Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory

This paper presents exact solutions for vibration of rectangular plates with an internal line hinge. The rectangular plate is simply supported on two parallel edges and the remaining two edges may take any combination of support conditions. The line hinge is perpendicular to the two simply supported parallel edges. The Lévy type solution method and the state-space technique are employed in connection with the first order shear deformation plate theory (FSDT) to study natural vibration of rectangular plates with an internal line hinge. In particular, exact vibration frequencies are obtained for rectangular plates of different aspect ratios and edge support conditions. The influence of the internal line hinge on the vibration behavior of rectangular plates is studied.     

Convergence studies on static and dynamic analyses of plates by using the U-transformation and the finite difference method

This paper presents the exact static and dynamic analyses of simply supported rectangular plates. The analytical solutions for displacements, buckling loads, natural frequencies and dynamic responses are obtained by using the double U-transformation method and the finite difference method. After an equivalent system with cyclic periodicity in two directions is established, the difference governing equation for such an equivalent system can be uncoupled by applying the double U-transformation. Then the exact analytical finite difference solutions, the exact error expressions and the exact convergence rates are derived. These results cannot be obtained if other methods are used instead.     

Stability and vibration of isotropic,orthotropic and laminated plates according to a higher-order shear deformation theory

A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.     

The symplectic eigenfunction expansion theorem and its application to the plate bending equation

This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.     

A comprehensive approach to the free vibration analysis of rectangular plates by use of the method of superposition

The authors have found the above techniques to constitute a powerful means for solving rectangular plate problems. At the time of writing, solutions for plates with two adjacent simply supported edges and two adjacent free edges have been obtained. The first 20 eigen-values for plates with all edges clamped have also been determined for a full range of aspect ratio and they are shown to be accurate to within less than one half of one percent. It will be appreciated that solutions for any combination of clamped-simply supported edge conditions can easily be obtained from the all-clamped solution by simply deleting appropriate solutions from the all-clamped combination. In Figure 2 contour lines for first mode vibration of a plate with two adjacent clamped and two adjacent simply supported edges is presented. The higher density of the contour lines along the simply supported edges will be noted.The method of superposition is currently being used by the authors to good advantage to obtain solutions of any desired degree of accuracy to all of the problems discussed. It is found to be easily utilized and unlike more complicated methods is readily comprehensible to the practicing engineer. Eigenvalues for all modes, aspect ratios, and boundary conditions are readily obtained. Modal shapes are expressed in terms of familiar analytic functions. Results of these studies will be made available in future publications.     

变厚度弹性圆薄板问题

一.引言 变厚度的圆薄板常常在机器零件的设计中遇到,例如蒸汽涡输机和活塞的膜片便是。 用平板小挠度理论求解变厚度平板问题并不是容易的事,因此到目前为止,这类的实际问题已解决的为数并不多。     

Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order

This paper presents exact solutions for free vibration of rectangular cross-ply laminated plates with at least one pair of opposite edges simply supported using refined kinematic theories of variable order. Exact natural frequencies are obtained using an efficient and unified formulation where the solving set of second-order differential equations of motion and related boundary conditions are expressed at layer level in terms of so-called fundamental nuclei having invariant properties with respect to the order of the plate theory. The nuclei are then appropriately expanded according to the number of layers and the order of the theory and the resulting equations are transformed into a first-order model whose solution is obtained by using the state space concept. In this way, the mathematical effort needed to derive analytical solutions is highly reduced. Both higher-order equivalent single-layer and layer-wise theories are considered in this study. Comparisons with other exact solutions are presented and useful benchmark frequency results for symmetric and un-symmetric cross-ply laminates are provided.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Vibration and buckling of rectangular plates under in-plane hydrostatic loading

Numerical solutions are presented for the fundamental natural frequency and mode shape of a rectangular plate loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and load magnitudes. All six possible combinations of simply supported and clamped edges are considered. The limiting conditions of unloaded vibration and buckling are discussed in detail, with emphasis on the preferred mode shape. Design curves and approximate formulae are presented which provide a simple means of determining the fundamental frequency parameter.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

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.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

The dynamic characteristics and stability of the moving thermoelastic coupling rectangular plate subjected to uniformly distributed tangential follower force are investigated. Based on the heat conduction equation containing the thermoelastic coupling term and the thin plate theory, the thermoelastic coupling differential equation of motion of the rectangular plate under the action of uniformly distributed tangential follower force is established. Dimensionless complex frequencies of the moving thermoelastic coupling rectangular plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor and dimensionless moving speed on the stability and critical load of the moving plate are analyzed. The results show that the divergence loads of the first order mode increase with the increase of the dimensionless thermoelastic coupling factor, and decrease with increasing the dimensionless moving speed.     

POWER FLOW ANALYSIS OF REINFORCED BEAM-PLATE COUPLED STRUCTURES

A power flow analysis method was developed to predict the vibrational responses of reinforced beam-plate coupled structures in frequencies ranging from medium to high. This analysis method was successfully applied to simply supported rectangular plates reinforced with multi-parallel beams by utilizing the power flow coupling relationships at the beam-plate junctions and the zero intensity conditions at the plate edges. Through numerical simulations, the power flow energy density and intensity fields of two different plates, consisting of single beam and eight beams, were compared with those of classical displacement solutions, and they show good agreement in terms of the global decay and the attenuation patterns of the energy density.     

A reduction method for problems of vibration of orthotropic plates

It is shown that the problem of vibration of an orthotropic plate can be reduced to that of another orthotropic plate by a simple co-ordinate transformation, and reduction formulae are obtained. To justify the reduction formulae, fundamental natural frequencies of orthotropic rectangular plates with various boundary conditions and of a clamped orthotropic elliptical plate are discussed. As an example, an exact natural frequency of a simply supported generally orthotropic skew plate with special flexural rigidities is obtained from that of a simply supported isotropic rectangular plate.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.     

Free vibration of super elliptical plates with constant and variable thickness by Ritz method

In this study free vibration of simply supported and clamped super elliptical plates is investigated. This class of plates includes a wide range of external boundaries varying from an ellipse to a rectangle. Although studies on the upper and lower bounds of these plate geometries, namely circle and rectangle, are quite extensive, contributions on the mid-shapes, especially for simply supported boundary edges are very limited. The Kirchhoff plate model with isotropic and homogeneous material is studied. The super elliptical powers are chosen from 1 to 10. The Ritz method is employed for the solution of the energy equations of the plates. The effects of Poisson's ratio, which should not be neglected for simply supported plates with curved boundaries, and the aspect ratio of the plate are both examined in detail. The effect of thickness variation is also considered in this study. In order to avoid long computational run times, physically pertinent trial functions are utilized. The frequency parameters obtained are presented and compared with published results for plate shapes that match the current cases.     

A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness

A new variable kinematic Ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented. Carrera's unified formulation and the versatile pb-2 Ritz method are properly combined to build a powerful yet simple modeling and solution framework. The proposed technique allows to generate arbitrarily accurate Ritz solutions from a large variety of refined two-dimensional plate theories by expanding so-called Ritz fundamental nuclei of the plate mass and stiffness matrices. Theoretical development of the present methodology is described in detail. Convergence and accuracy of the method are examined through several examples on thin, moderately thick, and very thick plates of rectangular, skew, trapezoidal and general quadrilateral shapes, with an arbitrary combination of clamped, free and simply supported edges. Present results are compared with existing three-dimensional solutions from open literature. Maximum and average differences of various higher-order plate theories and three-dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters.     

Vibrations of heated plates with two opposite edges simply supported

Vibration analysis of the family of rectangular plates with two opposite sides simply supported can be simplified by assuming mode shapes. In the present paper a vibration analysis of such plates which are heated so as to have a temperature varying in the direction parallel to these sides is presented. A steady state temperature which satisfies the Laplace equation is considered. Due to the assumption of mode shapes the governing plate differential equation, which in general is a function of the x and y co-ordinates, becomes a function of one co-ordinate. This equation is analyzed by a finite difference method and solved by a standard simultaneous iteration technique. The accuracy of the method is ascertained by comparing the results for some well known boundary conditions when there is no temperature effect with the standard solutions available in the literature. From the results an attempt has been made to correlate the natural frequency with the temperature. Plates of uniform thickness with different length to breadth ratios have been analyzed. The assumed linear temperature distribution satisfies the Laplace equation and the plate is free to expand in its plane at its edges so that no thermal stresses will be induced.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.