Study of deflection and damping in microbeam resonator based on microstretch thermoelastic theory

This article deals with the deflection and thermoelastic damping analysis in homogeneous, isotropic, micropolar microstretch generalized thermoelastic thin beam based on Euler–Bernoulli theory. Analytical expressions for deflection, thermoelastic damping, frequency shift, temperature distribution and microstretch functions have been obtained for various boundary conditions viz. clamped, simply supported and cantilever beam by using Laplace transform technique. The analytical results have been numerically analyzed with the help of MATLAB software in case of magnesium like materials. The computed results have been presented graphically in view of various boundary conditions.     

A two-temperature model for evaluation of thermoelastic damping in the vibration of a nanoscale resonators

In this work, the thermoelastic damping of a nano-scale resonator is analyzed by the generalized thermoelasticity theory based on two-temperature model (2TLS). The effect of two-temperature parameter and relaxation time in nano-scale resonator are investigated for beams under clamped conditions. Analytical expressions for deflection, temperature change, frequency shifts, and thermoelastic damping in the beam have been derived. The theories of coupled termoelasticity and generalized thermoelasticity with one relaxation time can extracted as limited and special cases of the present model. The numerical results have been presented graphically in respect of thermoelastic damping and frequency shift.     

矩形板的非线性热弹耦合振动

本文从完整的弹性力学方程出发,导出了板的非线性热弹耦合振动基本方程,分析了力和温度场的边界条件,并对夹支边界和混合边界情况作了具体的分析和数字计算,得到了热弹耦合效应对热弹耦合振动的振幅和频率的作用机理.所得结果对板动力学的理论研究和实际应用有较大参考价值.     

几何非线性平板的热力,动力耦合振动

本文根据由热力学第一、第二定律推出的能量守恒方程,进而导出平板几何非线性热力动力耦合振动有限元方程组,提出一种计算量少,精度高的校正显式积分格式,求解耦合方程组的时间历程问题。研究了热力动力耦合平板横向振动振幅频率的影响及热弹阻尼现象。     

Flexural vibrations of poroelastic plates

Summary The governing equations of flexural vibrations of thin, fluid-saturated poroelastic plates are derived in detail. The plate material obeys Biot's theory of poroelasticity with one degree of porosity, while the plate theory employed is the one due to Kirchhoff. These governing equations are compared with the corresponding ones for thermoelastic plates and a poroelastic-thermoelastic analogy for flexural plate dynamics is established in the frequency domain. The dynamic response of a rectangular, simply supported, poroelastic plate to harmonic load is obtained analytically-numerically and the effects of inertia as well as of porosity and permeability on the response is assessed.     

Investigation of thermoelastic damping in rectangular microplate resonator using modified couple stress theory

Thermoelastic damping is a significant energy lost mechanism at room temperature in micro-scale resonators. Prediction of thermoelastic damping (TED) is crucial in the design of high quality MEMS resonators. In this study the governing equations of motion and the thermal couple equation of a microplate with an arbitrary rectangular shape are derived using the modified version of the couple stress theory. Analytical expressions are presented for calculating the quality factor (QF) of TED in a rectangular microplate considering the plane stress and plane strain conditions. As a case study, a rectangular microplate resonator is considered with material property of gold that has a considerably high value of length-scale parameter in comparison with silicon and the effect of the length-scale parameter on the QF of TED is discussed in detail. The relation between QF and temperature increment for microplates with clamped boundary conditions based on plane stress and plane strain models are studied and results obtained by considering classical and modified couple stress theory (MCST) are compared. The effect of thickness of the plate on the rigidity ratio is studied and the critical thickness which is an important design parameter is obtained using the MCST for three boundary conditions. Variations of TED versus the plate thickness for various boundary conditions according to the classical and the modified couple stress theories are investigated.     

Analysis of free vibrations for Rayleigh — Lamb waves in a microstretch thermoelastic plate with two relaxation times

The propagation of free vibrations in a microstretch thermoelastic homogeneous isotropic plate subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Green and Lindsay (G—L) theories of thermoelasticity. The secular equations for the microstretch thermoelastic plate in closed form for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. At short wavelength limits, the secular equations for both modes in a stress-free thermally insulated and isothermal homogeneous isotropic microstretch thermoelastic plate reduce to the Rayleigh surface wave frequency equation. The results for symmetric and skew-symmetric wave modes are computed numerically and presented graphically. The theory and numerical computations are found to be in close agreement. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 36–46, January–February, 2009.     

Geometrically nonlinear forced vibrations of the symmetric honeycomb sandwich panels affected by the water

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with the four edges simply supported and one surface affected by the water are investigated in this paper using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose physical and mechanical properties are determined using the Gibson correlations. The effect of water acting on honeycomb panels can be described as added mass, additional damping and additional stiffness coefficients which are obtained by the semi-analytical fluid pressures. The basic formulation of nonlinear forced vibrations has been developed base on the third-order shear deformation plate theory and Green Lagrange nonlinear strain–displacement relation. The equilibrium equations have been obtained using the Hamilton’s principle. Effects of water velocity, height and height ratio on the nonlinear forced vibration response have been studied for the honeycomb sandwich panels.     

Buckling and vibrations of two-directional FGM Mindlin circular plates under hydrostatic peripheral loading

An analysis is presented for the effect of hydrostatic peripheral loading on the free axisymmetric vibrations of functionally graded moderately thick circular plates on the basis of first-order shear deformation theory. The mechanical properties of the plate material are supposed to vary according to a power-law in both the radial and transverse directions. The numerical solution of the governing differential equation derived by using Hamilton's energy principle for such simply supported and clamped boundary conditions has been obtained employing the harmonic differential quadrature method choosing zeros of Chebyshev–Gauss–Lobatto as the grid points. The effect of different parameters has been analyzed on the frequency parameter for the first three modes of vibration. The critical buckling loads for both the plates have been computed by putting the frequencies to zero. Three-dimensional mode shapes for particular plate have been plotted. Obtained results have been compared.     

The reference temperature dependence of Young’s modulus of two-temperature thermoelastic damping of gold nano-beam

This work is concerning with the study of the thermoelastic damping of a nanobeam resonator in the context of the two-temperature generalized thermoelasticity theory. An explicit formula of thermoelastic damping has been derived when Young’s modulus is a function of the reference temperature. Influences of the beam height and Young’s modulus have been studied with some comparisons between the Biot model and the Lord–Shulman model (L–S) for one- and two-temperature types. Numerical results show that the values of the thermal relaxation parameter and the two-temperature parameter have a strong influence on thermoelastic damping at nanoscales.     

Pressure dependence of the thermoelastic quotient for three glasses

The interrelationship between the mechanical work done on a material in the elastic range and changes in its thermodynamic properties, that is, between stress and strain, on the one hand, and temperature and entropy, on the other, is known as the Thermoelastic effect. The phenomenon is exactly adiabatic and is characterized by the thermoelastic quotient commonly referred to as thermoelastic constant. The thermoelastic effect can be used for stress analysis by monitoring the stress fluctuations by means of infrared radiometry, Also, it can be applied to study the anharmonicity in materials by measuring the temperature changes associated with adiabatic pressure changes, In this paper thermodynamic expressions are derived for the pressure derivative of the thermoelastic quotient under adiabatic as well as isothermal conditions, The derived expressions are applied to investigate the thermoelastic effect for the three glasses, namely, silica glass, soda-lime silica glass, and lead-silica glass, The isothermal pressure derivative of the thermoelastic quotient is evaluated for the three glasses. The isothermal volume derivative of the Gruneisen function is calculated.     

A finite element formulation for thermoelastic damping analysis

We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well‐known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8‐node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three‐dimensional 20‐node iso‐parametric thermoelastic element, which is suitable to model massive structures. For the 8‐node element the dissipation along the plate thickness has been taken into account by introducing a through‐the‐thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled‐field elements. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of free vibrations in axisymmetric functionally graded thermoelastic cylinders

The free vibrations of an axisymmetric functionally graded, transversely isotropic, thermoelastic hollow cylinder have been modeled and analyzed. The material has been assumed to be graded according to a simple power law in the radial coordinate. The Laplace transform method has been used to solve the problem. The complex Laplace transform parameter has directly been used to find the natural frequencies of free vibrations without performing inversion of the transforms. The frequency equations of free vibrations in a hollow cylinder have been solved by using the software Maple. The natural frequencies of the first ten modes for different values of the grading index have been computed numerically for zinc material. The numerical results for radial stress, circumferential stress, temperature change, frequency shift, and thermoelastic damping (inverse quality factor) have been presented graphically. The closed form solutions obtained here are interesting and allow further parametric studies of functionally graded structures. The inhomogeneity parameter is useful in design and it can be tailored for specific applications as a controller.     

On the procedure of evaluating the damping ability of the materials of specimens subjected to torsional and bending vibrations

We describe a method for the design and numerical analysis of compact mechanical vibrating systems with generators of vibrations on their inertial weights and establish the conditions of damping of unwanted vibrations in these systems aimed at guaranteeing the absence of favorable conditions for the introduction of systematic errors to experimental results. The designed and manufactured vibrating systems are dynamically balanced, and predicted values of the frequencies of their natural vibrations satisfactorily agree with the experimental data. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 145–152, July–August, 1997.     

Piezoelectric ceramic rectangular transducers in flexural vibration

Based on the equivalent elastic method and coupled vibration theory, an analytic method is presented to study the flexural vibration of rectangular transducers consisting of piezoelectric ceramic thin plates. By introducing a mechanical coupling coefficient, the flexural vibration of the piezoelectric ceramic rectangular thin plate is reduced to two simple, one-dimensional flexural vibrations of narrow piezoelectric ceramic strips. The resonance frequency equations for the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration are derived under the free and simply supported boundary conditions analytically. The relationship between the resonance frequency and the flexural vibrational order, the geometrical shape, and the dimensions of the piezoelectric ceramic rectangular thin-plate transducer is analyzed. It is demonstrated that the one-dimensional vibrational theory for the flexural vibration of a narrow piezoelectric ceramic strip and the stripe-mode flexural vibrational theory for the piezoelectric ceramic rectangular thin plate can be derived directly from the theory obtained in this paper. Experimental results show that the measured resonance frequencies of the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration under free-boundary conditions are in good agreement with the calculated results. The method presented in this paper can be used in the resonance frequency analysis of vibrating systems in coupled vibration.     

轴向拉力对微梁质量传感器精度和热弹性阻尼的影响

针对存在轴向拉力的矩形截面微梁谐振式质量传感器中的质量传感灵敏度、热弹性阻尼以及最小检测质量等问题进行了深入的研究。推导了质量传感器在存在轴向拉力情况下的检测灵敏度、热弹性阻尼以及最小检测质量的表达式。揭示了轴向拉力对质量传感器的工作性能的影响机理。结果表明:轴向拉力会提高质量传感灵敏度;轴向拉力会降低谐振器的热弹性阻尼;轴向拉力可以使得质量传感器捕获更微小的检测质量。     

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with completed clamped supported boundaries

Geometrically nonlinear forced vibrations of symmetric rectangular honeycomb sandwich panels with clamped supported boundaries at the four edges are investigated using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose parameters of physical and mechanical properties are calculated by the corrected Gibson’s formula. The basic formulation of nonlinear forced vibrations has been developed based on the classical plate theory (CPT) and the nonlinear strain–displacement relation. The equilibrium equations have been obtained using Hamilton’s principle. Effects of axial half-waves, height and height ratio on the nonlinear free vibration response have been investigated for honeycomb sandwich panels.     

Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Based on nonlocal theory, this article discusses vibration of CDVGS¹ systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.     

Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.     

Analysis of wave motion at the boundary surface of orthotropic thermoelastic material with voids and isotropic elastic half-space

The purpose of this research is to study the effect of voids on the surface wave propagation in a layer of orthotropic thermoelastic material with voids lying over an isotropic elastic half-space. The frequency equation is derived on the basis of the developed mathematical model under the boundary conditions for welded and smooth contacts. The dispersion curves giving the phase velocity and attenuation coefficient versus the wave number enable one to reveal the effects of voids and anisotropy for welded contact boundary conditions. The specific loss and amplitudes of the volume fraction, normal stress, and temperature change for welded contact are obtained and presented graphically for a particular model showing the voids and anisotropy effects. Some special cases are also deduced.