Large Amplitude Free Vibration of Elastically Restrained Tapered Beams Resting on Non-linear Elastic Foundation

Abstract

Large amplitude vibrational characteristics of variable section thin beams with edge rotations restrained by elastic torsional springs and supported on a cubic non-linear elastic foundation are studied. The motion equations and the corresponding boundary conditions are derived by employing Green’s strain together with von Kármán geometric non-linearity assumptions. The derived equations are discretized in the spatial domain using the differential quadrature method. The reliability and accuracy of the method are assessed through a comparative analysis of various available methods for beams with different geometrical parameters and boundary conditions. The study investigates the impact of various parameters on the non-linear to linear frequency ratios (NLFRs) of doubly linear and parabolic tapered beams. It is found that for double-linear tapered beams, the first three frequency ratios approach maximum values and then decrease by increasing the truncation factors. For double-parabolic tapered beams, first and third frequency ratios have maximum values, while the second frequency ratio increases initially and then remains constant. In addition, the transverse elastic coefficients depend on the shearing layer coefficient, especially for the doubly linear tapered beams. Also, in most cases, the frequency ratios decrease by increasing the transverse elastic coefficients. However, for the great values of shear layer elastic constant, the first NLFR of beams with a double-linear taper increases as the transverse elastic constants increase.