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.

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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

dmttmf2sindmttmN

mf2sinmA)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

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

(IN

/SE

C)

VELOCITY OF SYNTHESIZED PULSE

-0.5

0

0.5

0 0.05 0.10 0.15 0.20

TIME (SEC)

DIS

P (

INC

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/S

EC

)

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 (

INC

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