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Course Stability and Yaw Motion of a Ship in Steady Wind
Yasukawa, H.∗, Hirono, T.†, Nakayama∗, Y., Koh, K. K.‡
October 23, 2011
ABSTRACT
Concern for the safe navigation, the presence of an external force which directly induced onthe ship maneuvering behavior is important to be captured. This paper discusses the coursestability and yaw motion of ships traveling under steady wind condition in an analytical way.A course stability criterion and approximate formulas of yaw motion in steady wind includingaero/hydro-dynamic force derivatives on the ship are derived. To confirm the reliability of thecriterion and the formulas, the investigation results are introduced for the Pure Car Carrier(PCC) in steady wind. The results reveal that the course instability appears in head andfollowing wind directions is mainly caused by the influence of aerodynamic derivatives withrespect to the yaw restoring forces. However, the course instability can be easily improvedby using a steering control. In the range from head wind to beam wind, the yaw oscillationappears where the period is relatively long and the damping is small. The analytical formulasnewly derived are useful for a better understanding of the ship maneuvering behavior in steadywind.
1 INTRODUCTION
In most study, ship maneuvering is discussed in still water condition. However, the presence ofwind and wave forces which exerted on a ship will directly induce to the course stability. In theconcern of safe navigation at sea, it is indispensable to capture the ship maneuvering behaviorwith such external disturbances. Particularly, pure car carrier (PCC) and cruise passenger shipwhich have relatively large super structure are significantly influenced by wind. For the safety,it is required to capture the maneuvering characteristics of the ships in wind.
Generally, the following 3 methods are used for capturing the ship maneuvering character-istics in wind:
• experiment (free-running model test)
• numerical method (simulations on the ship maneuvering)
• theory (analytical solutions)
The execution of the free-running test in wind is not easy because there exists the restriction ofthe facility with respect to the wind generators and the cost for testing is expensive in general.In recent years, simulation approaches of ship maneuvering motion in wind have been developed
∗Department of Transportation and Environmental Systems, Hiroshima University†Mitsubishi Heavy Industries, Kobe Shipyard‡Universiti Teknologi Malaysia
1
by several researchers successfully towards capturing the motion characteristics in the variouswind conditions. Based on the nonlinear simulation method of the ship maneuvering motion,Ogawa[1] and Spyrou[2] discussed the course stability in wind. In those studies, the sign ofeigenvalue was addressed to analyze and discuss the course stability appropriately. However,their approaches were difficult to understand comprehensively the mechanism of the coursestability since the forces acting on a ship including wind forces were not explicitly described.
On the contrary, the advantage of the theory is to make possible capturing the essencethrough the plain formula or equation which was derived by linearization of the motion equa-tions. For example, the following formula is well known for course stability analysis in stillwater:
C ′ =N ′r
Y ′r −m′ −m′
x
− N ′v
Y ′v
(1)
If C ′ > 0, the ship is stable and if C ′ < 0, the ship unstable. Here, Y ′v , Y
′r , N
′v and N ′
r denotethe non-dimensionalized linear derivatives of hydrodynamic forces on maneuvering, and m′
and m′x are non-dimensionalized mass and added mass, respectively. However, the criterion for
course stability in wind has not been obtained analytically yet.In 1968, Eda[3] investigated the course stability of ships in steady wind by using an eigen-
value analysis of linearized motion equations under the assumption of small ship motions.However, his approach is also hard in capturing the mechanism of the course stability in windsince the eigenvalue analysis was made numerically. For a better understanding of the coursestability, it is considered that the theory on the ship maneuvering in wind should be developedand the investigations should be made in an analytical way.
In this paper, some analytical solutions are presented on the course stability and maneuver-ing characteristics of ships in steady wind. The linearized expression of the surge force, lateralforce and yawing moment acting on the ship due to wind is derived in general form and thecourse stability criteria in steady wind is analytically presented. In addition, the approximatesolution of yaw motion in steady wind is derived by extending Nomoto’s K−T model[4]. As anapplication example, the investigation results on the course stability and yaw motion in steadywind are presented for a PCC with 180m in ship length.
2 MOTION EQUATIONS AND EXPRESSION OF WIND FORCES
Two coordinate systems were used: O-X0Y0Z0 fixed in space and G-xyz fixed to ship hullwhere G means the center of gravity of the ship, as shown in Fig.1. Z0 and z axes are taken asvertical downward, and X0-Y0 and x-y planes coincide with the still-water surface. The headingangle ψ is defined as the angle between X0 and x axes. u denotes the velocity component ofx-direction, v the velocity component of y-direction and r the yaw rate around z-axis whichis defined as r ≡ ψ. U is the ship speed which is defined as U ≡ √
u2 + v2. β denotes thehull drift angle and δ the rudder angle. Moreover, UW is the absolute wind velocity and θWthe absolute wind direction. θW = 0deg means that the opposite direction to X0−axis fixed inspace, θW = 90deg means that the opposit direction to Y0−axis. UW and θW are assumed tobe constant in time and uniform in space.
Motion equations of maneuvering in ship fixed coordinate system are written as:
(m+mx) u− (m+my) vψ = Fx (2)(m+my) v + (m+mx) uψ = Fy (3)
(Iz + Jz) ψ = Mz (4)
where m means the ship’s mass, Iz is the yaw moment of inertia around z-axis. mx, my andJz denote added masses for x-, y-direction and added yaw moment of inertia, respectively. In
2
G
U
u
x
W
X0
Y0
y
-vβ
ψδ
r
O
θUW
Fig.1: Coordinate systems
the equations, u, v and ψ are unknown variables. Fx, Fy and Mz are the surge force, lateralforce and yawing moment acting on the ship, respectively, and these are expressed as:
Fx = XH +XP +XR +XA
Fy = YH + YR + YA
Mz = NH +NR +NA
⎫⎪⎪⎬⎪⎪⎭ (5)
Here, XH , YH and NH denote surge force, lateral force and yawing moment acting on the hullin obliquely moving and turning motions respectively. XP is the effective propeller thrust.XR, YR and NR denote hydrodynamic forces generated by steering, and XA, YA and NA arethe aerodynamic forces due to wind.
XH , YH and NH are expressed by using hydrodynamic derivatives as follows:
XH = Xuuu2 +Xvvv
2 +Xvrvr +Xrrr2 (6)
YH = Yvv + Yrr + Yvvvv3 + Yvvrv
2r + Yvrrvr2 + Yrrrr
3 (7)
NH = Nvv +Nrr +Nvvvv3 +Nvvrv
2r +Nvrrvr2 +Nrrrr
3 (8)
where Xuu is the resistance coefficient. Xvv, Yv, Yr, Nv, Nr etc. are the hydrodynamicderivatives on maneuvering.
XR, YR and NR are expressed by using hydrodynamic derivatives as follows:
XR = Xδδδ2
YR = Yδδ
NR = Nδδ
⎫⎪⎪⎬⎪⎪⎭ (9)
where Xδδ, Yδ and Nδ the rudder force derivatives.XA, YA and NA are expressed as follows[5]:
XA = (1/2)ρaAXV 2A CXA(θA)
YA = (1/2)ρaAY V 2A CY A(θA)
NA = (1/2)ρaAY V 2AL CNA(θA)
⎫⎪⎪⎬⎪⎪⎭ (10)
3
whereθA = tan−1(vA/uA) (11)V 2A = u2
A + v2A (12)
uA = u+ UW cos(θW − ψ) (13)vA = v + UW sin(θW − ψ) (14)
where ρa denotes the air density. AX and AY are front and side profile areas of the shipin air respectively. VA denotes the relative wind velocity and θA the relative wind direction.CXA, CY A and CNA are the wind force coefficients which are expressed as function of θA.
3 LINEARIZATION OF MOTION EQUATIONS
Next, linearizing the equations mentioned above, compact and useful equations are derived.u, v and ψ are assumed to be expressed as sum of steady and unsteady components as
follows:
u = u0 +Δu (15)v = v0 +Δv (16)ψ = ψ0 +Δψ (17)
Here, u0, v0 and ψ0 mean the steady state condition of the ship moving in wind, and ψ0 isassumed to be given. The unsteady components Δu, Δv and Δψ are assumed to be sufficientlysmall, namely, O(ε) where ε is of small quantity.
3.1 Linearization of lateral force and yawing moment due to wind
Consider the linearization of eq.(11) which refers to relative wind direction. uA and vA areexpanded as follows:
uA = u0 +Δu+ UW (cosϕ+Δψ sinϕ) +O(ε2) (18)vA = v0 +Δv + UW (sinϕ−Δψ cosϕ) +O(ε2) (19)
whereϕ = θW − ψ0 (20)
θA is expressed by Taylor expansion with respect to Δu, Δv and Δψ as follows:
θA = θA0 + θA1Δu+ θA2Δv + θA3Δψ +O(ε2) (21)
where
θA0 = tan−1{v0 + UW sinϕu0 + UW cosϕ
}(22)
θA1 = − v0 + UW sinϕ(u0 + UW cosϕ)2 + (v0 + UW sinϕ)2
(23)
θA2 =u0 + UW cosϕ
(u0 + UW cosϕ)2 + (v0 + UW sinϕ)2(24)
θA3 = − U2W + UW (v0 sinϕ+ u0 cosϕ)
(u0 + UW cosϕ)2 + (v0 + UW sinϕ)2(25)
4
The square term of relative wind velocity V 2A, eq.(12), is linearized as follows:
V 2A = VA0 + VA1Δu+ VA2Δv + VA3Δψ +O(ε2) (26)
where
VA0 = (u0 + UW cosϕ)2 + (v0 + UW sinϕ)2 (27)
VA1 = 2 (u0 + UW cosϕ) (28)
VA2 = 2 (v0 + UW sinϕ) (29)
VA3 = 2UW (u0 sinϕ− v0 cosϕ) (30)
Here, the wind force coefficient at θA is represented as C(θA). C(θA) can be expressed byTaylor expansion around θA = θA0 as follows:
C(θA) = C(θA0) +∂C(θA0)∂θA
(θA1Δu+ θA2Δv + θA3Δψ) +O(ε2) (31)
Pulling together the equations mentioned above, linearized aerodynamic surge force, lateralforce and yawing moment (XA, YA, NA) are expressed as follows:
XA = XA0 +XA1Δu+XA2Δv +XA3Δψ +O(ε2) (32)
YA = YA0 + YA1Δu+ YA2Δv + YA3Δψ +O(ε2) (33)
NA = NA0 +NA1Δu+NA2Δv +NA3Δψ +O(ε2) (34)
where
XA0 =12ρaAXVA0CXA(θA0) (35)
XAi =12ρaAX
[VA0
∂CXA(θA0)∂θA
θAi + VAi CXA(θA0)]
(i = 1, 2, 3) (36)
YA0 =12ρaAY VA0CY A(θA0) (37)
YAi =12ρaAY
[VA0
∂CY A(θA0)∂θA
θAi + VAi CY A(θA0)]
(i = 1, 2, 3) (38)
NA0 =12ρaAY LVA0CNA(θA0) (39)
NAi =12ρaAY L
[VA0
∂CNA(θA0)∂θA
θAi + VAi CNA(θA0)]
(i = 1, 2, 3) (40)
Here, XA0, YA0 and NA0 denote the steady values of surge force, lateral force and yawingmoment acting on the ship due to wind. XAi, YAi, NAi are the derivatives with respect to surgeforce, lateral force and yawing moment due to wind, which can be regarded as aerodynamicderivatives. There exist derivatives with respect to not only lateral velocity v but also surgevelocity u and yaw angle ψ. The aerodynamic derivatives are defined as function of θA0 andchange with steady solutions u0, v0 and ψ0.
5
3.2 Linearization of lateral force and yawing moment due to water
Next, consider the linearization of hydrodynamic forces acting on the ship due to water. Fromeqs.(6), (7) and (8), the linearized surge force, lateral force and yawing moment (XH , YH , NH)are expressed as follows:
XH = X0u20 + 2X0u0Δu+Xvvv
20 + 2Xvvv0Δv +Xvrv0Δψ +O(ε2) (41)
YH = Yvv0 + Yvvvv30 +
(Yv + 3Yvvvv2
0
)Δv +
(Yr + Yvvrv
20
)Δψ +O(ε2) (42)
NH = Nvv0 +Nvvvv30 +
(Nv + 3Nvvvv
20
)Δv +
(Nr +Nvvrv
20
)Δψ +O(ε2) (43)
3.3 Linearized motion equations and the non-dimensionalization
Here, for better understanding of the equations, we employ the following notations: (XAu,XAv,XAψ) ≡(XA1,XA2,XA3), (YAu, YAv, YAψ) ≡ (YA1, YA2, YA3) and (NAu, NAv, NAψ) ≡ (NA1, NA2, NA3).Pulling together the equations mentioned above, linearized motion equations are finally ex-pressed as follows:
(m+mx)Δu− (m+my) v0Δψ = X0u20 +Xvvv
20 + 2X0u0Δu+ 2Xvvv0Δv +Xvrv0Δψ
+XA0 +XAuΔu+XAvΔv +XAψΔψ +Xδδδ2 +XP (44)
(m+my)Δv + (m+mx)u0Δψ = Yvv0 + Yvvvv30 +
(Yv + 3Yvvvv2
0
)Δv
+(Yr + Yvvrv
20
)Δψ + YA0 + YAuΔu+ YAvΔv + YAψΔψ + Yδδ (45)
(Iz + Jz)Δψ = Nvv0 +Nvvvv30 +
(Nv + 3Nvvvv
20
)Δv +
(Nr +Nvvrv
20
)Δψ
+NA0 +NAuΔu+NAvΔv +NAψΔψ +Nδδ (46)
Rudder angle δ is also assumed to be expressed as sum of steady component δ0 and unsteadycomponent Δδ. Additionally, PD control with respect to yaw angle is employed for Δδ asfollows:
Δδ = −G1Δψ −G2Δψ (47)
where G1 and G2 are control gains. Δδ is assumed to be sufficiently small.Eqs.(44) and (45) are non-dimensionalized by dividing using (1/2)ρLdU2, and eq.(46) is
non-dimensionalized by (1/2)ρL2dU2, where L and d denote ship length and draught respec-tively. Here, it is assumed that X ′
P is be expressed as only steady component and the unsteadycomponent is negligible. Then, separating these equations into steady and unsteady motionterms, the following equations are obtained:
Steady motion term:
X ′0u
′20 +X ′
vvv′20 +X ′
A0 +X ′δδδ
20 +X ′
P = 0 (48)
Y ′vv
′0 + Y ′
vvvv′30 + Y ′
A0 + Y ′δδ0 = 0 (49)
N ′vv
′0 +N ′
vvvv′30 +N ′
A0 +N ′δδ0 = 0 (50)
Unsteady motion term:
(m′ +m′
x
)Δu′ −
(m′ +m′
y
)v′0Δψ
′ =(2X ′
0 +X ′Au
)Δu′
+(2X ′
vvv′0 +X ′
Av
)Δv′ +X ′
vrv′0Δψ
′ +X ′AψΔψ − 2X ′
δδδ0(G′
1Δψ +G′2Δψ
′) (51)
6
(m′ +m′
y
)Δv′ +
(m′ +m′
x
)u′0Δψ
′ =(Y ′v + 3Y ′
vvvv′20 + Y ′
Av
)Δv′
+(Y ′r + Y ′
vvrv′20
)Δψ′ + Y ′
AuΔu′ + Y ′
AψΔψ − Y ′δ
(G′
1Δψ +G′2Δψ
′) (52)
(I ′z + J ′
z
)Δψ′ =
(N ′v + 3N ′
vvvv′20 +N ′
Av
)Δv′ +
(N ′r +N ′
vvrv′20
)Δψ′
+N ′AuΔu
′ +N ′AψΔψ −N ′
δ
(G′
1Δψ +G′2Δψ
′) (53)
In the equations, symbol with prime means non-dimensional value. From eqs.(48), (49) and(50), non-dimensionalized steady surge force u′0(= u0/U), steady lateral velocity v′0(= v0/U)and offset rudder angle δ0 are obtained. Based on the steady solution, unsteady motion ofthe ship is defined in eqs.(51), (52) and (53). Calculating eigenvalues of the matrix madefrom eqs.(51), (52) and (53), and checking sign of the real part, the course stability can beinvestigated.
4 COURSE STABILITY IN STEADY WIND EXCLUDING
SURGE COUPLING EFFECT
4.1 Simplification of equations
To obtain analytical solution for deciding the course stability of ships in steady wind, we makethe further simplifications of the equations mentioned above. Then, we employ the followingassumptions:
• Non-dimensional steady forward velocity u′0 is 1.0. When the ship moves in head windunder the condition where the main engine output is constant, for instance, the shipspeed will drop due to the wind drag. However, in case that the ship has enough marginin the engine output, the captain would keep the expected ship speed by increasing thepropeller revolution. Here, it is assumed that the expected speed can be kept in steadywind.
• Surge coupling effect on the course stability is negligible. Actually, the surge couplingeffect is not so significant when the wind velocity is not so high. The calculation resultswill be presented in section 6.4.
Under the assumptions mentioned above, the motion equations to be dealt with become onlyeqs.(52) and (53) for sway and yaw.
4.2 Course stability criterion
Eqs.(52) and (53) with u′0 = 1 and Δu′ = 0 are the linearized coupled vibration equations withrespect to Δv′ and Δψ. The solution is assumed to be as follows:
Δv′ = C1eσt, Δψ = C2e
σt
Then, the following characteristic equation is obtained as
D1σ3 +D2σ
2 +D3σ +D4 = 0 (54)
where
7
D1 =(I ′z + J ′
z
) (m′ +m′
y
)(55)
D2 = − (I ′z + J ′
z
) (Y ′v + Y ′
Av + 3Y ′vvvv
′20
)−
(m′ +m′
y
) (N ′r +N ′
vvrv′20 −G′
2N′δ
)(56)
D3 =(N ′r +N ′
vvrv′20 −G′
2N′δ
) (Y ′v + Y ′
Av + 3Y ′vvvv
′20
)−
(Y ′r + Y ′
vvrv′20 −m′ −m′
x −G′2Y
′δ
) (N ′v +N ′
Av + 3N ′vvvv
′20
)−
(N ′Aψ −G′
1N′δ
) (m′ +m′
y
)(57)
D4 =(N ′Aψ −G′
1N′δ
) (Y ′v + Y ′
Av + 3Y ′vvvv
′20
)−
(Y ′Aψ −G′
1Y′δ
) (N ′v +N ′
Av + 3N ′vvvv
′20
)(58)
By applying Hurwitz’s method to eq.(54), the conditions for the stable solution can be obtainedas follows:
D1,D2,D3,D4 > 0D ≡ D2D3 −D1D4 > 0
}(59)
4.3 Consideration
To discuss the conditions for the course stability in steady wind, we consider the sign ofD1,D2,D3,D4 and D in case of no rudder control (G′
1 = G′2 = 0).
• D1 is of no relation to aerodynamic derivatives and becomes inevitably positive.
• D2 includes Y ′Av which is aerodynamic derivative expressing the lateral motion damping
due to wind. It changes according to wind velocity and direction, but is always negative.Also, Y ′
v , Y′vvv , N
′r and N ′
vvr are normally negative. As a result, D2 becomes alwayspositive.
• In considering D3. Rewriting the condition of D3 > 0, the following formula is obtained:
N ′r +N ′
vvrv′20
Y ′r + Y ′
vvrv′20 −m′ −m′
x
− N ′v +N ′
Av + 3N ′vvvv
′20
Y ′v + Y ′
Av + 3Y ′vvvv
′20
−N ′Aψ
(m′ +m′
y
)(Y ′v + Y ′
Av + 3Y ′vvvv
′20
) (Y ′r + Y ′
vvrv′20 −m′ −m′
x
) > 0 (60)
In a broad sense, the 1st term of left hand side is the acting point of turning motion andthe 2nd term is the acting point of oblique motion. The 3rd term is the particular termwhen dealing with the problem in steady wind, which considerably influences the coursestability. If the ship itself is course stable, sum of the 1st and the 2nd terms becomespositive. Since the denominator of the 3rd term is positive, the sign of D3 depends onN ′Aψ value. N ′
Aψ which means restoring moment derivative with respect to yaw anglechanges to positive/negative value as according to different wind velocity and direction.Negative N ′
Aψ contributes to stable course direction while positive N ′Aψ contributes to
course instability.
When considering no wind situation condition, all the aerodynamic derivatives and non-dimensionalized steady lateral velocity v′0 become zero in eq.(60). The following formulais obtained:
C ′ =N ′r
Y ′r −m′ −m′
x
− N ′v
Y ′v
> 0
This is the well-known criterion for course stability in still water, eq.(1).
8
• In considering D4. The condition of D4 > 0 is rewritten as follows:
N ′Aψ
Y ′Aψ
<N ′v +N ′
Av + 3N ′vvvv
′20
Y ′v + Y ′
Av + 3Y ′vvvv
′20
for Y ′Aψ > 0 (61)
N ′Aψ
Y ′Aψ
>N ′v +N ′
Av + 3N ′vvvv
′20
Y ′v + Y ′
Av + 3Y ′vvvv
′20
for Y ′Aψ < 0 (62)
Eq.(61) indicates that acting point of hydrodynamic force on the obliquely moving hullshould be located in front of the acting point of aerodynamic force. This is the conditionfor ship to automatically move to the weather side in head wind condition. The conditionsof eqs.(61) and (62) depend on the positive/negative value and the order of magnitudeof Y ′
Aψ and N ′Aψ, which change according to different wind velocity and direction.
• In considering D. D is rewritten as follows:
D ≡ − (I ′z + J ′
z
) (Y ′v + Y ′
Av + 3Y ′vvvv
′20
) [(N ′r +N ′
vvrv′20
) (Y ′v + Y ′
Av + 3Y ′vvvv
′20
)−
(Y ′r + Y ′
vvrv′20 −m′ −m′
x
) (N ′v +N ′
Av + 3N ′vvvv
′20
)− Y ′
Aψ
(m′ +m′
y
) (N ′v +N ′
Av + 3N ′vvvv
′20
)/
(Y ′v + Y ′
Av + 3Y ′vvvv
′20
)]−
(m′ +m′
y
) (N ′r +N ′
vvrv′20
)D3 (63)
The condition of D > 0 is satisfied when D3 is positive and the sign of [ ] in eq.(63)becomes positive. The time where the sign of [ ] is positive is when Y ′
Aψ < 0.
Pulling together the considerations mentioned above, in case of a certain degree of strongwind, the condition that the ship becomes course stable in wind is Y ′
Aψ < 0 and N ′Aψ < 0. To
check the course stability of the ship in detail, numerical calculation of D3,D4 and D may beuseful. In the equations mentioned above, the increase of non-dimensionalized lateral velocityv′0 contributes to the increase of motion damping and the improvement of course stability, forthe value of nonlinear derivatives Y ′
vvv and N ′vvr are normally negative.
Next, consider the influence of rudder steering control. It is noted that both G′1 and G′
2 arepositive, Y ′
δ is negative, and N ′δ is positive. G′
2 contributes to the increase of damping of turningmotion and G′
1 contributes to the changes of yaw restoring forces as increase of (Y ′Aψ −G′
1Y′δ )
term and the decrease of (N ′Aψ −G′
1N′δ) term. The changes make D3 and D4 positive.
5 YAW MOTION OF SHIPS IN STEADY WIND
In this chapter, next, we consider the yaw motion of ships in steady wind. Then, the surgecoupling effect is not taken into consideration.
5.1 Linearized equation of yaw motion
Eliminating Δv′ from eqs.(52) and (53) with u′0 = 1 and Δu′ = 0, the following equation isobtained:
D1r′ +D2r
′ +D3r′ +D4Δψ = q1Δδ
′ + q2Δδ (64)
where r′ = Δψ′, and
q1 = N ′δ(m
′ +m′y) (65)
q2 = Y ′δ
[N ′v + 3N ′
vvvv′20 +N ′
Av
]−N ′
δ
[Y ′v + 3Y ′
vvvv′20 + Y ′
Av
](66)
9
The characteristic equation of eq.(64) is the cubic equation as shown in eq.(54). The typeof the solutions (eigenvalues) is categorized as follows:
• 3 real roots are obtained (this case is named as non-oscillation mode)
• 1 real root and 2 complex roots (including complex conjugate root) are obtained(this case is named as oscillation mode)
Then, the categorization depends on the sign of the following function Δ:
Δ = −4D33D1 +D2
3D22 − 4D4D
32 + 18D4D3D2D1 − 27D2
4D21 (67)
Non-oscillation mode if Δ > 0 and the oscillation mode if Δ < 0. It is a special feature thatthe oscillation mode appears in the yaw motion of ships in steady wind.
When considering steady state condition in eq.(64), the following solution is obtained as:
Δψ =q2D4
Δδ
= − Y ′δ (N
′v + 3N ′
vvvv′20 +N ′
Av) −N ′δ(Y
′v + 3Y ′
vvvv′20 + Y ′
Av)Y ′Aψ(N ′
v + 3N ′vvvv
′20 +N ′
Av) −N ′Aψ(Y ′
v + 3Y ′vvvv
′20 + Y ′
Av)Δδ (68)
The steady solution expressed as eq.(68) is for heading angle, not yaw rate. Namely, it meansthat there does not exist the ship steady turning condition in wind within the framework of thelinearized theory mentioned above. Then, a parameter representing the rudder effectivenessin stead wind is q2/D4, which is determined by hydrodynamic derivatives in oblique motion(Y ′v , N ′
v etc.), rudder force derivatives (Y ′δ and N ′
δ) and aerodynamic derivatives with respect torestoring forces due to heading change (Y ′
Aψ and N ′Aψ). Here, Y ′
Aψ, N ′Aψ and v′0 are expressed as
the function of wind velocity and direction, so that q2/D4 is also the function of wind velocityand direction. When the absolute values of Y ′
δ and N ′δ increase, q2/D4 increases as shown in
eq.(68), namely the rudder effectiveness is improved. On the other hand, when the absolutevalues of Y ′
ψ and N ′ψ increase with increase of the wind velocity, q2/D4 decreases, namely the
rudder effectiveness becomes worse.
5.2 An approximate expression of yaw motion
Let us consider the yaw motion of ships in steady wind by solving eq.(64). Then, the eigenvaluesare required by solving the characteristic equation (54). However, it is difficult to obtain theeigenvalues analytically since the characteristic equation is a cubic equation with the steadysolutions. Here, describing a method to find the eigenvalues easily, an approximate solution ofthe yaw motion in steady wind is presented.
Here, the following approximations are employed:
• D1 is zero. This approximation acts to eliminate the motion component caused by eigen-value σ1 whose frequency is the highest. This is essentially the same as the approximationused in Nomoto’s K − T model[4].
• Steady solutions such as v′0 and δ0 are neglected.
• The term of δ is neglected.
It will be verified that the approximations employed here are practically acceptable throughthe calculation example in section 6.6.
Then, eq.(64) is written as:
D′2Δψ
′ +D′3Δψ
′ +D′4Δψ = q′2Δδ (69)
10
whereD′
2 = − (I ′z + J ′
z
) (Y ′v + Y ′
Av
) − (m′ +m′
y
)N ′r (70)
D′3 = N ′
r
(Y ′v + Y ′
Av
) − (Y ′r −m′ −m′
x
) (N ′v +N ′
Av
) −N ′Aψ
(m′ +m′
y
)(71)
D′4 = N ′
Aψ
(Y ′v + Y ′
Av
) − Y ′Aψ
(N ′v +N ′
Av
)(72)
q′2 = Y ′δ
(N ′v +N ′
Av
) −N ′δ
(Y ′v + Y ′
Av
)(73)
Here, no rudder control is assumed as G′1 = G′
2 = 0.When defining Q ≡ D′2
3 − 4D′2D
′4, Q > 0 means the non-oscillation mode, and the solution
of eq.(69) is analytically expressed as follows:
Δψ
Δδ=
q′2D′
4
[1 − σ2
σ2 − σ3eσ3t′ +
σ3
σ2 − σ3eσ2t′
](74)
whereσ2 = −D
′3 +
√Q
2D′2
(75)
σ3 = −D′3 −
√Q
2D′2
(76)
where t′ indicates the non-dimensional time defined as t′ = tU/L. When t′ = 0 is set, r′ = 0and ψ = 0 are imposed as the initial condition. In the non-oscillation mode, the yaw motion isexpressed as sum of 2 exponential functions. The motion changes monotonously.
Q < 0 means the oscillation mode, and the solution of eq.(69) is analytically expressed asfollows:
Δψ
Δδ=
q′2D′
4
[1 − eσ3Rt
′{
cos(σ3It′) − σ3R
σ3Isin(σ3I t
′)}]
(77)
whereσ3R = − D′
3
2D′2
(78)
σ3I =√−Q2D′
2
(79)
where σ3R is the real part while σ3I is the imaginary part of the complex root σ3. When t′ = 0,r′ = 0 and ψ = 0 are imposed as the initial condition. In the oscillation mode, the yaw motionis expressed as a product of sin / cos function and an exponential function. The motion changesoscillatory with damping.
6 CALCULATION EXAMPLE
Using the equations mentioned above, course stability and yaw motion are investigated invarious wind velocity and direction. In the calculation example, the ship is assumed to moveto the direction of ψ0 = 0deg.
6.1 Studied ship
A Pure Car Carrier (PCC) which has normally a large wind pressure area is employed as anexample in the calculations. Table 1 shows the principal particulars of the PCC[6]. In thetable, L, B and d denote the length of perpendicular, breadth and draught respectively. xGis the length from midship to the center of buoyancy and Cb is the block coefficient. Disp. is
11
referred to the displacement. AX and AY , which denote the front and lateral profile areas inair respectively, were determined by referring to Yoshimura’s paper[6]. Added mass coefficientsand hydrodynamic derivatives are shown in Table 2, referring to the captive model test resultsconducted by Yoshimura[7]. The linear derivatives were intentionally adjusted so as to makethe ship course stable in still water condition since the original ship was slightly course unstable.
Table 1: Principal particulars of PCC
Lpp(L) 180.00mB 32.20md 8.20mxG -2.53mCb 0.547
Disp. 26,650tonAX 859.0m2
AY 4386.7m2
Table 2: Added mass coefficients and hydro-dynamic derivatives on maneuvering of PCC
m′x 0.0093
m′y 0.1353
J ′z 0.0089
X ′0 -0.0156
X ′vv -0.071
X ′vr 0.060
X ′δδ 0.0Y ′v -0.289Y ′r 0.0343
Y ′vvv -1.55Y ′vvr -0.655Y ′δ -0.05
N ′v -0.0879
N ′r -0.0556
N ′vvv -0.173
N ′vvr -0.627N ′δ 0.025
Wind force coefficients which are needed to calculate the aerodynamic derivatives are esti-mated using the method proposed by Fujiwara et al.[8][9]. Fig.2 shows the wind force coeffi-cients. θA=0deg, 90deg and 180deg are referred to head wind, beam wind and following wind,respectively.
0 30 60 90 120 150 180−1
−0.5
0
0.5
1
CXA
θA(deg)0 30 60 90 120 150 180
0
0.2
0.4
0.6
0.8
1
1.2
−CYA
θA (deg)0 30 60 90 120 150 180
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
CNA
θA (deg)
Fig.2: Wind force coefficients of PCC
6.2 Steady solutions
Calculation for the steady condition of ships in steady wind was first carried out. Here, itis assumed that the ship can maintain the expected ship speed in steady wind and u′0 is 1.0.
12
When δ0 gets eliminated from eqs.(49) and (50), the following equation is obtained:
v′0(N′δY
′v − Y ′
δN′v) + v′30 (N ′
δY′vvv − Y ′
δN′vvv) = −N ′
δY′A0 − Y ′
δN′A0 (80)
Eq.(80) is the cubic equation with respect to v′0 and has up to 3 different real roots in maximum.However, it has been mathematically proven that the real root is only one, due to (N ′
δY′vvv −
Y ′δN
′vvv) < 0 and (N ′
δY′v − Y ′
δN′v) < 0. Therefore, v′0 can be obtained by numerically solving
eq.(80) under the proper initial condition.Fig.3 shows the calculated steady solution, offset rudder angle δ0 and hull drift angle β0(=
−v′0) in various wind velocities and directions. Absolute value of the offset rudder angle becomeslarge with the increase of wind velocity (UW /U). The maximum offset angle at UW /U = 5 isclose to 35deg which is maximum rudder angle for normal ships and occurs at near θW = 120deg.The maximum hull drift angle is around 11deg at near θW = 60deg.
0 30 60 90 120 150 1800
10
20
30
40−δ0 (deg)
θW (deg)
Uw/U = 4
Uw/U = 3
Uw/U = 2
Uw/U = 5
0 30 60 90 120 150 1800
2
4
6
8
10
12β0 (deg)
θW (deg)
Uw/U = 5
Uw/U = 4
Uw/U = 3
Uw/U = 2
Fig.3: Steady solution: offset rudder angle and hull drift angle of PCC in wind
6.3 Aerodynamic derivatives
The calculated results of aerodynamic derivatives are shown in Fig.4.These change accordingto wind velocity and direction. X ′
Au is the derivative which represents air resistance changeversus surge velocity and the value is generally negative. As shown in Fig.4, however, thepositive value range appears in the wind direction of 90deg< θW < 120deg. In the beam windrange, the air resistance is reduced. When comparing (X ′
Au,X′Av,X
′Aψ) with (Y ′
Au, Y′Av, Y
′Aψ),
the absolute values of aerodynamic derivatives for X ′A are one digit smaller as a whole. It is
expected that surge coupling effect is not so large. Y ′Av is the lateral damping force derivative
and always negative. The order of magnitude of Y ′Av is 10% of Y ′
v at most. Therefore, totaldamping characteristic due to wind with respect to lateral motion does not change much.Generally, N ′
Av is negative in value in head wind condition and positive in value in followingwind condition. This means that the acting point of the wind force component is located at thefore part in head wind and aft part in following wind, since Y ′
Av is always negative. Y ′Aψ and
N ′Aψ, the restoring force derivatives with respect to yaw motion, are important when discussing
the course stability in wind. By its nature, negative N ′Aψ stabilizes its yaw motion, and positive
value diverges the motion. As N ′Aψ is positive when θW is in between 0∼50deg and between
140∼180deg, it is anticipated that course instability appears in these ranges.
13
0 30 60 90 120 150 180−0.006
−0.005
−0.004
−0.003
−0.002
−0.001
0
0.001
XAu’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.002
−0.001
0
0.001
0.002
XAv’
θW (deg)
Uw/U=5
Uw/U=4Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.03
−0.02
−0.01
0
0.01
XAψ’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.02
−0.01
0
0.01
0.02
YAu’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0
YAv’
θW (deg)
Uw/U = 5
Uw/U = 4
Uw/U = 3
Uw/U = 2
0 30 60 90 120 150 180−0.1
−0.08−0.06−0.04−0.02
00.020.040.060.080.1
YAψ’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.004
−0.003
−0.002
−0.001
0
NAu’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3 Uw/U=2
0 30 60 90 120 150 180−0.005
−0.004
−0.003
−0.002
−0.001
0
0.001
0.002
0.003
NAv’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3Uw/U=2
0 30 60 90 120 150 180−0.02
−0.01
0
0.01
0.02
0.03
NAψ’
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
Fig.4: Aerodynamic derivatives of PCC
6.4 Course stability
6.4.1 Surge coupling effect
Fig.5 shows stable and unstable zone on course stability in steady wind. Black area meansunstable zone. In the figure, wind velocity ratio (UW /U) is the vertical axis and the absolutewind direction (θW ) is the horizontal axis. Then, no steering control is assumed, namelyG′
1 = G′2 = 0. The left figure includes the surge coupling effect and the right figure excludes
the surge effect. They are virtually the same although there is a slight difference between bothfigures in the region where UW/U is large. Thus, the surge coupling effect is not so significant.
On the other hand, Eda[3] has indicated that the surge coupling effect on the course sta-bility in wind can not be ignored, and this is quite different from the present result. Edainvestigated the relatively large wind velocity region of UW /U ≥ 5 in his paper[3], while thepresent calculation is dealt with the relatively small wind velocity region of UW /U ≤ 5. It isconsidered that the reason for the difference with respect to the surge coupling effect is mainlycaused by difference of the wind velocity region dealt with.
6.4.2 Effect of steering control
Fig.6 shows stable and unstable zone on the course stability in steady wind with differenceof several steering control gains. Left figure is a result in case of no steering control, namely
14
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
Inc. surge effect
stableunstable
unstable
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
Exc. surge effect
stableunstable
unstable
Fig.5: Surge coupling effect on course stability of PCC in wind (unsteered ship)
G′1 = G′
2 = 0. This is the same as the left figure in Fig.5. The tendency of course stabilityin steady wind is summarized as: In following wind condition, instability appears in any windvelocity; In head wind condition where θW = 0 ∼ 30deg, instability appears at the range ofUW /U over 1.6. This tendency that the course instability appears in head and following windsagrees with the results by Eda[3] and Ogawa[1]. The studied PCC is course stable in no windcase. However, the stable region in case of UW /U = 0 can not be distinguished clearly inthe figure. This is because the course instability appears in the following wind condition eventhough the wind velocity is very small amount.
Middle figure in Fig.6 is the result in case of gentle control for G′1 = 0.3 and G′
2 = 0. Thissituation may be translated as human steering against the instability appears in wind. Thestable zone increases in head and following wind conditions even though the gain value is small.The effect of steering control is considerably large.
Right figure is the result where autopilot is employed, G′1 = G′
2 = 1. Unstable zone disap-pears. Thus, autopilot is useful for stabilizing the ship moving in steady wind. In summary,although a ship moving in steady wind falls into the course instability in head and followingwinds, the instability can be easily recovered by human steering or autopilot. It is consideredthat the ship in wind can travel safely only under the appropriate steering.
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
G1’ = 0, G2’ = 0
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
G1’ = 0.3, G2’ = 0
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
G1’ = 1.0, G2’ = 1.0
Fig.6: Effect of steering control gain on course stability of PCC in wind
15
6.5 Yaw motion in steady wind
Next, let us consider the yaw motion of PCC in steady wind. Here, no steering control isassumed.
6.5.1 Kind of yaw motion and eigenvalues
Fig.7 shows the region for oscillation mode of the PCC in various wind velocity and direction.Black area means the case of oscillation mode, while white area means non-oscillation mode.In the case of oscillation mode, a periodic yaw motion appears. When UW /U is over 0.2, thearea from head wind to beam wind becomes oscillation mode. Comparing Fig.7 with Fig.5,almost non-oscillation mode region from beam wind to following wind coincides with the courseunstable zone.
0 30 60 90 120 150 180
1
2
3
4
5
θW (deg)
UW
/U
non−oscillationmode
oscillationmode
Fig.7: Region of oscillation mode of PCC in wind
Next, characteristics of the yaw motion in wind is being investigated by calculating theeigenvalues of the characteristic equation, eq.(54). Here, the obtainable 3 eigenvalues areexpressed as σ1, σ2 and σ3. An eigenvalue is either a real root or a complex root. The valueof σ1, the real part of σ2 and σ3, and the imaginary part of σ3 are being shown. Fig.8 showsthe calculation results of the eigenvalues. In the figure, Re(σ3) and Im(σ3) mean σ3R and σ3I ,respectively.
σ1 exists only as a real root. In the case of no wind condition, σ1 is referring to the motiondamping that expressed as T ′
1 in Nomoto’s linearized model[4] where T ′1 = −1/σ1 is realized∗.
As σ1 is always negative, its motion component contributes only to stability. The absolutevalue of σ1 is one digit larger than that of σ2 and σ3, hence the motion damping due to σ1
becomes larger. σ1 doesn’t change much as per wind velocity and direction.σ2 exists in either a real root or a complex root. In case of no wind condition, σ2 is same
as the motion damping that expressed as T ′2 in Nomoto’s model, where T ′
2 = −1/σ2. When σ2
becomes a complex root, it has conjugate relation to σ3. The real part of σ2 is a negative valueexcept at the narrow region in head wind condition.
σ3 also exists as a real root or a complex root. σ3 is an important eigenvalue when dealingwith problem in wind. In case of D4 = 0, eigenvalue σ3 becomes zero and the characteristicequation changes to quadric equation. The real part of σ3 is a positive value in head windcondition and has a wide range of value from beam wind to following wind condition. Course
∗In this paper, T ′1 indicates the smaller one of the 2 coefficients (T ′
1, T′2) in Nomoto’s model[4].
16
0 30 60 90 120 150 180−5
−4
−3
−2
−1
0σ1
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
Uw/U=0
0 30 60 90 120 150 180−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1
00.10.2Re(σ2)
θW (deg)
Uw/U=5
Uw/U=4Uw/U=3
Uw/U=2
Uw/U=0
0 30 60 90 120 150 180−0.4−0.3−0.2−0.1
00.10.20.30.40.50.6Re(σ3)
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
0 30 60 90 120 150 180−0.5
−0.4
−0.3
−0.2
−0.1
0Im(σ3)
θW (deg)
Uw/U=5
Uw/U=4
Uw/U=3
Uw/U=2
Fig.8: Calculation result of eigenvalues of PCC in wind
instability appears in those regions. Thus, it is recognized that course stability of ships mostlydepends on the characteristics of σ3. The imaginary part of σ3, which represents the frequencyof yaw oscillation, appears only in the region from head wind to beam wind condition, agreeswith the region of oscillation mode in Fig.7. The absolute value peaks at around θW=45degand becomes large with the increase of wind velocity.
6.5.2 Approximate solutions of yaw motion
To confirm the validity of the approximation employed in section 5.2, approximate solutionof the eigenvalues calculated using eqs.(75), (76), (78) and (79) is compared with the exactsolution as shown in Fig.8. Fig.9 shows the comparison between the approximate and exactsolutions of eigenvalues for PCC in wind. The approximate solutions agree well with the exactsolutions, although the accuracy in the range where θW is 45～90deg is slightly inferior to otherwind conditions. The reason for the poor accuracy of approximate solutions in this range seemsto be because that the steady solutions are not being considered. As the effect level of thesteady solutions is almost proportional to the wind velocity, the accuracy of the approximationsbecomes worse with the increase of wind velocity. In this case, although there is a range wherethe accuracy becomes inferior with the increase of wind velocity, the approximate solution isstill acceptable to be used as a practical analysis method.
Next, time history of the yaw response in unit steering was calculated using eq.(77). Asshown in Fig.7, yaw oscillation occurs where θW is in the range of 0～90deg and UW /U is over0.5. In addition, Fig.9 shows that the absolute value of σ3I reaches the maximum value atθW around 45deg. Consequently, UW /U = 1, 2, 3 and 4 in θW = 45deg are selected as thewind condition in the calculations. Table 3 shows eigenvalues σ3R, σ3I , non-dimensionalizedyaw oscillation period T ′
ψ(= 2π/σ3I ), yaw oscillation period of actual ship Tψ, and the rudder
17
0 30 60 90 120 150 180−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1Re(σ2)
θW (deg)
Uw/U=3
Uw/U=2
Uw/U=1
Uw/U=0
ExactApprox.
0 30 60 90 120 150 180−0.2
−0.1
0
0.1
0.2
0.3
0.4Re(σ3)
θW (deg)
Uw/U=3
Uw/U=2
Uw/U=1
ExactApprox.
0 30 60 90 120 150 180−0.4
−0.3
−0.2
−0.1
0Im(σ3)
θW (deg)
Uw/U=3
Uw/U=2
Uw/U=1ExactApprox.
Fig.9: Comparison of exact and approximate eigenvalues of PCC in wind
effectiveness parameter in wind q′2/D′4. Then, ship speed U is presumed to be 15kn. Using
these values, time history of the yaw response can be calculated.Fig.10 shows the comparison of the yaw response in unit steering Δψ/Δδ in UW /U = 2,
3 and 4. Value of Δψ/Δδ becomes small as a whole with increase of wind velocity. Thismeans that course change becomes difficult in strong wind when the rudder angle is the same.This corresponds to decrease of q′2/D′
4 with increase of wind velocity. Further, significant yawoscillation appears in the time histories. The yaw amplitude becomes large and the oscillationperiod becomes long with decrease of wind velocity. Even in UW/U = 4, the period is 348s asshown in Table 3 and it is actually long. In addition, the yaw damping is small as shown inFig.10.
Table 3: Eigenvalues, period of yaw oscillation and rudder effectiveness parameter of PCC inwind (U = 15kn, θW = 45deg)
UW /U 1 2 3 4σ3R -0.0227 -0.0246 -0.0265 -0.0287σ3I 0.0981 0.2072 0.3137 0.4191T ′ψ 64.1 30.3 20.0 15.0
Tψ(s) 1494 708 468 348q′2/D′
4 49.5 11.4 5.03 2.75
0 50 1000
10
20
Δψ /Δ
δ
t’ (=tU/L)
Uw/U=2
Uw/U=4Uw/U=3
Fig.10: Comparison of time history of the yaw response in unit steering of PCC in wind(θW = 45deg)
18
7 CONCLUSION
Theoretical equations on the course stability and yaw motion of ships in steady wind werepresented. By means of the equations, the course stability and yaw motion of a Pure CarCarrier (PCC) in wind were investigated. The knowledge obtained through this study wassummarized as follows:
1. The derivatives with respect to the lateral force and yawing moment acting on the shipdue to wind (so-called aerodynamic derivatives) and the course stability criteria in steadywind were derived. In addition, an approximate solution of yaw motion in steady windwas presented.
2. The ship in steady wind has a tendency where its course instability appears in head andfollowing wind conditions. This phenomenon occurs predominantly due to the influenceof the aerodynamic derivatives with respect to the yaw restoring forces (namely, Y ′
Aψ andN ′Aψ). However, the course instability can be easily improved by using a steering control.
3. Motion equations in steady wind contain the additional term with respect to the yawrestoring force due to wind. The characteristic equation becomes the cubic equation, sothe additional motion component for ship maneuvering appears together with 2 motioncomponents in still water[4]. The reason why course instability appears in the rangefrom beam wind to following wind is because of this additional component that becomesunstable in those ranges.
4. The eigenvalue of the new motion component becomes a real root in the range frombeam wind to following wind and it makes the ship course unstable easily. When theeigenvalue is a complex root, yaw oscillation occurs. In case of the studied PCC, yawoscillation appears in the range from head wind to beam wind, and the frequency reachesthe maximum value at θW around 45deg. Period of the yaw oscillation is relatively longand the damping is small. Period becomes short with the increase of the wind velocity.
5. A parameter of the rudder effectiveness in stead wind is represented as q2/D4 definedin eq.(68), which is determined by hydrodynamic derivatives in oblique motion (Y ′
v , N′v
etc.), rudder force derivatives (Y ′δ and N ′
δ) and aerodynamic derivatives with respectto restoring forces due to heading change (Y ′
Aψ and N ′Aψ). With increase of the wind
velocity, q2/D4 decreases, namely the rudder effectiveness becomes worse.
REFERENCES
[1] Ogawa, A.: Calculations on the Steered Motion of a Ship under the Action of ExternalForces (Part 1) - Coursekeeping and Turning of a Ship in Uniform Wind and Flow, J.Society of Naval Architects of Japan, Vol.126 (1969), pp.107-120. (in Japanese)
[2] Spyrou, K.: Yaw Stability of Ships in Steady Wind, Ship Technology Research, Vol.42,No.1 (1995), pp.21-30.
[3] Eda, H.: Low-Speed Controllability of Ships in Wind, Journal of Ship Research, Vol.12,No.3 (1968), pp.181-200.
[4] Nomoto, K., Taguchi, T., Honda, K. and Hirano, S.: On the Steering Qualities of Ships,Technical report, International Shipbuilding Progress, Vol.4 (1957), pp.354-370.
[5] Isherwood, R. M.: Wind Resistance of Merchant Ships, RINA Trans. Vol.115 (1972),pp.327-338.
19
[6] Yoshimura, Y. and Nagashima, J.: Estimation of the Manoeuvring Behaviour of Ship inUniform Wind, J. Society of Naval Architects of Japan, Vol.158 (1985), pp.125-136. (inJapanese)
[7] Yoshimura, Y.: Mathematical Model for the Manoeuvring Ship Motion in Shallow Water,J. Kansai Society of Naval Architects, Japan, No.200 (1986), pp.41-51. (in Japanese)
[8] Fujiwara, T., Ueno, M. and Nimura, T. : Estimation of Wind Forces and Moments actingon Ships, J. Society of Naval Architects of Japan, Vol.183 (1998), pp.77-90. (in Japanese),
[9] Fujiwara, T. , Ueno, M. and Nimura, T. : An Estimation Method of Wind Forces andMoments Acting on Ships, Proc. Mini Symposium on Prediction of Ship Manoeuvring Per-formance, Japan Marine Dynamics Research Sub-Committee, Society of Naval Architectsof Japan (2001).
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