5.3 Initial Turning Ability
For the ship designs demonstrating dynamic course stability it is important to establish if the designs may, in fact, be too stable. This could result in a ship with an unacceptably slow response to the helm. To investigate excessive stability the method employed examines the time taken for a 10^{0} heading change for a 10^{0} applied helm angle in terms of shiplengths travelled [21]. This is applied as shown in Eq.(36) subject to the assumption shown in Eq.(37). Then, setting the values to 10^{0} and rearranging to form Eq.(38) an iterative solution is obtained and presented in Table 3; demonstrating good agreement.
The test corresponds to the IMO initial turning requirement [22]. This criterion requires the ship to initiate such a turn in less than 2.5 ship lengths and can be used to benchmark prospective designs.
T' = T'1 + T'2  T'3 (37)
Table 3 Initial Turning Characteristics
Ship Lengths 
Cruise Liner 
Ropax 
Estimated 
1.76 
1.28 
Model Tests 
1.80 
1.20 

5.4 Phase and Gain of the Transfer Function
It is not strictly necessary for the openloop system to exhibit dynamic stability only that the closedloop system be controllable. Both the phase and gain must be examined if the characteristics of the closedloop system are to be established [23]. Taking the Laplace transform of Eq.(31), integrating for heading response and including a term for the steering gear we obtain Eq.(39). The gain and phase are given by Eq.(40) and Eq.(41) respectively [23].
 G (jω)  dB = 20logK' + 10log ( 1 + T'3s )  20logω'
10log ( 1 + T'1s )  10log ( 1 + T'2s )  10log ( 1 + T'Es ) (40)
φ[deg] = 270 + tan^{1} ( T'3ω' )  tan^{1} ( T'1ω' )
 tan^{1} ( T'2ω' )  tan^{1} ( T'Eω' ) (41)
These results were used to evaluate the response characteristics of the OPTIPOD ship designs. The Nichols chart given in Fig. 10 shows the GainPhase relationship for the four designs. Again, both the Cargo and Supply ship plots indicate an unstable open loop system; passing through zero gain, as they do, to the left of 180°phase. However, the grey shaded regions on the plot indicate various design point for the closedloop system. If the turning point of the unstable curve should lay in the dark grey region this should indicate that the ship would be controllable even if the helmsman was taking orders from a pilot [23]. If the turning point should lay in the light grey region then the helmsman should be able to introduce sufficient phase and gain so as to stabilise the ship [23]. Clearly, this would indicate the Cargo ship to be a feasible design while the current Supply ship design would be uncontrollable.
Fig. 10 Nichols Chart for OPTIPOD Designs
6. CONCLUSION
The proposed semiempirical equations provide a very good estimation of the manoeuvring derivatives when compared with captive tests result. Further, they provide a much improved estimate when compared with the existing prediction models.
The proposed empirical equations provide a method of calculating the control and stabilising derivative for both azimuthing pods and pods equipped with control surfaces.
In combination, the hull and pod derivatives are demonstrated to accurately predict course stability characteristics of both directionally stable and directionally unstable pod driven ships. Further, estimates of the initial turning characteristics of the directionally stable designs are in very good agreement with the freerunning tests results.
Over all, the proposed methodology can be used,with some confidence, to evaluate the manoeuvring characteristics of pod driven ships at the preliminary design stage.
7. NOMENCLATURE
A Area(m^{2})
a Effective aspect ratio()
AE Effective area(m^{2})
AR Area of pod in propeller race(m^{2})
B Breadth(m)
C Stability criterion()
c Chord length(m)
Cb Block coefficient()
CL Life coefficient()
CPa Aft body prismatic coefficient()
cr Chord length at root(m)
ct Chord length at tip(m)
CWa Aft body waterplanearea coefficient()
Dp Propeller race coefficient()
FL Life force(N)
g Acceleration due to gravity(m/s^{2})
J Advance coefficient()
KM Propeller race coefficient()
KT Propeller thrust coefficient()
L Ship length between perpendiculars(m)
T Draught(m)
U Ship speed(m/s)
UR Propeller race flow velocity(m/s)
Va Advance velocity(m/s)
w Wake fraction()
I'z Nondimensional moment of inertia about the zaxis()
K' Nondimensional gain term()
m' Nondimensional mass()
N'δf Nondimensional partial derivative of yaw moment with respect control flap angel()
N'δp Nondimensional partial derivative of yaw moment with respect pod angel()
N'r Nondimensional partial derivative of yaw moment with respect to yaw rate()
N' Nondimensional partial derivative of yaw moment with respect to yaw acceleration()
N'v Nondimensional partial derivative of yaw moment with respect to sway velocity()
N' Nondimensional partial derivative of yaw moment with respect to sway acceleration()
r' Nondimensional yaw rate()
' First derivative of r'()
' Second derivative of r'()
t' Nondimensional time term(ship lengths)
T'E Nondimensional time constant for steering gear()
T'n n^{th} nondimensional time constant()
v' Nondimensional sway velocity()
' First derivative of v'()
x'g Nondimensional longitudinal centre of gravity()
X'UU Nondimensional partial derivative of surge force with respect to the square of the surge velocity()
Y'δf Nondimensional partial derivative of sway force with respect control flap angle()
Y'δp Nondimensional partial derivative of sway force with respect to pod angle()
Y'r Nondimensional partial derivative of sway force with respect to yaw rate()
Y' Nondimensional partial derivative of sway force with respect to sway acceleration()
Y'v Nondimensional partial derivative of sway force with respect to sway velocity()
Y' Nondimensional partial derivative of sway force with respect to sway acceleration()
α Angel of attack of propeller race(deg)
δ Angel of attack due to advance(deg)
δf Helm angle of control flap(deg)
δp Helm angle of pod(deg)
φ Phase angle(deg)
Ψc Change of heading(deg)
Λ Effective rack angle(deg)
ρ Fluid density(kg/m^{3})
σα Aft body shape parameter()
σ1 First control root()
σ2 Second control root()
ω' Nondimensional response frequency()
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