Derivation of Equivalent Power Spectral Density Specifications for
Swept Sine-on-Random Environments via Fatigue Damage Spectra
By Tom Irvine
December 30, 2014
Email: tom@vibrationdata.com
_____________________________________________________________________________________
Introduction
Launch vehicles may encounter a variety of mixed sine and random vibration environments during
powered flight. The random vibration is typically driven by turbulent boundary layers, shock waves, and
other aerodynamic flow effects. The sine vibration may be due to a thrust oscillation for the case of a
solid motor. Furthermore, the thrust oscillation frequency and amplitude may each vary with time. Both
the sine and random environments may thus be nonstationary.
Avionics components must be designed and tested to withstand the composite vibration environment. A
single power spectral density specification which envelops the complete environment is usually desired
for simplicity. Furthermore, the power spectral density (PSD) specification is assumed to have a
corresponding time history which is both stationary and Gaussian.
Traditional specification derivation methods involve assuming piecewise stationary flight data and
making a maximum envelope from the piecewise segments. A more discerning method is to use the
fatigue damage spectrum method to derive a stationary power spectral density which yields the equivalent
fatigue damage of the composite nonstationary flight data. This paper demonstrates this fatigue damage
enveloping method using the method from References 1 and 2. An overview of the method is given in
Appendix A.
Examples
The test item is assumed to be a single-degree-of-freedom system.
independent variable from 20 to 2000 Hz.
The natural frequency is an
The amplification factor is varied as Q=10 and 30. The fatigue exponent is varies as b=4 and b=6.4.
There are thus four combinations of Q and b.
A PSD envelope is developed using trial-and-error to cover all four cases.
envelope that of the time history for each of the four cases.
The FDS of the PSD must
The PSD is also optimized to give the least possible overall levels for a fixed number of breakpoints
while still satisfying the enveloping requirement. The calculation is performed using a function within
the Vibration Matlab GUI package. The function within this package is PSD Envelope via FDS.
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Each PSD envelope will have four breakpoints with an intermediate plateau, but other options could be
chosen.
Example 1
Flight Data No. 1
4
3
2
Accel (G)
1
0
-1
-2
-3
-4
42
44
46
48
50
52
54
56
58
60
62
64
Time (sec)
Figure 1. Example 1, Time History, Flight Accelerometer Data
The time history in Figure 1 is actual flight accelerometer data from a suborbital launch vehicle during
stage 2 burn. The vibration is predominantly sine sweep due to a solid rocket motor thrust oscillation.
There is also some background random vibration.
A PSD envelope is derived for this segment as shown in the following plots. The PSD duration is set as
20 seconds, but this value can be varied.
2
Waterfall FFT
Time (sec)
Frequency (Hz)
Figure 2. Example 1, Waterfall FFT, Flight Accelerometer Data
The waterfall FFT shows the thrust oscillation sine sweep beginning at 520 Hz and sweeping down to 450
Hz. The corresponding amplitude varies with time, perhaps due to coupling with a structural resonance.
The cause of the peaks at 280 Hz is unknown.
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Power Spectral Density Flight Data No. 1
2.0 GRMS Overall
0.01
2
Accel (G /Hz)
0.001
0.0001
0.00001
20
100
1000
2000
Frequency (Hz)
Power Spectral Density,
2.0 GRMS Overall, 20 seconds
Frequency
(Hz)
Accel
(G^2/Hz)
20
2.90E-05
454
7.23E-03
494
7.23E-03
2000
3.63E-04
Figure 3. Example 1, PSD Envelope
The PSD in Figure 3 envelops the swept sine time history in Figure 1 in terms of fatigue damage spectra.
The justification is shown via the fatigue damage spectra in Figure 4.
4
PSD
Time History
Fatigue Damage Spectra Q=10 b=4
Fatigue Damage Spectra Q=30 b=4
9
10
10
10
7
8
4
Damage (G )
10
4
Damage (G )
10
5
10
3
10
4
10
1
10
2
-1
10
10
10
0
20
100
1000 2000
20
100
1000 2000
Natural Frequency (Hz)
Natural Frequency (Hz)
Fatigue Damage Spectra Q=10 b=6.4
Fatigue Damage Spectra Q=30 b=6.4
12
10
14
9
10
10
11
6.4
Damage (G )
10
6.4
Damage (G )
6
10
6
10
3
10
0
5
10
2
10
10
-3
10
8
10
-1
20
100
1000 2000
10
20
Natural Frequency (Hz)
Figure 4.
100
1000 2000
Natural Frequency (Hz)
Example 1, FDS Comparison for Four Cases
The FDS of the PSD envelops that of the flight accelerometer time history for each of the four cases and
over the entire natural frequency domain. The two Q=30 cases drive the PSD envelope.
The PSD envelope is conservative within the stated assumptions.
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Example 2
Flight Data No. 2
20
15
10
Accel (G)
5
0
-5
-10
-15
-20
0
5
10
15
20
25
30
35
40
45
50
55
60
65
Time (sec)
Figure 5. Example 2, Time History, Simulated Flight Accelerometer Data
The time history in Figure 4 is actually a combination of flight from two different vehicles. The idea was
to create a data set where several different effects occurred, including random, sine, and sine sweep.
A PSD envelope is derived for this segment as shown in the following plots. The PSD duration is set as
60 seconds, but this value can be varied.
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Waterfall FFT
Time (sec)
Frequency (Hz)
Figure 6. Example 2, Waterfall FFT, Simulated Flight Accelerometer Data
The waterfall FFT shows a mixture of vibration events. Random vibration due to aerodynamic effects
dominates the response from 10 to 30 seconds. A sine sweep occurs from 30 to 50 seconds due to a
simulated thrust oscillation. Sinusoidal spectra occur near the end of the record.
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Power Spectral Density Flight Data No. 2
3.6 GRMS Overall
2
Accel (G /Hz)
0.1
0.01
0.001
20
100
1000
2000
Frequency (Hz)
Power Spectral Density,
3.6 GRMS Overall, 60 seconds
Frequency
(Hz)
Accel
(G^2/Hz)
20
0.0012
195
0.0117
751
0.0117
2000
0.0013
Figure 7. Example 2, PSD Envelope
The PSD in Figure 5 envelops the swept sine time history in Figure 5 in terms of fatigue damage spectra.
The justification is shown via the fatigue damage spectra in Figure 8.
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PSD
Time History
Fatigue Damage Spectra Q=10 b=4
Fatigue Damage Spectra Q=30 b=4
10
11
10
10
9
4
Damage (G )
4
Damage (G )
8
10
6
10
4
10
100
10
1000 2000
100
1000 2000
Natural Frequency (Hz)
Fatigue Damage Spectra Q=10 b=6.4
Fatigue Damage Spectra Q=30 b=6.4
14
10
15
11
10
10
6.4
8
10
5
10
2
10
20
Natural Frequency (Hz)
Damage (G )
6.4
5
3
20
10
Damage (G )
7
10
10
2
10
10
20
12
9
10
6
10
3
100
1000 2000
10
20
Natural Frequency (Hz)
100
1000 2000
Natural Frequency (Hz)
Figure 8. Example 2, FDS Comparison for Four Cases
The FDS of the PSD envelops that of the flight accelerometer time history for each of the four cases and
over the entire natural frequency domain. The (Q=30, b=6.4) case is the driver.
The PSD envelope is conservative within the stated assumptions.
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References
1. ASTM E 1049-85 (2005) Rainflow Counting Method, 1987.
2. T. Irvine, Optimized PSD Envelope for Nonstationary Vibration, Revision A, Vibrationdata,
2014.
3. David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response Spectra,
Shock and Vibration Bulletin, No. 51, May 1981.
4. Halfpenny & Kim, Rainflow Cycle Counting and Acoustic Fatigue Analysis Techniques for
Random Loading, RASD 2010 Conference, Southampton, UK.
5. Halfpenny, A frequency domain approach for fatigue life estimation from Finite Element
Analysis, nCode International Ltd., Sheffield, UK.
APPENDIX A
Measured Time History Fatigue Damage Spectra

x
m = mass
c = damping coefficient
k = stiffness
m
k
c
y
Figure A-1. Single-degree-of-freedom System Subject to Base Acceleration
The first step is to calculate the time domain responses of an array of single-degree-of-freedom
(SDOF) systems to the base input per Reference 3.
An individual SDOF system is shown in Figure A-1. The natural frequency fn is an independent
variable. The amplification factor Q and the fatigue exponent b are fixed for a given case but are
varied between cases.
Again, the amplification factor is varied as Q=10 and 30 in this paper. The fatigue exponent is varies as
b=4 and b=6.4. There are thus four combinations of Q and b.
Only the fn and Q values are needed for this intermediate calculation.
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A rainflow cycle count is made for each response time history as a function of fn and Q, using
the method in Reference 1.
A relative damage index D can be calculated for each case using
m
D   Ai n i
b
(A-1)
i 1
where
Ai
is the response amplitude from the rainflow analysis
ni
is the corresponding number of cycles
b
is the fatigue exponent
Note that the amplitude convention for this paper is: (peak-valley)/2.
The fatigue damage spectrum expresses the relative damage as a function of natural frequency
for each of the four cases.
Optimized PSD Envelope Derivation
The next step is to generate a candidate base input PSD over the frequency domain from 20 to
2000 Hz. The PSD may have an arbitrary number of breakpoints, but simplicity is better. So
generate a PSD with four points. The PSD coordinates may be arbitrary using random numbers,
as long as the starting and ending frequencies are 20 and 2000 Hz, respectively.
Then the response PSD can be calculated for each candidate input PSD. This is done for each
natural frequency fn of interest and for each Q value. Again, the fn values will be varied in steps
from 20 to 2000 Hz.
The Dirlik rainflow cycle counting method is then performed for each response PSD per the
method in Reference 4 and 5.
The candidate base input PSD is then scaled so that each of its four fatigue damage spectra
envelops those of the measured acceleration time history, while keeping the PSD levels as small
as possible.
This process can be repeated for a few hundred candidate PSDs in order to find the least one
which still satisfies the fatigue damage enveloping requirement for the four cases.
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