Vibrationdata A Time Domain CurveFitting Method for Accelerometer

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

Objective Vibrationdata 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 2

Launch Vehicle Environments Vibrationdata 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 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 3

Vibrationdata Variables y(t) Amplitude Function A Amplitude constant wn Natural frequency x Damping ratio f Phase angle t Time AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 4

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

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

Notes Vibrationdata 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 7

Example 1: Pegasus Drop Transient Vibrationdata 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 8

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

Example 1: Numerical Results Amplitude A 0. 92 Natural Frequency fn 9. 56 Hz Damping x 1. 2% Phase f 6. 108 rad Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 10

Example 2: M 57 A 1 Motor Resonance Vibrationdata The M 57 A 1 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 M 57 A 1 has a distinct pressure oscillation. The oscillation frequency sweeps downward from 530 Hz to 450 Hz over a 16 -second duration. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 11

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

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

Vibrationdata Example 2: Numerical Results Amplitude A 0. 82 G Oscillation Frequency fn 488. 2 Hz Phase f 1. 048 rad AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 14

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

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

Example 3: Numerical Results Parameter Amplitude Oscillation Frequency Phase Vibrationdata Dominant Signal Harmonic 1. 5 G 0. 71 G 12. 5 Hz 37. 4 Hz 0. 854 rad 3. 672 rad The data reveals the dominant forcing frequency and a 3 X harmonic. This data could be used to troubleshoot the anomaly. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 17

Example 4: Launch Vehicle Transportation Vibrationdata 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 18

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

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

Vibrationdata Example 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 21

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

Conclusion Vibrationdata 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. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 23