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VEHICLE DYNAMICS MODELING & SIMULATION

MOHD AZMAN ABDULLAH JAZLI FIRDAUS JAMIL AHMAD ESMAEL MOHAN

To cite this book:

M.A Abdullah, J.F Jamil, A.E Mohan, Vehicle Dynamics Modeling & Simulation,

Malacca, Centre for Advanced Research on Energy (CARe), Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka, ISBN 978-967-0257-78-5, 2016

First published 2016 (27 th September 2016)

Copyright © 2016 by Centre for Advanced Research on Energy (CARe)

All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, electronic, mechanical photocopying, recording or otherwise, without the prior permission of the Publisher

Reprinted 2018 (2 nd Edition)

Edited and Reprinted February, 2019 (3 rd Edition)

ISBN: 978-967-0257-78-5

Published and Printed in Malaysia by:

Centre for Advanced Research on Energy,

Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka,

Hang Tuah Jaya 76100, Durian Tunggal, Melaka, MALAYSIA

Fax: +6062346884 | Email: care@utem.edu.my

As part of anti-plagiarism policy, we hereby acknowledge all various sources of facts and figures that we used in this book to give the understanding to the students Most

of the facts and figures we could not find the sources We would like to thank all legal owners of the facts and figures

This book is not for sale You may donate minimum of USD 7.00 to the author as a token of appreciation for compilation of the notes And/or at least cite this book in your publications

3 Lateral Vehicle Dynamic Modeling And Simulation

4 Longitudinal Vehicle Dynamic Modeling And Simulation

5 Tire Dynamic Modeling And Simulation

6 Mechanical System Modeling Tutorial Using Matlab

7 Vehicle Dynamic Modeling Tutorial Using Matlab/Simulink

8 Vehicle Dynamic Simulation Tutorial Using CarSim

9 Vehicle Dynamic Modeling Verification Tutorial

CHAPTER 1 INTRODUCTION

1.1 Dynamics of Vehicle

The dynamics of vehicle can be modelled and simulated for the purpose of study and analysis The dynamics behaviours of the vehicle are also can be observed through simulation This method in analysing and studying the dynamics performances of vehicle via simulation are due to the constraints (cost, time and safety) of other approaches such as actual vehicle experiment Vehicle can be classified to ground, fluid and inertia vehicles Ground vehicles are supported by the ground with maximum speed of 600 km/h, guided with constrained to move along a fixed path example railway

vehicle (Fig 1.1) and tracked levitated vehicle and non-guided (Fig 1.2) where the

vehicle can move in any direction on the ground example road and off-road vehicles Fluid vehicles can be operated to the maximum speed of 3,000 km/h example marine

craft (Fig 1.3), hydrofoils and ships that move on and under water Inertia vehicles can

be operated to the maximum speed of 50,000 km/h, example aircraft (Fig 1.4) and

spacecraft

Fig 1.1: Guided ground vehicle

Fig 1.2: Non-guided ground vehicles

Fig 1.3: Fluid vehicle

Fig 1.4: Inertia vehicle

The vehicle dynamics system interaction (Fig 1.5) consists of the inputs such as visual

(from the driver of camera), ground elevations and surface irregularities which interact with the tires and aerodynamic loads which act on the body The driver may apply inputs for direction through steering system and acceleration and braking through gas and brake pedal respectively The driver behavior is also can be modelled for simulation purposes The outputs evaluation of the vehicle are measured in term of performance, handling and ride

Fig 1.5: Vehicle dynamics interaction

In fundamental approach to vehicle system modeling, vehicle dynamics is concerned with movement of vehicles The movements include acceleration, braking, ride & cornering The vehicle dynamic behavior is determined by the forces imposed on the vehicle from the tires, gravity, and aerodynamics The vehicle and its components are studied to determine what forces will be produced by each of these sources at a particular maneuver and how the vehicle will respond to these forces For that purpose

it is essential to establish a rigorous approach to modeling the systems and the conventions that will be used to describe motions

In the lumped mass vehicle analysis, a vehicle is made up of many components distributed within its exterior envelope For most of the analysis on vehicle all components are assumed to one together For example, under braking, the entire vehicle slows down as a unit; thus it can be represented as one lumped mass located

at its center of gravity (CG) with appropriate mass and inertia properties For longitudinal vehicle & lateral motions (acceleration, braking and cornering), one mass (total vehicle mass) is sufficient For vertical vehicle motion, vehicle body (sprung mass) is treated as lumped mass and wheel (unsprung mass) is treated as another lumped mass

The basic vehicle axis system (Fig 1.6, Table 1.1) consist of 3 translational motions

and 3 moments which are the longitudinal motion through x-axis where the direction to the front of the vehicle is the positive value, the lateral motion through the y-axis where the direction to the right side of the vehicle is the positive value, the vertical motion through z-axis where the upward direction is the positive value, the roll moment about the x-axis where clockwise direction is positive, the pitch moment about the y-axis where clockwise is positive and the yaw moment about the z-axis where clockwise direction is positive

Fig 1.6: Vehicle axis system

Table 1.1: Vehicle motions Axis Translational

velocity

Angular displacement

Angular velocity

Force component

Moment component

x u (forward) , roll p or ∅̇ F x M x

y v (lateral) , pitch q or θ̇ F y M y

z w (vertical) , yaw r or 𝜑̇ F z M z

From the top view of the vehicle, the earth fixed coordinate system (Fig 1.7) consist

of the reference x-y-axis and the vehicle projected x-y-axis (or the vehicle x and y axis) The angle between the reference x-axis and the vehicle projected x-axis is heading angle The projection of instantaneous velocity direction is the direction of the vehicle

is not the same as the vehicle x-axis The steer angle is the angle of the front wheel during cornering

Vertical

Longitudinal Lateral

Figure 1.7: Earth fixed coordinate system

1.2 Tire Axis System

Fig 1.8 shows the tire axis system The slip angle, α is the angle between the direction

of the wheel (or tire) and the direction of motion This slip angle occurs when the vehicle

is sliding or drifting Mathematical model of vehicle dynamic system is a description in terms of mathematical relations and represents an idealization of an actual physical system The predicting performance from a model is called analysis



Fig 1.8: Tire Axis System

Fig 1.9: Physical system

(1

f f f f r r r r z

L C

L C

Fig 1.12: Simulation

1.3 Vehicle Modeling

Modeling steps starting with understanding of system function, problem definition and input and output variables identification Simplified schematics using basic elements are drawn The mathematical model is developed to identify reference point and positive direction through free body diagram (FBD) for each basic element and elemental equations with the application of physical laws The validation of the model

is by comparing simulation results with physical measurements

The advantages of vehicle system modeling and simulation (Fig 1.13) are they do not

require a physical vehicle and test track They can treat new designs and technologies without prototype The test in the simulation can be repeated In actual testing, a lot of time is required to setup and run the test Furthermore, in actual testing, the test vehicle must be modified or rebuild is there is need to change the vehicle parameters The simulation also supports driver safety where it does not need for a driver to perform risky maneuver tests such as J-turn, fishhook, etc Vehicle simulation also helps to gain insight and understanding of the vehicle system

Fig 1.13: Tripped rollover simulation of a European large van

Models for Vehicle System Dynamics

Appropriate models should be used for a given vehicle application or analysis High number of DOF with nonlinear models produce very detail results The models are complex with subsystems, require high maintenance, long development time, extensive data collection for parameters and the motions can be animated Examples software for this type of modeling are CarSim, CarMaker, ADAMS/Car

Low number of DOF models can be linear or nonlinear models and they are simple models The models are made up of differential equations and/or look-up tables They have short development time, few parameters required, fast computational time and used for parametric studies The models can be developed in Matlab/SIMULINK software

Development of Vehicle Model

Vehicle model is developed to capture the essential features of dynamics (DOF that are dominant) For analysis of vehicle handling, there are 3 primary motions that should

be included; longitudinal motion (translational along x-axis), lateral motion (translational along y-axis) and yaw motion (rotation about z-axis) The model has minimum parameters associated with a data collection cost and minimizing the data will minimize the total cost of the model The smaller the model the less data is required and that reduces uncertainty error in the model

Tire Model

Tire model determines the longitudinal and lateral tire forces based on the tire normal

load, slip ratio and slip angle (Fig 1.14)

Fig 1.14: Tire slip angle

Vertical Vehicle Dynamics Models

Vertical vehicle dynamics or ride dynamics is the study of the vertical motion of the vehicle due to the input from the road or other vibration sources Purpose of study of ride dynamics is to control vibration of vehicle so the passenger sensation of

discomfort does not exceed a certain level In order to study ride dynamics, the

following models will be developed

1 DOF quarter car model (base excitation vibration model)

2 DOF quarter car model

2 DOF bounce and pitch model

4 DOF pitch plane ride model

4 DOF roll plane ride model

7 DOF full car ride model

Human body model

Driver model

Lateral Vehicle Dynamics Models

Lateral vehicle dynamics (Fig 1.15) or vehicle handling is the study of the vehicle

motion due to the steering and environmental inputs (wind gust, road disturbance, etc.) that affect its direction of motion

Fig 1.15: Lateral model

Longitudinal Vehicle Dynamics Models

Longitudinal vehicle dynamics (Fig 1.16) or vehicle performance is the study of the

longitudinal motion of the vehicle under acceleration or braking input

Fig 1.16: Longitudinal model

Driver Model

Driver model (Fig 1.17) determines the hand wheel steering angle based on the

lateral vehicle position error

Fig 1.17: Driver model

Computer Simulation Tools

Several computer simulation software’s can be used to analyze and simulate the

dynamic of vehicle

Fig 1.18: CarSim

Fig 1.19: CarSim simulation

Fig 1.20: CarSim results

Fig 1.21: IPG CarMaker

Fig 1.22: IPG CarMaker interface

Fig 1.23: IPG CarMaker simulation

Fig 1.24: ADAMS/CAR

Fig 1.25: ADAMS/CAR powertrain simulation

(Fig 2.1) or ride dynamics is the study of the vertical motion of the vehicle due to the

input from the road or other vibration sources Purpose of study of ride dynamics is to control vibration of vehicle so the passenger sensation of discomfort does not exceed

a certain level

Fig 2.1: Vertical vehicle dynamics system

The function of suspension system (Fig 2.2)

 To provide good ride quality by isolating vehicle body from road disturbance

 To ensure good road holding

 To support vehicle static weight

 To ensure good handling

Excitation

Sources

Vertical Vehicle Dynamics

Mechanical Vibration

Human Perception

Touch Vision Hearing

Fig 2.2: Vehicle suspension system

In order to study ride dynamics, various models have been developed

1DOF quarter car model (base excitation vibration model)

2DOF quarter car model

2DOF bounce and pitch model

4DOF pitch plane ride model

4DOF roll plane ride model

7DOF full car ride model

Human body model

2.2 1DOF Quarter Car Model

1DOF model (Fig 2.3) is the simplest model to demonstrate the vibration isolation

provided by suspension system The vehicle body (sprung mass) is allowed to move only in vertical direction

Control of contact force variation

Isolating of body from roadway unevenness

Control of body relative motion

Fig 2.3 1DOF quarter car vehicle model Exercise 1: If the input to the vehicle model is shown in Fig 2.3 is road vertical profile,

determine the transmissibility between the vehicle body vertical displacement and road vertical profile Write a program using Matlab to obtain the transmissibility ratio plot Answer:

2 2

2 2

)2()1(

)2(1

r r

r Z

From the plot shown in Fig 2.4, it can be seen that amplification occurs at resonance

(close to r = 1) For r < 1.414, the transmissibility ratio is less then unity, therefore, amplification occurs Smaller damping ratio causes higher transmissibility ratio The displacement transmissibility ratio is unity at r = 1.414 For r > 1.414, the transmissibility ratio is less then unity, therefore, vibration isolation occurs Smaller damping ratio causes lower transmissibility ratio

Fig 2.4: Transmissibility between vehicle body vertical response and road input

Simulink model 1 DOF quarter car is shown in Fig 2.5

Parameters are given as

r

l

V Z

z 2

sin

(2.2)

Fig 2.5: 1DOF quarter car SIMULINK model

1DOF quarter car model responses are shown in Fig 2.6

Fig 2.6: 1 DOF Quarter car model responses

sin Trigonometric Function

To Workspace

Subtract1

1 s Integrator1

1 s Integrator 1/mb

Gain4

ks Gain3

Zra Gain2 2*pi*Vx/lw

Gain1

cs Gain

du/dt Derivative Clock

Add

2.3 2DOF Quarter Car Model

2DOF quarter car model represents quarter of a car The quarter car model is used for modeling the vertical vibration of a vehicle, taking 1/4th of the sprung mass and incorporating associated unsprung mass (effective tire, axle, etc.) Each sprung and unsprung masses are allowed to move in vertical direction The mathematical (vibration) model is derived by applying Newton’s law to each mass, identifying the forces acting on each mass 2DOF quarter car model schematic and free body diagram

are shown in Fig 2.7

Fig 2.7: (a) Schematic diagram of quarter car (b) Free body diagram

Suspension spring force

)( w b

b b b w s b w

Wheel mass, m w: Applying Newton’s 2nd law, assuming upward direction as positive

w w d s

w w b w s b w s w r

Exercise 2: Derive the expression for the undamped natural frequencies for the 2DOF

quarter car vehicle model

Answer:

w b

t s w b s w t b b

s w

t n

m m

k k m m k m k m m

k m

k

2

22

2

2 2 2 2

2

1

n n

f 



2 2

2

1

n n

Exercise 3: Determine the vehicle body and wheel natural frequencies using the

following vehicle parameters:

Vehicle body mass [454.5 kg]

Wheel mass [45.45kg]

Suspension spring stiffness [22 kN/m]

Suspension damper coefficient [2.4 kNs/m]

Tire stiffness [176 kN/m]

Answer:

Hz m

k f

b

s

5.454

220002

12

k f

w

w w

45.45

1760002

12

to the vehicle body natural frequency is

b

w n

3 performance criteria for ride performance are

Vibration Isolation – can be evaluated by the vehicle body vertical displacement response

Suspension Travel – can be evaluated by the relative vertical displacement between wheel and vehicle body

Road Holding – can be evaluated by the tire deflection response (critically depends

on the normal force acting between the tire and the road surface (dynamic tire

deflection))

Exercise 4: Obtain expression for the following transmissibility ratio;

 Transmissibility between the vehicle body vertical displacement response and road input

 Transmissibility between the suspension travel response and road input

 Transmissibility between the tire deflection response and road input

k t

Vibration Isolation

Fig 2.8: Transmissibility between the vehicle body vertical displacement response

and road input for different ratios of wheel to vehicle body mass

Fig 2.9: Transmissibility between the vehicle body vertical displacement response

and road input for different ratios of tire to suspension spring stiffness

Fig 2.10: Transmissibility between the vehicle body vertical displacement response

and road input for different damping ratios Suspension Travel

Fig 2.11: Transmissibility between the suspension travel response and road input for

different ratios of wheel to vehicle body mass

Fig 2.12: Transmissibility between the suspension travel response and road input for

different ratios of tire to suspension spring stiffness

Fig 2.13: Transmissibility between the vehicle body vertical displacement response

and road input for different damping ratios

Road Holding

Fig 2.14: Transmissibility between the tire deflection response and road input for

different ratios of wheel to vehicle body mass

Fig 2.15: Transmissibility between the tire deflection response and road input for

different ratios of tire to suspension spring stiffness

Fig 2.16: Transmissibility between the tire deflection response and road input for

different damping ratios

2.4 2DOF Bounce and Pitch Model

In this model, the vehicle body is allowed to move in vertical and pitch direction

Fig 2.17: 2DOF bounce and pitch model Exercise 5: Derive the expression for the undamped natural frequencies for the

2DOF pitch and bounce model

2.5 4DOF Pitch Plane Ride Model

In this model, the vehicle body is allowed to move in vertical and pitch direction and each wheel is allowed to move in vertical direction

Fig 2.18: 4DOF half car pitch plane ride model Exercise 6: Derive the equations of motion for the 4DOF half car pitch plane ride

model as shown in Fig 2.18

Answer:

Equation1:

𝑀𝑏𝑍̈𝑏 = 𝐹𝑠𝑓 + 𝐹𝑑𝑓 + 𝐹𝑠𝑟 + 𝐹𝑑𝑟

𝑀𝑏𝑍̈𝑏 = 𝑘𝑠𝑓(𝑍𝑤𝑓 − 𝑍𝑏 + 𝑙𝑓𝜃) + 𝐶𝑠𝑓(𝑍̇𝑤𝑓 − 𝑍̇𝑏 + 𝑙𝑓𝜃)̇ + 𝑘𝑠𝑟(𝑍𝑤𝑟 − 𝑍𝑏 − 𝑙𝑟𝜃) + 𝐶𝑠𝑟(𝑍̇𝑤𝑟 − 𝑍̇𝑏 − 𝑙𝑟𝜃)̇

2.6 4DOF Roll Plane Ride Model

In this model, the vehicle body is allowed to move in vertical and roll direction and each wheel is allowed to move in vertical direction

Fig 2.19: 4DOF half car roll plane ride model Exercise 7: Derive the equations of motion for the 4DOF half car roll plane ride model

as shown in Fig 2.19 Refer to Chapter 9 for the equations

2.7 7DOF Full Car Ride Model

In this model, the vehicle body is allowed to move in vertical, pitch and roll direction and each wheel is allowed to move in vertical direction

0.5t 0.5t

zb

zbl

Fig 2.20: 7DOF full car ride model Exercise 8: Derive the equations of motion for the 7 DOF full car ride model as

arr b

wrr srr arl b

wrl

srl

afr b

wfr sfr afl b

wfl

sfl

b b arr srr arl srl afr sfr afl sfl

b b b

Z

M

F t

w Z Z k F t

w Z Z

k

F t

w Z Z k F t

w Z Z

k

Z M F

F F F F F F

F

Z M F

)22(

)22(

)22(

wfr sfr afl b

wfl sfl

arr b

wrr srr arl b

wrl srl

p p

I

w F t

w Z Z k F t

w Z Z k

w F t

w Z Z k F t

w Z Z

k

I M

)22(

2

)22(

)22(

wrr srr afr b

wfr sfr

arl b

wrl srl afl b

wfl sfl

r r

I

t F t

w Z Z k F t

w Z Z k

t F t

w Z Z k F t

w Z Z

k

I M

)22(

2

)22(

)22(

Equation 4:

wfl wfl afl

b wfl sfl wfl rfl

tfl

wfl wfl wfl

Z M F t

w Z Z k Z Z

k

Z M F

)

Equation 5:

wfr wfr afr

b wfr sfr wfr rfr

tfr

wfr wfr wfr

Z M F t

w Z Z k Z

Z

k

Z M F

)

Equation 6:

wrl wrl arl

b wrl srl wrl rrl

trl

wrl wrl wrl

Z M F t

w Z Z k Z Z

k

Z M F

)

Equation 7:

wrr wrr arr

b wrr srr wrr rrr

trr

wrr wrr wrr

Z M F t

w Z Z k Z

Z

k

Z M F