optimized psd envelope for nonstationary vibration · 2019. 3. 25. · optimized psd envelope for...
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© The Aerospace Corporation 2010 © The Aerospace Corporation 2012
Optimized PSD Envelope for
Nonstationary Vibration
Tom Irvine
Dynamic Concepts, Inc.
NASA Engineering & Safety Center (NESC)
3-5 June 2014
Vibrationdata
2
Introduction - Nonstationary Flight Data
-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
• Liftoff Vibroacoustics
• Transonic Shock Waves
• Fluctuating Pressure at Max-Q
3
.
Conclusions
• 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!
Avionics Components
4
.
Conclusions
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
Rainflow Fatigue Cycles
5
.
Conclusions
References by Year
• Endo & Matsuishi, Rainflow Cycle Counting Method, 1968 • T. Dirlik, Application of Computers in Fatigue Analysis (Ph.D.), University of Warwick, 1985
• ASTM E 1049-85 (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, 2012 - Time Domain Method
6
Conclusions
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
7
Conclusions
Relative Damage Index
• A relative fatigue damage index can be calculated from the rainflow cycles using a Miners-type summation
• The damage index D becomes the Fatigue Damage Spectrum (FDS) metric as a function of: natural frequency, amplification factor Q and fatigue exponent b
i
m
1i
bi
nAD
where
A i is the acceleration response amplitude from the rainflow analysis
n i is the corresponding number of cycles
b is the fatigue exponent
8
Conclusions
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 G2/Hz level, etc.
• Or minimize weighted average of these metrics
9
Conclusions
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
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Conclusions
Dirlik Method
• Sample base input and SDOF response
Dirlik method calculates rainflow cycle cumulative histogram from response PSD. 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.
0.0001
0.001
0.01
0.1
1
10
100 100020 2000
Response 11.2 GRMSInput 6.1 GRMS
FREQUENCY (Hz)
AC
CE
L (
G2/H
z)
POWER SPECTRAL DENSITY fn=200 Hz Q=10
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Conclusions
• 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
SDOF Response Time Domain
Response Acceleration
Base Acceleration
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Conclusions
• Derive a 60-second PSD to envelope the flight data
• Consider 800 candidate PSDs formed by random number generation, with four coordinates each
Sample Flight Data
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Conclusions
Case Q b
1 10 4
2 10 9
3 30 4
4 30 9
Q & b Values for Fatigue Damage Spectra
Natural Frequencies: 20 to 2000 Hz
All cases will be analyzed for each successive trial.
Permutations
For Reference Only
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Conclusions
0.001
0.01
0.1
100 100020 2000
FREQUENCY (Hz)
AC
CE
L (
G2/H
z)
POWER SPECTRAL DENSITY ENVELOPE 3.3 GRMS OVERALL
Freq
(Hz)
Accel
(G^2/Hz)
20 0.0018
31 0.0019
211 0.0168
2000 0.0024
PSD Envelope, 3.3 GRMS, 60 sec
The PSD with the least overall GRMS which envelops the flight data via fatigue damage spectra
Optimized PSD
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Conclusions
102
104
106
108
1010
100 100020 2000
PSD EnvelopeMeasured Data
NATURAL FREQUENCY (Hz)
DA
MA
GE
IN
DE
XFATIGUE DAMAGE SPECTRA Q=10 b=4
FDS Comparison 1
16
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Conclusions
103
105
107
109
1011
100 100020 2000
PSD EnvelopeMeasured Data
NATURAL FREQUENCY (Hz)
DA
MA
GE
IN
DE
XFATIGUE DAMAGE SPECTRA Q=30 b=4
FDS Comparison 2
17
.
Conclusions
102
105
108
1011
1014
1017
100 100020 2000
PSD EnvelopeMeasured Data
NATURAL FREQUENCY (Hz)
DA
MA
GE
IN
DE
XFATIGUE DAMAGE SPECTRA Q=10 b=9
FDS Comparison 3
18
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Conclusions
104
107
1010
1013
1016
1019
100 100020 2000
PSD EnvelopeMeasured Data
NATURAL FREQUENCY (Hz)
DA
MA
GE
IN
DE
XFATIGUE DAMAGE SPECTRA Q=30 b=9
FDS Comparison 4
19
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Conclusions
0.0001
0.001
0.01
0.1
1
100 100020 2000
Maximum Envelope of 2.5-sec Segments, 2.0 GRMSFatigue Damage Spectrum, Optimized, 3.3 GRMS
FREQUENCY (Hz)
AC
CE
L (
G2/H
z)
POWER SPECTRAL DENSITY
PSD Comparison
Maximum Envelope is traditional piecewise stationary method, but its PSD need further simplification.
20
.
Conclusions
Conclusions
• 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
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.
Conclusions
Software
• Software and papers for applying this method are freely given at:
http://vibrationdata.wordpress.com/
• Will also cover in a future Shock & Vibration Webinar
© The Aerospace Corporation 2010 © The Aerospace Corporation 2012
Thank you
Tom Irvine
Email: [email protected]
Vibrationdata