AN ANALYSIS OF VEHICLE VIBRATION WITH UNCERTAIN SYSTEM PARAMETERS AN ANALYSIS OF VEHICLE VIBRATION WITH UNCERTAIN SYSTEM PARAMETERS

study an application of fuzzy arithmetic for analysis of randomly excited structures with uncertain or inexact system The purpose of the to investigate possibilities of a fuzzy technique in a vehicle dynamic analysis. The of the problem will using the program a special INTLAB.


Introduction
Engineering models of carrying structures are inevitably based on idealization of reality. Therefore, various degrees of idealization will provide different results. Generally, each engineering model has to consist of three fundamental components coherent to each other [1,8]: • loading (harmonic, periodic, stochastic etc.), • structural model (geometry, material properties etc.), • response (displacements, accelerations, stresses, cumulative damage etc.).

Fig. 1 Engineering models of structures
Uncertainty of each computational component (in loading or structural parameters) can be characterised by probability theory or fuzzy sets approach. Generally, uncertainties are often classified as imprecision, modelling and random [7]: • imprecision is due to vagueness in a characterizing performance in terms such as "good" or "unacceptable" • modelling uncertainty is due to reality idealization by the modeled structure and further simplifications in the computational models, • random uncertainty reflects variations in the operating environment and lack of the designer control.
Several methods include safety factors, "worst-case scenario", probabilistic methods and fuzzy set based methods. This study focuses only on the treatment by fuzzy uncertainties, using fuzzy set theory and numerical analysis.
Fuzzy set based methods use possibility distributions to model uncertainties and assess safety. The possibility distributions are estimated using numerical data or expert opinion. In theory, probabilistic methods should be more effective for problems involving only random uncertainties, because they account for more information about these uncertainties than the other methods. However, to be applied, probabilistic methods may require more information than is available. On the other hand, fuzzy techniques require less data than probabilistic methods [1,2].
The theory of fuzzy sets was formulated by Zadeh [9]. A fuzzy set x is the set with boundaries that are not sharply defined. A function, called membership function (MSF), signifies the degree to which each member of a domain X belongs to the fuzzy set x [11]. For a fuzzy variable x ʦ Ͻx1, x2Ͼ, (or x ʦ x), the membership function is defined as (x). If (x) ϭ 1, x is definitely a member of the x [5,6,11]. If (x) ϭ 0, x is definitely not a member of the x. For every x with 0 Ͻ (x) Ͻ 1, the membership is not certain. Typical membership functions of fuzzy sets are shown in figures 1 and 2.
By fuzzy technique, the complete information about the uncertainties in the model can be included and one can demonstrate how these uncertainties are processed through the calculation procedure in MATLAB [1,2].

Numerical study of random vibration of a fuzzy vehicle computational model
Vehicle dynamic models are often characterized by uncertain system parameters. A main goal of this example will be to analyse the influence of "uncertain" mass, damping, stiffness parameters to structural response and the mark of ride quality in a chosen point. The input will be expressed by a fuzzy random function (fuzzy function of the behaviour power spectral density).
Let's consider a 7-DOFs railway vehicle model (Fig. 4). The geometry of the model is following: L ϭ 24.5 m, a 1 ϭ 12 m,  The following assumptions were employed to derive this model: • The inputs of the system are two track undulations in the vertical direction. • The unevenness of the track at the left rail and at the right rail are presented in Fig. 5 by using the power spectral density [3]. The motion of the wheels is restricted to the vertical direction only.
It is assumed that the model is linearized around the operating state and that the coordinates y ϭ [y 1 , z1 , x1 , y 2 , x2 , y 3 , x3 ] T are measured from the equilibrium state. The equations of motion will be interpreted as follows Matrices M, B, K, T b , T k are presented in Appendix. Fuzzification of the structural parameters has been realised by multiplication of matrices M, B, K by the fuzzy number x presented in Fig. 6.
Applying the fundamental principles of correlation theory we can solve the equations of motion in the frequency domain using the known input fuzzy power spectral density (Fig. 5) as follows [4] The fuzzy PSD function of the vertical motion of the point A is

Conclusion
The paper discusses the possibility of fuzzy arithmetic application in stochastic structural analysis. The use of fuzzy arithmetic provides a new possibility of the appraisal of machineries quality. Due to this numerical approach we can more authentically analyse mechanical, technological, service and economic properties of investigated structures.
In this paper, we have investigated the possibilities of stochastic solution for a simple vehicle computational model with fuzzy structural parameters. We have evaluated power spectral density and standard deviation of the vertical displacement, velocity and acceleration in selected point of the carriage (See Fig. 4). The fuzzy technique enables to appreciate the results in a broader context. This work has been supported by the VEGA grant No. 1/0280/03.