Vibration Analysis Of Circular Plates Research Articles

Model refinements with experimental validation in piezoelectric plate actuators

The axisymmetric vibration analysis of circular plates with piezoelectric layers and step changes in plate thickness is conducted. For the chosen configurations, it is necessary to solve a coupled system of differential equations for extensional and flexural vibration to obtain accurate modal results. The adopted computational approach is based upon the classical transfer matrix method. Instead of employing analytical solutions in the form of e.g. Bessel functions, matrix exponentials of the system matrix are directly computed in the transfer matrices for finding highly accurate solutions. The developed approach has enabled the complete quantitative modal analysis in piezoelectric actuators. The procedure is firstly verified against available analytical results for natural frequencies and modes shapes in circular homogeneous plates, and by using finite element simulations for plates with asymmetric step changes in thickness. Ultimately, the developed approach is validated against experimental results of relevance to applications in piezoelectric synthetic jet actuation. The obtained numerical results for modal parameters and frequency response functions are in excellent agreement with the available analytical results, and with previous and newly presented in the paper experimental data, when considering the inherent uncertainty in parameter values. Based on extensive experimental results, realistic damping is introduced in the model of piezoelectric plate actuators. Importantly, it is shown that solutions of high accuracy can be obtained by a more general coordinate system than that traditionally linked to the plate neutral plane(s).

Investigation of dynamic behaviour of circular plates resting on Winkler and Pasternak foundations

Investigating the dynamic behaviour of circular plate resting on elastic foundations are very important in designing of structural systems. This study examines the free vibration analysis of circular plate resting on Winkler and Pasternak foundations. The governing nonlinear partial differential equation is transformed to Duffing equation based on von Karman geometric nonlinear principle and the nonlinear to linear frequency ratios are obtained while the linear natural frequencies are determined using Galerkin of weighted residual method. Also, the accuracy and reliability of the approximate solutions obtained are demonstrated by comparing the obtained results with available results reported in the literature. The analytical solutions obtained are used for examining the effect of elastic foundations on the dynamic behaviour of the circular plate. From the results, it is observed that, increasing elastic foundation parameter increases the natural frequency. As the nonlinear foundation increases, the nonlinear vibration frequency ratio decreases. The nonlinear Winkler foundation attenuates the amplitude of vibration of the circular plate. It is hoped that, the present study will contribute to the existing knowledge of classical theory of vibration.

Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method

The free vibration analysis of a circular plate made up of a porous material integrated by piezoelectric actuator patches has been studied. The plate is assumed to be thin and its shear deformations have been neglected. The porous material properties vary through the plate thickness according to some given functions. Using Hamilton's variational principle and the classical plate theory (CPT) the governing motion equations have been obtained. Simple and clamped supports have been considered for the boundary conditions. The differential quadrature method (DQM) has been used for the discretizations required for numerical analysis. The effect of some parameters such as thickness ratio, porosity, piezoelectric actuators, variation of piezoelectric actuators-to-porous plate thickness ratio, pores distribution and pores compressibility on the natural frequency, radial and circumferential stresses has been illustrated. The results have been compared with the similar ones in the literature.

Vibration analysis of circular plates in contact with fluid: A numerical approach

A mathematical formulation to obtain the non-dimensionalized added virtual mass incremental factor for uniform circular plate in contact with fluid is presented. Based on Rayleigh’s quotient, the natural frequencies can be evaluated using the added virtual mass incremental factors. A suitable approximation is used to describe the effect of non-dimensionalized added virtual mass incremental factors in the vibration analysis of circular plates. The numerical results obtained from the present formulation are compared with the known results reported in the literature and a good agreement was found. The maximum error percentage has been predicted from the present formulation is 0.005%.

Vibration analysis of thin circular plates with multiple openings by the assumed mode method

Circular plates with openings are used in many engineering structures, especially as swash bulkheads in cargo tanks of liquefied gas carriers. In this article, an approximate procedure is worked out for the vibration analysis of circular plates with different boundary conditions and a different number and size of openings with an arbitrary arrangement within the plate area. The assumed mode method, recently used in the vibration analysis of rectangular plates, is applied in this case. Following the basic idea, the effect of the openings is accounted for by subtracting the potential and kinetic energy of the cut-out plate parts corresponding to the total plate energies without openings. The procedure is illustrated with numerical examples related to the vibration analysis of annular plates and circular plates with openings. The results are evaluated through their comparison with an analytical and finite element method solution.

An analytical solution for free vibration analysis of circular plates in axisymmetric modes based on the two variables refined plate theory

This paper presents, for the first time, an analytical solution for free vibrations of an isotropic circular plate in axisymmetric modes based on the two variables refined plate theory. This theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Governing equations are derived using Hamilton’s principle and an analytical method on the basis of using Bessel functions is introduced to solve them. By this procedure, final form of the governing equations is obtained in matrix form. These equations are solved for classical boundary conditions and comparison studies are performed to verify the validity of the present results. It is found that the results obtained using RPT and TSDT are close to each other. As a benchmark, numerical results are presented in a dimensionless form for various values of thickness to radius ratio.

Free Vibration Analysis of circular plates with holes and cutouts

Circular plates with holes are extensively used in mechanical components. The existence of a hole in a circular plate results in a significant change in the natural frequencies and mode shapes of the structure. Especially if the hole is located eccentrically, the vibration behavior of these structures is expected to deviate significantly from that of a plate with a concentric hole. These holes usually cause the change of natural frequency as well as the decrease of load carrying capacity. It is important to comprehend the associated effects in the work of mechanical design or flight control of the structure. Therefore, in this study, an experimental method to determine the modal characteristics of a plate with multiple holes and slots are used is verified by the finite element analysis (FEA) with ANSYS. Also, the relationship between parameter variations and vibration modes is investigated. These results can be used as guidance for the modal analysis and damage detection of a circular plate with a hole.

A simple and unified displacement field for three-dimensional vibration analysis of prestressed circular plates

In this study a simple displacement field is proposed for finite element vibration analysis of circular plates. The displacement field is a combination of a conventional field, which is very popular for three-dimensional vibration analysis of circular plates or cylinders, and another field, which is most general for axisymmetric vibration. The investigation here is intended, through mathematical formulation and examples, to verify the existence of coupled axisymmetric vibration modes. These modes cannot be found by conventional displacement field, but can easily be revealed by the present model. Validity and accuracy of the proposed displacement field to show all the prominent vibration modes of circular plates or cylinder-like structures is also demonstrated by comparing results using the present method with those of a three-dimensional finite element analysis and experiments.

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical–analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.

독립좌표연성법을 이용한 편심 된 원형 구멍을 갖는 원판의 자유진동해석

This paper is concerned with the free vibration analysis of a circular plate with an eccentric circular hole by the Independent coordinate coupling method(ICCM). It was proved in the previous study that the ICCM can accurately predict the natural frequencies and mode shapes of the annular plates and can also be used for the free vibration analysis of the simply-supported circular plate with an eccentric circular hole. In this study, the clamped and free boundary conditions were considered for the circular plate. The numerical results show that the ICCM can be used effectively for the free vibration problem of circular plate with an eccentric hole compared to the finite element method.

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the cor- responding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solu- tion to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circu- lar holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analyt- ical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent prob- lem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational

Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs

In this paper, a semi-analytical approach is proposed to solve natural frequencies and natural modes for circular plates with multiple circular holes by using the indirect formulation in conjunction with degenerate kernels and Fourier series. All the kernels in the indirect formulation are expanded into degenerate form. By uniformly collocating points on the boundary, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the indirect boundary integral formulations. The results of the annular plate, as a special case, are compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with multiple circular holes, the results are also compared with those of finite element method (FEM) using ABAQUS. Besides, the effect of eccentricity of the hole on the natural frequencies is also considered. Good accuracy, high rate of convergence and computational efficiency are the main features of the present method due to the semi-analytical procedure.

Free vibration analysis of circular plates using generalized differential quadrature rule

The free vibration of solid circular plates has been studied using the generalized differential quadrature rule (GDQR). The effects such as stepped thickness, intermediate concentric ring supports, and elastic restraints have been considered. Discrete system equations for natural frequency analysis were derived based on the GDQR. Issues related to the implementation of the boundary and compatibility conditions were addressed. Two regularity conditions at the solid circular plate center were formularized, since they were not quite clear in the present literature. Some wrong applications in the implementation of both differential quadrature methods and regularity conditions at the solid circular plate center have been pointed out. Excellent results have been obtained with the presented examples.

HYDROELASTIC VIBRATION OF A CIRCULAR CONTAINER BOTTOM PLATE USING THE GALERKIN METHOD

The free vibration of a flexible thin plate placed into a circular hole and elastically connected to the rigid bottom slab of a circular cylindrical container filled with fluid having a free surface is studied. The liquid is assumed to be incompressible, inviscid and irrotational. The effect of the free surface wave is also taken into account in the analysis. First of all, the exact expression of velocity potential of the liquid movement is derived by a combination of the superposition method and the method of separation of variables. With the help of the Fourier–Bessel series expansion, part of the unknown coefficients in the solution is determined by the consistency condition between the liquid movement and the plate vibration, in the form of integrals associated with the dynamic deflection of the plate. Then, the Galerkin method is applied to derive the eigenfrequency equation of the fluid–plate interaction. Finally, the effects of various parameters and the free surface wave on eigenfrequencies of the fluid–plate system are discussed. As a consequence, the accuracy of the nondimensional added virtual mass incremental (NAVMI) factor solution has also been evaluated by comparing with the more accurate Galerkin solution. It is shown that the proposed method is also applicable to the vibration analysis of circular plates in contact with an infinite liquid by only taking a finite but larger size of liquid to replace the infinite liquid in the computation.

Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule

The generalized differential quadrature rule proposed recently by these authors is applied here to the free vibration analyses of solid circular plates with radially varying thickness and elastic restraints. The thickness can vary radially in any given continuous form, and it varies exponentially and linearly in this work. Two regularity conditions corresponding to the plate center are expressed in explicit formulae, since they have been inexactly or even wrongly expressed in the literature. Such errors are pointed out in the application of the differential quadrature techniques and in the expression of the regularity conditions at the plate center. Numerical results are presented for a number of plates, illustrating the versatility and accuracy of the approach.

Application of Maple V to the nonlinear vibration analysis of circular plate with variable thickness

This paper is concerned with the application of Maple V to the nonlinear vibration problems of circular plates with variable thickness. In this paper, the nonlinear equations of plates of variable thickness to the dynamic case can be solved by using the computer algebra systems method. Details of solution expressions and numerical results are given in computer algebra systems forms, for two kinds of boundary conditions, which are the clamped edge and the supported edge. The numerical results show that the solutions of the paper contain other cases when the plates are of uniform thickness. The effect of various thickness parameters has been investigated in detail. In addition, a Runge–Kutta method is used to solve the free vibration and the maximum deflection response to a uniformly distributed step load of plates with variable thickness. It is shown that the adoption of variable thickness plate would be useful in engineering design.