Nonstationary Dynamics of Elliptic Isotropic Conical Shells Under Distributed Loads*
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
NONSTATIONARY DYNAMICS OF ELLIPTIC ISOTROPIC CONICAL SHELLS UNDER DISTRIBUTED LOADS* V. F. Meish1, Yu. A. Meish2, and M. A. Belova3
We considered a solution to the problem of the forced vibrations of a truncated elliptic conical shell under a distributed impulsive load. A linear version of the equations of the Timoshenko type theory of conical shells is obtained in a non-orthogonal curvilinear coordinate system. To solve the problem, we elaborated a numerical algorithm based on the finite-difference approximation of the initial equations in the spatial and time coordinates. An example of the dynamical behavior of the conical shell is studied numerically. Keywords: conical shell, elliptical cross-section, non-orthogonal curvilinear coordinate system, Timoshenko type theory, forced vibrations, numerical solution Introduction. An extensive number of publications (articles, reviews, monographs) are devoted to the problems of the dynamic behavior of homogeneous and inhomogeneous (in thickness) shell structures of different spatial configurations. Almost all dynamic problems were solved for canonical shells (cylindrical, spherical, conical, etc.) in an orthogonal curvilinear coordinate system [2]. The numerical solution of the dynamic problems of the theory of multilayer reinforced shells of different geometries is presented in [15]. The stress–strain state of a discrete-reinforced ellipsoidal shell under nonstationary loading is considered in [13]. The paper [14] addressed the nonstationary vibrations of elliptic cylindrical sandwich shells within the framework of Tymoshenko’s model of shells and rods [2]. Problems of the dynamics of conical shells were mainly solved for elliptic shells. In particular, such problems are considered in [6–8, 10, 17, 19]. Conical shells under dynamic loads were considered in [5, 18] using a nonorthogonal coordinate system [1, 4, 5, 11]. Cases of the geometrically nonlinear theory of thin shells in a nonorthogonal coordinate system are presented in [1, 3, 9]. In this paper, we consider the problem of the dynamic behavior of a truncated elliptic conical shell under an impulsive load. The equations of vibrations of a conical shell in a nonorthogonal coordinate system are given. To solve this problem, a numerical algorithm is developed, which is based on the finite-difference approximation of the initial equations in space and time coordinates. Numerical results are given for the case of the dynamic behavior of a truncated elliptic conical shell under a distributed impulsive load. 1. Problem Statement. We consider the problem of nonstationary deformation of a truncated elliptic conical shell under a distributed internal impulsive load. The equations of the midsurface of the shell have the following parametric form:
1
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, Ukraine 03057; e-mail: [email protected]. 2National Transport University, 1 Mykhaila Omelianovycha-Pavlenka St.,
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