Effects of slots on thermoelastic quality factor of a vertical beam MEMS resonator

Thermoelastic damping is one of the dominant mechanisms of structural damping in vacuum-operated microresonators. A three dimensional numerical model based on the finite element method is used for simulating thermoelastic damping in clamped–clamped microelectromechanical beam resonators. In this regards, both simple and slotted beam are considered. To understand the effect of slot positions and sizes on the resonator performance, resonant frequency and thermoelastic quality factor are calculated for both simple and slotted beams for a wide range of beam length from 10 to 400 µm. Punching slots in the resonator beam reduces the stiffness and mass of the beam which affect the resonant frequency. In addition thermo-mechanical coupling mechanisms of the resonator are affected by the slots which improve the thermoelastic quality factor. For most of the beam lengths, it is shown that the slots at the beam-anchor interface region, where the strain is high, are more effectively enhanced the thermoelastic quality factor than one at the centre of the beam region. However, the highest resonance frequency is achieved with the slots at the center region.     

Stability of beamlike lattice trusses

A simple procedure is presented for predicting the buckling loads associated with general instability of large repetitive beamlike trusses. The procedure is based on replacing the original lattice structure by an equivalent continuum beam model and obtaining analytic solutions for the beam model. The continuum beam model accounts for warping and shear deformation in the plane of the cross section and is characterized by its strain energy and potential energy due to initial stresses from which the governing differential equations are derived. The high accuracy of the buckling predictions of the proposed continuum beam is demonstrated by means of numerical examples.     

Thermoelastic damping in fine-grained polysilicon flexural beam resonators

The design and fabrication of polysilicon flexural beam resonators with very high mechanical quality factors (Q) is essential for many MEMS applications. Based on an extension of the well-established theory of thermoelastic damping in homogeneous beams, we present closed-form expressions to estimate an upper bound on the attainable quality factors of polycrystalline beam resonators with thickness (h) much larger than the average grain size (d). Associated with each of these length scales is an independent damping mechanism; we refer to them as Zener and intracrystalline thermoelastic damping, respectively. For representative polysilicon beam resonators (h = 2 /spl mu/m; d = 0.1 /spl mu/m) at 300 K, the predicted critical frequencies for these two mechanisms are /spl sim/7 MHz and /spl sim/14 GHz, respectively. The model is consistent with data from the literature in the sense that the measured values approach, but do not exceed, the calculated thermoelastic limit. From the viewpoint of the maximum attainable Q, our model suggests that single-crystal silicon, rather than fine-grained polysilicon, is the material of choice for the fabrication of flexural beam resonators for applications in the gigahertz frequency range.     

Viscothermoelastic micro-scale beam resonators based on dual-phase lagging model

The analytical expressions for thermoelastic damping and frequency shift of coupled dual-phase-lagging generalized visco-thermoelastic thin beam have been established. The numerical illustration has been carried out for thermoelastic damping with the help of MATLAB programming software. We have used mechanical and thermal parameters of Silicon Nitride under different beam dimensions and boundary (clamped and simply supported) conditions.

     

Quality factor in trench-refilled polysilicon beam resonators

In this paper, thermoelastic damping (TED) in trench-refilled (TR) polysilicon microelectromechanical beam resonators is studied as a mechanism for limiting quality factor (Q) at low frequencies. An approximate model based on Zener's theory is developed and verified by numerical simulations in FEMLAB. According to the proposed model a double-dip characteristic is expected for the quality factor versus frequency curve of TR beam resonators. To verify the model experimentally, equal-width TR micro-resonators are fabricated in different length to cover a broad range of frequencies. Frequency response of these devices agrees well with our model. By using the theoretical and numerical models developed in this paper, an upper bound for the quality factor in TR beam resonators or any similar structure such as TR polysilicon gyros can be predicted.     

Interpolation/penalization applied for strength design of 3D thermoelastic structures

With design independent loads and only a constrained volume (no local bounds), the same optimal design leads simultaneously to minimum compliance and maximum strength. However, for thermoelastic structures this is not the case and a maximum volume may not be an active constraint for minimum compliance. This is proved for thermoelastic structures by sensitivity analysis of compliance that facilitates localized determination of sensitivities, and the compliance is not identical to the total elastic energy (twice strain energy). An explicit formula for the difference is derived and numerically illustrated with examples. In compliance minimization for thermoelastic structures it may be advantageous to decrease the total volume, but for strength maximization it is argued to keep the total permissible volume. Linear interpolation (no penalization) may to a certain extent be argued for 2D thickness optimized designs, but for 3D design problems interpolation must be included and not only from the penalization point of view to obtain 0–1 designs. Three interpolation types are presented in a uniform manner, including the well known one parameter penalizations, named SIMP and RAMP. An alternative two parameter interpolation in explicit form is preferred, and the influence of interpolation on compliance sensitivity analysis is included. For direct strength maximization the sensitivity analysis of local von Mises stresses is demanding. An applied recursive procedure to obtain uniform energy density is presented in details, and it is shown by examples that the obtained designs are close to fulfilling also strength maximization. Explicit formulas for equivalent thermoelastic loads in 2D and 3D finite element analysis are derived and applied, including the sensitivity analysis.     

Simulation and experimental analysis of thermo-mechanical behavior of microresonators under dynamic loading

Simulation, finite element analysis and experimental investigations of the dynamical response of a microresonator under electrostatic actuation are presented in this paper. The scope of this paper is to characterize the influence of thermo-mechanical behavior of the material on the frequency response, amplitude and velocity of oscillations under continuous actuation. The effect of the thermoelastic damping on vibrating structures is experimentally investigated by measuring the loss in amplitude and velocity of oscillations as a function of time and the changes in quality factor. The tests are performed in ambient conditions and in vacuum in order to separate the extrinsic damping of beam by the intrinsic effect given by the thermoelastic damping. The vibrating structure under investigation is a polysilicon clamped–clamped beam.     

Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation

This paper presents a formulation of isotropic finite strain thermoelasticity and addresses some aspects of its numerical implementation. On the theoretical side, a Eulerian setting of isotropic thermoelasticity is discussed, based on the Finger tensor as a strain measure. Novel aspects are a direct representation of the Eulerian thermoelastic moduli in terms of the Finger tensor and a rigorous decomposition of the thermoelastic response functions into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. An algorithmic procedure for the computation of the stress and thermoelastic moduli, based on a representation of the free energy in terms of eigenvalues of the unimodular part of the Finger tensor, is developed and applied to model problems of strictly entropic and modified entropic thermoelasticity. Furthermore, two algorithms for the solution of the coupled problem are discussed, based on operator splits of the global field equations of thermoelasticity. The paper concludes with some representative numerical simulations of thermoelastic processes in rubber-like materials.     

Self-buckling of micromachined beams under resistive heating

Self-buckling behavior of micromachined beams under resistive heating is described by an electromechanical model with experimental verifications. This model consists of both electrothermal and thermoelastic analyses for beam-shape polysilicon microstructures that are fabricated by a standard surface micromachining process. When an input electrical current is applied, Joule-heating effects trigger the thermal expansion of beam structures and cause mechanical buckling. The standard testing devices are clamped-clamped bridges, 2-μm wide, 2-μm thick, and 100-μm long. It is found that a minimum current of 3.5 mA is required to cause beam buckling. Under an input current of 4.8 mA, a lateral deflection of 2.9±0.2 μm at the center of the bridge is measured with a computer image processing scheme. The experimental measurements are found to be consistent with analytical predictions. A discussion of modeling considerations and process variations is presented     

Strength optimized designs of thermoelastic structures

For thermoelastic structures the same optimal design does not simultaneously lead to minimum compliance and maximum strength. Compliance may be a questionable objective and focus for the present paper is on the important aspect of strength, quantified as minimization of the maximum von Mises stress. With compliance defined as the product of resulting displacements and their corresponding total loads, then for thermoelastic problems compliance is different from total elastic energy. An explicit formula for this difference is derived and numerically illustrated in the optimized examples. As an alternative to mathematical programming, which with a large number of both design variables and strength constraints, is found non-practical, we choose simple recursive iterations to obtain uniform energy density and find by examples that the obtained designs are close to fulfilling also strength maximization. In compliance minimization it may be advantageous to decrease the total volume, but for strength maximization it is argued that it is advantageous to keep the total permissible volume. With the thermoelastic analysis presented directly in a finite element formulation, simple explicit formulas for equivalent thermoelastic loads are appended.     

孔隙热弹性梁的非线性气弹性特性

根据作者由Hamilton变分原理导出的一个孔隙热弹性梁的非线性数学模型和气弹性原理中的一阶修正线性活塞理论,本文首先给出了位于高速或者超高速流动中两端固定的平面孔隙热弹性梁的控制微分方程和定解条件,其中基本未知量是梁的轴向和横向位移以及孔隙百分比和温度变化引起的“力矩”.为了考察孔隙热弹性梁在横向载荷和气弹性载荷联合作用下的非线性力学特性,采用微分求积方法对问题进行空间离散,得到一组关于时间的非线性常微分方程,然后在给定初始条件下采用变步长Runge-Kutta方法对方程组进行数值求解,由此研究了孔隙热弹性梁的气弹性特性,考察了参数的影响,得到了一些有益的结论.     

Two-Port Electromechanical Model for Bulk-Piezoelectric Excitation of Surface Micromachined Beam Resonators

A small-signal model that describes the energy exchange between surface micromachined beams and bulk-lead zirconium titanate (PZT) actuators attached to the silicon substrate is presented. The model includes detection of acoustic waves launched from electrostatically actuated structures on the surface of the die, as well as their actuation by bulk waves generated by piezoelectric ceramics. The interaction is modeled via an empirical equivalent circuit, which is substantiated by experiments designed to extract the model parameters. The equivalent model is valid for cases where the beam resonance frequency is much smaller than the thickness mode resonance of the PZT/silicon stack. In this paper, the resonance frequency of the beams ranges between 200 and 300 kHz. As energy transfer between bulk-PZT and electrostatic actuated beam resonators must be reciprocal for small signals, this paper uses the extracted equivalent model to describe the physical sources of error that account for discrepancies in reciprocity.$hfill$[2008-0131]      

动态轴向载荷下谐振梁振动响应分析

在航空航天飞行控制中,为实现关键参数的高精度高动态测量,急需发展具有快速响应特性的谐振式传感器。谐振式传感器本质上是输入与谐振器振动状态之间的映射。这种映射一般通过跟随输入的轴向载荷调制谐振梁的固有频率实现。高动态应用中的核心问题是动态轴向载荷下谐振梁的振动响应。利用基本的微元力学平衡关系建立了动态轴向力作用下谐振梁振动行为的数学模型。此模型比Mathieu方程的适用面更广,在一般假设下更难以进行解析或数值求解。为此引入了等效电路方法进行模型求解。通过对等效电路的仿真,得到了谐振梁在多种典型动态载荷下的振动响应。动态轴向载荷对于谐振梁的作用具有强烈的非线性和独特的规律,值得进一步深入研究探讨。     

激光脉冲作用引起的梁形微谐振器的振动

在激光脉冲作用下,有两个因素需要考虑:一个是热传导的非傅立叶效应,另一个是温度场与应变场的耦合导致能量耗散,使物体的机械能转化为热能,并且这是不可逆的.本文综合考虑上面两个因素,研究了激光脉冲作用下微米尺度梁谐振器的热弹性耦合问题.采用Fourier变换与Laplace变换相结合的方法求解梁的横向振动,并分析不同环境温度与能量吸收深度的影响.     

Heating and melting deformable models

We develop physically-based graphics models of non-rigid objects capable of heat conduction, thermoelasticity, melting and fluid-like behaviour in the molten state. These deformable models feature non-rigid dynamics governed by Lagrangian equations of motion and conductive heat transfer governed by the heat equation for non-homogeneous, non-isotropic media. In its solid state, the discretized model is an assembly of hexahedral finite elements in which thermoelastic units interconnect particles situated in a lattice. The stiffness of a thermoelastic unit decreases as its temperature increases, and the unit fuses when its temperature exceeds the melting point. The molten state of the model involves a molecular dynamics simulation in which ‘fluid’ particles that have broken free from the lattice interact through long-range attraction forces and short-range repulsion forces. We present a physically-based animation of a thermoelastic model in a simulated physical world populated by hot constraint surfaces.     

Application of Mathematical Modeling to Determine the Thermoelastic Characteristics of Nano-Reinforced Composites

A two-level mathematical model is constructed to describe the thermomechanical interaction between structural elements of a composite (nanoclusters formed by randomly distributed anisotropic single-walled carbon nanotubes and matrix particles) and an isotropic medium possessing the desired thermoelastic characteristics. This model was first employed to obtain the thermoelastic properties of a nanocluster by the self-consistency method and then the same technique was used to describe the thermomechanical interaction of nanoclusters with an isotropic matrix of the composite. A comparative analysis of the calculated dependences for the elastic moduli of the composite and its thermal coefficient of linear expansion was carried out with two-sided estimates of these characteristics based on the dual variational formulation of the thermoelasticity problem. For comparison, the results of a numerical experiment are also used. The presented relationships make it possible to predict the thermoelastic properties of promising composites reinforced by nanoclusters.     

Dynamic analysis of geometrically nonlinear truss structures

Beam-like truss structures undergoing large deflections when subjected to static and dynamic loadings are studied by using the matrix method and equivalent continuum models. For the matrix method, an incremental procedure with equilibrium iterations are used. Equivalent continuum beam models are derived based on the properties of a typical substructure of the truss. Solutions obtained by using both methods are compared for a number of examples.     

Certain discussions in the finite element formulation of nonlinear vibration analysis

The finite element formulation and equivalent linearization technique used in the study of nonlinear vibrations of beams have been reexamined based on the earlier comments. Errors present in the equivalent linearization procedure, in the substitution of inplane boundary conditions at the element level instead of at the system level, and in the use of different connotation for the frequency, are discussed. A simply-supported beam with immovable ends is considered as an example to evaluate these errors.A formulation in terms of transverse displacement alone is also presented based on the assumption of average nonlinear stretching force.     

A nodal analysis method for electromechanical behavior simulation of bow-tie shaped microbeams

In order to perform mixed-domain simulation of electrically actuated bow-tie shaped doubly-fixed microbeams, a nodal model for the trapeziform beam and corresponding equivalent circuit are developed by HSPICE. The nonlinearities including mid-plane stretching and electrostatic forcing are considered. In the modeling, the Galerkin residual method is used to discretize the partial differential equations of the trapeziform beam. It proves accurate in comparison with FEA simulation results and available experimental data. In particular, as a lumped behavioral model, it is executed much faster than FEM program. The proposed model could be expected to be useful in the optimum design of related MEMS devices.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.