Effect of Tangential Velocity Slip on Stability and Dynamic Response of a Flexible Rotor Supported on Porous Bearing
5.3 Effect of Slip on Nonlinear Dynamic Response of a Flexible Rotor
The non-linear dynamic response of a rotor-bearing system under unbalance condition is very much important. In the present section an unbalanced rotor supported by two hydrodynamic porous oil bearings with slip is considered. Waterfall diagrams are generated to study the dynamics of the rotor-bearing system. Further, Bifurcation diagrams, Poincaré maps, time response, journal trajectories and FFT-spectrum etc.
are obtained to study the non-linear dynamics of the rotor-bearing system. The effect of velocity slip on the dynamic response of the rotor-bearing system is then analyzed.
The effect of slip coefficients on the dynamics of rotor-bearing can be studied by changing its value while other parameters are kept constant as shown in Figs. 5.2 through 5.5. Waterfall diagrams are obtained for four different values of slip coefficients, α=0.0, 0.1, 10.0 and 100.0 when / 2R L=0.0125,E/ (ρgL) 2.7 10= × 6,
d /
e C=0.25, md=1.5, L DB/ =1.0 and β=0.03 . It has been observed from Figs. 5.2 through 5.5 that for all four values of slip coefficients oil-whirl starts at m=1.8. No significant change in the dynamics of rotor-bearing system is observed as the slip coefficient is varied. Thus, it may be inferred that slip has little effect on the dynamic behaviour of the rotor-bearing system.
Fig. 5.2: Waterfall diagram of rotor during run-up with, α=0.0
Fig. 5.3: Waterfall diagram of rotor during run-up with α=0.1
Fig. 5.4: Waterfall diagram of rotor during run-up with, α=10.0
Fig. 5.5: Waterfall diagram of rotor during run-up with α=100
Similar to the previous analysis the effect of slip is further analyzed for higher bearing feeding parameter value of β =1.0. Waterfall diagrams are obtained for α= 0.0, 0.1, 10 and 100 when the other non-dimensional parameters are as follows:
/ 2 0.0125
R L= ,E/ (ρgL) 2.7 10= × 6, e Cd / =0.25, md=3.0, L DB/ =1.0 and β=1.0.
The corresponding waterfall diagrams are shown in Figs. 5.6 through 5.9. It may be observed that for all values of slip parameter, α, oil-whirl starts at m= 1.4. Thus, it may be concluded that slip has no or little effect on the dynamic behaviour of the system under unbalance excitation. The same trend was observed in case of β =0.03
Fig. 5.6: Waterfall diagram of rotor during run-up with α=0.0
Fig. 5.7: Waterfall diagram of rotor during run-up with α=0.1
Fig. 5.8: Waterfall diagram of rotor during run-up with α=10.0
Fig. 5.9: Waterfall diagram of rotor during run-up with α=100.0
The effect of the ratio of slip on the appearance of oil-whirl has been is tabulated in Table 5.1. It has been observed from Table 5.1 that for all values of ,αoil-whirl appears at same value of mass parameter. Also, it may be again observed that oil- whirl appears at lower mass parameter as β increases.
Table 5.1: Effect of slip parameter on appearance of oil-whirl α Mass parameter at which oil whirl appears during
run-up
β =0.03 β =1.0
0 1.8 1.4
0.1 1.8 1.4
10 1.8 1.4
100 1.8 1.4
In this analysis, the effect of the slip coefficient on the bifurcation characteristics is also studied for four different values of slip, α=0.0, 0.1, 10 and 100.0, when R/2L=0.0125, β =0.03andE/(ρgL) 2.7 10= × 6. Figures 5.10, 5.11 5.12 and 5.13 show the bifurcation diagrams for α= 0.0, 0.1, 10 and 100.0 respectively.
Fig. 5.10: Bifurcation diagram of rotor with α=0.0
Fig. 5.11: Bifurcation diagram of rotor with α=0.1
Fig. 5.12: Bifurcation diagram of rotor with α=10
Fig. 5.13: Bifurcation diagram of rotor with α=100
It has been observed that for all the four values ofα, bifurcation takes place at m=2.0. Thus it is evident that velocity slip does not have any significant effect on the bifurcation characteristics of the rotor-bearing system.
Trajectory, time response, Poincaré map and FFT-spectrum of rotor with / 2 0.0125
R L= , md=1.5, E/(ρgL) 2.7 10= × 6, β =0.03 and α=100 are shown in Figs. 5.14-5.17 respectively. The trajectories and response is asynchronous at m=2.0 but afterward becomes regular (Figs. 5.14-5.15). It has also been observed that for the above rotor-bearing parameters the motion is quasi-periodic throughout the range of m=2.0-5.0 as seen in Fig. 5.16. Further, the rotor whirl motion is sub-synchronous (Fig. 5.17).
Fig. 5.14: Trajectory at (a) m=2.0,
(b) m=3.0, (c) m=4.0, (d) m=5.0 Fig. 5.15: Time response (a)m=2.0, (b) m=3.0, (c) m=4.0, (d) m=5.0
Fig. 5.16: Poincaré map at (a) m=2.0,
(b) m=3.0, (c) m=4.0, (d) m=5.0 Fig. 5.17: FFT-spectrum at (a)m=2.0, (b) m=3.0,(c) m=4.0, (d) m=5.0 The effect of slip on bifurcation characteristics can be again analyzed at higher value of β =1.0. Bifurcation diagrams are plotted for four values of slip parameters α= 0.0, 0.1, 10 and 100 and the corresponding figures are shown in Figs. 5.18 through
5.19 respectively. The bifurcation diagrams are obtained for the following parameters:
/ 2 0.0125
R L= ,E/ (ρgL) 2.7 10= × 6, e Cd / =0.25, md=3.0, L DB/ =1.0 and β=1.0.
It may be seen from the bifurcation diagrams that for all values of slip parameter, α, the rotor-bearing system undergoes Hopf bifurcation at m=1.4. Thus, even at high value of bearing feeding parameter, β=1.0 slip coefficient does not have any significant effect on the bifurcation characteristics of rotor-bearing system.
Fig. 5.18: Waterfall diagram of rotor during run-up with α=0.0
Fig. 5.19: Waterfall diagram of rotor during run-up with α=0.1
Fig. 5.20: Waterfall diagram of rotor during run-up with α=10.0
Fig. 5.21: Waterfall diagram of rotor during run-up with α=100.0
Table 5.2 summarizes the effect of slip parameter on bifurcation characteristics of rotor-bearing system. It may be seen from Table 5.2, bifurcation occurs at same mass parameter as the slip parameter increases for all values of bearing feeding parameter.
Also, like previous cases, with increase in bearing feeding parameter, bifurcation takes place at lower mass parameter.
Table 5.2: Effect of slip parameter on bifurcation
α Mass parameter at which bifurcation takes place
β =0.03 β =1.0
0 2.0 1.4
0.1 2.0 1.4
10 2.0 1.4
100 2.0 1.4