Dynamic Characteristics of Flexible Rotor System of High Speed Motor with Magnetic Bearings

High speed motor with magnetic bearings has been developing due to its advantages of small size, high power density, no mechanical wear and low maintenance cost. In order to reduce vibrations, dynamic characteristics of the flexible rotor system of high speed motor for operation rotation speed 200000r/min with magnetic bearings are investigated. Results show that high speed motor may pass through some critical speeds and it is accurate to consider the motor shaft as a flexible rotor. Meanwhile, it is necessary to take measures to reduce the amplitude of resonance region of high speed motor such as increasing system damping. The critical speed increases with the increase of the bearing stiffness, which is a good way to remove the critical speed below working rotation speed.


Introduction
High speed motors have many advantages including high power density and reliability, small size, light weight, which can meet the special requirements of high-end equipment. The contactless characteristic of the magnetic bearing enables the rotor to operate at a very high speed. Its advantages of no lubrication and low bearing loss can make the equipment have lower maintenance cost and longer service life. Therefore, high-speed magnetic bearing motors have be paid more attention by researchers.
High speed motor usually refers to the motor whose speed exceeds 10000r / min. Among all kinds of motors, induction motor, permanent magnet motor and switched reluctance motor are the main ones to achieve high speed (1) . The maximum speed of high-speed induction motor is 180000r/min, and the power is 10kW. Adopting magnetic bearing and solid rotor structure, the linear speed can reach 219m/s, and the efficiency is about 85%. When the high speed induction motor is running at high speed, the conventional laminated rotor cannot bear the huge centrifugal force. Special high strength lamination or solid rotor structure is required. Because of the poor tensile strength of permanent magnet material for high speed permanent magnet motor, the problem of rotor strength is more prominent.
Many studies on the dynamic characteristics of flexible rotor magnetic bearings have been done. Smith (2) used the finite element method to analyze the causes of the rotor damage of a high speed permanent magnet synchronous motor with a speed of 30000 r/min and a power of 100kW. It is suggested that the friction and shear stress between the sheath and the permanent magnet can be reduced by using lubricant. Yon (3) proposed a laminated binding structure. Arumugam (4) determined the magnetic stainless steel structure of surface machining by finite element analysis and calculation, which meets the structural, thermal and magnetic. Lin (5) developed a comprehensive model to calculate the thermal characteristics of bearings under different preloads. It is found that the heating problem of high-speed motor in operation is related to the bearing preload. Huynh (6) analyzed the stator loss, rotor eddy current loss and wind friction loss of high speed permanent magnet motor. The analytical expressions of each loss are put forward. In order to ensure that the rotor has enough strength, the rotor of high-speed motor is mostly slender. Therefore, compared with the constant speed motor, the possibility of the rotor system of the high speed motor approaching the critical speed is greatly increased. Palazzo (7) used active magnetic bearing to control flexible rotor to study on dynamic characteristics, control of flexible rotor. The influence of support structure on the dynamics of rotor system was studied by Cavalca (8) . Eissa (9) studied the dynamic response of the rotor supported by active magnetic bearing under harmonic excitation. Besides, in order to avoid resonance, the critical speed of rotor system with magnetic bearing must be accurately predicted. But the determination of bearing stiffness increases the difficulty of accurate prediction.
The authors have developed a high-speed magnetic bearing motor with operation rotation speed 200000r/min. Dynamic characteristics of rotor system of high speed motor with magnetic bearings are investigated. Magnetic bearing is equivalent to elastic support and a finite element model of the motor shaft is established. The critical speeds of rotor system are obtained by numerical method. Besides, operating modal analysis is carried out with rotation speed 200000r/min. The influence of the support stiffness of magnetic bearings on the dynamic characteristics provides the improvement methods.

Dynamic modeling
In the magnetic bearing, the magnetic poles are generally simplified as shown in Figure 1. It consists of controller, power amplifier, sensor, rotor and electromagnet.
When the rotor is in the middle of the gap, in order to establish the magnetic field, there is equal current I 0 (Bias current) in the upper and lower pole coils. If the rotor is offset y, the air gap between the rotor and the upper magnet becomes: where k yy is defined as the force-displacement stiffness coefficient. It is the same as that of spring stiffness coefficient in general system. k yi is defined as force current stiffness coefficient.
when the higher order small quantity is omitted, the increment of dynamic force is 0 y y y yy yi When i 0 <<I 0 , the effect of removing I 0 can be ignored. In this case, Eq. (4) can be simplified as In general, the static feedback current i 0 is much smaller than the bias current I 0 . Therefore, Eq. (6) is used to describe the stiffness coefficient of the electromagnetic bearing. Fig. 2(a) gives the dynamic model of the rotor system of magnetic bearing motor. As shown in Fig. 2(b), the spring element is used to simulate the elastic support of the bearing.

Effect of support stiffness on critical speed
In order to remove the critical speed below the operating speed, Table 1 compares the effect of different support stiffness. It can be seen that the critical speed increases with the increase of the bearing stiffness. The third critical speed is larger than the operating rotating speed for support stiffness 7 1 10 N / m  . That means the rotor will pass through two critical speeds when the shaft reaches the working speed, which reduces the risk of resonance.