TABLE OF CONTENTS In compressors the crank shaft is a relatively vulnerable part. Via the crank the rotation of the shaft is transformed into an oscillating motion of the piston (or vice versa as in a combustion engine). In this a periodic force is exerted on the shaft, which may give rise to torsional oscillations in the shaft. If the amplitudes of these oscillations are too large, the axis is in danger of getting ruptured. We study this phenomenon using a mathematical model, which, though simple, still contains the essential features of realistic compressors. The model describes a shaft with one crank and one piston, The shaft is driven by a powerful engine and we assume that this happens with constant angular velocity ?, irrespective of the load of the crank. In practice this assumption is quite reasonable, because most compressors are equipped with a heavy flywheel. Friction will be ignored; in practice the losses due to friction are very small. The system is sketched in Fig. XII.23.
Clearly this is a mechanical system with kinematic constraints, so we have a DAE associated with it. The system is conveniently described within the Lagrangian formalism dealt with in XI.2. We denote the rotation angle of the crank by ?(t). The torsion ?(t) may deviate from the rotation angle ?t of the engine. For the derivation of the equation of motion we need an expression for the kinetic energy. It is the...
