VM152

VM152
2D Nonaxisymmetric Vibration of a Stretched Membrane

Overview

Reference:S. Timoshenko, D. H. Young, Vibration Problems in Engineering, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pp. 438-439, article 69.
Analysis Type(s):
Mode-frequency Analysis (ANTYPE = 2)
Static Analysis, Prestress (ANTYPE = 0)
Element Type(s):Axisymmetric-Harmonic Structural Shell Elements (SHELL61)
Input Listing:vm152.dat

Test Case

A circular membrane under a uniform tension S is allowed to vibrate freely. The edge of the membrane is simply supported. Determine the natural frequencies fi,j for the first mode of vibration (j = 1 = no. of nodal circles, including the boundary) for the first three harmonic (i = 0,1,2 = no. of harmonic indices). Also determine the next highest axisymmetric frequency f0,2. See VM153 for a 3D solution of this problem.

Figure 213: Circular Membrane Problem Sketch

Circular Membrane Problem Sketch

Material PropertiesGeometric PropertiesLoading
E = 30 x 106 psi
υ = 0.0
ρ = 0.00073 lb-sec2/in4
α = 1 x 10-5 in/in-°F
a = 3 in
t = 0.00005 in
S = 0.1 lb/in of boundary
ΔT = -6.6666°F

Analysis Assumptions and Modeling Notes

A total of 9 elements is selected for meshing. The prestress is induced by cooling the membrane. The necessary temperature difference, ΔT, is calculated from S = -E αt(ΔT). Modal analysis is solved using Block-Lanczos eigensolver.

Results Comparison

TargetMechanical APDLRatio
fo,1, Hz (L.S. 1, ITER 1)211.1211.21.000
f1,1, Hz (L.S. 2, ITER 1)336.5336.51.000
f2,1, Hz (L.S. 3, ITER 1)450.9451.01.000
f0,2, Hz (L.S. 1, ITER 2)484.7484.71.000

Figure 214: Mode Shape Displays

Mode Shape Displays

Window 1 - f0,1; Window 2 - f1,1; Window 3 - f2,1; Window 4 - f0,2;