Perturbation Technique
M. Zamanian; M. Hadilu; B. Firouzi
Abstract
In this paper, a comparison is made between direct and indirect perturbation approaches to solve the non-linear vibration equations of a piezoelectrically actuated cantilever microbeam. In this comparison, the equation of motion is considered according to Euler-Bernoulli theory with considering the non-linear ...
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In this paper, a comparison is made between direct and indirect perturbation approaches to solve the non-linear vibration equations of a piezoelectrically actuated cantilever microbeam. In this comparison, the equation of motion is considered according to Euler-Bernoulli theory with considering the non-linear geometric and inertia terms resulted from shortening effect. In the direct perturbation approach, the multiple scales method is directly applied to the partial differential equation of motion. In the indirect approach, the multiple scales perturbation technique is applied to the discretized equation of motion. It is shown that, if the equation of motion is discretized using one non-uniform microbeam mode shape as a comparison function, then the results of indirect perturbation approach will be identical to those of the direct perturbation approach. Moreover, it is observed that discretization using one uniform microbeam mode shape as a comparison function results in a different output. The concept of non-uniform microbeam mode shape is the linear mode shape of the microbeam by considering the geometric and inertia effects of the piezoelectric layer.
Control
M. Zamanian; S. A. A. Hosseini
Abstract
This article studied static deflection, natural frequency and nonlinear vibration of a clamped-clamped microbeam under discontinues electrostatic actuation. The electrostatic actuation was induced by applying AC-DC voltage between the microbeam and electrode plate. In contrast to previous works, it was ...
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This article studied static deflection, natural frequency and nonlinear vibration of a clamped-clamped microbeam under discontinues electrostatic actuation. The electrostatic actuation was induced by applying AC-DC voltage between the microbeam and electrode plate. In contrast to previous works, it was assumed that length of the electrode plate was smaller than that of the microbeam. In addition, it was assumed that a layer whose length was equal to that of the electrode plate was deposited on the lower side of the microbeam. Equation of motion was derived using Newton's second law. The static deflection due to the DC electrostatic actuation and the natural frequency about this position were obtained using the Galerkin method. Nonlinear vibration of the system due to the AC electrostatic actuation was obtained using the multiple scale perturbation method. Variations of static deflection, pull-in voltage, natural frequency and frequency response of vibration about the static deflection of microbeam with respect to variations of second layer length, second layer thickness, electrode plate length and value of electrostatic actuation were also studied. It was shown that, depending on the value of these parameters, static deflection and natural frequency of vibration about static deflection increased or decreased. Moreover, it was demonstrated that, depending on the value of these system parameters, nonlinear vibration of the system due to the AC electrostatic actuation might be realized as a softening or hardening behavior.
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