Abstract

This study deals with using the modal expansion of vibrations of an elastic structure such as a thin circular plate. The effect of the resonant frequency shift is included as a result of loading from surrounding water. It is shown that rough a priori estimations of the resonant frequencies, using the nondimensionalized added virtual mass incremental (NAVMI) factors and the Helmholtz equation (we take into account the radiation of acoustic waves), allow to correctly determine the modes that must be taken into account in order to obtain physically correct numerical results. The heavy fluid loading (e.g., by water) results in a strong lowering of the resonant frequencies compared to the light fluid loading (e.g., by air) or no fluid loading (in vacuum). Consequently, if too few modes are included in numerical calculations, the results will be incorrect. This can be easily seen for frequencies above the coincidence frequency, when the radiation efficiency, instead of tending to unity, drops to zero or values much less than unity. It is, however, possible to avoid such danger since the use of the NAVMI factors and the Helmholtz equation allows obtaining correct numerical results even above the frequency of coincidence. The presented method allows releasing the frequency limitation resulting from the use of the approximation for frequencies close to zero together with the use of the Laplace equation, as is the case in the studies of other authors. The results seem to be of practical importance, especially for the higher frequencies.

References

1.
Amabili
,
M.
, and
Kwak
,
M.
,
1996
, “
Free Vibrations of Circular Plates Coupled With Liquids: Revising the Lamb Problem
,”
J. Fluids Struct.
,
10
(
7
), pp.
743
761
.
2.
Kwak
,
M. K.
,
2010
, “
Free Vibration Analysis of a Finite Circular Cylindrical Shell in Contact With Unbounded External Fluid
,”
J. Fluid Struct.
,
26
(
3
), pp.
377
392
.
3.
Wu
,
H.
, and
Zhou
,
S.
,
2014
, “
Free Vibrations of Sensor Diaphragm With Residual Stress Coupled With Liquids
,”
J. Appl. Phys.
,
115
(
8
), p.
084303
.
4.
Dong
,
M.
,
Ma
,
S.
, and
Wang
,
S.
,
2018
, “
The Dynamic Characteristics of Micro-diaphragms Subjected to Thermal Stress When Coupled With a Fluid
,”
J. Appl. Phys.
,
124
(
12
), p.
125306
.
5.
Ma
,
S.
,
Dong
,
M.
, and
Wang
,
S.
,
2018
, “
Mode Dependent Fluid Damping in Pre-stressed Micro-diaphragm Resonators
,”
J. Appl. Phys.
,
124
(
23
), p.
235305
.
6.
Kim
,
S.-M.
, and
Kwak
,
M. K.
,
2022
, “
Coupled Vibration and Sloshing Analysis of the Circular Plate Resting on the Free Surface of a Fluid-Filled Cylindrical Tank
,”
J. Sound Vib.
,
536
, p.
117131
.
7.
Levine
,
H.
, and
Leppington
,
F.
,
1988
, “
A Note on the Acoustic Power Output of a Circular Plate
,”
J. Sound Vib.
,
121
(
2
), pp.
269
275
.
8.
Kinsler
,
L.
,
Frey
,
A.
,
Coppens
,
A.
, and
Sanders
,
J.
,
2000
,
Fundamentals of Acoustics
, 4th ed.,
John Wiley and Sons
,
New York
, p.
548
.
9.
Thorp
,
W. H.
,
1965
, “
Deep-Ocean Sound Attenuation in the Sub- and Low-Kilocycle-Per-Second Region
,”
J. Acoust. Soc. Am.
,
38
(
4
), pp.
648
654
, 10.
10.
Sommerfeld
,
A.
,
1964
,
Partial Differential Equations in Physics
,
Academic Press
,
New York
, p.
335
.
11.
Rdzanek
,
W.
,
2018
, “
Sound Radiation of a Vibrating Elastically Supported Circular Plate Embedded Into a Flat Screen Revisited Using the Zernike Circle Polynomials
,”
J. Sound Vib.
,
434
, pp.
92
125
.
12.
Olver
,
F.
,
Lozier
,
D.
,
Boisvert
,
R.
, and
Clark
,
C.
,
2010
,
NIST Handbook of Mathematical Functions
,
Cambridge University Press
,
New York
, pp.
1
951
.
13.
Rdzanek
,
W.
, and
Szemela
,
K.
,
2019
, “
Sound Radiation by a Vibrating Annular Plate Using Radial Polynomials and Spectral Mapping
,”
J. Acoust. Soc. Am.
,
146
(
4
), pp.
2682
2691
.
14.
Aarts
,
R.
, and
Janssen
,
A.
,
2009
, “
Sound Radiation Quantities Arising From a Resilient Circular Radiator
,”
J. Acoust. Soc. Am.
,
126
(
4
), pp.
1776
1787
.
15.
Rdzanek
,
W.
,
2016
, “
The Acoustic Power of a Vibrating Clamped Circular Plate Revisited in the Wide Low Frequency Range Using Expansion Into the Radial Polynomials
,”
J. Acoust. Soc. Am.
,
139
(
6
), pp.
3199
3213
.
16.
Fahy
,
F.
, and
Gardonio
,
P.
,
2007
,
Sound and Structural Vibration. Radiation, Transmission and Response
, 2nd ed.,
Academic Press Elsevier
,
New York
, p.
633
.
17.
The mpmath Development Team
,
2023
, “Math: A Python Library for Arbitrary-Precision Floating-Point Arithmetic (Version 1.3.0),” http://mpmath.org/
18.
Wiciak
,
J.
,
2007
, “
Modelling of Vibration and Noise Control of a Submerged Circular Plate
,”
Arch. Acoust.
,
32
(
4S
), pp.
265
270
.
19.
Waller
,
M.
,
1938
, “
Vibrations of Free Circular Plates. Part 1: Normal Modes
,”
Proc. Phys. Soc.
,
50
(
1
), pp.
70
76
.
20.
Waller
,
M.
,
1938
, “
Vibrations of Free Circular Plates. Part 2: Compounded Normal Modes
,”
Proc. Phys. Soc.
,
50
(
1
), pp.
77
82
.
21.
Waller
,
M.
,
1938
, “
Vibrations of Free Circular Plates. Part 3: A Study of Chladni’s Original Figures
,”
Proc. Phys. Soc.
,
50
(
1
), pp.
83
86
.
22.
Askari
,
E.
,
Jeong
,
K.-H.
, and
Amabili
,
M.
,
2013
, “
Hydroelastic Vibration of Circular Plates Immersed in a Liquid-Filled Container With Free Surface
,”
J. Sound Vib.
,
332
(
12
), pp.
3064
3085
.
23.
Rao
,
S.
,
2007
,
Vibrations of Continuous Systems
,
Wiley
,
NJ
, p.
720
.
24.
McLachlan
,
N.
,
1955
,
Bessel Functions For Engineers
,
Clarendon Press
,
Oxford
, p.
239
.
25.
Rdzanek
,
W.
, and
Engel
,
Z.
,
2000
, “
Asymptotic Formulas for the Acoustic Power Output of a Clamped Annular Plate
,”
Appl. Acoust.
,
60
(
1
), pp.
29
43
.
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