# Diagnosing the Machine Set Rotor Using Its Relative Vibrations
## 4 Measurements of Relative Vibration
### Measurements of Torsional Vibration Relative Acceleration of Machine Set Rotors  
The acceleration of relative torsional vibration can be measured in each cross-section of the rotor, but in the case of the piston combustion engines, the convenient and practically the only place is the end of the crank shaft. The acceleration of relative torsional vibration can be measured by various heads equipped with various sensors, e.g., tachometric generator, incremental encoder or Ferraris acceleration sensor. The incremental encoder is well known [52]. The principle of the Ferraris sensor was described in Ref. [53]. The advantages and disadvantages of the incremental encoder and the Ferraris acceleration sensor are indicated in Ref. [54].  
The head with a Ferraris sensor should have a holder connected with the body, a sensor attached to the holder, and a measuring disc mounted on the rotor, co-operating with the sensor. The measuring disc is made of an electric conductor and generally mounted on a small shaft, an extension of the rotor shaft.  
The solution presented in Refs. [51; 55] proposes that the Ferraris sensor be mounted rotationally without clearance on the measuring disc shaft and connected with the body by a torsionally rigid element, e.g., bellows coupling.  
The tested machine set was composed of a single-crank, three-cylinder piston air compressor driven by electric motor via belt transmission. According to the definition of the machine set rotor given in Section 2, two rotors of the machine set should be distinguished here: the rotor of a compressor with a pulley and the rotor of an electric motor with a pulley. The driving moment produced by the electric motor is not constant during one revolution of the motor rotor. For a bipolar stator, the force acting across the gap between the stator and the rotor is a pulsating force, with a variable component of frequency equal to double frequency of the current in the power supply network [56]. As a result of the slip between the belt and the rotor pulleys, there is a continuous change of phase shift between the motor rotor and the compressor rotor.  
Measurements were recorded of the torsional vibration of the compressor free end and phase angles of the compressor and motor rotors. The torsional vibration and phase angle of the compressor rotor was measured by a simplified, workshop-made head described in Ref. [51].  
It should be expected that the measured signal of torsional vibration will be the sum of true torsional vibration and the runout signal. The time waveforms of vibration acceleration signals and the angular marker are shown in Figures 8 and 9.  
Figure 8: Time waveforms of the signals: upper run—waveform of the compressor shaft marker; bottom run—torsional vibration acceleration of the piston compressor free end.  
The difference was observed in time waveforms of subsequent periods of torsional vibration acceleration of the compressor (Figure 8). It follows from Figure 9 that the period of the compressor rotor is not a multiple of the engine rotor period, and the phase shift between the rotors changes on a continuous basis.  
The time waveforms of 16 periods of torsional vibration were subjected to synchronous averaging. The averaged waveform is shown in Figure 10. The synchronously averaged amplitude--frequency spectrum in the 0-500 Hz range was made, Figure 11. Besides, 16 amplitude--harmonic order spectra were synchronously averaged, Figure 12.