Keywords: fluid-structure interaction problem, added virtual mass incremental factor, non-uniform rectangular plate, optimized Kantorovich method.

The fluid-structure interaction problem is dealt with in this paper. In particular, the vibrational behavior of non-uniform rectangular plates in contact with fluid will be investigated. It is well known that one can evaluate the natural frequencies of the non-uniform plate in contact with fluid by using the added virtual mass incremental (AVMI) factor. The numerical experiments on the determination of added virtual mass for water-contacting rectangular plates with non-uniform thickness is performed in this study.

The analytical approach for the addressed problem was initiated by Lord Rayleigh [1]. Lamb [2] calculated the change in natural frequency of a thin circular plate fixed along its boundary and placed in the aperture of an infinitely rigid wall in contact with water, and then Powell and Roberts [3] verified the theoretical results of Lamb's work by conducting experiments. McLachlam [4] extended Lamb's work and Peake and Thurston [5] generalized the work done by Lamb and McLachlam. Kwak and Kim [6,7] investigated the above problem and solve the mixed boundary problem by using Hankel Transform. M. Amabili and M. K. Kwak [8] performed the experiment verification about the validity of making the assumption that wet mode shapes are equivalent to dry ones. Amabili [9] researched into the effect of finite fluid depth on the hydroelastic vibrations of circular and annular plates. Amabili and Kwak [10] considered the effect of surface wave on the vibration of circular plates placing on a free fluid surface. Chang and Liu [11] studied the free vibration behavior of a rectangular isotropic plate in contact with liquid by using the double Fourier Transform. Amabili [12] discussed the solution to the fully coupled problem of the vibrations of circular plates resting on sloshing liquid. Later on, Chang and Liu [13] performed the forced vibration analysis of a rectangular composite plate in contact with fluid as well as discussing the variation of AVMI factor for different plates with various widths and length.

The optimized Kantorovich method, which has been proposed recently will be conducted to detect the natural vibration characteristics of the plate in the air. On the fluid-structure interface, some techniques will be adopted to deal with the relationships between the velocity potential and the mode shapes, and then by imposing the Fourier transform, one can derive the formulations of the reference kinetic energy for both the fluid and the plate itself. After these two energies have been obtained, the added virtual mass incremental factor (AVMIF) could be obtained rapidly and the added virtual mass can thus be acquired. Furthermore, the vibrational response of plates with different combination of boundary conditions will be investigated respectively.

1

Lord Rayleigh 1877 Theory of sound (two volumes). Dover, New York, second edition, 1945 re-issue.

2

Lamb, "On the vibrations of an elastic plate in contact with water", Proceedings of the Royal Society London, Series A 98, pp.205-206, 1920. doi:10.1098/rspa.1920.0064

3

J. H. Powell and J. H. T. Roberts, "On the frequency of vibration of circular diaphragms", Proceedings of the physical Society (London) 35, pp.170-182, 1923. doi:10.1088/1478-7814/35/1/321

4

N. W. McLachlam, "The accession to inertia of flexible discs vibrating in a fluid", Proceedings of the physical Society (London) 44, pp.546-555, 1932. doi:10.1088/0959-5309/44/5/303

5

W. H. Peake and E. G. Thurston, "The lowest resonant frequency of a water-loaded circular plate", Journal of the Acoustical Society of America 26(7), pp.166-168, 1954. doi:10.1121/1.1907302

6

M. K. Kwak and K. C. Kim, "Vibration of circular plates in contact with water", Journal of Applied Mechanics, 58, pp.481-483, 1991. doi:10.1115/1.2897209

7

M. K. Kwak and K. C. Kim, "Axisymmetric vibration of circular plates in contact with fluid", Journal of Sound and Vibration, 146(3), pp.381-389, 1991. doi:10.1016/0022-460X(91)90696-H

8

M. Amabili and M. K. Kwak, "Free Vibration of Circular Plates Coupled with Liquids: Revising the Lamb Problem", Journal of Fluids and Structures, 10, pp.473-761, 1996. doi:10.1006/jfls.1996.0051

9

M. Amabili, Effect of finite fluid depth on the hydroelastic vibrations of circular and annular plates, Journal of Sound and Vibration, 193(4), pp.909-925, 1996. doi:10.1006/jsvi.1996.0322

10

M. Amabili and M. K. Kwak, Vibration of circular plates on a free fluid surface: effect of surface waves, Journal of Sound and Vibrations, 226(3), pp.407-424, 1999. doi:10.1006/jsvi.1998.2304

11

T-P Chang and M-F Liu, "On the natural frequency of a rectangular isotropic plate in contact with fluid", Journal of Sound and Vibration, 236(3), pp.547-553, 2000. doi:10.1006/jsvi.2000.2955

12

M. Amabili, Vibrations of circular plates resting on sloshing liquid: solution of the fully coupled problem, Journal of Sound and Vibration, 245(2), pp.261-283, 2001. doi:10.1006/jsvi.2000.3560

13

T-P Chang and M-F Liu, "Vibration Analysis of Rectangular Composite Plates in contact with fluid", Journal of Mechanics of Structures and Machines, 29(1), pp.101-120, 2001. doi:10.1081/SME-100000005

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