VM59

VM59
Lateral Vibration of an Axially-loaded Bar

Overview

Reference:

S. Timoshenko, D. H. Young, Vibration Problems in Engineering, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 374, article 59.

Analysis Type(s):

Linear Perturbed Modal Analysis (ANTYPE = 2)

Element Type(s):

3D 2 node beam (BEAM188)

Input Listing:

vm59.dat

Test Case

A square cross-sectioned bar of length and weight per unit length γA is pinned at its ends and subjected to an axial compressive force F. Determine the stress σ and the axial displacement δ of the bar under these conditions. Determine the first three natural frequencies f_i of lateral vibration of the bar.

Figure 83: Axially-Loaded Bar Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

γ = 0.281 lb/in³

ρ = γ / g = 0.000727973 lb-sec²/in⁴

b = h = 2 in

= 80 in

F = 40,000 lb

Analysis Assumptions and Modeling Notes

Enough elements are selected so that the same model can be used to adequately characterize the dynamic analysis. Linear perturbed modal analysis is performed to include the prestress effects generated in the base static solve..

Results Comparison

		Target	Mechanical APDL	Ratio
Static	Deflection, in	-0.026667	-0.026667	1.000
Static	Stress , psi	-10,000	-10,000	1.000
Modal	F = 40,000 lb
	f₁ , Hz	17.055	17.013	0.998
	f₂ , Hz	105.32	104.82	0.995
	f₃ , Hz	249.39	246.94	0.990
	F = 0 lb
	f₁ , Hz [see VM50 ]	28.766	28.761	1.000