Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Vibration of skew plates by using B-spline functions

This paper deals with the free vibration of skew plates by the Rayleigh-Ritz method with B-spline functions as co-ordinate functions. Convergence of the solutions is investigated in a few typical cases and is found to be satisfactory. The accuracy of the results is compared with the existing results based on other numerical methods and found to be in good agreement.     

Vibration of skew plates resting on point supports

This paper deals with the vibrations of skew plates resting on point supports. The spline element method is used. To demonstrate the accuracy of the method, several examples are solved, and results are compared with those obtained by other approximate methods. The effects of skew angles and locations of point supports on natural frequencies of isotropic skew plates are also investigated.     

Vibration of thick plates carrying concentrated masses

The method for obtaining the natural frequencies and orthogonality relation for combined dynamical systems in which the Green functions of the vibrating subsystems are used is applied to a thick plate carrying concentrated masses. The effects of transverse shear and rotary inertia of each mass is accounted for. It is demonstrated that as the plate thickness goes to zero the results of thin plate analysis are obtained. The Green functions for both thin and thick vibrating plates are derived by modal analysis in the form of infinite series. The advantages and disadvantages of this representation are discussed. An example involving a simply supported isotropic square plate carrying a single concentrated mass at its center is provided.     

Vibration of simply supported-clamped skew plates at large amplitudest

The large amplitude, free, flexural vibration of orthotropic skew plates simply supported along two opposite edges and clamped along the other two are investigated on the basis of an assumed mode shape. The relationship between the amplitude and period is studied for both isotropic and orthotropic skew plates for various aspect ratios and skew angles under two in-plane edge conditions. It is found that the modal equation reduces to the Dufling type equation from which the period of non-linear vibration is found to decrease with increasing amplitude, exhibiting hardening type of non-linearity. The validity of the Berger approximation is investigated for the problem under consideration and this approximation is shown to give reasonably good results.     

Vibration analysis of plates of arbitrary shape—A new approach

The so-called finite strip method combined with the deflection contour method has proved highly successful in the analysis of bending of thin elastic plates of arbitrary shape. Here the same technique is used to obtain the fundamental frequency of plates of arbitrary shape. The method of approach is much simpler than the conventional finite element method since it requires less programming effort and a reduction in both memory space and time on the computer. Several representative plate problems of irregular boundaries are treated by the proposed method. For all cases, comparison of the results are made with other known solutions and the agreement appears to be excellent.     

Vibration analysis of annular-like plates

The existence of eccentricity of the central hole for an annular plate results in a significant change in the natural frequencies and mode shapes of the structure. In this paper, the vibration analysis of annular-like plates is presented based on numerical and experimental approaches. Using the finite element analysis code Nastran, the effects of the eccentricity, hole size and boundary condition on vibration modes are investigated systematically through both global and local analyses. The results show that analyses for perfect symmetric conditions can still roughly predict the mode shapes of “recessive” modes of the plate with a slightly eccentric hole. They will, however, lead to erroneous results for “dominant” modes. In addition, the residual displacement mode shape is verified as an effective parameter for identifying damage occurring in plate-like structures. Experimental modal analysis on a clamped-free annular-like plate is performed, and the results obtained reveal good agreement with those obtained by numerical analysis. This study provides guidance on modal analysis, vibration measurement and damage detection of plate-like structures.     

A variational approach to free vibration analysis of shear deformable polygonal plates with variable thickness

This paper presents a simple and general variational approach for the study of the free vibration behaviour of polygonal isotropic plates with variable thickness. The Reissner-Mindlin plate theory is used to take into account the effects of shear deformation and rotary inertia in the analysis. Moreover, this theory allows obtaining greater accuracy of frequency coefficients corresponding to vibration higher modes, even for the thin plates.The governing eigenvalue equation is obtained employing the Ritz method. The plate geometry is approximated by using non-orthogonal triangular co-ordinates, while sets of independent polynomials, expressed in these co-ordinates, are employed to approximate the displacement and rotation fields. The developed algorithm allows obtaining approximated analytical solutions for plates with different aspect ratios, thickness variation and boundary conditions, including edges elastically restrained by both translational and rotational springs. Therefore, a unified program has been easily implemented. Convergence and comparison analyzes are carried out to verify the reliability and accuracy of the numerical solutions. Finally, sets of parametric studies are performed and the results are given in graphical and tabular form.     

Free vibration analysis of moderately thick asymmetric piezoelectric adaptive cantilever beams using the distributed transfer function approach

A semi-analytical distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with a single surface-bonded piezoelectric patch. The asymmetric piezoelectric adaptive structure is decomposed into three segments; the first and third segments are bare beam parts before and after the patch, while the second segment contains the beam part with attached piezoelectric patch bonded to its upper surface. The theoretical formulation assumes first-order shear deformation kinematics and linear electric potential through the patch thickness with an electrode equipotential physical condition, and uses the extended Hamilton?s principle to derive the equations of motion and electromechanical boundary conditions. The latter, together with the continuity and equilibrium conditions at the segments interfaces, are then transformed into a first-order state space equation that is solved using the DTF approach. The electrodes of the piezoelectric patch are considered either in short-circuit (SC) or open-circuit (OC); this leads to two free-vibration problems to be solved for the corresponding SC and OC frequencies, from which the Electro-Mechanical Coupling Coefficient (EMCC) is post-treated. Four benchmarks from the open literature are simulated in order to validate the proposed approach. Very satisfactory correlations are obtained for all examples with maximum errors less thank 5 percent in all results. For future reference, an additional benchmark is proposed to assess the influence of the patch-to-composite host width ratio on the effective modal EMCC. It was found that the latter is mode-dependent (as expected) and decreases with increasing the former.     

Free vibration and buckling analysis of clamped skew sandwich plates by the Galerkin method

Galerkin's variational method has been used in the past by several investigators [1–3] to solve bending problems of clamped skew plates. In this paper the suitability of the Galerkin method for solution of problems of buckling under the action of in-plane forces and of free vibration of skew plates is studied. The method is first applied to investigate the problems for clamped rectangular sandwich plates. After the validity of the method has been established, the method is then extended to analyze similar problems for clamped skew sandwich plates. The governing differential equations for the skew sandwich plates are obtained by transforming the corresponding differential equations in Cartesian coordinates into skew co-ordinates. The parameters considered herein for the buckling and free vibration behaviour of the skew sandwich plates are the aspect ratio of the plate, Poisson's ratio, skew angle and various shearing stiffnesses of the core. Simplicity and quick convergence is the advantage of the method in comparison with other much more laborious numerical methods requiring extensive computer facilities.     

金属－压电陶瓷中厚度层合环板的动力分析

本文分析了电策动下,由半径不同的压电陶瓷和金属构成的中厚度层合环板的振动问题.在考虑中厚度层合环板变形特征的基础上,得到了其动力学基本方程.利用半解析单元法,得到了中厚度层合环板在各种边界条件下的振型和固有频率.此外本文还讨论了环板的振动转化为紧贴其上的转子的连续转动的现象.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.     

Vibration analysis of composite annular plates by an integral equation technique

In layered plates, coupling between bending and stretching makes the analysis complicated. A numerical method, namely an integral equation technique which has been used for solving static and dynamic problems, is extended to vibration analysis of layered annular plates which are fixed at the inner and outer radii. The results obtained are compared with those of other investigators and a parametric study is made of the effect of layer thickness, the layers being made up of two different materials.     

具有固支边的强厚度层合板的一种新解法

在文[1~3]的基础上,通过引入边界位移函数,对具有固支边的强厚度层合板建立了在任意荷载作用下的状态方程,给出静力问题的解析解。算例表明,该法具有收敛快,边界应力连续等特点。     

A sector finite element for dynamic analysis of thick plates

A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissner's thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.     

Dynamic analysis of circular and sector thick,layered plates

The finite element method is extended to the free vibration analysis of laminated thick plates with curved boundaries. Two elements are developed on the basis of Mindlin's thick plate theory in which the effects of thickness-shear deformation and rotary inertia are included. Both elements are derived in polar co-ordinates and can be joined together to handle annular as well as circular laminated anisotropic plate problems. Since axisymmetry has not been assumed, variations in material properties in the tangential direction can be dealt with. Numerical results are presented to demonstrate the influence of geometrical shape as well as that of thickness-shear deformation on the free vibrations of both homogeneous and layered plates. Comparisons between the numerical results obtained and those presented by other investigators confirm the accuracy of the new elements. The elements also can be used in the analysis of rectangular plates by assuming very large radii and very small subtended angle values.