VM79

VM79
Transient Response of a Bilinear Spring Assembly

Overview

Reference:

W. T. Thomson, Vibration Theory and Applications, 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, pg. 150, fig. 5.6-1.

Analysis Type(s):

Mode-Superposition Transient Dynamic Analysis (ANTYPE = 4)

Element Type(s):

Combination Elements (COMBIN40)

Gap Condition (GP)

Input Listing:

vm79.dat

Test Case

A mass supported on a nonlinear spring is subjected to an impulsive force F(t) and thereafter undergoes free vibration. The spring stiffness is characterized by the force-deflection curve shown below. Determine the maximum deflection y_max of the mass. Compare results with those of VM74 .

Figure 110: Bilinear Spring Assembly Problem Sketch

Material Properties

Geometric Properties

m = 0.5 lb-sec²/in

k₁ = 200 lb/in

y_o = 0.75 in

see time history in Figure 110: Bilinear Spring Assembly Problem Sketch

Analysis Assumptions and Modeling Notes

The node locations are arbitrarily selected. The final time of 0.105 sec allows the mass to reach its largest deflection. A static solution is done at the first load step. A gap condition with a spring constant of -k₁ is applied in parallel with k₁ to produce a combined spring stiffness of zero at gap closure. The integration time step (0.0025 sec) is based on 1/125 of the period to allow the step changes in acceleration to be followed reasonably well and to produce sufficient printout for the theoretical comparison. The impulse is applied over one integration time step.

Results Comparison

Time = 0.09 sec	Target	Mechanical APDL	Ratio
Y_max, in	-1.0417	-1.04047	0.999

Table 1: Comparison of Mechanical APDL Linear and Bilinear Spring Results

Time, sec	0.040	0.070	0.085	0.105
y, in (linear)[1]	-0.68122	-0.97494	-0.99604	-0.88666
y, in (bilinear)	-0.68122	-0.98672	-1.0383	-1.0020

From test case VM74 output. Positive displacement direction is reversed for comparison.