1. Learning from the Past, Looking to the Future An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration Tom Irvine Dynamic Concepts, Inc Email: tirvine@dynamic-concepts.com Page: 1

2. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future This project is an informal collaboration between: Page: 3 NESC NASA KSC Dynamic Concepts Space-X Falcon 9 Liftoff In the Spirit of the National Aeronautics and Space Act of 1958

4. Learning from the Past, Looking to the Future Page: 4 Ares 1-X, Prandtl–Glauert Singularity, Vapor Condensation Cone at Transonic Lift-off Vibroacoustics Transonic Shock Waves Fluctuating Pressure at Max-Q Random Vibration Environments

5. Learning from the Past, Looking to the Future Page: 5 Launch Vehicle Avionics Flight Computers Inertial Navigation Systems Transponders & Transmitters Receivers Antennas 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. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 7 MEFL Maximum Expected Flight Level Given as a base input PSD for avionics PSD Power Spectral Density Gives acceleration energy as a function of frequency. Can be calculated from Fourier transform. SDOF Single-degree-of-freedom Spring-mass system. Simplified model for avionics. SRS Shock Response Spectrum Gives peak response of SDOF systems to time history base input. VRS Vibration Response Spectrum Gives overall response of SDOF systems to a PSD base input. SDOF System Some Preliminaries...

8. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 9 RESPONSE (fn = 30 Hz, Q=10) RESPONSE (fn = 80 Hz, Q=10) RESPONSE (fn = 140 Hz, Q=10) Base Input: Half-Sine Pulse (11 msec, 50 G) SRS Example

10. Learning from the Past, Looking to the Future Page: 10 SRS Q=10 Base Input: Half-Sine Pulse (11 msec, 50 G) NATURAL FREQUENCY (Hz)

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

12. Learning from the Past, Looking to the Future Page: 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 = 1  ( std dev) assuming zero mean Histogram of instantaneous values is Gaussian, normal distribution, bell-shaped curve

13. Learning from the Past, Looking to the Future Page: 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 fn is the natural frequency.

14. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 15 Response Power Spectral Density Curves SDOF Systems Q=10 Next, calculate the overall level from each response curve. Again, this is the square root of the area under each curve. Each peak is Q 2 times the base input at the natural frequency, for SDOF response.

16. Learning from the Past, Looking to the Future Page: 16 Later in the presentation, peak vibration response and accumulated damage will be plotted against natural frequency. Vibration Response Spectrum SDOF Systems Q=10

17. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 18 Sample Time History

19. Learning from the Past, Looking to the Future Page: 19 Rainflow Cycle Counting Rotate time history plot 90 degrees clockwise Rainflow Cycles by Path PathCycles Stress Range A-B0.53 B-C0.54 C-D0.58 D-G0.59 E-F1.04 G-H0.58 H-I0.56 Rainflow Plot Stress

20. Learning from the Past, Looking to the Future Page: 20 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 Derive MEFL from Nonstationary Random Vibration

21. Learning from the Past, Looking to the Future Page: 21 Frequency (Hz) Power Spectral Density Accel (G^2/Hz) Maximum Envelope of 3 PSD Curves 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. Learning from the Past, Looking to the Future Page: 22 Nonstationary Random Vibration Liftoff Transonic Attitude Control Max-Q Thrusters Rainflow counting can be applied to accelerometer data.

23. Learning from the Past, Looking to the Future Page: 23 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 Background Reference

24. Learning from the Past, Looking to the Future Page: 24 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 1.Three damping cases with Q=10, 25 & 50 ( 5%, 2% & 1%) 2.Two fatigue exponent cases with b=4 & 6.4 (slope from S-N curve) 3.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. Objective

25. Learning from the Past, Looking to the Future Page: 25 The alternate damage method in this paper builds upon previous work by addressing an additional concern as follows: 1.Consider an SDOF system with a given natural frequency and damping ratio 2.The SDOF system is subjected to a base input 3.The base input may vary significantly with frequency 4.The response of the SDOF system may include non-resonant stress reversal cycles Objective (cont.)

26. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 27 Alternate Method Steps Peak Response The peak response is enveloped as follows. 1.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. 2.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. Learning from the Past, Looking to the Future Page: 28 (temporary assumption) 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: Alternate Method Steps (cont.) where  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. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 30 As an Aside (cont)… Integrate the Rayleigh Probability Density Function Probability * total peaks = 1 peak where A is the absolute amplitude of the local peaks. Total number of peaks = fn T

31. Learning from the Past, Looking to the Future Page: 31 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. As an Aside (cont)… Out of all the peaks, only one is expected >  So assume : maximum peak 

32. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 33 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. Alternate Method Steps (cont.) * This is not “true fatigue” which would be calculated from stress. Rather it is a fatigue- like metric for accumulated response acceleration cycles.

34. Learning from the Past, Looking to the Future Page: 34 Alternate Method Steps (cont.) 3.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. 4.Calculate the fatigue damage D for each of 270 rainflow responses for each of the two fatigue exponents as follows: where A i is the acceleration amplitude from the rainflow analysis n i is the corresponding number of cycles b is the fatigue exponent This step is performed using program: fatigue_threeq.cpp. Steps 3 through 4 are then repeated for the flight accelerometer data.

35. Learning from the Past, Looking to the Future Page: 35 Example: Nonstationary Random Vibration Duration (sec)DescriptionEnvelope Type 0 to 2LaunchSRS 2 to 60AscentPSD 60 to 68Attitude Control SystemSine The data could be divided into segments as shown in the table. But the entire signal will be used for the following example.

36. Learning from the Past, Looking to the Future Page: 36 Shock Response Spectra Taken over the entire duration of the nonstationary data. Time domain calculation.

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

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

40. Learning from the Past, Looking to the Future Page: 40 Derive Power Spectral Density Trial-and-error derivation (cont.) 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 Candidate PSD Freq (Hz) (G^2/Hz) Response Spectra for given Q Natural Frequency (Hz) Peak (G) Scale PSD by uniform factor so that its VRS envelops flight data for each Q Candidate VRS Flight Data SRS

41. Learning from the Past, Looking to the Future Page: 41 Derived Power Spectral Density 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 The lowest-level PSD whose VRS envelops the Flight Data SRS for three Q cases. Again, the PSD was derived by trial-and- error

42. Learning from the Past, Looking to the Future Page: 42 Response Spectra Comparison, Part I Response Spectra Comparison, Part I The 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. Response Spectra Comparison, Part I The 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. Learning from the Past, Looking to the Future Page: 43 Numerical Simulation Synthesize a time history to satisfy the Damage Potential PSD. Verify that the PSDs match.

44. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 45 SDOF Response Time History Comparison (fn=189 Hz, Q=10)

46. Learning from the Past, Looking to the Future Page: 46 SDOF Response Time History Comparison (fn=280 Hz, Q=10)

47. Learning from the Past, Looking to the Future Page: 47 Fatigue Response Spectra Comparison Three Q cases, b=6.4

48. Learning from the Past, Looking to the Future Page: 48 Fatigue Response Spectra Comparison Three Q cases, b=6.4 Fatigue Response Spectra Comparison Three Q cases, b=4

49. Learning from the Past, Looking to the Future Page: 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. Learning from the Past, Looking to the Future Page: 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: o Compare results of alternate Damage-Potential method with the DiMaggio method and with the customary piecewise stationary method o Extend method to multi-degree-of-freedom systems

51. Learning from the Past, Looking to the Future Page: 51 As an industry representative to NESC Load & Dynamics… I am here to serve you! Please contact: Tom Irvine Dynamic Concepts, Inc Email: tirvine@dynamic-concepts.com Phone: 256-922-9888 x343