Thesis

PREDICTION OF STEERING SYSTEM

PERFORMANCE VARIATION DUE TO

MANUFACTURING ERRORS

A Thesis Submitted

in Partial Fulfillment of the Requirements

for the Degree of

MASTER OF TECHNOLOGY

June, 2009

By

SHUBHAM GOEL

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY

KANPUR-208016 (INDIA)

i

Statement of Thesis Preparation

Thesis title “Prediction of Steering System Performance Variation due to

Manufacturing Errors”

1. Degree for which submitted : Master of Technology

2. The thesis guide was referred to

for thesis preparation : Yes

3. Specifications regarding thesis

format have been closely followed : Yes

4. The contents of the thesis were

organized according to the guidelines : Yes

(Signature of Student)

Name : Shubham Goel

Roll No : Y4177423

Department : Mechanical Engineering

iii

ACKNOWLEDGEMENTS

I would like to thank my research advisor, Dr. N S Vyas for his great guidance, directions,

constructive criticism, patience, kindness and caring heart. It was he who molded a

simple student like me into a researcher, with his encouragement and inspirational ideas I

continued to do the research in the field of automotive manufacturing errors and had an

opportunity to work closely with Dr. Anand Ramani who is senior researcher at GM

India.

I am extremely thankful to Mr. Shailesh, Mr. Kshitij and Mr S Shukla for providing me

invaluable help and suggestions throughout the thesis work. I am highly grateful to Mr. R

Chandar, Mr. L Trivedi and Mr. M Mohisin for all their help and support related to

laboratory facilities.

I further extend my gratitude to my colleagues and friends Rajesh, Adarsh, Shantanu,

Neha, Prasoon, Deep, Avani, Abhishek, Rahul, Praneet, Biplap, Nikhel, Tukesh and Dona

for providing a lively work environment and for making my stay at IITK a memorable

one.

Above all, I am blessed with such caring parents. I extend my deepest gratitude to my

parents and my brother for their invaluable love, affection, encouragement and support.

India Institute of Technology, Kanpur Shubham Goel

June, 2009

iv

ABSTRACT

The present study aims at understanding the effect of manufacturing errors in the steering

mechanism on the car steering feel. Manufacturing errors can cause a deviation in

steering kinematics and steering torque from the designed results. They may occur in the

upper steering system, pinion rack interface and the lower steering system. For the

present scope of work we chose to study the errors in the rack and pinion only.

Manufacturing errors in a rack and pinion type steering system is studied because most of

the modern cars are dominantly fitted with this type of a mechanism. To study the effect

of the error on the steering system a basic physics based model of a simple steering

system, capturing the admissible errors without any external power support is developed

and the results are validated using computer simulation software ADAMS CAR. Then the

prominent errors like center distance variation between rack and pinion, pinion helical

angle variation, backlash/lead error, pinion installation error (axis misalignment), rack

bend error, pinion tooth thickness error, pinion tooth profile error, pressure angle error

and pitch circle runout error are modeled for a rack and pinion system. With such a

model, it is possible to perform Monte Carlo simulations using appropriate statistical

distributions for various errors and predict the error band and distribution on steering

kinematics and steering torque.

v

CONTENTS

TITLETITLE PAGE

CERTIFICATECERTIFICATECERTIFICATE ii

ACKNOWLEDGEMENT ACKNOWLEDGEMENT ACKNOWLEDGEMENT iv

ABSTRACTABSTRACTABSTRACT v

CONTENTSCONTENTSCONTENTS vi

LIST OF FIGURESLIST OF FIGURESLIST OF FIGURES ix

LIST OF TABLESLIST OF TABLESLIST OF TABLES xiii

NOMENCLATURENOMENCLATURENOMENCLATURE xiv

1. INTRODUCTIONINTRODUCTION 1

1.1 Rack and Pinion type Steering System1.1 Rack and Pinion type Steering System 1

1.2 Literature Review1.2 Literature Review 2

1.3 Current Work1.3 Current Work 5

1.4 Methodology Adopted1.4 Methodology Adopted 6

1.4.1 Steering System Model 1.4.1 Steering System Model 6

1.4.2 Manufacturing Errors 1.4.2 Manufacturing Errors 6

1.5 Simulation and Results1.5 Simulation and Results 8

2. MODELING OF THE STEERING SYSTEMMODELING OF THE STEERING SYSTEM 9

2.1 Steering System2.1 Steering System 9

2.1.1 Universal Joint 2.1.1 Universal Joint 10

2.1.2 Rack and Pinion Assembly 2.1.2 Rack and Pinion Assembly 13

2.1.3 Tie Rod and the Knuckle 2.1.3 Tie Rod and the Knuckle 14

2.1.4 Integrated Assembly 2.1.4 Integrated Assembly 17

2.2 Validation2.2 Validation 20

2.2.1 ADAMS Modeling 2.2.1 ADAMS Modeling 22

2.2.2 Test Simulations 2.2.2 Test Simulations 22

2.3 Results 2.3 Results 24

2.4 Remarks2.4 Remarks 25

3. KINEMATICS OF GEAR MESHINGKINEMATICS OF GEAR MESHING 26

3.1 Kinematics of Gear Meshing3.1 Kinematics of Gear Meshing 26

vi

3.1.1 Velocity Ratio 3.1.1 Velocity Ratio 26

3.1.2 Constant velocity ratio for a rack and pinion 3.1.2 Constant velocity ratio for a rack and pinion 28

3.1.3 Condition for constant velocity ratio 3.1.3 Condition for constant velocity ratio 32

3.1.4 Involute tooth profile for a rack and pinion 3.1.4 Involute tooth profile for a rack and pinion 34

3.1.5 Properties of involute tooth 3.1.5 Properties of involute tooth 35

3.1.6 Torque transfer 3.1.6 Torque transfer 37

3.2 Helical rack and pinion3.2 Helical rack and pinion 38

3.2.1 Basic helical rack 3.2.1 Basic helical rack 39

3.2.2 Helical pinion 3.2.2 Helical pinion 41

3.2.3 Velocity ratio of a helical rack and pinion 3.2.3 Velocity ratio of a helical rack and pinion 44

3.2.4 Torque transfer for a helical rack and pinion 3.2.4 Torque transfer for a helical rack and pinion 45


4. MODELING OF THE GEAR ERRORSMODELING OF THE GEAR ERRORS 47

4.1 Center distance variation4.1 Center distance variation 48

4.1.1 Velocity ratio 4.1.1 Velocity ratio 48


4.1.3 Backlash 4.1.3 Backlash 53

4.2 Pinion helix angle variation4.2 Pinion helix angle variation 55



4.3 Backlash error4.3 Backlash error 57



4.4 Pinion installation angle error4.4 Pinion installation angle error 60

4.4.1 Pure shaft misalignment 4.4.1 Pure shaft misalignment 60

4.4.2 Pure tilt 4.4.2 Pure tilt 66

4.4.3 Misalignment error 4.4.3 Misalignment error 72

4.5 Rack bend error4.5 Rack bend error 81

4.6 Tooth thickness error4.6 Tooth thickness error 81




vii

4.7 Pressure angle error4.7 Pressure angle error 84




4.8 Pinion pitch circle runout4.8 Pinion pitch circle runout 92





5. CONCLUSION AND FUTURE WORKCONCLUSION AND FUTURE WORK 97

REFERENCESREFERENCESREFERENCES 98

viii

LIST OF FIGURES

FIGURE DESCRIPTION PAGE

1.1 Steering and the suspension system 2

1.2 Steering System modeled using ADAMS 7

2.1 Steering system 10

2.2 Universal Joint 10

2.3 Schematic of the Universal Joint 11

2.4 Plot between driven shaft angle and driving shaft angle 12

2.5 Plot between driven shaft speed and driving shaft speed 12

2.6 Rack and pinion assembly 13

2.7 Rack and pinion assembly schematic 14

2.8 Ball joint on the tie rod 15

2.9 Tie rod and knuckle assembly 15

2.10 Schematic of the steering system 17

2.11 Car wheel angular displacement 21

2.12 ADAMS model of the car steering system 21

2.13 Joints between parts in the ADAMS model 23

2.14 Comparison of Analytical and ADAMS simulations 24

2.15 Difference between Analytical and ADAMS simulations 25

3.1 A gear pair 27

3.2 Common normal at the contact point 29

3.3 Pitch point of a rack and pinion 31

ix

3.4 Two teeth meshing 32

3.5 Meshing diagram of a pinion and a basic rack 34

3.6 Gear with involute teeth 35

3.7 Involute gear teeth 36

3.8 Force body diagram for the rack 38

3.9 Helical gear and rack 39

3.10 Basic helical rack 40

3.11 Helix through point A0 42

3.12 Transverse section through the basic rack 43

3.13 Developed cylinder of radius R and length z 44

3.14 Tooth surface of a helical rack 45

4.1 Center distance variation 49

4.2 Rack tooth contact 50

4.3 Velocity ratio 52

4.4 Torque transfer 53

4.5 Backlash in center distance variation 54

4.6 Backlash 54

4.7 Yoke nut assembly 55

4.8 Plane of contact in the helix angle error 56

4.9 Backlash for two gears) 58

4.10 Pure shaft misalignment 61

4.11 Section of the pinion surface in the plane B2

!B2 62

4.12 Section of the pinion surface in the plane B1!B1

61

x

4.13 Cross section of the pinion tooth in the XZ plane 63

4.14 Pure shaft misalignment error 66

4.15 Pure tilt misalignment 64

4.16 Section of the pinion surface in the plane A2

!A2

68

4.17 Section of the pinion surface in the plane A1!A1

68

4.18 Cross section of the pinion tooth in theYZ plane 68

4.20 Axis misalignment error 72

4.21 Orientation of the pinion axis 73

4.22 Section of the pinion surface in the plane C2

!C2 74

4.23 Section of the pinion surface in the plane C1!C1 74

4.24 Cross section of the pinion tooth in the XZ plane 74

4,25 Cross section of the pinion tooth in theYZ plane 75

4.26 Axis misalignment for! = 0.25° 77

4.27 Axis misalignment for! = 0.75° 78

4.28 Torque transfer for Axis misalignment at! = 0.25° 80

4.29 Torque transfer for Axis misalignment at! = 0.75 80

4.30 Rack bend error 81

4.31 Pinion tooth thickness 82

4.32 Rack tooth thickness 82

4.33 Rack with pressure angle error 84

4.34 Variation in rack pressure angle 85

4.35 Velocity ratio for pressure angle variation 87

4.36 Torque transfer for pressure angle variation 88

xi

4.37 Tooth thickness relations 90

4.38 Change in backlash due to pressure angle variation 92

4.39 Pitch circle runout 93

4.40 Radian position of the pinion teeth 93

4.41 Change in backlash due to pitch circle runout 96

xii

LIST OF TABLES

TABLE DESCRIPTION PAGE

2.1 Dimensions for the car steering system model 23

xiii

NOMENCLATURE

Chapter 2

! Angle of the driving shaft; angle between the tie rod and the Y-axis

!Angle of the driven shaft; angle between the knuckle geometry and the Y-

axis

! Angle between the shaft axis

!1

Angular velocity of the driving shaft

!2

Angular velocity of the driven shaft

!2

Angular acceleration of the driven shaft

x Rack displacement

rp pinion pitch circle radius

!d

steering wheel angular displacement

rp pitch radius of the pinion

! rack displacement relative to the rack casing

Lt

length of the tie rod

Lk

length of the Knuckle

dy distance between the rack case and the knuckle along the Y direction

dx distance between the rack case and the knuckle along the X direction

LR

Length of the rack

LRC

Length of the rack case

Chapter 3

(vr)average Average rack velocity

(! )average Average pinion angular velocity

p Pitch of the rack

xiv

N number of pinion teeth

n̂nr

Unit vector along the common normal

!p Profile angle of the rack

!vAr Velocity of the rack

vr

Speed of the rack

vn

Ar rack velocity component along the common normal

!vA velocity of the point A on the pinion

! angular velocity of the pinion

vn

A velocity component at point A along the common normal

(Xn̂i+Yn̂

j) vector from the center C to the point A

RP

Pinion pitch circle radius

Rl

radial distance of the point I from the axis of the gear

!l

Pressure angle at point I

!l

Roll angle at point I

m Mass of the rack

a Acceleration of the rack

Nt

Normal reaction along the common normal

F Force on the rack

T Torque

!r

Rack helix angle

!R

Pinion helix angle at radius R

Chapter 4

!C Change in center distance

!p Pressure angle

xv

! Pinion rotation

!pi Instantaneous profile angle of the pinion at the point of contact

INV!pi

Involute of !pi

Rl

Radius at the point of contact

X x component of CP

Y y component of CP

! Angular velocity of the pinion

vr

Rack velocity

!Blc

Change in backlash due to center distance variation

! Angle of intersection between the rack reference plane and the pinion axis

! Angle between the pinion axis and the transverse cross section plane

!r Radian position of the point of contact on the pinion

!! Angular velocity vector of the pinion

!vA Velocity of the point of contact on the pinion

vn

A Component of !vA along the common normal

vn

Ar Velocity component of the point of contact on the rack along the common

normal

N Number of teeth on the pinion

!T Torque experienced by the pinion

!Blt

Change in backlash due to tooth thickness variation

!"p Change in rack pressure angle

!Blo

Change in backlash due to pressure angle variation

!Bpr Change in backlash due to pinion pitch circle runout

Rt

Distance between the points on the periphery of the operating pitch circle

and the gear center

er

Eccentricity in the position of the pinion axis

xvi

xvii

CHAPTER-1

INTRODUCTION

The car steering system is a widely studied system by automobile manufacturers

and research institutes across the globe. Current research shows that the modern

vehicle may sport a steering wheel which will not be physically connected with

the car wheel through linkages, but the driver will still receive haptic feedback

from a complex array of sensors and control system feedback. The steering feel

based on a rack and pinion type steering mechanism is still important because

most midrange consumer vehicles will continue to bear a rack and pinion type

steering mechanism.

1.1 Rack and Pinion type Steering System

The rack and pinion steering gear has become increasingly popular for today’s

small cars. Is is simpler, more direct acting, and may be straight mechanical or

power assisted in operation. Figure 1.1 shows the schematic of a rack and pinion

steering system. As the steering wheel and the shaft are turned, the rack moves

from one side to another. This pushes or pulls on the tie rods, forcing the knuckle

to pivot about the kingpin axis. This turns the wheel to one side or the other so

that the car is steered. The steering gear and the tie rod are visible in the figure.

The universal joint is at the upper end of the steering shaft and the flexible

coupling at the lower end. In small cars, rack and pinion steering is quick and

easy. It provides the maximum amount of road feel as the tires meet irregularities

in the road.

1

Figure 1.1 Steering and the suspension system

1.2 Literature Review

The torque transfer and displacements in the steering sub system and the rack and

pinion gear assembly have been the subjects of extensive research by world’s

major automobile companies and research institutions. Various theoretical

models for the steering sub system have been developed to analyze different

effects like stability, steering feel, torque performance, disturbance rejection,

noise rejection and road feel.

The problem of analyzing the effect of manufacturing errors in the rack and

pinion gear on the steering feel can be divided into two parts. The first would be

building a model of the steering system to see how torque and displacement is

transfered from one component to the other. Salaani et. al. [1] have developed a

real-time steering system torque feedback model which is used in the National

Advanced Driving Simulator(NADS). They have presented a detailed

mathematical model of the steering physics form low-speed stick-slip to high-

2

speed states. On-center steering weave handling and aggressive lane change

inputs have been used to validate the basic mathematical predictions. The

validations are objective and open loop, and were done using field experiments.

On the other hand Badawy et. al. [2] of Delphi Saginaw Steering Systems have

developed a model of the Electric Power steering system to analyze various

closed loop effects such as torque performance, disturbance rejection, noise

rejection, road feel and stability. The modeling has been achieved with both

simplicity and usability taken into account. A kinematic model of a rack and

pinion type steering linkage has been developed by Simionescu [3]. They have

mainly dealt with the synthesis and analysis of a translational input, double loop

rack and pinion type mechanism employed in the steering of rigid axle vehicles.

Even today rigid axles can be found in heavy jeeps but the clear disadvantage is

the large non-spring mounted masses. Ansarey et. al. [4] have presented a general

method for the optimization of vehicle steering linkages. The investigation is

focused on the geometrical parameters of a rack and pinion steering system, and

their contribution on the handling characteristics. A novel method is proposed to

set the optimized geometry of the steering system, in particular its joint

placements, by using a genetic based approach.

Gillespie [5] suggested a lumped parameter model for a two dimensional

representation of a steering mechanism. The model used composite stiffness

between the steering gearbox and the road wheels. He developed the modeling

equations based on the geometry of the mechanism and the relationship between

the forces and displacements.

Another important aspect of the steering system model is understanding the

steering feel i.e. the force generated by the power steering and vibration from

different road surfaces. Adams [6], Baxter [7], Engelman [8] and Sugitani et. al.

[9] have made an attempt to quantify the driver steering feel. Shimizu [10]

discussed on effects of speed and steering angle on driver/vehicle system

performance. The steering feel is largely dependent on the traction or the

3

longitudinal and lateral forces occurring in the tires. Dugoff et. al. [11] have

developed a model and expressions for the same.

The second part of the problem of understanding the effect of manufacturing

errors in the rack and pinion steering gear on the steering feel is to work out and

model these manufacturing errors and their effect on parameters like torque

transfer and the displacement function. Kamble et. al. [12] have made an attempt

to model the rack and pinion steering gear using ADAMS modeling software.

Effects of machining errors, assembly errors and tooth modifications on loading

capacity, load-sharing ratio and transmission error of a pair of spur gears by using

special-developed finite element method has been studied by Shuting [13]. It is

found that misalignment error of gear shafts on the plane of action of the spur

gears exerts great effects on tooth surface contact stress and tooth root bending

stress while misalignment error of gear shafts on the vertical plane of the plane of

action almost exerts no effects on tooth surface contact stress and tooth root

bending stress. It is also found that machining errors and lead crowning have

greater effects on tooth surface contact stress and tooth root bending stress. A

simplified model for wear simulation of helical gears has been developed by

Flodin et. al. [14]. In this model the helical wheel is treated as several thin spur

gear plates with a common axis of rotation. These plates are oriented at small

angular displacement corresponding to the helical angle. Ajmi [15] has proposed

an original model aimed at simulating the quasi-static and dynamic behavior of

solid wide-faced gears. Tooth shape deviations and alignment errors have been

considered, and solutions are sought by simultaneously solving the equations of

motion and the contact problem between the teeth. Analysis of tooth contact and

load distribution of helical gears with crossed axes has been attempted by Zhang

[16]. The approach is based on a tooth contact model that accommodates the

influence of tooth profile modifications, gear manufacturing errors and tooth

surface deformation on gear mesh quality.

Litvin et. al. [17] have studied the meshing and contact of enveloping gear tooth

surfaces. Ideally such surfaces are in line contact at every instant. Due to gear

4

misalignment cased by errors of assembly (change of crossing angle, center

distance, axial displacement of gears, etc.), the surface starts to contact each other

at every instant at a point but not a line. They have proposed a computerized

approach for simulation of meshing and contact, determining (i) the path of

contact point on gear tooth surfaces, (ii) deviation of the transmission function

from the theoretical one (transmission errors), and (iii) the bearing contact that is

formed as a set of contact ellipse. In another paper [18] the authors have

proposed approaches for computerized simulation of meshing of aligned and

misaligned involute helical gear.

Blankenship et. al. [19] have developed a new model that describes mesh force

transmissibility in a helical gear pair.

1.3 Current Work

The steering feel depends on many factors, ranging from the road conditions to

the diameter of the steering wheel. It has been observed that even a slight change

in the orientation of the pinion axis dramatically affects the steering feel.

Typically, the wheel’s resistance increases along with the steering angle, but after

production of power assisted steering wheels the steering feel is sometimes

dramatically different. Some modern steering wheel may transmit a feel for the

road by twitching noticeably when a wheel encounters road imperfections. This

is considered advantageous by many driving enthusiasts, but it seems to be the

exception rather than the rule, especially with front-wheel drive. So, every

component, its dimensions and orientation transforms the steering feel. In the

current work we try to understand how manufacturing errors in the rack and

pinion (a small subsystem) of the steering affect the feel.

For the modern consumer the steering feel is a vital factor before making a

purchase and hence different car companies are trying to get a better

understanding of which parameters affect the steering feel and how. The idea is

5

to provide the consumer with a desirable steering feel which the supplier can

design and decide.

1.4 Methodology Adopted

The problem of analyzing the manufacturing errors in the rack and pinion gear of

a steering system is attempted in the following order:

1. Modeling of the steering system

2. Modeling of the manufacturing errors

3. Simulation of the system

Each process is talked about in greater detail in the following subsections.

1.4.1 Steering System Model

The fundamental physics based model of the steering system is developed.

Components modeled include the steering wheel, steering column, universal

joint, intermediate shaft, lower shaft, pinion, rack, tie-rod, knuckle and the wheel.

Values of the model parameters like dimensions, masses and inertias are

measured from a small Indian family car. The model is then checked by

performing simple studies using test inputs using ADAMS as shown in Figure

1.2. For trial simulations, angular displacement is provided at the steering wheel

and the output parameters are recorded.

1.4.2 Manufacturing Errors

The next part is modeling the manufacturing errors in the rack and pinion. The

errors in the rack and pinion are modeled analytically and basic kinematic and

torque transfer equations are developed to study the effect of the errors on the

steering kinematics and feel. The most common type of rack and pinion used for

6

car steering mechanism, a helical pinion with a straight rack is modeled for the

analysis. The errors that are primarily focused on are:

a) Center distance variation

b) Pinion helix angle variation

c) Backlash

d) Pinion installation angle (axes misalignment)

e) Rack bend

f) Tooth thickness

g) Pressure angle

h) Pinion pitch circle runout

These analytical models are developed and then run through numerical

simulations.

Figure 1.2 Steering System modeled using ADAMS

7

1.5 Simulation and Results

For the model simulation, the input excitation is taken as a time-varying

displacement excitation at the steering wheel. The angular displacement of the

tire about the kingpin axis is taken as the output of the model variation. The value

is obtained for the complete travel of the steering wheel from on-center position

on either side. The model simulation is automated to accept input values in a

parametric format. Automation of model simulation is enabled such that error

values are accepted as input to the simulation process The automation code is

written using common scripts. The automation spans the entire process of:

a) Reading input values from a text file for each simulation

b) Performing simulations for each combination of input values and

c) Generating output in the required format.

Both inputs and results from simulations are made available in a spreadsheet

form to facilitate post-processing.

Details of the steering system model and its validation is discussed in Chapter 2.

The kinematics of gear meshing and equations for torque transfer have been

mentioned in Chapter 3. Chapter 4 gives information on the modeling of gear

errors and their effect on various parameters. Conclusions and scope for future

work is listed in Chapter 5.

8

CHAPTER-2

MODELING OF THE STEERING SYSTEM

To develop a fundamental physics based model of the steering system all the sub

components of the system are examined and then structured analytically. After

that they are all integrated together to observe parameters like steering ratio,

displacement transfer and torque transfer. To validate these results a vehicle

dynamics software ADAMS is used.

2.1 Steering System

A schematic of the steering system is shown in Figure 2.1. The steering system

comprises of a steering wheel turning a steering column. The steering column is

connected to an intermediate shaft through a universal joint. The universal joint

transmits torque to a lower shaft through another universal joint. A pinion at the

end of the lower shaft mates with the rack and converts the column rotary motion

into translatory motion of the rack. For modeling purposes, the rack can be

visualized as two similar sections on either side of the pinion. A ball joint is used

to connect the end of the rack to a tie rod, which connects to a knuckle through

another revolute joint. The kingpin axis is aligned with the global Z axis. The

knuckle carries the road wheel which turns due to the translatory motion of the

rack.

This complete system had been divided into three subcomponents for analysis,

namely the universal joint, the rack & pinion assembly and the tie rod & knuckle

assembly.

9

Figure 2.1 Steering system

2.1.1 Universal Joint

A universal joint, U joint, Cardan joint, Hardy-Spicer joint, or Hooke's joint is a

joint in a rigid rod that allows the rod to 'bend' in any direction, as shown in

Figure 2.2. It is commonly used in shafts that transmit rotary motion. It consists

of a pair of hinges located close together, oriented at 90° to each other, connected

by a cross shaft.

Figure 2.2 Universal Joint

10

Relation between different parameters for a Universal joint as shown in Figure

2.3 have been stated in [20] and are as follows:

tan! =tan"

cos# (1.1a)

!2=

!1cos"

1# sin2 " sin2$1

(1.1b)

!2="#

1

2sin

2 $ cos$ sin%&

(1" sin2 $ cos2&)2 (1.1c)

Figure 2.3 Schematic of the Universal Joint

where

! = angle of the driving shaft

! = angle of the driven shaft

! = angle between the shaft axis

!1= angular velocity of the driving shaft

!2= angular velocity of the driven shaft

!2= angular acceleration of the driven shaft

Figure 2.4 and 2.5 show the variation in driven shaft angular displacement and

driven shaft angular velocity with change in driving shaft angular displacement.

11

Figure 2.4 Plot between driven shaft angle and driving shaft angle

Figure 2.5 Plot between driven shaft speed and driving shaft speed

12

2.1.2 Rack and Pinion Assembly

A rack and pinion is a pair of gears which converts rotational motion into linear

motion. A circular pinion engages the teeth on a flat bar - the rack as shown in

Figure 2.6. Rotational motion applied to the pinion will cause the rack to move,

up to the limit of its travel. For example, in a rack railway, the rotation of a

pinion mounted on a locomotive or a railcar engages a rack between the rails and

pulls a train along a steep slope.

Figure 2.6 Rack and pinion assembly

The rack and pinion arrangement is commonly preferred in the steering

mechanism of cars or other wheeled, steered vehicles. Even though this

arrangement provides a lesser mechanical advantage than other mechanisms such

as recirculating ball, it has much less backlash and greater feedback, or steering

"feel".

The relation between the angular displacement of the pinion and the rack

translation as shown in Figure 2.7 is given by:

x = rp.!l (1.2)

13

Figure 2.7 Rack and pinion assembly schematic

where

x = rack displacement

rp

= pinion pitch circle radius

!l= driving shaft rotation

Under ideal conditions the rack displacement transfer is only a function of the

pinion rotation but in the presence of manufacturing errors it also become a

function of other parameters like the rack profile angle, center distance variation,

pinion installation angle etc.

2.1.3 Tie Rod and the Knuckle

The tie rod is part of the steering mechanism in a vehicle. A tie rod is a slender

structural rod that is used as a tie and is capable of carrying tensile loads only as

shown in Figure 2.8. It is a rod with a "ball and socket" at one end that is

connected to the steering arm or the knuckle. The other end is connected to the

rack. When the steering wheel moves, causing the rack to move, the ball and

socket allows the wheel to turn. Stud swing from side to side allows the tie rod to

function as the vehicle moves up and down. When two tie rods are used, their

length are kept adjustable, allowing the wheels to be aligned. Proper tie rod

14

function is important, as excessive movement can contribute to toe change,

which can effect tire wear and car stability.

Figure 2.8 Ball joint on the tie rod

The knuckle is mounted about the kingpin axis. It is mounted between the

shockers and the stub axel. The car wheel is installed on the knuckle and as it

turns about the kingpin axis it turn the vehicle.

In Figure 2.9 kingpin axis is at the point O along the Z-axis which is

perpendicular to the XY plane.

Figure 2.9 Tie rod and knuckle assembly

15

where

x = lateral rack displacement

Lx= lateral distance between the pinion and the king pin axis

!x= lateral distance between rack and the king pin axis

d = longitudinal distance between the rack and kingpin axis

Lt= length of the tie rod

Lk= length of the knuckle arm

!t= acute angle made by the tie rod

!k= acute angle made by the knuckle arm

and:

x = Lx!"

x (1.3)

Equating the distance along the X-axis, we get:

!x = Ltsin"

t# L

ksin"

k (1.4a)

Equating the distance along the Y-axis, we get:

d = Ltcos!

t+ L

kcos!

k (1.4b)

Eliminating!tfrom equations 1.4a and 1.4b. We first square the two equations to

get the expression:

(d ! Lkcos"

k)2= L

t

2cos

2"t (1.5)

(!x+ L

ksin"

k)2= L

t

2sin

2"t (1.6)

and then add them to get:

d2+ !

x

2+ L

k

2" 2dL

kcos#

k+ $!

xLksin#

k= L

t

2 (1.7)

The terms can be rearranged and we get the expression:

!x

2+ (d

2+ L

k

2" L

t

2) = 2L

k(d cos#

k" !

xsin#

k) (1.8)

Equation 1.8 directly relates the change in the angle at the knuckle!kto the

displacement of the rack !x.

16

2.1.4 Integrated Assembly

We want to get a relation between the angular displacement at the steering wheel

and the rotation at the tire. For this relation components including the steering

wheel, steering column, universal joint, intermediate shaft, lower shaft, pinion,

rack, tie-rod, knuckle and wheel are to be modeled together. Their relations and

interactions with each other have been formulated in the preceding sections.

Figure 2.10 gives a schematic for the integrated assembly. In the figure all links

between the steering wheel and the pinion are in the YZ plane and all the links

between the rack and the tire are in the XY plane.

Figure 2.10 Schematic of the steering system

17

where

!d

= steering wheel angular displacement

!1= intermediate shaft angular displacement

!l= lower shaft angular displacement

!1= acute angle between intermediate and steering shaft

!2

= acute angle between lower and intermediate shaft

rp

= pitch radius of the pinion

! = rack displacement relative to the rack casing

Lt= length of the tie rod

Lk= length of the Knuckle

dy = distance between the rack case and the knuckle along the Y direction

dx= distance between the rack case and the knuckle along the X direction

! = angle between the tie rod and the Y-axis

! = angle between the knuckle geometry and the Y-axis

! is also equal to the rotation of the tire as the rotation of the tire is equal to the

rotation of the knuckle. To find the relation between !d

and ! , relations are found

between the following variables and then these systems are compiled together.

Relation 1, between !dand !

l:

A universal joint connects the steering shaft and the intermediate shaft. From

equation 1.1a

tan!1=tan!

d

cos"1

(1.9)

Another universal joint connects the intermediate shaft and the lower shaft. Their

angular displacements are related as

18

tan!l=tan!

1

cos"2

(1.10)

By substituting !1from equation 1.9 into equation 1.10, we get a relation between

between !d

and !l

tan!l=

1

cos"1cos"

2

tan!d (1.11)

Relation 2, between !land ! :

From equation 1.2 the relation between a rack displacement and pinion angular

displacement is

! = rp ."l (1.12)

Relation 3, between ! and ! :

Equating the distance along the X axis

Lt cos! + Lk cos" = dy (1.13)

Equating the distance along the Y axis

Ltsin! " L

kcos# = d

x" $ (1.14)

Eliminating! from 1.13 and 1.14

Lt2= dy ! Lk cos"#$ %&

2

+ (dx ! ' ) + Lk sin"[ ]2

= dy2+ Lk

2+ (dx ! ' )

2 ! 2dyLk cos" + 2(dx ! ' )Lk sin" (1.15)

Rearranging the terms of equation 1.15

dy2+ Lk

2+ (dx ! " )

2 ! Lt2= 2Lx dy cos# ! (dx ! " )sin#$% &' (1.16)

dy cos!

dy2+ (dx " # )

2"

(dx " # )sin!

dy2+ (dx " # )

2=dy2+ Lk

2+ (dx " # )

2 " Lt2

2Lk dy2+ (dx " # )

2 (1.17)

To simplify equation 1.17 we define a new function! as:

19

sin! =dy

dy2+ (dx " # )

2, cos! =

(dx " # )

dy2+ (dx " # )

2 (1.18)

Substituting! in equation 1.17

sin(! "#) =dy2+ Lk

2+ (dx " $ )

2 " Lt2

2Lk dy2+ (dx " $ )

2

%

&

''

(

)

**

(1.19)

Taking the sine inverse of equation 1.19

! = " # sin#1 dy2+ Lk

2+ (dx # $ )

2 # Lt2

2Lk dy2+ (dx # $ )

2

%

&

''

(

)

**

(1.20)

! = sin"1 dy

dy2+ (dx " # )

2

$

%

&&

'

(

))" sin"1 dy

2+ Lk

2+ (dx " # )

2 " Lt2

2Lk dy2+ (dx " # )

2

$

%

&&

'

(

))

(1.21)

Combining equation 1.11 and 1.12 we get

! = rp " tan#1 tan$d

cos%1cos%

2

&

'(

)

*+ (1.22)

Hence the relation between ! and!dis defined using equations 1.21 and 1.22.

Using the data in Table 2.1 the angular displacement of the car tire! is plotted vs

angular displacement of the steering wheel, as shown in Figure 2.11. The steering

wheel is rotated from -360˚ to 360˚. We can see from the figure that the plot is

not an exact straight line and has a wave form. This is due angular displacement

transfer function of the universal joint.

2.2 Validation

The fundamental physics based model of the steering system is checked by

performing simple studies using test inputs and comparing the data with test runs

from a ADAMS model of the steering system with the same dimensions as

20

shown in Figure 2.12. For the trial simulations the model is simulated by an

angular displacement inputs at the steering wheel.

Figure 2.11 Car wheel angular displacement

Figure 2.12 ADAMS model of the car steering system

21

2.2.1 ADAMS Modeling

The components modeled in ADAMS are the steering wheel, steering shaft,

intermediate shaft, lower shaft, pinion, rack, rack case, tie rod, knuckle and the

tire. The steering and intermediate shaft and the lower and intermediate shaft are

connected via universal joints. All shafts are fixed to the ground using revolute

joints. A translation joint is defined between the rack and the rack case which is

locked to the ground. The displacement transfer function at the pinion is defined

using a couple joint. The rack and the tie rod are connected using a revolute joint

and so are the tie rod and the knuckle. The constraint joints defined between

different parts are also shown in Figure 2.13. It is possible to use revolute joint

because we assume that the rack, tie rod and the knuckle geometry are in the

same plane. The axis of the revolute joint is normal to this plane, i.e. along the

global Z direction. We are able to make this assumption because we are looking

at the steering sub system alone, without the suspension system and other

geometrical constraints from separate sub systems. The suspension system, caster

angle and the camber angle cause a component of the translation to exist along

the global Z direction.

2.2.2 Test Simulations

Data used to model for the test simulation are taken from a small Indian family

car. Data attached in Table 2.1. The fundamental physics model is simulated

analytical. An ADAMS model of the steering system build using the same data is

also simulated. The steering is rotated from -360° to +360°. Angular

displacement of the Knuckle is recorded for the two simulations and compared.

22

TABLE 2.1 Dimensions for the car steering system model

!1

!2

rp

Lt

Lk

dy

dx

LR

LRC

Symbol Value Detail

20° Acute angle between intermediate and steering shaft

25° Acute angle between lower and intermediate shaft

5.5 mm Pitch radius of the pinion

350mm Length of the tie rod

135mm Length of the Knuckle

150mm Distance between the rack case and the knuckle along the Y direction

400mm Distance between the rack case and the knuckle along the X direction

500mm Length of the Rack

400mm Length of the Rack case

Figure 2.13 Joints between parts in the ADAMS model

Key:

23

2.3 Results

The ADAMS model and the analytical model of the steering system give the

angular displacement of the Knuckle as output. As the tire is mounted on the

knuckle, the angular displacement of the knuckle can be taken as the angular

displacement of the tire. The data from the two simulations is compared in Figure

2.14. It can be seen from Figure 2.15 that the variation in the results from the two

plots is of the order of 10-1 degrees.

Figure 2.14 Comparison of Analytical and ADAMS simulations

24

Figure 2.15 Difference between Analytical and ADAMS simulations

2.4 Remarks

The model for the steering system in cooperates all the important linkages

between the steering wheel and the tire, which are responsible for the motion

transfer. The suspension system also causes a change in the steering ratio, but has

been left out from the model, keeping in mind the objective of analyzing the

effect of manufacturing errors at the rack and pinion.

25

!

CHAPTER-3

KINEMATICS OF GEAR MESHING

Gearing is a special division of mechanical engineering concerned with the

transmission of power and motion between rotating shafts. Gearing is usually the

best and the most economical means of achieving this transmission. Gears not

only transmit motion and enormous power satisfactorily, but they can do so with

very uniform motion, an important consideration in many applications.

3.1 Kinematics of Gear Meshing

In this section we look at the equations that relate the velocities of meshing

components and the torque and force transfer between them. We also find out the

factors that influence the velocity ratio and the torque transfer function.

3.1.1 Velocity Ratio

Consider two gears meshing together, as shown in Figure 3.1. The teeth of each

gear pass in and out of mesh with those of the other gear. The meshing occurs in

the region between the gear centers C1 and C2.

Let N1and N

2be the number of teeth on the two gears respectively and let nbe the

number of teeth in mesh during time interval T . Then the average speed of

rotation for the two gears can be expressed as:

(!1)average

= (n

N1

)2"

T (3.1a)

(!2)average

= "(n

N2

)2#

T (3.1b)

26

where

(!1)average

= average angular velocity of gear 1

(!2)average

= average angular velocity of gear 2

Figure 3.1 A gear pair

Combining equations 3.1a and 3.1b we get

N1(!

1)average

= "N2(!

2)average

. (3.2)

Equation 3.2 in true for all gears, what ever the shape of the teeth. However if the

tooth shape is arbitrary, the gear will not run smoothly. There will be periodic

vibrations, with the magnitude dependent on the profile of the gear tooth. The se

vibrations lead to fatigue cracks, resulting in early failure of the gear. Such gears

are not appropriate for use in precision machinery or high speed application. To

avoid early tooth breakage and smooth operation it is important to choose a tooth

profile which allows the gear to maintain a constant angular velocity ratio. The

requirement for angular velocity can be described by equation 3.3.

N1!1= "N

2!2 (3.3)

where

!1= angular velocity of gear 1


27

3.1.2 Constant velocity ratio for a rack and pinion

We will first look at the condition for constant velocity ratio for a rack and pinion

system as mentioned in [21]. When two gears mesh the smaller one is usually

called the pinion. As the rack is considered to be a gear of infinite radius any gear

meshing with the rack is called a pinion.

During any time interval T , the number n of the rack teeth passing through the

meshing area is equal to the number of pinion teeth which pass through it. Thus,

average values of rack velocity and pinion angular velocity can be expressed as:

(vr)average

=np

T (3.4)

(! )average

= (n

N)2"

T (3.5)

where

(vr)average

= average rack velocity

(! )average

= average pinion angular velocity

p = pitch of the rack

N = number of pinion teeth

Relation between the average rack velocity and the average pinion angular

velocity can be obtained by combining equation 3.4 and 3.5.

1

p(v

r)average

=N

2!(" )

average (3.6)

Equation 3.6 is analogous to equation 3.2. As with a pair of gears, the satisfactory

operation of a rack and pinion requires that the relation between vr

and !

remains constant. Hence, the tooth shape should be such that it satisfies the

following equation:

vr

p=N!

2" (3.7)

28

Figure 3.2 shows the rack and pinion tooth profile. Point Ar on the rack and point

A on the pinion are the point of contact. As the rack and the pinion are solid

bodies the velocity of the two points along the common normal must be the

same. The unit vector along the common normal can be written as:

n̂nr= ! sin"

pn̂i! cos"

pn̂j (3.8)

where

n̂nr

= unit vector along the common normal

!p= profile angle of the rack

n̂i , n̂ j , n̂k = mutually perpendicular unit vectors

Figure 3.2 Common normal at the contact point

The direction of n̂iand n̂

jare perpendicular and parallel to the rack reference line

respectively and n̂kis perpendicular to the plane of motion. The velocity of Ar and

its component along the common normal is given by:

!vAr = vrn̂ j (3.9a)

vnAr = n̂nr .

!vAr = ! cos"pvr (3.9b)

where

29

!vAr = velocity of the rack

vr= speed of the rack

vn

Ar = rack velocity component along the common normal

The vector from the center C to the point A is (Xn̂i+Yn̂

j). The velocity of point A

and its component along the common normal can be expressed as:

!vA=! n̂k " (Xn̂i +Yn̂j ) = #!Yn̂i +!Xn̂j (3.10a)

vnA= n̂nr .

!vA=!Y sin"p #!X cos"p (3.10b)

where

!vA = velocity of the point A on the pinion

! = angular velocity of the pinion

vn

A = velocity component at point A along the common normal

It is possible to equate the normal velocity component of Ar and A, given by

equations 3.9b and 3.10b, and substitute equation 3.7 to express the condition

required for constant velocity ratio. The equation obtained (3.11) must be

satisfied by X and Y , the coordinates of the contact point.

Y

(X !Np

2")

= cot#p (3.11)

The following points can be interpreted from equation 3.11. There is a fixed point

P, at a distance (Np / 2! ) from the center C on the line through C perpendicular to

the rack reference line, such that the slope of the line PA is equal to (! / 2 "#p) as

shown in Figure 3.3. This means that the line PA makes an angle (! / 2 "#p) with

the nidirection, and is therefore the common normal at the point of contact. The

common tangent makes an angle !p

with the nidirection. The position of point P

is shown in the Figure 3.3.

30

Figure 3.3 Pitch point of a rack and pinion

The result just derived is called the condition for constant velocity ratio for a rack

and pinion system. It can be restated in the following way. The condition that

must be satisfied by the tooth profiles of a rack and pinion, in order that the

relation between rack velocity and pinion angular velocity remains constant, is

that the common normal at the contact point should at all times pass through a

fixed point P. The position of P is at a distance (Np / 2! ) from the pinion center C,

on the perpendicular from C towards the rack reference line.

The point P is called the pitch point. The circle passing through P with center C is

called the pinion pitch circle, and its radius RP

is equal to the length CP,

Rp=Np

2! (3.12)

Hence using equation 3.12 relation between rack velocity and pinion angular

velocity can be restated as:

vr= R

p! (3.13)

31

3.1.3 Condition for constant velocity ratio

To understand the condition for constant velocity ratio for two gears, as

mentioned in [23] we look at the portions of the two teeth, one on gear 1 (the

pinion) and the other on the gear 2, as shown in Figure 3.4. The two teeth come

in contact at a point Q, and the wheels rotate in the direction shown in the figure.

In Figure 3.4:

TT = common tangent to the curves at the point of contact Q

MN = common normal to the curves at the point of contact Q

O1M and O2N = perpendicular to MN

QC = direction of movement of Q with respect to gear 1

QD = direction of movement of Q with respect to gear 2

Figure 3.4 Two teeth meshing

The component of velocities of the point Q on the gear 1 and 2 along the

common normal MN must be equal if the teeth are to remain in contact.

32

v1cos! = v

2cos" (3.14)

or (!1"O

1Q)cos# = (!

2"O

2Q)cos$ (3.15)

(!1"O

1Q)O1M

O1Q

= (!2"O

2Q)O2N

O2Q

(3.16)

as cos! =O1M

O1Q

and sin! =O2N

O2Q

(3.17)

!"1..O1M ="

2.O

2N (3.18)

!1

!2

=O2N

O1M

(3.19)

also from similar triangles O1MP and O

2NP , we get

O2N

O1M

=O2P

O1P

(3.20)

using equations 3.19 and 3.20,

!1

!2

=O2N

O1M

=O2P

O1P

(3.21)

where

v1= velocity of the point Q on the gear 1

v2

= velocity of the point Q on the gear 2



! = angle between MN and QC

! = angle between MN and QD

From equation 3.21 we see that the angular velocity ratio is inversely

proportional to the ratio of the distance of P from the centers O1 and O2 or the

common normal to the two surfaces at the point of contact Q intersects the line of

centers at point P which divides the center distance inversely in the ratio of

angular velocities. Therefore in order to have constant angular velocity ratio for

all positions of the gear, P is a fixed point called the pitch point for the two gears.

33

3.1.4 Involute tooth profile for a rack and pinion

From the condition for constant velocity ratio the shape of the pinion should be

such that the normal to the tooth profile at point A passes though P as shown in

Figure 3.5. This can be directly inferred from the condition for constant velocity

ratio for a rack and pinion. We can construct a perpendicular from the pinion

center to the line of action, and the foot of this perpendicular is labelled E. The

pinion circle with center C and radius equal to CE is known as the base circle.

Rb = Rp cos!p (3.22)

where

Rb

= radius of the base circle

Figure 3.5 Meshing diagram of a pinion and a basic rack

Using the base circle, the property of the tooth shape can be stated as: The shape

of the tooth profile must be such that the normal at the contact point touches the

base circle. As the pinion rotates, the contact point moves along the the pinion

tooth, and therefore at each point of the profile the normal to the profile must

touch the base circle. A curve with this property is known as an involute of the

34

base circle and is shown in Figure 3.6. Hence the tooth profile for a conjugate

gear meshing with a straight tooth faced rack is an involute.

Figure 3.6 Gear with involute teeth

3.1.5 Properties of involute tooth

As described in [22] the involute tooth form is easily generated, and its use

permits a variation of the center distance without affecting the uniform velocity

ratio, one of its most valuable features.

Figure 3.7 shows an involute curve generated from a circle of radius Rb. Rb is the

base circle radius of the gear. Point I is any point along the involute profile. The

radius of curvature at point I is given by

!l= (R

l

2 " Rb

2)0.5 (3.23)

where

!l= radius of curvature at point I

Rl= radial distance of the point I from the axis of the gear

35

Figure 3.7 Involute gear teeth

Using the right angled triangle OBI, pressure angle !l

at any point can be found

from

!l= cos

"1 Rb

Rl

#

$%&

'( (3.24)

From this expression, we see that the pressure angle varies all along the involute.

When referred to the pressure angle of a gear, the reference is generally to the

pressure angle at the pitch point.

The roll angle can be found by considering the fact that the arc length between

point A and B is the same as !l

since !l

is the “length of the string” unwound

from the base circle to produce the involute up to point I. Thus we can write:

!l="l

Rb

(3.25)

where

!l= roll angle

36

substituting equation 3.23 in 3.25

!l=(R

l

2" R

b

2)0.5

Rb

(3.26)

Also by noting that

tan!l="l

Rb

(3.27)

we may write

!l= tan("

l) (3.28)

The angle ! is defined as the difference between the roll angle and the pressure

angle;

!l= "

l#$

l= tan($

l) #$

l (3.29)

This function is often called the involute of the pressure angle.

INV (!l) = tan(!

l) "!

l (3.30)

3.1.6 Torque Transfer

Gears are generally used to transfer motion and torque. We have already had a

look at motion transfer. For torque transfer we will look at the rack and pinion

assembly as shown in Figure 3.5. To find the torque transmitted to the pinion

from the rack we assume there is a force F being applied on the rack. The rack is

a standard straight sided rack with pressure angle !pand mass m . Acceleration of

the rack is a . When the rack meshes with a gear with involute teeth, at the point

of contact P the normal reaction Ntwill act along the common normal at point P.

Another normal reaction Ngacts on the rack because of the contact with the rack

casing. Any line normal to the involute tooth of the gear is tangent to the base

circle. Hence, the perpendicular distance of the common normal or the normal

37

reaction vector from the center C of the pinion is Rb. The torqueT is calculated

about the pinion center P.

Figure 3.8 Force body diagram for the rack

From the force body diagram shown in Figure 3.8 we can write:

F ! Ntcos"

p= ma (3.31)

Nt=F ! ma

cos"p

(3.32)

For the pinion

T = Rb! N

t (3.33)

T = RbF ! macos"p

#

$%

&

'( (3.34)

3.2 Helical rack and pinion

A helical gear has teeth in the form of helix around the gear. Two such gears may

be used to connect two parallel shafts in place of spur gears. The pitch surfaces

are cylindrical as in spur gearing, but the teeth instead of being parallel to the

axis, wind around the cylinder helically like a screw thread. The teeth of helical

gears with parallel axis have line contact, as in spur gearing. This provides

gradual engagement and continuous contact of the engaging teeth.

38

3.2.1 Basic helical rack

Figure 3.9 Helical gear and rack

As stated by Colbourne in [21], Figure 3.9 shows the basic helical rack, used to

define the tooth surface of a helical gear. Just as the basic rack of a spur gear has

teeth which are straight-sided, the basic rack in Figure 3.10 has teeth whose faces

are flat planes. The angle between the gear axis and the direction of the rack teeth

is !r, and is called the basic rack helix angle. A plane cutting through the rack,

perpendicular to the gear axis is known as a transverse section of the rack, and a

plane cutting perpendicular to the rack teeth, in other words perpendicular to n! ,

is called the normal section.

Figure 3.10 also shows the transverse section and the normal section through the

basic rack. The distances in the two sections between corresponding points of

adjacent teeth are called the transverse rack pitch ptr

and the normal rack pitch

pnr

. It can be seen from triangle A1A2A3 that there is a relation between the two

pitch, given by:

pnr= p

trcos!

r (3.35)

39

Figure 3.10 Basic helical rack

The pressure angles shown in the transverse and normal sections in Figure 3.10

are called the transverse rack pressure angle !tr

and the normal rack pressure

angle !nr

. They can be expressed in terms of the tooth dimensions as follows:

tan!tr=ht

H (3.36)

tan!nr=hn

H (3.37)

whereH is the tooth depth, and htand h

nare the lengths shown in Figure 3.10 in

the transverse and normal tooth sections. These two lengths are related as

follows:

hn= h

tcos!

r (3.38)

40

From the last three equations we obtain a relation between the two pressure

angles and the helix angle.

tan!nr= tan!

trcos"

r (3.39)

The rack base pitches in the two sections are defined as the distances between

adjacent tooth profiles, measured in each case along the common normal. The

transverse base pitch ptbrand the normal base pitch pnbrare shown in Figure 3.10,

and are related to the rack pitch ptr

and pnr

in the following manner:

ptbr = ptr cos!tr (3.40)

pnbr = pnr cos!nr (3.41)

We have specified the basic rack by means of the following sever quantities ptr,

pnr

, ptbr , pnbr ,!tr

,!nr

, and!r. However, we have shown that there are four

relations between the quantities, given by the equations 3.35, 3.39, 3.40 and 3.41.

Hence only three quantities can be used to specify the basic helical rack.

3.2.2 Helical pinion

We now study the geometry of a gear with N teeth, whose tooth shape is defined

as being conjugate to the basic rack shown in Figure 3.10. If we consider a single

transverse plane through both the gear and the basic rack, the tooth profile of the

gear must be conjugate to that of the gear. Hence, the gear tooth profile in the

transverse plane can be found by means of the spur gear geometry described in

section 3.1.2. The profile of the spur gear is therefore an involute defined by a

basic rack with pitch ptr

and pressure angle !tr

. The radius of the pitch circle and

the base circle are similar to the ones mentioned in equations 3.12 and 3.22

Rp=Np

tr

2! (3.42)

Rb = Rp cos!tr (3.43)

41

It is clear that at every transverse cross section of the gear we obtain a standard

pitch circle, each with the same radius. The cylinder containing all these circles,

in other words the cylinder of radius Rb, is the pitch cylinder of the gear. When it

is meshed with its basic rack it is called the standard pitch circle. It is used as a

reference cylinder, in exactly the same manner as the standard pitch circle of a

spur gear. In particular, many of the quantities which define the shape of the

teeth, such as the pressure angles and the tooth thicknesses, are specified by their

values on the standard pitch cylinder.

In our study of the tooth shape of a helical gear, we are not considering a rotation

of the gear, but a rotation of the tooth profile as we move axially along the gear.

Therefore from Figure 3.11 and 3.12 we can see that for a profile rotation of !"

the rack tooth profile is displaced by a distance of z tan!r. Where z is the pinion

thickness. The equivalent displacement on the pinion is at the standard pitch

cylinder. Hence

!" =z tan#

r

Rb

(3.44)

Figure 3.11 Helix through point A0

42

Figure 3.12 Transverse section through the basic rack

Another important parameter for studying the helical gear is the helical angle !R

at some radial distanceR . A cylinder of radius R and length z is developed into

a rectangle. As shown in Figure 3.13 A0 is any point on the cylinder at plane

z = 0 , and A is a point at plane z on the helix through A0. Hence, we can write

tan!R=R" A # R" Ao

z (3.45)

! tan"R=R#$

z (3.46)

Using equation 3.44 we can rewrite 3.46 as

tan!R

R=tan!

r

Rb

(3.47)

43

Figure 3.13 Developed cylinder of radius R and length z

3.2.3 Velocity ratio of a helical rack and pinion

If we look at a transverse cross section of a helical rack meshing with a helical

pinion, it is exactly identical to a spur gear of the same pitch radius meshing with

a conjugate rack. This will be true for any transverse cross section of a helical

gear meshing with the conjugate rack. Hence the condition for constant velocity

ratio as shown in section 3.1.2 holds for a helical rack and pinion system.

Therefore we can state that:

vr= R

p! (3.48)

where

Rp

= pitch radius of the pinion


vr= velocity of the rack

and the pitch radius is given by the equation 3.12.

44

3.2.4 Torque transfer for a helical rack and pinion

For torque transfer in a helical rack and pinion assembly we consider a rack with

pressure angle !p

and rack helix angle !r

. To find the torque transmitted to the

pinion from the rack we assume there is a force F being applied on the rack.

When the rack meshes with a gear with involute teeth, at the point of contact P

the normal reaction Ntwill act along the common normal at point P. This common

normal AP is perpendicular to the helical tooth face. In Figure 3.14 the force

components are:

AP = resultant force (normal force)

AB = radial component

BP = perpendicular to the tooth face and tangential to the pitch circle

BD = axial component

DP = tangential to the pitch cylinder and perpendicular to the axis

Figure 3.14 Tooth surface of a helical rack

Balancing the force along the direction of rack travel. The component of normal

reaction in the direction of rack travel is Ntcos!

rcos"

p. Hence:

F ! Ntcos"

rcos#

p= ma (3.49)

Nt=

F ! ma

cos"rcos#

p

(3.50)

Torque on the gear along the axial direction works out to be:

45

T = RbF ! ma

cos" r cos#p

$

%&

'

() cos" r (3.51)

T = RbF ! macos"p

#

$%

&

'( (3.52)

which is the same expression as the one for a spur gear meshing with a rack.

3.3 Remarks

Any transverse cross section of a helical pinion and a conjugate rack will have

the profile of a spur gear with a conjugate rack. The velocity ratio and torque

transfer function will be same for a helical gear and a spur gear.

46

CHAPTER-4

MODELING OF THE GEAR ERRORS

Manufacturing errors in the steering system assembly occur in the upper steering

system, pinion-rack interface and the lower steering system. For the present

scope of work, only gear errors (at the pinion-rack interface) are considered. The

errors which have been modeled are:



c) Backlash


e) Rack bend

f) Tooth thickness

g) Pressure angle

h) Pinion pitch circle runout (It defines the runout of the pitch circle. It is

the error in radial position of the teeth. Most often it is measured by

indicating the position of a pin or ball inserted in each tooth space around

the gear and taking the largest difference. Alternately, particularly for

fine pitch gears, the gear is rolled with a master gear on a variable center

distance fixture, which records the change in the center distance as the

measure of teeth or pitch circle runout.)

The base model of an ideal rack and pinion is modified to capture the

manufacturing errors. In order to capture the above errors, gear tooth profile and

gear contact are modeled. We look at variation of output parameters like the

velocity ratio and the torque transfer with respect to variation in input parameters

like the profile angle, center distance variation etc.

47

4.1 Center distance variation

The center distance error is a variation in the distance between the pinion center

and the rack reference line. This variation is caused because of the following

reasons:

1) Tolerance in the bearing housing

2) Clearance in the supporting bearings

3) Temperature variation during operation. Expansion of the gear teeth and the

use of dissimilar materials with varying coefficients of expansion in the

gearbox affect the center distance variation and backlash.

To analyze the center distance variation error for a rack and pinion assembly we

vary the center distance by !C and look at its effect on the velocity transfer ratio,

torque transfer function and the backlash.

For a rack and pinion assembly the point of contact on the pinion is where the

slope of the involute is equal to the rack profile angle as shown in Figure 4.1. As

the rack profile angle does not change with center distance variation, the point of

contact on the pinion does not change. Keeping this in mind we calculate the

velocity ratio.

4.1.1 Velocity ratio

The condition for constant velocity for a rack and pinion states that the common

normal at the point of contact should pass through a fixed point. The position of

this fixed point comes out be at a distance (Np / 2! ) from the pinion center, on the

perpendicular from the center to the rack reference line, as shown in section

3.1.2. The point of contact on the pinion does not change with center distance

variation hence the the position of the point of contact on the perpendicular from

48

the center to the rack reference line remains fixed. The velocity ratio remains

constant for the error as show by the following derivation.

Figure 4.1 Center distance variation

We equate equation 3.9b and 3.10b to get a relation between the rack velocity

and pinion angular velocity. On equating we get:

! cos"pvr=#Y sin"

p!#X cos"

p (4.1)

To derive the term for velocity ratio we rearrange equation 4.1 to get:

vr

!=X cos"

p#Y sin"

p

cos"p

(4.2)

To show that the velocity ratio is constant for center distance variation we

compute and substitute the values ofX andY in equation 4.2.

49

At the point of contact for a rack and pinion the tangent at the rack and pinion

surfaces should have the same slope. The profile angle of the rack is !p

through

out the rack tooth profile. For a pinion that has rotated by an angle ! as shown in

Figure 4.2, the angle that the tangent to the pinion surface at the point of contact,

makes with the X axis works out to be !pi"# + INV!

pi. Equating the two to find

!pi

, the instantaneous profile angle of the pinion at the point of contact, we get:

!p= !

pi"# + INV!

pi (4.3)

Figure 4.2 Rack tooth contact

Substituting the value of the involute function as mentioned in equation 3.30 in

4.3, we get:

!p= !

pi"# + tan!

pi"!

pi (4.4)

!p= tan!

pi"# (4.5)

!pi= tan

"1(!

p+#) (4.6)

50

The radius of the point of contactRlfor the pinion can be determined by using

equation 3.24. From the equation we have:

cos!pi =Rb

Rl (4.7)

Rl =Rb

cos!pi

(4.8)

X and Y are the x and y components of CP respectively. Length of CP is Rl

and

the inclination of CP with the X axis is (! " INV#pi) . Hence:

X = Rl cos(! " INV#pi ) (4.9)

Y = Rl sin(! " INV#pi ) (4.10)

Substituting equation 4.9 and 4.10 in 4.2 we get:

vr

!=Rl cos(" # INV$pi )cos$p # Rl sin(" # INV$pi )sin$p

cos$p

(4.11)

Which can be simplified to:

vr

!=Rl cos(" # INV$pi + $p )

cos$p

(4.12)

From equation 4.3 we know:

! " INV#pi+ #

p= #

pi (4.13)

which holds for center distance variation.

Substituting equation 4.8 and 4.13 in equation 4.12 we get:

vr

!=

Rb

cos"pi

cos"pi

cos"p

(4.14)

vr

!=

Rb

cos"p

(4.15)

Rband !

premain constant for center distance variation. Hence the velocity ratio

remains constant with change in center distance as shown in Figure 4.3. The

51

velocity ratio function has been calculated for a pinion with base circle radius of

12.68 mm, pressure angle of 20˚and a pinion rotation range of -360˚ to +360˚.

The value of velocity ratio comes out to be the same as given by equation 4.15.

Figure 4.3 Velocity ratio

4.1.2 Torque transfer

Next we look at the effect of center distance variation on torque transfer. In

section 3.2.3 we saw that the torque transfered depends on the force applied, the

base radius of the pinion and the pressure angle. When there is a variation in the

center distance the point of contact with respect to the rack surface changes, but

with respect to the pinion remains the same. Hence the pressure angle and the

base radius does not change. Therefor center distance variation does not have a

affect on the torque transfer. The expression for torque transfer remains the same

as equation 3.52. The value of torque has been plotted for a pinion rotation from

-360˚ to +360˚. The value is calculated for a pinion with base circle radius of

12.68 mm and pressure angle of 20˚. We make the assumption that the rack

52

translates with a constant velocity, and a force of 1N is acting on it. We can see

from Figure 4.4 that the torque transfer function remains constant.

Figure 4.4 Torque transfer

4.1.3 Backlash

The backlash along the pitch circle is defined as the angle by which the pinion is

free to rotate if the position of the rack is fixed. Here we look at the change in the

value of backlash because of the presence of center distance variation. Variation

in backlash !Blc

is equal to the change in the rack tooth thickness at the new pitch

plane. Hence:

!Blc= T

r" T

ro (4.16)

where

Tr= rack tooth thickness at the original pitch plane

Tro

= rack tooth thickness at the new pitch plane

53

From the Figure 4.5 we can write:

!Blc = 2!C tan"p (4.17)

Figure 4.5 Backlash in center distance variation

This function is plotted in Figure 4.6 for a center distance variation of 2 mm and

a pinion with profile angle of 20˚.

Figure 4.6 Backlash

54

4.2 Pinion helix angle variation

The pinion helix angle by default is defined at the pinion pitch cylinder and is the

same as the rack helix angle. The contact between the helical pinion and the rack

is a line contact because the pinion helix angle and the rack helix angle are the

same. If there is a variation in the pinion helix angle and the axis of rotation

remains the same, then the line contact become a point contact. This point

contact is along the edge of the pinion if the rack tooth thickness is more and

along the edge of the rack if the pinion tooth thickness is more. Typically in a car

steering system the pinion tooth thickness is more than the rack tooth thickness.

This rack and pinion system with the point contact will only be able to rotate if

there is sufficient clearance for the gear to function. We assume that the center

distance between the pinion center and the rack reference plane is variable. We

are able to make this assumption because of the presence of yoke nut assembly in

the rack and pinion system of modern cars as shown in Figure 4.7. In the yoke

nut assembly a spring is mounted below the rack, which pushes the rack towards

the pinion.

Figure 4.7 Yoke nut assembly

55

This construction allows the distance between the rack and pinion to vary, hence

giving clearance to the rack and pinion to mesh even if there are errors present,

which require a larger clearance. In a car steering system the outer edge of the

rack comes in contact with the perimeter of the pinion cross section along the

transverse plane of the rack edge as shown in Figure 4.8. The contact is only in

the transverse plane along the edge of the rack. The motion is similar to that of a

spur gear meshing with a rack. This spur gear and rack system has the same

dimensions as that of the transverse cross section of the original helical rack and

pinion extruded. The stresses generated because of the line contact becoming a

point contact are very high and can lead to deformation of the pinion and the rack

teeth.

Figure 4.8 Plane of contact in the helix angle error


The velocity ratio function in presence of helix angle error does not change, as

the motion transfer is in the transverse plane along the edge of the rack and the

gear assembly behaves like a spur rack and pinion in that plane. Hence the

velocity transfer function is given by:

56

vr

!=

Rb

cos"p

(4.18)

It is independent of the helix angle of the pinion. Hence if the basic meshing

condition as described in chapter 3 are met the velocity ratio remains constant.


The contact force which was spread along a line earlier is now limited to a point.

The torque transfer as mentioned in section 3.2.3 is not a function a the helix

angle of the pinion, but the normal force is. Hence the torque transfered remains

the same, as stated by equation 3.52. The normal force equation deduced in

section 3.2.3 is for an ideal helical gear. When there is pinion helix angle error,

the force transfer is at the point contact and the transfer is like that for a spur

gear. Hence the torque transfer equation is the same as equation 3.34 and the

normal force as equation 3.32.

4.3 Backlash error

Backlash, lash or play is the clearance between the mating components as shown

in Figure 4.9. For a pair of gears, backlash is the amount of lost motion due to

clearance or slackness when movement is reversed and contact is reestablished.

In other words it is the difference between the tooth space and the tooth

thickness, as measured along the pitch circle. This gap means that when a gear-

train is reversed, the driving gear must be turned a short distance before all the

driven gears start to rotate. Theoretically, the backlash should be zero, but in

actual practice some backlash is allowed to prevent jamming of the teeth due to

manufacturing errors, deflection under load and differential expansion between

the gears and the housing.

57

Figure 4.9 Backlash for two gears

Backlash is undesirable in precision positioning applications such as machine

tool tables. It can be minimized by tighter design features such as ball screws

instead of lead screws, and by using preloaded bearings. A preloaded bearing

uses a spring or other compressive forces to maintain bearing surfaces in contact

despite reversal of direction. In a car rack and pinion system the backlash is

minimized by the use of the yoke nut assembly.

Backlash is created mainly because of two deviations. Deviation from the ideal

tooth profile and a change in the operating center distance. The change in

pressure angle and tooth thickness are the main contributors of the deviation

form the ideal tooth profile. The total backlash is defined as:

b = bt+ b

c (4.19)

where:

b = total backlash

bt= backlash due to change in tooth thickness

bc= backlash due to change in operating center distance

Backlash due to change in tooth thickness is measured along the pitch circle and

is defined by:

bt=t

i-ta (4.20)

58

where:

ti= tooth thickness on the pitch circle for ideal gearing (no backlash)

ta= actual tooth thickness

Backlash, measured on the pitch circle, due to change in operating center is

defined by:

bc = 2(!c) tan"p (4.21)

where:

!c = difference between actual and ideal operating center distances

!p= pressure angle


The velocity ratio function for a helical rack and pinion with backlash clearance

remains constant. When the teeth are in mesh the velocity is transfered at a

constant rate. When the pinion is moving through the clearance area i.e the teeth

are not in contact the velocity ratio becomes zero, as the rack does not move with

the rotation of the pinion. This condition arises for a brief rotational displacement

of the pinion when it starts rotating in the opposite direction. Hence the velocity

ratio when the teeth are meshing is given by:

vr

!=

Rb

cos"p

(4.22)

59


The torque transfer function for a rack and pinion system with backlash does not

change. When the teeth are in mesh the torque is transmitted as mentioned in

section 3.2.4. When the teeth loose contact while operating in the clearance zone,

no torque is transmitted.

4.4 Pinion installation angle error (axes misalignment)

In a car steering system the rack and pinion are installed in the car rack case. The

case is built with a specific orientation for the pinion axis. Some times the

installed orientation of the pinion axis makes an angle with the ideal position of

the axis. This is called the pinion installation angle error or the axes

misalignment error. To better understand the error, only two positions (pure shaft

misalignment and pure tilt) of the pinion are modeled first as shown in section

4.4.1 and 4.4.2.

When the shaft position misaligns, the line contact between the rack and pinion

tooth surfaces becomes a point contact. As the case with a car rack and pinion,

the pinion tooth thickness is more hence the point lies on the outer edge of the

rack. As the motion transfer is in the plane along the edge of the rack, for the

velocity ratio and torque transfer calculations a pinion cross section in this plane

is considered to be in mesh with the rack.

4.4.1 Pure shaft misalignment

In pure shaft misalignment or the intersecting axis misalignment the pinion axis

lies in the plane that is perpendicular to the rack reference plane and to the

transverse cross section through the rack. In this plane, the angle of intersection

between the rack reference plane and the pinion axis is ! , as shown in Figure

60

4.10. B1!B1

is the plane of contact or the plane in which motion transfer between

the rack and pinion takes place. Plane B1!B1is along the rack edge and plane B

2!B2is

perpendicular to the pinion axis.

Figure 4.10 Pure shaft misalignment

The tangent on the pinion surface at the point of contact makes an angle !" with

the X axis in the B1!B1 plane, as shown in Figure 4.12.

This is a projection of the angle! that the tangent makes with the X axis in the

B2

!B2plane, on the pinion surface at the point of contact, as shown in Figure 4.11.

Looking at an infinitesimally small section of the pinion surface at the point of

contact in the B1!B1plane and its projection in the B

2!B2plane, we can find the

relation between !" and ! . For a infinitesimally small section, the pinion tooth

61

curve can be approximated to a line. Hence for some dimensions l and h , as

shown in Figure 4.11, we can write:

tan! =l

h (4.23)

From Figure 4.13 we know that the projection of h in theB1!B1plane is h / cos! .

Hence looking at Figure 4.12 we can write:

tan !" =l cos#

h (4.24)

Figure 4.11 Section of the pinion surface in the plane B2

!B2

Figure 4.12 Section of the pinion surface in the plane B1!B1

62

Figure 4.13 Cross section of the pinion tooth in the XZ plane

From equation 4.23 and 4.24 we can write:

tan !" = cos# tan" (4.25)

From section 4.1.1 we know that:

! = "pi#$ + INV"

pi (4.26)

and for meshing of the rack and pinion:

!" = #p (4.27)


tan!p= cos" tan(!

pi#$ + INV!

pi) (4.28)

Rearranging terms in equation 4.28 to get:

tan!1 tan"

p

cos#$%&

'()= "

pi!* + INV"

pi (4.29)

tan!1 tan"

p

cos#$%&

'()= "

pi!* + tan"

pi!"

pi (4.30)

tan!1 tan"

p

cos#$%&

'()= tan"

pi!* (4.31)

The instantaneous profile angle at the point of contact for the pinion can be

expressed as:

63

!pi= tan

"1 # + tan"1 tan!

p

cos$%&'

()*

+

,-

.

/0 (4.32)

Velocity ratio

The velocity ratio function is calculated as show in section 3.1.2. We will equate

the velocities at the point of contact along the common normal and derive the

term for velocity ratio. The radial position of the point of contact on the pinion

with respect to the pinion axis in the global coordinate system is given by the

vector:

!r = (X cos!n̂i +Yn̂j + X sin!n̂k ) (4.33)

Where:

n̂i , n̂ j , n̂k = unit vectors along the X, Y and Z direction respectively

and X andY are the distance of the point of contact from the pinion axis in the

B2

!B2plane along the X andY axis respectively. The angular velocity of the pinion

can be stated as:

!! = (! cos"n̂

k#! sin"n̂

i) (4.34)

The velocity

!vA of the point of contact on the pinion is:

!vA=!! "!r (4.35)

!vA= (! cos"n̂k #! sin"n̂i ) $ (X cos"n̂i +Yn̂j + X sin"n̂k ) (4.36)

!vA=!X cos2 "n̂ j #!Y cos"n̂i #!Y sin"n̂k +!X sin

2 "n̂ j (4.37)

!vA= !"X cos#n̂i +"Xn̂j !"Y sin#n̂k (4.38)

The component of the velocity along the common normal is given by:

vn

A=!nnr!!vA (4.39)

vnA= (! sin"pn̂i ! cos"pn̂ j ) # (!$X cos%n̂i +$Xn̂j !$Y sin%n̂k ) (4.40)

vnA=!Y cos" sin#p $!X cos#p (4.41)

64

vnA=! (Y cos" sin#p $ X cos#p ) (4.42)

The velocity component of the rack along the common normal is given by:

vnAr= ! cos"pvr (4.43)

where vris the magnitude of rack velocity. We can equate the velocity component

of the rack and the pinion along the common normal to get:

vr

!=X cos"

p#Y cos$ sin"

p

cos"p

(4.44)

The value of X andY can be found as shown in section 4.1.1. The term forX andY

is given by:

X = Rl cos(! " INV#pi ) (4.45)

Y = Rl sin(! " INV#pi ) (4.46)

where:

Rl =Rb

cos!pi

(4.47)

!pi= tan

"1 # + tan"1 tan!

p

cos$%&'

()*

+

,-

.

/0 (4.48)

Substituting equation 4.45, 4.46 and 4.47 in 4.44 we get the function for velocity

ratio, which can be expressed as:

vr

!=

Rb cos(" # INV$pi )cos$p # sin(" # INV$pi )cos% sin$p&' ()

cos$p cos$pi

(4.49)

where !pi

is given by equation 4.48. The velocity ratio turns out to be a function

of! . ! has a range of 2! / N , where N is the number of teeth on the pinion.

The variation in velocity ratio is plotted for a pinion with base circle radius of

12 .68 mm, meshing with a rack having profile angle 25˚ and rotating from -20˚

to 20 ̊is show in Figure 4.14. The pinion has 5 teeth. The plot has been drawn for

65

five values of! . We can see from the figure that the variation in velocity ratio

increases with increase in the magnitude of! .

Figure 4.14 Pure shaft misalignment error

4.4.2 Pure tilt

In pure tilt or skew axis misalignment error, the axis of the pinion lies in a plane

parallel to the rack reference plane as show in Figure 4.15. The angle between the

pinion axis and the transverse cross section plane is! .

The tangent at the point of contact makes an angle !" with the X axis in the A1!A1

plane as shown in Figure 4.17. This is a projection of the angle! that the pinion

makes with the rack in the A2

!A2 plane, as shown in Figure 4,16. A similar

approach to section 4.4.1 is followed here. Analyzing a infinitesimally small

section at the point of contact between the rack and the pinion we find the

66

relation between ! and !" .

Figure 4.15 Pure tilt misalignment

For some dimensions hand l , from Figure 4.18 we can write:

tan! =l

h (4.50)

tan !" =l

hcos# (4.51)

Using equation 4.50 and 4.51 we get:

tan !" =tan"

cos# (4.52)

67

Figure 4.16 Section of the pinion surface in the plane A2

!A2

Figure 4.17 Section of the pinion surface in the plane A1!A1

Figure 4.18 Cross section of the pinion tooth in theYZ plane

68

In planeA2

!A2:

! = "pi#$ + INV"

pi (4.53)

and in plane A1!A1:

!" = #p (4.54)


tan!p=tan !

pi"# + INV!

pi( )cos$

(4.55)

Equation 4.55 can be rewritten as:

tan!1(tan"

pcos# ) = "

pi!$ + INV"

pi (4.56)

tan!1(tan"

pcos# ) = "

pi!$ + tan"

pi!"

pi (4.57)

tan!1(tan"

pcos# ) = tan"

pi!$ (4.58)

The term for the instantaneous profile angle can be expressed as:

!pi= tan

"1 # + tan"1tan!

pcos$( )%& '( (4.59)

Velocity ratio

The same approach is followed for calculating the velocity ratio as the one in

section 4.4.1. We equate the velocity component along the common normal at the

point of contact for the rack and pinion and rearrange the term to get the

expression for velocity ratio. The radial position vector of the point of contact on

the pinion with respect to the pinion axis is given by:

!r = (Xn̂i +Y cos! n̂ j + X sin! n̂k ) (4.60)

The angular velocity of the pinion is given by:

!! = (! cos" n̂k #! sin" n̂ j ) (4.61)

The velocity of the point of contact on the pinion can be stated as:

!vA=!! "!r (4.62)

69

!vA= (! cos" n̂k #! sin" n̂ j ) $ (Xn̂i +Y cos" n̂ j + X sin" n̂k ) (4.63)

!vA=!X cos"nj #!Y cos

2"ni +!X sin"nk #!Y sin

2"ni (4.64)

!vA= !"Yni +"X cos#nj +"Y sin#nk (4.65)

The component of !vA along the common normal is given by:

vn

A=!nnr!!vA (4.66)

vnA= (! sin"pn̂i ! cos"pn̂ j ) # (!$Yni +$X cos%nj +$Y sin%nk ) (4.67)

vnA=!Y sin"p #!X cos"p cos$ (4.68)

vnA=! (Y sin"p # X cos"p cos$ ) (4.69)

Velocity component of the rack along the common normal is given by:

vnAr= ! cos"pvr (4.70)

Equating the velocity component of the pinion and the rack along the common

normal, we get:

vr

!=X cos"

pcos# $Y sin"

p

cos"p

(4.71)

Term forX andY can be stated as:

X = Rl cos(! " INV#pi ) (4.72)

Y = Rl sin(! " INV#pi ) (4.73)

where:

Rl =Rb

cos!pi

(4.74)

!pi= tan

"1 # + tan"1tan!

pcos$( )%& '( (4.75)

By substituting equation 4.72 ,4.73 and 4.74 into 4.71 we get the expression for

velocity ratio, which can be stated as:

vr

!=

Rb cos(" # INV$pi )cos$p cos% # sin(" # INV$pi )sin$p&' ()

cos$p cos$pi

(4.76)

70

where!pi

is given by equation 4.75. Hence the velocity ratio varies with! and! ,

where! has a range of 2! / N .

This variation in velocity ratio is plotted against pinion rotation, for a pinion with

base circle radius of 12 mm and meshing with a rack having profile angle 25˚.

The pinion has 5 teeth and is rotated from -20˚ to 20˚. The plot is show in Figure

4.19.

The plot is made for five value of! . We can see from the figure that the

deviation in velocity ratio from the constant increases with increase in the

magnitude of! .

Figure 4.19 Pure tilt misalignment error

71

4.4.3 Misalignment error

In section 4.4.1 and 4.4.2 we looked at only two positions (pure shaft

misalignment and pure tilt) of the pinion and derived the expression for velocity

ratio for the two cases. In reality both the cases occur together. In such a case we

derive the expression for velocity ratio for a pinion axis which makes an angle!

with the transverse cross section plane of the rack and an angle !with the rack

reference plane, as shown in Figure 4.20. This rotation is similar to the

positioning of an object in the spherical coordinate system. The positioning of the

pinion axis has been show in Figure 4.21. The angles ! and! have been

exaggerated for clarity. We look at two cross section of the pinion as shown in

Figure 4.20.

Figure 4.20 Axis misalignment error

72

Figure 4.21 Orientation of the pinion axis

The cross section plane C2

!C2is perpendicular to the pinion axis. In this plane the

pinion cross section looks like a spur gear with involute teeth. The other cross

section planeC1!C1is along the outer edge of the rack as show by Figure 4.20. This

is the plane in which the rack and pinion teeth come in contact. The tangent to

the pinion surface at the point of contact makes an angle !µ with the X axis in the

C1!C1plane as shown by Figure 4.23. !µ is a projection of the angleµ that the

tangent makes with the X axis in the C2

!C2plane as shown in Figure 4.22.

For some dimensions hand l , for a infinitesimally small section of the pinion

surface at the point of contact, from Figure 4.24 and 4.25 we can write:

tanµ =l

h (4.77)

73

and

tan !µ =l cos"

hcos# (4.78)

Figure 4.22 Section of the pinion surface in the plane C2

!C2

Figure 4.23 Section of the pinion surface in the plane C1!C1

Figure 4.24 Cross section of the pinion tooth in the XZ plane

74

Figure 4.25 Cross section of the pinion tooth in theYZ plane

Using equation 4.77 and 4.78 we get:

tan !µ =cos"

cos#tanµ (4.79)

Section 4.1.1 gives us the relation:

µ = !pi"# + INV!

pi (4.80)

and at the rack surface:

!µ = "p (4.81)

Substituting equation 4.80 and 4.81 in 4.79, we get:

tan!p=cos"

cos#tan(!

pi$% + INV!

pi) (4.82)

Equation 4.82 can be simplified to get:

tan!1tan"

p

cos#cos$

%&'

()*= "

pi!+ + INV"

pi (4.83)

tan!1tan"

p

cos#cos$

%&'

()*= "

pi!+ + tan"

pi!"

pi (4.84)

tan!1tan"

p

cos#cos$

%&'

()*= tan"

pi!+ (4.85)

The term for!pi

can be expressed as:

!pi= tan

"1 # + tan"1tan!

p

cos$cos%

&'(

)*+

,

-.

/

01 (4.86)

75

Velocity ratio

A similar approach to section 4.4.1 is adopted to calculate the velocity ratio. We

equate the velocity component of the rack and pinion along the common normal

at the point of contact. The position of the point of contact on the pinion with

respect to the pinion axis in the global coordinate system is given by the vector:

!r = X cos!n̂i +Y cos" n̂ j + (X sin! +Y sin" )n̂k( ) (4.87)

The angular velocity of the pinion can be stated as:

!! = (! cos" cos#n̂k $! sin#n̂i $! sin" n̂ j ) (4.88)

and velocity at the point of contact as:

!vA=!! "!r (4.89)

!vA= (! cos" cos#n̂k $! sin#n̂i $! sin" n̂ j ) %

(X cos#n̂i +Y cos" n̂ j + (X sin# +Y sin" )n̂k ) (4.90)

!vA=! (cos" cos2 #X + sin

2 #X + sin" sin#Y )n̂ j

$! (cos2" cos#Y + sin" sin#X + sin2"Y )n̂i

+! (sin" cos#X $ cos" sin#Y )nk

(4.91)

The component of !vA along the common normal is given by:

vn

A=!nnr!!vA (4.92a)

vnA= (! sin"pn̂i ! cos"pn̂ j ) #

!vA (4.92b)

vnA=! (sin"p cos

2# cos$Y + sin"p sin# sin$X + sin"p sin2#Y

% cos"p cos# cos2 $X % cos"p sin

2 $X % cos"p sin# sin$Y ) (4.93)

We can equate equation 4.93 to the component of the rack velocity along the

common normal to get the term for the velocity ratio.

vr

!=

+ cos"pcos# cos2 $X + cos"

psin

2 $X + cos"psin# sin$Y

% sin"pcos

2# cos$Y % sin"psin# sin$X % sin"

psin

2#Y

&

'(

)

*+

cos"p

(4.94)

where

X = Rl cos(! " INV#pi ) (4.95)

76

Y = Rl sin(! " INV#pi ) (4.96)

Rl =Rb

cos!pi

(4.97)

!pi= tan

"1 # + tan"1tan!

p

cos$cos%

&'(

)*+

,

-.

/

01 (4.98)

Hence the velocity ratio turns out to be a function of ! ,! and ! .

In Figure 4.26 and 4.27 the velocity ratio is plotted for ! = 0.25 ̊ and 0.75˚

respectively. The plot is for a pinion with base circle radius 12 .68 mm, meshing

with a rack having profile angle 25˚ and rotating from -20˚ to 20˚. The pinion has

5 teeth and each plot is drawn for five values of! .

Figure 4.26 Axis misalignment for! = 0.25˚

77

Figure 4.27 Axis misalignment for! = 0.75˚

Torque transfer

Like the motion transfer, the torque transfer also takes place along the plane of

the rack edge. To find the torque transfer we find the normal reaction force

experienced by the pinion at the point of contact. The magnitude of the normal

reaction force which acts along the line of contact is given by:

Nt=F ! ma

cos"p

(4.99)

as stated in section 3.1.6

The normal reaction force vector is given by:

!N

t=

F ! macos"

p

#

$%%

&

'(((! sin"

pn̂i! cos"

pn̂j) (4.100)

The radial vector from the pinion axis to the normal reaction vector is given by:

!r = X cos!n̂i +Y cos" n̂ j + (X sin! +Y sin" )n̂k( ) (4.101)

where:

78

X = Rl cos(! " INV#pi ) (4.102)

Y = Rl sin(! " INV#pi ) (4.103)

Rl =Rb

cos!pi

(4.104)

!pi= tan

"1 # + tan"1tan!

p

cos$cos%

&'(

)*+

,

-.

/

01 (4.105)

There for torque

!T is given by:

!T =!r !!Nt

=F " macos#p

$

%&&

'

())

cos#p (X sin* +Y sin+ )n̂i " sin#p (X sin* +Y sin+ )n̂ j+(Y cos+ sin#p " X cos* cos#p )n̂k

$

%&

'

()

(4.106)

The torque also comes out to be a function of the pinion rotation! ,! and ! .

This variation in torque is plotted for two values of! and a variation in the value

of! as shown in Figure 4.28 and 4.29. The plot is made for a pinion with base

circle radius of 12 mm and meshing with a rack having profile angle 25˚. The

pinion has 5 teeth and is rotated from -20˚ to 20˚.

79

Figure 4.28 Torque transfer for Axis misalignment at! = 0.25˚

Figure 4.29 Torque transfer for Axis misalignment at! = 0.75˚

80

4.5 Rack bend error

The rack bend error as the name suggests is a bend in the rack. Some times

during the manufacturing and transportation the rack bends slightly. This

deviation causes a change in the tooth height with respect to the rack reference

line as show in Figure 4.30. This deviation is similar to a change in the center

distance. If there is an elevation in the tooth height, the rack tooth comes closer

to the pinion axis. This can be thought of as a reduction in the center distance

between the rack and pinion at that point. Hence the rack bend error does not

affect the velocity ratio and the torque transfer function. Even though the velocity

ratio remains constant there will be sudden spikes and dips in the velocity vs time

plot when the pinion looses and comes in contact with a rack tooth with a change

in altitude of the tooth height.

Figure 4.30 Rack bend error

4.6 Tooth thickness error

The tooth thickness error is a variation in the rack tooth thickness at the rack

reference line (Figure 4.32) or a variation in the pinion tooth thickness at the

pinion pitch circle (Figure 4.31). In a rack and pinion, we model the case when

with a variation in the tooth thickness the profile angle remains the same. In the

case of a pinion, there is only a tangential shift in the tooth surface with respect

to the pinion center as shown in Figure 4.31.

81

Figure 4.31 Pinion tooth thickness

Figure 4.32 Rack tooth thickness


For tooth thickness variation the position of the point of contact on the rack and

pinion does not change because there is no change in the pressure angle of the

rack or the profile angle of the pinion and the center distance remains the same.

Hence, the velocity components along the common normal at the point of contact

are the same. As the velocity components are the same the velocity ratio remains

constant and is given by:

82

vr

!=

Rb

cos"p

(4.107)


The torque transfer is dependent on the pressure angle, the position of the point

of contact and the normal reaction force. As all three of these quantities remain

constant, the term from the torque is the same as derived in section 3.2.3 and is

given by:

T = RbF ! macos"p

#

$%

&

'( (4.108)

4.6.3 Backlash

The change in the value of backlash is given by the sum of change in the pinion

tooth thickness at the pitch circle and the change in rack tooth thickness at the

pitch plane. Hence:

!Blt = [Tp " Tpn ]+ [Tr " Trn ] (4.109)

where

Trn

= new rack tooth thickness at the pitch plane

Tr= ideal rack tooth thickness at the pitch plane

Tpn

= new pinion tooth thickness at the pitch circle

Tp= ideal pinion tooth thickness at the pitch circle

Which is equal to:

!Blt = "t p + tr (4.110)

where

tr= decrease in the rack tooth thickness

83

tp

=increase in the pinion tooth thickness

trand t

pcan also be seen in Figure 4.31 and 4.32.

4.7 Pressure angle error

Consider a rack as shown in Figure 4.33 with pressure angle!p. When there is a

deviation !"p

in the pressure angle from the specified value then this type of

error is called the pressure angle error. The deviation in the pressure angle can be

positive or negative. A deviation which increases the pressure angle is treated as

positive and vice versa. In this section we will look at the effect of change of

pressure angle on the velocity ratio, torque transfer and the backlash function.

Figure 4.33 Rack with pressure angle error

4.7.1 Velocity Ratio

Figure 4.34 shows the rack tooth and pinion tooth profile with pressure angle

error. As per section 3.1.2 Ar and A are the point of contact on the rack and pinion

respectively. The vector along the common normal nnr

works out to be:

n̂nr= ! sin("

p+ #"

p)n̂

i! cos("

p+ #"

p)n̂

j; (4.111)

84

where

n̂i , n̂ j , n̂k = mutually perpendicular unit vectors

The velocity vector v̂Ar at Ar works out to be:

v̂Ar = vrn̂ j (4.112)

and its component along the common normal is:

vnAr = n̂nr .v̂

Ar = ! cos("p + #"p )vr (4.113)

where

vr= speed of the rack

vn

Ar = rack velocity component along the common normal

Figure 4.34 Variation in rack pressure angle

The vector from the center C to the point A is expressed as (Xn̂i+Yn̂

j). The

velocity vector at point A comes out to be:

v̂A=! n̂k " (Xn̂i +Yn̂j ) = #!Yn̂i +!Xn̂j (4.114)

and its component along the common normal can be expressed as:

vnA= n̂nr .v̂

A=!Y sin("p + #"p ) $!X cos("p + #"p ) (4.115)

85

where

v̂A = velocity of the point A on the pinion


vn

A = velocity component at point A along the common normal

We can equate the velocities of the rack and pinion along the common normal to

get:

! cos("p+ #"

p)v

r=$Y sin("

p+ #"

p) !$X cos("

p+ #"

p) (4.116)

and rearrange the terms to get the velocity ratio, expressed as:

vr

!=X cos("

p+ #"

p) $Y sin("

p+ #"

p)

cos("p+ #"

p)

(4.117)

We can equate the angle that the tangent at the point of contact on the pinion

makes with the X axis and the rack profile angle, to get:

!p+ "!

p= !

pi#$ + INV!

pi (4.118)

After substituting equation 3.30 in 4.118 we can express the term for

instantaneous profile angle!pi

of the pinion at the point of contact as:

!pi= tan

"1(!

p+ #!

p+$) (4.119)

The radius of the point of contact Rlfor the pinion is worked out using equation

3.24; thus we can write:

Rl =Rb

cos!pi

(4.120)

The x and y components of CP can be written as:

X = Rl cos(! " INV#pi ) (4.121)

Y = Rl sin(! " INV#pi ) (4.122)

respectively. Substituting equation 4.121 and 4.122 in 4.117 we get the term:

vr

!=Rl cos(" # INV$pi + $p + %$p )

cos($p + %$p ) (4.123)

86

which can be further simplified to:

vr

!=

Rb

cos("p + #"p ) (4.124)

using equation 4.118 and 4.120.

From equation 4.124 we see that the value of velocity ratio changes with the rack

pressure angle but it remains constant with respect to the angle of rotation as seen

in Figure 4.35. The velocity ratio does not vary with pinion rotation but as the

variation in pressure angle increases the value of this constant increases. The plot

is drawn for a pinion with base circle radius of 12.68 mm. We assume that the

rack is translating with a constant velocity.

Figure 4.35 Velocity ratio for pressure angle variation

87


The torque transfer equations in the presence of pressure angle error are similar

to the the ones derived in section 3.1.6. In the presence of pressure angle

variation the direction of the normal reaction force acting between the teeth of

the rack and pinion changes by !"p. Hence the torque transfer equation becomes:

T = RbF ! ma

cos("p + #"p )

$

%&

'

() (4.125)

With increase in pressure angle the value of torque transmitted increase as shown

in Figure 4.36, but does not change with pinion rotation. The plot is made for a

pinion with base circle radius of 12.68 mm.

Figure 4.36 Torque transfer for pressure angle variation

88

4.7.3 Backlash

The backlash along the pitch circle is defined as the angle by which the pinion is

free to rotate if the position of the rack is fixed in space. Here we look at the

change in backlash because of the change in pressure angle. As the pressure angle

changes the radius of the pitch circle changes as:

!Rp + Rp =Rb

cos("p + !"p ) (4.126)

where

!Rp

= change in the pinion pitch circle

The change in backlash is the sum of the change in tooth thickness of the pinion

at the new pitch circle and the change in the rack tooth thickness at the new pitch

plane. The tooth thickness of the pinion at the operating pitch circle has been

derived with respect to the tooth thickness at the ideal pitch circle. Considering

Figure 4.37, we have:

!Tp=Tp

2Rp

(4.127)

!Tpo

= !Tp" (!

po"!

p) (4.128)

!Tpo

=Tpo

2("Rp+ R

p)

(4.129)

where

!p= involute of the ideal pitch circle

!po

= involute of the operating pitch circle

Tp= tooth thickness of the pinion at the ideal pitch circle

Tpo

= tooth thickness of the pinion at the operating pitch circle

89

Figure 4.37 Tooth thickness relations

Hence:

!p= tan"

p#"

p= INV"

p (4.130)

!po= tan("

p+ #"

p) $ ("

p+ #"

p) = INV ("

p+ #"

p) (4.131)

Substituting equation 4.127, 4.129, 4.130 and 4.131 into equation 4.128, we

obtain:

Tpo

2(!Rp+ R

p)=Tp

2Rp

" [INV (#p+ !#

p) " INV#

p] (4.132)

Simplifying, to get:

Tpo= 2(!R

p+ R

p)

Tp

2Rp

+ INV"p# INV ("

p+ !"

p)

$

%&&

'

())

(4.133)

90

The tooth thickness of the rack at the operating pitch plane depends on the

change in the height of the point of contact on the rack with respect to the rack

reference plane. The pitch plane moves by a distance !Rp

, hence the tooth

thickness at the operating pitch plane is given by:

Tro= T

r+ 2!R

ptan("

p+ !"

p) (4.134)

where

Tro

= rack tooth thickness at the operating pitch plane

Tr= rack tooth thickness at the ideal pitch plane

The change in backlash !Blo

because of the pressure angle error is given by:

!Blo = [Tp " Tpo ]+ [Tr " Tro ] (4.135)

!Blo = Tp " 2Rb

cos(#p + !#p )

$

%&

'

()

Tp

2Rp

+ INV#p " INV (#p + !#p )*

+,,

-

.//

*

+,,

-

.//+

"2Rb

cos(#p + !#p )"

Rb

cos#p

$

%&

'

() tan(#p + !#p )

*

+,,

-

.//

(4.136)

The variation of!Blo

with pressure angle is shown in Figure 4.38. The curve in

red shows the variation due to the change in rack thickness as the operating pitch

plane and the curve in sky blue due to change in pinion tooth thickness at the

operating pitch circle. In dark blue is the total change in backlash variation.

These calculations are done for a pinion of base circle radius 12.68 mm and tooth

thickness of 2.355 mm. The pressure angle of a error free rack is taken to be 20˚.

91

Figure 4.38 Change in backlash due to pressure angle variation

4.8 Pinion pitch circle runout

In pitch circle runout error the central axis of the gear does not coincide with the

axis of the shaft. Say this displacement is eras shown in Figure 4.39. The teeth of

the pinion get radially displaced. The tooth thickness at the operating pitch circle

depends on the function which relates the distanceRtbetween the points on the

periphery of the operating pitch circle and the gear center. From the Figure 4.40

using cosine law on the triangle !OpPO we can write:

92

Figure 4.39 Pitch circle runout

Rt= e

rcos(180 !"

t) + R

op

2! e

r

2sin

2(180 !"

t) (4.137)

Figure 4.40 Radian position of the pinion teeth

where

O = geometric center of the gear

93

Op = position of the gear axis

Rp

= radius of the pitch circle

Rop

= radius of the operating pitch circle

Q = point on the circumference of the operating pitch with minimum

value of Rt

!t= radial position of a point on the circumference of the operating pitch

circle with respect to the point Q

The tooth thickness at a distance Rp+ !R

pis given by equation 4.133, and the

pressure angle by equation 4.126. Here

Rt= R

p+ !R

p (4.138)

Hence the tooth thickness at the operating pitch circle for a tooth located at !tcan

be written in terms of Rtas:

Tpo = 2(Rt )Tp

2Rp

+ INV!p " INV cos"1 Rb

Rt

#

$%&

'()

*++

,

-..

(4.139)


Because of the tooth thickness variations there will be phases when the rack will

loose contact with the pinion. However, the velocity ratio remains constant as

neither the pressure angle on the point of contact nor the point of contact on the

pinion changes.

94


The torque transfer function remains the same as stated in section 3.2.3. The

pitch circle runout does not affect the slope of the common normal or its distance

from the pinion center.

4.8.3 Backlash

The pitch plane of the rack does not change with pitch circle run out. The change

in backlash because of the change in the tooth thickness of the pinion at the pitch

circle can be given by:

!Bpr= T

p" T

po (4.140)

!Bpr = Tp " 2(Rt )Tp

2Rp

+ INV#p " INV cos"1 Rb

Rt

$

%&'

()*

+,,

-

.//

(4.141)

where Rtis given by:

Rt= e

rcos(180 !"

t) + R

op

2! e

r

2sin

2(180 !"

t) (4.142)

The variation in backlash is plotted for a pinion with 18 teeth with base circle

diameter 12.68 mm and pressure angle 20 ̊ as shown in Figure 4.41. The tooth

thickness of the pinion is 2.355 mm. From the figure we see that the backlash

depends on the radial position of tooth on the pinion the rack is meshing with.

For a tooth at!t= 0 ̊the value of change in backlash is negative and as we move

towards !t= 180˚ the value of backlash keeps increasing. Around !

t= 90˚ there

also comes a point when the change in the value of backlash is almost zero. This

is because around this range the value ofRtapproaches R

p.

95

Figure 4.41 Change in backlash due to pitch circle runout

4.9 Remarks

Is this chapter we see that velocity ratio and torque transfer varies with change in

the pinion installation angle and pressure angle. We also derived the expressions

for change in backlash in the case of variation in center distance, tooth thickness,

pressure angle and pitch circle runout.

96

CHAPTER-5

CONCLUSIONS AND FUTURE WORK

The objective of the present work was to develop a fundamental physics based

model for a rack and pinion type steering system and also model the prominent

manufacturing errors to see their effect on parameters like velocity ratio, torque

transfer function and backlash.

In the current work a model for the steering system has been developed which

incorporates all the important linkages connecting the steering wheel to the tire.

These linkages include the steering wheel, steering column, universal joint,

intermediate shaft, lower shaft, pinion, rack, tie-rod, knuckle and the wheel. This

model reports the angular displacement of the tire given a variation at the

steering wheel. In this model the suspension system is assumed to be a rigid

joint, as the primary focus of the model is to test it for various manufacturing

errors at the rack and pinion. A steering system model has also been developed

using ADAMS to validate the analytical model.

The velocity ratio function, torque transfer function and the backlash have been

analytically modeled for the following errors at the steering rack and pinion:



c) Backlash


e) Rack bend

f) Tooth thickness

g) Pressure angle

h) Pinion pitch circle runout

97

From the variation in these errors we see that the pressure angle and pinion

installation angle are the two prominent errors which affect the velocity ratio and

the torque transfer function. The effect of center distance, tooth thickness,

pressure angle and the pitch circle runout on change in the backlash has also been

studied.

In the present study the steering system model only takes angular variation at the

steering wheel and the part geometries as the input parameters. This model can

be extended to take variation in the torque and forces at the tire and report the

torque experienced by the driver at the steering wheel. This would enable us to

better understand the steering feel. Also manufacturing and orientation errors in

other part like the connecting rods, universal joints, knuckle and the tie rod can

be modeled to develop a more sophisticated and complete system. These errors

complete with their ADAMS model can be developed into a standard package to

get a feedback on the steering feel for any car model with rack and pinion type

steering mechanism.

98

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101