VM219

VM219
Frequency Response of a Prestressed Beam using APDL MATH Commands

Overview

Reference:R. D. Blevins, Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold Co., pg. 144, equation 8-20.
Analysis Type(s):Modal (ANTYPE=2)
Element Type(s):
2D Elastic Beam Elements (BEAM188)
Electromechanical Transducer Elements (TRANS126)
Input Listing:vm219.dat

Test Case

A beam is in series with an electromechanical transducer. With a voltage applied to the beam, determine the prestressed natural frequencies of the beam.

Figure 358: Element Plot

Element Plot

Material PropertiesGeometric PropertiesLoading

Ex = 169E3 μN/μm2

DENS = 2332E-18 kg/μm3

Beam Stiff = 112666.667 μN/μm

Gap Stiff = 20.022 μN/μm

L = 150μm

W = 4 μm

H = 2 μm

Initial Gap = 1 μm

Gap = 0.999 μm

Volt = 150.162V

Analysis Assumptions and Modeling Notes

The beam is created using BEAM188 elements which are used in series with one TRANS126 element. The node that connects the transducer and beam is free to move in the X-direction, while the end node of the beam and the ground node of the transducer are constrained in both the X and Y directions. It should also be noted that the transducer gap stiffness is much less than the beam stiffness and will not have any effect on the frequency solution. WRFULL command is used to write the assembled global matrices onto a .full file. *EIGEN command is then used to perform the modal analysis with unsymmetric matrices imported from .full file.

Results Comparison

TargetMechanical APDLRatio
Frequency (mode 1) 351699.332 351609.996 1.000
Frequency (mode 2) 1381162.303 1379724.174 1.001
Frequency (mode 3)3096816.0343089542.9861.002