Response of Two Quadratically-Coupled Oscillators to a Principal Parametric Excitation

Abstract

The authors study the dynamics of two oscillators coupled with quadratic nonlinearities in the case of two-to-one internal resonance when the higher mode is subjected to a principal parametric excitation. They use the method of multiple scales to obtain an approximate solution to the equations of motion and investigate theoretically its stability. Then, they verify the analysis experimentally. The authors use a cantilever steel beam and an analog second-order circuit to represent the two oscillators. The interaction between the two systems is achieved by fitting the beam with piezoceramic actuators and a strain gage and coupling the beam with the circuit through electronically generated quadratic nonlinearities. They subject the first mode of the beam to a principal parametric excitation and tune the frequency of the circuit to approximately one-half the frequency of the first mode of the beam. The theoretical and experimental results indicate that the system exhibits complicated responses, such as jumps, the saturation phenomenon, types I and II intermittency, as well as periodic, periodically, and chaotically modulated motions.