Dynamic Modeling of the Active Magnetic Bearing System Operating in Base Motion Condition

Active magnetic bearing (AMB) has been increasingly applied in high-speed rotating machinery applications because of the inherent merit of providing contactless magnetic force to levitate rotor. Generally, rotor AMB systems are applied in the static base situation whereas the recent work has also demonstrated their applicability in moving base conditions. Compared with traditional static base condition, the base motion condition is more complicated for rotor AMB system. For a better controller design and dynamic analysis, the dynamic modeling of AMB system operating in base motion condition is critical. However, the dynamics modeling of AMB system working in base motion condition has been rarely reported. Particularly, the rotational base motions are always neglected. Herein, in this work, based on a double-gimbal system model, we propose and develop a dynamic modeling method for AMB system considering both translational and rotational base motions. The derivation reveals that the translational base motion is equivalent to the external force applied to the rotor and the rotational base motion is equivalent to the external torque applied to the rotor. Furthermore, making use of the proposed dynamic model, we also analyze the dynamics of rotor displacement in the impact base motion condition from the aspects of excitation amplitude, pulse width and supporting parameters.


I. INTRODUCTION
Active magnetic bearing (AMB) is an electromagnetic device that provides magnetic force to suspend shaft other than conventional bearings relying on mechanical forces [1]. With the inherent distinguished features including absence of mechanical wear, elimination of lubrication, long life expectation, tunable stiffness and damping, as well as high attainable rotating speeds, AMBs have been increasingly applied in compressors, bearingless motors and other high-speed rotating machinery applications [2]- [4].
Typically, most of the AMB systems are applied in the stationary applications. The recent advances in AMB systems has also permitted their capability in base motion conditions such as hybrid electric vehicles, submarine propulsion systems, jet engines and spacecraft control moment gyro [5]- [9]. In general, the AMB system is open loop unstable and consequently feedback control is critical for The associate editor coordinating the review of this manuscript and approving it for publication was Mark Kok Yew Ng . levitation. Compared with conventional static base condition, the base motion conditions are more complicated as aggregate vibration of internal rotor dynamic vibrations and external base vibrations are both required to be considered. Thus, it is challenging for the controller design of AMB system operating in base motion conditions. For conventional rotor bearing system, the system dynamics under base motion condition has already trigged attention from both academia and industry [10]- [15]. However, to date, the research work regarding AMB controller design are mainly based on static base conditions and only a few has focused on controller design linked to base motion conditions. For instance, Kasarda et al. [16] modified the PID controller gains on a non-rotating test rig with a single magnetic bearing mounted on a shaker to detect the effect of sinusoidal base motion. Cole et al. [17] designed controllers that were capable of attenuating vibration arising from either directly applied to the rotor or transmitted through the bearings owing to base motion. Kang [18], Kang et al. [19] presented an optimal base acceleration feedforward controller VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ using filtered-X least-mean-square (LMS) algorithm and sliding mode controller to reduce the base motion response.
Sim et al. [20] modeled the base motion as the dynamic disturbances and designed controllers to compensate the disturbances. The experimental results indicated that the performance of the proposed controllers for the AMB system is satisfactory in compensating for the disturbances due to the base motion. Marx and Nataraj [21] developed a novel method which combined PD feedback with feedforward optimal control to suppress rotor motion caused by base excitation. Zhu [22] experimentally analyzed the dynamic behavior of AMB system with impact load and found that the PID controller has limited capability in suppressing the impact load. Zhang [23] developed a coupled dynamic model for rotor AMBs while only base translational motion is considered. Soni et al. [24] evaluated the parametric stability of active rotor AMB system with a novel control law subjected to periodic base motion. Additionally, attempts have been made in the maglev train system due to the similar dynamic equation as compared with AMB system. For instance, Han [25] established a dynamic modeling method of maglev train. Sun et al. [26], [27] proposed the adaptive sliding mode control and adaptive neural-fuzzy robust position control algorithm for maglev train disturbance suppression.
Despite the effort in controller design for AMB system operating in base motion condition, it should be noted that most of the proposed designs were based on experimental results. The dynamics modeling of AMB system working in base motion condition has been rarely reported and rotational base motions are always neglected. In this regard, it is essential to develop the dynamic modeling of AMB system operating in base motion condition for a better controller design and dynamic analysis.
Herein, in this work, we aim to fill this gap. For the first time, a dynamic modeling method for AMB system operating in base motion condition is developed and presented wherein both translational and rotational base motions are both considered. Additionally, making use of the proposed dynamic model, we also analyze the dynamics of rotor displacement in the impact base motion condition as impact base motion may cause serious damage to AMB system. The remainder of the paper is organized as follows. Section 2 describes the AMB system adopted in this paper. Section 3 presents the modeling method of AMB system considering base motion. The effect of the impact base motion on the dynamics are presented in Section 4. Conclusions are drawn in Section 5.

II. DESCRIPTION OF THE AMB SYSTEM
In this work, the AMB system was designed and built as a home-made research platform, as shown in Fig. 1. The length of rotor is 336mm. Two radial and two thrust AMBs provide magnetic levitation force for the rotor. The decentralized PID control is employed to stabilize the rotor AMB system.   Table 1 lists the radial AMB parameters employed in this system. Since the maximum speed of this test rig is far more behind the rotor first bending critical speed, the rigid rotor model is adopted in the following mathematic modeling derivation.

III. MODELING ROTOR AMB SYSTEM WITH BASE MOTION
A. MODELING THE TRANSLATIONAL BASE MOTION Fig. 2 presents the rotor center of mass motion considering base motion in vertical direction, wherein x is rotor displacement, u is base motion displacement, m is the rotor mass, k is the stiffness and c is the damping provided by AMB. The system equation of motion can be written as, Given that the stator is fixed on the base, the sensor detects the relative displacement δ between the rotor and the base, which could be written as, Substitute Eq. (2) into Eq. (1) and obtain Eq. (3) indicates that upon the translational base excitation, the motion of rotor relative to the base is equivalent to a single degree of freedom forced vibration system. The stimulated vibration excitation depends on the acceleration of base vibration and the magnitude of the equivalent base force is mü. Fig. 3 shows the equivalence of translational base excitation.  To express the translational base excitation, the rotor coordinate system is defined as O r x r y r z r , which is fixed to the rotor and moving with the rotor. The base coordinate system is defined as O b x b y b z b . As shown in the Fig. 4, in order to improve the magnetic force load capacity, the magnetic poles in test rig adopted here are inclined 45 degrees to the horizontal plane. Therefore, the rotor translational equation of motion in both x and y directions can be expressed as where F is the other generalized forces except base, U x and U y are the equivalent base forces in x and y direction, Eq. (4)(5) indicate that the translational base motion applied to the system can be equivalent to the external force applied to the rotor.

B. MODELING THE ROTATIONAL BASE MOTION
For the rotor (Fig. 5), the z direction is the rotational axis, therefore, it is only necessary to study the rotational base motion in x and y direction. When the translational base motion is not considered, the rotational base motion system comprised of the rotor and the base can be regarded as a double-gimbal system [8], [28], as shown in Fig. 5.
The rotation of the base in the x and y directions can be equivalent to the superposition of the two gimbals rotation. Before the derivation of rotational motion, the reference coordinate systems and transformation matrixes need to be established firstly. As shown in Fig. 5, O j x j y j z j is the outer gimbal coordinate system, O g x g y g z g is the inner gimbal coordinate system, O i x i y i z i represents the inertial coordinate system.
The outer gimbal coordinate system O j x j y j z j coincides with the inertial coordinate system O i x i y i z i under initial conditions and has only rotational freedom about the inertial frame y i axis. The rotational angle and angular velocity of the outer gimbal are denoted as θ j. and σ j . Therefore, the transformation matrix between the outer gimbal coordinate system and the The inner gimbal coordinate system O g x g y g z g coincides with the inertial coordinate system under initial conditions and has only rotational freedom about the outer gimbal x g axis. The rotation angle and angular velocity of the inner gimbal are denoted as θ i. and σ i . Therefore, the transformation matrix between the inner gimbal coordinate system and the outer gimbal coordinate system is  The rotor base coordinate system O b x b y b z b is fixed to the inner gimbal coordinate system with 45 degrees inclination to the horizontal plane (Fig.4). Therefore, the transformation matrix between the base coordinate system and the inner gimbal coordinate system is To establish the rotor rotation dynamics equation, a standard reference coordinate system is needed for transforming other coordinate systems to this standard system. The standard coordinate system, defined as O r x f y f z f and named as inner ring coordinate system, presents the coordinate system wherein the rotor inclines at an angle (α, β) relative to the base. The inner ring and the rotor coordinate systems can separate the relative angular from spin motion of the rotor. The rotor coordinate system O r x r y r z r describes the spin of the rotor and the inner ring coordinate system O r x f y f z f illustrates the angular motion of the rotor. Under initial conditions, the inner ring system coincides with the base coordinate system. Therefore, the transformation matrix between the inner ring coordinate system and the base coordinate system is In the rotor coordinate system O r x r y r z r , due to the rotor spin around the z r axis, the transformation matrix between the rotor coordinate system and the inner ring coordinate system is where is the rotor spin speed. The rotor absolute rotational motion includes three parts, (1) the rotor rotates relatively to the base, which is described using α, β, ; (2) the base rotates relatively to the outer gimbal, which is described using θ i. ; (3) the outer gimbal rotates relatively to the inertial frame, which is described using θ j . For the rotational base motion, since only the x and y direction are considered, the equation of motion can be established in the inner ring coordinate system.
The rotor angular velocity ω br relative to the base system is, whose projection in the inner ring system can be written as where and The inner gimbal angular velocity ω ig relative to the inertial coordinate system is, whose projection in the inner ring system can be written as where Therefore, combing Eq. (12) and Eq. (16), the projection of the rotor absolutely angular velocity in the inner gimbal coordinate system is For the rotor AMB system, given that rotor is restricted in the narrow air gap between the rotor and the touch-down bearing, the rotor rotational angle (α, β) is closed to zero. To simplify the following derivation, the base rotational angle (θ i , θ j ) is assumed to be very small, therefore, Substitute Eq. (19) into Eq. (18) and obtain The moment of inertia component for the rotor is denoted as, Therefore, the rotor angular momentum (22) and the first order differential of Eq. (22) is For the rotor AMB system, there is no external torque in the axial direction according to the Euler equation. The external torque in the radial direction of the rotor is determined by the electromagnetic force and its distance from the center of mass. Therefore, the external torque M received by the rotor and the angular velocity ω of the inner ring system rotor can be written as and The expression of Euler equation is Substituting Eq. (22)(23)(24)(25) into Eq. (26) yields where Eq. (27)(28) indicates that the rotational base motion can be equivalent to the external torque applied to the rotor.

C. MODELING THE ROTOR UNBALANCE
Vibrations induced by mass unbalance is inevitable. The unbalance can be divided into static and dynamic unbalance. Assuming that the position of the unbalanced mass point is q g and rotor centroid position is q c = [x, y, α, β], the mass unbalance component can then be denoted as, where e and ϕ are the amplitude and phase of static unbalance, respectively, ε and γ are the amplitude and phase of dynamic unbalance, respectively. Therefore, the unbalance force can be written as For the rotor adopted in this AMB system, the length of the rotor is much larger than the radial size, therefore J z can be negligible and Eq. (30) can be simplified as,

D. EQUATION OF MOTION OF THE ROTOR CONSIDERING BASE MOTION
Combing the above derivation, the equation of motion of the rotor considering base motion is and the matrix expression of Eq. (32) can be expressed as where f is magnetic force provided by the AMB, M is the mass matrix, G is the gyroscopic matrix, F b is the equivalent generalized force matrix generated by basic vibration, B is the matrix relevant to magnet force. These vectors and matrixes can be written as, In the rotor AMB system, the data obtained from the displacement sensor is adopted for calculation. However, as shown in Fig. 6, due to the sensor and actuator noncollocation, the transform relation between the centroid displacement q c , the actuator displacement q d and the sensor results q h should be established firstly. Table 2 lists the parameters of the rotor.  The displacement of the actuator is denoted as and the displacement of the sensor is denoted as The transformation matrix for the centroid displacement q c is where and The magnetic force provide by the AMBs could be modeled as stiffness and damping force, where K =diag[k x1 k y1 k x2 k y2 ] is the equivalent stiffness, C =diag[c x1 c y1 c x2 c y2 ] is the equivalent damping [29]. Substitute Eq. (41-44) into Eq. (33) and obtained, Eq. (45) is the AMB system dynamic model considering the base motion.
In order to perform the subsequent dynamic analysis, we consider the influence of the impact base motion on the rotor displacement amplitude. Given that the base impact is more likely to induce rotor instability, the rotor would touch the touch bearings. Thus, the displacement of the rotor at touch-down bearing q s is selected to establish the dynamic equation. The equation is similar to the displacement at the sensor, the derivation is as follows,

IV. THE EFFECT OF IMPACT BASE MOTION FOR AMB SYSTEM
From the time domain, the base motion could be classified into continuous excitation and short-time impact excitation. The impact base motion may cause serious damage to AMB system since the momentum is changing in a very short time. Therefore, we adopt the impact base motion in the following dynamic analysis. The displacement data x s1 for AMB1 at touch-down bearing position in x direction is selected for analysis.

A. COMPARISON OF TRANSLATIONAL AND ROTATIONAL IMPACT FOR AMB SYSTEM
For the translational base motion, semi-sinusoidal and rectangular impact excitation forms are adopted. In the case of rotational base motion, sinusoidal impact excitation form is adopted. The translational excitation applied to the translational freedom degree equation of rotor acts as force and the rotational excitation applied to the rotational freedom degree equation of rotor plays the role of torque. The differential Eq. (48) is evaluated numerically using the fourth-order Runge-Kutta algorithm with 0.001s step and the transient response is obtained. Table 3 lists the parameters for impact base motion simulation. The time domain displacement response is analyzed between 0 and 2s and the impact excitation is applied at 1s. The displacement x s1 obtained under translational excitation is illustrated in Fig. 7. The blue line represents the semi-sinusoidal impact excitation response and the red line is the rectangular impact excitation response. It can be seen that the application of base motion can significantly stimulate the increase of displacement. The maximum amplitudes corresponding to semi-sinusoidal and rectangular excitation are 0.09mm and 0.11mm, respectively.  Fig. 8 are the displacement x s1 upon the application of semi-sinusoidal translational with/without rotational excitation. It is found that the rotational excitation has small effect on the amplitude of displacement. However, on the other hand, the rotational excitation can yield a phase lag. Fig. 9 compares the displacement x s1 upon the application of rectangular translational with/without rotational excitation. Clearly, one can see that the resultant displacement is mainly dependent on rectangular translational excitation. The rotational base motion appears to slightly enhance the amplitude of displacement.

B. EFFECT OF IMPACT AMPLITUDE AND WIDTH
A systematic investigation regarding the effect of amplitude and width corresponding to translational and rotational impact base motion on displacement x s1 is performed here. As the first step, we analyze the influence of amplitude and width corresponding to translational impact base motion on displacement x s1.
Presented in Fig. 10 are the displacement x s1 versus amplitude and width upon the semi-sinusoidal translational excitation. Here, the translational amplitude increases from 1g to 12g by 1g and the width increases from 2ms to 40ms by 2ms. Other parameters such as AMB stiffness and damping which listed in table 3 remain unchanged. From Fig.10, it can be concluded that rotor displacement increases linearly in proportion to the impact amplitude. Meanwhile, the width fluctuation seems to have negligible effect on the rotor displacement, particularly in the cases wherein the width is over 8ms. Fig. 11 gives the displacement x s1 versus amplitude and width upon the rectangular translational excitation. Similar to the changing-trend mentioned above, the rotor displacement  appears to remain almost unchanged with increasing width and raise linearly in proportion to the impact amplitude. The displacement x s1 versus amplitude and width upon the semi-sinusoidal translational excitation and rotational excitation are manifested in Fig. 12. Apparently, with the increase of amplitude, a sudden collapse in displacement occurs in the width range of 20-40ms. This may be attributed to the fact that the rotational and translational excitation cancel out in the opposite phase. The collapse position is related to the pulse width of the excitation.
Accordingly, displacement x s1 versus amplitude and width upon the rectangular translational excitation and rotational excitation are illustrated in Fig. 13. Interestingly, the changing trends of x s1 versus amplitude and width is similar to these found in Fig.11 wherein only rectangular translational excitation is applied. Thus, it is deduced that the rotational excitation has little effect on the rotor displacement responses.  In the second step, the influence of amplitude and width corresponding to rotational impact base motion on displacement x s1 is discussed. Shown in Fig. 14 and Fig. 15 are the displacement x s1 versus amplitude and width upon the rotational impact base motion with semi-sinusoidal translational excitation and rectangular translational excitation, respectively. The rotational excitation amplitude increases from 1 π to 120 π and the width increases from 0ms to 40ms. Other parameters such as AMB stiffness and damping which listed in table 3 remain unchanged. From Fig.14 and Fig.15, it can be concluded that the rotational impact amplitude and width have rather small effect on the rotor responses.

C. EFFECT OF EQUIVALENT STIFFNESS AND DAMPING
For the AMB system, the electromagnetic force parameters, equivalent stiffness and damping are critical factors for the rotor dynamics analysis. In this work, the effect of stiffness and damping on displacement x s1 upon the base motion is explored.    Fig. 17 display the obtained displacement x s1 versus stiffness and damping upon semi-sinusoidal and rectangular translational excitation, respectively. The stiffness increases from 1 × 10 6 N/m to 9 × 10 6 N/m by 0.5 × 10 6 N/m and the damping increases from 400 N·s/m to 2000 N·s/m by 100 N·s/m. Other parameters which are plotted in table 3 remain unchanged. As can be seen in Fig.16 and Fig.17, the displacement performs a consistently decline with the increasing of the equivalent stiffness. When the stiffness is of relatively low level, the displacements upon semi-sinusoidal and rectangular translational excitation are almost comparable. It also should be noted there that, compared with the equivalent stiffness, the equivalent damping has much weaker effect on the displacement, particularly in the cases wherein rectangular translational excitation is applied.  The effect of stiffness and damping on displacement upon the combination of translational and rotational base motion is also evaluated here. Fig.18 and 19 show the displacement x s1 versus stiffness and damping upon rotational base motion with semi-sinusoidal and rectangular translational excitation, respectively. Clearly, upon the combination of translational and rotational base motion, the amplitude of displacement can be strengthened as compared with those upon simply translational base motion. However, the changing trends of displacements with increasing stiffness and damping are similar to those observed in cases wherein only translational excitations are applied.

V. CONCLUSION
In this work, a dynamic modeling method for AMB system operating in base motion condition is developed and presented wherein both translational and rotational base motions are both considered. The derivation indicates that the translational base motion is equivalent to the external force applied to the rotor and the rotational base motion is equivalent to the external torque applied to the rotor. Then, considering the influence of rotor unbalance mass and equivalent electromagnetic force, the dynamic equation of AMB system operating in base motion condition is established. Furthermore, making use of the proposed dynamic model, we analyze the dynamics of rotor displacement in the impact base motion condition from the aspects of excitation amplitude, pulse width and supporting parameters.
The results reveal the predominant role of translational excitation, particularly the rectangular translational excitation in affecting the performance of AMB system. The rotation excitation can only slightly enhance the overall displacement amplitude. Upon the application of individual translational excitation, the rotor displacement amplitude shows a linear correlation with the impact amplitude while it is rarely affected by the pulse width of excitation. In the case wherein individual rotational excitation is applied, neither impact amplitude nor pulse width of excitation has obvious effect on the rotor displacement amplitude. However, the coupling effect of translational and rotational excitations on rotor displacement amplitude cannot be ignored. Moreover, it can be concluded that the support stiffness is the main factor to restrain the rotor displacement, and the support damping has little effect on the rotor displacement amplitude.