Modal analysis of circular plates with radial side cracks and in contact with water on one side based on the Rayleigh–Ritz method

In this paper, free vibrations of the baffled circular plates with radial side cracks and in contact with water on one side are investigated based on the Rayleigh–Ritz method. The completely free, simply supported and completely clamped boundary conditions are considered. Corner functions are introduced to describe the singularities at the crack tip. The motion of water is expressed by the velocity potential and the interaction between the water and the plate is derived in the form of an integral equation including the dynamic deformation of the cracked plate. The convergence studies are carried out and the numerical results show that the distinctions between the dry and wet mode shapes will be increased obviously excluding the first symmetric and antisymmetric modes when cracks appear. When the approximate methods based on the assumption that the wet modes are identical with the dry modes are adopted to calculate the eigenfrequencies, the errors of the results for cracked circular plates are larger than those for intact ones. The influences of the water on the symmetric and antisymmetric modes are different evidently, and the greatest reduction ratio of eigenfrequency and least difference between dry and wet mode are relative to the first symmetric mode. The verifications based on numerical simulation show that the proposed method is adequate for the investigation of free vibration of baffled circular plates with radial side cracks and in contact with water on one side.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

On the crossing points of the Lamb modes and the maxima and minima of displacements observed at the surface

This article elaborates on the crossing points of the frequency–wavenumber branches for the symmetric and anti-symmetric Lamb modes in a homogeneous plate. It is shown both theoretically as well as experimentally that at these crossing points either the normal or the longitudinal components of modal displacement attain an extreme value, i.e. a maximum or it vanishes. This behavior is assessed herein using a method due to Mindlin, who showed that the dispersion curves for a plate with mixed boundary conditions – which are associated with uncoupled shear and dilatational modes – provide bounds to the spectral lines of the free plate. Therefore, a subset of the crossing points of the symmetric and antisymmetric Lamb modes for a free plate coincide with the crossing points for a plate with mixed boundary conditions.     

EXACT BUCKLING AND VIBRATION SOLUTIONS FOR STEPPED RECTANGULAR PLATES

This paper is concerned with the determination of exact buckling loads and vibration frequencies of multi-stepped rectangular plates based on the classical thin (Kirchhoff) plate theory. The plate is assumed to have two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The proposed analytical method for solution involves the Levy method and the state-space technique. By using this analytical method, exact buckling and vibration solutions are obtained for rectangular plates having one- and two-step thickness variations. These exact solutions are extremely useful as benchmark values for researchers developing numerical techniques and software for analyzing non-uniform thickness plates.     

Sound radiation from shear deformable stiffened laminated plates

Sound radiation from shear deformable stiffened laminated plates is studied theoretically. The equations of motion for the composite laminated plate are derived on the basis of the first-order shear deformation plate theory. Two sets of parallel stiffeners interact with the laminated plate only through the normal line forces. By using the Fourier wavenumber transform and the stationary phase method, the far-field sound pressure is described analytically. Sound pressure given by the first-order shear deformation plate theory and the classical thin plate theory is compared, and the differences of sound pressure are shown in the high frequency range for an isotropic plate. Sound pressure and the transverse displacement spectra are presented to illustrate the effects of force location, stiffeners and angle-ply layers. Sound radiation from symmetric and antisymmetric composite plates with multiple loadings is also investigated.     

A comprehensive approach to the free vibration analysis of rectangular plates by use of the method of superposition

The authors have found the above techniques to constitute a powerful means for solving rectangular plate problems. At the time of writing, solutions for plates with two adjacent simply supported edges and two adjacent free edges have been obtained. The first 20 eigen-values for plates with all edges clamped have also been determined for a full range of aspect ratio and they are shown to be accurate to within less than one half of one percent. It will be appreciated that solutions for any combination of clamped-simply supported edge conditions can easily be obtained from the all-clamped solution by simply deleting appropriate solutions from the all-clamped combination. In Figure 2 contour lines for first mode vibration of a plate with two adjacent clamped and two adjacent simply supported edges is presented. The higher density of the contour lines along the simply supported edges will be noted.The method of superposition is currently being used by the authors to good advantage to obtain solutions of any desired degree of accuracy to all of the problems discussed. It is found to be easily utilized and unlike more complicated methods is readily comprehensible to the practicing engineer. Eigenvalues for all modes, aspect ratios, and boundary conditions are readily obtained. Modal shapes are expressed in terms of familiar analytic functions. Results of these studies will be made available in future publications.     

Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions

In this paper, the free vibrations of rectangular Mindlin plates with variable thickness in one or two directions are investigated. The thickness variation of the plate is continuous and can be represented by a power function of the rectangular co-ordinates. A wide range of tapered rectangular plates can be described by giving various index values to the power function. Two sets of new admissible functions are developed, respectively, to approximate the flexural displacement and the angle of rotation due to bending of the plate. The eigenfrequency equation is obtained by using the Rayleigh-Ritz method. The complete solutions of displacement and angle of rotation due to bending for a tapered Timoshenko beam (a strip taken from the tapered Mindlin plate in some direction) under a Taylor series of static load have been derived, which are used as the admissible functions of the rectangular Mindlin plates with taper thickness in one or two directions. Unlike conventional admissible functions which are independent of the thickness variation of the plate, the static Timoshenko beam functions presented in this paper are closely connected with the thickness variation of the plate so that higher accuracy and more rapid convergence can be expected. Some numerical results are furnished for both truncated Mindlin plates and sharp-ended Mindlin plates. On the basis of convergence study and comparison with available results in literature, it is shown that the first few eigenfrequencies can be obtained with quite satisfactory accuracy by using only a small number of terms of the static Timoshenko beam functions.     

Absorption of zero Lamb waves in an elastic layer by mechanical resonators with friction

The propagation of zero Lamb waves in a plate with mechanical resonators attached to its surfaces is studied. It is shown that, in a plate that is single-mode for symmetric (antisymmetric) waves, the zero symmetric (antisymmetric) Lamb wave can be absorbed by mechanical resonators with a certain friction. The single-mode (for symmetric or antisymmetric waves) plate can be not thin compared to the wavelength of the transverse wave.     

An approximate solution for the bending vibrations of a combination of rectangular thin plates

The appropriate method often used for calculating the bending vibration of a single rectangular plate is extended to calculate the bending vibrations of a global system of combinations of rectangular plates with elastically supported and damped non-coupled edges. Two examples, a series of T-combinations and an L-combination of rectangular thin isotropic plates, are considered and the input and transfer mobilities due to point excitation derived. Numerical results are presented for the case of combinations of concrete plates and the effects of varying the material damping of the plates and edge damping are investigated.The eigenfrequencies of an L-combination of plates with one plate of very high bending stiffness are calculated and results compare well with the eigenfrequencies of a single plate calculated by means of the classical Ritz-Rayleigh method.     

Free vibrations of a simply supported,rectangular plate: An exact elasticity solution

An exact, three-dimensional solution for the free vibrations of simply supported, rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper. The solution, obtained in a semi-inverse fashion as was the solution of the elastostatic problem for such plates, satisfies all of the boundary conditions of the problem in a pointwise manner. It is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi-inverse solution. Other modes of oscillation may exist in the three-dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes. The two types of modes found are analogous to the flexural modes of classical plate theory and the thickness-twist modes, here called breathing modes, of Mindlin plate theory. Some numerical results are given which indicate that the predictions of Mindlin plates are uncannily good approximations to the flexural frequencies given by the present, three-dimensional analysis even for very thick plates. However, the predictions of Mindlin plate theory for the thickness-twist, or breathing, frequencies are not nearly so good. These discrepancies are discussed briefly in an appendix.     

Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

A reduction method for problems of vibration of orthotropic plates

It is shown that the problem of vibration of an orthotropic plate can be reduced to that of another orthotropic plate by a simple co-ordinate transformation, and reduction formulae are obtained. To justify the reduction formulae, fundamental natural frequencies of orthotropic rectangular plates with various boundary conditions and of a clamped orthotropic elliptical plate are discussed. As an example, an exact natural frequency of a simply supported generally orthotropic skew plate with special flexural rigidities is obtained from that of a simply supported isotropic rectangular plate.     

Bending,vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model

Based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously, the non-linear (in the von Kármán sense) equations of motion for thick multilayered orthotropic plates are developed. Successively, the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates. In order to assess the accuracy of the proposed theory, the sample problem of the bending, free undamped vibration and buckling of a three-layered, symmetric cross-ply, square plate simply supported on all edges is investigated. For purposes of comparison, numerical results from the exact elasticity theory, the classical lamination (Kirchhoff) theory and the shear deformation theory (Timoshenko and Mindlin) with three different values of the shear correction factor are also presented. It is found that the proposed approach is very efficient in predicting the global responses (deflection, natural frequencies and buckling loads) of thick multilayered plates and models effects, such as the distortion of the deformed normals, not attainable from the classical lamination theory, as well as the shear deformation theory.     

Natural frequencies and mode shapes of flexural vibration of plates: laser-interferometry detection and solutions by Ritz's method

This paper presents an experimental and theoretical study of flexural symmetric vibration modes of a linear elastic plate. A laser interferometer is used as detector of the free vibration of a rectangular parallelepiped-shaped aluminium plate. The vibration spectrum gives the lowest natural frequencies of the sample. Assumption that the vibration of the plates may be described by some approximate theories is proven to be inconsistent. The Ritz method, with products of powers of the co-ordinates as basis functions, is applied to obtain the lowest flexural natural frequencies. Three-dimensional solutions are obtained, unlike those provided by simpler theories. The experimental results are compared with the numerical predictions and a good agreement is obtained. Finally, forced motion is applied to the centre of the plate and the out-of-plane and in-plane displacement components for the first symmetric mode are measured. A good fit of the calculated values to the experimental values is found.     

Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory

This paper presents exact solutions for vibration of rectangular plates with an internal line hinge. The rectangular plate is simply supported on two parallel edges and the remaining two edges may take any combination of support conditions. The line hinge is perpendicular to the two simply supported parallel edges. The Lévy type solution method and the state-space technique are employed in connection with the first order shear deformation plate theory (FSDT) to study natural vibration of rectangular plates with an internal line hinge. In particular, exact vibration frequencies are obtained for rectangular plates of different aspect ratios and edge support conditions. The influence of the internal line hinge on the vibration behavior of rectangular plates is studied.     

Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution

In this paper, free vibration of beams, annular plates, and rectangular plates with free boundaries are analyzed by using the discrete singular convolution (DSC). A novel method to apply the free boundary conditions is proposed. Detailed derivations are given. To validate the proposed method, eight examples, including the free vibrations of beams, annular plates and rectangular plates with free boundaries are analyzed. Two kernels, the regularized Shannon's kernel and the non-regularized Lagrange's delta sequence kernel, are tested. DSC results are compared with either analytical solutions or/and differential quadrature (DQ) data. It is demonstrated that the proposed method to incorporate the free boundary conditions is simple to use and can yield accurate frequency data for beams with a free end and plates with free edges. Thus, the proposed method for applying the boundary conditions extends the application range of the DSC.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2×2 symmetric operator matrices is proved. The result is applied to 4×4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.     

Free vibration analysis of corner-supported rectangular plates with symmetrically distributed edge beams

Analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams, or strips, attached to the plate edges. The method of superposition is employed. Equations governing reactions at plate-beam interfaces are developed in dimensionless form. The approach is comprehensive in that both lateral and rotational stiffness, and inertia, of the beam are incorporated into the analysis. For illustrative purposes computed eigenvalues and mode shapes are presented for two plate-beam systems of realistic geometries. It is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution. This appears to be the first comprehensive analytical study of this problem of industrial interest.