Vehicle model response in frequency domain

The offered article deals with one of the possibilities of numerical analysis of the vehicle response in frequency domain. It works with quarter model of vehicle. For the selected computational model of vehicle it quantifies the Frequency Response Functions (FRF) of both force and kinematic quantities. It considers the stochastic road profile. The Power Spectral Density (PSD) of the road profile is used as input value for the calculation of Power Spectral Density of the response. Al calculations are carried out numerically in the environment of program system MATLAB. When we know the modules of FRF or the Power Response Factors (PRF) of vehicle model the calculation of vehicle response in frequency domain is fast and efficient.


Introduction
A roadway is a typical structure subject to the dynamic effects of moving vehicles. The problem of rough surface road profiles and its influence on vehicle unwanted vibrations due to kinematic excitations is still a subject of research among automotive manufacturers and research groups, whose objective is to minimize their effects on the driver and passengers. Unevenness on the surface of the pavement is the main source of kinematic excitation of vehicle. The real load acting on the roads is variable in time and in frequency composition. This should be known for the solution of many engineering tasks as lifetime, design, reliability, structure development, micro tremor, etc [1][2][3]. This article presents simplest way how to obtain the Power Spectral Density (PSD) of input signal (in our case it is PSD of road unevenness), how to calculate the dynamic properties of analysed system (specifically the Power Response Factors) and PSD of output signals of individual quantities.

Computational model of vehicle
The discrete (lumped mass) computational models of vehicles can be created on three qualitative different levels: 1Dquarter model, 2Dplane model and 3Dspatial model. Every model has its advantages and disadvantages and under certain assumptions, it can be used for the solution of real practical problems. Quarter model of vehicle is shown in Fig. 1. It can be used to model the effects of half of one axle of the vehicle on a traffic road and under certain simplifying assumptions to model the effects of the entire vehicle. The disadvantage of this model is that it can only model the heave effect of the upper mass on the contact forces between the wheel and the runway. It cannot model the pitch effect not the roll effect of the suspended mass on the contact force between the wheel and the runway.
Dependence between elastic forces in joining elements (in the sense of action of mass objects on jointing members) and its deformations is described by the equation where [k] is the stiffness matrix of joining elements. Dependence of damping forces on the velocity of deformations The system of equations of motion describing the vibration of the computational model of vehicle is then expressed by the relation where [m] is mass matrix. After substituting for {r(t)} = [r1(t), r2(t), u (t)] T , we obtain the system of ordinary differential equations describing the behaviour of the computational model and the equation for contact force The function r1(t) describes the time course of the vertical deflections of the upper mass m1, the function r2(t) describes the time course of vertical deflections of the lower mass m2 and the function u(t) describes the time course of the road profile. The stiffness and damping constants are denoted as k1, k2, b1 [4].

Frequency response function
Frequency response function FR(p), where p = i.ω is a complex number, is defined as a ratio of steady-state response and harmonic excitation. If the excitation is harmonic with uniting amplitude, then After substituting (11) to (12) The graphic representation of frequency response is the frequency characteristic. The graphic representation of the absolute value (modulus) of frequency response is amplitude characteristic. The phase characteristic is the graphic representation of argument (phase) of frequency response in dependence on the frequency of harmonic excitation. The function 2 ) ( p FR is called power response factor (PRF).
The Laplace integral transform for the passing from time to frequency domain can be used. The Laplace picture of some function r(t) will be denoted as Lr(t) = R(p). In this case    i p is the complex number. The function r(t) and their derivatives with the respect of time will be transformed as follows By the Laplace transform of equations of motion and by defining the frequency responses we obtain the system of 2 equations in complex form for the function i r (i = 1, 2). The matrix formulation is as follows Similarly, the frequency response of a dynamic component of contact force is defined as The frequency spectrum of the output signal can be obtained by multiplying the frequency response function of the system by the frequency spectrum of the input signal By introducing the power spectral densities of the input signal ) ( u PSD and output it can be written [4] )  Table 1. For numerical simulations of vehicle movement along the road, it is necessary to generate a random longitudinal road profile based on the known power spectral density of unevenness ) (Ω S u . There is a problem here. During measurement of random unevenness, only amplitude characteristics are evaluated and the phase characteristics are not evaluated. At the realization of reverse process one (phase) characteristic is missing. To generate a random road profile the following formula can be used The angle φi is the phase shift angle randomly divided into the interval (0; 2π). It is generated according to the uniform distribution.
The stochastic road profile generated by (21) for category B is shown in Fig. 3. It represents the source of kinematic excitation of vehicle. The Power Spectral Density of the road profile is shown in Fig. 4.

Solution of the dynamic response in frequency domain
Solution of dynamic response in frequency domain can be carried out in many ways. The simplest way is to obtain the Power Spectral Density of input signal, in our case the Power Spectral Density of road unevenness. Then to calculate the Frequency Response of analysed dynamical system specifically the Power Response Factors of individual quantities. And simply multiply the Power Response Factors of individual quantities by the Power Spectral Density of input signal, in our case by the Power Spectral Density of the road profile. The results thus obtained are shown in Fig. 5.

Conclusion
Vehicle vibration induced by its motion along stochastic road profile can be solved in time or in the frequency domain. When we know the modules of FRF or the Power Response Factors (PRF) of vehicle model the calculation of vehicle response in the frequency domain is fast and efficient. In PSD of road profile predominate low frequencies (long waves). Upper and lower masses m1 and m2 sensitively respond to these low frequencies. The dynamic component of contact force Fd sensitively responds to frequencies close to the second natural frequency of vehicle computational model, frequency interval approximately from 7 to 11 Hz.
This paper was supported by the Grant National Agency VEGA of the Slovak Republic (grant No. 1/0006/20).