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bd Systems, Inc. Advanced Technology Division. Waveform Reconstruction via Wavelets October 2005 Revision B. bd Systems, Inc. Advanced Technology Division 600 Boulevard South, Suite 304 Huntsville, Alabama 35802 (256) 882-2650 (256) 882-2683 Fax. Objective. - PowerPoint PPT Presentation
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11
bd Systems, Inc.Advanced Technology Division
Waveform Reconstruction via Wavelets Waveform Reconstruction via Wavelets
October 2005October 2005
Revision BRevision B
bd Systems, Inc.Advanced Technology Division
600 Boulevard South, Suite 304Huntsville, Alabama 35802
(256) 882-2650(256) 882-2683 Fax
22
ObjectiveObjective
Match both measured time domain signal and Match both measured time domain signal and corresponding SRS for shaker shock testing, corresponding SRS for shaker shock testing, using a series of wavelets.using a series of wavelets.
33
BackgroundBackground
Aerospace & military components are subjected to shock tests Aerospace & military components are subjected to shock tests
to verify their integrity with respect to shock environments.to verify their integrity with respect to shock environments.
The specification format may be:The specification format may be:
• Drop onto hard surface from prescribed heightDrop onto hard surface from prescribed height
• MIL-S-901 shock machineMIL-S-901 shock machine
• Classical pulse such as half-sineClassical pulse such as half-sine
• Shock Response Spectrum (SRS)Shock Response Spectrum (SRS)
SRS is the most common format for launch vehicles.SRS is the most common format for launch vehicles.
44
SRS AnimationSRS Animation
Natural Frequencies (Hz):
0.063 0.125 0.25 0.50 1.0 2.0 4.0
SoftMount
HardMount
Animation File: HS_SRS.avi Click on image to begin.
Base Input: 1 G, 1 sec Half-sine Pulse
55
Shaker ShockShaker Shock
• The shock test may be performed on a shaker if the shaker’s frequency and The shock test may be performed on a shaker if the shaker’s frequency and amplitude capabilities are sufficientamplitude capabilities are sufficient
• A time history must be synthesized to meet the SRS specification A time history must be synthesized to meet the SRS specification
• Typically damped sines or waveletsTypically damped sines or wavelets
• The net velocity and net displacement must be zeroThe net velocity and net displacement must be zero
66
SRS SynthesisSRS Synthesis
• A series of wavelets can be synthesized to satisfy an SRS A series of wavelets can be synthesized to satisfy an SRS specification for shaker shockspecification for shaker shock
• Wavelets have zero net displacement and zero net velocityWavelets have zero net displacement and zero net velocity
• Damped sines require compensation pulseDamped sines require compensation pulse
• Assume control computer accepts ASCII text time history file for Assume control computer accepts ASCII text time history file for shock test in following examplesshock test in following examples
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Wavelet EquationWavelet Equation
m
mdmdm f2
Ntttfor
dmttmf2sindmttmNmf2
sinmA)t(mW
Wm (t) = acceleration at time t for wavelet m
Am = acceleration amplitude f m = frequency t dm = delay
Nm = number of half-sines, odd integer > 3
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Sample WaveletSample Wavelet
-50
-40
-30
-20
-10
10
20
30
40
50
0
0 0.02 0.04 0.06 0.080.012
9
8
7
6
5
4
3
2
1
TIME (SEC)
AC
CE
L (G
)WAVELET 1 FREQ = 74.6 Hz
NUMBER OF HALF-SINES = 9 DELAY = 0.012 SEC
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InnovationInnovation
• A wavelet series may also be used to reconstruct a time historyA wavelet series may also be used to reconstruct a time history
• This is done using brute-force curve fitting with random number This is done using brute-force curve fitting with random number generationgeneration
• The resulting series satisfies both the time history and the SRSThe resulting series satisfies both the time history and the SRS
1010
Example 1: Single Time HistoryExample 1: Single Time History
-300
-200
-100
0
100
0 0.05 0.10 0.15 0.20-100
0
100
200
300
Wavelet Synthesis
Measured Data
TIME (SEC)
AC
CE
L (G
)
AC
CE
L (G
)
STS-6 B08D7127 FORWARD IEA, LONG
1111
Wavelet Series with 3 of 60 ComponentsWavelet Series with 3 of 60 Components
-100
0
100-100
0
100-100
0
100-100
0
100
0 0.05 0.10 0.15 0.20
Wavelet 3, 44.7 Hz
Wavelet 2, 80.8 Hz
Wavelet 1, 74.6 Hz
Wavelet Series
TIME (SEC)
AC
CE
L (G
)
SYNTHESIZED WAVELET SERIES AND THREE COMPONENTS
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More Synthesized Pulse Time HistoriesMore Synthesized Pulse Time Histories
-100
-50
0
50
100
0 0.05 0.10 0.15 0.20
TIME (SEC)
VE
L (I
N/S
EC
)
VELOCITY OF SYNTHESIZED PULSE
-0.5
0
0.5
0 0.05 0.10 0.15 0.20
TIME (SEC)
DIS
P (I
NC
H)
DISPLACEMENT OF SYNTHESIZED PULSE
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Example 1: SRS of Wavelet SeriesExample 1: SRS of Wavelet Series
1
10
100
1000
10 100 1000 2000
SynthesisMeasured
NATURAL FREQUENCY (Hz)
PE
AK
AC
CE
L (G
)
SRS Q=10 STS-6 B08D7127 FORWARD IEA, LONG
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Example 1: ConclusionExample 1: Conclusion
• Wavelet time history can be performed as shaker shock, Wavelet time history can be performed as shaker shock, satisfying both time history and SRS satisfying both time history and SRS
• Add safety margin if appropriateAdd safety margin if appropriate
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Multiple WaveformsMultiple Waveforms
• The reconstruction method can be extended for the case where The reconstruction method can be extended for the case where multiple measurements are taken in the same axis over a number of multiple measurements are taken in the same axis over a number of accelerometer locations or flightsaccelerometer locations or flights
• Spatial and flight-to-flight variation are both concernsSpatial and flight-to-flight variation are both concerns
1616
Example 2. Four Measured Time HistoriesExample 2. Four Measured Time Histories
-100
0
100-100
0
100-100
0
100-100
0
100
0 0.05 0.10 0.15 0.20
Signal 4
Signal 3
Signal 2
Signal 1
TIME (SEC)
AC
CE
L (G
)
MEASURED ACCELERATION TIME HISTORIES
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P95/50 SRSP95/50 SRS
10
100
1000
10 100 1000 2000
Signal 4Signal 3Signal 2Signal 1P95/50
NATURAL FREQUENCY (Hz)
PE
AK
AC
CE
L (G
)
SHOCK RESPONSE SPECTRUM Q=10 P95/50 ENVELOPE
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Example 2: Composite Signal DerivationExample 2: Composite Signal Derivation
• Add the four signalsAdd the four signals
• Shift time scale and invert amplitudes as necessary to achieve highest Shift time scale and invert amplitudes as necessary to achieve highest GRMS valueGRMS value
• Use brute force random number generationUse brute force random number generation
• Scaling in next steps will compensate for potential constructive and Scaling in next steps will compensate for potential constructive and destructive interference in composite pulse destructive interference in composite pulse
1919
Example 2: Composite SignalExample 2: Composite Signal
0 0.05 0.10 0.15 0.20
TIME (SEC)
AC
CE
LCOMPOSITE SHOCK PULSE - ARBITRARY SCALE
2020
Example 2: Synthesis and Scaling StepsExample 2: Synthesis and Scaling Steps
• Synthesize a wavelet time history to match composite pulseSynthesize a wavelet time history to match composite pulse
• Re-scale wavelet parameters so the wavelet SRS satisfies the Re-scale wavelet parameters so the wavelet SRS satisfies the P95/50 SRSP95/50 SRS
• Brute force random number generation is used for each stepBrute force random number generation is used for each step
• Final time history should “reasonably resemble” the composite of Final time history should “reasonably resemble” the composite of the four original signalsthe four original signals
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Example 2: Resulting Time HistoryExample 2: Resulting Time History
-150
-100
-50
0
50
100
150
0 0.05 0.10 0.15 0.20
TIME (SEC)
AC
CE
L (G
)
ACCELERATION WAVELET SYNTHESIS OF COMPOSITE PULSE SCALED TO MEET P95/50 SRS
2222
Example 2: Time History ComparisonExample 2: Time History Comparison
-150-100
-500
50100150
0 0.05 0.10 0.15 0.20
TIME (SEC)
AC
CE
L (G
)
ACCELERATION SCALED WAVELET SYNTHESIS
0 0.05 0.10 0.15 0.20
TIME (SEC)
AC
CE
L
COMPOSITE SHOCK PULSE - ARBITRARY SCALE
2323
Example 2: More Time HistoriesExample 2: More Time Histories
-150-100
-500
50100150
0 0.05 0.10 0.15 0.20
TIME (SEC)
VE
L (IN
/SE
C)
VELOCITY WAVELET SYNTHESIS OF COMPOSITE PULSE
-1.0
-0.5
0
0.5
1.0
0 0.05 0.10 0.15 0.20
TIME (SEC)
DIS
P (I
NC
H)
DISPLACEMENT WAVELET SYNTHESIS OF COMPOSITE PULSE
2424
Example 2: SRS ComparisonExample 2: SRS Comparison
10
100
1000
2000
10 100 1000 2000
Scaled Wavelet Synthesis of Composite Pulse3 dB Tolerance Bands about P95/50 SRS
NATURAL FREQUENCY (Hz)
PE
AK
AC
CE
L (G
)
SHOCK RESPONSE SPECTRA Q=10
2525
Example 2: ConclusionExample 2: Conclusion
• Synthesized wavelet series can be performed as a shaker shock Synthesized wavelet series can be performed as a shaker shock
• Both composite pulse and P95/50 SRS are satisfiedBoth composite pulse and P95/50 SRS are satisfied
2626
Future ResearchFuture Research
• Improve brute force methods with convergence algorithmsImprove brute force methods with convergence algorithms
• Optimize waveforms to minimize peak velocity and displacement Optimize waveforms to minimize peak velocity and displacement while still meeting other goalswhile still meeting other goals
• Address mechanical impedance and force limiting concernsAddress mechanical impedance and force limiting concerns