The turbine rotor of a ship has a mass of 30,000 kg, a radius of gyration of 600

Question

The turbine rotor of a ship has a mass of 30,000 kg, a radius of gyration of 600 mm, and
rotates at 2400 rpm in a clockwise direction when viewed from aft(tail or stern of the ship).
The ship pitches (moves up and down) through a total angle of 15o, 7 ½ o above and 7 ½o
below the horizontal, the motion being simple harmonic and having a period of 12 s.
Fig. 1 shows a side view of the ship.
Determine i) the maximum gyroscopic couple on the holding down bolts on the turbine,
[186.5 kN m]
ii) the direction of yaw (the turning of ship or the correct course of the ship) as the
bow (front end of ship) rises.
[ The ship yaws to starboard, i.e. to the right looking forward of the ship]

Explanation / Answer

Answer (i)

W = 30000 x 9.8 = 294000 Newton
radius of gyration = k = 0.6 meter
w = 2 x pi x 2400/60 = 251.327 rad/s

Amplitude of pitching motion = = 15 degrees = (pi/180) x 15 = pi/12 rads

Time period = 12 seconds

Angular frequency of oscillation = wo = 2 pi /12 = 0.5236 rad/s

Maximum Precessional Speed = wp = x wo = (pi/12) x 0.5236 = 0.1371 rad/s

I = W/g x k^2 = 294000/9.8 x 0.6^2 = 10800 kg-m^2


Maximum Gyroscopic COuple = C = Iw(wp) = 10800 x 251.327 x 0.1371 = 372134.86 N-m

Now,

the maximum gyroscopic couple on the holding down bolts on the turbine = C/2 = 186067.43 N-m

                                                                                                           = 186.067 kN-m

Answer (ii)

Reaction couple will have a clockwise sense whens seen from the a ft

Hence,
        The ship yaws to starboard, i.e. to the right looking forward of the ship

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