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VibrationdataVibrationdata

A Time Domain, Curve-Fitting Method for Accelerometer Data Analysis

By Tom Irvine

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VibrationdataVibrationdataObjective

Demonstrate a time-domain, curve-fitting method for analyzing accelerometer data.

The method is innovative in that it uses random number generation to determine the characteristics of the measured data.

These characteristics include the amplitude, frequency, phase angle, and damping ratio of the signal's components.

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VibrationdataVibrationdataLaunch Vehicle Environments

The Time-Domain, Curve-Fitting Method can be Applied to Data from:

• Transportation Shock and Vibration

• Launch Shock

• Aerodynamic Flow Excitation

• Motor Pressure Oscillation

• Stage Separation Events

• Anomalies

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VibrationdataVibrationdataVariables

y(t) Amplitude Function

A Amplitude constant

nNatural frequency

Damping ratio

Phase angle

t Time

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VibrationdataVibrationdataCandidate Functions for Data Curve-fit

)tsin(A)t(y n

)tnsin()tnexp(A)t(y

Pure Sine

Series of Pure Sinusoids

Lightly-damped Sine

n

1iiii)tsin(A)t(y

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VibrationdataVibrationdataApplication Method

• The curve-fitting method generates random numbers for each of the variables.

• It then compares the resulting trial function with the measured data.

• This is done in a trial-and-error manner, implemented via a computer program.

• The final function is the one that produces the least error when subtracted from the measured signal.

• This method tends to be more appropriate for brief, transient signals rather than longer signals. It can be used for a longer signal, however, if the signal is divided into segments.

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VibrationdataVibrationdataNotes

The time-domain, curve-fitting method is intended to supplement frequency domain methods, particularly the Fourier transform.

Each method has its own strengths, as shown in the following examples.

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VibrationdataVibrationdataExample 1: Pegasus Drop Transient

Consider the Pegasus launch vehicle mounted underneath an L-1011. The most significant event for the payload is the drop transient from the carrier aircraft.

The Pegasus vehicle is like a free-free beam subjected to an initial displacement that varies along its length.

During the five-second free-fall interval, the initial strain energy is released, causing the Pegasus vehicle to experience a damped, transient oscillation.

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VibrationdataVibrationdataExample 1: Damped Sine Data

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

-0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Synthesized DataFlight Data

TIME (SEC)

NO

RM

ALI

ZE

D A

CC

ELE

RA

TIO

N

MEASURED DROP TRANSIENT AT PAYLOAD INTERFACE OF A PEGASUS LAUNCH VEHICLE

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VibrationdataVibrationdataExample 1: Numerical Results

Amplitude A 0.92

Natural Frequency

fn 9.56 Hz

Damping 1.2%

Phase 6.108 rad

)tnsin()tnexp(A)t(y

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VibrationdataVibrationdataExample 2: M57A1 Motor Resonance

The M57A1 motor is a solid-fuel motor originally developed as a third stage for the Minuteman missile program.

This motor has since been used on a variety of suborbital vehicles, such as target vehicles.

The M57A1 has a distinct pressure oscillation.

The oscillation frequency sweeps downward from 530 Hz to 450 Hz over a 16-second duration.

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VibrationdataVibrationdataExample 2: Frequency Variation

440

450

460

470

480

490

500

510

520

530

128 130 132 134 136 138 140 142 144 146 148

TIME (SEC)

FRE

QU

EN

CY

(Hz)

FREQUENCY vs. TIME SUBORBITAL TARGET VEHICLEM57A1 MOTOR RESONANCE AVIONICS MODULE SKIN

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VibrationdataVibrationdataExample 2: Time History

-4

-3

-2

-1

0

1

2

3

4

138.00 138.02 138.04 138.06 138.08 138.10

Synthesized SignalMeasured Data

TIME (SEC)

AC

CE

L (G

)

SUBORBITAL TARGET VEHICLEM57A1 MOTOR OSCILLATION AVIONICS MODULE SKIN

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VibrationdataVibrationdataExample 2: Numerical Results

Amplitude A 0.82 G

Oscillation Frequency

fn 488.2 Hz

Phase 1.048 rad

)tnsin(A)t(y

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VibrationdataVibrationdataExample 3. Flight Anomaly

-4

-3

-2

-1

0

1

2

3

4

87.0 87.5 88.0 88.5 89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.5

TIME (SEC)

AC

CE

L (G

)

LAUNCH VEHICLECONTROL SYSTEM OSCILLATION AT STAGE 1 BURN-OUT

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VibrationdataVibrationdataExample 3: Segment

-4

-3

-2

-1

1

2

3

4

0

90.0 90.5 91.0

Synthesized DataFlight Data

TIME (SEC)

AC

CE

L (G

)

LAUNCH VEHICLECONTROL SYSTEM OSCILLATION AT STAGE 1 BURN-OUT

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VibrationdataVibrationdataExample 3: Numerical Results

Parameter Dominant Signal

Harmonic

Amplitude 1.5 G 0.71 G

Oscillation Frequency 12.5 Hz 37.4 Hz

Phase 0.854 rad 3.672 rad

The data reveals the dominant forcing frequency and a 3X harmonic. This data could be used to troubleshoot the anomaly.

n

1iiii)tsin(A)t(y

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VibrationdataVibrationdataExample 4: Launch Vehicle Transportation

A suborbital launch vehicle is being integrated at a missile assembly building (MAB) at Vandenberg AFB.

The distance from the MAB to the launch pad is 20 miles. The assembled launch vehicle will be mounted horizontally on a custom trailer for transportation from the MAB to the pad.

The launch vehicle must withstand the lateral loading that occurs as the tractor-trailer crosses over potholes, railroad tracks, and joints at bridges.

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VibrationdataVibrationdataExample 4: Time History

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-0.6

0

0 2 4 6 8 10 12 14-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

0

Synthesized Signal, Right Scale

Measured Data, Left Scale

TIME (SEC)

AC

CE

L (G

)

AC

CE

L (G

)

VAFB TRANSPORTATION TESTLAUNCH VEHICLE STAGE 2 VERTICAL

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VibrationdataVibrationdataExample 4: Synthesis Equation

n

1iiiii )t̂sin()t̂exp(A)t(y

Steps:

Synthesize the first damped sinusoid.

Subtract it from the signal.

Synthesize the next damped sinusoid.

Repeat these steps until n sinusoids are synthesized.

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VibrationdataVibrationdataExample 4: Numerical Results

Component Amplitude (G)

Frequency(Hz)

Phase (rad)

Damping Delay (sec)

1 0.109 5.22 4.925 0.5% 0.776

2 0.109 5.06 6.311 1.2% 0.881

3 0.040 2.53 5.979 0.6% 0.078

4 0.045 2.64 0.929 1.3% 4.638

5 0.012 1.18 0.517 0.2% 1.438

The synthesis consisted of 30 damped sinusoids. Only the top five are shown for brevity.

The sinusoids near 5 Hz were due to launch vehicle bending modes. The spectral components near 1 Hz and 2.5 Hz were primarily due to the trailer suspension, with the launch vehicle acting as a rigid-body.

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VibrationdataVibrationdataExample 4: Fourier Transform

0

0.01

0.02

0.03

0.04

0 1 2 3 4 5 6 7 8 9 10

FREQUENCY (Hz)

ACC

EL (G

)

FOURIER TRANSFORM MAGNITUDETRANSPORATION VIBRATION LAUNCH VEHICLE STAGE 2 VERTICAL

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VibrationdataVibrationdataConclusion

The time-domain, curve-fitting method presented in this report is a simple, powerful tool for analyzing accelerometer signals.

It can be used to identify amplitude, frequency, damping, and other parameters.

Interested parties may contact the author for copies of the software used in the previous examples.