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.     

Thermoelastic vibration analysis of Mems/Nems plate resonators with voids

In this paper, the effect of voids, relaxation times, thermomechanical coupling, surface conditions, and plate dimensions on energy dissipation induced by thermoelastic damping in microelectronics mechanical systems (MEMS)/ nanoelectronics mechanical systems (NEMS) resonators are investigated. Closed form expressions for the transverse vibrations of a homogenous isotropic, thermoelastic thin plate with voids, based on Kirchhoff theory have been derived. The exact solutions for the free vibrations of plates under clamped-simply supported (CS) and simply supported-simply supported (SS) conditions are obtained. Analytical expressions for deflection, temperature change, frequency shifts, and thermoelastic damping in the plate have been derived. Some numerical results with the help of MATLAB programming software in case of silicon nitride and magnesium like material have also been presented.     

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.     

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.     

Generalized thermoelastic diffusion in a nanoscale beam using eigenvalue approach

In this paper, the effect of mass diffusion in a thermoelastic nanoscale beam in context Lord and Shulman theory is studied. The analytical solution in the Laplace domain is obtained for lateral deflection, temperature, displacement, concentration, stress and chemical potential. The both ends of the nanoscale beam are simply supported. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. The results obtained are presented graphically for the effect of time and mass diffusion to display the phenomena physical meaning.     

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.     

冷弯薄壁型钢梁-OSB板组合楼盖静力挠度及振动性能试验研究

对冷弯薄壁型钢梁-OSB板组合楼盖足尺模型进行改变边界条件的振动试验, 研究组合楼盖在四边支撑和梁端简支、四边支撑和梁端固结、两边支撑和梁端简支、两边支撑和梁端固结四种不同边界条件时静力荷载下的挠度及人沿平行和垂直楼盖梁方向行走、脚跟冲击、沙袋从空中落下、激励锤冲击不同动力荷载工况下的自振频率。研究表明:边界条件的变化会引起组合楼盖挠度和自振频率的变化, 可以通过改变边界条件来提高组合楼盖的振动性能;边界条件不同时组合楼盖在1kN集中荷载作用下跨中挠度均小于2mm, 在动力荷载作用下除个别情况外自振频率均高于15Hz, 建议组合楼盖在1kN集中荷载作用下跨中挠度限值设为2mm, 振动舒适度要求较高时自振频率限值设为15Hz。另外, 采用等效刚度法提出的理论公式能较好地预测不同边界条件时组合楼盖在静力荷载下的挠度及动力荷载下的自振频率。     

反对称角铺设层合矩形板在不同边界条件下的后屈曲问题

本文以板的无量纲最大挠度为摄动参数,应用摄动法研究了在几种支承情况下,反对称角铺设层合矩形板的后屈曲问题,并以梁的特征函数所构成的广义付里叶级数作为高阶摄动偏微分方程的近似解,本文方法简捷,计算收敛性好、数据可靠。     

定常温度热弹性梁的精化理论

首先给出了定常温度热弹性Biot通解的一种新的简化形式,它看起来与各向同性弹性力学的Papkovich-Neuber通解十分相似。不作预先假设,从热弹性理论出发,利用Biot通解和Lur’e算子方法构造了梁的精化理论,得出了自由表面热弹性梁的三个精确方程:四阶方程、超越方程和温度方程。由一般的各向同性弹性梁推广到热弹性梁,导出了在反对称载荷和介质温度作用下热弹性梁的近似控制微分方程。     

Dynamic stability behavior of damped laminated beam subjected to uniformly distributed subtangential forces

The dynamic stability behavior of damped laminated beam with various boundary conditions subjected to the uniformly distributed subtangential forces is investigated using the finite element formulation. The formal engineering approach of the mechanics for the thin-walled laminated beam based on kinematic assumptions consistent with Vlasov beam theory is used. An extended Hamilton’s principle is employed to obtain the mass-, damping-, elastic stiffness-, geometric stiffness matrices, and the load correction stiffness matrix due to the subtangential forces, respectively. The method for the evaluation of critical values for divergence and flutter of the nonconservative systems is briefly introduced in case of considering and neglecting damping effects. Throughout numerical examples, the influence of various parameters on the dynamic stability behavior of the nonconservative laminated beam is newly investigated: (1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber orientation, (2) the effect of boundary condition on the instability region of the divergence-flutter system, and (3) the influence of external and internal damping on the flutter load.     

矩形板的中等大挠度两邻边铰支两邻边夹紧正交各向异性

利用Galerkin方法分析了von-Karman型两邻边铰支两邻边夹紧正交各向异性矩形板。所设的位移函数为梁振动函数,它不仅能精确地满足边界条件,而且具有正交的特性,从而把复杂的非齐次非线性偏微分方程组化为一组非线性代数方程组。通过非线性方程组的线性化和可调节参数的修正迭代解法找出问题的解。实践证明,梁振动函数的收敛很快,只须取出级数的前几项即可满足精度要求。最后求出了不同复合材料的挠度和应力值。      

反对称角铺设层合矩形板在不同边界条件下的后屈曲问题

本文以板的无量纲最大挠度为摄动参数,应用摄动法研究了在几种支承情况下,反对称角铺设层合矩形板的后屈曲问题,并以梁的特征函数所构成的广义付里叶级数作为高阶摄动偏微分方程的近似解,本文方法简捷,计算收敛性好、数据可靠。     

Thermoelastic damping in a micro-beam resonator using modified couple stress theory

In this paper, analytical expressions for the quality factor (QF) of thermoelastic damping (TED) applying modified couple stress theory (MCST) are presented for plane stress and strain conditions. For gold and nickel micro-beam resonators, which have a considerable length-scale parameter as case studies, the effect of the length-scale parameter on the QF of TED has been discussed in details. Results for the QF of TED show that when the beam thickness is close to the length-scale parameter of the material, the MCST diverges from the classical theory; otherwise, the two theories converge to each other. Also critical thickness variation due to applying MCST is studied.     

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.     

边界约束对梁抗爆动力响应的影响(Ⅰ)–理论研究及分析

边界约束的差异会直接影响结构的抗爆动力响应及承载能力,文中建立了复杂约束条件下抗爆梁在弹性阶段和塑性阶段的解析计算方法,并计算分析了竖向弹性与阻尼约束、水平约束刚度、抗弯约束、荷载形式以及屈服弯矩动力强化系数对动力响应的影响。计算表明：竖向弹性与阻尼约束会引起附加惯性力,能够明显降低结构在弹塑性阶段的位移动力系数。水平约束和抗弯约束影响结构的动态响应主要在塑性阶段,水平约束使梁截面在变形过程中产生横向压力,抗弯约束直接限制刚体转动,均有效降低了梁位移动力系数,相对提高结构的承载力。相同约束刚度和荷载峰值条件下,平台荷载下结构的位移动力函数均高于三角形荷载下位移动力函数,说明动荷载的作用时间越长,对结构承载越不利。另外考虑屈服弯矩的动力增强系数时,可提高结构的抗爆潜力。     

On the modeling of tilted fixed-guided flexible beams under tension

We derive explicit solutions for a fixed-guided slender suspension beam that is initially straight and tilted with respect to the moving direction of its sliding end. The beam experiences substantial axial forces during the tension, resulting in a nonlinear boundary value problem. We consider sliding end displacements in the direction that cause longitudinal tension along the beam. We first propose an exact approach, leading to analytical solutions for various physical variables such as the transverse force and deflection profile, in terms of the axial force and the positive real solution of a third-order algebraic equation. We also propose an alternative approximate solution based on a second-order equation, which provides closed-form analytical solutions for the physical variables. We also introduce analytical validation techniques for the underlying assumptions. Consistency with nonlinear finite-element analysis is also addressed. Moreover, the results of the approximate method are represented by dimensionless formulas, generating charts to predict solutions for arbitrarily assigned beam parameters. Magnitudes of the normal and shear stress values are also included to consider the effects of yield and shear strengths as the limiting factors at large deflection conditions.     

Aeroelastic analysis of bearingless rotors with a composite flexbeam

An aeroelastic analysis of bearingless rotors is investigated using large deflection beam theory in hover and forward flight. The bearingless configuration consists of a single flexbeam with a wrap-around torque tube and pitch links located at the leading edge and trailing edge of the torque tube. The outboard main blade, flexbeam, and torque tube are all assumed as an elastic beam undergoing arbitrary large displacements and rotations, that are discretized into beam finite elements. In the bearingless rotors, a flexbeam has various sections made of laminate. The sectional elastic constants of a composite flexbeam, including the warping deformations, are determined from a refined cross-sectional finite element method. Numerical results of the static deflections and the aeroelastic modal damping are presented for various configurations of a composite flexbeam and are compared with previously published experimental results and theoretical values obtained from a modal analysis using a moderate deflection-type beam theory.     

Dynamic response of a cracked beam subject to a moving load

Summary The dynamic response of a beam with a single-sided crack subject to a moving load on the opposite side is analyzed using Euler beam theory and the assumed mode method. The beam is modeled as two separate beams divided by the crack. Two different sets of admissible functions which satisfy the respective geometric boundary conditions are assumed for these two fictitious sub-beams. The rotational discontinuity at the crack is modeled by a torsional spring with an equivalent spring constant for the crack. The transverse deflection at the crack is matched by a linear spring of very large stiffness. Results of numerical simulations are presented for various combinations of constant axial velocity of the moving load and the crack size.     

On the size-dependent behavior of functionally graded micro-beams

In this paper, the size-dependent static and vibration behavior of micro-beams made of functionally graded materials (FGMs) are analytically investigated on the basis of the modified couple stress theory in the elastic range. Functionally graded beams can be considered as inhomogeneous composite structures, with continuously compositional variation from usually a ceramic at the bottom to a metal at the top. The governing equations of motion and boundary conditions are derived on the basis of Hamilton principle. Closed-form solutions for the normalized static deflection and natural frequencies are obtained as a function of the ratio of the beam characteristic size to the internal material length scale parameter and FGM distribution functions of properties. The results show that the static deflection and natural frequencies developed by the modified couple stress theory have a significant difference with those obtained by the classical beam theory when the ratio of the beam characteristic size to the internal material length scale parameter is small.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.