Torsional vibration question

Figure Q4 shows the layout of a yacht power unit in which a 4 cylinder, 2 stroke diesel engine is represented by J. Its flywheel J is integral with the pinion J3, which meshes with the main gear J4. This in turn drives the propeller J5 via the propeller shaft of diameter 100 mm. The gear box ratio (propeller speed/engine speed) is G = 0.5. The Table Q4 give the various moments of inertia and torsional stiffnesses. (a) An engineer gave rough estimates of the two lowest non- zero natural frequencies of torsional vibration as 33 Hz and 70 Hz and these were borne out by testing. Explain how these rough estimates could be quickly made based on simple calculations. [10 marks] (b) Determine the corresponding mode shape at 33 Hz, based on a more thorough calculation using a Holzer tabulation. [10 marks] (c) Estimate the maximum torsional stress in the propeller shaft when vibrating at a frequency near to 33 Hz, if the amplitude of torsional vibration at the propeller is 0.1 degrees. [7 marks] (d) At what engine speeds in the range 200-400 rev/min would you expect torsional resonances to occur due to engine excitation of the above natural frequencies? [6 marks] Moment of Inertia (kg m) engine J 15 flywheel J 28 pinion J3 4 gear J4 8 propeller J5 15 J 4 712 ONY J3 Table Q4. 100mm Torsional stiffnesses (MN m rad) 2 T12 0.6 T45 45 J5 Figure Q4 shows the layout of a yacht power unit in which a 4 cylinder, 2 stroke diesel engine is represented by J. Its flywheel J is integral with the pinion J3, which meshes with the main gear J4. This in turn drives the propeller J5 via the propeller shaft of diameter 100 mm. The gear box ratio (propeller speed/engine speed) is G = 0.5. The Table Q4 give the various moments of inertia and torsional stiffnesses. (a) An engineer gave rough estimates of the two lowest non- zero natural frequencies of torsional vibration as 33 Hz and 70 Hz and these were borne out by testing. Explain how these rough estimates could be quickly made based on simple calculations. [10 marks] (b) Determine the corresponding mode shape at 33 Hz, based on a more thorough calculation using a Holzer tabulation. [10 marks] (c) Estimate the maximum torsional stress in the propeller shaft when vibrating at a frequency near to 33 Hz, if the amplitude of torsional vibration at the propeller is 0.1 degrees. [7 marks] (d) At what engine speeds in the range 200-400 rev/min would you expect torsional resonances to occur due to engine excitation of the above natural frequencies? [6 marks] Moment of Inertia (kg m) engine J 15 flywheel J 28 pinion J3 4 gear J4 8 propeller J5 15 J 4 712 ONY J3 Table Q4. 100mm Torsional stiffnesses (MN m rad) 2 T12 0.6 T45 45 J5