Variational-difference method in the problem of the vibrations of laminated plates

Piskunov, V. G.; Fedorenko, Yu. M.; Stepanova, A. E.

doi:10.1007/bf00847068

Int Appl Mech

1992

DOI: 10.1007/bf00847068

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Variational-difference method in the problem of the vibrations of laminated plates

V. G. Piskunov¹,

Yu. M. Fedorenko²,

A. E. Stepanova³

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Cited by 3 publications

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“…There is no reason to prioritize some coordinate direction for approximation and it is natural to consider both variables of the domain as equivalent in this regard. In the methods of reduction to algebraic equations, this can be implemented in finite-difference and finite-element approaches, the Ritz-Galerkin projective and variational methods [13,16]. If we are to adhere to the reduction to one-dimensional problems, then the use may be made of the Kantorovich-Vlasov method generalized to both variables of the domain.…”

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Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

Bespalova

2008

Int Appl Mech

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A generalized Kantorovich-Vlasov method is used to solve stationary problems for shallow shells with rectangular planform and arbitrary boundary conditions. The efficiency of the approach is illustrated by examples Introduction. Whether a theoretical study of the elastic deformation of thin-walled structures is effective is greatly dependent on the method chosen to solve relevant two-dimensional problems.For shells of revolution with an arbitrarily shaped meridian, it is natural to use classical Fourier trigonometric series followed by the numerical solution of one-dimensional problems. Such an approach automatically accounts for the periodicity of the solution in the circumferential direction and reduces the problem to a system of ordinary differential equations for each harmonic of the series. According to [5], it is also expedient to reduce problems for cylindrical shells with arbitrary cross-section to one-dimensional ones. With certain boundary conditions, it is possible to set up systems of approximating functions that would identically satisfy these conditions and to approximately reduce the two-dimensional problem to a coupled (or uncoupled) system of ordinary differential equations. For example, a trigonometric-series solution is known for the case of hinged boundary conditions. For clamped boundary conditions, the Filonenko-Borodich cosine binomial, Horvay polynomials, spline functions etc. may be used as approximating functions. Such a dimension reduction method based on the Kantorovich-Vlasov integral or collocation approaches combined with the numerical solution of one-dimensional problems is successfully developed by Grigorenko and his disciples [3,[6][7][8][9][19][20][21].For the above-mentioned classes of shells, a rational direction for approximation is obvious. It is convenient to choose a variable such that boundary conditions for it can be satisfied identically and the geometrical parameters are relatively simple (constant or zero curvature). The situation is different for shallow shells with rectangular plan, doubly variable thickness and curvature, and arbitrary boundary conditions. There is no reason to prioritize some coordinate direction for approximation and it is natural to consider both variables of the domain as equivalent in this regard. In the methods of reduction to algebraic equations, this can be implemented in finite-difference and finite-element approaches, the Ritz-Galerkin projective and variational methods [13,16]. If we are to adhere to the reduction to one-dimensional problems, then the use may be made of the Kantorovich-Vlasov method generalized to both variables of the domain. Such a generalization was used in [4,10,12,18,24] in the two-dimensional case and in [2,14] for higher dimensions. The present paper generalizes the Kantorovich-Vlasov method to stationary problems for shallow shells.

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Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

Bespalova

2008

Int Appl Mech

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Determining the natural frequencies of highly inhomogeneous shells of revolution with transverse strain

Bespalova

Urusova

2007

Int Appl Mech

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A method of studying the natural vibrations of highly inhomogeneous shells of revolution is developed. The method is based on a nonclassical theory of shells that allows for transverse shear and reduction. By separating variables, the two-dimensional problem is reduced to a sequence of one-dimensional eigenvalue problems. The inverse iteration method is used to reduce these problems to a sequence of inhomogeneous boundary-value problems solved by the orthogonal sweep method. The capabilities of the method are illustrated by solving certain representative problems and comparing their solutions with those obtained using the three-dimensional theory of elasticity, the classical theory of shells, and the refined Timoshenko model Keywords: highly inhomogeneous shells of revolution, natural frequencies, transverse shear, reduction, solution technique, inverse iteration method, three-dimensional theoryIntroduction. General analysis of the frequency spectrum of elastic structures, as any linear system, has two interrelated subtasks: qualitative analysis of the behavior of this spectrum and quantitative dynamic analysis of a specific class of structures [1-3, 5, 7, 9, 10, 11, 16]. The present paper is concerned with the latter task and naturally continues the developments intended to determine the natural frequencies and modes of the elastic objects mentioned in [3]. Most studies in this area are based on two shell models: classical Kirchhoff-Love model and refined Timoshenko model. These models made it possible to accurately calculate the lowest frequencies for a wide class of thin and medium-thickness shells made of shearable materials. However, these models may fail to ensure required accuracy for shells highly inhomogeneous across the thickness, such as laminated shells with layers whose mechanical properties differ by more than an order of magnitude. In this connection, this paper proposes a method to calculate the dynamic characteristics of shells based on a nonclassical model that allows for all transverse strains. The capabilities of the method will be demonstrated by solving representative problems for a wide scope of assumptions on the inhomogeneity of shells across the thickness. The frequencies obtained by the three-dimensional theory of elasticity and by two-dimensional shell models of different degrees of accuracy will be compared. The nonclassical shell model that accounts for transverse shears and reduction was earlier tested by solving stress-strain problems for laminated shells under local loads and contact problems for shells of revolution on a rigid foundation [12,14,15]. Problem Formulation and SolutionTechnique. The subject of study is the class of inhomogeneous shells of revolution described, as three-dimensional bodies, in an orthogonal curvilinear coordinate system s, q, V (s is the meridional arc length, q is the cross-sectional central angle, and V is the thickness coordinate reckoned from some coordinate surface V = const). Shells of this class include an arbitrary number J of isotropic and ort...

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Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review)

2012

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Variational-difference method in the problem of the vibrations of laminated plates

Cited by 3 publications

References 1 publication

Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

Determining the natural frequencies of highly inhomogeneous shells of revolution with transverse strain

Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review)

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