Natural Frequencies Of Rectangular Plates Research Articles

Natural frequencies of thin rectangular plates using homotopy-perturbation method

• Homotopy perturbation method yields natural frequencies of thin rectangular plates. • An irregular-shaped plate can be analysed using an equivalent rectangular plate. • A guitar soundboard was used as a specific case of an irregular-shaped plate. • Results compared well with those from numerical as well as analytical solutions. The homotopy perturbation method (HPM) was developed to search for asymptotic solutions of nonlinear problems involving parabolic partial differential equations with variable coefficients. This paper illustrates that HPM be easily adapted to solve parabolic partial differential equations with constant coefficients. Natural frequencies of a rectangular plate of uniform thickness, simply-supported on all sides, are obtained with minimum amount of computation. The solution is shown to converge rapidly to a combination of sine and cosine functions. Truncating the series solution by using only the first three terms of the sine and cosine functions as compared to the exact solution results in an absolute error not exceeding 2 × 10 −4 and 9×10 −4 for the trigonometric functions respectively. HPM is then applied to solve the nonlinear problem of a rectangular plate of variable thickness. A direct expression for the eigenvalues (natural frequencies) of the rectangular plate is obtained as compared to determining its eigenvalues by solving the characteristic equation using the conventional method. Comparison of results for the frequency parameter with existing literature show that HPM is highly efficient and accurate. Natural frequencies of a simply-supported guitar soundboard were obtained using an equivalent rectangular plate with the same boundary condition.

A dierential quadrature procedure for free vibration of rectangular plates involving free corners

The Di erential Quadrature Method (DQM) is one of the most powerful approximation methods for analyzing the free vibration of rectangular plates. It is easy to use and straightforward to implement. However, in spite of its many advantages, the conventional DQM has some limitations in determining the natural frequencies of rectangular plates involving free corners. This is because it is very dicult to implement the free corner boundary condition in conventional DQM. As a result, the method may exhibit some convergence problems and this may lead to erroneous and oscillatory results for natural frequencies of rectangular plates involving free corners. To overcome this diculty, this paper presents a simple DQM formulation in which all the natural boundary conditions, including the free corner boundary condition, are implemented in an easy manner. Its accuracy and eciency are demonstrated through the vibration analysis of rectangular plates with di erent combinations of free edges and free corners. Numerical results prove that the proposed method can produce much better accuracy than the conventional DQM while exhibiting a monotonic convergence behavior with respect to the number of sampling points. Furthermore, unlike the conventional DQM, solutions of the proposed method are not very sensitive to the sampling point distribution.

Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

Vibration of a rectangular plate with a central power-law profiled groove by the Rayleigh–Ritz method

The prediction of natural frequencies of rectangular plates where a profiled indentation is present is made using a Rayleigh–Ritz variational energy method. Panels with holes are often found for access cables and access gaps, and it is shown that the application of damping to the profile leads to a more efficient method of reducing vibration than covering a whole rectangular plate, which is advantageous where weight saving is necessary.

A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners

Abstract It is well known that the classical global approximation methods such as the conventional Ritz method and the conventional differential quadrature method (DQM) have some difficulty in determining the natural frequencies of rectangular plates involving free corners. The mixed Ritz-DQM formulation, which has been recently developed by the present author, was also shown to have such difficulty. This is because it is very difficult to implement the free corner boundary condition in these methods. To overcome this difficulty, this paper presents a mixed Ritz-DQM formulation in which the free corner boundary condition is implemented in a simple and easy manner. First, we present a new scheme for implementing multiple boundary conditions in the DQM discretized equations. By the help of this scheme, we then show that the free corner boundary condition can also be easily implemented in the final matrix equations of the Ritz-DQM approach. Finally, we validate the effectiveness of the proposed approach through numerical experiments. Numerical results show the advantages of the proposed method over some versions of the DQM in terms of accuracy and performance.

Free vibration analysis of shear-deformable Al alloy plates using FEM with first order shear deformation theory

This paper mainly presents free vibration analysis of an isotropic thin plate using first order shear deformation theory. This analysis has been held for four types of aluminium (Al) alloys namely Al 95, Al 96, Al 98 and Al 995. The stiffness and mass matrices are calculated using first order shear deformation theory for two varying thicknesses of the selected materials with the application of Kirchhoff plate theory. This method is useful for getting accurate natural frequencies of rectangular plate due to its lateral strain consideration in plates. The calculated frequencies from the above numerical method have been compared with the frequency values of simply supported rectangular plate from finite element analysis commercial package ANSYS. The numerical values and values from ANSYS correlate well with each other.

Resonant frequencies of rectangular plates immersed in fluids

The response of a fluid-loaded structure is a subject of considerable interests in many areas of engineering. This includes applications in aerospace industry for composite structures, design of offshore and building structures partially immersed in water, and the design of rectangular micro-plates for bio-sensing devices. For many decades, it has been well recognized that the resonant frequencies of structures in contact with fluid decrease significantly if the fluid-loading on the structure is very heavy. For somewhat lighter fluid-loading, the structural properties are important, and the in vacuo resonant frequencies are usually needed in determining the response due to an external driving force. This paper exploits a simple empirical model to estimate the effect of fluid-loading on the resonant frequency of a rectangular plate. Simple formulas were developed to rapidly compute the natural frequencies of rectangular plates loaded with different fluids on both sides. The rectangular plates with simply supported or clamped edges were considered in the present analysis. Particularly, a square plate with a general boundary condition varying from simply supported to clamped edges was examined. Empirical formulas were derived to calculate the change in the resonant frequencies for baffled plates in all cases.

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.

Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads

Hydroelastic vibration and buckling analysis of horizontal rectangular plates resting on Pasternak foundation and subjected to linearly varying in-plane loads is investigated for different boundary conditions. Fluid is assumed to be inviscid and incompressible. To analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions. Convergence study is first carried out to confirm the stability of the present method. Then, a few comparisons are made for limited cases between present results and the available data in the literature. Finally, parametric study is conducted to highlight the effects of loading factors, fluid levels, foundation stiffness parameters, aspect ratios, thickness to width ratios and different boundary conditions on the critical buckling loads and wet natural frequencies of rectangular plates.

Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method

In this study Free vibration analysis of vertical rectangular Mindlin plates resting on Pasternak elastic foundation and fully or partially in contact with fluid on their one side is investigated for different combinations of boundary conditions. The plate is assumed to be one of vertical rectangular walls of a container in contact with fluid. In order to analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions in a Cartesian co-ordinate system. The method of separation of variables and the method of Fourier series expansion is used to model fluid and to obtain the exact expression of the motion of fluid in the form of integral equations. The fluid domain is finite in depth and width but infinite in the length direction. To demonstrate the accuracy of the present solution, convergence study is first carried out and then a few comparison studies are carried out with the available data in the literature. Finally, natural frequencies of rectangular plates are presented in tabular and graphical forms for different fluid levels, foundation parameters, aspect ratios, thickness to width ratios and boundary conditions.

Preservation of the fundamental natural frequencies of rectangular plates with mass and spring modifications

Preservation of the fundamental natural frequencies of rectangular plates with mass and spring modifications

BUCKLING AND VIBRATION OF ORTHOTROPIC PLATES WITH AN INTERNAL LINE HINGE

This paper presents the exact buckling loads and vibration frequencies of orthotropic rectangular plates with internal line hinge and having two opposite edges simply supported while the other two edges may have any combination of free, simply supported, and clamped conditions. An analytical method that uses the Lévy solution method and the domain decomposition technique is employed to determine the buckling loads and natural frequencies of rectangular plates with internal line hinge.

The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

This paper introduces the discrete singular convolution algorithm for vibration analysis of rectangular plates with mixed boundary conditions. A unified scheme is proposed for the treatment of simply supported, clamped and transversely supported (with nonuniform elastic rotational restraint) boundary conditions. The robustness and reliability of the present approach are tested by a number of numerical experiments. All results agree well with those in the literature.

On natural boundary conditions and DQM

It is well known that the classical global approximation methods such as the conventional Ritz method and the conventional differential quadrature method (DQM) have some difficulty in determining the natural frequencies of rectangular plates involving free corners. The mixed Ritz-DQM formulation, which has been recently developed by the present author, was also shown to have such difficulty. This is because it is very difficult to implement the free corner boundary condition in these methods. To overcome this difficulty, this paper presents a mixed Ritz-DQM formulation in which the free corner boundary condition is implemented in a simple and easy manner. First, we present a new scheme for implementing multiple boundary conditions in the DQM discretized equations. By the help of this scheme, we then show that the free corner boundary condition can also be easily implemented in the final matrix equations of the Ritz-DQM approach. Finally, we validate the effectiveness of the proposed approach through numerical experiments. Numerical results show the advantages of the proposed method over some versions of the DQM in terms of accuracy and performance.

A COMPARATIVE STUDY OF VIBRATING LOADED PLATES BETWEEN THE RAYLEIGH-RITZ AND EXPERIMENTAL METHODS

Natural frequencies of rectangular plates carrying a concentrated mass are obtained by employing a set of trigonometric beam functions in the Rayleigh-Ritz method. Two models of the energy method using one-term and 100-term model are used in the study. Results obtained from the analytical study using the energy method are compared with those measured experimentally. It is found that the 100-term analytical model can generally predict well the experimental frequencies of a plate carrying an arbitrarily placed concentrated mass, whereas the one-term analytical model is only good for estimating the first-mode frequency of a plate carrying a centrally placed mass.

Natural frequencies of rectangular plates

Natural frequencies of rectangular plates

Effect of transverse shear deformation and rotary inertia on the natural frequencies of rectangular plates with cutouts

A numerical method based on the Rayleigh principle is applied in conjunction with the simplified Reissner-Mindlin theory which involves only a single variable, the transverse displacement, for predicting the fundamental frequencies of rectangular plates with rectangular cutouts. The predicted results are compared with the finite elements results and the reported works available in the literature. It is concluded that the effect of rotary inertia can only be neglected for orthotropic plates with high degrees of orthotropy. For isotropic plates, the effect of rotary inertia becomes more pronounced with respect to the effect of transverse shear deformation with increased cutout size.

Prediction of natural frequencies of rectangular plates with rectangular cutouts

A simple numerical method based on the Rayleigh quotient is presented for predicting the natural frequencies of the fundamental and some higher modes of a rectangular plate with an arbitrarily located rectangular cutout. The edges of the cutout are parallel to the edges of the complete plate. The nodal patterns of these higher modes are pre-selected based on the symmetry of the mode shapes about the geometrical axes. The method is applied to the study of linear vibration of rectangular plates simply supported along one pair of opposite edges with any other boundary conditions at the remaining edges. The predicted results compare favourably with the corresponding results produced by other methods whenever available.

Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates

The simulation of classical edge conditions by finite values of elastic edge restraints is a problem frequently encountered in experimental vibration analysis. The support fixtures utilized in such studies always introduce undesired flexibilities or stiffnesses causing the support conditions to deviate from ideal classical ones. There are, also, some analytical studies in which finite values of flexibilities are used in conjunction with a specific distribution over the edge to represent classically supported edges with point supports. Thus one may encounter in the simulation of classical edge conditions a small but finite flexibility where it is required to be zero, or a large but finite flexibility where it is required to be infinity. The present study attempts to answer the question “How large can a small value of flexibility be and yet represent zero flexibility and how small can a large value be and yet represent infinite flexibility in the simulation of classical boundary conditions?”. The effects of finite values and distributions of the elastic edge restraint on the natural frequencies of rectangular plates are examined and guidelines are provided in terms of their ranges and limits for their use in representing classical edge conditions. Based on an analysis of the errors in the edge forces and moments the values of the flexibility coefficients suitable for stress analysis are also identified.

Vibration analysis of elastically restrained rectangular plates with concentrated mass

A numerical method developed earlier for determining natural frequencies of rectangular plates has been extended to include edges elastically restrained in translation and also to consider the effects of concentrated masses inside the plate. Many of the results are presented for the first time. Comparison of the results with those available showed good agreement.