样条扇形单元

本文在样条分片插值及样条矩形单元的基础之上进而讨论极坐标中二次及三次样条分片插值及样条(圆环)扇形单元.以用于求解圆(环)域与(圆环)扇形域上的各类问题.圆环扇形单元(r≠0)是样条分片插值在极坐标中简单的推广应用,但扇形单元则不然.本文根据扇形单元在r=0处的特殊性对各位移插值函数作了合理的处理,使得该单元即体现了r=0处的几何特性又可以消除该处应变、应力的奇异性.文中给出了用样条(圆环)扇形单元求解平面问题及薄板弯曲问题的数值算例用以说明该单元的效能.     

圆弧形波纹膜片的矩阵联乘解法

将圆弧形波纹膜片看作几段圆环壳和中心圆板的组合构件,本文利用钱伟长教授的圆环壳一般解[1]及圆薄板小参数振动理论[2],导出环壳和板的传递矩阵和连接矩阵,用矩阵联乘法求得线性精确解和非线性解,计算结果同W.A.Wildhack的实验[3]是吻合的.     

圆环曲线的几何画线

某机械厂承担了一批水管的铸造业务,为了便于检查和清洗管道,使用单位要求在90°弯头上开一个检查孔,盖上盖子后,管內仍保持圆滑无阻.这个孔盖的周边曲线该怎么画线?这一问题颇有实际意义.所要求的曲线,显然是平面切割圆环的相贯线,我们叫它圆环曲线.本文介绍了圆环曲线的直角坐标方程及该曲线的几何画线方法.     

磁场中旋转运动圆环板主共振分岔及混沌研究

研究了磁场中旋转运动圆环板的磁弹性主共振及分岔、混沌问题.通过Hamilton(哈密顿)原理推得磁场中旋转运动圆环板的横向振动方程,并采用Bessel(贝塞尔)函数作为振型函数进行Galerkin(伽辽金)积分,得到磁场中旋转运动圆环板的无量纲非线性振动常微分方程.利用多尺度法展开,得到静态分岔方程、对应的转迁集与分岔图,以及物理参数作为分岔控制参数时的分岔图.利用Mel’nikov(梅利尼科夫)方法,对系统混沌特性进行研究,得到外边夹支内边自由边界条件下异宿轨破裂的条件;通过数值计算,得到外激振力幅值作为分岔控制参数时系统的分岔图与指定参数条件下系统响应图.结果表明,磁场扼制多值现象的产生;激振频率、转速、磁感应强度越小,激振力幅值越大,系统的异宿轨越容易发生破裂,从而引发混沌或概周期运动.     

受弹性点支的任意形状的膜的振动

本文提供了一个求解受弹性点支的任意形状的膜的振动的新方法.将弹性点支反力看作是作用于膜上的未知外力,求出了包含有未知反力的运动方程的精确解,利用弹性点支处位移和反力的线性关系导出频率方程.最后以受弹性点支的圆膜为例给出了其频率方程的具体计算公式,并数值计算了受两个对称弹性点支的圆膜的固有振动频率.     

参量对正交异性矩形膜结构非线性振动的影响

本文主要研究了参量对正交异性矩形膜结构非线性无阻尼振动的影响。首先用解析法求得正交异性矩形膜结构在大挠度下的自由振动频率近似解析解,然后利用MATLAB编程分别计算在不同的初始预张力作用下膜面振动频率随膜面密度变化和在不同边长尺寸下膜面振动频率随膜面初始位移变化,最后分析得到了各参量对膜面振动频率的影响规律。     

任意梯度分布功能梯度圆环的热弹性分析

对沿径向任意变化的材料参数的功能梯度圆环进行了热弹性分析.与以前关于该问题的分析不同,既不需要预先给定具体的梯度变化形式,也不需要对结构进行细分.给出一种新的有效解法将问题转换为求解Fredholm积分方程,从而通过Fredholm积分方程的解给出热应力和位移的分布情况.最后通过算例分析了内外表面受不同温度作用时,材料参数呈现梯度变化对圆环的应力和位移变化的影响,计算结果表明某些特定的材料梯度可有效缓解圆环内的热应力分布.该文得到的结果对功能梯度圆环在结构安全设计方面有重要的理论指导意义.     

中细柔性圆环壳整体弯曲的一般解及在波纹管计算中的应用(Ⅰ)——基本方程和一般解

讨论了圆环壳在横向载荷作用下的整体弯曲问题,提出了中细柔性圆环壳的概念、基本方程和一般解。本文称粗细比(子午线曲率半径与子午线曲率中心到回转轴的距离之比)小于1/3的圆环壳为中细圆环壳;方程的推导基于E.L.Axelrad的柔性壳理论;所得的解含有处理边界问题所必须的积分常数并且在全域内处处收敛;本解能方便地用于有关波纹管的计算。全文共分4个部分:(Ⅰ)基本方程和一般解;(Ⅱ)Ω型波纹管的计算;(Ⅲ)C型波纹管的计算;(Ⅳ)U型波纹管的计算。     

矩阵方程ATXA=D的条件数与向后扰动分析

讨论矩阵方程ATXA=D,该方程源于振动反问题和结构模型修正.本文利用Moore-Penrose广义逆的性质,给出该方程解的条件数的上、下界估计.同时,利用Schauder不动点理论给出该方程的向后扰动界,这些结果可用于该矩阵方程的数值计算.     

中细柔性圆环壳整体弯曲的一般解及在波纹管计算中的应用（I）—基本方程和一般解

讨论了圆环壳在横向载荷作用下的整体弯曲问题，提出了中细柔性圆环壳的概念、基本方程和一般解。本文称粗细比（子午线曲率半径与子午线曲率中心到回转轴的距离之比）小于1／3的圆环壳为中细圆环壳；方程的推导基于E.L.Axelrad的柔性壳理论；所得的解含有处理边界问题所必须的积分常数并且在全域内处处收敛；本解能方便地用于有关波纹管的计算。全文共分4个部分：（I）基本方程和一般妥；（Ⅱ）Ω型波纹管的计算；（Ⅲ）C型波纹管的计算；（Ⅳ）U型波纹管的计算。     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

弹性振动的闭环系统

细长体的飞行器在飞行中考虑了既有刚性运动又有弹性振动的运动,由于刚性运动对弹性振动的影响,通过安装在飞行器上面的仪表所测得的角速度作为反馈信号输入到控制器,由控制器输出端输出信号到执行机构来实现反馈控制,把刚性运动飞行器、弹性振动飞行器同时考虑作受控对象,这里我们研究了由刚性飞行器、弹性飞行器和控制器三者形成的闭环系统的弹性振动问题,得到了求闭环系统的频率和振型的公式,设计控制器使得闭环系统渐近稳定的条件和能控性、能观测性的条件。     

计及弯曲刚度的印刷运动薄膜横向振动控制研究

研究了不同边界条件下,计及弯曲刚度的轴向运动薄膜横向振动的主动控制问题.建立计及弯曲刚度的印刷运动薄膜的计算模型.利用有限差分法,对轴向运动薄膜的振动微分方程进行离散,推导出轴向运动矩形薄膜横向振动控制系统的状态方程.采用次最优控制法,对不同边界条件下轴向运动矩形薄膜横向振动进行主动控制研究.计算结果表明:采用次最优控制法能够在短时间内迅速、有效地降低运动薄膜的振动强度,并使之衰减趋近于0.作动器作用在固定位置点处时,对运动薄膜施加控制后,四边简支边界条件下的控制效果好.作动器作用在不同位置点处时,两种边界条件下中心点处的控制效果最好.计算证明次最优控制法能够有效地抑制印刷过程中计及弯曲刚度的轴向运动薄膜的横向振动,从而提高印刷套印精度,保证精密印刷质量.     

两对边简支中间有任意多个单向弹性线支矩形板横向振动的一个解析解法*

本文给出了两对边简支另两对边任意支承的中间有任意多个单向弹性线支矩形板横向振动的一个新的解析解法、将弹性线支反力看作是作用于板上的待求外力,求得了含有来知的弹性线支反力的两对边简支矩形板的运动方程的解析解;利用弹性线支反力与板横向位移之间的线性关系导出频率方程;频率方程及振型函数的表述均与已有方法不同.     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.     

矩形网格扁壳结构的非线性振动

本文运用作者已建立的矩形网格扁壳的非线性弹性理论,求解了该类结构的非线性振动问题。通过采用横向挠度（网格节点横向位移）和力函数的某种（广义）Fourier级数形式的设定解,由试函数的加权得到解中系数之间的关系和决定时间未知函数的振动方程,然后利用正则摄动法和迦辽金法推导出结构自由振动和谐和激励作用下结构非线性受迫振动的幅频关系,并给出了计算实例。     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Normal modes of oscillations of a rotating membrane with a rigid central insert (an application of Heun functions)

The eigenvalues problem eigenvalues for the equation of the transverse oscillations of a homogeneous annular membrane with a rigid insert, rotating with a constant angular velocity about its central axis, is considered. Exact analytical expressions for the eigenfunctions in terms of special functions (local Heun functions), as well as normalization integrals, are found. An explicit expression for the time-invariant shape of the membrane during regular precession of its rotation axis is obtained.     

The nonlocal frequency behavior of nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon

Considering the potential applications of the buckled structures as nanomechanical mass sensors, this paper presents an analytical method to solve the frequency shifts of the first-order transverse and longitudinal vibration modes when a mass attaches on the surface of a buckled nanoribbon based on the nonlocal elastic theory and the Lagrange's motion equation. The first-order transverse and longitudinal vibration modes of the buckled nanoribbon are introduced. A comparison between the analytic solution and the finite element analysis (FEA) is presented. Then, the effects of the compressive strain, the magnitude and location of attached mass, the nonlocal parameter, and attached mass numbers on the frequency shifts are presented. From example calculations, it is seen that the magnitude of attached mass increases the frequency shifts of the first-order modes, except the first-order transverse vibration mode at the location Z1=0. The frequency shifts for the first-order transverse and longitudinal vibration modes are different, and could be used as important principles in mass sensing. What's more, the compressive strain and the nonlocal parameter play significant roles on the sensing process of the buckled nanoribbon. The results could serve as useful references for the fabrications and applications of buckled structures based nanomechanical mass sensors.     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.