A hydraulic turbine generates 30 MW of electric power when rotating at 250 rpm.

Question

A hydraulic turbine generates 30 MW of electric power when rotating at 250 rpm. The torsional vibration characteristics of the turbine are found to be unacceptable. The hollow circular generator shaft must therefore be replaced with a solid circular generator shaft. For these two shafts, (a) the lengths are the same, (b) they are made of the same material, (c) the maximum torsional shear stresses are the same and, (d) the transmitted torques are the same. The percentage increase in weight brought about by this change of design of the generator shaft is:

Given: d(o) = 350 mm
d(i) = 300 mm

a) 32%
b) 48%
c) 63%

Explanation / Answer

the maximum torsional shear stresses are the same and,the transmitted torques are the same

so the moment of inertia of both the shaft must be same.

let tha hollow cylinder has mass m and   

d(o) = 350 mm => r0=350/2 = 175 mm
d(i) = 300 mm => ri = 300/2=150 mm
moment of inertia of the hollow cylinder I1 = 1/2 * m( r02 + ri2)

I1 = 0.5 m( 1502 + 1752) =26562.5 m

for the solid cylinder

let the mass of the solid cylinder is m' and the diameter of the solid cylinder will be 350 mm for the fit.

moment of inertia of the solid cylinder I2 = 0.5 m' r2

=> I2 = 0.5 * 1752 m' = 15312.5 m'

  I1 =I2

=> 26562.5 m = 15312.5 m'

=> m' = 1.63m

% increase in the mass = (m' - m)/m *100 = 63%

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