Dynamic analysis of a nonlinear parametrically excited system using electromagnets

Abstract

Parametrically excited systems, where defining system parameters vary periodically with an independent variable (time), are a popular research topic in engineering. Cable-stayed bridges, free hanging marine flexible risers, planetary gear systems and other engineering structures are often subject to parametric excitation. Due to the high amplitude of responses as a result of parametric amplification, parametric excitation can be disastrous if not account. Parametric amplification in mechanical and electrical systems can be exploited for designing vibration energy harvesters and electrical filters.

This thesis contains various work on Linear and Nonlinear Parametrically Excited (LPE) and (NPE) systems. The system of interest is a clamped-free cantilever beam which is modelled as a single degree of freedom system. An electromagnetic system is used to generate time-periodic stiffness and control nonlinearities. The forces applied from the electromagnetic system are found analytically in order to compare this work to similar systems and to conduct parametric studies. The nonlinear electromechanical coupling, electrical damping, and the induced current is implemented in the analytical model. The free responses of LPE and NPE systems are investigated analytically with the method of averaging and harmonic balance, with particular attention paid to the stability of these systems. The effect of cubic and cubic parametric nonlinearity on the NPE systems is demonstrated through some analytical and experimental investigations. This study on the NPE system is employed to show the effect of time-periodic stiffness and stiffness nonlinearities on attenuating or amplifying the response. Increasing the response amplitude of amplifiers and filters with an electromagnetic system can be achieved by tuning the system at the parametric resonance. Furthermore, the electromagnetic system can be configured to reduce the electrical damping, or to control the nonlinearities and consequently increase the parametric amplification.

The responses and stability of the NPE system subject to a harmonic base excitation are investigated analytically and experimentally. Unlike previous studies, the parametric excitation is independent of the base excitation. A careful selection of system parameters, such as parametric amplitude, relative phase and cubic parametric nonlinearity, can result in significant parametric amplification, and can prevent the jump between stable solutions. Parametric attenuation can also be achieved by controlling the phase difference between the base and the parametric excitation. This study has successfully demonstrated the importance of nonlinearity in parametrically excited systems.

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ORCID for Emiliano Rustighi: orcid.org/0000-0001-9871-7795

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