Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2658-6
Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance∗ A. NIKPOURIAN1 , M. R. GHAZAVI1,† ,
S. AZIZI2
1. Mechanical Engineering Department, Tarbiat Modares University, Tehran 14115-111, Iran; 2. Mechanical Engineering Department, Urmia University of Technology, Urmia 57166-93187, Iran (Received Apr. 9, 2020 / Revised Jul. 1, 2020)
Abstract The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied. The microarch is subjected to a combination of direct current (DC) and alternating current (AC) electric voltages. Thin piezoelectric layers are thoroughly bonded on the top and bottom surfaces of the microarch. The piezoelectric actuation is not only used to modulate the stiffness and resonance frequency of the resonator but also to provide the suitable linear frequency ratio for the activation of the internal resonance. The size effect is incorporated by using the so-called modified strain gradient theory. The system is highly nonlinear due to the co-existence of the initial curvature, the mid-plane stretching resulting from clamped anchors, and the electrostatic excitation. The eigenvalue problem is solved to conduct a frequency analysis and identify the possible regions for activating the internal resonance. The effects of the piezoelectric actuation, the electric excitation, and the small-scale effect are investigated on the internal resonance. Exclusive nonlinear phenomena such as Hopf bifurcation and hysteresis are identified in the microarch response. It is shown that by applying appropriate piezoelectric actuation, one is able to activate microarch internal resonance regardless of the initial rise level of the microarch. It is also disclosed that among all the parameters, AC electric voltage has the greatest effect on the energy exchange between the interacting modes. The results can be used to design resonators and internal resonance based micro-electro-mechanical system (MEMS) energy harvesters. Key words microarch resonator, internal resonance, multiple time scales method, micro-electro-mechanical system (MEMS), piezoelectric actuation, electrostatic excitation Chinese Library Classification O341 2010 Mathematics Subject Classification
1
74F20
Introduction
Thanks to its distinguished intrinsic characteristics, micro-electro-mechanical system (MEMS) is becoming more prevalent and grabbing exceeding attention among researchers. Lightweight, small size, low cost, and low power consumption are just a few of its advantages. ∗ Citation: NIKPOURIAN, A., GHAZAVI, M. R., and AZIZI, S. Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2658-6 † Corresponding author, E-mail: [email protected] ©Shanghai University and Springer-Verlag GmbH Germany, part of Springer Natu
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