ch6

1

Chapter 6 Maneuvering Theory

An alterna)ve to the seakeeping formalism is to use maneuvering theory to describe the mo)ons of marine cra9 in surge, sway and yaw. In maneuvering theory, frequency-dependent added mass and poten)al damping are approximated by constant values and thus it is not necessary to compute the uid memory eects.

6.1 Rigid-Body Kine)cs 6.2 Poten)al Coecients 6.3 Nonlinear Coriolis Forces due to Added Mass in a Rota)ng Coordinate System 6.4 Viscous Damping and Ocean Current Forces 6.5 Maneuvering Equa)ons

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

Copyright Bjarne Stenberg/NTNU

2

MRB!" ! CRB!!"!rigid-body forces

!MA "!r ! CA!!r"!r ! D!!r"!rhydrodynamic forces

! g!#" ! gohydrostatic forces

# $ ! $wind ! $wave

#

#

M!! " C!!"! " D!!"! " g!"" " go # # " #wind " #wave

!r ! ! ! !c

In the case of ocean currents

Maneuvering equa)ons of mo)on (no ocean currents)

Chapter 6 Maneuvering Theory

!c ! ocean current velocity


This model can be simplied if the ocean currents are assumed to be constant and irrota)onal in {n} such that (see Sec)on 8.3): According to Property 8.1, it is then possible to represent the equations of motion by relative velocities only:

!" c !!S!#b/nb " 03!303!3 03!3

!c #

M !!r " C!!r"!r " D!!r"!r " g!"" " go # # " #wind " #wave # M ! MRB "MAC!!r" ! CRB!!r" " CA!!r"


3

6.1 Rigid-Body Kinetics

MRB !! " CRB!!"! # "RB

! ! !u,v,w,p, q, r"!

It is common to assume that conventional marine craft has homogeneous mass distribution and xz-plane symmetry so that: Let the {b}-frame coordinate origin be set in the centerline of the cra9 in the point CO, such that yg = 0. Under the previously stated assump)ons the 3-DOF model sa)ses:

Rigid-Body Kine)cs (Chapter 3)


Ixy ! Iyz ! 0 #

MRB !m 0 00 m mxg0 mxg Iz

, CRB!!" !0 !mr !mxgrmr 0 0mxgr 0 0

#

4

Seakeeping Analysis: The seakeeping equa)ons of mo)on are formulated as perturba)ons: about an iner)al equilibrium frame such that:

!MRB ! A"!#$!" ! B"!#!# ! C! " fcos"!t# # ! ! !" ! !!x,!y,!z,!",!#,!$"! #

forced oscilla)ons

The matrices A(),B() and C represents a hydrodynamic mass-damper-spring system which varies with the frequency of the forced oscilla)ons.

6.2 Potential Coefficients


The poten)al coecients are usually represented as frequency-dependent matrices for 6-DOF mo)ons. The matrices are:

A(w) added mass B(w) poten-al damping

where w is the wave excita)on frequency of a sinusoidal (regular) wave generated by a wave maker or the ocean.

5

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 x 10 6

A22

(w)

frequency (rad/s)

0 1 2 3 4 5 6 7 0 1 2 3 4 5 x 10 6

B22

(w)

frequency (rad/s)

6.2 Potential Coefficients

A() added mass

B() poten)al damping

!MRB ! A"!#$!" ! B"!#!# ! C! " fcos"!t# #

This introduces uid memory eects which can be interpreted as ltered poten)al damping forces. These forces are retarda)ons func)ons that can be approximated by state-space models. It is standard to include the uid memory eects in seakeeping models while 3-DOF classical maneuvering theory neglects the uid memory by using a zero-frequency assump)on.

As stressed in Chapter 5 the seakeeping model should be represented by Cummins equa)on in the )me domain to avoid the frequency-dependent matrices.

A!!" ! constant !!B!!" ! 0

# #

For underwater vehicles opera)ng at water depths below the wave-aected zone (20 m), the poten)al coecients will be independent of the wave excita)on frequency:

Poten)al coecients for a ship


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6.2.1 3 DOF Maneuvering Model Deni-on 6.1 (Zero-Frequency Models for Surge, Sway and Yaw). The horizontal mo)ons (surge, sway and yaw) of a marine cra9 can be described by a zero-frequency model: Discussion: When applying a feedback control system to stabilize the motions in surge, sway

and yaw, the natural periods will be in the range of 100-200 s. This implies that the natural

frequencies are in the range of 0.03-0.10 rad/s which is quite close to the zero wave excitation frequency.

Also note that viscous damping forces will dominate the potential damping terms

at low frequency and that fluid memory effects can be neglected at higher speeds.

This is a common assump)on in ship maneuvering. It is convenient to represent the equa)ons of mo)on without using frequency-dependent matrices since this reduces model complexity.

MA ! A!1,2,6"#0$ !A11#0$ 0 00 A22#0$ A26#0$0 A62#0$ A66#0$

Dp ! B !1,2,6"#0$ ! 0

#

#


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6.2.2 6-DOF Coupled Motions One limita)on of the zero-frequency assump:on is that it cannot be applied to heave, roll and pitch. For 2nd-order mass-damper-spring systems the domina)ng frequencies are the natural frequencies. Solu-on: Formulate frequency-independent models in heave, pitch and roll at their respec)ve natural frequencies and not the zero frequency Deni-on 6.2 (Natural Frequency Models for Heave, Roll and Pitch) The natural frequencies for the decoupled mo)ons in heave, roll and pitch are given by the implicit equa)ons: The poten)al coecients Aii and Bii (i=3,4,5) are computed at the natural frequencies such that

These equa)ons are decoupled damped oscillators. However, the natural frequencies can also be computed for 6-DOF coupled mo)ons by using the frequency-dependent modal analysis (Sec)on 4.3.2).

!heave !C33

m " A33!!heave", !roll ! C44Ix " A44!!roll"

, !pitch ! C55Iy " A55!!pitch" #

!m ! A33!!heave"" ! B33!!heave"w ! C33z " 0

!Ix ! A44!!roll""p# ! B44!!roll"p ! C44" " 0

!Iy ! A55!!pitch""q# ! B55!!pitch"q ! C55# " 0

#

#

#


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6.2.2 6-DOF Coupled Motions Frequency-Independent (Constant) Added Mass and Damping Matrices:

MA !

A11!0" 00 A22!0"

. . .0

A26!0"

. . .A33!!heave" 0 0

0 A44!!roll" 00 0 A55!!pitch"

. . .

0 A62!0" . . . A66!0"

Dp !

0 00 0

. . .00

. . .B33!!heave" 0 0

0 B44!!roll" 00 0 B55!!pitch"

. . .

0 0 . . . 0

#

#

DV ! diag!B11v ,B22v ,B33v ,B44v ,B55v ,B66v" # Diagonal elements computed from the )me constants and natural periods of the system (see Sec)on 6.4)

The matrix elements as a func)on of frequency are computed using a hydrodynamic program. Key assump-on: No coupling between surge-sway-yaw and heave-roll-pitch subsystems.


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6.3 Nonlinear Coriolis Forces due to Added Mass in a Rotating Coordinate System Added Mass: Hydrodynamic added mass can be seen as a virtual mass added to a system because an accelera)ng or decelera)ng body must move some volume of the surrounding uid as it moves through it. Moreover, the object and uid cannot occupy the same physical space simultaneously. The uid kine)c energy can be used in energy formula)on to derive the expressions for the Coriolis and centripetal matrix

CA!!r"

depends on which reference frames that are considered: 1) Nonlinear Lagrangian Theory

considers the mo)on of a rota)ng frame {b} with respect to {n}

2) Seakeeping Theory, the body frame {b} rotates about {s}. It is common to consider only the linearized version denoted by

CA!!r"

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

10

Alterna)ve approach to the Newton-Euler formula)on: Lagrangian mechanics

Euler-Lagranges Equa)on (only for generalized coordinates):

Kirchho's Equa)ons in Vector Form (uses only kine)c energy / velocity)

L ! T ! V

ddt

!L! !! "

!L!! " J

"!!!""

ddt

!T!!1

! S!!2!!T!!1

" "1

ddt

!T!!2

! S!!2!!T!!2

! S!!1!!T!!1

" "2

#

# T ! 12 !!M!

!1 ! !u, v,w"!

!2 ! !p,q, r"!!1 ! !X,Y, Z"!

!2 ! !K,M,N"!

kine)c energy

Dierence between kine)c and poten)al energy

Generalized coordinates = 6 DOF

Quaternions are not generalized coordinates

Body veloci)es are not generalized coordinates ! ! !u,v,w,p, q, r"!

! ! !N,E,D,!,",#"! #

! ! !N,E,D,!,"1,"2,"3"! #


6.3 Nonlinear Coriolis Forces due to Added Mass in a Rotating Coordinate System

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The matrix represents linearized forces due to a rota)on of {b} about the seakeeping frame {s}. Instead of using {s} as the iner)al frame, we will assume that {n} is the iner)al frame and that {b} rotates about {n}. The added mass matrix in {b} sa)ses:

Fluid Kine)c Energy The concept of uid kine)c energy can be used

to derive the added mass terms. Any mo)on of the vessel will induce a mo)on in

the otherwise sta)onary uid. In order to allow the vessel to pass through the uid, it must move aside and then close behind the vessel.

Consequently, the uid mo)on possesses kine)c energy that it would lack otherwise (Lamb 1932).

TA ! 12 !!MA!


MA ! !

Xu" Xv" Xw" Xp" Xq" Xr"Yu" Yv" Yw" Yp" Yq" Yr"Zu" Zv" Zw" Zp" Zq" Zr"Ku" Kv" Kw" Kp" Kq" Kr"Mu" Mv" Mw" Mp" Mq" Mr"Nu" Nv" Nw" Np" Nq" Nr"

CA!

MA ! A!!" ! constant #



12

ddt

!TA!u ! r

!TA!v "q

!TA!w "XA

ddt

!TA!v ! p

!TA!w "r

!TA!u "YA

ddt

!TA!w ! q

!TA!u "p

!TA!v "ZA

ddt

!TA!p ! w

!TA!v "v

!TA!w "r

!TA!q "q

!TA!r e " KA

ddt

!TA!q ! u

!TA!w "w

!TA!u "p

!TA!r "r

!TA!p "MA

ddt

!TA!r ! v

!TA!u "u

!TA!v "q

!TA!p "p

!TA!q "NA #

ddt

!T!!1

! S!!2!!T!!1

" "1

ddt

!T!!2

! S!!2!!T!!2

! S!!1!!T!!1

" "2

#

#

6.3 Nonlinear Coriolis Forces due to Added Mass in a Rotating Coordinate System Kirchho's Equa)ons

TA ! 12 !!MA!

MA ! !

Xu" Xv" Xw" Xp" Xq" Xr"Yu" Yv" Yw" Yp" Yq" Yr"Zu" Zv" Zw" Zp" Zq" Zr"Ku" Kv" Kw" Kp" Kq" Kr"Mu" Mv" Mw" Mp" Mq" Mr"Nu" Nv" Nw" Np" Nq" Nr"

Property 6.1 (Hydrodynamic System Iner-a Matrix) For a rigid-body at rest or moving at forward speed U 0 in ideal uid, the innity frequency hydrodynamic system iner)a matrix is posi)ve denite

MA ! MA! " 0


13

XA ! Xu" u" # Xw" !w" # uq" # Xq" q" # Zw" wq # Zq" q2

# Xv" v" # Xp" p" # Xr" r" ! Yv" vr ! Yp" rp ! Yr" r2

! Xv" ur ! Yw" wr# Yw" vq # Zp" pq ! !Yq" ! Zr" "qr

YA ! Xv" u" # Yw" w" # Yq" q"# Yv" v" # Yp" p" # Yr" r" # Xv" vr ! Yw" vp # Xr" r2 # !Xp" ! Zr" "rp ! Zp" p2

! Xw" !up ! wr" # Xu" ur ! Zw" wp! Zq" pq # Xq" qr

ZA ! Xw" !u" ! wq" # Zw" w" # Zq" q" ! Xu" uq ! Xq" q2

# Yw" v" # Zp" p" # Zr" r" # Yv" vp # Yr" rp # Yp" p2

# Xv" up # Yw" wp! Xv" vq ! !Xp" ! Yq" "pq ! Xr"qr

KA ! Xp" u" # Zp"w" # Kq" q" ! Xv"wu # Xr"uq ! Yw" w2 ! !Yq" ! Zr" "wq # Mr"q2

# Y "p v" # Kp" p" # Kr" r" # Yw" v2 ! !Yq" ! Zr" "vr # Zp" vp ! Mr" r2 ! Kq" rp# Xw" uv ! !Yv" ! Zw" "vw ! !Yr" # Zq" "wr ! Yp"wp ! Xq" ur# !Yr" # Zq" "vq # Kr"pq ! !Mq" ! Nr" "qr

MA ! Xq" !u" # wq" # Zq" !w" ! uq" # Mq" q" ! Xw" !u2 ! w2" ! !Zw" ! Xu" "wu# Yq" v" # Kq" p" # Mr" r" # Yp" vr ! Yr" vp ! Kr" !p2 ! r2" # !Kp" ! Nr" "rp! Yw" uv # Xv" vw ! !Xr" # Zp" "!up ! wr" # !Xp" ! Zr" "!wp # ur"! Mr"pq # Kq" qr

NA ! Xr"u" # Zr"w" # Mr"q" # Xv"u2 # Yw" wu ! !Xp" ! Yq" "uq ! Zp"wq ! Kq" q2

# Yr" v" # Kr"p" # Nr" r" ! Xv" v2 ! Xr" vr ! !Xp" ! Yq" "vp # Mr" rp # Kq" p2

! !Xu" ! Yv" "uv ! Xw" vw # !Xq" # Yp" "up # Yr"ur # Zq"wp! !Xq" # Yp" "vq ! !Kp" ! Mq" "pq ! Kr"qr #


Hydrodynamic added mass forces and moments in 6 DOF The expressions are complicated and not to suited for control design

Hydrodynamic so9ware programs such as WAMIT, VERES, and SEAWAY can be used to compute the added mass terms

The model can be more compactly wriren using the added mass system iner:a matrix and the added mass Coriolis and centripetal matrix

MA

CA!!"


14

The added mass Coriolis and centripetal matrix is found by collec)ng all terms that are not func)ons of body accelera)ons (Sagatun and Fossen 1991)

Property (Hydrodynamic Coriolis and centripetal matrix) For a rigid-body moving through an ideal uid the hydrodynamic Coriolis and centripetal matrix can always be parameterized such that it is skew-symmetric:

by dening:

Example (Fossen 1991):


CA!!" ! !CA!!!", "! # "6

CA!!" !03!3 !S!A11!1 " A12!2"

!S!A11!1 " A12!2" !S!A21!1 " A22!2

CA!!" !

0 0 0 0 !a3 a2

0 0 0 a3 0 !a1

0 0 0 !a2 a1 0

0 !a3 a2 0 !b3 b2

a3 0 !a1 b3 0 !b1

!a2 a1 0 !b2 b1 0

a1 ! Xu"u # Xv"v # Xw" w # Xp" p # Xq" q # Xr" ra2 ! Yu"u # Yv" v # Yw" w # Yp" p # Yq" q # Yr" ra3 ! Zu"u # Zv" v # Zw" w # Zp" p # Zq" q # Zr" rb1 ! Ku"u # Kv" v # Kw" w # Kp" p # Kq" q # Kr" rb2 ! Mu"u # Mv" v # Mw" w # Mp" p # Mq" q # Mr" rb3 ! Nu"u # Nv" v # Nw" w # Np" p # Nq" q # Nr" r


15

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

6.4 Viscous Damping and Current Loads Hydrodynamic damping for marine cra9 is mainly caused by:

Poten-al Damping: Radia:on-induced damping .The contribu)on from the poten)al damping terms

compared to other dissipa)ve terms like viscous damping are usually negligible.

Viscous damping: Linear skin fric:on due to laminar boundary layer theory and pressure varia)ons are important when considering the low-frequency mo)on of the vessel. Hence, this eect should be considered when designing the control system. In addi)on to linear skin fric)on, there will be a high-frequency contribu)on due to a turbulent boundary layer (quadra:c or nonlinear skin fric:on).

Wave DriV Damping: Wave dri9 damping can be interpreted as added resistance for surface vessels advancing in waves. This type of damping is derived from 2nd-order wave theory. Wave dri9 damping is the most important damping contribu)on to surge for higher sea states. This is due to the fact that the wave dri9 forces are propor)onal to the square of the signicant wave height Hs.

Damping Due to Vortex Shedding: D'Alambert's paradox states that no hydrodynamic forces act on a solid moving completely submerged with constant velocity in a non-viscous uid. In a viscous uid, fric)onal forces are present such that the system is not conserva)ve with respect to energy. This is commonly referred to as interference drag. It arises due to the shedding of vortex sheets at sharp edges.

LiVing Forces Hydrodynamic li9 forces arise from two physical mechanisms. The rst is due to the linear circula)on of water around the hull. The second is a nonlinear eect, commonly called cross-ow drag, which acts from a momentum transfer from the body to the uid.

16


6.4.1 Linear Viscous Damping Property 6.3 (Hydrodynamic Damping Matrix) For a rigid-body moving through an ideal uid the hydrodynamic damping matrix will be real, nonsymmetric and strictly posi)ve damping matrix Linear Damping (Seakeeping Theory)

D!!r" ! 0, ! ! " !6 #

x!D!!r"x ! 12 x!#D!!r" " D!!r"

! $x #0 ! x " 0 #

D!DP " DV

! !

Xu 0 0 0 0 00 Yv 0 Yp 0 Yr0 0 Zw 0 Zq 00 Kv 0 Kp 0 Kr0 0 Mw 0 Mq 00 Nv 0 Np 0 Nr

#

#

! Xu ! B11v! Yv ! B22v! Zw ! B33v " B33!!heave"! Kp ! B44v " B44!!roll"Mq ! B55v " B55!!pitch"! Nr ! B66v

# # # # # #

Viscous damping (to be added manually)

Poten)al damping (from hydrodynamic SW)

17

!M ! A"!" ! B!# ! C! $ 0

!" ! B!M ! A"

2"#

!# ! C!M ! A" !

#2

$ 0

B $ M ! A2"#

#

#

#

#


6.4.1 Linear Viscous Damping Rule-of-Thumb The linear viscous damping terms can be specied as three )me constants in surge, sway and yaw for the mass-damper systems. Addi)onal damping can be added in heave, roll and pitch.

(Tsurge,Tsway,Tyaw!

(!!heave,!!roll,!!pitch!

2nd-order system with and without spring

!M ! A"!" ! B!# $ 0M ! AB

T

!" ! !# $ 0

B $ M ! AT

#

#

#

# B ! 2!"!M " A"

B11v !m " A11!0"Tsurge

!8!"surge#m " A11!0"$

Tn,surge

B22v !m " A22!0"Tsway

!8!"sway#m " A22!0"$

Tn,swayB33v ! 2#"heave#heave#m " A33!#heave"$

#

#

#

B44v ! 2"! roll"roll!Ix # A44""roll#$

B55v ! 2"!pitch"pitch!Iy # A55""pitch#$

B66v !Iz # A66"0#Tyaw

!8#!yaw!Iz # A66"0#$

Tn,yaw

#

#

#

18

6.4.1 Linear Viscous Damping


mx! " !d " Kd"x# " Kpx $ 0 #

2!"n ! d " Kdm

"n2 !Kpm

#

#

T ! md #

Examples: A PD-controlled ship in surge, sway and yaw with natural periods: Tn = 100 s and Tn = 150 s and rela)ve damping ra)o: = 0.1 have )me constants in the range:

Rela)onship between Natural Period and Time Constant for PD controlled Marine Cra9

In open loop, Kp = Kd = 0, the )me constant becomes:

In closed loop, we have:

Tn ! 2!"n #

Kd/m ! 1/T

Closed loop:

1T !

Kdm " 2!

2"Tn "

4"!Tn #

T =Tn2

T =Tn2

159.2s T 238.7s

19


6.4.2 Nonlinear Surge Damping ITTC resistance

where density of water S wered surface of the hull k form factor giving a viscous correc)on (typically 0.1 for ships in transit) CF at plate fric)on from the ITTC (1957) line CR residual fric)on due to hull roughness, pressure resistance, wave making resistance Reynolds number:

=110m/s is the kinema)c viscosity at 20oC

ur ! u ! uc! u ! Vc cos!!c ! "" #

Rn !urLpp! #

X ! X |u|u|ur|ur # X |u|u| ! !12 !S!1 " k"Cf # 0 #

Nota)on of SNAME (1950)

X ! ! 12 !S!1 " k"Cf!ur"|ur|ur

Cf!ur" ! 0.075!log10Rn ! 2"2CF

" CR

#

#

20


6.4.2 Nonlinear Surge Damping Modied ITTC resistance

For low-speed maneuvering, gives to lirle damping compared to the quadra)c drag formula where CX > 0 is the current coecient and Ax is the frontal project area

X |u|u| ! ! 12 !S!1 " k"Cf # 0 #

Modied Resistance Curve

X |u|u| ! 12 !AxCx #

Cfnew!ur" ! Cf!urmax" " AxS CX ! C!ur

max" exp!!!ur2" #

Modied resistance curve Cfnew(ur) and Cf(ur) as a func)on of ur when CR = 0. The zero speed value Cfnew(ur) 0) = (Ax/S)CX where CX = 0.16 is the current coecient.

21


6.4.3 Cross-Flow Drag Principle For rela)ve current angles |c - | 0, where c is the current direc)on, the cross-ow drag principle may be applied to calculate the nonlinear damping force in sway and the yaw moment (Fal)nsen 1990): where Cd2D(x) is the 2-D sec)onal drag coecient and T(x) is the dra9.

2-D drag coecient Cd2D(x)

Y ! ! 12 ! "! Lpp2

Lpp2 T!x"Cd2D!x"|vr " xr|!vr " xr"dx

N ! ! 12 ! "! Lpp2

Lpp2 T!x"Cd2D!x"x|vr " xr|!vr " xr"dx

#

#

vr ! v ! vc! v ! Vc sin!!c ! "" #

Strip theory means that the ship is cut in sec)on along the x-axis. The integral is evaluated by summing up the contribu)on from each sec)on

Sec-on #

22


6.4.3 Cross-Flow Drag Principle

where Cd2D is the constant 2-D drag coecient and T is the dra9. The 2-D drag coecients Cd2D can be computed as a func)on of beam B and length T using Hoerner's curve. This is implemented in the Matlab MSS toolbox as:

vr ! v ! vc! v ! Vc sin!!c ! "" #

Y ! ! 12 !TCd2D "

!Lpp

2

Lpp2 |vr " xr|!vr " xr"dx

N ! ! 12 !TCd2D "

!Lpp

2

Lpp2 x|vr " xr|!vr " xr"dx

#

#

Cd ! Hoerner!B, T"

Hoerners Approxima)on (2-D theory using constant drag coecients)

23


6.4.3 Cross-Flow Drag Principle 3-D Theory (Curve Fiung to Maneuvering Coecients) A 3-D representa)on of: elimina)ng the integrals can be found by curve ung these expressions to 2nd-order modulus terms to obtain a maneuvering model in sway and yaw (Fedyaevsky and Sobolev 1963)

Y ! ! 12 ! "! Lpp2

Lpp2 T!x"Cd2D!x"|vr " xr|!vr " xr"dx

N ! ! 12 ! "! Lpp2

Lpp2 T!x"Cd2D!x"x|vr " xr|!vr " xr"dx

#

#

Y ! Y|v|v |vr|vr " Y|v|r|vr|r " Yv|r|vr|r|"Y|r|r|r|rN ! N |v|v |vr|vr " N |v|r|vr|r " Nv|r|vr|r|"N |r|r|r|r

# #


Copyright The US Navy

24

In hydrodynamics it is common to assume that the hydrodynamic forces and moments on a rigid body can be linearly superimposed. The "hydrodynamic mass-damper-spring" contribu)on to the hydrodynamic forces and moments can also be explained as: Forces on the body when the body is forced to oscillate with the wave excita:on frequency and there are no incident waves (Fal:nsen 1990)

Hydrodynamic Forces Added mass MA due to the iner)a of the surrounding uid. The corresponding

Coriolis and centripetal matrix due to added mass is due to the rota)on of {b} with respect to {n} is denoted CA

Radia)on-induced poten)al damping DP due to the energy carried away by generated surface waves.

Viscous damping caused by skin fric)on, wave dri9 damping, vortex shedding, li9/drag etc. The resul)ng hydrodynamic force is wriren as:

6.5 Maneuvering equations

!hyd ! !MA"# ! CA!"r""radded mass

! DP"rpotential damping

" !visc # !r ! ! ! !c!c ! !u,vc,wc,0,0,0"!

Hydrosta)c Forces Restoring forces due to Archimedes (weight and buoyancy)

!hs ! !g!"" ! go #


25

6.5.2 Nonlinear Maneuvering Equations

Rigid-Body Kine)cs including External Forces

MRB !! " CRB!!"! # "RB # !RB ! !hyd " !hs " !wind " !wave " ! #

!visc ! ! DV"rlinear

viscous friction

! Dn!"r""rnonlinear

viscous damping

#

!hyd ! !MA"# ! CA!"r""radded mass

! DP"rpotential damping

" !visc #

!hs ! !g!"" ! go #

Nonlinear Maneuvering Equa)ons of Mo)on

Hydrodynamic added mass and poten)al damping (mass and damping) Viscous forces (damping) Hydrosta)c forces (spring)


MRB!" ! CRB!!"! !MA "!r ! CA!!r"!r ! D!!r"!r ! g!#" ! go # $ ! $wind ! $wave #

26

The poten)al coecient matrices can be computed using a hydrodynamic code at a constant frequency or by curve ung to experimental data. This results in constant (frequency-independent) matrices in the following form: If nonlinear damping is modeled using the ITTC resistance law and cross-ow drag formulae, the following expression is obtained for the damping matrix:

6.5.2 Nonlinear Maneuvering Equations


Dn!!r" ! !

X |u|u|ur | 0 00 Y |v|v |vr | " Y |r|v |r| 00 0 Z |w|w|wr|0 0 00 0 00 N |v|v |vr | " N |r|v |r| 0

0 0 00 0 Y |v|r|vr | " Y |r|r|r|0 0 0

K |p|p|p| 0 00 M |q|q|q| 00 0 N |v|r|vr | " N |r|r|r|

MA ! !

Xu" 0 0 0 0 00 Yv" 0 Yp" 0 Yr"0 0 Zw" 0 Zq" 00 Kv" 0 Kp" 0 Kr"0 0 Mw" 0 Mq" 00 Nv" 0 Np" 0 Nr"

D ! !

Xu 0 0 0 0 00 Yv 0 Yp 0 Yr0 0 Zw 0 Zq 00 Kv 0 Kp 0 Kr0 0 Mw 0 Mq 00 Nv 0 Np 0 Nr

27

The linearized maneuvering equa)ons in surge, sway and yaw are wriren as: If ocean currents are neglected, this reduces to: Expanding the matrices give: No)ce that surge is assumed to be decoupled from the sway-yaw subsystem.

6.5.3 Linearized Maneuvering Equations


MRB!" ! CRB! !rigid-body forces

!MA "!r ! CA!!r ! D!rhydrodynamic forces

# # ! #wind ! #wave #

!MRB !MA"M!" ! !CRB! ! CA! ! D"

N! " # ! #wind ! #wave #

CRB! ! UMRBL

!

0 0 00 0 mU0 0 mxgU

#

CA! ! UMAL

!

0 0 00 0 "Yv"U0 0 "Yr"U

#

m ! Xu! 0 00 m ! Yv! mxg ! Yr!0 mxg ! Nv! Iz ! Nr!

u!v!r!

"

!Xu 0 00 !Yv !m ! Yv! "U ! Yr0 !Nr !mxg ! Yr!"U ! Nr

uvr

#

!1!2!6

#

ch6

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