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From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Analysis of crankshaft torsional vibration and disturbing torque caused by piston inertia.

Identifier  ExFiles\Box 6\1\  01-page008
Date  1st September 1926 guessed
  
contd :-

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Consideration of these results indicates that the phenomenon is a pure torsional vibration of the crankshaft and the disturbing force is mainly dependent upon the piston inertia.
We have xx constructed an inertia torque diagram for this engine at 3000 r.p.m. for three cylinders shewing the resultant fluctuation in torque at the crankshaft due to the piston masses alone. The magnitude for six cylinders would be twice that shewn due to pairs of cylinders being in phase. The resultant torque is of sine wave form which is satisfied by an equation of the form T = A sin 3θ

Where T = Torque.
θ = Crankshaft angle.
A = Constant.

In this particular case the constant equals 118.3 where T is in lbs.ft.

We see that the disturbing torque alternately assists and opposes the rotation of the crankshaft and passes through three complete cycles in each revolution.

Due to the inertia of the flywheel preserving nearly constant xxx angular velocity we should expect this force to tend to give slight angular deflection alternately positive and negative to the crankshaft. The deflections would naturally not be expected to assume any appreciable amplitude unless the natural period of oscillation of the complete crankshaft synchronised with the inertia torque impulses. The speed of synchronisation would be its critical speed and were it not for the internal molecular friction and damping, would build up to

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