VM85

VM85
Transient Displacements in a Suddenly Stopped Moving Bar

Overview

Reference:

S. Timoshenko, D. H. Young, Vibration Problems in Engineering, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 305, problem 3.

Analysis Type(s):

Mode-Superposition Transient Dynamic Analysis (ANTYPE = 4)

Element Type(s):

3D Spar (or Truss) Elements (LINK180)

Gap Condition (GP)

Input Listing:

vm85.dat

Test Case

A steel bar moving along the X-axis with a constant velocity v_o is suddenly stopped at the end X = 0. Determine the displacement at the free end and the axial stress σ_x near the stopped end of the bar at time , where a is the speed of sound in the bar.

Figure 120: Moving Bar Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

= 0.00073 lb-sec²/in⁴

= 10000 in

A = 1 in²

s_o = 0.64 in

v_o = 100 in/sec

Analysis Assumptions and Modeling Notes

The speed of sound in the bar is = 202,721 in/sec, hence time = 0.0493288 sec. A static solution is done at the first load step. The final time of 0.06 sec allows the bar to impact (at t_o = 0.0064 sec) and reach its maximum deflection (at t = t_o + t₁). The gap stiffness (k = 30,000,000 lb/in) is arbitrarily selected high enough to minimize the elastic contact deformation but low enough to also allow a practical integration time step size.

The integration time step size (ITS = 0.0001 sec) is based on the shortest period (during contact) to allow the abrupt changes in acceleration to be followed reasonably well, and to produce sufficient printout for the theoretical comparison.

The initial velocity is produced by a force = 1,825,000 lb acting over 4 ITS. The force is halved at the first and last node (nodes 1 and 17) since these nodes are only attached to one element instead of two elements like the other nodes. A "coasting" period of 60 ITS is allowed before the gap (s_o = 0.64 in) closes at impact. An expansion pass is done at t = t_o + t₁ to obtain the stress solution. POST26 is used to get the displacement solution and displays versus time.

Results Comparison

		Target[1]	Mechanical APDL [2]	Ratio
Mode-Superposition Transient Dynamic	d, in (t = 0.05573 sec)	4.9329	-	-
	d, in (t = 0.0544 sec)	-	4.7733	0.968
	d, in (t = 0.0557 sec)	-	4.7474	0.962
Expansion Pass	stress_x , psi (t = 0.05573 sec)	14799.	-	-
Expansion Pass	stress_x , psi (t = 0.0557 sec)	-	14819	1.001

t is time before contact (0.0064 sec) included.
d is relative displacement between the ends of the bar.

Assumes an infinitely rigid stop.
Uses a high, but finite stiffness for the stop.
Peak displacement.
Displacement at the time closest to the theoretical time point given.
From Element 1.

Figure 121: Displacements at Center and Ends of Bar

Figure 122: Velocities at Center and Ends of Bar