VM84

VM84
Displacement Propagation along a Bar with Free Ends

Overview

Reference: S. Timoshenko, D. H. Young, Vibration Problems in Engineering, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 311, problem 2.
Analysis Type(s): Mode-Superposition Transient Dynamic Analysis (ANTYPE = 4)
Element Type(s): 3D Spar (or Truss) Elements (LINK180)
Input Listing: vm84.dat

Test Case

A drill stem is a steel bar 4000 ft long. Considering it as a bar with free ends, find the displacement  δ of the end (x =  ) at t = τ/2 produced by a force F1 suddenly applied to this end.

Figure 117: Bar with Free Ends Problem Sketch

Bar with Free Ends Problem Sketch

Material PropertiesGeometric PropertiesLoading
E = 30 x 106 psi
γ = 0.278 lb/in3
ρ = γ / g = 0.0007202 lb-sec2/in4
A = 2 in2
g = 386 in/sec2
= 4000 x 12 = 48,000 in
F1 = 6000 lb
(see load history in Figure 117: Bar with Free Ends Problem Sketch)

Analysis Assumptions and Modeling Notes

A static solution is done at the first load step. The fundamental period of vibration (τ) is 0.47937 sec. The final time of 0.24 sec includes 1/2 of the fundamental period of vibration. The integration time step (0.005 sec) is based on 1/10 of the shortest period, to allow the initial step change in acceleration to be followed reasonably well, and to produce sufficient printout for the theoretical comparison. POST26 is used to extract results from the solution phase.

Results Comparison

TargetMechanical APDLRatio
Displacement, in[1]4.80004.84041.008

  1. Target results are for t = 0.23969 sec (t/2). Mechanical APDL results are reported from the closest time point, t = 0.240 sec.

Figure 118: Displacement vs. Time Graph

Displacement vs. Time Graph

Figure 119: Velocity vs. Time Graph

Velocity vs. Time Graph