Plotting digital simulation errors for a spring-mass-damper system and using these error curves to select type of integration, feedback update method, and number of samples per cycle at resonance reduces excessive number of samples per cycle and unnecessary iterations

Knox, L. A.

English

NASA Technical Reports Server

February 1971 Brief 71-10003 
NASA TECH BRIEF 
Marshall Space Flight Center 
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commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
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 Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Digital Simulation Error Curves for a Spring-Mass-Damper System 
The problem:	 A = () is the normalized integration gain. 
	
Reduce the excessive number of samples per cycle
	 WD is the effective angular resonance frequency. 
and the unnecessary iterations which increase cost
	
wp is the programmed angular resonance frequency. 
and computation time on digital simulations of
	 cp is the programmed damping ratio. 
continuous data systems. 	 T is the sampling interval in seconds. 
XIN	 XZ1	 AXZ1	 A2XZ1 
2 +
 
WD
 
rL_.i	 r — --, 
: eTs:	 HeTS 
	
2pW 0	 T	 A2Xl 
A* 
Spring-Mass-Damper Simulation 
AXZ 1	 A2XZ1	 A2X	 XZ1	 AXZ1	 A* 
LU 
'LQ+ L___
eTS[1+TS] t ZIU  
1=1,1	 0 R	 eTS [i +Ts+T2s2)J	 Time Advance For 
Time Advance for X
Figure 1 
The solution: 
Plot the digital simulation errors for a spring-
mass-damper system. Using these error curves, select 
the type of integration, the feedback update method, 
and the number of samples per cycle at resonance. 
How it's done: 
Block diagrams of a spring-mass-damper simulation 
and the preferred time advance update method are 
shown in Figure 1, where:
Z	 is a transfer function for I seconds time delay. 
e	 is a transfer function for T seconds time advance. 
is a constant for time advance update. 
I is the number of iterations for iteration update. 
X, *, and 3 represent the mass displacement, 
velocity, and acceleration, respectively. 
S is the Laplace operator. 
N R is the number of samples per cycle at res-
onance. 
The effective damping ratio is:
	 D	 p + ER 
(continued ocrIeaf) 
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	 Government assumes any liability resulting from the use of the Aeronautics and Space Administration. Neither the United States
	 information contained in this document, or warrants that such use Government nor any person acting on behalf of the United States
	 will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19710000003 2020-03-17T02:17:54+00:00Z
1.04 
1.03 
1.02 
1.01 
1 1.00 
WD 
WP
.98 
.97 
.03 
.02 
.01 
ER 0 
—.01 
- .02 
- .03 
.03 
.02 
.01 
0 
ER
- .01 
- .02 
- .03 
- .04 810 14 20	 40 60 100 200 
Trapezoidal Integration 
.03 
02 
E1RO  
INR 
2	 810 14 20	 40 60 100 200 
Figure 2 Simpson's Integration 
where ER is the damping bias error. This error 
is plotted (Figure 2) for trapezoidal integration and 
Simpson's 1/3 formula, where NR is the number of 
samples per cycle at resonance. Curves are shown 
for the iteration update method where .1 = I, 
and for the time advance update method where 
1/K = 0, 1/3, and 1/2. 
The normalized resonance frequency, WD/ Cop, 
shown in Figure 3, is for trapezoidal integration and 
Simpson's 1/3 formula. 
For the curves for trapezoidal integration, the time 
advance update method with 1/K = 1/3 has smaller 
errors at 20 samples per cycle than the one-itera- 
tion update method at 30 samples per cycle. Thus, 
for the same accuracy, the time advance update 
method will solve the simulated problem in half 
the time of the one-iteration update method. This is 
based on an added time per iteration of T/4, i.e., 
one-quarter of the computation time without itera- 
tion.
iI\ 
 
=^Iri 2 
6 10	 20	 40	 80 
—.04
2	 810 14 20
	 40 60 100	 200 
Figure 3 Simpson's Integration 
Note: 
Requests for further information may be directed 
to:
Technology Utilization Officer 
Marshall Space Flight Center 
Code A&TS-TU 
Huntsville, Alabama 35812 
Reference: B71-10003 
Patent status: 
No patent action is contemplated by NASA. 
Source: Lewis Alfred Knox of 
International Business Machines Corp. 
under contract to
Marshall Space Flight Center 
(M F S-20770)
1=2 No U p -Date 
=oINR
Trapezoidal Integration 	
NR-
I7:i
is
. 
Brief 71-10003	 Category 09 


Digital simulation error curves for a spring-mass-damper system

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