Nonlinear Forced Vibration of Piezoelectric and Electrostatically Actuated Nano/Micro Piezoelectric Beam

In this study, the nonlinear vibration analysis of nano/micro electromechanical (NEMS/MEMS) piezoelectric beam exposed to simultaneous electrostatic and piezoelectric actuation. NEMS/MEMS beam actuate with combined DC and AC electrostatic actuation on the through two upper and lower electrodes. An axial force proportional to the applied DC voltage is produced by piezoelectric layers present via a DC electric voltage applied in the direction of the height of the piezoelectric layers. The governing differential equation of the motion is derived using Hamiltonian principle based on the Eulere-Bernoilli hypothesis and then this partial differential equation (PDE) problem is simplified into an ordinary differential equation (ODE) problem by using the Galerkin approach. Hamiltonian approach has been used to solve the problem and introduce a design strategy. Phase plane diagram of piezoelectric and electrostatically actuated beam has plotted to show the stability of presented nonlinear system and natural frequencies are calculated to use for resonator design. The result compare with the numerical results (fourth-order Runge-Kutta method), and approximate is more acceptable and results show that one could obtain a predesign strategy by prediction of effects of mechanical properties and electrical coefficients on the stability and forced vibration of common electrostatically actuated micro beam

Effect of mass diffusion on the damping ratio in micro-beam resonators

AbstractPresent investigation is focused on studying the effect of mass diffusion on the quality factor of the micro-beam resonators. Equation of motion is obtained using Hamilton’s principle and also the equations of thermo-diffusive elastic damping are established using two dimensional non-Fourier heat conduction and non-Fickian mass diffusion models. Free vibration of a clamped–clamped micro-beam with isothermal boundary conditions at both ends, and also a cantilever micro-beam with adiabatic boundary condition assumption at the free end, is studied using Galerkin reduced order model formulation for the first mode of vibration. Mass diffusion effects on the damping ratio are studied for the various micro-beam thicknesses and temperatures and the obtained results are compared with the results of a model in which the mass diffusion effect is ignored. In addition to the classic critical thickness of thermoelastic damping, a new critical thickness concerning mass diffusion is introduced

Multi-Physical Study of MEMS Resonators and Oscillators

First, this research presents experimental and theoretical investigation of the response of micro scale dual-plate thermal-piezoresistive resonators (TPR) and self-sustained oscillators (TPO) to different gases and pressures. It is demonstrated that the resonant frequency of such devices follows particular trends in response to the changes in the surrounding gas and its pressure. A mathematical model has been derived to explain the damping dependent frequency shift characteristic of TPO. The solution of the model indicates that the stiffness of the actuator beam decreases as the value of damping coefficient drops at lower gas density caused by the change in the gas molecular mass or pressure. When operated in the TPR mode of the same device, however, the frequency shift of the same silicon structure is mainly a function of gas thermal conductivity. The two different sensing mechanisms are confirmed by the measurement results showing opposite frequency shift for the TPR and TPO in helium-nitrogen mixtures. In pressure tests, frequency shifts as high as -2300ppm were measured for a TPO by changing the air pressure from 84kPa to 43kPa. Second, the effect of geometry on thermoelastic damping (TED) in micro beam resonators is evaluated using an eigenvalue finite element formulation and its corresponding customized MATLAB program. The vented clamped-clamped (CC) and clamped-free (CF) beams with square-shaped vents along their center lines, are both analyzed. The quality factor and resonant frequency are obtained as functions of various geometrical parameters including the location, number and size of the vents. The numerical results reveal that the addition of vent sections in the clamped end region can significantly enhance the quality factor under TED. The maximum improved quality factor as high as 3,801 and 2,257 times as those of the solid CC and CF beams are realized. The methodology presented in this work provides a useful tool in the design optimization of micro beam resonators against TED. Third, a new method to compensate the TED by taking the advantage of piezoresistive effect is proposed. Such method is implemented by applying an electrostatic field through the MEMS beam resonator with negative piezoresistive coefficient. In the case of vibration, the stretched part of the beam generates higher electrical power thus higher temperature and vice versa. Such temperature distribution can compensate the opposite thermoelastic temperature to suppress TED. The work principle is described by a set of coupled differential equations and then solved by an eigenvalue finite element method. The numerical result indicates that the TED in beam resonators can be completely suppressed when the strength of electrical field reaches a critical value, namely CEF. The value of the CEF is further analyzed by parametric studies on various material properties and geometric factors