Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam

On the basis of the Euler-Bernoulli hypothesis, nonlinear static and dynamic responses of a viscoelastic microbeam under two kinds of electric forces [a purely direct current （DC） and a combined current composed of a DC and an alternating current] are studied. By using Taylor series expansion, a governing equation of nonlinear integro-differential type is derived, and numerical analyses are performed. When a purely DC is applied, there exist an instantaneous pull-in voltage and a durable pull-in voltage of which the physical meanings are also given, whereas under an applied combined current, the effect of the element relaxation coefficient on the dynamic pull-in phenomenon is observed where the largest Lyapunov exponent is taken as a criterion for the dynamic pull-in instability of viscoelastic microbeams.     

Analyzing the effects of jump phenomenon in nonlinear vibration of thin circular functionally graded plates

In this paper, the nonlinear vibration of a thin circular functionally graded material plates is studied. The plate thickness is constant, and the material properties of the plate are assumed to vary continuously through the thickness. The governing equations and boundary conditions are extracted. The assumed-time-mode method is used to analyze these equations. The time variable is eliminated by assuming a harmonic response for nonlinear vibration and using Kantorovich time averaging technique. Utilizing shooting and Runge–Kutta methods, the set of first-order nonlinear differential equations are solved. The effect of volume fraction index in free and forced vibration response and jump phenomenon is studied. The results show that jump phenomenon occur according to volume fraction index and uniform temperature in the special frequencies of forced vibration response.     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

On energy pumping, synchronization and beat phenomenon in a nonideal structure coupled to an essentially nonlinear oscillator

In this paper, we investigated the effectiveness of the linear electromechanical vibration absorber (LEVA) and a nonlinear electromechanical vibration absorber (NEVA) in the vibration attenuation for nonideal structures (NIS). This electromechanical damping device consists of an electrical system coupled magnetically to a mechanical structure under a nonideal excitation. An analysis of the effects of the parameters of coupling and of nonlinear coefficients with increasing of constant torque of the DC motor is presented.     

Chaotic vibration and resonance phenomena in a parametrically excited string-beam coupled system

The nonlinear behavior of a string-beam coupled system subjected to parametric excitation is investigated in this paper. Using the method of multiple scales, a set of first order nonlinear differential equations are obtained. A numerical simulation is carried out to verify analytic predictions and to study the steady-state response, stable solutions and chaotic motions. The numerical results show that the system behavior includes multiple solutions, and jump phenomenon in the resonant frequency response curves. It is also shown that chaotic motions occur and the system parameters have different effects on the nonlinear response of the string-beam coupled system. Results are compared to previously published work.     

基于压电振动能量俘获的弯曲结构损伤监测研究

建立了含裂纹损伤的曲梁压电能量俘获系统在强迫振动下的动力学模型. 基于Prescott型压电曲梁力电耦合振动方程的解析解和裂纹截面处的连续性条件, 求解了含裂纹损伤的压电曲梁的格林函数. 根据线性叠加原理, 对含裂纹的力电耦合模型的系统方程解耦, 得到强迫振动下含裂纹损伤的曲梁压电俘能器的输出电压. 在得到模型的强迫振动解析解后, 提出逆方法检测结构中的裂纹损伤, 这一检测方法适用于处于振动状态下的结构. 在数值计算中, 令裂纹深度为零, 通过对比本文的解析解与现有文献中的解析解, 验证了本文解的有效性. 分别分析了含裂纹损伤的压电曲梁的电压响应与裂纹深度、裂纹位置、材料的几何参数以及阻尼之间的关系. 研究结果表明: 裂纹的存在对曲梁式压电俘能器的影响比直梁式更加复杂; 裂纹出现时, 损伤曲梁在健康曲梁的一阶频率值处一定会出现波动并被激励出二阶频率, 此时的二阶频率是开路中健康压电曲梁的一阶频率值; 通过对电压响应的检测可以确定的损伤裂纹的深度和在结构中出现的位置范围; 利用振动问题的解来检测压电曲梁的健康状况是可行且准确的.      

Dynamics and control of a self-sustained electromechanical seismographs with time-varying stiffness

This paper is concerned with the nonlinear dynamics and vibration control of an electromechanical seismograph system with time-varying stiffness. The instrument consists of an electrical part coupled to mechanical one and is used to record the vibration during earthquakes. An active control method is applied to the system based on cubic velocity feedback. The electromechanical system is subjected to parametric and external excitations and modeled by a coupled nonlinear ordinary differential equations. The method of multiple scales is used to obtain approximate solutions and investigate the response of the system. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters have been reported.     

Forced axisymmetric vibrations and self-heating of thermoviscoelastic cylindrical shells with piezoelectric actuators

The coupled problem of the forced axisymmetric vibrations and self-heating of electrothermoviscoelastic cylindrical shells with piezoceramic actuators under monoharmonic electromechanical loading is solved. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The coupled nonlinear problem of electrothermoelasticity is solved by using a time-marching method with discrete orthogonalization at each time step (to integrate the equations of elasticity) and an explicit finite-difference method (to solve the heat-conduction equations). An analysis is made of the effect of the boundary conditions at the shell ends, the dimensions of the piezoactuator, and the self-heating temperature on the actuator voltage and the effectiveness of active damping of the forced vibrations of the shell under uniform transverse monoharmonic pressure     

Linear,nonlinear dynamics,and sensitivity analysis of a vibratory ring gyroscope

The linear and nonlinear dynamic responses of a vibratory ring gyroscope are investigated in this study focusing on the response mechanism of such a vibratory gyroscope. It is found that the nonlinear equations governing the drive and sense directions are coupled through both inertial linear and geometric nonlinear terms. Nonlinear responses are studied based on the full coupled nonlinear dynamic equations. The varying amplitude on the sense direction is analyzed for different input angular rates. The effect of nonlinearity on the ring gyroscope system is performed by comparing the results of nonlinear responses to those of linear responses. The contributions of some parameters to the amplitude responses and gyroscope sensitivity are analyzed, the conclusions of which provide guidelines to improve the sensitivity of the vibratory ring gyroscopes.     

A comparison of different semi-analytical techniques to determine the nonlinear oscillations of a slender microbeam

The purpose of this work is to investigate the dynamic behaviour of an electrically-actuated microbeam. The electromechanical model is based on the strain-gradient elasticity theory and it gives proper account of the nonlinear geometric term due to the mid-plane stretching and of an applied axial load. The free nonlinear vibrations are studied with the energy balance method and the homotopy analysis method Liao (Commun Nonlinear Sci Numer Simul 14(4):983, 2009), thus carrying out a thorough analysis with regard to the nonlinear terms. The analysis is based on a single-degree-of-freedom model, where the nonlinear electric force acting on the beam is approximated by the Chebyshev method and a fringing field correction term is considered as well. A numerical solution, obtained by a 4th order Runge Kutta algorithm, is also proposed as a benchmark for all the semi-analytical results. Major attention is paid to verify the agreement between the different methods and the their accuracy in the pull-in regime.     

Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.     

Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation

In this study, nonlinear static and dynamic responses of a microcantilever with a T-shaped tip mass excited by electrostatic actuations are investigated. The electrostatic force is generated by applying an electric voltage between the horizontal part of T-shaped tip mass and an opposite electrode plate. The cantilever microbeam is modeled as an Euler–Bernoulli beam. The T-shaped tip mass is assumed to be a rigid body and the nonlinear effect of electrostatic force is considered. An equation of motion and its associated boundary conditions are derived by the aid of combining the Hamilton principle and Newton's method.An exact solution is obtained for static deflection and mode shape of vibration around the static position. The differential equation of nonlinear vibration around the static position is discretized using the Galerkin method. The system mode shapes are used as its related comparison functions. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances.In addition, effects of mass inertia, mass moment of inertia as well as rotation of the T-shaped mass, which were ignored in previous works, are considered in the analysis. It is shown that by increasing the length of the horizontal part of the T-shaped mass, the amount of static deflection increases,natural frequency decreases and nonlinear shift of the resonance frequency increases. It is concluded that attaching an electrode plate with a T-shaped configuration to the end of the cantilever microbeam results in a configuration with larger pull-in voltage and smaller nonlinear shift of the reso-nance frequency compared to the configuration in which the electrode plate is directly attached to it.     

Bifurcation and chaos analysis of torsional vibration in a PMSM-based driven system considering electromechanically coupled effect

The rotor of PMSM-based electromechanically driven system is a typical electromechanically coupled system. In this paper, we analyzed the nonlinear magnetic interaction torque of permanent magnet synchronous motor (PMSM) and deduced the nonlinear electromechanically coupled equation of PMSM-based electromechanical driven system using Lagrange–Maxwell theory. We determined the equation of the movement of the dynamic system from his asymmetric double well potential. The unperturbed system was classified to several categories based on the shapes of potential functions and phase portraits. An analytical criterion for homoclinic chaos is written in terms of the system parameters by means of Melnikov’s method. Detailed numerical studies including phase portrait, Poincare map, and bifurcation diagram confirm the analytical prediction and reveal the effect of excitation amplitude and damp on the system transition to chaos. The conclusion can provide reference for deeply research the dynamic behaviors of mechanical drive system.     

Electromechanical Vibrations and Dissipative Heating of Viscoelastic Thin-Walled Piezoelements

The results of studying the electromechanical response of thin-walled viscoelastic piezoactive elements under harmonic loading are generalized. The nonlinear electrothermoviscoelastic problem for a harmonically deformed body is formulated in a simplified form with regard for the facts that the mechanical, thermal, and electric fields are coupled, the material is physically nonlinear, and its properties depend on temperature. Classical and refined electromechanical models of single-layer and multilayer shells and plates under general and harmonic loading are reviewed. The models consider that the electromechanical characteristics of the material depend on temperature and physical and geometrical nonlinearities. Methods for solving nonlinear coupled electrothermoviscoelastic problems are discussed. Analytical and numerical solutions are given to specific quasistatic and dynamic electrothermoviscoelastic problems for thin-walled elements such as rods, plates, and shells of various shapes under harmonic electric loading. The effect of dissipation, the temperature dependence of the material properties, and physical and geometrical nonlinearities on the harmonic and parametric vibrations and stability of piezoelectric elements is studied     

Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic pads under single-frequency electromechanical loading

The paper deals with the coupled problem of flexural vibrations and dissipative heating of a viscoelastic ring plate with piezoceramic actuators under monoharmonic electromechanical loading. The temperature dependence of the complex characteristics of passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoviscoelasticity is solved by an iterative method. At each iteration, orthogonal discretization is used to integrate the equations of elasticity and an explicit finite-difference scheme is used to solve the heat-conduction equation with a nonlinear heat source. The effect of the dissipative heating temperature, boundary conditions, and the thickness and area of the actuator on the active damping of the forced vibrations of the plate under uniform transverse harmonic pressure is examined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 99–108, February 2008.     

Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low,medium and large DC-voltages

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, most widely used theoretical approaches, although opposite extremes with respect to complexity, are nonlinear lumped-mass and finite-element models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Finite-element simulations, on the other hand, are adequate for static analysis, but inadequate for dynamic simulations. A third approach is that of a reduced-order modeling which has gained significant attention for single-element micro-electromechanical systems (MEMS), yet leaves an open amount of fundamental questions when applied to MEMS arrays. In this work, we employ a nonlinear continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the array’s behavior in regions of its internal one-to-one, parametric, and several internal three-to-one and combination resonances, which correspond to low, medium and large DC-voltage inputs, respectively. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weakly nonlinear system in the afore mentioned resonance regions, respectively. Analytically obtained results of a two-element system are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic behavior of the two- and three-beam systems reveal several in- and out-of-phase co-existing periodic and aperiodic solutions. Stability analysis of such co-existing solutions enables construction of a detailed bifurcation structure. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays. Furthermore, the results of this present work motivate future experimental work and can serve as a guideline to investigate the feasibility of new MEMS array applications.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

Experimental investigation of vibration attenuation using nonlinear tuned mass damper and pendulum tuned mass damper in parallel

The present paper experimentally and numerically explores the response attenuation of a hardening Düffing oscillator using a nonlinear tuned mass damper (NTMD) and an adaptive-length pendulum tuned mass damper (APTMD). The three degrees-of-freedom system is excited by harmonic ground motions. The cubic nonlinearity of the primary structure is obtained using an adaptive passive stiffness (APS) device. When an NTMD is used alone, a high amplitude detached resonance branch in the lower frequency range is identified in the experiment, which validates the results reported in earlier numerical research. In order to attenuate this high amplitude resonance branch, an APTMD with an adaptive frequency realized by means of a variable pendulum length is used in parallel with the NTMD. In the experiment, length of the APTMD is adjusted such that its natural frequency matches the dominant frequency of the harmonic ground motions. Results indicate that the high amplitude resonance branch in the case of an NTMD alone can be greatly attenuated using the APTMD, and significant attenuation of the structural responses over a large frequency range can be obtained. In addition, the APTMD can prevent the occurrence of the “jump phenomenon” existing in the forcing response curve of the nonlinear dynamic system, thereby protecting the primary nonlinear structure effectively when the forcing amplitude varies. Therefore, the present paper provides an effective and viable solution to control the hazardous bifurcations in a Düffing oscillator-NTMD dynamic system.     

基于多尺度方法的平动圆柱贮箱航天器刚——液耦合动力学研究

以充液航天器为工程背景,借助多尺度方法研究刚--液耦合动力学系统非线性动力学特性.利用多维模态方法,将描述横向外激励下圆柱贮箱中液体非线性晃动的自由边界问题转换为液体模态系数相互耦合的有限维非线性常微分方程组.推导液体晃动产生的作用于贮箱壁的晃动力和晃动力矩的解析表达式,进而建立航天器刚体部分平动和液体晃动耦合的非线性动力学方程组.应用多尺度方法对刚--液耦合系统的动力学特性进行解析分析,通过固有频率的特征方程求解耦合系统固有频率,推导外激励频率接近耦合系统第一阶固有频率时液体晃动稳态解的幅值频率响应方程.结合数值方法,研究了液体晃动稳态解的幅值频率响应曲线和激励--幅值响应曲线.结果表明, 随充液比变化,液体晃动稳态解的幅值频率响应曲线会发生软、硬弹簧特性转换现象和"跳跃"现象;幅值频率响应曲线的软、硬弹簧特性转换点受重力加速度和弹簧刚度系数影响;以上所得研究结果表明,考虑非线性效应时的刚--液耦合系统动力学特性与传统的线性系统模型所显示的动力学特性具有本质区别.本文的研究工作对进一步分析充液航天器刚--液耦合非线性动力学特性具有重要参考价值.      

Nonlinear Longitudinal Vibrations of Transversally Polarized Piezoceramics: Experiments and Modeling

Nonlinear behavior of piezoceramics at strong electric fields is a well-known phenomenon and is described by various hysteresis curves. On the other hand, nonlinear vibration behavior of piezoceramics at weak electric fields has recently been attracting considerable attention. Ultrasonic motors (USM) utilize the piezoceramics at relatively weak electric fields near the resonance. The consistent efforts to improve the performance of these motors has led to a detailed investigation of their nonlinear behavior. Typical nonlinear dynamic effects can be observed, even if only the stator is experimentally investigated. At weak electric fields, the vibration behavior of piezoceramics is usually described by constitutive relations linearized around an operating point. However, in experiments at weak electric fields with longitudinal vibrations of piezoceramic rods, a typical nonlinear vibration behavior similar to that of the USM-stator is observed at near-resonance frequency excitations. The observed behavior is that of a softening Duffing-oscillator, including jump phenomena and multiple stable amplitude responses at the same excitation frequency and voltage. Other observed phenomena are the decrease of normalized amplitude responses with increasing excitation voltage and the presence of superharmonics in spectra. In this paper, we have attempted to model the nonlinear behavior using higher order (quadratic and cubic) conservative and dissipative terms in the constitutive equations. Hamilton's principle and the Ritz method is used to obtain the equation of motion that is solved using perturbation techniques. Using this solution, nonlinear parameters can be fitted from the experimental data. As an alternative approach, the partial differential equation is directly solved using perturbation techniques. The results of these two different approaches are compared.