Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

Vibration of composite plates with internal supports

The free vibration of rectangular laminated composite plates with arbitrary support conditions along the edges, internal line supports and discrete point supports are studied using the Rayleigh-Ritz method. Polynomial approximation functions are selected to satisfy all essential boundary conditions along the edges of the plate and to vanish along line supports parallel to the co-ordinate axes. Straight line supports at an angle from the co-ordinate axes and curved line supports are modeled by introducing several point supports along the line. Zero displacement constraints at the point support locations are enforced using the Lagrange multiplier technique. The plate constitutive equations are expressed in terms of stiffness invariants and the fundamental natural frequency is maximized by selecting the appropriate lay-up. Several examples are presented to illustrate the versatility of the approach and provide results not previously available. The influence of the number of plies in the laminate, lay-up, material properties and plate aspect ratios are investigated.     

Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions

This paper studies the free vibrations of point-supported rectangular plates with variable thickness using the Rayleigh–Ritz method. The domain of the plate is bounded by x=αa′, a′ (0α<1); y=βb′, b′ (0β<1) in the Cartesian coordinate system. The thickness of the plate varies continuously and is represented by a power function (x/a′)^s(y/b′)^t. Varieties of tapered rectangular plates can be described by giving s and t different values. A set of static tapered beam functions which are the solutions of a tapered beam (a unit width strip taken from the particular plate under consideration in one or the other direction parallel to its edges) under a Taylor series of static loads, are developed as the admissible functions for the vibration analysis of point-supported rectangular plates with variable thickness in one or two directions. The eigenfrequency equation is derived through the Rayleigh–Ritz approach, supplemented by the zero deflection conditions at the point-supports. A very simple program in common use has been compiled. The convergence study shows a small computational cost and the comparison with known solutions for point-supported rectangular plates with uniform thickness demonstrates the accuracy of the present method. Finally, some new numerical results are given, which may serve as the benchmarks for future research on the aforementioned problem.     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

Elastic large deflection analysis of isotropic rectangular Mindlin plates

Governing equations for the large deflection analysis of isotropic rectangular Mindlin plates are introduced and their solution using the DR algorithm is briefly outlined. Two computer programs, based respectively on interlacing and non-interlacing finite-differences, have been developed for the numerical solution of these equations. The programs have been verified by analysing a variety of thin and moderately thick plate problems for which alternative solutions are available. Sample results comparisons are presented in order to quantify the accuracy of the DR results. The non-interlacing finite-difference DR program is then used to compute new results for uniformly loaded square moderately thick plates with simply supported, clamped and combined simply supported and clamped edges.     

Vibration analysis of rectangular plates with edge V-notches

A method is presented for accurately determining the natural frequencies of plates having V-notches along their edges. It is based on the Ritz method and utilizes two sets of admissible functions simultaneously, which are (1) algebraic polynomials from a mathematically complete set of functions, and (2) corner functions duplicating the boundary conditions along the edges of the notch, and describing the stress singularities at its sharp vertex exactly. The method is demonstrated for free, square plates with a single V-notch. The effects of corner functions on the convergence of solutions are shown through comprehensive convergence studies. The corner functions accelerate convergence of results significantly. Accurate numerical results for free vibration frequencies and nodal patterns are tabulated for V-notched square plates having notch angle α=5° or 30° at different locations and with various notch depths. These are the first known frequency and nodal pattern results available in the published literature for rectangular plates with V-notches.     

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC-Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.     

Pseudospectral analysis of buckling and free vibration of rectangular Mindlin plates

A study of buckling and free vibration of rectangular Mindlin plates is presented. The analysis is based on the pseudospectral method, which uses basis functions that satisfy the boundary conditions. The equations of motion are collocated to yield a set of algebraic equations that are solved for the critical buckling load and for the natural frequencies in the presence of the in-plane loads. Numerical examples of rectangular plates with SS-C-SS-C boundary conditions are provided for various aspect ratios and thickness ratios, which show good agreement with those of the classical plate theory when the thickness ratio is very small. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from the University of Michigan, Ann Arbor in 1992 and joined the Dept. of Mechanical and Design Engineering of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-free Galerkin method

A mesh-free Galerkin method for the free vibration analysis of unstiffened and stiffened corrugated plates is introduced in this paper, in which the corrugated plates are simulated with an equivalent orthotropic plate model. To obtain the corresponding equivalent elastic properties for the model, a constant curvature state is applied to the corrugated sheet. The stiffened corrugated plates are treated as composite structures of equivalent orthotropic plates and beams, and the strain energies of the plates and beams are added up by the imposition of displacement compatible conditions between the plate and the beams. The stiffness matrix of the whole structure is then derived. The proposed method is superior to the finite element methods (FEMs) because no mesh is needed, and thus stiffeners (beams) do not need to be placed along the mesh lines and the necessity of remeshing when the positions of the stiffeners change is avoided. To demonstrate the accuracy and convergence of the proposed method, several numerical examples are analyzed both with the proposed method and the finite element commercial software ANSYS. Examples from other research are also employed. A good agreement between the results for the proposed method, the results of the ANSYS analysis, and the results from other research is observed. Both sinusoidally and trapezoidally corrugated plates are studied.     

Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb-2 Ritz method

Free vibration analysis of rectangular plates with internal columns and elastic edge supports is presented using the powerful pb-2 Ritz method. Reddy's third order shear deformation plate theory is employed. The versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial and a basic function are taken as the admissible functions. Substituting these displacement functions into the energy functional and minimizing the total energy by differentiation, leads to a typical eigenvalue problem, which is solved by a standard eigenvalue solver. Stiffness and mass matrices are numerically integrated over the plate using the Gaussian quadrature. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. Many numerical results for reasonable natural frequency parameters of rectangular plates with different combinations of elastic boundary conditions and column supports at any locations are presented, which can be used as a benchmark for future studies in this area.     

Vibrations of rectangular plates with internal cracks or slits

This work applies the famous Ritz method to analyze the free vibrations of rectangular plates with internal cracks or slits. To retain the important and useful feature of the Ritz method providing the upper bounds on exact natural frequencies, the paper proposes a new set of admissible functions that are able to properly describe the stress singularity behaviors near the tips of the crack and meet the discontinuous behaviors of the exact solutions across the crack. The validity of the proposed set of functions is confirmed through comprehensive convergence studies on the frequencies of simply supported square plates with horizontal center cracks having different lengths. The convergent frequencies show excellent agreement with published accurate results obtained by an integration equation technique, and are more accurate than those obtained by a previously published approach using the Ritz method combined with a domain decomposition technique. Finally, the present solution is employed to obtain accurate natural frequencies and mode shapes for simply supported and completely free square plates with internal cracks having various locations, lengths, and angular orientations. Most of the configurations considered here have not been analyzed in the previously published literature. The present results are novel, and are the first published vibration data for completely free rectangular plates with internal cracks and for plates with internal cracks, which are not parallel to the boundaries.     

Free vibrations of continuous rectangular plates on oblique supports

The Finite Strip Method has been employed in the vibration problem of continuous rectangular plates on oblique supports. The structure has been divided into quadrilateral finite strips. The various properties of a quadrilateral finite strip have been derived using the displacement approach. The results are obtained for a two span rectangular plate with an oblique support at various angles and compared with other solutions. The results obtained due to various layouts of finite strips in the structure have been compared.     

Dynamic analysis of Coriolis flow meter using Timoshenko beam element

Coriolis mass flow meter (CFM) is used to measure the rate of mass flow through a pipe conveying fluid. In the present work, the Coriolis effect produced in the pipe due to a lateral excitation is modeled using the finite element (FE) method in MATLAB©. The coupled equation of motion for the fluid and pipe is converted to FE equations by applying Galerkin technique. The pipe conveying fluid is excited at its fundamental natural frequency. The time lag observed between symmetrically located measurement points which are equidistant from the point of excitation, is utilized to predict the mass flow rate. The results predicted by the present code is validated using the experimental, and numerical results published in the literature. The main contribution is the development of a FE model, using three node Timoshenko beam element to analyse the dynamics of fluid conveying pipes subjected to external excitation. The direction of the Coriolis force is perpendicular to the plane containing the velocity of flow vector and angular velocity vector of the pipe. Hence a three dimensional FE model is essential. This model can include curved geometry, damping, velocity and gyroscopic effects for three dimensional flexible tubes. The reduced integration used for overcoming shear locking in two node elements, will result in the formation of spurious modes leading to an incorrect prediction of natural frequencies and velocity. These modes will not occur while using three node elements. Influence of spatial as well as temporal discretisation on the time lag and frequency are also discussed. The sensitivity analysis shows that the time lag varies linearly with the mass flow rate.     

Vibration analysis of a beam with partially distributed internal viscous damping

In this paper, governing equations of vibration for a beam with distributed internal viscous damping are established by using Timoshenko beam theory and Hamilton's principle. Then, the transfer matrix method is applied to obtain the frequency equations for the beam. The results reveal, when the internal viscous damping fully distributes along the beam, that the natural frequency decreases with the increasing damping and drops to a zero value at a certain critical damping. While the damping is locally distributed, damped frequency, mode shape and transient response time are affected most significantly by locating the damped segment at the position with maximum bending moment. The flexural amplitudes and phase angles of a beam excited by the resonant harmonic load can be effectively predominated by tuning the damping value.     

Vibration and buckling characteristics of weld-bonded rectangular plates using the flexibility function approach

Structures with a combination of spot welds and adhesive bonding, often referred to as weld-bonded structures, are likely to see increasing usage in automotive and other engineering structures. The present study considers a representative weld-bonded rectangular plate having simple supports on two opposite edges and weld-bonded support conditions with periodic spot welds along the other two edges. The study shows that the flexibility function approach for modeling free edges with point supports [Bapat AV, Venkatramani N, Suryanarayan S. Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates. Journal of Sound and Vibration 1988;120(1):127–40; Bapat AV, Venkatramani N, Suryanarayan S. A new approach for the representation of a point support in the analysis of plates. Journal of Sound and Vibration 1988;120(1):107–25; Bapat AV, Venkatramani N, Suryanarayan S. The use of flexibility functions with negative domains in the vibration analysis of asymmetrically point-supported rectangular plates. Journal of Sound and Vibration 1988;124(3):555–76; Bapat AV, Suryanarayan S. Free vibrations of periodically point-supported rectangular plates. Journal of Sound and Vibration 1989;132(3):491–509; Bapat AV, Suryanarayan S. The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges. Journal of Sound and Vibration 1989;128(2):203–33; Bapat AV, Suryanarayan S. Free vibrations of rectangular plates with interior point supports. Journal of Sound and Vibration 1989;134(2):291–313; Bapat AV, Suryanarayan S. Importance of satisfaction of point-support compatibility conditions in the simulation of point supports by the flexibility function approach. Journal of Sound and Vibration 1990;137(2):191–207; Bapat AV, Suryanarayan S. A fictitious foundation approach to vibration analysis of plates with interior point. Journal of Sound and Vibration 1992;155(2):325–41; Bapat AV, Suryanarayan S. A theoretical basis for the experimental realization of boundary conditions in the vibration analysis of plates. Journal of Sound and Vibration 1993;163(3):463–78], used in the direct series solution of the governing differential equations, can be employed very effectively to study the vibration and buckling characteristics of the weld-bonded rectangular plates. This is done by using a flexibility function constructed in terms of Fourier components to model the weld-bonded edge that represents the finite uniform flexibility of the adhesively bonded segment of the weld-bonded edge along with zero flexibility at the spot welds modeled as discrete point supports. A detailed convergence study shows that by a proper choice of the number of terms used to represent the flexibility function and the number of terms in the Levy sine series for the solution of the plate displacement, accurate results can be obtained for vibration and buckling characteristics. This paper also includes a parametric study undertaken to show the effect of plate geometry, number of spot welds and adhesive joint parameters. The paper also discusses how such parametric studies can be of use to the designer in arriving at an optimal joint configuration of weld-bonded rectangular plates from linear elastic buckling and free vibration considerations.     

Vibration of nonuniform carbon nanotube with attached mass via nonlocal Timoshenko beam theory

This paper studies the vibrational behavior of nonuniform single-walled carbon nanotube (SWCNT) carrying a nanoparticle. A nonuniform cantilever beam with a concentrated mass at the free end is analyzed according to the nonlocal Timoshenko beam theory. A governing equation of a nonuniform SWCNT with attached mass is established. The transfer function method incorporating with the perturbation method is utilized to obtain the resonant frequencies of a vibrating nonlocal cantilever-mass system. The effects of the nonlocal parameter, taper ratio and attached mass on the natural frequencies and frequency shifts are discussed. Obtained results indicate that the sensitivity of the frequency shifts on the attached mass increases when the length-to-diameter ratio decreases. Tapered SWCNT possesses higher fundamental frequencies if the taper ratio becomes larger.     

Frequency solutions for circular plates with internal supports and discontinuous boundaries

The paper presents a study of the free-flexural vibration analysis of circular plates continuous over point supports, partial internal curved supports, and with mixed-edge boundary conditions. An approximate model which combines the advantages of the Rayleigh-Ritz and the Lagrangian multiplier methods is developed for analyzing this class of circular plate problems. The Rayleigh-Ritz method is used to formulate plates with classical boundary conditions, such as free, simply-supported or clamped, while the Lagrangian multiplier method is used to handle plates with point supports, partial internal curved supports and mixed-edge boundary conditions. The admissible pb-2 Ritz function consists of the product of a two-dimensional polynomial and a basic function. The basic function is defined by the product of the equations of the prescribed piecewise-continuous boundary shape each raised to the power of 0, 1 or 2, corresponding to free, simply-supported or clamped edge, respectively. The set of functions automatically satisfies all the kinematic boundary conditions of the plate at the outset. The geometric boundary conditions associated with the internal supports and discontinuous edges are simulated using a sufficient number of closely-spaced point constraints. Numerical results for several selected plate problems are presented to demonstrate the various features and accuracy of the present method.     

Vibration of initially stressed beam with discretely spaced multiple elastic supports

Vibration behavior of an initially stressed beam on discretely spaced multiple elastic supports has been studied and a theoretical formulation of the system is derived using the variational principle. Unlike beams on an elastic foundation, discretely spaced supports can distort the beam mode shapes when the supports have rather large stiffness, i.e. usually expected beam modes cannot be obtained, but rather irregular mode shapes are observed. Conversely, irregular modes can be recovered by changing initial stress. Since support location is closely associated with the dynamic characteristics, this work also discusses eigenvalue sensitivity with respect to the support position and some numerical examples are investigated to illustrate the above findings.     

Elastic small deflection analysis of annular sector Mindlin plates

An analytical method is developed for the bending response of annular sector Mindlin plates with two radial edges simply supported, and exact solutions are presented in the form of Levy-type series. Several different boundary conditions on the two circular edges are considered, viz. simply supported-simply supported, clamped-clamped and free-free. Numerical results for the case of uniform loading are presented to indicate the effect of shear deformation on the deflections and stress resultants at various points in the plate. Twisting stress couple and transverse shear stress resultant distributions along and near the edges of the plate are illustrated graphically, and the principal differences between the results predicted by Mindlin's plate theory and classical thin plate theory are discussed in detail. Results obtained with the present exact analysis may serve as references for approximate solutions and, especially, as a ‘shear locking’ test for thick plate finite element analysis.     

Postbuckling analysis of rectangular orthotropic plates

The postbuckling analysis of orthotropic simply supported rectangular plates of symmetric cross-section is studied and curves of load-shortening and effective widths are presented for various cases of orthotropic plates having different elastic properties. The results obtained are compared with results already published.