SIMULTANEOUS COMBINATION, PRINCIPAL PARAMETRIC AND INTERNAL RESONANCES IN A SLENDER BEAM WITH A LUMPED MASS: THREE-MODE INTERACTIONS

The non-linear dynamics of a parametrically base-excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined. The mass and the position of the attached element are so adjusted that the system exhibits 1:3:5 internal resonance by virtue of which a combination resonance of typeω1 +ω3occurs simultaneously in the system. The method of multiple scales (mms) is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state, periodic, quasi-periodic and chaotic responses of the system for different control parameters, namely, frequency and amplitude of base excitation, damping, internal detuning, etc. The results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion. Poincaré section and Lyapunov exponents are used to characterize the chaotic responses.