Two-Mode Interaction of a Beam with a Nonlinear Boundary Condition

Abstract

In order to investigate modal interactions in a subharmonic resonance of a beam with a nonlinear boundary condition, we consider a beam constrained by a nonlinear spring to a harmonic excitation. The resonance conditions considered are ωn ≈ 3ωm and Ω ≈ 3ωn , where 3ωm and 3ωn are the natural frequencies and Ω is the excitation frequency. This nonlinear problem is governed by a linear partial differential equation, initial conditions and a nonlinear and inhomogeneous boundary condition. The method of multiple scales is used to transform the problem into a system of autonomous ordinary differential equations for amplitude and phase variables. The steady-state responses and their stability are determined by use of this system. In order to check the validity of the analytical solution we solve the initial and boundary value problem by means of a finite difference analysis.