IN DEGREE PROJECT MECHANICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Dynamic Analysis of an Automotive Power Transfer unitTowards prediction of TE and housing vibrations

NIKHIL MAHARSHI KOSARAJU

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

TRITA -ITM-EX 2019:

566

www.kth.se

Dynamic Analysis of an AutomotivePower Transfer unit

Towards prediction of TE and housing vibrations

Nikhil Maharshi Kosaraju

Master of Science Thesis TRITA-ITM-EX 2019:566KTH Industrial Engineering and Management

Machine DesignSE-100 44 STOCKHOLM

August 2019

Master of Science Thesis, TRITA-ITM-EX 2019:566

Dynamic Analysis of anAutomotive Power Transfer unit

Nikhil MaharshiApproved

2019-08-28

Examiner

Ulf Sellgren

Supervisor

Ulf SellgrenCommissioner

GKN Automotive Koping AB

Contact Person

Stefano Orzi

Abstract

This work describes the use of Multi-Body Simulation (MBS) to create a virtual prototype of a geared drivecalled Power transfer unit (PTU). PTU is a subsystem of the all-wheel drive driveline responsible for transferof power between front and rear axles in an Automobile. The objective of the developing the prototype is tosimulate the dynamic behavior of the PTU. Focus is on predicting the gear transmission error(TE) and gearboxhousing vibration level. A Hypoid gear set, bearings, tubular shaft and housing are the major components inthe PTU. This work is carried out at GKN Automotive which specializes in development of Automotive Allwheel drive systems. When developing such geared systems one important characteristic analyzed is the noiseand vibration it generates. And for companies like GKN it is desirable to predict these characteristics as earlyas possible for two reasons, to avoid late design changes and to speed up the product development cycle. Toachieve this, a validated virtual model which is computationally efficient is desired.

The methodology followed contains of two facets, development of the MBS model and validation of the developedmodel with physical testing. An integrated MBS-FEM approach is used, an FE modal reduction technique isused to create flexible components with which a virtual prototype is built and simulated in an MBS tool MSCADAMS c©. Gear contact and bearings are defined using an analytical approach which considers the nonlinearstiffness and damping. A dynamic analysis and system level modal analysis is performed to predict the TE,housing vibrations and PTU modal parameters. Experimental modal analysis and physical testing on test rigare performed to measure the actual values of the above predicted outputs. Parameters like damping, contactstiffness of the model are then tuned to achieve correlation.

When comparing test and prediction, close correlation is seen in the TE and for housing vibration a similartrend is observed with some deviations. Predicted TE is heavily dependent on gear contact parameters. On themodal parameter comparison, a correlation of five modes and mode shapes below 2500Hz is seen which showsthe validity of the MBS model. Parameter studies are performed to study the effect of bearing damping andpreload on housing vibrations and TE. It is observed that an optimum value of preload and damping is essentialto avoid unnecessary vibrations. In conclusion, the model with some fine tuning of damping parameters can beused for virtual noise and vibration analysis of the PTU.

Keywords: Power transfer unit (PTU), Multi-Body Simulation (MBS), Transmission Error, Gear Noise-Vibration, Modal Analysis

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Examensarbete, TRITA-ITM-EX 2019:566

Dynamisk analys av en vinkelvaxel

Nikhil Maharshi KosarajuGodkant

2019-08-28

Examinator

Ulf Sellgren

Handledare

Ulf SellgrenUppdragsgivare

GKN Automotive Koping AB

Kontaktperson

Stefano Orzi

Sammanfattning

Detta arbete beskriver anvandningen av berakningsmetoden Multi-Body Simulation (MBS) for att skapa envirtuell prototyp av en vinkelvaxel (Power Transfer Unit, PTU ). PTU ar ett delsystem for fyrhjulsdrift som harfunktionen att overfora kraft mellan fram- och bakaxlar i en bil. Malet med att utveckla modellen ar att simuleraPTUns dynamiska beteende. Fokus ligger pa att berakna vinkelvaxelns transmissionsfel och vibrationsnivaerpa vaxelladans hus. De vikitgaste komponenterna i PTUn ar hypoidvaxeln med kronhjul och pinjong, roraxel,lager och hus. Detta arbete har utforts pa GKN Automotive som ar specialiserade pa utveckling av drivsystemfor fyhjulsdrivna bilar. Ljud och vibrationer ar viktiga egenskaper att ta hansyn till under utvecklingen.

For foretag som GKN ar det onskvart att kunna berakna dessa egenskaper sa tidigt i projektet av tva skal:dels for att undvika sena konstruktionsforandringar och dels att paskynda produktutvecklingscykeln. For attuppna detta behovs en validerad virtuell modell som ar berakningseffektiv.

Den metod som anvants innehaller tva delar: utveckling av MBS-modellen och validering av den utvecklademodellen med fysisk testning. En integrerad MBS-FEM -mettod har anvants. Det innebar att en FE-modalreduktionsteknik andvands for att skapa flexibla komponenter med vilka en virtuell prototyp byggs och simulerasi ett MBS-verktyg (MSC ADAMS (c) ). Lager och kuggkontakt i vaxeln definieras med hjalp av en analytiskmetod som beaktar den olinjara styvheten och dampningen. En dynamisk analys och modalanalys pa systemnivahar utforts for att berakna TE, husvibrationer och PTUns modala parametrar. Experimentell modalanalys ochtestning i rigg gjorts for att mata motsvarande varden som har beraknats. Parametrar som dampning ochkontaktstyvhet har sedan justerats for att uppna korrelation.

Vid jamforelse av test och forutsagelse ses en god korrelation i TE och for husvibrationer observeras en liknandetrend, med vissa avvikelser. Beraknat TE ar starkt beroende pa parametrar for kuggkontakten i vaxeln.Vid jamforelse av modala parametrar ses en god korrelation under 2500 Hz mellan fem moder i matning ochberakning vad galler frekvens och modform, vilket visar MBS-modellens giltighet. Parameterstudier har utfortsfor att studera effekten av lagerdampning och forbelastning pa TE och husvibrationer. Ett optimalt varde paforbelastning och dampning ar viktigt for att undvika onodiga vibrationer. Sammanfattningsvis kan modellenmed viss finjustering av dampningsparametrar anvandas i virtuell ljud- och vibrationsanalys av PTU.

Nyckelord: Vinkelvxlen (PTU), Multi-Body Simulering (MBS), transmissionsfel (TE), Vxelljud och vibra-tioner, Modal analys

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AcknowledgementThis thesis work was carried out at GKN Automotive, Koping AB, as part of the Master of Science degreein Engineering Design at KTH Royal Institute of Technology, Stockholm. My time at GKN where I was a partof the NVH team has been an enriching learning experience. I am heartfully grateful for the opportunity andthe people who supported/guided me during the period of work.

I would like to thank my manager Magnus lofberg for giving me the opportunity to carry out this work.And the NVH team Stefano Orzi my supervisor and Eva Lundberg for being such an amazing team to bepart of and the help throughout the work. From defining the objective to support with problems and guidingme when I was stuck. I cannot thank enough Puneeth and Shivanand, colleagues from the calculation teamfor their constant support. I express my gratitude to thetesting and design team mainly Christer, Marcus,Christian, Ghirmai and Rafal for the support with the work. I would like to also thank Ulf Sellgren, mysupervisor from KTH, for his support planning the work and valuable feedback whenever needed. I also wouldlike to thank all my colleagues for providing with such a humble and dynamic environment to work in.

And all this would not have been possible without the support of my family who encouraged me and providedthe opportunity to purse the Masters. Thanks to my friends who have been a great support constantlythroughout my studies.

Nikhil MaharshiStockholm, Sweden

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NomenclatureABBREVIATIONS

AWD All-Wheel Drive

CAD Computer Aided Design

CAE Computer Aided Engineering

CMS Component Mode Synthesis

DOF Degree of Freedom

EMA Experimental Modal Analysis

FEA Finite Element Analysis

FEM Finite Element Method

FFT Fast Fouriers Transform

FRF Frequency Response Function

GMF Gear Mesh Frequency

MBS Multi-Body Simulation

MNF Modal Neutral File

NV Noise-Vibration

NVH Noise Vibration and Harshness

PSD Power Spectral Density

PTU Power Transfer Unit

RBE Rigid body element

RDU Rear Drive Unit

RPM Revolutions per minute

TE Transmission Error

vii

Table of contents

Abstract i

Sammanfattning iii

Acknowledgement v

Nomenclature vii

1 INTRODUCTION 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Description of analyzed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Objective and research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 De-limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 FRAME OF REFERENCE 92.1 Hypoid Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Gear Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Hypoid gear contact ratio and Mesh stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Transmission error (types and focus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 TE Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Noise, Vibration and Harshness (NVH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Noise and frequency content in gear drives . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 NVH Transfer path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Bearing dynamic characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Methods to perform Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Fast Fourier Transform(FFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Multi Body Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Work flow in ADAMS c© . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 MBS Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.3 Flexible bodies in ADAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.4 Inertial modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.5 Bearing modelling in ADAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.6 Gear contact modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 METHODOLOGY 253.1 Overview of methodology employed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Models Built in Adams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Development of Dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 CAD clean up and Mesh Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Model Development in ADAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Analysis in ADAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.4 Linear Modes Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Physical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Test Rig layout and measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Model Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1

4 RESULTS & DISCUSSION 394.1 Transmission error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Physical Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 MBS results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.3 TE at different speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Housing Vibration level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Power Spectral Density (PSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Constant Speed region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Modal Analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Bearing Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Effect of bearing parameters of PTU dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5.1 Impact of bearing Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5.2 Impact of bearing preload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5.3 Impact of preload on TE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 CONCLUSIONS 515.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 FUTURE WORK 54

Bibliography 56

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1. INTRODUCTIONThis chapter introduces the work and describes the background, purpose, limitations and the methodology usedin the presented project.

In automobiles, apart from the combustion engine, gear transmission systems are one of the dominant sourcesof noise and vibration (N-V). As components become lighter and engines become quieter, the N-V from gearedsystems become more evident. Specifically gear whine, an unwanted vibro-acoustic phenomenon arising frommeshing gears is a tonal noise, which is perceived as unpleasant by human ear. The source of this whine overthe years has been observed to be a result of something called Transmission error (TE) in the gears. TEcan be thought of as, an error in the gear ratio caused by several reasons which results in oscillating forces i.e.vibration on the gear teeth. This further translates through different components to reach the housing whereit can become structure borne to cause body vibration or air borne noise. During design of such systems it isdesirable to assess the Noise Vibration and Harshness (NVH) behavior as early as possible to make feasibledesign changes. Traditionally experimental and physical testing methods are heavily employed to assess the NVbehavior. However, as Product development cycles become shorter, computer based numerical modelling andsimulations paired with physical testing are being used widely.

A validated simulation model is very useful as it can be used to perform several design studies, thus reducing costsand time involved in prototype building and testing. However computational costs for computer simulationscan be high as well, thus depending on the objective of testing and the level of accuracy required the necessarycomputer simulations need to be planned. When it comes to Computer Aided Engineering (CAE) baseddynamic analysis, Finite Element Analysis (FEA) and Multi Body Dynamic Simulation (MBDS) are usedheavily. FEA is usually robust in performing structural analysis to analyze stresses and strain in deformablecomponents. However, as complexity of systems increases with several connections the FEA model becomescomputationally intense and is not scalable. On the other hand, MBS is heavily used to verify kinematics ofrigid body systems and to calculate dynamic loads. This method is robust for analysis of multi body systemswhere local component flexibility is not critical. However for vibration analysis, a system level model withflexible components is desired. Thus an integrated MBS-FEM approach will be used in this work to developa dynamic model of the Power transfer unit(PTU).

1.1 Background

GKN Automotive develops, builds and supplies an extensive range of automotive driveline products andsystems. The site in Koping, Sweden where this work is carried out specializes is developing All wheel drive(AWD) systems. The purpose of an AWD is to transmit the power from transmission to all four wheels eitherfull time or on demand. The layout can be seen in Fig.1.1 with respective parts labelled. It typically consistsof a power transfer unit (PTU) with connect/disconnect facility, a rear drive unit (RDU), a propeller shaftconnecting PTU-RDU and side shafts with CV joints connecting PTU/RDU to wheels. The overall efficiencyand performance of the units is of utmost importance to deliver customers a sound product. One importantaspect thoroughly looked at is the NVH performance of the drive units, as to how they behave standalone andwhen mounted in a vehicle. Noise and vibration is something directly perceived by the end customer and apoor NVH behaviour reflects to poor qualtiy of design and manufacturing. Thus care is taken to analyse theperformance and ensure that NVH requirements are fulfilled.

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Figure 1.1: Automobile All Wheel Drive Layout

1.1.1 Description of analyzed system

This work is focused on developing a dynamic model to analyze the noise and vibration behaviour of GKNAutomotive’s standard PTU. A brief overview of the system is explained here. Fig.1.2 shows the sectionedview of the PTU with the components listed.

Figure 1.2: Power Transfer Unit

As seen from the section above the PTU is a subsystem in an AWD driveline. The main function of the PTUis to transmit power from the gear box to the rear drive unit based on torque requirement on rear wheels. Acontrol unit is used to establish a connection between the front and rear drives which makes the decision ofwhen and what amount of power needs to be supplied. A clutch system allows for connect and disconnect ofthe transfer of power to rear wheels. When the clutch is engaged, the input power from gear box goes throughinput shaft to tubular shaft. A hypoid gear set consisting of a crown gear and pinion is the core of the PTUto transmit power between perpendicular shafts. So, power flows through the crown gear to the pinion whichis connected via splines to a companion flange. The companion flange is then fixed to the propeller shaft. Thepinion and tubular shaft are supported by two tapered roller bearings each in the housing. In this thesis work,the clutch is not modelled to reduce complexity and the system is always analyzed in a connected state.

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1.2 Purpose

The common industrial practice for NVH testing relies on Experimental testing and Finite element(FE) baseddynamic analysis. In general, a system level FE based dynamic analysis is either computationally heavy (notscalable) or involves assumption of system linearization to perform quick frequency based simulations. Howeveran accurate yet computationally efficient and scalable dynamic model is desirable to provide NVH insightsin early design phases. Realistic NVH analysis during early stages is desirable to avoid costly changes laterand reduce costs in prototype making. Carrying forward from previous work, MBS will be used to model thedynamic behavior of a PTU. Previous work done in this line concluded in proposing integration of FE andMBS using component mode synthesis a modal reduction technique to capture the best of both domains andenable modelling a realistic representation of the drive unit. An attempt was made to predict Transmissionerror using the MBS model which showed good results. However bearings were modelled as bushings making alinear assumption and transient behavior was not considered. So a next step in that direction work is improvingthe model, by capturing necessary non-linearities and making the model usable by validating with testing.

A system such as drive unit involves several non-linearities such as bearings, gear contact behavior which arecomplicated to model. So a detailed study is needed to model a simple yet effective model to capture importantparameters while also being efficient computationally.

1.2.1 Objective and research questions

The goal of the thesis is to develop an MBS model of the PTU with gear set, bearings and housing usingintegrated MBS and FE models. The developed model shall predict Transmission error and housing vibrations.The effects of bearing characteristics on the systems dynamic performance will be studied.

The primary research question that will guide the thesis will be: ”How can we represent a power transfer unitwith an integrated MBS-FEM model that considers the bearing non-linearity’s to predict transmission error andvibration levels?”

The primary research question is further decomposed in to several questions:

• How can a gear set be designed with preliminary data to predict TE?

• How can we include gear the effects of gear imperfections in the dynamic analysis?

• How can we include the effects of bearing nonlinearities in a MBS model?

• What are the effects of bearing properties such as stiffness, damping, and preload on the TE, vibrationlevels?

• How can we correlate MBS model results with physical test data?

1.3 De-limitations

Statistician George Box quotes ”Essentially all models are wrong, but some are useful”. This describes thereality that any system existing in real world cannot be represented exactly. Some reasonable assumptions aremade and care is taken that the model behavior and results are still in some way relevant. In this work severalsuch assumptions are made while following the mantra often quoted by my Professor Ulf Sellgren ”the developedmodel should be as simple as possible and at the same time as complex as necessary”. The delimitations are asfollows:

• The preload in the bolted connections is not considered. Usually it has an impact on the housing stiffness.With reference from literature [1], where the effect of including and not including bolt preload on a windturbine gearbox housing are studied. It is noted that there is an effect, and the difference is about 3-4% change in natural frequencies. The noted range seemed acceptable and is justified for the reducedcomplexity in modelling. However, it needs to be considered, if optimizing bolt preload for vibrationreduction is of interest.

• Friction in gear contact is not considered in the model, to reduce the computational time. Frictioncalculations are highly discontinuous and can cause numerical convergence problems. [2]

• Predicting noise levels in cabin is not within the scope of this thesis.

• Lubrication in gears will not be considered to reduce complexity. Effects of temperature are not considered.

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• Analytical modelling or reasoning of bearing behaviour will not be dealt with as it will be a specific studyin itself. Knowledge about it will be gained from current research and be used to make conclusions.

• This study will be limited to studies on a specific model of GKN driveline PTU; an effort is put to producegeneral analysis methodology which can be applied to other systems.

1.4 Method

The overview of followed methodology used to carry out this thesis work is illustrated in the Fig.1.3. Themethodology employed contains of four major blocks which are represented with different colors in the figure.And an explanation of each block is provided below,

1. Literature Study and Defining objectives: Based on initial requirements a thorough literature studyis performed to gain familiarity in the area of NVH and dynamic analysis of geared systems. Relevantliterature is collected and used as reference further in the work. Then keeping in mind the output required,the objectives and research questions for the work are defined. The objective is to develop a dynamicmodel of the PTU which can predict TE and housing vibrations.

2. Modelling: To be able to develop a simple yet effective model, the problem at hand and different stepsin the modelling procedure are thoroughly studied. Acceptable assumptions are set and the model is builtin stages, increasing complexity in steps. The overview of the process is described below.

• Identify the components which are to be included in the analysis.

• Mesh the parts and perform modal reduction to generate an MNF file for each component.

• Import and assemble the components in MSC ADAMS c© by defining respective connections such asjoints, bearings, contact etc. Define motions and forces in the system for the dynamic simulation.Create measures and requests for all output that needs to be measured.

• Perform a dynamic analysis to predict TE and vibrations on housing.

• Perform a Linear modes analysis to predict the mode shapes and natural frequencies of the system.

Figure 1.3: Methodology used in this work

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3. Validation: This is an important step in the process, where physical testing is performed to measure theTE, housing vibration and modal parameters. These measured values will act as a reference to correlateand tune the simulation model.

• Testing objectives and test cycles need to be defined to measure the desired output.

• Measurements on a test rig are required to measure TE and vibration

• Experimental modal analysis needs to be performed to measure modal parameters of the PTU.

4. Tuning and Correlation: Based on comparison between test and simulation results, tuning of respectiveparameters should be performed.

• Gear contact parameters need to be tuned to predict TE accurately.

• Flexible body damping and bearing damping coefficients need to be tuned to correlate the vibrationresults.

• Mass and stiffness of the systems should be tuned to have a decent correlation of modal parameters.

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2. FRAME OF REFERENCEThis chapter presents the theoretical reference frame that is necessary for the performed model development.Fundamentals of gears, vibration, multi-body simulation are discussed.

2.1 Hypoid Gears

Gears are one of the most widely used components to transmit and alter rotary torque, speed and power asdesired. Bevel gears are used to transfer power between non parallel shafts. The angle between gears can be90◦ or any other angle [3] as seen in Fig.2.1. The gear set is composed of a pinion the smaller gear and thecrown gear larger one.

Figure 2.1: Bevel gear set [left ]; Bevel gear set different shaft angles [right ] [4]

Broadly, bevel gears can be classified as: [5]

1. Straight

2. Spiral

3. Skewed

4. Hypoid

The first three types are classified mainly by difference in the teeth geometry. Fig.2.2 shows the variation inteeth geometry. The hypoid gearset is similar to a spiral bevel gearset except for the difference that the pinionaxis is offset above or below the gear axis as seen in the right Fig.2.2. This means, unlike other bevel gearsthe axis of mating gears does not intersect. The hypoid offset can either be positive or negative, if the pinionis displaced in the direction of spiral angle it is positive and displacement opposite to spiral angle is a negativeoffset.

Hypoid gears are widely employed in Automobiles to transmit power between perpendicular shafts to transmitpower between front and rear axles. The possibility to offset the pinion arrangement allows lowering the driveshaft of the automobile thus allowing lowering the car body improving its stability and creating more cabinspace. This offset arrangement also facilitates an increased sliding contact when compared to a correspondingbevel gearset resulting in relatively lower noise levels.

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Figure 2.2: Types of Bevel gear [left ]; Hypoid Offset [right ][5]

Owing to their commonalities the terminology of bevel gears applies to hypoid gears. The gear geometry willnot be discussed in detail as it is not majorly used in this work. General terminology is depicted in the Fig.2.3for reference. A detailed explanation of hypoid gear geometry and manufacturing can be found in [5].

Figure 2.3: Bevel Gear Terminology [6]

2.1.1 Gear Ratio

It is a measure of how fast the driven gear spins relative to the driver gear. The formula is as follows

ωA

ωB=rBrA

=NB

NA(2.1)

Where N is the number of teeth, ω is the angular velocity and r is the radius of respective gears input gear Aand output gear B.

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2.1.2 Hypoid gear contact ratio and Mesh stiffness

Contact ratio can be understood as number of gear tooth sharing the applied load in a given gear mesh position.At a given point of time the contact condition in the hypoid gear pair is usually spread across three teeth ascan be seen in Fig.2.4 [7]. Depending on load, tooth geometry and pinion roll angle, the contact ratio changes.Which means the average number of teeth that are simultaneously engaged keeps varying.

Gear mesh stiffness is a property of the gear material which resists the deformation. Its value depends on gearmaterial, gear tooth curvature, loading on gear tooth, contact ratio and angular position of the gear. Usuallythey vary periodically. With varying contact ratio and tooth bending, there is a dynamic stiffness instead of aconstant value. This results in varying forces on the gear teeth, generating a Transmission error(TE) in-turncausing vibrations. The gear mesh stiffness can be calculated as follows: [7]

Km =F

el − eo(2.2)

Where, Km is the mesh stiffness of gear pair, F is the contact force along line of action and el, eo are thetranslational loaded and unloaded TE.

Figure 2.4: Hypoid Gear Tooth contact region

2.2 Transmission error (types and focus)

As the legend has it, Transmission Error(TE) is one of the important excitation sources of vibration in geardrives. And, has its origins in assembly and manufacturing imperfections and compliance of gears and gearteeth, shafts, bearings and housing flexibility [1]. It is a type of motion error. In an ideal case, if the gearswere perfectly rigid without any geometrical modifications, the gears would transmit the rotational motionperfectly, which means that a constant speed at the input shaft would result in a constant speed at the outputshaft. However as a result of friction, gear body and tooth flexibility and other inevitable deviations, thereare variations in the speed ratio. This variation has been observed to cause oscillating forces on the gear teethresulting in noise and vibration.

TE is defined as [8] ”The difference between the actual position of the output gear and the position it wouldoccupy if the gear drive were perfectly conjugate.”

It is the difference between the theoretical angular position of the driven gear and its actual angular position.A schematic description can be seen in Fig.2.5. In equation form it is expressed as follows:

TEang = θ1 − (rb2rb1

θ2) (2.3)

Where, rb2, rb1 are the Gear base radii,

θ2,θ1 are the Angular position of gears

TE is commonly expressed in micro radians and a sample TE signal measured in time can be seen in Fig.2.6(a).To acquire the frequency content of the TE signal a Fast Fourier transform (explained in section 2.4.2) isapplied to convert it into frequency domain which then gives us amplitudes at particular frequencies as seenin Fig.2.6(b). Usually it is the Gear mesh frequency(GMF) and its harmonics which contribute to noise and

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Figure 2.5: Illustration of Transmission Error

vibration [9]. Gear mesh frequency is the rate at which gear teeth mate together, so the frequencies are speeddependent. It can be calculated as follows: [10]

Gear mesh frequency(GMF ) = number of teeth× shaft speed (2.4)

The shaft speed is in rotations per second(rps). If there are multiple shafts, shaft speed can be defined withreference to input shaft speed. Gearboxes are excited strongly at gear mesh frequency and its harmonics. Forexample see Fig.2.6(b), it is a sample plot from TE measurement at 60 rpm in the PTU, where number of teethon pinion is 15. Peaks can be seen at 15Hz and its multiples which are the gear mesh frequencies.

(a) In time domain

(b) In Frequency Domain

Figure 2.6: TE sample signals

2.2.1 TE Types

TE measurement is classified based on the speed-load conditions as can be seen in Tab.2.1 and also if it ismeasured on a gear set level or full system level.

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Table 2.1: Types of Transmission error [9]

Load (Torque)Low High

SpeedLow

StaticUnloaded

StaticLoaded

HighDynamicUnloaded

DynamicLoaded

1. Static unloaded TE: It is measured at low speed and low torque on the gear-set level. It is commonlyused to understand the load independent errors. So the main contributors to this TE are manufacturingerrors and gear geometrical errors such as pitch error, surface defects etc. Usually it is used for gearquality inspection.

2. Static loaded TE: This is measured at low speed and high torque which means contribution of variablemesh stiffness of gears i.e. tooth bending and gear body deflections are considered.

3. Dynamic unloaded and loaded TE: From a noise and vibration prediction perspective this is the mostrelevant form of TE measured. Measurement is done on a system level when the gears are mounted in thegearbox. This is to consider the dynamic properties of the entire system with contributions from gears,shafts, bearings and housing. Physically it is complicated to measure, as there are several factors involvedowing to test rig control or shaft accessibility of the gearbox. Thus a validated simulation model couldprove useful in analyzing Dynamic Transmission error (DTE). And this is one of the main output to bepredicted in this work.

The static TE and dynamic TE will be referred to as STE and DTE in the rest of the document.

2.3 Vibration

Vibrations are oscillations of a system about an equilibrium position. This section provides an introductionto several concepts related to vibrations and its measurement. The focus is around automobiles and gearedtransmissions.

2.3.1 Noise, Vibration and Harshness (NVH)

NVH in the field of Automobile engineering is the search for the source of noise and vibrations and also acomplete analysis of how it is perceived in terms of feeling and hearing. The motive is not only to limit thesecharacteristics but sometimes to achieve a desired type of sound and vibration (brand recognition). The growingimportance towards environmentally friendly transport solutions has led to drastic changes in vehicle designs.Examples include using light weight structures, advanced after treatment systems, downsized combustion enginesand usage of alternative propulsion systems. These new changes pose a wide range of NVH problems whichhave to be addressed. Although it is a complex field with many aspects to consider, the sources of NVH in anautomobile can be classified as:

1. Aerodynamic

2. Mechanical

3. Electrical

The basic definitions and NVH related terminology are discussed below. [11]

• Noise: Any unpleasant or unexpected sound created by a vibrating object

• Vibration: Any objectionable repetitive motion of an object, back-and-forth or up-and-down

• Harshness: An aggressive suspension feel or lack of give in response to a single input. Harshness isgenerally used to describe the severity and discomfort associated with unwanted sound and/or vibration,especially from short duration events.

• Frequency: It is the number of complete cycles that occur in one second. Sound and vibration wavesare measured in Hz or cycles per second (CPS)

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• Order: It is the number of events per revolution. Each rotating component emits a response and Orderstrack the relationship between this response from the system, the RPM, and the frequency of rotation. Ithelps to identify how an individual component contributes to the overall level.

• Resonance: Resonance is the tendency of a system to respond to a compelling force oscillating at, ornear, the natural frequency of the system. All objects have natural frequencies and experience maximumresponse at the point of resonance.

As our focus of efficiency and comfort increases, the importance of verifying the systems NVH performancebecomes very important, both from regulations and customer/user point of view [12]. More vibrations orunwanted noise reflect poor design and manufacturing.

2.3.2 Noise and frequency content in gear drives

In automobiles the gearbox is one of the dominant sources of noise and vibration. Tab.2.2 shows the percentagecontribution of several systems in an Automobile to the total noise and vibration generated.

Table 2.2: % contribution of different sources in an Automobile to total Noise & Vibration [13]

No. Source % Contribution1 Engine 23 to 302 Exhaust System 25 to 353 Intake System 5 to 154 Fan & Cooling System 7 to 155 Transmission 12 to 156 Tires 10 to 17

Before analyzing a gear drive, having an idea of possible vibrational frequencies and noise is important. Ingeneral, the NV generating sources can be associated with the basic components such as gears, bearings, shafts,internal oil pumps and clutches. The basic spectrum is usually broken down into a combination of the followingeffects: [14]

1. Low harmonics of the shaft speed

2. Harmonics of the fundamental tooth

3. Meshing frequency and their side bands

4. Sub-harmonic components

5. Hunting tooth frequency components. Ghost (or strange) components

6. Periodicity in signals measured on planetary gearbox

7. Gear rattle

8. Components originating from faults in rolling-element bearings

Gear Whine: It is a high frequency tonal noise which is often perceived as unpleasant. TE is said to be themain cause of this noise. This noise is emitted from gears that are in mesh and the sound is characterized asvibrations with frequencies same as the gear mesh frequency and its multiples [15]. The noise is periodic andthus is perceived as a tonal noise. A tonality is caused when the amplitude of a small frequency is significantlyhigher than its neighboring frequencies. Sound of a whistle is a typical example. The human ear is very sensitiveto tonal noises; thus such a noise is perceived as highly unpleasant. Thus, gear whine even though is not theloudest source; its tonal nature renders it unpleasant to human ear.

2.3.3 NVH Transfer path

The force variations caused by TE on the gear teeth are the source of vibration. The vibrations induced byTE flow through the gears, shaft and bearings to the housing where they become air or structure borne, seeFig.2.7. Two ways usually exist to lower the drive unit’s vibration or noise:

1. Optimizing the gear design

2. Modification of vibration transfer path such that it is attenuated or not amplified as it travels from sourceto listener. [1]

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Figure 2.7: NVH Transfer path in a gear drive

By looking at the transfer path it is understood that when developing a dynamic model it is acceptable to leavesome components out of the model to reduce complexity. It is also observed that an accurate representation ofthe bearings and housing is important.

2.3.4 Bearing dynamic characteristics

Bearings are a common and important component in any rotating system which allows relative motion betweentwo bodies. Roller bearings are widely employed in automotive systems owing to their ability to take upaxial and radial loads while also accommodating minor misalignment’s. In the PTU, cylindrical tapered rollerbearings(TRB)(see Fig.2.8) are used to support the pinion and tubular shaft in the housing. TRBs consist ofan outer ring and an inner ring assembly with roller and cage. TRB can take axial loading only in one directionthus it is usually paired with another TRB. More about bearings can be found in [3]

Figure 2.8: Tapered roller bearing terminology and parts

When analyzing the dynamics of a system, bearings play an important role in how the vibration flows, forexample from gears to the housings. Bearings exhibit a non-linear behavior, as in the stiffness and damping ofthe bearings vary with applied load and speed which makes it a challenge to model/understand its behavior.Thus, it is important to understand which of the bearing properties affect the system behavior. The maincharacteristics of bearings affecting gearbox dynamics are:

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1. Bearing Stiffness

2. Bearing radial internal clearance

3. Bearing damping

4. Bearing Preload

Akerblom and Ulf in [12] studied the influence of bearing preload on gearbox noise level. They concluded thatthere is an impact of preload and it is higher at lower torques. Opitz showed the effect of different bearingtypes on gearbox noise. Results show TRB’s reduced noise level by 3 db when compared to ball bearings.And an increase in axial preload has been seen to reduce noise level by 1db for both bearings. Flemingsinvestigations on gear-shaft-bearing systems lead to conclusions that increase in bearing damping reduces thedynamic transmission error. Thus the effect of these parameters on the system vibration needs to be taken intoconsideration.

2.4 Modal Analysis

In simple terms modal analysis is the process of describing a structure in terms of its natural characteristicswhich are natural frequency, mode shapes and damping. These characteristics are termed as dynamic propertiesof a structure. [16]

• Natural frequency: A frequency at which the system will amplify the effect of an applied load. It is aninherent property of the system which depends on the mass and stiffness.

• Mode shapes: It defines the shape the structure would deform into at a particular natural frequency.

• Damping: It defines how quickly vibrational energy is dissipated in the system. Higher the damping,lower the vibrations.

To understand the basic process, lets take the example of a freely supported flat plate as seen in Fig.2.9 [16].A sinusoidally varying force is applied at one corner of the plate, the rate of oscillation (frequency) of the forcecan be varied while peak magnitude is constant. The response is measured using an accelerometer mounted onthe plate. As the rate of oscillation of the force is varied in time, it is observed that measured response amplifiesas well as reduces as seen in Fig.2.10. So whenever the rate of oscillation of the force gets closer to one ofthe plate natural frequencies the response is amplified. And by measuring the response at multiple points, theshape in which the plate deflects at a particular frequency can be measured.

Figure 2.9: Plate excitation

The time data shows us where vibrations are increased in time but is usually hard to make other conclusions.Thus, it is often transformed into frequency domain using a mathematical function called FFT to get a frequencyresponse function(FRF). A sample FRF plot can be seen in Fig.2.10. Peaks in the FRF indicate naturalfrequencies and the sharpness of the peak indicated the respective damping.

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Figure 2.10: Response (Time, Frequency domain) and Mode shapes [16]

2.4.1 Methods to perform Modal Analysis

There are two ways to perform a modal analysis, experimental and analytical. In this work both methods areemployed. Analytical method is used to create flexible bodies for a dynamic simulation and also to predictthe PTUs modal properties. While the experimental method is used to physically measure the PTUs modalparameters to validate the simulation model. A brief representation of both the methods is given below.

Experimental modal analysis(EMA)

The objective of an EMA is to deduce the dynamic properties which govern the vibration behaviour of a givenstructure. The overview of the process is described in Fig.2.11. It starts with defining the motive of theanalysis and what the main interests are in terms of frequency range, output to measure etc. Then either aroving hammer or roving accelerometer test can be performed. In this work a roving hammer test method isused where a hammer is used to excite the structure while response is measured using accelerometers mountedon the structure. For test setup based on observation, several points are marked on the structure discretizingit. A force input is given at these points one after the other and a response is measured through accelerometersmounted on the structure. Frequency response function as to how a force input translates to vibration outputon the structure is the main output measured. The collection of results gives us an FRF matrix from which themodal parameters are deduced. This method is employed heavily in the aerospace and automobile industriesto check the dynamic properties of the system.

Figure 2.11: Overview of EMA procedure [17]

Finite element analysis-based simulation

Dynamic analysis using finite element method can be used to predict the vibration characteristics of a structureunder design. The structure is represented by a theoretical collection of springs and masses and then a set ofmatrix equations can be written that describes the whole structure. Next a mathematical algorithm is appliedto the matrices to extract the natural frequencies and mode shapes of the structure. This technique is usedto predict the modal parameters before a structure is manufactured to find potential issues and address themearly in the design process. This solution is also called an Eigen frequency analysis or simply modal analysis.

2.4.2 Fast Fourier Transform(FFT)

The Fourier transform(FT) is an important mathematical tool used to transform a signal in time domain intofrequency domain. In general, measurements are made in time domain and the output signal is a concoction

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of several frequencies (from components, background noise etc.). This signal is not readable, so an FT trans-formation converts the signal to frequency domain by separating into several frequency components. Most ofthe times the measured signal is not a continuous function but is instead a series of discrete values. And a FastFourier Transform (FFT) is a mathematical algorithm, to apply the Fourier transform to discrete signals. Inthis thesis, FFT will be used to transform the measured acceleration data. This way the important frequenciesinfluencing system vibration can be deduced. The basic theory behind FT is discussed below.

According to Fourier series theory, a periodic signal can be expressed as an infinite series of sine and cosineterms, or alternatively an infinite series of complex exponential terms [15]. The frequency of these terms is aninteger multiple of the fundamental frequency of the original signal also called as the harmonics of the signal.Thus according to Fourier series a periodic signal with period T is expressed as follows:

f(t) = a0 +

∞∑n=1

(an cosω0nt+ bn sinω0nt) (2.5)

Where, ω0=2π/T is the fundamental frequency and an, bn are constants defined as [15]

an =2

T

∫ T

0

f(t) cosω0ntdt, bn =2

T

∫ T

0

f(t) sinω0ntdt (2.6)

2.5 Multi Body Simulation

Dynamics is a branch of mechanics dealing with study of forces and their effects on motion. Dynamic behaviorof a system means how it behaves as a result of time varying loads and motions (boundary conditions). It isvery important to analyze, since it reflects system robustness and efficiency.

Multi-body dynamics (MBD) is a specific field of engineering dynamics dealing with the mathematical modelling,the simulation and the analysis of structural systems made of sub-components (rigid or flexible bodies) intercon-nected through links (called joints) [18]. MBS is using the power of computer technology and numerical methodsto solve MBD problems and gets insights into system behavior. It is usually employed as a tool in early andlate design phases to evaluate the system kinematics and dynamics.

MSC ADAMS c© abbreviated as Automatic Dynamic analysis of mechanical systems is one of the widely usedtools for MBS and is the one used in this work. For simplicity the tool will be referred to as ADAMS in the restof the document. It enables users to build virtual prototypes and simulate them. For a simulated model if thekinematics, dynamics or load transmission are not what is required, quick modifications can be made and testedagain. This enables reducing time and costs involved in building several prototypes. Since it is a mathematicalmodel of the reality, there are always assumptions and errors involved. Thus it is important to build a physicalprototype and test it before finalizing the design.

2.5.1 Work flow in ADAMS c©

Below the basic MBS work flow, formulation of equations and types of analysis as briefly discussed. Flexiblebody analysis will be discussed in detail as it is the focus in this work. Fig.2.12 describes in general the overviewof building and simulating a model in ADAMS c©.

Build

• Bodies: Firstly, the geometry of the parts required can either be created using basic shapes or it can beimported form CAD files. Each part in the model initially has six degrees of freedom which means it cantranslate and rotate about three axes.

• Joints: In reality Joints are used to connect one part with respect to another and they define how onepart moves relative to another. ADAMS provides several types of ideal, primitive and realistic joints toconnect the components together. Joints can be categorized as lower and higher pairs. Lower pair meansa joint with area contact between two bodies, while higher pair means a line or point contact.

– Lower: revolute, cylindrical, spherical etc.

– Higher: gear, cam

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• Forces and Motion: Several types of forces and input motion can be applied to components to replicatereality. The input applied can either be constant or varying with time.

Figure 2.12: ADAMS general workflow [2]

• Simulation: After a model is built, we can run different types of simulations on it. When the model issimulated on an MBS program, the displacements, velocity and accelerations of the parts and how loads aretransferred is calculated. There are two types of simulations which can be run, kinematic or dynamic. Inkinematic problems results are calculated independent of the effect of forces, based on constraint equationsand equations of motion. While in a dynamic simulation the effect of forces are taken into account anda systems of Differential algebraic equations (DAEs) are solved. Eq.2.7 shows a sample of the equationsinvolved in the analysis. H(q)=0 is the equation reflecting the kinematic constraints, while the other isan equation of motion derived using Euler’s laws. These two combined form a system of DAEs, whichare then solved using numerical methods. Several solver options are available and an apt one needs to beselected depending on problem at hand. For more detailed understanding of how equations are setup fordifferent simulation [19] can be referred to.

Mq +HTq (q)λ = Qe(q, q, t)

H(q) = 0(2.7)

Where M is the mass matrix, q is the acceleration of the parts, HTq is the Jacobian of constraint equation,

Q is the external forces. H(q,t) and λ is a matrix of forces between bodies necessary to uphold frictionfree constraints [19].

In summary, at each time step the position, velocity and acceleration of each component is calculated, takinginto consideration effect of forces. This result is then implemented in the equation of motion to calculate theinternal forces.

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2.5.2 MBS Classification

Broadly analysis in MBS is classified as:

Rigid body dynamics:

Parts are assumed to be un-deformable, usually provides a quick solution to calculate the forces and motionsin a system. However an assumption is made that forces are low enough to not cause an elastic deformation inthe parts. If in reality there is a huge deformation it needs to be considered as it changes the force distribution.

Flexible body dynamics:

When there is considerable deformation in the parts or when flexibility of a part is important for the analysis,flexible body dynamics analysis is performed.

The classification of bodies in a MBS environment is as follows: Tab.2.3 [20]

Table 2.3: Classification of structural bodies in MBS

Deformable Un-deformableNo mass and inertia Rigid link Massless flexible body

With inertia Rigid body Flexible body

2.5.3 Flexible bodies in ADAMS

The general practice in MBS is to model mechanical systems as rigid bodies connected by joints as it is timeefficient. This approach works mostly when the deformations in the components are not significant. Howeverthe contribution of component flexibility to the dynamics response of the system is lost. In this work since themain focus is to analyze the vibrations of the systems generated due to TE which in turn is generated due tocompliance in several components, the flexible behavior is important. Steve Pliz [20] gives insights into the needto use flexible bodies.

However, considering a fully discretized component similar to an FE model with several degrees of freedom(DOFs) is not feasible in the MBS. It makes flexible body analysis time consuming. Thus to integrate theaccurate results of flex body dynamics and the time efficiency of rigid body dynamics innovative methods weredeveloped. The methods developed use a modal reduction technique, which is a FEM based modal analysismethod. In [21] U Sellgren describes the limitations of FEM and rigid body dynamics and how an integration ofboth results in a time efficient and accurate simulation. The scalability issue of a FEM model due a large numberof nodal DOFs is discussed along with the primitiveness of a rigid body simulation. Component mode synthesis(CMS) is a method where FE analyses is used to solve, a boundary value problem which determines the elasticbehavior of a component. In FE, total number of DOFs of a component depends on the discretization and meshrefinement used. So as number of components increase DOFs increase drastically. To reduce this DOFs and stillcapture the flexible behavior of a component CMS is used. Ulf discusses several methods available to performthis reduction along with the method used in ADAMS. In this work the Craig Bampton method is used whichis briefly discussed below. The below concepts explain how the FE reduction is achieved.

Modal Superposition:

All deformations of a part in the physical world can be expressed as the combination of different mode shapes.Fig.2.13 illustrates how a complex shape in the physical domain is expressed as a combination of simpler modeshapes.

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Figure 2.13: Concept of Modal Superposition [22]

In an FE model the discretization results in a finite number of nodal DOFs, translating to infinite number offlexible body DOFs. In order to represent this large number of nodal DOF in reasonable number of modalDOF, modal superposition is used. Eq.2.8 describes the relation between deformations and mode shapes.

u =

M∑i=1

φiqi (2.8)

Where M is the number of mode shape, u is the Linear nodal deformation of the FE mode, q is the Modalcoordinates, φ is the Mode shape.

Component Mode Synthesis (CMS):

Craig Bampton method: Component mode synthesis is targeted for dynamic simulations in the mechanicaldomain [21]. In Craig Bampton approach, the physical DOFs of a flexible body is condensed to a set of physicalDOFs representing external mating features and a set of Modal DOFs which represent the internal dynamicproperties of the body. It enables users to select a subset of nodal DOFs which are not subject to modelsuperposition. This is to retain the behaviour of attachment points to later on connect different bodies in anMBS environment. The set of retained DOFs are called boundary DOFs/interface DOFs. To achieve this, thewhole FE model divided into boundary DOFs and Interior DOFs. Then 2 sets of modes are calculated: [20]

• Constraint modes: These modes are obtained by giving each boundary DOF a unit displacement whilefixing the other DOFs, see Fig.2.14. This represents the external mating features of the component.

Figure 2.14: Constraint Modes

• Fixed boundary normal modes: The boundary DOFs are fixed and modes are calculated by deducingthe Eigen value solution, see Fig.2.15. These modes represent the internal dynamic properties of thebody.

Figure 2.15: Fixed Boundary normal Modes

This Craig bampton reduction analysis can be performed on most of the commercial FEM tools. The output ofthis analysis is a Modal Neutral File (.mnf file), which contains the modal information of the component.This file can be imported into ADAMS as a flexible body.

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2.5.4 Inertial modelling

ADAMS flex menu is used to import the generated mnf file into ADAMS. When importing, the type of inertialmodelling and damping needs to be specified. During a simulation, nine inertia invariants are used to calculatethe time varying mass matrix of a flexible body. The inertia invariants are computed from the nodes of thefinite element model based on information about each nodes mass, its undeformed location and its participationin the component modes [2]. The modal formulation of flexible bodies depends on which of these invariants isactive. ADAMS provides four custom formulations, a brief description about each is provided below:

• Rigid body: In this formulation, Adams Flex disables invariant 6, which is the modal mass. Thus aflexible body is considered rigid and all modes are ignored during simulation. It is close to simulatingrigid body dynamics.

• Constant: In this formulation, Adams Flex disables invariants 3, 4, 5, 8 and 9. The flexible body’s inertialproperties are unaffected by deformation. Which means Adams View does not account for changes in themoment of inertia due to deformation. Although simulation time is reduced, this effects the resultsdrastically and must be used with caution.

• Partial coupling: In this formulation, Adams Flex disables invariants 3, 4, 5 and 9. The effect of bodydeformation on moment of inertia is considered. The Invariants 5 and 9 provide a second-order correctionto the flexible body inertia tensor. These invariants impose the greatest computational overhead on theevaluation of the flexible body equations of motion. Disabling these invariants can reduce CPU timesignificantly while having minor impact on results in most cases.

• Full coupling: In this formulation, Adams Flex enables all of the invariants except for invariants 3 and4. This method is used when full accuracy is required. However the simulation time increases drasticallyas well.

Partial Coupling is the default option and is the one used in this work. It has significant computational efficiencyover the more accurate Full Coupling formulation. However it should be verified, that your model does notrequire Full Coupling.

2.5.5 Bearing modelling in ADAMS

There are four methods to model bearings in ADAMS. [2]

1. Joint method: The bearings are represented using an ideal kinematic joint. In this method no stiffness,damping or preload can be specified and the constrained Degree of freedom will have no relative motion.Such a bearing joint can be used when analyzing STE on gear set level.

2. Compliant method: A force is used to represent the bearing. It can be looked at as a bushing connectionwhere a linear stiffness and damping force profiles can be applied.

3. Detailed method: A six component force (gforce) is used to represent the bearings. Using advancedanalysis technology from drive-train simulation software KISSsoft, stiffness is calculated at every time stepbased on the positions and velocities at the bearing location. The damping component resists the velocityand is modelled as a damping coefficient multiplied by velocity. The damping component is determinedby multiplying the square root of the instantaneous stiffness by a damping factor. This method providesa far more accurate representation of the bearing compliance than the compliant method.

The detailed method is used in this work and the basic theory is discussed below:

Based on the input geometry an analytical model of the bearing is generated. The inner ring is fixed and rotateswith the shaft while the outer ring is assumed fixed to housing. The load distribution on the single rotatingelement is determined from the non-linear stiffness between rotating element and bearing (inner/outer) ring(see Fig.2.16). Calculation is based on DIN ISO 281 [23]. The calculated bearing stiffness matrix K is a 5x5matrix (3 linear and 2 rotational stiffness’s). The normal force per roller is calculated as per Eq.2.9. So asforces in bearing vary the deformation in the rollers varies. The normal force and in-turn stiffness of the bearingis calculated at every simulation time step.

Q = cL × δ109

with, cL = 35948 × Lwe89

(2.9)

Where, Q is the normal force on roller element of length Lwe.

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Figure 2.16: Model of Roller Bearing according to DIN ISO 281, Supplement 4 [23]

Damping factor or damping ratio is dimensionless quantity, calculated as the ratio of actual damping to criticaldamping. And in this method of bearing representation it is given as an input.

2.5.6 Gear contact modelling

One of the important interfaces between components to model in geared drives is the contact. In reality forhypoid gears the contact is spread over multiple teeth. And forces from the contact are one of the maincontributors to the dynamic response. Hence a decent representation is desired. In ADAMS c© an IMPACTfunction can be used to model the contact between two flexible bodies. [2] The contact is modelled as a nonlinearspring force with stiffness and damping coefficient. So the force or stiffness is calculated based on the penetrationdepth between the two bodies. The analytical formulation is as follows:

ContactForce =

{Max(0, k(g)

e − STEP (g, 0, 0, dmax, cmax) × g, for 0 < g

0, for0 ≥ g

Where, k is the stiffness, g is the penetration between the two bodies, e is the force exponent, cmax is thedamping coefficient, dmax is the depth (g) at which to apply maximum damping coefficient

Figure 2.17: Illustration of impact function

The equation implies:

• 0 ≥ g, no penetration occurs and force is zero

• 0 < g, a separating force is generated as per function

The contact force generated is a function of stiffness and damping. Stiffness component is generated basedon material stiffness and amount of penetration. While damping component is applied as a function of userspecified damping ratio, velocity of the bodies and penetration depth. The contact parameters such as stiffnessis considered based on a Hertzian formulation of stiffness derived as a result of material property and componentcurvature. Although the nonlinear contact based on penetration depth is used, the time varying stiffness of gearcontact is not included due to modelling complexity.

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3. METHODOLOGYThis chapter describes the model development procedure in an MBS environment and physical testing processemployed to validate the model.

Firstly, a work flow is defined and the most important components and parameters to be modelled are identified.To understand the necessary components to model and identify parameters that influence PTUs dynamicbehavior the transfer path of vibration is looked at (see sec.2.3.3). Considering the involvement in the vibrationtransfer path, only the major components such as gear-set, tubular shaft, bearings and housing cover-plate areincluded in the model. The properties of the components are described in Tab.3.1.

Table 3.1: Component Material Properties

Components MaterialYoung’s

Modulus [Gpa]Density

[kg/mmˆ3]Mass[kg]

Crown Gear Steel 208 7.85e-6 2.04Pinion Steel 208 7.85e-6 1.4

Tubular Shaft Steel 208 7.85e-6 1.4Housing Aluminium Alloy 70 2.75e-6 3.62

Cover Plate Aluminium Alloy 70 2.75e-6 0.85

The PTU is a time variant system which means several properties of the system such as gear stiffness, bearingstiffness, positioning of gear and pinion teeth are subject to the loading conditions on the unit. Modellingthis kind of a system with FE models usually demands high computational power and time. The multi bodyapproach on the other hand is robust to work with multiple bodies and interactions between components.However, the MB analysis is usually performed with a rigid body assumption which makes the model too crudefor a NVH analysis. Thus a flexible body modelling approach has been developed and integrated with the MBtool to perform transient simulations on time variant systems. The flexibility of a component is representedin MBS as an MNF. The modal behavior of the part is captured using condensation methods to reduce thenumber of DOFs while still retaining its behavior. FEM tool SimLab c© is used to mesh the geometry and createinterface nodes. MSC Marc c© is used to perform the modal analysis and generate the MNF file. Dynamicanalysis is performed in MSC ADAMS c©.

3.1 Overview of methodology employed

As described in sec.1.4, there are two main blocks followed in parallel. One is the numerical model developmentand simulation, the other is physical testing. The overview of model development is described as a flowchart inFig.3.1. The main highlights of testing are described later in the sec.3.3. The main steps in model developmentare CAD modifications and FE meshing, followed by FE modal reduction where an MNF file is generated foreach component(sub-structuring), followed by building the model in ADAMS.

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Figure 3.1: Steps in developing the Multi body simulation model

3.1.1 Models Built in Adams

The model is built on different levels to understand the level of detail required to predict the TE and vibrationsof the drive unit. (See Fig.3.2)

1. Gear set with primitive joints (left)

2. Gearset with Bushings (middle)

3. Gearset with Bearings (right)

A crawl, walk, run approach is used, by increasing the complexity in stages. The gear set level model is usefulin predicting STE, while the later models predict the dynamic transmission error. In this report due to spacelimitations the model development procedure and results for the full model with bearings will only be considered.For reference of model development for first two, [24] can be referred.

Figure 3.2: Models Built

3.2 Development of Dynamic model

The implementation of this work involves 3 stages:

3.2.1 CAD clean up and Mesh Preparation

Firstly only the necessary components which are the pinion, crown gear, tubular shaft, and housing and coverplate are separated from rest of assembly. Next the CAD of these components is simplified by removing certainfeatures like splines, logos, detailed features and other manufacturing details. These features do not affect theresults of a modal analysis but can increase the computational time and time required to mesh. The finalcomponents are exported from CATIA and are imported into SIMLAB. (.CATpart) format is used to maintaingeometry integrity during the file exchange.

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Meshing:

As the motive is to perform a modal analysis, a coarse mesh will suffice. However, care has to be takenwhen meshing. It is necessary to capture the stiffness of the components accurately, so at least 2 elements areneeded along the thickness on the thin faces especially housing and tubular shaft. A mesh convergence studyis performed to study how the natural frequency changes based on mesh size. A compromise is made betweenmesh size and accuracy to not make the file too huge. Fine mesh is used in areas where contact is defined suchas gear tooth faces. A relatively coarser mesh is applied on rest of the areas as model reduction is used. Themesh parameters used are as follows:

Table 3.2: Mesh Parameters

Components Element Type Element SizeCrown Gear &

PinionTET 10 1

Tubular Shaft TET 10 1.5Housing &Cover Plate

TET 10 2.5

Rigid body elements (RBE2) are created wherever an attachment point is needed. It basically is a single nodecontrolling a set of selected surface nodes. Thus, they are created at connection points on all the components.Fig.3.3 shows the meshed components.

Figure 3.3: Meshed Components with RBE2 elements

MNF Generation

Next is setting up the problem in an FE solver to generate the flexible bodies. MSC MARC is the tool used inthis work. The motive here is to perform a modal reduction and produce an ADAMS modal neutral file (MNF).The file contains modal data such as frequencies, stiffness and mode shapes. The process involves 3 main steps:

1. Defining the material properties and Units, which should be the same as to be used in the Multi-Bodysimulation environment

• Material properties are defined as in Tab.3.1. MMKS (mm, kg and second) units system is followedthroughout the modelling procedure.

2. Selecting the set of interface nodes

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• Go to Boundary conditions- New structural and select DOF set nodes option. Then tick all DOFsand select all previously defined RBE2 elements. This ensures that these nodes are not involved inthe modal reduction and the DOFs are retained. Constraint modes are calculated at these points asexplained in 2.5.3

3. Setting up the ADAMS Craig Bampton Analysis

• The frequency range of interest/ number of modes to include is specified. In our case the first 20modes are included

• Extended precision is removed to reduce file size and an mnf is requested as output. A .dat file isexported.

• The .dat is then run on a cluster on MARC solver which outputs a model neutral file (MNF) whichhas the dynamic data for the respective component.

This process is repeated for all the components separately and individual MNF files are generated.

3.2.2 Model Development in ADAMS

Importing flexible components

Firstly, the unit system is set to MMKS, such that units are consistent when importing the previously generatedMNFs into ADAMS environment. The components are imported individually through the ADAMS flex ribbon.At this step the damping and inertial modelling of the individual components can be defined. For initialmodelling generalized damping is set. Which means a damping of 1 % for frequencies up to 100Hz, 10 % up to1000 Hz and 100 % for higher frequencies is applied. And partial inertial coupling is set to enable respectivemodal formulation of the flexible bodies. More information about inertial coupling is discussed in Sec.2.5.4.Later, after results from testing are obtained the flex body damping is tuned.

Defining Connectors between the components

Different types of connections are available in ADAMS to assemble the components into a system. Primitivejoints (idealized), contacts and bearings are the ones used in this work. Ideally in the real world most connectionssuch as bolt joints, welds exhibit some stiffness and damping. However based on our motive of analysis a decisionhas to made if it is critical to take into account these factors. The connections defined in numerical model arelisted in Tab.3.3. For the crown gear and tubular shaft in reality there is a weld connection, so a fixed joint isused to connect them in the model. Between the crown gear and pinion a contact is defined. The pinion andtubular shaft are connected to housing via bearings. All bolted connections are modelled as fixed joints. Moreabout the connections is discussed in the following sections.

Table 3.3: Connections defined in the MBS model

Components Connection CommentsCrown Gear - Tubular Shaft Fixed Joint

Crown Gear - Pinion ContactIMPACT

function based

Tubular Shaft - Housing Two BearingsPreload

is applied

Pinion - Housing Two BearingsPreload is

appliedHousing - Ground Fixed JointsPinion - Ground Revolute Joint

Housing - Cover Plate Fixed Joints

Primitive joints

All the connections for flexible bodies are made by connecting the respective RBE interface nodes. The crowngear is connected to the tubular shaft as shown in Fig.3.4 by defining a fixed joint at the respective RBE nodes.A revolute joint is used to connect the pinion with respect to the ground at the location pointed in the figure.This connection represents the PTUs connection with the prop shaft.

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Figure 3.4: Rotational and Fixed joint

For all the bolted joints between Housing-ground and Housing-Cover plate fixed connections are used to connectrespective RBE center nodes(see Fig.3.5). Ideally preload needs to be defined between the housing and coverplate as it affects the housing stiffness. However from literature review [1] a decision has been made to assumeidealized fixed joints. Since the variation of stiffness and in-turn the modal parameters is about 2-3 %, whichis an acceptable range for the reduced complexity. Nonetheless for a more accurate representation it needs tobe included.

Figure 3.5: Fixed Joints Housing-Ground and Housing-Coverplate

Contact

An impact function based contact is defined between the gears. The analytical formulation is explained inSec.2.5.6. Basically the geometry engine of the software detects penetration between the bodies and based oncontact parameters defined a contact force is generated. Damping is applied as a function of velocity of the gearbodies. Contact stiffness, damping, force exponent and penetration depth are the contact parameters whichneed to be input, see Fig.3.6. Based on material property and gear curvature a Hertzian contact stiffness ofabout 2.2E5 is calculated. Damping is usually applied as a percentage of stiffness and in this case about 0.1 %of stiffness is applied which is 10 N-s/mm. A penetration depth of 0.01 is set and force exponent is set as 2.2.The last two values are tuned based on simulation performance. Several iterations were run to understand theinfluence of each parameter and the below conclusions are made.

• Contact stiffness can be defined based on material and geometry

• CAD surfaces need to be smooth which otherwise result in force spikes

• In the initial position the bodies should not be intersecting largely

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• Damping as .1 % stiffness is a good value, much higher or lower tends to increase the simulation time

Figure 3.6: Contact Parameters in ADAMS

Bearing Modelling

Bearings play an important role in the vibration transfer path, the vibrations generated from gear TE transmitsto housing through bearings. And capturing their nonlinear behavior is important to predict housing vibrationlevels. There are three methods to implement the bearings. ADAMS provides several type of connectors suchas bushings, analytically modelled bearings (as described in sec.2.5.5) and FEM based bearings. A/MachineryBearing module is used in this work. The bearing is represented as a six component force, a dynamic stiffnessand damping is calculated depending on loading conditions. As input, the bearing type, geometry (ring, roller),preloads, damping are given. Four Bearings are created using the bearing module at locations B11, B12 onpinion and B21, B22 on TBS as can be seen in Fig.3.7.

Figure 3.7: Bearing positions

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The advanced method is used where in the bearing geometry is specified to represent the TRBs. Since the TRBstake axial loading in only one direction, while creating them there are two things to take care of, one is the loadbearing direction of the bearing and the other is the direction of preload. In the model a constraint is definedwhich represents the housing step so respective constraints are set for the four bearings. The radial, axial andbending damping ratio’s are specified as 0.001 (dimensionless quantity) as recommendation from literature.Then the required bearing outputs requests are selected such as bearing forces, accelerations, dynamic stiffness,and damping. The inner race is fixed to the shaft at respective location and the outer race is fixed to thehousing. The analytical formulation is discussed in section Sec.2.5.5.

Creating motions and forces

To simulate the conditions of the PTU in an automobile, forces and motion are applied. These conditionsapplied replicate the test rig boundary conditions which will be discussed in Sec.3.3.1. A motion or speed inputis applied at the pinion as a point motion at location where the revolute joint is defined and a resistive torqueis applied at the tubular shaft at position B21 as seen in Fig.3.8.

Figure 3.8: Input Conditions (Speed and Torque)

Creating Measures

Angular displacement measures are created to measure rotation of pinion and tubular shaft. And the TE formulais applied to the signals to calculate and plot TE. To measure rotation of pinion and ring gear markers are used.At locations B11 and B21 the rotation is measured with respect to a fixed ground marker. Then the TE formulais applied by subtracting both the signals to obtain TE. Requests are created for bearing accelerations, velocitiesand forces. Markers are created on the housing to measure the vibration at seven locations, see Fig.3.9. Amarker is created on the flexible body and several surrounding nodes are attached to it, such that an averagedresponse can be measured.

Figure 3.9: Marker positions where vibration is measured

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Setting solver parameters and output

The solver settings are critical, as they determine the accuracy of the predicted results and simulation time.ADAMS offer different solvers to solve the differential algebra equation for the multi body dynamic simulation.More knowledge about these can be found from [2] & [19]. For practical purposes it is not discussed in detail.HHT solver is used in this work. And it is observed that the displacements calculated are more accurate thanthe acceleration. So it is better to use the displacement signal to extract accelerations. The settings used inthis work can be seen in the Fig.3.10

Figure 3.10: Solver Settings

Short simulations can be run with interactive simulation, while long simulations are run from the commandprompt. The command call adams2018 ru-s filename.acf is used.

Analysis of results:

The analysis of measured signals is performed in the ADAMS post processor window. The tools to perform a2D and 3D FFT are used to convert time signal to frequency domain. Curve edit toolbar is used to performmath operations on curves.

3.2.3 Analysis in ADAMS

Now as the virtual prototype of the PTU is built, virtual tests can be performed on it. Two types of simulationsare performed in ADAMS. Firstly a dynamic analysis is performed with the intention to predict TE and housingvibration levels. Second type is a systems level linear modes analysis to predict un-damped natural frequencies.The conditions and test cycles used to simulate are discussed briefly in the following sections.

Dynamic Analysis

The test cycles are categorized as TE cycles and Speed cycles. In the TE cycles the TE is measured at aconstant speed and torque. So it is tested at 3 speeds and torque levels to look at TE at different speed and

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load conditions. There are three cycles at three speeds 60, 300 and 600 rpm respectively. For each speed, TEis measured at three torque levels 15,100 and -15 Nm. The test cycle is seen in Fig.3.11

(a) (b)

(c)

Figure 3.11: TE Test cycles 3 speeds

The speed cycle as seen in Fig.3.12 ramps speed from 0 to 2000 rpm, then remains constant for some timeafter which it coasts down to 0 rpm. Torque is maintained at 100 Nm throughout the cycle.

Figure 3.12: Speed Ramp

3.2.4 Linear Modes Analysis

Apart from transient dynamic simulation a Normal modes analysis and Forced response analysis is performed inADAMS to predict the modal parameters such a natural frequency and mode shapes as explained in sec.2.4.1.The vibration plugin is used which linearizes the model about a point in time and solves for the Eigen solution.Due to this approach the simulation is very fast.

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3.3 Physical Testing

The testing is carried out at on a newly installed test rig at the Physical Test department at GKN Automotive.The test rig is capable of testing an entire driveline configuration under several speed and load conditions. Inthis study the testing is focused on the PTU. The main purpose of testing is to collect relevant data to validateand tune the simulation model. The following paragraphs detail the testing setup and process followed.

To achieve the above mentioned purpose two types of testing were performed:

1. Test rig measurements

• Objective: Measure Transmission Error and housing vibration levels, by running different test cycles

2. Experimental modal analysis

• Objective: Perform an experimental modal analysis on the PTU while it is mounted in the test rig.Measure modal parameters such as natural frequencies, damping and mode shapes

3.3.1 Test Rig layout and measurement setup

Fig.3.13 depicts the test rig layout. The PTU is mounted in a configuration resembling the end of line(EOL)testing. EOL is the final NVH inspection performed on the assembly line. The housing is mounted onto afixture on the test bed using bolts. A motor which controls speed is connected to the pinion via companionflange and a motor which controls torque is connected to the input shaft on the crown gear side. The motors arecontrolled via a control system to run the desired test cycle. The outputs are measured via two TransmissionError(TE) modules and accelerometers mounted on the PTU. The TE module TE1 measures crown gear angulardisplacement and torque, while TE2 measure the pinion angular displacement. These measurements will beused to calculate the TE. The tri-axial accelerometers mounted on two positions measure the housing vibrationlevels in all the directions X, Y and Z. All the sensors are connected to a SIEMENS data acquisition system(SCADAS), which in turn is connected to a computer where the test setup is made. Simcenter Signature TestingAdvanced is the computer program used to setup the sensor channels and acquire data.

Figure 3.13: Test Rig and Measurement Setup Schematic

A list of the equipment used and the physical setup can be seen in tab.3.4 and fig.3.14 respectively.

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Table 3.4: Test Rig Equipment description

Equipment DescriptionAccelerometers(Acc1; Acc2)

2 Tri-axialAccelerometers

TE Module (TE1)1 Torque Transducer;

1 Speed EncoderTE Module (TE2) 1 Speed Encoder

Data Acquisition System 1 (Siemens SCADAS)Motor M1 Controlled in SpeedMotor M2 Controlled in Torque

Figure 3.14: Picture of PTU mounted in Test Rig

3.3.2 Experimental Modal Analysis

In this testing, the PTU is mounted in the same configuration as described above. However the test rig ison stand still. A modal analysis as described in sec.2.4 is performed. Roving hammer test is performed,which means that an impact hammer as seen in right of Fig.3.15 is used to excite the PTU and the responseis measured from several accelerometers mounted on the PTU. Four Accelerometers are used in this test tomeasure the response at different locations. Three are located on the PTU and one on the fixture. The one onfixture is to deduce any test rig related natural frequencies that might be present. The test plan and equipmentspecifications are described in the table below. For a more detailed description of EMA [16] can be referred.

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Figure 3.15: Accelerometer positions of PTU and Impact Hammer

19 points are excited in total. A crude geometry as seen in Fig.3.16 is built in the Simcenter software tovisualize measured mode shapes. Then the Modal Analysis plugin is used to analyze the measured FRF’.

Figure 3.16: Geometry Built in TestLab

3.4 Model Tuning

Before comparing the data from test with simulation a further check is performed to compare the mass of thephysical unit and calculation model. Since the calculation model contains only the major components there isa weight difference which can change the modal parameter in turn the noise vibration behaviour of the model.Tab.3.5 shows the weight difference.

Table 3.5: PTU weight (in numerical model/Physical unit with and without oil)

Weight (Kg)Simulation Model 9.4

Physical unit 15.38Physical unit

(w/o oil)15.08

The weight difference is mainly due to lack of weight of the bearings, lubricant, input shaft and other auxiliarycomponents such as seals bolts etc. An initial comparison between EMA and MBS based modal analysisrevealed that modal parameters differ by a significant value. Hence to cater to this, dead weights are added to

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the housing at the four bearing centers. These masses do not have any inertia and the weight is distributedas per the spread of weight in the PTU. The MBS model weight is made equal to the physical weight. Thisapproach is not the best, but still is a simple way of tuning the weight of the model. Further it is observedthat the modal parameters shift closer to values predicted from test hence validating the approach. Dampingfor individual components is estimated using the 3db method and the parameters are further tuned using testresults.

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4. RESULTS & DISCUSSIONIn this chapter the results measured from the physical testing and results predicted by the MBS model arepresented. The correlation between both is discussed and causes for deviations are explored. Then the resultsfrom bearing parameter study are presented giving insights into how bearing damping, preload could affect TEand housing vibrations.

4.1 Transmission error

The results are deduced from the TE test cycles as presented in sec.3.2.3. TE is calculated as differencebetween the theoretical angular rotation of the pinion (gear ratio times ring gear rotation) and the actualmeasured angular rotation of the pinion. In simulation it is calculated from the measures created, while in thetest it form the data collected from speed encoders.

4.1.1 Physical Testing Results

TE is presented in micro-radians. The table4.1 presents TE results measured for 60 rpm speed at 3 torquelevels. The TE measurements on test rig do not yield the same value every single time, the repeatability ispoor. This could be due to several factors arising from the PTU itself and the rig control. So each cycle is runseveral times and the spread of TE in the unit is measured. Then for comparison sake an average of the valuesis taken.

Table 4.1: TE measured from Test rig at 60 rpm (Pinion Speed)

TE (urad)1st Order (15 Hz)

Torque (Nm)@ Pinion

Test measuredrange

TestAverage

15 (4.5-20) 15.5100 (7.9-15.8) 8.3-15 (26-34) 29.1

TE at -15 Nm torque is the highest, while it is the lowest at 100 Nm. This behaviour seems reasonable becausethe high torque acting on the PTU increases the stiffness in the components, lowering the fluctuations.

4.1.2 MBS results

The same TE test cycles are simulated in the MBS model and TE is predicted. A comparison of the predictedresults to the measured in presented in Tab.4.2. It can be seen from the results that, the predicted values fallin the same range as measured from test. Trying to achieve a perfect co-relation of test and measured values isnot realistic, since in measurements there is a deviation each time TE is measured. Thus a co-relation wherethe predicted TE falls in the same range as measured is reasonable. And the similar trend of low TE at 100Nmand highest TE at -15Nm can also be seen which implies that the behaviour is accurate.

Table 4.2: TE predicted from simulation compared with test

TE (urad)1st Order (15 Hz)

Torque (Nm)@ Pinion

Test measuredrange

TestAverage

Predicted

15 (4.5-20) 15.5 18.9100 (7.9-15.8) 8.3 6.3-15 (26-34) 29.1 22

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A plot of the TE signal for the 60rpm test cycle is seen in Fig.4.1. Since we have a speed of 60 rpm and thenumber of teeth on pinion is 15, the TE peaks are seen at 15 and its multiples.

Figure 4.1: FFT of TE signal (TE cycle 1)

From correlation and tuning the model it is observed that the TE value is highly dependent on gear contactparameters that are input. Several parameter studies were performed to tune the values. Force exponent andpenetration depth are the main parameters which need to be tuned. Damping effects the smoothness of thesignal, however an increase in damping value increases the simulation time.

4.1.3 TE at different speeds

The TE predicted for speeds 300 and 600 rpm at three torque levels and a comparison of all three speeds can befound in the Tab.4.3. Since TE test data is not available for these speeds, the simulation predicted values arecompared. A general trend is seen that magnitude of TE is slightly increasing with increase in speed. Howeverfrom other tests performed at gear set level, the reverse has seem to be observed. That with increasing speed TEdecreases. This difference could be due to the fact that friction and lubrication in gear contact is not consideredin numerical model. While in reality the higher speeds mean lowered friction and better lubrication conditionin the gears.

Table 4.3: TE predicted at different speeds from simulation

Transmission Error @ Different SpeedsTE (urad) - 1st Order

Torque (Nm)@ Pinion

60 rpm(15 Hz)

300 rpm(75 Hz)

600 rpm(150 Hz)

15 18.9 22.7 23.8100 6.3 7.4 7.9-15 22 26.6 32.4

The reason TE cannot be easily measured at high speeds is due to rig limitations in-terms of torque and speedcontrol. This can be overcome by setting the optimum control parameters for the rig. And also there arelimitations on the speed encoders, as they are required to have a very high resolution to capture TE at highspeeds. In this scenario a simulation model is advantageous to test several variations of cycles.

4.2 Housing Vibration level

Vibration level indicates, for a given test cycle or operating condition of the PTU, what is the amplitude ofvibration of the housing. To test and measure this, the speed ramp cycle is run on the test rig and also simulatedin the MBS model. It is a speed ramp from 0 to 2000 rpm, then stays constant at 2000 rpm for some timeand then coasts down to 0 rpm. Torque is maintained at 100 Nm throughout the cycle. In the test Twoaccelerometers placed on the housing are used to measure the vibration on the housing surface. The positionswhere accelerometer are placed on the housing can be seen in Fig.4.2. Markers are created in the simulationmodel at same location to measure the predicted response.

The response from ramp portion and the constant speed portion of the signal are looked at separately. Thisis done because the vibrational tendency is different during both cases and analyzing them separately gives

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different insights. The ramp portion gives information of effect of gear mesh and also what natural frequenciesare excited during the run up. While in the constant speed portion the contribution to vibration is mainly dueto gear mesh frequencies and other bearing frequencies.

Figure 4.2: Accelerometer Positions on PTU for vibration measurement

4.2.1 Power Spectral Density (PSD)

The vibration is looked in-terms of a Power spectral density(PSD). It is a measure of the energy content of thevibration signal and shows the mean square acceleration per unit bandwidth. The average acceleration of thesignal at any frequency is defined by the shape of the PSD plot. So the time data measured is converted intoPSD curves. Results from both accelerometers are presented. Each plot has three curves X, Y, Z representingaccelerations measured in respective directions. The directions of the axes X, Y and Z with respect to PTU canbe seen in fig.4.2. In the following sections, for each case the measured and simulation predicted response arepresented in the form of plots and are compared. So in total, response at two accelerometers for two conditions(speed ramp and constant speed region) will be looked at.

4.2.2 Constant Speed region

Accelerometer 1- Const. Speed (2k rpm @100 Nm)

By comparison between measured response (see Fig.4.3) and simulation predicted response(see Fig.4.4) it canbe observed that the vibration profile shows a decent correlation. Firstly peaks are seen at 500, 1000, 1500 Hzwhich are the gear mesh frequency and harmonics for 2000 rpm speed of pinion. A higher magnitude is seenat 1000 and 1500 in both. However the amplitude in direction Y in test result is much higher. The amplitudesfor X direction lie in the same range of about 0.15. However on Z, from the test the vibrations are highlydamped compared to predicted values. This implies the damping needs to be tuned further to achieve a bettercorrelation.

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Measured:

Figure 4.3: From Test: PSD-Accelerometer 1 Constant Speed (2000 rpm)

Predicted:

Figure 4.4: From Simulation: PSD-Accelerometer 1 Constant Speed (2000 rpm)

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Accelerometer 2- Const. Speed (2k rpm @100 Nm)

The PSD response from test and simulation for the test cycle section can be seen in Fig.4.5 and Fig.4.6.

Measured:

Figure 4.5: From Test: PSD-Accelerometer 2 Constant Speed (2000 rpm)

Predicted:

Figure 4.6: From Simulation: PSD-Accelerometer 2 Constant Speed (2000 rpm)

Between the above two plots it is observed that a very high amplitude of vibration is observed in reality in Zdirection, while that is not the case in the simulation. However the maximum amplitude occurring is at 1000Hz, which is common to both cases. This could be the due to the effect of difference in bearing damping valuesor some modal stiffness differences in the model.

Accelerometer 1- Ramp Portion Speed(2k rpm @100 Nm)

From Fig.4.7 and Fig.4.8 it is observed that during a speed ramp peak vibration is seen around 1400 Hz. Againthere is a good correlation in the general behavior, but if the amplitudes are observed the simulation predicted

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values are very low which means the damping of the particular mode is higher when compared to reality.

Measured:

Figure 4.7: From Test: PSD-Accelerometer 1 Speed Ramp (0 to 2000 rpm)

Predicted:

Figure 4.8: From Simulation: PSD-Accelerometer 1 Ramp portion (0 to 2000 rpm)

Trials were made to tune damping accordingly but it was realized that a proper Design of experiments(DOE)is needed to decouple several contributing factors. And in the test plot, highest peak is in Y direction while insimulation the magnitude in Y at the corresponding frequency is low. The reason for this difference has notbeen understood.

Accelerometer 2- Ramp Portion Speed(2k rpm @100 Nm)

By comparing Fig.4.9 and Fig.4.10 which are the measured and predicted vibration response plots respectively,a single peak is seen at 1279 Hz in the measured signal which is not at all seen in the response predicted bysimulation. In the simulation predicted vibration, peaks are mainly at 500 and 1000 Hz corresponding to GMF.

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There could be two reasons for this deviation either the modal properties of the numerical model are not accurateor there could be a test rig related frequency.

Measured:

Figure 4.9: From Test: PSD-Accelerometer 2 Speed Ramp (0 to 2000 rpm)

Predicted:

Figure 4.10: From Simulation: PSD-Accelerometer 2 Ramp portion (0 to 2000 rpm)

To understand this difference the experimental modal analysis (EMA) results were looked at. These results arediscussed in the next section. However by looking at FRF’s from accelerometer placed on fixture, a test rigrelated natural frequency is observed at 1270 Hz. The plot of FRF’s from fixture accelerometer can be seen inFig.4.11 which justifies the difference, since it cannot be replicated in the current model.

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Figure 4.11: Peak at 1270 HZ- FRF’s from Accelerometer on Test rig fixture

4.3 Modal Analysis results

Tab.4.4 shows results from both experimental and numerical modal analysis. From the test and simulationmodel, several natural frequencies are calculated for the system in the range of 0 to 3000 Hz. Some seen intest aren’t predicted by simulation, while some observed in simulation aren’t seen in test. This could be dueto several reasons such as, not all modes could be excited in the EMA and the simulation model has someassumptions and compromises when it comes to support stiffness’s and damping, bolt preloading and also meshrefinement. However five natural frequencies and mode shapes under 3000 Hz show close correlation. Thisvalidates the simulation model, proving that it is reliable to predict the dynamics of the system, althoughdepending on level of accuracy required the detailing and assumptions in the model need to be handled.

Table 4.4: Comparison of measured and predicted natural frequencies

Natural Frequencies (Hz)

EMA TestSimulationPredicted

441 4441155 11821802 18002537 2549

The respective mode shapes of the system at these frequencies are shown in Fig.4.12. For EMA a crudegeometry is built it observe the deformation behaviour.

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Figure 4.12: Correlated Mode Shapes

4.4 Bearing Accelerations

Since the model is validated to a certain degree it can be used to study the effect of design changes andbearing properties on the effect of systems dynamics. Firstly, a study can be made to observe at which ofthe bearings is the highest vibration seen. The speed ramp cycle is run and results are presented in a plot inFig.4.13. Accelerations in direction Z are plotted for all bearings. Tubular shaft bearings B21 and B22 seemto have higher vibrational levels than pinion bearings. And depending on requirements, if needed the bearingparameters can be tuned to achieve optimum behavior.

Figure 4.13: Comparison of vibration at different bearings

4.5 Effect of bearing parameters of PTU dynamics

As discussed in sec.2.3.4, the important bearing parameters which affect system dynamics are its stiffness(dependant on preload), damping and internal clearance. In this work the dynamic model is used to observethe change in dynamic behaviour as result of change in bearing preload and damping. Three cases with low,standard and high were run for each parameter separately.

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4.5.1 Impact of bearing Damping

Bearing damping is an important property which effects what amount of vibration actually flows through thebearings into the housing. Bearings offer damping in three directions: radial, axial and bending(torsional). Arotational damping is present as well in the form of friction torque. In this case a single value of damping isassigned to all three damping factors. And the effect of damping on housing vibration is tested for three levels.The values for three levels are listed in Tab.4.5.

The test cycle is a speed ramp from 0 to 2000 rpm in 7 seconds at a torque of 100 Nm. The vibration fromall three cases are measured and plotted as PSD curve in Fig.4.14. From the plot it can be seen that higherdamping reduces the vibration transferred by approximately 55 % at first order GMF. However for low andstandard damping the behaviour is almost similar. Here it has to be noted that the high value 0.5 that istested is unreasonable to achieve in reality for TRBs. However it is tested to demonstrate the effect of increaseddamping. Fluid film bearings and other types, offer higher damping relatively and more about it is discussedin [12].

Table 4.5: Levels of Bearing Damping tested

Tested Levels Damping RatioStandard 0.001

High 0.5Low 0.00001

Figure 4.14: PSD comparison showing effect of bearing damping on housing vibration

4.5.2 Impact of bearing preload

Bearings are preloaded to eliminate the play between the rings and rollers. For tapered roller bearing it canbe specified as an offset between inner and outer ring. Higher the offset, higher is the preload. Three levels ofpreload were tested as presented in Tab.4.6. The test cycle is a speed ramp from 0 to 2000 rpm in 7 seconds.The vibrations on the housing are measured for the three cases and are plotted in Fig.4.15.

From the plot it can be observed that for a low preload the vibration energy transmitted to housing increasesdrastically at GMF. However the standard and high preload shows pretty much the same vibration transmission.This intends that an optimized value of bearing preload was already chosen. It also indicated that blindlyincreasing the preload to a very high level does not guarantee a reduction in vibration. And a very high preloadis often not good for the bearing as well. Thus an optimum value from a bearing operational perspective andalso a Noise, vibration perspective needs to be set. And this tool could be useful in testing if a selected preloadvalue serves the purpose.

Table 4.6: Levels of Bearing Preload tested

BearingsPreload Levels(um) B11 B12 B21 B22

Standard 25.5 15.5 41 51High 40.5 30.5 56 66Low 10.5 0.5 26 36

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Figure 4.15: PSD comparison showing effect of bearing preload on housing vibration

4.5.3 Impact of preload on TE

Preload changes the stiffness of the bearing and thus in-turn affects the meshing point in gear contact. Theeffect of different preload on TE is tested by running the 60 rpm TE cycle for three levels of preload as describedin above section. The results are presented as a bar graph in Fig.4.16. The variation at three different torquelevels is plotted. At each torque level three levels of preload high, low and standard are tested. From the results,it is observed that TE varies slightly as preload changes for all three torque levels. At 15 Nm, for a low aswell as higher preload level TE increases compared to the standard preload. Which indicates an optimum valuefrom a preload perspective is already set. On the 100 Nm and -15 Nm case, although the change is small, thetrend is an increase in preload shows decrease in TE.

Figure 4.16: Bar graph showing effect of Bearing preload on TE (3 torque levels)

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5. CONCLUSIONSIn the beginning of the work research questions were defined to serve as a guide post. And in the end the answerto these questions was the main conclusion. The discoveries and learning’s are discussed.

5.1 Discussion

The developed model can be used to predict TE and housing vibrations. Since, the predicted results correlatewell with test results and reasonable trends are observed, the model can be used and developed further withconfidence. As seen in chapter 4, there are deviations when comparing the results for TE and housing vibrations.In the TE measurement from test rig, the repeatability is not very accurate, this could be due to a multitudeof factors including small variations in bearing preload, test rig setup and control parameters. A better controlof the rig itself and other noise parameters in needed to ensure a more accurate measurement. Sensors withhigher resolution can be used to measure TE at high speed, such that results can be compared with simulation.And in the numerical model the time step resolution can be increased to further understand the high frequencycontent of TE, which is mainly due to gear tooth imperfections and is the cause for gear whine. In the predictedTE, for the 60 rpm cycle results (see Tab.4.1) it can be observed that on low torque TE is underestimatedwhile on high torque it is overestimated slightly when compared with Test average. The reason for this need tobe understood to see if the contact parameters need more tuning.

The modal results match well, with five modes correlating between test and simulation under 3000 Hz. However,some modes do not correlate, to address this a further detailed EMA with a greater number of points anddirections could be done to excite all modes of the structure. And for the numerical model, the stiffness andmass need to be tuned. So, taking further steps to improve mesh refinement, include bolt preload beforeimporting into ADAMS and distributing the mass more evenly needs to be considered. Overall, a degree ofcorrectness is observed, thus the model can be used with confidence to perform a dynamic analysis to predictthe TE and dynamic response of the PTU.

5.2 Conclusions

• How can we represent a power transfer unit with an integrated MBS-FEM model that considers thebearing nonlinearities to predict transmission error and vibration levels?

– The method to develop a dynamic model using the integrated approach is shown in Sec.3.2. Firstdevelopment of flex bodies is discussed which are then assembled in the MBS environment. Bearingnon-linearities are considered using an analytical approach where the stiffness and damping in thebearings are calculated as a function of conditions in the system. The developed model is capable ofpredicting the transmission error and housing vibration levels.

• How can a gear set be designed with preliminary data to predict TE?

– Trials were made to develop gears directly in the MBS environment by using A/Machinery moduleby inputting the gear geometrical details. However it was noticed that it is suitable for rigid gearformulation. The conversion of those rigid bodies into flexible bodies is not yet supported by ADAMSdue to limited meshing capabilities. Thus an external tool is needed to develop the flexible gearbodies.

• How can we include gear the effects of gear imperfections in the dynamic analysis?

– Firstly a fine mesh is required on the gear teeth surface to capture the geometry. Then the resolutionof simulation time step can be increased to about thousandth of a second or more to capture thehigh frequency content of the measured signal. If needed a specific imperfection can be created inthe CAD geometry to understand its effect on the vibrations.

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• How can we include the effects of bearing nonlinearities in a MBS model?

– Out of other options, ADAMS Machinery Bearing module was used to take into consideration thebearing non-linearites in the MBS model. It is basically modelled as a six-component force, andthe instantaneous stiffness and damping for the bearing are calculated based on loads and operatingconditions in the model. Stiffness is calculated as described in Sec.2.5.5 and damping is deducedas square root of instantaneous stiffness. This method gives a simple yet effective representation ofthe bearings in the model. However, if focus is on in-depth bearing analysis then another approachneeds to be followed.

• What are the effects of bearing properties such as stiffness, damping, and preload on the TE, vibrationlevels?

– The plots describing the effects of bearing preload and damping on TE and vibration are presentedin sec.4.5. It has been observed that bearing preload significantly affects the magnitude of vibrationtransferred. From the study it has been observed that low preload could result in unintended vibrationin the PTU. However a very high preload does not mean vibration is reduced further. Results indicatean optimum threshold value should be set, where vibration in minimum and preload is not too high aswell. And regarding the effect on TE, a general trend has been identified that higher preload tends tolower TE. However the change is very small. And damping tests indicate higher the damping betterit is to reduce vibration.

• How can we correlate MBS model results with physical test data?

– Two approaches were taken in this work to validate the model and correlate the results. Physicaltests were run on a test rig and TE, housing vibration were measured. The measured values wereused to tune the contact and damping parameters in the MBS model. The TE predicted falls in thesame range and the vibrational levels show a similar tendency with some deviation. Some reasonsfor the deviation were found while the other still need to be explored. And an Experimental ModalAnalysis was performed to compare the natural frequencies and mode shapes. Comparing both testresults with the numerical model a good correlation was seen on all levels. A very good correlationof five natural frequencies and mode shapes is observed below 3000 Hz. These correlations renderthe validity of the simulation model. So it can be used with some level of confidence to predict theTE and study the dynamic behaviour of the system under different conditions.

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6. FUTURE WORKThere is ample scope for improvement and refinement of the model in order to increase the degree of usefulnessand reliability.

Depending on the motive to use the model it can be detailed in several directions.

• The ADAMS model can be linked with an Acoustics calculations tool to predict noise emitted and alsosee vibration transfer to car body.

• The RDU which contains more components can be modelled and tested to see the method’s adaptability.

• Clutch components can be added as rigid bodies to give a more realistic mass distribution and then therewill be a possibility to test other connect/disconnect test cycles.

• In the current work a manual comparison of modes and mode shapes was done and good correlation isseen. However to increase the confidence on validation, a Modal Assurance Criterion (MAC) study wheremeasured and predicted modes are compared mathematically can be performed.

• Depending on if impact of bearing micro-geometry on dynamic behavior is important then ADVANCEDBearing AT can be used to model the bearings.

• Test data is available and the tunable parameters are identified. A Design of Experiment (DOE) canbe performed to tune the contact parameters, bearing damping, component damping to a much betterdegree.

• For future testing a better PID control on the test rig is suggested for low torque holding.

• Bolt preload can be included, a method can be followed by preloading the bodies initially in an FE toolbefore importing to MBS.

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