1. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 1 A Time Domain, Curve-Fitting Method for Accelerometer Data Analysis By Tom Irvine

2. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 2 Objective 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.

3. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 3 Launch 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

4. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 4 Variables y(t) Amplitude Function A Amplitude constant nn Natural frequency  Damping ratio  Phase angle t Time

5. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 5 Candidate Functions for Data Curve-fit Pure Sine Series of Pure Sinusoids Lightly-damped Sine

6. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 6 Application 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.

7. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 7 Notes 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.

8. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 8 Example 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.

9. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 9 Example 1: Damped Sine Data

10. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 10 Example 1: Numerical Results AmplitudeA0.92 Natural Frequency fn9.56 Hz Damping  1.2% Phase  6.108 rad

11. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 11 Example 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.

12. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 12 Example 2: Frequency Variation

13. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 13 Example 2: Time History

14. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 14 Example 2: Numerical Results AmplitudeA0.82 G Oscillation Frequency fn488.2 Hz Phase  1.048 rad

15. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 15 Example 3. Flight Anomaly

16. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 16 Example 3: Segment

17. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 17 Example 3: Numerical Results ParameterDominant Signal Harmonic Amplitude1.5 G0.71 G Oscillation Frequency12.5 Hz37.4 Hz Phase0.854 rad3.672 rad The data reveals the dominant forcing frequency and a 3X harmonic. This data could be used to troubleshoot the anomaly.

18. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 18 Example 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.

19. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 19 Example 4: Time History

20. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 20 Example 4: Synthesis Equation Steps: Synthesize the first damped sinusoid. Subtract it from the signal. Synthesize the next damped sinusoid. Repeat these steps until n sinusoids are synthesized.

21. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 21 Example 4: Numerical Results ComponentAmplitude (G) Frequency (Hz) Phase (rad) DampingDelay (sec) 10.1095.224.9250.5%0.776 20.1095.066.3111.2%0.881 30.0402.535.9790.6%0.078 40.0452.640.9291.3%4.638 50.0121.180.5170.2%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.

22. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 22 Example 4: Fourier Transform

23. Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 23 Conclusion 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.