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Derivation of Equivalent Power Spectral Density Specifications for Swept Sine-on-Random Environments via Fatigue Damage Spectra
By Tom IrvineDecember 30, 2014
Email: [email protected]
_____________________________________________________________________________________
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. The natural frequency is an independent variable from 20 to 2000 Hz.
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. The FDS of the PSD must envelope that of the time history for each of the four cases.
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
-4
-3
-2
-1
0
1
2
3
4
42 44 46 48 50 52 54 56 58 60 62 64
Time (sec)
Acc
el (G
)
Flight Data No. 1
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.
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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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Waterfall FFT
Frequency (Hz)
Time (sec)
0.00001
0.0001
0.001
0.01
100 100020 2000
Frequency (Hz)
Acc
el (G
2 /Hz)
Power Spectral Density Flight Data No. 1 2.0 GRMS Overall
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.
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PSD Time History
10-1
101
103
105
107
109
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G4 )
Fatigue Damage Spectra Q=10 b=4
100
102
104
106
108
1010
100 100020 2000
Natural Frequency (Hz)D
amag
e (G
4 )
Fatigue Damage Spectra Q=30 b=4
10-3
100
103
106
109
1012
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G6.
4 )
Fatigue Damage Spectra Q=10 b=6.4
10-1
102
105
108
1011
1014
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G6.
4 )Fatigue Damage Spectra Q=30 b=6.4
Figure 4. 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
-20
-15
-10
-5
0
5
10
15
20
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Time (sec)
Acc
el (G
)Flight Data No. 2
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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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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Waterfall FFT
Frequency (Hz)
Time (sec)
0.001
0.01
0.1
100 100020 2000
Frequency (Hz)
Acc
el (G
2 /Hz)
Power Spectral Density Flight Data No. 2 3.6 GRMS Overall
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
102
104
106
108
1010
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G4 )
Fatigue Damage Spectra Q=10 b=4
103
105
107
109
1011
100 100020 2000
Natural Frequency (Hz)D
amag
e (G
4 )
Fatigue Damage Spectra Q=30 b=4
102
105
108
1011
1014
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G6.
4 )
Fatigue Damage Spectra Q=10 b=6.4
103
106
109
1012
1015
100 100020 2000
Natural Frequency (Hz)
Dam
age
(G6.
4 )
Fatigue Damage Spectra Q=30 b=6.4
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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m
k c
x
y
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
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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m = massc = damping coefficientk = stiffness
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
D =∑i=1
m
A ib ni
(A-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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