A new stochastic equivalent linearization implementation for prediction of geometrically nonlinear vibrations

Abstract

In this paper, the problem of random vibration of geometrically nonlinear MDOF structures is considered. The solutions obtained by application of two different versions of a stochastic linearization method are compared with exact (F-P-K) solutions. The formulation of a relatively new version of the stochastic linearization method (energy-based version) is generalized to the MDOF system case. Also, a new method for determination of nonlinear stiffness coefficients for MDOF structures is demonstrated. This method in combination with the equivalent linearization technique is implemented in a new computer program. Results in terms of root-mean-square (RMS) displacements obtained by using the new program and an existing in-house code are compared for two examples of beam-like structures. ~~ Introduction Resurgent interest in high speed fight vehicles and the daily operation of the aging commercial and military aircraft fleets necessitate the further development of sonic fatigue technology to understand the fatigue mechanisms and to estimate the service life of aerospace structures subjected to intense acoustic and thermal loads. Efforts to extend the performance and flight envelope of high speed aerospace vehicles have resulted in structures which may behave in a geometrically nonlinear fashion to the imposed loads. Such behavior can have a significant effect on fatigue life. Further improvements in vehicle performance and system design are hampered by the limited understanding of the physical nature of geometrically nonlinear *NRC Postdoctoral

A Combined Modal/Finite Element Analysis Technique for the Dynamic Response of a Non-Linear Beam to Harmonic Excitation

“…One of the most recent works which employ a similar philosophy to the method presented here is that of Muravyov [11] and Muravyov et al [12]. This approach utilizes a multi-degree-of-freedom modal co-ordinate system.…”

“…Assume also that the linear modal sti!ness is known and has been moved to the left-hand side of equation (10). If equation (12), which initially contains all of the possible terms which may occur in the sti!ness restoring force approximating function, is evaluated at all of the N¹ data points, then the results may be represented in a matrix form as…”

“…This avoids any potential numerical sensitivity problems that may be associated with using a prescribed displacement solution. In reference [12], two di!erent stochastic linearization schemes are compared for the case of a beam-like problem.…”

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“…Wang and Zhang [41], Elishakoff and Zhang [42], Zhang, Elshakoff and Zhang [43] (see also the recent study of Murayov, Turner, Robinson and Rizzi [44]) utilized a potential energy of the system as a parameter with respect to which the linearization should be performed. This criterion is derived here in much simpler manner as follows.…”

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“…RE-DERIVATION OF ENERGY CRITERIA Wang and Zhang [9], Elishako! and Zhang [10], Zhang et al [11] (see also the recent study of Murayov et al [12]) utilized a potential energy of the system as a parameter with respect to which the linearization should be performed. This criterion is derived here in much simpler manner as follows.…”