15.NA July 2018 Q.8(B)

A ship of 12000 tonne displacement has a rudder 15 m² in area, whose centre is 5 m below the waterline. The meta centric height of the ship is 0.3 m and the centre of buoyancy is 3.3 m below the waterline. When travelling at 20 knots the rudder is turned through 30°. Find the initial angle of heel if the force Fn perpendicular to the plane of the rudder is given by: Fa=577 A v² sin𝜶 N, Allow 20% for the race effect.

         If you are noticing some error in problems kindly comment below.Thanks

Given

Δ = 12000 t
A = 15 m²
GM = 0.3 m
COB = 3.3 m below water line
V = 20 knots
α = 30⁰

To find

1. Initial angle of heel , θ

Solution

Consider that rudder is turned by an angle α and let
Fn = Rudder force normal to the plane of rudder
Ft = Transverse rudder force

Consider Triangle ABD
Sin α = Fn / F
Fn = F sin α --------------------------(1)

Consider Triangle ABC
Cos α = Ft / Fn
Ft = Fn cos α -------------------------(2)

Substituting Fn value from (1)

Ft = F cos α sin α

But in the question above it is given that Rudder force Fn = 577 A x V2 x Sin α x N

So substituting in (2) ---------------->

Ft = 577 A x v² x Sin α x cos α  N

race is the additional flow past the rudder that is cause by the propeller turning at a faster speed than the vessel is moving through the water

Consider that race effect is 20%. This will affect ship speed.

So ship speed = V x 1852/3600

Considering race effect
Ship speed = 1.2 x V x 1852/3600
                   = 1.2 x 20 x 1852/3600
                v = 12.35 knots

Ft = 577 A x v² x Sin α x cos α N
    = 577 x 15 x 12.35² x sin 30 x cos 30  N
    = 571.61 kN

NL = Distance from COB to centre of the rudder
      = 5 - 3.3
      = 1.7 m

Heeling moment = Ft x NL x cos θ ----------------------------(3)
                           = 571.61 x 1.7 x cos θ
                           = 971.7 x cos θ kN m

Righting moment = Δ x g x GM x sin θ -----------------------(4)
                                                  
When heel is steady 

Heeling moment = Righting moment

 Ft x NL x cos θ = Δ x g x GM x sin θ
   971.7 x cos θ = 12000 x 9.81 x 0.3 x sin θ
   971.7 x cos θ = 35316 x sin θ
     sin θ / cos θ = 0.0275
                tan θ = 0.0275

Angle of heel θ = 1⁰ 57¹