Excitation of Parametric Resonance in Micro Beams By Joule's Heating

Abstract eng:
In this work we investigate the feasibility of excitation of parametric resonance in micro/nano beams by means of Joule’s heating of the beam’s material. A time-dependent voltage applied to the beam’s ends generates an alternate electric current through the beam, which results in a time-dependent heating of the beam’s material and appearance of a time-dependent compressive axial force acting on the beam. By choosing appropriate values of the frequency and magnitude of the voltage, efficient parametric excitation of the beam can be achieved. The main advantage of the suggested approach is its simplicity and compatibility with the existing fabrication processes and common configurations of micro devices. The beam is modeled in the framework of the Euler-Bernoulli theory. The influence of the thermo-elastic effects on the heat generation is neglected and it is assumed that the contribution of the Joule’s heating is dominant. The thermo-mechanical coupling is introduced through the thermal expansion of the beam’s material due to the heating. The time-harmonic heat-transfer problem is solved analytically. An approximate single degree of freedom (lumped) model of the beam is built by means of Galerkin decomposition. The results provided by the lumped model are verified through comparison with the numerical data obtained by the Finite Differences method. In the framework of the lumped model, the stability properties of the beam are described using the canonical damped Mathieu equation. It was found that within the range of the frequencies below the fundamental frequency of the beam a frequency region always exists such that parametric resonance is possible for the excitation voltages below the critical voltage, corresponding to the Euler buckling of the beam.

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