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

Authors: Jianhua Zhao; Xiaochen Wu; Fang Han; Xuchao Ma; Weidong Yan; Yingna Liang; Dianrong Gao; Guojun Du
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

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 ε2>0 or δ2>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.


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## Body

## 1. Introduction

Hydrostatic bearing is introduced into electromagnetic bearing to form MLDSB, and its supporting form is electromagnetic suspension with hydrostatic pressure auxiliary. It has the advantages of nonmechanical contact, high supporting capacity and stiffness, and operation stability and then the service life can be efficiently improved. So it is suitable for deep-sea exploration, hydropower generation, and other domains, especially medium-low speed, heavy load, and frequent starting occasion [1].MLDSB is composed of a bracket, a motor, a coupling, a shaft, two journal bearings, an axial bearing, a motor, and a journal motor as shown in Figure1.Figure 1
Overall view of MLDSB.MLDSB can take full advantage of electromagnetic bearing and hydrostatic bearing. On the premise of not affecting the electromagnetic force, the hydrostatic force can be added into the MLDSB to realize real-time double supporting, and its bearing capacity and supporting stiffness are improved greatly [2].The radial element of MLDSB is composed of step shaft, magnetic sleeve, supporting chamber, magnetic pole, inlet pipe, shell, coil, outlet, and so on as shown in Figures2 and 3 [3].Figure 2
Full profile of radial bearing.Figure 3
Real products of radial bearing.The control principle of MLDSB is shown in Figure4. 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.Figure 4
Single DOF support regulation principle of MLDSB.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 amplitude-frequency 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 supporting system, and then the reference can be provided for the design and supporting stability of MLDSB.


## 2. 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 Figure5), the dynamic model can be established [11].(1)ẋ=y,ẏ=1m∑i=12−1i+1F1+F2,Figure 5
Force diagram of single DOF supporting system.whereF1=Fq,0y−1/1+l/h0+−1ix/h0cosθ3, F2=Fi,01+−1i+1ic/i0/1+−1ix/h0cosθ2.The design parameters of MLDSB are shown in Table1.Table 1
Design parameters of MLDSB.
SymbolsVariable namesSizesALength of supporting cavity100 mmBWidth of supporting cavity20 mmaWidth of sealing belt6 mmbWidth of sealing belt4 mmB¯Flow coefficient4.3611μDynamic viscosity1.3077×10−3Pa•smMass of rotor100 kgμ0Permeability of air4π×10−7H/mAeArea of supporting cavity1504 mm2AbExtrusion area1056 mm2AsArea of pole1000 mm2θAngle22.5°NNumber of turns60A1Area of pole1000 mm2i0Initial current0.5 ~ 2.0AqFlow of supporting cavity0~2×10−4m3/sh0Oil film thickness10~90μmlGalvanized layer thickness0 ~90μmKpProportionality coefficient-70KdDifferential coefficient0.03The design and operation parameters include flowq0 of bearing cavity, coil current i0, oil film thickness h0, and galvanized layer thickness l.


### 2.1. Bifurcation Range of Parameter Group (i0, q0)

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 practical meaning of parameters, the bifurcation range of the system parameters is obtained. Assume that h0=30μm, l=0μm. According to the definition of singularity, the existence condition of singularity is ẋ=0 and ẏ=0 [12], then(2)y=0,−1m∑i=01−1iF1+F2=0.The design parameters in Table1 were substituted into equation (2), and the coordinates of the singularities were obtained as follows:(3)x0,y0=0,0,x1,1,y1=ε1,0,x1,2,y1=−ε1,0,x2,1,y2=ε2,0,x2,,2,y2=−ε2,0,where,εi=5.2697×10−28/184i0−63α+−1i−1β,i=1,2(4)α=3.6052×1024q0+5.4219×1020i02−6.3562×1019,β=2.9397×1041i04−4.0261×1041i03+3.9094×1045i02q0+2.0677×1041i02+8.0312×1045i0q0−4.7198×1040i0−3.2081×1045q0+4.0401×1039+1.2997×1049q02.From equation (3), it can be seen that there is a zero singularity (0, 0) and four nonzero singularities (x1,1, 0), (x1,2, 0), (x2,1, 0), and (x2,2, 0) in MLDSB system.The relationships amongx1,1, q0, and i0 are shown in Figure 6, and x1,1=−x1,2. The displacement x1,1 and x1,2 are beyond the actual thickness range of the oil film, so the singularities x1,1,0andx1,2,0 can be ignored without practical meaning [13].Figure 6
Relationships amongx1,1, i0, and q0.According to equation (3), static bifurcation occurs and singularity x2,1,0 and x2,2,0 exists when ε2>0, and then the range of parameter combinations is shown in Figure 7.Figure 7
Bifurcation area diagram of electric current and flow.


### 2.2. Bifurcation Range of Parameter Grouph0,l

Similarly, assume thati0=1.2A, q0=5.4370×10−8m3/s. The parameters in Table 1 are substituted into equation (2) to obtain the singularities as follows.(5)x0,y0=0,0,x1,1,y1=δ1,0,x1,2,y1=−δ1,0,x2,1,y2=δ2,0,x2,2,y2=−δ2,0,whereδi=−0.5h02h0+l2/γ2α7+−1i+11.33×1036β3,i=1,2(6)α7=4.67×1027h0l4+4.67×1027h04l+9.34×1026h05+9.34×1026h02l3+9.34×1026h03l2,β2=−9.01×10−17h07l3−1.56×10−16h06l4−1.89×10−16h05l5−1.58×10−16h04l6−9.01×10−17h03l7,γ2=1.16×1028h06l+3.49×1028h05l2+5.81×1028h04l3+5.81×1028h03l4+3.49×1028h02l5+1.16×1028h0l6.Similarly, from equation (5), it can be seen that there are only three meaningful singularities 0,0, x2,1,0, and x2,2,0, static bifurcation occurs and singularity x2,1,0 and x2,2,0 exists when ε2>0, and then the range of parameter combinations is shown in Figure 8.Figure 8
Bifurcation scope diagram.


## 2.1. Bifurcation Range of Parameter Group (i0, q0)

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 practical meaning of parameters, the bifurcation range of the system parameters is obtained. Assume that h0=30μm, l=0μm. According to the definition of singularity, the existence condition of singularity is ẋ=0 and ẏ=0 [12], then(2)y=0,−1m∑i=01−1iF1+F2=0.The design parameters in Table1 were substituted into equation (2), and the coordinates of the singularities were obtained as follows:(3)x0,y0=0,0,x1,1,y1=ε1,0,x1,2,y1=−ε1,0,x2,1,y2=ε2,0,x2,,2,y2=−ε2,0,where,εi=5.2697×10−28/184i0−63α+−1i−1β,i=1,2(4)α=3.6052×1024q0+5.4219×1020i02−6.3562×1019,β=2.9397×1041i04−4.0261×1041i03+3.9094×1045i02q0+2.0677×1041i02+8.0312×1045i0q0−4.7198×1040i0−3.2081×1045q0+4.0401×1039+1.2997×1049q02.From equation (3), it can be seen that there is a zero singularity (0, 0) and four nonzero singularities (x1,1, 0), (x1,2, 0), (x2,1, 0), and (x2,2, 0) in MLDSB system.The relationships amongx1,1, q0, and i0 are shown in Figure 6, and x1,1=−x1,2. The displacement x1,1 and x1,2 are beyond the actual thickness range of the oil film, so the singularities x1,1,0andx1,2,0 can be ignored without practical meaning [13].Figure 6
Relationships amongx1,1, i0, and q0.According to equation (3), static bifurcation occurs and singularity x2,1,0 and x2,2,0 exists when ε2>0, and then the range of parameter combinations is shown in Figure 7.Figure 7
Bifurcation area diagram of electric current and flow.


## 2.2. Bifurcation Range of Parameter Grouph0,l

Similarly, assume thati0=1.2A, q0=5.4370×10−8m3/s. The parameters in Table 1 are substituted into equation (2) to obtain the singularities as follows.(5)x0,y0=0,0,x1,1,y1=δ1,0,x1,2,y1=−δ1,0,x2,1,y2=δ2,0,x2,2,y2=−δ2,0,whereδi=−0.5h02h0+l2/γ2α7+−1i+11.33×1036β3,i=1,2(6)α7=4.67×1027h0l4+4.67×1027h04l+9.34×1026h05+9.34×1026h02l3+9.34×1026h03l2,β2=−9.01×10−17h07l3−1.56×10−16h06l4−1.89×10−16h05l5−1.58×10−16h04l6−9.01×10−17h03l7,γ2=1.16×1028h06l+3.49×1028h05l2+5.81×1028h04l3+5.81×1028h03l4+3.49×1028h02l5+1.16×1028h0l6.Similarly, from equation (5), it can be seen that there are only three meaningful singularities 0,0, x2,1,0, and x2,2,0, static bifurcation occurs and singularity x2,1,0 and x2,2,0 exists when ε2>0, and then the range of parameter combinations is shown in Figure 8.Figure 8
Bifurcation scope diagram.


## 3. Singularity Characteristics of Single DOF Supporting System

### 3.1. Singular Points Characteristic under Parameter Groupsi0,q0

Equation (1) was rewritten in the following form.(7)Px,y=y,Qx,y=−1m∑i=01−1iF1+F2.Jacobian matrixA of x and y can be obtained from equation (7).(8)A=∂P∂x∂P∂y∂Q∂x∂Q∂y=01∂Q∂x∂Q∂y.The characteristic equation of matrixA [14] can be expanded as follows:(9)A−λE=λ2−pλ+q=0,wherep=∂Q/∂y, q=−∂Q/∂x.


#### 3.1.1. Characteristics of Zero Singularity

The singularity0,0 was substituted into equation (9) to obtain as follows.(10)p=415.99i0−597.17,q=−2.8349×106i02+970644i0+5.655×1010q0,Δ=p2−4q.According to equation (10), the changing curves of p, q, and △ can be obtained as shown in Figure 9 (q coincides with △ to form a curve).Figure 9
Positive and negative boundary ofp, q, and △.According to Figure9, the coordinate system is divided into four regions:(1)
In area 1,p<0, q<0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


#### 3.1.2. Characteristics of Nonzero Singularities

Similarly, singularityx2,0 was substituted into equation (9) to obtain curves of p, q, and △ as shown in Figure 10.Figure 10
Positive and negative boundary ofp, q, and △.According to Figure10, 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).(1)
In area 1,p>0, q>0, △<0, and the singularities are unstable focus(2)
In area 2,p<0, q>0, △<0, and the singularities are stable focus(3)
In area 3,p<0, q>0, △>0, and the singularities are stable focus(4)
In area 4, nonzero singularities are not existing(5)
In area 5, nonzero singularities are not existing


### 3.2. Singularity Characteristics under Parameter Grouph0,l

Similarly, the singularity characteristics under the parameter grouph0,l are analyzed.


#### 3.2.1. Characteristics of Zero Singularity

Singularity0,0 was substituted into equation (9) to obtain p, q, and △ as follows:(11)p=4.49×10−7h0+l2−1.61×10−11h03,q=5.99×10−4h0+l2−1.10×10−7h0+l3+2.49×10−12h04,Δ=p2−4q.According to equation (11), the curves of p, q, and △ can be obtained as shown as Figure 11 (q coincides with △ to form a curve).Figure 11
Positive and negative boundary ofp, q, and △.According to Figure11, the coordinate system is divided into four regions:(1)
In area 1,p<0, q>0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


#### 3.2.2. Characteristics of Nonzero Singularities

The singularityx2,0 was substituted into equation (9) to obtain the curves of p, q, and △ as shown in Figure 12.Figure 12
Positive and negative boundary ofp, q, and △.As can be seen from Figure12, the coordinate system is divided into four areas:(1)
In area 1, the singularities are not existing(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p<0, q>0, △<0, and the singularities are stable focus(4)
In area 4, the singularities are not existing


## 3.1. Singular Points Characteristic under Parameter Groupsi0,q0

Equation (1) was rewritten in the following form.(7)Px,y=y,Qx,y=−1m∑i=01−1iF1+F2.Jacobian matrixA of x and y can be obtained from equation (7).(8)A=∂P∂x∂P∂y∂Q∂x∂Q∂y=01∂Q∂x∂Q∂y.The characteristic equation of matrixA [14] can be expanded as follows:(9)A−λE=λ2−pλ+q=0,wherep=∂Q/∂y, q=−∂Q/∂x.


### 3.1.1. Characteristics of Zero Singularity

The singularity0,0 was substituted into equation (9) to obtain as follows.(10)p=415.99i0−597.17,q=−2.8349×106i02+970644i0+5.655×1010q0,Δ=p2−4q.According to equation (10), the changing curves of p, q, and △ can be obtained as shown in Figure 9 (q coincides with △ to form a curve).Figure 9
Positive and negative boundary ofp, q, and △.According to Figure9, the coordinate system is divided into four regions:(1)
In area 1,p<0, q<0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


### 3.1.2. Characteristics of Nonzero Singularities

Similarly, singularityx2,0 was substituted into equation (9) to obtain curves of p, q, and △ as shown in Figure 10.Figure 10
Positive and negative boundary ofp, q, and △.According to Figure10, 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).(1)
In area 1,p>0, q>0, △<0, and the singularities are unstable focus(2)
In area 2,p<0, q>0, △<0, and the singularities are stable focus(3)
In area 3,p<0, q>0, △>0, and the singularities are stable focus(4)
In area 4, nonzero singularities are not existing(5)
In area 5, nonzero singularities are not existing


## 3.1.1. Characteristics of Zero Singularity

The singularity0,0 was substituted into equation (9) to obtain as follows.(10)p=415.99i0−597.17,q=−2.8349×106i02+970644i0+5.655×1010q0,Δ=p2−4q.According to equation (10), the changing curves of p, q, and △ can be obtained as shown in Figure 9 (q coincides with △ to form a curve).Figure 9
Positive and negative boundary ofp, q, and △.According to Figure9, the coordinate system is divided into four regions:(1)
In area 1,p<0, q<0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


## 3.1.2. Characteristics of Nonzero Singularities

Similarly, singularityx2,0 was substituted into equation (9) to obtain curves of p, q, and △ as shown in Figure 10.Figure 10
Positive and negative boundary ofp, q, and △.According to Figure10, 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).(1)
In area 1,p>0, q>0, △<0, and the singularities are unstable focus(2)
In area 2,p<0, q>0, △<0, and the singularities are stable focus(3)
In area 3,p<0, q>0, △>0, and the singularities are stable focus(4)
In area 4, nonzero singularities are not existing(5)
In area 5, nonzero singularities are not existing


## 3.2. Singularity Characteristics under Parameter Grouph0,l

Similarly, the singularity characteristics under the parameter grouph0,l are analyzed.


### 3.2.1. Characteristics of Zero Singularity

Singularity0,0 was substituted into equation (9) to obtain p, q, and △ as follows:(11)p=4.49×10−7h0+l2−1.61×10−11h03,q=5.99×10−4h0+l2−1.10×10−7h0+l3+2.49×10−12h04,Δ=p2−4q.According to equation (11), the curves of p, q, and △ can be obtained as shown as Figure 11 (q coincides with △ to form a curve).Figure 11
Positive and negative boundary ofp, q, and △.According to Figure11, the coordinate system is divided into four regions:(1)
In area 1,p<0, q>0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


### 3.2.2. Characteristics of Nonzero Singularities

The singularityx2,0 was substituted into equation (9) to obtain the curves of p, q, and △ as shown in Figure 12.Figure 12
Positive and negative boundary ofp, q, and △.As can be seen from Figure12, the coordinate system is divided into four areas:(1)
In area 1, the singularities are not existing(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p<0, q>0, △<0, and the singularities are stable focus(4)
In area 4, the singularities are not existing


## 3.2.1. Characteristics of Zero Singularity

Singularity0,0 was substituted into equation (9) to obtain p, q, and △ as follows:(11)p=4.49×10−7h0+l2−1.61×10−11h03,q=5.99×10−4h0+l2−1.10×10−7h0+l3+2.49×10−12h04,Δ=p2−4q.According to equation (11), the curves of p, q, and △ can be obtained as shown as Figure 11 (q coincides with △ to form a curve).Figure 11
Positive and negative boundary ofp, q, and △.According to Figure11, the coordinate system is divided into four regions:(1)
In area 1,p<0, q>0, △<0, and the singularities are stable focus(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p>0, q<0, △>0, and the singularities are saddle points(4)
In area 4,p<0, q<0, △>0, and the singularities are saddle points


## 3.2.2. Characteristics of Nonzero Singularities

The singularityx2,0 was substituted into equation (9) to obtain the curves of p, q, and △ as shown in Figure 12.Figure 12
Positive and negative boundary ofp, q, and △.As can be seen from Figure12, the coordinate system is divided into four areas:(1)
In area 1, the singularities are not existing(2)
In area 2,p>0, q>0, △<0, and the singularities are unstable focus(3)
In area 3,p<0, q>0, △<0, and the singularities are stable focus(4)
In area 4, the singularities are not existing


## 4. Phase Trajectories and Attractivity of Single DOF Supporting System

### 4.1. Phase Trajectories and Attractivity under Parameter Groupsi0,q0

(1)
Assume thati0=1.0A, q0=0.3×10−4m3/s, phase trajectories, and x−t curves can be obtained by fourth-order Runge-Kutta method [15] as shown in Figures 13 and 14Figure 13
Phase trajectories underi0=1.0A, q0=0.3×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 14
x−t curves under i0=1.0A, q0=0.3×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)The phase trajectories of initial point (1.5×10−5, -0.02) and (1.5×10−5, 0.02), respectively, surround and approach stable focuses (0.76×10−5, 0) and (−0.76×10−5, 0). Both of them reach balance after 0.03 s adjustment, and the rotor can be suspended stably as shown in Figures 13 and 14. However, the balance position is not the expected center of rotation [16].(2)
Assume thati0=1.0A, q0=1×10−4m3/s, and the phase trajectories and x−t curves [17] are shown in Figures 15 and 16Figure 15
Phase trajectories underi0=1.0A, q0=1×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 16
x−t curves under i0=1.0A, q0=1×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)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 thati0=1.6A, q0=0.25×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 17 and 18Figure 17
Phase trajectories underi0=1.6A, q0=0.25×10−4m3/s.
(a)
(x0=−1.5×10−5, v0=0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 18
x−t curves under i0=1.6A, q0=0.25×10−4m3/s.
(a)
(x0=−1.5×10−5, v0=0.02)(b)
(x0=1.5×10−5, v0=0.02)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.(4)
Assume thati0=1.8A, q0=1.2×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 19 and 20Figure 19
Phase trajectories underi0=1.8A, q0=1.2×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 20
x−t curves under i0=1.8A, q0=1.2×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)According to Figures19 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 andx−t curves as shown in Figures 21 and 22.Figure 21
Phase trajectories underi0=1.6A, q0=1.0×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.005)(b)
(x0=0.5×10−5, v0=0.005)Figure 22
x−t curves under i0=1.6A, q0=1.0×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.005)(b)
(x0=0.5×10−5, v0=0.005)According to Figures21 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].(5)
Assume thati0=1.8A, q0=1.6×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 23 and 24Figure 23
Phase trajectories underi0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 24
x−t curves under i0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)According to Figures23 and 24, the rotor oscillates in equal amplitude, and phase trajectories form a limit cycle [19].In order to verify the stability of the limit cycle, the initial point inside the limit cycle was selected to simulate the phase trajectories andx−t curves as shown in Figures 25 and 26.Figure 25
Phase trajectories underi0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.02)(b)
(x0=0.5×10−5, v0=0.02)Figure 26
x−t curves under i0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.02)(b)
(x0=0.5×10−5, v0=0.02)According to Figures25 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 ofi0 and q0, different initial positions of the rotor are attracted to different stable singularities. Basins of attraction under different parameter groups are shown in Figure 27. The red and green areas are, respectively, attracted to singularity x2,1,0 and x2,2,0.Figure 27
Basins of attraction under different parameters.
(a)
(1.2 A,0.4×10−4m3/s)(b)
(1.5 A,0.4×10−4m3/s)(c)
(1.8 A,0.4×10−4m3/s)(d)
(1.5 A,0.6×10−4m3/s)(e)
(1.8 A,0.6×10−4m3/s)(f)
(1.8 A,0.8×10−4m3/s)According to Figure27, singularity positions to which rotor is eventually attracted depend not only on initial velocity and initial displacement but also on coil current i0 and flow of bearing cavity q0. When i0 is small, the attraction of singularities will change greatly with i0 change slightly. When q0 is large, the attraction of singularities will change greatly with q0 change slightly.


### 4.2. Phase Trajectories and Attractively under Parameter Grouph0,l

(1)
Assume thath0=5.0×10−5m, l=5.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 28 and 29Figure 28
Phase trajectories underh0=5.0×10−5m, l=5.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)Figure 29
x−t curves under h0=5.0×10−5m, l=5.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)According to Figures28 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].(2)
Assume thath0=8.0×10−5m, l=2.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 30 and 31Figure 30
Phase trajectories underh0=8.0×10−5m, l=2.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)Figure 31
x−t curves under h0=8.0×10−5m, l=2.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)According to Figures30 and 31, the phase trajectories of initial point (−1.5×10−5, -0.02) and (1.5×10−5, 0.004), respectively, surround and gradually approach different limit cycles after multiple periodic adjustments. There is no stable singularity. The rotor cannot be suspended stably, and Hopf bifurcation may occur in the system [21].(3)
Assume thath0=8.0×10−5m, l=1.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 32 and 33Figure 32
Phase trajectories underh0=8.0×10−5m, l=1.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)Figure 33
x−t curves under h0=8.0×10−5m, l=1.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)According to Figures32 and 33, the phase trajectories of initial point (−1.5×10−5, -0.02) and (1.5×10−5, 0.004), respectively, surround and gradually approach different limit cycles after multiple periodic adjustments. There is no stable singularity. The rotor cannot be suspended stably, and Hopf bifurcation may occur in the system [22].(4)
Assume thath0=8.0×10−5m and l=1.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 34 and 35Figure 34
Phase trajectories underh0=6.0×10−5m, l=0.5×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)Figure 35
x−t curves under h0=6.0×10−5m, l=0.5×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)According to Figures34 and 35, the phase trajectory of initial phase point (−1.50×10−5, -0.02) gradually approaches stable focus (2.49×10−5, 0), while initial phase point (1.50×10−5, 0.02) gradually approaches stable focus (−2.49×10−5, 0). Both of them reach balance after 0.4 s adjustment, and the rotor can be suspended stably. However, the balance position is not the expected center of rotation [23].To sum up, the final balance position is affected not only by the initial phase point but also by oil film thicknessh0 and galvanized layer thickness l. And final balance position of the same initial phase point will be different under the combined influence of different h0 and l. Therefore, it is necessary to simulate basins of attraction under different h0 and l conditions.With the change ofh0 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.Figure 36
Basin of attraction under different parameters.
(a)
(5.0×10−5m, 2.0×10−6m)(b)
(5.0×10−5m, 4.0×10−6m)(c)
(6.0×10−5m, 2.0×10−6m)(d)
(6.0×10−5m, 4.0×10−6m)(e)
(7.0×10−5m, 2.0×10−6m)(f)
(7.0×10−5m, 4.0×10−6m)According to Figure36, the basins of attraction are distributed symmetrically. The phase points in the red region will be eventually attracted to the nonzero singularity x2,1,0, while the green region is attracted to singularity x2,2,0 [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 h0.


## 4.1. Phase Trajectories and Attractivity under Parameter Groupsi0,q0

(1)
Assume thati0=1.0A, q0=0.3×10−4m3/s, phase trajectories, and x−t curves can be obtained by fourth-order Runge-Kutta method [15] as shown in Figures 13 and 14Figure 13
Phase trajectories underi0=1.0A, q0=0.3×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 14
x−t curves under i0=1.0A, q0=0.3×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)The phase trajectories of initial point (1.5×10−5, -0.02) and (1.5×10−5, 0.02), respectively, surround and approach stable focuses (0.76×10−5, 0) and (−0.76×10−5, 0). Both of them reach balance after 0.03 s adjustment, and the rotor can be suspended stably as shown in Figures 13 and 14. However, the balance position is not the expected center of rotation [16].(2)
Assume thati0=1.0A, q0=1×10−4m3/s, and the phase trajectories and x−t curves [17] are shown in Figures 15 and 16Figure 15
Phase trajectories underi0=1.0A, q0=1×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 16
x−t curves under i0=1.0A, q0=1×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)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 thati0=1.6A, q0=0.25×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 17 and 18Figure 17
Phase trajectories underi0=1.6A, q0=0.25×10−4m3/s.
(a)
(x0=−1.5×10−5, v0=0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 18
x−t curves under i0=1.6A, q0=0.25×10−4m3/s.
(a)
(x0=−1.5×10−5, v0=0.02)(b)
(x0=1.5×10−5, v0=0.02)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.(4)
Assume thati0=1.8A, q0=1.2×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 19 and 20Figure 19
Phase trajectories underi0=1.8A, q0=1.2×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 20
x−t curves under i0=1.8A, q0=1.2×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)According to Figures19 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 andx−t curves as shown in Figures 21 and 22.Figure 21
Phase trajectories underi0=1.6A, q0=1.0×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.005)(b)
(x0=0.5×10−5, v0=0.005)Figure 22
x−t curves under i0=1.6A, q0=1.0×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.005)(b)
(x0=0.5×10−5, v0=0.005)According to Figures21 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].(5)
Assume thati0=1.8A, q0=1.6×10−4m3/s, and phase trajectories and x−t curves are shown in Figures 23 and 24Figure 23
Phase trajectories underi0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)Figure 24
x−t curves under i0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=1.5×10−5, v0=−0.02)(b)
(x0=1.5×10−5, v0=0.02)According to Figures23 and 24, the rotor oscillates in equal amplitude, and phase trajectories form a limit cycle [19].In order to verify the stability of the limit cycle, the initial point inside the limit cycle was selected to simulate the phase trajectories andx−t curves as shown in Figures 25 and 26.Figure 25
Phase trajectories underi0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.02)(b)
(x0=0.5×10−5, v0=0.02)Figure 26
x−t curves under i0=1.8A, q0=1.6×10−4m3/s.
(a)
(x0=0.5×10−5, v0=−0.02)(b)
(x0=0.5×10−5, v0=0.02)According to Figures25 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 ofi0 and q0, different initial positions of the rotor are attracted to different stable singularities. Basins of attraction under different parameter groups are shown in Figure 27. The red and green areas are, respectively, attracted to singularity x2,1,0 and x2,2,0.Figure 27
Basins of attraction under different parameters.
(a)
(1.2 A,0.4×10−4m3/s)(b)
(1.5 A,0.4×10−4m3/s)(c)
(1.8 A,0.4×10−4m3/s)(d)
(1.5 A,0.6×10−4m3/s)(e)
(1.8 A,0.6×10−4m3/s)(f)
(1.8 A,0.8×10−4m3/s)According to Figure27, singularity positions to which rotor is eventually attracted depend not only on initial velocity and initial displacement but also on coil current i0 and flow of bearing cavity q0. When i0 is small, the attraction of singularities will change greatly with i0 change slightly. When q0 is large, the attraction of singularities will change greatly with q0 change slightly.


## 4.2. Phase Trajectories and Attractively under Parameter Grouph0,l

(1)
Assume thath0=5.0×10−5m, l=5.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 28 and 29Figure 28
Phase trajectories underh0=5.0×10−5m, l=5.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)Figure 29
x−t curves under h0=5.0×10−5m, l=5.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)According to Figures28 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].(2)
Assume thath0=8.0×10−5m, l=2.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 30 and 31Figure 30
Phase trajectories underh0=8.0×10−5m, l=2.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)Figure 31
x−t curves under h0=8.0×10−5m, l=2.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)According to Figures30 and 31, the phase trajectories of initial point (−1.5×10−5, -0.02) and (1.5×10−5, 0.004), respectively, surround and gradually approach different limit cycles after multiple periodic adjustments. There is no stable singularity. The rotor cannot be suspended stably, and Hopf bifurcation may occur in the system [21].(3)
Assume thath0=8.0×10−5m, l=1.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 32 and 33Figure 32
Phase trajectories underh0=8.0×10−5m, l=1.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)Figure 33
x−t curves under h0=8.0×10−5m, l=1.0×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 4.00×10−3)According to Figures32 and 33, the phase trajectories of initial point (−1.5×10−5, -0.02) and (1.5×10−5, 0.004), respectively, surround and gradually approach different limit cycles after multiple periodic adjustments. There is no stable singularity. The rotor cannot be suspended stably, and Hopf bifurcation may occur in the system [22].(4)
Assume thath0=8.0×10−5m and l=1.0×10−5m, and the phase trajectories and x−t curves are shown in Figures 34 and 35Figure 34
Phase trajectories underh0=6.0×10−5m, l=0.5×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)Figure 35
x−t curves under h0=6.0×10−5m, l=0.5×10−5m.
(a)
(−1.50×10−5, -0.02)(b)
(1.50×10−5, 0.02)According to Figures34 and 35, the phase trajectory of initial phase point (−1.50×10−5, -0.02) gradually approaches stable focus (2.49×10−5, 0), while initial phase point (1.50×10−5, 0.02) gradually approaches stable focus (−2.49×10−5, 0). Both of them reach balance after 0.4 s adjustment, and the rotor can be suspended stably. However, the balance position is not the expected center of rotation [23].To sum up, the final balance position is affected not only by the initial phase point but also by oil film thicknessh0 and galvanized layer thickness l. And final balance position of the same initial phase point will be different under the combined influence of different h0 and l. Therefore, it is necessary to simulate basins of attraction under different h0 and l conditions.With the change ofh0 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.Figure 36
Basin of attraction under different parameters.
(a)
(5.0×10−5m, 2.0×10−6m)(b)
(5.0×10−5m, 4.0×10−6m)(c)
(6.0×10−5m, 2.0×10−6m)(d)
(6.0×10−5m, 4.0×10−6m)(e)
(7.0×10−5m, 2.0×10−6m)(f)
(7.0×10−5m, 4.0×10−6m)According to Figure36, the basins of attraction are distributed symmetrically. The phase points in the red region will be eventually attracted to the nonzero singularity x2,1,0, while the green region is attracted to singularity x2,2,0 [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 h0.


## 5. 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 whenε2>0 or δ2>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 currenti0, flow q0, oil film thickness h0, 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



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*Source: 6646235-2021-03-26.xml*