Influences of Assault Blocks on Rotor Vibrations in Amb System

In an active magnetic bearing (AMB) system, the rotor always rotates at extremely high speed which always accompany with huge vibrations and noises. Most of the former researches associated with reducing the rotor vibrations are mainly focused on the control methods of AMB. A new method of installing series of assault blocks in the rotor is proposed to reduce the rotor vibrations. Firstly, the dynamic models of rotor supported by AMB considering the influences of assault blocks are established. Then, both dynamic simulations with and without assault blocks are carried out separately using the real-time AMB support dynamic stiffness. The rotor vibration displacements are mainly analyzed. Finally , relevant experiments are made to verify the theoretical results. Both theoretical and experimental results validated the advantages of using assault blocks.


INTRODUCTION
AMB have many advantages over conventional mechanical bearings, such as active control, no mechanical contact, no friction, no lubrication and high speed.The rotor in AMB system always rotates at extremely high speed to improve its working performances.Because of the rotor unbalance, the rotor will has certain vibration synchronous with rotor frequency.Most of the former researchers tried to modify the control system to reduce the unbalance displacement vibrations.
Jing, et al. injected an additional force to AMB system to restrict the unbalance force [1].Reference [2] proposed an adaptive auto-centering approach, which can realize on-line identification of rotor unbalance and force compensation.Taguchi, et al proposed a vibration control system to deal a sudden sinusoidal disturbance [3].Mizuno, et al., proposed pole-zero cancellation compensation to realize displacement, current and force regulations [4].A LQR scheme for vibration control in AMB system to reduce the unbalance vibrations is proposed by Arias, et al. [5].Moreover, iterative learning control is applied in the unbalance compensation for AMB system [6][7][8][9][10].However, most of the existing methods to reduce the rotor unbalance vibrations in AMB system are based on the control system, which may to some extent influence the AMB system stability In this paper, a series of assault blocks are added to the rotor system.Both AMB mathematical model and rotor dynamic model are established.And based on hertz contact theory, the contact model between assault block and rotor is established.Then influences of those installed assault blocks on the rotor vibrations are theoretically and experimentally analyzed.

AMB STIFFNESS ANALYSIS
Accurate AMB support stiffness is necessary to obtain accurate rotor vibration characteristics.In the control diagram, i cx1 is the control current; v s is displacement sensor output; v r is input of PID controller, which is linear with rotor vibration displacement x 1 ; x 0 is the balanced air gap, here x 0 = 1.25e-4 m; G A,S are the linear amplifier power gain and sensor gain respectively, here G A = 0.4 A/V and G S = 20000 V/m.According to the control diagram, the relationship between control current and rotor vibration displacement can be expressed as: The resultant force acting on the rotor in one direction can be linearized as Where and K ix1 = µ 0 AN 2 I 0 / x 0 2 is the current stiffness, they can be calculated using the parameters listed in Table 1.As the complete structure symmetric of the radial magnetic bearings, the displacement stiffness and current stiffness of each degree are equal.
The transfer function of PID controller is ( ) The parameters in the formula are listed in Table 1.
The stiffness and damping of AMB can be deduced from formula (3): The parameters in the formula are listed in Table 1.
Then the AMB dynamic stiffness can be written as [10].
Using the parameters of AMB presented in Table 1, the AMB dynamic stiffness can be obtained.As the AMB thrust force is in the rotor axial direction, and the rotor does not exist unbalance in axial direction, the axial subsystem can be separated from the other four radial DOFs.Fig. (3) gives the rotor force analysis in the four radial directions, and the dimensions are in millimeters.Where O is the rotor mass center; L a and L c are the distances between radial magnetic bearing and rotor barycenter, radial displacement sensor and rotor barycenter respectively; θ x and θ y are the rotor rotational displacement around axis X and axis Y respectively.Each magnetic force is defined as F n (n = xaj or yaj, j = 1, 2 denote left and right side, respectively); each counterforce from one hand assault blocks is defined as F m (m = xsj or ysj).The magnetic force can be calculated using equation 6. While, detailed collision model between rotor and assault blocks should be established to calculate each counterforce.
Where M is the rotor mass matrix, X o is the displacement vector of rotor barycenter, G is the rotor gyro torque matrix, A and S are the introduced parameter matrix, F a , F S , F c are electromagnetic force, assault block counterforce and centrifugal force matrix respectively.
Where m r is the rotor mass; and J is rotor transverse moment of inertia (MOI).
Where x o and y o are the rotor barycenter displacements in X and Y direction respectively.
Where J z is the rotor polar MOI; Where e r is the rotor unbalance, g x and g y are the gravity acceleration in X and Y directions.
The real time support force of magnetic bearings can be calculated using the dynamic stiffness and displacement detected by the relevant displacement sensor.So the displacement vector X o of the rotor barycenter should be converted to the rotor displacement vector X c at position of the radial displacement sensors.
Where X c = x c1 y c1 x c2 y c2 !" # $ T and In the same way, the displacement vector X s at the two ends of assault blocks can be derived as Where X s = x s1 y s1 x s2 y s2 !" # $ T and Then the rotor motion equation ( 7) can be rewritten as

Contact Model
Taking the left end for example, the contact model between assault block and rotor is established, as shown in Fig. (4).During normal operation, the collisions between assault blocks and rotor play the role of reducing vibrations.There are four assault blocks at one end, the position angles are !11 、 !21 、 ! 31 and !41 , respectively.The contact angles between each impact normal direction and X axes are !11 , !21 , ! 31 and !41 , respectively.The coordinate position of the assault block center is (x s! 1 , y s! 1 ) , where != 1,2,3,4 .And the relevant coordinate position of the assault hole center in the rotor is (x h! 1 , y h! 1 ) , where != 1,2,3,4 .
Based on the hertz contact theory [10], the impact force between the !th assault block and hole can be written as follows: Where the contact stiffness K s depends on material property and contact geometry: here K s = 2.4E+9 N/m; the penetration depth !s" 1 = r s" 1 # (R h # R s ) , where R h and R s are the radiuses of assault hole and assault block respectively, and ( ) ( ) According to the geometrical relationship, the impact force can be decomposed into two component forces in X and Y directions.
Then the motion equation of the ς th assault block can be expressed as In the same way, the impact forces and motion equations of the other assault blocks can be obtained.Considering all the impact forces, the assault block counterforce vector can be rewritten as

SIMULATION RESULTS
Using a variable time-step fourth order Runge-Kutta [12] integration algorithm, both of the numerical solutions using assault blocks and without assault blocks are respectively obtained.The detailed parameters are listed in Table 2.

Fig. (5a)-(b)
show simulation displacements of the left journal from 0 to 0.1 second after AMB works, and the rotor rotating speed is 12 000 r/min.The following conclusions can be obtained: (1) because of the rotor gravity, the vibration center does not locate at the established origin of coordinates; (2) the use of assault blocks can effectively reduce the rotor vibration displacements, and the vibration amplitude has been reduced by about 35%; (3) compared with the rotor without assault blocks, the vibration amplitudes of the rotor installed assault blocks are not equal for the series of internal collisions.Fig. (7) shows the output waveforms of left radial sensor in X direction with and without assault blocks at the rotor rotational speed 12 000 r/min.The displacement sensor sensitivity is 20 mv/µm.It can be seen that the rotor radial displacements with assault blocks are obviously smaller, and the peak to peak amplitude has been reduced about 30%.Also, the rotor vibration amplitudes with assault blocks are not equal.

CONCLUSION
(1) The rotor vibration displacements are achieved through solving those established mathematical models.And the model accuracy is verified by the following experiments.
(2) Assault blocks are proposed to be installed in the rotor system to reduce the rotor vibrations.Both simulation results and experiment results prove their effects.And the peak to peak amplitude has been reduced by about 30%~35%.
As a result, except for the methods of modifying AMB controller system, the rotor vibrations can also be reduced by installing proper assault blocks in the rotor.

Fig. ( 2 )
Fig. (2) gives the rotor three-dimensional model with assault blocks.Those assault blocks are located in the outside of the rotor for clear statement.In practice, the eight assault blocks are installed in the relevant holes of the rotor.
Rigid rotor model is established to analyze the dynamic responses.According to Fig. (3), the motion formula can be expressed as:

Fig. ( 6 )
Fig. (6) shows the experimental facilities necessary for measurement.Three boards NI-9215 from National Instruments are chosen to collect the rotor displacement signals, and the relevant collection software is established on the platform of LABVIEW.Those collected signals are saved in computer for further analysis.
(a) Rotor without Assault Blocks.(b) Rotor Installed Assault Blocks.