Statistical characteristics of sources of vehicle kinematic excitation

The longitudinal and transverse road profiles represent the functions of a random variable from a mathematical point of view. It is appropriate to use methods of probability theory and mathematical statistics for their description. The unevenness of the runway surface is the main source of the vehicle's kinematic excitation. This paper describes the statistical properties of the mapped road profiles. It shows a way of categorizing road surface quality based on the power spectral density of unevenness. The interrelationships between the individual points of the profile and the profiles with one another are evaluated by correlation functions.


Introduction
The unevenness of the runway surface is the main source of the vehicle's kinematic excitation. Unevenness must first be mapped and then mathematically described. Longitudinal or transverse road profiles represent functions of a random variable from a mathematical point of view [1]. To describe them, it is appropriate to use methods of probability theory and mathematical statistics and to quantify their basic numerical characteristics. The files thus obtained serve as inputs for various numerical simulations of processes of vehicle motion along the roads and the interaction of vehicles with the runway [2]. In numerical simulations, a random profile of the runway surface is often numerically generated to meet certain desired parameters. For the description of the unevenness in the frequency domain, the power spectral densities are used and the interrelationships of the individual profiles are evaluated by correlation functions. Applications in practice are different [3][4][5][6][7][8].

Mapping of stochastic road profile
Precise leveling was used to map the longitudinal profiles of the runway surface. Leica NA 2002 was used as a leveling instrument, Fig. 1. This instrument uses digital electronic image processing to determine heights and distances and automatically records the results to a REC module that can be transported to a computer. It ensures the elimination of the human factor in reading and processing the measured data.

Basic statistical characteristic of the road profiles
For further statistical analysis, each record was re-sampled to contain N = 2 14 = 16384 samples. A cubic spline approximation was used to calculate the intermediate values. All other analyses were performed from these re-sampled files. The basic statistical characteristics for left (ul) and right (ur) road profiles are put into Table 1.  In general, it is claimed that the density distributions of unevenness are governed by the Gaussian law of normal distribution [10]. To assess this, Fig. 4 and 5 compared the histograms of unevenness in the left and right tire traces and the Gaussian law of normal distribution.

Power spectral densities
The random road profile u(x) is assumed as a stationary ergodic function with zero mean value and normal distribution. The quality of the road surface can be assessed using a number of criteria. In accordance with ISO 8608 [11], the pavements can be classified into eight categories A -H on the basis of Power Spectral Densities, (PSD). The real power spectral densities obtained by processing the measured data can be approximated by the relationship The power spectral densities calculated for road profiles in Fig. 3 are plotted in log-log scale in Fig. 6. The road profiles can be classified as profiles lying on the border between categories B and C.

Correlation functions
The statistical dependencies between individual points of a random profile, or between two random profiles, describe the correlation functions [12]. The auto-correlation function for one random profile u(x) is described by the relationship where ξ is lengthwise displacement in the X-axis direction, ξ = x2 -x1. An important feature of the auto-correlation function is that its zero point value is equal to the dispersion Du This means that the normed auto-correlation function ρu(ξ) has a function value for ξ = 0 equals 1 The normed auto-correlation functions of the left and right road profiles for 16 380 numbers of lags are plotted in Fig. 7. The normed cross-correlation function for two random profiles ul(x) and ur(x) is described by the relationship The normed cross-correlation functions of the left and right road profiles for 16 380 numbers of lags are plotted in Fig. 8.

Conclusion
In the solution of the problem of vehicle roadway interaction, the road unevenness represents the main source of kinematical excitation of vehicle. Workers involved in the construction and maintenance of roads map the random profile of the road surface. The quality of the road surface can be assessed using a number of criteria. In accordance with ISO 8608 [11], roadways can be classified into 8 categories A -H, on the basis of power spectral densities. For numerical simulations of vehicle movement along the road, it is necessary to perform the opposite process than to measure random irregularities on the road. It is necessary to generate a random longitudinal road profile based on the known power spectral density of the unevenness. There is a problem here. During the measurement of random unevenness, only amplitude characteristics are evaluated and the phase characteristics are not evaluated. At the realization of the reverse process one (phase) characteristic is missing. This deficiency is solved by generating a phase shift angle φi as a random number based on a uniform distribution function. By smoothing the power spectral density shown in the log-log scale a line is obtained. It can be very good approximated by equation (1). On the basis of power spectral density ( 0 ), for the reference wave number Ω0 = 1.0 [rad/m], the quality of roads surface can be classified into eight categories A -H. But actually on highways and roads of the 1 st and 2 nd classes, only categories A ÷ E come into consideration. The statistical dependencies between individual points of a random profile, or between two random profiles, describe the correlation functions. The results of the correlation analysis show that there is little statistical dependence between individual points of the random profile. It is clear to see from the correlation functions that the random profile contains a number of periodically repeating components.
This work was supported by the Grant National Agency VEGA of the Slovak Republic, project number 1/0006/20.