Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

Abstract

Abstract. The objective of this paper is to determine dynamic instability areas of moderately thick beams and frames. The effect of moderate thickness on resonance frequencies is considered, with transverse shear deformation and rotatory inertia taken into account. These relationships are investigated using the Timoshenko beam theory. Two methods, the harmonic balance method (HBM) and the perturbation method (PM) are used for analysis. This study also examines the influence of linear dumping on induced parametric vibration. Symbolic calculations are performed in the Mathematica programme environment. The majority of studies reported in the literature use numerical analysis for determining resonance areas. Only a few researchers have adopted an analytical approach.The purpose of this article is to propose two methods for determining parametric resonance areas for moderately thick beams and frames under various support conditions. These methods, never before used for this purpose, are the harmonic balance method (HBM) and the perturbation method (PM). The latter of these two methods represents a surprisingly simple tool for achieving the goal, as its results are very close to those obtained from the standard harmonic balance method which is far more troublesome in application.An analytical approach is used to analyze simply supported moderately thick beams, and a numerical technique, based on the finite element method with the use of physical shape functions [32] is applied to other beams and frames. The analysis covers the effect of moderate thickness on resonance frequencies and the influence of type of fixing used in the beams and frames on the instability areas. For this purpose, shear deformation and rotatory inertia are taken into account. The Timoshenko beam theory is applied to examine how these factors affect resonance frequency values. The results are compared with those obtained with the Bernoulli-Euler beam theory.In addition, the effect of linear dumping of induced parametric vibration is considered. Analysis of a simply supported beamThe following assumptions are adopted in physical and numerical modeling:• The beam is made of an isotropic homogenous linear elastic material with Young's modulus E, shear modulus G and Poisson's ratio v.

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