A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.     

Nonlinear free vibration of nanotube with small scale effects embedded in viscous matrix

In this paper, the nonlinear free vibration of the nanotube with damping effects is studied. Based on the nonlocal elastic theory and Hamilton principle, the governing equation of the nonlinear free vibration for the nanotube is obtained. The Galerkin method is employed to reduce the nonlinear equation with the integral and partial differential characteristics into a nonlinear ordinary differential equation. Then the relation is solved by the multiple scale method and the approximate analytical solution is derived. The nonlinear vibration behaviors are discussed with the effects of damping, elastic matrix stiffness, small scales and initial displacements. From the results, it can be observed that the nonlinear vibration can be reduced by the matrix damping. The elastic matrix stiffness has significant influences on the nonlinear vibration properties. The nonlinear behaviors can be changed by the small scale effects, especially for the structure with large initial displacement.     

四角点简支双稳态层合板振动特性研究

双稳态层合板作为一种柔性可变体结构,为实现振动状态下的构型调控,将对其开展固有振动特性的 研究．本文以四角点简支约束的矩形非对称铺设双稳态板作为研究对象,运用最小势能原理、一阶剪切变形理 论、以及冯卡门几何非线性位移应变关系,得到双稳态板的两种稳态构型．首次给出了适合于四点简支的6参 数构型函数与17参数构型函数,研究对比发现对于稳态构型采用17参数的构型函数结果相对更精确;对于应 用线性位移应变关系的基于任一稳态构型下振动固有特性而言,这两种构型函数得到的结果基频相差2 %,其 它阶次频率相差也不大．随后采用6参数构型函数得到的稳态构型研究了几何尺寸、铺层数分别对两种稳态构 型固有振动特性的影响．本研究对于进一步研究该类结构动态跳变机理具有重要意义．     

纤维曲线铺放的复合材料层合板的自由振动分析

假设曲线纤维的方向角沿板的长度方向按照线性规律变化,导出了纤维曲线铺设时的参考路径,将参考路径沿板的宽度方向平移可得一种曲线纤维增强复合材料单层板,当这种纤维曲线铺设的单层板对称铺放时即可得相应的曲线纤维增强复合材料层合板．基于弹性薄板的小挠度理论,建立了曲线纤维增强复合材料层合板自由振动问题的基本方程,采用微分求积法进行数值求解,得到了层合板的自振频率及相应的振型．与已有文献计算结果的比较,验证了本文计算结果的正确性．通过数值算例分析了微分求积法求解本问题时的收敛性,研究了纤维铺放路径和边界条件的不同对曲线纤维增强复合材料层合板频率及振型的影响,研究结果可为该种结构的设计提供一定的参考．     

基于车桥耦合理论的梁式桥阻尼比识别研究

提出一种基于车桥耦合动力学理论的梁式桥阻尼比识别方法. 首先按照动力学理论将测试车设计为单自由度体系, 然后利用安装在测试车上的传感器采集信号, 从测试车与桥梁接触点响应信号中得到梁式桥响应的信号, 基于车桥耦合动力学原理滤波处理得到包含梁式桥第一阶频率的信号, 最后假定梁式桥阻尼比值, 通过假定的梁式桥阻尼比值获取假定的梁式桥第一阶振型, 不断循环直至假定的阻尼比值下计算的第一阶振型最大值点居中, 即为识别的梁式桥真实阻尼比. 本文首先从车桥耦合动力学理论推导上说明了该方法的可行性, 然后考虑在不同车速与非恒定车速、路面粗糙度、环境噪音等影响因素下进行车桥耦合动力学模型分析, 最后通过实桥试验进行了初步验证. 研究结果表明: 该方法能一定程度上克服外界不利因素的影响, 达到识别梁式桥阻尼比的目的, 为识别梁式桥阻尼比提供一种更优方法, 其具有参数设置较少、操作简单方便以及更高测试精度等优点, 同时有助于推动基于车桥耦合的车桥耦合动力学理论技术在梁式桥模态参数识别工作中的实际工程应用.      

Modeling vibration behavior of delaminated composite laminates using meshfree method in Hamilton system

Free vibration analysis of composite laminates with delaminations is performed based on a three-dimensional semi-analytical model established by introducing the local radial point interpolation method (LRPIM) into a Hamilton system. The governing equation is derived with a transfer matrix technique and a spring layer model based on a local weak-form equivalent to the modified Hellinger-Reissner variational principle. Main superiority of the present model is that the scale of the governing equation involves only the so-called state variables at the top and bottom surfaces, and is insensitive to the thickness and the layer number of the composite laminates. Several numerical examples for analyzing the vibration frequencies and mode shapes of delaminated composite beams and plates are given to validate the model. The results are in good agreement with the pre-existing results.     

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES 1)

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.     

Mindlin 矩形微板的热弹性阻尼解析解

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.      

Free vibration and stability of an axially moving thin circular cylindrical shell using multiple scales method

This paper investigates the linear free vibration of axially moving simply supported thin circular cylindrical shells with constant and time-dependent velocity considering the effect of viscous structure damping. Classical shell theory is employed to express strain-displacement relation. Linear elasticity theory is used to write stress–strain relation considering Hook’s Law. Governing equations in cylindrical coordinates are derived using the Hamilton principle. Equilibrium equations are rewritten with the help of Donnell–Mushtari shell theory simplification assumptions. Motion equations for displacements in axial and circumferential directions are solved analytically concerning to displacement in the radial direction. As the displacement in the radial direction is the combination of driven and companion modes, the third motion equation is discretized using the Galerkin method. The set of ordinary differential equation obtained from the Galerkin method is solved using the steady-state method, which in practice leads to the prediction of the exact frequencies of vibration. By employing multiple scale method the critical speed values of a circular cylindrical shell and several types of instabilities are discussed. The numerical results show that by increasing the mean velocity, the system always loses stability by the divergence instability in different modes, and the critical speed values of lower modes are higher than those of higher modes. As well as the unstable regions for the resonances between velocity function fluctuation frequencies and the linear combination of natural frequencies is gained from the solvability condition of second order multiple scale method. The accuracy of the method is checked against the available data.

     

阻尼耗能减振框架结构地震作用振动方程求解及地震反应分析Calculation of vibration equation and analysis of seismic response for damping energy dissipation frame structure

研究了设置耗能阻尼器框架的地震作用振动方程求解及该结构的地震反应。框架结构设置耗能阻尼器后,振动方程的阻尼矩阵不再对振型具有正交性,本文对该振动方程给出了状态方程直接积分法,并与传统强制解耦法进行了比较分析,结果表明强制解耦法对结构第一阶振动反应求解偏差较小,而对于高阶振动反应差别较大,并且强制解耦法高估了阻尼器的减震效果。继而采用状态方程直接积分法对设置有粘滞阻尼器的框架结构进行了地震反应分析,探讨了阻尼器位置对框架结构地震反应的影响,结果表明设置有阻尼器的楼层减振效果明显,未设置阻尼器的楼层减振效果差别较大,甚至有可能出现楼层层间侧移增大的现象,由此提出耗能阻尼器应在结构中合理设置。     

Thermoelastic damping in micro-beam resonators

Thermoelastic damping is recognized as a significant loss mechanism at room temperature in micro-scale beam resonators. In this paper, the governing equations of coupled thermoelastic problems are established based on the generalized thermoelastic theory with one relaxation time. The thermoelastic damping of micro-beam resonators is analyzed by using both the finite sine Fourier transformation method combined with Laplace transformation and the normal mode analysis. The vibration responses of deflection and thermal moment are obtained for the micro-beams with simply supported and isothermal boundary conditions. The vibration frequency is analyzed for three boundary condition cases, i.e., the clamped and isothermal, the simply supported and isothermal, and the simply supported and adiabatic. The analytic results show that the amplitude of deflection and thermal moment are attenuated and the vibration frequency is increased with thermoelastic coupling effect being considered. In addition, it can be found from both the analytic results and the numerical calculations that these properties are size-dependent. When the thickness of the micro-beam is larger than its characteristic size, the effect of thermoelastic damping weakens as the beam thickness increases. The size-effect induced by thermoelastic coupling would disappear when the thickness of the micro-beam is over a critical value that depends on the material properties and the boundary conditions.     

Non-linear dynamic analysis of symmetric and antisymmetric cross-ply laminated orthotropic thin shells

In this paper, the governing equations for non-linear free vibration of truncated, thin, laminated, orthotropic conical shells using the theory of large deformations with the Karman-Donnell-type of kinematic nonlinearity are derived. Applying superposition principle and Galerkin’s method, these equations are reduced to a time dependent non-linear differential equation. The frequency-amplitude relationship for the laminated orthotropic thin truncated conical shell is obtained using the method of weighted residuals. In the particular case, we can obtain the similar relationships for the single-layer and laminated orthotropic cylindrical shells, also. The influence played by geometrical parameters of the conical shell and physical parameters of the laminate (i.e. material properties, staking sequences and number of layers) on the non-linear vibration behavior of the conical shell is examined. It is noticed that the non-linear vibration of shells is highly dependent on laminate characteristics and, from these observations, it is concluded that specific configurations of laminates should be designed for each kind of application. Present results are compared with available data for special cases.     

结构阻尼时域本构模型及其应用

阻尼合金作为一种新型结构功能材料在不少领域已获应用，由于其阻尼值较大且随频率呈复杂交化关系，传统的线性粘性阻尼理论或经典的非频变结构阻尼理论难以精确地描述其耗能行为。本文应用粘弹性阻尼理论，根据阻尼合金储能模量和损耗因子在频域的实测数据．应用最优化方法拟合出标准线性体模型中的本构参数；根据积分形式的三参量本构关系和变形体虚功原理，推导出了有限元形式的动力学方程；讨论了三参数初值的选取；对包含卷积积分的有限元动力学方程通过数学推导将其化为三阶线性微分方程组，再转化为标准状态变量方程，应用数值求解。数值计算实例证明了所提方法的正确和有效性。     

Free vibration analysis of stiffened laminated plates

Based on the semi-analytical solution of the state-vector equation theory, a novel mathematical model for free vibration analysis of stiffened laminated plates is developed by separate consideration of plate and stiffeners. The method accounts for the compatibility of displacements and stresses on the interface between the plate and stiffeners, the transverse shear deformation, and naturally the rotary inertia of the plate and stiffeners. Meanwhile, there is no restriction on the thickness of plate and the height of stiffeners. To demonstrate the excellent predictive capability of the model, several examples are analyzed numerically. The model presented in this paper can also be easily modified to solve the problems of stiffened piezolaminated plates and shells, or plates and shells with piezoelectric material patches.     

Modified H-R mixed variational principle for magnetoelectroelastic bodies and state-vector equation

Based upon the Hellinger-Reissner (H-R) mixed variational principle for three-dimensional elastic bodies, the modified H-R mixed variational theorem for magnetoelectroelastic bodies was established. The state-vector equation of magnetoelectroelastic plates was derived from the proposed theorem by performing the variational operations. To lay a theoretical basis of the semi-analytical solution applied with the magnetoelectroelastic plates, the state-vector equation for the discrete element in plane was proposed through the use of the proposed principle. Finally, it is pointed out that the modified H-R mixed variational principle for pure elastic, single piezoelectric or single piezomagnetic bodies are the special cases of the present variational theorem.     

架空输电线防振锤阻尼特性实验研究

本文利用梁的振动理论对防振锤的阻尼特性进行研究。通过正弦激励下的共振驻留法,分别对不同型号以及同种型号不同锤头质量的防振锤进行振动实验,应用能量平衡原理得到了防振锤的阻尼特性曲线,分析了不同模态响应在阻尼耗能中的贡献。考虑不同类型阻尼对于减振性能的作用,在Lazan阻尼假设的基础上,利用经验公式推导了阻尼系数计算公式。结果表明,钢绞线变形大小直接影响防振锤的阻尼值大小;阻尼对减小共振频率附近的受迫振动幅值作用明显,验证了防振锤的减振性能;同时锤头质量的变化可以影响阻尼曲线的变化。以上结论可为防振锤的防振设计和应用提供参考。     

基于三阶剪切变形理论的悬臂输流管道自由振动

采用三阶剪切变形理论,结合有限元法研究了悬臂输流管道的自由振动问题．利用虚功原理建立了输流管系统的有限元方程,同时将悬臂端弹性支承以势能的形式引入到系统方程中,求解了系统前三阶的复频率．分别探讨了流体速度和弹簧刚度对系统复频率实部和虚部的影响,重点分析了弹簧刚度与前三阶固有频率间的关系．在弹性支承刚度为零的特例下,对比了本文结果与Timoshenko梁理论的结果,证明了本文方法的可靠性．研究发现系统固有频率的实部恒为负值,表明一端带有弹性支承的约束形式有利于提高悬臂输流管道自由振动的稳定性;流体的流动对管道振动起到了阻尼作用,在流动速度足够大的情况下,各阶振动固有频率均趋于零;当弹簧刚度为无穷大,且流体速度足够大时,输流管道将发生失稳．     

Bending and vibration of composite laminated plates

Hybrid-stress finite element method is applied for analysis of bending and vibration of composite laminated plates in this paper. Firstly, based on the modified complementary principle, a rectangular hybrid-stress plate bending element is presented which applies to analysis of laminates. Inside the element, different stress parameters are assumed according to different layers. The boundary displacements are determined by means of the assumption of YNS theory on the boundary of elements. The element formed in this way not only can take effects of transverse shear deformation and local warping into account, but also has less degrees of freedom. Then, problems of bending and vibration of laminates are solved by using this element, and the numerical results are compared with the exact solutions. This shows that the results obtained in the paper are very close to the exact results.     

旋转薄壁圆柱壳振型进动的非线性振动特性

选取在工程上常用的悬臂旋转薄壁圆柱壳为研究模型,首先推导出考虑阻尼的振型进动因子,然后根据Donnell's简化壳理论建立考虑科氏力,阻尼与几何大变形的非线性波动方程,采用Galerkin方法对波动方程进行离散化,得到模态坐标中相互耦合的三阶非线性微分方程组.应用Runge-Kutta法求解获得非线性幅频特性曲线,分析了不同模态组合下系统主模态(m=1,n=6,k=1)的共振响应.应用谐波平衡法对系统三阶非线性微分方程组解析分析,与数值解比较验证了解析解的正确性和有效性.最后分析了动力系统的运动稳定性.结果表明,节径数n和频率倍数k对于主模态共振响应的影响很小,而轴向半波数m对主共振的影响则相对较大,因此只需选取相邻的两个轴向模态(M=2)即可较为简洁,准确的描述主共振响应;谐波平衡法可以很好的解决三阶微分方程组的非线性问题,并且能够达到较为满意的精度.     

Nonlinear harmonic response characteristics and experimental investigation of cantilever hard-coating plate

The nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hard-coating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate. Firstly, based on the theories of von Karman and Reddy’s three-order shear deformation, the nonlinear dynamic equations of hard-coating plate are built by Hamilton variation principle. Secondly, to obtain nonlinear governing equation of hard-coating plate under transverse load, these equations are discretized in Galerkin method. The system averaged equations with 1:3 internal resonances are obtained by the method of multiple scales, and the multi-periodic responses behavior of cantilever hard-coating plate under transverse loading could be presented. Finally, the vibration response experiment of hard-coating plate is conducted, and the multi-periodic responses are also present for the hard-coating plate with three-to-one internal resonance. Besides, through the vibration response experiment of uncoated titanium alloy plate, the damping characteristic of hard coating is further analyzed.