Vibration Analysis in the Presence of Uncertainties Using Universal Grey System Theory

Abstract

The uncertainty present in many vibrating systems has been modeled in the past using several approaches such as probabilistic, fuzzy, interval, evidence, and grey system-based approaches depending on the nature of uncertainty present in the system. In most practical vibration problems, the parameters of the system such as stiffness, damping and mass, initial conditions, and/or external forces acting on the system are specified or known in the form of intervals or ranges. For such cases, the use of interval analysis appears to be most appropriate for predicting the ranges of the response quantities such as natural frequencies, free vibration response, and forced vibration response under specified external forces. However, the accuracy of the results given by the interval analysis suffers from the so-called dependency problem, which causes an undesirable expansion of the intervals of the computed results, which in some case, can make the results unacceptable for practical implementation. Unfortunately, there has not been a simple approach that can improve the accuracy of the basic interval analysis. This work considers the solution of vibration problems using universal grey system (or number) theory for the analysis of vibrating systems whose parameters are described in terms of intervals or ranges. The computational feasibility and improved accuracy of the methodology, compared to interval analysis, are demonstrated by considering one and two degrees-of-freedom (2DOF) systems. The proposed technique can be extended for the uncertainty analysis of any multi-degrees-of-freedom system without much difficulty.