An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

Tom Irvine Dynamic Concepts, Inc Email: [email protected]

Learning from the Past, Looking to the Future

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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

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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

Learning from the Past, Looking to the Future

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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

0 -1 2 ARES 1-X FLIGHT ACCELEROMETER DATA IAD601A 1 -2 0 20 40 60 TIME (SEC) 80 100 120

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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.

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LCROSS vibration tests at the NASA Ames Research Center

• 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

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Some Preliminaries . .

.

Maximum Expected Flight Level MEFL

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 VRS Shock Response Spectrum

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

Vibration Response Spectrum

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

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SDOF System

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K 1 M 1 C1

Shock Response Spectrum Model

..

X 1 ..

X 2 ..

X 3 K 2 M 2 K 3 M 3

. . . .

K L M L C 2 C 3 C L ..

X L ..

Y (Base Input) f n 1 < f n 2 < f n 3 <

. . . .

< f n L • 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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SRS Example Base Input: Half-Sine Pulse (11 msec, 50 G)

100 100 RESPONSE (fn = 30 Hz, Q=10) 50 50 0 0 -50 -50 -100 0 0.01

0.02

0.03

TIME (SEC) 0.04

0.05

0.06

RESPONSE (fn = 140 Hz, Q=10) 100 -100 0 0.01

0.02

0.03

TIME (SEC) 0.04

0.05

0.06

100 RESPONSE (fn = 80 Hz, Q=10) 50 50 0 0 -50 -50 -100 0 0.01

0.02

0.03

TIME (SEC) 0.04

0.05

0.06

-100 0

Learning from the Past, Looking to the Future

0.01

0.02

0.03

TIME (SEC) 0.04

0.05

0.06

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200 100 50 ( 30 Hz, 55 G ) ( 80 Hz, 82 G ) ( 140 Hz, 70 G ) 20 10 5 10 100

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1000

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0.1

Typical Power Spectral Density Test Level

0.01

Corresponding time history shown on next slide.

0.001

10 100 FREQUENCY (Hz) 1000 2000 • • • The overall level is 6.1 GRMS. This is the square root of the area under the curve.

GRMS value = 1 s ( std dev) assuming zero mean The amplitude unit is G^2/Hz, but this is really GRMS^2/Hz

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30 20 10 0 -10 -20 -30 0 TIME HISTORY, 1-sec SEGMENT, STD DEV = 6.1 G 0.5

TIME (SEC) 1.0

80000 60000 40000 20000 0 -30 -20 HISTOGRAM -10 0 ACCEL (G) 10 • • • • • 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 s ( std dev) assuming zero mean Histogram of instantaneous values is Gaussian, normal distribution, bell-shaped curve

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20 30

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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:

x y  1   1   2   2 2   2   2   2 where   f / f n where f is the base excitation frequency and fn is the natural frequency .

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

 x  GRMS  f n ,    i N   1  1   1   i 2  ( 2  i 2   2 ) 2    i  2 Y A PSD ( f i )  f i , • • • • 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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0.1

0.01

10

Response Power Spectral Density Curves SDOF Systems Q=10

fn = 300 Hz fn = 200 Hz fn = 100 Hz Base Input 1 0.001

20 100 FREQUENCY (Hz) 1000 Next, calculate the overall level from each response curve. Again, this is the square root of the area under each curve.

2000

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Each peak is Q 2 times the base input at the natural frequency, for SDOF response.

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100 10

Vibration Response Spectrum SDOF Systems Q=10

( 300 Hz, 13.7 GRMS) ( 200 Hz, 11.1 GRMS) ( 100 Hz, 6.4 GRMS) 1 20 100 NATURAL FREQUENCY (Hz) 1000 2000

Later in the presentation, peak vibration response and accumulated damage will be plotted against natural frequency.

Learning from the Past, Looking to the Future

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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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-2 -3 -4 -5 -6 0 1 0 -1 6 5 4 3 2

Sample Time History

STRESS TIME HISTORY 1 2 3 4 TIME 5

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6 7 8

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0 1 2 3 4 5 6 7 8 -6 G -5 -4 C A E -3 I -2 -1 0 STRESS Stress 1 B 2 F H D 3 4 5 6

Rainflow Cycle Counting

Rotate time history plot 90 degrees clockwise Rainflow Cycles by Path Path Cycles A-B B-C C-D D-G E-F G-H H-I 0.5

0.5

0.5

0.5

1.0

0.5

0.5

Stress Range 3 4 8 9 4 8 6

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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

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-1 -2 2 1 0 TIME (SEC) Segment 1 Segment 2 Segment 3

Piecewise Stationary Enveloping Method Concept

Calculate PSD for Each Segment Power Spectral Density Accel (G^2/Hz) Maximum Envelope of 3 PSD Curves

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

Frequency (Hz)

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10

Nonstationary Random Vibration

FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE 5 0 -5 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 TIME (SEC) Liftoff Transonic Attitude Control Max-Q Thrusters 70 Rainflow counting can be applied to accelerometer data.

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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

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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 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.

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Objective (cont.)

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

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0.1

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

PSD SDOF (fn=280 Hz, Q=10) RESPONSE TO FLIGHT DATA OVERALL LEVEL = 1.1 GRMS Non-resonant Response Resonant Response 0.01

0.001

0.0001

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.

0.00001

10 100 300 1000 2000 FREQUENCY (Hz)

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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.

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Alternate Method Steps (cont.)

The enveloping is performed in terms of the n s 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: Expected Peak  s 2 ln ( fn T ) (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.

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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

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As an Aside (cont)…

P  s  A       s A s 2 exp    A 2 2 s 2  dA Integrate the Rayleigh Probability Density Function where A is the absolute amplitude of the local peaks. P  s  A     exp  1 2  2 Total number of peaks = fn T exp  1 2  2 fn T  1 Probability * total peaks = 1 peak

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As an Aside (cont)…

  2 ln ( fn T ) Out of all the peaks, only one is expected > s So assume : maximum peak   s  s 2 ln ( 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.

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Alternate Method Steps (cont.)

Expected Peak

 s

2 ln ( fn T )

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.

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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.

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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: A i n i b where D  i m   1 A i b n i is the acceleration amplitude from the rainflow analysis is the corresponding number of cycles 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.

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10

Example: Nonstationary Random Vibration

FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE 5 0 -5 -10 -5 0 5 10 15 20 25 30 35 TIME (SEC) 40 45 50 55 60 65 70 Duration (sec) 0 to 2 2 to 60 60 to 68 Description Launch Ascent Attitude Control System Envelope Type SRS PSD Sine The data could be divided into segments as shown in the table.

But the entire signal will be used for the following example.

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100 Q = 50 Q = 25 Q = 10

Shock Response Spectra

SRS FLIGHT DATA 10 1 0.1

10 100 NATURAL FREQUENCY (Hz) 1000 2000 Taken over the entire duration of the nonstationary data. Time domain calculation.

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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

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Derive Power Spectral Density

Trial-and-error derivation

Randomly Generated Candidate PSD Base Input Response PSD Given fn & Q Freq (Hz) 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 s 2 ln ( fn T )

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Freq (Hz)

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

Typically > 3 s

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Derive Power Spectral Density

Trial-and-error derivation (cont.)

Family of Response PSDs Freq (Hz) VRS of Candidate PSD for given Q Freq (Hz) All fn of interest at given Q Again, peak values are determined via : s 2 ln ( fn T ) Natural Frequency (Hz)

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Derive Power Spectral Density

Trial-and-error derivation (cont.)

Candidate PSD Scale PSD by uniform factor so that its VRS envelops flight data for each Q Response Spectra for given Q Candidate VRS Flight Data SRS Freq (Hz) Natural Frequency (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= s Program: envelope_srs_psd_three_q.cpp 2 ln ( fn T )

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Derived Power Spectral Density

DAMAGE-POTENTIAL POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS 0.1

0.01

•

The lowest-level PSD whose VRS envelops the Flight Data SRS for three Q cases.

0.001

•

Again, the PSD was derived by trial-and error

0.0001

10 100 FREQUENCY (Hz) 1000 2000 • The n s 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

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1000 100 10 1 0.1

10 1000 100 10 1 0.1

10 RESPONSE SPECTRA Q = 50 Damage Potential Flight Data

The Damage Potential PSD envelops the corresponding SRS curves in terms of peak potential VRS uses

Peak  s 2 ln ( fn T ) 100 NATURAL FREQUENCY (Hz) 1000 2000 RESPONSE SPECTRA Q = 25 Damage Potential Flight Data This will be verified in the time domain in

This will be verified in the time domain in upcoming slides.

RESPONSE SPECTRA Q = 10 1000 Damage Potential Flight Data 100 10 1 100 NATURAL FREQUENCY (Hz) 1000 2000 0.1

10 100 NATURAL FREQUENCY (Hz) 1000 2000

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20 SYNTHESIZED TIME HISTORY FOR DAMAGE POTENTIAL PSD OVERALL LEVEL = 3.3 GRMS 10 0 -10 -20 0 5 10 15 20 25 30 35 TIME (SEC) 40 45 50 55 60 POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS 0.1

Synthesis Damage Potential 0.01

Numerical Simulation

Synthesize a time history to satisfy the Damage Potential PSD.

Verify that the PSDs match.

0.001

0.0001

10 100 FREQUENCY (Hz) 1000 2000

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1000 SHOCK RESPONSE SPECTRA Q=50 Damage Synthesis Flight Data 100 10 1 10 100 NATURAL FREQUENCY (Hz) 1000 2000 SHOCK RESPONSE SPECTRA Q=25 1000 Damage Synthesis Flight Data 100

Response Spectra Comparison, Part II

Verification in the time domain for three Q cases Relaxed reliance on Peak  s 2 ln ( fn T ) because experimental proof that Damage Synthesis envelops Flight Data 1000 SHOCK RESPONSE SPECTRA Q=10 Damage Synthesis Flight Data 100 10 10 1 10 100 NATURAL FREQUENCY (Hz) 1000 2000 1 10 100 NATURAL FREQUENCY (Hz) 1000 2000

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

30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -5 0 5 SDOF RESPONSE fn = 189 Hz Q=10 10 15 20 25 Flight Data Damage Potential Synthesis 30 35 40 45 50 55 60 65 90 80 70 0 -10 -20 -30 70 -40 60 50 40 30 20 10

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

30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -5 0 5 SDOF RESPONSE fn = 280 Hz Q=10 10 15 20 25 Flight Data Damage Potential Synthesis 30 35 40 45 50 55 60 65 90 80 70 0 -10 -20 -30 70 -40 60 50 40 30 20 10

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10 18 10 14 10 10 10 6 10 2 10 FATIGUE DAMAGE Q=50 b=6.4

Damage Synthesis Flight Data 10 18 10 14 10 10 10 6 10 2 10 100 NATURAL FREQUENCY (Hz) 1000 2000 FATIGUE DAMAGE Q=25 b=6.4

Damage Synthesis Flight Data 100 NATURAL FREQUENCY (Hz) 1000 2000

Fatigue Response Spectra Comparison

Three Q cases, b=6.4

10 17 10 14 10 11 10 8 10 5 10 2 10 FATIGUE DAMAGE Q=10 b=6.4

Damage Synthesis Flight Data 100 NATURAL FREQUENCY (Hz) 1000 2000

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10 13 10 10 10 7 10 4 10 1 10 10 14 10 11 10 8 10 5 10 2 10 FATIGUE DAMAGE Q=50 b=4 Damage Synthesis Flight Data

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

Three Q cases, b=4 100 NATURAL FREQUENCY (Hz) 1000 2000 FATIGUE DAMAGE Q=25 b=4 FATIGUE DAMAGE Q=10 b=4 Damage Synthesis Flight Data 10 11 10 9 10 7 10 5 10 3 10 1 10 Damage Synthesis Flight Data 100 NATURAL FREQUENCY (Hz) 1000 2000

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100 NATURAL FREQUENCY (Hz) 1000 2000

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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

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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

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As an industry representative to NESC Load & Dynamics…

I am here to serve you!

Please contact: Tom Irvine Dynamic Concepts, Inc Email: [email protected]

Phone: 256-922-9888 x343

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