Nonlinear Response of a Parametrically Excited Buckled Beam A. M. ABOU-RAYAN, A. H. NAYFEH, D. T. MOOK Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 U.S.A. and M. A. NAYFEH Department of Electrical Engineering, University of Maryland, College Park, MA, U.S.A. (Received: 8 April 1992; accepted: 17 August 1992) Abstract. A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented. Key words: Chaos, buckled beam, parametric resonance, bifurcations. 1. Introduction Excitations of vibratory systems are frequently classified according to where the forcing function appears in the equation of motion. If the forcing function appears as a nonhomogeneous term, the excitation is said to be external. But, if the forcing function appears as a coefficient, the excitation is said to be parametric. In structural systems, parametric excitations occur when such parameters as stiffness, inertia, and damping vary with time. Because the apparent stiffness of a beam is influenced by an axial force, a time-varying axial load produces a parametric excitation. For a buckled beam the equation of motion for a single-mode approximation is governed by the Duffing equation with a negative linear stiffness and a softening-type cubic nonlinearity. Although the Duffing oscillator is simple in form, its response displays such typically nonlinear phenomena as bifurcations, fundamental, subharmonic, and superbarmonic resonances [1], narrow-band random behavior, and chaos. Externally excited Duffing oscillators, for the case of primary resonance and low levels of excitation, have been investigated by many authors. Using the method of harmonic balance, Tscng and Dugundji [2] analyzed the response of a fixed-fixed buckled beam and obtained periodic responses. They determined the stability of these responses by solving a corresponding variational Hill-type equation. Using magnetic forces to excite a beam, Moon [3] found chaos experimentally and proposed a criterion, based on the amplitude and frequency of the excitation, Nonlinear Dynamics 4: 499-525, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands