Influence of Design Parameters on Static Bifurcation Behavior of Magnetic Liquid Double Suspension Bearing

Magnetic liquid double suspension bearing (MLDSB) includes electromagnetic system and hydrostatic system, and the bearing capacity and stiffness can be greatly improved. It is very suitable for the occasions of medium speed, heavy load, and starting frequently. Due to the mutual coupling and interaction between electromagnetic system and hydrostatic system, the probability and degree of static bifurcation are greatly increased and the operation stability is reduced. And flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness are the key parameters to ensure operation safe and stable, which has an important influence on the static bifurcation behavior. So this article intends to establish the coupling model of MLDSB to reveal the range of parameter combination in the case of static bifurcation. The influences of different parameter groups on the singularity characteristics, phase trajectory, 
 
 x
 −
 t
 
 curves, and suction basin of the single DOF bearing system are analyzed. The result shows that there are nonzero singularities and static bifurcation occurs when 
 
 
 
 ε
 
 
 2
 
 
 >
 0
 
 or 
 
 
 
 δ
 
 
 2
 
 
 >
 0
 
 . As the flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness changes in turn, the singularities will convert between stable focus, unstable focus, stable node, and saddle point, and then the stable limit cycle may be generated. The attractiveness of singularity will change greatly with the flow of the bearing cavity and coil current changes slightly in the case of small current or large flow. The minimal change of galvanized layer thickness will lead to the fundamental change of the final stable equilibrium point of the rotor, while the final equilibrium point is slightly affected by the oil film thickness. This study can provide a reference for the supporting stability of MLDSB.


Introduction
The control principle of MLDSB is shown in Figure 4. Constant-flow supply model is adopted in the hydrostatic supporting system, and proportional velocity regulating valves connecting the upper and lower supporting cavities are connected by differential module. PD control is adopted in electromagnetic supporting system, and upper and lower coils are connected through differential connection module. When the rotor is radial offset by external interference, its displacement is adjusted by the hydrostatic supporting system and electromagnetic system, and then it returns to the equilibrium position gradually.
Due to the coupling and interference between the nonlinear electromagnetic system and the nonlinear hydrostatic system, the probability and complexity of static bifurcation are improved, and the reliability and operation stability of MLDSB can be decreased sharply [4]. So many scholars had studied the static bifurcation behavior of electromagnetic suspension bearing and achieved fruitful results.
Wang et al. [5] established the mathematical model of two DOF system under harmonic base motion based on the magnetic force model which can induce bistable phenomena. By the Routh-Hurwitz criterion, the static bifurcation of equilibration points is analyzed for the dimensionless governing equations. The results show that the amplitudefrequency curves of the system are in hard characteristic, while the amplitude variations of the displacement of the piezoelectric cantilever beam with mass ratio and stiffness ratio are in soft characteristic.
Hai and Liu [6] used a unified time-delayed feedback control method to control the spatial static bifurcation of 2-D discrete dynamical systems. The results show that this method can determine and then control the spatial static bifurcation of 2-D discrete dynamical systems by transferring the existing bifurcation or by producing a new fork-shaped, trans-critical, or saddle node bifurcation.
Pu and Hu [7] studied the magneto-elastic principal resonance bifurcation and chaos of rotating annular plates in magnetic fields. The results show that the magnetic field deters the occurrence of multivalue phenomena. With the decreasing of the external force frequency, the rotating speed, and the magnetic induction, and with the increasing of the external force, the system's heteroclinic orbits break more easily, meanwhile, chaos or almost periodic motion of the system is induced.
Luo et al. [8] studied the nonlinear vibration of the rotor system supported by two bearing and excited by electromagnetic force. The results show that the trends of motion of the rotor system in the two displacements are similar. The motions of periodic, pe9 + riod-doubling, and quasiperiodic alternately appear in the operation of the rotor system, and low oil film viscosity is more secure for the rotor system.
Zhao [9] analyzed the nonlinear dynamics phenomenon of cantilever crane. The Lagrange method was used to build the system equation, and the singularity theory was taken to get the bifurcation condition and the normal form of the problem.
Peng [10] established a nonlinear vibration model of parametrically excited stiffness, using the multiple-scale method to solve the amplitude-frequency equation of 1/2 harmonic resonance about the system. The reason that the strip surface appears chatter or vibration marks was found that is caused by the dynamic behaviors such as period, period-3 motion, and chaos of the rolling mill.
To sum up, the current researches only focus on the bifurcation behavior of electromagnetic suspension bearing, while the structure of MLDSB is essentially different from the conventional electromagnetic suspension bearing, and its internal supporting mechanism and behavior law of static bifurcation is more complex. Meanwhile, the design and operation parameters (flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness) are the premise and foundation of high-performance bearing and stable suspension of MLDSB and have a significant influence on the static bifurcation behavior of MLDSB. So the coupling model of MLDSB is established to explore the internal influence of design and operation parameters on the singularity characteristics, phase trajectory, and suction basin of a single DOF   International Journal of Aerospace Engineering supporting system, and then the reference can be provided for the design and supporting stability of MLDSB.

Bifurcation Range of Design Parameters
Taking the vertical single DOF supporting system as the research object (it includes upper and lower supporting units, rotors, and so on as shown in Figure 5), the dynamic model can be established [11].
The design and operation parameters include flow q 0 of bearing cavity, coil current i 0 , oil film thickness h 0 , and galvanized layer thickness l.
2.1. Bifurcation Range of Parameter Group (i 0 , q 0 ). From the definition of bifurcation, it can be seen that the system will not bifurcate when equation (2) has a unique solution, while the system will bifurcate when equation (2) has multiple solutions [12]. The boundary point of static bifurcation can be obtained by solving equations. And then based on the   3 International Journal of Aerospace Engineering practical meaning of parameters, the bifurcation range of the system parameters is obtained. Assume that h 0 = 30 μm, l = 0 μm. According to the definition of singularity, the existence condition of singularity is _ x = 0 and _ y = 0 [12], then The design parameters in Table 1 were substituted into equation (2), and the coordinates of the singularities were obtained as follows: x 0 , y 0 ð Þ= 0, 0 ð Þ, From equation (3), it can be seen that there is a zero singularity (0, 0) and four nonzero singularities (x 1,1 , 0), (x 1,2 , 0), (x 2,1 , 0), and (x 2,2 , 0) in MLDSB system.
Jacobian matrix A of x and y can be obtained from equation (7).
The characteristic equation of matrix A [14] can be expanded as follows: where p = ∂Q/∂y, q = −∂Q/∂x.

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According to Figure 10, the coordinate system is divided into five regions (the curve q is similar to a part of the curve △, the above two can be approximately regarded as a curve).
As can be seen from Figure 12, the coordinate system is divided into four areas: (1) In area 1, the singularities are not existing   International Journal of Aerospace Engineering can be suspended stably as shown in Figures 13 and 14. However, the balance position is not the expected center of rotation [16].

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International Journal of Aerospace Engineering The phase trajectories of initial point (1:5 × 10 −5 , -0.02) and (1:5 × 10 −5 , 0.02) both surround and approach the stable focus (0, 0). Both of them reach balance after 0.05 s adjustment, and the rotor can be suspended stably as shown in Figures 15 and 16. The balance position is the expected center of rotation, and the phase trajectories of different initial points eventually approach the same stable focus.
(3) Assume that i 0 = 1:6 A, q 0 = 0:25 × 10 −4 m 3 /s, and phase trajectories and x − t curves are shown in Figures 17 and 18 The phase trajectory of point (−1:5 × 10 −5 , 0.02) rapidly approaches stable focus (0:76 × 10 −5 , 0), while the phase trajectory of point (2:63 × 10 −5 , 0) gradually surrounds and approaches stable focus (2:63 × 10 −5 , 0). Both of them reach balance after 0.0012 s adjustment, and the rotor can be suspended stably as shown in Figures 17 and 18. However, the balance position is not the expected center of rotation, and the phase trajectories of different initial points eventually approach different stable focus. According to Figures 19 and 20, the rotor oscillates in equal amplitude, and the phase trajectories form a limit cycle.
In order to verify the stability of the limit cycle, the initial point inside the limit cycle was selected to simulate the phase trajectories and x − t curves as shown in Figures 21 and 22.
According to Figures 21 and 22, the phase trajectories of the inner initial point gradually diverge and finally approach a limit cycle, so the rotor cannot be suspended stably in this case [18]. According to Figures 23 and 24, the rotor oscillates in equal amplitude, and phase trajectories form a limit cycle [19].

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In order to verify the stability of the limit cycle, the initial point inside the limit cycle was selected to simulate the phase trajectories and x − t curves as shown in Figures 25 and 26.
According to Figures 25 and 26, the phase trajectories of inner initial points gradually diverge and finally approach a limit cycle, so the rotor cannot be suspended stably in this case.
With the gradual change of i 0 and q 0 , different initial positions of the rotor are attracted to different stable singu-larities. Basins of attraction under different parameter groups are shown in Figure 27. The red and green areas are, respectively, attracted to singularity ðx 2,1 , 0Þ and ðx 2,2 , 0Þ.
According to Figure 27, singularity positions to which rotor is eventually attracted depend not only on initial velocity and initial displacement but also on coil current i 0 and flow of bearing cavity q 0 . When i 0 is small, the attraction of singularities will change greatly with i 0 change slightly. When q 0 is large, the attraction of singularities will change greatly with q 0 change slightly.  According to Figures 28 and 29, the phase trajectories of initial point (−1:5 × 10 −5 , -0.02) and (1:5 × 10 −5 , 0.02) both surround and gradually approach the stable focus (0, 0). Both of them reach balance after 0.12 s adjustment, and the rotor can be suspended stably. And the balance position is the expected center of rotation [20].
To sum up, the final balance position is affected not only by the initial phase point but also by oil film thickness h 0 and galvanized layer thickness l. And final balance position of the same initial phase point will be different under the combined influence of different h 0 and l. Therefore, it is necessary to simulate basins of attraction under different h 0 and l conditions.
With the change of h 0 and l, the characteristics of the basin of attraction will be changed as well. Basins of attraction under different parameters are shown in Figure 36.

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International Journal of Aerospace Engineering According to Figure 36, the basins of attraction are distributed symmetrically. The phase points in the red region will be eventually attracted to the nonzero singularity ðx 2,1 , 0Þ, while the green region is attracted to singularity ðx 2,2 , 0Þ [24]. Moreover, the final stable balance position of the rotor changes greatly with l change slightly, while the final equilibrium point is slightly affected by h 0 .

Conclusion
The paper presents the static bifurcation behavior of MLDSB affected by design parameters. The singularity characteristics, phase trajectory, x − t curves, and suction basin of the single DOF bearing system are analyzed to verify the accuracy of the theoretical calculation. The conclusions are as follows.
(1) Nonzero singularities exist and static bifurcation occurs when ε 2 > 0 or δ 2 > 0 (2) With flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness changes, in turn, the singularities will convert between stable focus, unstable focus, stable node, and saddle point (3) As current i 0 , flow q 0 , oil film thickness h 0 , and zinc layer thickness l change, the phase trajectory may form a stability limit cycle, and the system oscillates at constant amplitude, which makes it impossible to achieve stable suspension (4) The attractiveness of singularity will change greatly when the flow of bearing cavity and coil current change slightly in the case of small current or large flow (5) The minimal change of galvanized layer thickness will lead to the fundamental change of the final stable equilibrium point of the rotor, while the final equilibrium point is less affected by the oil film thickness

Data Availability
The [DATA TYPE] data used to support the findings of this study are included within the article.