Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonators: Comparison Between Boundary Value Problem Model and Reduced Order Model

2018 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value ◽  
Mode Shapes ◽  
Voltage Response ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

Dynamic Analysis of Electrostatically Actuated Nanobeam Based on Strain Gradient Theory

2015 ◽  
Vol 15 (04) ◽  
pp. 1450059 ◽  
Author(s):  
Ehsan Maani Miandoab ◽  
Hossein Nejat Pishkenari ◽  
Aghil Yousefi-Koma
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Response ◽  
Strain Gradient ◽  
Strain Gradient Theory ◽  
Mode Shapes ◽  
Gradient Theory ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Electrostatically Actuated

In this study, dynamic response of a micro- and nanobeams under electrostatic actuation is investigated using strain gradient theory. To solve the governing sixth-order partial differential equation, mode shapes and natural frequencies of beam using Euler–Bernoulli and strain gradient theories are derived and then compared with classical theory. Galerkin projection is utilized to convert the partial differential equation to ordinary differential equations representing the system mode shapes. Accuracy of proposed one degree of freedom model is verified by comparing the dynamic response of the electrostatically actuated micro-beam with analogue equation and differential quadrature methods. Moreover, the static pull-in voltages of micro-beams found by one DOF model are compared with the reported data in literature. The main advantage of proposed method based on the Galerkin method is its simplicity and also its low computational cost in analyzing the dynamic and static responses of micro- and nanobeams. Additionally, effect of axial force, beam thickness and applied voltage are analyzed. The results obtained based on strain gradient theory, are compared with classical and modified couple stress theories which are the special cases of the strain gradient theory. It is shown that strain gradient theory leads to higher frequency and lower amplitude in comparison with two other theories.

Reduced Order Model of Nanoelectromechanical Systems to Include Casimir Effect

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Roberto J. Zapata ◽  
Martin W. Knecht
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Second Order ◽  
Reduced Order Model ◽  
Order Partial Differential Equation ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order

This paper deals with electrostatically actuated nanoelectromechanical (NEMS) cantilever resonators. The dynamic behavior is described by a second order partial differential equation. The NEMS cantilever resonator device is actuatedby an AC voltage resulting in a vibrating motion of the cantilever. At nano scale, squeeze film damping, Casimir force, and fringing effects significantly influence the dynamic behavior or the cantilever beam. The second order partial differential equation is solved using the Reduced Order Model (ROM) method. The resulting time dependent second order differential equations system is then transformed into a first order differential equations system. Numerical simulations were conducted using Matlab solver ode15s.

Voltage Response of Parametric Resonance of MEMS Circular Plates Under Hard Excitations

2019 ◽  
Author(s):  
Julio S. Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Voltage Response ◽  
Circular Plates ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

Abstract This work deals with the voltage response of parametric resonance of electrostatically actuated microelectromechanical (MEMS) circular plates under hard excitations. Method of Multiple Scales (MMS) and Reduced Order Model (ROM) method using two modes of vibration are used to predict the voltage-amplitude response of the MEMS circular plates. ROM is solved using AUTO 07p, a software package for continuation and bifurcation. MMS used in this paper has one term in the electrostatic force being considered significant. This is the way MMS is used to model hard excitations. MMS shows results similar to those of ROM at lower amplitudes and lower voltages. The differences between the two methods, MMS and ROM, are significant in high amplitudes for all voltages, and the differences are significant in all amplitudes for larger voltages. Significant differences can be noted in the effect of different parameters such as the detuning frequency and damping on the voltage response. ROM AUTO 07p is calibrated using ROM time responses in which the ROM is solved using the solver ode15s in Matlab.

Voltage Response of Circular Plate MEMS Resonators Under Superharmonic Resonance

2018 ◽  
Author(s):  
Martin Botello ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Circular Plate ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Voltage Response ◽  
Voltage Amplitude ◽  
Amplitude Response ◽  
Partial Differential ◽  
Superharmonic Resonance

The superharmonic resonance of second order of microelectro-mechanical system (MEMS) circular plate resonator under electrostatic actuation is investigated. The MEMS resonator consists of a clamped circular plate suspended over a parallel ground plate under an applied Alternating Current (AC) voltage. The AC voltage is characterized as hard excitation, i.e. the magnitude is large enough, and the operating frequency is near one-fourth of the natural frequency of the resonator. Reduced Order Model (ROM), based on the Galerkin procedure, transforms the partial differential equation of motion into a system of ordinary differential equations in time using mode shapes of vibration of the circular plate resonator. Three numerical methods are used to predict the voltage-amplitude response of the MEMS plate resonator. First, the Method of Multiple Scales (MMS) is directly applied to the partial differential equation of motion which is this way transformed into zero-order and first-order problems. Second, ROM using two modes of vibration is numerical integrated using MATLAB to predict time responses, and third, the AUTO 07P software for continuation and bifurcation to predict the voltage-amplitude response. The nonlinear behavior (i.e. bifurcation and pull-in instability) of the system is attributed to the inclusion of viscous air damping and electrostatic force in the model. The influences of various parameters (i.e. detuning frequency and damping) are also investigated in this work.

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

ROM of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Resonators

2015 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Second Order ◽  
Voltage Response ◽  
Order Model ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Method Effects ◽  
Ground Plate

This paper deals with the voltage-amplitude response (or voltage response) of superharmonic resonance of second order of MEMS resonator sensors under electrostatic actuation. The system consists of a MEMS flexible cantilever above a parallel ground plate. The AC frequency of actuation is near one fourth the natural frequency. The voltage response of the superharmonic resonance of second order of the structure is investigated using the Reduced Order Model (ROM) method. Effects of voltage and damping voltage response are reported.

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