Vibration analysis of rectangular Mindlin plates with internal line supports using static Timoshenko beam functions

In this paper, the vibration characteristics of rectangular Mindlin plates with internal line supports in one or two directions are studied by using the Rayleigh–Ritz method. The static Timoshenko beam functions are employed as admissible functions, which are composed of a set of transverse deflection functions and a set of rotation-angle functions due to bending. The static Timoshenko beam functions are derived from a point-supported strip of unit width taken from the particular plate under consideration in a direction parallel to the edge of the plate and acted upon by a series of static sinusoidal loads distributed along the length of the strip. It can be seen that the suggested approach is very simple mathematically, and each of the beam functions is only a sine or cosine function plus a polynomial function of not more than the third order. A unified program can be easily prepared, because the changes in boundary conditions, number and locations of internal line supports and thickness ratio of the plate will result in a corresponding change of only the coefficients of the polynomials. Both high accuracy and low computational cost have been verified by convergence and comparison studies. In addition, it can be seen that the admissible functions presented in this paper can also properly describe the discontinuity conditions of the shear forces at the line supports. Therefore, accurate results can be expected for the analysis of dynamic response and internal force distribution of the plate.     

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.     

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 analysis of rotating composite cantilever plates

A modeling method for the vibration analysis of rotating composite cantilever plates is presented in this paper. The coupling effects between inplane motions and the bending motion are considered and explicit mass and stiffness matrices are derived for the modal analysis. Numerical results are obtained and some of them are compared to those of a commercial program to confirm the accuracy of the present method. Numerical results show that the coupling effects become important only when laminates are stacked up unsymmetrically. Incidentally, natural frequencies loci veering, loci crossing, and associated mode shape variations are observed.     

Vibration of symmetrically laminated cantilever trapezoidal composite plates

The paper presents a general numerical method for vibration analysis of symmetrically laminated cantilever thin trapezoidal composite plates. The Rayleigh-Ritz method, along with displacements assumed in the form of 2-D orthogonal polynomials, is used to solve the problems. These 2-D orthogonal polynomials are generated using a proposed recurrence formula. Several numerical examples are presented to demonstrate the accuracy, efficiency and versatility of the method. Natural frequencies and mode shapes for the trapezoidal plates are obtained for different numbers of layers, fibre orientations, and ratios. These results, to the author's knowledge, are first known in the open literature. Thus the present study may provide very useful information to the engineers who work in this area.     

Vibration and buckling of generally laminated composite plates with arbitrary edge conditions

A method is developed for the analysis of free vibration and buckling of generally laminated composite plates having arbitrary edge conditions, such as clamped, simply supported or free. The procedure is an extension of the Ritz method utilizing a strain energy functional containing both bending and stretching effects and accommodating arbitrary ply stacking sequences. Displacement functions are taken in the form of polynomials, and an algorithm for satisfying the geometric boundary conditions is presented. Numerical results are compared with those of other researchers in order to establish the correctness and effectiveness of the method. Some additional new results are also presented.     

Vibration of antisymmetric angle-ply composite annular plates of variable thickness

Journal of Mechanical Science and Technology - Based on first order shear deformation theory, free vibrational behaviour of antisymmetric angle-ply composite annular plate is investigated. The...     

Three-dimensional vibration of laminated composite plates and cylindrical panels with arbitrarily located lateral surfaces point supports

On the basis of fully three-dimensional elasticity considerations, this paper presents a free vibration analysis of simply supported, cross-ply laminated plates and cylindrical panels that are subjected to an arbitrary number of lateral surfaces point supports. The analysis is based on a recursive approach suitable for the vibration analysis of corresponding unconstrained structural elements. By means of dynamic equilibrium considerations, the reaction of the point supports is imposed by using the Lagrange multipliers method. This yields the eigendeterminant of a constrained panel by appropriately coupling the response of a suitably large number of natural vibration modes of the corresponding unconstrained structural element.     

Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses

Natural frequencies and buckling stresses of angle-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Numerical results are compared with those of the published existing theories. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. The present global higher-order approximate theories can predict the natural frequencies, buckling stresses and modal stresses of thick multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers.     

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.     

Vibration analysis of delaminated composite beams and plates using a higher-order finite element

In order to analyze the vibration response of delaminated composite plates of moderate thickness, a FEM model based on a simple higher-order plate theory, which can satisfy the zero transverse shear strain condition on the top and bottom surfaces of plates, has been proposed in this paper. To set up a C⁰-type FEM model, two artificial variables have been introduced in the displacement field to avoid the higher-order derivatives in the higher-order plate theory. The corresponding constraint conditions from the two artificial variables have been enforced effectively through the penalty function method using the reduced integration scheme within the element area. Furthermore, the implementation of displacement continuity conditions at the delamination front has been described using the present FEM theory. Various examples studied in many previous researches have been employed to verify the justification, accuracy and efficiency of the present FEM model. The influences of delamination on the vibration characteristic of composite laminates have been investigated. Especially the variation of ‘curvature of vibration mode’ (i.e., the second-order differential of deflections in vibration mode) caused by delamination has been studied in detail to provide valuable information for the possible identification of delamination. Furthermore, two approaches have been investigated to detect a delamination in laminates by employing this information.     

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.     

Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory

This paper deals with the free vibration of open, laminated composite, circular cylindrical panels having a rectangular plan-form and all their edges free of external tractions. The material arrangement of the shell panels considered may vary from this of the single isotropic (or special orthotropic) layer to that of a general angle-ply lay-up. The analysis is based on the application of the Ritz approach on the energy functional of the Love-type version of a unified shear deformable shell theory. A through-thickness parabolic distribution of the transverse shear deformation is mainly assumed but, for comparison purposes, numerical results that are based on the assumptions of the classical Love-type shell theory are also presented. The Ritz method is a powerful analytical technique since, provided that a complete set of trial functions is employed, it can provide the exact solution of the problem considered in infinite series forms. The mathematical formulation is therefore presented in a general form, appropriate for any set of basis functions. The variational approach is, however, finally applied in conjunction with a complete functional basis made of the appropriate admissible orthonormal polynomials.     

Vibration of stepped thickness plates

The vibration characteristics of stepped thickness plates of rectangular geometry with simply supported edges are investigated through an exact analysis. To illustrate the effect of sudden variation of thickness in the step form on the dynamic characteristics of plates, numerical calculations have been made for various combinations of thickness step, step-length ratio and side ratio. Natural frequencies for the first eight to nine modes are presented in the form of tables. Nodal lines are also plotted for a few selected cases. Several interesting features such as that of the thickness step splitting the degenerate frequencies of uniform rectangular plates into distinct ones, and the “frequency crossing” of some of the modes are observed.     

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.     

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.     

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.     

Local surface defects of composite plates

A static three-dimensional model of an elasticoplastic solid is considered, with the dependence of the material’s parameters on temperature taken into account. The effects of maximum indentation depth and its in-plane sizes are investigated, as are those of plate thickness and temperature on deformations and stresses at each plate’s point, in order to discover heating patterns that would enable operation of the plates without degradation.     

Dynamic response with mass variation of laminated composite twisted plates

In this study, the dynamic response of free vibration of laminated composite twisted (skewed) plate with mass variation via cutout and additional mass is investigated. The mathematical model employs cubic variation of the thickness coordinates in the displacement field. The mathematical formulation also includes the effects of transverse shear and rotatory inertia. A realistic parabolic distribution of transverse shear strains through the plate thickness is assumed and the use of the shear correction factor is avoided. A C⁰ finite element formulation of the mathematical model is developed to analyze the dynamic behavior of laminated composite twisted plate with cutout and additional mass. Given that no results on the present problem of free vibration of laminated composite twisted plate with cutout and additional mass are available in literature, novel results for different twist angles of laminated composite plates along with mode shapes are presented by varying the geometry, boundary conditions, and ply orientations.     

Geometric analysis of composite cutting tools with replaceable polyhedral plates

The geometry of tools with replaceable polyhedral plates is analyzed as a function of the cutting-plate orientation. Formulas are proposed for the geometric analysis of existing and new tool designs.