Vibration Of Rectangular Thin Plates Research Articles

Free transverse vibration analysis of thin rectangular plates locally suspended on elastic beam

Free transverse vibration of thin rectangular plates locally suspended on deformable beam is presented using generalized differential quadrature method. The plate is completely free at all edges except a local region which is attached to a thin beam with rectangular cross section. The other side of the beam is fixed and the whole system is subjected to free transverse vibrations. According to classical plate theory and Euler–Bernoulli beam assumption, two coupled partial differential equations of the system are obtained. The governing equations and solution domain are discretized based on the generalized differential quadrature method and natural frequencies of the plate attached to the beam are obtained. Accuracy of the predictions is investigated using four simplified cases which show reasonably good agreement. For the general case, however, due to lack of data in the literature, predictions are compared with finite element results, which also demonstrate close agreement.

Free Vibration of Rectangular Plates with Attached Discrete Sprung Masses

A direct approach is used to derive the exact solution for the free vibration of thin rectangular plates with discrete sprung masses attached. The plate is simply supported along two opposite edges and elastically supported along the two other edges. The elastic support can represent a range of boundary conditions from free to clamped supports. Considering only the compatibility of the internal forces between the plate and the sprung masses, the equations of the coupled vibration of the plate-spring-mass system are derived. The exact expressions for mode and frequency equations of the coupled vibration of the plate and sprung masses are determined. The solutions converge steadily and monotonically to exact values. The correctness and accuracy of the solutions are demonstrated through comparison with published results. A parametric study is undertaken focusing on the plate with one or two sprung masses. The results can be used as a benchmark for further investigation. The solution provided in the paper is general and includes several special cases, such as the plate with classical boundary conditions, the plate attached with discrete rigid masses, the plate supported by discrete springs and the plate restricted by rigid vertical point-supports.

Free Vibration of Rectangular Plates with Attached Discrete Sprung Masses

A direct approach is used to derive the exact solution for the free vibration of thin rectangular plates with discrete sprung masses attached. The plate is simply supported along two opposite edges and elastically supported along the two other edges. The elastic support can represent a range of boundary conditions from free to clamped supports. Considering only the compatibility of the internal forces between the plate and the sprung masses, the equations of the coupled vibration of the plate-spring-mass system are derived. The exact expressions for mode and frequency equations of the coupled vibration of the plate and sprung masses are determined. The solutions converge steadily and monotonically to exact values. The correctness and accuracy of the solutions are demonstrated through comparison with published results. A parametric study is undertaken focusing on the plate with one or two sprung masses. The results can be used as a benchmark for further investigation.The solution provided in the paper is general and includes several special cases, such as the plate with classical boundary conditions, the plate attached with discrete rigid masses, the plate supported by discrete springs and the plate restricted by rigid vertical point-supports.

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2×2 symmetric operator matrices is proved. The result is applied to 4×4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.

Two modes nonresonant interaction for rectangular plate with geometrical nonlinearity

The vibrations of thin rectangular plate with geometrical nonlinearity are analyzed. The models of plate vibrations with different numbers of degrees-of-freedom are derived. It is deduced that two degrees-of-freedoms are enough to describe low-frequency nonlinear dynamics of plates. Nonlinear normal modes are used to analyze the system dynamics. If vibrations amplitudes are increased, single-mode plate vibrations are transformed into two mode ones. In this case, internal resonance conditions are not observed. Such transformation of vibration is described using Kauderer–Rosenberg nonlinear normal modes.

Stochastic bifurcation of rectangular thin plate vibration system subjected to axial inplane gaussian white noise excitation

A stochastic nonlinear dynamical model is proposed to describe the vibration of rectangular thin plate under axial inplane excitation considering the influence of random environment factors. Firstly, the model is simplified by applying the stochastic averaging method of quasi-nonintegrable Hamilton system. Secondly, the methods of Lyapunov exponent and boundary classification associated with diffusion process are utilized to analyze the stochastic stability of the trivial solution of the system. Thirdly, the stochastic Hopf bifurcation of the vibration model is explored according to the qualitative changes in stationary probability density of system response, showing that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simple way on the potential applications of stochastic stability and bifurcation analysis.

Symplectic eigenvector expansion theorem of a class of operator matrices arising from elasticity theory

This paper deals with the completeness of the eigenvector system of a class of operator matrices arising from elasticity theory, i.e., symplectic eigenvector expansion theorem. Under certain conditions, the symplectic orthogonality of eigenvectors of the operator matrix is demonstrated. Based on this, a necessary and sufficient condition for the completeness of the eigenvector system of the operator matrix is given. Furthermore, the obtained results are tested for the free vibration of rectangular thin plates.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectic eigenfunction expansion method.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simply-supported (S) and clamped (C) boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

STUDY ON THE FLEXURAL VIBRATION OF RECTANGULAR THIN PLATES WITH FREE BOUNDARY CONDITIONS

An analytical method is presented for the flexural vibration of rectangular thin plates with free boundary conditions. Based on the apparent elasticity method, the flexural vibration of a rectangular thin plate is reduced to two one-dimensional flexural vibrations of slender rods. Then the two-dimensional flexural vibration of a rectangular thin plate with free boundary conditions is considered as the coupling of these two equivalent flexural vibrations with different equivalent elastic constants. It should be noted that these two equivalent flexural vibrations are different from the traditional one-dimensional flexural vibrations of slender rods. They are coupled to each other by the introduced mechanical coupling coefficient. The analytical solutions for the isotropic rectangular thin plate in flexural vibration are derived and the resonance frequency equation is obtained. The natural vibrational mode is analyzed and the frequency spectra are calculated. It is found that the normal modes and the natural frequencies of the rectangular thin plate in flexural vibration are abundant. Theoretical analyses show that one-dimensional flexural vibration of a slender rod based on the classical elementary flexural theory, as well as the stripe mode vibration of a rectangular thin plate is a limiting vibrational mode of rectangular thin plates. Experiments show that the measured resonance frequencies are in good agreeement with the calculated results, and the displacement nodal line pattern is also observed experimentally.

GEOMETRICAL NON-LINEAR, STEADY STATE, FORCED, PERIODIC VIBRATION OF PLATES, PART I: MODEL AND CONVERGENCE STUDIES

A model for the steady state, geometrically non-linear, periodic vibration of thin rectangular plates under harmonic external excitation is presented. The equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method (HFEM). These equations are transformed into the frequency domain by the harmonic balance method (HBM) and are solved by a continuation method. The convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates.

GEOMETRICAL NON-LINEAR, STEADY STATE, FORCED, PERIODIC VIBRATION OF PLATES, PART II: STABILITY STUDY AND ANALYSIS OF MULTI-MODAL RESPONSE

The steady state, geometrically non-linear, periodic vibration of rectangular thin plates under harmonic external excitations, is analyzed using the hierarchical finite element and the harmonic balance methods. Modal coupling due to internal resonance is detected and the consequent multi-modal and multi-frequency response is demonstrated. The stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions and using Floquet's theory.

A General Solution For The Free Vibration Of Rectangular Plates With Arbitrarily Distributed Lateral And Rotational Elastic Edge Support

In this paper, the superposition method is exploited to obtain a solution for the free vibration of thin rectangular plates resting on elastic edge supports of arbitrarily distributed stiffnesses. Stiffnesses are represented in series form, thereby permitting the specification of any distributions, including concentrated stiffnesses and stiffnesses with discontinuities. Convergence is found to be very rapid for simple stiffness distributions such as the parabolic distribution utilized in illustrative cases. This appears to be the first comprehensive treatment of this general problem of plates on elastic edge supports.

Free Vibration Analysis of Rectangular Plates with Nonuniform Lateral Elastic Edge Support

The method of superposition is utilized to obtain a solution for the free vibration of thin rectangular plates resting on non-uniform lateral elastic edge supports. The stiffness of the elastic supports may have any desired distribution along the edges, including discontinuities and local concentrations. Convergence is found to be rapid. Graphical results are plotted for square plates in order to verify that proper frequency limits are approached as the edge stiffness approach limits of zero and infinity. Results are tabulated for square and nonsquare plates in order that other researchers will have data against which they can compare their results.

Remarks of some problems for rectangular thin plates with free edges on elastic foundations

For the bending, stability and vibrations of rectangular thin plates with free edges on elastic foundations, in this paper we give a flexural function which exactly satisfies not only all the boundary conditions on free edges but also the conditions at free corner points. Applying energy variation principle, we give equations defining parameters in flexural function, stability equation, frequency equation, and general formulae of minimum critical force and minimum eigenfrequency as well.

A general analytical solution for elastic vibration of rectangular thin plates

A general solution of differential equation for lateral displacement lunction of rectangular elastic thin plates in free vibration is established in this paper. It can be used to solve the vibration problem of rectangular plate with arbitrary boundaries. As an example, the frequency and its vibration mode of a rectangular plate with four edges free have been solved.

The flexural vibration of line supported rectangular plate systems

Orthogonally generated, polynomial functions are used in the Rayleigh-Ritz method to study the problem of the flexural vibration of thin rectangular plates which may be continuous over several supports in one or two directions or which may be arranged to form a box-like structure. It is demonstrated that the method may be used to tackle a variety of such plate problems, the difficulty of application not being significantly greater than that for the study of single-span plates.

Stability and vibration of thin rectangular plates by simplified mixed finite elements

Rectangular finite elements based on a Reissner type variational statement for plate bending are applied to stability and free vibration of rectangular plates. The finite elements are constructed from the weak statements of the two normal moments-displacement relations and the moment equilibrium equation in terms of the two normal moments. These finite elements are algebraically simple and yield better accuracies for the critical loads and natural frequencies when compared to conventional finite elements. Linear and quadratic rectangular finite elements are used to calculate frequencies and buckling loads of rectangular plates with various edge conditions.

Vibrations of thin rectangular plates with arbitrarily oriented stiffeners

Free vibration of plates with arbitrarily oriented stiffeners are studied using high precision plate bending and stiffener elements. Good convergence of frequency values for coarse mesh is demonstrated. Natural frequencies of square plates with various arrangement of stiffeners are determined for both simply supported and clamped boundary conditions.

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.