VM55

VM55
Vibration of a Stretched Circular 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, eq. 182.
Analysis Type(s):
Linear Perturbed Modal Analysis (ANTYPE = 2)
Element Type(s):2-Node Finite Strain Axisymmetric Shell (SHELL208)
Input Listing:vm55.dat

Test Case

A circular membrane of radius a, thickness t, and weight per unit area w is simply supported along its edge and subjected to an in-plane radial load of T lb/unit length of circumference. Determine the radial stress σr in the membrane and the first three natural frequencies fi of lateral vibration of the membrane.

Material PropertiesGeometric PropertiesLoading
E = 30 x 106 psi
w = 0.0028178 lb/in2
a = 15 in
t = 0.01 in
T = 100 lb/in

Analysis Assumptions and Modeling Notes

The in-plane radial load F is calculated as: F = 2 πa T = 9424.778 lb. The mass density ρ is calculated as w/gt = 0.00073 lb-sec2/in4, where g = 386 in/sec2. Enough elements are defined to adequately characterize the dynamic analysis. Linear perturbed modal analysis is performed on the prestressed structure using Block Lanczos eigensolver to determine the first three natural frequencies.

Results Comparison

 TargetMechanical APDLRatio
StaticStressr , psi10,000.10,000.1.000
Modalf1 , Hz94.40694.4641.001
f2 , Hz216.77217.1751.002
f3 , Hz339.85341.5101.005