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。      

文克尔地基上纵横弯曲[1]变截面梁主振型函数之正交性

导出梁的主振型函数之正交性条件     

高阶剪切变形理论下两邻边铰支两邻边夹紧复合材料层板的几何非线性分析

首先用虚位移原理推导出以位移形式表达的Reddy型高阶剪切变形理论的复合材料层板的非线性控制方程。选定的5个位移函数均满足两邻边铰支两邻边夹紧边界条件。用Galerkin方法把无量纲化之后的控制方程组转化为非线性代数方程组。稳定化双共轭梯度法用于求解稀疏线性方程组,可调节参数的修正迭代法用于求解非线性代数方程组。最后求出了不同复合材料的挠度和弯矩值并同Kirchhoff及Reissner-Mindlin板的结果进行了比较。     

基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程.提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数.随后将...     

有任意脱层复合材料梁的非线性动力稳定性

基于弹性理论建立了有脱层复合材料梁的基本方程式，研究了有任意脱层的考虑横向剪切变形的复合材料梁的非线性动力稳定性，对脱层梁进行分区处理，方便地描述了脱层长度，脱层位置，利用振型函数作为位移函数的形函数，采用增量谐波平衡法对基本方程式进行求解，考虑了不同脱层位置和不同脱层长度对脱层梁的非线性动力稳定性的影响，得出了各种情况下的动力不稳定性区域。     

四边弹性梁支承的矩形板非线性弯曲

本文给出四边弹性梁支承的矩形板在横向荷载作用下的非线性弯曲的重级数解。通过把板内的横向位移展成重傅里哀级数,在边界上的位移展成单傅里哀级数以及把应力函数展成广义傅里哀级数,使边界条件转化成了无穷线性代数方程,控制微分方程转化成了无穷非线性代数方程。文后给出了四边弹性梁支承和对边简支对边弹性梁支承的矩彤板非线性弯曲的计算结果。     

旋转梁振动分析的ODE求解器法

本文提出用标准的常微分方程（ＯＤＥ）求解器直接求解旋转梁的挥舞振动固有频率和振型，主要作法是通过建立旋转梁的挥舞振动方程综合运用ＯＤＥ变换技巧，将其化为标准的非线性问题，再输入求解器求解，本法简单明了且直接可靠，可由小到大求得各阶频率和振型，并使计算结果几乎可达到所需的任意精度。     

波纹膜片的非线性稳定

应用轴对称旋转扁壳的基本方程,研究了在任意载荷作用下各种边界条件的波纹膜片的非线性稳定问题。采用格林函数方法,将扁壳的非线性微分方程组化为非线性积分方程组。再使用展开法求出格林函数,即将格林函数展开成特征函数的级数形式,积分方程就成为具有退化核的形式,从而容易得到非线性代数方程组。应用牛顿法求解非线性代数方程组时,为了保证迭代的收敛性,选取位移作为控制参数,逐步增加位移,求得相应的载荷。作为算例,首先研究了带中心平台三波纹膜片的局部失稳现象,然后讨论了由于缺陷的存在,波纹膜片有可能出现的极值点失稳,这是一种类似扁球壳的总体失稳现象。解答可供波纹膜片的设计参考。     

非线性转子—轴承系统瞬态响应求解的分块直接积分法

提出了一种求非线性转子－轴承系统瞬态响应的分块直接积分法，该方法把隐式直接积分法需迭代求解的高维非线性代数方程降到一个低维非线性代数方程，从而比普通直接积分法的求解时间减少了许多。     

变截面开口薄壁杆约束扭转振动的主振型函数之正交性

导出杆的主振型函数之正交性条件。     

Multi-level residue harmonic balance solution for the nonlinear natural frequency of axially loaded beams with an internal hinge

In this article, an analytical approach, namely, multi-level residue harmonic balance is introduced and developed for the nonlinear free vibration analysis of axially loaded beams with an internal hinge. The main advantage of this method is that only one set of nonlinear algebraic equations is required to be solved for obtaining the zero level solution while the high accuracy of the higher level solutions can be obtained by solving a set of linear equations. The new approximate analytical solution method is developed for solving the governing differential equations. The accuracy and efficiency of the proposed method are verified by a numerical method. In the comparison, the results obtained from the proposed method well agree with those from other methods. The effects of vibration amplitude, axial force, and hinge location on the fundamental frequencies of various beam cases are investigated. The optimum and worst hinge locations are also studied.     

圆柱壳几何大变形非线性频率求解的渐近摄动法

为解决圆柱壳在工作状态中由几何大变形而引起的弱非线性振动问题，将渐近摄动法引入求解考虑几何非线性的薄壁圆柱壳振动频率。首先，应用Donnell＇s简化壳理论获得了考虑几何大变形情况下具有位移三次项的非线性频率方程，把位移及频率以非线性参数的幂级数形式展开，并令同次幂的非线性项系数相等，由此得到非线性频率一次近似值与初始振幅的一系列耦合代数方程，引入Galerkin＇s方法对非线性频率方程进行解耦正交并忽略其中的永年项，考虑了对应实数根，各阶频率对应的振幅间不存在相互耦合的内共振现象，最终在引入小参数后用摄动法求出了非线性频率的一次近似解。计算结果表明，几何非线性使薄壁圆柱壳产生硬化，其非线性频率升高，并同时讨论了线性、非线性频率与节径数及初始位移之间的关系。     

复合载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程，研究了波纹扁壳在复合载荷作用下的非线性受迫振动问题。采用格林函数方法，将扁壳的非线性偏微分方程组化为非线性积分微分方程组。再使用展开法求出格林函数，即将格林函数展开为特征函数的级数形式，积分微分方程就成为具有退化核的形式，从而容易得到关于时间的非线性常微分方程组。针对单模态振形，得到了谐和激励作用下的幅频响应。作为算例，研究了正弦波纹扁球壳的非线性受迫振动现象。得到的解答可供波纹壳的设计参考。     

非线性振动分析的切比雪夫谱元法

为了进一步探索Chebyshev时间谱元法求解非线性的振动问题,从Bubnov-Galerkin方法出发,在第二类Chebyshev正交多项式极点处;用重心Lagrange插值来构造节点基函数及其特性,推导了非线性振动问题的伽辽金谱元离散方案,借助Newton-Raphson法求解非线性方程组。对于非线性单摆,还需要将二分法和重心Lagrange插值结合求解角频率。以Duffing型非线性振动和非线性单摆振动问题为例,验证了此方法具有现实可行和高精度的优点。     

On the anisotropic piezoelastic contact problem for an elliptical punch

Summary This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media.     

Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method

Present investigation is concerned with the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions. The analysis is based on the classical and first order shear deformation beam theories. Material properties of the beam vary continuously in the thickness direction according to the power-law exponent form. Trial functions denoting the displacement components of the cross-sections of the beam are expressed in simple algebraic polynomial forms. The governing equations are obtained by means of Rayleigh–Ritz method. The objective is to study the effects of constituent volume fractions, slenderness ratios and the beam theories on the natural frequencies. To validate the present analysis, comparison studies are also carried out with the available results from the existing literature.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.