Vibration of sector plates on elastic foundations

Journal of Sound and Vibration, 2000

European Journal of Mechanics - A/Solids, 2009

The validity and the range of applicability of the classical plate theory (CPT) and the first-order shear deformation plate theory, also called Mindlin plate theory (MPT), in comparison with three-dimensional (3-D) p-Ritz solution are presented for freely vibrating circular plates on the elastic foundation with different boundary conditions. In order to achieve this purpose, a study of the 3-D elasticity solution is carried out to determine the free vibration frequencies of clamped, simply supported and free circular plates resting on an elastic foundation. The Pasternak model with adding a shear layer to the Winkler model is used for describing the elastic foundation. In addition to being employed the p-Ritz algorithm, the analysis is based on the linear, small strain and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in a cylindrical polar coordinate system is used to arrive eigenvalue equation yielding the natural frequencies for the circular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature and the MPT. Furthermore, the effect of the foundation stiffness parameters, thickness-radius ratio, and different boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the circular plates is investigated. Afterwards, the validity and the range of applicability of the results obtained on the basis of the CPT and MPT for a thin and moderately thick circular plate with different values of the foundation stiffness parameters are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution. Finally, the phenomenon of mode shape switching is investigated in graphical forms for a wide range of the Winkler foundation stiffness parameters.

Journal of Sound and Vibration, 1990

International Journal of Mechanical Sciences, 2014

This work describes a study of vibration characteristics of thin circular plates with elastic edge support and resting on partial Winkler-type elastic foundation. The foundation is described by the Winkler model, which is called as single parameter foundation. The exact analytical method is used to derive the frequency equation of the circular plate with elastic edge support-conditions and resting on partial elastic foundation system. Parametric investigations on the behavior of circular plates with elastic edge support and resting on partial elastic foundation have been carried out with respect to various values of transverse stiffness parameter, foundation parameter for a variety of boundary conditions. Extensive data is tabulated so that pertinent conclusions can be arrived at on the influence of translational edge restraint, and the foundation modulus parameters of the Winkler foundation on the natural frequencies of uniform isotropic circular plates. A comparison of the results obtained here in this paper with those available in the literature shows an excellent agreement. J m1 = J n 1 (kb) J n+1 (kb); J m 2 = J n 2 (kb) + J n+2 (kb); J m 3 = J n 3 (kb) J n+3 (kb) Y n1 = Y n 1 (kb) Y n+1 (kb); Y n2 = Y n 2 (kb) + Y n+2 (kb); Y n 3 = Y n 3 (kb) Y n+3 (kb) I p1 = I n 1 (kb) + I n+1 (kb); I p2 = I n 2 (kb) + I n+2 (kb); I p 3 = I n 3 (kb) + I n+3 (kb) K q1 = K n 1 (kb) + K n+1 (kb); K q2 = K n 2 (kb) + K n+2 (kb); K q 3 = K n 3 (kb) + K n+3 (kb) Case (iii) : Considering the case (iii), for k < , for region (II, n ), the solution to Eq. (2) is given by Eq. (11) and for this case the boundary conditions Eqs. , , and give us the following:

1999

This research deals with the linear elastic behavior of simply supported thick square plates resting on two-parameter elastic foundations using finite difference technique. Mindlin's thick plate theory is modified to include the effect of two-parameter elastic foundations. The elastic foundation is represented by Winkler compressional resistance and elastic membrane (Filonenko-Borodich model) and loaded by uniformly distributed load. The computer program (FDATPEF) (Finite Difference Analysis of Thick Plates on Two-Parameter Elastic Foundations) coded in Fortran_77 for the analysis of thick plates on two-parameter elastic foundations was written Several important parameters are incorporated in the analysis in order to study their effects on deflections and bending moments. The studied parameters are plate thickness, vertical subgrade reaction and membrane force. It was found that for a square thick plate, as the membrane force increases the deflection and the bending moment decreases. The effect of increasing vertica subgrade reaction on the behavior is found to be lesser than the effect of membrane force for the assumed values.

Journal of Sound and Vibration, 1986

An analysis is presented here for the design of engineering systems where composite materials and non-uniform structures are used. A simple model for free vibrations of a non-homogeneous isotropic elastic circular plate of linearly varying thickness resting on an elastic foundation is considered. The governing differential equation of motion is solved by the Frobenius method. The frequencies, deflections and moments corresponding to the first five modes of vibration are computed for clamped and simply supported edge conditions for various values of taper constant, non-homogeneity parameter and foundation modulus.

In this research, free vibration of rectangular functionally graded (FG) plates with in-plane exponentially non-homogeneous material is investigated. Young's modulus and mass density are assumed to vary between a metal-rich and a ceramic-rich zone along one in-plane direction of the plate. The governing differential equation is derived for the case, and a truncated Taylor series expansion technique is utilized to calculate natural frequencies. A Levy-type solution is obtained for plates having two simply supported edges parallel with the material gradient direction. Results for normalized natural frequency are compared with the 4th order Runge-Kutta method, and when possible with exact solution , showing an accurate agreement. Furthermore, a comprehensive parametric study is carried out to determine the effects of different boundary conditions, aspect ratios, and material variations on the free vibration of FGM plates. Nomenclature í µí± 1 , í µí± 2 : non-homogeneity parameters of module of elasticity and density, respectively. C, F, S: Clamped, free, and simply supported edges of the plate, respectively. í µí°·(x), í µí°¸(x), í µí¼Œ(x): flexural rigidity, module of elasticity and density of the material. í µí°· 0 , í µí°¸0 , í µí¼Œ 0 : Reference values of D(x), E(x) and ρ(x) at x = 0, respectively. N: number of terms in the truncated Taylor series expansion. í µí±‹(x); Displacement (shape) function along x. í µí±¥̅ , í µí±¦ ̅, í µí±¤ ̅: Non-dimensional x and y direction, and deflection of the plate along z direction, respectively. í µí¼”, í µí»½: Natural frequency and normalized natural frequency, respectively. í µí±Ž, í µí±, í µí¼†, ℎ: Length along x and y direction, aspect ratio, and thickness of the plate, respectively.

International Journal of Mechanical Sciences, 1990

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature. NOTATION a, b the sides of plate aij coefficient Ci coefficient D flexural rigidity = Eh3/12(1v 2) D O bending and twisting rigidities of orthotropic plate D11 Elh3/12(1-VI2V21) Dr2 v12E2h3/12(1-Y12V21) D22 E2h3/12(1-v12vzl) D66 Gh3/12 D'11 DI~/x/Dl~D22 D'I 2 D12/~fDllD22 D'22 D22/x/DllD22 D'66 D66/N/D11D22 f(~, ~/) generating function G shear modulus of elasticity h thickness of plate m number of terms used in the doubly infinite deflection series Tma x maximum kinetic energy Um,x maximum strain energy W(~, ~/) deflection function x, y Cartesian coordinates ~b(~, ~/) orthogonal plate function x/a r 1 y/b x/~ nondimensional frequency parameter v, v12, v21 Poisson's ratios ~0(¢, r/) edge function co angular frequency side ratio a/b

International Journal of Non-linear Mechanics, 2006

This paper is concerned with a new improved formulation of the theoretical model previously developed by Benamar et al. based on Hamilton's principle and spectral analysis, for the geometrically non-linear vibrations of thin structures. The problem is reduced to a non-linear algebraic system, the solution of which leads to determination of the amplitude-dependent fundamental non-linear mode shapes, the frequency parameters, and the non-linear stress distributions. The cases of C-S-C-S and C-S-S-S rectangular plates are examined, and the results obtained are in a good qualitative and quantitative agreement with the previous available works, based on various methods. In order to obtain explicit analytical solutions for the first non-linear mode shapes of C-S-C-S RP 2 and C-S-S-S RP, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. For beams and fully clamped rectangular plates, has been slightly modified, and adapted to the above cases, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values up to 0.75 and 0.6 for the first non-linear mode shapes of C-S-C-S RP and C-S-S-S RP, respectively.