Development of a non-Gaussian closure scheme for dynamic systems involving non-linear inertia
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The main objective of this investigation is to develop a non-Gaussian closure scheme adapted for the analysis of random response statistics of nonlinear dynamic systems which are subjected to parametric random excitations. The scheme is based on the asymptotic expansion of the non-Gaussian probability density. The technique is found to have two main advantages. The first is that it resolves an observed contradiction of results obtained by other techniques. The second is that it explores new response characteristics not predicted by other methods. The scheme is applied to a number of dynamic systems possessing single and two degrees-of-freedom and various types of nonlinearities. Numerical solutions are obtained and their validity is examined according to certain criteria based on the preservation of moment properties and Schwarz's inequality. Higher order terms are considered to examine the convergence of the Edgeworth expansion. It is found that the inclusion of the third and fourth order semi-invariants is adequate for the series convergence. The response of nonlinear single degree-of-freedom systems exhibits the occurrence of a jump in the response statistics at a certain excitation level which is mainly governed by the system linear damping factor. This new feature may be attributed to the fact that the non-Gaussian closure more adequately models the nonlinearity, and thus results in characteristics that are similar to those of deterministic nonlinear systems. The method is also used to determine the random response of an elevated water tower subjected to a random ground motion. The tower is represented by a two-degree-of-freedom system with cubic nonlinear coupling. In the neighborhood of the internal resonance condition r= w2/w1 =1/3, where w1 and w2 are the normal mode frequencies of the system, the nonlinear modal interaction takes the form of energy exchange between the two modes. Unlike the Gaussian solution, the non-Gaussian closure solution is found to achieve a strictly stationary response in the time domain. The response mean squares are presented as functions of the internal resonance detuning parameter r = 1/3 + 0(e), where Â£ is a small parameter, for various system parameters. Unbounded response mean squares are found to take place at regions above and below certain values of the internal resonance r=l/3. For regions well remote from the exact internal tuning the system exhibits the features of the linear response.