Geometrically Nonlinear Forced Transver Vibrations Analysis of Tapered Euler-Bernoulli Beams

Abstract

Tapered beams like structures are widely used in many fields including mechanical and civil engineering, such as high-rise buildings, robot arms, etc. The objective of this paper is to study the geometrically nonlinear free and forced transverse vibrations of tapered beams with a constant width and a linearly varying depth. The theoretical model is based on the Euler-Bernoulli beam theory and the Von Kármán assumptions for geometrical nonlinearity. The motion is assumed to be harmonic and the transverse displacement function of the nonlinear beam is expanded as a series of linear modes, determined by solving the linear problem in terms of Bessel functions satisfying the boundary conditions. The discretized expressions for total beam strain and kinetic energies are derived, and by application of Hamilton's principle, the problem is reduced to a non-linear algebraic system solved using a previously developed approximate method (the so-called second formulation). The effect of the linear variation of depth on the non-linear behaviour of the beam is examined and then illustrated. Using the single-mode approach, the non-linear dynamic behaviour of the tapered beam is studied in the forced case. The effects of the excitation level of the applied harmonic force are investigated and illustrated for various scenarios.