WANG, ZHAO - Modeling and Analysis of Gear Rattle in Automotive Transmissions

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ABSTRACT

Title of dissertation: Modeling and Analysis of Gear Rattle in

Automotive Transmissions

Weijie Zhao, Doctor of Philosophy, 2002

Dissertation directed by: Professor Yu M. WangDepartment of Mechanical Engineering

In recent years, attainment of high quality in automotive vibration and sound

has become a major effort in automotive vehicle refinement programs. In Europe

and Asia, the vast majority of passenger cars are equipped with manual trans

missions and diesel engines. Compared with automatic transmissions, manual

shifts offer lower cost, better fuel efficiency, and more a greater sense of being in

control of the car. However, unlike automatic transmissions, manual transmis

sions do not have the high viscous damping inherent to a hydrodynamic torque

converter to suppress the impacting of gear teeth oscillating through their gear

backlash. Therefore, a significant level of noise can be produced by the gear

rattle and transm itted both inside the passenger compartment and outside the

vehicle. Gear rattle, idle shake, and other noise generated by low frequency vibra

tion phenomena in the automobile driveline have become an important concern


to automobile manufacturers in their pursuit of an increase in perceived sound

quality.

Gear rattle is produced by the impacting of gear teeth through their un

loaded mesh backlash as a response to engine torque fluctuations. Not only is

rattle noise audibly objectionable, but it may also be misconstrued as an impend

ing transmission failure leading to warranty returns. The complexity of torsional

vibrations and various nonlinearities of the manual transmission present many

challenges for the analysis of torsional vibration characteristics and gear rattle

behavior. Despite intensive research in the past, numerical difficulties in handling

nonlinearity have prohibited the development of general criteria in transmission

design to alleviate the rattle.

The objective of this dissertation is to investigate and develop a complete

modeling method considering all the components of the powertrain, with robust

numerical techniques for study of the gear rattle phenomenon. The aim of the

modeling and analysis effort is to conduct parametric studies and provide design

guidelines for powertrain development and refinement. The dissertation focuses

on the following:

First, based upon a comprehensive understanding of the powertrain sys

tem, a decoupled torsional vibration model is developed. This model separates

the system into two parts with a baseline model and a rattle model to simplify

the analysis. The validity of the decoupled model is determined by implementing

and comparing it with a full model.


Second, a numerical technique based on Finite Elements in Time domain

(FET) is derived and implemented for the analysis of rattle dynamics. The

numerical integration algorithm is a key component for efficient and accurate

numerical investigations. The FET algorithm is compared with the Stiff ODE

algorithms of MATLAB to show its efficiency and effectiveness.

Third, with the developed decoupled model and numerical tools, a para

metric study is conducted for design applications. These parametric studies yield

the effects of the key design parameters on the effective indices of rattle dynam

ics. This allows the designer to evaluate trade-offs among various designs without

resorting to expensive and inefficient palliative measures.


MODELING AND ANALYSIS OF GEAR RATTLE IN AUTOMOTIVE

TRANSMISSIONS

by

Weijie Zhao

Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment

of the requirements for the degree of Doctor of Philosophy

2002

Advisory Committee:

Professor Yu M. Wang, Chair/Advisor Professor Balakumar Balachandran Professor Amr Baz Professor Sung W. Lee Professor Gregory Walsh


UMI Number 3055646

Copyright 2002 by Zhao, Weijie

All rights reserved.

___ (B)

UMIUMI Microform 3055646

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©Copyright by

Weijie Zhao

2002


DEDICATION

dedicated to my wife, mom, and Katherine

ii


ACKNOWLEDGMENTS

I would like to express my most sincere gratitude to Dr. Yu Wang, my advisor.

His advice, guidance, and ability have helped me throughout this four year period

of study and research. I will remember well the strong example he has set as a

teacher, researcher, and advisor.

I would also like to thank Daimler-Chrysler for funding this project. Per

sonally, I am thankful to Denis C. Wieczorek, Jeff Ward, and Ray Shaver, for

supplying the transmission data and allowing us to use the facilities of Daimler-

Chrysler Corporation. The summer internship of 1998 was of great help for the

progress of this project. I would also like to thank Basil Joseph of New Venture

Gears, for his consistent help from modeling stage to analysis stage.

I would like to take this opportunity to thank Dr. Lee, Dr. Baz, Dr.

Balachandran, and Dr. Walsh for being member of my advisory committee. Their

professional, insightful advice and direction make this dissertation be integrated.

I would also like to extend my thanks to all my fellow colleagues who

provided a wonderful work environment and all my friends for their support.

1U


TABLE OF CONTENTS

Dedication ii

Acknowledgments iii

List of tables vi

List of figures vii

1 Introduction 11.1 Introduction......................................................................................... 11.2 Literature review for gear rattle study ............................................ 11.3 Numerical algorithms for stiff p ro b lem ............................................ 81.4 Scope of dissertation w o rk ................................................................. 9

2 Modeling For Powertrain Gear Rattle 122.1 Introduction......................................................................................... 122.2 System modeling and problem formulation...................................... 12

2.2.1 The powertrain components and rattle sources.................. 132.2.2 Torsional vibration for rattle analysis.................................. 16

2.3 Dynamics of a gear pair with back lash ............................................ 212.4 Vibration and rattle analysis.............................................................. 25

2.4.1 Baseline torsional vibration ................................................. 262.4.2 Gear rattle m odeling............................................................. 27

2.5 Model verification................................................................................ 282.5.1 Equations of motion for full and decoupled m o d e l 282.5.2 Analysis resu lts ...................................................................... 32

3 Numerical Algorithms for Gear Rattle Analysis 383.1 Introduction......................................................................................... 383.2 ODE stiff s u i t e ................................................................................... 38

3.2.1 The ODElSs P ro g ra m .......................................................... 393.2.2 The ODE2Ss P ro g ra m .......................................................... 40

3.3 Finite element in time dom ain ........................................................... 413.3.1 Formulation of Finite element in time d o m a in .................. 41

iv


3.3.2 Primal form of F E T ............................................................. 423.3.3 Mixed form of F E T ................................................................. 433.3.4 Finite element approxim ation............................................... 453.3.5 The evaluation of G and Jacobian Matrix ......................... 48

3.4 Strategy for choosing numerical algorithms.................................... 50

4 Application to NT350 Transmission 524.1 Introduction....................................................................................... 524.2 Flexible shaft model versus rigid shaft m odel................................. 524.3 Time-varying meshing stiffness function.......................................... 55

4.3.1 Stiffness function of one helical tooth p a i r .......................... 554.3.2 Synthesized meshing stiffness of a meshing gear pair . . . . 56

4.4 Effect of time-varying meshing stiffness............................................ 584.5 Gear rattle indices ........................................................................... 614.6 Analysis for the first speed configuration....................................... 63

4.6.1 Baseline torsional v ib ra tio n .................................................. 644.6.2 Rattle analysis for unladen pairs of g e a rs ............................. 64

4.7 Comparison of rattle results: ODEs vs. F E T ................................. 68

5 Parametric Studies and Design Application 715.1 Introduction....................................................................................... 715.2 Characterization and reduction of gear ra t t le ................................. 725.3 Rattle m o d e s ...................................................................................... 735.4 Effect of drag torque and backlash on r a t t l e .................................. 76

6 Summary and Discussion 826.1 S u m m a ry .......................................................................................... 826.2 Contributions.................................................................................... 836.3 Future w ork ....................................................................................... 84

BIBLIOGRAPHY 86

v


LIST OF TABLES

2.1 Rattle Indices : Decoupled model vs. Full model

VI


LIST OF FIGURES

2.1 Illustration of Dodge Neon powertrain with NT350 transmission . 13

2.2 Engine excitation torque fluctuation................................................ 14

2.3 Time-varying gear meshing stiffness................................................ 16

2.4 The powertrain model for the first s p e e d ....................................... 17

2.5 The cause and effect diagram of gear rattle system........................ 18

2.6 Torsional vibration model for the first s p e e d ................................. 19

2.7 Various stiffness characteristics of powertrain parts ..................... 20

2.8 A meshing gear p a i r ........................................................................ 21

2.9 Geometric relation of a meshing gear pair with backlash ............ 23

2.10 The physical model of a pair of unladen gears with backlash . . . 27

2.11 Powertrain model with 4 D O F s ...................................................... 29

2.12 The force characteristics of c lu tc h ................................................... 30

2.13 The motion of driving gear from decoupled m o d e l........................ 33

2.14 The motion of driving gear from full m odel.................................... 33

2.15 The motion of driven gear from decoupled model ....................... 34

2.16 The motion of driven gear from full m o d e l.................................... 35

2.17 Rattle result from decoupled model a n a ly s is .................................. 36

vii


2.18 Rattle result from full model analysis............................................. 37

4.1 The baseline response with flexible s h a f t ....................................... 54

4.2 The baseline response with rigid shaft............................................. 54

4.3 Time varying stiffness with contact ratio = 1 .8 8 ............................ 57

4.4 Time varying stiffness with contact ratio = 2 .7 9 ............................ 58

4.5 The baseline response with constant contact stiffness................... 59

4.6 The baseline response with time-varying contact stiffness............. 60

4.7 The rattle response with constant contact stiffness......................... 60

4.8 The rattle response with time-varying contact stiffness ................ 61

4.9 Baseline response: Angular displacem ent........................................ 64

4.10 Baseline response: Angular velocity ................................................. 65

4.11 Rattle response of Gear pair 2 for the first speed (ODE15s) . . . . 66




4.15 Rattle response of Gear pair 2 for the first speed (F E T ).............. 68




5.1 Double side impact rattle m o d e ....................................................... 73

5.2 Single side impact rattle m ode.......................................................... 74


5.3 Irregular rattle m o d e ........................................................................ 75

5.4 Rattle indices versus drag to r q u e ............................................. 77

5.5 Rattle indices versus back lash ....................................................... 77

5.6 Rattle response with drag torque = 0.0 Nm ........................... 78

5.7 Rattle response with drag torque = 0.05 N m .................................. 78




ix


Chapter 1

Introduction

1.1 Introduction

Transmission rattle has long been an issue that draws the attention of researchers

in automotive industry. In this chapter, a literature review for transmission rattle

modeling and analysis will be conducted, followed by a brief introduction to the

development of the Finite Element in Time Domain algorithm. Finally, modeling

and analysis methods for the powertrain rattle will be presented.

1.2 Literature review for gear rattle study

The power transmission system of an automobile is highly nonlinear. The dy

namics of gear rattle is affected by a number of design parameters: amount of

backlash, mesh frequency, stiffness and damping of clutch, and load and inertia in

the transmission [1, 2]. However, no comprehensive analytical models and solu

tions exist which can facilitate the design process. Traditional “tuning” of clutch

torsional stiffness has been adopted as a means of reducing idle rattle. Without

an appropriate driveline system model, such a tuning can be time consuming.

1


Furthermore, it becomes difficult to consider both neutral and in-drive rattles

[3].

A common solution procedure for dynamic analysis of a nonlinear system is

numerical time integration. However, there are some difficulties in this approach.

For example, it is usually very time consuming to obtain the system response

with respect to a system parameter or parameters, especially in parameter re

gions where multi-valued response exists. Furthermore, integration with digital

computers may encounter numerical difficulties for some types of integration al

gorithms, particularly when the nonlinearity becomes very strong [2].

An alternative solution procedure for dynamic analysis of a nonlinear sys

tem is harmonic balance method. The approach is known to be limited by its

ability of solving the resulting nonlinear algebraic equations [4]. It is found that

the strength of nonlinearity, the number of harmonics sought in the solution,

and the relative magnitudes of higher harmonics with respect to the fundamental

are, among others, the major factors that are instrumental for the success of the

method. If an impulse-like force, for example, generated by a rigid stop is applied

to the system, a large number of higher harmonics are inevitably required, giving

rise to a large number of system equations. In this case, it is usually difficult to

achieve convergence in numerical solution.

Many researchers have tried to overcome the nonlinearity issue by making

assumptions and approximations that linearize the system equations. Y. Cai

and T. Hayashi [5] made linear approximations for the vibration of a pair of

2


spur gears. They have achieved this by treating the variable portion of the time-

varying stiffness as an equivalent exciting force. With this linearization, analytical

solutions to the equations are obtained. R. Singh, et al. [6], while analyzing

the neutral gear rattle in a powertrain, has also linearized the nonlinearities to

facilitate the problem solving and the avoidance of the numerical difficulties. The

powertrain is approximated to a 4 degrees of freedom system.

K. Umezewa, et al. [7], has made many studies on vibrations of helical gear

pair. The torsional vibration of the system was considered as a one DOF system.

While, C. Padmanabhan and R. Singh [8] gave cases in which two degree of

freedom systems could be approximated to two single degree of freedom systems.

Research has also been conducted to get nonlinear response of a system.

G.W. Blackenship and A. Kahraman [9] have tried to deal with clearance type

of nonlinearities in studying the forced response of a system. They also dealt

with the periodic stiffness variation. Newton-Raphson and Gaussian elimination

methods are used to solve the system equations. Experimental validation using

a rattling gear pair is provided. R.J. Comparin and R. Singh [10] have studied

nonlinear response of a gear pair with clearance type nonlinearity. They have

identified single and double sided impact modes. It is also shown that piecewise

linear assumption is not adequate. As an extension of previous work, R. J. Com

parin and R. Singh [12], have studied coupled nonlinear response of a multi-degree

of freedom system. In this study, only the first harmonic is considered.

Computer simulation is used to obtain rattle produced by gear pairs. Y.

3


Cai [13] has analyzed vibrations of helical gears. A computer simulation for

torsional vibration for a pair of helical gears is developed. Both nonlinear tooth

separation and time varying mesh stiffness are considered. A simply modified

stiffness function is employed for his calculations. The results of simulation agree

well with experimental and theoretical calculations. A. Laschet [14] has provided

a computer simulation method to evaluate gear noise. Results in both time and

frequency domains are illustrated. He attributes a peak in frequency domain to

a high noise level at that frequency. A. Szadkowski [15] has also developed a

mathematical model and a computer simulation for idle gear rattle.

The powertrain model has also been a favored area of research in the past.

Steve Meisner and Brian Campbell [3] have outlined the development of a power

train modeling. The model for various components of the powertrain is discussed,

and also verified with experimental data. C. Padmanabhan, et al. [16], has de

tailed three different stages of a powertrain modeling.

F. Pfeiffer has done a lot of work on gear rattle analysis. He along with

W. Prestl [17] has proposed a model for gear rattle by considering the impact on

a tooth of a rattling gear. They have found that rattle noise could be explained

by impact theories. F. Pfeiffer and A. Kunert [18, 19] have proposed rattling

models based on deterministic and stochastic processes. This probabilistic view

has reduced the computational time but further research is required. The main

drawback of this approach is that it can’t be easily extended to multi-degree of

freedom systems. F. Pfeiffer, along with K. Karagiannis [20] has investigated

4


gear rattle phenomenon theoretically and experimentally.

The interaction of various nonlinearities of powertrain is another interest

ing area for research and study. A. Kahraman and R. Singh [21] have studied

the interaction between time varying meshing stiffness and backlash in a geared

system. The existence of a strong correlation between these two nonlinearities

has been found. They have also studied resonance associated with forced excita

tion. The excitation force they have used consists of a mean torque with periodic

fluctuation. Digital simulation techniques are employed to solve the equations.

They have also studied the nonlinear dynamics of a spur gear system with mul

tiple clearances. They have considered both backlash and radial clearances and

studied the interactions. They [22] have also tried to give insights on chaotic

behavior.

T.E. Rook and R. Singh [23] also have studied rattle noise in gear pairs

with multiple clearances, such as idle gear. They have analyzed spectral interac

tions and proposed the concept of effective stiffness to simplify the solution. C.

Padmanabhan and R. Singh [24] have studied the influence of mean and varying

load components on an oscillator with clearance type nonlinearity. They conclude

that mean load has greater influence when the system has time varying nonlinear

parameters. They also notice a coupling effect between amplitude of response

and mean load. C. Padmanabhan and R. Singh [8] have investigated two degree

of freedom systems and possible resonance interactions.

Some research has also been done on the effect of a particular parameter

5


on actual rattle noise. C. Padmanabhan and R. Singh [25] have studied the

influence of clutch on rattle noise. A. Szadkowski [15] also has observed a strong

link between clutch and rattle noise. N.N. Powell and S.A. Amphlett [26] have

studied the transmission rattle using a suitable model. Their main objective is to

establish a relationship between backlash and gear noise. They have found that

there exists a critical backlash above which that noise actually reduces.

K. Umezewa, et al. [27], has analyzed the effect of contact ratio on vi

bration of a helical gear pair. Experimentally, they have determined that rattle

performances are good if contact ratio is greater than 2. I. Nakagawa, et al. [28],

has also proved that contact ratio plays an important role in gear noise. They

determine that pressure angle and bias contact also have influence on gear noise.

A. Kahraman and R. Singh [29] have studied impact of nonlinearities due to back

lash on vibrations of a spur gear pair. They have considered both external and

internal excitations. Harmonic balance method and digital simulation techniques

are used to solve the nonlinear equations. Furthermore, they have observed cases

of no impacts, single, and double sided impacts.

Some researchers have specifically concentrated on actual noise reduction

and have come up with suggestions to improve rattle performance of a powertrain.

Amo Sebulke [30] has detailed the advantages of dual-mass flywheel over the

conventional flywheel. It is suitable to counteract rattle noise and behaves as a

low pass filter. It isolates the rotational irregularities and vibrations from the

complete driveline. A. Laschet [31] has listed various causes of rattling noise

6


and operation ranges when each of them becomes serious. He also outlines a

construction of a vibration model and a computer simulation. He has observed

that major contributor to noise comes from powertrain and hence it characterizes

overall noise of the system.

H. J. Drexl [32] has discussed the role of conventional torsional dampers in

noise reduction and also presented alternative methods. The alternative methods

suggested are a twin mass flywheel system with a torsional damper between

the masses and hydrodynamic couplings for decoupling torsional vibrations by

means of a slip. M. Ill [33] has studied the possible ways to reduce gear noise by

changing contact ratio, pressure angle, gear finish etc. He also considers vibration

of housing, bearing, and their transmission characteristics. C. Padmanabhan, R.

Singh [25], and A. Szadkowski [15] have come up with the clutch spring values

to be used in order to reduce rattle noise.

Efforts have been made in the past to set up rattle criteria for rattle severity

evaluation. R. Singh, H. Xie and R.J. Comparin [6] have come up with a rattle

criterion based on the angular acceleration of the input gear. 0 . Johnson and

N. Hirami [34] have made experimental setup to determine the sources of gear

rattle and tried to objectively measure noise. Analysis of gear motion and casing

vibrations is deemed to be a good basis for rattle evaluation. C. Padmanabhan,

Todd E. Rook and R. Singh [16] have proposed various rattle indices to objectively

evaluate noise level. F. Pfeiffer and W. Prestl [17] have proposed a rattle index

based on the coefficient of restitution of the teeth impacts.

7


1.3 Numerical algorithms for stiff problem

In this paper, besides the ODE stiff-suite algorithms from MATLAB, an alterna

tive numerical integration algorithm, Finite Element in Time domain (FET),

is implemented for solving of transient and steady state response of geared-

transmissions with backlash. The method employs the technique by describing

the system as a boundary value problem. This requires solution of a set of highly

sparse algebraic equations, which describe response in terms of a set of temporal

nodes with all spatial degrees of freedom of the system.

The ODE stiff algorithms are suitable for the nonlinear system with stiff

characteristics, which is defined by the high ratio of maximum and minimum

eigenvalue of the system matrix [35]. Conventional ODE algorithms fail to eval

uate solutions for stiff system. Powertrain system is highly nonlinear, and in

evitably is a ’stiff’ system [36, 37].

The Finite Element in Time domain (FET), based on Hamilton’s principle,

can be employed to solve the dynamic response of system in which solution for

all spatial degrees of freedom at all time steps within a given time interval of

interest is sought via a set of algebraic equations [38, 39, 40, 42]. Some recent

development has shown that this approach offers several potential advantages over

the numerical solution of ordinary differential equations in time domain, such as

the flexibility in formulating the problem directly from system energy expressions,

the greater accuracy at specific time points of interest, and the use of adaptive

8


finite elements to improve computational efficiency and accuracy [38, 41, 42, 43].

Previously, the application of FET using primal form has been studied by

Y. Wang and his group [44, 45, 46] for periodic response prediction as well as for

modeling and analysis of automotive transmission rattle.

FET provides several potential advantages over finite difference based nu

merical methods. Through propagation of the time elements within the period of

the steady-state response, or with given initial state for transient case, a possible

solution for both cases exists. In addition, it is possible for the usage of adaptive

finite elements to improve computational efficiency and accuracy.

The periodic solution can be readily available by assembling a number of

time elements and imposing the appropriate periodic boundary constraint rela

tions, in contrast with time-stepping methods, which require initial conditions to

start the integration procedure but cannot impose periodic boundary conditions.

The second advantage is the straightforward determination of stability of

periodic solution. Since the corresponding transition matrix for analysis of the

stability of small perturbations about the periodic solution is a by-product of the

FET procedure, Floquet’s theory of stability can be readily applied without any

special effort.

1.4 Scope of dissertation work

The object of this dissertation is to develop a suitable modeling method as well as

the robust numerical algorithm, thus to provide a powerful tools for parametric

9


study and application to powertrain design.

In chapter 2, a decoupled model for powertrain system is developed based

on the understanding of powertrain components’ characteristics. This system

model must be well defined before any analysis can be implemented.

The decoupled model of powertrain system includes two parts: (1) baseline

model, which is a multi-DOF system including all parts of powertrain system,

even unladen floating gears yielding rattle noise whose equivalent moment of

inertia is added to their meshing splines; (2) rattle model, which includes several

pairs of unladen gears meshing, and can be solved independently. The responses

of spline gears from baseline model will be fed to the rattle model as inputs and

will thus attempt to seek rattle responses.

MATLAB ODE stiff algorithms and/or FET method could be used for

solution of both baseline multi-DOF model and SDOF rattle model. Chapter 3

discusses ODE stiff suite algorithms briefly. Major efforts have been put on to

the derivation of FET mixed form algorithm and software implementation under

MATLAB environment.

By using this model with ODE stiff-suite and FET algorithm, it is possible

to identify transmission problems before manufacturing, and to achieve an opti

mal design of powertrain for rattle performance. In chapter 4, after discussion of

several related issues, such as shafts rigidity and time varying meshing stiffness

of unladen gears, analysis for one of speed configurations has been performed.

It can be noticed that several elements introduce nonlinearities into the

10


system, within baseline and rattle models. They come from the multi-stage stiff

ness and dry friction of clutch, time-varying meshing stiffness of pairs of laden

and unladen gears, as well as backlash of unladen meshing gears. The effects of

backlash as well as drag torque on rattle are studied in Chapter 5 with decoupled

model developed and numerical algorithms implemented. The analytical results

allow designer to evaluate trade-offs of various designs without recourse to ex

pensive or inefficient palliative measures. These benefits are the motivations for

development of this gear rattle simulation methodology. Such a simulation tool

entails a reliable numerical technique for solving the dynamic response for gear

rattle.

This dissertation has developed the capability for modeling and simula

tion of the rattle dynamics in manual transmissions. The work in this disserta

tion provides comprehensive understanding of torsional vibration and gear rattle

characteristics with respect to major design parameters. Furthermore, modeling

and numerical algorithms can be integrated into vehicle powertrain development

programs. It is expected to make a significant contribution to shortening the

development cycle of new powertrain models.

11


Chapter 2

M odeling For Powertrain Gear R attle

2.1 Introduction

Gear rattle has long been a main concern in refinement of vehicle powertrain sys

tem. Many efforts have been put on to this issue. Other than previous modeling

and analysis methods, a new methodology is developed here.

In this chapter, system modeling is covered in detail before vibration and

rattle analysis can be performed, so as to implement modeling and analysis for

rattle.

2.2 System m odeling and problem formulation

Without an adequate system model, any analysis would be impossible. Based

on the understanding of powertrain components as well as their characteristics,

a new modeling method is proposed here.

12


Ry WheelInput Shaft

Gear Pairs

(

Synchronizers

Figure 2.1: Illustration of Dodge Neon powertrain with NT350 transmission

2.2.1 T he powertrain com ponents and rattle sources

Before the modeling method is discussed, the components of the powertrain are

worthy of a detailed look.

The powertrain contains many parts. It consists of input/output shafts,

pairs of gears, synchronizers, clutch, and some other necessary parts, as illustrated

in Figure 2.1. For a certain speed, all pairs of speed gears are meshing together,

but only a specific pair is engaged to transmit power through the function of

synchronizer.

It is obvious that it could only be modeled as a multi-DOF system, and in

evitably with high nonlinearity due to the nonlinearity of the clutch. Nonlinearity

of unladen gear pairs in meshing will be considered in rattle modeling .

For a typical powertrain layout, the essential elements in powertrain dy-

13


X

X,D

One Engine Revolution

t

Figure 2.2: Engine excitation torque fluctuation

namics include the following:

E ngine Excitation: The engine is the principal source of torsional exci

tation in driveline. Rotational fluctuations of the engine crankshaft are m ainly

caused by the firing pulses, combustion non-uniformity, and unbalanced inertia

forces [49]. The dominant component of the engine torsional signature is the

ignition firing, which shows up twice per engine revolution in a standard four-

cylinder engine, or in the second-order. At wide open throttle, the engine torque

fluctuation is nearly constant regardless of engine speed at a special amplitude

of about 10% (25% for diesel engine) of the mean engine torque, as illustrated in

Figure 2.2 .

C lu tch : The clutch plays an important role in transmission rattle behavior

and also in drivability. It typically consists of several strong nonlinearities: multi

stage torsional spring rate, dry friction damping, and preloads on the torsional

14


springs.

Currently, critical clutch characteristics are often evaluated by a timing

process using audible subjective rations by experienced engineers in order to de

termine the best combination of the nonlinearities which will “dampen” particular

rattle/vibration problems. It is a practical yet limited approach.

Transmission: The transmission includes three major types of compo

nents: input and output shafts, speed gears, and synchronizers. In modem trans

missions, all forward speed gears are helical gears with spline pinions. At any

speed shift, e.g., the 1 st shift, all the pinions are in rotation and are continu

ously meshing with their respective gears; but only a particular pair of gears of

the shift (e.g., 1 st shift) transmit power, when engaging the respective synchro

nizer. Three synchronizers are used in New Venture Gear’s NT350 transmission

for achievement of high gear shift quality.

These power-transmitting gears are laden gears and, together with the

shaft-synchronizer assemblies, they provide the baseline torsional vibration char

acteristics. The shaft-synchronizer assemblies are essentially linear with torsional

stiffness and the moment of inertia. The laden gear pairs exhibit a time-varying

meshing stiffness shown in Figure 2.3, due to the conjugate action of the involute

helical gear teeth. The details of contact stiffness function for a pair of meshing

gears will be covered later in Chapter 4.

In the in-drive mode, 4 pairs of unladen gears, however, cause the rattle

problem. Without transmitting any load but in meshing, the gear and the spline

15


300Average Stiffness

Single-pair meshing

200Two-pair meshing

CB00

Meshing Period

Meshing duration

Stan of meshing End of meshing

Figure 2.3: Time-varying gear meshing stiffness

teeth may be driven across the backlash, causing impacts and rattle noises. The

unladen gear pairs are all potential rattle sources. Rattle may also occur at idle

when the vehicle is at rest, the transmission is in neutral, while the clutch is

engaged. In the T350 transmission, gears for the 1st and 2nd speeds axe driven

by their spline pinions on the input shaft idle, becoming potential neutral rattle

sources.

2.2.2 Torsional vibration for rattle analysis

Gear rattle phenomena of a manual powertrain is typified by complex interactions

between the torsional vibration characteristics of the driveline and the coupled

vibro-acoustic response of unladen gears. The first step of our research effort is

to determine an appropriate model for efficient simulation while retaining all of

16


Figure 2.4: The powertrain model for the first speed

the essential dynamic characteristics. The primary objective is to examine the

dynamic interactions between the rattle gears and the key driveline components.

An insight into this process can be gained by examining the path of torque

transmission in the system. Again taking the 1st gear shift as an example as

shown in Figure 2.4.

The engine power is transmitted to the input shaft through the flywheel

and the fully engaged clutch. Then after velocity reduction through the first-

speed gears, the power is transmitted to the output shaft and subsequently to

the differential. The engine torque fluctuation generates torsional vibrations in

the components on the power transmission path. The speed-reduction gears are

continuously loaded and their teeth maintain continuous meshing in spite of the

17


ENGINEVss ~ Toque RmhaSan

CLUTCH BacUashiTme-Va/yngSffiessAMhStege

Springs

Torsional Vbratona ton _______ \

\ UNLADEN \ \GEARS —

INPUT SHAFT

LADEN GEARSTorsional Vbrafon

Tm-VaryingUeshSShass

OUTPUT SHAFT

Figure 2.5: The cause and effect diagram of gear rattle system

presence of backlash. Therefore, the laden components of the power path are

largely linear, except for the multi-stage stiffness of the clutch, which is piecewise

nonlinear. On the other hand, with the employment of all spline pinions and

synchronizers, other 4 gear pairs axe also meshing but without loading. Through

their backlash, the unladen gears will yield vibro-impact motions and rattle noise.

The relationship between the torsional vibration and gear rattle is illus

trated in the cause-and-effect diagram of Figure 2.5. The laden components of

the transmission form the b a s e l i n e s y s t e m , while the unladen gears become

a p p e n d a g e s to the baseline system. The engine torque output excites the laden

baseline system, which in turn vibrates the unladen gears. Thus, gear rattle phe

nomenon is the result of the interaction between the baseline torsional vibrations

and the vibro-impact characteristics of the unladen gears.

18


PwerTrjnsm iwon Pith

" ►

Figure 2.6: Torsional vibration model for the first speed

Therefore, a physical model of the transmission system is obtained accord

ing to the gear configuration of each shift. Figure 2.6 demonstrates the model

for 1st shift, or speed.

The baseline laden system consists of several degrees of freedom, with linear

torsional stiffness in drive shafts, time-varying contact stiffness between laden

gears, and piecewise linear torsional stiffness in the clutch, and the 4 unladen

appendages are strongly nonlinear because of the backlash, as shown in Figure

2.7. The number of degree of freedom of the system model depends on the model

chosen and whether it is a rigid shaft or flexible shaft, which will be covered later.

In the model, the inertial properties of the differential, C V joints, and drive shafts

are lumped to the end of the output shaft.

19


Clutch

U nladen G ear B acklash

G ear B acklash

Figure 2.7: Various stiffness characteristics of powertrain parts

An engine torque applied at the flywheel is of the form,

r { t ) = rm + Tf ( t ) (2 .1)

with a fluctuation period T m r j .

In general the fluctuation amplitude is about 10% of the mean torque for

gasoline engine, while at 25% for diesel engine. The mean torque is crucial for

the baseline system, because the clutch stiffness value depends on it, thereby

affecting the dynamics of the system.

20


Driving Gear

Base Circle 1

Pitch Circle 1Line of Action

Pitch Circle 2Base Cricle 2b2

Driven Gear

Figure 2.8: A meshing gear pair

2.3 Dynamics o f a gear pair w ith backlash

Before starting the discussion of the gear rattle model, the geometry and dynam

ics of a meshing gear pair with backlash will be covered first.

The basic structure of a meshing gear pair is shown in Figure 2.8. The

shafts of the two gears are assumed to be rigid and the only compliance considered

in this model is the compliance of the gear teeth. The measurement of mesh

compliance, teeth contact stiffness, will be examined later. The effect of backlash

between two meshing gears is considered in this section.

Backlash is the gap between mating teeth measured along the circumfer

ence of the pitch circle. Manufacturing tolerances preclude a zero backlash, as

21


all teeth cannot be exactly the same dimensions, and all must mesh without

jam m ing. So, there must be some small difference between the tooth thickness

and the space width. As long as the gear set is run with a non-reversing torque,

the backlash should not be a problem. However, whenever the torque changes

sign, or amplitude, the teeth will move from contact on one side to the other.

The backlash gap will be traversed and the teeth will impact with noticeable

noise and vibration. As well as increasing stresses and wear, backlash can cause

undesirable position error in some applications.

Backlash, causes discontinuous phenomena and impact effects on dynamics,

brings one uncertainty to the dynamic model of the gear pair system.

Backlash, 6, is defined as the clearance measured along the line of action

of a gear pair as shown in Figure 2.9. The clockwise direction of 0\ on the pinion

is defined to be positive, while the counterclockwise direction of 02 and on the

driven gear be positive.

The neutral position of a gear pair is defined as the position where the

centerline of a tooth in the drive gear 1 and the center of a tooth space on the

driven gear 2 are both coincident with the centerline of the two gear centers. The

approach portion is the part from the first point of contact to the pitch point on

the line of action and the recess portion is the part from the pitch point to the

last point of contact. Front side contact occurs when the leading edge of gear 1

meshes with the trailing edge of gear 2, and back side contact occurs when the

trailing edge of gear 1 meshes with the leading edge of gear 2, shown in Figure

22


Driving Gear

Base Circle 1

Pitch Circle 1Line o f Action

Ba< klashPitch Circle 2

Base Cricle 2b2

Driven Gear

Figure 2.9: Geometric relation of a meshing gear pair with backlash

2.9.

The dynamics of such a system can be divided into three cases according to

whether the two meshing gears are under front side contact, separation, or back

side contact. What follows is that the frictional forces at the point of contact

and at the journal bearings are neglected. The normal forces along the line of

action for meshing gears will be modeled as a combination of linear elastic and

damping forces.

Front side contact

When r iOi —^ 2 ^ 2 > b, the leading edge of gear 1 contacts with the trailing

23


edge of gear 2. The equations of motion can be written as,

{hQ_\ = n - Fnrbi I2®2 = 12 + Fnrb2

(2.2)

where

Fn = k (q -b )+ c q (2.3)

denotes the normal contact force with,

{<7 = rbiOi - rb202 q = rbxdi - rb2&2

(2.4)

denote the relative motion between the gears and the relative speed along the

line of action, respectively.

Separation

When b > rb\0\ - rb2Q2 > - b, separation occurs and there is no contact

force between two gears. Therefore, the equations of motion are given by

{& :;Back side contact

When rb\6 \ — r 1,262 < —b, the trailing edge of gear 1 meshes with the

leading edge of gear 2. The equations of motion are given as

hQ\ = Ti - Fnrbi I2 §2 = t 2 + Fnrb2

(2 .6 )

where

Fn = k(q + b) + cq (2.7)

24


denotes the normal contact force with,

/ q = rw 0i - rb262 ft>1 q = r j l - r b2e2 ^

denote the relative motion between the gears and the relative speed along the

line of action, respectively.

For all these three cases, the contact stiffness k will be covered later. While

damping coefficient c is selected as proportional damp with a known damping

ratio.

2.4 Vibration and rattle analysis

The physical models of the powertrain with manual transmission described above

consist of 5 separate models for the 5-speed in-drive modes and a neutral model

for the at-idle mode. The second phase of the research development is the numer

ical modeling and analysis of the dynamics of each model. The specific goal is to

understand (i) characteristics of the baseline torsional vibration of the laden sys

tem, (ii) rattle vibrations of unladen pairs of gears, and (iii) interactions between

the baseline system and the rattling appendages.

It has been widely recognized in the literatures of the field of study that

the unladen gears undertake the main role of rattle impacts [3]; Their vibration

have little effects on the motion of the baseline system. This fact can be utilized

to study the overall system behavior more effectively.

By an appropriate lumping of unladen gear inertia, vibration of the laden

system can be characterized by its response to the engine excitation. The result-

25


ing baseline torsional response then becomes the excitation to the unladen gears.

By focusing on the dominant effects, the suitable model for effective numerical

analysis will be obtained. This would also greatly facilitate the tasks of paramet

ric studies and design optimization. These analyses will be accomplished through

the use of a powerful numerical method proposed in the following chapter.

2.4.1 B aseline torsional vibration

The first goal of numerical analysis is to characterize the baseline vibration, i.e.,

the response of the laden system to engine excitation. This is an important under

standing for clutch design, in which torsional springs are employed to attenuate

the baseline vibration response. As shown in Figure (2.4) for the 1 st shift, the

baseline system is essentially linear, except for the multi-stage stiffness of the

clutch and the time-varying meshing stiffness could be described as

MQ + CQ + KQ + F(6 ) = r (2.9)

where 0 is rotation vector representing the n degrees of freedom of the baseline

system. M, K , C denote the mass, stiffness, and damping matrix of the system

respectively. While F(9), denotes the nonlinear force due to the multi-stage

spring of the clutch.

Here, it should be pointed out that the gear meshing stiffness of laden pair

of gear k(t) is of the dimension of linear stiffness, which requires concern when

it is assembled into the stiffness matrix K , in which the entries are all of the

dimension of rotational stiffness. The effect of the time-varying meshing stiffness

26


X(t'/•

(a) Rattle gear pair model Time-Varying-Piecewise-Linear

Figure 2.10: The physical model of a pair of unladen gears with backlash

of the laden pair of gears will be studied in the later chapter.

The equation of the baseline system can be solved by several different ways,

such as the ODE method, or the numerical method which will be fully discussed

later, finite element method in time domain of course.

2.4.2 Gear rattle m odeling

The unladen gears are the sources of rattle noise and their vibro-impact behavior

is the focus of our dynamics study. Any pair of the unsynchronized gears with

backlash is modeled as a SDOF system with the motion of driving gear being

known, as shown in Figure 2.10. For the most general case, the gear backlash is

defined by a piecewise meshing load between meshing gear teeth, and the meshing

stiffness is time-varying.

From the previous derivation, equation of motion for driven gear I2 with

27


respect to the linear relative displacement q — x(t) — X (t) = d2r2 — 6 \r i,

"^ 9 + fn (t , q) = - \ h r \ - Td/ r 2 (2.10)r2J r 2z

where rd denotes the drag torque applied on the driven gear, and f n{t,q) refers

to the nonlinear force between the pair of meshing gears due to backlash, which

is defined as,

U (t, q) =k(t) (q — b) + cq if q — b > 0;

0 ii —b < q <b] (2.11)k(t) (q + b) + cq if q + b < 0.

Thus the independent variable is q, with the given Q\. It becomes the

equation of motion for SDOF system that could be solved by various numerical

methods.

2.5 M odel verification

The decoupled model neglects the effect of the backlash for the baseline model

analysis first. Then, the responses of the driving gears of the rattle pairs are

fed as the input for the rattle analysis. This treatment simplifies the model

by solving them in two steps, which dramatically reduces the computation time

and maintains the necessary accuracy. The validation of the decoupled model is

provided by comparing the results from this decoupled model with that of the

so-called full model.

2.5.1 Equations o f m otion for full and decoupled m odel

The layout of the physical model used for the illustration of modeling verification

is shown in Figure 2.11. It consists of the flywheel, input and output shafts, laden

28


94

Wheel

Input ShaftClutch

2nd

Output Shaft

Figure 2.11: Powertrain model with 4 DOFs

gears, as well as one pair of rattle gears with the driving gear on the input shaft.

As for the purpose of illustration and simplicity, the rigid shaft model is

employed. The configuration and parameters are of the first speed, and rattle

pair of gears is the 2nd pair.

By carrying out the free body diagram analysis, the equations of the

motion for this system without any simplification is given below,

h \Q \ + T d = t

72202 — Tel + k i • T bu ( ^ i l l * 02 ~ 7*621 * 0 3 ) _ fn * 7*612 — 0 ^73303 — k\ • Tin (fill ' 02 — 7*621 ' 03) = 0

„ 7t404 + / n 7*622 = Td

where ra denotes the clutch torque.

The clutch is piece-wise linear due to the multi stage springs. Its charac

teristics are illustrated in Figure 2.12,

29


Coast side

-13

Degree

Drive side

Figure 2.12: The force characteristics of clutch

Thus the clutch torque can be expressed as,

rd =

K \ (0\ — 02 — Pi) + K2 ■ Pi i f 0\ — 6 2 > Pi ;K 2 (0i — 62) i f 0 i > 0i — 02 > 02]Kz (01 — 02 — 0 2 ) + ^ 3 * 02 i f 0 2 > 01 — 02 > 0 3 ’,

„ K 4 (01 — 02 — ^ 3 ) + K 2 • 02 + Kz ■ (0 3 — 0 2 ) i f 0 3 > 01 ~ 02-(2.13)

where K\ to K 4 denote the rotational stiffness corresponding to the different

stages of the clutch spring and of the rotational stiffness dimension.

By introducing the linear difference, q, between the rattle pair of gears,

9 — r b22 • 04 ” r 612 • 02 (2.14)

The contact force, /„ between the rattle pair of gears, can be given as,

(2.15)

Where k and b are the contact stiffness, and the backlash of the rattle pair of

k(q — b)+cq if 9 — 6 > 0; fn = { 0 if —6 < q < b;

k{q + b) +cq ifg + 6 < 0 .

30


gears respectively. Here the effect of the time-varying stiffness is put aside, and

the constant meshing stiffness is considered.

By neglecting the backlash between the rattle gears, the equations of

motion (2.12) for the full model above could be simplified, which then yields the

decoupled model.

The contact force /„ would be independent of backlash, of the form /„ =

k ■ q + cq. With the fourth equation of eqn.(2.12), considering no drag torque,

Assuming that the deformation between the rattle gears is small compared

to the motion of themselves, the gear law holds here, which is r612 • 02 = rb22 ■ 04-

It could be extended, and yield,

(2.16)

7-612 * @2 = 7*622 * 04 (2.17)

Thus 04 could be expressed as,

(2.18)7*622

with this, the second equation of the equ.(2 .12),

72202 — Td + k\ ■ rb 11 (rb 11 • 02 — 7*621 * #3) + 744 • • 02 = 0 (2.19)7*#,oo

The equations of motion then can be written,

7 ll0 1 + T d = T

( j %2 + 7m • 02 — T d + fci • r b 11 ( f i l l * 02 — 7*621 • 03) = 0

73303 — k \ • Till (r ill • 02 — 7"i21 • 03) = 0 7m04 + (kq + cq) rb22 = 0

(2 .20)

31


The second equation of the equ.(2.20) is now decoupled from the 4th DOF,

04. Thus the first three equations coupled together could be solved individually,

which has nothing to do with the backlash.

The fourth equation, including the effect of backlash, is of the form,

^4404 + fnTva — 0 (2 .21 )

This is the single DOF system with the solution of 02 decided from the

first three equations.

Through these procedures, the system is decoupled into two independent

parts, in which the first part is represented by the first three equations, reflecting

the baseline, and the second part, the SDOF equation, reflecting the rattle pair

of gears.

2.5.2 A nalysis results

Simulation has been performed with the geometric parameters and material prop

erties of powertrain provided by Chrysler Corp. and New Venture Gear Inc. The

results of the 2nd rattle gear pair for Speed 1 configuration are given and dis

cussed in detail to demonstrate the side by side comparison between the full and

decoupled model.

Figure 2.13 and Figure 2.14 show the driving motion of the rattle pair,

from the decoupled model and full model, respectively. The difference is that

the acceleration of driving gear from the decoupled model does not have the

’’hit back” effect which is indicated from full model analysis. This difference

32


Motion of 01

6

4

2

00 0.1 0.40 2 0 3 0.7 1Time Sec

20

5 10

f 0a>-1 0

0 0.1 02 0.4 0.7Time Sec

4000

£ 2000

0 0.1 0.4 0.6 0.7Time Sec

Figure 2.13: The motion of driving gear from decoupled model

Motion of 81

0.1 05 Time Sec

0 2 0.4 0.60 3 0.7

20

-200 50.1 02 0.4 0.7

Time Sec

-4 0 0 0 '------------1----------- 1_______ 1_______ 1------------ 1------------1_______ 1_______ i i_______0 0.1 02 0 3 0.4 0 5 0 5 0.7 0 5 0 5 1

Time Sec

Figure 2.14: The motion of driving gear from full model

33


Motion 02

0.1 0.4 0.5 Time Sac

0.6 0.7

5 10

-50.10 02 0.4 0.6 0.7 0.9

Time Sec10000

“ 5000

mu

-50000.1 0.4 0.60 3 0.7

Time Sec

Figure 2.15: The motion of driven gear from decoupled model

comes from the neglect of the backlash, which causes the absence of impact force

from the beginning of impact, at either drive side or coast side. All the while,

displacement and the velocity of the driving gears are quite similar. Additionally,

The ’’hit back” effect is very small compared to the driving gear acceleration itself.

Thus, the premise that the effect of the rattle gear on the baseline torsional

vibration can be neglected still stands.

Figure 2.15 and Figure 2.16 show the responses of driven gear from de

coupled model and full model respectively.

By studying the rattle, it can be seen from Figure 2.17, and Figure 2.18,

as the decoupled model neglecting the backlash for the stage of evaluation of

driving gear motion, the effect of hit back of driving gear would not be taken into

34


Motion of 82

23

0.1 03 Time Sac

03 0.4 0.6 0.7 0.8

20

€■cc*•5

-100.1 02 OS

Time Sec03 0.4 0.7 0.8 0.9

x 10*1eg

- 10 0.1 0 2 03 0.4 OS 0.6 0.7 0.8 0 3 1

Time Sec

Figure 2.16: The motion of driven gear from full model

account, when unladen pair of gears are hitting. This effect causes the contact

force between the rattle pair to be somewhat amplified. It is evidenced in the

figures of the contact forces for two models. By checking the rattle indices defined

in Chapter 4 for both cases, the rattle index from the decoupled model is about

10% higher than that from the full model which represents the real case.

In order to check whether this amplification effect is consistent for the

different speed configurations and different pairs of rattle gears, by neglecting

the backlash for the decoupled model, the side by side comparison has been

carried out for the different speeds. The rattle indices, which will be covered in

Chapter 4, are shown in the Table 2.1.

35


x 10"* Displacement rgeart Veiodty rgeart

0 0 2 0.4 0.6 0.8Time Sec

Excitation

■o 4

0.4 0.6Time Sec

0 0 2 0.4 0.6 0.8Time Sec

Contact force100

zcu.

-50

-100

0.6 0.80 0.4Time Sec

Figure 2.17: Rattle result from decoupled model analysis

Rattle Indices Decoupledmodel Fullmodel RatioSpeedl,pair2 0.8741 0.7904 1.1059Speed2,pairl 0.5130 0.4672 1.0980Speed3,pairl 0.4387 0.3948 1.1112SpeedA,pair\ 0.3525 0.3202 1.1007SpeedS, pairl 0.6932 0.6325 1.0959

Table 2.1: Rattle Indices : Decoupled model vs. Full model

36


x 10 Displacement gearl Velocity: gearl

I I I0 2 0.4 0.6 0.8

Time Sec

Contact force

5 3

0.4 0.6Time Sec

r i r r r r

0.4 0.6TmeSec

Figure 2.18: Rattle result from full model analysis

From these results, it can be said that the rattle index from the decoupled

model is 9.8% to 11.1% higher than that from the full model. Also, it can be

evident that the amplification factor is consistent. This makes the decoupled

model available for the rattle evaluation.

The introduction of the decoupled model simplifies the analysis and saves

the computation time of the simulation dramatically. The hourly computation

with the full model is reduced to minutes with the introduction of the decoupled

model. This modeling method is adequate and efficient.

37


Chapter 3

Numerical Algorithms for Gear Rattle Analysis

3.1 Introduction

The numerical algorithms employed for baseline and rattle model solution are

the stiff suite of Ordinary Differential Equation, as well as the Finite Element in

Time Domain.

In this chapter, the characteristics of ODE stiff suite are discussed and

the formulation and derivation of FET are covered in detail.

3.2 ODE stiff suite

Any initial value problem can be expressed as,

y = F{t,y) (3.1)

on a time interval [to, t/\, given initial values y(t0) = yo-

For many years, MATLAB has had only two ODE solvers available, ode23

and ode45 [37]. Even though they employ fairly simple algorithms, they have

38


proved remarkably effective. SIMULINK provides some additional methods, but

they are not easily accessible to MATLAB users.

When the system equation is claimed as ’stiff’, it means that the ratio of

maximum and minimum eigenvalue of the system matrix is very high [35]. The

algorithm used for the solution of this kind of system must be adapted for this

’stiff’ problem. The special ODE suite algorithm is aimed for this issue.

MATLAB presented their ODE stiff suite recently. A new family of for

mulas for the solution of stiff problems called the numerical differential formulas,

NDF’s, are devised. These formulas are more efficient than the backward differ

entiation formulas, BDF’s, although the higher order formulas are somewhat less

stable. There are two formulas available now, 0DEl5s, and ODE23s [36].

3.2.1 The ODE15s Program

The code ODE15s is a quasi-constant step size implementation of the NDF’s

in terms of backward differences. Options allow integration with the BDF’s and

integration with a maximum order less than the default of 5. It is natural to form

and factor the iteration matrix every time the step size or order is changed. The

rate of convergence will not be achieved in four iterations. Should this happen and

the Jacobian not be current, a new Jacobian matrix will be formed. Otherwise

the step size is reduced.

The scheme for reusing the Jacobian means that when the Jacobian is con

stant, 0DE15s will normally form a Jacobian only once in the whole integration.

39


Also, the code will form very few Jacobians when applied to a problem that is

not stiff. This ensures its efficiency, compared to the other codes, for non-stiff or

stiff problems.

3.2.2 T he ODE23s Program

Meanwhile the code of ODE23s uses the linearly implicit formulas for stiff

systems rather than the ODE15s, which employs the implicit formulas for stiff

systems.

The code ODE23s provides an alternative to ODE15s for the solution of

stiff problems. It is especially effective at crude tolerances, when a one-step

method has advantages over methods with memory, and when Jacobians have

eigenvalues near the imaginary axis. It is a fixed order method of such simple

structure that the overhead is low except for the linear algebra, which is relatively

fast in MATLAB. The integration is advanced with the lower order formula, so

ODE23s does not do local extrapolation.

The current version of ODE23s forms a new Jacobian at every step for

several reasons. A formula of order 2 is most appropriate at crude tolerances. At

such tolerances solution components often change significantly in the course of a

single step, so it is often appropriate to form a new Jacobian. In MATLAB the

Jacobian is typically of modest size of sparse and its evaluation is not very time

consuming compared with the evaluation of F. Lastly, evaluating the Jacobian

at every step enhances the reliability and robustness of the code.

40


For non-autonomous problems ode23s requires an approximation of dF/dt

in addition to the approximation of dF/dy. For the convenience of the user and

to make the use of all the codes the same, approximating of this partial derivative

numerically is always chosen.

3.3 Finite element in tim e domain

The details of the formulation of FET are discussed. The equations of motion

are the most general ones, which could be applied to any case. From this point

of view, it can be seen that the formulation is suitable for any kind of problem,

Single DOF or Multi DOF systems, linear or nonlinear systems.

In general a /V-DOF system with non-linear forces applied can be described

as the following equation of motion,

Mq + Cq + Kq + g{q) = f(t) (3.2)

where M, C, K, denote the mass, damping and stiffness matrix of the system,

respectively, g(q) represents the non-linear forces applied to the system, and

/( t) , the external excitation forces. If the linear system is considered, g(q) term

will be deteriorate into linear form and can be considered by adding one more

term onto system stiffness matrix.

3.3.1 Form ulation o f F in ite elem ent in tim e dom ain

The finite element method in time domain is based on a weak form of Hamilton’s

law. Commonly known as Hamilton’s weak principle, it can be expressed in its

41


general displacement form as

f tf (6 L + 6 W)dt = 5qr -p\ (3.3)

3.3.2 Prim al form o f FET

A displacement formulation may be devised by resorting to the principle of virtual

work, thus requiring a displacement field compatible with the deformations. This

implies that the equation

is satisfied, and that the displacement boundary conditions are satisfied as well.

When this relationship is enforced, it leads to the following displacement form,

where only x is the independent field. If the position vector x is stated as a

function of a suitable number of generalized coordinates q and of time t, it yields,

where L denotes the Lagrangian function, and Q are in general non-conservative

generalized forces.

This equation is well known as ” Hamilton’s law of varying action”, which

becomes Hamilton’s principle if the test functions axe chosen so as to vanish

at the boundaries. In the following, this form is referred to with the name of

’primal form’, since it deals only with one independent and thus the primal field

(3.4)

[38, 42, 44).

42


It is shown that the analogies between these weak forms and the well-

known weak forms of elastostatics, are not restricted to a slight resemblance.

Particularly, the locking phenomenon [42]which may be observed in solid me

chanics in pure displacement formulations, has a corresponding analog even in

pure displacement formulations for dynamics, namely the primal form.

This remark sets forth the need to develop an alternative weak form where

the independent fields are represented by generalized coordinates and momenta,

thus establishing what will be referred to as a ’mixed form’. This second approach

seems to be much more alluring even from the point of view of Hamiltonian

mechanics: the phase space of a system is represented giving the same dignity

and the same order of approximation to its two components, the generalized

coordinates q and momenta p. A single field formulation has not this kind of

parallelism in the treatment of q and p since the momenta are introduced by

means of the time derivatives of the generalized coordinates, thus negatively

affecting its numerical behavior [42].

3.3.3 M ixed form o f FET

The form employed for the formulation of finite element model in this thesis is

the mixed form, which has some advantages, as discussed before, over the primal

form.

A Legendre transformation can be applied to the Lagrangian function L,

transforming the velocities into momenta and the Lagrangian function into the

43


Hamiltonian function, thus leading to the sought-for mixed form.

There are several kinds of mixed forms available. The one employed here

is

£ " ( i f p + t f q - S H + 6 f Q ) d t = S f ? |!T* (3.7)

The independent fields in this case axe p and q. Prom definition, p and q

are

9LP = -gr = Mq

q = M~lp (3.8)

respectively. While the Hamiltonian function

H = pTq - L

= \ jF M ~ lp + \qTKq (3.9)

and,

Q = F - C q

= F — CM~lp (3.10)

thus,

__ d K .m = a^p+V ?

= pTM -1Sp + qTKSq (3.11)

Substituting into equation 3.7 yields,

Su+1 [ ^ P + SjFq — jFM~l8 p - qTK 8 q+

SqT (F - CM ~lp)\ dt = 5qTp6 |£+l (3.12)

44


3.3.4 F in ite elem ent approxim ation

In order to develop a finite element approximation, the time interval (U, U+1) is

subdivided into a certain number of time nodal points, which is the order of the

element, k.

This procedure gives rise to two different possibilities. One is an implicit

step-by-step self-starting integration formula; the other is an assembly process

developed to obtain a solution over a time period of interest.

Let the trial functions be

q = N - { q }

p = N - { p } (3.13)

while the test functions

Sq = N ■ {££}

5p = N-{Sp} (3.14)

5q = N • {£?}

5p = • {5p} (3.15)

where N, N axe the shape functions. They can be with the same or different

order. Here the form of polynomials is employed. The order k of the shape

function could be chosen according to the accuracy required, considering the

computation time it would take.

Substituting the trial and test functions into the mixed form formula, Equa-

45


tion (3.12) becomes

J 8 qTN T Np + 8pTN TNq — pTM 1 ® N T N 8 p — ( f K ® N TN 8 q

+8 f (F - C M -'p ) ® J\Tr tf] dt = S f j f ® N1" |£+1 (3.16)

where ® denotes the right Kronecker product of two matrices or vectors, and

F = f( t) ~ 9 (q, t)

includes the excitation force as well as the nonlinear force.

(3.17)

Furthermore, it yields,

J 8 p[ ( - M 1 ® N TNpi + N TNqi) dt +

f ti+l Sqf {[Nt N - C M - 1 ® N TN) p i - K ® I^N q ^ j dt +

[ + dgf /(£) ® NTdt - [ + 8 qf g{q, t) ® N Tdt = 8 qTpb ® N T |j’+1 Ju Ju 1

(3.18)

It could be rearranged into matrix form as follows,

( S p J j g ) | /£ +l -A / ' 1 ® f F N d t £ +l - J ^ N d tf i +l (N N - C M ' 1 ® JS^N) dt £ +1 - K ® I ^ N d t

{ * J + { /£* ' m ® i f * J + { S i? ' - 9 (1 , t) ® i T d t }}

= (8 p J , 8 qJ) | 5 . } (3-19)

Since \ 8 p{, 8 (g) are arbitrary, the finite element approximation at the

element level is given as,

A n A 12

A 21 A22 { * } + { g, } + { p<} { s , } <3 20>

46


where

A u = [ +1 - M ~ l ® N TNdtJk

Au = r +l —N r NdtJu

421 = ( N N - CM ' 1 ® J ^ N ) dt

A22 = J " ' 1 —K ® N TNdt (3.21)

and

Gi = f + - g(q, t) ® N Tdt Ju

Pi = f( t ) ® NTdt (3.22)Jti

From the previous equation, it can be derived that

•An Pi + Ai2?» = 0 (3.23)

Pi = -A n 1 Ai2ft (3.24)

By substituting into the previous equation, we have

[A22 — A2iAnJA12] {ft} + Gi + Pi = Bi (3.25)

Introducing the new matrix notation At- = [A22 — A21 Ajj1 A12] , it yields

Ai{qi} + Gi(qi) + Pi = Bi (3.26)

Through the system matrix assemblage for the Ne elements, the system equation

for the whole period of interest will be,

A{q} + G(q) + P = B (3.27)

47


where

A = C = £ C (t=lNe

i=l

p = £ « , « = £ * (3.28)i=l «=1

and

^ { f t I PATwi+lt Pi; 0 PNen*k+i PNc*N*k+N }

* = No. o / DOF (3.29)

As for the solution of Equation (3.27), the standard Newton-Raphson

procedure is employed. The solution for either the initial value problem, or

steady state problem can be carried out, by imposing the corresponding boundary

conditions.

3.3.5 T he evaluation o f G and Jacobian M atrix

G matrix, and Jacobian matrix, J , are required for the solution sought proce

dure of Newton-Raphson iteration.

For simplifying of the derivation, here only a SDOF system is considered.

The general formulation for any kind of system, single or multiple non-linear

DOFs, backlash type of non-linearity or other type, could be expanded with the

similar derivation.

For a single DOF system with non-linear force depicted as in the previous

chapter,' kc {q - b) + cq if q - b > 0;

g(q, t) = - 0 if - b < q < 6;fcc (q + b) + cq if q + 6 < 0.

(3.30)

48


the G matrix could be evaluated as,

G = Y^Gi (3.31)t=i

with GiAfU+iu

Gi = [ 1 -g(q, t) • N Tdt, (3.32)

For an interval [<j, t2\ within fo, ii+1], if g > 6,

INTGGj = - N7' [kc { q - b ) + cq] dt,

= - J N TkcNqdt + j N TkJ)dt - J + N TcNqdt (3.33)

and if — 6 < q < b, then

INTGGj = - [ ^ l N t -0dt = 0 (3.34)

if 9 < ~b,

INTG G j = - [ +l N 7" [kc (q + b) + cq] dt, (3.35)Jtir t+1 m — r »+i m r i+1 m 7

= - N kcNqdt — / N Tkcb d t - N TcNqdt (3.36) Jti Jti Jl

For all the intervals, the Gt could be computed as the summation of the

INTGGj, as

Gi = INTGGj (3.37)i=i

While for the Jacobian matrix J ,

N,tJ = (3-38)

i= 1

49


For certain element i,

Ji = ^ (3-39)dq

= (3.40)h dq

Similarly, for an interval [ti, Z2] within [f,-, k+i]>

if \q\ > b,

IN TG Jj = - r+l N TkcNdt - N Tctidt (3.41)Jti Ju

and if —b < = q < = b, then

IN TG Jj = 0 (3.42)

Ji of element i, over the [£i? £,+i] time span could be evaluated as,

Ji = j r i NTGJj (3.43)j=i

With the G and J matrix ready, the Newton-Raphson iteration algorithm could

be performed for the solution from this FET formulation.

3.4 Strategy for choosing numerical algorithms

From previous study, it can be seen that ODE stiff suite, as well as FET are

suitable for attle, clearance type computation. By considering efficiency and

availability, ODE stiff suite is employed for both baseline and rattle analysis,

while FET algorithm is employed for rattle analysis only.

Due to the advantage of adaptive step in ODE suite, the input for rattle

evaluation has already been adaptively stepped from baseline responses. This

50


makes the following rattle analysis be free of step choosing burden. Either ODE

stiff suite or FET algorithm works very well for the solution of rattle model.

Because of the adaptive time step of ODE stiff algorithm, it is suitable for

the baseline model, which is of multi-DOFs. As the baseline response will be

fed to rattle model, with this adaptive time step, both ODE stiff algorithm and

FET algorithm can be used. It can be seen from later analysis, results from both

algorithms agree very much.

51


Chapter 4

Application to NT350 Transmission

4.1 Introduction

Before the analysis can be carried out, several issues related to rattle analysis are

covered. They are rigidity of shafts, contact stiffness function of a meshing gear

pai’ and indication of rattle severity. Based on the methods of modeling and

analysis, simulation for the powertrain with NT350 transm ission is performed.

All simulations are carried out with the geometric parameters and material

properties of powertrain provided by Chrysler Corporation and New Venture Gear

Inc.

4.2 Flexible shaft m odel versus rigid shaft model

The baseline model consists of shafts, gears, and other components. Compared

with other parts of the powertrain, shafts are the most rigid ones. Thus for the

modeling, we are given the choice between rigid shaft and flexible shaft, for the

baseline model.

52


Before any decision can be made, these two kinds of models are built and

studied. With rigid shaft, the number of DOF for the system equation is 3. While

for flexible shaft, it is cut into number of segments and a total of 29 DOFs are

employed for the system model.

Since shafts are the parts only included in the baseline model, the analysis

is carried out with the baseline model. For the rigid shaft model, the responses

are solved with the system equation directly. While for the flexible shaft model,

the responses of the baseline are computed with the superposition of the first

several modes of the system, rather than all modes. This is due to the fact that

only first several modes are significant, and the solution of a 29 DOFs system

equation could take days to get.

By studying the frequencies of the modes found from the flexible shaft

model, it can be convinced that only first three modes are necessary for the

response calculation. As for the fourth mode, its frequency is too high compared

to the 3rd mode, and unlikely to be excited.

Figure 4.1 and Figure 4.2 illustrate the baseline responses of one driving

gear. It can be said that the responses from these two different models are fairly

close by checking the figures.

With this conclusion of the availability of the rigid shaft model, the model

of the baseline could be simplified, becoming easy to use for the analysis. Nev

ertheless, modal analysis is always available for the analysis, if the flexible shaft

model should be presented. With the first several modes considered for the re-

53


0 5 0.6 0.7 08 0 90 J3

1

>-2

01 05 0.6 0.7 08 09

20cI

Ol 0.6 0.70.3 08

Figure 4.1: The baseline response with flexible shaft

1

0•1

-2 01 05 0.6 0.70 0804

I>-3 01 05 0.6 0.7 09

10

-20

-300.6 0.7 08 09

Figure 4.2: The baseline response with rigid shaft

54


sponse calculation, the analysis time will be lowered dramatically, and close to

the time required for the rigid shaft model.

Within the rattle modeling and analysis software package, both of the rigid

and flexible shaft models are implemented. For the flexible model, the number

of modes employed for the response calculation can be selected. The user can

choose either one for the analysis, or both for the verification purpose.

4.3 Time-varying meshing stiffness function

It is well known that helical gears can transmit power quietly with low levels

of vibration compared to spur gears. The meshing stiffness function of a helical

tooth pair as proposed by Y. Cai [13] is employed for the evaluation of the time

varying meshing stiffness of a gear pair.

4.3.1 Stiffness function o f one helical too th pair

The stiffness function of a helical tooth pair k(t), can be calculated as follows,

t - ( e t z ) 2 3k(t) = kp ■ exp(Ca ) (4-1)1.125 x

where kp represents the stiffness of the pitch point, which will be discussed later;

t is the total contact ratio, while q* is the transverse contact ratio; and tz is

the meshing period passing one transverse base pitch. Especially since, t is the

meshing time from the start to the end of the meshing on the line of action, it is

clear that t equals from 0 to etz.

55


The coefficient Ca has a linear relationship with the helix angle /3q,

Ca = 0.322 • (A) - 5) + [0.23(6/H) - 23.26] (4.2)

where b is the face width, and H the total tooth depth.

For a pair of meshing helical teeth, kp could be evaluated as,

*** co + C7i ( i r + i ) + c 2( ^ - + ^ ) ( *

where Zy\ and Zy2 are equivalent numbers of teeth, and coandc2 are coefficients.

Specifically, Cq is the deflection of a helical rack pair which can be obtained as,

= ______________ 2i25______________00 [-0.166 x (bH) + 0.08] (ft - 5) + 44.5 ( }

And the coefficients ci,C2, and C3 are obtained by a least-square approximation,

which are of the value,

ci = -0.00854

c2 = -0.11654 (4.5)

c3 = 2.9784

Equation (4.1) is called the modified stiffness function for a helical tooth pair,

which is derived from the function of a helical rack tooth pair.

4.3.2 Synthesized meshing stiffness o f a meshing gear pair

The meshing stiffness of a meshing gear pair is needed for the analysis. With the

stiffness function of a pair of teeth ready, the meshing stiffness function for a pair

of meshing gears could be computed by synthesizing the pairs of m eshing teeth.

56


Time wytng corteti sttffnm250

200

150

Tooth par's stiffness100

SOEndatMMhjngStart of

0.05 0.1 0.15 0.26md 0.25 0.3

Figure 4.3: Time varying stiffness with contact ratio = 1.88

Two cases are studied, for a pair of meshing gears, with different contact

ratio, while having the same other parameters, such as face width 6, normal

module rim, etc. The total contact ratio for the first pair of gears is less than 2,

approximately at 1.88. While for the second pair of gears, it is larger than 2, at

2.79 for this case.

Figure 4.3 and Figure 4.4 illustrate the time varying meshing stiffness over

the meshing position, or time, for two different pairs of gears. With the low con

tact ratio e = 1.88, the contact stiffness function demonstrates more fluctuation

of its amplitude, as Figure 4.3 shows.

The meshing stiffness for the higher contact ratio, e = 2.79, shown in

Figure 4.4, varies less than its amplitude compared to that of the lower contact

ratio case. Thus, it can be seen that the helical gear pairs with high contact ratio

57


250

200

150

Tooth100

50

Start o( ling

0.1 0.3 d rad

0.4 0 S

Figure 4.4: Time varying stiffness with contact ratio = 2.79

possess the attribute of less fluctuation of stiffness over the meshing period. This

could explain why the pair of helical gears with high contact ratio transmits the

torque and motion quietly.

4.4 Effect o f time-varying meshing stiffness

The contact stiffness of a pair of meshing gears changes throughout its rotation

cycle. While the meshing of pair of gears continuous until the meshing ends, the

meshing stiffness of the teeth is not a constant. This phenomenon is also known

as parametric excitation.

The effect of this time-varying contact stiffness is studied with the meshing

stiffness function discussed previously. For the baseline system, as the backlash

does not contribute anything to the response, the result from the averaged contact

stiffness and tha t from the time-varying contact stiffness are almost the same, as

58


DOF30 4

0.3

0.1 -

0.1 as02 0.3 0.4 0.6 07 0.8 OJ0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 06 0 7 0.8

3001 i t ■ r i r i - t » »

.100 ------1___ t i l l ____i____I____I____ l___0 0.1 02 0.3 0.4 05 0.6 0.7 0.8 0.9 1

Tim*

Figure 4.5: The baseline response with constant contact stiffness

shown in Figure 4.5 and Figure 4.6.

The effect of the time-varying contact stiffness as well as the backlash on

the rattle model has been studied. A pair of meshing gears with backlash is

used for the evaluation of the effect of the time-varying meshing stiffness. The

responses for the average contact stiffness and the time-varying contact stiffness

are shown as in Figure 4.7 and Figure 4.8.

When comparing these two figures, it is evident that the responses of the

driven gear, either for the one with constant contact stiffness, or for the one with

time-varying contact stiffness, are quite similar. The conclusion can be drawn

that the effect of the time-varying contact stiffness on the rattle responses is quite

limited for this speed configuration, with high contact ratio.

As all five pairs of gears of NT350 transmission are helical gears with the

59


OOFS0.4 r

f OJ - §0.2 - 50.1 -

0.1 02 05 0.6 0.90.6 0 7

0.4

0.3

0.1

0.1 02 0.5 0.90.3 0.6 07 0.6300

3 200M£ 100

- to o 1 0.1 05Tim*

02 0.3 04 0.6 0.6 0.90.7

Figure 4.6: The baseline response with time-varying contact stiffness

x iq*-4 Dtoptacernam Qaar Vetocty gear

E*

-2

0 0.1 02 0.3 0.4Time Sac

PirMlmi

1.5

TJ£«0.5

0.1 0.3

0.2

0.1If3

- 0.1

- 0.2

0.30 0.40.1Time Sec

Contact force

zcLL

0.30 040.1Time Sec

Figure 4.7: The rattle response with constant contact stiffness

60


K10"* Otaptactmanl :gmr V«tocty «•m

0.1

>-ai

0.3 0.4o 0.1o o.i 0.30.2 Tim* See

Exctotion Contact fare*

zcu.

-10

-20

0.2 0.3 0.40 01

190.5

0.40.1 0.3

Figure 4.8: The rattle response with time-varying contact stiffness

contact ratio higher than 2.5, the time-varying contact stiffness has much less

effect on the rattle responses. Yet for spur gear with low contact ratio, it could

be a completely different scenario.

4.5 Gear rattle indices

The main concern of this thesis is to study the effects of various components or

system designs on the rattle reduction. Thus, the measurement of rattle severity

is required, which is known as the rattle indices. Previously, various rattle indices,

in order to evaluate the transmission and drive-train design, have been developed

based on intuition.

Due to vibro-impacts, impulsive transients occur in a periodic fashion over

one cycle of the acceleration time history. An increase of noise level caused by

61


this phenomenon is a strong function of the acceleration peak amplitude and the

decay rate.

Using the time response signals, the indices, E\ based on the ratio of mean-

square values of the accelerations, and E? based on the ratio of the peak to peak

values, are developed to measure the effectiveness of any design from the rattle

point of view,

denotes the mean square value of the acceleration of the driven gear, and the

flywheel, respectively, with T = 27ru;, corresponds to the period of the flywheel;

Further, two more indices, based on the response signals, are developed

specifically to assess the sound perception. One is based on the mean-square

energy, Es, and the other is based on the energy contained within the initial

sharp pulse, £4,

(4.7)

(4.6)

Where,

(4.8)

(4.9)

while 6 2 ,P, and 0/iP are the peak to peak value of the accelerations.

(4.10)

62


e a = 8& t (4.11)

where i refers to the position of the gears, p refers to the peak pulse, and St, the

pulse duration.

The index E\ is chosen for analysis within this paper. Prom the rattle

analysis, the angular velocities of all the rattle gears can be obtained. Then the

accelerations are computed by a forward difference method. The same procedure

is carried out for the acceleration of the flywheel. The ratio of the mean square

value of the driven gear to that of the flywheel is the rattle index for that specific

gear pair. The overall rattle index for certain speed could be calculated as the

root square value of all the unladen pairs of gears at that speed.

4.6 Analysis for th e first speed configuration

With the modeling method proposed, and several related issues studied, the

modeling and analysis procedure is performed for the speed 1 configuration. The

shafts are considered flexible. Thus the baseline model is of 29 DOFs, with the

multi-stage spring clutch. The first three modes of the system are included for the

response calculation. At this stage, the proportional Raleigh damping is applied

for the model.

The torque excitation is of the form as,

r(£) = 150 + 40sin(38.8f) Nm (4.12)

where torque fluctuation is about 25 percent of the mean torque. The system is

63


S pM dl Gm t2 SpM dl Gmt325

20

O

-5

Tim*

-«

SpMdl Gmt4 SpMdl GmiS

?

o

-8

Tim*

-8

Time

Figure 4.9: Baseline response: Angular displacement

assumed at rest for the initial condition. The backlash of unladen pairs of gears

is 1.25 x 10-4m.

4.6.1 B aseline torsional vibration

The responses of the baseline for the 2nd, 3rd, 4th and 5th pinion positions are

illustrated as Figure 4.9 and Figure 4.10.

The rigid model is also studied, and gives the consistent result with that

of the rigid shaft model. It is evident that both models are available.

4.6.2 R attle analysis for unladen pairs o f gears

For speed 1 configuration, other 4 unladen pairs of gears in meshing without

transmitting any load yield the rattle. For the 2nd pair of gears, the driving

pinion is on the input shaft, while for the rest 3 pairs of unladen gears, the

64


g -0.5

-10

3Tim*

-3

Figure 4.10: Baseline response: Angular velocity

pinions are all on the output shaft. The corresponding excitations for rattle

analysis can be taken from the input shaft response for the 2nd pair, and the

output shaft for the remaining 3 pairs, with the baseline response evaluation.

The responses of the rattle pairs from MATLAB ODElbs are shown in

Figure 4.11, Figure 4.12, Figure 4.13, and Figure 4.14, respectively. Each figure

consists of 4 sub plots, they are the linear displacement between gears, relative

velocity, rotation of the pinion, as well as the contact force between gears.

It is evident, from the rattle responses, that all pairs of unladen gears

undergo drive-coast rattle, while the 2nd pair of gears undergoes the rattle more

severely. This could be explained from the responses of the pinion position on the

input or output shaft. The torsional vibration of the pinion on the input shaft is

more severe, compared to that of the pinions on the output shaft.

65


x ^q-* Dtoptaoement :geer2 Vetocty 9«ar2

Eii.Uthjn

20

® 10

Nonlinear force

^ V W riV iV,sv

1 1.5 2Time See

zs

Figure 4 .11: Rattle response of Gear pair 2 for the first speed (ODE15s)

*10 Di^tacomoft :gaar3 Velocity .gears

• 0.005

5 - 0.005

- 0.01

-0 015

0.1

-2

-5

-6

Noofinear force

1 2 Time See

z

-20 0.5 15 2 2.51

Time See

Figure 4.12: Rattle response of Gear pair 3 for the first speed (ODEl5s)

66


f 0.005

> -0.005

-0.015

Excftatton

1 1.5 2Tima Sec

Nonlinear force

T)so

21 30

ze(L.

0 0.5 2.51 1.5 2Tima Sac Tima Sac

Figure 4.13: Rattle response of Gear pair 4 for the first speed (ODE15s)

Vetodty :QearS

0.005

- 0.01

Tuna Sac

Exctatbn Nonfinear force0•1

-2

-5

-621 30

Tima Sac

432

Z 1 c “■ 0

-1-2-3

0.5 1 1.5 2Tima Sac

2.5

Figure 4.14: Rattle response of Gear pair 5 for the first speed (ODEl5s)

67


V«tocty :gMr2

Emfritfan Nonflneer farce

20

© 10

0 0.5 1 1.5 2 Z5Time See

Figure 4.15: Rattle response of Gear pair 2 for the first speed (FET)

As the fluctuation of engine excitation is about 25% of the mean torque,

the rattle level is somewhat high. Ii is interesting that rattle level is proportional

to the fluctuation of engine excitation.

4.7 Comparison of rattle results: O D Esvs. FET

For the rattle evaluation, both algorithms of FET and ODE stiff suite are em

ployed. Figure 4.15, Figure 4.15, Figure 4.15, and Figure 4.15, show the responses

of gear pair 2 to gear pair 5 from FET algorithm.

From the side to side comparison of the results from ODEs and FET, it

can be seen that they agree very well. Taking a close look at the graphs for rattle

gear pair 2 of speed 1 configuration, Figure 4.11 and Figure 4.15, there is a slight

deviation from one to another. This slight difference is due to different algorithms,

68


Diaptaeameft VModty :qmi3

0.005

- 0.01

•4.0151 1.5 2

Time See

NonOnaar force

-3

2 310

32 ;

10•1

0.5 2 2.50 1 1.5Time Sac Time Sac


xIO Dtaptacamert rgaar* Vetoctty :gaar4

0.5

2.50.5 1.5 210Tima Sac

Exctatton0•1

-2

-3

-6-6

21 30

0.015

-0.005

-0.015

Nonlinear force

42

0

0 0.5 15 2 2.51Time See Tima Sac


69


I- 0.01

Time S*c

ExdMion Nonlinear force

- i

0 2 31 1 1.5 2Time Sec

2.5


in which different schemes of convergence and time steps are employed. This

difference can also be noticed from the contact force graphs.

Checking with other three sets of response graphs for same rattle gear

pairs, similar observations can be made. The responses agree very well from two

different rattle analysis algorithms with very slight differences.

It is noteworthy that FET algorithm runs about 10% faster than ODE

stiff suite for the same time period simulation.

With the responses from baseline model and rattle model, the rattle indices

can be computed, for each pair of the unladen gears, as well as overall rattle

indices for a specific speed.

70


Chapter 5

Parametric Studies and Design Application

5.1 Introduction

With ODE stiff-suite and FET algorithms developed, it is possible to perform

parametric study. The numerical algorithms allow us to examine the influence of

key parameters of the transmission on gear rattle. Subsequently, it provides the

guidance to achieve a quiet transmission design.

The key parameters of a drivetrain include backlash, clutch characteristics,

drag torque, inertia and rigidity properties of driveline, engine angular accelera

tion signature, etc. It is not possible to study all of these effects on rattle. In this

chapter, the characteristics for rattle reduction, issues about clutch design, and

powertrain design are discussed. Additionally the detailed parametric studies are

performed on the parameters of drag torque and backlash.

71


5.2 Characterization and reduction of gear rattle

The primary interest of the parametric study is rattle noise level. The rattle noise

intensity is known to be directly related to the kinetic energy loss during meshing.

The energy loss index neglects the effect of the noise transmission path and gives

no indication of the potential customer annoyance by gear rattle. However, an

investigation of the effect on meshing energy index of changes to various design

parameters is a useful first step that provides an interesting insight into possible

different rattle mechanism.

First, unsynchronized gear pairs will be studied, focusing on their major

design parameters: backlash and drag torque. Past theoretical and experimental

works have identified some general effects with these parameters. For instance,

the energy index increases with increasing backlash until a given threshold value

of backlash, at which the energy index may reduce significantly. The drag torque

can prevent impacting at the meshes, but breaking mesh does not always give

rise to an audible noise.

A reason for the complex relationship is that the complex motion possible

in the rattle gears demonstrates different modes of rattle behaviors. In one mode,

alternating impacts occur with the drive and coast sides of the unladen gear

periodically. This is typical for low backlash and low drag with an impact contact

at every engine firing event. In another mode, impacts occur with only the drive

of mesh due to an increased drag. Other modes of motion including irregular

72


xW*15

as

-as

in-1.5

2 2.1 2.2 2.4 2.6 2.7 2.6 2.9

Figure 5.1: Double side impact rattle mode

motion are also known to exist.

5.3 Rattle modes

The gear rattle shows some certain patterns called rattle modes under differ

ent configurations or operating conditions. It is helpful to have the potential

rattle modes studied, and to familiarize ones self with them before a complete

understanding of gear rattle characteristics can be achieved.

In general, rattle modes can be categorized into following types, from

previous study, as well as modeling and analysis performed in this paper.

Double side impact

This is the most popular mode of rattle. The gear rattles are mainly of this mode.

Figure 5.1 shows the characteristics of this rattle mode.

73


x 10"4

1.5

0.5

-1.5

0.55 065 0.7 075 Tims. See

0.6 0.6 0.85 0.9 0.95

Figure 5.2: Single side impact rattle mode

The meshing unladen pair of gears bounce within positive and negative

backlash range. The impact demonstrates the multiple impact characteristics.

After several impact on one side, for example during driving or coast side, the

driven gear is pushed back to the other side. This shows why a restitutive model

for the rattle is not adequate enough.

Single side impact

This is another mode of rattle, which is likely to occur. Figure 5.2 shows this

kind of rattle mode.

This rattle mode is more likely to happen when the powertrain is engaged

at low speed, or when the drag torque applied on driven gear of unladen pair of

gears is high, holding back the driven gear from coast side impact. Under this

rattle mode, the rattle index is low.

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jc IO -*

1.5

0.5

-1.5

0 0.3 0.40.1 0.2 0.6 0.7 0.8 0.9 1

Figure 5.3: Irregular rattle mode

Irregular mode

This is not the mode likely to occur during steady state operation of the vehicle.

It usually occurs dining vehicles starting period. Figure 5.3 illustrates this rattle

mode.

As reflected by its name, the irregular characteristics of this mode are

demonstrated from the previous figure. The driven gear penetrates between pos

itive and negative clearance very fast, with even greater momentum compared

to that of the double side impact mode. The deflection of the meshing teeth is

greater. And within one impact, the meshing gears are separated all the way to

the opposite side impact. This causes a lot of rattle, at relatively high levels.

The rattle modes discussed here are the first of their types to be presented.

Although there are a few published papers that discuss rattle modes and give

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some illustrated patterns, such detailed rattle characteristics have not been given

before.

5.4 Effect of drag torque and backlash on rattle

In order to check the effect of some key parameters on rattle indices, the analysis

is performed by changing drag torque and backlash within a certain allowed range.

The change of rattle behavior and rattle indices is tracked, so as to determine in

which way that rattle phenomenon can be alleviated.

A very detailed analysis is provided for the first speed configuration. The

drag torque applied is changing from 0 to 0.2JVm. While the backlash changes

from 90 %— to 110% of the current value.

Figure 5.4 shows rattle indices versus change of drag torque, and Figure

5.5 shows rattle indices versus change of backlash. It can be seen that rattle level

decreases when drag torque increases, while the effect of backlash change is not

monotonic. As evident in the figure of rattle indices versus backlash, the change

of rattle indices is not as great as that of drag torque change. It can be seen that

100 % percent of backlash is corresponding to the lowest rattle index.

Rather than only looking at the rattle indices which is a scalar value, it

is important to examine the rattle in detail. The rattle indices figures for fixed

backlash, while changing drag torque from 0 Nm to 0.2 Nm, are shown from

Figure 5.6 to Figure 5.10.

The figures tell us that rattle pattern is affected upon increase of drag

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Raffle index vs. Drag Torque1.4

♦ 105% Backlash p 110% Baditah

02

0.160.02 0.08 Drag Torque N.m

0.1401 0.12

Figure 5.4: Rattle indices versus drag torque

Rattle index vs . Backlash

0.5 N

1.2

5 0.

06

0.2

0.96 096 1 1.02Backtab, (1 - I25e-4m )

0.9 0.92 1.06 1.06

Figure 5.5: Rattle indices versus backlash

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0 0 2 0 4 0 6 a sTime See

o 02 a4 as asTime Sac

frucftaton Contact force

SDCS

02 080 4 0 6Time Sec

s

Figure 5.6: Rattle response with drag torque = 0.0 Nm

« 10- OUptenement rqae rl

0 0 2 0 4 OS OSTime Sec

ExdtMlon

19

0802

Veiodty :geart

0 0 2 0.4 OS 08Time Sec

Contact foroe

100

z

0 0.602 08Time See


78


Veiocty g**rt

Eicftfoon

1e

02 0 4 06Tim* See

06

Contact fora*

SO

zeu.

-60

-100

0 6 0 80Tkn*S*e


Dtsptaoement rg w l

EicltMton

1

02 06

Vtiodty Q**fi

04 0.6 0.8Tbn*S*e

Contact fora*

Tim*S*e


79


Dispteovnofit Valocty g n r l

EiatMton

02 0 4 0 6Tims Sac

08

r0 2 0 4 0 6 08

Urn* Sac


torque. When drag torque is not presented, the driven gear of the unladen mesh

ing pair of gears bounces within the backlash, which is so called a double-side

impact mode. This is the mode that corresponds to the high rattle indices. As

drag torque increasing, rattle mode gradually changes to that of a driving side

mode, which yields low rattle indices. If the drag torque reaches a certain value,

there will be no rattle. The separation of unladen pair gears is not likely to hap

pen. This is not the case for the real application because drag torque is related to

the viscosity of lubricant oil directly. There are many concerns about the choice

of its attributes, like the power and efficiency of transmission, etc.

From the results presented, there is significant proof that certain amount

of drag torque is quite helpful in alleviating rattle. But, due to the limitation of

other aspects, it is not feasible to apply great amount of drag torque onto the

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unladen driven gears.

The effects of backlash do not behave like that of drag torque. Given the

curves derived from the evaluation, its effect on rattle changes less, compared

to the change resulted by drag torque, which demonstrates a non-monotonic

behavior.

By looking through other configuration of speeds, similar behaviors are

observed. But due to the different rattle mode possessed of different speeds,

they demonstrate different ways of rattle mode changing, and thereby the rattle

indices. Manoj’s paper [47] has discussed the details of all five speeds rattle

analysis results.

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Chapter 6

Summary and Discussion

6.1 Summary

A decoupled model for powertrain torsional dynamics analysis is developed in

this dissertation. It includes the major powertrain components, such as flywheel,

clutch, transmission and other parts. The concept of separation of the baseline

and rattle model significantly simplifies the process of numerical analysis simula

tion. This decoupled model has been verified with a comparison to a full model.

The numerical results have shown that the decoupled model is effective for rattle

analysis with high efficiency and decent accuracy.

The effects of the shaft rigidity have been investigated. It can be seen that

the rigid shaft assumption reduces the computation time of numerical integration

for the baseline model simulation dramatically.

A mixed form formulation of Finite Element in Time domain has been

derived and implemented. From the simulations performed, it turns out that

FET mixed form runs about

By introducing a time-varying meshing stiffness function into the rattle

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model, the effect of interaction of backlash and time varying meshing stiffness is

shown to be insignificant.

Different rattle modes are identified for a specific speed configuration. Our

findings verify the modes provided from previous research. During one rattle

period multiple impacts may occur for either double or single side mode.

A parametric study has been performed with two major parameters of the

powertrain: drag torque and backlash. With drag torque increasing, a rattle

index decreases as drag torque prevents impacts on the coastal side. This may

change rattle mode from double side impact into single side impact. The effect of

backlash on rattle is not monotonic. It turns out that the current backlash level

in the T350 transmission is very close to its optimal value from the rattle point

of view.

6.2 Contributions

In this dissertation, the research work has concentrated on modeling and anal

ysis of rattle phenomena in an automotive powertrain equipped with a manual

transmission.

The development of a decoupled model is based on an understanding of

powertrain component characteristics. The decoupled model makes use of the

assumption of neglecting the effect of unladen driven gears on driving gear. This

model is examined through a comparison with a full model. The results indi

cate that the decoupled model is accurate for practical applications with great

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efficiency.

The Finite Element in Time domain (FET) numerical method was first

proposed about three decades ago, but its application has not been widely stud

ied. In this dissertation, a FET mixed form formulation is derived in detail and

is implemented for the analysis of powertrain rattle. Through numerical com

parisons with stiff ODE algorithms, the effectiveness and efficiency of the FET

method have been verified.

As noted, rattle phenomena are more prominent for certain speeds of op

eration, less for others. With understanding of the general concept of rattle

characteristics for the existing transmission, the parametric study can be initi

ated. By checking the effect of key parameters, such as backlash and drag torque,

on the rattle mechanism and its level, a guide to the design of clutch and trans

mission with better rattle characteristics is possible, thus moving the design for

the powertrain into a new and more objective phase.

6.3 Future work

It is our ambitious goal to develop a complete modeling and analysis package

for powertrain rattle. We have covered many issues related to physical modeling

and numerical simulation. There still remain topics for further study.

If there arises a chance for continued study, good topics might be damping

model and dry friction characteristics of the clutch. The model developed in this

dissertation employs proportional damping for both baseline and rattle models.

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The possibility of using other types of damping and including hysteresis charac

teristics of the clutch could be interesting topics. With study and understanding

of these two aspects, the capacity of the model for powertrain rattle analysis will

be greatly deepened.

Another important topic is experimental investigation. This dissertation

studied analysis and numerical simulations only. Experimental results will be

the final validation of the modeling and analysis methodology developed in this

dissertation.

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WANG, ZHAO - Modeling and Analysis of Gear Rattle in Automotive Transmissions

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