Random processes of bounded variation are generated by using of randomized sinusoidal model and nonlinear filter model. In the randomized sinusoidal model, random noises ar\e introduced in phase angles; while in the nonlinear filter model, a set of nonlinear Itô differential equations are employed. In both methods, the spectral density of a modeled random process can be matched by adjusting model parameters. However, the probability density of the process generated by the randomized sinusoidal model has a fixed shape, and cannot be adjusted. On the other hand, the nonlinear filter model covers a variety of profiles of probability distributions.
When investigating a dynamical system under random excitations, it is important that each excitation process should be modeled properly to resemble its measured or estimated statistical and probabilistic properties. In many cases, Gaussian distribution is assumed for convenience of analysis. However, the range of Gaussian distribution is unbounded; namely, there exists the probability of having very large values. This violates the very nature of a real physical quantity, which is always bounded. In the reliability analysis of a physical system, the allowable failure probability is usually very small. Thus, the assumption of Gaussian distribution may affect the reliability estimation significantly.
Based on such a consideration, one versatile model for bounded random processes has been proposed [1] and [2] by using a sinusoidal function with a constant amplitude, a constant average frequency, and a random phase varying as a Wiener process. Such a random process is bounded by the constant amplitude assigned in the model. It is capable of modeling a random process possessing a one-peak spectrum which can be either broad band or narrow band. It has been used, for example, to investigate a straight pipe with a slug flow of a two-phase fluid [3], a structure with a spatially disordered traveling parametric excitation [4], and log-span bridges in turbulent winds [5]. However, the probability density of such a random process cannot be adjusted. Moreover, it has a singular shape with a minimum value at the zero and approaching infinity at the boundaries, which may be inappropriate in many practical cases. It is known that the shape of the probability distribution of the excitation process may have a significant effect on the response of certain types of systems, especially when the system behaviors during the transient state are of interest [6] and [7]. For such cases, the sinusoidal model may be not applicable.
Another type of bounded random processes were generated using nonlinear filters [8], in which the Itô type stochastic differential equations are employed with the drift coefficient adjusted to match the spectral density and the diffusion coefficient adjusted to match the probability density. Although the procedure is capable of generating both unbounded and bounded random processes, it is especially suitable to model bounded random processes with different types of probability distributions. However, the general procedure introduced in Ref. [8] is limited to one-dimensional Itô differential equations which could only model low-pass random processes, namely, processes with spectrum peaks at zero frequency. This is a severe restriction since random processes in many engineering problems have spectra with peaks at nonzero frequencies or multiple peaks.
In this article, the randomized sinusoidal model is reviewed first. It is then extended to model a process with more than one peak in its spectrum. The probability distribution of such a process is also investigated. To extend the method of nonlinear filters to model random processes with spectrum peaks at nonzero frequencies and with multiple spectrum peaks, multi-dimensional Itô stochastic equations are constructed. The spectral densities of the processes can be adjusted by parameters in drift coefficients, and the probability distributions can be of different shapes adjusted by parameters in diffusion coefficients. In both methods, parameters can be selected to model processes with either narrow band or broad band.
Bounded random processes can be generated by two different models: the randomized sinusoidal model and the nonlinear filter model. Both models are capable of generating processes of spectra with multiple peaks and with either narrow or broad band widths. The randomized sinusoidal model is simply to implement by introducing random noises in phase angles, but the probability distributions of the generated processes are of singular shapes and cannot be adjusted. Thus it is suitable for cases in which effects of the probability distribution are not important. In the nonlinear filter model, the drift terms in the Itô equations are adjusted to match the spectral density, and the diffusion terms are determined according to the boundary of the random process and the shape of its probability density. Since it is capable of covering a variety of probability distribution profiles, it may be used for cases in which the probability distribution plays important role, such as system transient behaviors. Nevertheless, it is more difficult to implement in comparison with the randomized sinusoidal model.
