THERMOELASTIC DAMPING IN A MICRO-BEAM RESONATOR TUNABLE WITH PIEZOELECTRIC LAYERS

In this paper,thermoelastic damping (TED) in a micro-beam resonator with a pair of piezoelectric layers bonded on its upper and lower surfaces is investigated.Equation of motion is derived and the ther...     

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

Present 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.     

Thermoelastic damping in micro-beam resonators

Thermoelastic damping is recognized as a significant loss mechanism at room temperature in micro-scale beam resonators. In this paper, the governing equations of coupled thermoelastic problems are established based on the generalized thermoelastic theory with one relaxation time. The thermoelastic damping of micro-beam resonators is analyzed by using both the finite sine Fourier transformation method combined with Laplace transformation and the normal mode analysis. The vibration responses of deflection and thermal moment are obtained for the micro-beams with simply supported and isothermal boundary conditions. The vibration frequency is analyzed for three boundary condition cases, i.e., the clamped and isothermal, the simply supported and isothermal, and the simply supported and adiabatic. The analytic results show that the amplitude of deflection and thermal moment are attenuated and the vibration frequency is increased with thermoelastic coupling effect being considered. In addition, it can be found from both the analytic results and the numerical calculations that these properties are size-dependent. When the thickness of the micro-beam is larger than its characteristic size, the effect of thermoelastic damping weakens as the beam thickness increases. The size-effect induced by thermoelastic coupling would disappear when the thickness of the micro-beam is over a critical value that depends on the material properties and the boundary conditions.     

考虑尺度效应的微梁静态弯曲数值分析

基于修正偶应力理论及表面弹性理论,本文提出了一种新的双曲线剪切变形梁模型,用于均匀微尺度梁的静态弯曲分析。该理论可以直接利用本构关系获得横向剪切应力,满足梁顶部和底部的无应力边界条件,避免了引入剪切修正因子。根据广义Young-Laplace方程建立了梁的内部与表面层的应力连续性条件,单一的变量场可以描述梁的位移模式。通过在位移场中考虑表面层厚度以及表面层的应力连续条件,可以使新模型能够更准确地预测微尺寸和表面能相关的尺度效应。通过Hamilton原理推导出了梁的控制方程和边界条件。应变能除了考虑经典弹性理论,还要考虑微结构效应和表面能。Navier-type的解析解适用于简支边界条件,而基于拉格朗日插值的微分求积法(DQEM)可以研究在不同边界条件下的力学响应。把该数值解与Navier方法得出的解析解作了对比,得出:微尺度梁在考虑表面能或微尺寸效应、不同载荷和梁高变化下的响应一致;当不考虑微结构相关性和表面能效应时,该模型退化为经典的欧拉梁模型。     

Analysis of nonlinear dynamic behavior of electrically actuated micro-beam with piezoelectric layers and squeeze-film damping effect

An analytical model based on a nonlinear deflection equation and the Reynolds equation is proposed to describe the dynamic behavior of an electrically actuated micro-beam with two piezoelectric layers. The proposed model takes explicit account of the fringing field effect, the axial stress effect, the residual stress effect, and the squeeze-film damping effect between the micro-beam and the lower electrode. The nonlinear governing equation of the micro-beam is solved using a hybrid computational scheme comprising the differential transformation method and the finite difference method. The validity of the analytical model and numerical solution procedure is demonstrated by comparing the result obtained for the pull-in voltage of a micro-beam actuated by a DC voltage only with that presented in the literature. It is shown that the nonlinear dynamic response of the micro-beam can be controlled using a combined driving scheme consisting of both the magnitude and the frequency of the AC actuating voltage and a DC driving voltage. The effects of the AC/DC actuating conditions, micro-beam geometry parameters, and squeeze-film damping force on the center-point displacement of the micro-beam are systematically examined. In addition, the actuating conditions which ensure the stability of the micro-beam are identified by means of phase portraits and Poincaré maps. In general, the results show that the analytical model and hybrid numerical scheme provide a feasible means of analyzing the dynamic response of a variety of electrostatically-actuated microstructures.     

Viscous fluid damping in a laterally oscillating finger of a comb-drive micro-resonator based on micro-polar fluid theory

Viscous damping is a dominant source of energy dissipation in laterally oscillating micro-structures. In micro-resonators in which the characteristic dimensions are compa-rable to the dimensions of the fluid molecules, the assumption of the continuum fluid theory is no longer justified and the use of micro-polar fluid theory is indispensable. In this paper a mathematical model was presented in order to predict the viscous fluid damping in a laterally oscillating finger of a micro-resonator considering micro-polar fluid theory. The coupled governing partial differential equations of motion for the vibration of the finger and the micro-polar fluid field have been derived. Considering spin and no-spin boundary conditions, the related shape functions for the fluid field were presented. The obtained governing differential equations with time varying boundary conditions have been trans-formed to an enhanced form with homogenous boundary conditions and have been discretized using a Galerkin-based reduced order model. The effects of physical properties of the micro-polar fluid and geometrical parameters of the oscillat-ing structure on the damping ratio of the system have been investigated.     

Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.     

Bifurcation analysis of an electro-statically actuated micro-beam in the presence of centrifugal forces

In this paper, the influence of centrifugal forces on the stability of an electro-statically actuated clamped–clamped micro-beam has been investigated. The non-dimensional governing static and dynamic equations have been linearized using the step by step linearization method (SSLM), then, a Galerkin-based reduced order model has been used to solve the linearized equations. For constant value of a bias DC voltage and different values of angular velocity the equilibrium points of the corresponding autonomous system including stable center points, unstable saddle points and singular points have been obtained using the equivalent mass-spring model. Subsequently the bifurcation diagram has been depicted using the obtained fixed point. The static pull-in voltage value for different values of angular velocity and the static pull-in angular velocity for different values of bias voltage have been calculated. The obtained results are validated using results of previous studies and a good agreement has been observed. The effect of the centrifugal force on the fixed points has been studied using the phase portraits of the system for different initial conditions. Moreover, the effects of centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different angular velocities.     

结构阻尼时域本构模型及其应用

阻尼合金作为一种新型结构功能材料在不少领域已获应用，由于其阻尼值较大且随频率呈复杂交化关系，传统的线性粘性阻尼理论或经典的非频变结构阻尼理论难以精确地描述其耗能行为。本文应用粘弹性阻尼理论，根据阻尼合金储能模量和损耗因子在频域的实测数据．应用最优化方法拟合出标准线性体模型中的本构参数；根据积分形式的三参量本构关系和变形体虚功原理，推导出了有限元形式的动力学方程；讨论了三参数初值的选取；对包含卷积积分的有限元动力学方程通过数学推导将其化为三阶线性微分方程组，再转化为标准状态变量方程，应用数值求解。数值计算实例证明了所提方法的正确和有效性。     

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.     

各向异性对平板裂纹扩展路径的影响研究

近场动力学理论克服了经典连续介质力学在模拟裂纹扩展方面的不足。本文基于微梁键构建了一般各向异性近场动力学平面模型,并探讨了各向异性对于裂纹扩展路径的影响。首先,建立一般各向异性的微梁键,并基于插值法建立了一般各向异性近场动力学平面模型的应变能密度表达式;然后,利用应变能密度互等原理求解出近场动力学平面本构模型的一般各向异性参数;最后,通过带初始切口的平板断裂实验研究了不同各向异性参数对于平板裂纹扩展路径的影响。     

Effect of strain-gradients of surface micro-beams on frequency-shift of a quartz crystal resonator under thickness-shear vibrations

With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator(QCR) system,consisting of a QCR and surface micro-beam arrays.The frequency shift that was induced by micro-beams with consideration of strain-gradients is discussed in detail and some useful results are obtained,which have important significance in resonator design and applications.     

基于不同板理论的嵌于柔性层中薄膜的皱曲分析

论文对于柔性层-薄膜-柔性层三层结构系统,基于经典板理论、一阶剪切变形理论和高阶剪切变形理论,分别推导给出薄膜皱曲的控制方程.对于两个柔性层,则把它们处理成具有有限厚度的平面应变弹性体.针对上下柔性层固支边界或自由边界条件,利用线性扰动方法得到柔性层对薄膜的横向压力差,最终获得确定薄膜具有周期性正弦型皱曲的临界载荷方程...     

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.     

Effects of local thickness defects on the buckling of micro-beam

A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity. By introducing the local thickness defects function of the micro-beam, the variable coe-cient differential equations of the buckling problem are obtained with the variational principle. Combining the eigensolution series of the complete micro-beam with the Galerkin method, we obtain the critical load and buckling modes of the micro-beam with defects. The results show that the depth and location of the defect are the main factors affecting the critical load, and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam. The effect of defect location on buckling is related to the axial gradient of the rotation angle, and defects should be avoided at the maximum axial gradient of the rotation angle. The model and method are also applicable to the static deformation and vibration of the micro-beam.     

Behavior explanation and a new model for nonlinear viscous fluid dampers with a simple annular orifice

Viscous fluid dampers have been used in many building and bridge construction projects for earthquake damage mitigation. Previous study has shown that silicone oil properties, such as the fluid shear-thinning and relaxation effects, play important roles for the annular-orificed fluid damper behavior, and the Navier-Stokes equations based on these mechanisms were developed. In the current study, attempts are made to explain the effects of frequency, damper dimensions, and viscosity of silicone oil on the damper stiffness behavior using the developed equations. It is found that the developed equations successfully explain the observed phenomena. To avoid the complicated fluid dynamics analyses for damper parameters, such as the damping factor and the velocity power exponent, a new four-parameter equation considering both the fluid shear-thinning and stiffness effects, with a form similar to the widely used two- or three-parameter equation is proposed. The results of the new model successfully capture the damper behavior both at low and high frequencies and show an advantage that better consistent results can be obtained in the velocity range for the building and bridge applications.     

Dynamic analysis of a dielectric elastomer-based microbeam resonator with large vibration amplitude

A non-linear vibration equation with the consideration of large amplitude, gas damping and excitation is developed to investigate the dynamic performance of a dielectric elastomer (DE)-based microbeam resonator. Approximate analytical solution for the vibration equation is obtained by applying parameterized perturbation method (PPM) and introducing a detuning variable. The analysis exhibits that active tuning of the resonant frequency of the resonator can be achieved through changing an applied electrical voltage. It is observed that increasing amplitude will increase the natural frequency while it will decrease the quality factor of the resonator. In addition, it is found that the initial pre-stretching stress and the ambient pressure can significantly alter the resonant frequency of the resonator. The analysis is envisaged to provide qualitative predictions and guidelines for design and application of DE-based micro resonators with large vibration amplitude.     

Stability analysis of pipes conveying fluid with fractional viscoelastic model

Divergence and flutter instabilities of pipes conveying fluid with fractional viscoelastic model has been investigated in the present work. Attention is concentrated on the boundaries of the stability. Based on the Euler–Bernoulli beam theory for structural dynamics, viscoelastic fractional model for damping and, plug flow model for fluid flow, equation of motion has been derived. The effects of gravity, and distributed follower forces are also considered. By transferring the equation of motion to the Laplace domain and using the Galerkin method, the characteristic equations are obtained. By solving the eigenvalue problem, frequencies and dampings of the system have been obtained versus flow velocity. Some numerical test cases have been studied with viscoelastic fractional model and the effect of the fractional derivative order and the retardation time is investigated for various boundary conditions.

     

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES 1)

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.     

Mindlin 矩形微板的热弹性阻尼解析解

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.