Large amplitude free flexural vibrations of thin plates of arbitrary shape

A finite element formulation is developed for analyzing large amplitude free flexural vibrations of elastic plates of arbitrary shape. Stress distributions in the plates, deflection shape and nonlinear frequencies are determined from the analysis. Linearized stiffness equations of motion governing large amplitude oscillations of plates, quasi-linear geometrical stiffness matrix, solution procedures, and convergence characteristics are presented. The linearized geometrical stiffness matrix for an eighteen degrees-of-freedom conforming triangular plate element is evaluated by using a seven-point numerical integration. Nonlinear frequencies for square, rectangular, circular, rhombic, and isosceles triangular plates, with edges simply supported or clamped, are obtained and compared with available approximate continuum solutions. It demonstrates that the present formulation gives results entirely adequate for many engineering purposes.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Mathematical observations in structural dynamics

In dynamic structural analysis, the basic relations between forces and displacements for a beam element subjected to axial, torsional or flexural vibration are obtained either by solving the appropriate equation of motion or by using an approximate method. The exact equation leads to the dynamic stiffness matrix while the approximate method results in the superposition of elastic and inertial forces represented respectively by the stiffness and mass matrices.The common procedure in finding the natural frequencies is to set the determinant of the dynamic stiffness matrix for the system equal to zero. The approximate method leads to an eigenvalue type problem while the exact method results in a transcendental equation of trigonometric and hyperbolic functions. The natural frequencies in a region of interest are found by a systematic search in the determinntal function.The purpose of this paper is to show that the search technique cannot be applied for certain values of the argument at which the determinantal function is not defined. It is proved that the natural frequencies of any isolated member in the system are critical values for the determinantal function. A practical method is given to obviate the difficulty in order to find the natural frequencies from the determinant, including the critical values at which the dynamic stiffness matrix is not defined. Also, as part of this investigation, the mathematical relation is established between the dynamic stiffness matrix derived by the approximate finite element method and the results obtained from the exact Bernoulli-Euler equation for flexural vibration or the wave equation for axial or torsional vibration.     

Procedures for the forced,damped vibration analysis of structural frames using distributed parameter models

The paper summarises an approach to the forced vibration analysis of skeletal elastic structures, making use of “exact” matrix technique. The analysis is based on the modal superposition method, which has been developed in the context of models that retain characteristics of both distributed mass and stiffness. The free vibration analysis is performed using generalised dynamic stiffness functions appropriate to a Rayleigh-Timoshenko-Euler beam-column element. The contributions to dynamic displacements and dynamic internal forces by inertia effects during forced vibration are properly accounted for, and all results are obtained from explicit solutions to the relevant equations at all stages. Numerical integration or other approximate procedures are not required. It is the objective of the present paper to summarise procedures, numerical examples being available elsewhere (e.g. [1–3]). However, the paper is concluded with a practical example illustrating results obtainable by the techniques described below.     

Dynamics of frameworks by continuous mass method

The continuous mass matrix method has been extended to include the forced vibration in the dynamic analysis of plane and space frameworks. The forcing forces may be continuous of discontinuous functions of time but all the forcing forces acting simultaneously on a framework must have the same time variation in order that the modal analysis can be applied. The damping has been neglected. The concept of code numbers in the case of static loading has been extended to the dynamics of structures. The validity of the two orthogonality conditions of the modal shapes has been proved for the continuous mass matrix method so that the modal analysis could be applied easily. The set of simultaneous equations of motion has been converted to equivalent one-degree-of-freedom systems. In the case where the forcing forces have different time variation functions a numerical analysis can be performed. Illustrative sample problems have been solved and the results are given in tabular form.     

不同位置四点支承矩形薄板的自由振动特性

研究不同位置四点支承条件下矩形薄板的自由振动特性.首先,在板结构模型的不同位置上引入横向约束弹簧,并设定人工弹簧的刚度值以模拟出四点支承的边界条件.然后,基于二维改进傅里叶级数表示结构的位移容许函数,其中改进部分的正弦附加项可解决以往位移函数在边界上可能存在的求导不连续问题.建立矩形板系统能量对应的泛函,令其取驻值建立线性方程组.最后,求解矩阵特征值问题得到点支承矩形板自由振动频率等参数,给出不同位置四点支承条件下矩形薄板的振动特性.所应用二维改进傅里叶级数法中,位移函数基于改进傅里叶级数展开时的附加项能够提高结果的精度和收敛速度.研究结果为不同位置点支承矩形板的自由振动问题提供一定的参考.     

Finite element displacement method for large amplitude free flexural vibrations of beams and plates

A finite element method to determine the nonlinear frequency of beams and plates for large amplitude free vibrations is presented. The equation of motion is characterized by the basic stiffness, mass, geometrical stiffness and the associated inplane force matrices. The procedures for solving the system equations of motion are discussed, and the explicit formulations of the geometrical stiffness and inplane force matrices of a rectangular plate element are given. Examples of large amplitude free oscillations for rectangular plates and beams with various boundary conditions are given. Characteristics of convergence are investigated. In all the cases where comparisons with previous investigations are made, good agreement has been obtained. It indicates that the present method will give results entirely adequate for engineering purposes.     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

A finite element method for the free vibration of plates allowing for transverse shear deformation

An isoparametric quadrilateral plate bending element is introduced and its use for the free vibration analysis of both thick and thin plates is examined. Plates of rectangular planform and of orthotropic materials are analysed and excellent results are obtained. The element performance is assessed by comparison with well established analytical and numerical solutions based on Mindlin's thick plate theory, three dimensional elasticity solutions and solutions based on thin plate theory. The ease with which the element may be implemented is stressed. The use of an eigenvalue economiser which produces considerable economy in the computer solution is demonstrated. Various mass lumping schemes and numerical integration rules used in the construction of the element mass matrix are also examined.     

Finite element free vibration of eccentrically stiffened plates

An isoparametric stiffened plate element is introduced for the free vibration analysis of eccentrically stiffened plates. The element has the ability to accommodate irregular boundaries. Moreover, the formulation considers shear deformation, hence, the formulation is applicable to both thick and thin plates. In the present formulation, the stiffeners can be placed anywhere within the plate element and they need not necessarily follow the nodal lines. In addition, the effects of lumped and consistent mass matrices on natural frequencies of stiffened plates are investigated. The effects of several parameters of the stiffener—eccentricity, shape, torsional stiffness etc.—on the natural frequencies of the stiffened plates are studied.     

Modal spectral element formulation for axially moving plates subjected to in-plane axial tension

The use of frequency-dependent spectral element matrix (or dynamic stiffness matrix) in structural dynamics is known to provide very accurate solutions, while reducing the number of degrees-of-freedom to resolve the computational and cost problems. Thus, in the present paper, the modal spectral element is formulated for thin plates moving with constant speed under a uniform in-plane axial tension. The concept of the Kantorovich method is used to formulate the modal spectral element matrix in the frequency-domain. The present modal spectral element is then evaluated by comparing its solutions with exact analytical solutions as well as with FEM solutions. The effects of the moving speed and the in-plane tension on the dynamic characteristics of a moving plate are investigated numerically.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.     

The unbonded contact problem of a plate on the elastic half space

This paper is concerned with a finite element iterative procedure for determination of the contact region between the plate and the elastic half space. A special conforming triangular plate bending element having ten terminals is used together with the stiffness matrix of the half space, the development of which is based on the Boussinesq solution. Fast convergence and effectiveness of this method is demonstrated on examples of the circular, square and triangular plates.     

Asymmetric vibration and stability of circular plates

The asymmetric vibration and stability of circular and annular plates using the finite element method is discussed. The plate bending model consists of one-dimensional circular and annular ring segments using a Fourier series approach to model the problem asymmetries. Using displacement functions which are the exact solutions of the static plate bending equation, the stiffness coefficients corresponding to the 1st and nth harmonics are used in closed form. By assuming that the static displacement function closely represents the vibration and stability modes, the mass and stability coefficient matrices for an annular and circular element are also constructed for the 1st and nth harmonics. Several numerical examples are presented to demonstrate the efficiency and accuracy of the finite element model with that of classical methods.     

弹性支撑下风电机组传动系统结构动力分析

考虑水平轴风力发电机组齿轮箱弹性支撑的柔性连接特性,基于集中质量思想和拉格朗日方法,建立风力发电机传动系统多体动力学模型,研究了齿轮箱弹性支撑对传动系统结构动力学特性的影响.利用动力学模型和模态分析方法,得到了由弹性支撑耦合到系统后的模态频率,并获取了在该模态激励下的模态动能分布.采用变参数方法进行传动系统模态对齿轮箱弹性支撑刚度变化的敏感性分析,利用模态叠加法进行齿轮箱体的动响应分析.数值求解结果和分析表明,考虑齿轮箱弹性支撑的传动系统某阶固有频率即为弹性支撑下齿轮箱体振动主模态;弹性支撑线刚度对传动系统低频率固有模态存在一定影响;齿轮箱体振动分析时应考虑1阶和2阶的低频模态较为合理.本研究工作对传动链系统方案可靠性设计和抑制传动链振动的加阻控制提供了一定理论基础.     

A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges

On the basis of three-dimensional elasticity, this paper presents a free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries. The analysis is based on a recursive solution suitable for three-dimensional vibration analysis of simply supported plates. Clamped boundary conditions are imposed by suppressing the edge displacements of a number of planes which are parallel to the mid-plane. This is achieved by coupling a number of different vibration modes of the corresponding simply supported plate using a Lagrange multiplier method. A satisfactory solution can be obtained by choosing suitably larger numbers of the coupled vibration modes and the fixed planes across the thickness of the plate. Numerical results are presented to show the convergence of the solution. Results are also obtained for either isotropic or cross-ply laminated plates having different combinations of simply supported and clamped boundaries.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

A method for system identification using the optimality criteria optimization approach

An algorithm similar to the optimality criteria approach used in structural optimization is presented for identifying stiffnesses of structural members by using vibration test data. A set of equivalent static inertia forces are obtained from the vibration analysis using d'Alembert's principle and are used to solve the multiple displacement constraint problem. The displacement constraint values are specified based on the measured experimental modal displacement data at critical locations. The algorithm is used to find the changes needed in the stiffnesses of the elements and the distribution of nonstructural mass of the nominal analytical model to correlate the analytical and experimental data. The algorithm alternates between the vibration analysis and static analysis to find the equivalent load vector and modify the stiffnesses. The identified stiffness properties of the structural elements can be used to control and study the dynamic response of the structure.