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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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: Wengang@chalmers.se