Damping In Micromechanical Resonators Research Articles

Significance of memory-dependent derivative approach for the analysis of thermoelastic damping in micromechanical resonators

Thermoelastic damping is becoming a leading factor for determining the quality of the micromechanical resonators in terms of their sensitivity. The present study is devoted to the analysis of thermoelastic damping of a micro-beam resonator by employing the concept of three-phase-lag (TPL) theory of thermoelasticity in the context of the memory-dependent derivative (MDD). The memory-dependent derivative is characterized by its time delay and the kernel function that can be chosen freely with imposed restrictions. Analytical expressions of thermoelastic damping, attenuation as well as frequency shift have been derived. Computational results for the prominent parameters of micromechanical resonators such as thermoelastic damping, attenuation, and frequency shift are obtained. The influence of the constituents of memory-dependent derivative, i.e. time-delay and kernel functions, on the parameters of the micro-beam resonator has been analyzed through graphical results. Furthermore, numerical results are compared with the results obtained in the absence of a memory-dependent derivative through numerical simulations. The significant role of the memory-dependent derivative has been identified and it has been found that the insertion of the memory-dependent derivative is capable of providing accurate results as compared to the results obtained in its absence.

Evaluation of thermoelastic damping in micromechanical resonators with axial pretension: An analytical model accounting for two-dimensional thermal conduction

It has been reported that application of tensile axial stress can simultaneously increase quality factor and resonant frequency for micromechanical resonators. In this study, we formulate an analytical model for evaluating thermoelastic damping in micromechanical resonators based on the thermal energy method, in which thermal conductions in both thickness direction and axial direction are considered. An explicit expression for thermoelastic damping in the form of infinite series has been obtained. The proposed analytical model is further validated by finite element analysis. Results of the present study demonstrate that the 2D model needs to be adopted in order to accurately evaluate thermoelastic damping of micromechanical resonators with axial pretension. In addition, the 2D model proposed in the present study eliminates the inherent inconsistency entailed in the 1D model.

A study of thermoelastic damping in micromechanical resonators under unified generalized thermoelasticity formulation

In this article, a unified formulation for the generalized coupled thermoelasticity theories by employing an appropriate system of partial differential equations as the governing system is presented to investigate thermoelastic damping of a microbeam resonator. The generalized coupled thermoelasticity theories namely: the extended thermoelasticity proposed by Lord and Shulman, the thermoelasticity without energy dissipation (thermoelasticity type-II) and the thermoelasticity with energy dissipation (thermoelasticity type III) in a unified way by introducing the unified parameters. An explicit formula of thermoelastic damping has been derived in a unified way and numerical results for effects of the beam height, relaxation time parameter on thermoelastic damping of the microbeam resonator have been studied and compared.

Effects of phase-lag on thermoelastic damping in micromechanical resonators

The present article attempts to investigate thermoelastic damping (TED) of a microbeam resonator by employing the three-phase-lag (TPL) thermoelasticity theory proposed by Roychoudhuri. An explicit formula of TED has been derived and the effects of the beam height, the phase-lag parameters on TED of the microbeam resonator have been studied. Effects of beam height and phase-lags on TED have been shown with the numerical results. A comparison of the results with the corresponding results of the theory of thermoelasticity of type GN-III is also presented. It has been observed that GN-III predicts a high-quality factor of the resonator’s sensitivity as compared to TPL model.

Thermoelastic damping in micromechanical resonators operating as mass sensors

This paper presents an analytical study on evaluation of thermoelastic damping (TED) in micromechanical resonators operating as mass sensors by means of the thermal energy approach. Both quasi-one-dimensional model and two-dimensional model are proposed and analytical expressions of TED in infinite series form for micro-beam resonator-based mass sensors are derived. We apply the proposed models to examine effects of various factors on TED, such as the position of attached mass particle, boundary conditions, geometry of resonators and mode shape of vibration. Numerical results show that thermoelastic damping of micro-beam resonator-based mass sensors increases obviously with the increase of the mass ratio of the attached mass to mass of the resonator. Compared with results of TED obtained by 2-D model accounting for the two-dimensional thermal conduction in micro-beam resonators, the quasi-1-D model overestimates TED in the high frequency range, and underestimates TED in the low frequency range. Noticeable differences are observed in the results of TED between quasi-1-D model and 2-D model. We also obtain the expressions of mass sensitivity and minimum detectable mass imposed by thermomechanical noise process for resonant mass sensors of bridge and cantilever configurations.

An Entropy Based Analytical Model for Thermoelastic Damping in Micromechanical Resonators

In this paper, we present an analytical model for thermoelastic damping (TED) in micromechanical resonators, which is based on entropy generation, a thermodynamic parameter measuring the irreversibility in heat conduction. The temperature field of thin beam with small vibration is obtained by solving governing equations of linear thermoelasticity. The analytical solution is derived from the entropy generation equation. This method of entropy generation can provide an accurate estimation of the quality factor in flexural resonators. The results are compared with Zener’s approximation and LR (Lifshitz and Roukes) method. It is shown that the analytical model described in this paper is valid to estimate the quality factor due to thermoelastic damping.

Thermoelastic Damping in Micromechanical Resonators with a Proof Mass and a Network of Suspension Beams

Predicting thermoelastic damping is crucial for the design of high Q micromechanical resonators. In the past, for microresonators which consist of a proof mass and a network of suspension beams, some experiments showed that Zener's model [Phys. Rev. 52 (1937) 230; Phys. Rev. 53 (1938) 90] and Lifshitz and Roukes' model [Phys. Rev. B 61 (2000) 5600] can give a reasonable prediction, and others experiments showed that the two models fail to give a reasonable prediction. Few works give a reasonable and detailed explanation for this phenomenon. In this paper, a general proof is presented that shows Lifshitz and Roukes' model is valid for microresonators with a proof mass support by a network of suspension beams if all suspension beams are operated at pure bending vibration and all suspension beams have the same thickness. The accuracy of Lifshitz and Roukes' model is verified by comparing its results with the experimental results available in the literature.

Thermoelastic Damping in Micromechanical Resonators with a Proof Mass and a Network of Suspension Beams

Predicting thermoelastic damping is crucial for the design of high Q micromechanical resonators. In the past, for microresonators which consist of a proof mass and a network of suspension beams, some experiments showed that Zener's model [Phys. Rev. 52 (1937) 230; Phys. Rev. 53 (1938) 90] and Lifshitz and Roukes' model [Phys. Rev. B 61 (2000) 5600] can give a reasonable prediction, and others experiments showed that the two models fail to give a reasonable prediction. Few works give a reasonable and detailed explanation for this phenomenon. In this paper, a general proof is presented that shows Lifshitz and Roukes' model is valid for microresonators with a proof mass support by a network of suspension beams if all suspension beams are operated at pure bending vibration and all suspension beams have the same thickness. The accuracy of Lifshitz and Roukes' model is verified by comparing its results with the experimental results available in the literature.

A thermal-energy method for calculating thermoelastic damping in micromechanical resonators

In this paper, a thermal-energy method is presented for calculating thermoelastic damping in micromechanical resonators. In this method, thermoelastic damping is interpreted as the generation of thermal energy per cycle of vibration and consequently is expressed in terms of entropy—a thermodynamic parameter measuring the irreversibility in heat conduction. As compared with a commonly used complex-frequency method, this thermal-energy method does not involve complex values and thus can be implemented in ANSYS/Multiphysics, a finite element modeling software, with fast speed. Based on the governing equations of linear thermoelasticity, the mathematical expressions are first derived for thermoelastic damping in micromechanical resonators made from isotropic and anisotropic materials, respectively. Through two sequential numerical simulations: uncoupled elastic modal simulation and transient heat conduction, the numerical values for these expressions are then calculated in ANSYS/Multiphysics for micromechanical resonators taking different structural geometries. This method is verified using the well-known theoretical solution to thermoelastic damping in a beam resonator and experimental data. As a result, the developed thermal-energy method can calculate thermoelastic damping in micromechanical resonators with any complex structural geometry and made from isotropic and/or anisotropic materials.

Theory of Thermoelastic Damping in Micromechanical Resonators With Two-Dimensional Heat Conduction

Analysis of thermoelastic damping (TED) is an important component of the design of low-loss vacuum-operated micro- and nanomechanical resonators used in microelectro- mechanical systems (MEMS). The quasi-1-D theories developed by Zener in 1937, and subsequently improved by Lifshitz and Roukes in 2000, are now widely used in MEMS design. This paper presents an exact theory for TED with 2-D heat conduction that enables a detailed evaluation of the accuracy of the quasi-1-D theories. A Green's function method is used to solve the 2-D heat- conduction equation, and an expression for TED is derived in the form of an infinite series. The effects of beam geometry, length-to- thickness aspect ratio, natural frequency, flexural mode shapes, and structural boundary conditions on TED are investigated for the representative case of single-crystal silicon microbeam resonators. The errors in the exact quasi-1-D theory range from 2% to 80% depending upon the aspect ratio and the mode shape. Implications for the use of the quasi-1-D and 2-D theories in MEMS design are discussed. [2007-0199].

Engineering MEMS Resonators With Low Thermoelastic Damping

This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices