learning from the past, looking to the future an alternate damage potential method for enveloping...

NESC ACADEMY WEBCAST

Learning from the Past, Looking to the Future

An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

Tom IrvineDynamic Concepts, IncEmail: [email protected]

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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.


Learning from the Past, Looking to the Future

This project is an informal collaboration between:

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• NESC• NASA KSC• Dynamic Concepts• Space-X

Falcon 9 Liftoff

In the Spirit of the National Aeronautics and Space Act of 1958



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

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ARES 1-X FLIGHT ACCELEROMETER DATA IAD601A



Launch Vehicle Avionics Flight 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.



• 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



MEFL Maximum Expected Flight LevelGiven as a base input PSD for avionics

PSD Power Spectral DensityGives acceleration energy as a function of frequency. Can be calculated from Fourier transform.

SDOFSingle-degree-of-freedomSpring-mass system. Simplified model for avionics.

SRSShock Response SpectrumGives peak response of SDOF systems to time history base input.

VRSVibration Response SpectrumGives overall response of SDOF systems to a PSD base input.

SDOF System

Some Preliminaries . . .



Shock Response Spectrum Model

. . . .Y (Base Input)..

M1 M2 M3 ML

X..

1 X..

2X..

3 X..

L

K1 K2K3 KL

C1 C2 C3 CL

fn1 < << < . . . .fn2fn3

fnL

• 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.



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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



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( 140 Hz, 70 G )

( 80 Hz, 82 G )

( 30 Hz, 55 G )

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SRS Q=10 BASE INPUT: HALF-SINE PULSE (11 msec, 50 G)SRS Q=10 Base Input: Half-Sine Pulse (11 msec, 50 G)




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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 mean• The amplitude unit is G^2/Hz, but this is really GRMS^2/Hz

Corresponding time history shown on next slide.



• 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

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HISTOGRAM



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:

21 2x2y 221 2

wherenf/f

where f is the base excitation frequency and fn is the natural frequency.



,f)f(Y,fxN

1iiiPSDA

21

)2(1nGRMS

2i

22i

2i

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



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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.



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( 300 Hz, 13.7 GRMS)

( 200 Hz, 11.1 GRMS)

( 100 Hz, 6.4 GRMS)


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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, Japan



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STRESS TIME HISTORY

Sample Time History



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RAINFLOW PLOT Rainflow Cycle Counting

Rotate time history plot 90 degrees clockwise

Rainflow Cycles by Path

Path CyclesStress Range

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B-C 0.5 4

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D-G 0.5 9

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Rainflow Plot

Stress



• 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



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-2

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Segment 1Segment 2

Segment 3

Would use shorter segments if we were doing this in earnest.



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FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE

Nonstationary Random Vibration

Liftoff Transonic Attitude Control

Max-Q Thrusters

Rainflow counting can be applied to accelerometer data.



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



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



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.)



Typical SDOF Response to Previous Flight Accelerometer Data (nonstationary time history)

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PSD SDOF (fn=280 Hz, Q=10) RESPONSE TO FLIGHT DATAOVERALL LEVEL = 1.1 GRMS

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.



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.



(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 duration

This step is performed using program: envelope_srs_psd_threeq_single.cpp.

)Tfn(ln2PeakExpected



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



As an Aside (cont)…

Integrate the Rayleigh Probability Density Function

Probability * total peaks = 1 peak

dA2

Aexp

AAP

2

2

2

2

2

1expAP

where A is the absolute amplitude of the local peaks.

1Tfn2

1exp 2

Total number of peaks = fn T



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)…

)Tfn(ln2

Out of all the peaks, only one is expected > ls

So assume : maximum peak )Tfn(ln2




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.

)Tfn(ln2PeakExpected



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.




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:

i

m

1i

bi nAD

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.



Example: Nonstationary Random Vibration

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FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE

Duration (sec) Description Envelope Type

0 to 2 Launch SRS

2 to 60 Ascent PSD

60 to 68 Attitude Control System Sine

The 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

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)SRS FLIGHT DATA

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



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



Derive Power Spectral DensityTrial-and-error derivation

)Tfn(ln2

Randomly Generated Candidate PSD Base Input

Freq (Hz)

(G^2

/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 )s = GRMS assuming zero mean.

The peak is typically assumed to be 3 .s

But a better estimate is

Repeat this calculation for all fn & Q values of interest.

Typically > 3s )Tfn(ln2



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

)Tfn(ln2

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



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

)Tfn(ln2

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



Derived Power Spectral Density

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DAMAGE-POTENTIAL POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS

• 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



Response Spectra Comparison, Part I 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.

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Damage PotentialFlight Data


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RESPONSE SPECTRA Q = 25

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RESPONSE SPECTRA Q = 10

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.

)Tfn(ln2 Peak



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SYNTHESIZED TIME HISTORY FOR DAMAGE POTENTIAL PSD OVERALL LEVEL = 3.3 GRMS

Numerical Simulation

Synthesize a time history to satisfy the Damage Potential PSD.

Verify that the PSDs match.

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SynthesisDamage Potential

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Damage SynthesisFlight Data


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SHOCK RESPONSE SPECTRA Q=10

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

)Tfn(ln2 Peak



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Damage Potential Synthesis

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



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

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DFATIGUE DAMAGE Q=50 b=6.4

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Fatigue Response Spectra Comparison Three Q cases, b=6.4



Fatigue Response Spectra Comparison Three Q cases, b=6.4

Fatigue Response Spectra Comparison Three Q cases, b=4

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FATIGUE DAMAGE Q=10 b=4



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



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



As an industry representative to NESC Load & Dynamics…

I am here to serve you!

Please contact:

Tom IrvineDynamic Concepts, Inc

Email: [email protected]

Phone: 256-922-9888 x343

learning from the past, looking to the future an alternate damage potential method for enveloping...

Documents

future page

response fn

base input psd

psd base input

srs shock response spectrum

peak response of sdof

time history base input

base excitation