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Mathematics and Computers in Simulation 61 (2003) 77–88

Simulation of pavement roughness and IRI based on power spectral density Lu Sun∗ Transportation Systems Engineering Program, ECJ Hall 6.10, Department of Civil Engineering, The University of Texas at Austin, Austin, TX 78712, USA Received 1 June 2001; accepted 22 June 2001

Abstract The international roughness index (IRI) for evaluating pavement surface roughness is simulated using the power spectral density (PSD) of pavement surface fluctuation. Quarter-car models recommended by the World Bank for measuring pavement roughness are adopted to simulate vehicle response. Surface roughness in time domain is generated based on 36 known PSDs of roughness. Newmark sequential integration method is employed to conduct the simulation of quarter-car motion. Statistical analysis of system output shows that the IRI is linearly correlated with the standard deviation of relative vertical velocity between the axle and sprung mass. A linear regression equation is obtained, based on which further analysis is conducted to represent the IRI in terms of PSD roughness. It is found that if PSD roughness is expressed as a polynomial function, the IRI can be simply calculated by means of the square root of the sum of the weighted regression coefficients of PSD roughness. The weighting coefficients are provided and can be conveniently used in practice. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: IRI; PSD; Computer simulation; Roughness; Random vibration

1. Introduction In highway industry, many statistics for evaluating pavement surface roughness have been developed since 1960s [1–6]. One of the most popular statistics is the international roughness index (IRI), which was developed and recommended by the World Bank to evaluate pavement roughness [1,7–9]. Today, the IRI has been a widely used indicator for evaluating the ride quality of pavement in highway industry. It basically measures the output of a specified test vehicle as it traverses a tested pavement section at a constant speed 80 km/h. The test vehicle is simplified as a quarter-car and calibrated before conducting the test. The IRI is defined as the overall relative velocity between the axle and sprung mass of the quarter-car [1,4,8]. ∗

Present address: Transportation and Operations Research Program, Department of Civil Engineering. The Catholic University of America, Washington D.C., 20064, USA. Tel.: +1-202-319-6671; fax: +1-202-319-6677. E-mail address: [email protected] (L. Sun). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 8 6 - X

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In the vehicle industry and research community, a significant statistic for characterizing pavement roughness is power spectral density (PSD) [1,9–11]. Historically, PSD roughness has been used primarily to evaluate vehicle response, suspension optimization and control, dynamic pavement loading and energy consumption [12–16]. As a direct statistic of roughness, PSD roughness is different from the IRI in that the former has been routinely adopted by vehicle manufactures for automobile design purpose for many years. However, the IRI is the most commonly used statistic for evaluating roughness in highway transportation agencies. It is believed that if a relationship can be found between the IRI and the PSD roughness, it will be much easier and produce more benefits for both highway and vehicle industries to compare their criteria and further to improve their production designs. This paper describes a simulation-based investigation for predicting IRI using known PSD roughness. In Section 2 we introduce the method. In Section 3 a quarter-car model is established to model the “Golden” car used for developing IRI. In Section 4 Newmark sequential integration method is introduced to simulate vehicle vertical vibration under surface roughness excitation. In Section 5 statistical analysis is performed. A good linear correlation relationship between the IRI and standard deviation of relative vertical velocity between axle and sprung mass is found. In Section 6 stochastic process theory is employed to represent IRI in terms of square root of the sum of the weighted regression coefficients of PSD roughness. The weighting factors are also provided in the paper. In Section 7 conclusions are drawn.

2. Surface roughness simulation 2.1. Description of PSD roughness Numerous measurements indicate that roughness can be modeled as a zero mean Gaussian isotropic random field in the spatial domain and will become a normal stationary ergodic random process in the time domain [17,18]. The height, ξ , of surface profile, representing pavement roughness, is a function of spatial distance, x, along the pavement. From the Wiener–Khintchine theory, the following forms constitute a couple of Fourier transform [17]:
1 ∞ Sξ (Ω) = Rξ (X)e−iΩX dX (1a) 2π −∞
∞ Sξ (Ω)eiΩX dΩ (1b) Rξ (X) = −∞

where X represents the distance of two points along the pavement, Sξ (Ω) is the PSD roughness in terms of wave number, Ω, which represents spatial frequency, and Rξ (X) is the spatial auto-correlation function and is defined as: Rξ (X) = E[ξ(x)ξ(x + X)] in which E[·] represents the expectation of a random process that can be estimated from:
1 X E[ξ(x)] = lim ξ(x) dx X→∞ X 0

(2)

(3)

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From stochastic process theory, PSDs of roughness, Sξ (ω), functions of angular frequency (or circle frequency) ω, are described by:
1 ∞ Sξ (ω) = Rξ (τ )e−iωτ dτ (4a) 2π −∞
∞ Rξ (τ ) = Sξ (ω)eiωτ dω (4b) −∞

in which τ represents time lag, Rξ (τ ) is temporal auto-correlation function. Assume that a vehicle moves along a pavement of length X at constant speed v. The relation of length and speed gives that X = vτ , where τ is the time it takes the vehicle to traverse the pavement section. Substituting this equation into Eq. (1a), we have:

1 ∞ v ∞ −i(ω/v)vτ Sξ (Ω) = Rξ (vτ )e d(vτ ) = Rξ (τ )e−iωτ dτ = vSξ (ω) (5) 2π −∞ 2π −∞ where ω = v. 2.2. Roughness generation One effective procedure for simulating stochastic processes is to use the combination of triangular series. The principal concept involved in this procedure is to represent a stochastic process in terms of the sum of a number of cosine functions. Initially, Shinozuka [19] applied this method to the simulation of random processes. Assume that the PSD roughness represented by the angular frequency of a pavement section is known as Sξ (ω). This method says that the temporal random excitation formed by a rough pavement surface can be expressed by means of: ξ(t) =

M  Ak cos(ωk t + Φk )

(6)

k=1

where M is a positive integer and Φ k is an independent random variable with uniform distribution at range [0, 2π). Also, the discrete frequency ωk is given by:   (7a) ωk = ωl + k − 21 ω in which frequency interval ω = (ωm − ωl )/M and [ωl , ωm ] is the range of frequency where Sξ (ω) has significant values. The amplitude Ak in Eq. (7a) and (7b) is represented by:   (7b) Ak = 2Sξ (ωk )ω = 2Sξ (Ωk )Ω Applying the central limit theorem, one can prove that as M → ∞, ξ (t) given by (7a) and (7b) will approach an ergodic Gaussian stationary process and PSD of ξ (t) will approximate Sξ (ω). This combination model applies to the simulation of stochastic process with any form of PSD function. However, higher computational efforts are involved in using common algorithms when calculating (7a) and (7b). To improve the computational efficiency, fast Fourier transform has been applied to this problem by Schueiler and Shinozuka [20]. We leave the FFT algorithm to the paper of Shinozuka [21].

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The generation of Φ k is implemented by using linear congrenutial generator (LCG). From LCG method [18,22], an independent random variable uk with uniform distribution U (0, 1) can be generated by: uk+1 = (auk + c)bmod(m )

(8)

Law and Kelton [22] recommended that c = 0, a = 630,360,016 and m = 231 − 1 = 2,147,483,647 are appropriate parameters that can lead to random variable with better statistical properties. Once independent random variables with U (0, 1) distribution are achieved, multiplying a constant coefficient 2π to uk can generate the random variable with U (0, 2π) distribution. The procedure for generating roughness is as follows: generate independent random variables uk ∼ U (0, 1) using Eq. (8); calculate random variable Φk = 2πuk so that Φk ∼ U (0, 2π); determine vehicle speed, v; convert PSD roughness Sξ (Ω) to Sξ (ω) through the equality (5); choose respective lower and upper bound of angular frequency ωl and ωm such that Sξ (ω) has significant values and must be taken into account within the frequency range [ωl , ωm ]; (6) determine N and calculate frequency interval, ω; (7) compute temporal roughness excitation ξ (t) by means of Eq. (6).

(1) (2) (3) (4) (5)

2.3. Expressions of PSD roughness Based on the analysis of a number of measurements conducted on variety of pavement surface condition, Dodds [12] presented a series function of PSD roughness. These functions area obtained by fitting the measured PSD roughness using regression technique. Eq. (9) provides the form of the regressed PSD roughness from Dodds [12]  −γ Ω Sξ (Ω) = Sξ (Ω0 ) (9) Ω0 where the fix-datum wave number Ω 0 is set as 1/2π cycle/m. The measurement shows that various values exist for exponential γ and the so-called roughness coefficient Sξ (Ω 0 ), ranging from 1.5 to 3.0 for γ and from 2 × 10−6 m3 per cycle to 2048 × 10−6 m3 per cycle for Sξ (Ω 0 ). These different values reflect the components of wavelength in elevation fluctuation and surface condition. Eq. (9) is used as PSD roughness later on to generate pavement profile. 3. Quarter-car model The most popular response-type device for measuring pavement roughness at normal highway speeds in the US is the Mays Ride Meter (MRM), which was developed for the Texas Highway Department in the 1960s [4,5]. This device was modeled by one-fourth of a vehicle “traveling” over pavement profile data. The IRI, roughness summary statistics, is defined as one of the response of quarter-car model at speed 80 km/h (22.222 m/s) where the vehicle parameters are as recommended by the Highway Safety Research Institute (HSRI). A quarter-car model with two degrees of freedom on a rough pavement is shown in Fig. 1. The motion equations controlling this suspension system are given by ordinary differential Eqs. (10a)

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Fig. 1. The HSRI quarter-car model.

and (10b): mt y¨t + cs (y˙t − y˙s ) + ct (y˙t − ξ˙ ) + ks (yt − ys ) + kt (yt − ξ ) = 0

(10a)

ms y¨s − cs (y˙t − y˙s ) − ks (yt − ys ) = 0

(10b)

the pavement profile is modeled as a one-dimensional random field and is represented by the height of pavement surface irregularities ξ ; ys and yt are absolute displacements of sprung mass and unsprung mass, respectively. One and two dots on the above symbols are first and second derivative processes, respectively, i.e. the velocity process dy/dt = y˙ and the acceleration process d2 y/dt 2 = y. ¨ The parameters of the HSRI quarter-car model are kt /ms = 653.0 s−2 , ks /ms = 62.3 s−2 , mt /ms = 0.15, cs /ms = 6.0 s−1 and ct /ms = 0.0 s−1 [1,4,8]. Let zt (t) = yt − ξ,

zs (t) = ys − yt

(11)

in which zt (t) and zs (t) are relative displacements of sprung mass and unsprung mass, respectively. Substituting these two equations into Eq. (15) we have: mt ξ¨ + mt z¨ t = ct z˙ t + kt zt − cs z˙ s − ks zs

(12a)

ms ξ¨ + ms z¨ t + ms z¨ s = cs z˙ s + ks zs

(12b)

If we rewrite Eqs. (12a) and (21b) in the matrix form, we get ¨ ˙ M{Z(t)} + C{Z(t)} + K{Z(t)} = {R(t)} where



−mt 0 M= , −ms −ms 

mt ξ¨ (t) , {R(t)} = ms ξ¨ (t)

 C=

−cs , cs

ct 0

{Z(t)} =

zt (t) , zs (t)

(13)  K=

kt 0

˙ {Z(t)} =

−ks ks



z˙ t (t) , z˙ s (t)

¨ {Z(t)} =



z¨ t (t) z¨ s (t)

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4. Sequential integration method In theory, various computational algorithms such as, Euler’s method and Runge–Kutta methods, are available for solving the differential Eq. (15) [23]. These methods can be classified into two categories: sequential integration and superposition of mode [24]. Here the sequential integration method is used for numerically calculating (15). One of the sequential integration methods is Newmark algorithm, which has been used to solve many dynamic problems [24]. This algorithm is also applied in this paper. ˙ + t)} and {Z(t + t)} by means of the Taylor’s expansion with different orders Representing {Z(t of accuracy, we have: ... ˙ + t)} = {Z(t)} ˙ ¨ {Z(t + {Z(t)}t + 21 {Z (t)}(t)2 (14) ... 2 ˙ ¨ {Z(t + t)} = {Z(t)} + {Z(t)}t + 21 {Z(t)}(t) (15) + 16 {Z (t)}(t)3 Assume linear acceleration relation exists within time interval [t, t + t], i.e. ... 1 ¨ ¨ {Z (t)} = [Z(t + t)} − {Z(t)}] t

(16)

Substituting Eq. (16) into (14) and (15) we obtain: ¨ ¨ + t)}] ˙ + t)} = {Z(t)} ˙ + {Z(t {Z(t + 21 t[{Z(t)}

(17)

¨ ¨ + t)}](t)2 ˙ + 16 {Z(t {Z(t + t)} = {Z(t)} + {Z(t)}t + [ 13 {Z(t)}

(18)

and

The Newmark method requires ˙ + t)} = {Z(t)} ˙ ¨ ¨ + t)}]t {Z(t + [(1 − α){Z(t)} + α{Z(t

(19)

¨ ¨ + t)}](t)2 ˙ + β{Z(t {Z(t + t)} = {Z(t)} + {Z(t)}t + [( 21 − β){Z(t)}

(20)

and

where α and β are constants. Obviously, if α = 1/2 and β = 1/6, the result (19) and (20) given by the Newmark method is equivalent to Eqs. (22) and (23), or the so-called Wilson-θ method, with θ = 1 [24]. From Eqs. (19) and (20) we can obtain: ¨ + t)} = a0 {Z(t + t)} − a0 {Z(t)} − a2 {Z(t)} ˙ ¨ {Z(t − a3 {Z(t)}

(21)

˙ ¨ ˙ + t)} = a1 {Z(t + t)} − a1 {Z(t)} − a4 {Z(t)} − a5 {Z(t)} {Z(t

(22)

and

where 1 , β(t)2 1 a3 = − 1, 2β

a0 =

α , βt α a4 = − 1, β a1 =

1 , βt   t α a5 = −1 2 β

a2 =

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Substituting Eqs. (21) and (22) into (13) we have: ¯ + t)} ¯ K{Z(t + t)} = {R(t

(23)

where ¯ = K + a0 M + a1 C K

(24)

˙ ¨ ¯ + t)} = {R(t + t)} + M[a0 {Z(t)} + a2 {Z(t)} + a3 {Z(t)}] {R(t ˙ ¨ + C[a1 {Z(t)} + a4 {Z(t)} + a5 {Z(t)}]

(25)

and

¨ + t)} and {Z(t ˙ + t)} can be calculated respectively from Solve Eq. (25) for {Z(t + t)}. Then, {Z(t Eqs. (21) and (19) or (22). From the perspective of computational accuracy and stability, α = 1/2 and β = 1/4 are more efficient [24], and, therefore, are used in the paper.

5. Implementation of system simulation So far, we have demonstrated the feasibility of performing simulation to generate the HSRI quarter-car response using PSD roughness. We have identified this problem as a steady-state simulation problem. In the following sections, we will investigate the relationship between the IRI statistic and other statistics of roughness, such as the standard deviation of quarter-car response based on simulated data. Five independent replication runs of simulation coming from different seeds 927711160, 364849192, 2049576050, 638580085 and 547070247 of Φ k are generated. Each run with datasets of 4000 ξ (t), given that M = 400, is generated according to the steps provided in Section 3. The vehicle speed is set at 22.222 m/s (80 km/h) as required by the IRI test [1,8]. The time interval, t, of quarter-car simulation is taken as 0.001 s. Therefore, the sample interval of pavement profile is equal to 0.0222 m. When using Eq. (23) for iterative calculation, all initial conditions are set as zeros. For each run, pavement profiles with total length about 1 km are simulated. A sample of roughness generated is shown in Fig. 2, in which γ and Sξ (Ω 0 ) are 2.0 × 10−6 and 4 × 10−6 m3 per cycle, respectively.

Fig. 2. Simulated PSD roughness.

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Let T be the time it takes a vehicle to traverse a pavement section of length L at constant speed v = 22.222 m/s. The IRI is defined by the average of the absolute value of z˙ s (t) over time T = nt, i.e. 1 IRI = L




T

0

n

n

1  1 |˙zs (t)| dt = |˙zs (ti )|t = |˙zs (ti )| vnt i=0 vn i=0

(26)

As another indirect statistic of roughness, the standard deviation, σZ˙ s , of z˙ s (t) can be estimated from:   n  1   [˙zs (t) − z¯˙ s (t)]1/2 (27) σZ˙ s = n − 1 i=0 in which z¯˙ s (t) can be estimated by: n

1 z˙ s (t) z¯˙ s (t) = n i=0

(28)

Another set of different seeds (1973272912, 281629770, 20006270, 1280689831, 2096730329, 1933576050, 913566091, 246780520, 1363774876, 604901985, 1511192140, 1259851944, 824064364, 150493284, 242708531, 75253171, 1964472944, 12022299975, 233217322, 1911216000, 726370533, 403498145, 993232223, 1103205531, 762430696, 1922803170, 1385516923, 76271663, 413682397, 7264666604, 336157058, 1432650381, 1120463904, 595778810, 877722890 and 1046574445) are respectively used in Eq. (8) to generate 36 independent sets of random numbers. These 36 sets of random numbers are then used to generate different pavement profile based on different PSD roughness functions given by Eq. (9), where γ takes the values of 1.5, 2.0, 2.5, 3.0 and Sξ (Ω 0 ) takes the values of 4, 8, 16, 32, 64, 128, 256, 512, 1024 × 10−6 m3 per cycle. Therefore, there are a total of 36 pavement profiles generated. For each profile dataset, 2000 data points are simulated in the time domain and only the last 1500 data are used for steady-state analysis. The batch mean method [22] is employed to perform steady-state analysis of the data from only one run. Ten batches are selected and each batch contains 270 data of interest (i.e. n = 270). The quantities studied here are standard deviation σZ˙ s and the IRI. The formula used to estimate the mean of the quantity  ¯¯ ¯ ¯ of interest is Y¯¯ = 10 k=1 Yk /10, in which Yk (k = 1, 2, . . . , 10) is the batch mean of k-th batch and Y the mean of the quantity of interest. Since different seeds are chosen in the LCG model to generate independent random number, the results of the 36 runs are also mutually independent.

6. Result analysis Figs. 3 and 4 show a time history of relative velocity and acceleration between axle and sprung mass as the quarter-car vehicle traverses a pavement with γ = 2.0 and Sξ (Ω0 ) = 4 × 10−6 m3 per cycle. Define the ratio IRI/σZs . Based on regression, it is found: IRI = 0.03602σZ˙ s

(29)

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Fig. 3. The time history of the relative velocity between axle and sprung mass.

Eq. (28) shows that a strong linear correlation exists between the IRI and σZ˙ s . According to the stochastic process theory, it is well known that the following relation holds:
∞ 2 σZ˙ s = ω2 |HZs (ω)|2 Sξ (ω) dω (30) −∞

in which HZs (ω) is the frequency response function and is defined as the response as the excitation term ξ in Eq. (12a) and (12b) takes the value eiωt . In conjunction with Eq. (5), Eq. (28) gives
∞ 2 2 Ω 2 |HZs (vΩ)|2 Sξ (Ω) dΩ (31) σZ˙ s = 2v 0

Let PSD roughness be represented in the form of: Sξ (Ω) =

M 

Sj+ Ω j

j =0

Fig. 4. The time history of the relative acceleration between axle and sprung mass.

(32)

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or Sξ (Ω) =

M  Sj− Ω −j

(33)

j =0

where M is the highest term that is included in regression analysis of PSD roughness. Because both expressions can be conveniently implemented using linear regression analysis (for the later formula, one just needs to convert Ω to another argument q = Ω −1 ), the coefficients Sj+ and Sj− can be estimated by means of statistical regression. Then, the IRI can be finally given by either one of the following formulae: 1/2  M  (34) IRI =  Bj+ Sj+  j =0

or

 1/2 M  IRI =  Bj− Sj− 

(35)

j =0

The appropriate selection of Eqs. (33) and (34) depends upon which of the Eqs. (31) and (32) is adopted in regression analysis. In practice, the integration of (30) is virtually not from zero to infinity. Certain lower-bound wave numbers Ω s (or named start-up frequency) and upper-bound wave number Ω c (i.e. the so-called cut-off frequency), respectively corresponding to the aforementioned ωl and ωm , exist for validating PSD roughness. Suitable determination of Ω s and Ω c involves the reasonable consideration of pavement geometric characteristics and pavement surface material properties. It is noted that Ω = 2π/λ, in which λ is the wavelength of pavement profile. Since the geometric variation in elevation along the longitudinal direction exists, the lower-bound Ω s can be determined by choosing longer wavelengths λs = 100 or 50 m, which leads to Ωs = 0.02π cycles/m or Ωs = 0.04π cycles/m. For upper bound of wave number, we respectively choose two short wavelengths as λs = 0.5 and 10 m. This leads to two cut-off frequencies Ωc = 4π cycles/m and Ωc = 2π cycles/m. The lower-bound frequency of Ωs = 0.02π cycles/m implies that longitudinal slopes or surface fluctuations with wavelength exceeding 100 m in highway geometric alignment will be filtered while processing the data to eliminate the influence of longitudinal slope on the IRI. The upper-bound frequency of Ωc = 4π cycles/m or Ωc = 2π cycles/m simply means that surface variation of pavement materials and some distresses, such as small cracking and small potholes with spatial scale less than 0.5 or 1.0 m, will also be filtered by integrating Eq. (30) from Ω s to Ω c . We believe these upper and lower bounds are reasonable choices for accounting the effect of roughness on the IRI. Computation efforts have been taken to calculate the coefficient Bj+ and Bj− with the highest order M = 5. The results for the HSRI quarter-car model are obtained and provided for Bj+ and Bj− in Table 1. From Table 1 we can see that lower and upper bounds [Ωs , Ωc ] do not have significant influences on coefficients Bj− (i = 0, 1, . . . , 5) if the representation of PSD roughness is based on Eq. (32). However, this is not true for PSD roughness based on regression equation of form (31). Based on different start-up and cur-off frequency, we conduct some numerical computations. The results show that in the case of common lower-bound frequencies, Ω s does not have significant influence

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Table 1 Coefficients for Eqs. (32) and (33) [Ωs , Ωc ]

[0.02π, 4π]

[0.04π, 2π]

B0+ B1+ B2+ B3+ B4+ B5+

78.20821 317.4197 1632.365 10615.96 83603.62 750880.1

68.23027 232.0303 871.5755 3560.098 15622.37 72801.7

B0− B1− B2− B3− B4− B5−

78.20821 24.1014 9.777339 6.44466 8.512586 19.35002

68.23027 22.88958 9.624391 6.419998 8.460586 18.83203

on Bj+ , Bj− and thus the IRI. As far as the cut-off frequency is concerned, Ω c does affect Bj+ , Bj− and the IRI significantly. This is simply because the IRI is measured at a higher vehicle speed of 80 km/h. Thus, surface fluctuations with short wavelength can be converted by the quarter-car to a frequency around one of the resonance frequency of the vehicle. This will amplify its effect on the IRI and, consequently, the coefficient Bj+ and Bj− . The results of Eqs. (33) and (34) are quite simple and can be conveniently used in practice for converting PSD roughness to the IRI. It is also worth noting that all procedures and coefficients provided in this paper are SI unit based. In other words, the IRI given by Eqs. (33) and (34) are dimensionless. Also, the SI units, that is cycle/m for Ω and m3 per cycle for Sξ (Ω) must be adopted in the regression of Eqs. (31) and (32). Hence, the coefficients in Table 1 cannot be directly used if the non-SI units are adopted for any regression analysis. To use the coefficients in Table 1, the involved unit transformation must be taken into consideration. We also want to point that one should be aware that the highest order in Eqs. (31)–(34) may not be chosen as high as provided in this paper (i.e. M = 5). On the contrary, one may use any order of polynomial, say M = 3, to fit the curve of PSD roughness, which is usually obtained either from realistic measurement or from theoretical calculation. As long as the highest order of the polynomial selected in the regression process is within the range of [0,5], all of the coefficients provided in Table 1 can certainly be used for the purpose of transferring the PSD roughness to the IRI by directly computing Eq. (33) or (34).

7. Concluding remarks For a pavement section of which PSD roughness is known, the IRI of that pavement section can be obtain using the simulation procedure developed in this paper. The procedure contains simulating pavement profile from the PSD roughness; employing the Newmark sequential integration method to solve the continuous motion equation of quarter-car model and; computing the IRI by applying statistical procedures to output analysis.

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Based on simulation and regression analysis, a strong linear correlation is found between the IRI and the standard deviation of relative vertical velocity between axle and sprung mess. If the PSD roughness is represented in the form of polynomial function of wave number, the IRI can be simply predicted by the square root of the sum of the weighting coefficients of the regressed PSD roughness (i.e. Eqs. (33) and (34)). The weights are provided in Table 1. This allows highway agencies to convert PSD roughness of a given pavement section to the IRI, which makes the standards in both the highway agency and the vehicle manufacturing industry comparable. References [1] T.D. Gillespie, M.W. Sayers, L. Segel, Calibration of response-type pavement roughness measuring systems, National Cooperative Highway Research Program, Res. Rep. No. 228, Transportation Research Board, National Research Council, Washington, DC, 1980. [2] M.S. Janoff, J.B. Nick, P.S. Davit, G.F. Hayhoe, Pavement roughness and rideability, National Cooperative Highway Research Program, Res. Rep. No. 275, Transportation Research Board, National Research Council, Washington, DC, 1985. [3] M.S. Janoff, Pavement roughness and rideability field evaluation, National Cooperative Highway Research Program, Res. Rep. No. 308, Transportation Research Board, National Research Council, Washington, DC, 1988. [4] W.R. Hudson, et al., Root-mean-square vertical acceleration as a summary roughness statistics, in: T.D. Gillespie, M. Sayers (Eds.), ASTM special Technical Publication, No. 884, 1985. [5] R. Hass, W.R. Hudson, J. Zniewski, Modern Pavement Management, Krieger Publishing Company, Malabar, FL, 1994. [6] L. Sun, Theoretical investigations on vehicle-ground dynamic interaction, Final Report prepared for National Science Foundation of China, Southeast University, Nanjing, China, 1998. [7] T.D. Gillespie, et al., Effects of heavy-vehicle characteristics on pavement response and performance, National Cooperative Highway Research Program, Res. Rep. No. 353, Transportation Research Board, National Research Council, Washington, DC, 1993. [8] M. Sayers, Development, implementation, and application of the reference quarter-car simulation, in: T.D. Gillespie, M. Sayers (Eds.), ASTM special Technical Publication, No. 884, 1985. [9] J.A. Marcondes, M.B. Snyder, S.P. Singh, Predicting vertical acceleration in vehicles through pavement roughness, J. Transport. Eng., ASCE 118 (1) (1992) 33–49. [10] J.K. Hedric, et al., Predictive models for evaluating load impact factors of heavy trucks on current pavement conditions, Interim Report to USDOT Office of University Research under contract DTES5684-C-0001, 1985. [11] L. Sun, X. Deng, Predicting vertical dynamic loads caused by vehicle-pavement interaction, J. Transport. Eng., ASCE 124 (5) (1998) 470–478. [12] C.J. Dodds, The laboratory simulation of vehicle service stress, J. Eng. Ind., ASME 2 (1974) 391–398. [13] P.F. Sweatman, A study of dynamic wheel forces in axle group suspensions of heavy vehicles, Special Report No. 27, Australia Pavement Research Board, 1983. [14] K. Wei, et al., Vehicle Dynamics, Renming Jiaotong Publication Inc., Beijing, China, 1988. [15] L. Sun, X. Deng, Transient response for infinite plate on Winkler foundation by a moving distributed load, Chin. J. Appl. Mech. 14 (2) (1997) 72–78. [16] L. Sun, B. Greenberg, Dynamic response of linear systems to moving stochastic sources, J. Sound Vib. 229 (4) (2000) 957–972. [17] D.E. Newland, An Introduction to Random Vibration and Spectral Analysis, 2nd ed., Longman, New York, NY, 1984. [18] W.Q. Zhu, Random Vibration, Academic Press, Beijing, China, 1992. [19] M. Shinozuka, Digital simulation of random processes and its applications, J. Sound Vib. 25 (1972) 111–128. [20] G.I. Schueiler, M. Shinozuka, Stochastic Methods in Structural Dynamics, Martinus Nijhoff Publishers, 1987. [21] M. Shinozuka, Digital simulation of random processes in engineering mechanics with the aid of FFT technique, in stochastic problems in mechanics, in: S.T. Ariaratnam, H.H.E. Leipholtz (Eds.), University of Waterloo Press, Waterloo, 1974. [22] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 3rd ed., McGraw-Hill, New York, 1999. [23] L. Meirovitch, Elements of Vibration Analysis, 2nd ed., McGraw-Hill, New York, 1986. [24] H.Q. Chen, Numerical Analysis and Programming for Dynamic Problem, Hehai University Press, Hehai, 1984.