Abstract
In the case of circular symmetry, we conduct numerical and analytical studies of Chladni modes of an elastic plate floating on the liquid surface and cantilevered at the center to a vertical support. Using the theory of long waves in shallow water and the approximation of the Euler beam vibrations for the bounded and unbounded water basins, we obtain the dependence of the natural and quasinatural frequencies of Chladni figures on the geometric parameters of the plate and the vibration region in presence of the bottom unevenness.
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References
L. A. Tkacheva, “Eigenvibrations of a Flexible Platform Floating on Shallow Water,” Prikl. Mekh. Tekhn. Fiz. 41 (1), 173–181 (2000) [J. Appl. Mech. Techn. Phys. 41 (1), 159–168 (2000)].
I. V. Sturova, “Impact of anUnsteady External Load on an Elastic Circular Plate Floating on ShallowWater,” Prikl. Mat. Mekh. 67 (3), 453–463 (2003).
A. A. Korobkin and T. I. Khabakhpasheva, “Construction of Exact Solutions of a Problem Concerning a Floating Plate,” Prikl. Mat. Mekh. 71 (2), 321–328 (2007).
I. V. Sturova, “Effect of Bottom Topography on the Unsteady Behaviour of an Elastic Plate Floating on ShallowWater,” Prikl. Mat. Mekh. 72 (4), 588–600 (2008) [J. Appl. Math. Mech. 72 (4), 417–426 (2008)].
V. V. Alekseev, D. A. Indeitsev, and Yu. A. Mochalova, “Vibration of a Flexible Plate in Contact with the Free Surface of a Heavy Liquid,” Zh. Tekhn. Fiz. 72 (5), 16–21 (2002) [Technical Physics 47 (5), 529–534 (2002)].
V. V. Meleshko and S. O. Papkov, “W. Ritz (1909) to Our Time,” Akustichnii Visnik 12 (4), 34–51 (2009).
M. H. Meylan, “An Application of Scattering Frequencies to Hydroelasticity,” in Proceedings of the 11th Offshore and Polar Engineering Conference (Stavanger, Norway, June 17–22, 2001) (International Society of of fshore and Polar Engineers, Cupertino, CA, 2001), pp. 385–391.
E. A. Grigolyuk and I. T. Selezov, Mechanics of Rigid Deformable Bodies, Vol. 5: Nonclassical Theories of Vibration of Rods, Plates, and Shells (VINITI, Moscow, 1973) [in Russian].
S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).
J. J. Stoker, Water Waves: The Mathematical Theory with Applications (Wiley-Interscience, New York, 1958; Inostrannaya Literatura, Moscow, 1959).
V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1988) [in Russian].
G. Zilman and T. Miloh, “Hydroelastic Buoyant Circular Plate in Shallow Water: a Closed Form Solution,” Appl. Ocean Research 22, 191–198 (2000).
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Original Russian Text © A.G. Greshilov, S.V. Sukhinin, 2017, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2017, Vol. XX, No. 4, pp. 31–40.
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Greshilov, A.G., Sukhinin, S.V. Chladni figures of a circular plate floating in the bounded and unbounded water basins with the cantilevered central support. J. Appl. Ind. Math. 11, 49–57 (2017). https://doi.org/10.1134/S1990478917010069
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DOI: https://doi.org/10.1134/S1990478917010069