VM153

VM153
3D 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, pg. 439, article 69.
Analysis Type(s):
Linear Perturbation Modal Analysis (ANTYPE = 2)
Static Analysis (ANTYPE = 0)
Element Type(s):
4-Node Finite Strain Shell Elements (SHELL181)
Input Listing:vm153.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 two modes of vibration (j = 1, 2 = no. of nodal circles, including the boundary) for the first two harmonics (i = 0, 1 = no. of harmonic indices). See VM152 for a 2D solution of this problem.

Figure 215: 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 30° sector is used with cyclic symmetry to model the membrane. The prestress is induced by uniform cooling. The temperature difference, ΔT, is calculated from S = E αt (ΔT). The low angle edge of the sector is defined as a component for cyclic symmetry analyses. Block Lanczos is used in the modal analysis to extract the first four frequencies.

Results Comparison

TargetMechanical APDLRatio
fo,1, Hz211.1211.31.001
fo,2, Hz484.7486.51.004
f1,1, Hz336.5338.11.005
f1,2, Hz616.1626.61.017