Rayleigh-Ritz Method

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Shanu Sharma - One of the best experts on this subject based on the ideXlab platform.

Saleh M Hassan - One of the best experts on this subject based on the ideXlab platform.

  • original articlenumerical computation of bcops 1 in two variables for solving the vibration problem of a cf elliptical plate
    Journal of King Saud University - Science, 2010
    Co-Authors: Saleh M Hassan
    Abstract:

    Boundary characteristic orthogonal polynomials in xy-coordinates have been built up over an elliptical domain occupied by a thin elastic plate. Half of the plate boundary is taken clamped while the other half is kept free. Coefficients of these polynomials have been computed once and for all so that an orthogonal polynomial sequence is generated from a set of linearly independent functions satisfying the essential boundary conditions of the problem. Use of this sequence in Rayleigh–Ritz Method for solving the free vibration problem of the plate makes it faster in convergence and leads to a simplified system whose solution is comparatively easier. Three-dimensional solution surfaces and the associated contour lines have been plotted in some selected cases. Comparison have been made with known results whenever available.

  • transverse vibrations of elliptical plate of linearly varying thickness with half of the boundary clamped and the rest simply supported
    International Journal of Mechanical Sciences, 2003
    Co-Authors: Saleh M Hassan, M Makary
    Abstract:

    Abstract The Rayleigh–Ritz Method has been applied to study the problem of transverse vibrations of elliptical plate with half of the boundary clamped and the rest simply supported. The thickness of the plate is varying linearly in the space coordinates. For the same thickness variation, the tabulated results in case of mixed boundary conditions lies between those reported for the two extreme cases when the entire boundary is either clamped or simply supported. Convergence of the results is indicated by increasing the order of approximation. Some results for a circular plate of linearly varying thickness with half of the boundary clamped and the rest simply supported have been obtained as special case. Comparisons have been made with the known results. Three-dimensional mode shapes have been drawn for some selected cases.

  • transverse vibration of a circular plate with arbitrary thickness variation
    International Journal of Mechanical Sciences, 1998
    Co-Authors: Bani Singh, Saleh M Hassan
    Abstract:

    Abstract Rayleigh–Ritz Method has been employed to obtain approximations to frequencies and mode shapes of circular plates with variable thickness. The boundary is either clamped, simply supported or completely free. The main distinguishing feature of the present investigations is that the thickness approximation is done by measuring thickness at a suitable set of sample points and then using interpolation to get the approximating polynomial. Thus, unlike other Methods already available in literature where either linear or quadratic variation of thickness has been examined, here one can have a polynomial of arbitrary degree depending upon the number and locations of the sample points. The results have been tabulated in a large number of cases and three-dimensional mode shapes have been plotted for some selected cases. Comparison has been made with available results. A short tables are also given to depict the rate of convergence with the order of approximation.

  • transverse vibration of triangular plate with arbitrary thickness variation and various boundary conditions
    Journal of Sound and Vibration, 1998
    Co-Authors: Bani Singh, Saleh M Hassan
    Abstract:

    Abstract The Rayleigh–Ritz Method has been employed to obtain the numerical solution of the vibration problem of a triangular plate with arbitrary thickness variation and various boundary conditions at the three edges. The thickness has been approximated by a polynomial in natural co-ordinates which have been used everywhere as they greatly simplify the calculations. Successive approximations have been worked out until the first three frequencies and mode shapes converge to at least three significant figures. The results are tabulated for selected cases and are compared with known results for uniform and linear thickness variation. Three-dimensional mode shapes have been drawn using the tools for computer graphics.

A K Gupta - One of the best experts on this subject based on the ideXlab platform.

Fulei Chu - One of the best experts on this subject based on the ideXlab platform.

  • free vibration analysis of rotating functionally graded cnt reinforced composite cylindrical shells with arbitrary boundary conditions
    Composite Structures, 2019
    Co-Authors: Zhaoye Qin, Xuejia Pang, Babak Safaei, Fulei Chu
    Abstract:

    Abstract In this paper, a general approach is provided for the free vibration analysis of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells with arbitrary boundary conditions. General formulations are derived based on the first-order shear deformation theory, the Donnell-type kinematic assumptions, and the artificial spring technique. Coriolis and centrifugal effects due to rotation are taken into account in the shell model. By employing Chebyshev polynomials as admissible functions, the Rayleigh-Ritz Method is employed to derive the equations of motion for rotating FG-CNTRC cylindrical shells. The approach proposed is validated by comparing the present results with those reported in literature. The traveling wave motions of rotating FG-CNTRC shells are investigated. The effects of geometric parameters, volume fraction of carbon nanotubes, and boundary conditions on shell vibrations are also evaluated.

  • free vibrations of cylindrical shells with arbitrary boundary conditions a comparison study
    International Journal of Mechanical Sciences, 2017
    Co-Authors: Zhaoye Qin, Fulei Chu
    Abstract:

    Abstract In this paper, the free vibration characteristics of cylindrical shells with arbitrary boundary conditions are investigated. The Sanders shell theory is used to calculate the elastic strain energy. Artificial springs are implemented at the ends of the shells to represent the arbitrary boundary conditions. The shell displacements are expanded by three different sets of formulations, namely, the modified Fourier series, the Orthogonal polynomials, and the Chebyshev polynomials. A unified solution for the three different types of expansion functions is developed using the Rayleigh-Ritz Method. The unified solution is validated by comparing with the available results in the literature. The accuracy, convergence rate, and computational efficiency of the three expansion functions are compared. Based on the comparison studies, the Chebyshev polynomials of high computational efficiency are selected to investigate the influence of boundary conditions on the free vibration characteristics of cylindrical shells.

Arun Gupta - One of the best experts on this subject based on the ideXlab platform.

  • effect of thermal gradient on vibration of non homogeneous orthotropic trapezoidal plate of linearly varying thickness
    Ain Shams Engineering Journal, 2013
    Co-Authors: Arun Gupta, Shanu Sharma
    Abstract:

    Abstract In this paper, thermal effect on vibration of non-homogeneous orthotropic trapezoidal plate of linearly varying thickness is studied through a numerical Method. Rayleigh Ritz Method is used to evaluate the fundamental frequencies. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. Frequency corresponding to first two modes of vibration is calculated for the orthotropic trapezoidal plate having C–S–C–S edges for different values of thermal gradient, taper constant and aspect ratio. The proposed Method is applied to solve orthotropic trapezoidal plate of variable thickness with C–S–C–S boundary conditions. Results are shown by figures for different values of thermal gradient, taper constant and aspect ratio for first two modes of vibrations.

  • study of thermally induced vibration of non homogeneous trapezoidal plate with varying thickness and density
    American Journal of Computational and Applied Mathematics, 2013
    Co-Authors: Arun Gupta, Pragati Sharma
    Abstract:

    The goal of present investigation is to study the effect of thermal gradient on the vibrations of non-homogeneous trapezoidal plate whose thickness varies parabolically and density varies linearly. Effect o f other plate parameters such as non-homogeneity constant, taper constant and aspect ratios have also been studied. C-S-C-S boundary condition with two term deflection is taken into consideration. Rayleigh-Ritz Method is used to find the solution of the problem. Results are calculated with great accuracy and presented in tabular form. Co mparison of the results with published data has also shown.

  • study of the effect of the linear temperature be haviour on a non homogeneous trapezoidal plate of parabolically varying thickness
    International Journal of Acoustics and Vibration, 2013
    Co-Authors: Arun Gupta, Pragati Sharma
    Abstract:

    An analysis is presented for studying the effect of the linear temperature behaviour on the transverse vibration of a non-homogeneous trapezoidal plate of varying thickness on the basis of classical plate theory. The nonhomogeneity of the plate material is assumed to arise due to the variation in density, which is assumed to vary parabolically. The thickness of the plate also varies parabolically. A two-term deflection function is performed to solve the equation of motions using the Rayleigh-Ritz Method. The frequency equation is derived when two edges of the plate are simply supported and two are clamped, which is called clamped simply-supported clamped simplysupported. Effects of the non-homogeneity with other plate parameters—such as aspect ratio, taper constant, and thermal gradient on the first two modes of vibration—have been analysed. Results are presented in graphical form. 1

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