An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

1. An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration Tom IrvineDynamic Concepts, Inc Email: tirvine@dynamic-concepts.com

2. The purpose of this presentation is to introduce a customizable framework for enveloping nonstationary random vibration using damage potential. Please keep the big picture in mind. The details are of secondary importance.

3. This project is an informal collaboration between: • NESC • NASA KSC • Dynamic Concepts • Space-X In the Spirit of the National Aeronautics and Space Act of 1958 Falcon 9 Liftoff

4. Random Vibration Environments • Lift-off Vibroacoustics • Transonic Shock Waves • Fluctuating Pressure at Max-Q Ares 1-X , Prandtl–Glauert Singularity, Vapor Condensation Cone at Transonic

5. Launch Vehicle Avionics Flight ComputersInertial Navigation Systems Transponders & Transmitters ReceiversAntennas Batteries etc. Image is from a SCUD-B missile. Would rather show image of US launch vehicle avionics, but cannot because such images are classified, FOUO, proprietary, no-show to foreigners, etc.

6. Launch vehicle avionics components must be designed and tested to withstand random vibration environments • These environments are often derived from flight accelerometer data of previous vehicles • The flight data tends to be nonstationary LCROSS vibration tests at the NASA Ames Research Center

7. Some Preliminaries . . . SDOF System

8. Shock Response Spectrum Model • The shock response spectrum is a calculated function based on the acceleration time history. • It applies an acceleration time history as a base excitation to an array of single-degree-of-freedom (SDOF) systems. • Each system is assumed to have no mass-loading effect on the base input.

9. Base Input: Half-Sine Pulse (11 msec, 50 G) RESPONSE (fn = 30 Hz, Q=10) SRS Example RESPONSE (fn = 80 Hz, Q=10) RESPONSE (fn = 140 Hz, Q=10)

10. SRS Q=10 Base Input: Half-Sine Pulse (11 msec, 50 G) NATURAL FREQUENCY (Hz)

11. Typical Power Spectral Density Test Level Corresponding time history shown on next slide. • The overall level is 6.1 GRMS. This is the square root of the area under the curve. • GRMS value = 1s ( std dev) assuming zero mean • The amplitude unit is G^2/Hz, but this is really GRMS^2/Hz

12. The time history is stationary • Time history is not unique because the PSD discards the phase angle • Time history could be performed on shaker table as input to avionics component • GRMS value = 1s ( std dev) assuming zero mean • Histogram of instantaneous values is Gaussian, normal distribution, bell-shaped curve

13. Response of an SDOF System to Random Vibration PSD Do not use Miles equation because it assumes a flat PSD from zero to infinity Hz. Instead, multiply the input PSD by the transmissibility function: where where f is the base excitation frequency and fnis the natural frequency.

14. Response of an SDOF System to Random Vibration PSD (cont.) • Multiply power transmissibility by the base input PSD • Sum over all input frequencies • Take the square root • The result is the overall response acceleration

15. Response Power Spectral Density Curves SDOF Systems Q=10 Each peak is Q2 times the base input at the natural frequency, for SDOF response. Next, calculate the overall level from each response curve. Again, this is the square root of the area under each curve.

16. Vibration Response Spectrum SDOF Systems Q=10 Later in the presentation, peak vibration response and accumulated damage will be plotted against natural frequency.

17. Rainflow Fatigue Cycles Endo & Matsuishi 1968 developed the Rainflow Counting method by relating stress reversal cycles to streams of rainwater flowing down a Pagoda.ASTM E 1049-85 (2005) Rainflow Counting Method Develop a damage potential vibration response spectrum using rainflow cycles. Goju-no-to Pagoda, Miyajima Island, Japan

18. Sample Time History

19. Rainflow Cycle Counting Rotate time history plot 90 degrees clockwise RainflowPlot Stress

20. Derive MEFL from Nonstationary Random Vibration • The typical method for post-processing is to divide the data into short-duration segments • The segments may overlap • This is termed piecewise stationary analysis • A PSD is then taken for each segment • The maximum envelope is then taken from the individual PSD curves • MEFL = maximum envelope + some uncertainty margin • Component acceptance test level > MEFL • Easy to do • But potentially overly conservative

21. Power Spectral Density Maximum Envelope of 3 PSD Curves Accel (G^2/Hz) Frequency (Hz) Piecewise Stationary Enveloping Method Concept Calculate PSD for Each Segment Segment 1 Segment 2 Segment 3 Would use shorter segments if we were doing this in earnest.

22. Nonstationary Random Vibration Liftoff Transonic Attitude Control Max-Q Thrusters Rainflow counting can be applied to accelerometer data.

23. Background Reference • S. J. DiMaggio, B. H. Sako, and S. Rubin, Analysis of Nonstationary Vibroacoustic Flight Data Using a Damage-Potential Basis, Journal of Spacecraft and Rockets, Vol, 40, No. 5. September-October 2003. • This is a brilliant paper but requires a Ph.D. in statistics to understand. • Need a more accessible method for the journeyman vibration analyst, along with a set of shareable software programs, including source code • Use same overall approach as DiMaggio, Sako & Rubin, but fill in the details using brute-force numerical simulation • Alternate method will be easy-to-understand but bookkeeping-intensive • But software does the bookkeeping

24. Objective • The goal of this presentation is to derive a Damage Potential PSD which envelops the respective responses of an array of SDOF systems in terms of both peak level and fatigue. • This must be done for • Three damping cases with Q=10, 25 & 50 ( 5%, 2% & 1%) • Two fatigue exponent cases with b=4 & 6.4 (slope from S-N curve) • A total of ninety natural frequencies, from 10 to 2000 Hz in one-twelfth octave steps • The total number of response permutations is 540, which is rather rigorous. This is needed because the avionics components’ dynamic characteristics are unknown.

25. Objective (cont.) • The alternate damage method in this paper builds upon previous work by addressing an additional concern as follows: • Consider an SDOF system with a given natural frequency and damping ratio • The SDOF system is subjected to a base input • The base input may vary significantly with frequency • The response of the SDOF system may include non-resonant stress reversal cycles

26. Typical SDOF Response to Previous Flight Accelerometer Data (nonstationary time history) Non-resonant Response Resonant Response Existing damage potential methods tend to assume that the response is purely resonant. The alternate method given in this paper counts the cycles as they occur for all frequencies.

27. Alternate Method Steps • Peak Response • The peak response is enveloped as follows. • Take the shock response spectrum of the flight data for three Q values and for the ninety frequencies. This is performed using program: qsrs_threeq.cpp. • Derive a Damage Potential PSD which has a VRS that envelops the SRS curves of the flight data for the three Q cases. This is performed using trial-and-error via program: envelope_srs_psd_three_q.cpp.

28. Alternate Method Steps (cont.) The enveloping is performed in terms of the n value which is the maximum expected peak response of an SDOF system to the based input PSD, as derived from the Rayleigh distribution of the peaks. The following equation for the expected peak is taken: (temporary assumption) where s is the standard deviation of response fn is the natural frequency T is the duration This step is performed using program: envelope_srs_psd_threeq_single.cpp.

29. As an Aside… Rayleigh Distribution Probability Density Function The Rayleigh distribution is a distribution of local peak values for the narrowband response time history of an SDOF system to a broadband, stationary, random vibration base input

30. As an Aside (cont)… Integrate the Rayleigh Probability Density Function where A is the absolute amplitude of the local peaks. Total number of peaks = fn T Probability * total peaks = 1 peak

31. As an Aside (cont)… Out of all the peaks, only one is expected > ls So assume : maximum peak  Assumes ideal Rayleigh distribution for narrowband SDOF Response to stationary input. Some “hand-waving” due to secondary effects of non-resonant cycles, damping, etc. Again, the maximum peak formula is used only temporarily.

32. Alternate Method Steps (cont.) Note that a longer duration T for the Damage Potential PSD allows for a lower base input PSD & corresponding time history amplitude. Furthermore this method seeks the minimum PSD for a set duration which will still satisfy the peak envelope requirement. The optimization is done via trial-and-error.

33. Alternate Method Steps (cont.) • Fatigue Check* • The peak response criterion tends to be more stringent than the fatigue requirement. But fatigue damage should be verified for thoroughness. • The fatigue damage for the Damage Potential PSD is performed as follows. • Synthesize a time history to satisfy the Damage-Potential PSD. This is performed using program: psdgen.cpp. The time history is non-unique because the PSD discards phase angles. • Calculate the time domain response for each of the three Q values and at each of the ninety natural frequencies. This is performed using program: arbit_threeq.cpp. * This is not “true fatigue” which would be calculated from stress. Rather it is a fatigue-like metric for accumulated response acceleration cycles.

34. Alternate Method Steps (cont.) • Taken the rainflow cycle count for each of the 270 response time histories. Note that the amplitude and cycle data does not need to be sorted into bins. This step is performed using program: rainflow_threeq.cpp. • Calculate the fatigue damage D for each of 270 rainflow responses for each of the two fatigue exponents as follows: This step is performed using program: fatigue_threeq.cpp. Steps 3 through 4 are then repeated for the flight accelerometer data. where

35. Example: Nonstationary Random Vibration The data could be divided into segments as shown in the table. But the entire signal will be used for the following example.

36. Shock Response Spectra Taken over the entire duration of the nonstationary data. Time domain calculation.

37. Derive Power Spectral Density • Derive a base input PSD so that the peak response of the SDOF system will envelope the Flight Data SRS at each corresponding natural frequency and Q factor • Select PSD duration = 60 seconds • But could justify using longer duration

38. Derive Power Spectral Density Trial-and-error derivation Response PSD Randomly Generated Candidate PSD Base Input (G^2/Hz) (G^2/Hz) Given fn & Q Freq (Hz) Freq (Hz) Repeat this calculation for all fn & Q values of interest. The overall GRMS is the square root of the area under the curve. Std dev (1s) = GRMS assuming zero mean. The peak is typically assumed to be 3s.But a better estimate is Typically > 3s

39. Family of Response PSDs Derive Power Spectral Density Trial-and-error derivation (cont.) (G^2/Hz) Freq (Hz) (G^2/Hz) VRS of Candidate PSD for given Q All fn of interest at given Q Freq (Hz) Peak (G) Again, peak values are determined via: Natural Frequency (Hz)

40. Derive Power Spectral Density Trial-and-error derivation (cont.) Response Spectra for given Q Candidate PSD Scale PSD by uniform factor so that its VRS envelops flight data for each Q Candidate VRS Peak (G) (G^2/Hz) Flight Data SRS Natural Frequency (Hz) Freq (Hz) • Perform the above process for a few thousand scaled candidate PSD functions to derive minimum PSD which satisfies the VRS/SRS comparison. • Derived & optimized PSD via trial-and-error using peak= • Program: envelope_srs_psd_three_q.cpp

41. Derived Power Spectral Density • The lowest-level PSD whose VRS envelops the Flight Data SRS for three Q cases. • Again, the PSD was derived by trial-and-error • The n VRS of the Damage Envelope PSD is shown for three Q values along with the flight data SRS curves on the next slide • Need to verify via numerical simulation for peak & fatigue

42. Response Spectra Comparison, Part IThe Damage Potential PSD envelops the corresponding SRS curves in terms of peak response for three Q cases. Damage potential VRS uses This will be verified in the time domain in upcoming slides.

43. Numerical Simulation Synthesize a time history to satisfy the Damage Potential PSD. Verify that the PSDs match.

44. Response Spectra Comparison, Part II Verification in the time domain for three Q cases Relaxed reliance on because experimental proof that Damage Synthesis envelops Flight Data

45. SDOF Response Time History Comparison (fn=189 Hz, Q=10)

46. SDOF Response Time History Comparison (fn=280 Hz, Q=10)

47. Fatigue Response Spectra Comparison Three Q cases, b=6.4

48. Fatigue Response Spectra Comparison Three Q cases, b=4

49. Conclusions • Successfully derived a MEFL PSD using the alternate Damage-Potential method • Could reduce MEFL PSD level by using a longer duration • Peak requirement tended to be more stringent than fatigue for the case considered • The alternate Damage-Potential method is intended to be another tool in the analyst’s toolbox • Each flight time history is unique • The derivation of PSD envelopes by any method requires critical thinking skills and engineering judgment • Other approaches could have been used such as using an SRS to cover peak response and damage potential to cover fatigue only

50. Conclusions (cont.) • C/C++ source code & related tutorials available from Tom Irvine upon request • Response acceleration was the amplitude metric used in this presentation • The method could also be used with relative displacement and pseudo velocity • Future work: • Compare results of alternate Damage-Potential method with the DiMaggio method and with the customary piecewise stationary method • Extend method to multi-degree-of-freedom systems