Dual-phase-lag thermoelastic damping models for micro/nanobeam resonators

Thermoelastic damping (TED) affects the quality factors of vacuum-operated micro/nanobeam resonators significantly. In this work, by adopting the non-Fourier theory of dual-phase-lag (DPL) model, an analytical formula of TED in micro/nanobeam resonators with circular cross-section is first developed. Moreover, for micro/nanobeam resonators with rectangular cross-section, the series-form type of DPL-TED model is also proposed and compared with the modified existing model. The characteristics of TED spectra with the single-peak, dual-peak, and multiple-peak phenomena are explored. The simulation results reveal that the ratio of dual-phase-lag times and the characteristic dimension of beams such as the radius and thickness have significant influences on TED behaviors. In addition, temperature distributions in micro/nanobeams exhibit an apparent distinction under the DPL non-Fourier effect.     

Nonlinear thermoelastic analysis of layered structures

The nonlinear thermoelastic behavior of orthotropic layered slabs and cylinders including radiation boundaries, temperature-dependent material properties, and stress-dependent layer interface conditions is investigated. A one-dimensional finite element formulation employing quadratic layer and linear interface elements is used to perform the analyses. The transient heat conduction portion of the program is temporally discretized via an implicit linear time interpolation algorithm which includes Crank-Nicolson, Galerkin, and Euler backward differencing. The nonlinear heat conduction equations are iteratively evaluated using a modified Newton-Raphson scheme. Direct iteration between heat conduction and stress analysis is employed when stress-dependent interface thermal resistance coefficients are utilized. Verification problems are presented to demonstrate the accuracy of the finite element code.     

An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate

A steady state thermoelastic mixed boundary value problem for an isotropic thick plate is considered in this paper. The faces of an external circular crack situated in the mid-plane of the plate are opened up by the application of temperature while the bounding surface of the plate are maintained at a constant zero temperature. Solution valid for large values of the ratio of the plate thickness to the diameter of the crack has been obtained. Expressions for various quantities of physical interest are derived by finding iterative solutions of the equations and the results are shown graphically.     

Thermoelastic damping in flexural vibration of bilayered microbeams with circular cross-section

Predicting of thermoelastic damping (TED) is crucial in the design of micro-resonators with composite structures. Several analytical models were developed to evaluate TED in bilayered and three-layered microbeams in the past. However, the previous models focus on the microbeams with rectangular cross-section. This paper aims to study the TED in a bilayered microbeam with circular cross-section. The temperature field is approximated by using sine series and Bessel series in the circular cross-section. An analytical full model for TED in flexural vibration of bilayered microbeam is presented in the form of an infinite series. A simple model is also developed by retaining only the first term. The simulation results of the present model have a good agreement with those of the finite element method (FEM). The results indicate that well-resolved two peaks of TED are typically observed in the bilayered microbeams in which the value of k₂/C₂ is much less or higher than that of k₁/C₁. The present model is not a rapidly converging infinite series for most of the bilayered microbeams. The present model is more suitable for the long slender beams.     

Deflection analysis of micro-scale microstretch thermoelastic beam resonators under harmonic loading

In this paper, the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler–Bernoulli beam resonators, due to time harmonic load have been investigated. The axial ends of the beam are assumed to be at either clamped-clamped, simply supported-simply supported or clamped-free conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The numerical simulation has been carried out with the help of MATLAB software for magnesium like material. The graphical representations and observations have been discussed for deflection of beam under various boundary conditions and for distinct considered values of time and space as well.     

Finite element analysis of thermoelastic field in a rotating FGM circular disk

This study focuses on the finite element analysis of thermoelastic field in a thin circular functionally graded material (FGM) disk subjected to a thermal load and an inertia force due to rotation of the disk. Due to symmetry, the FGM disk is assumed to have exponential variation of material properties in radial direction only. As a result of nonuniform coefficient of thermal expansion (CTE) and nonuniform temperature distribution, the disk experiences an incompatible eigenstrain which is taken into account. Based on the two dimensional thermoelastic theories, the axisymmetric problem is formulated in terms of a second order ordinary differential equation which is solved by finite element method. Some numerical results of thermoelastic field are presented and discussed for an Al₂O₃/Al FGM disk. The analysis of the numerical results reveals that the thermoelastic field in an FGM disk is significantly influenced by temperature distribution profile, radial thickness of the disk, angular speed of the disk, and the inner and outer surface temperature difference, and can be controlled by controlling these parameters.     

FINITE ELEMENT ANALYSIS OF FREE VIBRATION OF LIQUID－FILLED AXISYMMETRIC TANKS

ＦＵＲＵＤＥ（符儒德）；ＵＳＨＩＪＩＭＡＴＥＲＵＯ（牛岛照夫）（ＤｅｐａｒｔｍｅｎｔｏｆＣｏｍｐｕｔｅｒＳｃｉｅｎｃｅ＆ＩｎｆｏｒｍａｔｉｏｎＭａｔｈｅｍａｔｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＥｌｅｃｔｒｏ－Ｃｏｍｍｕｎｉｃａｔｉｏｎｓ，Ｔｏｋｙｏ１８...     

Effect of geometrical nonlinearity on MEMS thermoelastic damping

Although the thermoelastic damping is a direct consequence of thermodynamic relations, its influence in macroscopic systems can be considered as negligible. This is not the case in microscopic devices, where this effect can be, for instance, one of the main causes that limit the quality factor of MicroElectroMechanical Systems (MEMS) resonators. Besides showing why it is important to introduce the thermoelastic damping in the design of MEMS, we present simulation results that demonstrate the differences that appear when a realistic nonlinear analysis is considered instead of using a linear approximation. Thus, we show how the nonlinearities due to large displacements impact the oscillation frequency as well as the decay time of displacements and energy for a MEMS cantilever.     

Thermoelastic damping effect analysis in micro flexural resonator of atomic force microscopy

In the design of high-Q micro/nano-resonators, dissipation mechanisms may have damaging effects on the quality factor (Q). One of the major dissipation mechanisms is thermoelastic damping (TED) that needs an accurate consideration for prediction. Aim of this paper is to evaluate the effect of TED on the vibrations of thin beam resonators. In particular, we will focus on cantilever beam resonator used in atomic force microscopy (AFM). AFM resonator is actually a cantilever with a spring attached to its free end. The end spring is considered to capture the effect of surface stiffness between tip and sample surface. The coupled governing equations of motion of thin beam with consideration of TED effects are derived. In general, there are four elastic equations that are coupled with thermal conduction equation. Based on accurate assumptions, these equations are simplified and the various boundary conditions have been used in order to validate the computational procedure. In order to accurately determine TED effects, the coupled thermal conduction equation is solved for the temperature field by considering three-dimensional (3-D) heat conduction along the length, width and thickness of the beam. Weighted residual Galerkin technique is used to obtain frequency shift and the quality factor of the thin beam resonator. The obtained results for quality factor, frequency shift and sensitivity change due to thermo-elastic coupling are presented graphically. Furthermore, the effects of beam aspect ratio, stress-free temperature on the quality factor and the influence of the surface stiffness on the frequencies and modal sensitivity of the AFM cantilever with and without considering thermo-elastic damping effects are discussed.     

Nonlinear vibration analysis of tapered Timoshenko beams

The nonlinear flexural vibration analysis of tapered Timoshenko beams is conducted. The equations of motion for tapered Timoshenko beams are established in which the effects of nonlinear transverse deformation, nonlinear curvature as well as nonlinear axial deformation are taken into account. The nonlinear fundamental frequencies of tapered Timoshenko beams with two simply supported or clamped ends are presented.     

Finite element modeling of the nonlinear oscillations in axisymmetric acoustic resonators

Oscillation of a gas in closed resonators has gained considerable interest in the past years. In this paper, the nonlinear equations governing the behavior of the gas oscillations inside the resonator are formulated in a weak form and then modeled using the finite element method. The pressure ratios, predicted by the proposed model, are in close agreement with the exact solutions available for simple geometries such as cylindrical, exponential and linearly varying area resonators. The presented comparisons validate the accuracy of the finite element model and emphasize its potential for predicting the performance or resonators of more complex geometries which are necessary for generating high pressures from the standing waves. Also, gas flow through the boundaries of the resonator is implemented in the proposed model. The presented finite element model presents an invaluable tool for designing a new class of acoustic compressors which can be used, for example, in refrigeration and vibration control applications.     

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonlinear analysis on the vibration of elastic plates

We consider the vibration of elastic thin plates under certain reasonable assumptions. We derive the nonlinear equations for this model by the Hamilton Principle. Under the conditions on the hyperbolicity for the initial data, we establish the local time well-posedness for the initial and boundary value problem by Picard iteration scheme, and obtain the estimates for the solutions.     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

A novel matrix method for coupled vibration and damping effect analyses of liquid-filled circular cylindrical shells with partially constrained layer damping under harmonic excitation

A novel matrix method is further developed for a liquid-filled circular cylindrical shell with partially constrained layer damping (CLD), which consists of treating liquid domain with Bessel function approach, and shell domain with transfer matrix equation based on a new set of first order matrix differential equation. In order to indicate its advantage to the finite element method (FEM), free vibration analysis on such an empty shell under clamped–clamped boundary are carried out by using the present method together with FEM. Meanwhile, coincident result is yielded for the liquid-filled shell by adopting the present method with by other transfer matrix method. Finally, a series of valuable numerical results are obtained by this method.     

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic circular cylindrical shells

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic (FGMETE) circular cylindrical shell are carried out in the present work. The Hamilton principle, higher order shear deformation theory, constitutive equation considering coupling effect between mechanical, electric, magnetic, thermal are considered to derive the equations of motion and distribution of electrical potential, magnetic potential along the thickness direction of FGMETE circular cylindrical shell. The influences of various external loads, such as axis force, temperature difference between the bottom and top surface of shell, surface electric voltage and magnetic voltage, on the buckling response of FGMETE circular cylindrical shell are investigated. The natural frequency obtained by present method is compared with results in open literature and a good agreement is obtained.     

Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro coriolis flowmeters based on strain gradient theory

Micro coriolis flowmeters are extensively used in fluidic micro circuits and are of great interest to many researchers. Straight and curved coriolis flowmeters are common types of coriolis flowmeters. Therefore in the present work, the out-of- plane vibration and stability of curved micro tubes are investigated to study the dynamic behavior of curved coriolis flowmeters. The Hamilton principle is applied to derive a novel governing equation based on strain gradient theory for the curved micro tube conveying fluid. Lagrangian nonlinear strain is adopted to take into account the geometric nonlinearity and analyze hardening behavior as a result of the cubic nonlinear terms. Linear stability analysis is carried out to investigate the possibility of linear instabilities. Afterwards, the first nonlinear out-of-plane natural frequency is plotted versus fluid velocity to determine the influence of nonlinear terms and hardening behavior on stability of the system. The influence of the length scale parameter is studied by comparison of the results for classical, coupled stress and strain gradient theory. Finally the phase difference between two points at upstream and downstream is plotted versus fluid velocity. Linear relation between the phase difference and fluid velocity is noticed, thus the curved coriolis flowmeter can be calibrated to measure flow rate by measuring the phase difference between two points.     

Thermal deflection of an inverse thermoelastic problem in a thin isotropic circular plate

In this paper an attempt has been made to solve the inverse problem of thermoelasticity in a thin isotropic circular plate by determining the unknown temperature gradient, temperature distribution and the thermal deflection on the edge of the circular plate. The results are obtained in terms of series of Bessel’s function and illustrated numerically.