Modern Physics Letters B Vol. 31, Nos. 19–21 (2017) 1740014 (6 pages) c World Scientific Publishing Company
DOI: 10.1142/S0217984917400140
Mod. Phys. Lett. B 2017.31. Downloaded from www.worldscientific.com by UNIVERSITY OF TEXAS AT AUSTIN on 08/27/17. For personal use only.
Dynamic responses of the rotor supported by a new type zero-clearance catcher bearing
Yi-Li Zhu∗ and Zhong-Qiao Zheng Department of Electrical Engineering, Changzhou Institute of Technology, Changzhou 213032, China ∗
[email protected] Received 9 September 2016 Published 5 April 2017
Catcher bearings (CB) are required to support the rotor rotating for some time when a failure event of active magnetic bearing (AMB) system occurs. For this purpose, a new type zero-clearance catcher bearing (NTZCB) is proposed. The influences of different parameters of NTZCB on the rotor dynamic responses are theoretically and experimentally analyzed. The results indicate that choosing relatively soft spring and heavy moveable supporting pedestal can effectively buffer the rotor vibrations, which makes it possible for the rotor to keep rotating with the support of the CB system for a long time. Keywords: Dynamic response; new type zero-clearance catcher bearing; moveable supporting pedestal; active magnetic bearing; catcher bearings.
1. Introduction In active magnetic bearing (AMB) system, the catcher bearings (CB) are indispensable to temporarily support the rotor from directly impact the stators.1 The researches of CB are mostly focused on the rotor dynamic responses after rotor drop. Ishii and Kirk proposed that an optimum damping can effectively prevent destructive backward whirl.2 Sun et al. presented a detailed ball bearing model for magnetic suspsension CB.3 Kirk et al. provided the test results for 38 rotor drops with varying rotor speed, unbalance amplitude and location.4 Other researches focused on new type of CBs, such as zero clearance auxiliary bearing5 and hybrid air foil auxiliary bearing.6 Some special applications such as the aero engines requires CB to replace AMB to support the rotating rotor for a certain time when AMB is in a failure condition. A new type zero-clearance catcher bearing (NTZCB) is proposed for this purpose. Its performances during stably supporting the high-speed rotating are theoretically and experimentally analyzed. 1740014-1
Y.-L. Zhu & Z.-Q. Zheng Moveable Axial MB Supporting pedestal New type CB
Electromagnet
Foundation bed
Rotor
θy
y
y xp1 θ Rb
yb1 xb1 x
Rotor δ b 0 = 0.125 mm
δ b0
x ω
mr g
Displacement sensor
Fxa
z Fya
( xs , ys )
Amplifier
Inverter Fixed supporting pedestal
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Axial MB
Ls = 90 mm La = 116.5 mm
Lb = 173mm
Spring
δ d0 = 0.25 mm
θx
Or
( xb , yb )
xp2
Radial MB
CB
Fxb
Fyb
Rotor
Radial MB
Fig. 1.
AMB control system
Structure principle diagram of the analyzed test rig.
2. Structure of AMB System The designed structure principle diagram of AMB system is shown in Fig. 1. The whole AMB system test bench is horizontally placed. The NTZCB is installed in the left end of the system. And the structure of NTZCB is also presented in the figure. During testing, the two ends of the rotor are supported by NTZCB and radial magnetic bearing (MB) respectively. Excitation of the electromagnet produces a magnetic force to eliminate the gap between the outer race of the ball bearing and the moveable supporting pedestal (MSP). Each MSP is mounted on a fixed ball linear guide rail pair and collected to the foundation bed by a spring.
3. Theoretical Analysis Models 3.1. Rotor dynamic model Rigid rotor model is established to analyze the dynamic responses. According to Fig. 1, the motion formula can be expressed as: ¨ + GB X˙ = AF − F c , M BX
(1)
where M is the rotor mass matrix, mr is the rotor mass (2.14 kg), and J is the rotor transverse moment of inertia (MOI) (1.6 × 104 kg · mm2 ); B is the introduced displacement transformation matrix; X is the displacement vector, X = [xb yb xs ys ]T ; G is the rotor gyro torque matrix; A is the introduced force parameter matrix; F and Fc are the external force and centrifugal force vector respectively. Those matrices are deduced as: 1 B= Lb + Ls
Ls
0
Lb
0 0 −1
Ls
0
1
0
0
1
0
Lb , −1 0 1740014-2
0 0 G= 0
0
0
0
0
0
0
0
0
−ωJz
0
0 , ωJz 0
Dynamic responses of the rotor supported by a NTZCB
1
0 A= 0 −Lb F = Fxb
0
0
1
1
0
Lb
0
0
La
1 , −La 0
Fyb
Fxa
Fya
T
,
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F c = mr er ω 2 cos(ωt) mr er ω 2 sin(ωt) + mr g
0
0
T
,
where Fxb and Fyb are the support force from the NTZCB; Fxa and Fya are the support force from the radial MB; Jz is the rotor polar MOI (380 kg · mm2 ), er is the rotor unbalance (2 µm), g is the gravity acceleration. 3.2. NTZCB support model As the concave radius of MSP is smaller than the outer arc radius of the ball 1, 2, 3, 4 as shown bearing, there are only four edges of the MSP (marked by in Fig. 1) that can contact the outer race of the ball bearing. In order to analyze conveniently, the deformation of the ball bearing is neglected, and it is supposed that the MSP only has displacement in the x-direction. Based on the improved 1 and the outer race of Hertz contact theory,7 the contact force between the edge the ball bearing can be expressed as: 3 10/9 Fnx1,ny1 = Kc ςx1,y1 1 + κς˙x1,y1 , (2) 2 where ςx1 = xp1 − xb1 − δb0 and ςy1 = yb1 ; parameter κ depends on the material of the two contacting bodies, here κ = 0.08. The contact stiffness between the MSP and the outer race can be calculated by Kc = 7.85 × 1010 L8/9 c ,
(3)
where Lc is the contact length. The contact force between other edge and outer ring can also be calculated referring to formula (2). And Fxb and Fyb are the resultant force of the four contact forces. 3.3. AMB support model The real time support force of MB can be calculated using the dynamic stiffness and displacement detected by the displacement sensor. Using the methods mentioned in Ref. 8, the support stiffness and damping of the radial MB can be calculated combining the relevant AMB parameters. K = 12,000 Kec Re[Gc (jω)] − Kd , (4) 12,000 Kec Im[Gc (jω)] C = ω 1740014-3
Y.-L. Zhu & Z.-Q. Zheng
where Kd and Kec are the displacement stiffness coefficient and voltaic stiffness coefficient, respectively, and Kd = 1.25×106 N·m−1 , Kec = 166.5 N·A−1 . The conkD s troller transfer function Gc (s) = kP + ksI + 1+τ , where the proportional coefficient Ds kP = 2.3, the integral coefficient kI = 1.5, the differential coefficient kD = 1.4e − 4, and the time constant τD = 1.4e − 5. Then the AMB dynamic stiffness can be written as: p Kd = K 2 + (Cω)2 . (5)
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3.4. MSP dynamic model Taking the left MSP shown in Fig. 1 for example, its motion equation can be written as: mp1 x ¨p1 + ks xp1 = Fm − Fnx1 − Fnx2 ,
(6)
where mp1 is the mass of MSP (0.175 kg), ks is the stiffness of the spring (1 × 103 N/m), the magnetic force Fm ≈
2 Bm S µ0 ,
and the flux density Bm =
µ0 N i 4(δd0 −δb0 )
when Bm is less than the material saturation induction (1.5 T), once Bm ≥ 1.5 T, 2 Fm ≈ 1.5µ0 S , and N is the number of the coils (300), i is the excitation current (1.5 A), µ0 is the air permeability, S is the magnetic pole area (130 mm2 ). 4. Experiment Rig Figure 2 shows the photograph of the test rig. The AMB control system realizes stable suspension of the right end of the rotor and the inverter drives the rotor to rotate at a set speed. Eddy current sensors in the motor were adopted to measure the rotor real-time vibration displacements. The rotor vibration displacements were first collected by the data acquisition cards based on the LABVIEW environment, then those data were further analyzed based on MATLAB.
Data acquisition system based on LABVIEW environment AMB system controller Inverter
Fig. 2.
Magnetic levitation motor
NTZCB
Photograph of the test rig.
5. Result Analysis Figure 3 presents simulation results of the influences of various parameters of NTZCB on the peak-to-peak value of xb . It can be seen that the support of NTZCB makes the rotor vibration amplitude to decrease with the increase of rotor rotating 1740014-4
18 16 14 1
12
26
1: mp1 = mp2 = 0.075 kg 2: mp1 = mp2 = 0.125 kg 3: mp1 = mp2 = 0.175 kg 4: mp1 = mp2 = 0.225 kg
2 3
10 4
8 6
Peak-to-peak value of xb xbp/µm
Peak-to-peak value of xb xbp/µm
Dynamic responses of the rotor supported by a NTZCB
5
10 15 20 25 Rotor rotational speedθ /(kr/min)
18 2
10
3 4
30
6
xs 0.01
10 15 20 25 Rotor rotational speed θ /(kr/min)
30
Influence of various parameters of NTZCB on peak-to-peak value of xb .
xb
0
5
(b) Stiffness of supporting spring
0.02 0.03 time t/s
0.04
0.05
Vibration displacement x/µm
Vibration displacement x/µm
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4 3 2 1 0 -1 -2 -3 -4
1
14
(a) Mass of MSP Fig. 3.
1: ks = 1.5×105 N/m 2: ks = 1.0×105 N/m 3: ks = 5.0×104 N/m 4: ks = 1.0×104 N/m
22
xb
6 4 2 0 -2 -4 -6
xs 0
(a) Simulation results Fig. 4.
0.01
0.02 0.03 time t/s
0.04
0.05
(b) Experimental results
Vibration displacements of the rotor two ends.
speed. In the selected parameter ranges, the vibration amplitude decreases with the increase of MSP mass while increases with the increase of the stiffness of the spring in NTZCB. Figure 4 presents the vibration displacements of the rotor two ends when the rotor rotates at the speed of 30,000 r/min. Both simulation results and experimental results show that the NTZCB can support the rotor rotating with relatively small vibration amplitude. And the vibration amplitude of the left end supported by NTZCB is larger than that of the right end supported by radial MB. The simulation results show that the vibration displacement wave of xb is the superposition of two waves of different frequencies because of the mounted two springs in NTZCB, which is also revealed by the experimental results. 6. Conclusions The NTZCB is proposed to support rotor continuously rotating in an AMB system. Both theoretical simulations and experiments are carried out to analyze the dynamic responses, and the results show that the NTZCB can safely support the rotor rotating at high speed with relatively small vibration amplitude and makes the rotor vibration amplitude decreases with the increase of rotor rotating speed. The mounted springs in the NTZCB can influence the vibration displacement wave 1740014-5
Y.-L. Zhu & Z.-Q. Zheng
of the rotor. It is advisable to choose relatively softer spring and heavier MSP to decrease the rotor vibration amplitude. Acknowledgments
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This research is supported by National Nature Science Foundation of China (51405040) and Jiangsu Natural Science Funds (BK20151182), and the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (TAPP, No. PPZY2015B129). References 1. 2. 3. 4. 5. 6. 7. 8.
F. D. Pinckney and J. M. Keesee, J. Tribol. 35 (1992) 561. T. Ishii and R. G. Kirk, J. Rotating Mach. Veh. Syst. 35 (1991) 191. G. Sun, A. Palazzolo, A. Provenza and G. Montague, J. Sound Vib. 269 (2004) 933. R. G. Kirk, E. E. Swanson and F. H. Kavarana, in Proc. 4th Int. Symp. Magnetic Bearings, Zurich, Switzerland (1994), pp. 207–212. M. Salehi and H. Heshmat, J. Tribol. 43 (2008) 435. D. Kim and M. K. Varrey, J. Tribol. 55 (2012) 529. J. Jones, Contact Mechanics (Cambridge University Press, Cambridge, 2000). L. Zhao and H. Cong, Tsing Hua Univ. (Sci&Tech) 39 (1999) 96.
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