 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
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| In English (Mech. Solids): | | 4760 |
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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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