M.Sc Student | Joffe Aharon |
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Subject | Parametric Amplification in Electrostatic Resonators |

Department | Department of Mechanical Engineering |

Supervisor | Professor David Elata |

Full Thesis text |

In this work, the steady-state dynamic response of a parametrically amplified linear resonator is investigated. A parametrically amplified linear resonator is essentially a classic linear resonator, which is driven by a harmonic driving force, but simultaneously, it is also driven by a time-modulated stiffness which constitutes the parametric excitation.

The dynamic response is explored using numerical integration, leading to valuable insights. When the frequency of parametric amplification is identical to the natural frequency of the system, it generates an increased sensitivity at the linear resonance. In addition, non-resonant parametric amplification is shown to generate additional functionality by adding a second peak at a non-resonant frequency. The distribution of energies in response components with distinct frequencies is shown numerically using a 3-D mapping. This reveals that the response components should be considered in frequency pairs.

An analytic solution of the steady-state response is developed based on the insights obtained from the numerical solutions. This analytic solution is a simple and efficient approximation that considers a finite number of response frequencies. The algebraic set of equations forms a sparse matrix, which can be efficiently solved using Matlab. The convergence and accuracy of this solution are presented, validating the analytic approximation of the system. This approximate analytic solution drastically reduces computation time relative to the numerical integration.

The influence of each system parameter, such as: the quality-factor; the amplitude of stiffness modulation, and the frequency of stiffness modulation, is explored. It is shown that the linear relation between the quality-factor and the peak amplitude of motion, known in linear resonators, is maintained in the parametrically amplified system for both the primary and secondary peaks.