An assessment of approximate and efficient methods for the dynamic analysis of non-linear beams

Abstract

A literature survey on the field of non-linear vibrations of beams and thin panels has been performed with the aim of identifying efficient and accurate techniques which might be used in the aerospace industry. Attention is then focused on beam systems to allow candidate methods to be assessed. Emphasis is placed on the study of geometrically non-linear vibration of isotropic slender beams with simply-supported end conditions when exposed to external excitation. The Buffing's equation has been derived and its applicability in describing non-linear beam vibrations is justified. The characteristics of non-linear harmonic vibration of the beam are studied. The non-linear response is first approximated by the Harmonic Balance Method based on the Buffing's equation and then compared with three sets of published results. The single- degree-of-freedom (SDOF) Buffing's equation is solved by a time-domain numerical method. An ANSYS® Finite Element Analysis is carried out to simulate the vibration problem. These solutions are compared with the Harmonic Balance results. The non-linear response of a simply-supported beam to uniformly distributed random white-noise pressure is further studied. The non-linear root-mean-square displacement re- sponses are approximated by applying the Birect Equivalent Linearisation Method to the SBOF Buffing's equation. The accuracy of the approximation is assessed by comparing the results with those obtained by the numerical integration of the SDOF Buffing's equation with simulated Gaussian white noise. The total non-linear displacement response due to the first two modes of vibrations for the simply-supported beam subjected to various excitations are computed-by solving both the uncoupled and coupled Buffing's equations by a step-by-step numerical integration scheme. The differences between the uncoupled and coupled responses are analysed. Conclusions are drawn about the precision of the various methods of analysis and the importance of mode-coupling in non-linear beam vibrations.

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