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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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. 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

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. 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).



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