A theoretical and experimental investigation of parametrically excited nonlinear mechanical systems

Abstract

The response of one- and two-degree-of-freedom (SDOF and 2DOF) systems with quadratic and cubic nonlinearities to fundamental, principal, and combination harmonic parametric excitations is investigated theoretically and experimentally. The method of multiple scales (MMS) is used to determine the equations that describe to first and second order the amplitude- and phase-modulations with time. These equations are used to determine the fixed points and their stability. The perturbation results are verified by integrating the governing equations on a digital computer. The analytical results are in excellent agreement with the numerical solutions. In the SOOF systems with quadratic and cubic nonlinearities, the large responses that oscillate about three equilibrium positions are investigated on the digital and analogue computers. The analogue computer is used to generate a bifurcation diagram in the excitation amplitude versus excitation frequency domain. The digital computer is used to obtain Poincare maps of strange attractors, to investigate larger amplitude responses, and to show the transition to a fractal basin of attraction. The system exhibits 2T, 3T, 4T, ST, 6T, 7T, 8T, 12T, 16T, and ∞T period-multiplying bifurcations.

The response of a flexible cantilever beam with a concentrated mass to principal parametric base excitation of the first bending mode is analyzed theoretically. The model takes into account the geometric nonlinearities due to large displacements. Galerkin's method is used to reduce the fourth-order nonlinear POE to a second-order ODE having periodic coefficients and cubic nonlinearities. The MMS is used to determine steady-state responses and their stability. Experiments are performed on metallic and composite beams; the results show good qualitative agreement with the theory. Chaotic responses are observed in the response of the composite beam.

The response of 2DOF systems with quadratic nonlinearities to a combination parametric resonance in the presence of 2:1 internal resonances is investigated using the MMS. The first-order perturbation solution predicts qualitatively the stable steady-state solutions and illustrates the quenching and saturation phenomena. The reduced equations also predict a transition from periodic to quasi-periodic responses (i.e., Hopf bifurcation). 359 pages, 110 figures.