processes

Article Impact-Rubbing Dynamic Behavior of Magnetic-Liquid Double Suspension under Different Protective Bearing Forms

Jianhua Zhao 1,2, Lanchun Xing 1, Sheng Li 1, Weidong Yan 1, Dianrong Gao 1 and Guojun Du 2,*

1 Fluid Power Transmission and Control Laboratory, Yanshan University, Qinhuangdao 066004, China; [email protected] (J.Z.); [email protected] (L.X.); [email protected] (S.L.); [email protected] (W.Y.); [email protected] (D.G.) 2 College of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao 066004, China * Correspondence: [email protected]

Abstract: The magnetic-liquid double suspension bearing (MLDSB) is a new type of suspension bearing, with electromagnetic suspension as the main part and hydrostatic supports as the auxiliary part. It can greatly improve the bearing capacity and stiffness of rotor-bearing systems and is suitable for a medium speed, heavy load, and frequent starting occasions. Compared with the active electromagnetic bearing system, the traditional protective bearing device is replaced by the hydrostatic system in MLDSB, and the impact-rubbing phenomenon can be restrained and buffered. Thus, the probability and degree of friction and wear between the rotor and the magnetic pole are reduced drastically when the electromagnetic system fails. In order to explore the difference in the dynamic behavior law of the impact-rubbing phenomenon between the traditional protection device and hydrostatic system, the dynamic equations of the rotor impact-rubbing in three kinds of  protection devices (fixed ring/deep groove ball bearing/hydrostatic system) under electromagnetic  failure mode are established, and the axial trajectory and motion law of the rotor are numerically Citation: Zhao, J.; Xing, L.; Li, S.; simulated. Finally, the dynamic behavior characteristics of the rotor are compared and analyzed. Yan, W.; Gao, D.; Du, G. Impact- The results show that: Among the three kinds of protection devices (fixed ring/deep groove ball Rubbing Dynamic Behavior of bearing/hydrostatic system), the hydrostatic system has the least influence on bouncing time, impact- Magnetic-Liquid Double Suspension rubbing force, and impact-rubbing degree, and the maximum impact-rubbing force of MLDSB is Bearing under Different Protective greatly reduced. Therefore, the protective bear is not required to be installed in the MLDSB. This Bearing Forms. Processes 2021, 9, 1105. study provides the basis for the theory of the “gap impact-rubbing” of MLDSB under electromagnetic https://doi.org/10.3390/pr9071105 failure, and helps to identify electromagnetic faults.

Academic Editor: Ján Pitel’ Keywords: magnetic-liquid double suspension bearing; protecting bearing; hydrostatic system; impact-rubbing dynamics; electromagnetic failure Received: 23 March 2021 Accepted: 22 June 2021 Published: 25 June 2021

Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in The magnetic-liquid double suspension bearing (MLDSB) is a new type of suspension published maps and institutional affil- bearing, with electromagnetic suspension as the main part and hydrostatic supports as iations. the auxiliary part. With this, the bearing capacity, operation stability, and service life of rotor-bearing systems can be greatly improved. MLDSB is suitable for hydroelectric power, deep-sea exploration, and other fields, especially those that feature a medium speed, heavy load, and frequent starting occasions. Copyright: © 2021 by the authors. The MLDSB Test Table includes a variable speed motor, coupling, radial bearing, axial Licensee MDPI, Basel, Switzerland. bearing, axial loading motor, radial loading motor, step shaft, and frame, as shown in This article is an open access article Figure1[1]. distributed under the terms and The radial bearing includes a rotor, magnetic sleeve, supporting cavity, magnetic pole, conditions of the Creative Commons oil inlet/return hole, shell, and coil, as shown in Figures2 and3[ 1]. The magnetic pole and Attribution (CC BY) license (https:// magnetic guide sleeve were treated with chromium plating to prevent them from being creativecommons.org/licenses/by/ corroded due to their immersion in oil for a long time [2,3]. 4.0/).

Processes 2021, 9, 1105. https://doi.org/10.3390/pr9071105 https://www.mdpi.com/journal/processes Processes 2021, 9, x FOR PEER REVIEW 2 of 15 Processes 2021, 9, x FOR PEER REVIEW 2 of 15 Processes 2021, 9, x FOR PEER REVIEW 2 of 15

ProcessesProcesses2021 2021, ,9 9,, 1105 x FOR PEER REVIEWMotor Coupling Radial Bearing Axial Motor 22 of 1515 Motor Coupling Radial Bearing Axial Motor Motor Coupling Radial Bearing Axial Motor

Motor Coupling Radial Bearing Axial Motor

Axial Bearing Step Shaft Radial Motor Bracket Axial Bearing Step Shaft Radial Motor Bracket FigureAxial Bearing1. MLDSB TestStep ShaftTable. Radial Motor Bracket Figure 1. MLDSB Test Table. Figure 1. MLDSB Test Table. AxialThe Bearing radial bearingStep Shaft includesRadial Motora rotor, Bracketmagnetic sleeve, supporting cavity, magnetic The radial bearing includes a rotor, magnetic sleeve, supporting cavity, magnetic pole,The oil inlet/returnradial bearing hole, includes shell, and a rotor, coil, asmagnetic shown insleeve, Figures supporting 2 and 3 [1].cavity, The magneticmagnetic Figurepole,Figure oil 1.1. MLDSBMLDSBinlet/return TestTest Table. Table.hole, shell, and coil, as shown in Figures 2 and 3 [1]. The magnetic pole,pole andoil inlet/returnmagnetic guide hole, sleeve shell, were and treatedcoil, as wishownth chromium in Figures plating 2 and to 3 prevent [1]. The them magnetic from pole and magnetic guide sleeve were treated with chromium plating to prevent them from polebeing and corroded magnetic due guide to their sleeve immersion were treated in oil wi forth a chromium long time plating[2,3]. to prevent them from beingThe corroded radial duebearing to their includes immersion a rotor, in oilmagnetic for a long sleeve, time supporting[2,3]. cavity, magnetic beingpole, oilcorroded inlet/return due to hole, their shell, immersion and coil, in oilas shownfor a long in Figurestime [2,3]. 2 and 3 [1]. The magnetic Threaded Outlet Hole Hole poleThreaded and magnetic guideOutlet sleeve Hole were treated with chromium plating to prevent them from ThreadedHole OutletInter Hole Hole Hole Plating beingRotor corroded due to theirInter Hole immersion in oil for a long time [2,3]. InterCoil Hole Plating Rotor Plating Rotor Coil Threaded OutletMagneticCoil Hole Hole Inlet Hole MagneticPole Liquid InterMagnetic Hole Film Inlet Hole PoleMagnetic PlatingLiquid InletRotor Hole Pole Liquid Coilsleeve Film Plating Stator Magnetic Film Magneticsleeve Plating StatorMagneticsleeve Plating Stator FigureInlet Hole 2. Cutaway view of thePole Radial Unit. Liquid Figure 2. Cutaway view of the Radial Unit. Film Figure 2. Cutaway view of the MagneticRadial Unit. Figure 2. Cutaway view of the Radialsleeve Unit. Plating Stator Magnetic FigureMagneticSleeve 2. Cutaway view of theCoil RadialOutlet Unit. Magnetic Sleeve Coil Outlet Sleeve Coil Outlet

Magnetic Shell Magnetic Sleeve Coil OutletShell MagneticPole Shell MagneticPole Pole Threaded ThreadedHoleShell Threaded MagneticCoil Hole Hole CoilPole Coil ThreadedInlet HoleInlet Inlet FigureCoil 3. Photo of the Radial Unit. Figure 3. Photo of the Radial Unit. FigureFigure 3.3. Photo of thethe RadialRadial Unit.Unit. The regulation principleInlet of MLDSB is shown in Figure 4 [1]. The PD control and con- The regulation principle of MLDSB is shown in Figure 4 [1]. The PD control and con- stantTheThe pressure regulation supply principle principle mode areof of MLDSB MLDSBadopted is is inshown shownthe electromagnetic in inFigure Figure 4 [1].4[ 1The system]. The PD control PDand control hydrostatic and andcon- stantFigure pressure 3. Photo ofsupply the Radial mode Unit. are adopted in the electromagnetic system and hydrostatic constantstantsystem, pressure respectively, pressure supply supply to mode realize mode are arereal-time adopted adopted regulation in in the the electromagnetic electromagnetic of the rotor [1]. system system and hydrostatichydrostatic system,system, respectively,respectively, toto realizerealize real-timereal-time regulation regulation of of the the rotor rotor [ 1[1].]. system, respectively, to realize real-time regulation of the rotor [1]. The regulation principleU of MLDSB is shown in Figure 4 [1]. The PD control and con- 01 Power stant pressure supplyU mode are adopted in the electromagnetic system and hydrostatic 01 Amplifier 1 i0-icy UcyU01 Power Reference Ur Power + Amplifier 1 i0-icy system, respectively,ControllerU cyto realize real-time regulation of the rotor [1]. Position Ur Amplifier 1 i0-icy Reference + - Ucy Power Reference Ur Controller y Position + -Uy U02 Amplifier 2i0+icy Position Controller Power x U- U01 Power i +iy1 y y Differential 02 AmplifierPower 2 0 cy y x Uy U02 Amplifier 2i0+icy Measurement Amplifier 1 i0yy-i12cy x DifferentialUcy y Reference Ur+ Differential Throttle 1 MeasurementControllerPressure y2 Position Measurement q1 - Gauge PowerThrottlevalve 1 y2 Pressure y Uy Pressure U02q1AmplifierThrottlevalve 12 i0+icy Relief GaugePump q1 Throttle x Gauge q2 valve 1 y ReliefValve Differential Throttlevalve 2 1 Pump q ReliefValve MeasurementPump 2 Throttlevalve 2 y2 q2 Valve Pressure Throttlevalve 2 q Figure 4. Single DOFGauge control1 systemvalve 1 of MLDSB. Figure 4. Single DOF control system of MLDSB. FigureFigureRelief 4.4. Single DOFDOFPump controlcontrol systemsystemThrottle ofof MLDSB.MLDSB. Valve q2 valve 2

Figure 4. Single DOF control system of MLDSB.

Processes 2021, 9, 1105 3 of 15

Electromagnetic system failure, which is caused by electromagnetic coil corrosion and power amplifier circuit fault [4,5], will lead to friction and excessive wear between the rotor and magnetic pole, causing the structural strength and reliability of the electromagnetic bearing to be severely reduced. Thus, the fixed ring/deep groove ball bearings and other protective devices are equipped with AMB (Active Magnetic Bearings) to improve the operation stability. In recent years, many scholars at home and abroad have deeply studied the impact- rubbing dynamics of AMB and achieved many results. Professor JARROUX [6] studied the rotor drop dynamics numerically and experi- mentally when AMB unexpectedly stops. The finite unit method and three kinds of TDB models were used to simulate rotor drops in the time domain and measure rotor drop responses, according to the displacement and transmission load. Professor PATPICK [7] studied the control scheme of AMB when intermittent fault or overload occurs. The results showed that proper forced phase synchronization can destabilize the synchronous forward friction response. The rotor under contact rubbing and slight rotational motion is beneficial to the system and does not interfere with the main electromagnetic bearing control loop. A robust control strategy to ensure suspension recovery was proposed by Professor PESCH [8]. The specific AMB control laws were found by using model-based optimiza- tion synthesis to explain and prevent electromagnetic levitation failure, caused by large deformations, using control contact force saturation. The dynamic model was established by Professor Zhao [9] to study the dynamic characteristics of rotor drops, and a PID control system for a magnetic bearing was designed with contact possibilities. The control strategy was theoretically demonstrated to be effective in returning a rotor to the contact-free levitation position and avoid further fierce contacts under several circumstances involving external disturbances. The falling behavior of the spindle structure of a vertical axis wind turbine was simulated by Professor Wu [10] The movement track of the spindle in the axial direction was compared to visually illustrate the influence of different protective bearing structures on the movement track of the magnetic suspension spindle in the falling process. The situation of the AMB falling on the self-eliminating clearance protection bearing after failure was studied by Professor Xu [11–14]. Compared with the traditional roller bear- ing, the self-eliminating clearance protection bearing significantly reduced the vibration of the rotor after falling, and the falling of the high frequency speed was more reliable. Professor Zdzislaw [15] studied H∞ and H2 control of rigid rotor movement, which is supported in magnetic bearings, and investigated the robust control of magnetic bearings. The paper proposes robust control with a multi-objective controller to achieve good robust stability when the model of a plant is uncertain. Finally, the success of the robust control is verified by numerical simulation results. Compared with the traditional AMB, the hydrostatic system is used in MLDSB here to replace the protection devices, so that it not only effectively supports the rotor, but can also buffer the friction fault of the rotor. In this way, the wear degree and probability between the rotor and the magnetic pole can be reduced. However, the protection mechanism of the rotor/stator by the static pressure system in this new bearing system is not clear yet, and the difference between the static pressure system and the traditional protection bearing needs to be explored. Therefore, the dynamic equations of the rotor impact-rubbing in three kinds of protection devices (fixed ring/deep groove ball bearing/hydrostatic system) under electromagnetic failure mode are established in this paper, and the axial trajectory, motion law, and dynamic behavior characteristics of the rotor are numerically simulated and analyzed. Processes 2021, 9, 1105 4 of 15

2. Rotor Impact-Rubbing Model during Electromagnetic Failure The time of rotor impact-rubbing is very short, so the assumptions are as follows [16,17]: (1) There is a local elastic collision between the magnetic pole and the magnetic jacket, and the deformation is elastic deformation; (2) The winding magnetic flux leakage, edge magnetic flux, loss, core material saturation, and coupling effect between magnetic poles are ignored; (3) The tiny gaps between the poles are ignored; (4) The inertial force and viscous pressure characteristics of the liquid are ignored; (5) The gravity of the inner ring of deep groove ball bearings is ignored; (6) The deep groove ball bearing is simplified into a spring damping system with mass.

2.1. Impact-Rubbing Dynamics Model of the “Rotor-Fixed Ring” The mechanical model of rotor impact-rubbing on the fixed ring protection device in electromagnetic failure mode is shown in Figure5, and the mathematical equations of the supporting system are shown as follows [18].  .. FMx,j + Fx = mx  .. FMy,j + Fy − mg = my (1)  ..  Jθ = −Ftr

 . 2 2 2  +(− )j +   µ0SN cosϕ j+1 i0 1 (kpxy kdx x)  FMx,j = 2 ∑ (−1) j  = h0+2l+(−1) xcosϕ  j 1   " 2 # 2 −mg(h +2l) .  i +(− )j 0 +k y+k y  (1a) 2  0 1 2 py dy   µ SN2 ϕ +  2µ0i0SN cosϕ   F = 0 cos (−1)j 1  My,j 2 ∑ + +(− )j  j=1  h0 2l 1 ycosϕ    

. .    e1 p 2 2 xx+yy p 2 2   k1 x + y − h1 + c1 √ x + y − h1 ≥ 0  F = x2+y2  n p   0 x2 + y2 − h < 0  1 Ft = f1Fn (1b)  √−Fn  Fx = (x − f1y)  x2+y2  −Fn  Fy = √ ( f1x + y)  x2+y2

where FMx and FMy are the electromagnetic supporting forces in the x and y directions [19]. Fx and Fy are the components of the impact-rubbing force in the x and y directions. m is the rotor mass. g is the gravity acceleration. J is the inertia moment around the centroid of the rotor. θ is the angular displacement. Fn and Ft are the radial and tangential impact-rubbing forces [20]. R is the radius of the rotor. ϕ is the angle between the magnetic pole and the center line of the rotating shaft. h0 and h1 are the initial unilateral clearance between the rotor and magnetic pole and the inner ring of the protection device, respectively. µ0 is the vacuum permeability. l is the coating thickness. i0 is the bias current. S is the area of the magnetic poles. N is the number of coil turns. kpx and kdx are the scale and differential coefficients in the x direction. kpy and kdy are the proportional and differential coefficients in the y direction. e1, k1, c1, and f 1 are the contact coefficient, stiffness, damping, and friction coefficient of the rotor in contact with the inner ring of the protection device, respectively. Processes 2021, 9, x FOR PEER REVIEW 4 of 15

2. Rotor Impact-Rubbing Model during Electromagnetic Failure The time of rotor impact-rubbing is very short, so the assumptions are as follows [16,17]: (1) There is a local elastic collision between the magnetic pole and the magnetic jacket, and the deformation is elastic deformation; (2) The winding magnetic flux leakage, edge magnetic flux, eddy current loss, core material saturation, and coupling effect between magnetic poles are ignored; (3) The tiny gaps between the poles are ignored; (4) The inertial force and viscous pressure characteristics of the liquid are ignored; (5) The gravity of the inner ring of deep groove ball bearings is ignored; (6) The deep groove ball bearing is simplified into a spring damping system with mass.

2.1. Impact-Rubbing Dynamics Model of the “Rotor-Fixed Ring”

Processes 2021, 9, 1105 The mechanical model of rotor impact-rubbing on the fixed ring protection device5 of 15in electromagnetic failure mode is shown in Figure 5, and the mathematical equations of the supporting system are shown as follows [18].

y Fixed Ring

Rotor x o α o1 Ft

Fn c1

kn1

Figure 5. Mechanical model of the “rotor-fixed ring” protection device. Figure 5. Mechanical model of the “rotor-fixed ring” protection device.

Electromagnetic system failure in𝐹, the+𝐹 paper =𝑚𝑥 includes the failure of solely the upper unit and the failure of the upper and lower𝐹, +𝐹 units. −𝑚𝑔=𝑚𝑦 The mathematical model of the impact-(1) rubbing dynamics is shown in Equation𝐽𝜃 =−𝐹 (1), which𝑟 just requires the following changes:

( F = 0 Upper unit failure 𝜇 𝑆𝑁 𝑐𝑜𝑠My,1 𝜑 𝑖 + (−1) 𝑘𝑦+𝑘𝑥 ⎧𝐹 = (−1) (2) , = = ⎪ FMy2 FMy 0 Upper/Lowerℎ +2𝑙+(−1) 𝑥𝑐𝑜𝑠𝜑 units failure ⎪ ,1 ,2 −𝑚𝑔(ℎ +2𝑙) (1a) 2.2. Impact-Rubbing⎨ Dynamics Model of the⎧ “Rotor-Deep𝑖 + (−1) Groove Ball Bearing”+𝑘 𝑦+𝑘 𝑦⎫ 𝜇 𝑆𝑁 𝑐𝑜𝑠 𝜑 2𝜇 𝑖 𝑆𝑁 𝑐𝑜𝑠 𝜑 ⎪ 𝐹, = (−1) The mechanical⎪ model2 of the rotor impact-rubbingℎ +2𝑙+ on( the−1) deep𝑦𝑐𝑜𝑠𝜑 groove ball bearing ⎨ ⎬ under electromagnetic⎩ failure mode is shown⎩ in Figure6, and the mathematical model⎭ of the supporting system is shown as follows [18]. 𝑥𝑥+𝑦𝑦 ⎧ 𝑘 𝑥 +𝑦 −ℎ +𝑐 .. 𝑥 +𝑦 −ℎ ≥0 𝐹 =  + = 𝑥 +𝑦 ⎪  FMx,j Fx1 mx ⎪  ..  FMy,j + 0Fy1 − mg = my 𝑥 +𝑦 −ℎ <0 ⎪  .. 𝐹 =𝑓𝐹   Jθ = −Ft1r .. ((3)1b) ⎨ −𝐹 F − F = m x 𝐹 =  (𝑥−𝑓x2 𝑦)x1 b b ⎪  .. 𝑥 +𝑦 Fy2 − Fy1 = mby ⎪  .. b −𝐹 ⎪𝐹 = (J𝑓bθ𝑥+𝑦b = )(Ft2 − Ft1)(h1 + r) ⎩ 𝑥 +𝑦   e1 . . . .  q (x−x )(x−x )+(y−y )(y−y ) q   2 2 b b b b 2 2  Mx k1 (xMy− xb) + (y − yb) − h1 + c1 q (x − xb) + (y − yb) − h1 ≥ 0 where F and F are the electromagnetic supporting( − )2+( − )2 forces in the x and y directions [19].  Fn1 = x xb y yb  q Fx and Fy are the components of the impact-rubbing force (in− the)2 x+ (and− y)2 directions.− < m is  0 x xb y yb h1 0 the rotorFt1 = fmass.1Fn1 g is the gravity acceleration. J is the inertia moment around the centroid(3a) of  −Fn1  Fx1 = q [(x − xb) − f1(y − yb)] the rotor. θ− is 2+the− angular2 displacement. Fn and Ft are the radial and tangential impact-  (x xb) (y yb)  −Fn1  F = q [ f (x − x ) + (y − y )]  y1 2 2 1 b b  (x−xb) +(y−yb) . .  p e2 x x +y y  F = k x 2 + y 2 + c √b b b b  n2 2 b b 2 2 2  xb +yb   Ft2 = f2Fn2 F (3b) F = √ n2 (x − f y ) x2 2 2 b 2 b  xb +yb   F = √ Fn2 ( f x + y )  y2 2 2 2 b b xb +yb

where Fx1 and Fy1 are the components of the impact-rubbing force acting on the rotor in the x and y directions. mb is the mass of the bearing’s inner ring. Fx2 and Fy2 are the components of the impact-rubbing force acting on the inner ring of the bearing in the x and y directions. xb and yb are the displacements of the bearing’s inner ring in the x and y directions. Jb is the moment of inertia of the bearing’s inner ring around the centroid. θb is the angular displacement of the bearing’s inner ring. Fn1 and Ft1 are the radial and tangential impact-rubbing forces acting on the rotor. Fn2 and Ft2 are the radial and tangential impact-rubbing forces acting on the inner ring of the bearing. e2, k2, c2, and f 2 are the contact coefficient, stiffness, damping, and friction coefficient of the bearing’s inner ring in contact with the ball, respectively. Processes 2021, 9, x FOR PEER REVIEW 5 of 15

rubbing forces [20]. R is the radius of the rotor. φ is the angle between the magnetic pole and the center line of the rotating shaft. h0 and h1 are the initial unilateral clearance be- tween the rotor and magnetic pole and the inner ring of the protection device, respec- tively. μ0 is the vacuum permeability. l is the coating thickness. i0 is the bias current. S is the area of the magnetic poles. N is the number of coil turns. kpx and kdx are the scale and differential coefficients in the x direction. kpy and kdy are the proportional and differential coefficients in the y direction. e1, k1, c1, and f1 are the contact coefficient, stiffness, damping, and friction coefficient of the rotor in contact with the inner ring of the protection device, respectively. Electromagnetic system failure in the paper includes the failure of solely the upper unit and the failure of the upper and lower units. The mathematical model of the impact- rubbing dynamics is shown in Equation (1), which just requires the following changes: 𝐹 = 0 Upper unit failure , 𝐹 =𝐹 = 0 Upper/Lower units failure (2) , ,

2.2. Impact-Rubbing Dynamics Model of the “Rotor-Deep Groove Ball Bearing”

Processes 2021, 9, 1105 The mechanical model of the rotor impact-rubbing on the deep groove ball bearing6 of 15 under electromagnetic failure mode is shown in Figure 6, and the mathematical model of the supporting system is shown as follows [18].

y y r k2 h1 x · o x θ o(o1) o1 c2 c1 k1 θ· · θb

(a) Initial equilibrium stage. (b) Contact and rubbing stage. Processes 2021, 9, x FORFigure PEER 6.REVIEW Mechanical model of the protection device for the “rotor-deep groove ball bearing”. 6 of 15 Figure 6. Mechanical model of the protection device for the “rotor-deep groove ball bearing”.

The mathematical model of the𝐹 impact-rubbing+𝐹 =𝑚𝑥 dynamics in the deep groove ball ⎧ , bearing is shown in Equation (3), which𝐹 requires+𝐹 −𝑚𝑔=𝑚𝑦 the changes as shown in Equation (2). , 𝑥 𝑥 +𝑦 𝑦 ⎪ ⎧𝐹 =𝑘 𝑥 +𝑦 +𝑐 𝐽𝜃 =−𝐹𝑟 2.3. Impact-Rubbing Dynamics⎪ Model of the “Rotor-Hydrostatic𝑥 System”+𝑦 (3) ⎪ ⎨𝐹 −𝐹 =𝑚𝑥 ⎪𝐹 =𝑓𝐹 The mechanical model of the⎪ impact-rubbing𝐹 −𝐹 =𝑚𝑦 dynamics on the hydrostatic system 𝐹 (3b) under electromagnetic failure⎨𝐹 mode= ⎩𝐽 is𝜃 shown = (𝐹(𝑥 in−𝐹 −𝑓 Figure)(𝑦ℎ)7+𝑟, and) the mathematical model of 𝑥 +𝑦 the supporting system is shown⎪ as follows. ⎪ 𝐹 ⎪ (𝑥 −𝑥 )(𝑥−𝑥𝐹 )=+ (𝑦 −𝑦 )(𝑦−𝑦(𝑓𝑥)+𝑦)  .. ⎧ 𝑘 (𝑥−𝑥) + (𝑦−𝑦) −ℎ +𝑐 ⎩ +𝑥 +𝑦+ = (𝑥−𝑥) + (𝑦−𝑦) −ℎ ≥0  FLx ,j FMx,j Fx mx ⎪𝐹 = (𝑥−𝑥 ) + (𝑦−𝑦) .. FLy,j + FMy,j + Fy − mg = my (4) ⎪ where Fx1 and Fy1 are the components.. of the impact-rubbing force acting on the rotor in 0  (𝑥−𝑥) + (𝑦−𝑦) −ℎ <0 ⎪ the x and y directions. mb is theJθ mass= − Foftr the bearing’s inner ring. Fx2 and Fy2 are the com- 𝐹 =𝑓𝐹 ponents of the impact-rubbing force acting on the inner ring of the bearing in the x and(3a y) ⎨ −𝐹  ( ) 𝐹 = (𝑥−𝑥) −𝑓(𝑦−𝑦) 2 j ⎪ (𝑥−𝑥 ) + (𝑦−𝑦directions.) xb and yb are the displacements(−1) 2ps A eofcos theϕ bearing’s inner ring. in the x and y direc-  FLx,j = ∑ 3 − 2Ab AeRhj xxcosϕ ⎪  h j x i , −𝐹 tions. Jb is the moment ofj= inertia1 1+(β of−1 )the1+ bear(−1) ing’scosϕ inner ring around the centroid. θb is the h0 ⎪𝐹 = 𝑓(𝑥−𝑥) + (𝑦−𝑦) ( ) (4a) angular displacement of 2the bearing’sj inner ring. Fn1 and Ft1 are the radial and tangential ⎩ (𝑥−𝑥) + (𝑦−𝑦)  (−1) 2ps Aecosϕ .  F = n2 t2− 2A A R ycosϕ impact-rubbing forcesLy,j acting∑ on the rotor.h F y andi F3 are theb eradialhj,y and tangential impact-  j=1 1+(β−1) 1+(−1)j cosϕ rubbing forces acting on the inner ring of theh0 bearing. e2, k2, c2, and f2 are the contact coef- ficient, stiffness, damping, and friction coefµficient of the bearing’s inner ring in contact Rhj x = 3  , h j i with the ball, respectively. B h0+(−1) xcosϕ j = 1, 2 (4b) The mathematical modelR of =the impact-ruµ bbing dynamics in the deep groove ball  hj,y h j i3 bearing is shown in Equation (3), whichB h0+ requ(−1)iresycos ϕthe changes as shown in Equation (2). where F and F are the hydrostatic force in the x and y directions [21]. p is the oil supply 2.3. Impact-RubbingLx Ly Dynamics Model of the “Rotor-Hydrostatic System” s pressure. Ae and Ab are the effective bearing area and extrusion area of the supporting The mechanical model of the impact-rubbing dynamics on the hydrostatic system cavity, respectively. Rhj is the fluid resistance of the bearing cavity after loading. µ is the oil viscosity.under electromagneticB is the flow coefficient failure mode [22]. isβ shownis the throttlingin Figure 7, ratio. and the mathematical model of the supporting system is shown as follows.

Inlet Hole Upper Support Unit y1

φ Left y Right Support x Support Unit x2 ω mg Unit x1

Lower Electromagnetic Support Coil Unit y2 FigureFigure 7.7. MechanicalMechanical modelmodel ofof thethe protectionprotection devicedevice forfor thethe “rotor-“rotor- hydrostatic hydrostatic system”. system”.

The mathematical model of𝐹, the+𝐹 impact-rubbing, +𝐹 =𝑚𝑥 dynamics in the hydrostatic system is (4) shown in Equation (4), which requires𝐹, +𝐹, the+𝐹 changes −𝑚𝑔=𝑚𝑦 as shown in Equation (2). 𝐽𝜃 =−𝐹𝑟 ⎧ ⎧ ⎫ (−1) 2𝑝𝐴 𝑐𝑜𝑠 𝜑 ⎪𝐹, = −2𝐴𝐴𝑅ℎ,𝑥 𝑐𝑜𝑠 𝜑 ⎨ 𝑥 ⎬ ⎪ 1+(𝛽−1)1+(−1) 𝑐𝑜𝑠 𝜑 ⎩ ℎ ⎭ (4a) ⎨ ⎧ ⎫ ⎪ (−1) 2𝑝𝐴 𝑐𝑜𝑠 𝜑 𝐹, = −2𝐴𝐴𝑅ℎ,𝑦 𝑐𝑜𝑠 𝜑 ⎪ ⎨ 𝑦 ⎬ 1+(𝛽−1)1+(−1) 𝑐𝑜𝑠 𝜑 ⎩ ⎩ ℎ ⎭ 𝜇 𝑅 = ℎ, 𝐵ℎ + (−1) 𝑥𝑐𝑜𝑠𝜑 (4b) 𝜇 𝑗=1，2 𝑅ℎ, = 𝐵 ℎ + (−1) 𝑦 𝑐𝑜𝑠 𝜑 where FLx and FLy are the hydrostatic force in the x and y directions [21]. ps is the oil supply pressure. Ae and Ab are the effective bearing area and extrusion area of the supporting

Processes 2021, 9, 1105 7 of 15

3. Numerical Simulation of the Rotor Impact-Rubbing Process under Processes 2021, 9, x FOR PEER REVIEW 7 of 15 Electromagnetic Failure The initial design parameters of the fixed ring/deep groove ball bearing/hydrostatic system support are shown in Table1. cavity, respectively. Rhj is the fluid resistance of the bearing cavity after loading. μ is the oil viscosity.Table 1. Initial 𝐵 is design the flow parameters coefficient of the [22]. protection β is the device. throttling ratio. The mathematical model of the impact-rubbing dynamics in the hydrostatic system Rotor Mass isGravitational shown in Equation Acceleration (4), which requiresRotor the Radius changes as shownRotor in Equation Rotational (2). Inertia m/(kg) g/(m/s2) r/(m) J/(kg·m2)

103. Numerical 10 Simulation of the Rotor Impact-Rubbing 0.1 Process under Electromagnetic 0.05 Failure Angle TheBias initial current design parameters ofCoating the fixed thickness ring/deep grooveInitial ball unilateral bearing/hydrostatic clearance ◦ µ µ ϕ/( ) system supporti0/(A) are shown in Table 1. l/( m) h0/( m) 22.5 0.5 50 50 Table 1. Initial design parameters of the protection device. Magnetic area Vacuum permeability Coil number Stator contact coefficient Rotor Mass Gravitational Acceleration Rotor Radius Rotor Rotational Inertia S/(mm2) µ /(H/m) N/(dimensionless) e /(dimensionless) m/(kg) g/(m/s0 2) r/(m) 1 J/(kg·m2) 101080 10 4π × 10−7 0.150 10/9 0.05 Angle Bias current Coating thickness Initial unilateral clearance Statorφ/( damping°) Statori0/(A) stiffness Stator frictionl/(μm) coefficient Inner ring unilateralh0/(μm) clearance −10/9 c1/(N22.5 ·m/s) k10.5/(N/m ) f 1/(dimensionless) 50 h1/( 50µ m) Magnetic area Vacuum permeability Coil number Stator contact coefficient 1000 3.5 × 108 0.1 25 S/(mm2) μ0/(H/m) N/(dimensionless) e1/(dimensionless) Inner ring1080 rotational inertia 4πInner × 10−7 quality Inner ring friction50 coefficient Contact 10/9 coefficient 2 StatorJb /(kgdamping·m ) Stator stiffnessmb/(kg) Statorf 2friction/(dimensionless) coefficient Inner ering2/(dimensionless) unilateral clearance c1/(N·m/s) k1/(N/m−10/9) f1/(dimensionless) h1/(μm) × −4 1100010 3.5 × 100.018 0.1 0.007 1.5 25 InnerBall ring bearing rotational stiffness inertia InnerBall bearing quality damping Inner ringOil friction supply coefficient pressure f2/(di- ContactExtrusion coefficient area Jb/(kg·m2)− 3/2 mb/(kg) mensionless) e2/(dimensionless)2 k2/(N/m ) c2/(N·m/s) ps/(MPa) Ab/(mm ) 1 × 10−4 0.01 0.007 1.5 12 Ball bearing4 × 10stiffness Ball bearing damping800 Oil supply 0.05 pressure Extrusion 56 area k2/(N/m−3/2) c2/(N·m/s) ps/(MPa) Ab/(mm2) Oil viscosity Throttling ratio Discharge 4 × 1012 800 0.05 Effective bearing 56area A /(mm2) µ/(Pa·s) β/(dimensionless) coefficient/(dimensionless) e Oil viscosity Throttling ratio Discharge coefficient Effective bearing area Ae/(mm2) μ1.3/(Pa·s)× 10 −3 β/(dimensionless)2 B/(dimensionless) 0.71 416 1.3 × 10−3 2 0.71 416

3.1.3.1. Impact-Rubbing Impact-Rubbing Dynamics Dynamics Behavior Behavior of of the the Rotor-Fixed Rotor-Fixed Ring Ring Supporting Supporting System System 1.1. Impact-rubbingImpact-rubbing behavior behavior of of the the rotor rotor under under upper upper unit unit failure failure mode mode TheThe axis axis trajectory trajectory of of the the rotor rotor falling falling on on the the fixed fixed ring ring under under upper upper unit unit failure failure mode mode isis shown shown in in Figure Figure 88..

×10-5 4

2 /m y

0 Displacement -2

-4 -4-2 02 4 -5 Displacement x/m ×10 FigureFigure 8. 8. AxisAxis trajectory trajectory under under upper upper unit unit failure failure mode. mode.

According to Figure 8, the impact-rubbing phenomenon of the rotor can be divided into the bouncing stage and eddy stage, when the upper unit fails. When the rotor falls directly on the fixed ring, it firstly bounces and collides and then vortexes repeatedly for- ward/backward.

Processes 2021, 9, 1105 8 of 15

According to Figure8, the impact-rubbing phenomenon of the rotor can be divided Processes 2021, 9, x FOR PEER REVIEW 8 of 15 Processes 2021, 9, x FOR PEER REVIEWinto the bouncing stage and eddy stage, when the upper unit fails. When the rotor8 fallsof 15 directly on the fixed ring, it firstly bounces and collides and then vortexes repeatedly forward/backward. InIn order order to to further further analyze analyze the the bounce-eddy bounce-eddy situation situation of of the the rotor rotor falling, falling, the the rotor rotor In order to further analyze the bounce-eddy situation of the rotor falling, the rotor displacement,displacement, phase phase trajectory, rubbingrubbing force,force, and and electromagnetic electromagnetic force force were were extracted, extracted, as displacement, phase trajectory, rubbing force, and electromagnetic force were extracted, asshown shown in in Figure Figure9. 9. as shown in Figure 9.

-4 2 N ×10 / ×10 /N 8 ×10 2 N

-4 y / My -1 ×10 0 ×10

×10 8/N F -5 8 F y My

-1 0 0 ×10 8 FMy F -5 F

m·s 4 /m 0 ×10 / -1 FMy y y

m·s 4 6 FMy,1 /m v / -1 y -1 y 6 Max:880 N FMy,1 0v F -1 0 4 Max:880 N -2 My,2 -2 FMy,2 -2 -4 4 -2 -4 -3

Ve loc ity 2 -8 -3

Ve loc ity 2 Displacement -3 -8

Displacement -4 -3 -2.68 -2.60 ×10-5 0 02351 4 -5 02350 1 4 0235-4 1 4 -1 -2.68 -2.60 -1 -1 Displacement y/m ×10 Impact-rubbing Force 02351 Time t/s 4×10 02351 ×104 02351 ×104 Electromagnetic Force Time t/s Time t/s -1 -1 -1 Displacement y/m Impact-rubbing Force Time t/s ×10 ×10 Time t/s ×10 Electromagnetic Force Time t/s (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (d) Electromagnetic force (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (d) Electromagnetic force Figure 9. Operating law under upper unit failure. FigureFigure 9. 9.Operating Operating law law under under upper upper unit unit failure. failure. According to Figure 9, the bouncing stage mainly occurs within 0.07 s. The rotor dis- AccordingAccording toto FigureFigure9 9,, thethe bouncingbouncing stage ma mainlyinly occurs occurs within within 0.07 0.07 s. s.The The rotor rotor dis- placement, impact-rubbing force, and residual electromagnetic force shake violently, and displacement,placement, impact-rubbing impact-rubbing forc force,e, and and residual residual electromagnetic electromagnetic force force shake shake violently, violently, and theirand theiramplitudes amplitudes gradually gradually decrease. decrease. The maximum The maximum impact-rubbing impact-rubbing forceforce is 880 is N. 880 N. their amplitudes gradually decrease. The maximum impact-rubbing force is 880 N. TheThe eddy eddy stage stage mainly mainly occurs occurs after after 0.07 0.07 s. s. The The displacement displacement gradually gradually converges converges at at The eddy stage mainly occurs after 0.07 s. The displacement gradually converges at aboutabout −−2626 μµm,m, which which is is larger larger than than the the unilateral unilateral air air gap, gap, indicating indicating that that the the rotor rotor will will about −26 μm, which is larger than the unilateral air gap, indicating that the rotor will eventuallyeventually stagnate stagnate at at the the bottom bottom of of the the fixed fixed ring. ring. The The phase phase trajecto trajectoryry presents presents a a double- double- eventually stagnate at the bottom of the fixed ring. The phase trajectory presents a double- periodicperiodic closed circlecircle fromfrom the the outside outside to to the the inside, inside, which which indicates indicates the displacement the displacement tends periodic closed circle from the outside to the inside, which indicates the displacement tendsto converge. to converge. The variationThe variation of the of impact-rubbingthe impact-rubbing force/electromagnetic force/electromagnetic force force is similaris sim- tends to converge. The variation of the impact-rubbing force/electromagnetic force is sim- ilarto that to that of the of displacement,the displacement, respectively, respectively, converging converging at about at about 125 N 125 and N− and25 N,−25 and N, theirand ilar to that of the displacement, respectively, converging at about 125 N and −25 N, and theirvector vector sums sums are just are injust balance in balance with with the rotor’s the rotor’s gravity. gravity. their vector sums are just in balance with the rotor’s gravity. 2.2. Impact-rubbingImpact-rubbing behavior behavior under under upper upper and and lower lower unit unit failure failure 2. Impact-rubbing behavior under upper and lower unit failure The axis trajectory of the rotor under upper and lower unit failure mode is shown in TheThe axis axis trajectory trajectory of of the the rotor rotor under under upper upper and and lower lower unit unit failure failure mode mode is is shown shown in in Figure 10. FigureFigure 10 10..

×10-5 4 -5 4 ×10 /m

y 2 /m

y 2

0 0

Displacement-2

Displacement-2 -5 ×10 -5 -4 ×10 -4-4 -2 0 2 4 -4Displacement-2 0 x/m2 4 Displacement x/m Figure 10. Axis trajectory under upper and lower unit failure. FigureFigure 10. 10.Axis Axis trajectory trajectory under under upper upper and and lower lower unit unit failure. failure. Similarly, the impact-rubbing phenomenon of the rotor can be divided into the Similarly,Similarly, the the impact-rubbing impact-rubbing phenomenon phenomenon of the of rotorthe rotor can be can divided be divided into the into bounc- the bouncing stage and eddy stage, when the upper unit fails. When the rotor falls directly on ingbouncing stage and stage eddy and stage, eddy whenstage, thewhen upper the upper unit fails. unit Whenfails. When the rotor the rotor falls directlyfalls directly on the on the fixed ring, it bounces and collides firstly and then vortexes repeatedly forward/back- fixedthe fixed ring, ring, it bounces it bounces and collides and collides firstly firstl andy then and vortexesthen vortexes repeatedly repeatedly forward/backward. forward/back- ward. ward.Similarly, the rotor displacement, phase trajectory, rubbing force, and electromagnetic Similarly, the rotor displacement, phase trajectory, rubbing force, and electromag- force wereSimilarly, extracted, the rotor as shown displacement, in Figure 11phase. trajectory, rubbing force, and electromag- netic force were extracted, as shown in Figure 11. netic force were extracted, as shown in Figure 11.

Processes 2021, 9, 1105 9 of 15 Processes 2021, 9, x FOR PEER REVIEW 9 of 15

N

-4 2 /

-5 N /N ×10-4 2 / ×10-5 y ×10 /N 0 ×10 8 My FMy y ×10 ×10 -1 F 0 8 My FMy F -1 F 1 4 F 1 FMy,1 /m F , m·s 4 6 My 1 y /m / m·s -1 y 6 y / FMy,2 v

-1 y FMy,2 v 4 Max:802 N 0 4 0 -2 0

Ve loc ity 2

Ve loc ity 2 Ve loc ity-4

-3 Ve loc ity -5 -3 -4 ×10-5 -1 ×10 0 -1 0231 4 5 -2.64 -2.60 -2.56 02351 4 02351 4 0231 4 5-1 -2.64 -2.60 -2.56 02351 4 -1 02351 4 -1 Time t/s ×10-1 Displacement y/m Impact-rubbing Force Time t/s ×10-1 时间t/s ×10-1 Impact-rubbing Force Impact-rubbing Force Time t/s ×10 Displacement y/m Time t/s ×10 ForceElectromagnetic 时间t/s ×10 ForceElectromagnetic (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (d) Electromagnetic force

Figure 11. OperatingOperating law law of of the system under the failure of the upper and lower units.

AccordingAccording to to Figure Figure 11, 11 the, the bouncing bouncing stage stage mainly mainly occurs occurs within within 0.1 0.1s. The s. Therotor rotor dis- placement,displacement, impact-rubbing impact-rubbing force, force, and andresidual residual electromagnetic electromagnetic force force shake shake violently, violently, and theirand theiramplitudes amplitudes gradually gradually decrease. decrease. The maximum The maximum impact-rubbing impact-rubbing force force is 802 is N. 802 N. TheThe eddy eddy stage stage mainly occurs after 0.1 s. In this stage, the displacement gradually convergesconverges atat aboutabout −−26 µμm, whichwhich indicatesindicates the the rotor rotor will will eventually eventually stagnate stagnate at theat the bottom bot- tomof the of fixedthe fixed ring. ring. The The phase phas trajectorye trajectory presents presents a double-periodic a double-periodic closed closed circle circle from from the theoutside outside to the to inside,the inside, which which indicates indicates that that thedisplacement the displacement tends tends to converge. to converge. There There is no isresidual no residual electromagnetic electromagnetic force, force, andthe and variation the variation of the of impact-rubbing the impact-rubbing force force is similar is sim- to ilarthat to of that the displacement,of the displacement, converging converging at about at 100 about N, and100 N, the and vector the value vector is value just in is balance just in balancewith the with rotor’s the gravity.rotor’s gravity. 3.2. Impact-Rubbing Dynamics Behavior of the Rotor-Ball Bearings Supporting System 3.2. Impact-Rubbing Dynamics Behavior of the Rotor-Ball Bearings Supporting System 1. Impact-rubbing behavior law under upper unit failure 1. Impact-rubbing behavior law under upper unit failure InIn the the upper upper unit unit failure failure mode, mode, the the axis axis tr trajectoryajectory of of the the rotor falling in the deep groovegroove ball ball bearing is shown in Figure 12.12.

FigureFigure 12. 12. AxialAxial trajectory trajectory under under upper upper unit unit failure. failure. Similarly, the impact-rubbing phenomenon of the rotor can be divided into the bounc- Similarly, the impact-rubbing phenomenon of the rotor can be divided into the ing stage and eddy stage, when the upper unit fails. When the rotor falls directly on the bouncing stage and eddy stage, when the upper unit fails. When the rotor falls directly on inner ring of the ball bearing, it bounces and collides firstly and then vortexes repeatedly the inner ring of the ball bearing, it bounces and collides firstly and then vortexes repeat- forward/backward. edly forward/backward. Similarly, the rotor displacement, phase trajectory, rubbing force, and electromagnetic Similarly, the rotor displacement, phase trajectory, rubbing force, and electromag- force were extracted, as shown in Figure 13. netic force were extracted, as shown in Figure 13. According to Figure 13, the bouncing stage mainly occurs within 0.06 s. The rotor displacement, impact-rubbing force, and residual electromagnetic force shake violently, -4 2 N -5 / N -4 /N ×102 ×10 ×10 / ×10-5 y

0 /N ×10 6 ×10 ×10 My 0 ×10 and their amplitudes gradually decrease.y The maximum impact-rubbing force is 872 N. 0 F

6 8 My /m

F 0 -1 F

y 8 /m F

-1 4 FMy y -1 The4 eddy phase mainly occurs after 0.06 s, and the displacement-1 graduallyFMy converges m·s 6 -1 / 2 -1 FMy,1 m·s

y 6

/ 2 Max:872 N FMy,1 v at abouty −26 µm, which is larger than theMax:872 unilateral N air gap, indicating that the rotor will v 0 -2 FMy,2 0 4 FMy,2 -2 eventually stagnate at the bottom of4 the inner ring. The phase-2 trajectory presents a double- -2 2 -3 Displacement -3 periodic-4 closed circle from the outside2 to the inside, which indicates the displacement tends Ve loc ity -5 Displacement -3 -4 Ve loc ity ×10-5 -6 ×10 0 -4 02351 4 to converge.-6 The variation of the0 impact-rubbing02351 4 force/electromagnetic-402351 force4 is similar 02351 4 -1 -2.68 -2.64 -2.60 02351 4 -1 02351 4 -1 ×10-1 -2.68 -2.64 -2.60 Impact-rubbing Force ×10-1 ×10-1 Impact-rubbing Force Impact-rubbing Force Time t/s ×10 Displacment y/m Time t/s ×10 Force Electormagnetic Time −t/s ×10 Time t/s to that of theDisplacment displacement, y/m respectively convergingTime t/s at about Force Electormagnetic 126 N andTime t/s26 N, and their vector sums are just in balance with the rotor’s gravity. (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (d) Electromagnetic force

Processes 2021, 9, x FOR PEER REVIEW 9 of 15

N

-5 -4 2 / ×10 /N ×10 y ×10 0 8 My FMy -1 F F 1 4 FMy,1 /m

m·s 6 y /

-1 y FMy,2 v Max:802 N 0 4 -2 0

Ve loc ity 2

-3 Ve loc ity-4 -5 ×10 0 -1 0231 4 5 -2.64 -2.60 -2.56 02351 4 02351 4 -1 -1 -1 Time t/s ×10 Displacement y/m Impact-rubbing Force Time t/s ×10 时间t/s ×10 ForceElectromagnetic (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (d) Electromagnetic force

Figure 11. Operating law of the system under the failure of the upper and lower units.

According to Figure 11, the bouncing stage mainly occurs within 0.1 s. The rotor dis- placement, impact-rubbing force, and residual electromagnetic force shake violently, and their amplitudes gradually decrease. The maximum impact-rubbing force is 802 N. The eddy stage mainly occurs after 0.1 s. In this stage, the displacement gradually converges at about −26 μm, which indicates the rotor will eventually stagnate at the bot- tom of the fixed ring. The phase trajectory presents a double-periodic closed circle from the outside to the inside, which indicates that the displacement tends to converge. There is no residual electromagnetic force, and the variation of the impact-rubbing force is sim- ilar to that of the displacement, converging at about 100 N, and the vector value is just in balance with the rotor’s gravity.

3.2. Impact-Rubbing Dynamics Behavior of the Rotor-Ball Bearings Supporting System 1. Impact-rubbing behavior law under upper unit failure In the upper unit failure mode, the axis trajectory of the rotor falling in the deep groove ball bearing is shown in Figure 12.

Figure 12. Axial trajectory under upper unit failure.

Similarly, the impact-rubbing phenomenon of the rotor can be divided into the bouncing stage and eddy stage, when the upper unit fails. When the rotor falls directly on the inner ring of the ball bearing, it bounces and collides firstly and then vortexes repeat- Processes 2021, 9, 1105 edly forward/backward. 10 of 15 ProcessesProcesses 2021 2021, 9, ,9 x, xFOR FOR PEER PEER REVIEW REVIEW 1010 of of 15 15 Similarly, the rotor displacement, phase trajectory, rubbing force, and electromag- netic force were extracted, as shown in Figure 13.

Figure 13. Operating law under upper unit failure. Figure-4 13. Operating law under upper2 unit failure. N -5 / ×10 /N ×10 ×10 0 ×10 y 6 My 0

F 8 /m F -1 y 4 FMy -1 According to Figure 13, the bouncing stage mainly occurs-1 within 0.06 s. The rotor Accordingm·s to Figure 13, the6 bouncing stage mainly occurs within 0.06 s. The rotor

/ 2 FMy,1 y Max:872 N v displacement,0 impact-rubbing force, and residual electromagnetic force shakeFMy, violently,2 -2 displacement, impact-rubbing force,4 and residual electromagnetic-2 force shake violently, andand their their-2 amplitudes amplitudes gradually gradually decrease. decrease. The The maximum maximum impact-rubbing impact-rubbing force force is is 872 872 N. N. 2 -3

Displacement -3 -4 TheTheVe loc ity eddy eddy phase phase mainly mainly×10-5 occurs occurs after after 0.06 0.06 s, s, and and the the displacement displacement gradually gradually converges converges -6 0 -4 02351 4 02351 4 02351 4 -1atat about about-2.68 − 26−26 μ μm,-2.64m, which which -2.60 is is larger larger than than the the unilateral unilateral air-1 air gap, gap, indicating indicating that that the the -1rotor rotor will will ×10 Impact-rubbing Force ×10 ×10

Time t/s Displacment y/m Time t/s Force Electormagnetic Time t/s eventuallyeventually stagnate stagnate at at the the bottom bottom of of the the inner inner ring. ring. The The phase phase trajectory trajectory presents presents a adouble- double- (a) Displacement periodicperiodic( bclosed )closed Phase circle trajectorycircle from from the the (outside coutside) Impact-rubbing to to the the inside, inside, force which which(d) Electromagneticindicates indicates the the displacement forcedisplacement tendstends to to converge. converge. The The variation variation of of the the impact impact-rubbing-rubbing force/electromagnetic force/electromagnetic force force is is sim- sim- ilarilar to to thatFigure that of of 13.the theOperating displacement, displacement, law under respectively respectively upper unit converging failure.converging at at about about 126 126 N N and and − 26−26 N, N, and and theirtheir vector vector sums sums are are just just in in balance balance with with the the rotor’s rotor’s gravity. gravity. 2. Impact-rubbing behavior law under upper and lower unit failure 2.2. Impact-rubbing Impact-rubbing behavior behavior law law under under upper upper and and lower lower unit unit failure failure InInIn upper upper upper and and and lower lower lower unit unit unit failure failure failure mode, mode, mode, the the axis axis trajectory trajectory trajectory of of the the rotor rotor falling falling is is shown shown ininin Figure Figure Figure 14.14 14..

FigureFigureFigure 14. 14. 14. AxialAxial Axial trajectory trajectory trajectory under under under uppe upper upper rand and and lower lower lower unit unit failure. failure.

Similarly,Similarly,Similarly, thethe the impact-rubbingimpact-rubbing impact-rubbing phenomenon phenomenon phenomenon of theof of the rotorthe rotor rotor can becan can divided be be divided divided into the into into bounc- the the bouncingingbouncing stage stage and stage eddy and and eddy stage, eddy stage, whenstage, when thewhen upper the the upper upper unit fails.unit unit fails. Whenfails. When When the rotorthe the rotor rotor falls falls directlyfalls directly directly on theon on theinnerthe inner inner ring ring ofring the of of the ball the ball bearing,ball bearing, bearing, it bounces it it bounces bounces and and collidesand collides collides firstly firstly firstly and and thenand then then vortexes vortexes vortexes repeatedly repeat- repeat- forward/backward. edlyedly forward/backward. forward/backward. Similarly, the rotor displacement, phase trajectory, rubbing force, and electromagnetic Similarly,Similarly, the the rotor rotor displacement, displacement, phase phase tr trajectory,ajectory, rubbing rubbing force, force, and and electromag- electromag- force were extracted, as shown in Figure 15. neticnetic force force were were extracted, extracted, as as shown shown in in Figure Figure 15. 15. N N -5 -3 2 / -3 / -5 /N 2

×10 /N ×10 y ×10 ×10 My 0 ×10 1 8 y ×10 FMy My 0 -1 1 8 FMy F -1 F F /m F

/m 1 y 1 FMy,1 y FMy,1 m·s 6 m·s / 6 / -1 y

-1 y FMy,2 v FMy,2 v 0 0 Max:788Max:788 N N 4 4 0 -2-2 0 2 2 -1 Ve loc ity Displcement Displcement -1 -1 Ve loc ity Displcement -3Displcement -1 -1 -5 ×10 -3 ×10 -1 ×10 -5 ×10 ×10 ×10 0 0 -1-1 0230231 1 4 4 5 -2.66 -2.58 -2.50 023502351 1 4 4 023502351 1 4 4 5 -2.66 -2.58 -2.50 -1 -1 Impact-rubbing Force Impact-rubbing Force ×10 Time t/s Impact-rubbing Force Time t/s Time t/s Time t/s Displacement y/m Time t/s ×10 ForceElectromagnetic Time t/s Displacement y/m ForceElectromagnetic (a()a Displacement) Displacement (b) (b) Phase Phase trajectory trajectory (c ()c Impact-rubbing) Impact-rubbing force force (d ()d Electromagnetic) Electromagnetic force force

FigureFigure 15. 15. Operating Operating law law under under upper upper and and lower lower unit unit failure. failure.

AccordingAccordingAccording to toto Figure FigureFigure 15, 15,15 the, the the bouncing bouncing bouncing stage stage stage ma ma mainlyinlyinly occurs occurs occurs within within within 0. 0.0202 0.02s. s. Both Both s. Boththe the rotor rotor the displacementrotordisplacement displacement and and impact-rubbing impact-rubbing and impact-rubbing force force shak forceshake eviolently. shake violently. violently. The The amplitude amplitude The amplitude decreases decreases decreases gradu- gradu- ally,gradually,ally, and and the the and maximum maximum the maximum impact-rubbing impact-rubbing impact-rubbing force force value force value valueis is 788 788 isN. N. 788 N. TheTheThe vortex vortex vortex stage stage stage mainly mainly mainly occurs occurs after after 0.02 0.02 s. s. The The The displacement displacement displacement gradually gradually gradually converges converges at at aboutaboutabout −− 26−2626 μ µμmmm in inin this thisthis stage, stage,stage, indicating indicatingindicating that thatthat the thethe rotor rotorrotor will will will eventually eventually eventually stagnate stagnate stagnate at at the the the bottom bottom ofofof the the inner inner ring. ring. The The The phase phase trajectory trajectory presen presents presentsts a adouble-periodic double-periodic closed closed circle circle from from the the outsideoutsideoutside to to to the the the inside, inside, inside, which which which indicates indicates indicates that that the the displacement displacement tends tends to to converge. converge. There There is is no no residualresidualresidual electromagnetic electromagnetic electromagnetic force. force. The The The variation variation variation of of the the the impact-rubbing impact-rubbing impact-rubbing force force is is similar similar to to that that

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of the displacement, respectively converging at about 100 N, and the vector value is just ofof thethe displacement, displacement, respectively respectively converging converging at at about about 100 100 N, N, and and the the vector vector value value is just is just in in balance with the weight of the rotor. balancein balance with with the the weight weight ofthe of the rotor. rotor. 3.3.3.3. Impact-Rubbing Impact-Rubbing Dynamics Dynamics Behavior Behavior of of the the Rotor-Hydrostatic Rotor-Hydrostatic System System 3.3. Impact-Rubbing Dynamics Behavior of the Rotor-Hydrostatic System 1. 1. Impact-rubbing Impact-rubbing behavior behavior law law under under upper upper unit unit failure failure 1. Impact-rubbing behavior law under upper unit failure In upper unit failure mode, the axis trajectory of the rotor falling in the hydrostatic InIn upperupper unitunit failurefailure mode,mode, thethe axisaxis trajectorytrajectory ofof thethe rotorrotor falling falling in in the the hydrostatic hydrostatic system is shown in Figure 16. systemsystem isis shownshown inin FigureFigure 16 16..

-5 6 ×10-5 6 ×10 4

4/m y /m

y 2 2 0 0 -2

-2Displacement

Displacement -4 -4 -5 ×10-5 -6 ×10 -6 -6-4 -2 0 26 4 -6-4 -2Displacement 0 26 x 4/m . Displacement x/m . Figure 16. Axial trajectory under upper unit failure. FigureFigure 16.16.Axial Axial trajectorytrajectory underunder upperupper unitunit failure.failure. According to Figure 16, when the upper unit fails, the rotor directly falls to the mag- AccordingAccording to Figure 16,16 ,when when the the upper upper unit unit fails, fails, the therotor rotor directly directly falls fallsto the to mag- the netic pole with the contact and friction of a small tremor. magneticnetic pole pole with with the thecontact contact and andfriction friction of a ofsmall a small tremor. tremor. In order to analyze the impact-rubbing situation in the rotor-hydrostatic system, the InIn orderorder to analyze analyze the the impact-rubbing impact-rubbing situation situation in inthe the rotor-hydrostatic rotor-hydrostatic system, system, the rotor displacement, phase trajectory, rubbing force, electromagnetic force, hydraulic re- therotor rotor displacement, displacement, phase phase trajectory, trajectory, rubbing rubbing force, force, electromagnetic electromagnetic force, force, hydraulic hydraulic re- sistance, and hydrostatic force were extracted, as shown in Figure 17. resistance,sistance, and and hydrostatic hydrostatic force force were were extracted, extracted, as as shown shown in in Figure Figure 17. 17 .

-5 -6 2 /N ×10

×10 y -5 ×10-6 1.6 2 -1 /N F /m 0 1 ×10 ×10 ×10 y 1.6 159 N y -1 F /m 0 1 159 N y m·s 1.2 / y

m·s 1.2 v

-2 / y 0 -2 v 0.8 0 0.8 -4 -4 0.4

Ve loc-1 ity Displacement 0.4

Ve loc-1 ity Displacement -6 0.0 -6 02351 4 -5.19 -5 0.0 02351 4 -1 ×10-5 -1 02351 4 -5.19 Impact-rubbing Force 02351 4 Time t/s ×10-1 Displacement y×10/m Time t/s ×10-1 Impact-rubbing Force Time t/s ×10 Displacement y/m Time t/s ×10 (a) Displacement (b) Phase trajectory (c) Impact-rubbing force (a) Displacement (b) Phase trajectory (c) Impact-rubbing force N / ×10 14 2

N ×10 ×10 N F / My 14 2 Ly

0 / ×10 ×10 14 2×10 N F 2.1×10 Ly F My Ly 0 14 / 2 FLy,1 F (N·s/m)

F 2 2.1×10 -2 Ly FMy /

h FLy,1 F -2 (N·s/m) 2 FLy,2 /

FMy R -4 FMy,1 h Rh1 1 FLy,2 -4 FMy,1 R Rh1 FMy,2 R 1 -6 F 1 h2 -6 My,2 1 Rh2 -8 34 N 0 34 N -8 9 0 -10 2.0×109 -10 0 2.0×10 02351 4 02351 4 Force Hydrostatic 02351 4 -1 0 -1 -1 02351 Time t/s4 ×10 02351 4 Force Hydrostatic 02351 4

Electromagnetic ForceElectromagnetic ×10 ×10 Liquid Resistance Resistance Liquid Time t/s Time t/s Time t/s ×10-1 -1 -1

Electromagnetic ForceElectromagnetic ×10 ×10 Resistance Liquid Time t/s Time t/s (d) Electromagnetic force (e) Liquid resistance (f) Hydrostatic Force (d) Electromagnetic force (e) Liquid resistance (f) Hydrostatic Force Figure 17. Operating law under upper unit failure. FigureFigure 17. 17.Operating Operating lawlaw underunder upper upper unit unit failure. failure. According to Figure 17, the drop stage mainly occurs within 0.02 s. The rotor dis- AccordingAccording to to Figure Figure 17 17,, the the drop drop stage stage mainly mainly occurs occurs within within 0.02 0.02 s. The s. rotorThe rotor displace- dis- mentplacement decreases decreases rapidly, rapidly, which meanswhich themeans rotor the has rotor a high has speed,a high speed, resulting resulting in a rapid in a riserapid placement decreases rapidly, which means the rotor has a high speed, resulting in a rapid inrise the in hydrostatic the hydrostatic supporting supporting force FforceLy. The FLy. fluid The resistancefluid resistanceRh1 of R theh1 of upper the upper supporting support- rise in the hydrostatic supporting force FLy. The fluid resistance Rh1 of the upper support- uniting almost unit almost does does not change, not change, but thebut fluidthe fluid resistance resistanceRh2 Rofh2 theof the lower lower supporting supporting unit unit ing unit almost does not change, but the fluid resistance Rh2 of the lower supporting unit increasesincreases due due to to the the dynamic dynamic extrusion extrusion effect.effect. The residual residual electromagnetic electromagnetic force force FFMyMy de- increases due to the dynamic extrusion effect. The residual electromagnetic force FMy de- decreasescreases rapidly. rapidly. creases rapidly. TheThe impact-rubbing impact-rubbing contact contact stage stage mainly mainly occurs occurs after after 0.02 0.02 s, s, andand thethe rotorrotor reachesreaches a The impact-rubbing contact stage mainly occurs after 0.02 s, and the rotor reaches a a newnew equilibriumequilibrium state:state: thethe phasephase trajectorytrajectory shows shows the the rotor rotor displacement displacement can can achieve achieve new equilibrium state: the phase trajectory shows the rotor displacement can achieve

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about −51.9 μm and the rotor vibrates periodically at this position at a low speed. As the − µ contact depthabout between −51.951.9 the μ mrotorm and and and the the statorrotor rotor vibratesin vibratescreases periodically periodically gradually, theat at this impact-rubbing position at a force low speed. As the contact depth between the rotor and stator increases gradually, the impact-rubbing force Fn increases tocontact about depth159 N, between which can the balance rotor and the stator rotor inweightcreases with gradually, the residual the impact-rubbing electro- force Fn increasesMy to about 159 N, which can balanceLy the rotor weight with the residual electro- magnetic forceFn Fincreases of −93 toN aboutand the 159 hydrostatic N, which canforce balance F of 34the N. rotor The weight liquid resistancewith the residual of electro- magnetic force FMy of −93 N and the hydrostatich1 forceh2FLy of 34 N. The liquid resistance of the upper supportingmagnetic unit force and FMy the of lower−93 N supportingand the hydrostatic unit R and force R F Lyrespectively of 34 N. The stabilize liquid resistance of 9 the upper supporting14 unit and the lower supporting unit Rh1 and Rh2 respectively stabilize at 2.0 × 10 N·s/mthe upper and 2.1 supporting × 10 N·s/m. unit and the lower supporting unit Rh1 and Rh2 respectively stabilize at 2.0 × 109 N·s/m and 2.1 × 1014 N·s/m. 2. Impact-rubbingat 2.0 × 10 behavior9 N·s/m lawand under2.1 × 10 upper14 N·s/m. and lower unit failure. In upper2. and Impact-rubbingImpact-rubbing lower unit failure behavior mode, law the underaxis trajectory upper and of lowerthe rotor unit falling failure. in the hydrostatic systemInIn is upper shown and in Figurelower unit18. failure mode, the ax axisis trajectory of the rotor falling in the hydrostatic system is shown in Figure 1818..

6 ×10-5

4 6 -5

/m ×10 y 2 4 /m y 0 2

-2 0 Displacement -4 -2 -5 Displacement ×10 -6 -4 -6-4 -2 0 26 4 -5 Displacement x/m ×10 -6 -6-4 -2 0 26 4 Figure 18. Axial trajectoryDisplacement under uppe x/m r and lower unit failure. Figure 18. AxialAxial trajectory under uppe upperr and lower unit failure. According to Figure 18, when both the upper and lower units fail, the rotor directly falls to the magneticAccording pole and to collides Figure with18,18, when a little both tremor. the upper and lower units fail, the rotor directly Similarly,fallsfalls the to rotor the displacement,magnetic pole phaseand collides trajectory, with impact-rubbing a little tremor. force, electromag- netic force, hydraulicSimilarly, resistance, the rotor and displacement, hydrostatic force phase phase are traj trajectory, extracted,ectory, impact-rubbing impact-rubbing as shown in Figure force, force, electromag- electromag- 19. netic force, hydraulichydraulic resistance,resistance, and and hydrostatic hydrostatic force force are are extracted, extracted, as as shown shown in Figurein Figure 19. 19. -5 -6 ×10 /N ×10 ×10 y -1 F /m 0 1 6 66 N y -5 -6 /N m·s ×10 ×10 ×10 y / y -1 F /m 0 -2 v 1 6 66 N y 0 4 m·s / y

-2 v -4 0 2 4

Velocity -1 Displacement -6 -4 0 2 -5 02351 4 -5.09Velocity -1 02351 4 Displacement -1 ×10 -1 Time t/s-6 ×10 Displacement y/m Impact-rubbing Force Time t/s0 ×10 02351 4 -5.09 -5 02351 4 -1 ×10 -1 (a) Displacement Time t/s(b)×10 Phase trajectory Displacement( cy/m) Impact-rubbing Impact-rubbing Force force Time t/s ×10

N 13

/ 13

(a) Displacement ×10 (b)6.7×10 Phase trajectory N ×10 (c) Impact-rubbing force / My FMy 12 FLy Ly F

N 6 1 F 13 FLy,1 / F , 13 My 1 (N·s/m) / ×10 6.7×10 N ×10 / h

My F FLy My R 12 FLy,2 R F h1 8 Ly F My,2 6 1 4 F , F FLy,1 My 1 (N·s/m)

/ Rh2 0 h R FLy,2 FMy,2 R 4 h1 8 2 4 R 34 N 0 h2 2.0×109 0 4 -1 0 2 34 N 02351 4 02351 4 HydrostaticForce 902351 4 -1 -1 2.0×10 0 -1 Time t-1/s ×10 Time t/s ×10 ×10 Electromagnetic ForceElectromagnetic Resistance Liquid 0 Time t/s 02351 4 02351 4 HydrostaticForce 02351 4 (d) Electromagnetic force Time t(/se) Liquid×10-1 resistance Time t/s (f) Hydrostatic×10-1 Force ×10-1

Electromagnetic ForceElectromagnetic Resistance Liquid Time t/s (d) Electromagnetic force (e) Liquid resistance (f) Hydrostatic Force Figure 19. OperatingFigure law 19. Operatingunder upper law and under lower upper unit and failure. lower unit failure. Figure 19. Operating law under upper and lower unit failure. According to AccordingFigure 19, tothe Figure drop stage19, the mainly drop stageoccurs mainly within occurs 0.04 s. within In this 0.04stage, s. Inthe this stage, the rotor displacementrotorAccording displacement decreases torapidly, Figure decreases which 19, the rapidly, in dropdicates which stage the mainly indicatesrotor has occurs thea high rotor within speed, has 0.04 aresulting high s. In speed, this stage, resulting the in a rapid riserotorin in a the rapid displacement hydrostatic rise in the decreasessupporting hydrostatic rapidly, force supporting F whichLy. The force influiddicatesF resistanceLy. Thethe rotor fluid Rh 2has resistance of thea high upper Rspeed,h2 of theresulting upper supporting unit shows little change, but the fluidh resistance2 R of the lower supporting supporting unitin a shows rapid riselittle in change, the hydrostatic but the fluidsupporting resistance force R FLy of. The the fluid lowerh resistance2 supporting Rh2 of the upper unit increases rapidly due to the dynamic extrusion effect. The resultant electromagnetic unit increasessupporting rapidly due unit to theshows dynamic little change,extrusion but effect. the fluid The resultantresistance electromagnetic Rh2 of the lower supporting force FMy is 0.unitforce increasesFMy is 0. rapidly due to the dynamic extrusion effect. The resultant electromagnetic force FMy is 0.

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The collision stage mainly occurs after 0.04 s, and the rotor reaches a new equilibrium state. The phase trajectory shows the rotor displacement can achieve about −51.9 µm and the rotor vibrates periodically at this position at a low speed. With the increase of the contact depth between the rotor and the stator, the impact-rubbing force Fn increases to about 66 N, which can balance the rotor weight with the hydrostatic force FLy of 34 N. The liquid resistance of the upper supporting unit and the lower supporting unit Rh1 and Rh2 respectively stabilize at 2.0 × 109 N·s/m and 6.7 × 1013 N·s/m.

3.4. Comparison of the Three Protection Devices in Impact-RUBBING Behavior 1. Analysis of the impact-rubbing behavior under upper unit failure. According to Figures8, 12 and 16, it can be seen that under upper unit failure mode, the axis trajectories of the rotor falling in the fixed ring/deep groove ball bearings are basically the same. They both remain in the bouncing stage for a long time, followed by the forward/backward eddy stage, with a low value. Due to the dynamic extrusion effect of oil, only small tremors occur near the falling position of the rotor when it falls in the hydrostatic system, and there is no complex bouncing and eddy phenomena. According to Figures9, 13 and 17, it can be seen that the rotor displacement falling in the hydrostatic system is relatively stable in comparison with the fixed ring and deep groove ball bearing protection devices. The phase trajectory is no longer a double-periodic closed circle from the outside to the inside, but a periodic superposition with the same frequency. Compared with the fixed ring and deep groove ball bearings, the maximum impact force respectively decreases by 81.9% and 81.8% in the hydrostatic system. The variation of the residual electromagnetic force is gentler. 2. Analysis of the impact-rubbing behavior under upper and lower units. According to Figures 10, 14 and 18, it can be seen that under upper and lower unit failure mode, the axis trajectory of the rotor falling in the fixed ring firstly remains for a long time in the bouncing stage and then presents a low value for the forward/backward vortex phenomenon. A short bouncing phenomenon occurs in the case of falling in the deep groove ball bearing, and then presents a longer lasting time for the forward/backward vortex phenomenon. Due to the dynamic extrusion effect, only the small tremor occurs near the drop position in the case of falling in the hydrostatic system, and there is no complex bouncing and eddy phenomena. According to Figures 11, 15 and 19, compared with the fixed ring and deep groove ball bearing protection devices, the displacement of the rotor falling in the hydrostatic system changes more stably, and only a slight tremor occurs under upper and lower unit failure mode. The phase trajectories show the form of reciprocating motion with the same frequency. The maximum impact force is reduced by 91.8% and 91.6%, and the degree of impact of the bounce and eddy is greatly reduced. 3. Analysis of results. In summary, the influence of the three protection devices on rotor impact-rubbing under electromagnetic failure mode is shown in Tables2 and3.

Table 2. Influence of upper unit failure on impact-rubbing.

Protective Devices Bounce Time Eddy Max. Impact-Rubbing Force Impact-Rubbing Parts

Fixed ring <0.07 s yes 880 N rotor and fixed ring

Ball bearing <0.06 s yes 872 N rotor and inner ring

Hydrostatic system none none 159 N rotor and pole Processes 2021, 9, 1105 14 of 15

Table 3. Influence of upper and lower unit failure on impact-rubbing.

Protective Devices Bounce Time Eddy Max. Impact-Rubbing Force Impact-Rubbing Parts

Fixed ring <0.1 s yes 802 N rotor and fixed ring

Ball bearing <0.02 s yes 788 N rotor and inner ring

Hydrostatic system none none 66 N rotor and pole

4. Conclusions 1. The influences of the protection devices on the bouncing time, impact-rubbing force, and impact-rubbing degree are found to be as follows: the hydrostatic system shows the best results, followed by the deep groove ball bearing and then the fixed ring. 2. Compared with the fixed ring and deep groove ball bearing, the positions of the rotor impact-rubbing in the hydrostatic system are the rotor and magnetic pole, and the maximum impact-rubbing force is lower without bounce and eddy phenomena. 3. Compared with the bounce time and the maximum impact-rubbing force of the rotor in the three protective devices, it can be seen that there is no bounce phenomenon of the rotor in the hydrostatic system and the maximum impact-rubbing force is greatly reduced, so the protective bear is not required to be installed in the MLDSB.

Author Contributions: Conceptualization, J.Z.; Data curation, J.Z.; Methodology, J.Z.; Project admin- istration, J.Z.; Resources, J.Z. and W.Y.; Software, L.X.; Supervision, D.G. and G.D.; Writing—original draft, L.X.; Writing—review and editing, L.X. and S.L. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Nature Science Foundation of China (no. 52075468); General project of Natural Science Foundation of Hebei Province (E2020203052); Youth Fund Project of scientific research project of Hebei University (QN202013); Open Project Funding of Jiangsu Provincial Key Laboratory of Advanced Manufacture and Process for Marine Mechanical Equipment and Open Project Funding of Fluid Power Transmission Control Laboratory of Yanshan University. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

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