Vibration Analysis Of Rectangular Plates Research Articles

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method, the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method (ICCM). The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate. The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method. From the continuity condition of the vibration displacement function at the cutout, the transition matrix between the two coordinate systems is constructed, and the mass and stiffness matrices are completely obtained. As a result, the calculation is simplified and the computational efficiency of the solution is improved. In this paper, numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods, and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

A discrete differential geometry-based approach to buckling and vibration analyses of inhomogeneous Reddy plates

In this paper, a novel discrete differential geometry-based numerical procedure for buckling and vibration analyses of rectangular plates with non-uniform thickness is developed. In the proposed approach a plate is discretized using a finite number of rigid bars, lumped masses, and elastic rotational springs to simulate both bending and shear deformation responses, allowing the analysis of thick plates through the adoption of the Reddy’s third-order shear deformable plate theory. An interesting analogy between the proposed model and the central finite difference method for solving a set of partial differential equations is also highlighted, showing how the former can be seen as the physical model behind the mathematical representation of the latter. The numerical results presented show both the versatility and the accuracy of the proposed approach.

Free vibration analysis of rectangular plates with arbitrary elastic boundary conditions

boundary conditions are In this paper, an improved Fourier series method is presented for the free vibration analysis of rectangular plates with arbitrary elastic conditions. The stiffness value of the restraining springs is determined as required to simulate the arbitrary elastic boundary conditions. The exact solution of plates with arbitrary elastic boundary conditions is solved by the introduced supplementary func-tions. The matrix eigenvalue equation of plates is derived by using boundary conditions and the governing equations. Compared with exist methods, the presented method can be easily applied to most of plate vibration problems with different boundary conditions. To validate the accuracy of the presented method, numerical simulations with different boundary conditions are presented.presented.

Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

The paper is continuation of our efforts on application of the properly constructed sets of exponential functions as the trial (basic) functions in weighted residuals method, WRM, on example of classical tasks of structural mechanics. The purpose of this paper is justification of new method’s efficiency as opposed to getting new results. So, static deformation and free vibration of isotropic thin – walled plate are considered here. Another peculiarity of paper is choice of weight (test) functions, where three options are investigated: it is the same as trial one (Galerkin method); it is taken as results of application of differential operator to trial function (least square method); it equals to the second derivative of trial function with respect to both x and y coordinate (moment method). Solution is considered as product of two independent sets of functions with respect to x or y coordinates. Each set is the combination of five consequent exponential functions, where coefficient at first function is equal to one, and four other coefficients are to satisfy two boundary conditions at each opposite boundary. The only arbitrary value in this method is the scaling factor at exponents, the reasonable range of which was carefully investigated and was shown to have a negligible impact on results.
 Static deformation was investigated on example of simple supported plate when outer loading is either symmetrical and concentrated near the center or is shifted to any corner point. It was demonstrated that results converge to correct solution much quickly than in classical Navier method, while moment method seems to be a best choice. Then method was applied to free vibration analysis, and again the accuracy of results on frequencies and mode shape were excellent even at small number of terms. At last the vibration of relatively complicated case of clamped – clamped plate was analyzed and very encouraged results as to efficiency and accuracy were achieved.

Performance of similitude methods for structural vibration analyses of rectangular plates

Engineers often use similitude analyses to design small scale models for experimental tests or to design size ranges of mechanical structures such as drive technology systems. This paper is concerned with similitude analysis methods for vibration analyses of rectangular plates. If their geometry is scaled by different factors (distorted similitude), the scaling laws approximate the actual vibration responses with a certain accuracy only. This paper introduces a performance measure that reliably assesses how well the scaling laws approximate the actual vibration responses of rectangular plates. This measure, the so-called Mahalanobis distance, applies in a-posteriori analyses, where the vibration responses obtained from the scaling laws are compared to the actual ones. Numerical and experimental investigations on vibrating rectangular plates validate that the Mahalanobis distance is suitable to assess the performance of similitude analyses. The Mahalanobis distance can be linked to the geometrical properties of the rectangular plates in order to define a maximum permissible distortion of the geometry. Scaling laws approximate the vibration responses of the rectangular plates sufficiently well up to this maximum permissible distortion. Furthermore, the performance of two different state-of-the-art similitude analysis methods is compared. Both similitude analysis methods are found to perform well up to the maximum permissible amount of geometrical distortion.

Fractional Calculus Approach to Nonlocal Three-Dimensional Vibration Analysis of Plates

The nonlocal three-dimensional vibration analysis of rectangular plates is investigated within the framework of fractional calculus in the sense of the Caputo fractional derivative. To show the effect of the fractional derivative on nonlocality, the frequency spectra of the plates with different boundary conditions and symmetry modes are carried out for the different orders of the fractional derivative and different values of the length scale parameter . The vibration analysis is obtained by the Ritz energy method, whereas Chebyshev polynomials are used as admissible functions. The results of the frequency spectrum demonstrate that the nonlocal effect decreases and the results get closer to the values of the classical frequencies as the order of the fractional derivative approaches the classical derivative .

A generalized superposition method for accurate free vibration analysis of rectangular plates and assemblies.

A generalized superposition method is presented in this paper for free bending vibration analysis of single rectangular plates and assemblies of rectangular plates with arbitrary boundary conditions. The method was developed on the basis of the Levy method, the principle of superposition, and the uniform convergence of half-range Fourier cosine series for continuous functions in a closed interval. Numerical results, obtained using the proposed method for thin isotropic plates and plate assemblies, show that the proposed method is accurate and rapidly convergent. The proposed method can be extended to handle vibration of moderately thick plates made of isotropic and orthotropic materials.

Extension of Hencky bar-net model for vibration analysis of rectangular plates with rectangular cutouts

Comprising rigid bars and spring systems at the joints and in the cells, the Hencky bar-net model (HBM) has been shown to be a physical structural representation of the finite difference plate model (FDM). In this paper, the HBM is extended for the vibration analysis of rectangular plates with rectangular cutouts. This extension addresses the rotationally elastic and transverse spring stiffnesses for the HBM at the cutout corners. After verifying the HBM model by comparing the vibration results with existing solutions for some plate problems, the model is used to obtain some new free vibration solutions for plates having various boundary conditions including cracked corners modelled by free small rectangular cutouts.

Hencky bar-net model for buckling and vibration analyses of rectangular plates with non-uniform thickness

The Hencky bar-net model, developed from a finite difference plate model, is proposed for buckling and vibration analyses of rectangular plates with non-uniform thickness. The Hencky bar net comprises rigid bars joined by elastic rotational springs to allow for bending flexibility and diagonal springs in each panel to simulate the twisting effect. The non-uniform plate thickness can be readily handled by adjusting the spring stiffnesses. For generality, the edges of the plate are elastically restrained where the special cases are the classical boundary conditions of simply supported edge, clamped edge and free edge. The plate is subjected to uniaxial load or biaxial loads. In order to demonstrate the versatility and the accuracy of the Hencky bar model, some plate problems are analysed and the obtained results compared with existing results reported in the open literature.

Hencky Bar-Net Model for Vibration of Rectangular Plates with Mixed Boundary Conditions and Point Supports

This paper is concerned with the development of the Hencky bar-net model (HBM) for free vibration analyses of rectangular plates with mixed boundary conditions and point supports. The HBM is a two-dimensional discrete net system composed of rigid segments connected by frictionless hinges and rotational springs. In the model, bending is accommodated by rotational springs at each joint while the twisting by a diagonal spring system in each grid cell. The total mass of the plate is distributed as lumped mass at each joint and the continuous boundary stiffness of plate is simulated by springs located at the edge joints. Owing to the discrete property of HBM, it is able to readily handle any boundary conditions of plates including mixed boundary conditions and point supports. The HBM is herein used to solve some vibration problems of rectangular plates with mixed boundary conditions and point supports to demonstrate its accuracy and convenience for plate analyses.

Vibration analysis of rectangular plates resting on four rigid supports

PurposeThis paper aims to thoroughly investigate the free vibration characteristics of rectangular plates resting on symmetrically distributed four rigid supports by using a finite element (FE) method.Design/methodology/approachANSYS parametric design language was used to generate the FE models and to run the analysis. The FE models were initially validated and were then used to solve for the plate first natural frequency and mode shape. The effect of the plate aspect ratio and support location on the free vibration properties of the plate was thoroughly studied by conducting several FE runs. Finally, simple empirical formulas were developed to conveniently calculate the plate first natural frequency based on the geometric parameters and support type, as well as support locations. Those well-formulated equations were in a great match with the FE data.FindingsThe results showed that the plate first natural frequency and mode shape are highly affected by the plate size and support locations. Specifically, the natural frequency deceases as the plates becomes larger. Also, the bending behavior of the first mode is highly affected by the support location, which results in a significant change in the natural frequency of the plate structure. In addition, the presently formulated empirical formulas are faithfully able to predict the natural frequency of the plate based on the geometric parameters and support location.Originality/valueThis paper provides numerous new data on the vibration properties of the rectangular plate resting on rigid supports. Also, it provides a simple way to easily calculate the natural frequency of this plate problem, unlike the very limited and complicated analytical solutions available in literature.

Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: An analytical approach

In this paper, the free vibration analysis of rectangular plates composed of functionally graded materials with porosities is investigated based on a simple first-order shear deformation plate theory. The network of pores in assumed to be empty or filled by low pressure air and the material properties of the plate varies through the thickness. Using Hamilton's principle and utilizing the variational method, the governing equations of motion of FG plates with porosities are derived. Considering two boundary layer functions, the governing equations of the system are rewritten and decoupled. Finally, two decoupled equations are solved analytically for Lévy-type boundary conditions so as to obtain the eigenfrequencies of the plate. The effects of porosity parameter, power law index, thickness-side ratio, aspect ratio, porosity distribution and boundary conditions on natural frequencies of the plate are investigated in detail.

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.

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.

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.

Free vibration analysis of rectangular plates with central cutout

AbstractA nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Semi-analytical modeling of cutouts in rectangular plates with variable thickness – Free vibration analysis

This paper presents a new semi-analytical method for modeling rectangular plates with variable thickness and cutouts. The plate thickness is represented as a finite sum of multiplications of one-dimensional functions. The plate deflections are also assumed in the similar separable form and the variational extended Kantorovich method is applied. In order to enhance the accuracy of the solution, a multi-term formulation of the extended Kantorovich method is developed. It is shown that this representation is very general and it allows the description of a complex variation of the thickness including step thickness changes and cutouts. It is demonstrated that this approach avoids singularities at the cutout areas and it does not require assembly of predefined trial functions or computational domains satisfying plate geometry and boundary conditions. The presented method is applied for the free vibration analysis of rectangular plates with various rectangular cutouts and variable thickness. The accuracy and convergence of the solution is studied through comparisons with other semi-analytical methods (where applicable) and the results of finite element analysis.

Free and forced in-plane vibration of rectangular plates with non-uniform elastic boundary conditions

In this paper, the free and forced in-plane vibration analysis of rectangular plates are performed for the first time using an improved Fourier series method, in which the boundary restraining spring stiffness can vary in any functional pattern along each edge. Two-dimensional improved Fourier series displacement forms are constructed with four supplementary polynomials introduced into the standard 2-D Fourier series to make the field functions sufficiently smooth in the whole solving domain. Energy formulations are employed to describe the in-plane dynamics of plate system, in which the in-plane concentrated point force is taken into account in the form of work term. All the unknown Fourier series coefficients are then solved through the Rayleigh-Ritz procedure. Several numerical examples are given to demonstrate the correctness and effectiveness of the proposed model through the comparison with those calculated via finite element analysis (FEA). The results show that these two results can agree very well with each other for various non-uniform boundary conditions. Based on the established model, the in-plane vibration response is also studied. Some curves and contours are obtained to illustrate how the boundary restraining stiffnesses affect the in-plane point and transfer mobility of rectangular plate structure.

A new finite element formulation for vibration analysis of thick plates

A new procedure for determining properties of thick plate finite elements, based on the modified Mindlin theory for moderately thick plate, is presented. Bending deflection is used as a potential function for the definition of total (bending and shear) deflection and angles of cross-section rotations. As a result of the introduced interdependence among displacements, the shear locking problem, present and solved in known finite element formulations, is avoided. Natural vibration analysis of rectangular plate, utilizing the proposed four-node quadrilateral finite element, shows higher accuracy than the sophisticated finite elements incorporated in some commercial software. In addition, the relation between thick and thin finite element properties is established, and compared with those in relevant literature.

Forced vibration analysis of arbitrarily constrained rectangular plates and stiffened panels using the assumed mode method

This paper deals with numerical procedure for the vibration analysis of rectangular plates and stiffened panels subjected to point excitation force and enforced displacement at boundaries. The procedure is based on the assumed mode method, where natural response is determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. Mode superposition method is applied to calculate plate/stiffened panel frequency response. The Mindlin plate theory is adopted for a plate, while the effect of stiffeners having the properties of Timoshenko beams is taken into account by adding their potential and kinetic energies to the corresponding plate energies. The accuracy of the proposed procedure is justified by several numerical examples which include forced vibration analysis of plates and stiffened panels with different dimensions and framing sizes and orientations, having various combinations of boundary conditions. The results obtained by the developed in-house code are compared to those obtained by the finite element method (FEM) and experimental results from the relevant literature. The presented procedure is confirmed to be highly accurate.