Exact solutions for vibrating nonhomogeneous rectangular membranes with exponential density distribution

Presented herein are exact solutions for the vibration problems of nonhomogeneous rectangular membranes with an exponential density distribution and with a linear density distribution. These exact solutions are useful for verifying numerical methods and algorithms.     

On radially symmetric vibrations of non-uniform annular sandwich plates

Analysis and numerical results for the axisymmetric vibrations of annular sandwich plates with relatively stiff core of non-uniform thickness have been investigated using a refined theory. The face sheets are treated as membranes of constant thickness and the core is assumed to be solid as well as moderately thick of parabolically varying thickness. Due to this-type of variation in the thickness of the core, the face sheets take the shape of paraboloid of revolution and because of this, the face sheets membrane forces contribute to the bending and transverse shear stresses of the core of the plate. The equations of motion for such a plate have been derived using Hamilton’s energy principle. The frequency equations for three different combinations of boundary conditions; clamped at the inner edge and clamped or simply supported or free at the outer edge are obtained employing differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as radii ratio, taper parameter, thickness of the facings and core at the centre on the natural frequencies has been studied. Three-dimensional mode shapes for the specified plates have been illustrated. A comparison of the results with published work has been made.     

Exact solutions for vibration of a vertical heavy string with a tip mass

This technical note considers the free vibration problem of a vertical heavy, inextensible string/chain with a tip mass. The top end is fully restrained from moving in the vertical direction but it is partially restrained in the lateral direction by an elastic spring. The exact frequency characteristic equation for the aforementioned vibration problem is derived and the first five vibration frequencies are presented for various tip mass to string mass ratios and spring constants.     

Vibrations of orthotropic annular plates subjected to non-uniform boundary conditions over different sections of the inner and outer circumferences

The free vibrational response of homogeneous or symmetrically laminated polar orthotropic annular plates subjected to non-uniform boundary conditions on the inner and outer rims is scrutinized in detail, using a finite-difference method. Although the same sets of boundary conditions are considered, the angular location at which the altered boundary conditions are applied and the segments that they cover are allowed to be different on the inner and outer circumferences. The sensitivity of vibrational frequencies to the extent and relative location of the non-uniformities is investigated and shown to be strongly influenced by the material properties with the orthotropy ratio for the plate (defined here by D_rr/D_θθ) being the predominant parameter for indicating the sensitivity to non-uniformity. The behavior of higher modes of vibration is also discussed and the phenomenon of crossover, whereby two modes having the same frequency exist, is found to occur. Unlike the crossover previously reported in the literature, for which only higher modes of frequency were involved, in the current context, it was discovered that the phenomenon may also be associated with the lowest mode of vibration. The circumstances under which crossover transpires are dependent on the relative location of the boundary non-uniformities, the relative length over which the non-uniformities are applied and the orthotropy ratio.     

An approximate analytical procedure for natural vibration analysis of free rectangular plates

Natural vibrations of free rectangular plates are usually analysed by numerical methods since it is not possible to obtain the closed form analytical solution. In this paper a simple analytical procedure for estimation of natural frequencies of free thin rectangular plates, based on the Rayleigh's quotient, is presented. First, natural modes are assumed in the usual form as products of beam natural modes in longitudinal and transverse direction, satisfying the grillage boundary conditions. Based on a detailed FEM analysis some additional natural modes are recognized, which are defined as sum and difference of the cross products of beam modes. Their frequency spectra are very close and identical in some special cases manifesting in such a way double frequency phenomenon. These three families of natural mode shapes form a complete natural frequency spectrum of a free rectangular plate as a novelty. The reliable approximation of natural modes enables application of the Rayleigh's quotient for estimation of higher natural frequencies. Application of the developed procedure is illustrated in the case of a free thin square and rectangular plate. The obtained results are compared with those determined by FEM and also with more rigorous ones from the relevant literature based on the Rayleigh–Ritz method. The achieved accuracy is acceptable from the engineering point of view, and the procedure can be applied to improve the hydroelastic analysis of VLFS.     

A simple approximate method for free vibration analysis of framed tube structures

A simple approximate method is developed for determining natural frequency of tall buildings. Timoshenko's beam model, which considers the influence of shear and flexural deformation, was used in modelling framed tube structures. In this paper, natural frequency and mode shape of framed tube structures were calculated based on the flexural and shear rigidities along with the effects of rotational inertia. Dynamic model of Timoshenko's beam can be obtained by writing equilibrium equations of forces acting on an infinite element. The solution of the dynamic model was obtained by first applying the separation of variables to time and space, followed by the assumption of harmonic motion in time, the steady state eigen system was obtained. Natural frequencies of framed tube structures were calculated by solving the eigenproblem. A numerical example has been presented to demonstrate the ease of application and accuracy of the proposed method. The structure's fundamental frequency was computed using ETABS V9.0.0 (Computer and Structures, Berkeley, California, USA) and compared with the result obtained from proposed method, which shows that the percentage of error was low and acceptable. The proposed method can be adopted as an alternative procedure to evaluate the natural frequency of framed tube in the preliminary stages of structural design. Copyright © 2010 John Wiley & Sons, Ltd.     

A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates

Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented. The governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions. To cause the impact due to nonlinear terms more significant, in-plane immovable simply supported, clamped and different combinations of them are considered. The effects of different parameters on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Some new results for laminated composite skew plates with different mixed boundary conditions are presented and are compared with those obtained using the first order shear deformation theory based DQ (FSDT-DQ) method. Excellent agreements exist between the solutions of the two approaches but with much lower computational efforts of the present DQ methodology with respect to FSDT-DQ method.     

A meshfree model without shear-locking for free vibration analysis of first-order shear deformable plates

Difficulty in imposing essential boundary conditions in the standard element-free Galerkin method (EFG) is due to the lack of Kronecker’s delta function property of shape functions generated by moving least square approximation (MLS). In this paper, we further apply a meshfree model based on the moving Kriging interpolation method (MK) to free vibration analysis of first-order shear deformable plates. The deflection and two rotation field variables of plate are approximated by the MK method, which is employed to construct the shape functions having the delta function property. With this approach, the drawback in enforcement of the boundary conditions caused by the MLS is now avoided. The present formulation is based on the first-order shear deformation plate theory (FSDT) associated with an effective elimination of the shear-locking phenomenon completely, and hence the approach is applicable to both moderately thick and thin plates. Numerical examples considering various aspect ratios and different boundaries are examined and solutions on natural frequencies obtained by the present method are then compared with existing reference solutions, and very good agreements are observed.     

An analytical approach to free vibration analysis of multi‐outrigger–belt truss‐reinforced tall buildings

This paper deals with a new and simple mathematical model that may be used to determine natural frequencies and mode shapes of a multistory building that consists of a framed tube, a shear core and multi‐outrigger–belt trusses. The effect of outrigger–belt truss and shear core on a framed tube was modeled as a concentrated moment placed at outrigger–belt truss location, which acted in opposite direction of the rotation created by lateral loads. The analysis is based on a continuum approach, in which a tall building structure may be replaced by an idealized cantilevered beam to model the building's structural characteristics. Energy method and Hamilton's principle have been used to develop the governing equations. After applying separation of variables method to time and space variables, the resulting eigensystem was solved to obtain the building's natural modes and frequencies of vibration. A computer program has been developed in MATLAB (Mathworks Inc., CA, USA) environment, and a numerical example has been solved to demonstrate the accuracy of this method. Results obtained from the proposed mathematical model give a good understanding of a structure's dynamic characteristics. The method is simple to use yet reasonably accurate and hence suitable for quick evaluations during preliminary design stages. Copyright © 2011 John Wiley & Sons, Ltd.     

悬臂系统振动特性的研究

本文研究悬臂系统,包拓悬臂弹性梁及其附带的质块的振动特性.本文首先导出了该系统频率和振型的分析表达式,并利用计算机辅助设计软件,分别讨论了质块质量及弹簧常数对系统频率的影响.上述系统的不同参数,对系统的固有频率的影响是不同的,为此我们研究了在若干特定参数下,系统固有频率对质块位置的敏感度,并得到相应的结论.     

Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM

Large amplitude free vibration analyses of tapered Mindlin rectangular plates with elastically restrained against rotation edges are investigated using different differential quadrature method (DQM). The governing equations are based on the first-order shear deformation plate theory in conjunction with Green's strain and von Karman assumption. The spatial derivatives are discretized using DQM and the harmonic balance method is used to transform the resulting differential equations into frequency domain. A direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and their accuracy is demonstrated by comparing the results with those of the limiting cases, i.e. nonlinear free vibration analysis of plates with classical boundary conditions and also linear free vibration analysis of tapered plates. The effects of the elastic restraint coefficient at the edges and the geometrical parameters on the ratio of the nonlinear natural frequency to linear natural frequency of plates with linearly and bi-linearly varying thickness are studied.     

On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates

A comprehensive study on the use of a set of trigonometric functions, originally proposed by Beslin and Nicolas [Journal of Sound and Vibration 1997;202:633–55], as admissible solutions in the Ritz method for general vibration analysis of rectangular orthotropic Kirchhoff plates is presented. The approach is denoted here as Trigonometric Ritz method (TRM). Since its introduction, application of TRM was limited to a very few plate problems. The aim of this work is to extend the potential of the method on predicting natural flexural frequencies of plates with various complicating factors, including in-plane loads, elastically restrained edges, rigid/elastic concentrated masses, intermediate line and point supports or their combinations. Computational efficiency, stability, convergence and accuracy of the method are discussed and supported by extensive analysis. TRM-based solutions are compared with many reference cases available in the literature obtained with other methods or Ritz functions. Numerical results indicate that TRM exhibits good to excellent accuracy for all cases considered. New solutions are also presented for future comparison purpose.     

Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers

The free vibration behavior of quadrilateral laminated thin-to-moderately thick plates with carbon nanotube reinforced composite (CNTRC) layers is studied. The governing equations are based on the first-order shear deformation theory (FSDT). The solution procedure is based on transforming the governing differential equations from an arbitrary straight-sided physical domain to a regular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. Four different profiles of single walled carbon nanotubes (SWCNTs) distribution through the thickness of layers are considered, which are uniformly distributed (UD) and three others are functionally graded (FG) distributions. The fast rate of convergence of the presented approach is numerically demonstrated and to show its high accuracy, wherever possible comparison studies with the available results in the open literature are performed. Then, the effects of volume fraction of carbon nanotubes (CNTs), geometrical shape parameters, thickness-to-length and aspect ratios, different kinds of CNTs distribution along the layers thickness and different boundary conditions on the natural frequencies of laminated plates are studied.     

K6型单层球面网壳的自振特性分析

选用实际工程中应用较为普遍的K6型单层球面网壳作为重点研究对象，计算了多种工况（不同的矢跨比、荷载、支座条件）对网壳结构自振特性的影响，对K6型网壳结构的自振特性做了较为系统的研究，以得到K6型网壳结构的自振规律。     

Stability analysis of simply-supported rectangular plates under non-uniform uniaxial compression using rigorous and approximate plane stress solutions

The non-classical problem of buckling of a simply-supported rectangular plate due to various types of non-uniform compressive edge loads is analysed here. For each case, the elasticity solution for the internal in-plane stress field is obtained rigorously using a superposition of Airy's stress functions and also approximately using extended Kantorovich method. Subsequently, the convergent buckling loads are obtained using Galerkin's method. Results are presented to highlight the dependence of the total buckling load and the corresponding buckled shape on the edge load distribution, as well as to illustrate the applicability of the approximate plane stress solutions.     

Nonlinear vibration of thin plates with initial stress by spline finite strip method

The spline finite strip method and the incremental time–space finite element procedure are used to analyse large amplitude vibration of plates with initial stresses. Two improvements for the procedure are presented. The free vibration and the internal resonance of plates with initial stress as well as the forced vibration of plates with damping and initial stress are computed. The results compared favourably with those available in other publications.     

饱和单层土体一维热固结精确解

岩土体在非等温场下的固结响应一直是研究热点。基于考虑渗流的饱和土热固结控制方程,给出了一种三类任意边界条件下,饱和单层土体一维热固结精确解的求解方法。首先,利用函数φ将对温度和超静孔压的求解转换为对函数φ的求解,采用分离变量法,结合边界条件得到微分方程的特征函数。然后,将非齐次边界条件齐次化,利用微分方程的特征函数,采用系数变易法,给出非齐次边界条件下的精确级数解。最后,通过算例揭示土体边界的透水条件及温度变化对土体热固结响应影响显著。     

Exact solution for dynamic response of an elastic elliptical membrane

The elaborated method of eigenfunction expansion in terms of transcendental Mathieu and modified Mathieu functions is employed to present the first known exact time-domain series solution for transverse vibrations of a uniform elastic membrane of elliptical shape under arbitrary loading and initial conditions. Numerical calculations for displacement time response of an elliptical membrane of typical aspect ratio excited by various common types of transient loads are presented and discussed. Limiting cases are considered and good agreements with available results are obtained.     

Developing an element free method for higher order shear deformation analysis of plates

A procedure is presented to eliminate transverse shear locking in analysis of laminated composite plates using Element Free Galerkin (EFG) method based on higher-order transverse shear deformation theory (HSDT). In the procedure, derivatives of the transverse displacement are introduced as independent variables. Thus, a formulation requiring C⁰ continuity shape functions for approximation is proposed for higher-order transverse shear deformation analysis of plates. Shear locking is avoided considering reduced integration for shear stiffness matrix; a method which is implemented in FEM. Moving Least Squares (MLS) method is utilized for shape functions and the penalty method is implemented to impose approximate boundary conditions. The present solutions are verified with other higher-order shear deformable studies. Moreover, a comparison between the present solutions with those obtained by EFG procedure based on First-order transverse Shear Deformation Theory (FSDT) is performed.     

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C¹ continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C¹ continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.