Optimized PSD Envelope for Nonstationary Vibration
Revision A
By Tom Irvine
Email: tom@vibrationdata.com
July 2, 2014
____________________________________________________________________________
FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE
10
ACCEL (G)
5
0
-5
-10
-5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
TIME (SEC)
Figure 1.
Introduction
There is a need to derive a PSD envelope for nonstationary acceleration time histories,
including launch vehicle data, which may be similar to that shown in Figure 1.
A PSD can be derived using rainflow fatigue cycle counting along with a Miners-type relative
fatigue damage index as shown in Appendix A. The enveloping is then justified using a
comparison of fatigue damage spectra between the candidate PSD and the measured time
history, using the methods in References 1 through 4. The fatigue damage spectrum gives the
cumulative damage index as a function of natural frequency for duly-noted Q and b values.
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The derivation process can be performed in a trial-and-error manner in order to obtain the PSD
with the least overall GRMS level, or least PSD level, which still envelops the flight data in terms
of fatigue damage spectra. The semi-empirical Dirlik method can be used to calculate the
fatigue damage spectrum of each candidate PSD in the frequency domain, instead of using the
longer, time domain synthesis approach.
Furthermore, this can be done for a number of amplification factor Q and fatigue exponent b
permutations for the case where these values are unknown. This adds conservatism to the final
PSD envelope.
Again, the goal is to derive the minimum PSD which envelopes the measured data in terms of
fatigue. The PSD’s duration can be selected by the user. It may or may not be the same as that
of the measured data. The method will scale the PSD to compensate for either a shorter or
longer duration.
Statistical uncertainty factors or safety margins can then be added to the PSD envelope as a
separate, post-processing step, with consideration that sufficient conservatism may have
already been added via a family of Q and b values.
Example
Derive an optimized PSD with a 60-second duration to envelope the nonstationary, base input
acceleration time history in Figure 1. This is needed for a hypothetical avionics component
which is to be mounted near the measurement location on a future flight. The component
must be designed and tested accordingly.
Measured Time History Fatigue Damage Spectra

x
m = mass
m
c = damping coefficient
k
c
y
k = stiffness
Figure 2. Single-degree-of-freedom System Subject to Base Acceleration
2
The first step is to calculate the time domain responses of an array of single-degree-of-freedom
(SDOF) systems to the base input, using the method in Reference 5.
An individual SDOF system is shown in Figure 2. 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. The fn, Q and b are unknown for the avionics component, so
assume the permutations shown in Table 1. Only the fn and Q values are needed for this
intermediate calculation. The fatigue exponent b will be applied in a later calculation. The fn
values will be varied in steps from 20 to 2000 Hz.
Table 1. Q & b Values for Fatigue Damage Spectra
Case
Q
B
1
10
4
2
10
9
3
30
4
4
30
9
A rainflow cycle count is made for each response time history as a function of fn and Q, using
the method in Reference 1.
Next the fatigue damage spectrum is calculated for the set of response time histories using the
b values and the method in Appendix A.
These calculations yield fatigue damage spectra for each of the cases in Table 1, as shown later
in this report.
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.
3
The Dirlik rainflow cycle counting method is then performed for each response PSD using the
method in References 2 through 4.
The fatigue damage spectrum is calculated for each of the cases using the method in Appendix
A.
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.
Results
The base input time history in Figure 1 was read into to a C++ program: envelope_fds.cpp,
version 1.3. The program then calculated fatigue damage spectra for this time history for the
four permutations in Table 1.
Next the program generated 800 candidate PSDs then performed the fatigue damage spectra
and scaling steps previously outlined.
The run-time on a desktop PC with Windows 7, 64-bit, and 8192 MB Ram was about 75
minutes.
The PSD envelope with the least overall displacement, velocity and acceleration overall RMS
levels is given in Figure 3.
The fatigue damage spectra comparison is given in Figures 4 through 7.
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POWER SPECTRAL DENSITY ENVELOPE
3.3 GRMS OVERALL
2
ACCEL (G /Hz)
0.1
0.01
0.001
20
100
1000
2000
FREQUENCY (Hz)
PSD Envelope, 3.3 GRMS, 60 sec
Freq
(Hz)
20
Accel
(G^2/Hz)
0.0018
31
0.0019
211
0.0168
2000
0.0024
Figure 3.
5
FATIGUE DAMAGE SPECTRA
10
Q=10 b=4
10
DAMAGE INDEX
PSD Envelope
Measured Data
10
8
10
6
10
4
10
2
20
100
1000
2000
NATURAL FREQUENCY (Hz)
Figure 4.
6
FATIGUE DAMAGE SPECTRA
10
Q=30 b=4
11
DAMAGE INDEX
PSD Envelope
Measured Data
10
9
10
7
10
5
10
3
20
100
1000
2000
NATURAL FREQUENCY (Hz)
Figure 5.
7
FATIGUE DAMAGE SPECTRA
10
Q=10 b=9
17
DAMAGE INDEX
PSD Envelope
Measured Data
10
14
10
11
10
8
10
5
10
2
20
100
1000
2000
NATURAL FREQUENCY (Hz)
Figure 6.
8
FATIGUE DAMAGE SPECTRA
10
Q=30 b=9
19
DAMAGE INDEX
PSD Envelope
Measured Data
10
16
10
13
10
10
10
7
10
4
20
100
1000
2000
NATURAL FREQUENCY (Hz)
Figure 7.
Conclusions
An optimized PSD envelope was successfully derived for the measured data using the method
outlined in the main text. The PSD is very conservative for three of the four cases.
The results show that higher values for Q and b lead to a higher PSD envelope for the sample
data in Figure 1. This would probably be true for most real-world measured time histories, but
there could be some exceptions. Thus varying both Q and b is advised. Both of these values
are huge uncertainty factors for “black boxes.”
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References
1. ASTM E 1049-85 (2005) Rainflow Counting Method, 1987.
2. Halfpenny & Kim, Rainflow Cycle Counting and Acoustic Fatigue Analysis Techniques
for Random Loading, RASD 2010 Conference, Southampton, UK.
3. Halfpenny, A frequency domain approach for fatigue life estimation from Finite
Element Analysis, nCode International Ltd., Sheffield, UK.
4. T. Irvine, Experimental Verification of the Dirlik Fatigue Cycle Method, Vibrationdata,
2013.
5. David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response
Spectra, Shock and Vibration Bulletin, No. 51, May 1981.
6. T. Irvine, An Introduction to the Vibration Response Spectrum, Revision D,
Vibrationdata, 2009.
7. T. Irvine, A Method for Power Spectral Density Synthesis, Rev B, Vibrationdata, 2000.
APPENDIX A
A relative damage index D can be calculated 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.
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APPENDIX B
Dirlik Method
The nth spectral moment m n for a PSD is

0
m n   f n G(f ) df
(B-1)
where
is frequency
f
G(f)
is the one-sided PSD
The expected peak rate E[P] is
E[P] 
(B-2)
m4 m2
The Dirlik histogram formula N(S) for stress cycles ranges is
N(S)  E[P]  T  p (S)
(B-3)
where
T
is the duration
S
is the stress cycle range (peak-to-peak)
The function p(S) is
p (S) 
2
  Z2 

D1
 Z D Z
  D Z exp   Z 
exp     2 exp 
3
 2R 2 
 2 
Q
 Q
R2




2 m0
(B-4)
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The coefficients and variables are
D1 
D2 

2 xm  2
1 2

1    D1  D12
1 R
D3  1  D1  D 2
Z
Q
S
1.25   D 3 D 2 R 
D1
  x m  D12
R
1    D1  D12
m2
m1
m0
(B-7)
(B-9)
(B-10)
(B-11)
m0 m4
xm 
(B-6)
(B-8)
2 m0

(B-5)
m2
m4
(B-12)
A cumulative histogram of the peaks can then be calculated from equation (B-3).
The stress range of individual cycles can then be obtained by interpolating the cumulative
histogram.
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