2.3 Mathematical model for Propeller Thrust and Rudder Force
Mathematical model for thrust produced by propeller is expressed using thrust coefficient Kr defined as function advance coefficient JP:
tP : thrust reduction coefficient,
tP0 : thrust reduction coefficient in straight forward motion,
n : propeller revolution,
DP : propeller diameter,
C1, C2, C3 : constants,
wP : effective wake fraction coefficient at propeller location,
wP0 : effective wake fraction coefficient at propeller location in straightforward motion.
Mathematical models for terms on rudder forces are assumed as:
tR : coefficient for additional drag,
aH : ratio of additional lateral force,
x'H : non-dimensional distance between the center of gravity of ship and the center of additional lateral force (x'H=xH/L)
x'R : non-dimensional longitudinal distance between non-dimensional longitudinal distance between the center of gravity of ship and the center of lateral force (x'R=xR/L)
δ : rudder angle.
Following expressions are assumed for the normal force acting on rudder F'N with normal force coefficient CN:
F'N = (AR/Ld)CNU2RsinαR, (11)
AR : rudder area,
hR : rudder height,
KR : aspect ratio of rudder,
UR : effective rudder inflow speed,
αR : effective rudder inflow angle,
C : coefficient for starboard and port rudder,
wR : effective wake fraction coefficient at rudder location,
wR0 : effective wake fraction coefficient at rudder location in straight forward motion,
P : propeller pitch,
γ : flow straightening coefficient.
In Equation (12), the coefficients γ & wR0, originally represent flow straightening coefficient and effective wake fraction coefficient at rudder location in straightforward motion respectively. They have much influence upon ship manoeuvrability but it is very difficult to estimate these values precisely. Therefore the authors had assumed that these parameters are representative of interaction effects among hull, propeller & rudder and treated them differently from their original meaning (Kijima et al. ). Hereafter they are noted with subscript "E", that is γE & wR0E and they can be replaced with γ & wRO in Equation (12).