An Analytical Study for Nonlinear Free and Forced Vibration of Electrostatically Actuated MEMS Resonators

2019 ◽  
Vol 19 (07) ◽  
pp. 1950072 ◽  
Author(s):  
S. K. Lai ◽  
X. Yang ◽  
C. Wang ◽  
W. J. Liu
Keyword(s):  
Free Vibration ◽  
Lower Order ◽  
Forced Vibration ◽  
Analytical Approximation ◽  
Beam Model ◽  
Decomposition Approach ◽  
Dc Voltage ◽  
Electrostatically Actuated ◽  
Mems Resonators ◽  
Resonance Response

This work aims to construct accurate and simple lower-order analytical approximation solutions for the free and forced vibration of electrostatically actuated micro-electro-mechanical system (MEMS) resonators, in which geometrical and material nonlinearities are induced by the mid-plane stretching, dynamic pull-in characteristics, electrostatic forces and other intrinsic properties. Due to the complexity of nonlinear MEMS systems, the quest of exact closed-form solutions for these problems is hardly obtained for system design and analysis, in particular for harmonically forced nonlinear systems. To examine the simplicity and effectiveness of the present analytical solutions, two illustrative cases are taken into consideration. First, the free vibration of a doubly clamped microbeam suspended on an electrode due to a suddenly applied DC voltage is considered. Based on the Euler–Bernoulli beam theory and the von Karman type nonlinear kinematics, the dynamic motion of the microbeam is further discretized by the Galerkin method to an autonomous system with general nonlinearity, which can be solved analytically by using the Newton harmonic balance method. In addition to large-amplitude free vibration, the primary resonance response of a doubly clamped microbeam driven by two symmetric electrodes is also investigated, in which the microbeam is actuated by a bias DC voltage and a harmonic AC voltage. Following the same decomposition approach, the governing equation of a harmonically forced beam model can be transformed to a nonautonomous system with odd nonlinearity only. Then, lower-order analytical approximation solutions are derived to analyze the steady-state resonance response of such a problem under a combination of various DC and AC voltage effects. Finally, the analytical approximation results of both cases are validated, and they are in good agreement with those obtained by the standard Runge–Kutta method.

scholarly journals Bifurcation Type Change of AC Electrostatically Actuated MEMS Resonators due to DC Bias

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Multiple Scales ◽  
Dc Voltage ◽  
Electrostatically Actuated ◽  
Mems Resonators ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Type Change ◽  
Ground Plate ◽  
Dc Bias ◽  
And Shifts

This paper investigates the nonlinear response of microelectromechanical system (MEMS) cantilever resonator electrostatically actuated by applying a soft alternating current (AC) voltage and an even softer direct current (DC) voltage between the resonators and a parallel fixed ground plate. The AC frequency is near natural frequency. This drives the resonator into nonlinear parametric resonance. The method of multiple scales (MMS) is used to solve the dimensionless differential equation of motion of the resonator and find the steady-state solutions. The reduced order model (ROM) method is used to validate the results obtained using MMS. The effect of the soft DC voltage (bias) component on the frequency response is reported. It is shown that the DC bias changes the subcritical Hopf bifurcation into a cyclic fold bifurcation and shifts the bifurcation point (where the system loses stability) to lower frequencies and larger amplitudes.

scholarly journals Phononic Band Gap and Free Vibration Analysis of Fluid-Conveying Pipes with Periodically Varying Cross-Section

2021 ◽  
Vol 11 (21) ◽  
pp. 10485
Author(s):  
Hao Yu ◽  
Feng Liang ◽  
Yu Qian ◽  
Jun-Jie Gong ◽  
Yao Chen ◽  
...  
Keyword(s):  
Free Vibration ◽  
Band Gap ◽  
Cross Section ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Critical Parameters ◽  
Mode Shapes ◽  
Beam Model ◽  
The Impact ◽  
Fluid Conveying Pipes

Phononic crystals (PCs) are a novel class of artificial periodic structure, and their band gap (BG) attributes provide a new technical approach for vibration reduction in piping systems. In this paper, the vibration suppression performance and natural properties of fluid-conveying pipes with periodically varying cross-section are investigated. The flexural wave equation of substructure pipes is established based on the classical beam model and traveling wave property. The spectral element method (SEM) is developed for semi-analytical solutions, the accuracy of which is confirmed by comparison with the available literature and the widely used transfer matrix method (TMM). The BG distribution and frequency response of the periodic pipe are attained, and the natural frequencies and mode shapes are also obtained. The effects of some critical parameters are discussed. It is revealed that the BG of the present pipe system is fundamentally induced by the geometrical difference of the substructure cross-section, and it is also related to the substructure length and fluid–structure interaction (FSI). The number of cells does not contribute to the BG region, while it has significant effects on the amplitude attenuation, higher order natural frequencies and mode shapes. The impact of FSI is more evident for the pipes with smaller numbers of cells. Moreover, compared with the conventional TMM, the present SEM is demonstrated more effective for comprehensive analysis of BG characteristics and free vibration of PC dynamical structures.

The Lift Force on a Cylinder Vibrating in a Current

1990 ◽  
Vol 112 (4) ◽  
pp. 297-303 ◽  
Author(s):  
G. Moe ◽  
Z.-J. Wu
Keyword(s):  
Free Vibration ◽  
Forced Vibration ◽  
Cross Flow ◽  
Reference Frequency ◽  
Average Force ◽  
Vibrational Response ◽  
Uniform Current ◽  
Extensive Program ◽  
Vibration Tests ◽  
Lock In

This paper reports an extensive program of forced and free vibration tests on a single circular cylinder moving mainly perpendicularly to a uniform current. For both free and forced vibration tests, two cases were investigated: one in which the cylinder was restrained in the in-line direction and the other in which it was supported on suitable springs. The cross-flow vibrational response and hydrodynamic forces on the cylinder were measured. Large variations of motion frequency in the “lock-in” range were found from the free vibration tests. This leads to two different definitions of reduced velocity, namely, a so-called nominal reduced velocity based on one reference frequency and the true reduced velocity based on the actual vibration frequency. When different results are compared, the true reduced velocity should be used. The forced vibration tests showed, as may be expected, that the transverse force in the “lock-in” range on the average will add energy to the cylinder at moderate motion amplitudes and subtract energy at large amplitudes. Some conditions resulting in a steady-state vibration of a flexibly mounted cylinder were analyzed. The actual force traces also show very large and apparently random deviations from the average force amplitude. The results from the forced and the free vibration tests are consistent with each other if the true reduced velocity and reduced amplitude are the same.

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.

Large Amplitude Forced Vibration Analysis of Stiffened Plates under Harmonic Excitation

2012 ◽  
pp. 26-61
Author(s):  
Anirban Mitra ◽  
Prasanta Sahoo ◽  
Kashinath Saha
Keyword(s):  
Free Vibration ◽  
Forced Vibration ◽  
Large Amplitude ◽  
Vibration Analysis ◽  
Mathematical Formulation ◽  
Free Vibration Analysis ◽  
Harmonic Excitation ◽  
Stiffened Plates ◽  
Backbone Curve ◽  
Forced Vibration Analysis

Large amplitude forced vibration behaviour of stiffened plates under harmonic excitation is studied numerically incorporating the effect of geometric non-linearity. The forced vibration analysis is carried out in an indirect way in which the dynamic system is assumed to satisfy the force equilibrium condition at peak excitation amplitude. Large amplitude free vibration analysis of the same system is carried out separately to determine the backbone curves. The mathematical formulation is based on energy principles and the set of governing equations for both forced and free vibration problems derived using Hamilton’s principle. Appropriate sets of coordinate functions are formed by following the two dimensional Gram-Schmidt orthogonalization procedure to satisfy the corresponding boundary conditions of the plate. The problem is solved by employing an iterative direct substitution method with an appropriate relaxation technique and when the system becomes computationally stiff, Broyden’s method is used. The results are furnished as frequency response curves along with the backbone curve in the dimensionless amplitude-frequency plane. Three dimensional operational deflection shape (ODS) plots and contour plots are provided in a few cases.

Micromachined Snap-In Resonators

2012 ◽  
Vol 2012 (DPC) ◽  
pp. 001920-001935 ◽  
Author(s):  
Colin Stevens ◽  
Robert Dean ◽  
Chris Wilson
Keyword(s):  
Resonant Frequency ◽  
Proof Mass ◽  
Pressure Sensors ◽  
Mass System ◽  
Dc Voltage ◽  
Movable Electrode ◽  
Mems Resonators ◽  
Spring Force ◽  
In Series ◽  
Fixed Electrode

MEMS resonators have many applications, including micromachined gyroscopes, resonating pressure sensors and RF devices. Typically, MEMS resonators consist of a proof mass and suspension system that allows the proof mass motion in one or two directions. Micromachined actuators provide kinetic energy to the proof mass, usually at its resonant frequency. In the simplest resonators, the actuators are driven with an AC signal at or near the resonant frequency. In more complex resonators, the actuator-proof mass system is placed in an amplifier feedback circuit so that the electromechanical system self-resonates. MEMS parallel plate actuators (PPAs) are simple to realize, yet complex nonlinear variable capacitors. If a DC voltage is applied in attempt to move the proof mass greater than 1/3 of the electrode rest gap distance, the device becomes unstable and the electrodes snap into contact. A current limiting resistor is often placed in series with the PPA to limit short circuit current due to a snap-in event. Consider the effect of placing a large resistor, on the order on 10 meg-Ohms, in series with the PPA. Then apply a DC voltage across the resistor-PPA pair of sufficient voltage to cause snap-in. Once the electrostatic force (ES) exceeds the spring force (SF), the electrodes will accelerate toward each other. The capacitance between the electrodes swells as the separation distance shrinks. Since the large resistor limits the charging rate of the capacitor, the voltage across it drops. Once the SF exceeds the EF, the momentum of the movable electrode brings it into contact with the fixed electrode, discharging the capacitor. The movable electrode then accelerates away from the fixed electrode while the resistor slowly allows recharging. After recharging, the cycle repeats resulting in stable oscillation. This resonator requires only a DC power supply, a resistor and a MEMS PPA.

Solution of Free Vibration Equations of Euler-Bernoulli Cracked Beams by Using Differential Transform Method

2011 ◽  
Vol 110-116 ◽  
pp. 4532-4536 ◽  
Author(s):  
K. Torabi ◽  
J. Nafar Dastgerdi ◽  
S. Marzban
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Transform Method ◽  
Beam Model ◽  
Cracked Beam ◽  
Transform Method ◽  
Simple Method ◽  
Differential Transform

In this paper, free vibration differential equations of cracked beam are solved by using differential transform method (DTM) that is one of the numerical methods for ordinary and partial differential equations. The Euler–Bernoulli beam model is proposed to study the frequency factors for bending vibration of cracked beam with ant symmetric boundary conditions (as one end is clamped and the other is simply supported). The beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in both vertical displacement and rotational due to bending. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section by using DTM and analytical solution. The results show that DTM provides simple method for solving equations and the results obtained by DTM converge to the analytical solution with much more accurate for both shallow and deep cracks. This study demonstrates that the differential transform is a feasible tool for obtaining the analytical form solution of free vibration differential equation of cracked beam with simple expression.

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