scholarly journals Nonlinear Coupled Vibration of Electrically Actuated Arch with Flexible Supports

Micromachines ◽  
2019 ◽  
Vol 10 (11) ◽  
pp. 729 ◽  
Author(s):  
Ze Wang ◽  
Jianting Ren
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Single Mode ◽  
Coupled Vibration ◽  
Order Ordinary Differential Equation ◽  
Electrically Actuated ◽  
Second Mode ◽  
Flexible Supports ◽  
Stable Periodic ◽  
Phase Angles

The nonlinear coupled vibration of an electrically actuated arch microbeam has attracted wide attention. In this paper, we studied the nonlinear dynamics of an electrically actuated arch microbeam with flexible supports. The two-to-one internal resonance between the first and second modes is considered. The multiple scales method is used to solve the governing equation. Four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles were obtained. The equilibrium solution and its stability are determined. In the case of the primary resonance of the first mode, stable periodic motions and modulated motions are determined. The double-jumping phenomenon may occur. In the case of the primary resonance of the second mode, single-mode and two-mode solutions are possible. Moreover, double-jumping, hysteresis, and saturation phenomena were found. In addition, the approximate analytical results are supported by the numerical results.

Two-to-One Internal Resonances in Buckled Beams

1995 ◽  
Author(s):  
Wayne Kreider ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Buckled Beams ◽  
Second Mode ◽  
Quadratic Nonlinearities ◽  
Period Doubling Bifurcations

Abstract The vibrations of buckled beams with two-to-one internal resonances (ω2 ≈ 2ω1) about a static buckled position are analyzed. General boundary conditions and harmonic excitations (frequency Ω) in both the transverse and axial directions are considered. The analysis assumes a unimodal static buckled deflection, considers quadratic nonlinearities only, and determines the amplitude and phase modulation equations via the method of multiple scales. The following specific cases are treated: Ω ≈ 2ω2, Ω ≈ ω1 + ω2, and Ω ≈ ω1. From the modulation equations for a primary resonance of the second mode (i.e., Ω ≈ ω2), one-mode and two-mode stable equilibrium solutions are found in addition to dynamic solutions caused by Hopf bifurcations. In the region of dynamic solutions, a variety of phenomena are documented, including period-doubling bifurcations, intermittency, chaos, and crises.

scholarly journals Three-to-One Internal Resonance in MEMS Arch Resonators

Sensors ◽  
2019 ◽  
Vol 19 (8) ◽  
pp. 1888 ◽  
Author(s):  
Ze Wang ◽  
Jianting Ren
Keyword(s):  
Governing Equation ◽  
Internal Resonance ◽  
Phase Plane ◽  
Multiple Scales ◽  
Order Model ◽  
Order Ordinary Differential Equation ◽  
Microelectromechanical System ◽  
Reduced Order ◽  
First Order ◽  
Phase Angles

We present an investigation of the nonlinear dynamics of a microelectromechanical system (MEMS) arch subjected to a combination of AC and DC loadings in the presence of three-to-one internal resonance. The axial force resulting from the residual stress or temperature variation is considered in the governing equation of motion. The method of multiple scales is used to solve the governing equation. A four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles is obtained. The equilibrium solution and its stability of the modulation equations are determined. Moreover, we also obtain the reduced-order model (ROM) of the MEMS arch employing the Galerkin scheme. The dynamic response is presented in the form of time traces, Fourier spectrum, phase-plane portrait, and Poincare sections. The results show that when there is an internal resonance, the energy transfer occurs between the first and third modes. In addition, the response of the MEMS arch presents abundant dynamic behaviors, such as Hopf bifurcation and quasiperiodic motions.

Modal Interactions in a Thermally Loaded Annular Plate

2003 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Annular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Hopf Bifurcations ◽  
Modal Interactions ◽  
Harmonic Force ◽  
Large Component ◽  
Second Mode ◽  
Plate Equations

We investigate the nonlinear forced vibrations of a thermally loaded annular plate with clamped-clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the nonlinear von Ka´rma´n plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations which lead to aperiodic oscillations of the plate.

Vibration Suppression of a Saturation Control System Using Delayed Feedback Control

2012 ◽  
Vol 226-228 ◽  
pp. 82-86
Author(s):  
Yan Ying Zhao
Keyword(s):  
Feedback Control ◽  
Analytical Solutions ◽  
Vibration Suppression ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Single Mode ◽  
Original System ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Saturation Control

In the present paper, the delayed feedback control is applied to suppress the amplitude of the vibration of a beam. The method of multiple scales is employed to obtain the analytical solutions when the primary resonance and 1:2 internal resonance occur simultaneously. The predictions from analytical solutions agree with the numerical simulations well. The analytical results show that the amplitude of the beam of the saturation control is much larger than its amplitude of the single-mode motion. The effects of the delayed feedback control on amplitude of the beam are investigated when the original system is in the saturation control. There is a tunable range of the delay could be used to suppress the amplitude of the beam for a fixed value of the gain. The amplitude of the beam can be suppressed from 0.20 to 0.10 when the gain and the delay are chosen appropriate values.

On Nonlinear Forced Response of Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Single Mode ◽  
Forced Response ◽  
Thickness Variation ◽  
Softening Effect ◽  
Bending Vibration

This paper reports the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation. The case of nonlinear curvature, moderately large amplitudes, is considered. The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions. The frequency-response is analytically determined, and numerical results show a softening effect of the geometrical nonlinearities.

Investigation of 3:1 Internal Resonance of Electrostatically Actuated Microbeams With Flexible Supports

2020 ◽  
Author(s):  
Praveen Kumar ◽  
Mandar M. Inamdar ◽  
Dnyanesh N. Pawaskar
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time Integration ◽  
External Source ◽  
Mems Devices ◽  
Electrode Gap ◽  
Dynamical Equations ◽  
Electrostatically Actuated ◽  
Second Mode ◽  
Flexible Supports

Abstract Interaction between modes due to internal resonance has many applications in MEMS devices. In this paper, we investigate the modal interaction through 3 : 1 internal resonance of an electrostatically actuated microbeam with flexible supports in the form of rotational and transversal springs. The static displacement and the first three modal frequencies are obtained at the applied DC voltage by a reduced order model for a specified ratio of electrode gap and thickness. We then obtain the value of applied voltage for which 3 : 1 internal resonance exists for four different combinations of unequal end support stiffnesses. We calculate the coefficients of the coupled dynamical equations of first two modes for all the four cases and solve them by using numerical time integration and the method of multiple scales. We observe the interaction between the first and the second mode when each of the modes is independently excited by an external source. When the second mode is externally excited, interestingly, we also find that the undriven mode response amplitude is twice that of the driven mode.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

Nonlinear Forced Vibration Analysis of Smart Curved CNTs Conveying Fluid

2021 ◽  
Author(s):  
Vahid Mohamadhashemi ◽  
Amir Jalali ◽  
Habib Ahmadi
Keyword(s):  
Carbon Nanotube ◽  
Velocity Vector ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Maximum Amplitude ◽  
Beam Theory ◽  
Physical Parameters ◽  
Conveying Fluid

In this study, the nonlinear vibration of a curved carbon nanotube conveying fluid is analyzed. The nanotube is assumed to be covered by a piezoelectric layer and the Euler–Bernoulli beam theory is employed to establish the governing equations of motion. The influence of carbon nanotube curvature on structural modeling and fluid velocity vector is considered and the slip boundary conditions of CNT conveying fluid are included. The mathematical modeling of the structure is developed using Hamilton’s principle and then, the Galerkin procedure is employed to discretize the equation of motion. Furthermore, the frequency response of the system is extracted by applying the multiple scales method of perturbation. Finally, a comprehensive study is carried out on the primary resonance and piezoelectric-based parametric resonance of the system. It is shown that consideration of nanotube curvature may lead to an increase in nonlinearity. Implementing the fluid velocity vector in which nanotube curvature is included highly affects the maximum amplitude of the response and should not be ignored. Furthermore, different system parameters have evident impacts on the behavior of the system and therefore, selecting the reasonable geometrical and physical parameters of the system can be very useful to achieve a favorable response.

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