Study on Clearance-Rubbing Dynamic Behavior of 2-DOF Supporting System of Magnetic-Liquid Double Suspension Bearing

Abstract

As a new type of suspension bearing, Magnetic-Liquid Double Suspension Bearing (MLDSB) is mainly supported by electromagnetic suspension and supplemented by hydrostatic supporting. Its bearing capacity and stiffness can be greatly improved. Because of the small liquid film thickness (it is smaller 10 times than air gap), the eccentricity, crack, bending of the rotor, and the assembly error, it is easy to cause a clearance-rubbing fault between the rotor and stator. The coating can be worn and peeled, the operating stability can be reduced, and then it is one of the key problems of restricting the development and application of MLDSB. Therefore, the clearance-rubbing dynamic equation of 2-DOF system of MLDSB is established and converted into Taylor Series form and the nonlinear components are retained. Dimensionless treatment is carried out by dimensional normalization method. Finally, the rotor displacement response under different rotor eccentricity ratio and rotating speeds is numerically simulated. The studies show that the trajectory of the rotor is periodic elliptic without clearance-rubbing phenomenon when the eccentricity ratio is less than 0.2, while the rotor is greatly affected by the rotation speed and a variety of motions, such as single-period, quasi-period, double-period and chaos, are presented when greater than 0.3. Within the largest range of rotating speed and eccentricity ratio, the rotor presents the single-period trajectory, and then the number of Poincare mapping point is 1, without a clearance-rubbing fault. When the rotational speed is in the scope of (9, 13) krpm and the eccentricity ratio is in the scope of (0.27, 0.4), the number of Poincare mapping point is more than one, the maximum dimensionless rubbing force is −5.7, and then clearance-rubbing fault occurs. The research can provide a theoretical basis for the safe and stable operation of MLDSB.

1. Introduction

The active electromagnetic bearing (AMB) has many defects, such as the insufficient electromagnetic attraction caused by the magnetic pole magnetic saturation, the higher temperature rise in the magnetic pole/coil caused by the copper loss and the eddy current loss [1,2] (the faults of AMB are shown as Figure 1), the bearing characteristics of operation stability of AMB can be limited, and then it becomes “technical bottleneck”, which restricts the further development and application promotion of AMB [3,4].

The hydrostatic bearing concept is introduced into AMB to form the novel suspension bearing-Magnetic-Liquid Double Suspension Bearing (MLDSB) [5]. It is supported by the electromagnetic suspension and supplemented by the hydrostatic supporting, and it has the advantages of an electromagnetic system and hydrostatic system [6]. The bearing capacity and stiffness can be improved drastically [7].

MLDSB is composed of a bracket, motor, coupling, multi-diameter shaft, journal bearing unit, axial bearing unit, journal loading motor, axial loading motor, etc., as in Figure 2 [8].

The journal bearing unit is composed of a rotor, magnetic sleeve, bearing pocket, magnetic pole, inlet hole, shell, coil, etc., as in Figure 3 and Figure 4 [9]. The materials of magnetic pole and magnetic sleeve are cold-rolled non-oriented silicon-steel, and chromium layer is coated in order to solve the problem of matrix corrosion in the lubricants for a long time [10].

The adapting principle of MLDSB is shown as Figure 5. The magnetic suspension system adopts a PD control mode, and the hydrostatic system adopts a constant-pressure supply system [11].

With the development of a bearing-rotor system towards being large-scale and high-power, the design gap between the rotor and stator becomes smaller and smaller, which makes the liquid film thickness of MLDSB reduce by 10 times compared with the traditional electromagnetic air gap [12]. (Air gap = Liquid film + Chromium coating, Liquid film is (30, 50) μm, Air gap is (100, 300) μm).

Moreover, the clearance-rubbing fault between magnetic pole and magnetic sleeve can be caused easily by the eccentricity, crack, bending, improper assembly of rotor; the coating cracks and falls off, and then the operation stability and reliability of MLDSB can be reduced significantly [13].

At present, many scholars have researched the clearance-rubbing fault of a Bearing-Rotor system in detail, and achieved fruitful results. Braut et al. [14] proposed the method and experiment of using Variational Mode Decomposition (VMD) to diagnose partial rotor-stator rubbing. The results showed that the VMD method can effectively analyze the partial rubbing phenomenon in a constant and slightly variable speed operation. Bularevich et al. [15] analyzed regularities of the possible separation of additives and products of tribochemical reactions under sliding friction. The results showed that nonregular oscillations of the force of friction in the tribosystems under study are interpreted as the result of the rupture-healing of adsorbed antifriction layers. Kostyuk et al. [16] studied the rubbing between the absolute rigid rotor and the absolute rigid stator. The results showed that the system model has a simple analytic solution, which deepened the understanding of rubbing phenomenon of readers. Zhang et al. [17] researched nonlinear dynamics of a 5-DOF active magnetic bearings-rotor system and analyzed the rotor motion stability under different conditions. The results showed that there is the phenomenon of stable periodic motion, critical quasi-periodic motion and instable chaotic motion when the system is under different parameter conditions. Zhu et al. [18] analyzed the support damping and radial clearance of the flexible backup bearing on the transient behavior of the rotor. The results showed that the rolling element backup bearing with flexiblity and more damping can greatly improve the dynamic behavior of a rotor drop on rolling element backup bearing.

In sum, the traditional rubbing dynamics research method takes Bearing Body as the rigid supporting unit to analyze dynamic characteristics of a Bearing-Rotor system. MLDSB bears the severe Clearance-Rubbing compound impact load and becomes the weakest link for Bearing-Rotor system [19].

The Clearance-Rubbing fault is the external representation, and dynamic behavior and law is the internal mechanism of MLDSB. Therefore, the 2-DOF clearance-rubbing dynamics equation was established and transformed into Taylor series form with the nonlinear component. Dimensionless processing was adopted through the use of a Dimensional Unitary Method [20]. The displacement response of a rotor is numerical, simulated under different eccentricity ratios and speeds [21].

2. Rubbing Dynamic Model of 2-DOF Supporting System

2.1. Rubbing Force Model

Due to the small Clearance-Rubbing time, the assumptions can be shown as follows [22]:

(1)

There is local Elastic Collision and Coulomb Friction between magnetic pole and magnetic sleeve;

(2)

The small gap between two magnetic poles can be ignored;

(3)

The frictional thermal effect between magnetic pole and magnetic sleeve can be ignored;

(4)

The winding leakage, marginal magnetic flux, vortex loss, magnetic material saturation, coupling effect between magnetic poles can be ignored;

(5)

The inertial force and viscous pressure characteristics of the liquid are ignored.

Clearance-Rubbing phenomenon of MLDSB can be shown as Figure 6. o, o₁, P_T, P_N, α, w_s, e are, respectively, center of stator, center of rotor, tangential rubbing force, radial rubbing force, angle between rubbing point and x axis, rotation angle, radial displacement [23].

The mathematical expression of rubbing force P_N and P_T can be shown as follows [24]

$$\{\begin{array}{l}{P}_N=\left(e-h_0\right)k_c\\ P_T=f{P}_N\end{array}\begin{array}{c}\end{array}e=\sqrt{x^2+y^2}\ge h_0$$

(1)

where h₀ is liquid film thickness, k_c is radial stiffness of stator, f is friction coefficient.

Decompose it in the x and y axis to

$$\left\{\begin{array}{l}{f}_x\\ f_y\end{array}\right\}=-\frac{\left(e-h_0\right)k_c}{e}\left[\begin{array}{cc}1& -f\\ f& 1\end{array}\right]\left\{\begin{array}{l}x\\ y\end{array}\right\}$$

(2)

where f_x is rubbing force of x axis, f_y is rubbing force of y axis.

Equation (2) can be transformed into dimensionless form as follows [25]

$$\left\{\begin{array}{c}{f}_X\\ f_Y\end{array}\right\}=-\left(1-\frac{h_0}{e}\right)h_0{k}_c\left[\begin{array}{cc}1& -f\\ f& 1\end{array}\right]\left\{\begin{array}{c}X\\ Y\end{array}\right\}$$

(3)

where f_X is dimensionless rubbing force of x axis, f_Y is dimensionless rubbing force of y axis, X is dimensionless displacement of x axis, Y is dimensionless displacement of y axis.

2.2. Dynamic Equation of 2-DOF Supporting System

Due to the same supporting structure of x axis and y axis, the force diagram of a y-direction single DOF supporting system can be shown as Figure 7 [26].

The dynamic model of supporting system can be shown as follows [27]

$$\{\begin{array}{l}m\ddot{x}+\Delta F_x+\Delta P_x=f_x+u_x\\ m\ddot{y}+\Delta F_y+\Delta P_y=f_y+u_y\end{array}$$

(4)

where m is mass of rotor, ΔF_x and ΔF_y are magnetic force of x and y axis, ΔP_x and ΔP_y are hydrostatic force of x and y axis, u_x and u_y are the disturbing force of x and y axis.

(1) ΔF_x, ΔF_y can be transformed into an increment form as follows [28]

$$\{\begin{array}{c}\Delta F_x=\frac{{\mu }_{0}S{N}^2{i}_0^2\cos \theta }{2{\delta }_0^2}\left[{\left(\frac{1+\frac{F_{x0}{\delta }_0^2}{8k{i}_0^2}+\frac{P_x}{i_0}x+\frac{D_x}{i_0}\dot{x}}{1+\frac{x}{{\delta }_0}\cos \theta }\right)}^2-{\left(\frac{1-\frac{F_{x0}{\delta }_0^2}{8k{i}_0^2}-\frac{P_x}{i_0}x-\frac{D_x}{i_0}\dot{x}}{1-\frac{x}{{\delta }_0}\cos \theta }\right)}^2\right]\\ \Delta F_y=\frac{{\mu }_{0}S{N}^2{i}_0^2\cos \theta }{2{\delta }_0^2}\left[{\left(\frac{1+\frac{F_{y0}{\delta }_0^2}{8k{i}_0^2}+\frac{P_y}{i_0}y+\frac{D_y}{i_0}\dot{y}}{1+\frac{y}{{\delta }_0}\cos \theta }\right)}^2-{\left(\frac{1-\frac{F_{y0}{\delta }_0^2}{8k{i}_0^2}-\frac{P_y}{i_0}y-\frac{D_y}{i_0}\dot{y}}{1-\frac{y}{{\delta }_0}\cos \theta }\right)}^2\right]\end{array}$$

(5)

where i₀ is bias current, k is electromagnetic coefficient, k = μ₀SN²/4, θ is angle between electromagnetic force and y axis, δ₀ is air gap, μ₀ is air permeability, S is area of magnetic pole, N is number of coil, F_x0, F_y0 are external load of x, y axis, P_x, D_x are proportionality and differential coefficient of x axis, P_y, D_y are proportionality and differential coefficients of y axis;

(2) ΔP_x, ΔP_y can be transformed into increment form as follows [29],

$$\{\begin{array}{l}\Delta P_x=A\left(\frac{1}{1+\left(\beta -1\right){\left(1-\tilde{x}\right)}^3}-\frac{1}{1+\left(\beta -1\right){\left(1+\tilde{x}\right)}^3}\right)+2A_b{R}_{h_0}{A}_e{\cos }^2\theta \left(\frac{1}{{\left(1+\tilde{x}\right)}^3}+\frac{1}{{\left(1-\tilde{x}\right)}^3}\right)\dot{x}\\ \Delta P_y=A\left(\frac{1}{1+\left(\beta -1\right){\left(1-\tilde{y}\right)}^3}-\frac{1}{1+\left(\beta -1\right){\left(1+\tilde{y}\right)}^3}\right)+2A_b{R}_{h_0}{A}_e{\cos }^2\theta \left(\frac{1}{{\left(1+\tilde{y}\right)}^3}+\frac{1}{{\left(1-\tilde{y}\right)}^3}\right)\dot{y}\end{array}$$

(6)

where A is hydrostatic supporting coefficient, A = 2p_sA_ecosθ, p_s is system pressure, A_e, A_b are supporting area and extrusion area, R_h0 is initial hydraulic resistor, β is throttle ratio, $\tilde{x}=x\cos \theta /h_0$, $\tilde{y}=y\cos \theta /h_0$;

(3) Disturbing force can be shown as follows [30]

$$\{\begin{array}{l}{u}_x=m{e}_x{w}_s^2\cos \left(w_{s}t\right)+m{e}_y{w}_s^2\sin \left(w_{s}t\right)\\ u_y=-m{e}_x{w}_s^2\sin \left(w_{s}t\right)+m{e}_y{w}_s^2\cos \left(w_{s}t\right)\end{array}$$

(7)

where, e_x, e_y are eccentricity of x and y axis, w_s is rotation speed of rotor.

ΔF_x, ΔF_y, ΔP_x, ΔP_y can be transformed to Taylor Series with nonlinear part and substituted into Equation (4), and the dimensionless equation can be shown as follows [31,32]

$$\{\begin{array}{l}\ddot{X}=-2{\zeta }_x\dot{X}-{\beta }_{x}X-{\phi }_{x^2}{X}^2-{\phi }_{x\dot{x}}X\dot{X}+f_X+{\rho }_x\cos \left(w\tau \right)+{\rho }_y\sin \left(w\tau \right)\\ \ddot{Y}=-2{\zeta }_y\dot{Y}-{\beta }_{y}Y-{\phi }_{y^2}{Y}^2-{\phi }_{y\dot{y}}Y\dot{Y}+f_Y-{\rho }_x\sin \left(w\tau \right)+{\rho }_y\cos \left(w\tau \right)\end{array}$$

(8)

$$\{\begin{array}{l}{\varsigma }_x=\frac{1}{m{w}_n}\left(\frac{4D_x{i}_{0}k}{{\delta }_0^2}+2A_b{A}_e{R}_{h_0}{\cos }^2\theta \right)\begin{array}{c},\end{array}{\zeta }_y=\frac{1}{m{w}_n}\left(\frac{4D_y{i}_{0}k}{{\delta }_0^2}+2A_b{A}_e{R}_{h_0}{\cos }^2\theta \right)\\ {\beta }_x=\frac{8k\cos \theta }{m{w}_n^2{\delta }_0^2}\left(P_x{i}_0-\frac{\left(i_0^2+i_{x_0}^2\right)\cos \theta }{{\delta }_0}\right)+\frac{12p_s{A}_e\left(\beta -1\right)\cos \theta }{m{w}_n^2{\beta }^2{h}_0}\\ {\beta }_y=\frac{8k\cos \theta }{m{w}_n^2{\delta }_0^2}\left(P_y{i}_0-\frac{\left(i_0^2+i_{y_0}^2\right)\cos \theta }{{\delta }_0}\right)+\frac{12p_s{A}_e\left(\beta -1\right)\cos \theta }{m{w}_n^2{\beta }^2{h}_0}\\ {\phi }_{x^2}={\phi }_{x\dot{x}}=0\begin{array}{c},\end{array}{\phi }_{y^2}=\frac{F_{y_0}{h}_0\cos \theta \left(-2P_y+\frac{3i_0\cos \theta }{L}\right)}{i_{0}Lm{w}_n^2}\begin{array}{c}\begin{array}{c},\end{array}\end{array}{\phi }_{y\dot{y}}=\frac{-2D_y{F}_{y_0}{h}_0\cos \theta }{i_{0}Lm{w}_n}\\ U_x=\frac{e_x}{h_0}\begin{array}{c},\end{array}{U}_y=\frac{e_y}{h_0}\begin{array}{c},\end{array}w=\frac{w_s}{w_n}\begin{array}{c},\end{array}{\rho }_x=U_x{w}^2\begin{array}{c},\end{array}{\rho }_y=U_y{w}^2\end{array}$$

(9)

where w_n is nature frequency, U_x, U_y are eccentricity ratio of x and y axis.

3. Simulation of Clearance-Rubbing of 2-DOF Supporting System

The initial design parameters of MLDSB can be shown as

Table 1. Design parameters of MLDSB.

Parameters	Value	Parameters	Value
Bias current i0	1 A	Film thickness h0	50 μm
System pressure ps	0.1 Mpa	Angle θ	22.5°
Throttle ratio β	2 Dimensionless	Flow coefficient $\overline{B}$	0.7 Dimensionless
Lubricants viscosity μ	1.3 × 10−3 pa·s	Area of magnetic pole S	1080 mm2
Mass of rotor m	4 kg	Number of coil N	50 Dimensionless
Air permeability μ0	4π × 10−7 H/m	Hydraulic resistor Rh0	1.5 × 10−10 N·s/m
Supporting area Ae	416 mm2	Friction coefficient f	0.2 Dimensionless
Extrusion area Ab	56 mm2	Stiffness of rotor kc	6 × 107 N/m
Bearing bore diameter	61 mm	Bearing length	71 mm

.

U_y = 0.1, the influence of design parameters (eccentricity ratio U_x and rotation speed n) on rotor displacement and rubbing force can be shown as follows. In Section 3, all the parameters in the figure are dimensionless except for the rotation speed.

3.1. Influence of U_x on Rotor Displacement

Rotor displacement under different eccentricity ratios, U_x, can be simulated with the fourth-order Runge–Kutta method, as in Figure 8 [33].

According to Figure 8a,b, the rotor keeps stable and its displacement is single trajectory when U_x < 0.2. The displacement has a positive value under the slow rotation speed. As rotation speed increases, the displacement of rotor is translated from a positive value into a negative value, and it reaches the maximum value when n = 9.5 krpm. The rotor displacement reduces to equilibrium state when rotation speed increases again.

According to Figure 8c,d, the rotor changes in the order of single-period, quasi-period, double-period, chaos and single-period when U_x ≥ 0.3 and n ∈ (4.5, 18). As U_x increases, the unstable bifurcation interval and double-period interval increase, as in

Table 2. Unstable Rotation Speed with Different Eccentricity Ratio.

Eccentricity Ratio Ux	Stable Scope 1	Unstable Scope 2	Stable Scope 3
0.30	(4.5, 9.5) krpm	(4.5, 9) krpm	(12.5, 18) krpm
0.40	(4.5, 9) krpm	(9, 13) krpm	(13, 18) krpm

.

U_x ∈ (0.2, 0.3) is the transitional interval from a stable single-period motion to an unstable motion, and then it can be analyzed in detail in Section 3.3 in order to determine the critical U_x.

3.2. Influence of Rotation Speed on Rotor Displacement

Take U_x = 0.3, for example, where the bifurcation rule of rotor can be analyzed as in Figure 8c.

The displacement of rotor is the stable single curve when n ∈ (4.5, 9.5). Rotor changes are in the order of quasi-period, double-period, chaos when n ∈ (9.5, 12.5). The displacement of rotor becomes the stable single curve again when n ∈ (12.5, 18), as in

Table 3. Rotor Motion with different Rotation Speed.

Rotation Speed n	Motion Rule	Rotation Speed Test n
(4.5, 9.5) krpm	Single Period	5.5 krpm
(9.5, 10.5) krpm	Quasi Period	10 krpm
(10.5, 11) krpm	Double Period	10.7 krpm
(11, 12.5) krpm	Chaos	11.5 krpm
(12.5, 18) krpm	Single Period	13.5 krpm

.

According to Table 3, the displacement of the rotor shows the rich dynamic characteristics. Therefore, the rule of the rotor can be analyzed at the different rotational speeds, as shown in the Figures below.

According to Figure 9, when n = 5.5 krpm, the displacement of rotor is the regular sinusoidal motion, as shown in Figure 9a, the single discrete peak occurs in the range of one frequency, as shown in Figure 9b, the mass center of rotor is single closed elliptical trajectory, as shown in Figure 9c, the mapping of Poincare is single point, as shown in Figure 9d, and the dimensionless rubbing force is zero without clearance-rubbing phenomenon, as shown in Figure 9e.

According to Figure 10, when n = 10 krpm, the displacement of rotor is the quasi-period motion (the irregular small-amplitude sinusoidal motion), as shown in Figure 10a, the no common-divisor spectrum occurs in the range of double frequency, as shown in Figure 10b, the mass center of rotor is multi-closed elliptical trajectory, as shown in Figure 10c, the mapping of Poincare is closed loop, as shown in Figure 10d, and the dimensionless rubbing force is 0.8 with clearance-rubbing phenomenon, as shown in Figure 10e.

According to Figure 11, when n = 10.7 krpm, the displacement of rotor is the double-period motion (the irregular sinusoidal motion and the transitional state from quasi-period to chaos state), as shown in Figure 11a, the finite discrete spectrum occurs in the range of double frequency, as shown in Figure 11b, the mass center of rotor is 3-closed elliptical trajectory, as shown in Figure 11c, the mapping of Poincare is 3 points, as shown in Figure 11d, the dimensionless rubbing force is 1.5 with clearance-rubbing phenomenon, and the peak number of rubbing-force decreases, as shown in Figure 11e.

According to Figure 12, when n = 11.5 krpm, the displacement of rotor is the chaos motion, as shown in Figure 12a, the continuous spectrum occurs in the range of double frequency, as shown in Figure 12b, the mass center of rotor is multi irregular overlapped elliptical trajectory, as shown in Figure 12c, the mapping of Poincare is cloud-shaped closed graph, as shown in Figure 12d, the dimensionless rubbing force is 3.0 with clearance-rubbing phenomenon, and the peak number of rubbing-force decreases to the lowest level, as shown in Figure 12e.

According to Figure 13, when n = 13.5 krpm, the displacement of rotor is the single-period sinusoidal motion, as shown in Figure 13a, the single discrete spectrum occurs in the range of one frequency, as shown in Figure 13b, the mass center of rotor is the regular elliptical trajectory, as shown in Figure 13c, the mapping of Poincare is one point, as shown in Figure 13d, and the dimensionless rubbing force is zero without clearance-rubbing phenomenon, as shown in Figure 13e.

3.3. Influence of Rotor Displacement on Eccentricity Ratio and Rotation Speed

Maximum displacement, number of point and maximum rubbing-force are selected as the indexes of the x–t curve, Poincare mapping and clearance-rubbing phenomenon, and then the motion rule of rotor can be analyzed in the range of U_x ∈ (0.2, 0.3), as shown in the Figures below.

According to Figure 14, rotor presentsa single-period trajectory and its amplitude is in the range of (−0.6, 0.3) without clearance-rubbing fault within the largest range, as in the white zone. A series of complicated motions of rotor, such as single-period, double-period and chaos, occurs in turn, and its displacement scope is (−1, 1.5) when n ∈ (9, 13) krpm and U_x ∈ (0.27, 0.4), as in the color zone. The color depth presents the severity of bifurcating phenomenon as color zone of Figure 14b.

According to Figure 15, rotor presents a single-period state without bifurcating phenomenon, and the point number of Poincare mapping is one within the largest range, as in the white zone. A series of points of Poincare mapping occurs and its maximum is up to 15 when n ∈ (9, 13) krpm and U_x ∈ (0.27, 0.4), as in the color zone. The color depth presents the point number of Poincare mapping as the color zone of Figure 15b.

According to Figure 16, there isn’t clearance-rubbing phenomenon and its rubbing force is zero within the most range as white zone. The clearance-rubbing phenomenon occurs and its rubbing-force is −5.7 when n ∈ (9, 13) krpm and U_x ∈ (0.27, 0.4) as color zone. The color depth presents the magnitude of rubbing force as color zone of Figure 16b.

In sum, in order to ensure the stable operation of MLDSB, the rotor should be avoided in the range of n ∈ (9, 13) krpm and U_x ∈ (0.27, 0.4) for a long time.

4. Conclusions

The traditional AMB air gap is 0.1 mm. The bearing will fail when clearance-rubbing fault occurs. Therefore, protective bearings are often added. The hydrostatic bearing concept is introduced into AMB to form the novel suspension bearing MLDSB without protective bearings. In this paper, its operation characteristics and clearance-rubbing fault rules are studied.

When the system rotation speed is close to its natural frequency, it will cause strong resonance, and the eccentricity, crack, bending of the rotor and assembly error will lead to machine faults. Firstly, the clearance-rubbing dimensionless dynamic equation of 2-DOF system of MLDSB is established. Then, rotor displacement under different eccentricity ratios U_x and rotating speeds can be simulated with the fourth-order Runge–Kutta method. Finally, the results show that when the eccentricity ratio is greater than 0.27, the rotor displacement exceeds its liquid film limit during the whole acceleration process and clearance-rubbing fault occurs, so the rotation speed range of resonance is (9, 13) krpm. Simultaneously, the value of rotor eccentricity ratio is not required with a working speed lower than the natural frequency. In the process of rotor acceleration, when the working speed is greater than the natural frequency, its eccentricity should be strictly controlled below 0.27.