J. Mod. Transport. (2013) 21(3):135162 DOI 10.1007/s40534-013-0017-8
An overview of a unied theory of dynamics of vehiclepavement interaction under moving and stochastic load
Lu Sun
Received: 18 May 2013 / Revised: 12 July 2013 / Accepted: 19 July 2013 / Published online: 26 September 2013 The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract This article lays out a unied theory for dynamics of vehiclepavement interaction under moving and stochastic loads. It covers three major aspects of the subject: pavement surface, tirepavement contact forces, and response of continuum media under moving and stochastic vehicular loads. Under the subject of pavement surface, the spectrum of thermal joints is analyzed using Fourier analysis of periodic function. One-dimensional and two-dimensional random eld models of pavement surface are discussed given three different assumptions. Under the subject of tirepavement contact forces, a vehicle is modeled as a linear system. At a constant speed of travel, random eld of pavement surface serves as a stationary stochastic process exciting vehicle vibration, which, in turn, generates contact force at the interface of tire and pavement. The contact forces are analyzed in the time domain and the frequency domains using random vibration theory. It is shown that the contact force can be treated as a nonzero mean stationary process with a normal distribution. Power spectral density of the contact force of a vehicle with walking-beam suspension is simulated as an illustration. Under the subject of response of continuum media under moving and stochastic vehicular loads, both time-domain and frequency-domain analyses are presented for analytic treatment of moving load problem. It is shown that stochastic response of linear continuum media subject
to a moving stationary load is a nonstationary process. Such a nonstationary stochastic process can be converted to a stationary stochastic process in a follow-up moving coordinate.
Keywords Vehiclepavement interaction Random eld
Continuum medium Spectral analysis Greens function
Linear system
List of SymbolsFx; t Moving source
d Dirac delta function
v Source velocity Pt Source magnitude
h Impulse response function
H Frequency response function
S Power spectral density
R Correlation function
E Expectation
r2 Variancew Standard deviation G Greens function
u Response function of continuum media
1 Introduction
The investment of the United States in the nations transportation infrastructure alone (highways, bridges, railways, and airports) amounted to $7 trillion by 1999. To preserve infrastructure longevity in a cost-effective manner, the research in pavement design and infrastructure management has been growing rapidly in recent years. From a pavement design point of view, pavement response, damage, and performance are essentially the result of long-term
L. Sun (&)
Department of Civil Engineering, The Catholic University of America, Washington, DC 20064, USAe-mail: [email protected]
L. SunInternational Institute of Safe, Intelligent and Sustainable Transportation & Infrastructure, Southeast University, Nanjing 210096, China
123
136 L. Sun
vehiclepavement interaction. When vehicle speed is low, the dynamic effect of vehicular loads on pavements is insignicant. However, with the promotion of high-speed surface transportation in the world, this dynamic effect must be taken into account to develop more rational pavement design methods. For instance, real causative mechanisms that lead to fatigue damage of pavement material might be frequency dependent. From an infrastructure management point of view, vehiclepavement interaction has a profound impact on the way that existing technologies of structural health monitoring, environmental vibration mitigation, nondestructive testing and evaluation, and vehicle weight-in-motion are to be improved, innovated and implemented. For instance, modern high-speed surface transportation systems are normally accompanied by rises in levels of noise and vibration that may cause a signicant detrimental effect to the ecology. Vehicle pavement interaction-induced structure-borne and ground-borne vibrations emit and propagate toward some extent. Residents may experience hardship from uncomfortable vibration, and high-precision equipment may suffer from malfunctioning to irreparable damage. To mitigate noise and vibration in surrounding areas of roadway, it is necessary to investigate predominant frequencies of vehicle pavement interaction, the source of vibration, so as to develop effective noise and vibration countermeasures. The study of vehiclepavement interaction also plays a critical role in developing better inversion algorithms for nondestructive testing and evaluation of transportation infrastructure. In addition, taking into account dynamic effects of vehicle vibration caused by rough surface may considerably improve accuracy and reliability of weigh-in-motion systems, which measure a vehicles weight as it travels at a normal speed.
Vehiclepavement interaction is not only a central problem to pavement design, but also has a profound impact on infrastructure management, vehicle suspension design, and transportation economy. Figure 1 shows a central role of vehiclepavement interaction played in a wide variety of applications. From a vehicle-design point of view, a designer needs to consider vehicle vibration and controllability, which affect ride quality and vehicle maneuver. Vehiclepavement interaction also has a huge economic impact. As pavement performance deteriorates and roadway surface gets rougher, both operation costs (fuel, tire wear, and routine maintenance) and roadway maintenance cost of the vehicle increase dramatically, accompanied by decreasing transportation productivity. There is no doubt that there is a great and urgent need for fundamental research on vehiclepavement interaction due to rapid deterioration of huge highway infrastructure nationwide, tight maintenance budget, and the key role played by vehiclepavement interaction.
Since the American Association of State Highway Ofcials (AASHO) road test, the fourth power law has been widely used by pavement engineers to design highway and aireld pavement and to predict the remained life and cumulated damage of pavements [13]. Besides the damage caused by static loads, dynamic loads may lead to additional pavement damage. A consequence of a high power in the damage law is that any uctuation of pavement loading may cause a signicant increase in the damage suffered by pavement structures. A number of recent led measurements and theoretical investigations showed that vehicle vibration-induced pavement loads are moving stochastic loads [48]. The researchers concluded that vibrations of vehicles were related primarily to pavement surface roughness, vehicle velocity, and suspension types [913].
Estimation of pavement damage caused by dynamic loads varies anywhere from 20 % to 400 % [14]. The smaller estimates are based on the assumption that peak dynamic loads (and hence the resulting pavement damage) are distributed randomly over the pavement surface. The larger estimates are based on the assumption that vehicles consistently apply their peak wheel loads in the same areas of the pavement. Theoretical studies by Cole [15] and Hardy and Cebon [16] conrmed that for typical highway trafc, certain areas of the pavement surface always suffered the largest wheel forces, even when the vehicles had a wide range of different suspension systems, payloads, and speeds.
Complicated relationships exist among vehicle suspension, dynamic wheel loads, pavement response, and damage [1719]. On one hand, it has been known for years on how to manufacture automobiles operating properly on a variety of pavement surfaces. On the other hand, however, the effect of vehicle design on pavement has not been thoroughly studied. For instance, Orr [20] stated in his study that comparatively little was known about the inuence of suspension design on pavement in the automobile industry yet.
A number of obstacles exist in revealing vehiclepavement interaction. A theoretical foundation universally applicable to the involved specic problems is no doubt very attractive. It will not only provide a guide for experimental study and validation, but will also enable better design and maintenance of vehicles and pavements. As such, this article provides an overview of a unied theory for dynamics of vehiclepavement interaction under moving and stochastic loads. This article covers three major aspects of the subject: pavement surface, tirepavement contact forces, and response of continuum media under moving and stochastic vehicular loads developed by Sun and his associates (Fig. 2). The remainder of this article is organized as follows. Section 2 addresses mathematical
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An overview of a unied theory of dynamics 137
description of pavement surface roughness. Section 3 studies contact force generated due to vehiclepavement interaction, which is the source to both vehicle dynamics and pavement dynamics. Section 4 investigates pavement response under moving stochastic loads. Section 5 discusses difculties and deciencies in the existing research, and projects further research and applications. Section 6 makes concluding remarks.
2 Pavement surface
Irregularities in pavement surface are often called roughness. All pavement surfaces have some irregularities. The National Cooperative Highway Research Program denes roughness as the deviations of a pavement surface from a true planar surface with characteristic dimensions that affect vehicle dynamics, ride quality, and dynamic pavement loads [21]. Factors contributing to roughness, in a general sense, include vertical alignment, cracks, joints, potholes, patches and other surface distresses.
Newly constructed pavements may be poorly nished or have design features such as construction joints and thermal expansion joints, which can be main sources of vehicle vibration [22, 23]. Measurements show that pavement roughness can be modeled as a random eld consisting of different wavelength. Considerable effort has been devoted to describing pavement surface roughness [2426]. The
peak-to-valley measurements, the average deviation from a straight edge, and the cockpit acceleration are several distinct approaches that have been suggested for pavement surface characterization [27]. Practical limitations of these three approaches could be found in the study of Hsueh and Penzien [28].
Spectrum of a deterministic function referring to its Fourier transform reveals sinusoid components that a deterministic periodic or nonperiodic function is composed of. Spectrum analysis of a deterministic function f t can
be obtained if and only if this deterministic function satises Dirichlet condition and absolute integrability condition [29]:
Z
1 1
f t
j j dt\ 1: 1
Pavement surface as a random eld of elevation does not decay as spatial coordinates extend to innity; therefore, it does not satisfy inequality (1). For this reason, taking Fourier analysis to a sampled pavement surface does not make too much sense in theory. However, the correlation vector of random eld does decay as spatial coordinates extend to innity. Hence, applying Fourier transform to correlation vector of pavement surface, commonly known as power spectral density, does serve the purpose of spectrum analysis in theory [30]. As such, mathematical description of pavement surface takes place under the theoretical framework of spectral analysis of stochastic process.
Vehicle Maneuver & Control
Vehicle Suspension System
Stochastic & Dynamic Loads
Pavement Material Fatigue Mechanism
Vehicle Design Pavement Design
Transportation Economics Infrastructure Management
Ride Quality & Safety
Performance Modeling & Prediction
Vehicle Operation Cost
Nondestructive Testing & Evaluation
Transportation Productivity
Highway Maintenance Cost
Vibration Mitigation
Structural Health Monitoring
Vehicle Weigh-in-Motion
Fig. 1 A central role of vehiclepavement interaction played in various applications
Surface Vehicle Contact Structural
Dynamic
Response
Roughness Systems Forces Systems
Fig. 2 Dynamics of vehiclepavement interaction
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The following subsections consist of a general mathematical framework for describing pavement surface roughness, a review of statistical description of pavement surface roughness, a description of periodic joints of cement concrete pavement, and three different hypotheses when projecting a one-dimensional road prole to a two-dimensional pavement surface.
2.1 One-dimensional description
Given the manifest complexity exhibited in pavement surface as a random eld, making some assumptions becomes indispensable to simplify mathematical description of pavement surface. Commonly used assumptions on surface roughness are that surface roughness is an ergodic and homogeneous random eld with elevation obeying Gaussian distribution [3133]. The assumption of ergodicity makes sure that the temporal average of a sample of stochastic process equals to the statistical mean of stochastic process, which enable one to obtain the statistical characteristics of a stochastic process by measuring only a few samples. The assumption of homogeneity ensures random property of surface roughness is independent to the sites measured. When a vehicle travels at a constant speed, a homogeneous random led is converted to a stationary stochastic process. The assumption of Gaussian distribution ensures that transformation of such randomness through a linear system is still Gaussian.
Road prole along longitudinal direction (i.e., direction of travel) is often an exchangeable term of pavement roughness. The simplest model describes the pavement surface as a cylindrical surface dened by a single longitudinal prole n(x). Figure 3 shows an illustrative example of road prole. Instead of varying with time, the elevation n of the surface varies with respect to longitudinal distance x along the direction of travel. In the spatial domain, low-frequency components correspond to long wavelengths while high frequency components correspond to short wavelengths. Let n(x) be a zero mean, homogeneous,
Gaussian random eld. Its probabilistic structure can be completely described by either the autocorrelation function or the power spectral density (PSD).
According to the WinnerKhintchine theory [31], the following expressions constitute a pair of Fourier transform:
SnnX 2p 1 Z
1 1
RnnXe iXXdX; 2a
RnnX Z
1 1
SnnXeiXXdX; 2b where X represents the distance between any two points along the road. Wavenumber spectrum, SnnX, is the direct
PSD function of wavenumber X , which represents spatial frequency dened by X = 2p/k where k is the wavelength of roughness. Under the assumption of homogeneity, spatial autocorrelation function RnX is dened by
RnnX Enx1nx2 Enx1nx1 X 3 for any x1 and X. Here E is the expectation operator and
can be calculated by
Enx lim
X!1
1
2X
Z
X
X nxdx: 4
This expectation, differing from statistical mean of a random process, is temporal average of a sample of stochastic process.
2.1.1 PSD roughness
As a one-dimensional random eld, roughness can be characterized in the spatial domain and the wavenumber domain. An analytic form of PSD roughness is often desired for theoretical treatment [34, 35]. Spectral analysis of pavement roughness has been the subject of considerable research for years. Many function forms of PSD roughness have been proposed to characterize bridge deck surface [36], rail-track surface [33, 37], and aireld runway surface [24, 38].
Sayers et al. [39, 40] suggested a power function for PSD roughness:
SnnX AXa: 5
Early studies on PSD of longitudinal proles of runways and roads are a special case of (5) for a 2 [4144].
Although power function (5) is convenient for parameter estimation and design purpose, it creates mathematical difculties at X 0, where SnnX becomes innite. For
this reason, two distinct function forms have been proposed in later studies.
One form is to use rational functions: Sussman [45] for (6), Sussman [45], Snyder and Wormley [46], and Yadav and Nigam [47] for (7), Macvean [48] for (8), Bolotin (1984) for (9), and Gillespie [25, 26] for (10), to name a few.
(x
)
A B
X
1
x 2
x x
Fig. 3 One-dimensional road prole
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2.1.2 Effect of periodic joints of rigid pavement
Thermal expansion joints are commonly set on rigid pavement surface to reduce thermal stress in pavement. Thermal joints of rigid pavement can be a signicant source of excitation of vehicle vibration, especially when the joint sealing material gets lost. This could cause undesirable riding qualities at a high speed [43]. Figure 4 shows an idealized rigid pavement surface with periodic joints. An actual rigid pavement surface might be considered as a combination of this periodic prole and a random eld.
In the spatial domain, pavement roughness n(x) in Fig. 4 can be described by a periodic function within a period [-d/2,d/2].
nx
0; x 2 d=2; D=2
cx; x 2 D=2; D=2 :
0; x 2 D=2; d=2
SnnX
S0X2 a2 b2
X2 a2 b22 4X2a2
; 6
SnnX
S0 X2 a2
; 7
SnnX
S0
X2 a22
; 8
SnnX
4r2 p
aX20 X2 X20 4a2X2
; 9
SnnX S01 X20X 2 2pX 2: 10
Another form is to use piecewise functions. For instance, (11) was suggested by the International Organization for Standardization (ISO) to cover different frequency ranges [4952].
SnnX
CspX w1a for 0 X Xa CspX w2 for Xa X Xb;
0 for Xb\X
8
>
<
>
:
8
<
:
11
where Xa represents reference frequency, and Xb represents cut-off frequency. Iyengar and Jaiswal [33] provided a similar split power law in the form of (11) for rail-track in which Csp = 0.001(m-2 cycles/m) and w1 = w2 = 3.2.
Marcondes et al. [22, 23] proposed an exponential piecewise function:
Snn X
A1 exp kXp
for X X1;
13
Here spatial period along the longitudinal direction, d, represents slab length of rigid pavement, D and h, respectively, represent maximum width and depth of joint, c(x) is named shape function of joint, which is a symmetrical function and describes the shape of the joint. With this description, pavement roughness n(x) becomes an even function with period d.
A period function can be legitimately expanded using Fourier series, which is sometimes called frequency spectrum analysis of periodic function [53]. As such, coefcients of the expanded Fourier series of n(x) an and bn (n = 0,1,2,) are given by
an
1 d=2
12
In all aforementioned PSD roughness, A, A1, A2, p, q, k,
w1, w2, Csp, a b, S0, and X0 are the real and positive parameters estimated from eld test.
Although various PSD functions have been proposed for tting the measured PSD roughness, it has been found that most pavement proles including road surface, runway surface, and rail-track surface exhibit very similar trends of PSD curves. Some researchers have even reduced the PSD curves to only two families: one for rigid (cement concrete pavements), and the other for exible (asphalt concrete pavements) [25, 26].
A2 X X0
q for X1 X:
Z
d=2
d=2 nx cos
npx d=2dx
2 d
Z
D=2
D=2 cx cos
2npxd dx;
14a
bn 0; n 0; 1; 2; . . . (for nx is an even function): 14b
Wavenumber spectrum of roughness is thus expressed as
SnnXn a2n b2n
4 d2
" #2; 15a
)
Z
D=2
D=2 cx cos
2npxd dx
(x
x
h
d
Fig. 4 Periodic joints of rigid pavement surface
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140 L. Sun
n=d n 0; 1; 2; : 15b In which Xn represents discrete wavenumber and
SnnXn is discrete wavenumber spectrum. It is evident
from (18) that different shapes of joints determine different discrete spatial spectrum. Four types of shape function, namely rectangular curve, parabolic curve, cosine curve, and triangular curve have been observed and investigated by Sun [54].
Table 1 gives the discrete spatial spectrums of these four types of shape functions computed from (15a). The comparison between these discrete spectrums for specied rigid pavement with slab length 5 m and joint width 0.03 m is plotted in Fig. 5 where S hD=d, Xn 2pn=d,
n 0; 1; 2; , and D=d 0:006. From this gure it can be
seen that the spectrum of rectangular joint is the greatest in four types of joints, while the spectrum of triangular joints is the smallest. The ratio of the magnitude of four types of joints based on SnnX0 approximates to 4:1.78:1.62:1.
Because S2 = (hD/d)2, the effect of joint of rigid pavement with long slab length d is much less than that of the pavement with short slab length.
2.2 Two-dimensional description
One-dimensional random eld model of pavement surface is adequate for two-wheel vehicles, such as bicycles and
motor-cycles but inadequate for cars and trucks having two or more wheel per axle. Actual highway and aireld pavement consists of a two-dimensional surface of nite width, a nominal camber and grader. The elevation of pavement surface exhibit random uctuations about the nominal geometry and therefore should be more accurately treated as a two-dimensional random eld, n(x, y), with space coordinates x and y as the indexing parameters as shown in Fig. 6.
When ignoring those isolated large uctuations such as potholes, uctuations of pavement surface can be approximate by a homogenous, Gaussian random eld with a zero mean [49, 55]. Probabilistic structure of a two-dimensional random eld can be completely dened either by two-dimensional autocorrelation function
RACX; Y Enx1; y1nx2; y2 Enx1; y1nx1 X; y1 Y 16
or by two-dimensional PSD, SnnX; K, where X = x2 x1,
Y = y1-y2, X and K represent wavenumber in x-axis and y-axis directions, respectively. Here SnnX; K is dened as
a double-sided Fourier transform of autocorrelation function (16).
SX; K 2p 2 Z
1 1
Xn
Z
1 1
RX; Ye iXXKYdXdY:
17
Table 1 Discrete wavenumber spectrum resulting from periodic joints of rigid pavement
Joints Shape function Diagram Discrete spatial spectrum
Rectangular curve h SnnX0 4S2
SnnXn
sinDXn=2DXn=2 2SnnX0:
D2 x2 h SnnX0 16S2 9
SnnXn
9cosDXn=2 sinDXn=2.DXn=
2 2
Parabolic curve 4h
DXn=24
SnnX0:
Cosine curve h cos pD x SnnX0 16S2 p2
SnnXn
cosDXn=21 DXn=p2 2
SnnX0:
Triangular curve 2h
D x
j j h SnnX0 S2
SnnXn
41 cosDXn=2 2
DXn=24
SnnX0:
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An overview of a unied theory of dynamics 141
4.5
4.0
Discrete spatial spectrum (S2)
3.5
D C
y
Rectangular shape function
Parabolic shape function
Cosine shape function
Triangular shape function
Left Wheel Path
3.0
2 Y
A
B
C'
2.5
y
Right Wheel Path
1
2.0
1.5
1.0
X
x
x
Fig. 7 A plane view of two-dimensional isotropic random eld
0.5
x 2
1
0.0 0 5 10 15 20
Wave number (n)
Fig. 5 Discrete wavenumber spectrum resulting from periodic joints of rigid pavement
y
,
( y
x
)
y
B A
y
2
Y 1
y
Y
0
x
x x
X
1
2
Fig. 6 Two-dimensional random eld model of pavement surface
The determination of the two-dimensional autocorrelation function or two-dimensional PSD relies on the measured elevation of the entire pavement surface, which is a possible but formidable task in data acquisition, numerical computation and data storage for hundreds of thousands of miles of roadways. Efforts have, therefore, been made to simplify the two-dimensional random eld such that it can be uniquely generated from the one-dimensional random eld models by imposing certain hypothetic properties to pavement surface [49, 50, 55].
2.2.1 The hypothesis of isotropy
Consider a two-dimensional random eld model of pavement surface as shown in Fig. 7. There are two parallel wheel paths separated by a constant distance Y along the transverse direction. Longitudinal proles along each wheel path can be derived from the two-dimensional random eld n(x, y) as follows.
From (16) the autocorrelation functions between A and B and between D and C are, respectively, given by
RABX; 0 Enx1; y1nx1 X; y1 18a and
RDCX; 0 Enx1; y2nx1 X; y2 : 18bSince nx; y is homogenous,
RABX; 0 RDCX; 0 RnnX: 19 Crosscorrelation functions, RACX; Y and RCAX; Y,
are even functions of X and Y:
RACX; Y RCAX; Y RAC X; Y: 20
Now, assume that n(x, y) be an isotropic random eld [49, 56]. The property of isotropy requires that for any prole making an angle h with x-axis, the following condition holds:
Rq cos h; q sin h Rq; 0: 21From (16), (19), and (21), it follows that
RACX; Y RCAX; Y RAC X; Y RAC
X2 Y2
p
; 0 Rnn
X2 Y2
p
: 22
Since autocorrelation function and PSD form a Fourier pair, cross-PSD is given by
SACX SCAX 2p 1 Z
1 1
Rnn
X2 Y2
p
e iXXdX:
23
Equations (19) and (23) are general results for spectral analysis under the hypothesis of isotropy. When using the denition of a normalized cross-PSD developed by Dodds and Robson [49], cross-PSD can also be written as
SACX SCAX gXSnnX: 24 where gX is coherence function between direct-PSD and
cross-PSD of two parallel wheel paths, which is always not
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142 L. Sun
greater than 1 [57]. Heath [58] further presented a closed-form gX
gX 1
Y SnnX Z
Note that
RnnX s Z
1 1
0 Snn
SnnKeiKXsdK: 30 Replacing RnnX s in (29) by (30) we obtain
SACX SCAX
2p 1 Z
1 1
1
X2 f2
q J1Yfdf; 25 where J1 is the rst-order Bessel function and X 0.
2.2.2 The hypothesis of uncorrelation
Parkhilovskii [55] proposed the hypothesis of uncorrelation. It assumes that the two proles of two parallel wheel paths, n(x, y1) and n(x, y2) nx; y2, can be derived from two
uncorrelated random elds, that is
nx; y1 fx Ycx; 26a nx; y2 fx Ycx: 26b
It then follows that
SABX SDCX SffX Y2SccX; 27a SACX SDBX SffX Y2SccX: 27b
Thus, direct- and cross-PSD can be expressed in terms of direct PSDs of fx and cx. Robson [59] has examined
the physical and mathematical basis of Parkhilovskii model. He concluded that Parkhilovskii model can be made compatible with isotropic model for a prole-pair description of pavement (i.e., two parallel railway tracks), and may be used where isotropy assumption is not valid.
2.2.3 The Hypothesis of Shift
Sun and Su [60] proposed the hypothesis of shift to construct two-dimensional PSD using one-dimensional PSD. According to homogeneity in (19), autocorrelations of n(x, y1) and n(x, y) equal to each other. Sun and Su [60] assumed nx1 X; y1 Y nx1 X s; y1 Y. In other words,
the elevation of one wheel path is equal to the elevation of a parallel but shifted wheel path with a spatial lag s. With
RACX; Y Enx1; y1nx1 X; y1 Y Enx1; y1nx1 X s; y1 Y
RnnX s; 28
where spatial lag s is a parameter that can be estimated from eld test of autocorrelation functions of parallel wheel path. Some properties about the spatial lag can be s = 0 as Y = 0 where RACX; 0 RnnX and s ! 1 as
Y ! 1 where RACX; 1 0.
Under the hypothesis of shift, since PSD is the Fourier transform of correlation function, we have
SACX SCAX 2p 1 Z
1 1
Z
1 1
SnnKeiKXse iXXdXdK:
31
Reversing the order of integration and using the convergence concept of a generalized function we have the inner integral
eiKs
Z
1 1
SnnKeiK XXdX 2pdK XeiKs: 32 Here the following integral is used [61]
Z
1 1
eiZXdX 2pdZ: 33
Substituting (32) into (31) we get
SACX SCAX SnnXeisX: 34
Here the property of Dirac delta function (Lighthill 1958)
Z
1 1
f tdt t0dt f t0: 35 is used in the derivation of (34).
In summary, several used assumptions related to pavement surface are homogeneity, ergodicity and normal distribution. These have been supported by a number of measurements of pavement surface. However, there are still counterexamples demonstrating the existence of pavement surfaces that do not satisfy the homogeneous and ergodic properties [62]. In those situations, pavement surfaces are perceived by a vehicle as nonstationary stochastic processes [47, 63, 64]. It is worthwhile noting that nowadays with technology advancement, one can measure and obtain entire two-dimensional topology of pavement surface using remote sensing, line scanning laser, and synthetic aperture radar. For more accurate applications, it is no longer necessary to use hypothetical random eld model of pavement surface.
3 Contact force on vehiclepavement interface
Wheel loads in specications of highway and airport pavement design are presently treated as a static load pressure with uniform distribution [65]
Pr P0=pr20 for r
j j r0; 36
RnnX se iXXdX: 29
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An overview of a unied theory of dynamics 143
where r0 is the radium of circle distribution of wheel loads and P0 represents the total loads applied. Such a simplication of actual vehicle/aircraft load in design can be adequate when the speed of travel is low and pavement surface is smooth. However, when the speed of travel is high and pavement surface is uneven due to deterioration, inuence of dynamic load generated by vehiclepavement interaction can become very signicant. As a matter of fact, pavement damage is directly resulting from a long-term effect of dynamic trafc loading. Therefore, contact force on vehiclepavement interface needs to be considered in pavement design to more closely reect actual loading condition.
All pavement surfaces exhibit irregularities that cause vehicle vibration, which, in turn, results in moving stochastic contact forces on pavement [66]. At low speed of travel vehicle vibration is insignicant but at high speed of travel vehicle may vibrate signicantly due to poor pavement surface conditions. It does not only reduce the ride quality but also generates additional damage to vehicle and pavement structures beyond static load.
The interaction between vehicles and pavements has been investigated extensively by the automobile industry for improving ride quality and reducing the mechanical fatigue of vehicle [5, 25, 50, 6769]. Hedric [9], Abbo et al. [10], and Markow et al. [70] identied some critical factors that affect dynamic loads on pavement. These factors include vehicle and axle conguration, vehicle load, suspension characteristics (stiffness, damping), speed of travel, pavement roughness, faults, joint spacing and slab warping. From the perspective of pavement design and maintenance, however, little has been known about the effects of vehicle vibration on response, damage and performance of pavement structures [20, 54]. Although some efforts have been devoted to the measurement and prediction of dynamic wheel loads [4, 6, 11, 12, 71, 72], few of them provides a complete theoretical foundation for describing contact force induced by vehiclepavement interaction. This can be a crucial element leading to a more precise dynamic analysis of pavement structures [54, 73, 74].
Of three approaches for studying vibration-generated contact forces, namely, analytic, experimental, and numerical simulation methods, only the last two have been used widely. Experimental method offers real results, it is however very costly and limited by safety requirements. Moreover, results from experimental method only suits in some degree for specic test conditions (e.g., vehicle and pavement types, speed of travel) [54]. In many circumstances, numerical simulation associated with a limited eld tests serves as the most prevailing approach.
Numerical simulation has the advantage of capable of extrapolating experimental results over a range of test conditions where the experiment would be too dangerous
or too expensive. To study the contact forces between vehicle and pavement using numerical simulation, a vehicle must be simplied to a vehicle model so as to simulate the real operation conditions of the vehicle. Then, based on the vehicle model associated with measured or simulated pavement surface roughness, dynamic contact forces can be generated using a validated vehicle simulation program.
3.1 Pavement surface as a source of excitation to vehicle
As far as the vehicle vibration is concerned, pavement surface serves as a source of excitation when the vehicle travels along the road. As such, spatial uctuation of pavement surface gets converted to temporal random excitation of vehicle suspension system. In this article, we assume that the speed of travel is constant. Different stochastic process of excitation can be resulting from different hypotheses on two-dimensional random eld of pavement surface, which are present in this section.
3.1.1 General two-dimensional random eld
Let x x; y be the xed coordinates in which the spatial
random eld is dened. Let x0 x0; y0 be the moving coor
dinates connected with the moving vehicle and travel at the constant speed v (vector quantity) with respect to the xed coordinates x. Denote Rnx1; x2 the correlation function
that describes the spatial random eld nx. Denote
Rgx01; x02; t1; t2 the correlation function that describes the
temporal random excitation gx0; t. Sincegx0; t nx0 vt; 37 we have
Rgx01; x02; t1; t2 Rnx01 vt1; x02 vt2: 38 From the assumption mentioned above, nx is a
homogeneous random eld. Accordingly, its correlation function Rnx1; x2 becomes RnX whereX x2 x1.
Since vehicle velocity is a constant vector, the homogenous random eld gx0; t in spatial domain is converted to a
stationary stochastic process in time domain. In other words, (38) can be replaced by
RgX0; s RnX0 vs; 39 where X0 x02 x01 and s t2 t1. Now we dene SnX
as the spatial PSD of two-dimensional random eld, i.e.,
SnX 2p 2 ZR2 RnXe iXXdX: 40
The PSD of random excitation gx0; t can be expressed
as
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J. Mod. Transport. (2013) 21(3):135162
144 L. Sun
SgX; x SnXdx Xv: 41Equation (41) can be proved as follows. Since
RgX0; s Z
1 1
ZR2 SgX; xeiXX
0xsdXdx: 42 Applying Eqs. (41) to (42) gives
RgX0; s ZR2 SgXeiXx
0
Figure 8 shows a plan view of a vehicle with N tires. We now consider the cross-PSD of between i th and j th tires. It is clear that if i th and j th tires are located in the same side of the vehicle, either in the left side or in the right side, then we should have X0 = Xij and Y = 0. If i th tire is located in right side and j th tire is located in left side, or vise verse, then we should have X0 = Xij and Y = 0.
Given Fig. 7 condition, (48) can be written in terms of spatial PSD roughness:
where Xij xj xi i; j 1; 2; . . .; N. As for direct-PSD
excitation, we have Xij 0i j and Y0 = 0. From Eq.
(50) we get
SgigiXii; x
1 v
Z
1 1
dx Xveixsdx dX RnX0 vs: 43
SgigjXij; x
50
R
1
1 v 1 Snxv ; KeiKY0dK eixXij=v if i and j are located in the different side
R
1
1 v 1 Snxv ; KdK eixXij=v if i and j are located in the same side
This is the same as Eq. (39). It is necessary to represent the PSD of random excitation gx0; t in terms of angular
frequency x and spatial lag X0, i.e.,
SgX0; x
1
2p
Z
1 1
dK
; i 1; 2; . . .; N:
Z
1 1
Sn x
v ; K
RnX0 vse ixsds: 44 If the direction of vehicle velocity vector is the same as that of the x-axis, then the following transformation applies:
X0 vs X; Y0 Y and v v
j j: 45 Substituting (45) into (44), (9) is converted to
SgX0; x
1
2pv
RnXe ixX=vdXeixX0=v; 46 where X X; Y and X0 X0; Y0. Corresponding to
(40), spatial correlation function RnX is the inverse
Fourier transform of the spatial PSD SnX, i.e.,
RnX ZR2 SnXeiXXdX; 47
where X X; K. Combining Eqs. (47) and (46), we
obtain
SgX0; x
1 v
51
Since the correlation function RnX and PSD function
SnX of the two-dimensional random eld are both known
from (40) and (47), there is no obstacle to compute the excitation PSD of (40) and (41).
3.1.2 Two-dimensional random eld under hypothesis of isotropy
When pavement surface is treated as a two-dimensional random eld under hypothesis of isotropy [56, 58], its two-dimensional PSD can be represented in terms of one-dimensional PSD. In other words, in light of (26) and the denition of two-dimensional PSD (40), the two-dimensional PSD can be constructed from the following Fourier transform:
SnX 2p 2 ZR2 Rnn
Z
1 1
e iXXKYdXdY:
52
eixX0=v: 48 In the derivation of (13), the following equation is used.
Z
1 1
Z
1 1
Sn x
v ; K
eiKYdK
p
X2 Y2
Notice that
Z
1 1
49
Equation (48) is a general result applicable to any form of two-dimensional random eld. Under the restriction of the specied vehicle systems, we can represent (48) furthermore.
e iXxv XdX 2pd
xv X
:
SnXeiKY0dK
2p 2 Z
1 1
ZR2 Rnn
X2 Y2
p
e iXXKYeiKY0dXdYdK
2p 1 Z
1 1
Rnn
X2 Y20
q e iXXdX 53
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 145
N j N/2+1 left-hand side
Direction of travel
Y
0
1 i N/2 right-hand side
ij
X
Fig. 8 A plane view of an N-tire vehicle model
SgigjXij; x
Here, the general integration [61]
Z
1 1
and (30b), respectively. Without any difculty, it is straightforward to see that (40) and (41) can be further written as
and
SgigiXii;x
1 v
e iKY Y0dK 2pdY Y0 54
58
1 v Sffxv Y20Sccxv eixXij=v if i and j are located in the same side
1 v Sffxv Y20Sccxv eixXij=v if i and j are located in the different side
and the property of Dirac delta function [75]
Z
1 1
f xdx x0dx f x0 55 are used in the derivation of Eq. (53).
Comparing (53) with (27), it is clear that the left side of (53) is just the PSD given by SACX. Since it has been
proved that SACX gXSnnX where gX is an
ordinary coherence function and SnnX is one-dimensional
PSD roughness. Hence, (40) and (41) can be further expressed as
SgigjXij; x
1v Snnxv eixX
ij=v if i and j are located in the same side
1v gxvSnnxveixX
ij=v if i and j are located in the different side
( i 1;2;...;N:
59
3.1.4 Two-dimensional random eld under hypothesis of shift
When pavement surface is constructed from a two-dimensional random eld under hypothesis of shift, (40) and (41) can be written as
SgigjXij; x
1v Snnxv eixX
Sffxv Y20Sccxv if i is located in the right side
1 v
Sffxv Y20Sccxv if i is located in the left side
(
56
(
ij=v if i and j are located in the same side
1v SnnxveixX
ij
and
Sgigi Xii; x
s=v if i and j are located in the different side
60a
1v Snn
x v
;
and
SgigiXii; x
1v Snnxv if i is located in the right side
1v Snnxveixs=v if i is located in the left side
i 1; 2; . . .; N: 57
3.1.3 Two-dimensional random eld under hypothesis of uncorrelation
When pavement surface is treated as a two-dimensional random eld under hypothesis of uncorrelation [55], the integration of the left side of (53) becomes either SABX if i and j are located on the same side of the
vehicle, or SACX if i and j are located on the different
side of the vehicle, which are further given by (30a)
( i 1; 2; . . .; N:
60b
3.2 Vehicle models
Here, we consider vehicle models for simulating dynamic contact forces. Three kinds of vehicle models are commonly used nowadays, i.e., quarter-vehicle, half-vehicle,
123
J. Mod. Transport. (2013) 21(3):135162
146 L. Sun
where g and _
g are, respectively, the pavement surface elevation vector and its derivative process. Clearly, here we assume that the contact between tires and pavement surface is in the form of point distribution.
Without loss of generality, let components of the vehicle system directly contacting with ground in the moving coordinates x0 be numbered from 1 to N (see Fig. 7), where
N represents the total number of tires. Let
Zi Yi gi for i 1; 2; . . .; N
Zi Yi for i N 1; N 2; . . .; M
and full-vehicle models. Making a reasonable choice between a complex vehicle model and a simple vehicle model relies on the nature of the problem. The advantage of the complex vehicle model is that it can accurately predict the acceleration response at different locations of the vehicle. The complex vehicle model is thus suitable for studying inuences of vehicle vibrations on human body and the fatigue life of vehicles caused by vehicle vibrations. However, the complex vehicle model is highly sophisticated and requires detailed input and long execution times even for simple problems. Because vehicle sizes and loads vary greatly, it is more difcult to select representative parameter values for the complex vehicle model than for the simple vehicle model, which increases the difculty to compare computer simulation results conducted by different research groups. This should be always kept in mind.
In Sect. 3.1, pavement surface is converted into a stationary stochastic process as input to excite vehicle vibration when the vehicle travels at a constant speed. To proceed further, we also assume that vehicles suspension system is linear, under which the equations of motion of vehicle suspension systems for small oscillations can be derived using the Lagrange equations.
Let Y be a set of M independent generalized coordinates (e.g., the absolute displacement of components of vehicle systems) that completely specify the conguration of the system measured from the equilibrium position. Then, the kinetic energy T, potential energy U, and dissipative energy D, can be expressed as
T 12
_
64
Equation (63) can be rewrote as
M Z
f g C Z
f g K Z
f g Cf _
g
f g Kf g
f g F t
f g;
65
where M, K and C are mass, stiffness, and viscous damping matrices with respect to relative displacement
Z fZ1; Z2; . . .; ZMgT, respectively. Since the total number
of degree of freedom is assumed to be M, and g is N-dimensional temporal excitation vector representing pavement surface-induced displacements in coordinates x0, M, K; and C are M by M matrices, Cf ; and Kf are M by
N matrices, and force vector Ftis an M by 1 matrix.
Equations (63) and (64) represent a linear mathematical model of vehicle systems. Taking Fourier transform to both sides of Eq. (65), we have
x2M ixC K M Mf
gM 1
ixCf Kf M NfgN 1; 66
in which and are, respectively, the Fourier transform of Z and g, i.e.,
x
1
2p
YTMabs _
Y; 61a
U 12
YTKabs _
Y; 61b
D 12
_
Z
1 1
Z t
e ixtdt; 67a
x
YTCabs _
Y; 61c
where Mabs, Kabs and Cabs are mass, stiffness, and viscous
damping matrices with respect to absolute displacement vector Y fY1; Y2; . . .; YMgT, respectively. The Lagarange
equations of motion are
d dt
oT o _
_
1
2p
Z
1 1
g t
!
oD o _
Yj
oUoYj 0v v
j j j 1; 2; . . .; M:
62
Under the assumption that all components of the vehicle system be linear, (61) and (62) yield a set of simultaneous second-order differential equations with constant coefcients:
Mabs Y
n o
Yj
e ixtdt: 67b Multiplying the inverse matrix of x2M ixC K ,
it is straightforward to see
f
g x2M ixC K 1ixCf Kf fg: 68
The primary aim of establishing vehicle models is to nd the scalar response of Zt as well as
x. This can
be accomplished by using stochastic process theory of linear systems. The dynamic characteristics of the vehicle at angular frequency x are dened by M 9 N frequency response function (FRF) matrix, H(x), of the vehicle system, where H(x) = [H(x)ij], i 1; 2; . . .; M for the M
outputs and j = 1,2,.,N for the N inputs of pavement excitation to tires. These functions are dened as follows.
If a vertical displacement gjt eixt is applied to the
vehicle at jth tire, with all the other inputs being zero, then
Cabs Y
n o
Kabs Y
n o
Cabsf _
g
f g Kabsf g
f g Fabs t
;
63
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 147
the response of the vehicle is given by Ztij Hxijeixt. Based on Eq. (68), it is straightforward to see that FRF matrix can be given by
Hx x2M ixC K 1M MixCf Kf M N: 69
Let
A AR iAI x2M ixC K 1 70a
and
Q QR iQI ixCf Kf ; 70b
where AR and AI; respectively, represent real and imaginary parts of matrix A, QR and QI; respectively, represent real and imaginary parts of matrix Q. It is convenient to express FRF matrix in terms of real part HR and imaginary part HI
H x
HR iHI; 71 where
HR AR QR AI QI;
HI AI QR AR QI:
From the stochastic process theory, PSD response of a linear system and PSD excitation satisfy the following relationship [76]:
SZXij; x H xSgXij; xHTx
M M M N N N N M:
72a
or equivalently,
SZiZjXij; x X
N
l1
XNk1H ikxHjlxSgkglXkl; x; 72b where SZXij; x SZiZjXij; x M M,SgXij; x Sgigj Xij; x N N, H x and HTx are, respectively, the
conjugate matrix and the transposed matrix of FRF matrix Hx. It may be noticed that H x and HTx can be,
respectively, expressed in the form of real and imaginary parts by means of Eqs. (72a) and (72b)
H x HR iHI; 73a
HT x HRT iHIT: 73b
Similarly, we may write SZXij; x as
SZXij; x SRZXij; x iSIZXij; x 74 in which SRZXij; x and SIZXij; x are, respectively, the
real and imaginary parts of response spectral density SZx. If the pavement excitation spectral density Sgx is
a real spectrum matrix, then by expanding (72b) using (73a) and (73b), we have
SRZXij;x HR SgXij;x HRT HI SgXij;x HIT: 75a
and
SIZXij;x HR SgXij;x HIT HI SgXij;x HRT: 75b
If the pavement excitation spectral density Sgx is a
complex spectrum matrix, say
SgXij; x SRgXij; x iSIgXij; x 76 then by expanding Eqs. (72b) using (73a), (73b), and (76), we have
SRZXij;x HR SRgXij;x HI SIgXij;x HRT;
HR SIgXij;x HI SRgXij;x HIT; 77a
and
SIZXij;x HR SIgXij;x HI SRgXij;x HRT HR SIgXij;x HI SRgXij;x HIT: 77b
3.3 Dynamic load
3.3.1 Time-domain analysis
Figure 9 shows a sketch of the ith tire contacted with rough pavement surface. From Newtons second law of motion it is direct to know that the contact force between ith tire and pavement, Pit, can be given by
Pit kiZit ci
_
Zit; i 1; 2; ::; N; 78 where ki and ci, respectively, the ith tire spring stiffness and viscous damping, both being constant. According to stochastic process theory, if the input of a linear time-invariable system is a stationary random process, then the output of the system is also a stationary random process [76]. As was noted, since temporal random excitation of pavement surface Gaussian ergodic random process with zero mean, the response Zi(t) and its derivative process _
Ztt are both zero mean Gaussian stationary stochastic
processes. Taking expectation to both sides of (78), we get the mean function of dynamic contact forces, mP
mPit EPit kiEZit ciE
_
Zit 0: 79 It should be pointed out that the mean function mP(t)
here, not containing the static load of vehicle, only represents the statistical average of dynamic effect. If the static load is considered, then the complete mean function of dynamic contact forces,
Pi, is given by
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J. Mod. Transport. (2013) 21(3):135162
148 L. Sun
Pi mPit P0i mig; 80 where g is the acceleration due to gravity, P0i and mi are, respectively, the effective weight and effective mass distributed by the vehicle on ith tire. Without special statements, the derivation of formulation in the context is based on (79).
It is of interest to know the correlation function of dynamic contact forces between vehicle and pavement because this information is critical for analyzing the dynamic response of pavement structure under vehicle loads. According to the denition of correlation function and using Eq. (78), we have
RPiPjXij; s EPitPjt s kikjRZiZjXij; s cicjR
SPiPjXij; x
1
2p
Z
1 1
RPiPjXij; se ixtds; 86a
SZiZjXij; x
1
2p
Z
1 1
RZiZjXij; se ixtds; 86b
S _
Zi _
Zj
Xij; x
1
2p
Z
1 1
R _
Zi _
Zj
Xij; se ixtds; 86c
SZi _
Zj
Xij; x
1
2p
Z
1 1
RZi _
Zj
Xij; se ixtds; 86d
S _
ZiZj
Xij; x
1
2p
Z
1 1
R _
ZiZj
Xij; se ixtds: 86e Correlation function RPiPjs is the Fourier inverse
transform of PSD function SZiZjx:
RZiZjXij; x Z
1 1
_ Xij; s
kicjRZi
_ Xij; s cikjR_ZiZj
Xij; s: 81
It is known from stochastic differential principles that there exists exchangeability between expectation and square differential [31]. In other words, if a random process Z(t) is differentiable for any order, it is proved that
onm
otnosm RZZt; s RZ
nZm t; s; 82 where Zn dnZt=dtn. In addition, if Zt is a stationary
process, then (73a) becomes
1n
SZiZjXij; seixtdx: 87 Taking derivative to both sides of Eq. (87) gives [31, 182]
dndsn RZiZjXij; x in Z
1 1
onm
nZm s: 83
Applying Eq. (83) into Eq. (81) gives
RPiPjXij; s kikjRZiZjXij; s cicj
oos RZiZjXij; s cikj
xnSZiZjXij; seixtdx: 88 By combining Eqs. (83) and (88), we nd that
S _
Zi _
Zj
Xij; x x2SZiZjXij; x; 88a SZi _
Zj
Xij; x ixSZiZjXij; x; 88b S _
ZiZj
Xij; x ixSZiZjXij; x: 88c Substituting (88) into (84), we eventually get the PSD forces in terms of PSD of relative displacement response:
SPiPjXij; x kikj cicjx2 ixkicj cikj SZiZjXij; x: 89
Equation (89) applies to both direct-spectrum and cross-spectrum. In addition, as far as direct-spectrum is concerned, we can rewrite PSD forces in more concise form:
SPiPix k2i c2ix2SZiZix: 90
Standard deviation of contact forces can be expressed in terms of PSD forces. Applying Eq. (90) to the denition of standard deviation directly shows that
r2Pi Z
1 1
osnm RZZs RZ
o2os2 RZiZjXij; s
kicj
oos RZiZjXij; s:
84
It is clearly seen that autocorrelation function is obtained as i j
RPiPiXij; s k2iRZiZiXij; s c2i
o2os2 RZiZiXij; s; 85
and crosscorrelation function is obtained as i 6 j.
3.3.2 Frequency-domain analysis
Having obtained correlation function, it is not difcult to represent the power spectral density (PSD), which is dened as the Fourier transform of correlation function. From Eq. (81) we have
SPiPjXij; x kikjSZiZjXij; x cicjS
SPiPixdx Z
1 1
k2i c2ix2SZiZixdx:
91
Furthermore, if the standard deviations of displace response and velocity response, rZi and r _
Zi , are deployed,
_ Xij; x
kicjSZi
_ Xij; x cikjS_ZiZj
Xij; x; 86
Eq. (91) can be rewritten as
r2Pi k2ir2Zi c2ir2_Zi; 92
where
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 149
Direction of travel
Pavement Roughness i
k i
c )
(t
Zi Relative Displacement
Pi
(t
)
Fig. 9 A sketch of the ith tire on rough pavement surface
in which
r2Zi Z
1 1
SZiZixdx and r2_Zi Z
1 1
3.3.3 A Case Study of Walking-Beam Suspension System
As a case study, a walking-beam suspension system is shown in Fig. 10. Governing equations of this vehicles suspension system are also given by (65) in matrix form, where parameters and their values related to this walking-beam system are listed in Table 2 with m1 = 1,100 kg, m2 = 3,900 kg, I = 465 kgm2, s = 1.30 m, a1 = a2 =
0.5, k1 =k2 = 1.75 (MN/s), k3 = 1.0 (MN/s), c3 = 15.0 (kNs/m), and c1 = c2 = 2.0 (kNs/m).
M
0 0 m2 m1a2 m1a1 0 I=b I=b 0
2
64
3
75;
A two-dimensional isotropic random eld is used for numerical simulation. The one-dimensional PSD roughness proposed by the International Organization for Standardization (ISO) in the power form [49, 52]
SnnX S0X c; 94 with parameters S0 = 3.37 9 10-6 m3/cycle and c = 2.0 is adopted for numerical computation. Figure 11 shows the
PSD contact force of the right tire at the speed of 20 m/s.
In aforementioned study, the contact area between the tire and the pavement surface is assumed as a point contact. There is no difculty to extend this point contact to a distributed contact by considering the footprint as a weighted integration of contacting points [7779]. It should be noted that the effect of nonlinearity in vehicle suspension and variable speed of travel (e.g., acceleration, deceleration, etc.), and inhomogeneity in pavement surface on contact force between vehicle and pavement have not been addressed here, which have been considered in various studies [80].
4 Pavement response under moving stochastic loads
4.1 Background
A large class of time-dependent sources such as vehicle, submarines, aircraft, and explosion-induced waves belongs to moving sources. The study of response of media (pavement, runway, rail-track, bridge, air, and sea) to moving sources is named moving source problem (MSP), which is of particular interest to structural design, noise assessment, target detection, etc. [81]. A number of studies have been addressed to MSP in various elds of physics. For instance, the response of an ice plate of nite thickness caused by moving loads was discussed by Strathdee et al. [82]. Wells and Han [83] analyzed the noise generated by a moving source in a moving medium. As for the aspect of
x2SZiZixdx:
93
C
c3a2 c3a1 c3 c1 c3a2 c2 c3a1 c3 c1a2b c2a1b 0
2
64
3
75;
3
K
k3a2 k3a1 k3
k1 k3a2 k2 k3a1 k3
k1a2b k2a1b 0
2
4
5;
fFtg
0 c3a2 k3a2 0 c3a1 k3a1 m1a2 c3a2 k3a2 m1a1 c3a1 k3aa I=b 0 0 I=b 0 0
2
64
3
75
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
_
g1 _
g1 g1
g2 _
g2 g2
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
:
123
J. Mod. Transport. (2013) 21(3):135162
150 L. Sun
m2 3
y
k3 c3
y 1
y
2
m1 I
k2
c2 c1 k1
2
1
a1 s
a2
s
Fig. 10 A walking-beam suspension system
Table 2 Parameters used in the case study of walking-beam suspension system
Parameter Description
m1 Unsprung massm2 Sprung massI Moment inertia of unsprung mass
k1 Spring stiffness of right side tire k2 Spring stiffness of left side tire k3 Spring stiffness of suspensionc1 Damping of right side tirec2 Damping of left side tirec3 Damping of suspensions Effective width of the vehicle
y Absolute vertical displacement of center of unsprung mass
y1 Absolute vertical displacement of right side of unsprung mass
y2 Absolute vertical displacement of left side of unsprung mass
y3 Absolute vertical displacement of sprung mass
n1 Absolute height of pavement prole along right wheel path
n2nn Absolute height of pavement prole along left wheel path
s
elastodynamics, even more investigations are being done [8494]. One may refer to Sun [95] for detailed review of MSP.
Methods for solving MSP primarily include integral transformation, characteristic curve, and modal analysis [96100]. A common characteristic of previous studies is to utilize Galilean transform. The advantage of using Galilean transform is that the governing eld equations, usually partial differential equations, can be reconstructed in a moving coordination so that the effect of source velocity may be reected in parametric ordinary differential equations. However, since Galilean transform requires
that the source is steadily moving, the methods based on Galilean transform only apply to the steady-state solution. As for transient solution, a feasible way is to directly apply high-order integral transformation to the eld equations. Since velocity parameter is included in boundary or initial condition, there is intractable obstacle when integrating the eld equations. This can be the reason why most of the MSPs are solved only for steady-state solution.
As far as pavement response is concerned, some pioneers have dedicated their research to the subject of effects of moving loads on pavement structure. To perform the dynamic analysis of pavement structures, one of the indispensable considerations is the dynamic trafc loading, which is the excitation source of pavement structures. Since pavement loads are caused by vehicle vibration, it is necessary to include the vehicle in the investigations of pavement loads. Employing MIT heavy truck simulation program Hedric and his associates, Abbo and Markow [911], examined the inuence of joint spacing, step faulting, vehicle suspension characteristics, and vehicle velocity on pavement damage as dened by fatigue cracking in concrete slab. The response of pavement under their consideration is achieved using PMARP, a static nite element program for pavement analysis developed by the Purdue. The conclusions obtain provide a valuable approximation of pavement response.While PMARP is a static nite element program, the properties of inertia and damping of the pavement structure were not included in consideration.
An advanced pavement model called MOVE program was developed by Chen [101] and Monismith et al. [73] at UC-Berkeley. This model can take the motion of load into account by means of nite element methods and is an important dedication to pavement structural analysis. The vehicle load is assumed to be an innite line load to simplify the analysis from a three-dimensional problem to a two-dimensional problem. In addition, the model is based on the deterministic elastodynamics; therefore, neither pavement surface roughness nor vehicle suspension can be considered in the presented framework.
Since dynamic effects have been increasingly important in the prediction of pavement response, damage, and performance [14, 16, 18], it has become necessary to develop better mathematical models to account the effects of motion and uctuation of contact forces caused by various types of vehicle suspensions [73]. The author has carried out a number of studies on deterministic and stochastic MSPs over the last two decades by providing a general approach that can include surface roughness and vehicle suspensions in the response of continuum media [54, 74].
The complexity in MSP is the variable load position, which not only makes the representation of the eld
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 151
equation in time-invariant xed coordinates difcult but also increases the difculty of integration due to the appearance of speed parameter. In most of the previous MSP studies, some simple cases with deterministic conditions, for example, a constant load with uniform speed, are considered. However, the research on dynamic response of
continuum media to moving stochastic vehicle loads has not yet been studied in the literature.
4.2 Problem statement of MSP
4.2.1 Pavement as a continuum medium
Pavement structures are traditionally classied into two categories: rigid pavement and exible pavement. The former often refers to Portland cement concrete (PCC) pavement, the latter often refers to asphalt concrete (AC) pavement. This classication is not very strict because PCC pavements are not rigid, or are AC pavements exible in many situations. From the standpoint of structural nature PCC pavements behave more like a slab, while AC pavement acts more like a multi-layered system. Therefore, it is more reasonable to classify pavement on the base of mechanic properties. There have been developed many models to simulate the behavior of pavement structures. For instance, Hardy and Cebon [16, 18] simplied AC pavement into an EulerBernoulli beam. From the theoretical perspective, most of the proposed pavement models belong to continuum media [102, 183187].
14
2 )
6 sN
12
One-sided PSD loads (10
10
x
8
6
4
2
0 0 10 20 30 40 50
Frequency (Hz)
Fig. 11 PSD of contact forces against frequency (speed of travel = 20 m/s)
z
fixed coordinates follow-up coordinates
z
x
observation position
( 0
x
) real velocity vector
real source
x- equivalent
o y
observation position
imaginary velocity vector
o
y
imaginary source
x x
Fig. 12 Schematic sketch of a stationary coordinate and a moving coordinate
Fig. 13 Physical models of two general types of pavement structures
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152 L. Sun
Consider a linear medium with the region R or the boundary B being at rest initially. A load is applied when the medium is at rest and moves according to a given law of motion (Fig. 12). The MSP is to solve the response of the medium to the moving load. The governing equation of such a system thus belongs to linear partial differential equation. According to the theory of linear equation, the solution of the equation can be constructed by the integration of the fundamental solution of the equation or the so-called Greens function. It is usually described as the superposition principle or, equivalently, the reciprocal principle [103].
Besides the above-mentioned description of the continuum media, the following assumptions are made here. One assumption is that, comparing to the mass of vehicle, the mass of pavement structures (including surface layer, base, subgrade, and soil foundation) is large enough such that pavement vibration is much smaller than vehicle vibration. Another assumption is that wave velocity excited by a dynamic load in pavement structure is much faster than vehicles speed of travel. With these two assumptions, the couple effect of vehiclepavement interaction can be negligible.
4.2.2 Moving stochastic vehicle loads
Contact forces applied on pavements by vehicles are moving stochastic loads. The statistical characteristics of the dynamic contact forces have been discussed in detail previously. Dynamic contact forces follow with two aspects of meanings. One is that the location of the force is changing continuously with traveling of vehicle, and another is that the amplitude of the force is varying due to vehicle vibrations.
Without any loss of generality, a vehicle is assumed to be a linear system traveling along the x-axis at a constant speed. The contact forces between the vehicle and pavement can be given by a concentrated moving load:
Fx; t Pstotdx vtdy 95
where t and v are time variable and vehicle velocity, respectively, d() is the Dirac-delta function, which is dened by
dx x0
0; for x 6 x0 1; for x x0
Here P0, representing the static load applied at a tire when the vehicle is at rest, is a constant quantity, and P(t)
represents the dynamic portion of the stationary stochastic contact forces with a zero mean. Its correlation function, power spectral density and standard deviation are, respectively, denoted as RPPs, SPPx, and rP.
4.3 Representation theory of MSP
It is convenient to assume a three-dimensional conguration with observation variable x = (x,y,z), source variable n n; g; f, and time t 0. Suppose a linear differential
operator O describes the dynamic property of a physical system and appropriate interface and boundary conditions relate the eld quantities of specied problems. Obviously, for a concrete elastodynamic problem, the linear differential operator O is given by the well-known NavierStokes eld equations. The Greens function is then dened as the fundamental solution of the system. In other words, for the problem discussed here, the Greens function corresponds to the solution of the governing equations as the point source takes the form of a Dirac delta function in both spatial and temporal domains.
4.3.1 General Formulation
Without loss of generality, vanishing initial conditions are assumed here. According to the causality of a physical system, the Greens function Gx; t 0 for t\0. We may
then write
OGx n; t s dx ndt s: 98
Here, the initiation of the source is delayed by s. The causality of a physical system requires that for Greens function t s: Since the initial condition of the linear
medium is zero, the dynamic response could be expressed as
ux; t Z
t
1
and Z
1 1
f xdx x0dx
f x0: 96
In addition, Pstot, a function of time, represents
amplitudes of stochastic contact forces and can be expressed as two terms below:
Pstot P0 Pt: 97
ZS Fn; sGx; t s; ndnds; 99 where x x; y; z, n n; g; f, v vx; vy; vz,
dn dndgdf, and S is the region R or the boundary B. It is
know that the Greens function Gx; t; n corresponds to
the solution of the equation when a unit point impulse is applied at position n. Assume that the medium is innite in those dimensions of interest. It is known that the governing equations are linear. According to the reciprocal principle, the response of media at eld point x in the xed coordinates when the source lies at n in the xed coordinates is equal to the response of media at eld point x n in the
xed coordinates when the source lies at 0 in the same coordinate.
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 153
Dene the impulse response function (IRF) hx; t as the
solution of
Ohx; t s dxdt s: 100 According to the above mentioned analysis, we have
hx n; t Gx; t; n Gx n; t; 0: 101Substituting (101) into (99), we get
ux; t Z
t
1
quantity. Actually, this idea is straightforward demonstrated by putting the velocity variable v = 0 into (105). The result gives that
ux; t P0 Z
1
ZS Fn; shx n; t sdnds: 102 Furthermore, if the concrete form Fn; s in (103) is
considered, we may rewrite (102) as
Ux; t Z
t
1
Pstoshx vs; t sds: 103 in which vs vxs; vys; vzs. Also, the property of Dirac
delta function (102) is used here. If the transformation h
t s is used, then (103) can be expressed as
ux; t Z
1
0 hx; hdh: 106
Clearly, the response is a constant without depending on the time variable.
4.3.3 Stochastic analysis for a moving stochastic vehicle load
In this section, a stochastic moving source is analyzed. In the derivation of the GDI in (103), we require no special assumptions on Pstot. Therefore, if Pstot is a stochastic
process, and (103) becomes an integral in the sense of Stieltjes integration. Meanwhile, the response ux; t
becomes a stochastic process. We now consider the response of media to Ptdx vtdy.
As mentioned before, P(t) is a zero mean stationary process with autocorrelation function RPPs, PSD SPPx and
standard deviation rP.Taking the expectation of both sides of (103) and using the exchangeability of expectation and integration, we obtain the mean function of the response, i.e.,
ux; t Z
1
0 Pstot hhx vt vh; hdh: 104 Equations (103) and (104) are general results for MSP and are named generalized Duhamels integral (GDI). From the point of view of time history, we might regard a moving load as a series of impact on continuum media during a number of tiny time intervals. The integration of the response of the medium excited by each impulse is thus equal to the cumulative effect of the moving load. Although solving for the Greens function is still a nontrivial task, convolution (103) does provide a sound theoretical representation, which can be very powerful when combining with numerical computation such as nite element method.
4.3.2 Deterministic analysis for a moving constant load
The solution of the problem described here can be constituted using the solutions of two individual problems because of the superposition principle of the solution of linear equations. One problem deals with the deterministic response of the medium under the moving constant load P0dx vtdy. The other
problem deals with the random response of the medium under the moving stochastic load Ptdx vtdy. In this section,
the rst problem is analyzed. In the next section, the second problem is analyzed. The summary of the solution is provided in the sections followed. If the load is a constant with amplitude P0, the response are given by
ux; t P0 Z
1
0 EPt hhx vt vh; y; z; h dh: 107
It is not difcult to obtain the spatial-time correlation functions for the response, i.e.,
Rux1; x2; t1; t2
Z
t2
1
Z
t1
1
EPs1Ps2hx1 vs1; y1; z1; t1 s1 hx2 vs2; y2; z2; t2 s2 ds1ds2; 108
where Rux1; x2; t1; t2 is the correlation function of
response ux; t. Let x1 x2 x; then we obtain the
time autocorrelation function
Rux; t1; t2 Z
Z
1
1 0 EPt1 h1Pt2 h2
hx vt1 vh1; y; z; h1
hx vt2 vh2; y; z; h2 dh1dh2: 109
where hj tj sjj 1; 2. By substituting t1 t2 t
into (108) and (109), it is straightforward to nd second moment functions, i.e., the mean square functions of the random response.
It has been known that for a linear system with a stationary stochastic excitation at a xed position, the response of that system is still a stationary stochastic process [76]. However, this conclusion only applies to the xed source problem. For a linear system with a moving stochastic source, the random response of that system is a nonstationary stochastic process
0
0 hx vt vh; y; z; hdh: 105
It is obvious that the response is no longer a constant independent on the time t. If the source is a xed load, according to our example, the solution should be a static
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154 L. Sun
and
Rux vt x0; s Z
even if the random excitation Pt is a stationary stochastic
process [54, 74]. To show this, we inspect (107) and (109), which are apparently not stationary because time variable t is contained in the kernel function of the solution. In other words, in a situation where a source is moving with respect to a receiver a nonstationary signal will be recorded at the observer position, even when the source produces a stationary output. This effect is known as the Doppler shift. It is also useful to realize that although the system is a linear system, it essentially becomes a time varying system when a moving source is applied.
In many circumstances, there may be a demand to push the analysis further into the frequency domain. For instance, response information of amplitude distributions and frequency components of media to moving vehicle loads is needed for vehicle optimum control and pavement performance prediction. To fulll this purpose, it is necessary to use spectral analysis techniques to obtain the required information. As such, one may encounter difculties when performing Fourier spectral analysis technique because this technique has been devised primarily for stationary signals. Although some variations of Fourier spectral analysis technique have been introduced to dealing with nonstationary stochastic processes, Fourier spectral analysis are not ideal tools for nonstationary signals induced by MSP. This shortcoming has led Sun [104]to develop the so-called follow-up spectral analysis, by which the commonly used spectral analysis technique is still applicable with sound theoretical foundation.
Let coordinates oxyz and o0x0y0z0 be, respectively, xed coordinates and follow-up coordinates moving with the moving source. The relationship between the two coordinates is
x0 x vt: 110
Therefore, a moving source x0 vt in the xed coordinate
oxyz becomes a xed source x0 x0 in the follow-up
coordinate o0x0y0z0. Here x0 is a constant vector. Now consider the response of the medium at a moving eld point x vt x0 in the xed coordinates. Utilizing (104), we have ux vt x0; t Z
1
0 Pstot hhx x0 vh; hdh:
Z
1
1 0 RPPs h1 h2hx x0 vh1;h1 hx x0 vh2; h2dh1dh2: 113
According to the denition of a stationary stochastic process [31], (112), and (113) indicate that the response at a moving eld point x vt x0 in the xed coordinates
becomes a stationary process. It is evident that the moving eld point x vt x0 in the xed coordinates becomes a
xed eld point x x0 in the follow-up coordinates as
illustrated in Fig. 12. To show this, we replace x in (110) by x vt x0x0 x x0: 114
In other words, the response of a xed position x ? x0 in the follow-up coordinates to a moving stationary stochastic load possesses the stationary property. Thus, Fourier spectral analysis technique is still applicable here. It should be pointed out that the explanation of the stationary process ux vt x0; t is essentially different
from the commonly described stationary process. In general, the mean function of a commonly described stationary process refers to the time average of the random process, while the mean in (112) is indeed interpreted as a spatial average of the random response. The same explanation applies to the autocorrelation function shown in (113).
In light of the aforementioned explanation, there is no difculty to dene PSD in the follow-up coordinates. The following spectral analysis is performed in the follow-up coordinates. We may rewrite the autocorrelation function in the follow-up coordinates as
Rux0; s Z
0
Z
1
1 0 RPPs h1 h2hx x0 vh1;h1 hx x0 vh2; h2dh1dh2; 115
where x0 is expressed by (114). Let s 0 and put it into Eq.
(115), we obtain the mean square function of the random response in the follow-up coordinate
w2ux0; t Z
0
111
The mean function and autocorrelation function of the response described by (111) are, respectively, given by
Eux vt x0; t Z
1
Z
1
0 EPstot hhx x0 vh; h dh
EPstot h Z
1
0 hx x0 vh; hdh
P0 Z
1 0 w2ph2 h1hx x0 vh1; h1
hx x0 vh2; h2dh1dh2: 116
In addition, since Pt is a zero mean stationary process,
we have
r2P VarPt w2P RPP0 const; 117 where rp and VarPt are, respectively, the standard
deviation and variance of Pt. So we can rewrite (117) as
0
1
0 hx x0 vh; hdh: 112
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 155
2RPP0;
118
where ru and w2u are the standard deviation and variance of the response eld, respectively. Dene the relationship between the frequency response function and the impulse unit response function in the follow-up coordinates as
Hx0; x Z
1
r2ux0; t w2ux0; t Z
1
0 hx x0 vh; hdh
0 hx x0 vh; he ixhdh: 119
According to Wiener-Khintchine theory, the PSD and the autocorrelation function form a Fourier transform pair. Taking Fourier transform to both sides of (115) and noticing the (119), we obtain the expression of PSD in followup coordinates, i.e.,
Sux0; x; v Hx0; x
j j2SPPx: 120
Similarly, an expression for the time autocorrelation function can be obtained by taking the Fourier inverse transform of (120)
Rux0; s 2p 1 Z
1 1
Hx0; x
j j2SPPxeixsdx: 121 Hence, the mean square function is also given by
w2ux0 Rux0; s 2p 1 Z
1 1
Hx0; x
j j2SPPxdx:
122
4.4 Pavement Models
There are mainly two types of pavement structures: Port-lant cement concrete pavement and asphalt concrete pavement (Fig. 13). These pavement structures can be modeled by a beam, a slab, a layered medium on a half-space or rigid bedrock. A number of studies have been carried out lately by Sun and his associates using analytic method and analyticnumerical method [105163]. Beskou and Theodorakopoulos [164] provided a recent review on numerical methods for studying dynamic effects of moving loads on road pavements. Sun and Greenberg [74], Sun and Luo [125128], Sun et al. [153], Luo et al. [165] and Sun et al. [123] provide concrete examples of pavement models subject to moving loads.
The response of pavement systems under dynamic loads may be expressed in partial-differential equations. A generic description of the governing equations is:uux; t; h Px; t; 123 where u is a partial-differential operator, x x; y; z a
spatial vector in Cartesian coordinates, t is the time variable, Px; t is the applied dynamic load (i.e., the input),
which can be recorded by data acquisition system during laboratory experiments and eld tests, ux; t; h the pave
ment response vector (i.e., the output in the form of displacements, stresses, and strains), and h h1; h2; . . .; hn
is the parameter vector to be identied.
A pavement structure usually consists of a surface course, base courses and subgrade. Within each course, there may be several sub-layers made up of different materials. A two-dimensional Kirchhoff thin slab resting on a Winkler foundation is the common model for PCC pavements. The operator u for a Kirchhoff thin slab is u Dr2r2 K Co=ot qho2=ot2; 124
where D Eh3=121 l2 , and q, l and h are the density,
Poissons ratio and thickness of the slab, E is the Youngs elastic modulus, K is the modulus of subgrade reaction, C is the radiation damping coefcient, and r2 o2=ox2 o2=oy2 is the Laplace operator. The parameter vector is h E; h; l; K; C; q. For a multilayered exible
pavement system, the governing equation is controlled by a three-dimensional NavierStokess equation for each layer
G r2u k G rr u qf qo2u=ot2; 125 where r o=ox o=oy o=oz, f the body force vector,
the Laplace operator r2 o2=ox2 o2=oy2 o2=oz2, G G1 igd, k k1 igd in which i
1
p , gd is
the hysteretic damping coefcient, Lamb constants k and G the bulk modulus and shear modulus, respectively, k and G the complex counterparts of k and G, respectively. The subgrade may be articially divided into a number of thin layers. Within each layer the soil is characterized to be isotropic, homogenous and have the same structural and material properties, while these properties vary for different layers. Furthermore, physical nonlinearity may possibly be presented in asphalt surface layer and soil subgrade using nonlinear constitutive models involving viscoelasticity-viscoplasticity. Eq. (125) adopts the simplest model to account for viscoelasticity. A more generic model is the generalized viscoelastic model, which includes the Burgers model, Maxwell model and Kelvin model as its special cases. Fig. 14 presents a schematic plot of a list of viscoelastic models.
To solve a viscoelastic problem, elastic solutions is sought rst and then the correspondence principle is applied to convert the elastic solution into a viscoelastic solution. Two elastic/viscoelastic subgrade models will be studied: a half-space and a layer resting on bedrock. Clearly, the parameter vector h E; h; k; G; q varies
from layer to layer. When viscoelasticity is considered, more parameters will appear in the parameter vector. In principle, the adoption of a generalized viscoelastic model in forward dynamic analysis introduces no signicant
123
J. Mod. Transport. (2013) 21(3):135162
156 L. Sun
Fig. 14 A schematic plot of different viscoelasic models
difculty. The number of Kelvin components in this model can be estimated through a thorough investigation. With properly specied initial and boundary conditions, Eqs. (123)(125) constitute a complete mathematical description of the forward dynamic problem. Forward analysis aims to solve for the response ux; t; h provided that the
excitation Px; t and the parameter vector h are known.
These mathematical, physical models describe the behavior of different types of pavement systems and they will be studied in great depth.
Equation (123) belongs to a wave equation from a mathematical physics point of view. Its solution can be obtained in the form of a LebesgueStieltjes integral using proper integral transformation, depending upon the nature of the problem (e.g., steady-state vs. transient) and upon how the problem is formulated (e.g., in Cartesian or cylindrical coordinates). Let the Greens function (the fundamental solution) of Eq. (1) be Gx; t; h u 1dx; t , in which d
is the Dirac-delta function and u 1 denotes the inverse
operator of u . Let pavements be at rest prior to the NDE
test, leading to a vanishing initial condition. The solution of Eq. (123) under loading condition Px; t can be constructed
as
ux; t; h u 1Px; t ZS Gx; t; hdS; 126
where S is the region where Px; t is dened. Equation
(126) can also be equivalently represented in the transformed domain
n; x; h Tfux; t; h g Z
n; x; hd
; 127
where Tf g is a transformation operator,n; x; h and
n; x; h are the response and the Greens function in the
transformed domain, respectively, is the region in the transformed domain where the transformed dynamic load ~Pn; x is dened; n n; g; f and x are the counterpart
of spatial vector x x; y; z and time variable t.
The Thomas-Haskell method relates a transformed response at the bottom of a layer, in the form of a transfer matrix, to a corresponding quantity at the top of a lower layer. For the last half century, this method has served as the
cornerstone for numerous studies in multilayered elastic analysis. Another benchmark proposed by Kausel and Roesset contributes an alternative, in which the dynamic stiffness matrix is expanded in terms of wave number and approximated by taking terms only up to the second order of the wave number. The Thomas-Haskell method is more accurate than the Kausel-Roesset method but demands more computational effort. Forward dynamic analysis for multilayered viscoelastic media will not add signicant difculty because both methods are still applicable, though it is a demanding task. Since each method has its own strengths and weaknesses, both methods will be adopted in the project to tackle wave propagation through multilayered visco-elastic media. Equations (126) and (127) are amenable to numerical evaluation of the dynamic response in the time space domain and in the transformed domain, respectively, provided that Px; t and h are known. Without doubt, the
computation here involves intensive numerical evaluation of multifold integration of complex functions with unstable characteristics in time and space.
5 Discussion and Future Research
In this section, discussion and future research of the unied theory of dynamics of vehiclepavement interaction under moving and stochastic load is carried out from the following aspects: (a) nature of the problem, (b) modeling,(c) methodology, and (d) further extension and engineering application.(a) Nature of the problem. The setting of vehicle pavement interaction can be categorized either in a deterministic framework or a stochastic framework. Stochasticity may be presented in speed, magnitude and position of the loading, structural models of vehiclepavement system and constitutive models. A stochastic framework provides a more realistic setting but exhibit more complexity. The setting of vehiclepavement interaction can also be categorized either as a steady-state problem or as a transient problem. Except very few studies [104, 145], almost all existing literatures belong to a steady-state problem described in a deterministic framework.
123 J. Mod. Transport. (2013) 21(3):135162
An overview of a unied theory of dynamics 157
(b) Modeling. The modeling of vehiclepavement interaction involves the load model, the vehicle model and the pavement model. The load model addresses spatial distribution of the load (e.g., concentrated load, distributed load, multiple load, etc.), speed of the load (e.g., constant speed, varying speed, etc.), trajectory of the load (e.g., straight line, curve, etc.), and magnitude of the load (e.g., constant load, impact load, sinusoidal load, varying load, and random load). The vehicle model addresses vehicle suspension system, which involves mass distribution, dimension, and conguration of the vehicle. Quarter-vehicle model, half-vehicle model, and whole vehicle model have all been established with increasing complexity involving different number of spring and dashpot elements. Regardless of how many spring and dashpot elements are being used in the built vehicle model, almost all the literature studies tend to use linear suspension system due to its ease of computation when integrated with a pavement model. The pavement model addresses the simplication of structural system of pavement using beam model, slab model, and layered medium on a half space or bedrock. In terms of constitutive models of the material, almost all studies only considered linear elastic and linear viscoelastic materials when tackling dynamics of vehiclepavement interaction because of the complexity of the problem. Asphalt concrete pavement actually exhibits complex viscoelasticviscoplastic-damage properties [166172], which should be integrated into the subject. Also, it is rare to consider the coupling effect of vehiclepavement interaction, while this is not the case in trainrailway interaction because of the mechanism of the interaction and the relative mass difference between vehicles (e.g., car, truck, train, and airplane) and transportation infrastructure (e.g., highway, bridge, and railway). None of the studies has addressed vehiclepavement interaction using deteriorated pavement models as well as long-term pavement damage and failure due to vehiclepavement interaction.(c) Methodology. The methods for solving vehicle pavement interaction problem can be classied into analytic approach and numerical application, though the implementation of analytic approach still requires numerical computation. For vehicle dynamics, equation of motion is rst established as a set of linear differential equation system and solved in the frequency domain using frequency response function or in the time domain using numerical sequential integration. For pavement dynamics, analytic approach makes use of integral transform to treat wave propagation in continuum media, while numerical approach such as nite difference, nite element and boundary element methods makes use of discretization in time and in space. The advantage of analytic approach is that it provides insights for revealing physics of the wave propagation in continuum media and can be highly efcient
in terms of computation implementation, particularly when the spatial scale of the problem involves hundreds of kilometers (e.g., seismology) or the speed of the load is very high and close to various critical speeds of waves in media. The advantage of numerical approach is that they can deal with pavement having complex geometric structure as well as nonlinear constitutive model of the material.(d) Further extension and engineering application. The study of dynamics of vehiclepavement interaction provides a deep understanding for improving vehicle design (e.g., road-friendly vehicle suspension system), road transportation safety [173177], long-lasting pavement structures design [178181], ride quality and infrastructure asset management. An improved quantitative understanding on effects and mechanisms of various factors on dynamics of vehiclepavement interaction is the fundamentals for increased application and accuracy and reliability of structural health monitoring, nondestructive testing and evaluation, environmental vibration mitigation and weight-in-motion. It will also benet the nations transportation economy by reducing operation and maintenance costs of vehicles and transportation infrastructure as well as increasing transportation productivities.
6 Conclusions
Irregularities of pavement surface, from the small-scale unevenness of material on pavement surface to the large-scale undulating of vertical curve of a highway or an airport, all belong to spatial uctuation of pavement surface at different scale. Extensive study has been accomplished both domestically and internationally, toward measuring the physical aspects of pavement roughness, analyzing the resulting data, and evaluating the riding performance of pavements. Instrumentation and analysis technique including the spectral analysis approaches have been summarized in the article. A number of the PSD functions including the effect of thermal joints on PSD roughness have been presented here. These PSD functions are similar in their shapes and only different in their mathematical descriptions. Under the condition of constant speed of travel and linear vehicle suspension, it is proven that dynamic contact force between vehicle and pavement is a stationary stochastic process. Its mean function is given by (79) without the consideration of static loads or by (80) with the consideration of static loads. The correlation function, PSD forces, and standard deviation are, respectively, given by (84), (89) and (92). The concept and the methodology present in the article are not restricted to specic surface roughness and/or vehicle models. They are generally applicable to all kinds of linear vehicle models and measured pavement surface conditions. The response
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158 L. Sun
of linear continuum media to moving stochastic vehicle loads is analyzed herein. We show that there exists predicable relation among surface roughness, vehicle suspensions and speed, and the response of continuum media. The theory developed here is widely applicable to moving vehicle loads.
Acknowledgments This study is sponsored in part by the National Science Foundation, by National Natural Science Foundation of China, by Ministry of Communication of China, by Jiangsu Natural Science Foundation to which the author is very grateful. The author is also thankful to Professor Wanming Zhai, Editor Mr. Yao Zhou and Editor-in-Chief Yong Zhao for their invitations.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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The Author(s) 2013
Abstract
This article lays out a unified theory for dynamics of vehicle-pavement interaction under moving and stochastic loads. It covers three major aspects of the subject: pavement surface, tire-pavement contact forces, and response of continuum media under moving and stochastic vehicular loads. Under the subject of pavement surface, the spectrum of thermal joints is analyzed using Fourier analysis of periodic function. One-dimensional and two-dimensional random field models of pavement surface are discussed given three different assumptions. Under the subject of tire-pavement contact forces, a vehicle is modeled as a linear system. At a constant speed of travel, random field of pavement surface serves as a stationary stochastic process exciting vehicle vibration, which, in turn, generates contact force at the interface of tire and pavement. The contact forces are analyzed in the time domain and the frequency domains using random vibration theory. It is shown that the contact force can be treated as a nonzero mean stationary process with a normal distribution. Power spectral density of the contact force of a vehicle with walking-beam suspension is simulated as an illustration. Under the subject of response of continuum media under moving and stochastic vehicular loads, both time-domain and frequency-domain analyses are presented for analytic treatment of moving load problem. It is shown that stochastic response of linear continuum media subject to a moving stationary load is a nonstationary process. Such a nonstationary stochastic process can be converted to a stationary stochastic process in a follow-up moving coordinate.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer