The non-stationary response of vehicles on rough ground
Vehicles moving on rough surfaces are subject to inputs which may be conveniently regarded as a combination of deterministic and random processes. Although this general problem is briefly addressed, it is the latter class of inputs which is of concern in the present work. In general, the random component of the excitation is `perceived' by the vehicle as a non-stationary random process, due either to inhomogeneity (spatial non-stationarity) in the surface roughness, or to variations in the vehicle velocity, or both. Analysis of the response of vehicles to such processes is further complicated by the multiple degree of freedom nature of the problem and by inherent non-linearity in vehicle dynamic systems, rendering exact statistical analysis of such systems analytically intractable, and thus requiring numerical simulation which is a costly alternative. A unified, analytical approach to this problem is presented here combining the techniques of linear systems theory and the approximate method known as statistical linearization, to facilitate the approximate analysis of non-linear systems excited by non-stationary random processes. The basis of the method is the use of a `shaping filter' description for the ground roughness, i.e., the height profile is represented as the output of a white noise excited, linear filter in the spatial domain (extensive justification of this assumption is presented). The key to the present work is the linking of the space domain filter with a state-space model for the vehicle dynamics by a formal change of variable, i.e., space is regarded as a function of time, related via the (variable) velocity function. This yields a time variable filter formulation for the excitation process which may then be coupled into the dynamic equations. After some further manipulations (using results from the theory of generalisation functions) differential equations may be constructed for the propagation of the mean vector and zero-lag auto-covariance matrix. For linear systems these results are exact and have been extended to the multiple input (multiple wheel) case. This extension presents no conceptual difficulty although it is computationally considerably more involved. For non-linear dynamic systems the method of statistical linearization is adopted so that the above method of obtaining means and covariances applies. This technique is essentially the replacement of the true non-linear element by a linear one such that their output differences is in some way minimised. This results in linear coefficients depending on the instantaneous mean and variance of response, just those quantities calculated by the above method and so a coupled set of non-linear, deterministic differential equations are obtained, governing the propagation of the approximate means and covariances of response. In order to validate results obtained by this method, in the absence of analytical solutions, extensive use is made of Monte-Carlo simulation. A fundamental concept arising as a result of the formulation is that of the `covariance equivalence' of two random processes. This enables the frequency-time (evolutionary) spectral analysis of random processes having a `frequency modulated' structure, a task which was hitherto not possible. This concept has also found application in the field of acoustics in the study of moving noise sources.