A theoretical investigation of hybrid foil-magnetic bearings on operation mode and load sharing strategy

To guide the conceptual design of hybrid foil-magnetic bearings (HFMBs), this paper presents a theoretical investigation on operation mode and load sharing strategy. According to the inherent characteristics of GFBs and AMBs, seven possible work scopes are discussed to understand the operation mode of HFMBs. A numerical model coupling the calculations of the film pressure in GFBs and the magnetic forces in AMBs is conducted to predict the performance of HFMBs. The deflection of top foil and bumps is also included in the prediction model. The analysis, which is conducted for static and dynamic performance of HFMBs within three representative work scopes by varying the load sharing, uncovers the effects of the operation mode and load sharing strategy on bearing behavior. Results show that HFMBs performance including the supporting stiffness and stability can be obviously enhanced by adjusting the operating mode and load sharing strategy.

energy density, and high working efficiency require oil-free bearings to have more comprehensive performance and adaptability to severe environment. Gas foil bearings (GFBs) and active magnetic bearings (AMBs), as two innovative types of oil-free bearings, are able to meet the requirements of most high-speed machines and have been developed and investigated for several generations [1].
GFBs, as a type of compliant, self-acting hydrodynamic fluid film bearings using ambient gas as lubricant, have shown many advantages including inherent simple structure, no complex oil lubrication and sealing system, excellent high-speed operation characteristics and tolerance for high temperatures [2]. To date, GFBs have been successfully used in many oil-free machinery applications, such as air cycle machine in aircrafts [3], turbochargers [4,5], turbojet engine [6,7], and micro gas turbine engine [8][9][10]. However, there are still some issues that hinder the development of GFBs. A special friction-resistant coating are needed on the surface of top foil to extend the applicable bearing life [11]. Due to its poor damping property, extra coulomb damping is usually required to increase the high-speed stability of GFBs during the design of the compliant structure [12].
AMBs generate forces through magnetic fields. There is no contact between bearing and rotor, which permits operation with no lubrication and no mechanical wear [13]. Current industrial applications of AMBs include vacuum and cleanroom systems [14,15], machine tools [16], medical devices [17,18], turbo-machinery [19,20]. However, in high-speed applications, AMBs are easily affected by structural resonance, which is the primary reason leading to the control system instability [1].
Moreover, the lack of reliable and durable auxiliary bearing is also an issue that should be solved when AMBs are operated under high speeds [21].
Hybrid foil magnetic bearings (HFMBs) , as shown in Fig. 1, combine the above two oil-free bearing technologies. The unique coupling method enables HFMBs to make full use of the advantages of both bearings while effectively overcoming the disadvantages. However, as the two bearings are mutually dependent and influenced, to make them operate in a coordinated manner become extremely important for the design of HFMBs. That is to say the adjustment of the rotor by AMBs must be consistent with the working condition of GFBs.
Heshmat et al. [1,22] presented an experimental investigation on HFMBs and proposed the earliest control algorithms for the load assignment of on the two bearings, GFBs and AMBs. Swanson et al. [23] designed a test rig supported by a HFMB system to simulate the rotor dynamics of a small gas turbine engine. Jeong et al. [24] built an experiment set-up to analyze the performance of rigid rotor supported by combined smart bearings on critical speeds with PD control. Tian et al. [25] presented a searching algorithm to determine the steady-state working position of a hybrid foil-magnetic bearing. Pham and Ahn [26] conducted an experimental investigation on the parameter optimization of hybrid foilmagnetic bearings to support a flexible rotor. Yang et al [27] presented a theoretical method to calculate the load capacity, dynamic stiffness and damping coefficients of HFMBs under a specified load sharing factor λ with the predetermined operational state. Jeong and Lee [28] tested a rigid rotor supported by HFMBs to elucidate the effect of initial eccentric position on the vibration response of rotor. Jeong et al. [29] developed a 225 kW class turbo blower by including hybrid foil-magnetic bearings.
Tian and Sun [30] presented an adaptive control method to simplify the controller design and improve the performance of the rotor-HFMB system. Jeong and Lee [31] used a control algorithm to reduce the sudden imbalance vibration amplitudes of a rigid rotor which was operated at up to 12,000rpm. Basumatary et al. [32] presented a dynamic model that coupled dynamics of gas foil bearings and electromagnetic actuator to discusses the effect of electromagnetics actuators on the stability of a rotor supported on gas foil bearings.
However, HFMBs design remains largely empirical, in spite of a small amount of successful applications. Obviously, in the actual application of HFMBs, GFBs and AMBs will play different roles when the equipment is operated at different conditions. A sloppy design by just simply coupling the two kinds of bearings will not be sufficient to extract the full performance potential of HFMBs and lead to oversize bearing. GFBs and AMBs, based on different working principles, have their own unique and distinct characteristics. Thus, distributing the work of the two bearings reasonably according the system requirement can be definitely complex.
Although rotor systems supported by HFMBs were investigated for oil-free turbo-machineries and some empirical hybrid arrangements for HFMBs were obtained, there are few theoretical analyses to explore the optimal performance of HFMBs including active and passive components matching strategy. Based on the characteristics of GFBs and AMBs, the possible work scope of HFMBs with respect to load capacity and rotational speed is discussed.
A numerical model coupling the calculations of the film pressure in GFBs and the magnetic forces in AMBs is conducted to calculate the static and dynamic performance of HFMBs. Prediction results of the bearing performance under different operating conditions were given and the importance of matching design between GFBs and AMBs was illustrated.
This work is to be used as a preliminary design study to develop turbomachinery supported by HFMBs. 2

Characteristics of Hybrid Foil-Magnetic Bearings
HFMBs are configured as a nested design by inserting GFBs components in the radial gap between the AMB poles and the rotor, as shown in Fig. 2.
AMBs directly act on the rotor by magnetic force, while GFBs support the rotor through hydrodynamic pressure in the gas film. The performance of HFMBs is the result of the joint action of the two bearings.

Operation range of GFBs and AMBs
The structure of GFBs consist of a smooth top foil and a flexible corrugate bump foil, as shown in Fig. 2a. GFBs have no physical clearance between the top foil and rotor at rest. When the rotor rotates, the high gas pressure will be generated and cause the foils to provide elastic deflection and frictional damping. The main factors limiting the work scope of GFBs are load capacity and bearing stability, as shown in Fig. 3a. The bearing load capacity increases with the rise of the rotational speed until it reaches a certain value [33]. Meanwhile, GFBs will become unstable as the increase speed due to the sub-synchronous vibration (cross-coupled stiffness) [34][35][36]. The suitable work scope of GFBs is shown in Fig. 3a. Generating contact free magnetic field forces by actively controlling the dynamics of an magnetic is the principle of AMBs which is actually used most often among the magnetic suspensions [37]. Stiffness, damping, as well as load capacity, can be varied widely within physical limits, and can be adjusted to technical requirements. The dynamics of the contact-free hovering depends mainly on the implemented control law [38][39][40][41][42][43]. The suitable work scope of AMBs is shown in Fig. 3b.

Operation range of HFMBs
In Fig. 4, the work scope of HFMBs is divided into seven parts on the basis of the relationship between load capacity and speed of the two bearings. The specific load capacity of the bearing depends on the type of ferromagnetic material and the design of the bearing magnet. It will be about 2

/
N cm and can be as high as 2 40 / N cm [37]. The designed load capacity of AMBs is usually less than that of GFBs in a hybrid system in view of larger air gap and more compact structure. The function of seven work scopes is described briefly below. As the rotational speed is low, the gas pressure generated between the top foil and the rotor is not enough to support the bearing load. Therefore, it is suggested to use AMBs in this scope (no need to use HFMBs).

Scope 2:
As the increase of rotational speed, the rotor may be separately loaded by GFBs or AMBs. Different operation modes represent different load ways. That means in this scope HFMBs can suspend rotor by AMBs or GFBs individually, or the hybrid way.

Scope 3, 4:
With the further increase of rotational speed, GFBs supporting system will show linear instability. However, due to the addition of the active control system (AMBs), the dynamic performance of the bearing is enhanced, and the linear stable scope expands towards the high-speed direction. Note that the difference between scope 3 and scope 4 is that the electromagnetic forces from AMBs which are used to stabilize GFBs are applied along different directions.

Scope 5:
The load exceeds the load capacity of AMBs, thus the rotor can be supported by GFBs alone or HFMBs.

Scope 6:
As the rotational speed increases, GFBs steps into its linear unstable range and have the possibility of instability. By applying electromagnetic force to increase the bearing eccentricity, GFBs can be stabilized again. GFBs and AMBs are unable to support the rotor alone in scope 6.

Scope 7:
The bearing load exceeds both the load capacity of GFBs or AMBs. Thus, the only choice for the system is to use HFMBs to support the rotor.
In the static calculation process of GFBs, the specific bearing load and speed correspond to a certain eccentricity and attitude angle of the rotor under the action of gas film force. In the dimensionless process of Reynolds equation of compressible gas, the following dimensionless parameters are considered: Referring to Eq. (1), it should be noted that the gas film thickness depends not only on the initial eccentricity and attitude angle, but also on the elastic deformation of the supporting structure. Therefore, the gas film thickness equation can be obtained: is the compliant surface deformation. It can be seen from Fig. 6 that zero slopes at both ends are an appropriate approximation because the top foil is continuous and the slope at each bump support is close to zero. The deflection of the top foil at the center between two adjacent bumps can be found as [45]: Where, EI is the bending stiffness of the top foil segment. Note that the local deflection function above was found assuming each bump is rigid, hence the total top foil deflection along the radial direction can be calculated by adding the local deflection of top foil to the bump deflection. The bumps are considered as springs with a constant structural stiffness, so the normalized foil deformation is given by [46]: The solution of Reynolds equation must be based on the deflection of foil structure and the distribution of gas film pressure. At both ends of GFBs along the axial direction, the gas film boundary is connected with the atmosphere, which can be regarded as the same as the atmospheric pressure. The gas film at the starting point and the ending point is also connected with the atmosphere. The boundary condition of Eq. (1) is:

Active magnetic bearing model
The constant ix k (N/A) and xx k (N/m) in Eq. (7) The magnetic force along X direction of HFMBs can be written simplistically: The first part of Eq.

Hybrid foil magnetic bearing model
The schematic diagram of HFMBs system is shown in Fig. 8. The load of an off-centre operation HFMBs is divided into two parts. One is borne by the hydrodynamic pressure between the top foil and the rotor, while the other is borne by the magntic force generated by a pair of poles along the X-axis direction. In addition, AMBs control current plays a role in providing dynamic stiffness and damping and suppressing rotor vibration. In order to ensure that GFBs and AMBs do not interfere with each other, the rotor center of GFBs under specified load must be calculated first.
According to the rotor center, the bias current that need to be provided by AMBs is calculated. In order to study the effect of load sharing strategy on the performance of HFMBs, it is assumed that the PID controller gains will not change with the working state.
The calculation process of static and dynamic performance of HFMBs is shown in Fig. 9. According to the load assigned to GFBs, the finite    (12) The supporting characteristics of the electromagnetic system depend on the control system. By taking the Laplace transform of the electromagnetic force and substituting the transfer function of the controller, the generalized stiffness of the electromagnetic system can be obtained.
In order to simplify the calculation, the coupling effect between the force along X and Y direction in AMBs is neglected. Therefore, the It is the most commonly used scope on account of adoptable three different operation mode and excellent low speed suspension with AMBs.

Speed=35krpm
The magnetic force of AMBs can be the supplement of hydrodynamic pressure of GFBs for the reason of the rotor mass beyond the capacity of AMBs.

Speed=42krpm
Another way of improving stability of the bearing is using magnetic force to increase the rotor mass.  The distribution curve of gas film pressure of HFMBs (on the middle surface) and the variation curve of differential current are shown in Figs. 11a and 11b, separately, with different load ratios. It is obvious that the gas film pressure gradually decreases with the increase of AMBs load ratio. As a cooperative component, the differential current along the X direction increases linearly, while the differential current along the Y direction shows a parabola descent. The differential current in both directions is slightly greater than zero when AMBs load ratio is zero, which is caused by the eccentricity of rotor. When the rotor reaches the bearing center, the air film pressure is close to the atmospheric pressure in the circumferential direction. Owing to the X direction differential current generates a separate load on the rotor, the differential current along the X direction reaches its maximum, and the differential current along the Y direction drops to zero. The significance of static parameters lies in that the rotor center can indirectly reflect HFMBs dynamic property in real time so as to modify and adjust the load sharing ratio.

HFMBs versus AMBs load ratio
(rotational speed:30kprm, total load:8N) Direct stiffness coefficient and cross-coupled stiffness coefficient of the HFMB (rotational speed: 30krpm, bearing load: 8N) are shown in Fig. 12 with respect to load sharing ratio. The direct stiffness coefficient xx K of HFMBs decreases sharply with the increase of AMBs load ratio, and the direct stiffness coefficient yy K also decreases significantly, as shown in Fig. 12a. On the one hand, the gas film pressure of GFBs will decrease with the increase of AMBs load ratio, which will lead to a decrease in the dynamic stiffness coefficient of GFBs and finally cause the decrease of the total dynamic stiffness coefficient of HFMBs. On the other hand, the direct stiffness of AMBs increases with the increase of force/current factor and decreases with the increase of force/displacement factor. The increase of force/displacement factor with the increase of AMBs load ratio is slightly greater than force/current factor, therefore, the direct stiffness coefficient of AMBs will also decrease significantly. It is easy to understand the reason that direct dynamic stiffness coefficient of HFMBs along the X direction has a significantly stronger downward trend than that along the Y direction. In addition, the direct stiffness coefficient of the HFMBs along X direction is lower than that of GFBs when AMBs load ratio is exceeding 0.8, while the direct stiffness coefficient of GFBs along X direction is significantly higher than that of AMBs. Therefore, in terms of maximum stiffness of the bearing, the bearing stiffness coefficient of HFMBs is the best when AMBs load ratio is not more than 0.8, followed by GFBs, and finally AMBs. In addition, the direct stiffness coefficient of HFMBs provided by GFBs accounts for an increasing proportion, although the direct stiffness coefficient of GFBs itself is gradually decreasing. The direct stiffness coefficient along the Y direction of HFMBs is greater than that of AMBs than that of GFBs. The stiffness coefficient provided by GFBs gradually decreases, but the direct stiffness coefficient provided by AMBs is almost constant.
As can be seen from Fig. 12b, the cross-coupled stiffness coefficient of HFMBs decreases gradually with the increase of AMBs load ratio. Since the cross-coupled stiffness coefficient of AMBs is neglected, the instability of HFMBs can be attributed to GFBs. It is an understandable fact that the instability factor of HFMBs is reduced because GFBs bears less load. Both direct stiffness coefficient and cross-coupled stiffness coefficient of HFMBs decrease with the increase of AMBs load ratio. When AMBs shows negative stiffness coefficient, the influence of force/displacement factor is significantly greater than that of force/current factor which is a nature of AMBs itself. The direct stiffness of HFMBs is greatly affected by the PID controller gains, so the change of eccentricity and AMBS load require the corresponding PID controller gains. The maintenance of constant PID controller gains will lead to lower direct stiffness coefficient or even negative stiffness coefficient of the bearing.
However, the influence of PID controller gains on direct damping coefficient of HFMBs is not obvious. Compared with GFBs and AMBs, the HFMB (load ratio of AMB is less than 0.4) have the maximum stiffness coefficient along the X direction. The optimal load sharing ratio in scope 5 is around 0.4 of AMBs which is quite different from that of HFMBs in scope 2.

Operation point in scope 6
Static characteristics force provides a positive load, the rotor center gradually approaches the bearing center with the increase of AMBs load ratio, which is the same as other operation points. However, when the magnetic force provides a reverse load to enhance the rotor mass, the rotor center gradually moves away from the bearing center with the increase of AMBs load ratio. In both cases, the center trajectory of the rotor together forms a complete curve. In Fig. 16b and Fig. 16c, corresponding to Fig. 16a, the gas film pressure of HFMBs independently supporting the rotor is further increased due to the reverse loading of magnetic force. When the magnetic force act in opposite way, the differential current along the X direction also changes from positive to negative, while the differential current along the Y direction still shows a parabolic decline. It can be concluded that using the magnetic force to increase the rotor mass is actually an extended application of the original operation mode. comparison. As can be seen from Fig. 17a, when AMBs provide a positive effect, its direct stiffness coefficient curve changes with the load ratio similar to that of the first two operation points. The direct stiffness coefficient along X and Y direction decreases with the increase of AMBs load ratio. However, when AMBs act the opposite way, the direct stiffness coefficient along the X direction shows a short rise followed by a rapid decline, while the direct stiffness along the Y direction keeps rising. The maximum value of the direct stiffness coefficient along X direction occurs when AMBs load ratio is around -0.1. The reason is that the rise of differential current of AMBs will reduce the stiffness coefficient, even if it is an opposite effect of the magnetic force.

Dynamic characteristics
Therefore, there is always a maximum value for the direct stiffness coefficient of HFMBs along the X direction. The increase of AMBs load in the opposite way will continuously increase the stiffness coefficient of GFBs, but also decrease the stiffness coefficient of AMBs.
In addition, the range of direct stiffness coefficient available for HFMBs in this diagram is between -0.5 and 0.5. AMBs further expand the control scope of load sharing strategy of HFMBs by increasing the rotor mass in reality. From Fig. 17b and Fig. 17c, it can be seen that the cross-coupled stiffness coefficient of HFMBs decreases continuously with the increase of AMBs load ratio. Meanwhile, the direct damping coefficient of HFMBs along the X direction first decreases slightly and then remains stable with the increase of AMBs load ratio, while the direct damping coefficient along the Y direction remains basically unchanged. It can be reasonably inferred from the cross-coupled stiffness coefficient and direct damping coefficient that the optimal load sharing strategy for the stability of HFMBs is 0.5 for AMBs.

Conclusion
In order to guide the conceptual design of HFMBs, this paper presented a theoretical investigation on operation mode and load sharing strategy with the consideration of bearing stability. Because GFBs and AMBs have totally different physical properties and suitable work scope, the hybrid bearing, HFMBs, can be designed into seven possible operation scopes. By cross correlating Reynolds equation, foil structure deflection and magnetic force, a numerical model was presented. The bearing performance is not just determined by the rotational speed and load but also related to the operation mode and load sharing strategy.
Three representative work points were selected to analyze the effect of operation modes and load sharing strategies on HFMBs performance. The prediction results showed that the differential current of AMBs should be In the following work, a test rig supported by HFMBs will be built and the influence of PID controller gains on the performance of HFMBs will be discussed.