Nonlinear analysis of the forced response of a beam with three mode interaction

Abstract

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,ωn . Three mode interaction (ω2 ≈ 3ω1 and ω3 ≈ ω1 + 2ω2) is considered and its influence on the response is studied. The case of two mode interaction (ω2 ≈ 3ω1) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.