 | | 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
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| L. D. Akulenko, S. V. Nesterov, and A. L. Popov, "Natural frequencies of an elliptic plate with clamped edge," Mech. Solids. 36 (1), 143-148 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
1 |
Pages |
143-148 |
| Title |
Natural frequencies of an elliptic plate with clamped edge |
| Author(s) |
L. D. Akulenko (Moscow)
S. V. Nesterov (Moscow)
A. L. Popov (Moscow) |
| Abstract |
A modified Rayleigh-Ritz method is applied to obtain highly accurate
estimates for frequencies and shapes of lower vibration modes of
an elliptic plate with clamped edge. A relationship between
the spectrum of an elliptic plate and that of a circular plate
is established. The estimates obtained are compared with the numerical
results of other authors and with experimental data.
The investigation of natural vibrations of two-dimensional distributed
systems with an elliptic boundary is, undoubtedly, important for both
theory and applications (see, e.g., [1-3]). For applications,
of considerable importance is determining vibration frequencies
and shapes (especially those of lower modes) of tight membranes, elastic
plates, a heavy liquid in a basin, acoustic and electromagnetic waves
in wave-guides and resonators, as well as numerous other systems.
Theoretical interest is justified by the fact that the relevant
boundary value problems for an elliptic region
are natural generalizations of the corresponding problems for
a circle, which have been thoroughly studied. An attractive feature
of these problems is the fact that they admit separation of variables
by introducing the elliptic coordinates and can be tackled with analytical
and numerical-analytical methods applied to coupled one-dimensional
boundary value problems of the Sturm-Liouville type.
Commonly, natural vibration frequencies and shapes are calculated
by means of variational, Rayleigh-Ritz, Bubnov-Galerkin,
finite element, or finite-difference methods [1-5].
In the present paper, we suggest a modified Rayleigh-Ritz method
to calculate frequencies and shapes for lower vibration modes
of an elliptic plate. The method involves the introduction
of generalized polar coordinates and definition of the deflection
function that depends on the polar angle in a prescribed periodic
manner, with the number of radial nodal lines being fixed (as is
the case for a circular plate). The dependence of the deflection on
the radial coordinates is not prescribed in advance but is determined
by the solution of the corresponding Euler-Lagrange equation.
A similar approach was used in [3] for an elliptic membrane. |
| References |
| 1. | Ph. M. Morse and H. Feshbach, Methods of Theoretical Physics
[Russian translation], Izd-vo Inostr. Lit-ry, Moscow, Vol. 1, 1958;
Vol. 2, 1960. |
| 2. | M. D. O. Strutt, Lamé, Mathieu, and other related functions
in physics and engineering [Russian translation], Gostekhizdat Ukrainy,
Kharkov, Kiev, 1935. |
| 3. | L. D. Akulenko and S. V. Nesterov, "Free vibrations of
a homogeneous elliptic membrane," Izv. AN. MTT [Mechanics of Solids],
No. 1, pp. 179-190, 2000. |
| 4. | L. Collatz, Eigenvalue Problems [Russian translation],
Nauka, Moscow, 1968. |
| 5. | R. Courant and D. Hilbert, Methods of Mathematical Physics.
Volume 1 [Russian translation], Gostekhizdat, Moscow, 1951. |
| 6. | I. M. Babakov, Theory of Vibrations [in Russian],
Nauka, Moscow, 1965. |
| 7. | G. Pólya and G. Szegö, Isoperimetric Inequalities
in Mathematical Physics [Russian translation], Fizmatgiz, Moscow,
1962. |
| 8. | J. W. Strutt (Lord Rayleigh) Theory of Sound. Volume 1 [Russian
translation], Gostekhizdat, Moscow, Leningrad, 1940. |
| 9. | Y. Shibaoka, "On the transverse vibration of an elliptic plate
with clamped edge," J. Phys. Soc. of Japan, Vol. 11, No. 7,
pp. 797-803, 1956. |
| 10. | E. Kamke, Handbook on Ordinary Differential Equations
[Russian translation], Nauka, Moscow, 1971. |
| 11. | V. F. Zaitsev and A. D. Polyanin. Handbook on Ordinary
Differential Equations [in Russian], Nauka, Moscow, 1995. |
| 12. | A. A. Il'yushin and V. S. Lenskii, Strength of Materials
[in Russian], Fizmatgiz, Moscow, 1959. |
|
| Received |
10 April 2000 |
| << Previous article | Volume 36, Issue 1 / 2001 | Next article >> |
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