10 - responses in irregular waves

7/28/2019 10 - Responses in Irregular Waves

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7 – IRREGULAR WAVES

NUNO FONSECA - IST

The visible characteristic of the ocean waves is their irregularity:

• In the space

• In the time

confirmed by the analysis of wave elevation

measurements

Aerial photo of ocean waves

7.1 Introduction



Wave elevation time record (Fissel et al., 1999, OTC10794)

Aerial photo of ocean waves



The visible characteristic of the ocean waves is their irregularity in space and in

time

However:

In relatively large areas and for periods of approx. 20-30min the seastatemaintains the appearance

Which means:

The seastate maintains its statistical characteristics, it is statistically stationary

Confirmed from the analysisof wave elevation time records

THEN

For the seakeeping calculations the irregular waves can be represented as a:• Random (or stochastic) process

• With statistically stationary short term characteristics

• A long term model may be represented by a succession of short term

seastates



7.2 Generation and Propagation of Waves

Waves are generated by the interaction between the wind and the water surfacethrough:

• Frictional forces between the two fluids

• Local pressure field which changes in space and time

If the waves are of small amplitude then the propagation and dispersion of

waves is governed by the superposition principle



Generation of Waves

(a) The seastate results from multiple interactions between the wind and thefree surface, which vary in time and in space

The wave elevation felt at one point in space is the result from the sum ofall effects from all perturbations in the generation area to windward



(b) A local perturbation generates a wave system that radiates from the

perturbation point

At a long distance the wave system generated at a point look like 2D or long

crested

Due to angular dispersion the several wave systems come from different

directions and the combined system is 3D, or short crested



(c) The distance from the observation point to the boundary of the generation area

(to windward) is called fetch

(d) At a point fixed in space, the wave elevations increase with time. The time

interval since the storm initiation is called duration

(e) When the wave system reaches a statistically stable condition, then the seastate

is called fully developed

(f) If the observation point is well outside the storm generation area, then waves

look more 2D or regular and the seastate is called a swell



7.3 Short Term Model

A long term model may be represented by a succession ofshort term models

The figure shows a typical wave elevation record obtained in a point

fixed in space

Time vs Wave elevation



Assuming that all wave components advance in the same direction (long

crested waves), then the wave elevation in a point fixed in space may berepresented by the sum of harmonic and independent components (St. Denisand Pierson, 1953):

In this case x = y = 0 and:

are the wave amplitude and frequency of the component i

is the random phase angle

( ) ( )∑ −=i

iii t t ε ω ζ ζ cos

ii ω ζ ,

iε

(7.1)

Each harmonic component used to represent the irregular wave is:

( ) ( )iiii t t ε ω ζ ζ −= cos (7.2)



The harmonic wave components can be defined in terms of a function known

as variance spectrum or point spectrum:

( ) ( )δω ω ζ ii S t =2

(7.3)

point spectrum



The variance of an harmonic component may be calculated as:

Where T is the wave period. The former equation reduces to:

Combining (7.3) and (7.5) results in:

( ) ( ){ }∫−

−=2 /

2 /

22cos

1T

T

iiii dt t T

t ε ω ζ ζ

( ) 22

2

1ii t ζ ζ =

(7.4)

(7.5)

( )δω ω ζ ii S 2= (7.6)

Meaning that the amplitudes of the harmonic components of anirregular wave can be determined from its variance spectrum



The total variance of the irregular waves is a measure of the severity of the

seastate:

Since the individual harmonic components in (7.1) are independent and

random variables,then

the variance of the sum tends to sum of the variances (for a large number of

independent variables):

Or integrating equation (7.3) we have:

( )2t E ζ = (7.7)

( ) ( )∑==i

i t t E 22 ζ ζ

( ) ω ω d S E ∫∞

=0

(7.8)

(7.9)

The area under the variance spectrum gives the variance

of the seastate



Wave Energy

The energy of an harmonic wave per unit area is:

On the other hand equations (7.3) is (7.5) show that the variance of the

harmonic components within a frequency interval δ w is:

Which differs from the wave energy per unit area by a factor of ρ ρρ ρ g . This is thereason why the variance spectrum is sometimes referred as energy spectrum

2

2

1igζ ρ

( ) ( ) 22

2

1iiii S t ζ δω ω ζ ==



7.3 How to obtain the variance spectrum from wave records

(A)The wave elevation time records can be obtained from:

• Buoys

• Stationary Ships

• Systems based on laser or radarSatellites

Radars based onshore or on board of ships

(B) With the wave elevation records, the variance spectrum can be

obtained applying:

• The auto-correlation function calculated for various delays τ and applyingthe Fourier transform

• Using FFT programs (Fast Fourier Transform)

( ) ( )dt t t

T

∫ +⋅0

τ ζ ζ



7.4 Short Crested Waves

Equation (7.1) represents the wave elevation of long crested waves for a point inspace with coordinates x = y = 0. For an arbitrary point in space the waveelevation is:

long crested waves

The wave elevation for a wave system with its components travelling in different

directions µ is:

short crested waves

or setting x = y = 0:

( ) ( )[ ]∑ −++=i

iiii t y xk t y x ε ω µ µ ζ ζ sincoscos,, (7.10)

( ) ( )[ ]∑ ∑ −++=i j

iji j jiij t y xk t y x ε ω µ µ ζ ζ sincoscos,,

( ) [ ]∑∑ −=i j

ijiij t t y x ε ω ζ ζ cos,,

(7.11)

(7.12)



The variance of the short crested waves is:

where the amplitude of the harmonic components is:

The directional spectrum defines the seastate more correctly than the point

spectrum

( ) ( ) ω µ µ ω ζ ζ π

d d S E t i j

ij∑∑ ∫ ∫∞

===0

2

0

22,

2

1

( ) δωδµ µ ω ζ jiij S ,2= (7.14)

(7.13)



Directional spectrum



7.5 Characteristics of the point spectrum

Variance of the seastate:

A typical wave record measured in a point fixed in space is a continuous andirregular function.

Assuming that the process has a zero mean value, the variance is given by:

On the other hand the variance is given by the area bellow the spectrum:

( ) ( )∫−∞→

=2 /

2 /

22 1T

T T

dt t T

t Lim ζ ζ

( ) ( ) ω ω ζ d S t E ∫∞

==0

2

(7.15)

(7.16)



Gaussian properties:

Since the propagation of wave systems can be represented by the sum of a

large number of harmonic and statistically independent wave components

then

The wave elevation at one point has a normal or Gaussian distribution:

Using only the variance obtain the probability distribution of the wave elevation

( )

−=

E E p

2exp

2

1 2ζ

π ζ (7.17)



Wave elevation time record

Normaldistribution



-5 -4 -3 -2 -1 0 1 2 3 4 5

ζ(m)

0

200

400

600

N

Histograma - intervalos de 0.4m

Ajuste com distribuição normal

H S = 4.20m

T 0 = 11.5s

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

ζ(m)

0

100

200

300

400

N

Histograma - intervalos de 0.5m

Ajuste com distribuição normal

H S = 6.13m

T 0 = 11.5s

Comparison between histogram of measured wave elevationsand fitted normal distribution



Variance Spectrum

Typical variance wave spectrum

wave spectrum

0

2

4

0 0.5 1 1.5 2

ω (rad/s)

ζ S

( )sm2

=

=

m H

sT onda

S 20.4

5.11:

0

ω p

is the peak frequency or

modal frequency

is the peak period or modalperiod

pω

p

pT ω

π 2=



The nth order moment of the variance spectrum is defined as:

The moments of order 0, 2 and 4 represent respectively the:

( ) ω ω ω d S mn

n ∫∞

=0

(7.18)

( ) ( ) E d S t m === ∫∞

ω ω ζ 0

2

0

( ) ( ) ω ω ω ζ d S t m ∫

∞

==0

22

2&

Variance of the wave elevation displacement:

Variance of the wave elevation velocity:

Variance of the wave elevation acceleration: ( ) ( ) ω ω ω ζ d S t m ∫∞

==0

42

4&&



There are several parameters that can be estimated from the spectrum moments:

Average period of component waves:

Period corresponding to average frequencyof component waves:

Average period between zero upcrossings

( ) ( )

= ∫∫

∞∞

−

00

1 / ω ω ω ω d S Td S T

0

1

1

2

m

m

T −

− =

π

( )

1

0

0

1 / 2

−∞

= ∫ md S T ω ω ω π

1

01

2

m

mT

π =

2

02m

mT z π =

(7.19)

(7.20)

(7.21)



Average period between peaks (betweenmaxima or between minima):

Average wavelength between zeroupcrossings:

4

22 m

mT c π =

c zT T g

Lπ 2

= (7.23)

(7.22)



Typical wave record at a fixed point and several definitions



Probability distribution of maxima (and minima)

The wave elevation is a stationary process, with zero mean value andGaussian. If additionally the variance spectrum is narrow banded

then

The probability distribution of the maxima (and minima) can beapproximated to a Rayleigh distribution.

ProcessStationaryZero mean

Narrow banded

Rayleigh distributionof maximathen

( )

−=

E E

p

2

exp2ξ ξ

ξ (7.24)



Probability distribution of the maxima (and minima) – Rayleigh distribution:

Probability distribution of the double wave amplitude

( )

−=

E E p

2exp

2ξ ξ ξ (7.24)



The cumulative distribution gives the probability that the maxima is smaller

than a specific value ξ *:

The Rayleigh distribution is valid only for processes with a narrow banded

variance spectrum. The spectral broadness parameter is defined as:

ε = 0 the spectrum is narrow banded

In practical terms, narrow banded is assumed when ε < 0.6

( ) ( )

−−=<

∗∗

E p

2exp1

2ξ

ξ ξ (7.25)

2

1

−=

z

c

T

T ε (7.26)



Narrow banded and wide banded spectra



Picos da onda - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

p ( x < x

' )

Exp. positivos

Exp. negat ivos

Rayleigh

( )m pζ

=

=

m H sT onda

S 20.45.11:

0

Picos da onda - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 3 6 9

p ( x < x

' )

Exp. positivos

Exp. negat ivos

Rayleigh

( )m pζ

=

=

m H

sT onda

S 89.9

5.16:

0

Cumulative probability distribution of the maxima and minimaof the wave elevation



Another measure of the severity of the seastate is the estimate of the averagevalue of the 1/ n th highest maxima of the process. This is given by the centroidof the shaded area bellow:

The shaded area represents the probability that a maximum exceed ξ 1/n :

( ) ( )n

d p p

n

n

1

/ 1

/ 1 ==> ∫∞

ξ ξ ξ ξ ξ

(7.27)



The statistically mean value of the maxima above ξ 1/n is:

From equation (7.24) one obtains the probability density function of the waveheight Hw :

From equations (7.28) and (7.29) it is possible to calculate several statisticsrelated to the wave elevation namely:

Mean wave height: n =1

Mean wave amplitude: n =1

(7.28)( ) ξ ξ ξ ξ ξ

d pn

n

n ∫∞

=

/ 1

/ 1

( )

−=

E

H

E

H H p ww

w8

exp4

2

(7.29)

(7.31)

(7.30) E H w 5.2=

E 25.1=ζ



Mean value of the 1/3 highest waves, or the significant wave height:

n =3

Mean value of the 1/10 highest waves:

n =10

The former results are estimated assuming that the spectrum is narrowbanded.

In practical terms it is assumed that the spectrum is narrow banded if ε < 0.6,

which usually in errors smaller than 10%

The formulation and equations presented before are valid for many of the

ship responses.

(7.33)

(7.32)( ) E H w 0.43 / 1=

( ) E H w 1.510 / 1

=



7.6 Frequency of Encounter Wave Spectrum

In the previous sections the wave elevation and the corresponding variancespectrum were represented with respect to a reference system fixed in the space.

However the waves felt by the ship are modified due to the Doppler effect

associated to the forward speed.

In a reference system fixed in the space the, wave elevation of an harmonic waveis:

where: β is the ship heading with respect to the wave

ε ι is the random phase angle

A reference system fixed in the ship (with forward speed U ) is related with the

space fixed reference system by:

( ) ( )[ ]iiiii t y xk t y x ε ω β β ζ ζ +−+= sincoscos,, 0000

=

+=

y y

Ut x x

0

0

(7.34)

(7.35)



Substituting (7.35) into (7.34) results in the free surface elevation felt by the ship:

Which is the same as:

Since:

Finally the irregular wave elevation felt by the ship is:

(7.36)( ) ( )[ ]iiiiiei t U k yk xk t y x ε β ω β β ζ ζ +−−+= cossincoscos,,

( ) [ ]ieiiiiei t yk xk t y x ε ω β β ζ ζ +−+= sincoscos,,

β ω ω cosU k iiei −=

(7.37)

(7.38)

( ) [ ]∑ +−+=i

ieiiiie t yk xk t y x ε ω β β ζ ζ sincoscos,, (7.39)



The variance of the waves encountered by the ship is:

Where the spectral amplitudes of the wave spectrum in the frequency of

encounter are:

And since we have:

Then the relation between the spectra represented in the wave frequency and

encounter frequency is:

( ) ( )∑ ∫∞

===i

eeie d S E 0

22

21 ω ω ζ ζ

( ) eie d S ω ζ ω / 2

1 2=

δω β ω

δω

−=

g

U e

cos21

( )( )

g

U

S S e β ω

β ω cos2

1

,

−

=

(7.41)

(7.43)

(7.42)

(7.40)



7.7 Ocean Wave Data

To define the environment where the ship will operate it is necessary to:

(A) Know the probability of occurrence of the short term seastates –

Scatter Diagram (depends on the ocean area and the season)

(B) Use seastate variance spectra appropriate to the ocean area where

the ship operates (depends on the fetch, duration, etc.,)

Ocean wave statistics exist from the following sources:

(a) Visual estimates of wave conditions

(b) Wave measurements

(c) Hindcast wave climatology

b1) Point spectra fromwave measurements

b2) Directional spectra



(a) Visual estimates of wave conditions

These are visual estimates of the height and period of the waves by:

(1) Trained observers onboard weather ships

* Data exists only for the North Atlantique

(2) Officers onboard British ships

* This is the most extensive coverage available (Hogben and Lumb, 1967)

* Data from approximately 500 British ships travelling in all oceans

* Data collected between 1953 and 1961

The visual estimates of the wave height is approximately equal to thesignificant wave height

The visual estimates of the wave periods is approximately equal to the zeroupcrossing period

3 / 1

H H V

≅

zV T T ≅



(b) Wave Measurements

Data available is limited but more accurateb1) Point spectra from wave measurements

Point spectra can be computed if the time record of the wave elevation is of

sufficient duration. Wave elevation records can be obtained from:

• Ship borne wave meter

• Floating buoys with vertical accelerometers

• Laser technology

b2) Directional spectra

Directional spectra include information not only of the distribution of energyover the frequency range, but also the angular distribution of the direction ofpropagation. Directional spectra can be obtained from:

• Buoys fitted with an inertial measurement system to measure the wave

elevation and the slope in two perpendicular directions

• Systems installed on satellites

• Radar images



(b) Hindcast wave climatology

Calculate the wave spectra from wind data recorded during many years.

Both point spectra and directional spectra can be obtained



Scatter diagram for the Northern North Atlantic



Ocean Areas



7.8 Parametric Wave spectra

Parametric point wave spectra make it possible to obtain the variance wavespectrum as function of a small number of parameters.

(a) The Pierson Moskowitz Spectrum

• To represent wing generated waves, for fully developed waves and inareas without fetch limitations

• Depends of only one parameter, the wind speed, which can be directlyrelated to the wave peak period

• Not suitable for general design use



There are different spectral forms even if the seastate has the samesignificant wave height. Some of the most used spectral forms forseakeeping problems are:

(b) The JONSWAP spectrum

The JONSWAP (Joint North Sea Wave Project) spectrum resulted from the

analysis of extensive wave measurements in the North Sea

North sea is characterized by limited fetch in the generating area

Resulting wave spectra are more narrow banded than open sea spectra

( )

( )

=

−−

−

−22

24

52 p

p

p e

egS

σω

ω ω

γ ω α ω ω

ω β

JONSWAP spectrum

(7.44)

where:



where:

=⇒≥

=⇒≤≤

=⇒≤

−

5 0.5

0.56.3

5 6.3

15.175.5

γ

γ

γ

S p

H

T

S pS

S p

H T

e H T H

H T

S

p

25.1= β

( )[ ]γ α ln287.010609.54

2

−=

p

S

T

H

When γ = 1 in (7.44), sometimes the resulting spectra is calledPierson Moskowitz wave spectrum

periodpeak wavetheis Theightwavesinificanttheis

p

S H



(b) ISSC spectrum (International Ship Structures and Offshore Congress)

A two-parameter spectrum

Can be used to represent rising seas, falling seas and fully developed seas

( )

4

25.1

5

42313.0

−

= pe H

S pS ω

ω ω

ω (7.45)

frequencypeak wavetheis

heightwavesinificanttheis

pω

S H



Examples of measured and parametric wave spectra

Espectro da Onda

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

Experimental

Teórico (Bretshneider)ζ S

( )sm2

( ) Hz f e

=

=

m H

sT onda

S 20.4

5.11:

0

Espectro da Onda

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5

Experimental

Teórico (Bretshneider)ζ S

sm2

( ) Hz f e

=

=

m H

sT onda

S 13.6

5.11:

0

8 – SHIP RESPONSES TO IRREGULAR WAVES



The wave spectra that a ship encounters varies continuously in the space anin the time. On the long term and over large distances the waves are non-stationary

However it is assumed that:

In the neighbourhood of the ship the wave statistics change slowly enough

such that the reality can be approximated by a succession of short termprocesses that are stationary in the neighbourhood of the ship

THEN

On can focus on the ship responses to anexcitation that is random and Gaussian

8.2 Ship Responses to Irregular Waves



p p g

The ship responses to regular (and harmonic) waves can be interpreted asa linear transformation of the wave elevation

on the other hand

The theory of probability says that:

A linear transformation of a random, stationary and Gaussian process

results

Into another random, stationary and Gaussian process

(A)

(B)

(A) + (B) means that:

If the ship response is linear, then it is possible to relate the stationary andGaussian wave field with the stationary and Gaussian ship response

Stationary and Gaussianwave field

Stationary and Gaussianship response

simple relation

If the ship response is linear then it is possible to relate the stationary and



If the ship response is linear, then it is possible to relate the stationary andGaussian wave field with the stationary and Gaussian ship response:

In long-crested irregular waves

( ) ( ) ( )ω β ω β ω ζ S xS a

j j

2

,, =

wave spectrum

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0

We (rad/s)

S e ( m

^ 2 s )

Roll amplitude of the trans fer function

V=10kn, h ead.=120º

0

4

8

12

16

0.0 1.0 2.0 3.0

We (rad/s)

X 4 / a m

p ( º / m

Roll response spectrum

V=10kn, head .=120º

0

10

20

30

40

50

0.0 1.0 2.0 3.0

We (rad/s)

S x 4 ( g r a

u s ^ 2 * s

Responsespectrum

Amplitude oftransfer function

= *

2

2Wavespectrum= *

(8.1)

Simple Statistics of the Ship Responses



Simple Statistics of the Ship Responses

The variance of the ship response is:

And more generally the nth order moment of the variance spectrum is definedas:

( ) ( ) E d S t xm j j === ∫∞

ω ω 0

2

0

( ) ω ω ω d S m j

n

n ∫∞

=0

( ) ( ) ω ω ω d S t xm j j ∫∞

==0

22

2&

( ) ( ) ω ω ω d S t xm j j ∫

∞

==0

424 &&

(8.2)

(8.3)

(8.4)

(8.5)



The periods of the responses are calculated as presented for the wave elevation

by equations (7.19) to (7.23).

Important periods of the responses are:

Average period between zero upcrossings

Average period between peaks (between

maxima or between minima):4

22m

mT

c

π =

2

02mmT z π = (8.6)

(8.7)

Probability distribution of maxima (and minima) of the ship response



The linear ship response is a stationary process, with zero mean value

and Gaussian. If additionally the variance spectrum is narrow banded

then

The probability distribution of the maxima (and minima) can beapproximated to a Rayleigh distribution.

Process

Stationary

Zero mean

Narrow banded

Rayleigh distribution

of maximathen

( )

−=

E E p

2exp

2ξ ξ ξ (8.8)

The distribution of maxima and the statistics of maxima are similar to

those presented for the waves

8 3 S i i f M i



8.3 Statistics of Maxima

statistics of maxima are similar to those presented for the waveheight, although divided by two:

Mean response amplitude:

Mean value of the 1/3 highest response amplitudes, or thesignificant response amplitude:

Mean value of the 1/10 highest waves:

E x j 25.1=

( ) E x j 00.23 / 1=

( ) E x j 55.210 / 1

=

(8.10)

(8.11)

(8.9)

Th b bilit f d



The probability of exceedance

The cumulative distribution gives the probability that the maximum

of the response is smaller than a specific value x j*:

And finally the probability that a maximum is larger than a specific

value x j * is:

( )

( )

−−=<

∗

∗

E

x

x x p

j

j j 2exp1

2

( ) ( )

−=>

∗

∗

E

x x x p

j

j j2

exp

2

(8.12)

(8.13)



Picos da arfagem - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

p ( x

< x ' )

Exp. positivos

Exp. negativos

Rayleigh

( )m p3ξ

=

=

m H

sT onda

S 13.6

5.11:

0

Picos da arfagem - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

p ( x

< x ' )

Exp. pos itivos

Exp. negativos

Rayleigh

( )m p

3ξ

=

=

m H

sT onda

S 89.9

5.16:

0

Experimental data from a containership in head long crested waves with Fn = 0.25

Empirical cumulative distribution of the heave maxima and minima comparedto Rayleigh distribution



Picos do cabeceio - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

p ( x

< x ' )

Exp. pos itivos

Exp. negativos

Rayleigh

=

=

m H

sT onda

S 13.6

5.11:

0

( )graus p5ξ

Picos do cabeceio - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

p ( x

< x ' )

Exp. pos itivos

Exp. negativos

Rayleigh

=

=

m H

sT

ondaS 89.9

5.16

:0

( )graus p5ξ


Empirical cumulative distribution of the pitch maxima and minima comparedto Rayleigh distribution



Picos do MFV a meio-navio - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 3.0E+05 6.0E+05 9.0E+05

p ( x

< x ' )

Exp. alquebramento

Exp. contra-alquebramento

Rayleigh

( )KNm M p5

=

=

m H

sT onda

S 13.6

5.11:

0

Picos do ECV na secção 15 - dist. cumulativa

0.0

0.2

0.4

0.6

0.8

1.0

0 4000 8000 12000 16000

p ( x < x ' )

Exp. contra-alquebramento

Exp. alquebramento

Rayleigh

( )KN V p

3

=

=

m H

sT onda

S 13.6

5.11:

0


Empirical cumulative distribution of the vertical shear force and verticalbending moment maxima and minima compared to Rayleigh distribution

Expected Extreme Amplitudes



The statistical theory of extreme values combined with the narrow-band

assumption results in the estimates for the expected maximum in a

sample of N successive maxima:

However, if a large number of samples of the former sizes are taken, five

percent of them would be expected to contain maximum amplitudes

exceeding:

This involves the concept of confidence level

0

0

0

m4.45 ,01000

m3.85 ,1000

m3.25 ,100

=

=

=

N

N

N

0

0

45.4 ,1000

9.3 ,100

m N

m N

=

=

(8.14)

(8.15)

8.4 Assessment of Seakeeping Performance



(A) INTRODUCTION

Operability of a Ship: ship ability to carry out its mission safely

However

The effects of waves degrades the ability to carry out the mission

comparatively to calm water condition

Operability Index:

Measures the degradation of ship ability to carry out its mission

or in other words

Quantifies the seakeeping quality of the ship

The Operability Index accounts for:



The Operability Index accounts for:

o The ship mission, through the use of seakeeping criteria, which represent

acceptable limits of operation

o The hydrodynamic and inertia characteristics of the ship, through the use of a

seakeeping program to calculate the motions transfer functions

o The wave climate where the ship operates, through the use of the probability

distribution of short term seastates

The Seakeeping Criteria ensures that:

The ship is able to physically carry out its mission

The working conditions onboard are acceptable

The Seakeeping Criteria are usually related to absolute motions, relative motions,accelerations onboard, slamming and green water on deck.

(B) THEORY



( ) O

The method to calculate the operability index follows four steps:

1. Calculate the ship transfer functions

2. Calculate the ship responses to short term seastates

3. Select seakeeping criteria and calculate curves of maximum significant wave heightsin which the ship can operate

4. Calculate the percentage of time in which the ship is operational in a given ocean

area or route (operability index)

(b1) Transfer Functions

• Absolute motions

• Relative motions at selected positions

• Vertical and lateral accelerations atselected positions

Are calculated for all

frequency range and heading

range using a Strip Method

(b2) Ship Responses to Short Term Seastates



( ) p p

Pierson Moskowitz spectrum represents the short term seastates in terms of H s and T p

The response spectrum is calculated by:

( ) ( ) ( )ω ω ω 2 H S S w R =

And the variance of the response is

( ) ( ) ( ) 21

2

0

21

2

0

2 RS wS R R H d H S H d S σ ω ω ω ω ω σ === ∫∫

∞∞

(b3) Calculate curves of H Smax in which the ship can operate

If a seakeeping criterion is defined as a limiting root mean square of the response σ CR,

then the maximum allowed H s for a given mean wave period and ship heading is:

( )1

max , R

CR zS T H σ

σ β =

(8.16)

(8.17)

(8.18)

(a) If a seakeeping criterion is defined as a limiting root mean square of the response



(b) If a seakeeping criterion is defined as a limiting probability, pCR,of exceeding a

critical value r max, then the corresponding σ CR may be calculated assuming a Rayleigh

distribution of the peaks of the responses (see eq. 8.13):

( )CR

CR p

r

/ 1ln2

2max=σ

(a) If a seakeeping criterion is defined as a limiting root mean square of the response

σ CR, then the maximum allowed H s for a given mean wave period and ship heading is:

( )1

max , R

CR zS T H

σ

σ β =

(b4) Calculate the Operability Index

Compare de curves of H smax(T z , β ) with the scatter diagram for a given ocean area and

calculate the probability that the ship is operational in for that ocean area

(8.19)

(8.20)

(C) CALCULATION EXAMPLE



Containership Fishing Vessel

Area of operation: North Atlantique

L pp = 175 mC B = 0.57

Displ. = 24742 ton

Speed = 22 Kn

Area o f operation: Portuguese west coast

L pp = 20.1 mC B = 0.42

Displ. = 129.5 ton

Speed = 10 Kn

(c1) Seakeeping Criteria



Response Location (m) x ,y ,z Criterion

Roll - 4º (rms)

Green water on deck 90, 0, 11 5% (prob)

Vert. accel. at fwd. pp. 90, 0, 0 0.2g (rms)

Slamming 63.8, 0, -9.5 2.5% (prob)

Propeller emergence -83, 0, -4.5 12% (prob)

Vert. accel. at bridge -63, 0, 0 0.15g (rms)

Lat. accel. at bridge -63, 0, 27 0.10g (rms)

Response Location (m) x ,y ,z CriterionRoll - 6º (rms)

Green water on deck 10.8, 0, 4.4 5% (prob)

Slamming 10.8, 0, -2.0 3% (prob)

Vert. accel. at bridge 7.6, 0, 4.5 0.2g (rms)

Lat. accel. at bridge 7.6, 0, 4.5 0.1g (rms)

Propeller emergence -8.15, 0, 1.2 15% (prob)

Vert. accel. work. deck -8.5, 0, 2.5 0.2g (rms)

Lat. accel. work. deck -8.5, 0, 2.5 0.1g (rms)

Containership

Fishing Vessel

(c2) Ship Transfer Functions



Vertical Acceleration at the Bridge (V=22Kn)

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4

180º

150º

120º

90º

60º

30º

0º

( )2−

sa

V

ζ

ξ &&

g L pp /

Roll Amplitudes (V=22 Kn)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 1 2 3 4

150º

120º

90º

60º

30º

g L pp

/

ak 4

Calculate: (13 transfer functions) * (7 headings)

(c3) Maximum Allowed Significant Wave Heights



Max. Sig. Wave Heights (V =10 Kn, β = 180º)

0

4

8

12

0 5 10 15

Roll Water on deck

Slamming Vert. accel. at bridge

Lat. accel. at bridge Vert. accel. on deck

Lat. accel. on deck Propeller emergence

( )m H S

( )sT z


0

4

8

12

0 5 10 15

Roll Water on deck

Slamming Vert. accel. at bridge

Lat. accel. at bridge Vert. accel. on deck

Lat. accel. on deck Propeller emergence

( )m H S

( )sT z

Calculate response spectra and variances for: (13 responses) * (25 seastates,2s<T z<15s, H s=1m) * (7 headings)

Calculate ( ) 1max / , RCR zS T H σ σ β =

(c4) Calculate the Operability Index



Compare de curves of H smax(T z , β ) with the scatter diagram for a given ocean area andcalculate the probability that the ship is operational in for that ocean area


0

2

4

6

0 2 4 6 8 10

Roll

Water on deck

Slamming

Vert. accel. at bridgeLat. accel. at bridge

Vert. accel. on deck

Lat. accel. on deck

Propeller emergence

( )sT z

( )m H S


0

2

4

6

0 2 4 6 8 10

All criteria satisfied

( )m H S

( )sT z



14.5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0002 0.0002 0.0001

13.5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0000

12.5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0003 0.0002 0.0001 0.0001

11.5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0005 0.0004 0.0002 0.0001

10.5 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0007 0.0008 0.0007 0.0004 0.0001

9.5 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0014 0.0016 0.0011 0.0006 0.0002

8.5 0.0000 0.0000 0.0000 0.0000 0.0003 0.0015 0.0028 0.0029 0.0019 0.0009 0.0003

Hs(m) 7.5 0.0000 0.0000 0.0000 0.0001 0.0009 0.0035 0.0059 0.0054 0.0032 0.0013 0.0004

6.5 0.0000 0.0000 0.0000 0.0003 0.0026 0.0083 0.0121 0.0098 0.0051 0.0019 0.0005

5.5 0.0000 0.0000 0.0000 0.0010 0.0073 0.0193 0.0237 0.0165 0.0074 0.0024 0.0006

4.5 0.0000 0.0000 0.0001 0.0033 0.0195 0.0413 0.0414 0.0239 0.0091 0.0025 0.0006

3.5 0.0000 0.0000 0.0006 0.0106 0.0454 0.0726 0.0566 0.0260 0.0080 0.0018 0.0003

2.5 0.0000 0.0000 0.0025 0.0271 0.0771 0.0855 0.0479 0.0163 0.0038 0.0007 0.0001

1.5 0.0000 0.0002 0.0079 0.0421 0.0656 0.0430 0.0151 0.0034 0.0005 0.0001 0.0000

0.5 0.0000 0.0013 0.0076 0.0134 0.0084 0.0024 0.0004 0.0000 0.0000 0.0000 0.0000

3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5

Tz(s)

Global Wave Statistics – Area 8, annual statistics



Fishing Vessel operating in

the Portuguese W. Coast

(Annual statistics)

Containership operating in

the North Atlantique

(GW areas 8, 9, 15 an 16)

Head

(deg)F.Foz G.W. 16 Year Winter

180 0.88 0.61 0.86 0.74

150 0.91 0.72 0.82 0.70

120 0.96 0.74 0.90 0.8490 0.77 0.50 1.00 0.99

60 0.96 0.75 0.75 0.59

30 1.00 1.00 0.99 0.99

0 1.00 1.00 0.99 0.98

Aver. 0.93 0.76 0.90 0.83

Operability Indexes

10 - responses in irregular waves

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