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IssuesArchive of Issues2001-3pp.135-145

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L. D. Akulenko, I. I. Karpov, and S. V. Nesterov, "Natural vibrations of a rectangular membrane with sharply changing surface density," Mech. Solids. 36 (3), 135-145 (2001)
Year 2001 Volume 36 Number 3 Pages 135-145
Title Natural vibrations of a rectangular membrane with sharply changing surface density
Author(s) L. D. Akulenko (Moscow)
I. I. Karpov (Moscow)
S. V. Nesterov (Moscow)
Abstract A numerical-analytical method and a computational algorithm of accelerated convergence for determining natural vibration frequencies and shapes of a rectangular membrane with fixed boundary are presented. The mass per unit area and the surface tension of the membrane may sharply change to a great extent. To fix ideas, we present the computational results for the fundamental vibration mode of a square membrane with constant surface tension and an inhomogeneity of a special type. The inhomogeneity is modeled by two mutually orthogonal strips forming a cross-shaped figure or its modifications (a shifted or asymmetrical cross, an angle, or a T-shaped figure). The parameters determining the density (length, width, and place of intersection of the strips) were varied within a broad range. We have performed a numerical analysis of the desired characteristics of the membrane vibration, found interesting mechanical effects, and commented on these.

Analysis of various aspects of the transverse vibration of a plane membrane is of considerable interest both for theory and applications. This analysis is based, as a rule, on the efficient construction of solutions of the corresponding eigenvalue problems, which permits one to construct finite-dimensional dynamical models. Of particular interest is the problem of high-accuracy determination of frequencies and shapes for lower vibration modes, depending on the mechanical properties of the membrane. The lower modes determine the operating performance of systems containing a stretched membrane as the working surface. Higher modes usually have relatively small amplitudes and decay much faster than the lower modes. The classical results for homogeneous uniformly stretched membranes of various shapes (rectangular, circular, annular, sectorial, elliptic, triangular, etc.) are presented in [1-5].

The analysis and computation of frequencies and shapes of lower vibration modes in the general case, where both the mass per unit area and the surface tension change over the membrane surface, encounters significant difficulties. To roughly estimate these vibration characteristics, the Rayleigh-Ritz method [1, 3-5] and that of finite elements [6,7] are widely used. A number of problems have been stated and solved in terms of special functions, for example, problems for circular membranes with piecewise constant or other simple radial inhomogeneities [8].

In [9], an effective numerical-analytical method is suggested for calculating natural vibration frequencies and shapes for an inhomogeneous rectangular membrane with fixed boundary. This method uses the assumption that the surface density can be approximated by the sum of two functions, each depending on a single coordinate, whereas the residual is relatively small and can be taken into account by means of perturbation techniques. This class of membranes is fairly representative; the investigation of their vibration is of interest for methodology and applications. To check the operational performance of the method, we have carried out calculations for a number of model problems [9].

In the present paper, we present the solution of an important problem of vibration of a membrane whose density sharply changes to a great extent with respect to each of the coordinates and the inhomogeneity is cross-shaped. Such a model can be encountered in engineering applications (e.g., a space sail or a large-scale antenna). Inhomogeneities in the form of rectangular strips may be introduced for ensuring the desired shape of a structure or used as connecting members.
References
1.  R. Courant and D. Hilbert, Methods of Mathematical Physics. Volume 1 [Russian translation], Gostekhizdat, Moscow, 1951.
2.  A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow, 1966.
3.  F. M. Morse and G. Feshbach, Methods of Theoretical Physics. [Russian translation], Izd-vo Inostr. Lit-ry, Moscow, Volume 1, 1958; Volume 2, 1960.
4.  L. Collatz, Eigenvalue Problems [Russian translation], Nauka, Moscow, 1968.
5.  J. W. Strutt (Lord Rayleigh) Theory of Sound. Volume 1 [Russian translation], Gostekhizdat, Moscow, Leningrad, 1940.
6.  G. Strang and G. J. Fix, An Analysis of Finite Element Method [Russian translation], Mir, Moscow, 1977.
7.  O. C. Zienkiewicz, Finite Elements in Engineering [Russian translation], Mir, Moscow, 1975.
8.  R. S. Gupter and R. Ghosh, "Method of perturbation applied to the vibration problem of a circular membrane of varying density," J. Acoust. Soc. America, vol. 36, No. 6, pp. 1118-1120, 1964.
9.  L. D. Akulenko and S. V. Nesterov, "Vibration of an nonhomogeneous membrane," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 134-145, 1999.
10.  L. D. Akulenko and S. V. Nesterov, "Determination of vibration frequencies and shapes for inhomogeneous distributed systems with boundary conditions of the third kind," PMM [Applied Mathematics and Mechanics], Vol. 61, No. 4, pp. 547-555, 1997.
11.  L. D. Akulenko and S. V. Nesterov, "The effective solution of the generalized Sturm-Liouville problem," Doklady AN, Vol. 363, No. 3, pp. 323-326, 1998.
12.  A. I. Zhurov, I. I. Karpov, and I. K. Shingareva, Basic Maple. Application to Mechanics [in Russian], Preprint No. 536, Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 1995.
13.  I. I. Karpov and A. K. Platonov, "Accelerated numerical integration of equations of motion in celestial mechanics," Kosmicheskie Issledovaniya [Space Research], Vol. 10, No. 6, pp. 811-826, 1972.
14.  L. D. Akulenko and S. V. Nesterov, "Oscillations of the interacting systems with inhomogeneous distributed parameters," Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 15-25, 1999.
Received 17 June 1999
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