© The Aerospace Corporation 2010 © The Aerospace Corporation 2012

An Alternate Damage Potential Method for Enveloping Nonstationary Random Vibration

Tom Irvine

Dynamic Concepts, Inc.

NASA Engineering & Safety Center (NESC)

19–21 June 2012

2

Introduction

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

.

Random Vibration Environments

-2

-1

0

1

2

0 20 40 60 80 100 120

TIME (SEC)

AC

CE

L (

G)

ARES 1-X FLIGHT ACCELEROMETER DATA IAD601A

Ares 1-X , Prandtl–Glauert Singularity, Vapor Condensation Cone at Transonic

• Liftoff Vibroacoustics

• Transonic Shock Waves

• Fluctuating Pressure at Max-Q

4

Derive Maximum Expected Flight Level (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

So Let’s Derive a New Method!

5

.

New Method Background

This project is an informal collaboration between:

In the Spirit of the National Aeronautics and

Space Act of 1958

• NESC

• NASA KSC

• Dynamic Concepts

• Space-X

Falcon 9 Liftoff

6

.

References

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

7

.

References (cont.)

After writing this paper, I learned that Scot McNeill had previously presented a

similar paper at a SAVIAC conference in 2008:

Implementing the Fatigue Damage Spectrum and Fatigue Damage

Equivalent Vibration Testing

So I may have reinvented the wheel on this one.

But I whimsically noticed that Scot used two of my previous papers in his own

paper!

8

.

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.

9

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8

TIME

ST

RE

SS

STRESS TIME HISTORY

.

Sample Time History

10

.

Rainflow Cycle Counting

0

1

2

3

4

5

6

7

8-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

A

H

F

D

B

I

G

E

C

STRESS

TIM

E

RAINFLOW PLOT

Rotate time history plot 90 degrees clockwise

Rainflow Cycles by Path

Path Cycles Stress Range

A-B 0.5 3

B-C 0.5 4

C-D 0.5 8

D-G 0.5 9

E-F 1.0 4

G-H 0.5 8

H-I 0.5 6

11

Sample Time History to Demonstrate

Alternate Method

-10

-5

0

5

10

-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

TIME (SEC)

AC

CE

L (

G)

FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE

Liftoff Transonic Attitude Control

Max-Q Thrusters

Rainflow counting

can be applied to

the SDOF

responses for the

base input

accelerometer data.

12

.

Alternate Method for Damage Potential PSD

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.

13

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Innovations

The alternate damage method in this presentation 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

14

0.00001

0.0001

0.001

0.01

0.1

10 100 1000300 2000

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

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.

15

.

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.

2. Derive a Damage Potential PSD which has a VRS that envelops the SRS curves of

the flight data for the three Q cases.

16

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Enveloping

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:

)Tfn(ln2PeakExpected (temporary assumption)

where

s is the standard deviation of response

fn is the natural frequency

T is the duration

.

17

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Fatigue Check*

The peak response criterion may 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. 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 not “true fatigue” which would be calculated from stress.

Rather it is a fatigue-like metric for accumulated response acceleration

cycles.

18

.

Fatigue Check (cont.)

3. Take 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.

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

Steps 3 through 4 are then

repeated for the flight

accelerometer data.

where

A i is the acceleration amplitude from the rainflow analysis

n i is the corresponding number of cycles

b is the fatigue exponent

19

Introduction

0.1

1

10

100

10 100 1000 2000

Q = 50Q = 25Q = 10

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SRS FLIGHT DATA

Taken over the entire duration of the nonstationary data.

Time domain calculation.

20

.

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

21

.

Introduction

0.0001

0.001

0.01

0.1

10 100 1000 2000

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

DAMAGE-POTENTIAL POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS

• The lowest-level PSD

whose VRS envelops

the Flight Data SRS

for three Q cases.

• Again, the PSD was

derived by trial-and-

error

• 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 response & fatigue

22

.

Introduction

0.1

1

10

100

1000

10 100 1000 2000

Damage PotentialFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

RESPONSE SPECTRA Q = 50

0.1

1

10

100

1000

10 100 1000 2000

Damage PotentialFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

RESPONSE SPECTRA Q = 25

0.1

1

10

100

1000

10 100 1000 2000

Damage PotentialFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

RESPONSE SPECTRA Q = 10

)Tfn(ln2 Peak

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.

23

.

Introduction

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-10

0

10

20

0 5 10 15 20 25 30 35 40 45 50 55 60

TIME (SEC)

AC

CE

L (

G)

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

0.0001

0.001

0.01

0.1

10 100 1000 2000

SynthesisDamage Potential

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS

Numerical Simulation

Synthesize a time history to

satisfy the Damage Potential

PSD.

Verify that the PSDs match.

24

.

Introduction

1

10

100

1000

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SHOCK RESPONSE SPECTRA Q=50

1

10

100

1000

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SHOCK RESPONSE SPECTRA Q=25

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

1

10

100

1000

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

PE

AK

AC

CE

L (

G)

SHOCK RESPONSE SPECTRA Q=10

25

.

Introduction

102

106

1010

1014

1018

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=50 b=6.4

102

106

1010

1014

1018

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=25 b=6.4

102

105

108

1011

1014

1017

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=10 b=6.4

Fatigue Response Spectra

Comparison

Three Q cases, b=6.4

26

.

Introduction

102

105

108

1011

1014

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=50 b=4

101

104

107

1010

1013

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=25 b=4

101

103

105

107

109

1011

10 100 1000 2000

Damage SynthesisFlight Data

NATURAL FREQUENCY (Hz)

FA

TIG

UE

DA

MA

GE

D

FATIGUE DAMAGE Q=10 b=4

Fatigue Response Spectra

Comparison

Three Q cases, b=4

27

.

Conclusions

0.0001

0.001

0.01

0.1

10 100 1000 2000

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

DAMAGE-POTENTIAL POWER SPECTRAL DENSITY OVERALL LEVEL = 3.3 GRMS

• Successfully derived a MEFL PSD

using the alternate Damage-

Potential method

• 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

Software & Tutorials for this Damage Potential

Method are posted on Vibrationdata Blog.

Software is C/C++ with included source code.

© The Aerospace Corporation 2010 © The Aerospace Corporation 2012

Thank you

Tom Irvine

Email: tirvine@dynamic-concepts.com