3. DISCUSSION OF INSTABILITY
3.1 Unstable area
By taking the example of the propulsion system [4] data of which are shown in Table 1, the calculation result of the unstable area is shown by the solid line in Fig. 2. The axis of abscissa represents the torsional spring constant of the shaft while the ordinate Sommerfeld No.
It is noted that the unstable area complicatedly varies with the torsional spring constant of the shaft and the boundary line appears in two positions, one is in the region of low Sommerfeld No. and the other in high one. The former one increases as the spring constant increases, which means that the stability in the region of low Sommerfeld No. is strongly influenced by the spring constant of the shaft while the boundary line in high one is approximately constant. It is confirmed by another calculation that the gear tooth bending rigidity dominates the boundary line in high one.
It is well known that the stability is usually dominated only by the journal rpm. But the calculation result clearly shows that the stability varies with the torsional spring constant of the shaft when taking the coupled vibration of the self-exciting vibration of the bearing oil film and torsional vibration into consideration. The unstable area exists in such low rpm range as is inexplicable by the theory of the oil whirl phenomena.
The broken line shows the unstable area obtained by the computer simulation the equation of which is the same as Eq. (1) with consideration of the wheel gear but ignoring backlash. The stability is judged from the waveform aspect of the eccentricity in time domain when the constant external torque is given to the system, i.e., if the eccentricity is increased or fluctuates, the system is judged unstable. Both lines, solid and broken, are almost the same but a little difference is observed because of the different condition of the wheel gear, in which one is assumed fixed while the other is free.
Also, the chain line of Fig. 2 shows the result of computer simulation in the same condition but considering the backlash. In this case, the unstable area is changed compared with the other cases. There are tendencies that the unstable area is expanded toward the lower side in the region of high Sommer-feld No. and is shafted toward the right side in the region of low one. These phenomena are caused by the chattering of the gear due to the existence of the backlash which results in the decrease of the equivalent spring constant.
As the stable area is dominated mainly by the spring constant of the shaft which varies with the amplitudes of the torsional vibration in such non-linear system as has gear backlash, the unstable area is always changed with the amplitude of the torsional vibration. When the external exciting torque is small and vibration amplitude is small, the equivalent spring constant is decreased and unstable area is shifted leftward, which results in such phenomena as make the unstable area stable and vice versa. It was observed in the actual propulsion system that the unstable condition was changed to the stable and vice versa which made the severe fluctuations appear.
3.2 Examples of computer simulation on various conditions
Fig. 3, 4, 5 and 6 shows the example of waveforms of the alternating torque and eccentricity ratio in time domain calculated by the computer simulation, in which it is assumed that the external torque is sinusoidal curve with 5th order and constant amplitude.
Fig. 2 Unstable area of coupled vibration
Fig. 3 Calculated waveform (point a ignoring backlash)
Fig. 4 Calculated waveform(point b ignoring backlash)
