VM177

VM177
Natural Frequency of a Submerged Ring

Overview

Reference:

E. A. Schroeder, M. S. Marcus, "Finite Element Solution of Fluid Structure Interaction Problems", Shock and Vibration Symposium, San Diego, CA, 1975.

Analysis Type(s):

Full Harmonic Analysis (ANTYPE = 3)

Unsymmetric Matrix Modal Analysis (ANTYPE = 2)

Element Type(s):

3D Acoustic Fluid Elements (FLUID30)

Elastic Shell Elements (SHELL63)

2D Acoustic Fluid Elements (FLUID29)

3D 2 Node Beam (BEAM188)

4-Node Finite Strain Shell (SHELL181)

8-Node Structural Shell (SHELL281)

3D Acoustic 20-node Fluid Elements (FLUID220)

4-D Acoustic 10-node Fluid Elements (FLUID221)

Input Listing:

vm177.dat

Test Case

A steel ring is submerged in a compressible fluid (water). Determine the lowest natural frequency for x-y plane bending modes of the fluid-structure system.

Figure 265: Submerged Ring Problem Sketch

Material Properties

Geometric Properties

For steel:

E = 30 x 10⁶ psi

υ = 0.3

ρ = 0.0089 slugs per cubic inch (from reference)

For water:

C = 57480 in/sec

(speed of sound in water)

ρ = 0.001156 slugs per cubic inch (from reference)

a = 10 in

b = 30 in

t = .25 in

Analysis Assumptions and Modeling Notes

For this problem, the fluid is assumed as extending only to a finite radius b where the pressure P_o is zero and b is taken to be 30 inches. From the reference, this assumption should result in an error of less than 1% compared to the frequency for an unbounded fluid (b = ). The natural boundary conditions at the coordinate axes imply that δP / δx = 0 at x = 0 and δP / δy = 0 at y = 0.

This problem is solved using five separate analyses. The first uses 3D acoustic fluid elements (FLUID30) with quadrilateral shell elements (SHELL63), the second uses 2D acoustic fluid elements (FLUID29) with BEAM188, the third uses 3D acoustic fluid elements (FLUID30) with quadrilateral shell elements (SHELL181), the fourth uses 3D 20 node hexahedral acoustic fluid elements (FLUID220) with 8 node structural shell elements (SHELL281) and the fifth one uses 3D 10 node tetrahedral fluid elements (FLUID221) with 8 node structural shell elements (SHELL281).

For cases 1, 3, 4, and 5, due to fluid-structure coupling involving unsymmetric matrices, the natural frequency is determined by first performing a modal solve (ANTYPE = 2), and then the first mode is excited by performing a full harmonic (ANTYPE = 3) analysis with a frequency sweep. Monitoring the displacement of key nodes over the frequency range indicates the approximate desired natural frequency as the point where the nodal displacements indicate a resonant condition.

In the second case, the lowest natural frequency is obtained using an unsymmetric matrix modal analysis (ANTYPE = 2) corresponding to the frequency of mode one of the frequency data from the Lanczos unsymmetric eigensolver.

A preliminary finite element analysis (not shown here) was used to estimate a narrow range in which the 1st even bending mode frequency occurs. For cases 1, 3, 4, and 5, two unit loads are arbitrarily used to excite the desired bending mode of vibration.

Results Comparison

		Target[1]	Mechanical APDL	Ratio
FLUID30 and SHELL63	f₁, Hz	10.20	10.560 < f < 10.580	1.036
FLUID29 and BEAM188	f₁, Hz	10.20	10.63	1.042
FLUID30 and SHELL181	f₁, Hz	10.20	10.640 < f < 10.660	1.044
FLUID220 and SHELL281	f₁, Hz	10.20	10.270 < f < 10.290	1.008
FLUID221 and SHELL281	f₁, Hz	10.20	10.33 < f < 10.35	1.014

Solution from the reference under the same assumptions mentioned in the modeling notes.

Figure 266: Node 1 Displacement vs. Driving Frequency Near 1st Bending Mode Natural Frequency (Full Harmonic Analysis)

Figure 267: Real Displacement Component obtained from Case 1 (FLUID30 and SHELL63 Elements)

Figure 268: Imaginary Displacement Component obtained from Case 1 (FLUID30 and SHELL63 Elements)