bd systems, inc. advanced technology division
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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