environmental forces and moments

7/28/2019 Environmental Forces and Moments

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2Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Superposition of Wind and Wave Disturbances

For control systems design it is common to assume the principle of superposition whenconsidering wind and wave forces:

Ocean CurrentsThe effect due to ocean currents is simulated by introducing the relative velocity vector:

Chapter 8 - Environmental Forces and

Moments

M C D g g0 wind wavew

r c c

uc

vc

wc

0

0

0

MRB CRBrigid-body terms

MA r CArr Drrhydrodynamic terms

g gohydrostatic terms

w


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8.1.1 Wind Forces and Moments on

Marine Craft at Rest

Beaufort number Description of wind Wind speed (knots)

0 Calm 0-1

1 Light air 2-3

2 Light breeze 4-7

3 Gentle breeze 8-11

4 Moderate breeze 12-16

5 Fresh breeze 17-21

6 Strong breeze 22-27

7 Moderate gale 28-33

8 Fresh gale 34-40

9 Strong gale 41-48

10 Whole gale 49-56

11 Storm 57-65

12 Hurricane More than 65

The wind speed is usually specified in terms of Beaufort numbers

(Price and Bishop, 1974)


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Marine Craft at RestWind Coefficient Approximation for Symmetrical Ships

For ships that are symmetrical with respect to the xz- and yz-planes, the wind

coefficients for horizontal-plane motions can be approximated by:

which are convenient formulae for computer simulations.

Experiments indicate that:

cx {0.50, 0.90}cy {0.70, 0.95}

cn {0.05, 0.20}

These values should be used with care.

CXw cx cosw

CYw cy sinw

CNw cn sin2w


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Moving Marine Craft

wind 12aVrw

2

CXrwAFw

CYrwALw

CZrwAFw

CK

rwA

LwH

Lw

CMrwAFwHFw

CNrwALwLoa

Vrw urw2 vrw

2

rw atan2vrw,urw

urw u uw

vrw v vrw

uw Vw cosw

vw Vw sinw

The wind speedVwand its directionwcan be measured by a wind sensor (anemometer).

The wind coefficients (look-up tables) are computed numerically or by experiments in

wind tunnels.

Generalized force in terms ofrelative angle of attackgrw

and relative wind speed Vrw

:


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8.1.3 Wind Coefficients Based on Flow

over a Helmholtz-Kirchhoff PlateReference: Blendermann (1994)

The load functions are parameterized in terms

of four primary wind load parameters:

CXw CDlALwAFw

CDlAF

cosw

1 2

1 CDl

CDtsin22w

CYw CDt sinw1

21

CDl

CDtsin22w

CKw CYw

CNw sL

Loa 0.18 w

2

CYw

Type of vessel CDt CDlAF0 CDlAF

1. Car carrier 0.95 0.55 0.60 0.80 1.2

2. Cargo vessel, loaded 0.85 0.65 0.55 0.40 1.7

3. Cargo vessel, container on deck 0.85 0.55 0.50 0.40 1.4

4. Container ship, loaded 0.90 0.55 0.55 0.40 1.4

5. Destroyer 0.85 0.60 0.65 0.65 1.1

6. Diving support vessel 0.90 0.60 0.80 0.55 1.7

7. Drilling vessel 1.00 0.701.00 0.751.10 0.10 1.7

8. Ferry 0.90 0.45 0.50 0.80 1.1

9. Fishing vessel 0.95 0.70 0.70 0.40 1.1

10. Liquified natural gas tanker 0.70 0.60 0.65 0.50 1.1

11. Offshore supply vessel 0.90 0.55 0.80 0.55 1.2

12. Passenger liner 0.90 0.40 0.40 0.80 1.2

13. Research vessel 0.85 0.55 0.65 0.60 1.4

14. Speed boat 0.90 0.55 0.60 0.60 1.115. Tanker, loaded 0.70 0.90 0.55 0.40 3.1

16. Tanker, in ballast 0.70 0.75 0.55 0.40 2.2

17. Tender 0.85 0.55 0.55 0.65 1.1

CDl CDlAFwAFwALw

Matlab MSS toolbox

>> [w_wind,CX,CY,CK,CN] = blendermann94(gamma_r,V_r,AFw,ALw,sH,sL,Loa,vessel_no)


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0 30 60 90 120 150 180

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Angle of wind gw

(deg) relative bow

Wind Coefficients

0 30 60 90 120 150 180

0

0.2

0.4

0.6

0.8

Angle of wind gw

(deg) relative bow

0 30 60 90 120 150 180

0

0.2

0.4

0.6

0.8

Angle of wind gw

(deg) relative bow0 30 60 90 120 150 180

-0.1

-0.05

0

0.05

0.1

0.15

Angle of wind gw

(deg) relative bow

CX

CY

CK

CN

8.1.3 Wind Coefficients Based on Flow

over a Helmholtz-Kirchhoff Plate

CX is generated using CDl,AF(0) and CDl,AF() corresponding to head and tail winds.

Research vessel withAfw= 160.7 m,Alw= 434.8 m, sL = 1.48 m, sH = 5.10 m, Loa = 55.0 m,

Lpp= 48.0 m and B =12.5 m


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8.1.4 8.1.7 Wind Coefficients

8.1.4 Wind Coefficients for Merchant Ships (Isherwood, 1972)Isherwood (1972) has derived a set of wind coefficients by using multiple regression

techniques to fit experimental data of merchant ships. The wind coefficients are

parameterized in terms of eight parameters.

8.1.5 Wind Resistance of Very Large Crude Carriers (OCIMF, 1977)Wind loads on very large crude carriers (VLCCs) in the range 150 000 to 500 000 dwt

can be computed by applying the results of (OCIMF 1977).

8.1.6 Wind Resistance of Large Tankers and Medium Sized ShipsFor wind resistance on large tankers in the 100 000 to 500 000 dwt class the reader is

advised to consult Van Berlekom et al. (1974).

Medium sized ships of the order 600 to 50 000 dwt is discussed by Wagner (1967).

8.1.7 Wind Resistance of Moored Ships and Floating StructuresWind loads on moored ships are discussed by De Kat and Wichers (1991) while an

excellent reference for huge pontoon type floating structures is Kitamura et al. (1997).


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8.2 Wave Forces and MomentsA marine control system can be simulated under influence of wave-induced forces by

separating the 1st-order and 2nd-order effects:

1st-order wave-induced forces (wave-frequency motion)

Zero-mean oscillatory motions

2nd-order wave-induced forces (wave drift forces)

Nonzero slowly varying component

Wave forces are observed as a mean slowly varying component referred to as Low-

Frequency (LF) motions and an oscillatory component called Wave-Frequency (WF)

motions which must be compensated for by the feedback control system:

The mean component (LF) can be removed

by using integral action

The oscillatory component (WF) is usually

removed by using a cascaded notch and

low-pass filter (wave filtering)

wave wave1 wave2


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8.2 Wave Forces and MomentsWave Response Models:

1. Force RAOs

2. Motion RAOs

3. Linear state-space models (WF models)

The first two methods require that the RAO tables are

computed using a hydrodynamic program (depends on vessel geometry)

The last method is frequently used due to simplicity but it is only intended for testing of

robustness and performance of control systems, that is closed-loop analysis.

RAO = Response Amplitude Operator


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1. Force RAOs

2. Motion RAOs

8.2 Wave Forces and Moments

Marine

craftWave

spectrum

sea state

generalized

force

generalized

position

Marine

craft

Wave

spectrum

sea state

Generalized forcegeneralizedposition

Force

RAO

Motion

RAO


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8.2 Wave Forces and Moments3. Linear State-Space Models (WF models)

Marine

craft

Linear wave spectrum

approximation

White noise

Generalized force

generalized

position

Tunable

gain K

WF

position


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8.2 Wave Forces and Moments1. Force RAOs

2. Motion RAOs

3. Linear state-space models (WF models)

Wavespectrum

Motion RAO

Seastate

Waveamplitude

1st-order wave-induced positions

S( )

w

H ,Ts z Ak

Wave

spectrum

1st-order

Force RAO

Seastate

Waveamplitude

1st-order wave-induced force

2nd-orderForce RAO

2nd-orderwave drift force

S( )

wave1H ,Ts z Ak

Generalized

force

Generalized

position

Generalized

position


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8.2.1 Sea State Descriptions

For marine craft the sea can be characterized by the following wave spectrum parameters:

The significant wave height Hs(the mean wave height of the one third highest waves, also denoted as H1/3)

One of the following wave periods:- The average wave period, T

- Average zero-crossing wave period, Tz- Peak period, Tp (this is equivalent to the modal period, T)


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8.2.2 Wave Spectra

The process of wave generation due to wind

starts with small wavelets appearing on the

water surface. This increases the drag force,

which in turn allows short waves to grow.

These short waves continue to grow until

they finally break and their energy is

dissipated.

A developing sea, or storm, starts with high

frequencies creating a spectrum with a peak

at a relative high frequency.

A storm which has lasted for a long time is

said to create a fully developed sea.

After the wind has stopped, a low -frequency

decaying sea or swell is formed. These long

waves form a wave spectrum with a low peak

frequency.

S( )

Swelland tidalwaves

Developingsea

If the swell from one storm interacts with the

waves from another storm, a wave spectrum

with two-peak frequencies may be observed.

In addition, tidal waves will generate a peak at

a low frequency. Hence, the resulting wave

spectrum might be quite complicated in cases

where the weather changes rapidly


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8.2.2 Wave Spectra

Neumann Spectrum (1952)One-parameter spectrum:

Bretschneider Spectrum (1959)Two-parameter spectrum:

Pierson-Moskowitz Spectrum (1963)

Two-parameter spectrum:

Modified Pierson-Moskowitz (MPM) Spectrum (1964)For prediction of responses of marine craft and offshore

structures in open sea, the International Ship and Offshore

Structures Congress and the International Towing Tank

Conference (ITTC), have recommended the use of a

modified version of the PM-spectrum where

A 8.1 103 g2 constant

B 0.74g

V19.4

4

3.11Hs

2

S A5 expB4

S 1.250

4Hs2

45 exp 1.250/

4

S

C

6

exp2g2

2

V

2

lim1

S g2 5

Hs 2.06

g2

V19.42

A 43Hs

2

Tz4

, B 163

Tz4

Tz 0.710T0 0.921T1


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0 2 4 8 12 17 22 28 34 41 49 57 65

5

10

15

20

25

knots

m/s

Sea state 9

Sea state 8

Sea state 7

Sea state 6

Sea state 5

Sea state 4Sea state 3

Significant

waveheightH

s(m)

3020100

Windspeed V

Beaufortnumber1 2 3 4 5 6 7 8 9 10 1211

Hs= 2.06 V2/g2

8.2.2 Wave Spectra

Hs 2.06

g2

V19.42


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8.2.2 Wave Spectra

S 155Hs

2

T145 exp 944

T14

4 Y 0.07 for 5.24/T1

0.09 for 5.24/T1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100JONSWAP spectrum

w (rad/s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

Modified Pierson-Moskowitz (PM) spectrum

w (rad/s)o

= 0.4

Hs

Hs

S() (rad/s)

S() (rad/s)

JONSWAP SpectrumIn 1968 and 1969 an extensive

measurement program was carried

out in the North Sea, between the

island Sylt in Germany and Iceland.

The measurement program is known

as the Joint North Sea Wave Project

(JONSWAP) and the results from

these investigations have beenadopted as an ITTC standard in 1984.

Y exp 0.191T1 1

2

2


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8.2.2 Wave Spectra

Torsethaugen spectrum: upper plot shows only one peak

at = 0.63 rad/s representing swell and developing sea

while the lower plot shows low-frequency swell and newly

developing sea with peak frequency = 1.57 rad/s.

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

(rad/s)

0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

(rad/s)

Hs

= 3-10 m

To

= 10 s

Hs

= 3-10 m

To

= 4 s

S()/(Hs2T

o)

Hs

Hs

0

= 0.63 rad/s

o

= 1.57 rad/s

Torsethaugen SpectrumThe Torsethaugen spectrum (1996) is

an empirical, two-peaked spectrum

developed by Norsk Hydro.

Includes the effect of swell (low-

frequency peak) and newly developed

waves (high-frequency peak).

The spectrum is developed using

curve fitting of experimental data

from the North Sea.


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8.2.2 Wave Spectra

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

(rad/s)

S()/(Hs2T

0)

Modified Pierson-Moskowitz

JONSWAP for g= 3.3JONSWAP for g= 2.0Torsethaugen

Wave spectra plotted forsame wave height and

peak frequency.

Wave demo option in the

MSS toolbox script:

>> gncdemo

Computation and plotting

of wave spectra:

>> S = wavespec()


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8.2.3 Wave Amplitude Response Model

The relationship between the wave spectrum S(k)and the wave amplitudeAkfor a

wave component kis (Faltinsen 1990)

where is a constant difference between the frequencies. This relationship can be

used to compute wave-induced responses in the time domain.

Long-Crested Irregular SeaThe wave elevation of a long-crested irregular sea in the origin of {s} under the

assumption of zero speed can be written as the sum ofNharmonic components

wherekis the random phase angle of wave component number k. Since this

expression repeats itself after a time 2/a large number of wave componentsNare

needed. However, a practical way to avoid this is to choose krandomly in the interval

12

Ak2 Sk

k1

N

Akcosk k k1

N

2Sk cosk k

k 2

,k2


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8.2.3 Wave Amplitude Response Model

Short-Crested Irregular Sea

The most likely situation encountered at sea is short crested or confused waves. This isobserved as irregularities along the wave crest at right angles to the direction of the wind.

Short-crested irregular sea can be modeled by a 2-D wave spectrum

where = 0corresponds to the main wave propagation direction while a nonzero -valuewill spread the energy at different directions.

S, Sf

f 2 cos

2, /2 /2

0; elsewhere

k1

N

i1

M

2Sk,i cosk k

Spreading function

Wave elevation


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8.2.3 Wave Amplitude Response ModelExtension to Forward Speed using the Frequency of EncounterFor a ship moving with forward speed U, the peak frequency of the spectrum will be

modified according to:

eU,0, 0 0

2

g Ucos

k1

N

i1

N

2Sk,i cosk k

2

g Ucosi

eU,

k,

i

k

Wave Elevation using Frequency of Encounter

e is the frequency of encounter

b is the angle between heading and wave direction


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Normalized Force RAOsThe 1st- and 2nd-order wave forces for varying wave directions i and wave frequencies k

Processing Re-Im parts of the RAOs (ASCII file generated by hydrodynamic code)

8.2.4 Wave Force Response Amplitude

Operators

Fwave1dof

k,i wave1dof

k,i

gAkejwave1

do fk,i

Fwave2

dofk,i

wave2dof

k,i

gAk2ejwave2

do fk,i

Fwave1dof

k,i Imwave1dofk, i2 Rewave1dofk, i

2

Fwave1dof k,i atanImwave1dofk, i,Rewave1dofk, i

Fwave2dof

k,i Rewave2dofk,i

Fwave2dof

k,i 0

complex variables

(wave drift)


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No Spreading Function

With Spreading Function


Operators

wave1dof

k1

N

g Fwave1dof

k, Akcos eU,k,tFwave1dof

k, k

wave2dof

k1

N

g Fwave2dof

k, Ak2 coseU,k,t k

(wave drift)

Wavespectrum

1st-orderForce RAO

Seastate

Waveamplitude

1st-order wave-induced force

2nd-orderForce RAO

2nd-orderwave drift force

S( )

wave1

H ,Ts z Ak12

Ak2 Sk

wave1dof

k1

N

i1

M

g Fwave1dof k, i Akcos eU, k, it Fwave1

dof k, i k

wave2dof

k1

N

i1

M

g Fwave2dof k, i Ak

2 cos eU, k, it k

(wave drift)


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OperatorsThe motion RAOs are processed in the MSS Hydro toolbox by using:

wamit2vessel % read and process WAMIT data

veres2vessel % read and process ShipX (Veres) data

The data is represented in the workspace as Matlab structures

vessel.forceRAO.w(k) % frequencies

vessel.forceRAO.amp{dof}(k,i,speed_no) % amplitudes

vessel.forceRAO.phase{dof}(k,i,speed_no) % phases

where speed_no = 1 represents U = 0. For the mean drift forces only surge, sway and yaw

are considered (dof {1,2,3} where the 3rd component corresponds to yaw)

vessel.driftfrc.w(k) % frequenciesvessel.driftfrc.amp{dof}(k,i,speed_no) % amplitudes

It is possible to plot the force RAOs using:

plotTF

plotWD


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Wave-Frequency Motion (1st-order Motions)

are the motion RAO amplitude and phase

for frequency k and wave direction i

Total Motion

8.2.5 Motion Response Amplitude

Operators

(wave drift must be added manually in the equations of motion)

12

Ak2 Sk

wdof

k1

N

i1

M

wdof

k,i Akcos eU,k,it wdof

k,i k

Wavespectrum

Motion RAO

Seastate

Waveamplitude

1st-or er wave-induced positions

S( )

w

H ,Ts z Ak

y w

where |w1dof

k,i| and w1dof

k,i


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8.2.5 Motion Response Amplitude

OperatorsWave-Frequency Motions

The 1st-order wave-induced forces, wave1, are zero-mean oscillatory wave forces.In a linear system we have:

The 1st-order wave response can be evaluated as:

where the transfer function is a LP filter representing the vessel dynamics.

Moreover, the 1st-order wave response can be computed by LP filtering the generalized

forces wave1. The linear responses are usually represented by RAOs.

Notice that the motion RAOs depend on the vessel matrices MRB,A(),B() andCwhile

force RAOs are only dependent on the wave excitations.

MRB A B C wave1 cost Reejt

2MRB A jB C wave1

Hvjwave1 Hvj 2MRB A jB C

1

harmonic motions


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8.2.6 State-Space Models for Wave

Responses

J

M C D g g0 wind wave2

y w

w

KHssws

Approximation 2 (Wave Spectrum)A linear wave response approximation for Hs(s) is usually preferred by ship control

systems engineers. Easy to use and easy to test control systems.

No hydrodynamic SW is needed for the price of a tunable gainKwhich must be

tuned manually to match the response w

Approximation 1 (WF Position)

tunable gain

8 2 6 S S M d l f W


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Responses

x w1

x w2

0 1

02 20

xw1

xw2

0

Kww

yw 0 1 xw1

xw2

hs Kws

s2 20s 02

x w Awxw eww

ywcwxw

Linear Wave Spectrum Approximation (Balchen 1976)

PM-spectrum

State-Space Model

S A5 expB4

8 2 6 St t S M d l f W


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Responses

8 2 6 St t S M d l f W


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Responses

Example 8.2 (Linear Models for 1st- and 2nd-order Wave-Induced Forces)

A marine control system can be tested under influence of wave-induced forces byseparating the 1st- and 2nd-order wave-induced forces. For a surface vessel in 3 DOF

(dof{1,2,6}) the wave forces and moments are:

The wave drift forces di(i=1,2,3) are modeled as slowly varying bias terms (Wienerprocesses):

wave Xwave,Ywave,Nwave Xwave Kw

1s

s2 21e1

s e1 2

w1 d1

Ywave Kw2

ss2 22e

2s e

2 2w2 d2

Nwave Kw

6s

s2 26e6

s e6 2

w3 d3

d 1 w4

d 2 w5

d 3 w6

Here wi(i=1,2,...,6) are Gaussian white noise processes.


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8.3 Ocean Current Forces and Moments

Current Speed and Direction

The current speed is denoted by Vcwhile its direction relative to the moving craft is

conveniently expressed in terms of two angles: angle of attack c, and sideslip angle c

For computer simulations the current velocity can be generated by using a 1st-order Gauss-

Markov Process

V c Vc w Vmin Vct Vmax

xb

-b

xstabzb

yb

Uxflow


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Equations of Motion including Ocean Currents

rigid-body and hydrostatic terms hydrodynamic terms

MRB CRB g g0 MA r CArr Drr wind wave

r

c relative velocityc

uc,vc,wcvcb ,0,0,0

The generalized ocean current velocity of an irrotational fluid is:


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Property 8.1 (Irrotational Constant Ocean Currents)If the Coriolis and centripetal matrix CRB(r) is parameterized independent of linear velocity

n = [u, v, w]T, for instance by using (3.57):

and the ocean current is irrotational and constant, the rigid-body kinetics satisfies(Hegrens, 2010):

Equations of Relative VelocityProperty 8.1 can be used to simply the representation of the equations of motion.

Moreover,


MRB CRB MRB r CRBrrr

vbvcb

b/nb

CRB mS2 mS2Srg

b

mSrgbS2 SIb2

n2 = [p, q, r]T

M r Crr Drr g g0 wind wave

M MRB MA

Cr CRBr CAr


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8.3.1 3-D Irrotational Ocean Current Model

3-D Irrotational Ocean Current ModelA 3-D current model is obtained by transforming the current speed Vc from FLOW axes toNED velocities:

Transformation from NED to BODY

c uc,vc,wc,0,0,0

vcn Ry,c

Rz,c

Vc

0

0

vcn

Vc cosccosc

Vc sinc

Vc sinccosc

uc

vc

wc

Rbn

nbvc

n

xb

-b

xstabzb

yb

Uxflow


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2-D Ocean Current ModelFor the 2-D case, the 3-D equations with c= 0reduces to:

These expressions are used to compute current forces for ships using:

Current coefficients

Cross-flow drag

For underwater vehicles the 3-D representation should be used.

uc Vc cosc

vc Vc sinc

vcn

Vc cosc

Vc sinc

0

Vc uc2 vc

2


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39

m11 m12 0 0

m21 m22 0 0

0 0 1 0

0 0 0 1

v

r

V c

m11Vc sin c m11Vcrcos c d11Vc sin c d11v d12r

m21Vc sin c m21Vcrcos c d21Vc sin c d21v d22r

r

Vc

b1

b2

0

0

Ywind

Nwind

0

0

Ywave

Nwave

0

0

m11 sin c

m21 sin c

0

1

w

Example 8.3 (Maneuvering Model)Consider a linearized maneuvering model in state-space form:

Augmenting the current speed to this equation gives:

N i h h d l i li i,

V d

b


m11 m12 0

m21 m22 0

0 0 1

v

r

d11 d12 0

d21 d22 0

0 1 0

v vc

r

b1

b2

0

Ywind

Nwind

0

Ywave

Nwave

0

vc Vc sinc v c V c sinc Vcrcosc

Vc sinc wsinc Vcrcosc

environmental forces and moments

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