Optimized PSD Envelope for Nonstationary Vibration

Transcription

1 Optimized PSD Envelope for Nonstationary Vibration Tom Irvine Dynamic Concepts, Inc NASA Engineering & Safety Center (NESC) 3-5 June 2014 The Aerospace Corporation 2010 The Aerospace Corporation 2012 Vibrationdata

2 ACCEL (G) Introduction - Nonstationary Flight Data 2 ARES 1-X FLIGHT ACCELEROMETER DATA IAD601A Ares 1-X TIME (SEC) Liftoff Vibroacoustics Transonic Shock Waves Fluctuating Pressure at Max-Q 2

3 Avionics Components Electronic components in vehicles are subjected to shock and vibration environments Practical flight accelerometer time histories are nonstationary and non-gaussian A single PSD with an implied stationary, normal distribution time history must be derived for design and test purposes - use rainflow fatigue! 3

4 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 (2005) Rainflow Counting Method 4

5 References by Year Endo & Matsuishi, Rainflow Cycle Counting Method, 1968 T Dirlik, Application of Computers in Fatigue Analysis (PhD), University of Warwick, 1985 ASTM E (2005) Rainflow Counting Method, 1987 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 K Ahlin, Comparison of Test Specifications and Measured Field Data, Sound & Vibration, 2006 Scot McNeill, Implementing the Fatigue Damage Spectrum and Fatigue Damage Equivalent Vibration Testing, SAVIAC Conference, 2008 A Halfpenny & F Kihm, Rainflow Cycle Counting and Acoustic Fatigue Analysis Techniques for Random Loading, RASD Conference, 2010 T Irvine, An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration, Aerospace/JPL Spacecraft and Launch Vehicle Dynamic Environments Workshop, Time Domain Method 5

6 SDOF Model Assume component behaves as single-degree-of-freedom (SDOF) system Avionics are typically black boxes for mechanical engineering purposes! Unknowns Component natural frequency Amplification factor Q Fatigue exponent b Perform fatigue damage calculation on each response for permutations of the three unknowns This adds conservatism to the final PSD envelope The fatigue calculation can be performed starting with either a time history or PSD base input 6

7 Relative Damage Index A relative fatigue damage index can be calculated from the rainflow cycles using a Miners-type summation m b D A i i 1 ni where A i n i b is the acceleration response amplitude from the rainflow analysis is the corresponding number of cycles is the fatigue exponent The damage index D becomes the Fatigue Damage Spectrum (FDS) metric as a function of: natural frequency, amplification factor Q and fatigue exponent b 7

8 Enveloping Approach A PSD envelope can be derived for nonstationary flight data using rainflow cycling counting and the relative fatigue damage index The enveloping is justified using a comparison of Fatigue Damage Spectra between the candidate PSD and the measured time history The derivation process can be performed in a trial-and-error manner in order to obtain the PSD with the least overall GRMS level which still envelops the flight data in terms of fatigue damage spectra Could also seek to minimize overall displacement, velocity, peak G 2 /Hz level, etc Or minimize weighted average of these metrics 8

9 Enveloping Approach (cont) The Dirlik semi-empirical method can be used to calculate the FDS for each candidate PSD in the frequency domain The immediate output of the Dirlik method is a rainflow cycle probability density function (PDF) The rainflow PDF can be converted to a cumulative histogram The cumulative histogram can be converted into individual cycles with their respective amplitudes Compare the fatigue spectra of the candidate PSD to that of the flight data for each Q & b case of interest Scale candidate PSD so that it barely envelops the flight data in terms of FDS Include some convergence option along the way Select the candidate which has the least overall GRMS level, or some other criteria 9

10 ACCEL (G 2 /Hz) Dirlik Method 10 1 POWER SPECTRAL DENSITY fn=200 Hz Q=10 Response 112 GRMS Input 61 GRMS Dirlik method calculates rainflow cycle cumulative histogram from response PSD FREQUENCY (Hz) The Dirlik equation is based on the weighted sum of the Rayleigh, Gaussian and exponential probability distributions Uses area moments of the response PSD as weights Sample base input and SDOF response 10

11 SDOF Response Time Domain Response Acceleration Base Acceleration The response analysis for the nonstationary time history is performed using the Smallwood, ramp invariant digital recursive filtering relationship, for each fn & Q Perform rainflow cycle count on response time history Calculate the damage index D for each fn, Q & b The damage for each permutation is then plotted as function of natural frequency, as an FDS 11

12 Sample Flight Data Derive a 60-second PSD to envelope the flight data Consider 800 candidate PSDs formed by random number generation, with four coordinates each 12

13 Permutations Q & b Values for Fatigue Damage Spectra Case Q b For Reference Only Natural Frequencies: 20 to 2000 Hz All cases will be analyzed for each successive trial 13

14 ACCEL (G 2 /Hz) Optimized PSD 01 POWER SPECTRAL DENSITY ENVELOPE 33 GRMS OVERALL FREQUENCY (Hz) PSD Envelope, 33 GRMS, 60 sec Freq (Hz) Accel (G^2/Hz) The PSD with the least overall GRMS which envelops the flight data via fatigue damage spectra 14

15 DAMAGE INDEX FDS Comparison FATIGUE DAMAGE SPECTRA Q=10 b=4 PSD Envelope Measured Data NATURAL FREQUENCY (Hz) 15

16 DAMAGE INDEX FDS Comparison FATIGUE DAMAGE SPECTRA Q=30 b=4 PSD Envelope Measured Data NATURAL FREQUENCY (Hz) 16

17 DAMAGE INDEX FDS Comparison FATIGUE DAMAGE SPECTRA Q=10 b=9 PSD Envelope Measured Data NATURAL FREQUENCY (Hz) 17

18 DAMAGE INDEX FDS Comparison FATIGUE DAMAGE SPECTRA Q=30 b=9 PSD Envelope Measured Data NATURAL FREQUENCY (Hz) 18

19 ACCEL (G 2 /Hz) PSD Comparison 1 01 POWER SPECTRAL DENSITY Maximum Envelope of 25-sec Segments, 20 GRMS Fatigue Damage Spectrum, Optimized, 33 GRMS FREQUENCY (Hz) Maximum Envelope is traditional piecewise stationary method, but its PSD need further simplification 19

20 An optimized PSD envelope was derived for nonstationary flight data using the fatigue damage spectrum method The FDS case with both the highest Q & b values drove the PSD derivation for the sample flight data Still recommend using permutations because other cases may be the driver for a given time history The method can be used more effectively if the natural frequency, amplification factor, and fatigue exponent are known The method is flexible The PSD duration can be longer or shorter than the flight vibration duration Could require the candidate PSDs to each have a ramp-plateau-ramp shape A similar method could be used for deriving force & pressure PSDs 20

21 Software Software and papers for applying this method are freely given at: Will also cover in a future Shock & Vibration Webinar 21

22 Thank you Tom Irvine The Aerospace Corporation 2010 The Aerospace Corporation 2012 Vibrationdata