Analytical solution for vibration of functionally graded beams with variable cross-sections resting on Pasternak elastic foundations

Abstract

An efficient approach to obtain accurate solutions for free vibration of functionally graded beams (FGB) with variable cross-sections and resting on Pasternak elastic foundations is presented. The general expressions of displacements and stresses which thoroughly result from the dynamic differential equations and boundary conditions for FGB with arbitrarily variable cross-sections are obtained by using the separate variable method and Laplace transform on the basis of 2-D elastic theory. Meanwhile, the frequency equations for free vibration of FGB with variable cross-sections are derived by applying Fourier series expansion along the lower and upper boundary conditions of the FGB. Validity of the developed approach as well as its effectiveness and accuracy are verified through analysing several typical FGB examples, and this provides a potential alternative approach for analysing vibration of functionally graded components with variable cross-sections in case of ultra-high precision requirement in modern mechanical systems. The effect of geometric and mechanical parameters on vibration frequency and mode shapes of FGB with variable cross-sections resting on Pasternak elastic foundations is further analysed.