Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

Abstract

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded (AFG) non-uniform beam resting on an elastic foundation. The mathematical formulation is displacement based and derivation of governing equations is accomplished following Hamilton's principle. The foundation has been mathematically incorporated into the analysis as a set of linear springs. According to the basic assumption of the present method force equilibrium condition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently represented as a static one, which is solved by following a numerical implementation of the Broyden method. It is a method that utilizes the Jacobian matrix and subsequent correction of the initial Jacobian to solve a system of nonlinear equations. The large amplitude dynamic behaviour of the system in terms of non-dimensional frequency response curves is validated against established results and new results are furnished for a parabolic tapered AFG beam on a linear elastic foundation.

Nonlinear response of axially functionally graded Timoshenko beams on elastic foundation under harmonic excitation

“…Only a few articles are available in literature relating to elastic foundation supported AFG thin beams, where Euler-Bernoulli theory is utilised for mathematical formulation of the problem. Huang and Luo [42] presented a new and simple method to calculate the critical buckling loads of beams with axial inhomogeneity on elastic foundation, whereas, Lohar et al [43][44][45] investigated the dynamic behaviour of AFG beam on elastic foundation. Research work in the field of AFG Timoshenko beam is extremely rare.…”