Resistance To Ship Motion - A Mathematical Analysis

In our previous article we have discussed about what is water resistance in context of ship motion and the types of resistance. We also carried out a mathematical analysis of propeller forces. Now we shall discuss the measurement of resistance.

We have seen that there are two basic types of resistance. They are: 1. Frictional resistance and 2. Residuary resistance.

The idea of dividing the total resistance into two was put forward by Froude. He found that when he observed geometrically similar forms of ships at different speeds, the wave patterns appeared to be similar.

• There are three important observations given by Froude. They are:

1. For geometrically similar ships, " The speed of the ships is found to be directly related with the square root of their lengths (ie) v/√l = V/√L where, v and l are the length and speed of the ship or model, V and L are the speed and length of the ship respectively. This speed is called as corresponding speed.

2. For any two geometrically similar ships or between a ship and its model which runs at corresponding speed, their residuary resistance is directly related with the displacement (ie) Rr1 / l3 = Rr2 / L3 where, Rr1 and 2 are the residuary resistance of the model and the ship, l, L are the length of the model and the ship respectively.

3. Froude gave the formula for frictional resistance as

Rf = f S vn

where the symbols represent the following

f - coefficient depending upon the length of the surface,

S - wetted surface area of the ship in m²,

v - speed of the ship in m/s,

n = 1.825

The wetted surface area can be given by S = c√(Δl)

Where c- coefficient mostly = 0.28, Δ- Displacement in tonnes, l- length of the ship in m,

• The Froude's tank was constructed in 1871. It is used for measuring the total resistance of a geometrically similar model to that of a ship.
• The tank will be filled with freshwater and the model will be towed with the help of carriages and the resistance records are made by the dynamometer which is connected with the model and will be placed remote from the model.

• Measure the total resistance of the model by towing the model in the Froude's tank. Now the total resistance will be recorded in the dynamometer.
• The frictional resistance for the model is calculated by using the formula : Rf model = f S1 vn
• This value to be corrected with the density as the model runs at freshwater but practically the ship runs mostly at sea water. If 'x' is the resistance in fresh water then its resistance in seawater having a density (1025kg/m3 ) will be product of the 'x' and 1.025 (t/m3 )
• We have earlier discussed that the total resistance can be given by the sum of frictional resistance and the residuary resistance. Hence Residuary resistance of the model is given by: Rr1 = Rt1 - Rf model
• From the froude's law ( no: 2) Rr1 / l3 = Rr2 / L3 The value of the Residuary resistance of the ship can be calculated (Rr2)
• Now calculate the frictional resistance of the ship by using the formula:

R_f2 = f S₂ Vⁿ

• Now we can calculate the value of the total resistance by using the expression:

R_t2 = R_f2 + R_r2

• The propelling device for the ship must be able to do effective work in order to overcome this resistance. This is taken as effective power as the appendages are not considered.
• Effective power = Rt x v x 0.5144 in kNm/s
• The value 0.5144 is used to convert the speed of the ship from knots to m/s
• The use of calculating the Rt is to know how much power is required to overcome the resistance so that the engine power can be chosen accordingly.