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

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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

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

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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/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

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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)

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

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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%

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

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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)

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).

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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

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Thank you!

http://www.chalmers.se/math/EN/research/research-groups3561/spatio-temporal

Email: Wengang@chalmers.se