Vibration Of Rectangular Thin Plates Research Articles

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

Selected Problems of Random Free Vibrations of Rectangular Thin Plates with Viscoelastic Dampers.

The main motivation of this work was to present a semi-analytical extension of the correspondence principle in stochastic dynamics. It is demonstrated for the stochastic structural free vibrations of Kirchhoff–Love elastic, isotropic and rectangular plates supported by viscoelastic generalized Maxwell dampers. The ambient temperature of the plate affects the dampers only and is included in a mathematical model using the frequency–temperature correspondence principle. The free vibration problem of the plate–viscoelastic damper system is solved using the continuation method and also the Finite Element Method (FEM). The stochastic approach begins with an initial deterministic sensitivity analysis to detect the most influential parameters and numerical FEM recovery of the polynomial representation for lower eigenfrequencies versus these parameters. A final symbolic integration leads to the first four basic probabilistic characteristics, all delivered as functions of the input uncertainties.

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

Vibration of a rectangular plate using finite continuous Radon transform

In this study, the finite continuous Radon transform the vibrations of rectangular thin plates is demonstrated. Finite continuous Radon transform using graphical representation GNU Octave, version 5.2.0 and solution for different functions are also studied. In order to check the validity of formulation, the numerical results are established using finite continuous Radon transform. In the concluding section the comparative study with approximations is demonstrated.

Analysis of the free vibration of thin rectangular plates with cut-outs using the discrete singular convolution method

Abstract In this paper, free vibration of thin rectangular plates with cut-outs for different boundary conditions is analyzed by using the discrete singular convolution (DSC) method. First, the entire plate configuration is divided into appropriate sub-domains. The edges involved with cut-outs in each sub-domain are treated as free boundary conditions and the Taylor series expansion method is adopted for solving. Then, the convergence of the DSC method is tested and the frequency parameters were validated by comparing with the exact solutions by Leissa. At last, the influence of rectangular and U-shaped cut-outs structure parameters on the natural frequency is studied.

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.

The overall assessment of closed-form solution methods for free vibrations of rectangular thin plates

In this work, a new separation-of-variable (nSOV) method for the free vibration of Kirchhoff rectangular thin plates is developed based on the Rayleigh quotient variational principle. Using this nSOV method, all closed-form solutions can be obtained as long as not both directions include free edge, and the solution accuracy is improved compared with the separation-of-variable (SOV) method proposed previously by the present authors. In addition, as a natural extension of the nSOV method, an iterative SOV (iSOV) method is developed, which is suitable for rectangular thin plates with any kinds of homogeneous boundary conditions. Moreover, all closed-form solutions are presented in explicit forms, and the proposed methods and results are validated via theoretical and numerical comparisons.

Eigen vector buckling based boundary analysis method for free vibration of rectangular thin plates

The main research topic of this paper is the application of the finite element method boundary analysis theory to study the stability of plate in irregular shape. First of all, the basic theory of free vibration of elastic plate and the basic equations of elastodynamics are introduced, and the governing equation of small deflection of free vibration of plate is given. Secondly, the finite element method boundary analysis method is introduced in details, including the basic idea and analysis steps of finite element method boundary analysis, and some element types, so as to lay a theoretical foundation for better understanding of finite element method boundary analysis method and application of it to analyze the stability of plate. At last, finite element method boundary analysis is applied to analyze the stability of plate, among which, the structural analysis types in ANSYS software and its analysis steps are introduced, and buckling (eigenvalues) analysis in ANSYS software is mainly introduced, and the effective performance of algorithm is verified by applying ANSYS software.

A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates

The free flexural vibration of thin rectangular plates is revisited. A new, quasi‐exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam‐like expressions. Using the proposed quasi‐exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB®, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS® and in‐house conventional FEM programs using both 12‐ and 16‐DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi‐exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two‐dimensional structural components during the preliminary design stages, before a detailed design begins.

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.

Damping for large-amplitude vibrations of plates and curved panels, Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.

Application of a Weak Form Quadrature Element Method to Nonlinear Free Vibrations of Thin Rectangular Plates

Geometrically nonlinear free vibrations of thin rectangular plates are studied using the recently developed weak form quadrature element method (QEM). The nonlinear von Karman plate theory is employed to express the strain-displacement relations. The weak form description of the plate is formulated on the basis of the variational principle. The integrals involved in the variational description are evaluated by an efficient numerical integration scheme, and the partial derivatives at the integration points are approximated by differential quadrature analogs. A system of algebraic equations is eventually derived, and the nonlinear frequencies and mode shapes are extracted from solving the equations. The efficiency of the method is demonstrated by a convergence study. The accuracy of the method is illustrated by comparing the computed nonlinear to linear frequency ratios with those available in the literature. The influences of the nonlinearity on higher order frequencies and mode shapes are exhibited as well.

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Analytic free vibration solutions of rectangular thin plates with two adjacent corners point-supported are obtained for the first time by using an up-to-date symplectic superposition method, which demonstrates the ability to yield solutions via step-by-step derivations from the governing equations, without predetermining any forms of solutions. The essential difference between the current method and the other ones is that the solution procedure is conducted in the symplectic space rather than in the classical Euclidean space. This extends the scope of application of eigen expansion and paves the way for more analytic solutions of similar problems. Comprehensive highly accurate numerical results are presented to show the validity of the current solutions and to provide the benchmarks for future comparison and verification of other methods.

A symplectic analytical wave based method for the wave propagation and steady state forced vibration of rectangular thin plates

A semi-analytical method is used to investigate the wave propagation characteristics and steady state forced vibration response for rectangular thin plates. By way of a rigorous but simple derivation, the governing differential equations for transverse vibration of rectangular thin plates are first converted into Hamiltonian canonical equations. Following the method of separation of variables, a symplectic eigenproblem is formed whose solution gives analytically the dispersion equation and the wave mode shape. Using the wave modes, i.e. the wave propagation parameters and wave shapes, and combining the directly excited waves, the wave propagation within the plate and the wave reflection at the boundary, the forced response of the plate can be computed in the wave domain with high precision and high efficiency. The present method is based on the basic elasticity equations of the plate, and can give the symplectic analytical solutions for the wave modes for any combination of simple boundary conditions along the plate edges. The present method eliminates the limitation of the traditional analytical wave propagation method which can only obtain wave modes for plates with two opposite edges simply supported. In contrast to numerical wave propagation methods, the present method provides symplectic analytical solutions for the wave modes, and hence the computations are of high precision and well conditioned. Also, continuously distributed external forces can be easily considered. In the numerical examples, the wave propagation characteristics are analyzed for plates with three different combinations of boundary conditions, i.e. with two opposite edges either both simply supported, or both clamped, or one simply supported and the other clamped. The steady state forced responses are also computed for plates excited by point forces, as well as for line and area distributed forces, for the three combinations of boundary conditions. Comparison of the present results with analytical results, wave finite element and finite element results validates the high accuracy and high efficiency of the present method.

Physically and geometrically non-linear vibrations of thin rectangular plates

Static deflection as well as free and forced large-amplitude vibrations of thin rectangular rubber plates under uniformly distributed pressure are investigated. Both physical, through a neo-Hookean constitutive law to describe the non-linear elastic deformation of the material, and geometrical non-linearities are accounted for. The deflections of a thin initially flat plate are described by the von Kármán non-linear plate theory. A method for building a local model, which approximates the plate behavior around a deformed configuration, is proposed. This local model takes the form of a system of ordinary differential equations with quadratic and cubic non-linearities. The corresponding results are compared to the exact solution and are found to be accurate. Two models reflecting both physical and geometrical non-linearities and geometrical non-linearities only are compared. It is found that the sensitivity of the deflection to the physically induced non-linearities at moderate strains is significant.

Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

Nonlinear vibrations of thin rectangular plates are considered, using the von Karman equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.

Effect of boundary condition nonlinearities on free large-amplitude vibrations of rectangular plates

The effect of boundary condition nonlinearities on free nonlinear vibrations of thin rectangular plates is analyzed. The method for analysis of the plate vibrations with geometrical nonlinearity and the boundary condition nonlinearity is suggested. The nonlinear boundary conditions for membrane forces are transformed into linear ones using the in-plane stress function. Additional boundary conditions for the in-plane displacements vanishing on the clamped edge of the plate are imposed on the stress function. Simply supported and cantilever plates are analyzed. The backbone curves obtained by satisfying linear and nonlinear boundary conditions are compared. It is shown that the results of the calculations with nonlinear boundary conditions differ essentially from the data obtained without these boundary conditions.

Transverse Vibrations of Nonhomogeneous Rectangular Plates with Variable Thickness

The effect of nonhomogeneity on the free transverse vibration of thin rectangular plates of linearly varying thickness along one direction has been studied using two-dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. The first three frequencies have been computed for four different combinations of clamped, simply supported, and free edges. The effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on frequencies is illustrated. Mode shapes for specified plates are given.

Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

A new finite element for transverse vibration of rectangular thin plates under a moving mass

In this paper a new finite element which can be used in the analysis of transverse vibrations of the plates under a moving point mass is presented. In this technique, which allows for the inclusion of inertial effects of the moving mass, the load is replaced with an equivalent finite element. By means of using the relations between nodal forces and nodal deflections of 16 DOF conforming plate element with C(1) continuity, on the one hand, and shape functions, on the other hand, mass, stiffness, and damping matrices of the new finite element are determined by the transverse inertia force, Coriolis force and centrifuge force, respectively. This method was first applied on a simply supported beam so as to provide a comparison with the previous studies in the literature, and it was proved that the results were within acceptable limits. Second, it was applied on a cantilevered plate so as to determine the dynamic response of the planer entry plate of a high-speed wood-cutting machine.