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Mark Paulus-NUWC Keyport!
Slide 1
Enhancement of power law model for accelerated
life testing
Naval Undersea Warfare Center
Abhijit Dasgupta, PhD
University of Maryland
CALCE-Center for Advanced Life Cycle Engineering
Mark Paulus, PhD Principal Engineer, Advanced Test Development
Distribution Statement A: Approved for Public Release; Distribution is unlimited.
Mark Paulus-NUWC Keyport!
Slide 2
♦ Discuss the limitations of the Power Law Model when comparing different excitation vibration profiles
♦ Discuss the effects of a change in natural frequency and the resulting change in stress state
♦ Introduce the new semi-empirical model which accounts for the changing stress state
ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Outline
Mark Paulus-NUWC Keyport!
Slide 3
Test Specimen ♦ 1018 cold rolled mild steel
Ø annealed
♦ Full failure was defined when tip touched bar located ~1 in below tip at start
Full Failure Point
Mark Paulus-NUWC Keyport!
Slide 4 ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Excitation profiles
0.00001
0.0001
0.001
0.01
0.1
1
100 100020 2000
ED SMOOTH RS-40WHITE-MEDWHITE-LOWWHITE-HIGHED SIM RS-40
Frequency (Hz)
G2 /H
z
• 5 test specimens used for each profile RS-40 ED SIM RS-40 WHITE-HIGH Excitation WHITE-HIGH Response
Mark Paulus-NUWC Keyport!
Slide 5
0.00001
0.0001
0.001
0.01
0.1
1
100 100020 2000
ED SMOOTH RS-40WHITE-MEDWHITE-LOWWHITE-HIGHED SIM RS-40
Frequency (Hz)G
2 /Hz
ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Power Law Method
nkPSDPSDTTF 1)( =
Power Law using PSD
0
20
40
60
80
ED SIM R
S-40
ED SIM R
S-60
ED SMOOTH R
S-40
ED SMOOTH R
S-60
WHITE-HIG
H
WHITE-LOW
WHITE-MED
Profile
Tim
e to
Fai
lure Time-Predicted
Time-Measured
♦ Use measured TTF and initial PSD from WHITE-HIGH and WHITE-LOW, combined with the power law to predict TTF of other profiles.
Mark Paulus-NUWC Keyport!
Slide 8
0.00001
0.0001
0.001
0.01
0.1
1
100 100020 2000
ED SIM RS-40WHITE-HIGH
Frequency (Hz)
G2 /H
z
Stress State changes
.0349 G2/Hz 8 Grms-15 min
Fn@ t0
.347 G2/Hz 8 Grms- 31 min
Fn@ t1
Fn@ t2
Natural frequency shift affects time to failure
Fn@ t3
Fn@ t4
Mark Paulus-NUWC Keyport!
Slide 9
0
50
100
150
200
250
300
350
400
0 10 20 30Time(min)
Fn (H
z)
ED SIM RS-40WHITE-HIGH
Natural Frequency Shift During Failure
1ii
1iii tt
fnfnRFC
−
−
−
−=
Slope is Rate of Natural Frequency Change (RFC)
Mark Paulus-NUWC Keyport!
Slide 10
Accelerated Life Model Implementation
♦ Step 1- Subject a test item to a vibration profile. Measure the RFC(fn) and ξ(fn)
♦ Step 2- Compute the SDOF relative displacement
♦ Step 3- Perform maximum likelihood estimate (or equivalent) to determine C and p.
np
rmsCyRFC ω=
( ) ( )∫∞
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎠⎞⎜
⎝⎛ +−
=0 22224 )2(2
1)(),( dfffff
fwffynnn
PSDnrmsξπ
!!
∑=
Δ=
N
i in
in
fRFCf
TTF1 )(
Mark Paulus-NUWC Keyport!
Slide 11
Comparison to Power Law using PSD
♦ Use measured RFC and PSD from WHITE-HIGH to predict TTF of other profiles.
♦ Only need to test one excitation profile
nkPSDPSDTTF 1)( =
Power Law using PSD
0
20
40
60
80
ED SIM R
S-40
ED SIM R
S-60
ED SMOOTH R
S-40
ED SMOOTH R
S-60
WHITE-HIG
H
WHITE-LOW
WHITE-MED
Profile
Tim
e to
Fai
lure Time-Predicted
Time-Measured
Beam 2-SEL Model
01020304050607080
ED SIM
RS-40
ED SIM
RS-60
ED SMOOTH R
S-40
ED SMOOTH R
S-60
WHITE-HIG
H
WHITE-LOW
WHITE-MED
Profile
MTT
F (m
in)
Time-PredictedTime-Measured
Mark Paulus-NUWC Keyport!
Slide 12 ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Rate of Frequency Change
0
0.5
1
1.5
2
2.5
3
0 50 100 150 200 250 300 350
RFC
(Hz/
s)
Natural Frequency (hz)
White-High Meas.White-High Pred.
• Prediction of RFC is more accurate in the elastic region • Modeling can be done over any frequency change that is desired
Mark Paulus-NUWC Keyport!
Slide 13
A Note About Damping Factor
0
20
40
60
80
100
0 100 200 300fn(Hz)
Q(G/G)
Q=2*sqrt(fn)Q=linearQ=meas.
0
20
40
60
80
100
0 100 200 300fn(Hz)
Q(G/G)
Q=2*sqrt(fn)Q=linearQ=meas.
WHITE-HIGH WHITE-LOW
• Damping ratio can be related to quality factor by
• from Steinberg
• Measured value of Q from experiment
• Item dependent
• Amplitude and frequency dependent
• Linear approximation gives reasonable results
nfQ 2=
Q21
=ζ
Q21
=ζ
Mark Paulus-NUWC Keyport!
Slide 14
Comparison of Damping Models
ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Beam 2-SEL Model
01020304050607080
ED SIM
RS-40
ED SIM
RS-60
ED SMOOTH R
S-40
ED SMOOTH R
S-60
WHITE-HIG
H
WHITE-LOW
WHITE-MED
Profile
MTT
F (m
in)
Time-PredictedTime-Measured
010203040506070
MTT
F(m
in)
Profile
Beam 2 SEL Model - Linear Q
Time-PredictedTime-Measured
Mark Paulus-NUWC Keyport!
Slide 15
♦ Power Law model does not account for a change in natural frequency
♦ A change in natural frequency leads to a change in the stress state
♦ New semi-empirical model accounts for changing stress state which improves prediction accuracy
ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
Conclusions
Mark Paulus-NUWC Keyport!
Slide 16
Mark Paulus, PhD NUWC-Keyport
Principal Engineer- Advanced Test Development Further Reading:
Ø M.E. Paulus, A. Dasgupta and E. Habtour, Life estimation model of a cantilevered beam subjected to random vibration, Fatigue and Fracture of Engineering Materials. 2012. DOI:10.1111/j.1460-2695.2012.01693.x
Ø M.E. Paulus, A. Dasgupta, Semi-empirical life model of a cantilevered beam subject to random vibration, International Journal of Fatigue, 45 (2012) 82–90.
ASTR 2012 Oct 17-19, Toronto, Ontario, Canada
QUESTIONS?
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