An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

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An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

Tom IrvineDynamic Concepts, IncEmail: [email protected]: 1NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 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.NESC Academy WebcastLearning from the Past, Looking to the FutureThis project is an informal collaboration between:

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NESCNASA KSCDynamic ConceptsSpace-XFalcon 9 LiftoffIn the Spirit of the National Aeronautics and Space Act of 1958NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 4

Ares 1-X , PrandtlGlauert Singularity, Vapor Condensation Cone at Transonic Lift-off Vibroacoustics

Transonic Shock Waves

Fluctuating Pressure at Max-Q

Random Vibration Environments

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 5

Launch Vehicle AvionicsFlight ComputersInertial Navigation SystemsTransponders & TransmittersReceiversAntennasBatteriesetc.

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.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 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 CenterNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 7MEFL Maximum Expected Flight Level

Given as a base input PSD for avionicsPSD Power Spectral Density

Gives acceleration energy as a function of frequency. Can be calculated from Fourier transform.SDOFSingle-degree-of-freedom

Spring-mass system. Simplified model for avionics.SRSShock Response Spectrum

Gives peak response of SDOF systems to time history base input.VRSVibration Response Spectrum

Gives overall response of SDOF systems to a PSD base input.

SDOF SystemSome Preliminaries . . .

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 8Shock 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.


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


SRS Q=10 Base Input: Half-Sine Pulse (11 msec, 50 G)NATURAL FREQUENCY (Hz)NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 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 = 1s ( std dev) assuming zero meanThe amplitude unit is G^2/Hz, but this is really GRMS^2/Hz

Corresponding time history shown on next slide.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 12The time history is stationaryTime history is not unique because the PSD discards the phase angleTime history could be performed on shaker table as input to avionics componentGRMS value = 1s ( std dev) assuming zero meanHistogram of instantaneous values is Gaussian, normal distribution, bell-shaped curve

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 13Response 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.


Response of an SDOF System to Random Vibration PSD (cont.)

Multiply power transmissibility by the base input PSDSum over all input frequenciesTake the square rootThe result is the overall response accelerationNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 15

Response Power Spectral Density CurvesSDOF 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 Q2 times the base input at the natural frequency, for SDOF response.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 16

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


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, JapanNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 18

Sample Time HistoryNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 19

Rainflow Cycle CountingRotate time history plot 90 degrees clockwise Rainflow Cycles by PathPathCyclesStress RangeA-B0.53B-C0.54C-D0.58D-G0.59E-F1.04G-H0.58H-I0.56Rainflow Plot Stress NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 20The typical method for post-processing is to divide the data into short-duration segmentsThe segments may overlapThis is termed piecewise stationary analysisA PSD is then taken for each segmentThe maximum envelope is then taken from the individual PSD curvesMEFL = maximum envelope + some uncertainty marginComponent acceptance test level > MEFLEasy to doBut potentially overly conservative

Derive MEFL from Nonstationary Random Vibration

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 21Frequency (Hz)Power Spectral DensityAccel (G^2/Hz)Maximum Envelope of 3 PSD CurvesPiecewise Stationary Enveloping Method Concept

Calculate PSD for Each Segment

Segment 1Segment 2Segment 3Would use shorter segments if we were doing this in earnest.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 22

Nonstationary Random VibrationLiftoff Transonic Attitude Control Max-Q ThrustersRainflow counting can be applied to accelerometer data.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 23S. 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

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 24The 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 stepsThe total number of response permutations is 540, which is rather rigorous. This is needed because the avionics components dynamic characteristics are unknown.Objective

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 25The 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

Objective (cont.)

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 26Typical SDOF Response to Previous Flight Accelerometer Data (nonstationary time history)

Non-resonant ResponseResonant ResponseExisting 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.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 27Alternate Method StepsPeak ResponseThe 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.

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 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 s is the standard deviation of response fn is the natural frequencyT is the durationThis step is performed using program: envelope_srs_psd_threeq_single.cpp.


As an AsideRayleigh 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 inputNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 30As an Aside (cont)Integrate the Rayleigh Probability Density FunctionProbability * total peaks = 1 peak

where A is the absolute amplitude of the local peaks.

Total number of peaks = fn TNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 31Assumes 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 > ls

So assume : maximum peak

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 32Alternate 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.

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 33Fatigue 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.NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 34Alternate 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:

whereA i is the acceleration amplitude from the rainflow analysisn i is the corresponding number of cycles b is the fatigue exponentThis step is performed using program: fatigue_threeq.cpp. Steps 3 through 4 are then repeated for the flight accelerometer data.


Example: Nonstationary Random Vibration

Duration (sec)DescriptionEnvelope Type0 to 2LaunchSRS2 to 60AscentPSD60 to 68Attitude Control SystemSineThe data could be divided into segments as shown in the table.But the entire signal will be used for the following example.


Shock Response Spectra

Taken over the entire duration of the nonstationary data. Time domain calculation.

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 37Derive 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 durationNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 38

Derive Power Spectral DensityTrial-and-error derivation

Randomly Generated Candidate PSD Base InputFreq (Hz)(G^2/Hz)

(G^2/Hz)Freq (Hz)Response PSD Given fn & QThe 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 isRepeat this calculation for all fn & Q values of interest.Typically > 3s


Derive Power Spectral DensityTrial-and-error derivation (cont.)

Freq (Hz)(G^2/Hz)Natural Frequency (Hz)Peak (G)All fn of interest at given QAgain, peak values are determined via:

VRS of Candidate PSD for given Q(G^2/Hz)Freq (Hz)Family of Response PSDsNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 40

Derive Power Spectral DensityTrial-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 PSDFreq (Hz)(G^2/Hz)Response Spectra for given QNatural Frequency (Hz)Peak (G)Scale PSD by uniform factor so that its VRS envelops flight data for each QCandidate VRSFlight Data SRSNESC Academy WebcastLearning from the Past, Looking to the FuturePage: 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

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 42Response Spectra Comparison, Part IResponse 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.

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.


Numerical SimulationSynthesize a time history to satisfy the Damage Potential PSD.Verify that the PSDs match.


Response Spectra Comparison, Part IIVerification in the time domain for three Q casesRelaxed reliance on

because experimental proof that Damage Synthesis envelops Flight Data


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

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 46SDOF Response Time History Comparison (fn=280 Hz, Q=10)


Fatigue Response Spectra ComparisonThree Q cases, b=6.4

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 48Fatigue Response Spectra ComparisonThree Q cases, b=6.4

Fatigue Response Spectra ComparisonThree Q cases, b=4

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 49ConclusionsSuccessfully 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 analysts 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

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 50Conclusions (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

NESC Academy WebcastLearning from the Past, Looking to the FuturePage: 51As an industry representative to NESC Load & Dynamics

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Please contact:Tom IrvineDynamic Concepts, IncEmail: [email protected]: 256-922-9888 x343

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An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration