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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
Page 2
80oW 60oW 40oW 20oW 0o 12oN
24oN
36oN
48oN
60oN
Measured voyaged in the North Atlantic during 0.5 yearInvestigated vessel and the measurement location
Measurement location
1. Full‐scale measurements of mid‐section during half year
2. 14 voyages: 7 EU‐US and 7 US – EU
3. Rainflow (RFC) fatigue estimation as a reference
4. 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
Page 3
RFC analysis of measure stresses
Page 4
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
Total fatigue damage by RFC
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 s e a s tate s
Standard deviation
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 13-15
-10
-5
0
Difference Ratio (%)
3.7% underestimation
Fatigue difference due to removing small response
Our definition of whipping
Page 5
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/s1.52 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 res pons eCros s ing level u=4*s td
Spectrum
Investigation of whipping
Page 6
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)
Normalized response spectrum 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
EnergyEnergy
Whipping effect on Fatigue
Page 7
Fatigue components based on measurements1. Wave induced fatigue (WF): 72%,
2. Whipping contributed fatigue (HF): 24%,
3. Some other contributions: 4%.
Wave induced fatigue72%
stationary period<1%
Whipping contribution24%
Remove Std<10 MPa4%
Fatigue componentsWhipping contribution to total fatigue damage
Date of each voyage (identical with the column 1 in Table 1)080106 080117 080129 080209 090218 080301 080312 080321 080401 080411 080424 080603 080613
0
10
20
30
40
Difference Ratio (%) HF fatigue: Whipping contributed
24% fatigue !!
Gaussian assumption
Page 8
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 1010
No. of the pas s age s
Fatigue damage
1 2 3 4 5 6 7 8 9 10 11 12 130
10
20
30
40
50
Difference Ratio (%) High fre que ncy 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
Page 9
/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)
Energy density function
HDG = 0HDG = 40HDG = 90
Hσ() – RAOs (transfer function)
hs Significant stress range (energy)
fz frequency of the mean levelStrongly 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).
Page 10
Hs – significant wave height
Tz – crossing period of waves
U0 – ship forward speed
– heading angle
454
23
)2
(1exp4)(
z
z
s TT
HS
0
2
0 02
02 )()(),|(cos)/(
ddfSUHgUw
n
n
Model for fz
Page 11
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)
Fraction of fz among different approaches
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()(
2.41)( iH
gviHCtD s
iss
iii
nbj
M
j
nbj
nb DTD1
)(
Δt – time interval of one stationary sea state
vs – ship forward speed
i – ship heading angleHs – 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
Fatigue damage estimated
by the model
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.
Page 12
Page 13
Gaussian assumption for Extreme prediction
Rice’s formula for Gaussian crossings:
0
2
2)]([ u
zeTfuNE T
))((
))((21)(
txVtxVtf 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
Page 14
Thank you!
http://www.chalmers.se/math/EN/research/research-groups3561/spatio-temporal
Email: [email protected]