Free Vibration Of Rectangular Plates Research Articles

A deviatoric couple stress Mindlin plate model and its degeneration

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.

Free vibrations of rectangular plates

The paper investigates free vibrations of rectangular metal plates. The finite element method and analytical calculation are particularly used to determine the vibration frequency. The analytical calculation is based on the equation of motion of a thin rectangular plate. The asymptotic method is applied to determine the dynamic edge effect. As a result, the free vibration frequency is determined for the rectangular metal plate. The finite element analysis is performed in Lira and SolidWorks software packages. For this, a solid plate model with sensors is created to measure the free vibration frequency; the effective mass participation factor was determined.The plate vibration tests were conducted to confirm the results of analytical calculations. The method of smooth sinusoidal vibrations is used. Spectrum graphs of the plate vibrations are suggested based on the vibration acceleration of sensors. Errors are identified in the free vibration frequencies depending on the applied method. The paper does not consider frequencies with the low effective mass participation factor.

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.

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Free vibrations of rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges. An analytical and a semi-analytical method

The present study provides a novelty in treating rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges using two approaches. On one hand, the analytical solution of the plate vibration equation satisfying the rotational and translational spring boundary conditions is obtained for different plate aspect ratios leading to the corresponding modes and frequencies. On the other hand, the Rayleigh-Ritz method (RRM) is applied using plate functions defined as products of beam functions satisfying the beam appropriate end conditions in each direction. i.e. simply supported in the direction of the two opposite edges, and elastic end conditions in the other direction. The results obtained by the two methods coincide for all frequencies and modes examined, showing the effectiveness of the RRM applied to this problem in a general and systematic way. This makes the RRM ready for use in further studies concerning similar plates with added point masses or springs, or plate vibrating in the nonlinear regime. Also, useful numerical results corresponding to both the frequencies and mode shapes are presented for various plate modes, stiffness values and plate aspect ratios.

A modified Fourier-Ritz method for free vibration of rectangular plates with elastic constrains

1. Abdulkerim S., Dafnis A., Riemerdes H., 2019, Experimental investigation of nonlinear vibration of a thin rectangular plate, International Journal of Applied Mechanics, 11, 267-289. CrossRef Google Scholar

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

Nonlinear free vibration of rectangular plates reinforced with 3D graphene foam: Approximate analytical solution

This paper investigates nonlinear free vibration of three-dimensional graphene foam skeleton reinforced rectangular plates. The three-dimensional graphene foam skeleton can distribute in different patterns along the thickness direction of the plate. According to the mixing rule, the effective Poisson’s ratio, mass density and elastic modulus of three-dimensional graphene foam reinforced (3D-GFR) plates are described. In the framework of the von Kármán nonlinear plate theory, Hamilton’s principle is utilized to derive equations of motion. Then, analytical nonlinear frequencies of 3D-GFR plates are solved by using the Galerkin method and the harmonic balance method. Results show that 3D-GFR plates exhibit hardening nonlinearity. The effect of porosity coefficient on nonlinear vibration depends on the foam skeleton distribution. In addition, the nonlinear frequencies of 3D-GFR plates increase as the skeleton weight fraction rises.

Analytical approach to investigation of free vibration of thin rectangular plate immersed in fluid, resting on Winkler and Pasternak foundations

Dynamic behaviour of free vibration of rectangular plate is investigated. This study considered an analytical approach to investigation of free vibration of thin rectangular plate immersed in fluid, resting on Winkler and Pasternak foundations.The governing nonlinear partial differential equation is analyzed using two-dimensional differential transform method. The accuracy of the analytical solutions obtained is verified with existing results in literature and confirmed in excellent agreement. Thereafter, the analytical solutions are used for investigation of effect of elastic foundation, fluid and aspect ratio on vibrating plate.From the result, it is observed that, increase elastic foundation parameters increases natural frequency, increase aspect ratio increase natural frequency, influence of fluid decreases the natural frequency of the plate.Hopefully the present study will contribute to existing knowledge in field of vibration.

Buckling and free vibrations of rectangular plates with symmetrically varying mechanical properties – Analytical and FEM studies

The subject of the paper is a rectangular plate with symmetrically varying mechanical properties in the thickness direction. The nonlinear hypothesis of deformation of the straight line normal to the plate neutral surface is assumed. The field of displacements of the plate is formulated assuming this hypothesis. Based on the Hamilton’s principle three differential equations of motion are obtained. The system of equations is analytically solved. The critical loads and fundamental natural frequencies for exemplary plates are derived. Moreover, the FEM model of the plate in the ABAQUS system is developed, and analogical calculations for exemplary plates are carried out. The calculation results of these two methods are compared.

Free Vibration of Rectangular Plates with Porosity Distributions under Complex Boundary Constraints

Free vibration of rectangular plates with three kinds of porosity distributions and different boundary constraints has been performed by means of a semianalytical method. The distribution of porous varies along the thickness of the plate, in which the mechanical properties are defined by open‐cell metal foam. Regardless of boundary conditions, displacement admissible functions are represented by combination of standard cosine Fourier series and auxiliary sine series. The kinetic energy and potential energy of plates are also expressed on the basis of first‐order shear deformation theory (FSDT) and displacement admissible functions. Finally, the coefficients in the Fourier series which determine natural frequencies and modal shape are derived by means of the Rayleigh–Ritz method. Convergence and dependability of the current method are verified by comparing with the results of FEM and related literatures. In addition, some new results considering geometry parameters under classical and elastic boundary constraints are listed. The effects of geometry parameters, material parameters, and boundary constraints have been discussed in detail.

Чисельний аналіз вільних коливань прямокутних пластин на основі різних підходів

Чисельний аналіз вільних коливань прямокутних пластин на основі різних підходів

A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates.

A generalized solution procedure is developed for in-plane free vibration of rectangular and annular sectorial plates with general boundary conditions. For the annular sectorial plate, the introduction of a logarithmic radial variable simplifies the basic theory and the expression of the total energy. The coordinates, geometric parameters and potential energy for the two different shapes are organized in a unified framework such that a generalized solving procedure becomes feasible. By using the improved Fourier–Ritz approach, the admissible functions are formulated in trigonometric form, which allows the explicit assembly of global mass and stiffness matrices for both rectangular and annular sectorial plates, thereby making the method computationally effective, especially when analysing annular sectorial plates. Moreover, the improved Fourier expansion eliminates the potential discontinuity of the original normal and tangential displacement functions and their derivatives in the entire domain, and accelerates the convergence. The generalized Fourier–Ritz approach for both shapes has the characteristics of generality, accuracy and efficiency. These features are demonstrated via a few numerical examples.

Free Vibration of Embedded Porous Plate Using Third-Order Shear Deformation and Poroelasticity Theories

This research aims at studying free vibration of rectangular plate made of porous materials in whichY-foam,G-foam, andCoustoneare used and compared with each other. To obtain the Biot formulation of the constitutive equations for a porous material, linear poroelasticity theory is used. Young modulus and density of porous plate are different in transverse direction versus porosity. In order to increase the accuracy of results in comparison with classical plate and first-order shear deformation theories, Reddy’s theory was utilized in this research. Besides, five coupled equations of motion have been studied using Hamilton’s principle and are solved by differential quadrature method (DQM). Detailed results of this study show the significant effect of aspect ratio, thickness ratio, boundary conditions, and porosity on dimensionless frequency and deflection of porous plate. Results of this study can contribute to the design of pneumatic conveying, handling, and control systems.

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.

Study on the Use of Boundary Characteristic Orthogonal Polynomials in Determining the Fundamental Frequency of Rectangular Plates with Bidirectional Linear Thickness Variation

Free vibration of rectangular plates with linearly varying thickness is considered. Plates are important structural components used in aircrafts, bridges etc. and hence for their safe design, thorough vibration analysis is important. Many research and practical applications use plates of variable thickness due to economical usage of the material and increased strength. Vibration response of these types of plates is different from plates of uniform thickness, which makes their analysis critical. In this study, Boundary Characteristic Orthogonal Polynomials (BCOPs) in one and two dimensions have been used to obtain deflection shape function. The first member of the series is generated using the boundary conditions, in this case all edges clamped, which satisfies both the geometric boundary conditions and the natural boundary conditions. Gram Schmidt Orthogonalization for polynomials is used to generate the higher members of the shape function which only satisfy the geometric boundary conditions. Thickness variation considered for the plate is linear in both x and y direction. Natural frequencies were obtained by using Rayleigh-Ritz method. Natural frequencies were calculated by varying taper parameters for both directions and compared with those obtained with the case of uniform thickness. Natural frequencies were also found comparable with those obtained from Finite Element Analysis by using ANSYS.

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.

A New First-Order Shear Deformation Theory for Free Vibrations of Rectangular Plate

A new first-order shear deformation theory (FSDT) with pure bending deflection and shearing deflection as two independent variables is presented in this paper for free vibrations of rectangular plate. In this two-variable theory, the shearing deflection is regarded as the only fundamental variable by which the total deflection and bending deflection can be expressed explicitly. In contrast with the conventional three-variable first-order shear plate theory, present variationally consistent theory derived by using Hamiltonian variational principle can uniquely define the bending and the shearing deflections, and give two rotations by the differentiations of bending deflection. Due to more restrictive geometrical constraints on rotations and boundary conditions, the obtained natural frequencies are equal to or higher than those by conventional FSDT for the rectangular plate with at least one pair of opposite edges simply supported. This new theory is of considerable significance in theoretical sense for giving a simple two-variable FSDT which is variational consistent and involve rotary inertia and shear deformation. The relation and differences of present theory with conventional FSDT and other relative formulations are discussed in detail.

Large Amplitude Free Vibrations of Plates Resting on a Pasternak Type Elastic Foundation

In this paper following Berger's approximate plate theory for large deflections, large amplitude free vibrations of Rectangular Plate, isosceles right-angled triangular plate, Equilateral triangular plate and circular plate resting on a Pasternak-type elastic foundation have been discussed.