Chalmers University of TechnologyDepartment of Mathematical SciencesGöteborg, Sweden
Effect of whipping on ship fatigue- Gaussian VS non-Gaussian modelling
Wengang Mao, Igor Rychlik
Measurement data
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Measured voyaged in the North Atlantic during 0.5 year Investigated vessel and the measurement location
Measurement location
1. Full-scale measurements of mid-section during half year2. 14 voyages: 7 EU-US and 7 US – EU 3. Rainflow (RFC) fatigue estimation as a reference4. Total fatigue = wave induced fatigue (WF) + whipping fatigue (HF)
Scope of the project
1. From full-scale measurements:• Our definition of whipping;
• How much fatigue is contributed from whipping;
• Investigation of Narrow band approximation (NBA) and Non-Gaussian contributions.
2. General methodology (NBA) for the ship fatigue design
3. Non-Gaussian effect on extreme responses
4. Discussions
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RFC analysis of measure stresses
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1 2 3 4 5 6 7 8 9 10 11 12 130
1
2
3
4
5
6x 10
10
No. of the passages
To
tal f
atig
ue
da
ma
ge
by
RF
C Winter voyages
Summer voyages
Fatigue estimated for different voyages (RFC)
0 1000 2000 3000 4000 5000 6000 70000
5
10
15
20
25
30
35
40
45
50
No. of individual sea states
Sta
nd
ard
de
via
tio
n
Standard deviationof each sea state
Spikes
Measured stresses during one voyage (2 weeks)
Standard deviation of responses in each individual sea state
Overview of fatigue estimation
1. Fatigue in Winter from EU to US (Important);
2. Measurement errors should be cleaned
3. Large sea states for further investigation
4. Small sea states only contribute 3.7% fatigue damage
1 2 3 4 5 6 7 8 9 10 11 12 130
2
4
6x 10
10
No. of the passages
Fat
igue
dam
age
(RF
C)
1 2 3 4 5 6 7 8 9 10 11 12 13-15
-10
-5
0
Diff
eren
ce R
atio
(%)
Based on all sea statesSea states with Std>10MPa
3.7% underestimation
Fatigue difference due to removing small response
Our definition of whipping
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2 2.1 2.2 2.3 2.4 2.5 2.6
x 104
-20
0
20
40
60
80
100
120
140
160
Wave induced responseHigh frequency response
1 2 3 4 5 6 70
100
200
300
400
500
600
700
Spectral density
Frequency [rad/s]
S(w
) [m
2 s /
rad]
Wave inducedfp1 = 1.1 [rad/s]Whipping contrifp2 = 4.6 [rad/s]
Whipping: High frequency response
Separated signal with wave frequency & high frequency
Whipping Definition:
1. Total response = Wave induced + Whipping
2. Wave induced response: fz [0, 2] [rad/s]
3. Whipping response: fz [2, 8] [rad/s]
4. Measurement noise: fz > 8 rad/s 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7
x 104
-30
-20
-10
0
10
20
30
High frequency response
Crossing level u=4*std
Spectrum
Investigation of whipping
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0 2 4 6 8 10 1210
-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Radius frequency (rad/s)
No
rma
lize
d r
esp
on
se s
pe
ctru
m S
()
Wave induced response
Whipping response
Normalized spectrums of response in a voyageWave induced energy
97%
Energy of high frequency (whipping) response
3%
Whipping induced energy:
1. Three peaks of measured spectrum;
2. Last peak is treated as measurement noise;
3. Whipping ratio =
4. Average whipping ratio is less than 3%.
Total
whipping
Energy
Energy
Whipping effect on Fatigue
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Fatigue components based on measurements1. Wave induced fatigue (WF): 72%,2. Whipping contributed fatigue (HF): 24%,3. Some other contributions: 4%.
Wave induced fatigue72%
Splitting signal into stationary period
<1%
Whipping contribution24%
Remove Std<10 MPa4%
Fatigue componentsWhipping contribution to total fatigue damage
080106 080117 080129 080209 080218 080301 080312 080321 080401 080411 080424 080603 0806130
1
2
3
4
5
6x 10
10
Date of each voyage (identical with the column 1 in Table 1)
Fatig
ue da
mage
(
D
(t))
080106 080117 080129 080209 090218 080301 080312 080321 080401 080411 080424 080603 0806130
10
20
30
40
Diffe
rence
Rati
o (%
)
Total fatigue damage (WF + HF)
Wave induced fatigue (WF)
HF fatigue: Whipping contributed24% fatigue !!
Gaussian assumption
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Non-Gaussian effect on fatigue:
1. Simulated process (the same spectrum);
2. Largest fatigue difference 5%;
3. Identical average fatigue
4.Gaussian model is available for
applications.
1 2 3 4 5 6 7 8 9 10 11 12 130
2
4
6
8x 10
10
No. of the passages
Fatig
ue d
amag
e1 2 3 4 5 6 7 8 9 10 11 12 13
0
10
20
30
40
50
Diff
eren
ce R
atio
(%) High frequency whipping 32.8% by NBA !!
Non-Gaussian responses (whipping) contribute 24% of total fatigue!
1. Balanced between whipping contribution (30%) and NBA conservative part (33%);
2. For long-term fatigue estimation, NBA maybe a good choice!
Simulated Gaussian process
Gaussian model and NBA
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/47.0)2/1(2)/()( 32/sz
mmsz
nb htfmhtftDE
0
22 zf04 sh
NBA for Gaussian response
1. Energy of ship response 3%: (less influenced by whipping!)
2. hs stress range little influenced by whipping!
3. No measurement available Numerical analysis (linear).
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
5
6x 10
15
Angular frequency (rad/s)
En
erg
y d
en
sity fu
nctio
n
HDG = 0
HDG = 40
HDG = 90
Hσ() – RAOs (transfer function)
hs Significant stress range (energy)
fz frequency of the mean level
Strongly effected by noise (cut-frequency)
hs based on hydrodynamic numerical analysis
),,(/ UTfHhC zsisii
04 sh
A new simple fatigue model
Polar diagram of the constant C (linear relation between hs and Hs) in terms of the ship speed U (radial direction) and the heading angle (hoop direction).
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Hs – significant wave height
Tz – crossing period of waves
U0 – ship forward speed
– heading angle
4
54
23
)2
(1
exp4
)(
z
z
s T
T
HS
0
2
00
20
2 )()(),|(cos)/(
ddfSUHgUwn
n
Model for fz
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1. fz is strongly influenced by whipping (measurement)
2. fz is computed by linear numerical analysis: fz(waveship)
3. Approximated by the encountered wave frequency: fz(mod)
fz model (encountered wave frequency)
|)/()cos2(/1| 2zzz gTUTf
Vibration period of 2 seconds
Vibration of ship beam model (Hogging)
fz comparison by different approaches
40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
Observed significant response range (MPa)
Fra
ctio
n o
f fz
am
on
g d
iffe
ren
t a
pp
roa
ch
es
fz(mod)/fzfz(waveship)/fz
• Linear numerical underestimate fz;
• Expected value of fz computed from model is close to fz.
Estimation of Safety indexTotal fatigue damage during one voyage
25.23)(
64.17
)cos()(
2.4
1)( iH
g
viHCtD s
iss
iii
nbj
M
j
nbj
nb DTD1
)(
Δt – time interval of one stationary sea statevs – ship forward speed
i – ship heading angle
Hs – significant wave height
0 1 2 3 4 5 6 7 8
x 10-3
0
1
2
3
4
5
6
7
8x 10
-3
Fatigue damage estimated by Rainflow
Fa
tigu
e d
am
ag
e e
stim
ate
d b
y th
e m
od
el
The model works well for the measurement:•Errors for this ship are below 30%.;
•Errors are smeared out when compute the total damage;
•The proposed model depends on Hs.
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Page 13Page 13
Gaussian assumption for Extreme prediction
Rice’s formula for Gaussian crossings:
0
2
2)]([ u
zeTfuNE T
))((
))((
2
1)(
txV
txVtf z
1. Up-crossing rate of one stationary period (30min)
2. Up-crossing rate of half a year period
Crossing of half a year interval: E[Nhy
+(X100)]=3.6*10-4
Discussions
• Whipping contributed fatigue 30%• Whipping induced average energy 3%
• hs can be computed by linear analysis
• Simple NB fatigue model works well wrt RFC• Gaussian assumption will lead to large
underestimation of extreme response
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Thank you!
http://www.chalmers.se/math/EN/research/research-groups3561/spatio-temporal
Email: [email protected]