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Dynamic modelling of a Power Transfer

Unit of All-Wheel Drive vehicle in a 1-D

simulation environment

Shivanand Ambalavanan

Master of Science Thesis TRITA-ITM-EX 2018:367

KTH Industrial Engineering and Management Machine Design

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Master of Science Thesis TRITA - ITM - EX 2018:367

Dynamisk 1-D modell av en kraftenhet till en bil med allhjulsdrift

Shivanand Ambalavanan

Approved Examiner Supervisor

2018-06-11 Ulf Sellgren Stefan Bj¨orklund

Commisioner Contact Person

GKN Driveline K¨oping AB Eva Lundberg

Sammanfattning

En Power Transfer Unit (PTU) eller vinkelv¨axel i ett drivsystem f¨or allhjulsdrift ¨ar en v¨axell˚ada med en hypoid-v¨axel som drevsats. PTUn ¨ar placerad mellan fordonets trans-mission och kardanaxel och anv¨ands f¨or att f¨ordela momentet fr˚an drivsystemet mellan alla hjulen. De dynamiska egenskaperna hos vinkelv¨axeln ¨ar kopplade till de ljud och vibrationer som uppfattas i bilen, speciellt tonalt ljud som v¨axelvin. K¨allan till denna vibration kan relateras till transmissionsfelet i v¨axeln. Transmissionsfelet beror p˚a fak-torer som geometri, rotationshastighet och statiskt moment. Om fakfak-torernas inverkan kan identifieras skapar det m¨ojligheter att reducera felet genom designf¨or¨andringar. 1D-eller system-simulering ¨ar en f¨orenklad beskrivning av det dynamiska beteendet av sys-temet. Det ¨ar en flexibel metod som kan ge en uppskattning av systemets egenskaper i ett tidigt skede och kan anv¨andas i s˚av¨al tids- som frekvensdom¨onen.

Denna studie syftar till att bygga en 1-D system-modell av en PTU och studera dess dynamiska beteende. De typer av analyser och resultat som ¨ar m¨ojliga att f˚a fr˚an en dy-namisk 1-D modell av en specific produkt har utv¨arderats. Befintliga komponenter fr˚an mjukvarans bibliotek har anv¨ands f¨or att bygga en f¨orenklad modell med lumpade mas-sor av den fysiska systemet. Simuleringar har utf¨orts b˚ade i tidsdom¨anen och frekvens-dom¨anen.

System-modellen ¨ar mycket anv¨andbar f¨or modelling av hela system och av hur delarna v¨axelverkar i ett tidigt skede av produktutvecklingen. Ber¨akningen av niv˚an p˚a transmis-sionsfelets grundtonen st¨ammer v¨al med tillg¨angliga m¨atresultat. Rotationshastigheter-nas variation d˚a kopplingen kopplar i och ur vinkelv¨axeln illustrerar tydligt kopplingens inverkan p˚a dynamiken i systemet. Det var dessutom m¨ojligt att erh˚alla systemets tor-sionsegenfrekvenser och modformer fr˚an den linj¨ara frekvensanalysen.

Nyckelord: 1-D simulering, systemsimulering, LMS AMESim, transmissionsfel, hypoidv¨axel, vinkelv¨axel

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Master of Science Thesis TRITA - ITM - EX 2018:367

Dynamic modelling of a Power Transfer Unit of All Wheel Drive vehicle in 1-D simulation environment

Shivanand Ambalavanan

Approved Examiner Supervisor

2018-06-11 Ulf Sellgren Stefan Bj¨orklund

Commissioner Contact Person

GKN Driveline K¨oping AB Eva Lundberg

Abstract

A Power Transfer Unit (PTU) of an All-Wheel Drive system is a hypoid gearbox which is a driveline component, used to distribute power from the powertrain to all the wheels of a vehicle. The gearbox dynamics is closely related to the gearbox noise and vibration, especially tonal noise like gear whine. The source of this vibration is referred to as the transmission error in the unit. Transmission error is attributed to various geometrical and operating conditions, which if mapped mathematically, allows the designer to reduce the error by varying the design parameters. The demand in the automotive industry to reduce time to market is high. A lot of time can be saved if system performance can be assessed at the concept stage, even before the detailed design. This is where system-level simulation plays a key role. 1-D or system simulation technique studies the dynamic behaviour of the system in one dimension. This greatly simplifies the model and allows for the flexibility to get early estimates of the system behaviour with respect to time and frequency. Here, such a system model is built for a hypoid gear based driveline system. This work aims to build a 1-D system model of the PTU and study the dynamic behaviour of the response. The evaluation helps in understanding the capabilities on 1-D system model in simulating a specialised product dynamic characteristics. LMS AMESim was the commercial tool used in building the system model. Existing components from the software library were used to build a sketch of a simplified, lumped mass model of the physical system. The model was then simulated in both the time domain and frequency domain through a temporal and linear analysis respectively.

It is observed that the system model is very useful in early modelling of a system and its interactive effects. The fundamental harmonic of the transmission error is predicted well in the system model. The clutch connect/disconnect behaviour can also be seen in the rotary velocity response of the gear. The system eigenfrequencies and mode shapes were obtained from the linear analysis.

Keywords: 1-D simulation, System simulation, LMS AMESim, Transmission Error, Hypoid gear, Power Transfer Unit

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Acknowledgements

The research for this thesis work was carried out from January 2018 to May 2018, in partial fullfillment of the Master of Science degree in Engineering Design at KTH Royal Institute of Technology, Stockholm. I would like to thank KTH and GKN Driveline K¨oping AB, the two institutions involved in supporting this work.

I cannot thank enough, Magnus L¨ofberg, Manager of NVH and Vehicle Engineering at GKN Driveline, for giving me this opportunity to work on a very interesting and challenging thesis work. My supervisors at GKN Driveline, Eva Lundberg and Stefano Orzi, have been a pillar of strength and support, guiding me throughout this thesis with their invaluable insight and knowledge in this topic and I cannot thank them enough. Marco Schwab at GKN Driveline Germany has been very supportive and helped me with the software and I would like to thank him for his help. The Calculation, Test, Design and Vehicle Integration teams have been prompt with information whenever I needed it, especially in a thesis where extensive cross-functional data was required. I would like to extend my sincere gratitude to all of them.

Stefan Bj¨orklund, my supervisor at KTH, has been a constant support throughout the thesis work and I would like to thank him for visiting me at GKN Driveline in K¨oping and giving me his valuable inputs. I would also like to thank Ulf Sellgren, my examiner at KTH, for providing timely support with the thesis and giving his insights whenever needed.

I would like to thank my parents, Ambalavanan and Senthilnayaki, who are the reason for whatever I have achieved in my life. My sister Shivapriya and friends have been patiently putting up with me through my stressful days and I cannot thank them enough for their support. Finally, all praises to God!

Shivanand Ambalavanan

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Contents

Sammanfattning iii Abstract v Acknowledgements vii Nomenclature xi 1 Introduction 1 1.1 Background . . . 1 1.2 Purpose . . . 2 1.3 Delimitation . . . 3 1.4 Methodology . . . 4 2 Frame of Reference 5 2.1 Hypoid gears . . . 5

2.1.1 Bevel Gear Geometry . . . 6

2.2 Transmission Error . . . 8

2.2.1 Sources of Transmission Error . . . 9

2.2.2 Transmission Error and Gear Whine . . . 10

2.2.3 Measuring Transmission Error . . . 11

2.3 System Simulation . . . 12

2.3.1 System-level Modelling . . . 13

2.3.2 System Linearization . . . 14

2.4 System Modelling in LMS AMESim . . . 16

2.4.1 Submodels . . . 17

2.4.2 Libraries . . . 17

2.4.3 Numerical Solver . . . 17

2.4.4 Causality . . . 18

2.5 Driveline Dynamics . . . 18

2.5.1 Rigid multi-body simulation . . . 19

2.5.2 Finite element simulation . . . 19

2.5.3 Flexible multi-body technique . . . 19

3 Implementation 21 3.1 Model Simplification . . . 21

3.2 Sketch Building . . . 22

3.2.1 Bevel Gear . . . 22

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3.2.3 Physical Model . . . 25

3.3 Parameterization . . . 26

3.3.1 Bevel Gear Parameters . . . 26

3.3.2 Dog Clutch Parameters . . . 27

3.3.3 Inertia and Compliance Parameters . . . 27

3.4 Simulation . . . 28

3.4.1 Temporal Analysis . . . 29

3.4.2 Linear Analysis . . . 30

4 Results and Discussion 31 4.1 Transmission error comparison . . . 31

4.2 Clutch Transient Behaviour . . . 32

4.3 Linear Analysis Results . . . 33

5 Conclusion and Future Work 37 5.1 Conclusion . . . 37

5.2 Future Scope of Work . . . 38

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Nomenclature

ABBREVIATIONS

1-D One dimension

AWD All-Wheel Drive

CAD Computer Aided Design

DAE Differential Algebraic Equations

FFT Fast Fourier Transform

GUI Graphical User Interface

ODE Ordinary Differential Equation

PTU Power Transfer Unit

RDU Rear Drive Unit

TE Transmission Error

NOTATIONS

θP,G Angular displacement of pinion/gear

g Acceleration due to gravity

i Gear transmission ratio

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Chapter 1

Introduction

Gears are an integral part of power transmission systems, used in a wide range of applica-tions from simple toys to complex aircraft. The compact nature of geared transmissions and their high efficiency make them popular in engineered systems. However, just like most mechanical systems, gears are associated with mechanical losses, noise and vibra-tions. Gear efficiency and vibration are a major challenge for geared systems, and engi-neers are entrusted with improving gear design to maximise the efficiency and minimise the vibrations. As the demand for cleaner and quieter products increases across the globe, gear dynamics has become a focus area among researchers. Therefore, a lot of resources and effort is being diverted towards gear study, in order to understand gear dynamics and the associated vibrations. Modelling the dynamic behaviour of geared transmission systems accurately is the key to understanding the relationship between gear noise and dynamics, as well as determining ways to reduce the vibrations. This work is one such attempt at modelling a geared transmission system in a one-dimensional (1-D), system level, simulation environment.

This chapter familiarises the reader with the background, purpose and methodology adopted in this work.

1.1

Background

GKN Driveline K¨oping AB, a part of the GKN Plc. group, is involved in the design, engineering and manufacturing of driveline components for All-Wheel Drive (AWD) ve-hicles. The main purpose of an AWD driveline is to transfer power from the powertrain to opposite end of the vehicle, distributing torque between the front and rear wheels, depending on the traction requirements on each wheel. One such AWD system drive-line can be seen in figure 1.1. The main components of an AWD system can be summarised as:

• Power Transfer Unit (PTU) • Propeller Shaft

• Rear Drive Unit (RDU)

GKN Driveline K¨oping AB specialises in the engineering and production of PTU and RDU for a wide range of automobiles. The PTU and RDU are gearboxes which use a hypoid gearset to transmit power from the vehicle transmission gearbox to the rear

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wheels. The PTU is coupled with the primary transmission of the vehicle and transmits power to the RDU, through the propeller shaft. The RDU distributes the power to the rear wheels. Various types of PTU and RDU are available, depending on the customer requirements. In this work, one such PTU, with a Connect/Disconnect capability, will be modelled and studied. A Connect/Disconnect PTU uses a clutch to disconnect the

Figure 1.1: An AWD system (Source: GKN Driveline)

propeller shaft and thereby the RDU, when there is no need to power all four wheels of the vehicle. This feature is very useful when the vehicle is coasting on highways and there is no need for torque in rear wheels. Disconnecting the power to the rear wheels via the propeller shaft leads to lower fuel consumption and better efficiency. The rear wheels can be powered only during off-road conditions or other such road conditions when there is a traction demand. Figure 1.2 shows the cross-section of the PTU which is to be modelled in this thesis work. The various components of this PTU are as follows:

• Hypoid Pinion • Hypoid Ring Gear • Tubular shaft

• Dog Clutch (Connect/Disconnect) • Input Shaft

• Output Flange

These are the components which will be modelled and studied in this work, in order to study the dynamic behaviour of the overall system.

1.2

Purpose

In order to study the dynamic behaviour of the PTU, it is important to build an appro-priate mathematical model of the system and its response at various torque and speed.

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Figure 1.2: PTU with Connect/Disconnect capability (Source: GKN Driveline)

One-dimensional (1-D) simulation or system level simulation is an essential method of multi-physical system analysis. The purpose of this work is to model the PTU in a 1-D simulation environment and study its dynamic behaviour.

The fundamental research question for this thesis can be stated as:

“How can 1-D simulation be used to build a dynamic model of a Power Trans-fer Unit?”

This fundamental research question can be further decomposed into the following sub-questions:

• What is the source of excitation for gearbox vibrations?

• How does hypoid gear geometry affect the source of excitation?

• How can we model the various components of the PTU in 1-D simulation environ-ment?

• How is the contact in a gearset modelled in a 1-D model? • How is the dog clutch modelled in 1-D model?

• What is the torsional vibration response of the system?

• What are the eigen frequencies and torsional mode shapes of the various components in the system?

• How do the various response parameters obtained through the model correlate with the experimental data?

1.3

Delimitation

This work does not aim to find solutions to decrease gear whine or gear rattle in the PTU. Gear whine is the result of complex gear mesh contact behaviour and involves

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various physical mechanisms which contribute to the transmission error. Various studies have been carried out in this domain and this work only refers to those studies for better understanding of the gear mesh details.

This work doesn’t involve component level study of modal response characteristics but is a system level model to study the vibratory response.

Considering this is a 1-D model, only torsional vibratory behaviour of the system has been studied. The model has used various relevant approximations where necessary due to software limitations and data constraints.

The effect of bearing stiffness and transfer of vibrations to gearbox housing isn’t relevant to this study.

The dog clutch is used to study the transient effect on the torsional dynamics during its engagement and disengagement. Therefore, the clutch control in the model has been simplified drastically and is not of interest in this work.

1.4

Methodology

In order to answer the fundamental research question, the basic methodology used was to model the system in a 1-D environment and study the torsional vibration response. LMS AMESim is a commercially available system level simulation software package from Siemens PLM. AMESim has a wide range of validated submodels for physical components, which can be used to build a physical system. It is widely used in the automobile industry and was used for this thesis work. The main tasks involved in this work are:

1. Modelling the gearset level component in LMS AMESim

2. Validating the gearset level response with experimental data of static transmission error

3. Modelling the dog clutch in LMS AMESim

4. Modelling the overall drive system in LMS AMESim

5. Torsional vibration analysis of the system model

Extensive literature study to understand transmission error, gear vibrations, modelling of hypoid transmission error, 1-D simulation and usage of LMS AMESim was carried out to understand the physics behind the gear dynamic behaviour and build a model as accurate as possible.

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Chapter 2

Frame of Reference

This chapter provides a background to several topics and certain key terminology, which are used extensively through out the course of this work. The chapter introduces hypoid gears and it’s gear kinematics. Transmission error in gears is explained in detail followed by the basic concept of 1-D simulation and its benefits. The last section of this chapter discusses the available research in dynamic modelling of driveline.

2.1

Hypoid gears

Hypoid gears are non-parallel, conical gears with non-intersecting axes, used to transmit rotary motion, especially in automotive driveline components. They are primarily used to transmit power from the vehicle transmission to the opposite end of the vehicle and also to distribute the power to the two wheels at the opposite end. The main difference between a hypoid gear and a bevel gear is the non-intersecting axes i.e. an offset of the pinion shaft, as shown in figure 2.1. Thus, hypoid gears can be considered as a type of

Figure 2.1: Hypoid gears - Offset Axis [10]

bevel gear with different gear kinematics and tooth profile. The major reason for the popularity of hypoid gears over bevel gears in automobile application is the lower noise generation due to increased sliding contact and the flexibility to lower the propeller shaft of the vehicle due to the offset, thereby allowing more cabin room.

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2.1.1

Bevel Gear Geometry

Since hypoid gears are a type of bevel gears with the offset, most of the geometry defined for bevel gears applies for hypoid gears as well. Bevel gear geometry, unlike cylindrical gears, has a constantly changing macro geometry along the face width, due its conical shape. This makes it harder to define the geometry of bevel gears, when compared to cylindrical gears. The ISO 23509 standard ”Bevel and Hypoid Gear Geometry” stan-dardises the geometry definition. The name ”bevel gear” is used as a generic name to represent all the various types like spiral bevel and hypoid gears. The non-offset bevel gear pair can be represented by two cones, which roll over each other without any sliding, called as pitch cones. Figure 2.2 shows a section of a bevel gear pair with all the relevant nomenclature.

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A plane passing perpendicular to the axial plane and to the pitch plane, at the pitch point (shown in figure 2.3) is called the transverse plane. A few important geometry, defined in the transverse section are shown in figure 2.4.

Figure 2.3: Transverse and Normal Plane at Pitch Point [3]

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In addition, hypoid gears have an offset axis which needs to be defined by an extra set of parameters. These can be seen in figure 2.5. A few of the important geometry parameters, relevant to this thesis, are defined as follows:

• Face width - length of the teeth measured along the pitch cone element [2] • Mean pitch diameter - the mean diameter of the pitch cone circles in the case

of non-offset bevel gears

• Pitch angle - the angle between the pitch cone surface and the axis of gear/pinion • Generating Plane Gear - an imaginary plane gear, which rotates between the

pinion and gear, tangent to both pitch cones

• Spiral angle/Helix angle - the angle between the tangent to any point along the tooth flank and the line of tangent point to the apex of pitch angle, defined in the Generating Plane Gear

Figure 2.5: Hypoid Gear Nomenclature [2]

2.2

Transmission Error

An ideal gear would be characterised as perfectly rigid, with no geometrical modifications or errors during manufacturing. A pure rolling at contact, along with an assumption of no friction would mean no fluctuation in the forces which thereby would imply that gears would transmit torque perfectly and continuously at a constant gear ratio. However, in reality, non-constant stiffness, geometrical errors, intentional modification of tooth profile, friction etc. lead to a non-constant velocity ratio, which thereby leads to variations in the torque transmitted and thus vibrations are induced in the system. This excitation

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mechanism is considered a major source of gear noise and vibration and is referred to as transmission error (TE).

Transmission error was defined by Welbourn [16] as “the difference between the actual position of the output gear and the position it would occupy if the gear drive were perfectly conjugate.” A perfectly conjugate gear drive would have a constant velocity ratio. For such a conjugate gear pair, the relationship between the pinion and gear angular displacements will be given by

θP = i × θG

where,

θP = angular displacement of pinion,

θG= angular displacement of gear, and

i= gear ratio.

This can be demonstrated through a visualisation of a pair of cylindrical gears, as shown in Figure 2.6.

Figure 2.6: Transmission error definition [6]

Mathematically, transmission error can be denoted as:

T E = θG−

ZP

ZG

.θP (2.1)

where,

ZP= number of teeth on the pinion, and

ZG= number of teeth on the gear.

Transmission error can be expressed as angular displacements or linear displacements. Depending on the operating conditions, they can be further classified as shown in table 2.1

2.2.1

Sources of Transmission Error

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Akerblom [1], in his detailed literature survey of gear noise, classifies the cause of trans-mission error in to three main categories:

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Low Load High Load Low Speed Static Unloaded Static Loaded High Speed Dynamic Unloaded Dynamic Loaded

Table 2.1: Types of Transmission Error [1]

• Deflections

– Contact deformations in gear mesh – Gear teeth bending deflections – Gear blank deflections

– Shaft deflections

– Bearing and gearbox casing flexibility • Geometrical Errors

– Involute alignment deviations – Involute form deviations – Pitch errors

– Run-out

– Error in bearing position in the casing • Geometrical Modifications

– Lead crowning

– Helix angle modification – Profile crowning

– Tip relief and root relief

As it can be seen, there are various causes of transmission error in gears and it can be difficult to isolate a particular cause. Deflections occur due to the compliance in the com-ponents while geometrical errors occur mainly due to manufacturing errors. Geometrical modifications are intentional changes to the desired gear profile for smooth engagement and disengagement of gears. Figure 2.7 is an example of a typical transmission error signal with its components.

2.2.2

Transmission Error and Gear Whine

Gear noise can be categorised into different types based on the source of the noise and its acoustic nature. One such distinct gear noise is known as gear whine and it is emitted from the gears in mesh. Vibrations characterising gear whine have been observed to have the same frequency as the gear mesh frequency and its integer multiples [13]. This leads to periodicity in the waveform and is perceived as tonal noise to human ears. Tonal noises are known to be more annoying to human ears than random noise characteristics and therefore it is important to reduce gear whine in transmission systems.

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Figure 2.7: Typical TE signal with its components [1]

direction or position[14]. Depending on the gear profile, one of the variation predominates. In case of involute profiles, the variation in force occurs due to variation in the drive smoothness, which in turn in is due to variation of the profile from a true involute and varying elastic deflection of the teeth, which are all the causes of transmission error. This vibration created at the gear contact is then transferred to the housing through the shafts and bearing, thereby leading to airborne noise. Transmission error is identified as a major source of gear noise and particularly gear whine [8] [4]. Therefore, it is desirable to have low transmission error in the gear drive. In this work, transmission error is being used a measure of validity of the model built.

Figure 2.8: Typical TE measurement setup [1]

2.2.3

Measuring Transmission Error

Transmission error is generally measured using optical encoders. Having a pair of en-coders, one on each shaft, allows the measurement of the difference in angular displace-ment. A typical error measuring setup can be seen in figure 2.8. This difference in angular displacement of the two gear shafts, as defined in equation 2.1 gives us the trans-mission error in angular units. In order to display the error in linear units, the angular

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displacements are multiplied by the base circle radius or pitch circle radius. It is more interesting to study the frequency content of the transmission error signal than the raw signal. Therefore, the signal is transferred to the frequency domain using a FFT (Fast Fourier Transform). The tooth mesh frequencies and its harmonics are usually the source of gear vibrations [15]. Therefore, the TE amplitudes at these frequencies are most com-monly used for TE evaluation in gear sets. The TE values at the gear mesh frequency and its harmonics will be used for the validation of the model built in this work.

2.3

System Simulation

System simulation involves simulating a physical system and analysing its behaviour. It is a part of the broader system engineering discipline, which aims to model, analyse and simulate physical systems. The first step in system simulation is to create a mathemat-ical model of the physmathemat-ical system using existing physmathemat-ical laws. This model is a set of differential equations, which represent the physics of the system. Solving these equations at various instants of time gives the response of the system with respect to time. Thus, system engineering allows the following study to be carried out [5]:

• Study the transient response of the system for varying inputs • Study the stability and steady response of the system

• Modify the given system to behave in the desired manner using control mechanism

Figure 2.9: System V-cycle representation [7]

With an increasing demand in the automotive industry to reduce the time to market, a lot of impetus is placed on reducing the dependence on physical tests with calculations and by identification of system performance at an early stage. As can be seen in figure 2.9, modern products are becoming increasingly multi-disciplinary and a tool which can

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predict the system behaviour earlier in the product life cycle will significantly reduce the design iteration process. System simulation bridges this gap between concept and prototype since it does not require detailed design information, but gives a good estimate of the system response.

A system can be classified based on various factors like its dependence on previous states, nature of magnitudes or control arrangement into static/dynamic, continuous/discrete or open/closed loop respectively. Further, a dynamic system can also be considered as a process that responds to certain inputs and gives corresponding outputs. For continuous systems, this can be represented by figure 2.10. Such systems can be described as a black box, external description or as a system state, internal description [5]. The system state representation defines the dynamic system by a minimum set of variables, known as state variables, such that the knowledge of the initial values of state variables, along with the inputs, determines the system behaviour completely at any instant of time. This is the fundamental principle used in system modelling of a physical system to predict system behaviour.

Figure 2.10: Representation of a dynamic system model [5]

2.3.1

System-level Modelling

A system model is a simplified representation of the physical system, which is used to analyse the system behaviour. It can be a mental model with abstract description of the functioning or a mathematical model with detailed equations, graphs and diagrams, derived form the mental model. These equations, when built in a computer, gives a computer model of the system, which mimics the behaviour of the system and its rela-tionships. System models can be classified as [5]:

1. Linear/Non-linear - depending on whether the principle of superposition is appli-cable or not to the system dynamic equations

2. Distributed/Lumped - depending on the dependence of the variables on spatial coordinates

3. Time-varying/Stationary - depending on variation of system model parameters in time

4. Continuous/Discrete - depending on the continuous or discrete range definition of variables

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The method of developing a system model of a multi-physical system is by decomposing the overall system into simple systems and creating dynamic equations for each sub-system with an interface between the various sub-sub-systems. generalised variables are considered for this purpose where the variable flow (f ) in a dynamic system is caused due to a variable effort (e) to transfer the variable power (P ), which is the product of both variables. The variable displacement (q) is defined as the integral of flow while the variable momentum (p) is defined as the integral of effort. These relations can be represented as a tetrahedron, as shown in figure 2.11. The linkages between the generalised elements gives

Figure 2.11: Generalised variables used in system modelling [5]

rise to the concept of resistance (R), capacitance (C ) and inertia (L), which characterise the system components. Every component in a system model would be characterised as one of the above mentioned categories and thus have a characteristic behaviour. This would apply to any component from the mechanical, electrical, hydraulic or thermal domain. Table 2.2 gives the general non-linear as well as linear relationship between the generalised variables in direct and inverse form. In order to obtain the dynamic equations

Table 2.2: Relation between generalised variables [5]

General Relation Linear Relation Resistive Elements e = FR(f ); f = FR−1(e) e = Rf ; f = e/R = Ge

Capacitive Elements q = FC(e); e = Fc−1(q) q = Ce; e = q/C

Inertia Elements p = FI(f ); f = FI−1(p) p = If ; f = p/I

pertaining to the components, conservation laws are applied depending on the type of element. In general, the values of effort of connected elements in a loop is zero and the flow at a node of connection is zero as shown in figure 2.12 .

2.3.2

System Linearization

Most physical systems are linear in nature and thus would be represented by non-linear set of equations. However, solving non-non-linear equations is a complicated task which is why system linearization methods are commonly used. System linearization will derive a linear approximate system whose response will be similar to the real non-linear system. A method called the disturbance method is used for linearizing the system [5]. In this method, a small variation about a chosen point called operation point is used to determine the slope of the curve at that point and the response in assumed linear in the vicinity of this point. This has been depicted graphically in figure 2.13. The operation point chosen

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Figure 2.12: Conservation laws applied to obtain dynamic equations [5]

Figure 2.13: Graphical representation of system linearization [12]

is point a for the non-linear function Y = F (X). For small change in value of Xa and

Ya, the slope at the operating point is given by,

dY dX and the approximate linear relationship is given by

dY dX|a.x

The linearized system can be represented either as a transfer function or as a state space representation [A,B,C,D] [12]. The state space representation links the inputs to the outputs of the system through linear matrices [A,B,C,D]. The state equations are represented in terms of the state space vector matrix as follows:

∂ ¯x

∂t = Ax + Bu (2.2)

y = Cx + Du (2.3)

where,

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u: vector of the control variables (input vector) y: vector of observer variables (output vector)

LMS AMESim uses the above methodology to linearize the system in order to perform a linear analysis and determine the frequency domain information of the system.

2.4

System Modelling in LMS AMESim

LMS AMESim stands for Advanced Modelling Environment for performing Simulations of engineering systems [12]. It gives an intuitive, Graphical User Interface (GUI) based platform to build a system and monitor its behaviour throughout the simulation. The software allows 1-D simulation to model and analyse multi-domain, multi-level system and predict its behaviour. The architecture of LMS AMESim can be seen in figure 2.14. The user can build a model from the standard library of components available or create a

Figure 2.14: Architecture of LMS AMESim simplified [12]

own library of components. The software architecture allows flexibility to aid the design process of the system V-cycle. There are four major modes to be followed sequentially while building a model:

• Sketch mode • Submodel mode • Parameter mode • Simulation mode

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The sketch mode involves building the model using components of physical systems avail-able in the AMESim libraries or using custom built components. The submodel mode involves assigning suitable submodels or mathematical equations to these components. The parameter mode involves assigning the parameters in the equations defined in the submodel mode. The simulation mode is the final step where a simulation of the model built is performed. It can be a simulation with respect to time when a temporal analysis is chosen or it can be a linear analysis to study the frequency domain characteristics.

2.4.1

Submodels

The set of governing equations which define the dynamic behaviour of the engineering system is built into LMS AMESim as a code which is the model of the system. The model is built as a collection of components, each being defined by their corresponding equations, known as submodels. The AMESim library consists of a large number of such component icons and its submodels.

2.4.2

Libraries

The libraries in LMS AMESim contain the components and their icons of a vast range of physical engineering systems. These are classified into various categories like Mechanical, Signals and Controls, Hydraulics, Electrical etc. The user chooses the components needed to a build a physical system sketch using the components in these categories and then assigns submodels to these components in the Submodel mode.

2.4.3

Numerical Solver

When a system model is simplified to a 1-D model using certain assumptions, the partial differential equations governing the system behaviour is also reduced to ordinary differen-tial equations (ODEs) and differendifferen-tial algebraic equations (DAEs). LMS AMESim inte-grator solves or integrates these equations to give simulation results. When one performs a simulation in LMS AMESim, a computer code is generated which implements the math-ematical equations on which the submodel is based and the integration algorithm calls the submodel in a computationally efficient manner and advances the solution through time [12]. This can also be seen in figure 2.15. There are various numerical methods which are used to solve ODE equations. All of these methods fall into two main categories - explicit Runge-Kutta methods and linear multistep methods [11]. Various algorithms have been developed to implement these methods in the most efficient way. The algorithms used by AMESim are:

• Adams code - used for non-stiff problems.

• Gears method - the best known algorithm for stiff problems.

• LSODA - aims to get the best of Adams code and Gears method in a single algorithm In case of DAE, it uses only one algorithm, known as the DASSL (Differential Algebraic System Solver). It is an extension of Gears method and uses the highly stable backward differential formulae.

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Figure 2.15: Code structure generated by LMS AMESim [12]

2.4.4

Causality

Causality refers to the compatibility between components and is important in building a model in LMS AMESim. The multi-port physical modelling approach in AMESim means every component has a set of ports where data exchange takes place. As mentioned in section 2.3.1, the effort and flow variables are exchanged across these ports and the output at one port of a component corresponds with the input at the connecting port of the other component. This has to be maintained in order to build a model in AMESim [12]. Therefore, as shown in figure 2.16, a one port mass component can only take a force (effort) variable as input and the output would be displacement, velocity or acceleration (flow) variables. Similar causality rule applies to all components used to build a model in AMESim.

2.5

Driveline Dynamics

Driveline dynamics involves multiple degrees of freedom and involves a lot factors like teeth contact stiffness, shaft stiffness, bearing stiffness, misalignment etc. Therefore, it is difficult to choose a single modelling approach to capture the effect of all the factors. De-pending on the level of complexity required, three modelling methods are commonly used

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in the industry [9]. These are explained in the subsequent sections. System simulation to model the driveline dynamics is relatively a novel approach. The one dimensional nature of the model delimits the capabilities but offers a quick and early design information to the designer. This work explores the capabilities of such a model built for a driveline product system.

2.5.1

Rigid multi-body simulation

This is the most commonly used method where all the components of the driveline are modelled as rigid bodies. The deformation and stresses in the components are not con-sidered. Each component will have at least one degree of freedom. If the model has only torsional degree of freedom, this is known as torsional multi-body model. The number of degrees of freedom could increase up to six which will give a more elaborate information of the driveline dynamics.

2.5.2

Finite element simulation

This is the most detailed modelling and analysis method where the components are discretised into finite elements and the number of degrees of freedom are significantly large. The most accurate solutions could be expected in this method. However, due to the need for extensive computation time and effort, modelling large models using this method is difficult.

2.5.3

Flexible multi-body technique

This technique combines the finite element simulation and rigid body simulation to get a flexible multi-body simulation. The components have a reduced modal representation which includes static deformations and dynamic responses. This model aids in optimising solution accuracy with computation efficiency.

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Chapter 3

Implementation

This chapter describes the methodology adopted in order to develop the system model of the PTU. Building a system model involves creating a lumped-mass model of the physical system and determining the right parameters which will represent the physical system closely. The purpose of this model is give a reasonable estimate of the transmission error, early in the design cycle and study the corresponding torsional vibrations in the system. The basic steps involved in modelling the PTU can be summarised as:

• Model Simplification - discretization of the components involved in making the system

• Sketch Building - identification of equivalent submodels and system building • Parameterization - determining appropriate parameter values to every

compo-nents which will reproduce the physical system

• Simulation - system simulation of the model built and temporal or linear analysis, depending on the objectives

The further sections in this chapter will explain how the above steps were implemented in LMS AMESim software to model the PTU and simulate its behaviour.

3.1

Model Simplification

In this section, the PTU architecture will be explained along with the simplification performed to create a lumped mass model of the PTU. The PTU, as mentioned in chapter 1, is a hypoid gearbox which transfers rotary power from a power source (e.g. IC engine) to the rear wheels of the vehicle. In this case, PTU draws power from the main transmission gearbox of the vehicle and transfers it to the RDU through the propeller shaft. The various significant components in the PTU can are listed in section 1.1 and can be seen in figure 1.2. The input shaft couples to the main transmission and serves as the power input point to the system. A dog clutch connects the input shaft to the tubular shaft and serves as a connect/disconnect option depending on the need for power transfer to the rear wheels. The ring gear mounted on the tubular shaft meshes with the pinion gear and forms the hypoid gear pair, which transfers power in a 90◦ angle. The pinion gear is connected to the propeller shaft through the output flange. There are a total of five bearings supporting the shafts, two on the pinion, two on the tubular shaft and one on the input shaft assembly. Apart from these major components, there are seals, washers

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and breather nipples which are insignificant to the scope of this work.

In order to model this physical system in a 1-D environment, we need to identify the basic degree of freedom of this system i.e. rotation. The PTU consists of rotational components which transmit rotary power and hence this study will involve the dynamic behaviour of the PTU in the rotational degree of freedom. Bearings will not contribute to the torsional stiffness in the system, and hence will be excluded from this model. The remaining components, which contribute to the torsional stiffness in the system, are modelled as lumped mass parameters, which will be used to build the sketch in LMS AMESim. The schematic of the lumped mass model of the PTU can be seen in figure 3.1. This schematic is used in modelling the physical system in LMS AMESim and thereby

Figure 3.1: System Lumped Schematic

solve the torsion dynamic equations of the system. Initially, a gearset level model is built to compare the variation in teeth stiffness induced transmission error values with those obtained from experiments. Once the gearset level model was used to predict transmission errors within the same order of magnitude as experimental results, the overall PTU model is built.

3.2

Sketch Building

The sketch window (shown in figure 3.2) in LMS AMESim is used to build a model of the physical system. Based on the system schematic built in figure 3.1, a sketch of the system is built using components available in the inbuilt components library of LMS AMESim. Table 3.1 depicts the components of the PTU and their corresponding AMESim components used to build the sketch. Suitable submodels are applied to these components to ensure compatibility in terms of data exchange at the ports.

3.2.1

Bevel Gear

The bevel gear component, as the name indicates, is used to build a bevel gear in the AMESim environment. The component consists of two ports with variables exchange as rotary velocity and torque, as shown in figure 3.3. The component can be used as a static submodel with a gear ratio alone, or can be dynamic with contact stiffness and gear inertia specifications. In the static case, the torque and rotary velocity are computed as:

T2 = −T1

Rp2w

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Figure 3.2: LMS AMESim window layout

Table 3.1: PTU Components and their corresponding LMS AMESim Components

PTU Physical Components LMS AMESim Equivalent Hypoid Gearset Bevel Gear Component

Pinion Shaft Rotary Inertia Input Shaft Rotary Inertia Output Flange Rotary Inertia Pinion Shaft Compliance Rotary Spring Damper Tubular Shaft Compliance Rotary Spring Damper

Dog clutch Indexation Model

and ω1 = −ω2 Rp2w Rp1w where, T2-Torque at port 2 T1-Torque at port 1

Rp1w-Working radius of gear at port 1

Rp2w-Working radius of gear at port 2

ω1-Rotary velocity of gear at port 1

ω2-Rotary velocity of gear at port 2

In the dynamic case, teeth positions are computed with the help of rotary velocity input and gear inertia of gear at port 2. The relative velocity between the gears is then used to determine the relative linear displacement between two teeth. At contact, this displace-ment will be used as the compression of the spring-damper representing the contact and thus provides the contact force, depending on the type of contact chosen by the user. The fundamental gear dynamic equation is then used to compute the rotary acceleration

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of the gear, which upon integration will give the rotary velocity (see Appendix A). The gear teeth contact in AMESim is modelled as a spring-damper system. Various

Figure 3.3: Bevel Gear submodel in LMS AMESim

contact models are available and depending on the choice, the contact force computation varies. In this work, variable contact tooth stiffness was considered. This option takes into account both the non-linearity of teeth stiffness as well as evolution of the number of teeth in contact at a particular angle. The user sets a maximum and minimum tooth stiffness at the middle and start/end of tooth engagement, respectively and the contact stiffness is then approximated using a polynomial expansion. The transverse contact ratio input gives the information for effective teeth engagement in the gear mesh at a particular gear angle. The contact damping is assumed to be fixed at the critical damping. Since, the gear contact is modelled as spring-damper system and gear kinematics are not taken into consideration for computing gear acceleration, the use of a bevel gear submodel for a hypoid gear a valid approximation.

3.2.2

Dog Clutch

An indexation submodel in LMS AMESim, built for gear synchronizers, is used to model the dog clutch. The submodel represents an indexation phase between a sleeve and a ring or between a sleeve and idle gear. The component has four ports, as shown in figure. Two ports exchange rotary degree of freedom variables while the other two exchange linear degree of freedom data. The component allows the user to build a free

Figure 3.4: Indexation submodel in LMS AMESim

geometry of the ring and sleeve teeth in 2D. A plane section of the a meshed teeth pair is used to build the 2D section using the clutch dimensions available. A 2D contact detection algorithm in AMESim detects the contact between two teeth and computes the corresponding contact force by modelling it as a spring damper system. This way, teeth engagement, torque transfer and axial displacement of the the sleeve can be observed using the indexation submodel and represent the dog clutch behaviour in the model.

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Friction effects are optional though they have been ignored in this work. The major purpose of the dog clutch in the current model is to study the transient effects during the time of engagement and disengagement in the clutch.

3.2.3

Physical Model

The physical model of the PTU is built according to table 3.1. The sketch of the model built is as shown in figure 3.5. As shown in this figure, the bevel gear submodel represents the hypoid gearset. The submodel considers the inertia of the gear at port 2 inherently and therefore, the ring gear and tubular shaft inertia is built into the submodel. The

Figure 3.5: Physical model of PTU in LMS AMESim

input shaft inertia is connected to the bevel gear (tubular shaft/ring gear side) through a rotary spring and damper which represents the shaft torsional stiffness and stiffness in the splines interfacing the two parts. The input shaft is connected to a rotary velocity input source with a dog clutch supercomponent in between. A supercomponent is built as a collection of various other components with a specific function. In this case, the indexation component and other components for axial displacement of the sleeve, are modelled within a single supercomponent. Figure 3.6 depicts the components making up the supercomponent. On the pinion side, the bevel gear component is connected to the pinion rotary inertia. This inertia component is connected to the output flange rotary inertia through a rotary spring-damper component which represents the pinion shaft torsional stiffness and spine stiffness at the interface of the two parts. A constant torque input is given at the output flange inertia port, which is controlled through the clutch component, as a feedback loop. The torque is cutoff when the clutch disconnects to map the clutch connect/disconnect behaviour. The transmission error can be computed by finding the difference in angular displacement of the ring gear and the pinion, measured at the two angle sensors, placed in the driveline. The transmission error signal can then

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Figure 3.6: Dog clutch super component

be mapped in the frequency domain by performing an Fast Fourier Transform (FFT) and comparing the amplitude of the error at the fundamental and first harmonic frequency. This is used to validate the accuracy of the model.

3.3

Parameterization

Parameterization refers to the assignment of parameter values to the various submodels used to build the overall physical system. This in turn defines the parameter values of the governing equations of the various components, thereby defining their behaviour. Depending on the complexity of the submodel chosen, different parameters are required to define the behaviour of the system. The user defines these parameters based on existing literature, calculations, experimental data, manufacturer data etc. This section defines the parameters set for various components used to build the PTU model.

3.3.1

Bevel Gear Parameters

As mentioned in section 3.2.1, the bevel gear submodel was used to model the hypoid gearset of the PTU. A dynamic case with variable contact stiffness was used to define the gear behaviour and corresponding parameters were set in the model. Table 3.2 gives a list of the parameters defined for the bevel gear submodel. The gear geometry parameters like working pitch radius, transverse pressure angle, transverse contact ratio and helix angle were obtained from the gear generation software, which is used to design and manufacture the gears. The gear inertia, which in this case is the inertia of the ring gear side is a sum of the ring gear and tubular shaft inertia, obtained from CAD - software. The moment of inertia about the axis of rotation was considered. The teeth stiffness

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Table 3.2: Parameters of the bevel gear submodel

Parameter Unit

Working pitch radius (gear at port 1) mm Working pitch radius (gear at port 2) mm Gear inertia (gear at port 2) kg.m2 Working transverse pressure angle degree

Helix angle degree Number of teeth of gear at port 2 -Maximal teeth stiffness in the middle of engagement N/m Minimal teeth stiffness in the start and end of contact N/m

Correction factor for stiffness -Transverse contact ratio

-Contact damping Nm/(m/s)

was also obtained from the average tooth stiffness value provided by the gear simulation software. An upper and lower limit was set to obtain a similar average teeth stiffness and at the same time provide realistic transmission error results. The contact damping was calculated assuming a critical damping condition. The damping coefficient was assumed to be 0.025, which is the value for steel-steel contact [12]. Thus, the critical damping value was calculated using the equation of critical damping as,

H = 2 × ζ ×√K × I where,

H = Contact damping ζ = Coefficient of damping K = Contact stiffness I = Gear moment of inertia

3.3.2

Dog Clutch Parameters

The dog clutch is modelled in AMESim as explained in section 3.2.2. The sleeve (moving dog teeth) section geometry and the ring (input shaft teeth) section geometry is defined by five coordinate points and assigned to the submodel as data files. The shape editor feature in LMS AMESim is used to visualise and edit the geometry as needed. This is shown in figure 3.7. The geometry of the teeth cross section was obtained from part drawings of the dog clutch parts. The dog clutch control is explained in section 3.2.2. The spring stiffness of the moving dog ring was also obtained form the part drawing. The force exerted on the moving dog ring is equivalent to a cam force exerted on the moving dog ring and was calibrated to displace the sleeve (moving dog ring) by 3.6 mm which is the maximum displacement of the ring in the physical setup.

3.3.3

Inertia and Compliance Parameters

The list of rotary inertia parameters and rotary spring-damper parameters can be seen in table 3.3. The rotary inertia of the components were obtained from CAD software. The

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Figure 3.7: Clutch teeth visualisation in shape editor

torsional stiffness for the rotary spring-damper components were obtained from multi-body simulation data available with GKN.

Table 3.3: Inertia and Spring-Damper Parameters

Component Parameter Value Unit

Pinion Rotary Inertia Moment of inertia 0.001 kgm Viscous friction 0.001 nm

Input Shaft Rotary Inertia Moment of inertia 0.000554 kgm Viscous friction 0.001 nm

Output Flange Rotary Inertia Moment of inertia 0.000554 kgm Viscous friction 0.001 nm

Pinion Shaft Spring-Damper Stiffness 3159.05 Nm/deg Damping 0.1 Nm/(rev/min)

Tubular Shaft Spring-Damper Stiffness 1989.68 Nm/deg Damping 0.1 Nm/(rev/min)

3.4

Simulation

The model was simulated in the time domain and frequency domain using the temporal and linear analysis setting available in LMS AMESim. The time domain analysis lets us visualise the behaviour of the model as the time progresses. In linear analysis, the system is linearized about an operating point as explained in section and the eigen frequencies and mode shapes of the system is computed. This section explains the system input conditions and settings in the two different analyses.

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3.4.1

Temporal Analysis

The system has a rotary velocity input at the dog-clutch/ring gear end and a torque load on the pinion end. In order to compare the transmission error output of the model with that of the actual system, the system input needs to be kept the same in the simulation and experimental setups. The experimental inputs to the system is shown in table 3.4. Due to causality of the components, it is only possible to give a rotary velocity input to the

Table 3.4: Torque and rotary velocity inputs to actual system

Torque input at Ring Gear (30-50) N-m Rotary velocity at Pinion (50-70) RPM

clutch while the output flange inertia needs a torque input. Therefore, the input settings shown in table 3.4 where given on their opposite ends in the simulation model, corrected using the gear ratio. Thus, the rotary velocity and torque inputs to the simulation model can be seen in table 3.5. The rotary velocity is a constant value with an initial ramp up

Table 3.5: Torque and rotary velocity inputs to AMESim model

Torque input at Pinion (10-20) N-m Rotary velocity at Ring Gear (20-30) RPM

to one second. The torque load is also a constant value, however, it is governed by the clutch on-off state. The torque supply is cut-off when the clutch disconnects, in order to reduce the ring gear and pinion rotary velocity to zero as it would behave in the actual system upon clutch disconnect. These inputs can be visualised graphically in figure 3.8 and 3.9. The solver settings are described in table. These are the default settings in LMS

Figure 3.8: Rotary velocity input at clutch/ring gear end

AMESim and it was left for the software to choose the suitable solver. The simulation time and clutch control details are as shown in table 3.6.

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Figure 3.9: Torque input at pinion end

Table 3.6: Simulation parameters

Simulation time 15 s Clutch Disengagement at 5 s Clutch Re-engagement at 10s

3.4.2

Linear Analysis

In order to perform a linear analysis of the model, the linear analysis mode of the sim-ulation was enabled. The linearization time was set to zero second which will be the operating point. There were a total of four state variables, the rotary velocity component of each of the inertial components, which were set as state observers. The remaining state variables were set as free variables. These four state observers will be used to study the mode shapes of the PTU components in torsion. Linear analysis simulation was run for one second and the results were obtained.

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Chapter 4

Results and Discussion

The temporal and linear analysis performed on the PTU model built gave us results which simulate the system behaviour in the time and frequency domain respectively. The transmission error data was computed by calculating the difference in the angular displacements of the pinion side and the ring gear side, using the output from the angular sensors inserted in the model. This error and other results can be seen in the following sections.

4.1

Transmission error comparison

The transmission error measured across the pinion and ring gear is a static transmission error since teeth compliance and compliance in the shafts are the only cause of error in the transmission. This error is measured at a low rotary velocity and low torque load. The error signal is measured with the clutch in a continuously engaged condition and the corresponding raw signal needs to be divided into its frequency components. A FFT (Fast Fourier Transform) of this signal is performed for this purpose. A Hanning window is chosen as the most ideal for this purpose and the frequency content of the signal is obtained. The raw transmission error signal and it’s frequency content post FFT can be seen in figure 4.1. The FFT results show that the meshing frequency and its multiples have

Figure 4.1: Transmission error signal and FFT data (AMESim model)

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first harmonics of the signal. The meshing frequency of the gears at the set pinion speed is 1Hz. Thus, the fundamental and first harmonic are a common measure of determining the mesh quality of gears. These are compared with the experimentally measured values in order to understand the validity of the model. This comparison is shown in table 4.1. This shows that the model built predicts the transmission error amplitude of the

Table 4.1: Comparison of transmission error results - Model and Experiment

Transmission Error (in µrad) AMESim Model Experimental Results Fundamental Harmonic 13 13

First Harmonic 9 2

fundamental harmonic quite accurately as this frequency can be attributed primarily to the teeth compliance. The higher multiples however are over predicted and a relationship could not be established in this case. However, considering the complexity in a three-dimensional phenomenon like transmission error, the results are a good estimate of the teeth compliance involved in the error and thus validate the model for torsional analysis.

4.2

Clutch Transient Behaviour

The dog clutch engagement and disengagement could result in a transient state in the rotary velocity response. In order to observe this, the clutch was disengaged at t = 5 s and re-engaged at t = 10 s. This can be seen in figure 4.2 where the clutch mass linear displacement has been plotted with respect to time. At the time of re-engagement, one

Figure 4.2: Linear displacement of the clutch

can observe a momentary disturbance in the motion of the clutch. This is due to the phase where the sleeve is trying to engage with the ring since the sleeve is stationary in rotation at the time of re-engagement. This is a deviation from the physical system where the external control systems would synchronise the speed of the two parts at the

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time of engagement. Such control architecture, however, is not in the scope of this work and therefore not included here. The rotary velocity of the pinion and ring gear can be seen in figure 4.3. At t = 5s and t = 10s, on can see a transient acceleration in the rotary

Figure 4.3: Rotary velocity of pinion (left) and ring gear (right)

velocity response. This could transfer as a momentary vibration through the system which could translate to a airborne noise at the time of clutch connect and disconnect. The clutch rotary velocities also exhibit similar behaviour as can be seen in figure 4.4. The periodic disturbances in the response under steady operating conditions are the torsional vibrations in the system. A change in the damping values of the inertial and spring-damper components is shown to have a significant effect on the torsional vibrations. By tuning these parameters, the torsional vibrations could be reduced significantly.

4.3

Linear Analysis Results

The linear analysis of the PTU model gives us the frequency domain results of the sys-tem. Table 4.2 shows the eigenfrequencies and damping ratios of the syssys-tem. These are estimates based on the inertia of the components and the stiffness of the spring-damper components. The corresponding mode shapes of the components at each frequency can be seen in figures 4.5 - 4.7. The results depicted above were obtained at a linearization time of 0s assuming an equilibrium position. The results do change with respect to the linearization time and corresponding equilibrium position, which is hard to determine. As shown in the figures 4.5 - 4.7, the magnitude of rotary velocities of each component is shown in a bar plot to depict the mode shapes. Figure 4.5 represents the first eigen mode

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Table 4.2: Eigenfrequency and damping of system torsional modes

Torsional mode Frequency Damping Ratio

Oscillating Mode - 1 1747.76 Hz 0.033 Oscillating Mode - 2 2369.93 Hz 0.617 Oscillating Mode - 3 3680.63 Hz 0.571

of 1.7 kHz. The Y-axis of the bar plot represents the magnitude of rotary velocity in percentage while the X-axis represents the components in PTU, listed 1-4. The numbers represent the input shaft, ring gear/tubular shaft, pinion shaft and the output flange from 1-4. A positive magnitude of the velocity represents all the components vibrating in phase. This frequency is relatively underdamped as seen in the figure. Figure 4.6 repre-sent the second eigen mode of 2.3 kHz. The ring gear/tubular shaft and the output flange can be seen vibrating out of phase with the input shaft and pinion at this frequency. The input shaft magnitude is significantly higher compared to the other components. How-ever, this frequency is heavily damped. Figure 4.7 represents the third eigen mode of 3.6 kHz. This is also a heavily damped frequency in the system. The pinion shaft can be seen vibrating out of phase with the other components. These results however need to

Figure 4.5: Torsional first eigen mode - 1747 Hz

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Figure 4.6: Torsional second eigen mode - 2369 Hz

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Chapter 5

Conclusion and Future Work

This chapter summarises the work carried out in this thesis and describes the future prospects for this work.

5.1

Conclusion

In summary, a 1-D system level model of a Power Transfer Unit was built in LMS AMESim suite. The conclusions which can be made through this work are as follows:

• The transmission error in a gear drive is widely accepted as a major source of gearbox vibrations.

• 1-D system simulation in LMS AMESim can be used to perform a torsion dynamic analysis of a drivetrain/driveline of an entire vehicle. In this case, a particular driveline product was modelled and analysed

• The transmission error measured in 1-D is only due the teeth contact compliance as other complex factors like misalignment, shaft run-out, contact friction etc. are not modelled in the gearset.

• The dog clutch can be modelled well in 1-D and its behaviour can be studied. This work elaborates the method of modelling the mechanical aspects of a dog clutch. • The transmission error in a gearset is a complex 3-D phenomenon and is attributed

to multiple factors. 1-D simulation gives a reasonable estimate of this error, though it is not the best tool for this purpose

• The linear analysis of the system gives the eigenfrequecy, damping and mode shapes of the system in torsion. This estimate could be used earlier in the design stage to identify the frequencies which could cause resonance in the system and thus change the design parameters like stiffness earlier in the design cycle.

• Component level modelling in 1-D is useful only to study the behaviour of that component in a system and its interactions. 3-D geometry analysis is a more suitable method for independent component level optimisation.

• Correlation of transmission error values obtained in the model, with experimental results, is precise in the fundamental harmonic, though the higher harmonics deviate significantly.

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5.2

Future Scope of Work

This work was a primary initiative to map a component level product in a system envi-ronment. The work served as an entry into the world of system level simulation for the author and a lot of understanding regarding its functioning, capabilities and limitations were realised during the course of this work. Owing to the time and knowledge constraints in the project, some activities were not performed during the course of this thesis and hence have been suggested for future work. These are as follows:

• Only torsional dynamic behaviour could be mapped in the model built in this work. In order to study the vibration transfer through the shafts to the bearings and to the housing, a more detailed, component level analysis model is needed. This could be achieved by co-simulation with a multi-body dynamic model of the system. This way, even the linear accelerations of the housing vibrations could be obtained which can be measured easily on an actual system.

• Experimental data of the torsional behaviour of the PTU was not available at the time of this thesis. This could be performed through an experimental modal analysis or similar analyses which gives a torsional behaviour data of the system.

• The dog clutch modelled in this work is extensively simplified version of the actual clutch. The actual control of the clutch could be obtained from the control software as a Functional Mock-Up (FMU) unit and used in the 1-D environment or it can be built completely within AMESim.

• If the engine firing harmonics details are available, the model could be coupled with an engine submodel and thereby obtain the engine induced excitation in the system. • An hypoid submodel could be developed which considers the hypoid contact

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Bibliography

[1] Mats ˚Akerblom. Gear Noise and Vibration: A Literature Survey. 2001.

[2] Design Manual for Bevel Gears. Standard. Virginia,USA: American Gear Manu-facturers Association, 2003.

[3] Basics and Development of Spiral Bevel and Hypoid Gears - Training Report. Tech. rep. 2015.

[4] MR Beacham et al. Development of transmission whine prediction tools. Tech. rep. SAE Technical Paper, 1999.

[5] Javier Fernandez De Canete, Cipriano Galindo, and Inmaculada Garcia-Moral. Sys-tem Engineering and Automation: An Interactive Educational Approach. Springer Science & Business Media, 2011.

[6] Hiroaki Endo and Nader Sawalhi. “Gearbox Simulation Models with Gear and Bearing Faults”. In: Mechanical Engineering. Ed. by Murat Gokcek. Rijeka: InTech, 2012. Chap. 2.

[7] Getting Started with LMS Imagin.Lab Amesim 15 - Training Report. Tech. rep. 2016.

[8] Stephen L. Harris. “Dynamic Loads on the Teeth of Spur Gears”. In: Proceedings of the Institution of Mechanical Engineers 172.1 (1958), pp. 87–112.

[9] Bincheng Jiang. “Modeling of Dynamics of Driveline of Wind Stations: Implemen-tation in LMS Imagine AMESim Software”. In: (2010).

[10] Jan Klingelnberg. Bevel Gear. Springer, 2016.

[11] LMS Imagine.Lab Amesim - Integration Algorithms used in LMS Amesim. Tech. rep. 2015.

[12] LMS Imagine.Lab Amesim - Reference Guide. Tech. rep. 2015. [13] Ivar Nilsson. “Gear whine noise excitation model”. MA thesis. 2013. [14] J Derek Smith. Gear noise and vibration. CRC Press, 2003.

[15] P. J. Sweeney. Transmission error measurement and analysis. Tech. rep. 1994. [16] DB Welbourn. Fundamental knowledge of gear noise: a survey. Tech. rep. 1979.

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Appendix A

Bevel Gear Submodel

The bevel gear submodel (shown in figure A.1 ), TRBVG01A, is an assembly of bevel gears. Their behaviour can be static or dynamic. In the dynamic case, the contact stiffness and gear inertia needs to be defined. Friction effects and backlash can also be modelled. In the static case, the input velocity at port 2 is multiplied by the ratio of the radius of the two gears and then is output at port 1. In the dynamic case, the teeth positions are computed with the help of rotary velocity at port 2 and the velocity of the gear inertia. Teeth positions are then used to compute the contact force. The dynamic equilibrium equation is then used to compute the acceleration of the gear inertia. The resulting acceleration is integrated to obtain the gear velocity which is the output at port 1.

Figure A.1: Bevel gear assembly submodel [12]

The equations involved in this submodel are are shown below:

vrel = Rp2w.w2+ Rp1w.w (A.1)

The relative linear displacement x is computed from,

dx

dt = v1− v2 (A.2) The contact force F is computed by,

F = K.x + H. ˙x (A.3)

where, K is the contact stiffness and H is contact damping. This contact force is then used to compute the gear acceleration.

(54)

References

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