复合载荷作用下夹层圆板的非线性振动和屈曲

基于von Karman薄板理论,建立了均布载荷和周边面内载荷联合作用下夹层圆板非线性振动问题的控制方程和滑动固定边界条件,给出了相应静力问题的精确解及其数值结果．基于时间模态假设和变分法,得到了空间模态的控制方程,并使用修正迭代法求解该方程,得到夹层圆板幅频-载荷特征关系．讨论了两种载荷对夹层圆板振动特性的影响规律．当周边面力使夹层圆板的最低固有频率为零时,就可获得临界载荷的值．     

周边面内压力作用下夹层圆板的非线性振动

基于von Krmn薄板理论,讨论了滑动固定基础上周边面内压力作用下夹层圆板的非线性振动问题,应用变分法导出了该问题的非线性特征方程和边界条件,给出了其精确静态解,并使用修正迭代法求解了该方程,导出了夹层圆板振幅和非线性振频的解析关系式．当周边面力使夹层圆板的最低固有频率为零时,就可获得临界载荷的值．     

夹层圆板大挠度问题的进一步研究*

本文首次给出了在均布载荷作用下、具有滑动固定边界条件并计及表板抗弯刚度的夹层圆板的非线性问题的解.在求解非线性弯曲方程时,我们提出了修正幂级数方法.然后,把本文结果与刘人怀和施云方的结果作了比较.本文解在工程应用中可用作更精确的基础.     

夹层圆板的非线性弯曲

本文导出了具有软夹心的夹层圆板的非线性轴对称弯曲理论的基本方程和边界条件,并给出了表板很薄情况下的这些方程和边界条件的简化形式.作为算例,研究了在均布横向载荷作用下具有滑动固定边界条件的夹层圆板,使用修正迭代法,得到了相当精确的解析解.     

线载和弹性支承作用面内运动薄板磁固耦合双重共振

针对磁场环境中具有线载荷和弹性支承作用的面内运动薄板,给出了系统的势能、动能及电磁力表达式,应用Hamilton变分原理,推得面内运动条形板的磁固耦合非线性振动方程.考虑边界为夹支-铰支的约束条件,利用变量分离法和Galerkin积分法,得到了含简谐线载力和电磁阻尼力项的两自由度非线性振动微分方程组.应用多尺度法对主-...     

夹层圆板的大幅度振动

本文利用哈密顿原理导出了夹层圆板轴对称大幅度自由振动的基本方程,并给出了表板很薄情况下的简化形式.作为算例,利用修正迭代法求出了具有滑动固定边界条件夹层圆板对轴称大幅度自由振动的一种解析解,并由此导出了夹层圆板振幅和振频的解析关系式.     

剪切变形对直线型正交异性层合圆板大幅度受迫振动的影响*

本文研究了计及横向剪切变形的直线型正交异性层合圆板在简谐载荷q₀cosωt作用下的非线性受迫振动问题.采用伽辽金方法得到强振频率与振幅关系的解析解.最后,分析了横向剪切对板振动的影响,并给出了板的非线性自由振动的非线性周期对线性周期的比值.     

简支夹层矩形板的非线性弯曲

本文应用变分法导出了具有软夹心的夹层矩形板的非线性弯曲理论的基本方程和边界条件.然后,使用摄动法研究了均布横向载荷作作用下简支夹层矩形板的非线性弯曲问题,得到了相当精确的解析解.     

夹层矩形板的非线性振动

本文首先应用Hamilton原理建立了具有软夹心和薄表层的夹层矩形板的非线性振动理论,然后用方法对简支和铰支两种边界条件下的夹层矩形板的非线性自由振动问题进行了研究,获得了非线性周期与振幅的关系式。     

修正迭代法在波纹圆板非线性振动问题中的应用

在本文中,我们将修正迭代法成功地推广运用于全波纹圆板的非线性振动问题的研究,获得了全波纹圆板的非线性振频和振幅的解析关系式.本文还讨论了波纹圆板的几何参量对其振动特性的影响,本文结果对精密仪器弹性元件的设计具有一定的实际意义.     

Free vibration analysis of cracked postbuckled plate

In this paper, a methodology is introduced to address the free vibration analysis of cracked plate subjected to a uniaxial inplane compressive load for the first time. The crack, assumed to be open and at the edge is modeled by a massless linear rotational spring. The governing differential equations are derived using the Mindlin theory, taking into account the effect of initial imperfection. The response is assumed to be consisting of static and dynamic parts. For the static part, differential equations are discretized using the differential quadrature element method and resulting nonlinear algebraic equations are solved by an arc-length strategy. Assuming small amplitude vibrations of the plate about its buckled state and exploiting the static solution in the linearized vibration equations, the dynamic equations are converted into a non-standard eigenvalue problem. Finally, natural frequencies and modal shapes of the cracked buckled plate are obtained by solving this eigenvalue problem. To ensure the validity of the suggested approach an experimental setup and a numerical finite element model have been made to analyze the vibration of a cracked square plate with simply supported boundary conditions. Also, several case-studies of cracked buckled plate problem have been solved utilizing the proposed method, and effects of selected parameters have been studied. The results show that the applied load and geometric imperfection as well as the position, size and depth of the crack have different impact on natural frequencies of the plate.     

Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations

This paper presents a nonlinear free vibration analysis of corrugated circular plates with shallow sinusoidal corrugations under uniformly static ambient temperature. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton’s principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated plates are obtained successfully. The comparison with available published results shows that the numerical analysis here is of good reliability. A detailed parametric study is conducted involving the dependency of nonlinear frequency on the depth and density of corrugations along with the temperature change. Effects of these variables on the trend of nonlinearity are plotted and discussed.     

Electro-dynamic analysis of smart nanoclay-reinforced plates with integrated piezoelectric layers

Integrating engineering structures with piezoelectric layers as actuator and/or sensor offers smart sandwich structures with controllable static and dynamic deflections. In this paper, a smart sandwich plate consisting of a light nanoclay-reinforced composite core and two piezoelectric face sheets is considered. The static and dynamic behaviors of the proposed smart plate are obtained using their governing coupled electro-mechanical system of equations. In order to facilitate the governing equations, a mesh-free method based on moving least square (MLS) shape function and first order shear deformation (FSDT) is developed and implemented. Two morphologies of intercalated stack and exfoliated nanoclay dispersions are considered in the distribution of the nanoclay into the polymeric matrix. The effects of morphology and volume fraction of the nanoclay, time-dependent loading, and essential boundary condition on the static and dynamic behavior of the smart piezoelectric-integrated nanocomposite plates are examined. In the dynamic analysis, resonance and amplitude modulation phenomena are studied. It is observed that the use of nanoclay, especially with exfoliated morphology, improves the static and the free vibration responses of the smart sandwich plates. Moreover, the frequency of the applied mechanical load has a significant effect on the electro-dynamic response of the proposed smart sandwich plates.     

几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程,运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比,验证了公式的正确性．基频近似解表达式简单明了,物理意义明确,清楚地说明了初始荷载及相关因素对板自由振动基频的影响,直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度,提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.     

Analysis of the effects of a pulsed electromagnetic field on the dynamic response of electrically conductive composites

The effects of pulsed electromagnetic fields on the dynamic mechanical response of electrically conductive anisotropic plates are studied. The analysis is based on the simultaneous solving of the system of nonlinear partial differential equations that include equations of motion and Maxwell’s equations. Physics-based hypotheses for electro-magneto-mechanical coupling in anisotropic composite plates and dimension reduction solution procedures for the nonlinear system of the governing equations are presented. A numerical solution procedure for the resulting two-dimensional nonlinear system of the governing equations has been developed and consists of the sequential application of time and spatial integration and quasilinearization. The developed methodology is applied to the problem of the dynamic response of a long current-carrying unidirectional carbon fiber polymer matrix composite plate subjected to transverse impact load and in-plane pulsed electromagnetic load. The interacting effects of the pulsed electric current, external magnetic field, and mechanical load are studied.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: Possio integral equation approach

In this work, the static stability of a thin plate in axial subsonic airflow is studied using the framework of Possio integral equation. Specifically, we consider the cases when the plate’s leading edge is free and the plate’s trailing edge is either pinned or clamped. We formulate the problem under consideration using a partial differential equations (PDE) model and then linearize the model about the free stream velocity, density, and pressure, to enable analytical treatment. Based on the linearized model, we introduce a new derivation of a Possio integral equation that relates the pressure jump along the thin plate to the plate’s downwash. The steady state solution to the Possio equation is then used to account for the aerodynamic loads in the plate steady state governing equation resulting in a singular differential-integral equation which is transformed to a singular integral equation that represents the static aeroelastic equation of the plate. We verify the solvability of the static aeroelastic equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. By constructing solutions to the static aeroelastic equation and matching the nonzero boundary conditions at the trailing edge with the zero boundary conditions at the leading edge, we obtain characteristic equations for the free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. We also construct, analytically, flow speed intervals that correspond to static stability regions for free-clamped plates. Furthermore, we resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.