A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under...
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Fatigue Workshop Fatigue Workshop -- ““Broadband spectral fatigue: from Gaussian to nonBroadband spectral fatigue: from Gaussian to non--Gaussian, from Gaussian, from research to industry”research to industry”
A comprehensive approach to fatigueA comprehensive approach to fatigueunder random loading:under random loading:
nonnon--Gaussian and nonGaussian and non--stationary loading investigationsstationary loading investigations
ENDIFENDIFDipartimento di IngegneriaDipartimento di IngegneriaUniversità di Ferrara, ItalyUniversità di Ferrara, Italy
DIEGMDIEGMDip. Dip. IngIng. . ElettricaElettrica GestGest. . MeccanicaMeccanica
Università di Udine, ItalyUniversità di Udine, Italy
Denis BenasciuttiDenis Benasciutti Roberto Roberto TovoTovo
February 24February 24thth, 2010 , 2010 –– Paris (F)Paris (F)
OverviewReal service loading :
•• RandomRandom•• nonnon--GaussianGaussian•• nonnon--stationarystationary•• multimulti--axialaxial
Planned research activity steps
1. Stationary, Gaussianuniaxial loading
2. non-Gaussian loading
3. non-stationary loading
4. multi-axial loading
Int J Fatigue (2002, 2005)Prob Eng Mechanics (2006)
Prob Eng Mechanics (2005)Int J Fatigue (2006)
This presentation:
• Introduction & theoretical background
• Gaussian loadings
• non-Gaussian loadings. Case study: mountain-bike data, automotive application
• non-stationary loadings (only a brief introduction)
This presentation:
• Introduction & theoretical background
• Gaussian loadings
• non-Gaussian loadings. Case study: mountain-bike data, automotive application
• non-stationary loadings (only a brief introduction)
Fat Fract Eng Mat Struct (2007)"VAL 2" Conference (2009)
Int J Mat & Product Tech (2007)Fat Fract Eng Mat Struct (2009)
Fatigue analysis of random loadings
FATIGUE LIFE
??
TIME DOMAINTIME DOMAIN FREQUENCY DOMAINFREQUENCY DOMAIN
DAMAGE – FATIGUE LIFE
Force \ stress \ strain
Time
amplitude
n° cycles
CYCLE DISTRIBUTIONamplitude
n° cumulated cycles (log)
LOADING SPECTRUM
COUNTING METHOD(e.g. ‘rainflow’ counting)
DAMAGE ACCUMULATION RULE(e.g. Palmgren-Miner linear law)
Frequency
PSD
Force \ stress \ strain
Time
•• randomrandom•• uniaxialuniaxial•• stationarystationary
Stationary random loadingsStationary random loadings
GAUSSIAN• narrow-band
• broad-band
- Rayleigh amplitude PDF
- Wirsching & Light (1980)- Dirlik (1985)- Zhao & Baker (1992)- Tovo (2002), Benasciutti & Tovo (2005)- Markov approach (Rychlik)
NON-GAUSSIAN• narrow-band
- Hermite model (Winterstein 1988)- power-law model (Sarkani et al. 1994)
• broad-band- Yu et al. (2004)- Benasciutti & Tovo (2005)- Markov approach- trasformed model (Rychlik)
STATIONARY LOADING
time
s(t)
ω
G(ω)
Gaussian
non-Gaussian
40
22
20
11 ;
0
ii dωωGωλ
Spectral parameters :
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
Max u
min
v
Fatigue analysis of random loadings
For repeated measurements (in the same condition):
{C1 , C2 , ... , Cn1}
{C1 , C2 , ... , ... , Cn2}
{C1 , ... , Cnk}
Counted cycle: (u,v)
MEASURED LOADMEASURED LOAD
+ +
RAINFLOWRAINFLOW CYCLES CCYCLES Cii
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
minv
Max u
Is olines of h(u,v) joint PDF
s
m
u v
dydxy)h(x,v)H(u,
Cycle distribution in random loadingsCycle distribution in random loadings
v)h(u, joint PDF
CDF
u
v v][u,dv duv)h(u, Prob
u = s + m
v = s - m
ms
s
m)sm,h(s2m)(s,p ma,
-
ma,a dmm)(s,p(s)p
amp-mean PDF
amp. PDF
Loading spectrum and fatigue damageLoading spectrum and fatigue damage
s
a dx(x)p(s)F
fatigue loading spectrum
0
amm ds(s)pss
Ks(T)N(T)D
m
fatigue damage
(s)F
s
PDF , CDF
h(u,v) , H(u,v)
amp. PDF
pa(s)
damage
D
LES
S “I
NFO
RM
ATI
ON
”LE
SS
“IN
FOR
MA
TIO
N”
(Palmgren-Miner rule)
Gaussian random loadingsGaussian random loadings
rclccrfc hb)(1hbh
rclccrfc Hb)(1HbH
The method only works for : stationary Gaussian ((broad-band)
random loadings
Distribution of rainflow cycles :
22
21α2.11
212121app 1α
ααe)α(ααα11.112ααb2
‘rfc’ rainflow counting
‘lcc’ level-crossing counting‘rc’ range-counting
nonnon--Gaussian random loadingsGaussian random loadingsObserved loading responses are often :• stationary (or almost-stationary)• non-Gaussian• broad-band
Gaussian : sk = ku-3 = 0
kurtosisσ
])μ(ZE[ku 4Z
4Z
skewnessσ
])μ(ZE[sk 3Z
3Z
Characterisation of non-Gaussian loading Z(t) :
OUTPUTSYSTEMSYSTEMINPUT
non-Gaussian(wave or wind loads,
road irregularity)
linearnon-Gaussian
nonlinearGaussian
EXAMPLE: data measured on a mountainEXAMPLE: data measured on a mountain--bike on offbike on off--road trackroad track
A model for nonA model for non--Gaussian loadingsGaussian loadings
Transformed Gaussian model: Z(t) = G{ X(t) }
Inverse transformation: X(t) = g{ Z(t) }
Existing models Existing models :• Hermite (Winterstein 1988, 1994)• exponential (Ochi & Ahn, 1994)• power-law (Sarkani et al., 1994)• nonparametric (Rychlik et al., 1997)
sk=0.5 ku=5
non-Gaussian Gaussian
• time-independent (memory-less)• strictly monotonic
G(-) is strictly monotonic :
2) xp(t1) > xp(t2) → zp(t1) > zp(t2) relative position
1) xp(ti) → zp(ti)=G{ xp(ti) } peak-peak (valley-valley) link
tit1 t2
x(t) Gaussian
z(t) non-Gaussian
rainflow count : same peak-valley couplingrainflow count : same peak-valley coupling
Transformation of rainflow cyclesTransformation of rainflow cyclesA non-Gaussian cycle (zp , zv) will be transformed to a corresponding Gaussian cycle (xp , xv) :
zp
zvxv
xp
(xp, xv) (zp, zv) = G{ (xp,xv) } = ( G{xp}, G{xv} )
)x,(xH)g(z),g(zH)z,(zH vpG
rfc X,vpG
rfc X,vpnG
rfc Z,
g(g(--))
G(G(--))
• peaks and valleys in a random loading are random variables• transformation G(-) “shifts” probabilities
)x,(xHb)(1)x,(xHb vpG
rc X,vpG
lcc X, Gaussian case :
Analysis schemeAnalysis scheme
NON-GAUSSIAN DATA
•Compute skew and kurt
•Estimate transformation g(-)
NONNON--GAUSSIAN DOMAINGAUSSIAN DOMAIN
g(g(--))
G(G(--))
GAUSSIAN DOMAINGAUSSIAN DOMAIN
• Estimate power spectrum
ω
G(ω)
GAUSSIAN DATA
• Estimate ‘rainflow’ distribution
rclccvpG
rfcX, Hb)(1Hb)x,(xH
• non-Gaussian ‘rainflow’ distribution
)x,(xH)z,(zH vpG
rfc X,vpnG
rfc Z,
)z,(zh vpnG
rfc Z,
Possible analysesPossible analysesZ(t) stationary non-Gaussian loading :
neglect non-Gaussianity:
)z,(zh vpG
rfc Z,
include non-Gaussianity
)z,(zh vpnG
rfc Z,
Case study: MountainCase study: Mountain--bike databike dataData measurements on a Mountain-bike in a Off-road use:
• various cycling conditions (uphill, downhill, level road cycling);
• different surface conditions (asphalt, cobblestone, gravel);
• both seated and standing cycling conditions.
Each measurement is clearly non-stationary.
Possible analyses:Possible analyses: -- irregularity factor, irregularity factor, IFIF-- variancevariance-- timetime--varying spectrum (STFT)varying spectrum (STFT)
50 100 150 200 250 300 350 400 450 500 5500
0.2
0.4
0.6
0.8
1
50 100 150 200 250 300 350 400 450 500 550
-50
0
50
Time [s ]
twind = 16 secoverlap = 80 %
Irregularity factor, Irregularity factor, IFIF
50 100 150 200 250 300 350 400 450 500 5500
40
80
120
160
200
twind = 16 secoverlap = 80 %
VarianceVariance
TIME, secTIME, sec TRACKTRACK SURFACESURFACE
0 – 100 plane asphalt100 – 442 uphill gravel442 – 515 downhill cobblestn.515 – 570 plane cobblstn.+
asphalt
FORCE on the BICYCLE FORKFORCE on the FORCE on the BICYCLE FORKBICYCLE FORK
Frequency[Hz
50 100 150 200 250 300 350 400 450 500 5500
20
40
60
80
100
120
140
160
180
200
-300
-250
-200
-150
-100
-50
0
50Spectrogram (STFT)Spectrogram (STFT)
nonnon--Gaussian dataGaussian data
Each segment is non-Gaussian
EXAMPLE EXAMPLE –– Force on bicycle forkForce on bicycle fork
50 100 150 200 250 300 350 400 450 500 550
-50
0
50
Time [s ]
Extraction of stationary segments
Estimated fatigue cumulative spectrumEstimated fatigue cumulative spectrumComparison : experimental spectrum (from data)
non-Gaussian estimated spectrum Gaussian estimated spectrum (as if Z(t) were Gaussian).
0
10
20
30
40
50
1E-05 0.0001 0.001 0.01 0.1 1 10 100
Experimental loading spectrum
non-Gaussian estimation
Gaussian estimation
amplitude
cumulated cycles/sec
skZ = - 0.19kuZ = 4.54skX = 0.02kuX = 2.99
0
5
10
15
20
1 10 100 1000 10000 100000
amplitude
cumulated cycles
100 blocks
Estimate fatigue life over theservice period (100 blocks )
0
5
10
15
20
1 10 100 1000
amplitude
cumulated cycles
Automotive applicationAutomotive applicationIn cooperation with
C.R.F. (Centro Ricerche FIAT)C.R.F. (Centro Ricerche FIAT)Orbassano, Italy
Stress in the critical pointfor 1 block(1 block = 60 sec)
?1 block
0
5
10
15
20
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
observednon-GaussianGaussian
amplitude
cumulated cycles
100’000 blocks
Analysis of nonAnalysis of non--stationarity loadingsstationarity loadings
50 100 150 200 250 300 350 400 450 500 550
-50
0
50
Time [s ]
Examples: road-induced loads in vehicles on different roads, loads in trucks switching between loaded/unloaded condition, wind/wave actions on off-shore structures under variable sea states conditions
Example of a switching loading
It is difficult to develop general models which apply to all types of load non-stationarity encountered in practical applications.
Several types of service loadings may be modelled as a sequence of adjacent stationary segments or states (“switching loadings”). Variability of switches is controlled by an underlying random process (‘regime process’).
p
1i sii
p
1iipw dx)x(pN)s(L(s)L Ni n° rainflow cycles in i-th segment
pi(x) amplitude distribution
Loading spectrum for piece-wise variance :
Each loading spectrum Li(s) can be also estimated in the frequency-domain from PSD.
0 200 400 600 800 1000 1200 1400 1600 1800 2000-20
-10
0
10
20
time [sec]
Switching loading with constant mean valueSwitching loading with constant mean valueAdjacent load segments with:• equal mean value• constant variance• deterministic switching times
Adjacent load segments with:• equal mean value• constant variance• deterministic switching times
Benasciutti D., Tovo R.: Frequency-based fatigue analysis of non-stationary switching random loads.Fatigue Fract. Eng. Mater. Struct. 30 (2007), pp. 1-14.
s
Li(s)
segment “i”s
Lj(s)
segment “j”
...+... =loading spectrum for piece-wise variance stationary load
s
Lpw(s)
Lpw(s) = Li(s)+Lj(s)
GOAL: Estimate the overall loading spectrum by including transition cycles.
-20
-10
0
10
20
30
40
50
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
time [sec]
‘REGIME PROCESS’
s
L(s)
Overall loading spectrum
s
Lt(s)
PROBLEM UNDER STUDY: Switching loadings with variable mean value
GIVEN the statistical properties of:• each stationary loading segment;• the ‘regime process’.
...+... =s
Li(s)
segment “i”s
Lj(s)
segment “j”s
Lpw(s)
Lpw(s) = Li(s)+Lj(s)
Switching loading with variable mean valueSwitching loading with variable mean valueAdjacent load segments with:• different mean values• constant variance• random switching times
Adjacent load segments with:• different mean values• constant variance• random switching times
Loading spectrum for transition cycles
loading spectrum for piece-wise variance stationary load
Numerical exampleNumerical example
100
101
102
103
104
1050
5
10
15
20
25
30
cumulated cycles
ampl
itude
from simulation Lpw(s) [transition cycles excluded]
L(s) [transition cycles included]
0
10
20
30
40
50
60
70
80
Z(t)
0 50 100 150 200 250 300 350 400 450 500
10
30
60
time [sec]
Uk
simulatedsimulated samplesample
ComparisonComparison of of loadingloading spectraspectra
5000510m3
5500010m2F =
10105000m1
m3m2m1
“From-to” matrix of ‘regime process’
Final overview of the method Final overview of the method
multiaxial
uniaxial
Type of load
Fat Fract Eng Mat Struct (2007)"VAL 2" Conference (2009)broad-band
Gaussiannon-Gaussian
non-stationary(switching)
Int J Mat & Product Tech (2007)Fat Fract Eng Mat Struct (2009)broad-band
Gaussiannon-Gaussian
stationary
Prob Eng Mechanics (2005)Int J Fatigue (2006)broad-bandnon-Gaussian
Int J Fatigue (2002, 2005)Prob Eng Mechanics (2006)broad-bandGaussianstationary
BandwidthPDF
ENDIFENDIFDipartimento di IngegneriaDipartimento di IngegneriaUniversità di Ferrara, ItalyUniversità di Ferrara, Italy
DIEGMDIEGMDip. Dip. IngIng. . ElettricaElettrica GestGest. . MeccanicaMeccanica
Università di Udine, ItalyUniversità di Udine, Italy
Denis Benasciutti Roberto Denis Benasciutti Roberto TovoTovo
Thanks for your attention!
[email protected]@uniud.it [email protected]@unife.it
Thanks for your attention!
Definition of the stress quantities• The Cauchy stress tensor
'xx1 2
3s 'zz
'yy2 2
1s
'xy3s '
xz4s 'yz5s
t'Itt H
3ttt2
tt
t3
ttt2t
tt3
ttt2
t'
yyxxzzzyzx
yzxxzzyy
yx
xzxyzzyyxx
ttttttttt
t
zzzyzx
yzyyyx
xzxyxx
ttr31tσH
• Deviatoric and spherical parts
• Euclidean representationof deviatoric part
Projection by ProjectionDamage estimation
• Euclidean deviator representation
1s
3s
time
S1' S3'
time
• Projection on “principal”frame of reference
Ref erence Curve 2k
i
k2
p,ii
ρref
ρrefΓDΓD
• Partial Damage Estimation of each “projected” load history
p,iΓ
j
i,jp,ii DΓD
• Total Damage estimation by proper Partial Damage cumulating
Deperrois A. (1991)De Freitas M, Li B, Santos JLT. (2000)
Cristofori A., Susmel L., Tovo R.Int J Fatigue, Vol. 30 n. 9, pp. 1646-1658 2008