24
MATHEMATICAL MODELLING IN CAR SUSPENSION SYSTEM PANG AIK YEN UNIVERSITY TECHNOLOGY MALAYSIA
MATHEMATICAL MODELLING IN CAR SUSPENSION SYSTEM
PANG AIK YEN
UNIVERSITY TECHNOLOGY MALAYSIA
MATHEMATICAL MODELLING IN CAR SUSPENSION SYSTEM
PANG AIK YEN
A dissertation submitted in partial fulfillment of the
requirements for the award of the degree of
Master of Science (Engineering Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
DECEMBER, 2010
iii
To my beloved parents,
Pang Kee Yok and Lai Yook Hong,
my dearly sister and brother,
Pang Aik Niu and Pang Aik Kah,
to lovely Tiong Chan Xing,
for your everlasting love, care and support.
To my friends and coursemates,
for your wits, intelligence and guidance in my life.
Thank you all for everything.
iv
ACKNOWLEDGEMENT
First and foremost, I am appreciated that Associated Professor Dr.
Shamsuddin Bin Ahmad who kindly spare his time and energy to guide me to
complete my dissertation. During this period, he had given me valuable advice,
encouragement, guidance, and invaluable knowledge that have brought to the
completion of this dissertation.
A token of appreciation to lecturers who have been giving me much
information suggestion and comments in constructing the Maple programming. Also,
thanks to all my friends and course mates for their assistance, cooperation, guidance
and friendship throughout the completion of dissertation. Without their support and
guidelines, I might not gain spirit to complete this dissertation.
Last but not least, I would like to express my deepest grateful to my parents
and my siblings for their support mentally and physically. Thanks to those who have
contributed directly or indirectly.
v
ABSTRACT
A vehicle suspension system is defined as a mechanical system of springs or
shock absorbers connecting the wheels and axles to the chassis of a wheeled vehicle.
The purpose of a car suspension system is to improve the comfort of the car subject
to different road profiles. The main objective of this study is to formulate a basic
quarter car suspension systems subject to a sinusoidal road profile. In this study, the
suspension travel limit and magnitude of car’s vertical acceleration are validated
using the standard of Ford Scorpio car. The comfort of the car is studied by means
of the reduction of the body acceleration caused by the car body subject to sinusoidal
road profile and ‘Sleeping Policeman’ road profile. In order to increase the comfort
of a car, a refined model is formulated by including the effects of wheel and axle in
the quarter car suspension system. The results of the study is obtained and analyzed
by using Fourth Order Runge-Kutte Method via Maple 12.
vi
ABSTRAK
Suatu sistem gantungan kenderaan ditakrifkan sebagai sistem mekanikal
spring atau penyerap kejutan yang bergabung dengan tayar dan gandal kepada chasis
pada suatu kenderaan beroda. Tujuan sistem gantungan kereta adalah untuk
menambahbaik keselesaan kereta yang dinaiki apabila melalui jalan dengan
permukaan yang berbeza. Objektif utama kajian ini adalah untuk memodelkan satu
per empat sistem gantungan kereta mengikut profil jalan berbentuk sinus. Dalam
kajian ini, had penggantungan perjalanan dan magnitud cepatan mencangcang kereta
disahkan dengan mengguna piawaian kereta Ford Scorpio. Keselesaan kenderaan
bagi system gantungan kereta mengikut profil jalan ‘sleeping policeman’ dikaji dan
dibandingkan dengan sistem gantungan kereta mengikut profil jalan berbentuk sinus.
Dalam usaha meningkatkan keselesaan kereta yang dinaiki, sistem gantungan kereta
diperbaiki dengan mengambil kira kesan roda dan gandal dalam satu per empat
sistem gantungan kereta. Keputusan kajian diperolehi dan dianalisis menggunakan
Kaedah Runge-Kutte Peringkat Keempat dengan pengaturcaraan Maple 12.
vii
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES xi
LIST OF APPENDICES xiv
1 INTRODUCTION 1
1.1 Introduction
1.2 Background of the Problem
1.3 Statement of Problem
1.4 Purpose/Objective(s) of the Research
1.5 Significance of the Research
1.6 Scope of the Research
1
3
3
4
4
5
2 LITERATURE REVIEW 6
2.1 Brief Introduction on Car Suspension
2.2 The Definition of Suspension
2.3 The Suspension Systems
2.4 Vehicle Dynamics
2.5 Comparison of Active Suspension and Passive
Suspension
6
8
9
13
15
viii
2.6 Road Profile
2.6.1 Sinusoid Road Profile
2.6.2 “Sleeping Policeman” Road Profile
2.7 The Newton’s Law of Motion
2.8 Hooke’s Law
2.9 Fourier Series
22
22
24
25
26
27
3 METHODOLOGY 31
3.1 Examples of Car Suspension System
3.1.1 Method of Undetermined Coefficients
3.1.2 Fourier Series
3.2 Runge-Kutta Method
3.2.1 Second Order Runge-Kutta Method
3.2.2 Forth Order Runge-Kutta Method
3.2.3 Higher-Order System and Runge-Kutta Method
3.2.4 Runge-Kutta Method for Car Suspension System
3.6 Maple 12 Software
31
33
35
39
40
40
42
44
46
4 FORMULATION OF A QUARTER OF CAR
SUSPENSION MODEL
48
4.1 Factors to Set Up A Quarter of Car Suspension Model
4.2 Assumptions
4.3 Formulate a Quarter Car Suspension Model
4.4 Solution of a Quarter Car Suspension Model
4.4.1 Analytical Solution of a Quarter Car Suspension
Model
4.4.2 Interpretation and Validation of Graphs
4.5 Model Refinement
4.5.1 Solution of Model Refinement
4.5.2 Interpretation and Validation of Graphs
4.6 Summary
48
49
50
55
55
59
62
64
69
76
ix
5 SLEEPING POLICEMAN ROAD PROFILE 77
5.1 Sleeping Policeman Road Profile
5.2 Formulate a Model on Sleeping Policeman Road Profile
5.2.1 Interpretation and Validation of Graphs
5.3 Model Refinement on Sleeping Policeman Road Profile
5.3.1 Interpretation and Validation of Graphs
5.4 Summary
77
79
86
92
95
100
6 DISCUSSION
6.1 Discussion of Refined Model on Sinusoid Road
Profile and Sleeping Policeman Road Profile
6.2 Discussion on Car Suspension System
101
101
104
7 CONCLUSION
7.1 Conclusion
7.2 Recommendation for Future Research
110
110
111
REFERENCES 116-120
APPENDIX A
APPENDIX B1
APPENDIX B2
APPENDIX C
APPENDIX D1
APPENDIX D2
APPENDIX E
121
122-124
125-127
128-129
130-132
133-135
136-140
x
LIST OF TABLES
TABLES NO. TITLE PAGE
2.5.1
4.5.1
Active Suspension Parameters
Data of Refined Model
17
68
xi
LIST OF FIGURES
FIGURES NO. TITLE PAGE
2.1
2.3.1
2.3.2
2.3.3
2.3.4
2.4.1
2.4.2
2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
2.5.6
2.5.7
Car Suspension Systems
A Passive Suspension System
A Semi-Active Car Suspension System
A Low Bandwidth or Soft Active Suspension System
A High Bandwidth or Stiff Active Suspension System
Vehicle Response Due to Road and Steering Input
Different between Active and Passive Suspension
Active Suspension with Hydraulic Actuator for a Quarter
Car Model
Comparison of Active Suspension and Passive Suspension
A Half Car Model
Front Vertical Wheel Displacement
Rear Vertical Wheel Displacement
Front Body Acceleration
Rear Body Acceleration
8
10
11
11
12
14
14
16
18
19
20
20
21
21
xii
2.6.1
2.6.2
2.9
3.1.1
3.1.2
4.2
4.3.1
4.3.2
4.4.2.1
4.4.2.2
4.5.1
4.5.2
4.5.2.1
4.5.2.2
4.5.2.3
4.5.2.4
4.5.2.5
4.5.2.6
5.1.1
5.2.1.1
5.2.1.2
Sinusoid Road Profile
Sleeping Policeman Road Profile
Sine, Square, Triangle and Sawtooth Waveforms
Road Profile
Model of a Car
Sinusoid Road Surface Profile
Simple Car Body Model
Car Body Free Diagram
Displacement as a Function of Time
Acceleration as a Function of Time
Refined Car Body Model
Forces diagram for the Refined Car Body Model
Graph of against within
Graph of against within
Graph of against
Graph of against within
Graph of against within
Graph of against within
Sleeping Policeman Road Profile
Graph of Various Time within
Graph of Various Time within
23
24
28
31
32
50
50
52
60
60
63
63
70
71
72
73
74
75
78
87
87
xiii
5.2.1.3
5.2.1.4
5.2.1.5
5.2.1.6
5.3.1.1
5.3.1.2
5.3.1.3
5.3.1.4
5.3.1.5
5.3.1.6
6.1.1
6.1.2
6.1.3
6.2.1
6.2.2
6.2.3
7.3.1
7.3.2
7.3.3
Graph of Various Time within
Graph of Various Time within
Graph of Various Time within
Graph of Various Time within
Graph of against within
Graph of against within
Graph of against within
Graph of against within
Graph of against
against within
Graph of against
Graph of against
Graph of against
Graph of Various Time within for
Graph of Various Time within for
Active Suspension Model
Extended Car Body Model
Triangle Waveforms Road Profile
Sawtooth Waveforms Road Profile
88
89
90
90
95
96
97
98
99
99
102
102
103
106
106
109
112
114
115
xiv
LIST OF APPENDICES
APPENDICES NO. TITLE PAGE
A
B1
B2
C
D1
D2
E
Maple Algorithm of Section 4.4.2 (Fig 4.4.2.1, Fig 4.4.2.2)
Maple Algorithm of Section 4.5.2 (Fig 4.5.2.1, Fig 4.5.2.3,
Fig 4.5.2.5)
Maple Algorithm of Section 4.5.2 (Fig 4.5.2.2, Fig 4.5.2.4,
Fig 4.5.2.6)
Maple Algorithm of Section 5.2.1 (Fig 5.2.1.1-Fig 5.2.1.6)
Maple Algorithm of Section 5.3.1 (Fig 5.3.1.1,Fig 5.3.1.3,
Fig 5.3.1.5)
Maple Algorithm of Section 5.3.1 (Fig 5.3.1.2,Fig 5.3.1.4,
Fig 5.3.1.6)
Maple Algorithm of Section 6.1 (Fig. 6.1.1, Fig. 6.1.2, Fig.
6.1.3)
121
122
125
128
130
133
136
CHAPTER 1
INTRODUCTION
1.1 Introduction
In our modern societies, most of the passengers prefer a smooth journey with
advanced ride quality rather than a luxury or an excellent outlook design of vehicle.
Ride quality of a vehicle has been the first priority of customer needed. A vehicle's
suspension system is one of the decisive factors in determining the quality of ride.
Even the smoothest roads are full of variations in height and surface texture.
Combined with the car's natural shake as the engine runs and the constant buffeting
of air at high speeds, suspension makes driving a comfortable and safe undertaking.
Apart from the car’s tyres and seats, the suspension is the prime mechanism
that separates your bum from the road. It also prevents the car from shaking itself to
pieces. No matter how smooth the road is, it is a bad place to propel over a ton of
metal at high speed. So we rely upon suspension. People who travel on trains wish
that those vehicles relied on suspension too. But they do not and that’s why the ride
is so harsh. Actually it is harsh because trains have no lateral suspension to speak of.
So as the rails deviate from side-to-side slightly, so does the entire trains, and its
passengers. In a car, the rubber in car’s tyres helps with this little problem. In it is
2
most basic form, suspension consists of two components: springs and shock
absorbers.
A suspension system defined as a mechanical system of springs or shock
absorbers connecting the wheels and axles to the chassis of a wheeled vehicle. A lot
of the system's work is done by the springs. Under normal conditions, the springs
support the body of the car evenly by compressing and rebounding with every up-
and-down movement. This up-and-down movement, however, causes bouncing and
swaying after each bump and is very uncomfortable to the passenger. These
undesirable effects are reduced by the shock absorbers.
In this research, a basic model of car suspension system is formulated when a
car is moving over a ‘sinusoid’ road surface profile. In this model, the shock
absorber is modelled as a simple spring-dashpot system with spring stiffness and
dashpot constant. Most car suspension systems use springs in the form of a coil or a
series of leaves and are usually made of steel, although rubber and plastic composites
are possible. The dashpot which commonly known as damper is usually a hydraulic
device which is effectively a piston moving inside a housing containing fluid.
Next, a model refinement will be constructed by including wheel and axle in
the model with the same road surface profile to improve the car suspension system.
Aside of this, a model of car suspension will be formulated when a car moves over a
series of ‘sleeping policemen’ road surface profile. All the mathematical problems
will solve by analytical or by Maple programming.
3
1.2 Background of the Problem
A suspension system plays a vital role to smooth out the ride while
maintaining excellent control. The suspension system has two basic functions, to
keep the car's wheels in firm contact with the road and to provide a comfortable ride
for the passengers. This may sound like a simple job, but with acceleration comes
force, and force translates into raw energy. When a vehicle accelerates down a road,
bumps cause forward energy to be converted into vertical energy, which travels
through the frame of the vehicle. Without coil and leaf springs to absorb this, the
vertical energy would cause the vehicle to jump up off the road, reducing tire friction
and control. The car would then come bounding back down with even greater force,
making for a very uncomfortable and dangerous ride. As a consequence, the car
suspension system should be investigated and suspensions for car also have their
potential to improve vehicle performance.
1.3 Statement of Problem
This research is concerning on the following questions. How do formulate a
basic quarter car suspension model by a ‘sinusiod’ road profile? How do solve this
basic model by analytical methods? How do refine the basic quarter car suspension
model by including wheel and axle behavior? How do construct a mathematical
formulation of the Fourier series when a car moves over a series of ‘sleeping
policemen’ road profile? How do solve the mathematical by constructing Maple 12
programming? How do plot the graphs by Maple 12 software? How do
interpretation and validation of the graphical results of each model which obtain by
Maple 12 Programming?
4
1.4 Objectives of the Research
The objectives of this research are:
(i) To formulate a basic quarter car suspension model by a ‘sinusiod’ road
profile.
(ii) To refine the basic quarter car suspension model of ‘sinusiod’ road profile
by including wheels and axle behavior.
(iii) To formulate a car suspension model based on the situation when a car
moves over a series of ‘sleeping policeman’ road profile.
1.5 Significance of the Research
Nowadays, the automotive industries still put great efforts in producing
highly developed vehicles’ suspension system as it is one of the decisive factors in
determining the quality of ride. This research will give benefits to Mathematicians,
Physicists, Engineers and others field indirectly to improve the vehicles’ suspension
system and also improve vehicles’ performance.
The advantage that we gain is we will able to understand the application of
Mathematics in our real life. In this case the application of Mathematics is specific
to a car suspension system by different road profile. In addition, we can learn how to
plot graphs by using Maple software. We can also learn how to construct a Maple
programming to solve the problems of different equations in this research. As a
result, the problems can be solved to obtain approximate answers. By using Maple
software, it save time and the graph can be plotted immediately. Although it takes
5
time to familiar oneself with Maple computer programming system, but this time is
well-spent. After carry out this research, we gain a greater understanding and
appreciation of the power of mathematics.
1.6 Scope of the Research
The car suspension system is ideal to have a general understanding of the
functions of car suspension system. There are some engineers design good car
suspension systems which can be used in our real life to provide customers
comfortable and safe ride. However, this research construct a basic model for a car
suspension system where the shock absorber is modeled as a simple spring-dashpot
system with spring stiffness and dashpot constant. The model is obtained from the
application of Newton’s Second Law of Motion in a vertical direction of car body.
This research will focus on differential equation approach and linear ordinary
differential equation approach by applying on the car suspension system.
Mathematical formulations based on a quarter car suspension system and different
road surface profiles will be modeled. These mathematical models will be solved by
Maple 12 programming and the graphical results will be obtained. Maple will be
used to graphically and numerically simulate the solution to obtain approximate
answers. The assembly of the complete model and the general analysis of the
smoothness of solutions benefits greatly from Maple programming. The numeric
and graphical tools provided by Maple are well suited for solving the system for
specific sets of parameter values. Therefore, the car suspension system problem is
ideally suited for analysis with Maple.
116
REFERENCES
1. Stewart, M. and David, C. P. Learning Modelling With Derive. Prentice Hall.
1995. Chapter 6, pp. 157-173.
2. Yahaya, Md. S. Robust Control of Active Suspension System for a Quarter Car
Model. Journal of Control and Instrumentation Engineering Department,
Faculty of Electrical Engineering, University Malaysia. 2006.
3. Dirman, H. and Mohd. Fua’ad R. System Identification of Nonlinear Model of
A Quarter Car Passive Suspension with Backprogation Neural Network.
Journal of Control and Instrumentation Engineering Department, Faculty of
Electrical Engineering, University Technology Malaysia. 2005.
4. Vigneswaran, M. R. Computer Simulation of Passive and Active Suspension
System Using Simulink. Journal of School of Electrical Systems Engineering
Department, University Malaysia Perlis. April 2008.
5. Yahaya, Md. S. and Johari, H. S.O. Sliding Mode Control of A Hydraulically
Actuated Active Suspension. Journal Technology, 44(D): 37-48, University
Technology Malaysia. Jun 2006.
6. Yahaya, M. S. Modelling and Control of Active Suspension System Using
Proportional Integral Sliding Mode Control. Ph. D. Thesis. University
Technology Malaysia. 2004.
117
7. Kim C. and Ro, P. I. Reduced Order Modeling and Parameter Estimation for A
Quarter Car Suspension System. Journal of Automobile Engineering,
Department of Mechanical and Aerospace Engineering, North Carolina State
University, USA. Vol. 214, pp. 851-864, 2000.
8. Dirman H. and Mohd. Fua’ad R. System Identification of Nonlinear Model of A
Quarter Car Passive Suspension with Backprogation Neural Network. Journal
of Control and Instrumentation Engineering Department, Faculty of Electrical
Engineering, University Technology Malaysia. 2005.
9. Abu Bakar Adham Bin Hell Mee. Modeling and Controller Design for an
Active Car Suspension System Using Half Car Model. Journal of Control and
Instrumentation Engineering Department, Faculty of Electrical Engineering,
University Technology Malaysia. 2008/2009.
10. Marius-Constantin, O. S. P. and Nikos, E. M. Testing and Simulation of a
Motor Vehicle Suspension. International Journal of System Applications,
Engineering and Development. Issue 2, Volume 3, 2009.
11. Salem, M. M. M. and Ayman A.Aly. Fuzzy Control of a Quarter-Car
Suspension System. Journal of World Academy of Science, Engineering and
Technology. 2009.
12. James, W. B. and Ruel, V. C. Fourier Series and Boundary Value Problems,
Seventh Edition. Mc Graw Hill, Higher Education. America, New York. 2008.
Chapter 1, pg1-23.
13. Edwards, C.H. and David, E.P. Differential Equations and Boundary Value
Problems: Computing and Modelling. 2nd
Edition. New Jersey: Prentice Hall
International, Inc. 2000. Chapter 3, pp. 180-218.
118
14. Randall J. Swift, Stephen A. Wirkus. A Course In Ordinary Differential
Equations. United State of America: Taylor & Francis Group, LLC. 2007.
Chapter3-4, pp.153-286.
15. Butcher, J. C. Numerical Methods for Ordinary Differential Equations. Second
Edition. John Wiley & Sons Ltd, Chichester, England, 2008. Chapter 3,
pp.137-313.
16. William, E. B. and Richard, C.D. Elementary Differential Equations and
Boundary Value Problems. New York: John Wiley & Sons (Asia) Ptd Ltd.
2010. Chapter 10, pp. 584-611.
17. Edwards, C.H. and David, E.P. Elementary Differential Equations with
Boundary Value Problems. Sixth Edition. New Jersey: Prentice Hall
International, Inc. 2008. Chapter 2, pp. 162-170. Chapter 8, pp. 554-585.
18. Edwards, C.H. and David, E.P. Differential Equations and Boundary Value
Problems: Computing and Modelling. 2nd
Edition. New Jersey: Prentice Hall
International, Inc. 2000. Chapter 3, pp. 180-218.
19. Davis, J. H. Differential Equations with Maple: An Interactive Approach.
United States of America: Birkhauser Boston. 2001. Chapter 4, pp. 88-108.
20. Belinda, B. and Glenn, R. F. Mathematical Modeling with Case Studies: A
Differential Equations approach using Maple and Mathlab. 2nd
Edition. United
States of America: Chapman & Hall/CRC, Taylor & Francis Group. 2009.
21. Ang, K.C. Computational Methods with Maple. Singapore: Prentice Hall,
Pearson Education Asia Ptd Ltd. 2002.
22. Davis, J. H., Differential Equations with Maple: An Interative Approach.
United Stated of America: Birkhauser Boston. 2001.
119
23. Donald G. Numerical Solution of Ordinary Differential Equations for Classical,
Relativisitc and Nano Systems. Wiley-VCH. 2006. Chapter 2, pp. 11-34.
24. Heck, A. Introduction to Maple. Third Edition. New York: Springer Science &
Business Media. 2003.
25. Che Rahim Che Teh. Programming for Scientists and Engineers: Advanced
Maple Programming Mathematical Approach. CRCT, Department of
Mathematics, Faculty of Science, University Technology Malaysia: Esktop
Publisher. Jun 2008.
26. Darren, R. The Maple Handbook, Maple V Release 4. Springer-Verlag, New
York. 1996.
27. Darren, R. and Edgar, C. Maple O.D.E. Lab Book. Springer-Verlag, New York.
1996.
28. Darren, R. The Practical Approach Utilities for Maple, Maple V Release 3.
Springer-Verlag, New York. 1995.
29. Roelof, J. S. and Johan, F. K. Discovering Mathematics with Maple, An
Interactive Exploration for Mathematicians, Engineers and Econometricians.
Birkhauser Verlag, Basel Switzerland. 1999.
30. Mircea, V. S., Petre, P. T. and Ileana T. Mathematics and Its Applications
Ordinary Differential Equations with Applications to Mechanics. Springer, The
Netherlands. 2007.
31. Mohan, C. J. Ordinary Differential Equations, Modern Perpective. Alpha
Science, India. 2009.
120
32. Richard, H. Elementary Applied Partial Differential Equations with Fourier
Series and Boundary Value Problems, Third Edition. Prentice Hall, United
State of America. 1998. Chapter 2 and 3. Pg. 32-129.
33. Dodson, C. T. J. and Gonzalez, E. A. Experiments In Mathematics Using
Maple. Springer-Verlag Berlin Heidelberg New York. 1995.
34. Kent, H. and Robert, J. L. Discovering Calculus with Maple, Second Edition.
John Wiley & Sons, Inc. United States of America. 1995.
35. Micheal, K. Maple An Introduction and Reference. Addison-Wesley Longman
Ltd. United Stated of America. 1997.
