Free Vibration Of Circular Plates Research Articles

Highly Accurate Wavelet Solution for Bending and Free Vibration of Circular Plates Over Extra-Wide Ranges of Deflections

Abstract The wavelet multiresolution interpolation Galerkin method in which both the unknown functions and nonlinear terms are approximated by their respective projections onto the same wavelet space is utilized to implement the spatial discretization of the highly coupled and nonlinear Von Karman equation for thin circular plates with various types of boundary conditions and external loads. Newton’s method and the assumption of a single harmonic response are then used for solving the static bending and free vibration problems, respectively. Highly accurate wavelet solutions for an extremely wide range of deflections are finally obtained by the proposed method. These results for moderately large deflections are in good agreement with existing solutions. Meanwhile, the other results for larger deflections are rarely achieved by using other methods. Comparative studies also demonstrate that the present wavelet method has higher accuracy and lower computational cost than many existing methods for solving geometrically nonlinear problems of thin circular plates. Moreover, the solutions for large deflection problems with concentrated load support the satisfactory capacity for handling singularity of the proposed wavelet method. In addition, a trivial initial guess, such as zero, can always lead to a convergent solution in very few iterations, even when the deflection is as large as over 46 times thickness of plate, showing an excellent convergence and stability of the present wavelet method in solving highly nonlinear problems.

Free Vibration Analysis of Nonlinear Circular Plates Resting on Winkler and Pasternak Foundations

Dynamic behaviour of nonlinear free vibration of circular plate resting on two-parameters foundation is studied. The governing ordinary differential equation is solved analytically using hybrid Laplace Adomian decomposition method. The analytical solutions obtained are verified with existing results in literature. The analytical solutions are used to determine the influence of elastic foundation, radial and circumferential stress on natural frequency of the plate. Also, the radial and circumferential stress determined. From the results, it is observed that, increase in elastic foundation parameter increases the natural frequency of the plate. It is recorded that the modal radial and circumferential stress affect the extrema mode of the plate. It is hoped that the present study will contribute to the existing knowledge in the field of vibration analysis of engineering structures.

Free Vibration Analysis of Elliptic Plate by Conformal Mapping Method

Thin plate structure is very common in various engineering fields. So, it is very important to analyze its vibration characteristics in order to avoid the fatigue damage. Though the rectangular or circular plates are popular because of the low cost of their manufacturing, occasionally due to the stress imbalance which results from eccentric mass during manufacturing them, circular plates may become elliptic plates. In the view of above facts, it is also necessary to study the vibration of elliptic plates. However, the traditional approaches, such as Rayleigh-Ritz method and Mathieu function, have their own limitation so that they will be very awkward. Therefore, this article has brought up a new approach to the study of the vibration of thin plate, which is called conformal mapping method. This method could map a complicated curve plate (like elliptic plate) into a circular plate. Then the theory of free vibration of circular plates in complicated boundary conditions will be used to calculate the analytical solution of natural frequencies (accurate solution) and the corresponding modal vibration modes. This method is universal for any thin plates which have complicated curve shapes so that it will widen the horizon of the study of vibration of plates in practice and break through the limitation of above traditional methods.

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson’s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

Pressure-Based and Potential-Based Differential Quadrature Procedures for Free Vibration of Circular Plates in Contact with Fluid

The differential quadrature method (DQM) has been so far applied to a wide variety of fluid and/or structural problems. The results of many researchers reveal that the DQM is computationally efficient and is applicable to a large class of boundary value problems. However, there is little information about its applications to fluid-structure interaction problems. Therefore, the purpose of this paper is to provide some information in this area and to develop procedures based on the DQM for the numerical solution of fluid-structure interaction problems. First, the governing partial differential equations of motion of the structure and fluid are discretized separately using the DQM. Then, by applying the boundary condition at fluid-structure interface, the governing eigenvalue equations of the coupled system are obtained which can then be solved for the eigenvalues of the system. The applicability of the proposed procedures is shown herein through the free vibration analysis of thin circular plates in contact with a cylindrical fluid-filled cavity. Issues related to the implementation of the regularity conditions at the center of the circular plate and the central line of the cylindrical cavity are addressed. Two new regularity conditions are proposed for the circular cylindrical fluid domain. The accuracy and efficiency of the proposed procedures are demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate converged results can be obtained by the proposed procedures using a small number of grid points. Three new dimensionless parameters and variables are also introduced for the free vibration of the coupled system. The influences of these parameters on dynamic behavior of the system are studied.

Free Vibration of Circular Plates with Various Edge Boundary Conditions by 3-D Elasticity Theory

Free Vibration of Circular Plates with Various Edge Boundary Conditions by 3-D Elasticity Theory

Free vibration of a FGPM circular plate placed in a uniform magnetic field

In this paper, free vibration of a circular plate composed of a transversely isotropic functionally graded piezoelectric material (FGPM) placed in a uniform magnetic field is investigated. Material properties are assumed to depend on the thickness of the circular plate and they are expressed as the same exponential function of h. The problem of free vibration for the transversely isotropic FGPM circular plate with clamped and simply supported boundary conditions is solved by means of the state space method. Numerical examples and some useful discussions are given to demonstrate the significant influence of material inhomogeneity, and adopting a certain value of the graded index can optimize structures of the circular plate. This will be of particular importance in modern engineering design.

Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface

In the present study, a theoretical method is developed to investigate free vibrations of circular plates immersed in fluids and a series of experimental tests are presented to validate the model. The coupled governing equations of both hydroelastic vibration of the plate and liquid sloshing are solved by a semi-analytical procedure, simultaneously. The effect of the plate, used as a baffle, on suppression free surface waves is also considered. Plates with two different boundary conditions, free-edge and clamped edge, are studied. The fluid domain is non-convex because of the presence of the plate, which introduces a singularity in the formulation of the fluid velocity potential. Both the least square and Galerkin methods are applied to determine the unknown coefficients in the velocity potential. Natural frequencies and mode shapes are obtained using the Rayleigh–Ritz method, taking fluid–structure interaction into account. The present approach is validated by comparison to results of modal test on two different steel plates with the free edge and submerged in water, as well as comparison to those of a commercial finite element code. The results obtained from the present method agree with those obtained from modal test and the finite element analysis.

Asymmetric free vibration of circular plate in contact with incompressible fluid

Vibration of circular plates in contact with fluid has extensive applications in the industry. This paper derives added mass and frequencies for asymmetric free vibration of coupled system including clamped circular plate in contact with incompressible bounded fluid. Considering small oscillations induced by the plate vibration in the incompressible and inviscid fluid, velocity potential function is used to describe the fluid motion. Derivation uses Kirchoff’s thin plate theory. Two approaches are used to derive the free vibration frequency of the system. The solutions include an analytical solution employing Fourier–Bessel series and a variational formulation applied simultaneously on the plate and fluid. Strong correlation is found between free vibration frequencies of the two solutions. Finally the effect of fluid depth on the added mass and free vibration frequencies of the coupled system is investigated.

Free Vibration of Circular Plates with Elastic Edge Support and Resting on an Elastic Foundation

An analytical study is presented for the vibration characteristics of thin circular plates with rotational elastic edge support and resting on an elastic foundation. The Winkler model was used to describe this foundation. The exact analytical method is used to derive the frequency equation of the plate with elastic support and resting on this foundation. Parametric investigations of the vibration of circular plates with an restrained elastically edge against rotation and resting on an elastic foundation have been carried out with respect to various rotational stiffness parameters, foundation stiffness parameters, and boundary conditions. Twelve vibration modes are presented. The location of the stepped region with respect to rotational restraint changed very little and moved from the range of 0.014 26† [10.006 204 8‡]–1 134.227 [10.025 94] to 0.017 58 [14.001 133]–6 352.958 2 [14.658 356]. Here, † represents the rotational spring stiffness ratio, and ‡ represents the foundation stiffness ratio. Extensive data are tabulated so that pertinent conclusions can be arrived at on the influence of rotational edge restraint and the foundation parameters of a Winkler foundation on the natural frequencies of uniform thin isotropic circular plates.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

Analysis of Nonlinear Free Vibration of Circular Plates With Cut-Outs Using R-Function Method

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the R-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.

Free vibrations of circular plates with axisymmetric thickness variation

The integral equation method is developed for vibration analysis of solid circular plates with arbitrary thickness variation along the radius. General boundary conditions are considered. The problem is formulated in terms of displacements. The resulting fourth-order linear hyperbolic equation with variable coefficients is solved using the analogue equation method of Katsikadelis. According to this method the linear governing differential equation is replaced by a linear equation of a substitute beam with unit bending stiffness, under a fictitious load distribution. Numerical examples are presented for plates with various thickness variation laws, which illustrate the method and demonstrate its efficiency and accuracy.

Analytical study and numerical experiments of true and spurious eigensolutions of free vibration of circular plates using real-part BIEM

Following the success of the CHEEF method [Chen IL, Chen JT, Kuo SR, Liang MT. A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method. J Acoust Soc Am 2001; 109(3):982–98] and the real-part BEM [Kuo SR, Chen JT, Huang CX. Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM. Int J Numer Methods Eng 2000; 48:1401–22] for solving the membrane eigenproblem, we extend to the plate problem in this paper. The boundary integral equation method (BIEM) using only the real-part kernel instead of the complex-valued kernel is employed to solve the plate eigenproblem for saving half effort in computation. The spurious eigenvalue that resulted due to insufficient constraint is examined. To deal with this problem, a combined Helmholtz exterior integral equation formulation method (CHEEF) is employed to provide sufficient constraints to filter out spurious eigenvalues. The constraint equations of the transverse displacement, normal derivative and tangent derivative for the exterior collocating points are derived. If these constraint equations are properly chosen, one collocating point was sufficient to filter out all the spurious eigenvalues easily and efficiently, even for the repeated spurious eigenvalues. Finally, numerical experiments are performed to verify the analytical results.

Transverse vibration of thin solid and annular circular plate with attached discrete masses

Circular plates have many engineering applications. These are commonly found in spacecrafts, missiles, land based vehicles, underwater vessels and structures. In many situations, these plates carry loads at eccentric positions. Therefore, one must know the fundamental frequencies of the plates with different boundary conditions with point masses placed at any arbitrary position. Leissa [1] and Laura et al. [2] have reviewed analytically most of the early works in the area of free vibration of circular plate with concentrated mass attached at the centre of the plate. Bambill et al. [3] determined the fundamental frequency of vibration of circular plate carrying a concentrated mass at an arbitrary position with different boundary conditions using variational approach with Rayleigh Ritz formulation. The present study is concerned with the determination of the fundamental frequency of vibration of solid and annular circular plate with different boundary conditions and with point mass attached at an arbitrary position using finite element analysis (FEA). The fundamental frequency of vibration of clamped and simply supported solid circular plate with point mass attached at the centre determined by other researchers [1–3] are compared with the present study. A good agreement is obtained in the values of fundamental frequency. This methodology has been extended to determine the fundamental frequency of solid and annular plate with different boundary condition with varying point masses placed at an arbitrary position.

VIBRATION OF CIRCULAR PLATES ON A FREE FLUID SURFACE: EFFECT OF SURFACE WAVES

The effect of free-surface waves on free vibrations of circular plates resting on a free fluid surface is studied. The solution is achieved by using a perturbation technique and the Hankel transformation method which give a couple of dual integral equations of Titchmarsh type. The fluid is considered inviscid and incompressible and the velocity potential describes its motion. The Kirchhoff theory of plates is used to model the elastic thin plate. The theory is suitable for all axisymmetric plate boundary conditions. Numerical results are given in non-dimensional form for modes up to five nodal circles and diameters for clamped, simply supported and free-edge plates, to be ready-to-use in applications. The effect of free-surface waves on the plate's natural frequencies is significant only when bluging and sloshing modes of the system have close natural frequencies.

Three-dimensional elasticity solutions for free vibrations of circular plates: A polynomials-Ritz analysis

This paper presents accurate three-dimensional elasticity solutions for free vibration of circular plates. The derivation of a linear frequency equation based on an exact three-dimensional, small-strain, linearly elastic principle is detailed. The solution to this problem is made possible by using the Ritz method with a set of orthogonal polynomial series to approximate the spatial displacements of the circular plate in cylindrical polar coordinates. The perturbation of frequency responses due to the variations of boundary conditions and thickness is investigated. First known frequency parameters and three-dimensional deformed mode shapes are presented in vivid graphical forms. The accuracy of these results are verified by appropriate convergence studies and, when possible, are checked with existing solutions.

Method of initial functions for axially symmetric elastic bodies

Method of initial functions for an axially symmetric state of stress in elasto-dynamic problems has been formulated which gives a complete choice in prescribing the boundary conditions in terms of either stresses or displacements, or a combination of stresses and displacements. The general dynamic response of the elastic body has been derived in the form of a set of transcendental partial differential equations from which the initial functions can be evaluated in terms of prescribed boundary conditions. The method is applied, as an illustration, to the flexural theory of circular plate subjected to antisymmetric lateral loads. Numerical examples of free vibrations of circular plates are given. The results are compared with solutions from classical theory.

Free vibration of circular plates with elastic or rigid interior support

In this work reported in this paper the problem of free vibration of a circular plate with an elastic or rigid interior support distributed uniformly around the plate at an arbitrary radius has been considered. The case of an elastic or rigid point support at the plate center has also been studied. The frequency and mode shape equations have been formulated by using the modal expansion technique and the receptance method. Numerical results for natural frequencies are presented, and wherever it has been possible the results obtained are compared with results given in other papers in the literature.

Free vibration of circular plates with elastic edge supports using the receptance method

Application of the receptance method has been extended to the free vibration of circular plates restrained elastically on their boundaries. Different combinations of linear and torsional spring stiffnesses have been considered. For all these cases, frequency and mode shape equations have been obtained and numerical results for natural frequencies of vibration are presented. Solutions for some of the combinations investigated have not previously been reported in the literature. Wherever possible, the results obtained are compared with the existing values in the literature and a very good agreement is shown to exist.