Vehicle Tire Road Dynamics Handling Ride and NVH Tan Li All Chapter
Vehicle Tire Road Dynamics Handling Ride and NVH Tan Li All Chapter
Vehicle Tire Road Dynamics Handling Ride and NVH Tan Li All Chapter
Tan Li
Maxxis Technology Center, Maxxis
International—USA, Suwanee, GA, United States
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ISBN: 978-0-323-90176-5
1 Introduction 1
1.1 Background 1
1.2 Literature 2
1.3 Organization 3
References 12
sound pressure level at the rear seat can also be 3 dB higher than at the front seat. The
vibration levels can also be different on the floor, seat, and steering wheel, which is
usually not in the scope of ride vibrations focused on smoothness at the vehicle center
of gravity. It is also implied that NVH is a highly subjective performance, which is
why sound quality becomes more and more important for vehicle product quality.
Similarly, handling quality and ride quality in terms of subjective evaluations have
drawn higher attention recently.
In summary, there is a spectrum of vehicle performance areas: handling for low fre-
quency (<10 Hz), ride for middle frequency (10–100 Hz), and NVH for high frequency
(>100 Hz). Conventionally, “vehicle dynamics” only implied handling that is most crit-
ical relating to safety, later it merged with ride because comfort became important, now
NVH has drawn higher attention for product quality. All of the three areas are actually
dynamics, but we are not used to calling NVH dynamics yet. All three areas can be ana-
lyzed in frequency and time domain, though handling is usually analyzed in time
domain while NVH in frequency domain. In this book, both frequency domain analysis
and time domain analysis will be presented for all three areas. A summary comparison
between vehicle handling, ride, and NVH is shown in Table 1.1.
Generally, there are five major noise sources for a vehicle with internal combustion
engine (ICE): powertrain, intake system, exhaust system, aerodynamic turbulence
(wind), and tire-road interaction. For an electric vehicle (EV), the powertrain noise from
electric motor is much lower than ICE, and the intake/exhaust noise is removed. This
book mainly focuses on tire-road-induced NVH, which is dominant for regular vehicle
speeds. Traditionally, vehicle performance is limited by the ICE, as the engine deter-
mines the vehicle traction capability and generates most of the interior noise under reg-
ular conditions. However, electric motor with high torque and low noise can eliminate
those disadvantages of ICE. Hence, the performance of EVs will be largely dependent
on tires providing sufficient grip and minimizing road noise. Vehicle electrification can
also provide more spacious passenger compartment due to compact powertrain/
drivetrain system and lower height of center of gravity due to the battery arrangement
under the vehicle body, which improves ride comfort and roll stability. The advanced
suspension technology also puts tire in a more important role in improving vehicle com-
fort. Therefore, tire, as a component of vehicle system, will be critical for the major
vehicle performances including handling, ride, and NVH. The tire contact patch is
the only interaction between a vehicle and the road; the forces generated in this patch
dictate the movement and vibration of the vehicle. For the autonomous vehicle, due to
little or no control of the vehicle, occupants may have different perspectives regarding
handling (lower requirements for steering feel but higher requirements for safety/stabil-
ity) and higher standards for ride comfort and NVH.
1.2 Literature
A comprehensive search for published books on vehicle handling/ride/NVH has been
performed, as listed in Table 1.2. These are good references for readers to delve into
specific areas of interest. However, it can be seen that there is a need to put together a
book covering all the areas.
Introduction 3
1.3 Organization
To fill the knowledge gap between different areas, Chapter 2 presented the basic def-
initions and fundamentals related to vehicle handling, ride, and NVH.
In the generalized vehicle/tire dynamics, the steady state or the short-time transient
tire forces dominate the vehicle handling behavior (Chapter 3); then, the fluctuation of
the reaction forces influences the ride quality (Chapter 4); finally, the stabilized oscil-
lation of each vehicle/tire component relates to NVH performance (Chapter 5). How-
ever, there are no clear-cut boundaries between vehicle handling, ride, and NVH.
Table 1.2 List of relevant books (✓ denotes over 20%; denotes 5–20%; X denotes below 5%).
Scope System Depth Application
Unique
Author/Editor Year Publisher Title Handling Ride NVH Vehicle Tire Road Theory Practice Experiment Simulation feature
Handling series
Yong et al. [1] 1984 Elsevier Vehicle Traction ✓ X X ✓ ✓ ✓ ✓ X Vehicle-terrain
Science Mechanics: energy transfer,
Volume 3 FEM,
trafficability
Garg and 1984 Academic Press Dynamics of ✓ X X ✓ X X ✓ ✓ Railway vehicle
Dukkipati [2] of Elsevier Railway Vehicle
Systems
Haug [3] 1989 Allyn and Computer Aided X X X X X X ✓ X X ✓ Multibody
Bacon Kinematics and application to
Dynamics of vehicle
Mechanical
Systems: Basic
Methods
Gillespie [4] 1992 SAE Fundamentals of ✓ X ✓ X ✓ X X X Steering,
International Vehicle Dynamics rollover
Milliken and 1994 SAE Race Car Vehicle ✓ X X ✓ X ✓ ✓ ✓ X Race handling
Milliken [5] International Dynamics metrics,
aerodynamics
Genta [6] 1996 World Motor Vehicle ✓ X ✓ X ✓ X ✓ Vehicle model
Scientific Dynamics: datasets
Publishing Modeling and
Simulation
Dixon [7] 1996 SAE Tires, Suspension ✓ X X ✓ X ✓ X X Tire-suspension
International and Handling (2nd system
Edition)
Wong [8] 2001 John Wiley & Theory of Ground ✓ X ✓ X ✓ ✓ ✓ Tracked vehicle,
Sons Vehicles (3rd terramechanics
Edition)
Howard et al. [9] 2004 SAE Car Suspension ✓ ✓ X ✓ X ✓ ✓ Suspensions,
International and Handling (4th durability
Edition)
Dukkipati et al. 2008 SAE Road Vehicle ✓ ✓ X ✓ ✓ X ✓ X Theory of
[10] International Dynamics dynamics,
accident
Jazar [11] 2008 Springer Vehicle ✓ X ✓ X ✓ ✓ X Vehicle
Dynamics: kinematics
Theory and
Application
Wong [12] 2009 Butterworth- Terramechanics ✓ X X ✓ X ✓ ✓ ✓ Terrain
Heinemann of and Off-Road behavior,
Elsevier Vehicle computer-aided
Engineering: method
Terrain Behavior,
Off-Road Vehicle
Performance and
Design (2nd
Edition)
Popp and 2010 Springer Ground Vehicle ✓ X ✓ X X ✓ X X ✓ Multibody
Schiehlen [13] Dynamics
Pacejka [14] 2012 Butterworth- Tire and Vehicle ✓ ✓ X ✓ X ✓ X Tire models
Heinemann of Dynamics (3rd
Elsevier Edition)
Rajamani [15] 2012 Springer Vehicle Dynamics ✓ ✓ X ✓ X ✓ X Autonomous
and Control (2nd control
Edition)
Doumiati et al. 2012 John Wiley & Vehicle Dynamics ✓ X X ✓ X ✓ ✓ Tire variable
[16] Sons Estimation Using estimation
Kalman Filtering:
Experimental
Validation
Mastinu and 2014 CRC Press of Road and Off- ✓ X ✓ X ✓ X Comprehensive,
Ploechl (Eds.) Taylor & Road Vehicle man vehicle
[17] Francis System Dynamics
Handbook
Blundell and 2015 Butterworth- The Multibody ✓ X X ✓ ✓ X ✓ X X ✓ Data set for full
Harty [18] Heinemann of Systems vehicle
Elsevier Approach to
Vehicle Dynamics
(2nd Edition)
Abe [19] 2015 Butterworth- Vehicle Handling ✓ X X ✓ X ✓ ✓ X Handling
Heinemann of Dynamics: quality, rear-
Elsevier Theory and wheel steering
Application (2nd
Edition)
Continued
Table 1.2 Continued
Scope System Depth Application
Unique
Author/Editor Year Publisher Title Handling Ride NVH Vehicle Tire Road Theory Practice Experiment Simulation feature
Ride series
Henry and 1992 ASTM Vehicle, Tire, ✓ ✓ X ✓ ✓ ✓ ✓ ✓ ✓ X Road profile and
Wambold (Eds.) Pavement roughness
[33] Interface
Guglielmino 2008 Springer Semi-active X ✓ X ✓ X ✓ ✓ ✓ Suspension
et al. [34] Suspension control
Control:
Improved Vehicle
Ride and Road
Friendliness
Continued
Table 1.2 Continued
Scope System Depth Application
Unique
Author/Editor Year Publisher Title Handling Ride NVH Vehicle Tire Road Theory Practice Experiment Simulation feature
NVH Series
Ewins [38] 1984 Research Modal Testing: X X ✓ X X X ✓ ✓ ✓ X Modal analysis
Studies Press Theory and
Practice
Kinsler et al. 2000 John Wiley and Fundamentals of X X ✓ X X X ✓ X X X Acoustics
[39] Sons Acoustics (4th
Edition)
Sandberg and 2002 INFORMEX Tyre/Road Noise X X ✓ ✓ ✓ ✓ ✓ X Classic book on
Ejsmont [40] Reference Book TRN
Harrison [41] 2004 Butterworth- Vehicle X X ✓ ✓ X X ✓ ✓ ✓ X Interior and
Heinemann of Refinement: exterior noise
Elsevier Controlling Noise assessment and
and Vibration in control
Road Vehicles
Fahy and 2007 Academic Press Sound and X X ✓ X X X ✓ X X Vibroacoustics
Gardonio [42] of Elsevier Structural
Vibration:
Radiation,
Transmission and
Response
Fastl and 2007 Springer Psychoacoustics: X X ✓ X X X ✓ ✓ ✓ X Sound quality
Zwicker [43] Facts and Models
(3rd Edition)
Crocker (Ed.) 2008 John Wiley & Handbook of X X ✓ X X X ✓ ✓ ✓ ✓ Comprehensive
[44] Sons Noise and NV control
Vibration Control
Thompson [45] 2009 Elsevier Railway Noise X X ✓ X X X ✓ ✓ Railway NVH
Science and Vibration:
Mechanisms,
Modeling and
Means of Control
Wang (Ed.) [46] 2010 Woodhead Vehicle Noise and X X ✓ ✓ X X ✓ ✓ X Vehicle NVH
Publishing of Vibration refinement
Elsevier Refinement
Sheng [47] 2012 SAE Vehicle Noise, X X ✓ ✓ X X ✓ ✓ ✓ X NVH
International Vibration, and fundamentals
Sound Quality and evaluations
Kraft and White 2013 Woodhead Mems for X X X X X ✓ ✓ ✓ X Sensors
(Eds.) [48] Publishing of Automotive and
Elsevier Aerospace
Applications
Inman [49] 2014 Pearson Engineering X ✓ X X X ✓ ✓ FEM for
Vibration (4th vibration
Edition) systems
Fuchs et al. 2016 Springer Automotive NVH X X ✓ ✓ X X ✓ ✓ ✓ ✓ Vehicle
(Eds.) [50] Technology components
NVH
Bies and Hansen 2017 Spon Press of Engineering X X ✓ X X X ✓ ✓ Noise
[51] Taylor & Noise Control: engineering
Francis Theory and
Practice (4th
Edition)
Pang [52] 2018 John Wiley & Noise and X X ✓ ✓ X X ✓ ✓ ✓ Vehicle body
Sons Vibration Control NVH
in Automotive
Bodies
Continued
Table 1.2 Continued
Scope System Depth Application
Unique
Author/Editor Year Publisher Title Handling Ride NVH Vehicle Tire Road Theory Practice Experiment Simulation feature
Tire series
Hays and 1974 Springer The Physics of ✓ X X X ✓ ✓ ✓ ✓ X Tire traction
Browne (Eds.) Tire Traction:
[54] Theory and
Experiment
Kovac [55] 1978 Goodyear Tire Technology X X X X ✓ X X ✓ ✓ X Tire
(5th Edition) manufacture
Fleming and 1979 ASTM Tire X X X X ✓ X ✓ ✓ ✓ X Tire cord
Livingston Reinforcement
(Eds.) [56] and Tire
Performance
Clark [57] 1981 NHTSA Mechanics of ✓ X X ✓ X ✓ ✓ X First book on tire
Pneumatic Tires (bias)
Meyer and 1983 ASTM Frictional ✓ X X ✓ ✓ ✓ ✓ ✓ X Tire/Road
Walter (Eds.) Interaction of Tire traction
[58] and Pavement
Ridha and 1994 iSmithers Rapra Advances in Tyre X X ✓ X ✓ ✓ X Tire
Theves [59] Publishing Mechanics performance,
references
Rivers [60] 2001 Charles C Tire Failures and X X X X ✓ X X ✓ ✓ X Tire failure
Thomas Pub Evidence Manual:
Ltd For Traffic
Accident
Investigation
Evans [61] 2002 iSmithers Rapra Tyre X X X X ✓ X X ✓ X X Tire compound,
Publishing Compounding for references
Improved
Performance
Gent and Walter 2005 NHTSA The Pneumatic ✓ X ✓ X ✓ ✓ ✓ X Passenger car
(Eds.) [62] Tire radial tire
TRB [63] 2006 The National Tires and X X ✓ X X ✓ X X Fuel economy
Academies Passenger Vehicle
Press Fuel Economy:
Informing
Consumers,
Improving
Performance
Giapponi [64] 2008 SAE Tire Forensic X X X X ✓ X X ✓ X ✓ Tire forensic
International Investigation:
Analyzing Tire
Failure
Mark et al. 2013 Academic Press The Science and X X X X ✓ X ✓ X Rubber
(Eds.) [65] of Elsevier Technology of properties
Rubber (4th
Edition)
Leister [66] 2018 Springer Passenger Car X X X X ✓ X X ✓ ✓ X Tire/Wheel
Tires and Wheels: development
Development -
Manufacturing -
Application
Nakajima [67] 2019 Springer Advanced Tire X ✓ X ✓ ✓ ✓ ✓ Tire structural
Mechanics mechanics
Statistics for 67 reference books ✓ 34 13 15 38 21 4 60 52 22 22
5 12 5 7 23 8 1 7 20 16
X 28 42 47 22 23 55 6 8 25 29
Present book
Tan Li 2023 Elsevier Vehicle/Tire/ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Full system for
Road Dynamics: spectral vehicle
Handling, Ride, dynamics
and NVH
12 Vehicle/Tire/Road Dynamics
For example, the ride movement of the vehicle and tire has an effect on the distribution
of vehicle load and tire force and, thus, influences the handling; the ride vibration of
the vehicle and tire contributes to the noise generation (low-frequency NVH). That is
to say, ride dynamics is something bridging handling dynamics and NVH. In this
book, the chapter for handling dynamics provides elementary introduction to ride
dynamics, and the chapter for ride dynamics provides elementary introduction to
NVH, presenting better coherence and synergy between these three major areas of
vehicle/tire dynamics. Several topics on the dependence between handling, ride,
and NVH are discussed in Chapter 6.
Chapter 7 highlights the road effect on handling, ride, and NVH. Chapter 8 talks
about intelligent tire and autonomous EV where the vehicle/tire/road dynamics plays a
great role.
Accompanying the fundamental theories, case studies are given to facilitate com-
prehension throughout the chapters. Besides the experimental implementations, the
state-of-the-art approaches to simulating vehicle/tire dynamics are also presented
from the viewpoint of both industry and academia.
References
[1] R.N. Yong, E.A. Fattah, N. Skiadas, Vehicle Traction Mechanics, Vol. 3, Elsevier Sci-
ence, 1984.
[2] V.K. Garg, R.V. Dukkipati, Dynamics of Railway Vehicle Systems, Academic Press,
1984.
[3] E.J. Haug, Computer Aided Kinematics and Dynamics of Mechanical Systems: Basic
Methods, Allyn and Bacon, 1989.
[4] T.D. Gillespie, Fundamentals of Vehicle Dynamics, SAE International, 1992.
[5] W.F. Milliken, D.L. Milliken, Race Car Vehicle Dynamics, SAE International, 1994.
[6] G. Genta, Motor Vehicle Dynamics: Modelling and Simulation, World Scientific Publish-
ing, 1996.
[7] J.C. Dixon, Tires, Suspension and Handling, second Edition, SAE International, 1996.
[8] J.Y. Wong, Theory of Ground Vehicles, third Edition, John Wiley & Sons, 2001.
[9] G. Howard, J.P. Whitehead, D. Bastow, Car Suspension and Handling, fourth Edition,
SAE International, 2004.
[10] R. Dukkipati, J. Pang, M. Qatu, G. Sheng, S. Zuo, Road Vehicle Dynamics, SAE Inter-
national, 2008.
[11] R.N. Jazar, Vehicle Dynamics: Theory and Application, Springer, 2008.
[12] J.Y. Wong, Terramechanics and Off-Road Vehicle Engineering: Terrain Behaviour, Off-
Road Vehicle Performance and Design, second Edition, Butterworth-Heinemann of
Elsevier, 2009.
[13] K. Popp, W. Schiehlen, Ground Vehicle Dynamics, Springer-Verlag, Berlin Heidelberg,
2010.
[14] H.B. Pacejka, Tire and Vehicle Dynamics, third Edition, Butterworth-Heinemann, 2012.
[15] R. Rajamani, Vehicle Dynamics and Control, second Edition, Springer, 2012.
[16] M. Doumiati, A. Charara, A. Victorino, D. Lechner, Vehicle Dynamics Estimation Using
Kalman Filtering: Experimental Validation, John Wiley & Sons, 2012.
[17] G. Mastinu, M. Ploechl (Eds.), Road and Off-Road Vehicle System Dynamics Handbook,
Taylor & Francis, 2014.
Introduction 13
[18] M. Blundell, D. Harty, The Multibody Systems Approach to Vehicle Dynamics, second
Edition, Butterworth-Heinemann, 2015.
[19] M. Abe, Vehicle Handling Dynamics: Theory and Application, second Edition,
Butterworth-Heinemann, 2015.
[20] M. Meywerk, Vehicle Dynamics, John Wiley & Sons, 2015.
[21] J.P. Pauwelussen, Essentials of Vehicle Dynamics, Butterworth-Heinemann, 2015.
[22] W. Chen, H. Xiao, Q. Wang, L. Zhao, M. Zhu, Integrated Vehicle Dynamics and Control,
John Wiley & Sons, 2016.
[23] H. Taghavifar, A. Mardani, Off-Road Vehicle Dynamics: Analysis, Springer, Modelling
and Optimization, 2017.
[24] J. Balkwill, Performance Vehicle Dynamics: Engineering and Applications, Butterworth-
Heinemann, 2017.
[25] M. Guiggiani, The Science of Vehicle Dynamics: Handling, Braking, and Ride of Road
and Race Cars, second Edition, Springer, 2018.
[26] B. Maclaurin, High Speed Off-Road Vehicles: Suspensions, Tracks, Wheels and Dynam-
ics, John Wiley & Sons, 2018.
[27] H. Zhang, D. Cao, H. Du (Eds.), Modeling, Dynamics, and Control of Electrified Vehicles,
Woodhead Publishing, 2018.
[28] D. Schramm, M. Hiller, R. Bardini, Vehicle Dynamics: Modeling and Simulation, second
Edition, Springer, 2018.
[29] B.P. Minaker, Fundamentals of Vehicle Dynamics and Modelling: A Textbook for Engi-
neers with Illustrations and Examples, John Wiley & Sons, 2019.
[30] G. Rill, A.A. Castro, Road Vehicle Dynamics: Fundamentals and Modeling with
MATLAB®, second Edition, Taylor & Francis, 2020.
[31] D. Vangi, Vehicle Collision Dynamics: Analysis and Reconstruction, Butterworth-
Heinemann, 2020.
[32] S. Azadi, R. Kazemi, H.R. Nedamani, Vehicle Dynamics and Control: Advanced Meth-
odologies, Elsevier, 2021.
[33] J.J. Henry, J.C. Wambold (Eds.), Vehicle, Tire, Pavement Interface, ASTM, 1992.
[34] E. Guglielmino, T. Sireteanu, C.W. Stammers, G. Gheorghe, M. Giuclea, Semi-Active
Suspension Control: Improved Vehicle Ride and Road Friendliness, Springer, 2008.
[35] S. Yang, S. Li, L. Chen, Dynamics of Vehicle-Road Coupled System, Springer, 2015.
[36] G. Genta, A. Genta, Road Vehicle Dynamics: Fundamentals of Modeling and Simulation,
World Scientific Publishing, 2016.
[37] D.J. Inman, Vibration with Control, second Edition, John Wiley and Sons, 2017.
[38] D.J. Ewins, Modal Testing: Theory and Practice, Research Studies Press Ltd., 1984,
pp. 1–269.
[39] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics, fourth
Edition, John Wiley and Sons Inc., 2000.
[40] U. Sandberg, J.A. Ejsmont, Tyre/Road Noise Reference Book, INFORMEX, Kisa, Swe-
den; Harg, Sweden, 2002.
[41] M. Harrison, Vehicle Refinement: Controlling Noise and Vibration in Road Vehicles,
Butterworth-Heinemann, 2004.
[42] F. Fahy, P. Gardonio, Sound and Structural Vibration: Radiation, Transmission and
Response, Academic Press of Elsevier, 2007.
[43] H. Fastl, E. Zwicker, Psychoacoustics: Facts and Models, third Edition, Springer, 2007.
[44] M.J. Crocker (Ed.), Handbook of Noise and Vibration Control, John Wiley & Sons, 2008.
[45] D. Thompson, Railway Noise and Vibration: Mechanisms, Modelling and Means of Con-
trol, Elsevier Science, 2009.
14 Vehicle/Tire/Road Dynamics
[46] X. Wang (Ed.), Vehicle Noise and Vibration Refinement, Woodhead Publishing, 2010.
[47] G. Sheng, Vehicle Noise, Vibration, and Sound Quality, SAE International, 2012.
[48] M. Kraft, N.M. White (Eds.), Mems for Automotive and Aerospace Applications,
Woodhead Publishing, 2013.
[49] D.J. Inman, Engineering Vibration, fourth Edition, Pearson, 2014.
[50] A. Fuchs, E. Nijman, H.-H. Priebsch (Eds.), Automotive NVH Technology, Springer,
2016.
[51] D.A. Bies, C.H. Hansen, Engineering Noise Control: Theory and Practice, fourth Edition,
Spon Press of Taylor & Francis, 2017.
[52] J. Pang, Noise and Vibration Control in Automotive Bodies, John Wiley & Sons, 2018.
[53] X. Wang (Ed.), Automotive Tire Noise and Vibrations, Butterworth-Heinemann, 2020.
[54] D.F. Hays, A.L. Browne (Eds.), The Physics of Tire Traction: Theory and Experiment,
Springer, 1974.
[55] F.J. Kovac, Tire Technology, fifth Edition, Goodyear Tyre & Rubber Company, 1978.
[56] R.A. Fleming, D.I. Livingston (Eds.), Tire Reinforcement and Tire Performance, ASTM,
1979.
[57] S.K. Clark, Mechanics of Pneumatic Tires, U.S. Department of Transportation, National
Highway Traffic Safety Administration, 1981.
[58] W.E. Meyer, J.D. Walter, Frictional Interaction of Tire and Pavement, ASTM, 1983.
[59] R.A. Ridha, M. Theves, Advances in Tyre Mechanics, iSmithers Rapra Publishing, 1994.
[60] R.W. Rivers, Tire Failures and Evidence Manual: For Traffic Accident Investigation,
Charles C Thomas Pub Ltd, 2001.
[61] M.S. Evans, Tyre Compounding for Improved Performance, iSmithers Rapra Publishing,
2002.
[62] A.N. Gent, J.D. Walter, The Pneumatic Tire, National Highway Traffic Safety Adminis-
tration, 2005.
[63] Transportation Research Board, Tires and Passenger Vehicle Fuel Economy: Informing
Consumers, Improving Performance, The National Academies Press, 2006.
[64] T. Giapponi, Tire Forensic Investigation: Analyzing Tire Failure, SAE International,
2008.
[65] J.E. Mark, B.R. Erman, C.M. Roland (Eds.), The Science and Technology of Rubber,
fourth Edition, Academic Press, 2013.
[66] G. Leister, Passenger Car Tires and Wheels: Development - Manufacturing - Application,
Springer, 2018.
[67] Y. Nakajima, Advanced Tire Mechanics, Springer, 2019.
Definitions and fundamentals
2
The vehicle/tire/road dynamics in this book covers handling, ride, and NVH (noise,
vibration, and harshness). The fundamentals of vehicle handling are multibody
dynamics, which will be introduced in Section 2.1. The fundamentals of vehicle ride
are vibrations of discrete systems and will be discussed in Section 2.2. Control theory
is typically accompanied with vibrations and thus will be introduced in Section 2.3,
which is also the foundation for the autonomous vehicle that will be discussed in
Chapter 8. As higher level dynamics, NVH includes both structure-borne acoustics
(Section 2.4) and airborne acoustics (Section 2.5). In Section 2.6, the acoustic reso-
nance is discussed, which is also a common phenomenon in the vehicle/tire/road
system.
Prismatic or P or T 1 translation
translational
Universal U 2 rotations
Others Spherical S 3 rotations
The coordinates without prime symbol in Fig. 2.1 are in global reference frame,
whereas the coordinates with prime symbol are in local reference frame of each body.
The constraint equation vector is written as
ΦK ðq, tÞ
Φðq, tÞ≡ D ¼ 0, (2.2)
Φ ðq, tÞ
where ΦK(q, t) is the kinematic constraint vector based on the joints and ΦD(q, t) is the
driving constraint vector based on the actuators.
As shown in Fig. 2.2, the location of a rigid body i in the global XY plane can be
defined by the vector ri and angle of rotation φi. The vector rPi that locates point P of
rigid body i in global reference frame can be written as
where s0i P is the position vector in the local reference frame Xi0 Yi0 for point P. The
expression for the virtual displacement of point P can be obtained by taking the dif-
ferential of the above equation:
The equation derivations for the 3D multibody dynamics are omitted here, but a vehi-
cle’s front suspension system and the kinematic chain modeling are illustrated in
Fig. 2.3 for the reference.
Definitions and fundamentals 19
Fig. 2.2 Transformation from local coordinate system to global coordinate system.
There are typically two types of problems for a multibody system: kinematic prob-
lem and dynamic problem. The former studies the position or the motion of the mul-
tibody system, irrespective of the forces and reactions that generate it; the latter
involves the forces that act on the multibody system and its inertial characteristics
(mass, inertia tensor, the position of its center of gravity or CG). The governing equa-
tions are summarized in Table 2.2.
2.2 Vibrations
Vehicle ride comfort is associated with the oscillations of the cabin. This section
begins with the vibration of single-lumped mass without damping, then the vibration
of single-lumped mass with damping, and lastly the vibration of multiple-lumped
masses.
Fig. 2.3 Multibody system for MacPherson strut with dissolved lower wishbone: (A) structure,
(B) topology.
Modified from D. Schramm, M. Hiller, R. Bardini, Vehicle Dynamics: Modeling
and Simulation, 2nd ed., Springer, 2018. Reprinted with permission from Springer.
Kinematic Φ _ ðq, q,
_ tÞ ¼ Φq q_ + Φt ¼ 0 The single dot denotes first-order derivative with respect to time. The
velocity ν≡Φq q_ ¼ Φt subscript denotes first-order derivative with respect to that variable/
q_ ¼ Φ1 q Φt
vector
Kinematic Φ€ ðq, q,
_ q€ , tÞ ¼ Φq q € + Φq q_ q q_ + 2Φtq q_ + Φtt ¼ 0 The double dot denotes second-order derivative with respect to time
acceleration € ðq, q, _ q q_ + Φ
_t¼0
Φ _ q€ , tÞ ¼ Φq q €+Φ
γ≡Φq q € ¼ Φq q_ q q_ 2Φtq q_ Φtt
γ≡Φq q € ¼ Φ _ q_ Φ _t
h q i
€ ¼ Φ1
q q Φq q_ q q_ + 2Φtq q_ + Φtt
€ ¼ Φ1 _ _ +Φ _t
q q Φq q
" # " #
Dynamic T € M is mass matrix, QA is applied force vector, and λ is Lagrange
M Φq q QA
¼ multiplier vector. F and T with prime denote reaction forces and torque
Φq 0 λ γ
8 in local reference frame with origin at CG. F and T without prime denote
< F’ki ¼ ATi ΦkT ri λ k
reaction forces and torque in global reference frame
: T’ki ¼ s’PT
i Bi Φri Φφi λ
T kT kT k
8
< Fki ¼ Ai F’ki ¼ ΦkTri λ
k
: T ki ¼ T’ki ¼ s’PT
i Bi Φri Φφi λ
T kT kT k
22 Vehicle/Tire/Road Dynamics
d2 x
+ ω20 x ¼ 0
dt2 , (2.6)
k
02
ω ¼
m
where ω0 is called the undamped natural frequency of the system. To solve the above
differential equation, a solution in the form of power series is assumed as
x ¼ a0 + a1 t + a2 t2 + a3 t3 + :…, (2.7)
where a0, a1, a2 … are coefficients to be solved. Substituting Eq. (2.7) into Eq. (2.6)
yields
ω0 2 t2 ω0 4 t4 ω0 6 t6 ω0 3 t3 ω0 5 t5
x ¼ a0 1 + + … + a1 t + :…
2 24 720 6 120 , (2.8)
¼ a0 cos ðω0 tÞ + a1 sin ðω0 tÞ
where the constants a0 and a1 are determined by applying the boundary conditions (the
initial displacement/velocity conditions) to the problem. The solution can alterna-
tively be written in the form of complex number as
where e iω0t indicates a rotating vector at the angular velocity ω0 on the complex
plane, and A0 is the complex amplitude indicating magnitude and phase. Convention-
ally, the real part of the complex solution is used to represent the actual physical
behavior. However, the imaginary part associated with phase of the motion is needed
when adding it to another motion.
d2 x dx
m +c + kx ¼ 0, (2.10)
dt2 dt
or rewritten as
d2 x dx
+ 2ζω0 + ω20 x ¼ 0
dt2 dt
c c
ζ ¼ ¼ pffiffiffiffiffiffi , (2.11)
c0 2 mk
rffiffiffiffi
k
ω0 ¼
m
where ζ is called the damping ratio. The damping ratio is a dimensionless measure
describing how oscillations in a system decay after a disturbance, which can vary from
undamped (ζ ¼ 0), underdamped (ζ < 1, system oscillates with the amplitude gradu-
ally decreasing to zero at a slightly lower frequency than the undamped case) through
critically damped (ζ ¼ 1, amplitude returns to zero as quickly as possible without
oscillating) to overdamped (ζ > 1, amplitude exponentially decays without
oscillating).
The solution of the 1-DOF damped system is given by
x ¼ A0 eβt eiωd t
c
β¼
2mqffiffiffiffiffiffiffiffiffiffiffiffi : (2.12)
ωd ¼ ω 0 1 ζ2
The Q factor of a damped oscillator represents the ratio of energy stored to energy lost
per cycle and is calculated by
1
Q¼ : (2.13)
2ζ
24 Vehicle/Tire/Road Dynamics
Mnn €
xn1 + Cnn x_ n1 + Knn xn1 ¼ f n1 , (2.14)
where x is the vector of generalized coordinates (physical DOFs) and f is the vector of
the externally applied force. M, C, and K are mass, damping, and stiffness matrices,
which are constant for linear systems, whereas for nonlinear systems, the elements of
these matrices are functions of generalized displacements/velocities that are time
dependent. In this section, only linear system is considered.
First, we assume the system is undamped under free vibration, that is, C and f
are null
x + Kx ¼ 0:
M€ (2.15)
Typically, the matrices for M and K are nondiagonal, meaning the n-DOFs are
coupled and cannot be solved separately. To address this, a modal transformation
matrix Φ needs to be specified to convert the generalized coordinates x to the principal
coordinates p (modal DOFs), given by
x ¼ Φp: (2.16)
p + KΦp ¼ 0,
MΦ€ (2.17)
ΦT MΦ p € + ΦT KΦ p ¼ 0, (2.18)
|fflfflfflffl{zfflfflfflffl} |fflfflffl{zfflfflffl}
⁎ ⁎
M K
where λi ¼ ω2i is called eigenvalue (ωi is the natural frequency for the ith principal
coordinate).
Now we assume the system is damped under forced vibration. Substituting
Eq. (2.16) into Eq. (2.14) and premultiplying ΦT yields
ΦT MΦ p € + ΦT CΦ p_ + ΦT KΦ p ¼ ΦT f: (2.20)
|fflfflfflffl{zfflfflfflffl} |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl}
⁎ ⁎ ⁎
M C K
It is noted that C⁎ is not necessarily diagonal. If the system has proportional damping,
meaning
C ¼ αM + βK, (2.21)
p€i + 2ζ i ωi p_ i + ω2i pi ¼ f i
, (2.22)
2ζ i ωi ¼ α + βω2i
where fi is normalized/weighted external force. Thus, the solution for each normal
mode is obtained as
ðt
1
pi ðtÞ ¼ f i ðτÞ exp ½ζ i ωi ðt τÞ sin ωdi ðt τÞdτ, (2.23)
ωdi 0
qffiffiffiffiffiffiffiffiffiffiffiffiffi
where ωdi ¼ ωi 1 ζ 2i . The physical response of the system can be calculated by
modal transformation Eq. (2.16), which is known as the normal mode
summation method.
2.3 Control
A dynamics control system regulates the movements of mechatronic devices by con-
trolling the position, velocity, or force using some type of actuators (hydraulic, pneu-
matic, magnetic, or electric motors), the performance of which can be evaluated by
fast response, moderate overshoot, and minimal steady-state error [4]. A typical
motion control system is illustrated in Fig. 2.7, which includes sensors/estimators,
actuators, and controllers. This is analogous to a forced vibration system with specific
targets.
26 Vehicle/Tire/Road Dynamics
In this section, some mathematics for signal processing is introduced, which is also
the basis for NVH data analysis. The control design techniques based on state-space
representations are demonstrated.
ð
∞
1
gð ω Þ ¼ f ðtÞeiωt dt: (2.24)
2π
∞
ð
∞
Some commonly seen FTs are illustrated in Fig. 2.8. When the signal in the time
domain is periodic, the frequency domain is discrete, such as Fig. 2.8A. When the sig-
nal in the time domain is nonperiodic, the frequency domain is continuous, such as
Fig. 2.8B, C, and D.
Definitions and fundamentals 27
Fig. 2.8 Fourier transform and inverse Fourier transform for some commonly seen signals:
(A) sine wave, (B) impulse signal, (C) symmetric square wave, and (D) asymmetric
square wave.
Modified from Wikipedia “Fourier transform”.
ð
∞
Table 2.3 Partial list of functions and their Laplace transforms with zero initial conditions
and t > 0.
Laplace
Function Time domain x(t) s-domain X(s)
where f(t) ¼ F0cosωt is the sinusoidal force input on the system. Solving the algebraic
equation for X(s),
F0 s
X ðsÞ ¼ , (2.28)
ðms2 + cs + kÞðs2 + ω2 Þ
which can be inverse transformed to get the time response x(t), defined as
γ +ði∞
1 1
xðtÞ ¼ L ½XðsÞ ¼ XðsÞest ds, (2.29)
2πi
γi∞
where γ is a real number so that the contour path of integration is in the region of con-
vergence of X(s).
Eq. (2.27) can be rewritten as transfer function format,
ms2 + cs + k XðsÞ ¼ FðsÞ
+ , (2.30)
X ðsÞ 1
H ðsÞ ¼ ¼
FðsÞ ms2 + cs + k
where x1 is a real number. The probability density function is the derivative of CDF
with respect to x1
30 Vehicle/Tire/Road Dynamics
d
pð x 1 Þ ¼ Pðx1 Þ: (2.32)
dx1
The first-order statistical properties include mean x, mean square Ehx2i, and variance
σ 2, defined by
ð∞
x ¼ Eh x i ¼ xpðxÞdx
ð ∞ ∞
E x2 ¼ x2 pðxÞdx , (2.33)
∞
D E ð ∞
σ 2 ¼ E ðx xÞ2 ¼ ðx xÞ2 pðxÞdx
∞
where E is the expectation operator and means “average.” The second-order statistical
properties include autocorrelation ρxx(τ), cross-correlation ρxy(τ), auto-covariance
cxx(τ), and cross-covariance cxy(τ), defined by
ð
∞ ð
∞
orthogonal if
and independent if
2 3 2 3
Ehx1 ðtÞi x1
6 7 6 7
x ¼ EhxðtÞi ¼ 4 ⋮ 5 ¼ 4 ⋮ 5: (2.39)
EhxN ðtÞi xN
Similarly, the correlation matrix R(τ) and covariance matrix C(τ) associated with x(t)
are
2 D E D E3 2 3
E x1 ðtÞx1 ðt + τÞ ⋯ E x1 ðtÞxN ðt + τÞ ρ ðτÞ ⋯ ρ1N ðτÞ
6 7 6 11 7
D E 6 7 6 7
6 7 6 7
RðτÞ ¼ E xðtÞxT ðt + τÞ ¼ 6 ⋮ ⋱ ⋮ 7≡6 ⋮ ⋱ ⋮ 7
6 7 6 7
4 D E D E5 4 5:
E xN ðtÞx1 ðt + τÞ ⋯ E xN ðtÞxN ðt + τÞ ρN1 ðτÞ ⋯ ρNN ðτÞ
D E
CðτÞ ¼ E ðxðtÞ xÞðxðt + τÞ xÞT ¼ RðτÞ xx T
(2.40)
The components of x(t) are said to be uncorrelated if all off-diagonal elements (i.e.,
element-wise cross-correlations) of R(τ) are zero.
If the components of x(t) are zero-mean uncorrelated Gaussian random variables,
2 3
σ 211 ⋯ 0
RðτÞ ¼ CðτÞ ¼ 4 ⋮ ⋱ ⋮ 5δðτÞ ≡ QδðτÞ
, (2.41)
0 ⋯ σ 2NN
Q ¼ EhxxT i
Fig. 2.9 Control system: (A) feed-forward control (open loop) and (B) feedback control
(closed loop).
For tracking and disturbance rejection, that is, Y(s) R(s), it is desirable to have G(s)
H(s) ≫ 1, which can be achieved by turning up the compensator gain j G(0)j. However,
too much compensator gain can cause instability, which is even worse than the uncon-
trolled process. Instability occurs when not all the roots of the characteristic equation 1
+ G(s)H(s) ¼ 0 are located in the left-half plane (further left indicates faster response).
There are two common choices for classical controller/compensator designs: PID
controller GPID(s) and lead/lag controller GL(s), defined as
1
GPID ðsÞ ¼ kP + kI + kD s
|{z} s
|{z} |{z}
Proportional control Derivative control
(
Integral control
: (2.44)
s+b a>b Lead control
GL ðsÞ ¼ k ,
s+a a<b Lag control
For single input single output system, the plant H can typically be represented or esti-
mated by a transfer function format:
where m is the order of the numerator polynomial function, n is the order of the
denominator polynomial function, bi are the coefficients of the numerator polynomial
function, ai are the coefficients of the numerator polynomial function, and τ is the
transport delay.
where u(t) is input vector, y(t) is output vector, x(t) is state vector collecting gener-
alized coordinates and their derivatives, x_ ðtÞ is time derivative state vector, A is state
matrix representing the dynamics of the open-loop system, B is input matrix, C is out-
put matrix, and D is direct transmission matrix.
The state is a set of variables summarizing the current status of a system, which can
be positions, velocities, accelerations, forces, momentum, torques, pressure, voltage,
current, charge, and so on. For example, the state of the 1-DOF spring-mass oscillator
discussed in Section 2.2.1 (c.f. Fig. 2.4) can be the displacement x(t) and velocity x_ðtÞ
of the mass (Fig. 2.10)
x1 xðtÞ
x¼ ¼ : (2.47)
x2 x_ ðtÞ
The input of the system is u(t) ¼ F(t). Suppose the output of interest is the spring force
and the mass acceleration, that is,
" kx1 #
y1 kxðtÞ
y¼ ¼ ¼ u kx1 : (2.48)
y2 x€ðtÞ
m
Thus, the full state-space equations for this system can be written as
" # 2 3" # 2 3
x_ 1 ðtÞ 0 1 x1 ðtÞ 0
¼4 k 5 + 4 1 5 uðtÞ
x_ 2 ðtÞ 0 x2 ðtÞ
|{z}
|fflfflffl{zfflfflffl} |fflfflfflfflfflfflm
ffl{zfflfflfflfflfflfflffl}|fflfflffl{zfflfflffl} |fflffl{zffl
m ffl} u
x_ A
x
B
2 3" 2 3 : (2.49)
" # k 0
# 0
y1 ðtÞ x1 ðtÞ
¼4 k 5 + 4 1 5 uðtÞ
y2 ðtÞ 0 x2 ðtÞ
|{z}
|fflfflffl{zfflfflffl} ffl{zfflfflfflfflfflfflffl}|fflfflffl{zfflfflffl} |fflffl{zffl
|fflfflfflfflfflfflm m ffl} u
y x
C D
The order of a system N is the minimal number of state variables required to describe
it. The 1-DOF spring-mass oscillator needs to be described by a second-order differ-
ential equation, which should be reduced to two first-order differential equations for
state-space representation. Therefore, the order of the 1-DOF spring-mass oscillator is
N ¼ 2. The choice of state variables is not unique; one may instead choose the spring
force and momentum as the two state variables. Because x_ ðtÞ and y(t) are linear com-
binations of x(t) and u(t), this system is called linear system. In addition, due to the fact
that all the coefficients A, B, C, and D are constants, this system is called linear time
invariant (LTI) system. This control system discussed in this subsection is focused
on LTI system; nonlinear systems can be linearized about a specified operating point
(x0,u0).
With derivation omitted here, the solution of the state vector is given by
ðt h i
xðtÞ ¼ eAðtt0 Þ xðt0 Þ + eAðtτÞ B uðτÞdτ, (2.50)
t0
where the first term represents the response due to initial conditions at t0 and the sec-
ond term represents the forced response (a convolution integral where eAtB is the sys-
tem impulse response). By defining the state transition matrix indicating how the state
evolves from initial conditions,
Φðt, t0 Þ ¼ eAðtt0 Þ )
ðt : (2.51)
xðtÞ ¼ Φðt, t0 Þxðt0 Þ + ½Φðt, τÞBuðτÞdτ
t0
The solution for the state can also be calculated by Laplace transform method. By tak-
ing the Laplace transform of the state and output equations Eq. (2.46),
yðsÞ
HðsÞ ¼ ¼ C½sI A1 B + D
uðsÞ
, (2.53)
1 adj½sI A
½sI A ¼
jsI Aj
which is alternative to the transfer function format in Eq. (2.45). The eigenvalues of
the A matrix are the solutions of the characteristic equation jsI A j ¼ 0, that is, the
poles of the transfer functions. The matrix exponential eAt can be related to the state
matrix A through the inverse Laplace transform:
n o
L1 ½sI A1 ¼ eAt ¼ Φðt, 0Þ: (2.54)
Numerical methods can also be used to find the solution of the state [6].
The discrete state-space formulation can be deduced from the continuous one by
using the first-order Euler approximation formula
xk xk1
x_ k1 ¼ , (2.55)
ts
where k is the index of the discrete sequence and ts is the sampling time (a.k.a. sam-
pling interval or sampling period).
where G is a matrix of static (constant) gains. Substituting Eq. (2.56) into the state
equation Eq. (2.46) gives
x_ ðtÞ ¼ ½A BGxðtÞ
, (2.57)
yðtÞ ¼ ½C DGxðtÞ
Q ¼ BABA2 B…AN1 B
ð t1
: (2.58)
Pðt1 , t0 Þ ¼ eAðt1 τÞ BBT eA ðt1 τÞ dτ
T
t0
full state feedback controller except that BG (converging x) is replaced with KC (con-
verging e).
Similarly, the definition of observability is: the poles of an observer can be placed
arbitrarily if and only if the system is observable or the observability matrix has full
rank, rank(O) ¼ N,
h i
N1
O ¼ CT AT CT AT 2 CT … AT CT : (2.60)
Observability is a measure of how well the internal states of a system can be inferred
by knowledge of its inputs and external outputs [7].
The state feedback problem and the observer problem can be decoupled and solved
separately. It is also noted that there is not a unique solution for G or K; one just needs
a satisfactory or balanced solution that meets the response criteria (faster response typ-
ically associates with higher overshoot). One set of poles selected leads to one set of
solutions, determining the response performance such as convergence rate. As
increasing the convergence rate of the observer can recover the performance of the
full-state feedback controller, it is generally considered good design practice to place
the observer poles to the left of the closed-loop poles. However, increasing the con-
vergence rate of the observer typically increases the control effort, compensator gain,
and compensator bandwidth.
The augmented system dynamics are
" # " #" #
x_ A BG x
¼
x_
b KC ðA KC BGÞ bx
" # " #" # , (2.61)
y C DG x
¼
b
y 0 C DG b x
Fig. 2.12 Block diagram for integral control with exogenous inputs.
Modified from Prof. Steve Southward’s lecture notes on ME 5554 Applied Linear Systems,
Virginia Tech.
the reference inputs by minimizing the steady-state error between the outputs and the
reference outputs and (2) minimize the system response to external disturbances.
There are typically two types of disturbances, namely, process noise and measure-
ment (or sensor) noise. As shown in Fig. 2.12, process noise w affects both the
states x and the outputs y, whereas measurement noise θ only affects the outputs
y but not the states x.
The control law includes both integral control (typically effective to eliminate
steady-state tracking error) and full state feedback, given by
If the full state vector x is not available, the compensator using a linear observer (static
gains K) to estimate the states for state feedback can be developed, as discussed in
Section 2.3.5. The complete open-loop state equations with observer are
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