Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape,part II: Applications and experiments

The eigenfrequencies of ring-shaped polygonal plates, ring-shaped elliptical plates, rectangular plates with elliptical inner boundaries and circular plates with elliptical inner boundaries are obtained. The method developed by the author in the companion paper (Part I) is applied for finding the eigenfrequencies. To verify the present results, experimental tests have also been carried out for some important cases.     

Free vibration of annular sector plates with edges possessing different degrees of rotational restraint

Results for the natural frequencies of annular sector plates possessing different degrees of elastic restraint along the edges are presented. The analysis is based on a numerical method developed by the author. The functions in the circumferential direction satisfying the boundary conditions along the radial edges, which are required in the analysis, are indicated. To the best of the author's knowledge, no previous results exist for such plates.     

Dependence of X-ray diffraction patterns on the parameters of primary incident beam

X-ray diffraction patterns obtained from a plane-parallel crystalline plate with a spherical cavity and from a wedge-shaped crystal have been investigated depending on the collimation of the primary incident beam. It is shown that in the case of ribbon-like incident beam the sectional topograms, obtained from the wedge-shaped crystal, have the form of V-shaped or straight fringes depending on the orientation of reflecting planes relative to the surface of the X-ray ribbon-like beam. In the case of a wide parallel beam or an asymmetrically reflected wide monochromatic beam the parallel fringes are obtained from the wedge-shaped crystals, and the circular fringes are obtained from the plane-parallel crystalline plates with a spherical cavity. For the primary ribbon-like beam the V-shaped patterns acquire the form of circular fringes as a result of scanning the crystal with a spherical cavity and the X-ray film.     

Vibration of edge reinforced annular plates

The free transverse vibration of annular plates reinforced by circular rings along the simply supported outer and free inner boundaries is analyzed. A closed form solution is obtained by applying the appropriate forces, moments and motions of the circular edge-beams as boundary conditions on the differential equation for the free transverse vibration of plates. A study of the resulting natural frequencies compared with previous results for unreinforced edges indicates that edge-beams with even a relatively small stiffness have a significant effect upon the natural frequencies of the system.     

Free vibration of annular sector plates by an integral equation technique

A numerical method is presented for the free vibration analysis of polar orthotropic clamped annular sector plates. The results are compared with analytical and experimental values of other investigators. A parametric study has been done by varying the sector angle and radii ratio. The frequencies for isotropic and orthotropic cases are presented in the form of graphs.     

Receptances of non-proportionally and continuously damped plates-equivalent dampers method

A method for the dynamic analysis of continuously and non-proportionally damped plates is discussed. The method is quite general and suitable for various damping treatments, such as in multilayer plates with damping layers. The transverse vibrations of partially coated plates under harmonic excitation are analyzed by the proposed method. The results of the undamped modal analysis made by classical finite element methods are used in the suggested lumped parameter analysis. The receptance matrices of coated plates have been computed at undamped natural frequencies. The computational results have been verified by comparison with experimental values for partially and fully coated rectangular plates.     

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.     

Free vibration of rectangular plates with edges having different degrees of rotational restraint

A numerical method developed by the author has been used as a basis for determining natural frequencies of rectangular plates possessing different degrees of elastic restraints along the edges. The basic functions satisfying the boundary conditions along two opposite edges for such cases have been derived. Comparison of results with others that are available indicates excellent accuracy. Many new results have been presented.     

FEM Modal Analysis on the Gravity Support System of ITER

The three-dimensional finite element method model with 20 degree sector of the ITER overall gravity support system was built by the ANSYS software. The modal analysis of the gravity support system was made and first ten natural frequencies and vibration modes of the gravity support system were calculated by using Block Lanczos method. The results of modal analysis on ITER represent that the stiffness of flexible plates has influenced greatly for the natural frequency of the system.     

Three-dimensional free vibrations of thick skewed cantilevered plates

The natural frequencies of skewed cantilevered thick plates are determined by using the Ritz method. The present work is the first known three-dimensional study of the problem. Assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly, and which are mathematically complete. Accurate natural frequencies are calculated for skewed thick plates having arbitrary degrees of skewness. Detailed numerical studies reveal interesting trends concerning the variation of frequencies with increasing skew angle. Results obtained by using the present method are compared with those obtained by using three-dimensional finite elements.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

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.     

Axisymmetric vibrations of linearly tapered annular plates under an in-plane force

Analysis and numerical results are presented for the axisymmetric vibrations of circular annular plates with linear variation in thickness under the action of a hydrostatic in-plane force on the basis of the classical theory of plates. The governing differential equation with variable coefficients has been solved by Chebyshev collocation technique. The effect of inplane force on the natural frequencies of vibration has been investigated for two different boundary conditions and for different radii ratio and taper constant. Transverse displacements, moments and the critical buckling loads in compression with thickness variation have also been computed for the first two modes.     

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.     

Natural frequencies and vibration modes of laminated composite plates reinforced with arbitrary curvilinear fiber shape paths

The development of tow-placement technology has made it possible to control fiber tows individually and place fibers in curvilinear distinct paths in each layer of a laminated plate. This paper presents an analytical method for determining natural frequencies and vibration modes of laminated plates having such curvilinear reinforcing fibers. Spline functions are employed to represent arbitrarily shaped fibers, and Ritz solutions are used to derive frequency equations using series type shape functions. The strain energy is evaluated by numerical integration involving the fiber orientation angle, and is calculated using the derivative of the spline function in minute intervals. The results show that the natural frequencies obtained by the present method agree well with results from finite element analyses. The vibration mode shape contour plots of the plates are seen to reflect clear influences of the fiber shapes.     

Dynamic response of composite plates with cut-outs,part I: Simply-supported plates

The effect of square cut-outs on the natural frequencies of square, simply-supported composite plates is investigated. The forced and free dynamic response of plates with cut-outs is formulated. Laminations are assumed to be symmetric about the mid-plane and the plates are considered analytically as homogeneous anisotropic plates. In the method of solution it is assumed that the effect of the cut-out is equivalent to an external loading on the plate. For free vibration, the method leads to an infinite system of frequency equations. Depending upon the accuracy required, a suitable size of the system of frequency equations is selected. Results are given for square, simply-supported composite plates with centrally located square cut-outs for different modulus ratios. A comparison of results obtained from this method for isotropic plates with cut-outs with available literature is made and excellent agreement is obtained.     

Refined vibration and damping analysis of multilayered rectangular plates

Refined vibration and damping analysis of a general multilayered rectangular plate consisting of an arbitrary number of layers of orthotropic materials has been developed by considering extension, bending, in-plane shear and transverse shear deformations in all the layers and taking into account the rotary and longitudinal translatory inertias along with the transverse inertia of the plate. The solution for a multilayered plate with simply supported edges has been taken in series summation form and resonating frequencies and associated loss factors for plates with alternate elastic and viscoelastic layers have been evaluated by application of the correspondence principle of linear viscoelasticity. Results for three-, five- and seven-layered plates obtained by the present refined analysis are compared with the results obtained by conventional analysis of multilayered plates.     

Interferometry that combines the measurement and testing of the wedging of transparent plates

Several variants and schemes are proposed for implementation of the interference method that makes it possible to combine the measurement and testing of parameters of phase objects in the measurement testing of transparent wedge-shaped plates. The proposed variants are employed in the measurement testing of wedge-shaped glass plates.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.