Msme Thesis Tf

REDUCTION OF VEHICLE CHASSIS VIBRATIONS USING THE POWERTRAIN

SYSTEM AS A MULTI DEGREE-OF-FREEDOM DYNAMIC ABSORBER

A Thesis

Submitted to the Faculty

of

Purdue University

by

Timothy E. Freeman

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Mechanical Engineering

May 2004

ii

ACKNOWLEDGEMENTS

I would like to acknowledge the financial support provided to me through the

National Consortium for Graduate Degrees for Minorities in Engineering and Science,

Inc. during my tenure as a graduate student at Purdue University. I also would like to

thank General Motors for sponsorship of this research. This research would not have

been possible without the advanced vehicle platform upon which this research is based.

Additionally, I would like to thank the following employees of General Motors for

providing vital information on the subject. The people below had a direct impact on the

completion of this research:

John Zinser Gary Cummings Mary Wolos Angela Barbee-Hatter Elizabeth Pilibosian Ping Lee Craig Lewitzke Richard Smith Mel Richards James Vallance

In addition, I would like to thank Dr. D. E. Adams for serving on my advisory

committee, and for supplying testing and nonlinear analysis expertise. I also would like

to thank Dr. J. M. Starkey for serving on my examining committee on short notice. To

all these individuals, thank you for enabling me to complete my thesis.

Timothy E. Freeman

April 27, 2004

iii

TABLE OF CONTENTS

Page

LIST OF TABLES...............................................................................................................v

LIST OF FIGURES ........................................................................................................... vi

LIST OF SYMBOLS .......................................................................................................... x

ABSTRACT..................................................................................................................... xiv

CHAPTER 1: INTRODUCTION........................................................................................1

1.1: Overview of Powertrain Mounting Systems.....................................................1

1.1.1: Simple Elastomeric Mounts...............................................................3

1.1.2: Hydraulic Engine Mounts..................................................................4

1.1.3: Semi-Active (Adaptive) Hydraulic Mounts.......................................7

1.1.4: Active Hydraulic Mounts...................................................................8

1.2: Design Conflicts ...............................................................................................8

1.3: Thesis Statement .............................................................................................12

CHAPTER 2: PRELIMINARY ANALYSIS ....................................................................14

2.1: Nonlinear Powertrain to Ground (SDOF).......................................................15

2.2: Nonlinear Powertrain-Body (2DOF) ..............................................................24

CHAPTER 3: 13DOF VEHICLE MODELING/SIMULATION......................................28

3.1: Thirteen Degree-Of-Freedom .........................................................................28

3.1.1: Model Description ...........................................................................28

3.1.2: Calibration .......................................................................................37

3.1.3: Linear Stiffness Effects....................................................................42

3.2: Nonlinear Model Description .........................................................................46

3.3 Curve Fit Models .............................................................................................51

iv

Page

3.3.1 Frequency Dependent Curve Fit Model............................................52

3.3.2 Piecewise Nonlinear Curve Fit Model ..............................................56

3.3.3 Curve Fit Model Comparison ...........................................................59

CHAPTER 4: 15DOF VEHICLE MODELING/SIMULATION USING HYDRAULIC

POWERTRAIN MOUNTS ...............................................................................................61

4.1: Hydromount Model Description.....................................................................61

4.2: Individual Element Effects .............................................................................64

4.3: Hydromount Model Verification ....................................................................70

4.4: Implementing Hydromount Model .................................................................71

4.5: Fifteen Degree-Of-Freedom ...........................................................................73

CHAPTER 5: EXPERIMENTAL IDENTIFICATION OF LINEAR VEHICLE

VIBRATION MODEL ......................................................................................................78

5.1 Overview of Automated Model Development Approach................................78

5.2 Eleven Degree-of-Freedom Vehicle Model with Rear Wheel Constraints .....79

5.3 Approach for Hybrid Analytical / Experimental Model Development ...........83

5.4 Results of Hybrid Model Development using Direct Parameter Estimation...91

5.5 Determine Degree of Nonlinearity in Vehicle .................................................98

CHAPTER 6: SUMMARY..............................................................................................102

CHAPTER 7: CONCLUSIONS ......................................................................................105

LIST OF REFERENCES.................................................................................................106

APPENDIX A..................................................................................................................108

A.1 one.m.............................................................................................................108

A.2 one_fof_model ..............................................................................................111

A.3 two.m.............................................................................................................112

A.4 two_fof_model_disp .....................................................................................115

A.5 ssr13_linear_stiffen.m...................................................................................116

A.6 animate.m......................................................................................................125

A.7 ssr13_NL.m...................................................................................................129

A.8 ssr13_linear_fithz.m......................................................................................139

v

Page

A.9 ssr13_linear_fitdel.m...................................................................................146

A.10 leastsquare.m...............................................................................................153

A.11 ssr_15DOF.m ..............................................................................................155

A.12 DPEssrfinala.m ...........................................................................................166

A.13 caldata.m .....................................................................................................169

A.14 integdata.m..................................................................................................170

A.15 hpx.m...........................................................................................................171

A.16 generatecoord.m..........................................................................................172

A.17 sweeptf.m ....................................................................................................174

vi

LIST OF TABLES

Table Page

1.1: Properties of possible powertrain mount materials [7].............................................4

1.2: Powertrain Mouting Systems ..................................................................................................10

1.3: Ideal powertrain mount characteristics. ..................................................................11

3.1: 13DOF Vehicle model mode shapes. .....................................................................39

3.1: 13DOF Vehicle model mode shapes. (continued) ..................................................40

3.1: 13DOF Vehicle model mode shapes. (continued) ..................................................41

4.1: Summary of each elements effect on the hydraulic mount performance. .............70

5.1: Tri-axial sensor channel documentation for electro-hydraulic shaker

experiments on half-car vehicle testbed (channel number and name, voltage

range, low pass filter, high pass filter and source level). ....................................90

5.2: Tri-axial sensor channel documentation for electro-hydraulic shaker

experiments on half-car vehicle testbed (sensor calibration factors, serial

numbers and other test settings).............................................................................90

6.1: Model Summary ...................................................................................................104

vii

LIST OF FIGURES

Figure Page

1.1: Hydraulic mounts---Inertia track with decoupler [2]................................................6

2.1: Thesis work flow diagram ......................................................................................15

2.2: SDOF Model of nonlinear powertrain on ground (rigid base) ..............................16

2.3: Complete mass displacement time history..............................................................18

2.4: Input Force (f(t)) .....................................................................................................18

2.5: Steady state portion of mass displacement x(t) ......................................................18

2.6: SDOF Model Analytical-Numerical Comparison ..................................................19

2.7: SDOF analytical-numerical comparison SDOF (Zoom in) ....................................20

2.8: Higher frequency term effects on SDOF analytical results ....................................20

2.9: Higher frequency term effects on SDOF analytical results(zoom in) ....................21

2.10: effect on SDOF transmissibility .........................................................................22

2.11: Input amplitude effect on SDOF transmissibility...................................................22

2.12: Two degree-of-freedom system with nonlinear term .............................................24

2.13: 2DOF X2/Xb Transmissibility Response .................................................................26

2.14: 2DOF X1/Xb Transmissibility.................................................................................26

2.15: 2DOF X2/X1 Transmissibility ................................................................................27

3.1: SSR Side View........................................................................................................29

3.2: 13 DOF Vehicle Model Schematic.........................................................................29

3.3: Sponsor supplied SSR vertical front suspension road test......................................35

3.4: Sponsor supplied SSR vertical rear suspension road test .......................................36

3.5: Sponsor supplied SSR vertical steering column road test ..............................................36

3.6: Modal plot of calibrated 13DOF Model .................................................................38

viii

Figure Page

3.7: Calibrated 13DOF body transmissibility, upper--bounce DOFs

lowerRoll DOFs .................................................................................................38

3.8: Powertrain Natural Frequencies..............................................................................42

3.9: Powertrain mount nominal stiffness effects on mode plot. ....................................43

3.10: 13DOF Powertrain Transmissibility.......................................................................44

3.11: Front body FRFs (upper-- bounce, lowerroll) with varied linear

engine mount factor ...............................................................................................45

3.12: Middle body FRFs (upper-- bounce, lowerroll) with varied linear


3.13: Rear body FRFs (upper-- bounce, lowerroll) with varied linear


3.15: Linear Transmisibilites. ..........................................................................................47

3.15: Nonlinear Transmisibilites......................................................................................48

3.16: Nonlinear Restoring Force......................................................................................48

3.17: Nonlinear Effect on Front body . ............................................................................49

3.18: Nonlinear Effect on Middle body ........................................................................................50

3.19: Nonlinear Effect on Rear body . ...........................................................................................51

3.20: Sponsor Supplied SSR Mount Stiffness Data.........................................................52

3.21: Transmissibility using frequency dependent stiffness (0.1 mm peak

to peak deflection amplitude).................................................................................54

3.22: Transmissibility using Frequency Dependent Stiffness (1.0 mm

peak to peak deflection amplitude) ........................................................................54

3.23: Front Frequency Dependent FRFs.........................................................................55

3.24: Middle Frequency Dependent FRF........................................................................55

3.25: Rear Frequency Dependent FRF............................................................................56

3.26: Stiffness Curve fit at 10 Hz....................................................................................57

3.27: Stiffness Curve fit at 25 Hz....................................................................................58

3.28: Linear Interpolation Effect on Transmissibility ..............................................................58

3.29: Curve fit Model Front Body Comparison (Small amplitude)................................59

ix

Figure Page

3.30: Curve fit Model Middle Body Comparison (Large Amplitdue)............................60

4.1: Hydraulic Mount Model ........................................................................................62

4.2: Effect of Ks on mount properties...........................................................................64

4.3: Effect of Kv on mount properties ..........................................................................65

4.4: Effect of Kd on mount properties ..........................................................................66

4.5: Effect of Cs on mount properties...........................................................................67

4.6: Effect of Cv on mount properties ..........................................................................67

4.7: Effect of Cd on mount properties ..........................................................................68

4.8: Effect of Fluid mass on mount properties..............................................................69

4.9: Effect of Lever arm on mount properties...............................................................69

4.10: Transmissibility magnitude of X1/Xo.....................................................................70

4.11: Phase of mount X1/Xo ............................................................................................71

4.12: Output from Hydrofit Program ..........................................................................72

4.13: Fifteen Degree-of-freedom Vehicle Model ...........................................................74

4.14: Fifteen Degree-Of-Freedom Body FRF..............................................................................75

4.15: Fifteen Degree-of-Freedom Powertrain FRFs ..................................................................75

4.16: Fifteen Degree-of-Freedom Front FRFs .............................................................................76

4.17: Fifteen Degree-of-Freedom Middle FRFs .........................................................................77

4.18: Fifteen Degree-of-Freedom Rear FRFs ..............................................................................77

5.1: Front isometric view photograph of half-car electro-hydraulic shaker testbed

showing left-front tire and shaker wheel pan, shaker pedestal and left-rear

tire restraint. ...........................................................................................................80

5.2: Rear view photograph of half-car shaker testbed showing left and right rear

tire platform with lightly ratcheted restraining straps.. ..................................................80

5.3: Schematic of eleven degree-of-freedom transverse vibration model of vehicle

showing grounded assumption at rear spindles and excitation at left front tire

patch.......................................................................................................................82

x

Figure Page

5.4: (a) Schematic of thirteen accelerometer measurement degree-of-freedom

locations in half-car vehicle electro-hydraulic shaker tests; and (b)

photograph of two accelerometer mounting locations on wheelpan and

powertrain vehicle testbed. .....................................................................................................89

5.5: Magnitude of measured frequency response functions from 0-15 Hz

between responses z1, z2, z5, z6, z7 and z8 for left front 4 mm swept

wheel pan excitation, zlf. ........................................................................................94

5.6: Magnitude of synthesized frequency response functions from 0-15

Hz between responses z1, z2, z5, z6, z7 and z8 for left front 4 mm

random wheel pan excitation, zlf. ...........................................................................95

5.7: Absolute values of imaginary parts of 22 modal frequencies for

estimated eleven DOF model for 200, 400, 800, 1000, 2000 and

3000 time points showing convergence for Nt>1000. ...........................................95

5.8: Absolute values of real parts of 22 modal frequencies for estimated

eleven DOF model for 200, 400, 800, 1000, 2000 and 3000 time

points showing convergence for Nt>1000. ............................................................96

5.9: Magnitude of synthesized frequency response functions from 0-15

Hz between responses z1, z2, z5, z6, z7 and z8 for left front 4 mm

random wheel pan excitation, zlf, for different values of tire

damping with c=0.001, 0.002 and 0.02. ................................................................97

5.10: Spindle and Front Body Spectrogram....................................................................99

5.11: Left Middle and Rear Body Spectrograph...........................................................100

5.12: Left Powertrain and Transmission Spectrogram .................................................101

xi

LIST OF SYMBOLS

M mass

C viscous damping

K stiffness

m nonlinear cubic stiffness parameter

f(t) force as a function of time

Fo input force amplitude

x(t) displacement as a function of time

X1 displacement amplitude at input frequency

X2 displacement amplitude at 3 times the input frequency

Xb input displacement amplitude

w0 frequency of applied force or known base motion

? o phase shift between input and output

t time

x&& acceleration x& velocity Mp powertrain mass

Cp powertrain mount damping

Kp powertrain mount stiffness

KNOM Nominal powertrain mount static stiffness

[ ]A adjoint of a matrix angle of a complex number

determinant of a matrix; absolute value of a real number

xii

[ ] 1- inverse of a matrix

magnitude of a complex number

[ ] matrix

{ } vector

Dt sample time

q general angle

w frequency

wn undamped natural frequency

Dt sample time

xiii

ABSTRACT

Freeman, Timothy E., M.S.M.E., Purdue University, May, 2004. Reduction of Vehicle Chassis Vibrations Using the Powertrain System as a Multi Degree-Of-Freedom Dynamic Absorber. Major Professor: Dr. Douglas E. Adams, School of Mechanical Engineering.

The goal of this project is to reduce vehicle chassis vibrations using the

powertrain system as a multi degree-of-freedom dynamic absorber. In order to achieve

this goal using typical linear mount design techniques, the overall mount stiffness would

need to be much larger than the nominal stiffness. On the contrary, increases in mount

stiffness result in poor vibration isolation characteristics. This design trade-off between

vibration isolation and energy absorption has traditionally been overcome using active

mounts, which use sensor feedback to tune mount stiffness and damping properties to

reduce vibrations in ride at the given operating condition. The present research aims to

develop an alternative, passive nonlinear mount design, which effectively overcomes this

design trade-off without the expense of an active mounting system. For example,

nonlinear hardening mounts automatically adjust their stiffness characteristics to provide

good energy absorption at higher amplitudes and higher frequencies as well as good

vibration isolation at lower amplitudes and lower frequencies. In this work, multi degree-

of-freedom nonlinear models are developed for an advanced vehicle platform, the models

are studied using nonlinear vibration analysis and simulations are conducted to account

for frequency-dependent as well as amplitude dependent mount characteristics.

A 15 DOF model is developed as a tool that can be used to predict body

transmissibility response at two (or more) operating conditions such as idle and road

xiv

conditions. The model can be run at multiple conditions and can show the effect of the

current tuning of the hydraulic mount and suggest increases or decrease in amplitude

dependence in order to reduce body vibrations. In addition, a modified version of Direct

Parameter Estimation (DPE) is developed to construct accurate stiffness and damping

matrices. The mass, stiffness and damping matrices computed from DPE can be modified

and used in the 15 DOF model to speed up the 15 DOF model construction time. The

estimation of mass, stiffness and damping eliminate the need to calibrate the 15 DOF

model in order to match model modes to the vehicle modal tests.

1

CHAPTER 1: INTRODUCTION

The powertrain is a significant source of vibration in automobiles and possesses a

significant percentage of the total weight of the vehicle. The powertrain is also a

potential aid in reducing vehicle vibrations. Mounts that are carefully designed can

respond at the system level by coupling into the resonant frequencies of the vehicle

suspension, chassis and body to serve as a dynamic absorber to attenuate unwanted

vibration. Simultaneously, the mounts must also be designed to isolate the chassis and

body of the vehicle from the powertrain. Many different mounting configurations have

been developed to support the powertrain as the vehicle has progressed from a motor

carriage. Mounting systems must isolate unwanted frequencies from the vehicle chassis

and effectively support the powertrain.

1.1: Overview of Powertrain Mounting Systems

There is a great deal of ongoing research to model and/or simulate hydraulic

mount performance. More advanced models will give designers a good tool to achieve

specifications accurately. In addition, an accurate model that can capture the built-in

nonlinear effects of the mount will help the designer capitalize on these effects. Kim and

Singh [1] have done research in this area. His main objective was to develop a

simplified, yet reasonably accurate, low frequency nonlinear mathematical model of a

hydraulic mount with an inertia track. This work successfully identified the mount

nonlinearity, developed experimental methods to characterize non-linear fluid resistance

parameters, and developed and verified a nonlinear mathematical model from 1 to 50 Hz.

2

Coupling effects between engine mounting systems and vehicle flexion modes are

apparent in todays light and powerful vehicles. Other work has analyzed the effects of

vehicle cradle flexibility on the powertrain dynamic response. Most dynamic models for

the engine mount systems have been based on isolation theory, and vibration of the

foundations has been neglected [2]. It is necessary to model engine mounting systems

with flexible foundations in order to capture these vibration coupled problems [2]. In this

previous research, it was found that the coupling effects were substantial for frequencies

lower than idle speed but negligible for frequencies higher than the idle speed. In

addition, this work showed that the mount solution would be improved if the foundation

flexibilities are taken into account.

There have been many different design methods developed for reducing unwanted

vibrations. A survey [3] provided basic working principles for designing powertrain

mounts. This survey suggests the primary function of an engine mount, in addition to

supporting the weight of the engine itself, is to isolate the unbalanced disturbance forces

from the main structure of the vehicle. The survey also suggests the mounting system

should have low stiffness and damping to prevent vibration transmission through the

mount. The mounting system must also prevent large displacements of the powertrain

during shock excitations, which may be induced through sudden stops or accelerations.

Therefore, the elastic stiffness must be high enough to prevent powertrain and/or engine

component damage. Consequently, the mounts should exhibit high damping around 10

Hz and reasonably low damping above 15 Hz to reduce idle vibration [4]. There are

several different types of powertrain mounts in use today. Different mount types are used

for different powertrain mounting systems. Due to performance and economical reasons,

each system design has specific advantages and disadvantages.

3

1.1.1: Simple Elastomeric Mounts

Simple elastomeric (rubber compound) mounts have been used since 1930 and are

considered the most conventional. Elastomeric mounts in general have high stiffness

characteristics with high frequencies and lower stiffness with low frequencies. This

general trend complicates the design process since most mount applications need the

mount to exhibit low stiffness for high frequencies to improve idle vibrations. If the

stiffness is tuned to isolate during idle, the stiffness value may be too low to prevent large

low frequency shake. Furthermore if the mount is tuned for road or lower frequency

oscillation the mount may be too stiff to isolate the powertrain from the vehicle.

Subsequently, a compromise between the two specifications must be implemented to

optimize mount performance.

Elastomeric mounts can isolate powertrain vibrations in all directions by allowing

different stiffness characteristics in different directions. Some researchers have improved

the directional capabilities through shape optimization methods. Shape optimization,

optimizes the mounts physical dimensions in order to optimize isolation for different

conditions and directions. Kim and Kim [5] have achieved this shape optimization with

parameter optimization.

Current research is focused on identifying materials with high internal damping or

amplitude dependent damping and stiffness. Trial and error methods with various

materials have improved the performance of elastomeric mounts. Improvements for

temperature range and durability for different atmospheric conditions have also been

developed. Blended polymers have shown improved capabilities at achieving front

engine mount specifications [6]. Table 2.1 from Lewitzke and Lee [7] describes different

rubber/plastic materials that are used for isolation purposes.

4

Table 1.1 Properties of possible powertrain mount materials [7]. Elastomer Major properties

Applications

Natural Rubber or Polyisoprene (NR)

Available properties satisfy a broader range of engineering application than any other Elastomer family. Excellent tensile strength and tear resistance.

Powertrain mounts, suspension bushings, exhaust hangers, shock and strut mounts, front axle bushings, rear differential mounts.

Synthetic Isoprene (IR)

Similar to Natural Rubber. Slightly lower tensile strength and tear resistance.

Powertrain mounts, suspension bushings

Styrene-butadiene (SBR)

Reinforced or stiffer compounds offer properties only slightly lower than those of NR and IR, but more economical.

Powertrain mounts, jounce bumpers

Butyl or Polyisobutylene (IIR, CIIR)

Outstanding impermeability, chemically inert, excellent weathering resistance, high gum strength, high damping at moderate temperatures.

Cradle and body mounts, jounce bumpers, vibration dampers

Poly-butadiene (BR)

Properties range a little below NR and IR. Resilience and low temperature flexibility better than NR and IR.

Same as Natural Rubber

Neoprene Moderate solvent resistance. Excellent aging characteristics flame resistant. Approaches the broad engineering properties of NR and IR

Powertrain mounts, strut mounts

Poly-urethane Outstanding oil and solvent resistance. Good impermeability. Excellent aging. Resistance to oils and gasoline. Ozone resistant.

Body mounts, jounce bumpers, suspension bushings

Silicon (VMQ) Highest and lowest useful temperature range of all elastomeric compounds. Superlative aging properties. Radiation resistant. Reasonable oil resistance.

Powertrain isolators, exhaust hangers

1.1.2: Hydraulic Engine Mounts

Hydraulic engine mounts were developed and patented by Richard Rasmusen in

1962. Hydro mounts operate similar to a piston to force a fluid through a restricted

orifice between an upper and lower chamber to provide damping. Many different

components have been added within hydraulic mounts to serve different purposes. The

simplest component of a hydraulic mount is a restricted orifice to channel fluid flow

between the chambers. The orifice decreases the stiffness of the mount to some degree.

The orifice decreases compression of the fluid and allows it to flow from the upper

section of the mount. The mount will be capable of larger displacements for the same

applied force. The main improvement over the simple elastomeric mounting system is

5

their nonlinear stiffness and damping characteristics. The mount will exhibit smaller or

larger stiffness characteristics depending on amplitude and frequency of excitation.

Furthermore, the size or diameter of the orifice dramatically affects the mount

performance. The size of the orifice is another parameter available for design. The

added parameter gives the mount design another method to achieve optimum mount

performance. However, the restricted orifice is not as versatile at achieving different

performance characteristics as other types of hydraulic mounts.

Another component incorporated in hydraulic mounts is an inertia track. The

inertia track is a channel of specific length used to transport fluid between the upper and

lower chambers. The fluid flow through the inertia track enables the mount to provide

additional damping. Similar to the orifice, the inertia track incorporates frequency

dependence. The inertia track length and cross-sectional area are additional parameters

that can be changed in order to produce a desired response. Additionally, the use of a

decoupler incorporates amplitude dependence. A decoupler incorporates a small flexible

diaphragm between the upper and lower chambers. The decoupler allows the fluid to

remain in the upper chamber for small amplitude displacements. By forcing the fluid to

stay in the upper chamber the mount will provide less damping because of the lack of

fluid flow through the inertia track.

Figure 1.1 shows a schematic of a hydraulic mount, which is equipped with an

inertia track and a decoupler. The hydraulic mount is connected to the engine and chassis

through the mounting studs (1) and (2). The top element (3) made up of rubber material

supports the static engine weight. The upper chamber (4) and lower chamber (5) are

filled with the glycol fluid mixture of antifreeze and distilled water. A cyclic engine

motion causes oscillating fluid flow between the two chambers. A fraction of the

displaced fluid is accommodated by the decoupler (6) motion and the remaining portion

is forced to flow through the inertia track (7). The decoupler is supported by a rubber

membrane in the center of the mount. The rubber membrane allows for small deflections

of the decoupler causing small deflections in the mount before fluid is forced through the

inertia track. The decoupler is typically produced from duro 70 rubber. The compliant

thin rubber bellows (10) comprising the lower chamber is produced from duro 51 rubber.

6

The air breather (11) enables the rubber bellows to move freely without any air

compression effect. The canister (12) contains the inside parts mentioned above [8].

Figure 1.1: Hydraulic mounts---Inertia track with decoupler [8].

The most advanced hydraulic mount incorporates a simple orifice, inertia track

and decoupler. All of the components add beneficial complexity to the hydraulic

powertrain mount. A hydraulic mount that uses all of these components possesses both

amplitude and frequency dependent characteristics. In addition, each component can be

adjusted to modulate stiffness and damping frequency dependence. The diameter of the

orifice, the length and cross-sectional area of the inertia track and the maximum

decoupler deflection are design tools to develop the best mount for the given application.

7

1.1.3: Semi-Active (Adaptive) Hydraulic Mounts

Standard hydraulic mounts are normally tuned in order to suit a specific

application. This process can be long and costly. Furthermore, this retuning involves a

compromise in performance throughout multiple frequency ranges [9]. Semi-active

mounts are implemented in order to overcome this compromise. The benefit of a semi-

active control scheme is that it dissipates the vibration energy by changing the hydro

mounts damping properties using a low speed, low power actuator at a minimal cost [8].

The semi active mount controls the system properties of the mount in order to change the

performance. Damping is the controlled system parameter because it is implemented

most easily; however, low stiffness can also be achieved. Semi-active mounts are

controlled in an open loop manner.

The main types of semi active mount systems include Vacuum Actuation, Electro-

Rheological (ER) Fluid Activation and Magneto-Rheological (MR) Fluid Activation.

Each type uses a slightly different method to alter hydro mount stiffness and/or damping

but share the same objective. Vacuum actuation uses an electronic control module

controlled vacuum source to activate a valve. Depending on whether low stiffness or

high damping properties are desired, the valve can be opened or closed. When the valve

is open it allows fluid to bypass the inertia track creating an open passage for fluid to

freely flow between the upper and lower chambers providing a low stiffness trait. When

the valve is closed the fluid is forced through the inertia track resulting in higher

damping.

Electro-Rheological (ER) mounts also use hydraulic mounts. Unlike in vacuum

activation, the ER method uses ER fluids to change the properties of the fluid rather than

altering the path of the fluid. The fluid has small dielectric particles that are suspended

throughout the fluid. These particles increase the viscosity of the fluid when it exposed

to an electric field. The damping performance of the mount can be changed for different

operating conditions.

Similar to ER fluid mounts, MR fluid mounts also use a contaminant to alter the

fluids viscosity. Instead of reacting to electric fields, MR fluids react to magnetic fields.

8

Subsequently, the damping increases proportionally to magnetic fields created by current

induced wire coils in proximity of the mount.

1.1.4: Active Hydraulic Mounts

In active control, an active energy source should be continuously supplied to

counteract the continuously generated target energy source [9]. The primary control

method implements closed loop control, which requires the use of more equipment than

previous systems. Active mounts require the use of sensor(s) and an actuator(s) in

addition to the standard hydraulic mount. The actuator must be controlled by another

source such as the ECM (electronic computer module) according to specific senor values.

Active mount components work simultaneously in order to suppress the transmission of

disturbance forces. A sensor is mounted on the frame/chassis side of the mount to

measure vibrations. From the sensor readings, a force equal in magnitude and 180

degrees out of phase is applied to counteract unwanted vibrations. This mounting system

is often costly to implement due to the number of parts. Furthermore, the increase in

parts also decreases the reliability of the system because of possible sensor failures.

1.2: Design Conflicts

The optimum powertrain mount design depends on whether the vehicle is exposed

to road or idle conditions. Idle conditions are composed of small amplitude high

frequencies oscillations, whereas road conditions have larger amplitude oscillations at

lower frequencies. Because one goal in mount design is to suppress vibrations

throughout the vehicle due to engine dynamic imbalance forces, the powertrain mounts

should exhibit low stiffness. The low stiffness would most likely isolate the body from

the idle vibrations; however, excessively low stiffness can cause problems when the

vibrations are no longer of small amplitude at high frequency. For large amplitudes, the

powertrain must have adequate clearance; therefore, nonlinear structures or isolators are

9

needed. Hydro elastic powertrain mounts (hydraulic mounts) exhibit nonlinear

characteristics, which can be tuned to achieve better performance in vibration isolation.

Different mounting systems have advantages and disadvantages. As a result,

different mounting systems may perform better or meet various objectives to various

degrees. Table 1.2 gives a synopsis of the different types of mounting systems with their

respective design trade offs.

10

T

able

1.2

: Po

wer

trai

n M

ount

ing

Syst

ems.

11

Table 1.3 shows the optimum mount characteristics to satisfy idle and general

road vibration conditions. As indicated in the table, the stiffness and damping of the

mount should exhibit frequency dependence and nonlinearity. In addition, the mount

materials must be able to withstand automotive operating conditions. The materials must

withstand heat from the powertrain, fuel, any oils and/or fluids and road substances such

as road salt. Many of these substances can be corrosive. A mounts ability to achieve the

desired characteristics is limited by material capabilities.

Table 1.3 Ideal powertrain mount characteristics.

w/r/t frequency w/r/t displacement w/r/t frequency w/r/t displacement

High stiffness needed at low freq to provide engine support

Small amplitudes tend to be higher frequency.

High damping needed at low freq to prevent large engine displacement

Small amplitudes tend to be higher frequency.

Low stiffness needed at high freq to provide body isolation.

High amplitudes tend to be lower frequencies.

Low damping needed at high freq to provide body isolation

High amplitudes tend to be lower frequencies.

Mount material must withstand high temperatures and aggressive substances such as oils and fuels.

Want mounting system natural frequency below the engine disturbance frequency of engine idle speed to avoid excitation of mounting system resonance.

DampingStiffness

Other Requirements

Take into account foundation flexion modes. Foundation coupling has large effect in low frequency range.

Frequency

Stiff

ness

Frequency

Dam

ping

Deflection

Dam

ping

Deflection

Stif

fnes

s

The powertrain is normally the only source of vibration during idle. Because the

engine exhibits vibration due to firing pulsations and/or imbalance forces, the vibration is

proportional to engine speed. Problematic vibrations during idle occur at higher

frequencies than unwanted vibrations due to road inputs. In addition, the engine

oscillations are much smaller than road input amplitudes.

As stated earlier, problematic vibrations due to road inputs will normally have

larger amplitudes. Large powertrain displacements cause clearance issues for automotive

components. Current vehicles are packaged tightly to allow more usable space for the

occupant in the interior. The powertrain could possibly collide with other parts within

the engine compartment if the powertrain experiences large oscillations. Furthermore,

excessively large oscillations could cause the powertrain to hit the hood or other body

panels. Thus, road conditions require higher mount stiffness and/or damping to prevent

large oscillations.

12

Because low amplitude oscillations occur at higher frequencies and high

amplitudes occur at lower frequencies, stiffness and damping should roll off as a function

of frequency to provide the best idle isolation. Furthermore, the stiffness and damping

should be adequate to restrict motion for large amplitude displacements. In order for the

mount to exhibit both characteristics, the mount must have some degree of nonlinearity.

1.3: Thesis Statement

To minimize vehicle vibrations it is necessary to address both road and idle

conditions. Both conditions can be addressed by capitalizing on nonlinear and frequency

dependent dynamic response characteristics in hydraulic mounts. Hydro mounts have

nonlinear stiffness and nonlinear damping characteristics, which are frequency

dependent. Idle vibrations have low amplitudes of oscillations at relatively high

frequencies. Furthermore, severe road excitations that cause body resonance problems in

ride generally have larger amplitudes and occur at lower frequency than idle vibrations.

Consequently, hydro mounts should have low damping and stiffness characteristics at

low amplitudes and higher damping and stiffness at higher amplitudes in order to

simultaneously isolate the chassis from idle vibrations and absorb kinetic energy from the

chassis during ride.

The hypothesis of this work is that, because the powertrain mass is a significant

portion of a vehicles mass, it should be theoretically possible to use the powertrain as a

dynamic absorber by designing the nonlinear and frequency dependent mount

characteristics with ride vibrations in mind. The hardening nature of the nonlinear mount

is used to position powertrain resonant frequencies to coincide with problematic body

resonant frequencies in a tramp condition where the left front and right rear tires are

driven in phase. The hydro mount should allow the mount to perform well at idle and to

stiffen in order to place powertrain modes in optimal locations. In this way, the

powertrain is used as a multi degree-of-freedom dynamic absorber to reduce vehicle

chassis vibrations.

13

A suite of models is used to analyze the effects of amplitude and frequency

dependent mount properties. First, a single degree-of-freedom nonlinear powertrain

model to examine how hardening stiffness characteristics can be used to overcome trade-

offs in vibration isolation in idle and dynamic absorption is implemented. Second, a two

degree-of-freedom nonlinear chassis and powertrain model to examine the vibration

reduction possible when mounts have amplitude dependent stiffness properties is

employed. Next, a 13 degree-of-freedom model of the vehicle is used to examine linear

and nonlinear stiffness characteristics in the mount that are beneficial for reduction of

vibration in a vehicle in particular. Lastly a 15 degree-of-freedom model of the vehicle

including a hydro mount model is constructed to examine the effects of amplitude and

frequency dependent mount characteristics on vibration.

14

CHAPTER 2: PRELIMINARY ANALYSIS

To analyze the relationship between the powertrain, nonlinear mounts and a

vehicle platform the Chevrolet SSR was selected as the base test platform. The Chevrolet

SSR is a new advanced platform from General Motors. This platform will allow the

analysis to be applied on the latest technology and chassis proportions. Figure 2.1 shows

the path and methods used to demonstrate the feasibility of using the powertrain as

dynamic absorber. First, low order models including powertrain and powertrain-body

models were constructed to develop an understanding of the effect of nonlinearity on

frequency response and, consequently, an understanding of the effect of nonlinear mounts

on the vehicle behavior.

Second, simplified full vehicle models were used to determine the effect of

nonlinear mounts on different portions of the vehicle. The models were constructed

using linear and nonlinear mount subsystem models. The forced response of linear

models was analyzed with frequency response functions. The nonlinear models were

analyzed using a fourth order Runga-Kutta integration algorithm to numerically generate

response time histories. The time histories were then converted to frequency response

functions at the excitation frequency only near the primary resonances of the model.

Multiple versions of the linear and nonlinear models were used. Each version uses a

slightly different way to represent the nonlinearities and/or frequency dependence of

powertrain mounts. The nonlinear model attempts to capture the nonlinearity of the

powertrain mounts by either assuming the powertrain mounts have a cubic hardening

stiffness characteristic or piecewise nonlinear characteristics. A piecewise nonlinear

system for this research is defined as a system that behaves linearly for a specific

amplitude of deflection.

15

The objective of using this suite of models was to progress toward an

understanding of the behavior of the full vehicle with hydro mounts. For example, the

thirteen degree-of-freedom model with cubic stiffness in the mounts is useful for

conducting proof-of-concept simulations; however, mounts with purely cubic stiffness do

not exist and should not be used because they lack frequency dependence. On the

contrary, hydro mounts are utilized by the sponsor in production vehicles; therefore, the

fifteen degree-of-freedom model with hydro mount nonlinear and frequency dependent

stiffness characteristics is useful for conducting more practical simulations of the vehicle

test bed.

Vehicle Platform Chevy SSR

Linear Analysis Nonlinear Analysis

13 DOF

13 DOF Piecewise nonlinear K(w) @ each Delta x

15 DOF Freq Dependent (Hydraulic mount)

Cubic Stiffness Piecewise nonlinear K(Delta x) @ each w

Simulations

Low order models

13DOF nominal stiffness gain

Figure 2.1: Thesis work flow diagram.

2.1: Nonlinear Powertrain to Ground (SDOF)

The powertrain and powertrain mount dynamics alone provide information about

how the nonlinearity affects the powertrain in general. The powertrain connected to

16

ground simulates how imbalance and engine firing forces can transmit vibrations to the

chassis (ground plane) during engine idle. In this simplified model, the powertrain is

treated as a single degree-of-freedom (SDOF) model with one vertical forcing function.

This force includes firing pulsations and/or engine unbalance. Figure 2.2 shows the

configuration for this simulation. The component in Figure 2.2 represents the nonlinear

effect of the powertrain mounts. The force in the component is proportional to the cube

of the relative displacement between the powertrain and the base (ground).

Figure 2.2: SDOF Model of nonlinear powertrain on ground (rigid base).

The equation of motion (EOM) for this system is:

)(3 tfxKxxCxM =+++ m&&& , (2.1) where, K=100 N/mm, C=1 (N s)/mm and M=1 Kg.

The input force was assumed to be f(t)=Focos(? ot). Also the mass displacement was

assumed to be x(t)=X1cos(? ot+? o) at the excitation frequency only in order to understand

the limitations of this assumption on the response.

The force and response functions were then substituted into the EOM:

2

1 1 1

3 31 o o

cos( ) sin( ) cos( )

cos ( ) F cos( t)o o o o o o o o

o o

M X C X t KX t

X t

w w f w w f w f

m w f w

- + - + + +

+ + =, (2.2)

where through the use of the trigonometric identity:

2 1 1cos ( ) cos(2 2 )2 2o o o o

tw f w f+ = + + , (2.3)

the following substitution can be made in Equation (2.2):

K

x

M

f(t)

C

17

3 3 1cos ( ) cos( ) cos(3 3 )4 4o o o o o o

t t tw f w f w f+ = + + + , (2.4)

in addition to these other familiar trigonometric identities:

cos( ) cos( )cos( ) sin( )sin( )

sin( ) sin( )cos( ) cos( )sin( )o o o o o o

o o o o o o

t t t

t t t

w f w f w fw f w f w f

+ = -+ = +

. (2.5)

By substituting these forms into the EOM, combining similar trigonometric terms and

ignoring higher frequency terms for the time being, the following equation can be found:

( ) ( ) ( ) ( ) ( ) ( )2 3

1 1 1 1

cos sin cos cos sin sin 0

3,

4

o o o o o o o

o o

A B F t B A t

where A KX M X X and B C X

f f w f f w

w m w

+ - + - =

= - + = -. (2.6)

In order for the previous equation to be satisfied for all time, each trigonometric

coefficient must be equal to zero. Therefore, the following two equations must be

satisfied simultaneously:

( ) ( )cos sino o oA B Ff f+ = and (2.7)

( ) ( )cos sin 0o oB Af f- = . (2.8)

If Equation (2.7) is squared and added to the square of Equation (2.8), the following

result is obtained:

[ ]2

22 2 2 2 3 21 1 1 1

34o o o o

A B F KX M X X C X Fw m w + = - + + - = (2.9)

It can be shown that the FRF function of this system is of the form:

( )21

122 2 2

1 34o

o o

Xwhere NL X

F K M NL Cm

w w= =

- + +. (2.10)

Numerical simulations verify this analytical relationship for amplitudes of

displacement, Xo, relatively small. Simulink (Matlab toolbox) was used to simulate

displacement time histories such as the one shown in Figure 2.3. The frequency and

amplitude of the output were determined by performing Discrete Fourier Transforms

(DFTs) (the function fft in MATLAB) after the response reached steady state. A flat-

top window (P-301) was used to weight the time history prior to performing the DFTs in

18

order to prevent numerical leakage in the computation. Figure 2.5 shows the portion of

the time history that is used for the DFT.

1 2 3 4 5 6 7 8 9 10

-8

-6

-4

-2

0

2

4

6

8

x 10-3

Figure 2.3: Complete mass displacement time history.

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 2.4: Input Force (f(t)).

0 20 40 60 80 100 120-0.01

-0.005

0

0.005

0.01

0.015

time

Figure 2.5: Steady state portion of mass displacement x(t).

The magnitude of each nominally linear FRF was then determined by dividing the

displacement amplitude by the force amplitude. Although in Figure 2.6 it appears that

the analytical relation matches the numerical results perfectly, there are discrepancies

near resonance. Figure 2.7 shows a close up of the resonance region; it shows the effect

Inpu

t For

ce

Time (s)

Mas

s D

ispl

acem

ent x

(t)

Time (s)

Time (s)

Dis

plac

emen

t

19

of neglecting higher order frequency terms to develop the FRF relationship. It can be

concluded that neglecting the higher frequency component in the response amplitude

prediction will predict slightly larger responses near resonance than the actual nonlinear

model will exhibit. It is important to examine limitations in single-frequency

assumptions regarding the response because linear methods such as this are currently

practiced in the automotive industry by the majority of engineers. One of the objectives

of this thesis was to draw attention to such analysis limitations due to nonlinearities.

Figure 2.6: SDOF Model Analytical-Numerical Comparison.

If the displacement response is assumed to be X(t)=X1cos(? ot+? o)+X2cos

(3? ot+3? o) and the same procedure is used without neglecting higher frequency terms,

the nonlinear component of the FRF relationship becomes:

++= 21

22

21 4

323

43

XXXXNL m . (2.11)

Using these higher frequency terms, the analytical prediction is more accurate as shown

in Figure 2.8 and 2.9.

Mag

nitu

de x

/F

Frequency (Hz)

20

Figure 2.7: SDOF analytical-numerical comparison SDOF (Zoom in).

Figure 2.8: Higher frequency term effects on SDOF analytical results.

Mag

nitu

de X

o/F O

Frequency (Hz)

Mag

nitu

de x

/F

Frequency (Hz)

21

1.7 1.72 1.74 1.76 1.78

10-1.09

10-1.08

10-1.07

10-1.06

10-1.05

NumericalAnalytical (wot)Analytical (wot +3wot)

Figure 2.9: Higher frequency term effects on SDOF analytical results (zoom in).

Figure 2.10 shows the FRF as the nonlinear term, *x3, is increased in size in the

model by changing between 1000 4000 and 6000. Similar effects are observed in

Figure 2.11, when the amplitude of the excitation is increased. Because the cubic term is

dependent on displacement, it has little effect on the response (at the excitation

frequency) for excitation frequencies away from resonance. Note that this research

focuses on the behavior of nonlinear models near their primary resonant frequencies and

does not consider other types of nonlinear resonance (subharmonic, superharmonic,

combination, internal, etc.). Nonlinearity in this powertrain model is observed to affect

the behavior of the powertrain near resonance where large relative displacements occur.

Mag

nitu

de X

o/F O

Frequency (Hz)

22

Figure 2.10: effect on SDOF transmissibility, () =1, (--)=2, (----)=4 , ( ) =8.

Figure 2.11: Input amplitude effect on SDOF transmissibility, () input amplitude 0.05, (--) input amplitude 0.1, (----) input amplitude 0.3, ( ) input amplitude 0.5.

Increasing nonlinear coefficient

Mag

nitu

de X

/Fo

Frequency (Hz)

Increasing Input Amplitude

Mag

nitu

de X

/Fo

Frequency (Hz)

23

In this simplified powertrain model, the resonance frequency of the mount is

observed to move toward higher frequencies as either the nonlinearity in the mount or the

amplitude of the excitation (and response) are increased. This type of nonlinear tuning of

the powertrain mount stiffness is desirable in the present application because it can

potentially be used to overcome the trade offs discussed in Chapter 1 regarding design of

mounts in idle and ride. For example, when the powertrain responds with small

amplitudes of displacement at idle, the effective resonant frequency of the powertrain

remains low resulting in good isolation of the engine unbalance and firing forces. When

the powertrain responds with relatively larger amplitudes of displacement in ride, the

effective resonant frequency increases resulting in less deflection across the mount

(longer life) and potentially better vibration energy absorption capabilities at higher

frequencies. This latter aspect of powertrain mount nonlinear resonant tuning is

examined using various models in the following sections.

24

2.2: Nonlinear Powertrain-Body (2DOF)

The following two DOF system shown in Figure 2.12 was analyzed numerically

with nonlinearities incorporated. The two DOFs represent the body and powertrain of

vehicle. The input used in this model was an input displacement (Xb(t)) at the spindle.

xS, (Body)

xP

xb, (wheel spindle)

C P K P

Powertrain (MP)

K S C S

Sprung Mass(Ms)

Powertrain Mount

Vehicle Suspension

Figure 2.12: Two degree-of-freedom system with nonlinear term.

To examine the effects of the nonlinear term on the dynamics of the system,

nominally linear transmissibility functions were calculated using numerical simulations.

The terms nominally linear are used in this thesis to refer to the ratio of the response

(amplitude/phase) to the excitation (amplitude/phase) at the excitation frequency. In

other words, nominally linear transmissibility functions contain information about only

primary resonances in nonlinear systems. The simulations of the system were conducted

with Simulink (MATLAB) using EOMs, Equations (2.12) and (2.13):

( ) ( ) ( ) ( )3 sinS S S S S S P P S P P S P S b oM x C x K x C x x K x x x x X tm w+ + - - - - - - =&& & & & , (2.12)

( ) ( ) ( )3 0P P P P S P P S P SM x C x x K x x x xm+ - + - + - =&& & & . (2.13) where,

KS=200 N/mm KP=75 N/mm

CS=4 (N s)/mm CP=1 (N s)/mm

MS=10 Kg MP=1 Kg

25

The transmissibility plots in Figure 2.13 and 2.14 show the system behavior in the

presence of a cubic nonlinearity (m(xP-xS)3 ). Note that the nonlinearity primarily affects

the transmissibility functions near the second resonant frequency because the forced

response characteristics near the first resonant frequency correspond to the in phase

motion of the powertrain and body inertias. This in phase motion does not exercise the

powertrain mount and, therefore, does not elicit nonlinear behavior in the model. At the

second resonant frequency, the powertrain and body inertias move out of phase resulting

in more nonlinear behavior as the mount is exercised more effectively. Figure 2.15

demonstrates why the nonlinear mount is effective at selectively transmitting vibration

(kinetic) energy from the body to the powertrain as the amplitude of the excitation

(response) increases. In this figure, the powertrain motion exhibits a desirable attribute

as the degree of nonlinearity in the dynamics increases. As the excitation amplitude

(degree of nonlinearity) increases, the frequency at which the powertrain is an effective

absorber increases as well. Because this frequency of high energy absorption of the

powertrain increases with amplitude, it can be concluded that good isolation at idle when

the response amplitudes are small can be achieved simultaneously with good dynamic

absorption when the response amplitudes are relatively larger.

This property of varying degrees of nonlinear body and powertrain interactions is

important when considering how the powertrain can be designed as dynamic absorber.

Moreover, it is desirable to have nonlinear interactions between the powertrain and body

when vibrations occur in ride near the second resonance because these vibrations result in

a harsher ride. The objective in the remaining models is to examine how these

nonlinear interactions change as more degrees of freedom are added to the model. For

instance, the next section examines these nonlinear interactions between the powertrain

and body when the unsprung mass is included as well.

26

Figure 2.13: 2DOF X2/Xb Transmissibility Response () =5, (--) =10, (----) =30,

( ) =50.

Figure 2.14: 2DOF X1/Xb Transmissibility ()=5, (--) =10, (----)=30, ( )=50.

Mag

nitu

de T

rans

mis

sibi

lity

X2/

Xb

Frequency (Hz)

Tra

nsm

issi

bilit

y X

1/X

b (d

B)

Frequency (Hz)

27

Figure 2.15: 2DOF X2/X1 Transmissibility ()=5, (--)=10, (----)=30, ( )=50.

Tra

nsm

issi

bilit

y x

2/x1

Frequency (Hz)

28

CHAPTER3: 13 DOF VEHICLE MODELING/SIMULATION

3.1: Thirteen Degree-Of-Freedom

In order to develop insight into the effect of the powertrain on vehicle body ride

vibrations, a more complete vehicle model must be used. This model should describe the

most important aspects of vehicle ride without adding too much complexity making it

difficult to determine the source, cause or result of different mount nonlinearities and

frequency dependencies. Simplified models such as the one used in this section are

important in developing a better understanding of the vehicle; however, future work may

need to implement the mount design process discussed in this thesis in a more complete

vehicle model and in full vehicle tests to confirm these findings. A thirteen DOF model

was constructed. This model describes many of the key vehicle vibration resonant

frequencies without making it too difficult to extract general information about

powertrain mount design.

3.1.1: Model Description

The Chevrolet SSR, which is manufactured by General motors, was used as the

vehicle of interest for this study. Many of the nominal mass, inertia, stiffness and

damping properties of the vehicle were provided by the sponsor based on vendor

information (suspension, tire, etc.) and finite element models (inertia properties, etc.).

Based on these values and the dimensions of the vehicle itself the model shown below in

Figure 3.2 was developed.

29

Figure 3.1: SSR side view.

a a

b1 b2

Kbb, Cbb Ktb, Ctb

Ktb, Ctb Ktm

c

Kfs Cfs

Krss, Crss

Krs

Crt Krt

Cft Kft

Mp

Mfs

bMf, bIfx

Mrs

Kem Cem

Ipx

Ipy

Ctm

cMf, cIfx

aMf, aIfx

f f

x y

z

Cfs Kfs

Kbb Cft

Krs Krss, Crss

Mrs

Crt Krt

Cem

zrf(t)

zrr(t)

zlr(t)

d

e

z1

z2

z3

z4

z5,q5

z6,q6

z7,q7

z8,q8x ,q8y

Figure 3.2: 13 DOF vehicle model schematic.

Rear Front

Middle

Front Wheel spindle (Unsprung)

Rear Wheel spindle (Unsprung)

Tire Patch

Powertrain

Rear Body (Sprung)

Middle Body (Sprung)

Front Body (Sprung)

30

The model has 13 DOFs. Four of the DOFs describe the unsprung masses for the

wheels. Three main sections of the vehicle in the front, middle and rear are described

using six DOFs. Each of these three sections was permitted to roll and bounce. There

are bending and torsional stiffness elements between the sections. The last three DOFs

are used to describe powertrain bounce, pitch and roll movement. The powertrain is

supported by three simple lumped springs; two for engine support and an additional one

to support the rear of the transmission. The model uses proportional damping to describe

dissipation throughout the vehicle.

The vehicle model was constructed to provide a minimal but sufficient description

of the powertrain dynamics (ignoring lateral motions and twist). For example, it is

possible in the 13 DOF model to observe the front and rear sections of the body as they

each experience roll motions out of phase. This shape, normally referred to as torsion,

can be observed and documented. In addition, if each unsprung wheel mass has its own

DOF, then the model can describe wheel hop conditions (i.e., resonance of the spindle

relative to the vehicle chassis). This condition is of interest to vehicle dynamics groups

for performance aspects and could involve a design trade-off for ride performance.

In matrix form, the input-output equations of motion are [ ]{ } [ ]{ } [ ]{ } { }

{ } { }{ } {

}

1 2 3 4 5 5 6 6 7 7 8 8 8 13 1

13 1

where

0 0 0 0 0 0 0 0 0

T

x y

ft lf ft lf ft rf ft rf rt lr rt lr rt rr rt rr

T

z z z z z z z z

C z K z C z K z C z K z C z K z

q q q q q

+ + =

=

= + + + +

M R C R K R F

R

F

&& &

& & & &

(3.1)

where the damping is assumed to be proportional to the mass and stiffness of the system:

[ ] [ ] [ ]C M Kh n= + .

31

and [K] and [M] are the following:

[M]=

000000000000000000000000000000000000000000

000000000000000000000000000000000000000000

MfIfx

MfMrs

MrsMfs

Mfs

ba

a

IpyIpx

MpIfx

MfIfx

000000000000000000000000000000000000000000000000000000000000000000000000

cc

b, (3.2)

32

[K]=

++++

++++

++

000000000000

Krs)f(Krss-Krs)f(Krss00Krs)(Krss-Krs)(Krss-00

0000000000aKfs-aKfs00Kfs-Kfs-

KrsKrssKrt0000KrsKrssKrt0000KfsKft0000KfsKft

d)-(c Ktmb2)-(b1 d Kemd)-(-d Kem0)b2-(-b1 Kemb2)-(b1 Kem

Ktm-b2)-(b1 KemKem-Kem-000

Kbb-000Ktb-0

KtmKbb 2000Ktb)b2(b1 Kema Kfs 2b1)-(b2 Kema)-(a Kfs

Kbb-b1)-(b2*Kema)-(a*KfsKbbKem*2KfsKfs0000000Kfs*a-Kfs-0Kfs*aKfs-

22

222

+++++

++++

33

000000000

Krs)(Krss f 2KtbKrs)(Krss fKrs)(Krss f-Ktb-Krs)(Krss fKrs)(Krss f-Krs)(Krss 2Kbb0

Ktb-0KtbKtb0Kbb-000Ktb-000

Krs)(Krss f-Krs)(Krss-0Krs)(Krss fKrs)(Krss-0

000000

2 ++++++++++

+

++++

++

+

22

22

22

d)-(c Ktmd Kem 2B1)-(B2*d*KemKtm d)-(c-Kem d 2b1) -(b2 d Kem)b2(b1*Kemb1)-(b2 Kem

Ktm*d)-(c-d Kem 2b1)-(b2*KemKtmKem 2000000000

d)-(c Ktm0Ktm-b2)-(b1 d Kem)b2-(-b1 Kemb2)-(b1 Kem Kem d 2-b2)-(b1 KemKem 2-

000000000000

. (3.3)

34

Kft Front Tire StiffnessKrt Rear Tire StiffnessKfs Front Suspension StiffnesssKrs Rear Suspension StiffnessKbb Body Bending StiffnessKtb Body Torsional StiffnessKem Powertrain Mount StiffnessKtm Trans

-------- mission Mount Stiffness

Mfs Front UnSprung Mass(spindle)Mrs Rear UnSprung Mass(spindle)Mf Frame Mass(Body)Mp Powertrain MassIfx Frame Rotational InertiaIpx Powertrain Rotational InertiaIpy Powertrain R

--

----- otational Inertia

a Front Mass proportion Middle Mass proportion? Rear Mass proportion

---

The thirteen DOF model was programmed into MATLAB, which calculates 13

transmissibility equations based on a specified road input at the four tire patches. The

road excitation used in this research corresponds to the vehicle tramp excitation, in

which the left front and right rear tires are forced in phase and out of phase with the left

rear and right front tires. Because the model is linear, the law of superposition holds so

the response due to multiple inputs can be generated by adding the individual results for

each excitation applied separately. For example, the tramp excitation, which excites

torsional body modes in the vehicle, produces transmissibilities that are the sum of the

transmissibilities for the left front and right rear tires.

[ ] [ ] [ ] [ ]( ) ][12 DKCjMT -++-= ww (3.4)

where,

[ ] ( )j C+Kft; j C+Kft; j C+Kft; j C+Kft; 0; 0; 0; 0; 0; 0; 0; 0; 0D diag w w w w=

35

Because the chassis and body of the vehicle do not have the three lumped masses

as assumed in the model, the values of the bending and torsional stiffness coefficients

(body stiffness) between the three sections must be adjusted such that the model modes

match modal results supplied by the sponsor. If the vehicle damping is assumed to be

small in the model, then the imaginary portion of the transmissibility tracks the relative

motion of the 13 DOFs at each frequency of excitation. Matlab code animate.m in

Appendix A animates the model mode shapes. The subsequent section uses animate.m to

calibrate the 13 DOF model.

It is necessary to develop a frequency range of interest. A frequency range of

interest defines an area to gauge improvements. Figures 3.3 through 3.5 are provided by

the sponsor. Since the vehicle is convertible there is data for top up and top down

conditions. Each plot displays two curves corresponding to either top up or top

down condition. These plots show that the primary frequency range of interest appears

to be 10 to 15 Hz with large vertical accelerations occurring there in the suspension and

steering hub.

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Au

top

ow

er S

pec

tru

m (

dB

ref

1(m

2/s4

)/H

z)

Baseline Top Down Baseline Top Up

Figure 3.3: Sponsor supplied SSR vertical front suspension road test

Aut

opow

er S

pect

rum

dB

(m2/

s4)

Frequency (Hz)

36

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Au

top

ow

er S

pec

tru

m (

dB

ref

1(m

2/s4

)/H

z)

Baseline Top Down Baseline Top Up

Figure 3.4: Sponsor supplied SSR vertical rear suspension road test.

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Au

top

ow

er S

pec

tru

m (

dB

ref

1(m

2/s4

)/H

z)

Baseline Top Down Baseline Top Up GMUTS 6

Figure 3.5: Sponsor supplied SSR vertical steering column road test.

Aut

opow

er S

pect

rum

dB

(m2/

s4)

Frequency (Hz)

Aut

opow

er S

pect

rum

dB

(m2/

s4)

Frequency (Hz)

37

3.1.2: Calibration

Each of the body stiffness parameter values were determined in an ad hoc manner

using a Matlab code for simulating the mode shapes. The mode shapes were displayed

by observing each displacement in a synchronous motion for a specified input

configuration and frequency. See below for full calibration procedure. Figure 3.6 shows

the modal plot of the results of the calibration analysis. Figure 3.7 displays the

transmissibility functions for the 13 DOF model after the calibration procedure was

applied. Table 3.1 displays the modal vibration shapes at specific frequencies.

Calibration Procedure

1) Develop model with best estimate parameters.

2) Plot imaginary portion of FRF to produce a mode plot.

3) Determine mode shapes of each peak using animation script (animate.m)

4) Tune for torsional mode first.

5) Re-execute model with extreme (large) value of Kbb (Bending Stiffness).

6) Re-animate mode shapes.

7) Vary Ktb (Torsional Stiffness) to understand its effect.

8) Fine tune Ktb to place mode shapes in correct locations in the frequency spectrum.

9) Repeat steps 5-8 for bending mode. With Ktb extreme and vary Kbb.

10) Combine calibration factors.

11) Re-execute model

12) Verify mode shape locations with animations.

13) Repeat 7-11 if necessary

38

Figure 3.6: Modal Plot of Calibrated 13 DOF Model.

Figure 3.7: Calibrated 13 DOF body transmissibility, upper--bounce DOFs lowerRoll

DOFs.

Roll and Bounce

Torsion & Pitch

Suspension Torsion

Suspension

Powertrain Roll & Torsion

Imag

inar

y po

rtio

n of

FR

F

Powertrain Pitch

39

Table 3.1: 13 DOF Vehicle model mode shapes.

Frequency (Hz) Shape Screen Shot

2.8 Roll

3 Bounce

8.4 Powertrain Roll and Torsion

10Powertrian Pitch, Front suspension

and Bending

40

Table 3.1: 13 DOF Vehicle model mode shapes. (continued).


13.8 Torsion

18.1 Powertrain Pitch

19.6 Front suspension

20.3 Bending

41

Table 3.1: 13 DOF Vehicle model mode shapes. (continued).


28.3 Torsion

30.6 Bending

42

3.1.3: Linear Stiffness Effects

In order to determine how the system will respond to changes in nominal engine

mount stiffness, the 13 DOF model was run several times with different engine mount

stiffness values. The intention of these changes in mount stiffness was to shift the three

resonances (bounce, pitch and roll) of the powertrain in frequency. Theoretically, the

powertrain could absorb energy from the body at each of the powertrains natural

frequencies, which are listed below for the chosen parameters

( )

( ) (Roll) Hz 35.82/

21

(Pitch) Hz 3.172/2

(Bounce) Hz 8.112/2

22

3

22

2

1

=+

=

=+-

=

=+

=

pw

pw

pw

px

empn

py

emtmpn

p

tmempn

IbbK

IdKdcK

MKK

.

Figure 3.8: Powertrain Natural Frequencies.

To determine the effect of nominal stiffness changes, the forced response analysis

focused on the 10-15 Hz frequency range because the tramp resonance of interest is in

this frequency range. Shifts in the natural frequencies of the powertrain are evident in

Figure 3.9 as the linear mount stiffness varied from a factor of 1 to 2.2 times KNOM

(nominal stiffness value), 390 N/mm. Note the motion toward the right of the peak in the

shaded region. The motion of this peak is desirable for dynamic absorption by the

powertrain because it positions the powertrain resonance in the neighborhood of the

tramp resonance to be reduced. Figures 3.10 through 3.13 show the predicted

Bounce motion of Mp Rotational motion in x-y of Ipx,Ipy

43

transmissibility of the powertrain and each of the body modes as the engine mount

stiffness was also varied from a factor of 1 to 2.2 times KNOM, 390 N/mm.

Figure 3.9: Powertrain mount nominal stiffness effects on mode plot.

44

Figure 3.10: 13 DOF powertrain transmissibility (.)1.0*KNOM, (----)1.4*KNOM, (-.-

.)1.8*KNOM ( )2.2*KNOM.

Each of the transmissibility plots shows the frequency range of interest (10-15

Hz). In each of the three sections of the body (front, middle and rear), the bounce motion

is reduced in the frequency range of interest; however, the middle section of the body has

shown the most improvement. The larger mount stiffness appears to neither benefit nor

harm the responses of the body except in the rear torsional motion. The rear body roll

transmissibility in Figure 3.13 exhibits a shift of the lower frequency torsional mode into

the frequency range of interest. The front and middle sections do not exhibit this

prevalent peak. The middle section of the body (Figure 3.12) seemed to improve most

significantly from these linear changes, whereas the front section (Figure 3.11)

experienced some benefit. The rear section did not improve as much as the front and

middle sections. This variation toward the rear of the vehicle was anticipated because the

powertrain inertia is located in the front of the vehicle.

45

Figure 3.11: Front body FRFs (upper-- bounce, lowerroll) with varied linear engine

mount factor (.)1.0*KNOM, (----)1.4*KNOM, (-.-.)1.8*KNOM ( )2.2*KNOM.

Figure 3.12: Middle body FRFs (upper-- bounce, lowerroll) with varied linear engine


Reduction

Reduction

46

Figure 3.13: Rear body FRFs (upper-- bounce, lowerroll) with varied linear engine


3.2: Nonlinear Model Description

The same linear 13 DOF model described in section 3.1.1 was also used in this

section with some modifications, which involved a cubic nonlinear stiffness term in the

powertrain mounts. The nonlinear term was defined in the same manner as for the

nonlinear term in the low order models.

The linear algebra methods applied up to this point in the thesis were not applied

in this section because of the presence of the cubic nonlinearity. In order to integrate the

EOMs, a fourth order Runge-Kutta (R-K) ordinary differential equation (ODE) algorithm

was used. In this approach, the derivative of the nonlinear state variable function was

evaluated four times at each time step in order to predict the response at the subsequent

time step to fourth order accuracy. The fourth order R-K algorithm for a scalar state

function, f(tn,yn), as a function of the explicit variable of integration, time t, with time

Amplification Reduction

47

step, Dt, is listed below for reference and the MATLAB code (SSR_NL.m) is provided in

Appendix A:

( )

( )

( )

( )34

23

12

1

543211

, 2

,2

2,

2

, where,

6336

,For

kytttfk

ky

tttfk

ky

tttfk

yttfk

tOkkkk

yy

ytfdtdy

nn

nn

nn

nn

nn

+D+D=

+

D+D=

+

D+D=

D=

D+++++=

=

+

. (3.5)

To verify that the numerical nonlinear code was operating correctly was set zero

in order to create Figure 3.14. When is set to zero, the cubic nonlinear term is removed

forcing the model to operate linearly. The transmissibilities in Figure 3.14 match Figure

3.7 previously developed using linear algebra.

Figure 3.14: Linear transmissibilities (.)Front (----) Middle ( )Rear.

48

Figure 3.15: Nonlinear transmissibilities (.)Front (----) Middle ( )Rear

-1 -0.5 0 0.5 1 1.5-2000

-1500

-1000

-500

0

500

1000

1500

Displacement [mm]

Res

torin

g F

orce

[N]

nonlinearlinear

Figure 3.16: Nonlinear Restoring force

49

Figure 3.17 through Figure 3.19 compare the response before and after the

nonlinear term is turned on ( 0m ). In the frequency range of interest, 10-15Hz, each of

the three sections has some form of reduction either with roll or bounce. The front

section achieves reduction for bounce. This reduction may be due to a shift in the

powertrain roll resonance. In addition, the front achieves reduction in the frequency

range above approximately 22 Hz.

The linear front body bounce in Figure 3.14 has a anti-resonance just below 10

Hz. But the nonlinear front bounce response has an anti-resonance just above 10Hz;

therefore, an obvious shift in the powertrain mounts stiffness. In addition, the powertrain

roll mode in the linear analysis was located around 8 Hz. However, the nonlinear

response shows that its resonance may have shifted up in frequency to 10 Hz.

Figure 3.17: Nonlinear Effect on front body (upperBounce, lower--Roll) (....) Linear,

( ) Nonlinear.

Similar to the front section of the body, the middle experiences an even greater

bounce mode reduction from 10-24 Hz. But the nonlinear middle roll does not have as

good of reduction. Instead, there are two upper resonances that shift down in frequency.

The lower frequency of the two experiences reduction and the higher resonance is

50

amplified. As shown in Figure 3.19, rear body bounce has a significant reduction in the

range of 10-18 with the use of nonlinearity. Furthermore, a body roll anti-resonance

shifts down in frequency from 14.5Hz to 12.5 Hz. Like all three sections of the body the

8.4 Hz powertrain roll mode resonance is no longer apparent in the nonlinear model.

Overall, the 13 DOF model with cubic nonlinear powertrain mount stiffness

produced similar results with the nominal stiffness changes seen in section 3.1.3. This

model suggests that body transmissibility reduction is capable using nonlinear mounts.

Nevertheless, it is not realistic to assume powertrain mounts are capable of cubic

nonlinearity. Consequently, in the next section a more physically realizable mount

stiffness characteristic is implemented within the 13 DOF model.

Figure 3.18: Nonlinear Effect on middle body (upperBounce, lower--Roll) (....) Linear,

( ) Nonlinear.

51

Figure 3.19: Nonlinear Effect on rear body (upperBounce, lower--Roll) (....) Linear,

( ) Nonlinear.

3.3 Curve Fit Models

Powertrain mount frequency dependence was incorporated in an attempt to make

the 13 DOF model more realistic. As the modeling becomes more sophisticated, an

actual hydro mount model will be incorporated; however, the approach in this thesis is to

gradually add modeling detail so that the effects of each additional detail can be

understood separately. For example, the stiffness and damping of the powertrain mount

model was varied in this section as a function of frequency to mimic the frequency

dependence of the powertrain hydraulic mounts that are incorporated later in full. The

sponsor provided stiffness and damping characteristics as a function of frequency at

several different deflection amplitudes. Because curve fits can be used to incorporate

frequency depen

Msme Thesis Tf

Documents

active hydraulic mounts

nonlinear powertrainbody

hydraulic engine mounts

powertrain system

dof vehicle modelingsimulation

nonlinear analysis expertise

simple elastomeric mounts

list of figures