Dynamic stiffness analysis of laminated beams using a first order shear deformation theory

In this paper the exact vibration frequencies of generally laminated beams are found using a new method, including the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams, due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact dynamic stiffness matrix is derived, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. The natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.     

中厚圆柱壳自由振动的动力刚度法分析

该文阐述了将动力刚度法应用于中厚圆柱壳的自由振动分析。从考虑横向剪切变形和转动惯量的中厚壳理论出发,将圆柱壳的振动分解为一系列确定环向波数下的一维振动问题。用常微分方程求解器COLSYS求解该一维问题的动力刚度,通过Wittrick-Williams算法及导护型牛顿法求得该环向波数下结构的频率和振型。由于求解动力刚度时使用COLSYS对控制方程进行了精确求解,所以该文方法是精确方法。数值算例验证了中厚圆柱壳壳段固端频率计数J0计算方法的可靠性。综合表明：应用动力刚度法对中厚圆柱壳自由振动进行分析是可靠、精确的。     

非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵.采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响.动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状.针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果.     

On curved finite elements for the analysis of circular arches

This paper proposes a straightforward criterion to warrant the displacement functions being used in the finite element approximation of circular arches. The criterion was established by studying the natural shape function, i.e. the exact solution of the deformed shape, of the circular arch element. The exact stiffness matrix [ K ]_exact is derived from the natural shape and is confirmed to be the inverse of the well-known flexibility matrix [ F ]_exact in the curved beam theory. The present paper compares the inverse [ K ]^?1 of the stiffness matrix derived from the assumed displacement function with the [ F ]_exact. It is shown that the procedure also guarantees the implicit inclusion of rigid-body modes in the pertinent stiffness matrix [ K ]. Case studies on typical approximate displacement functions assure the appropriateness as well as the ease of application of the proposed method.     

Winkler地基上变厚度圆（环）板的非对称自由振动

本文提出了Winkler地基上变厚度圆(环)板非对称自由振动的传递矩阵法.应用贝塞尔函数理论,求得等厚度圆板和环板单元非对称自由振动传递矩阵的正确公式.然后将Winkler地基上的变厚度圆(环)板划分成一系列的等厚度的圆板和环板单元,应用传递矩阵原理得到变厚度圆(环)板的整体传递矩阵公式.最后给出了一些数值结果,表明板厚和地基模量变化对固有频率的影响.     

EXACT DYNAMIC STIFFNESS MATRIX FOR BEAMS OF ARBITRARILY VARYING CROSS SECTIONS

In this paper, the exact dynamic stiffness matrix is derived for the transverse vibration of beams whose cross-sectional area and moment of inertia vary in accordance to any two arbitrary real-number powers. This variation represents a very large class of arbitrary varying beams and thus, fills the void currently existing in this area of research. With this approach, most beams can be modelled by just one element, and for beams having abrupt profile changes or with very complex profiles, they can be divided into separate distinct parts, with each of the part modelled by just one element, and then assembled together. The method is exact; however, the accuracy of the results depends only on the solver used to solve the exact frequency equation. To demonstrate the procedure, beams of non-linearly varying circular and elliptical cross-sections, and a combination beam consisting of a linear-tapered section, a uniform section and a non-linearly varying-section are analysed for their natural frequencies. Since there are no known solutions for these structures, comparison with finite element results was made and very good agreement was observed. © 1997 by John Wiley & Sons, Ltd.     

动态刚度阵法的研究概况

动态刚度阵法被广泛的应用于工程结构振动分析中，尤其是在需要获得更高阶频率和更高精度的振动问题时。因为它不像传统的有限元方法和其它的近似方法，该方法通过极少的自由度就能较精确地计算出无数个固有频率和固有振型，所以也被称为精确方法。该方法的所有假设仅来自于建立该单元运动微分方程的经典理论，所形成的动态刚度阵是固有频率的超越函数，解决这类超越特征值问题的有效方法是Williams—Wittrick算法。     

An algorithm for exact eigenvalue calculations for rotationally periodic structures

An existing algorithm ensures that no eigenvalues are missed when using the stiffness matrix method of structural analysis, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. The algorithm permits efficient multi-level substructuring and gives ‘exact’ results when the member equations used are those obtained by solving appropriate differential equations. The present paper extends this algorithm to cover rotationally periodic (i.e. cyclically symmetric) three-dimensional structures which are analysed by using complex arithmetic to obtain a stiffness matrix which involves only one of the rotationally repeating portions of the structure. Nodes and members are allowed to coincide with the axis of rotational periodicity and the resulting modes are classified. Rigid body freedoms are accounted for empirically, and the ‘exact’ member equations and efficient multi-level substructuring of the earlier algorithm can be used when assembling the stiffness matrix of the repeating portion.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

Free vibrations of shear‐flexible and compressible arches by FEM

The purpose of this paper is to analyse free vibrations of arches with influence of shear and axial forces taken into account. Arches with various depth of cross‐section and various types of supports are considered. In the calculations, the curved finite element elaborated by the authors is adopted. It is the plane two‐node, six‐degree‐of‐freedom arch element with constant curvature. Its application to the static analysis yields the exact results, coinciding with the analytical ones. This feature results from the use of the exact shape functions in derivation of the element stiffness matrix. In the free vibration analysis the consistent mass matrix is used. It is obtained on the base of the same functions. Their coefficients contain the influences of shear flexibility and compressibility of the arch. The numerical results are compared with the results obtained for the simple diagonal mass matrix representing the lumped mass model. The natural frequencies are also compared with the ones for the continuous arches for which the analytically determined frequencies are known. The advantage of the paper is a thorough analysis of selected examples, where the influences of shear forces, axial forces as well as the rotary and tangential inertia on the natural frequencies are examined. Copyright © 2001 John Wiley & Sons, Ltd.     

沿轴向指数分布的功能梯度Timoshenko梁的频率精确解

通过对状态空间变量进行变量替换,求得了沿轴向指数分布的功能梯度Timoshenko梁的状态空间传递矩阵方程。通过传递矩阵法计算了多种边界条件下结构固有频率的精确解,并与解析解进行对比。通过分析梯度参数对结构固有频率与模态振型的影响,该计算结果表明频率与材料梯度变量之间的关系曲线是连续光滑的,并未出现部分文献中的跳跃现象,并且采用有限元法该计算结果进行验证。通过对比不同梁理论的计算结果,定量的分析了剪切刚度和转动惯量对结构固有频率的影响。计算结果表明,该方法物理概念清晰,降低问题求解难度的同时可以减少计算量。     

基于DQM的曲梁平面外固有振动特性及参数分析

基于DQM(DifferentialQuadratureMethod,微分求积法),对曲线梁的频率方程和边界条件进行离散,通过采用不等分网点划分和替换法边界处理,求解了单跨圆形曲线梁和单跨回旋缓和曲线梁的平面外固有振动特性,并将结果与精确解进行对比,验证了微分求积法的高效性,讨论了网点数对求解精度的影响。在此基础上,研究了弯扭刚度比、翘曲系数以及边界约束形式等因素对曲线梁振动频率的影响,分析对比了圆形和回旋缓和曲线梁振动特性的参数影响规律。研究表明,采用微分求积法可以方便、高效地求解曲线梁的固有动力特性;弯扭刚度比、翘曲系数以及边界约束条件对曲线梁振动频率有着较为显著的影响,随着边界约束的减少,两种形式曲线梁的基本频率均随之减小,对于多数约束情况,两种形式曲线梁振动特性的参数影响规律相类似,唯有一种悬臂形式下的回旋缓和曲线梁振动特性的参数影响规律表现出与其他约束情况下相反的规律。     

Bernoulli-Euler梁横向振动固有频率的轴力影响系数

给出了考虑轴力对于Bernoulli-Euler梁横向振动固有频率影响系数的高精度表达式。与动力刚度法推导等截面梁自由振动分析的动态刚度阵不同,首先获得承受常轴力的Bernoulli-Euler梁横向自由振动微分方程的通解,并通过位移边界条件消去待定常数,得到精确形函数;使用有限元方法,建立了使用精确形函数表达等截面Bernoulli-Euler梁动态刚度阵的微分格式,该微分格式精确刚度阵与动力刚度法得到的刚度阵完全一致。仿照Timoshenko对压弯梁静态挠度表达中取用轴力影响因子的方法,提出了Bernoulli-Euler梁横向振动固有频率的轴力影响系数表达式,结合Wittrick-Williams算法和动态刚度阵证明了当轴力在±0.5倍第1阶欧拉临界力之间变化时,轴力影响系数表达式最大误差不超过2%,且随固有频率阶次的提高,误差越来越小。     

Exact determinant for infinite order FEM representation of a Timoshenko beam–column via improved transcendental member stiffness matrices

Transcendental stiffness matrices for vibration (or buckling) have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods (FEM). Their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or critical load factors) can be found with certainty using the Wittrick–Williams algorithm. A very recently discovered analytical property is the member stiffness determinant, which equals the FEM stiffness matrix determinant of a clamped ended member modelled by infinitely many elements, normalized by dividing by its value at zero frequency (or load factor). Curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles. In this paper, the transcendental stiffness matrix for a vibrating, axially loaded, Timoshenko member is expressed in a new, convenient notation, enabling the first ever derivation of its member stiffness determinant to be obtained. Further expressions are derived, also for the first time, for unloaded and for static, loaded Timoshenko members. These new expressions have the advantage that they readily reduce to corresponding expressions for Bernoulli–Euler members when shear rigidity and rotatory inertia are neglected. Additionally, the total equivalence of the normalized transcendental determinant with that of an infinite order FEM formulation aids understanding and evaluation of conventional FEM results. Examples are presented to illustrate the use of the member stiffness determinant. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic stiffness vibration analysis using a high‐order beam model

This work presents the derivation of the exact dynamic stiffness matrix for a high‐order beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross‐section according to the high‐order theory, which include cubic variation of the axial displacements over the cross‐section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Using the dynamic stiffness matrix exact vibration frequencies for beams with various combinations of boundary conditions are tabulated and compared with results from the Bernoulli–Euler and Timoshenko beam models. Copyright © 2003 John Wiley & Sons, Ltd.     

A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations

An alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental, e.g. dynamic, stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program VICONOPT. The method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem. The procedure presented has five stages, including two accuracy checking stages which prevent unnecessary calculations. Numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method.     

Calculation of vibration frequencies by a hybrid element method based on a generalized complementary energy principle

Based on a generalized complementary energy principle the derivation of the element matrices is presented for calculation of natural frequencies. The degrees-of-freedom are not defined on nodal points but in an abstract way. No restrictions about the number of interpolation functions in the interior and at the boundaries of the element have been introduced. The exact solution of the discretized element equations leads to the dynamic stiffness matrix while the approximate solution results in a linear eigenvalue problem. Plate bending problems are used to study the convergence of frequencies depending on the degrees of interpolation functions within the element and on its boundaries and on the number of elements.     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

Axisymmetric vibration analysis of laminated hollow cylinders with ring stiffeners

This paper presents an axisymmetric vibration analysis of laminated hollow cylinders composed of monoclinic layers and stiffened by ring stiffeners. A successive approximation approach, which is based on a transfer matrix and then an equivalent stiffness matrix formulation, is used to enable three-dimensional solutions to be found. It is assumed that the ring stiffeners are attached to the lateral surfaces of the cylinder and only provide elastic supports in radial direction. These constraints are imposed by using Lagrange multipliers to couple the responses of a number of vibration modes of corresponding cylinders without stiffeners. Using this method the natural frequencies of a stiffened cylinder are found to be the eigenvalues of a constraint matrix and the predictions can be arbitrarily close to the exact three-dimensional solutions.     

随机参数齿轮-转子系统扭转振动固有频率分析

应用拓广的随机因子法分析了物理参数和几何参数均为随机变量的齿轮-转子系统的时变固有频率。将系统的刚度矩阵和质量矩阵分解为具有相同随机因子的矩阵之和的形式,再由求解系统固有频率的瑞利商公式出发,将系统频率展成部分频率分量之和的形式,利用求解随机函数数字特征的代数综合法求解系统固有频率的数字特征。通过算例分析了随机参数对系统固有频率的影响,并验证了方法的可行、有效和正确性。