Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

Analytic Solution of Interaction Between Surface Waves and Floating Elastic Plates with Elastic End-Restraints

Based on the dynamic theories of water waves and Mindlin plates, the analytic solution of interaction between smface waves and two-dimensional floating elastic plates with edge-restraint is constructed by use of the Wiener-Hopf technique. Firstly, without regard for elastic edge restraint, the wave-induced responses of elastic floating plate analyzed by the present method are in good agreement with the results from literature and experimental results. Therefore, it can be shown that the present method is valid. Secondly, three end-restraint cases （i.e., the left-end elastic restraints, the both-end elastic restraints, and the right-end elastic restraints） are proposed to reduce the vibration of floating plates, in which the spring is used to connect the sea bottom and the floating plate＇ s left （ or right ） edge. The relations between the spring stiffness and the parameters of wave-induced responses of floating plates are discussed. Moreover, the effective method to reduce the vibration of floating elastic plates can be obtained through comparison.

The wave and vibratory power transmission in a finite L-shaped Mindlin plate with two simply supported opposite edges

In this paper,wave and vibratory power transmission in a finite L-shaped Mindlin plate with two simply supported opposite edges are investigated using the wave approach.The dynamic responses,active and reactive power flow in the finite plate are calculated by the Mindlin plate theory （MPT） and classic plate theory （CPT）.To satisfy the boundary conditions and continuous conditions at the coupled junction of the finite L-shaped plate,the near-field and far-field waves are entirely contained in the wave approach.The in-plane longitudinal and shear waves are also considered.The results indicate that the vibratory power flow based on the MPT is different from that based on the CPT not only at high frequencies but also at low and medium frequencies.The influence of the plate thickness on the vibrational power flow is investigated.From the results it is seen that the shear and rotary inertia correction of the MPT can influence the active and reactive power at the junction of the L-shaped plate not only at high frequencies but also at low and medium frequencies.Furthermore,the effects of structural damping on the active and reactive power flow at the junction are also analyzed.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.

FLEXURAL WAVE PROPAGATION IN NARROW MINDLIN'S PLATE

Appling Mindlin＇s theory of thick plates and Hamilton system to propagation of elastic waves under free boundary condition, a solution of the problem was given. Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method. It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin. Based on the two kinds of theories, the dispersion curves show great difference in the region of short waves, and the cutoff frequencies are higher in Hamiltonian systems. However, the dispersion curves are almost the same in the region of long waves.

Multiresolution Finite Element Method Based on a New Locking-Free Rectangular Mindlin Plate Element

A locking-free rectangular Mindlin plate element with a new multi-resolution analysis (MRA) is proposed and a new finite element method is hence presented. The MRA framework is formulated out of a mutually nesting displacement subspace sequence whose basis functions are constructed of scaling and shifting on the element domain of basic full node shape function. The basic full node shape function is constructed by extending the split node shape function of a traditional Mindlin plate element to other three quadrants around the coordinate zero point. As a result, a new rational MRA concept together with the resolution level (RL) is constituted for the element. The traditional 4-node rectangular Mindlin plate element and method is a mono-resolution one and also a special case of the proposed element and method. The meshing for the monoresolution plate element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the rigorous mathematical basis. The analysis clarity of a plate structure is actually determined by the RL, not by the mesh. Thus, the accuracy of a plate structural analysis is replaced by the clarity, the irrational MRA by the rational and the mesh model by the RL that is the discretized model by the integrated.

Sympletic eigen-solution for clamped Mindlin plate bending problem

Hamiltonian system for the problem on clamped Mindlin plate bending was established by introducing the dual variables for the generalized displacements in this letter. By separation of variables, the transverse eigen-problem was derived based on the sympletic geometry method. With the solved sympletic eigen-values, the generalized sympletic eigen-solution was derived through eigenfunction expansion. An example of plate with all edges clamped was given. The sympletic solution system was worked out directly from the Hamiltonian system. It breaks the limitation of traditional analytic methods which need to select basis functions in advance. The results indicate that the sympletic solution method could find its more extensive applications.

Vibration reduction of floating elastic plates in surface waves

Based on dynamical theories of water waves and dynamics of Mindlin thick plates, the investigation of the wave-induced responses and the vibration reduction of an elastic floating plate are presented using the Wiener-Hopf technique. Without regard to the case of elastic connector, the calculated results obtained by the present method are in good agreement with those from the literature and the experiment. It can be shown that the present method is valid. Relations between the spring stiffness to be used to connect the sea bottom and the floating plate and the parameters of wave-induced responses of floating plates are investigated using the present method. Therefore, these results can be used as theoretical bases for the design stage of super floating platform systems.

A NEW HYBRID QUADRILATERAL FINITE ELEMENT FOR MINDLIN PLATE

In this paper a new quadrilateral plate element concerning the effect of transverse shear strain has been presented. It is derived from the hybrid finite element model based on the principles of virtual work. The outstanding advantage of this element is to use more rational trial functions of the displacements. For this reason, every variety of plate deformation can be simulated really whilc the least degrees of freedom is employed.A wide range of numerical tests was conducted and the results illustrate that this element has a very wide application scope to the thickness of plates and satisfactory accuracy can be obtained by coarse mesh for all kinds of examples.

Non-harmonic resonance of viscoelastic structures subjected to time-dependent exponentially decreasing transverse distributed loads

In this paper,the non-harmonic resonance of Bernoulli viscoelastic beams,Kirchhoff viscoelastic plates,Timoshenko viscoelastic beams,and Mindlin viscoelastic plates subjected to time-dependent exponentially decreasing transverse distributed load is investigated for the first time.The constitutive equations are expressed utilizing Boltzmann integral law with a constant bulk modulus.The displacement vector is approximated by employing the separation of variables method.The Laplace transformation is used to transfer equations from the time domain to the Laplace domain and vice versa.The novel point of the proposed method is to express,prove and calculate the critical time in which the displacement will be several times the displacement at time zero.In addition,this new method calculates the maximum deflection at the critical time,explicitly and exactly,without any need to follow the time-displacement curve with a low computational cost.Additionally,the proposed method introduces the critical range of time so that the responses are greater than the responses at time zero.

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines,which reveals an interesting coarse mesh superconvergence.Firstly,the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived,where the degeneration to Timoshenko beams is discussed as well.Subsequently,in accordance with these frequency error measures,the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration.On the other hand,the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy.Meanwhile,it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates,in comparison with the ultimate fourth order accuracy as the meshes are progressively refined.Furthermore,the mesh size threshold for the coarse mesh superconvergence is provided as well.The proposed theoretical results are consistently proved by numerical experiments.

Analytical modeling on the vibration response of a beam-stiffened Mindlin thick plate with free boundary conditions

This paper presents a new analytical solution for the vibration response of a beamstiffened Mindlin plate having a completely free boundary condition by utilizing a finite cosine integral transform.In the solution,the unknown coupling force and moments at the beam/plate interface and the unknown modal constants from the integral transform are determined by the continuity and compatibility conditions at the interface as well as the boundary conditions.It provides an easily implemented tool for exploring complex edge value problems for a class of higher-order partial differential equations represented by fully free‐stiffened Mindlin thick plates.The validity of the model is evaluated by comparing the calculated free and forced vibration responses of the beam‐stiffened plate with those calculated using a beamstiffened thin plate and those from finite element analysis.

ANALYSIS AND EXPERIMENT ON TRANSIENT DYNAMIC RESPONSE IN FINITE MINDLIN PLATE

The transient wave propagation in the finite rectangular Mindlin plate is investi- gated by the analytical and experimental methods. The generalized ray method （GRM） which has been successfully applied to study the transient responses of beams, planar trusses, space frames and infinite layered media is extended to investigate the transient wave propagation and early short time transient response in finite Mindlin plate. Combining the wave solution, the shock source and the boundary conditions, the ray groups transmitted in the finite rectangular plate can be determined. Numerical simulations and experiments are performed and compared with each other. The results show that the transient wave propagation and early short time transient responses in the finite plate can be studied using the GRM. The early short time transient acceler- ations are very large for the finite plate subjected to the unit impulse, while the early short time transient displacements are very small. The early short time transient accelerations under the unit impulse are much larger than those under the unit step impulse. The thickness and material characteristics have remarkable effects on the early short time transient responses.

Free Vibration Analysis of Sectorial Plates Using the Triangular Differential Quadrature Method

The triangular differential quadrature method was used to analyze the free vibrations of moderately thick sectorial plates. A triangular serendipity transformation was introduced to map the sectorial domain onto a unit isosceles right triangle. The first six non-dimensional frequencies of the sectorial plates were obtained for various combinations of clamped and simply supported boundary conditions. For sectorial plates with simply supported radial edges, the present results agree well with the available exact solutions and finite element solutions, demonstrating the effectiveness of the method.

Bandgap Analysis of Periodic Composite Microplates with Curvature-Based Flexoelectricity:A Finite Element Approach

A finite element model is proposed permitting prediction of elastic wave bandgaps of periodic composite microplates incorporating flexoelectric effect.In this model,we applied curvature-based flexoelectricity and Mindlin plate theories and derived a finite element formulation that has been implemented for bandgap analysis.The finite element model utilizes a three-node triangle element with 30 degrees of freedom satisfying Mindlin kinematics assumptions.It is based on a non-conforming interpolation scheme which provides nodal C^(1) continuity and ensures compatibility with curvature-based flexoelectricity.The approach accounts for microstructure effects and,owing to the triangular element topology,can be used to assist the design of microplates with complex microstructures.Validation of the approach is performed through comparison with both analytical and numerical models,in which the effect of flexoelectricity on the bandgap is studied based on cases demonstrating size dependence.

Construction and Application of Multivariable Finite Element for Flat Shell Analysis

Based on B-spline wavelet on the interval （BSWI） and the multivariable generalized variational principle, the multivariable wavelet finite element for flat shell is constructed by combining the elastic plate element and the Mindlin plate element together. First, the elastic plate element formulation is derived from the generalized potential energy function. Due to its excellent numerical approximation property, BSWI is used as the interpolation function to separate the solving field variables. Second, the multivariable wavelet Mindlin plate element is deduced and constructed according to the multivariable generalized variational principle and BSWI. Third, by following the displacement compatibility requirement and the coordinate transformation method, the multivariable wavelet finite element for fiat shell is constructed. The novel advantage of the constructed element is that the solving precision and efficiency can be improved because the generalized displacement field variables and stress field variables are interpolated and solved independently. Finally, several numerical examples including bending and vibration analyses are given to verify the constructed element and method.