b ∙ R5





(326a)

k119894b =

Kb


2 + Ri ∙ϕi

2

(326b)

120521 =

θ1

θ2

θ3

θ4

θ5

partθt1

partθt2

(327)

119824 =

Q1

Q2

Q3

Q4

Q5

Qt1

Qt2

(328)















120521119940 =

1205791

1205794

1205792

1205793

1205795

1205971205791199051

1205971205791199052

and 120521119938 =

1205794

1205791

1205792

1205793

1205795

1205971205791199051

1205971205791199052

(329a amp b)

119816119940 =

I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

and 119816119938 =

I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2

(329c amp d)



phase respectively




(330)











ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)





θm = 120495119846 ∙ sin(ω ∙ t) (331d)







I1 120782120782 119816120784120784

θ 1120521 119846

+ C11 119810120783120784119810120784120783 119810120784120784

θ 1120521 119846

+ K11 119818120783120784

119818120784120783 119818120784120784 θ1

120521119846 =

QCS ISG

119824119846 (332)

1









θm = 120495119846 ∙ sin(ω ∙ t) (333)



120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)








Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)



qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)





ω = 2 ∙ ωcs = 4 ∙ π ∙ n

60

(337)




result of (338)

θ 1CS = 650 ∙ sin(ω ∙ t) (338)

θ1CS = minus650

ω2sin(ω ∙ t)

(339a) where

Θ1CS = minus650

ω2

(339b)


























nISG = nCS ∙DCS

DISG

(341)


















1ISG

(345a)


ωISG2

(345b)




60

(346a)


ω =2 ∙ π

60∙ nCS ∙

DCS

DISG

(346b)



Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)


Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)




119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)

where

120495119847 = Θ1

120495119846 (350)





119827prime = 119827119844 ∙ 120521 (351)

119824 = 119818 ∙ 120521 (352)











119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3

35 Simulations
















well










4 ISG Pulley 6820 [24759 16664]





Pulley Forward

Connection Point

Backward

Connection Point

Wrap

Angle

ϕi (deg)

Angle of

Belt Span

βji (deg)

Length of

Belt Span

Li (mm)

1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828

2 Air

Conditioning [275299-5717] [220484 -115575] 101425 277528 14064

3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658

4 ISG [218374184225] [27951154644] 169554 58956 129513

5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949





















b = 0)






belt




Rigid Body Data

Pulley Inertia

[kg∙mm2]

Damping

[N∙m∙srad]

Stiffness

[N∙mrad]

Required

Torque

[Nm]




0 0

ISG 3000 0 0 5


0 0

Tensioner Arm 1 1500 1000 10314 0

Tensioner Arm 2 1500 1000 16502 0

Tensioner Arm

couple 1000 20626

Belt Data













respectively








tanφ = cb ∙ 2 ∙ π ∙ f (354)



















Rigid Bodies






ISG Pulley ΔΘ5


















pulley







Driving

Phase


Driving

Phase





Driving

Phase


Driving Phase








Driving Phase

















Driving Phase


Driving

Phase





Driving Phase


Driving

Phase





Driving

Phase


Driving Phase









353 Static Analysis






Driving

Phase








































36 Summary










































71


TWIN TENSIONER

41 Introduction










42 Methodology











value Maximum value

Coupled Spring

Stiffness kt

20626


Tensioner Arm 1

Stiffness kt1

10314


Tensioner Arm 2

Stiffness kt2

16502


Tensioner Pulley 1

Diameter D3 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 2

Diameter D5 007240 m 4344 ∙ 10

-3 m 00290 m 0116 m

Tensioner Pulley 1

Initial Coordinates

[0292761



Initial Coordinates

[012057

009193] m




Pulleys 1 amp 2

CS

AC

ISG

Ten 1

Ten 11

Region II

Region I
















arbitrarily































shown in Figure 43










Figure 44











Figure 45











Figure 46


















(N)

























(N)





47


in Figure 49



(N)













shown in Figure 410













(N)












(N)












(N)
























(N)

























(N)










44 Conclusion








































5

95





















Optimization 96








Parameter Symbol


Mode) [N]

Tension at


Crankshaft Mode [N]




[N]

Tension at




kt

465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16

28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19

kt1

465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28

28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34

kt2

465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18

28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22

D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564

28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688

D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77

28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93

[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794

28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239





Optimization 97

[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413

28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697





















Optimization 98





















Optimization 99





Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896

or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896

(51a)

where

119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)

119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)

Subject to


119886 le 1198833 le 119888

1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887

1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892



(52)





Optimization 100





local search method


















Optimization 101















performed [59]







Optimization 102














Trial

No


Objective

Function

Value [N]

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 314069 204844 165020 [02928 00703] [01618 01036]

2 3582241 103646 205284 198901 [03009 00607] [01283 00809]

3 3582241 126431 204740 43549 [03010 00631] [01311 01147]

4 3582241 180285 206230 254870 [03095 00865] [01080 01675]

5 3582241 74757 204559 189077 [03084 00617] [01265 00718]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]

Optimization 103


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -1572307 5592176 -00025 -49748

2 -4054309 3110174 -00002 -20213

3 -3930858 3233624 -00004 -38370

4 -1309751 5854731 -00010 -49525

5 -4092446 3072036 -00000 -17703

Original Design -322803 393645 16410 -4571



stopping conditions



Trial

No


Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]

1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]

2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]

3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]

4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]

Optimization 104

5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]


Trial No



Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2

1 -2584641 4579841 -02430 67549

2 -3708747 3455736 -00023 -41068

3 -1707181 5457302 -00099 -43944

4 -269178 6895304 00006 -25366

5 -2982335 4182148 -00003 -41134

Original Design -322803 393645 16410 -4571






54a and 54b below

Optimization 105


Function

Trial

No

Objective

Function

Value [N]

of

Function

Evals ( of

Iterations)

kt [N∙mrad]

kt1 [N∙mrad]

kt2 [N∙mrad]

[X3Y3]

[m] [X5Y5]

[m]


1 33509e

-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]


1 73214e

-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]

Original Design 20626 10314 16502 [02928 0087] [01206 00919]



Tensioner Arm 1

Displacement [deg]

Tensioner Arm 2

Displacement [deg]

T3 T4 Θt1 Θt2


1 -00003 7164479 -00588 -06213


1 -00000 7164482 15543 -16254

Original Design -322803 393645 16410 -4571






Optimization 106



533 Discussion


















Optimization 107


Twin Tensioner

















Optimization 108














slackest spans









Optimization 109

54 Conclusion















for the system






Optimization 110












111


61 Summary




















Conclusion 112



62 Conclusion



















Conclusion 113





















applicable findings

Conclusion 114
























Conclusion 115











116

REFERENCES





386-393 Oct 3 2004



0566















01-0523

References 117























wwwosdorgtr18pdf



38 pp 1525-1533 NovDec 2002

References 118






2002-01-1196






Sciences vol 12 pp 1053-1063 1970








406-413 1996







References 119









WO2005119089-A1 Jun 6 2005 2005


system Feb 27 2002






May 11 2000 1998










References 120






A1 Mar 21 2002 2000




German DE10159073-A1 Jun 12 2003 2001






DE10359641-A1 Jul 28 2005 2003












References 121



States US2001007839-A1 Jul 12 2001 2001





2003










2005




23 2007






References 122







123

APPENDIX A













10 11 ndash idlers



16 ndash bush




20 ndash pipe







Appendix A 124




epespacenetcom [34]


WO0026532-A1





epespacenetcom [34]

Appendix A 125




epespacenetcom [34]



14 ndash belt


18 19 ndash springs



25c ndash gang bolt




epespacenetcom [35]


Appendix A 126



16 ndash belt





epespacenetcom [36]


14 ndash tensioner




32 34 ndash arms

33 35 ndash pulley


Appendix A 127



epespacenetcom [37]








epespacenetcom [38]

Appendix A 128









epespacenetcom [39]





Appendix A 129


DE10159073-A1


epespacenetcom [40]


DE10159073-A1


epespacenetcom [40]

Appendix A 130


DE10159073-A1


epespacenetcom [40]


2 ndash idler


4 ndash swivel arm



7 ndash crankshaft











17 ndash base

18 ndash pylon

19 ndash hub



23 ndash [nil]







30 ndash system

31 ndash sensor



Appendix A 131



epespacenetcom [42]


10 ndash rod


13 ndash spring



Appendix A 132





12 ndash shaft




epespacenetcom [43]




Appendix A 133


9 ndash lever arm


16 ndash end stop





2 ndash belts





7 ndash idler

8 ndash idler

9 ndash scale beams



12 ndash camps



15 ndash arrow


Appendix A 134


























US20010007839-A1

Appendix A 135


epespacenetcom [46]


K - crankshaft





1 ndash belt pulley

4 ndash belt pulley







3 ndash AC Pulley

4a ndash 1st roller

4b ndash 2nd roller


6 ndash Belt



8 ndash WP Pulley

9 ndash IG Switch

10 ndash Engine



Appendix A 136

7 ndash ECU








18 ndash Water Pump





5 ndash tensioner

4 ndash belt pulley



Appendix A 137


epespacenetcom [49]



10 ndash actuator

10c ndash cylinder

12 ndash rod



22 ndash screw bolt



EP1658432 and WO2005015007




10 ndash starter



80 ndash pulley


206 - arm





138

APPENDIX B


Equation of Motion








Appendix B 139











Appendix B 140









(B5)

Appendix B 141





119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)




b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)


b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)


b ∙ [R3 ∙ θ 3 minus R4 ∙



b ∙ [R4 ∙ θ 4 minus R1 ∙



Tt = T3 = T4 (B13)

Appendix B 142





119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)

119816 =

I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1

(B15)








Appendix B 143

K =

(B16)

Kbi =Kb


2 + Ri ∙ϕi

2

(B17)

C =

(B18)

Appendix B 144

Appendix B 144

120521 =

θ1

θ2

θ3

θ4

partθt

(B19)

119824 =

Q1

Q2

Q3

Q4

Qt

(B20)










145

APPENDIX C

MathCAD Scripts


1 - CS

2 - AC

4 - Alt

3 - Ten1

5 - Ten 2

6 - Ten Pivot

1

2

3

4

5




XY2 224 6395( ) XY5 12057 9193( )

XY3 292761 87( ) XY6 201384 62516( )

Computations

Lt1 XY30 0

XY60 0

2

XY30 1

XY60 1

2

Lt2 XY50 0

XY60 0

2

XY50 1

XY60 1

2

t1 atan2 XY30 0

XY60 0

XY30 1

XY60 1

t2 atan2 XY50 0

XY60 0

XY50 1

XY60 1

XY

XY10 0

XY20 0

XY30 0

XY40 0

XY50 0

XY60 0

XY10 1

XY20 1

XY30 1

XY40 1

XY50 1

XY60 1

x XY

0 y XY

1

Appendix C 146



p 0 1 4

k p( ) p 1( ) p 4if

0 otherwise


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2


XYk p( ) 0

XYp 0

XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

p( ) if XYk p( ) 1

XYp 1




D

D1

D2

D3

D4

D5

Lf p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2

Lb p( ) XYk p( ) 0

XYp 0

2

XYk p( ) 1

XYp 1

2

1

mm

Dk p( )

2

Dp

2

2




Dp

2

Dk p( )

2

Dp

Dk p( )

if

atanLf p( ) mm

Dk p( )

2

Dp

2

Dp

Dk p( )

if

2D

pD

k p( )if

b p( ) atan

mmLb p( )

Dp

2

Dk p( )

2

Appendix C 147


XYf p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYb p( ) XYp 0

Dp

2 mmcos p( ) f p( )

XYp 1

Dp

2 mmsin p( ) f p( )

XYfb p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )

XYbf p( ) XYp 0

Dp

2 mmcos p( ) b p( )

XYp 1

Dp

2 mmsin p( ) b p( )



Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )

2 mmcos p( ) f p( )

XYk p( ) 1

Dk p( )

2 mmsin p( ) f p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


Dk p( )


k p( ) 1

Dk p( )

2 mmsin p( ) b p( )


XYfi

XYf 0( )0 0

XYf 1( )0 0

XYf 2( )0 0

XYf 3( )0 0

XYf 4( )0 0

XYf 0( )0 1

XYf 1( )0 1

XYf 2( )0 1

XYf 3( )0 1

XYf 4( )0 1

XYfi

6818

269222

335325

251552

108978

100093

89099

60875

200509

207158

x1 XYfi0

y1 XYfi1

Appendix C 148

XYbi

XYb 0( )0 0

XYb 1( )0 0

XYb 2( )0 0

XYb 3( )0 0

XYb 4( )0 0

XYb 0( )0 1

XYb 1( )0 1

XYb 2( )0 1

XYb 3( )0 1

XYb 4( )0 1

XYbi

47054

18575

269403

244841

164847

88606

291

30965

132651

166182

x2 XYbi0

y2 XYbi1

XYfbi

XYfb 0( )0 0

XYfb 1( )0 0

XYfb 2( )0 0

XYfb 3( )0 0

XYfb 4( )0 0

XYfb 0( )0 1

XYfb 1( )0 1

XYfb 2( )0 1

XYfb 3( )0 1

XYfb 4( )0 1

XYfbi

42113

275543

322697

229969

9452

91058

59383

75509

195834

177002

x3 XYfbi0

y3 XYfbi1

XYbfi

XYbf 0( )0 0

XYbf 1( )0 0

XYbf 2( )0 0

XYbf 3( )0 0

XYbf 4( )0 0

XYbf 0( )0 1

XYbf 1( )0 1

XYbf 2( )0 1

XYbf 3( )0 1

XYbf 4( )0 1

XYbfi

8384

211903

266707

224592

140427

551

13639

50105

141463

143331

x4 XYbfi0

y4 XYbfi1


XYf2i

XYf2 0( )0 0

XYf2 1( )0 0

XYf2 2( )0 0

XYf2 3( )0 0

XYf2 4( )0 0

XYf2 0( )0 1

XYf2 1( )0 1

XYf2 2( )0 1

XYf2 3( )0 1

XYf2 4( )0 1

XYf2x XYf2i0

XYf2y XYf2i1

XYb2i

XYb2 0( )0 0

XYb2 1( )0 0

XYb2 2( )0 0

XYb2 3( )0 0

XYb2 4( )0 0

XYb2 0( )0 1

XYb2 1( )0 1

XYb2 2( )0 1

XYb2 3( )0 1

XYb2 4( )0 1

XYb2x XYb2i0

XYb2y XYb2i1

Appendix C 149

XYfb2i

XYfb2 0( )0 0

XYfb2 1( )0 0

XYfb2 2( )0 0

XYfb2 3( )0 0

XYfb2 4( )0 0

XYfb2 0( )0 1

XYfb2 1( )0 1

XYfb2 2( )0 1

XYfb2 3( )0 1

XYfb2 4( )0 1

XYfb2x XYfb2i

0

XYfb2y XYfb2i1

XYbf2i

XYbf2 0( )0 0

XYbf2 1( )0 0

XYbf2 2( )0 0

XYbf2 3( )0 0

XYbf2 4( )0 0

XYbf2 0( )0 1

XYbf2 1( )0 1

XYbf2 2( )0 1

XYbf2 3( )0 1

XYbf2 4( )0 1

XYbf2x XYbf2i0

XYbf2y XYbf2i1

100 40 20 80 140 200 260 320 380 440 500150

110

70

30

10

50

90

130

170

210


250

150

y1

y2

y3

y4

y

XYf2y

XYb2y

XYfb2y

XYbf2y


Appendix C 150


XY15 XYbf2iT 4

XY12 XYfiT 0


XY21 XYf2iT 0

XY23 XYfbiT 1


XY32 XYfb2iT 1

XY34 XYbfiT 2


XY43 XYbf2iT 2

XY45 XYfbiT 3


XY54 XYfb2iT 3

XY51 XYbfiT 4




L1

L2

L3

L4

L5

mm


i

atan

XY121

XY211

XY12

0XY21

0

atan

XY231

XY321

XY23

0XY32

0

atan

XY341

XY431

XY34

0XY43

0

atan

XY451

XY541

XY45

0XY54

0

atan

XY511

XY151

XY51

0XY15

0

Appendix C 151


12 i0 2 21 i0 32 i1 2 43 i2 54 i3

15 i4 2 23 i1 34 i2 45 i3 51 i4

15

21

32

43

54

12

23

34

45

51


1 2 atan2 XY150

XY151

atan2 XY120

XY121

2 atan2 XY210

XY1 0

XY211

XY1 1

atan2 XY230

XY1 0

XY231

XY1 1

3 2 atan2 XY320

XY2 0

XY321

XY2 1

atan2 XY340

XY2 0

XY341

XY2 1

4 atan2 XY430

XY3 0

XY431

XY3 1

atan2 XY450

XY3 0

XY451

XY3 1

5 atan2 XY540

XY4 0

XY541

XY4 1

atan2 XY510

XY4 0

XY511

XY4 1

1

2

3

4

5

Lb length of belt

Lbelt Li1

2

0

4

p

Dpp


Kt 20626Nm


and 2)

Kt1 10314Nm


Kt2 16502Nm


Appendix C 152

C2 Dynamic Analysis



RaD

2

Appendix C 153

RaD

2


Ii0

0

1

2

3

4

5

6

10000

2230

300

3000

300

1500

1500

I diag Ii( ) kg mm2


1000kg

m3

CrossArea 693mm2


cb 2 KbM

Lbelt

Cb

cb

cb

cb

cb

cb

Cri

0

0

010

0

010

N mmsec

rad

Ct 1000N mmsec

rad Ct1 1000 N mm

sec

rad Ct2 1000N mm

sec

rad

Cr

Cri0

0

0

0

0

0

0

0

Cri1

0

0

0

0

0

0

0

Cri2

0

0

0

0

0

0

0

Cri3

0

0

0

0

0

0

0

Cri4

0

0

0

0

0

0

0

Ct Ct1

Ct

0

0

0

0

0

Ct

Ct Ct2

Rt

Ra0

Ra1

0

0

0

0

0

0

Ra1

Ra2

0

0

Lt1 mm sin t1 32

0

0

0

Ra2

Ra3

0

Lt1 mm sin t1 34

0

0

0

0

Ra3

Ra4

0

Lt2 mm sin t2 54

Ra0

0

0

0

Ra4

0

Lt2 mm sin t2 51

Appendix C 154

Kr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Kt Kt1

Kt

0

0

0

0

0

Kt

Kt Kt2

Tk

Kbi 0( ) Ra0

0

0

0

Kbi 4( ) Ra0

Kbi 0( ) Ra1

Kbi 1( ) Ra1

0

0

0

0

Kbi 1( ) Ra2

Kbi 2( ) Ra2

0

0

0

0

Kbi 2( ) Ra3

Kbi 3( ) Ra3

0

0

0

0

Kbi 3( ) Ra4

Kbi 4( ) Ra4

0



0

0

0

0

0



Tc

Cb0

Ra0

0

0

0

Cb4

Ra0

Cb0

Ra1

Cb1

Ra1

0

0

0

0

Cb1

Ra2

Cb2

Ra2

0

0

0

0

Cb2

Ra3

Cb3

Ra3

0

0

0

0

Cb3

Ra4

Cb4

Ra4

0

Cb1

Lt1 mm sin t1 32

Cb2

Lt1 mm sin t1 34

0

0

0

0

0

Cb3

Lt2 mm sin t2 54

Cb4

Lt2 mm sin t2 51

C matrix

C Cr Rt Tc

K matrix

K Kr Rt Tk



IA augment I3

I0

I1

I2

I4

I5

I6

IC augment I0

I3

I1

I2

I4

I5

I6

I1kgmm2 1 106

kg m2

0 0 0 0 0 0


T

IAT 1

T

IAT 2

T

IAT 4

T

IAT 5

T

IAT 6

T


T

ICT 1

T

ICT 2

T

ICT 4

T

ICT 5

T

ICT 6

T

Appendix C 155

RtA augment Rt3

Rt0

Rt1

Rt2

Rt4

RtC augment Rt0

Rt3

Rt1

Rt2

Rt4

Rta stack RtAT 3

T

RtAT 0

T

RtAT 1

T

RtAT 2

T

RtAT 4

T

RtAT 5

T

RtAT 6

T

Rtc stack RtCT 0

T

RtCT 3

T

RtCT 1

T

RtCT 2

T

RtCT 4

T

RtCT 5

T

RtCT 6

T

TkA augment Tk3

Tk0

Tk1

Tk2

Tk4

Tk5

Tk6

Tka stack TkAT 3

T

TkAT 0

T

TkAT 1

T

TkAT 2

T

TkAT 4

T

TkC augment Tk0

Tk3

Tk1

Tk2

Tk4

Tk5

Tk6

Tkc stack TkCT 0

T

TkCT 3

T

TkCT 1

T

TkCT 2

T

TkCT 4

T

TcA augment Tc3

Tc0

Tc1

Tc2

Tc4

Tc5

Tc6

Tca stack TcAT 3

T

TcAT 0

T

TcAT 1

T

TcAT 2

T

TcAT 4

T

TcC augment Tc0

Tc3

Tc1

Tc2

Tc4

Tc5

Tc6

Tcc stack TcAT 0

T

TcAT 3

T

TcAT 1

T

TcAT 2

T

TcAT 4

T


CHECK

KA augment K3

K0

K1

K2

K4

K5

K6

KC augment K0

K3

K1

K2

K4

K5

K6

CA augment C3

C0

C1

C2

C4

C5

C6

CC augment C0

C3

C1

C2

C4

C5

C6

Appendix C 156

Kacheck stack KAT 3

T

KAT 0

T

KAT 1

T

KAT 2

T

KAT 4

T

KAT 5

T

KAT 6

T

Kccheck stack KCT 0

T

KCT 3

T

KCT 1

T

KCT 2

T

KCT 4

T

KCT 5

T

KCT 6

T

Cacheck stack CAT 3

T

CAT 0

T

CAT 1

T

CAT 2

T

CAT 4

T

CAT 5

T

CAT 6

T

Cccheck stack CCT 0

T

CCT 3

T

CCT 1

T

CCT 2

T

CCT 4

T

CCT 5

T

CCT 6

T





H n( ) 1 n 750if

0 n 750if



I11mod n( ) Ic0 0

H n( ) 1if

Ia0 0

H n( ) 0if





K11mod n( )

Kc0 0


Ka0 0


C11modn( )

Cc0 0


Ca0 0


























Appendix C 157



2














EVmodn( )

t 0 0001 1



sin nmod 100( )1 t



sin nmod 800( )1 t





Pulley responses

Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1


qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm


qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm

Appendix C 158


engine speed n

n 60 90 6000 n( )4n

60 a n( )

2n Ra0

60 Ra3

mod n( ) n( ) H n( ) 1if

a n( ) H n( ) 0if







Ac n( ) 650 1 n( )Ac n( )

n( ) 2



Aa n( )42

I3 3

1a n( )Aa n( )

a n( ) 2

Qc0 4


I11mod n( )



H n( ) 1if

Qc0 H n( ) 0if


I11mod n( )



H n( ) 0if

0 H n( ) 1if

Q n( ) 48 n

Ra0

Ra3

48

18000


Appendix C 159


Qtt1mod n( )Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1 n( )

H n( ) 1if

Kt Kt1


Ct Ct1

UnitsOf Cr( )

mmod n( )4 1a n( )

H n( ) 0if

Qtt2mod n( )Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1 n( )

H n( ) 1if

Kt Kt2


Ct Ct2

UnitsOf Cr( )

mmod n( )5 1a n( )

H n( ) 0if

Appendix C 160


d n( ) 1 n( ) H n( ) 1if

1a n( ) H n( ) 0if

mod n( )

d n( )

mmod n( ) d n( ) 0 0

mmod n( ) d n( ) 1 0

mmod n( ) d n( ) 2 0

mmod n( ) d n( ) 3 0

mmod n( ) d n( ) 4 0

mmod n( ) d n( ) 5 0

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if

Tm n( ) j n( )Tcc

UnitsOf Tcc( )

Tkc

UnitsOf Tkc( )

mod n( )

H n( ) 1if

j n( )Tca

UnitsOf Tca( )

Tka

UnitsOf Tka( )

mod n( )

H n( ) 0if


Appendix C 161

C3 Static Analysis






Qc

Q4

Q2

Q3

Q5

Qt1

Qt2

Qc

5

2

0

0

0

0

J Qa

Q1

Q2

Q3

Q5

Qt1

Qt2

Qa

68

2

0

0

0

0

N m

cK22mod 900( )( )

1

N mQc A

K22mod 600( )1

N mQa

a

A0

A1

A2

0

A3

A4

A5

0

c1

c2

c0

c3

c4

c5

Tc Tk Ts Ta Tk a Ts

162

APPENDIX D



D11 TwinMainm









______________________________________________________________________________

clc

clear all


Initial Conditions

Kto = 20626

Kt1o = 10314

Kt2o = 16502

D3o = 007240

D5o = 007240

X3o =0292761

Y3o =087

X5o =12057

Y5o =09193





D3 = (D3o-060D3o)006D3o(D3o+060D3o)

D5 = (D5o-060D5o)006D5o(D5o+060D5o)

No of data points

s = 21

T = zeros(5s)

Ta = zeros(5s)

Parametric Plots

for i = 1s

Appendix D 163


end

figure






xlabel(Kt (Nmrad))









for i = 1s


end

figure






xlabel(Kt1 (Nmrad))









for i = 1s


end

figure






xlabel(Kt2 (Nmrad))


Appendix D 164








for i = 1s


end

figure






xlabel(D3 (m))









for i = 1s


end

figure






xlabel(D5 (m))









Mesh points

m = 101

n = 101

Appendix D 165

T = zeros(5nm)

Ta = zeros(5nm)


minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

midX3 = 0311641



for i = 1n

for j = 1m





lt= y lt= circle4

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X3Y3real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X3Y3real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar




Appendix D 166


figure


surf(X3Y3real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X3Y3real(Za3))

ZLim([50 500])

axis tight

colormap jet

colorbar





minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456

midX5a = 0131965

midX5b = 017729



for i = 1n

for j = 1m

if


(X5(ij)^2))))

[T(ij)Ta(ij)] =


elseif


146468))

[T(ij)Ta(ij)] =



024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))

Appendix D 167

[T(ij)Ta(ij)] =


else

T(ij) = zeros(511)

Ta(ij) = zeros(511)

end

end

end

figure

Z1 = squeeze(T(1))

surf(X5Y5real(Z1))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure

Z5 = squeeze(T(5))

surf(X5Y5real(Z5))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za4))

ZLim([50 500])

axis tight

colormap jet

colorbar





figure


surf(X5Y5real(Za3))

ZLim([50 500])

axis tight

Appendix D 168

colormap jet

colorbar













D2 Optimization












______________________________________________________________________________

clc

clear all


Kto = 20626

Kt1o = 10314

Kt2o = 16502

X3o = 0292761

Y3o = 0087

X5o = 012057

Appendix D 169

Y5o = 009193


Variable ranges

minKt = Kto - 1Kto

maxKt = Kto + 1Kto





minX3 = 0260200

maxX3 = 0317677

minY3 = -0056640

maxY3 = 0228456

minX5 = -0037093

maxX5 = 0212509

minY5 = 00362

maxY5 = 0228456





for optimization







provided






D22 confunTwinm






Appendix D 170

D23 objfunTwinm







171

VITA

ADEBUKOLA OLATUNDE














Documents

design & analysis of a tensioner for a belt-driven integrated starter