Various Bifurcation Phenomena in a Nonlinear Curved Beam Subjected to Base Harmonic Excitation

Abstract

Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation, which is governed by a coupled nonlinear equation with both quadratic and cubic nonlinearities, are investigated using the incremental harmonic balance (IHB) method. The nonlinear partial differential equation that governs the motion of the curved beam is given using Hamilton’s principle. A spatially discretized governing equation is derived using Galerkin’s method, yielding a set of second-order nonlinear ordinary different equations. A high-dimensional model that can take nonlinear model coupling into account is derived. Specific attention is paid to the different bifurcation phenomena of frequency responses and amplitude responses of the system without and with an anti-symmetric mode being excited. Numerical results reveal the rich and interesting diverse bifurcation phenomena that have not been presented in the existent literature on the nonlinear vibration of the curved beam system. Saddle-node, Hopf, and period-doubling bifurcations are observed without an anti-symmetric mode being excited. Besides, a symmetry-breaking bifurcation is observed with an anti-symmetric mode being excited. Furthermore, the phase portraits and bifurcation points obtained by the IHB method agree very well with those obtained by the numerical integration using the fourth-order Runge–Kutta method.