Research Papers

Natural Frequencies of Submerged Structures Using an Efficient Calculation of the Added Mass Matrix in the Boundary Element Method

[+] Author and Article Information
Luis E. Monterrubio

Engineering Department,
Robert Morris University,
6001 University Boulevard,
Moon, PA 15108
e-mail: monterrubio@rmu.edu

Petr Krysl

Structural Engineering Department,
University of California, San Diego,
9500 Gilman Dr., #0085,
La Jolla, CA 92093
e-mail: pkrysl@ucsd.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 28, 2017; final manuscript received September 26, 2018; published online November 13, 2018. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 141(2), 021008 (Nov 13, 2018) (9 pages) Paper No: VIB-17-1385; doi: 10.1115/1.4041617 History: Received August 28, 2017; Revised September 26, 2018

This work presents an efficient way to calculate the added mass matrix, which allows solving for natural frequencies and modes of solids vibrating in an inviscid and infinite fluid. The finite element method (FEM) is used to compute the vibration spectrum of a dry structure, then the boundary element method (BEM) is applied to compute the pressure modes needed to determine the added mass matrix that represents the fluid. The BEM requires numerical integration which results in a large computational cost. In this work, a reduction of the computational cost was achieved by computing the values of the pressure modes with the required numerical integration using a coarse BEM mesh, and then, interpolation was used to compute the pressure modes at the nodes of a fine FEM mesh. The added mass matrix was then computed and added to the original mass matrix of the generalized eigenvalue problem to determine the wetted natural frequencies. Computational cost was minimized using a reduced eigenvalue problem of size equal to the requested number of natural frequencies. The results show that the error of the natural frequencies using the procedure in this work is between 2% and 5% with 87% reduction of the computational time. The motivation of this work is to study the vibration of marine mammals' ear bones.

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Lin, Z. , and Liao, S. , 2011, “ Calculation of Added Mass Coefficients of 3D Complicated Underwater Bodies by FMBEM,” Commun. Nonlinear Sci. Numer. Simul., 16(1), pp. 187–194. [CrossRef]
Ghassemi, H. , and Yari, E. , 2011, “ The Added Mass Coefficient Computation of Sphere, Ellipsoid and Marine Propellers Using Boundary Element Method,” Pol. Maritime Res., 68(1), pp. 17–26. [CrossRef]
Geers, T. L. , 1978, “ Doubly Asymptotic Approximations for Transient Motions of Submerged Structures,” J. Acoust. Soc. Am., 64(5), pp. 1500–1508. [CrossRef]
Deruntz, J. A. , and Geers, T. L. , 1978, “ Added Mass Computation by the Boundary Integral Method,” Int. J. Numer. Methods Eng., 12(3), pp. 531–48. [CrossRef]
Antoniadis, I. , and Kanarachos, A. , 1987, “ Decoupling Procedure for the Modal Analysis of Structures in Contact With Incompressible Fluids,” Commun. Appl. Numer. Methods, 3(6), pp. 507–517. [CrossRef]
Sandberg, G. , 1995, “ A New Strategy for Solving Fluid-Structure Problems,” Int. J. Numer. Methods Eng., 38(3), pp. 357–370. [CrossRef]
Ugurlu, B. , and Ergin, A. , 2006, “ A Hidroelasticity Method for Vibrating Structures Containing and/or Submerged in Flowing Water,” J. Sound Vib., 290(3–5), pp. 572–596. [CrossRef]
Rajasankar, J. , Iyer, N. R. , and Rao, V. S. R. A. , 1993, “ A New 3-D Finite Element Model to Evaluate Added Mass for Analysis of Fluid-Structure Interaction Problems,” Int. J. Numer. Methods Eng., 36(6), pp. 997–1012. [CrossRef]
Kwak, M. K. , 1996, “ Hydroelastic Vibration of Rectangular Plates,” ASME J. Appl. Mech., 63(1), pp. 110–115. [CrossRef]
Everstine, G. C. , 1991, “ Prediction of Low Frequency Vibrational Frequencies of Submerged Structures,” ASME J. Vib. Acoust., 113(2), pp. 187–191. [CrossRef]
Fu, Y. , and Price, W. G. , 1987, “ Interactions Between a Partially or Totally Immersed Vibrating Cantilever Plate and the Surrounding Fluid,” J. Sound Vib., 118(3), pp. 495–513. [CrossRef]
Wilken, M. , Of, G. , Cabos, C. , and Steinbach, O. , 2009, “ Efficient Calculation of the Effect of Water on Ship Vibration,” Analysis and Design of Marine Structures, S. Guedes and P. K. Das , eds., Taylor & Francis, London, pp. 93–101.
Jensen, F. B. , Kupperman, W. A. , Porter, M. B. , and Schmidt, H. , 1994, Computational Ocean Acoustics, American Institute of Physics, New York.
Monterrubio, L. E. , and Krysl, P. , 2012, “ Efficient Calculation of the Added Mass Matrix for Vibration Analysis of Submerged Structures,” 11th International Conference on Computational Structures Technology, Stirlingshire, UK, Sept. 4–7, Paper No. 212 2012.
Pozrikidis, C. , 2002, A Practical Guide to Boundary Element Methods With the Software Library BEMLIB, Chapman & Hall/CRC Press, Boca Raton, FL.
Pozrikidis, C. , 1998, Numerical Computation in Engineering and Science, Oxford University Press, New York.
Pina, H. L. G. , Fernandes, J. L. M. , and Brebbia, C. A. , 1981, “ Some Numerical Integration Formulae Over Triangles and Squares With a 1/r Singularity,” Appl. Math. Modell., 5(3), pp. 209–11. [CrossRef]
Bathe, K. J. , 1982, Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, NJ.
Sundqvist, J. , 1983, “ An Application of ADINA to the Solution of Fluid-Structure Interaction Problems,” Comput. Struct., 17(5–6), pp. 793–807. [CrossRef]
Lindholm, U. S. , Kana, D. D. , Chu, W. H. , and Abramson, H. N. , 1965, “ Elastic Vibration Characteristics of Cantilever Plates in Water,” J. Ship Res., 9, pp. 11–22.
Price, W. G. , Randall, R. , and Temarel, P. , 1988, “ Fluid-Structure Interaction of Submerged Shells,” Naval Architecture and Offshore Engineering Conference, Guildford, Surrey, UK.
Randall, R. J. , 1990, “ Fluid-Structure Interaction of Submerged Shells,” Ph.D. dissertation, Brunel University, London.
Gilroy, L. E. , 1993, “ Finite Element Calculations of Cylinder Natural Frequencies,” Defense Research Establishment Atlantic, Ottawa, ON, Canada, Technical Communication No. 93.
Krysl, P. , 2012, “ FAESOR: Matlab Toolkit for Finite Element Analysis (Computer Software),” San Diego, CA, accessed July 25, 2018, http://hogwarts.ucsd.edu/~pkrysl/faesor
Monterrubio, L. E. , and Ilanko, S. , 2012, “ Sets of Admissible Functions for the Rayleigh-Ritz Method,” 11th International Conference on Computational Structures Technology, Stirlingshire, UK, Sept. 4–7, Paper No. 97 2012.
COMSOL, 2009, Acoustics Module User Guide, COMSOL, Inc., Burlington, MA.
ABAQUS, 2011, “ ABAQUS Version 6.11-2 User's Manual,” Simulia, Providence, RI.


Grahic Jump Location
Fig. 1

(Left) 24 × 24 finite element mesh used to obtain the dry modes and (right) 6 × 6 coarse boundary element mesh used to compute the pressure modes

Grahic Jump Location
Fig. 2

Modes of vibration of a submerged unconstrained plate. The figures correspond to mode 1 (top-left), mode 2 (top-right), mode 3 (center-left), mode 4 (center-right), and mode 5 (bottom-left).

Grahic Jump Location
Fig. 3

Modes of vibration of a submerged cantilever plate. The figures correspond to mode 1 (top-left), mode 2 (top-right), mode 3 (center-left), mode 4 (center-right), and mode 5 (bottom-left).

Grahic Jump Location
Fig. 4

Modes of vibration of the unconstrained cylinder presented by Price. Modes correspond in order to the modes presented in Table 7, from left to right (1,2), (1,3), (e1), (e2), (1,4), (2,3), (2,4), and (3,4). Plate modes have a zero pressure along the cylinder.

Grahic Jump Location
Fig. 5

(Right) finite element mesh used to obtain the dry modes and (left) coarse boundary element mesh used to compute the pressure modes of the ear-bone



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