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Abstract
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Engineering
Structural Engineering

Forced response of a vertical rotor with tilting pad bearings

Profile image of Jan-olov AidanpääJan-olov Aidanpää

2016

Abstract

This paper evaluates how tilting pad bearings can affect the forced response of a vertical rotor. The stiffness and damping for a tilting pad bearing is dependent on the position in the bearing (load on pad or load between pad) and the magnitude of the load. In a vertical machine the position of the shaft in the bearing will change for each time-step and produce a periodic stiffness and damping. This periodic stiffness has shown to produce higher frequency components depending on the number of pads. For the four pad bearing considered in this paper these frequency components are 3Ω and 5Ω in the stationary coordinate system. The aim of this study is to evaluate if the periodic coefficients in the bearing could excite the system and cause resonance problems. It is found that for high loads the system can be excited due to the bearing dynamics at Ω/ω n = 0.33 and Ω/ω n = 0.2. However at low loads this effect is negligible.

Key takeaways
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  1. Periodic bearing coefficients significantly influence rotor dynamics in vertical machines.
  2. The study evaluates resonance risk due to bearing dynamics at Ω/ω n = 0.33 and Ω/ω n = 0.2.
  3. Using a four-pad bearing, higher frequency components at 3Ω and 5Ω were observed in simulations.
  4. Load magnitude impacts periodic excitation; high loads amplify vibrations while low loads minimize them.
  5. Displacement changes at Ω/ω n = 0.33 indicate potential oversight in conventional monitoring methods.
Figures (13)
Figure 1. Fourier transform of the bearing load from earlie simulations.
Figure 1. Fourier transform of the bearing load from earlie simulations.
Figure 3. 4—pad bearing geometry, with coordinate system (€,7) rotated a degree’s from the fixed coordinate system (x,y).
Figure 3. 4—pad bearing geometry, with coordinate system (€,7) rotated a degree’s from the fixed coordinate system (x,y).
Table 1. The test rig rotor and bearing properties.
Table 1. The test rig rotor and bearing properties.
Figure 4. Stiffness as function of a at 30% eccentricity in the coordinate system (&,7) (left) and stiffness LOP and LBP as function of eccentricity (right).
Figure 4. Stiffness as function of a at 30% eccentricity in the coordinate system (&,7) (left) and stiffness LOP and LBP as function of eccentricity (right).
The bearing coefficients calculated is in the local coordi- nate system (&,77) and must be transformed using the follow- ing transformation [7]. and Kg and Cg is the bearing stiffness and damping matrices in the local coordinate system (€,77).
The bearing coefficients calculated is in the local coordi- nate system (&,77) and must be transformed using the follow- ing transformation [7]. and Kg and Cg is the bearing stiffness and damping matrices in the local coordinate system (€,77).
Figure 5. Campbell diagram for constant stiffness at 30% eccentricity assuming LOP.
Figure 5. Campbell diagram for constant stiffness at 30% eccentricity assuming LOP.
Where the rotational speed Q is in rad/s. In Figure 6 dis- placement in the bearing is presented and it can be seen that something changes at Q/w, = 0.2 and Q/w, = 0.33. Figure 6. Displacement in the bearing for Fy = 90 N This becomes more clear if studying the disc displacement in Figure 7, here resonance peaks are clearly visible at these frequency multiples. Recall from Figure 4, here it can be seen that the difference between LOP and LBP is large at eccentricities around 45% of the bearing clearance. According to Figure 6 the displacement in the bearing is around 50% of the bearing clearance at Q/w, = 0.2 and 40% at Q/w,, = 0.33. Since the difference between LOP and LBP is large in this region the frequency component at 3Q and 5Q is large and will excite the system.
Where the rotational speed Q is in rad/s. In Figure 6 dis- placement in the bearing is presented and it can be seen that something changes at Q/w, = 0.2 and Q/w, = 0.33. Figure 6. Displacement in the bearing for Fy = 90 N This becomes more clear if studying the disc displacement in Figure 7, here resonance peaks are clearly visible at these frequency multiples. Recall from Figure 4, here it can be seen that the difference between LOP and LBP is large at eccentricities around 45% of the bearing clearance. According to Figure 6 the displacement in the bearing is around 50% of the bearing clearance at Q/w, = 0.2 and 40% at Q/w,, = 0.33. Since the difference between LOP and LBP is large in this region the frequency component at 3Q and 5Q is large and will excite the system.
Figure 7. Displacement at the disc for Fo = 90 N. In Figure 8 the whirl ratio at the disc is shown, here the maximum and minimum whirl ratio is presented. As seen in Figure 8 the whirl interval is increased especially at the frequency ratio Q/w, = 0.33. This indicates some other type of dynamic behaviour of the system. In Figure 9 the displacement orbit at rotating speed 440 rpm is presented, this operating speed corresponds to Q/w, = 0.33 and it shows how the 4-pad bearing effect the dynamics. For low bear- ing loads the difference between LOP and LBP is small and the periodic behaviour in the bearing is low. This should de- crease the periodic excitation and remove the resonance peaks observed in Figure 7.
Figure 7. Displacement at the disc for Fo = 90 N. In Figure 8 the whirl ratio at the disc is shown, here the maximum and minimum whirl ratio is presented. As seen in Figure 8 the whirl interval is increased especially at the frequency ratio Q/w, = 0.33. This indicates some other type of dynamic behaviour of the system. In Figure 9 the displacement orbit at rotating speed 440 rpm is presented, this operating speed corresponds to Q/w, = 0.33 and it shows how the 4-pad bearing effect the dynamics. For low bear- ing loads the difference between LOP and LBP is small and the periodic behaviour in the bearing is low. This should de- crease the periodic excitation and remove the resonance peaks observed in Figure 7.
Figure 8. Maximum and minimum whirl ratio at the disc for Fo = 90 N.
Figure 8. Maximum and minimum whirl ratio at the disc for Fo = 90 N.
Figure 9. Displacement orbit at the disc for Fy = 90 N
Figure 9. Displacement orbit at the disc for Fy = 90 N
Figure 10. Displacement in the bearing for Fo = 23 N. Figure 12. Maximum and minimum whirl ratio at the disc for Fy = 23 N.
Figure 10. Displacement in the bearing for Fo = 23 N. Figure 12. Maximum and minimum whirl ratio at the disc for Fy = 23 N.
Figure 11. Displacement at the disc for Fp = 23 N Figure 13. Displacement orbit for the disc where Fy = 23 N
Figure 11. Displacement at the disc for Fp = 23 N Figure 13. Displacement orbit for the disc where Fy = 23 N
Figure 14. FFT at Q = 440 rpm where Fo = 90 N (black) and Fo = 23 N (ted).
Figure 14. FFT at Q = 440 rpm where Fo = 90 N (black) and Fo = 23 N (ted).

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References (6)

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