A radial clearance adjustable bearing reduces the vibration response of the rotor system during acceleration

Journal Pre-proof A radial clearance adjustable bearing reduces the vibration response of the rotor system during acceleration Lei Zhang, Hua Xu, Shenglun Zhang, Shiyuan Pei PII:

S0301-679X(19)30626-7

DOI:

https://doi.org/10.1016/j.triboint.2019.106112

Reference:

JTRI 106112

To appear in:

Tribology International

Received Date: 18 November 2019 Accepted Date: 8 December 2019

Please cite this article as: Zhang L, Xu H, Zhang S, Pei S, A radial clearance adjustable bearing reduces the vibration response of the rotor system during acceleration, Tribology International (2020), doi: https:// doi.org/10.1016/j.triboint.2019.106112. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

*Author Contributions Section Lei Zhang is the most important participant in this study. His work includes design and evaluation of adjustable bearings, dynamic analysis of rotor bearing systems, and design and construction of experimental test rig, and then collecting and analyzing experimental data. Conceptualization, Methodology, Software, Data curation, Writing-Original draft preparation, Writing- Reviewing and Editing. Hua Xu is the instructor of Lei Zhang and provides a lot of guidance and advice for this research work and provides financial support. Supervision. Shenglun Zhang assisted Zhang Lei to build the experimental test rig and collect and analyze experimental data. Validation. Shiyuan Pei provides guidance in the theoretical calculation of rotor dynamics. Validation, Investigation.

A radial clearance adjustable bearing reduces the vibration response of the rotor system during acceleration Lei Zhanga,b, Hua Xua,b, Shenglun Zhanga,b，Shiyuan Peia,b a Key Laboratory of Education Ministry for Modern Design & Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 710049, PR China b School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China ARTI C LE I N F O

Keywords: radial clearance adjustable bearing rotor dynamics vibration suppression

AB S TRACT

A radial clearance adjustable bearing is proposed in this study. The structure and working principles of the adjustable bearing are introduced. This adjustable bearing can change the dynamics characteristics of the bearing by adjusting the radial clearance. In this paper, a simple rotor bearing finite element model is used to study the vibration response of the rotor system. When the rotational speed does not reach the critical speed, reducing the radial clearance can effectively reduce the vibration of the rotor, and the vibration suppression effect can reach 67%. When the rotational speed approaches the critical speed, increasing the radial clearance can significantly reduce the resonance amplitude of the rotor, and the vibration suppression effect reaches about 37%. The amplitude response of the system decreases almost equally in the vertical and horizontal directions. In addition, when the rotational speed increases to twice the critical speed, the oil whip occurs under light load condition, and the oil film instability can be suppressed by increasing the radial clearance. The current study has achieved manual adjustment of the bearing’s clearance; furture work includes developing an intelligent controller that enables active adjustment of the clearance.

1. Introduction Due to the imbalance of the rotating machinery, the vibration amplitude will change during the acceleration process, showing complex dynamic characteristics. Conventional fixed pad bearings are designed for specific operating conditions. When the speed and other working conditions change, the bearing cannot make adjustment response according to the actual working conditions, which may lead to safety problems, such as excessive vibration of the rotor and bearing damage. As industrial and manufacturing industries now place higher demands on the accuracy and stability of rotating machinery, the focus of research is shifting to bearing lubrication and performance. In order to improve the stability of rotating machinery, many concepts of adjustable bearings and controllable bearings have been proposed, such as movable bearing pad bearings [1-2], electromagnetic bearings[3-8], and flexible sleeve bearings[9-10]. Various vibration control theories are also applied to vibration suppression, such as the optimal regulator theory (Nonami [11]), the disturbance observer theory (Reinig and Desrochers [12]), or the variable velocity feedback method (Nonami and Zhen [13], Yang and Sheu[14]), and the discontinuous spring characteristics method (Ishida and Liu [15]). Theoretical and experimental studies have proved that adjustable or controllable bearings can significantly improve the traditional fluid bearings, and some effective methods for

suppressing vibration have been recognized. Many studies on the suppression of the vibration of adjustable controllable bearings are presented in the literature. For example, using the oil injection pressure to control the dynamic coefficient of the bearing (Santos [16-18]); the external force is applied to the rotor or the bearing by the piezoelectric machine to suppress the vibration of the rotor [19-21]. A relatively simple method is to increase the bearing damping by elastic support of the leaf spring elements (Tallian and Gustafsson [22], Itou [23]). The dynamic vibration absorber theory is applied to the elastic support rotor to suppress the vibration of the rotor (Kirk and Gunter [24], Ota and Kanbe [25]). In addition, there are mechanisms for reducing the vibration of the rotating machine by absorbing vibration energy, such as, seal dampers [26], squeeze-film dampers [27], and hybrid squeeze-film dampers [28]. Active journal bearing devices that improve the dynamics of rotating machinery, include for example, hydraulic actuator journal bearings (Althaus and Ulbrich[29]), variable impedance bearings (Goodwin et al.[30]), deformable bushings (Kicinski and Materny[31]), and journal bearings with flexible sleeves (Sun et al. [32]). Chasalevris and Dohnal [33-34] propose a passive adjusting bearing mechanism which does not require any additional operation. When the oil film force exceeds a certain limit, the bearing pad will move to produce additional oil film. The working principle is based on a combination of vibration

suppression using discontinuous spring characteristics [15] and changes in bearing characteristics caused by additional fluid films [35-36]. Chasalevris and Dohnal’s [34] latest theoretical and experimental studies have found that the adjustable bearing is theoretically applicable to large rotor bearing systems and is then validated in manufacturing-related industries. Therefore, it is confirmed that the bearing pad adjustable bearing not only does not affect the normal operation of the rotor system, but also can significantly improve the stability of the rotor system. One of my research focuses in this study is the design of adjustable bearing structure and its working principle. Although the bearing is also an adjustable bearing, its structure and function are different from those proposed in the previous literature. The bearings designed in the literature adjust the clearance of the bearings in a passive manner. When the fluid film forces exceed a preload of an external spring, the bearing bush moves downward to increase the bearing clearance. The bearing designed in this article is a semi-active adjustment bearing which can adjust the bearing clearance arbitrarily according to the vibration of the rotor. At present, the academic community mainly performs the optimization of the elastic support system and active vibration control algorithm to suppress the vibration of the rotor. These vibration suppression methods are based on hydraulic or electromagnetic systems and have made some research results. This study designed a radial clearance adjustable bearing that can change the vibration response of the rotor system by adjusting the clearance. This type of

bearing relies on the mechanical structure to adjust the journal bearing clearance and enhance the bearing's ability to adapt to changing working conditions. This bearing concept inherits the characteristics of journal bearings and electromagnetic bearings. It has the advantages of large bearing capacity and large damping of the journal bearing, and also has the active control capability of the electromagnetic bearing. The purpose of this type of radial clearance adjustable bearing is to put forward a new concept that the bearing has the ability of independent adjustment and can make adjustment response according to working conditions to improve the stability of the rotor system. The vibration of the rotor system is reduced by adjusting the bearing radial clearance to ensure that the vibration response of the rotor is always within safe limits during acceleration. Designing the mechanical structure to adjust the bearing clearance is key to this research. The bearing radial clearance is very small (it is one thousandth of the bearing diameter), and it is very difficult to adjust it directly. Bearings designed in the previous literature only increase the bearing clearance and cannot control the amount of bearing clearance. In this study, the bearing clearance is precisely adjusted by the mechanical structure, so the clearance size of the bearing can be accurately set. The precise adjustment of the bearing clearance allows a more intuitive study of the effect of bearing clearance on rotor vibration. In future research, it is necessary to continuously optimize the adjustment structure and adjustment method, which will be the development

direction of bearing research. In this study, a radial clearance adjustable bearing structure is designed firstly, and then the influence of bearing clearance on the vibration response of the rotor system is studied under various working conditions. The purpose of these studies is to determine whether the adjustable bearing can improve the vibration of the rotor system and how to adjust the bearing clearance to improve the vibration of the rotating machine. This research process is an attempt to build a database to find a way to adjust the bearing clearance to improve the vibration of the rotating machinery. For different rotor vibration conditions, find the corresponding clearance adjustment method. 2. Design and operation flow of the radial clearance adjustable bearing

Fig. 2. Work flow chart of the radial clearance adjustable bearings

Fig. 1. Structural design of a radial clearance adjustable bearing and adjustable property

The radial clearance adjustable bearing design is shown in Fig.1. The bearing structure is designed with two bearing pads, and the vertical movement of the lower bearing bushing allows for the adjustment of the bearing clearance. The mechanical structure design utilizes a screw mechanism to convert rotational motion into horizontal sliding. The servo motor can accurately position the angle of the rotary motion, and then vertical motion of the bearing bushing can be amplified by the motion amplification mechanism to accurately adjust the bearing clearance. Since the bearing bushing is moved upward from its original position where it forms a circle, the top clearance of the adjustable bearing is smaller than the side clearance. Thus, the upper and lower pads can form an oil wedge, which can produce better operating stability.

The working function of the clearance adjustable bearing is to reduce the vibration response of the rotor system by adjusting the clearance. Fig. 2 is a working flow chart of the radial clearance adjustable bearing. The displacement sensor is mounted on the rotor. If the amplitude of the rotor is too large, the controller issues a command to control the rotation of the servo motor, and the servo motor drives the screw to rotate. The screw then converts the rotary motion into a horizontal motion to achieve a horizontal movement of the inclined iron. Because the inclined iron and the bearing’s lower pad are in inclined contact, the horizontal sliding of the inclined iron can be converted into the vertical sliding of the bearing’s lower pad. The adjustment of the bearing radial clearance is achieved by the series of motion transformations described above. As shown in Fig. 2, it is a closed-loop control process. If the amplitude of the rotor does not meet the requirements, the bearing clearance can be continuously adjusted through the above process until the rotor vibration is optimal.

to the limit position, the top clearance is the same as the side clearance, and the bearing becomes a conventional circular bearing (100%cr). When the bearing’s lower pad moves upward, in order to have good lubrication between the bearing and the journal, it is necessary to ensure that the top clearance of the bearing is not too narrow, so the minimum radial clearance is set to 30% cr (According to the mechanical design manual of the journal bearing).

Slope 1:125

(a) Front view of bearing device

(b) Top view of partly device. Fig. 3. View and description of the physical structure of the radial clearance adjustable bearing device.

Fig. 3 is a view and description of the physical structure of the adjustable bearing device. Creating a design feature for a sleeve bearing that allows for adjusting its diameter is a challenging task. The increment of adjustment for the radial clearance must be at least one order of magnitude smaller than its nominal value. The adjustment of the bearing radial clearance is achieved by the structural design as shown in Fig.3. The bearing ring is divided into two parts: the upper part is fixed with the housing while the lower part can slide up and down. The inclined iron connection is the lead screw and the bearing’s lower part. The inclined iron is used to convert the rotational motion of the lead screw into a vertical sliding of the lower pad. The slope of the plane in contact with the inclined iron and the bearing lower part is 1:125. The lead screw rotates one revolution to drive the inclined iron to move 1mm horizontally and the bearing’s lower pad to slide 0.008mm vertically. Through the mechanical structure design, the adjustment of the small clearance of the bearing can be accurately set. The effect of the bearing radial clearance on the system vibration response is the focus of this paper. In order to study the vibration suppression effect of the radial gap more comprehensively, the clearance adjustment range is set as wide as possible. When the bearing’s lower pad moves down

3. Description of working principle and dynamic characteristics of the radial clearance adjustable bearing The idea of radial adjustable bearing is to improve the stability of rotor system and make the vibration response of the rotor reach a desired result. The effect of bearing radial clearance on the vibration response of the rotor system is a complex dynamic problem. Bearing radial clearance affects the formation of oil film, and different thickness of oil film causes different dynamic characteristics of bearing, which will lead to complex vibration response of rotor system. The first step is to study the dynamic characteristics of the adjustable bearing. The working state diagram of the adjustable bearing is shown in Fig. 4. Since the radial clearance adjustable bearing is composed of two arc-shaped bearing pads, and the top clearance of the adjustable bearing is smaller than the side clearance, when the journal is operating under the load W, there are both a convergent gap and an open gap between the journal and both the upper and lower pads. When there is lubrication oil in the gap, two sections of oil bearing film are formed. The positional relationship between the journal center and the bearing center under normal operation is shown in Fig. 4. The oil film thickness equation is obtained based on the geometric relationship of the triangle O'O1M.

Fig. 4. Schematic diagram of the radial clearance adjustable bearing working state and bearing parameter definition.

R 2 = e1 + (r + h1 ) 2 − 2e1 (r + h1 ) cos ϕ1 2

(1)

2

2

Subtract e1 cos and sort it out:

ϕ from both ends of the above Eq. (1),

h1 = ( R 2 − e1 sin 2 ϕ1 ) − r + e1 cosϕ1 2

(2)

Eq. (2) is expanded using the McLaughlin series.

1 e 1 e1 4 4 ( ) sin φ1 −⋅⋅⋅] − r + e1 cosφ1 h1 = R[1− ( 1 )2 sin2 φ1 + 2 R 2× 4 R (3) 1 e1 1 e1 3 4 2 = cr + e1 cosφ1 − × esin φ1 + ( ) sin φ1 −⋅⋅⋅ 2 R 8 R e In the case of fluid lubrication, 1 ≤ 0.001 , so the R high-order small-quantity is omitted in Eq.(3). The oil film thickness of the lower bearing pad: (4) h1 ≈ cr + e1 cos ϕ1 ,  =  −  The oil film thickness of the upper bearing pad obtained by the analogy method is: (5) h2 ≈ cr + e2 cos ϕ 2 ,  =  −  + 

According to the geometric relationship of the triangle OO'O1 and the triangle OO'O2 in Fig. 4, the following expression can be obtained.

e1 = e 2 + (e') + 2ee' cosθ 2

e2 = e + (e') − 2ee' cosθ e sin θ θ1 = arcsin 2

2

e1

θ 2 = arcsin

e sin θ e2

(6) (7)

conditions. 6. The lubricant properties do not vary substantially throughout the oil film. 7. The shaft does not tilt in the bearing. Therefore, for the finite length bearing used in this paper, the fluid pressure distribution P should satisfy the dimensionless Reynolds equation shown in Eq. (11). ∂ ∂P d  ∂  ∂P ∂h h  +  (h )=3 + 6(εcosφ + εθ sinφ) (11 ∂φ ∂φ l ∂λ ∂λ ∂φ At every discrete time step the fluid pressure distribution of the adjustable bearing can be evaluated by solving Eq. (11) using the finite difference method (FDM). The Reynolds equation given in Eq. (11) is transformed into Eq. (12) to correspond to a finite difference grid. The fluid film pressure is evaluated at a finite difference grid of 20×20 intervals at the circumferential and the axial direction respectively. !( $ )"#$,' *!( $ )"+$,' ,(!( $ *!( $ ))",' "# ,' "+ ,' "# ,' "+ ,' % % % % (∆.)%

!( $ )",'#$ *!( $ )",'+$ ,(!( $ *!( $ ))",' ",'+ ",'# ",'+ /  ",'#% % % % (0) (∆1)% ! $ ,! $

3

"# ,' %

∆.

"+ ,' %



=

(12)

The coefficients in the equation are defined as: 23,4 = ℎ3*,4 

63,4 = ℎ 3,,4 

(8)

8 ∆    ℎ 3,4* 73,4 = 9 ∆:  8 ∆   ;3,4 =  ℎ 3,4, 9 ∆:  <3,4 = 23,4 + 63,4 + 73,4 + ;3,4

(9)

Equations 6, 7, 8, and 9 are introduced into equations 4, 5, respectively, to obtain an oil film thickness expression, see Eq. (10). e sin θ 2  2  cr + e + ( e ') + 2ee 'cosθ cos(ϕ − arcsin e )  1 (10) h = e sin θ 2 c + e2 + ( e ') − 2ee 'cosθ cos(ϕ − π + arcsin )  r e2 After calculating the oil film thickness of the adjustable bearing, the lubrication properties of the bearing can be evaluated using the Reynolds equation. It is assumed that the lubricating oil flowing in the bearing clearance meets the following assumptions. 1. The lubricant in the bearing gap is Newtonian fluid and the flow is laminar. 2. The viscosity and density of the fluid are constant throughout the film. 3. It is assumed that the lubricant is an incompressible fluid. 4. The lubricant pressure is zero at the edges of the bearing, there is no change of pressure in the direction of lubricant thickness. 5.The bearing is operating under steady running

+

=3,4 = 3∆ ℎ3*,4 − ℎ3,,4  



After replacement, the expression of Eq. (12) is obtained as in Eq. (13).

23,4 >3*,4 + 63,4 >3,,4 + 73,4 >3,4* + ;3,4 >3,4, − <3,4 >3,4 = =3,4

(13)

The Eq. (13) is solved by the over-relaxation iterative method to obtain the expression of the oil film pressure as in Eq. (14). This way, the pressure value of the central node (i, j) can be calculated according to the pressure values of the four nodes around the central node (i, j). ? is an over-relaxation iteration coefficient, and its value can generally be selected between 1 and 2. This paper selects 1.5. >3,4 (@) =

βB

23,4 >3*,4 (@,) + 63,4 >3,,4 (@) + 73,4 >3,4* (@,) + ;3,4 >3,4, (@) − =3,4 <3,4 − >3,4 (@,) C + >3,4 (@,) (14)

An iterative procedure is established after setting initial (E)

values of the pressure P3,4

at each node of the grid,

(E)

including boundaries, as P3,4 = 0 , for i=2,3,…20 and (I)

j=2,3,… 20. At the GH! iteration, the pressure P3,4 evaluated as

M (L) ∑O ,)",' (L+$) K 'N% ∑"N%K)",' O M ∑'N% ∑"N%K)",' (L) K

is

≤ 10, , for i=2,3,…20

and j=2,3,… 20. In this way the pressure at the boundaries will remain zero while the pressure of the internal grid will build up at every iteration until the relative error is less than or equal to 10-3. After calculating the fluid film pressure P, the pressure is integrated in the x and y directions to obtain the oil film force in the horizontal direction and the vertical direction of the bearing. The fluid film forces of lower bearing pad in horizontal and vertical direction are evaluated as in Eq. (15). P1 is the fluid film pressure of the bearing lower pad. Fx1 and Fy1 are the fluid film forces in the horizontal and vertical directions of the bearing lower pad, respectively. 

=S =

.

%$> sin( +  )8 8: E .$$  . − TE T. $% > cos( +  )8 8: $$

=Q = − R R

(15)

The fluid film forces of upper bearing pad in horizontal and vertical direction are evaluated as in Eq. (16). P2 is the fluid film pressure of the bearing upper pad. Fx2 and Fy2 are the fluid film forces in the horizontal and vertical directions of the bearing upper pad, respectively.  . =S = TE T. %% > cos( −  )8 8: 

%$

.

=Q = TE T. %% > sin( −  )8 8: %$

(16)

The fluid film force of the adjustable bearing is the sum of upper pad and lower pad, see Eq. (17). =Q = =Q$ + =Q% ，=S = =S$ + =S%



=Q = − R R E

=S =

.

%$.$$

.%%

+R R E



.%$

.

− TE T. $% > cos( +  )8 8: + $$

−  )8 8:

(17)

When there is a convergent area and an open area in the gap between the bearing bush and the shaft diameter, the position where the fluid film naturally ruptures in the open

Vℎ

U)W U.

/  U

X+V X U!

3ℎ[VZ[\ U

U.

Vℎ

U)` U.

3ℎ[Vsin U

U. U.

> sin( −  )8 8:

 . TE T. %% > cos( %$

U

U.

U

> sin( +  )8 8: 

area should be predicted. This fluid film rupture position can be determined by iterative calculations of the Reynolds boundary. In the process of calculating the fluid film pressure point by point from the starting edge to the ending edge, if a point is calculated as a negative pressure, it is taken as zero. The location of this point is the approximate location of the natural rupture of the fluid film. The negative pressures are not included in the numerical integration in Eq. (17). Then through continuous iteration correction, gradually approaching the location of the natural rupture of the fluid film. The calculations in this study use the double Reynolds boundary conditions to calculate the boundary of the fluid film, which means that both the initial and fracture boundaries of the fluid film satisfy the Reynolds boundary condition. The stiffness and damping coefficient of the adjustable bearing are calculated using common methods. The difference is that the dynamic coefficients of the upper and lower tiles need to be calculated separately, and then the vectors are added. The following is a method for calculating the stiffness coefficient of the lower bearing pad. Put the previously calculated fluid film thickness into Reynolds equation, and then get the differential equation of disturbance pressure by solving the partial derivative of Reynolds equation, see equation (18-21). Finally, the dynamic coefficient of fluid film is obtained by integrate the differential equation, see equation (22-25). The calculation process of the upper bearing pad dynamic coefficient is the same and will not be repeated here.

Vℎ Vℎ

U)Wa U.

/  U 0

U.

U1

U1

/  U 0

U1

0

U1

/  U

X+V X .

U)W U1

U)

U.

(ℎ

+ Z[\ℎX

X+V X



(ℎ

+ \^GℎX

X+V X

U!

U)`a U.

U.

0

U1

U.

(ℎ

/ 

+ V X Z[\

U)`

U)

(ℎ

Y

) = −3sin − Z[\ 0

/  0

U1

U1 U1

]

Y

U1 U1

+ (18)

!

U! U)

U!

U.

U!

U.

]

+ (19)

) = 6Z[\

(20)

) = 6sin

(21)

U)`a U1

U! U)

) = 3Z[\ − sin

+ V X sin

U)Wa

!



.

bQQ = − TE T. $% >c cos( +  )8 8: − TE T. %% >c cos( − $$

 )8 8: 

%$

(22)

.



.

bQS = − TE T. $% >c sin( +  )8 8: − TE T. %% >c sin( − $$

 )8 8: 

.

%$

(23) 

.

bSQ = − TE T. $% >d cos( +  )8 8: − TE T. %% >d cos( − $$

 )8 8:

%$

(24)



.



.

bSS = − TE T. $% >d sin( +  )8 8: − TE T. %% >d sin( − $$

 )8 8:

%$

(25)

When the bearing clearance is changed during the calculation, the thickness of the fluid film formed in the gap changes in response, and finally affects the dynamic coefficient of the bearing. Fig. 5 shows the calculated stiffness and damping coefficient as a function of rotational speed under different bearing clearance conditions.

d) Horizontal damping Fig. 5 The stiffness and damping coefficient varies with the rotational speed under different bearing clearance conditions.

a) Vertical stiffness

b) Horizontal stiffness

c) Vertical damping

The critical speed of the rotor system designed in this study is 1500 r/min, so it is only necessary to study the trend of stiffness and damping coefficient from 0 to 3000 r/min. From the change in the curve in Figure 5, the bearing clearance has a large effect on the stiffness and damping coefficient. When the radial clearance increases from 30%cr to 90%cr around the rotational speed of 1500r/min, the vertical stiffness decreases by more than two times and the horizontal stiffness decreases by more than one time. The damping coefficient also has a similar trend, which provides a great possibility for suppressing the critical speed resonance vibration of the rotor system. At other rotational speeds, the stiffness and damping coefficients have similar trends, and the reduction in bearing clearance results in greater stiffness and damping coefficients. This makes it possible to change the vibration characteristics of the rotor at any rotational speed. In addition, the clearance adjustable bearing designed in this study only changes the gap at the top of the bearing when the gap is adjusted, and the side gap does not change, so the dynamic coefficient of the vertical direction changes more than the horizontal direction. This is consistent with the direction of the load on the bearing, which is convenient for controlling the influence of the external load on the rotor system.

4. Dynamics of the radial clearance adjustable bearing in a simple rotor model 4.1 Dynamics model of rotor bearing system The adjustable bearings designed in this study are used to support a simple rotor system(Fig.6). The rotor system is a slender flexible shaft with a multi-stage quality disc installed in the middle to adjust the critical speed of the rotor system. Adjustable bearings are mounted on both sides of the rotor to adjust the stiffness and damping factor of the bearing. The physical and geometric properties of the flexible rotor and bearing are shown in Table 1.

between the two solid red lines is too large, which is the expected area for suppressing vibration.

The equation of motion of the adjustable bearing rotor system is as follows.

efg + hij + k(f, fa )lfa + mf = nop − nq

Fig. 6. A slender flexible rotor model supported by the radial clearance adjustable bearing Table 1 Physical and geometrical properties of the flexible rotor and adjustable bearing.

/ s r t /H sa

=

x s E u,vE+$ w ,v+$ yz{*|(s,sa )}~ rsa t + rv+$ (

(27)

Fig. 7 shows the vibration amplitude response of a rotor supported by a circular bearing during acceleration. For the rotating machine, the vibration amplitude of the rotor will change greatly during acceleration, and the dynamic characteristics of the rotor system will be different at different speeds. For example, when the rotor speed is close to the critical speed, the amplitude of the rotor is extremely large, and excessive vibration may affect the service life of the rotating machine. Alternatively, the external load changes randomly, resulting in a critical speed offset that will affect the stability of the rotor system. When the rotor is unstable, a reasonable method is proposed to reduce the vibration amplitude of the rotor, which is of great significance for the safe operation of the rotating machine. The following work is to study the influence of bearing clearance on the dynamics of the rotor system and to summarize the clearance adjustment method to reduce the vibration of the rotor.

Fig. 7. Vibration response of the rotor system supported by circular bearing during acceleration. The red dotted line indicates the safety threshold of the vibration amplitude. The amplitude of the vibration

(26)

Where M, K, and C are the mass, stiffness, and damping matrixs of the rotor system respectively. q is generalized displacement, Qub is a vector of unbalanced forces, Qg is a vector of gravitational forces, i is the speed of rotation of the rotor. The method for solving Eq. 26 is to solve in the time domain using a time stepping process. The differential equation is rearranged into a state space form (Eq.27), and then the fourth-order Runge-Kutta method is used to solve the equations in the time domain.

t

€ ,‚ )

4.2 The effect of adjustable bearing clearance on rotor vibration characteristics. The rotor exhibits different dynamic characteristics at different speeds, so it is necessary to separately study the influence of bearing clearance on rotor vibration at different speeds. The speeds below and above the critical speed and the critical speed are selected as the research speed points. The critical speed of the rotor system in this study was 1500 r/min. The speed of 1000r/min, 2000r/min and 1500r/min were selected for the study. Fig. 8 shows the vibration response of the journal at different bearing clearances at 1000 r/min. Abx and Aby respectively represent the vibration amplitude of the horizontal and vertical directions of the journal at the bearing position.

Fig. 8. The vertical and horizontal amplitude changes of the journal under different bearing radial clearances. (1000 r/min)

The calculations found (Fig. 8) that by introducing an

adjustable bearing, the amplitude of the journal is reduced by up to 69%. The degree of reduction in rotor amplitude is closely related to bearing radial clearance. The smaller of the bearing radial clearance, the stronger the vibration suppression effect, and the reduction of the amplitude in the horizontal direction and the vertical direction is almost the same. The bearing radial clearance is adjusted from 90% to 30% cr, the amplitude of the rotor is significantly reduced, the vertical direction is reduced from 0.023 mm to 0.007 mm, and the horizontal direction is reduced from 0.026 mm to 0.008 mm.

Fig. 9. The vertical and horizontal amplitude changes of the journal under different bearing radial clearances. (2000 r/min)

Then adjust the speed to 2000 r/min, the other working conditions are the same as 1000 r/min, calculate the vibration amplitude of the journal. The calculation results are shown in Fig. 9. The amplitude response of the journal is approximately the same as the amplitude response at 1000 r/min, and the vibration suppression effect can reach 67%. The amplitude response of the journal in the horizontal and vertical directions is significantly reduced. The smaller the bearing radial clearance, the smaller the vibration amplitude response of the journal. Below is a graph of the vibration response of the rotor mid-position and bearing position when the rotor passes the first critical speed at an acceleration of 1.5rad/s2. The bearing radial clearance is set to 90% cr, 70% cr, 50% cr, 30% cr (other parameters are exactly the same), and the vibration response of the acceleration process of the rotor is solved

respectively. The responses of the rotor disk as a function of the rotational speed are shown in Fig. 10. The responses of the journal as a function of the rotational speed are shown in Fig. 11. The responses amplitude highlights that introducing an adjustable bearing the maximum deflection of the rotor disk passing the critical speed is lowered by up to 36%. The maximum deflection depends on the values of the stiffness and damping. For the sample cases in Fig. 10, the larger bearing radial clearance result in a stronger vibration suppression effect due to the smaller values of the stiffness and damping (see Fig.5). The bearing radial clearance is adjusted from 30%cr to 90%cr, the amplitude of the rotor passing the critical speed is significantly reduced, decreasing from 1.35mm to 0.82mm in the horizontal direction and from 2.25mm to 1.45mm in the vertical direction. The corresponding vertical response is presented in Fig.10(b) and shows that the vibration response suppression effect is of the same level than for the horizontal response. When the critical speed is approaching, the bearing clearance is adjusted from 30%cr to 90%cr, resulting in a slight increase in the amplitude response of the journal (Fig. 11). Trend of vibration amplitude change in bearing position is opposite to that of mass disk position in the process of clearance adjustment. It is necessary to further analyze the cause of this phenomenon and determine a reasonable clearance adjustment method. When the bearing radial clearance is adjusted from 90% cr to 30% cr, the bearing stiffness and damping coefficient are increased, a larger oil film force wraps the journal, and the amplitude of the journal position is reduced. However, the amplitude of the mass disk position is increased. If the inverse adjustment clearance is adjusted from 30% cr to 90% cr, the oil film thickness will increase, and the amplitude of the mass disk position will decrease. Although the amplitude of the journal will increase at this time, the bearing clearance will also increase. The increase in the journal amplitude is much smaller than the bearing clearance and does not affect the lubrication characteristics of the bearing. From the above calculation and analysis, the most reasonable adjustment method at the critical speed is to increase the bearing radial clearance.

5. Experimental investigation of the adjustable bearing 5.1 Design and description of experimental equipment To verify the previous simulation calculation, a rotor bearing test bench (Fig.12) is designed which is in good agreement with the numerical analysis model. Therefore, the dynamic characteristics of the test bench are the same as the numerical model. The test bench can realize arbitrary adjustment of radial clearance and has the ability to verify the influence of radial clearance adjustment on the dynamic characteristics of the rotor system. Table 1 lists the main physical and geometric properties of the bench.

Fig. 12. Physical diagram of radial clearance adjustable bearing test bench and data acquisition system

The design of the radial clearance adjustable bearing is a very important part of the design process of the test bench. To meet the experimental requirements, the clearance of the bearing needs to have the largest possible adjustment range. In this study, the clearance of bearings is designed to vary from 30% cr to 90% cr. The bearing lower pad can be moved up and down by a distance of 112µm, which is used for the adjustment of the bearing clearance. The following three equations are listed by the above conditions. ƒO"M$ ƒO„…

= 100%

ƒO"M% ƒO„…

Z‡3I − Z‡3I = 112

= 30%

(28) (29)

Solve the above equation to get the Side clearance Cmax = 160µm ， Top clearance Cmin = 48~160µm . Table 2 shows the radial clearance adjustable bearing design parameters. In the following experimental study, the clearance is used as the adjustment parameter, and the

experimental study is carried out under different clearance by adjusting the moving distance of the bearing lower pad. Table 2 Parameters of radial clearance adjustable bearing Bearing width l = 30mm Bearing diameter d = 40mm Clearance ‰ = 90%cr, 70%cr, 50%cr, 30%cr Top clearance cmin= 144mm, 112mm, 80mm, 48mm Side clearance cmax= 160 5.2 Accelerated process experiment This experiment is to study the effect of bearing clearance on the vibration response of the rotor during acceleration. According to the purpose of the experiment, combined with the actual situation of the experimental bench, the experimental steps are as follows. Firstly, the bearing radial clearance is adjusted to 30%cr, and the speed is accelerated in a linear manner from 0 to 3000r/min at a constant acceleration of 1.5rad/s2. The displacement signals of the bearing position and the mass disk position are collected respectively. Then, the radial

clearance is adjusted to 50%cr, 70%cr, 90%cr, respectively, and the vibration displacement of the rotor is repeatedly collected. Finally, the speed is reduced from 3000r/min to 0 r/min in 30 seconds. The following are the experimental results obtained using the above experimental equipment. Fig. 13 is the response of the rotor disk as a function of the rotational speed under four different cases. Fig. 14 is the response of the journal as a function of the rotational speed under four different cases. The experimental results emphasize that the introduction of adjustable bearings, the resonance amplitude of the rotor disk can be reduced by up to 40%. For the experimental case in Fig. 13, larger values for the clearance result in a stronger vibration suppression effect due to the smaller values of the stiffness and damping (see Fig. 5). The radial clearance is adjusted from 30%cr to 90%cr. The resonance amplitude value of the rotor disk is significantly reduced from 1.4mm to 0.9mm in the horizontal direction and from 2.1mm to 1.2mm in the vertical direction. The vibration suppression in the vertical direction is the same as or better than the horizontal direction.

When the bearing radial clearance is adjusted from 30%cr to 90%cr, the amplitude of the journal is slightly increased. This is because the increase of bearing radial clearance leads to the decrease of bearing stiffness and damping, the journal is wrapped by a smaller oil film force, and eventually causing the amplitude of the journal to become larger. When the clearance is adjusted to 90%cr, the resonance amplitude of the journal is the largest, and the maximum amplitude is 40 microns, which is much smaller than the bearing radial clearance of 144 microns. This indicates that even if the amplitude of the journal increases as the radial clearance increases, the amplitude of the vibration is also within the stability threshold. The experimental results are consistent with the theoretical calculations. When the radial clearance is adjusted from 30% cr to 90% cr, the amplitude response of the mass disk is significantly reduced, and the amplitude response of the journal is slightly increased. However, the amplitude of the journal is much smaller than the bearing clearance and does not affect the normal operation of the rotor system. Therefore, when the rotor system passes the critical speed during the acceleration process, a reasonable adjustment method is to increase the bearing radial clearance. 5.3 Fixed speed experiment The theoretical calculation results Fig. 11 show that the smaller bearing radial clearance result in a stronger vibration suppression effect at a speed range of 1800-2500 r/min. This speed range away from the critical speed, the rotor behaves as a rigid rotor. The following experiment will adjust the value of the clearance at 2000r/min to verify the previous theoretical calculation. Fig. 15 and Fig. 16 is the experimental result of a fixed

speed of 2000r/min. The bearing radial clearance is adjusted to 90%cr, 70%cr, 50%cr, 30%cr, and the experimental data of the vibration displacement of the rotor are collected. From the experimental results in Fig. 15-16, it is found that the axis orbit of the rotor is elliptical, and the smaller values for the clearance result in a smaller rotor vibration amplitude at 2000r/min. The bearing clearance is adjusted from 90%cr to 30%cr, the axis orbit gradually decreases. When the clearance is adjusted to 30%cr, the rotor amplitude response is the smallest, and the amplitude suppression degree can reach more than 70%. (The experimental results of the rotational speed of 1000 r/min and 2000 r/min showed the same trend and are ignored here.) In Fig. 15, the axis orbit appears as a small circle inside the large circle when the clearance decreases to 70%cr. The analysis results in Fig. 16 show that when the clearance is adjusted to 70%cr, the system has a new vibration component besides the vibration caused by the mass imbalance, and the frequency is half of the co-frequency vibration. This vibration component is due to the oil film whirl. Oil film whirl has little effect on the normal operation of the rotor and usually does not destroy the normal lubrication of the bearing, but it is a precursor to the instability of the rotor bearing system. If the rotational speed of the rotor system increases in this case, oil film whirl may develop into oil film oscillation. This may cause devastating damage to the rotating machine. In addition, the phenomenon of oil film whirl appears with the decrease of clearance (see Fig.15), and the clearance continues to decrease the axis orbit continues to decrease. The results show that the rotor vibration can be effectively suppressed, even if oil film whirl occurs. However, it is worth noting that due to the existence of the half-frequency vibration frequency, the rotor bearing

system has a great safety hazard. The above experimental results are completely consistent with the simulation results, verifying the correctness of the theoretical calculations. When the rotor passes the critical speed, increasing bearing radial clearance can suppress the vibration of the rotor. When the rotor is running at a high speed or a low speed away from the critical speed, reducing the clearance can suppress the vibration of the rotor.

Vertical displacement amplitude, mm Horizontal displacement amplitude, mm

5.4 Control of oil film instability by adjustable bearing The operating speed of the steam turbine is generally in a high-speed range. When the rotor speed exceeds the critical speed, there may be some stability problems in the working state of the rotor, so the stability design of the rotor system is an important part of the high-speed turbine design. The function of the bearing design work is to increase safety and enhance stable operation of the rotor system, but the traditional fixed-pad bearing design can only be implemented in specific working conditions. When the operating parameters such as the rotation speed and load change, the fixed-bearing cannot be adjusted according to the actual working conditions. This can lead to safety issues such as oil film instability and excessive rotor vibration. The radial clearance adjustable bearing proposed in this paper can adjust the bearing dynamics under continuous conditions to solve the above problems. During the acceleration process, if there is a half-frequency vibration component (i.e., oil whirl) in the vibration signal of the rotor, which is when the rotor speed continues to increase to twice the critical speed, the half-frequency vibration and the critical speed coincide, resulting in oil whirl resonance. The phenomenon of oil whirl resonance is called oil whip. The transition from oil whirl to oil whip is called oil film instability. The oil whip may occur when the speed is close to twice the critical speed and under light load conditions. To investigate oil instability, the rotor's own weight was chosen as the load, and the motor driven the rotor speed to twice the critical speed, which led to the appearance of oil film whip (Fig. 17). 150 100 50 0 -50 -100 -150 150 0

5

10

15

20

25

30

35

40

100 50 0 -50 -100 -150 0

10

20

30

40

Time t, s

Fig. 17. The journal vibration response of the adjustable bearing position during acceleration. During the experiment, the radial

clearance was not adjusted, so the oil whirl occurred at the 20-seconds mark, and the emergency stop was engaged at the 30-second mark.

For the first accelerated test, the bearing clearance was constant and maintained at 30% cr. The experimental results in Fig. 18 show that the rotor amplitude increases when the rotational speed is close to the first-order critical speed, and the amplitude can be recovered after exceeding the critical speed. However, when the speed is accelerated to twice the critical speed, the rotor amplitude is extremely increased, and the amplitude does not decrease after exceeding twice the critical speed. In the second acceleration experiment (Fig.18), the oil film whip was controlled by increasing the bearing radial clearance (30%cr to 60%cr) at nearly twice the critical speed. In this test, when the rotational speed is close to twice the critical speed, the rotor amplitude also begins to increase, but at this time the bearing radial clearance is increased, causing the rotor working position to decrease and the rotor amplitude to return to a stable working state.

Fig. 18. The journal vibration response of the adjustable bearing position during acceleration. During the experiment procedure, the radial clearance was adjusted from 30%cr to 60%cr at the 23 seconds.

In studying the process of controlling the oil film instability, five special time points were selected to extract experimental data (see Fig. 19), draw the axis orbit of the journal at these time points as shown in Fig. 20, and perform a spectrum analysis as shown in Fig. 21. At 16 seconds, the oil whip has not yet occurred. At 20 seconds, the oil whip occurs. At 23 seconds, the bearing radial clearance is adjusted. At 25 seconds, the oil whip begins to decrease. At 32 seconds, the rotor returns to the stable working state.

20 15 10 5 0

0

100

200

300

frequency (Hz)

t =32 s Fig. 21. Vibration amplitude of journal at five time points. The sequence number 1 represents the synchronous vibration component, 2 represents the half-frequency vibration component, and 3 represents the bearing movement excitation.

Fig. 19. The journal vibration response of the adjustable bearing position during acceleration. The radial clearance was adjusted at 23 seconds and the experimental data was extracted for analysis at 16, 20, 25, and 32 seconds.

t=16 s

t =20 s

t =25 s

t =23 s

t =32 s

Fig. 20. Axis orbit of journal at the locations of the adjustable bearings at 5 time points 20

20

15

15

10

10

5

5

0

0

100

200

300

0

0

frequency (Hz)

t=16 s 20

15

15

10

10

5

5

0

100

200

frequency (Hz)

t =23 s

200

300

t =20 s

20

0

100

frequency (Hz)

Analysis of the experimental data (see Fig. 20 and Fig. 21) revealed that the amplitude of the journal was very small at 16 seconds, but a half-frequency vibration component appeared. This indicates that there is already oil film whirl at 16 seconds, but its vibration is small and does not affect the normal operation of the rotor at low speeds. At 20 seconds, the amplitude of the journal starts to increase, and there are two large vibration frequency components: one component of the synchronous vibration and the other half-frequency vibration component. This indicates that the rotational speed reaches twice the critical speed at the 20th second. At this time, the half-frequency vibration frequency coincides with the critical speed frequency, causing the oil whirl to develop into an oil whip, and the amplitude starts to increase At 23 seconds, the orbit of the axis increases significantly, and three vibration components appear. The first frequency component caused by the movement of the bearing pad, the second is the half frequency vibration component, and the third is the synchronous vibration component. At this point, the half-frequency vibration dominates, and the journal amplitude increases sharply. This indicates that there is a severe oil whip vibration, which will cause great damage to the normal operation of the rotor system. At 25 seconds, the orbit of the axis begins to decrease, and the half-frequency vibration component begins to decrease. This shows that adjusting the bearing clearance can improve the stability and control the phenomenon of oil whip. At 32 seconds, the amplitude response of the rotor returns to a steady state, and only the synchronous vibration component exists, which indicates that the oil whip is suppressed.

300

0

0

100

200

frequency (Hz)

t =25 s

300

Conclusions The radial clearance adjustable bearing has the ability to change the stiffness and damping coefficient of the oil film. A reasonable adjustment of the clearance can reduce the vibration of the rotor. a) When the rotor speed is far from the critical speed, the smaller the bearing radial clearance results in a stronger vibration suppression effect. This vibration suppressing effect can reach 67%.

b) When the rotor speed approaches the critical speed, the larger the bearing radial clearance, the stronger the vibration suppression effect. This vibration suppression effect can reach about 37%. The amplitude response of the shaft is reduced by almost the same extent in both the vertical and horizontal directions. c) When the oil film instability occurs in the rotor bearing system, increasing the bearing radial clearance can reduce the half-frequency vibration to suppress the oil film whirl. The attenuation of the half-frequency vibration component can significantly reduce the amplitude response of the rotor, causing the rotor system to return to a steady state. The above studies show that the radial clearance adjustable bearing can reduce the rotor vibration response during different stages of acceleration. Therefore, a database for adjusting the bearing clearance can be established to make a quick adjustment response according to the vibration state of the rotor, which is very helpful for the stable operation of the rotor bearing system. The current research has attained the ability to manually adjust the radial clearance. Future research is to enable the adjustable bearing to automatically adjust the bearing radial clearance, and to achieve the adjustment of the bearing clearance by using the time-frequency controller. Acknowledgements This study was funded by the project of National Natural Science Foundation of China (Grant No. 51575421). References [1] Chasalevris A, Dohnal F. A journal bearing with variable geometry for the suppression of vibrations in rotating shafts: Simulation, design, construction and experiment. Mech Syst Sig Proc 2015;52–53:506–28. [2] Chasalevris A, Dohnal F. Improving stability and operation of turbine rotors using adjustable journal bearings. Tribology International, 104 (2016), 369-382. [3] El-Shafei A, Dimitri AS. Controlling journal bearing instability using active magnetic bearings. ASME J Eng Gas Turbine Power 2010;132(1):1–9. [4] Dohnal F, Markert R. Enhancement of external damping of a flexible rotor in active magnetic bearings by time-periodic stiffness variation. J Syst Dyn 2011;5:856– 65. [5] Furst S, Ulbrich H. An active support system for rotors with oil-film bearings. In: Proceedings of the 4th international conference on vibration in rotation machinery. IMechE; 1988. p. 61–8. [6] Althaus J, Ulbrich H. A fast hydraulic actuator for active vibration control. ImechE 1992;C432/045:141–2. [7] Zhu KY, Xiao Y, Rajendra AU. Optimal control of the magnetic bearings for a flywheel energy storage system[J]. Mechatronics, 2009, (19): 1221-1235.

[8] Sivrioglu AS. Adaptive control of nonlinear zero-bias current magnetic bearing system[J]. Nonlinear Dynamics, 2007, (48): 175-184. [9] Krodkiewski JM, Sun L. Modelling of multi-bearing rotor system incorporating an active journal bearing. J Sound Vib 1998;210:215–29. [10] Krodkiewski JM, Cen Y, Sun L. Improvement of stability of rotor system by introducing a hydrodynamic damper into an active journal bearing. Int J Rotor Mach 1997;3:45–52. [11] Nonami K. Bend vibration control of rotor system with active type bearing, JSME Transactions 1985; 51(470): 2463–2471. [12] Reinig K, Desrochers A. Disturbance accommodating controllers for rotating mechanical system, ASME Journal of dynamic systems, measurement and control 1986;108 (3): 24–31. [13] Nonami K, Zhen L, Model control with variable feedback for structural vibration systems (2nd Report, application of a variable velocity feedback method to flexible rotor systems), Transactions JSME (Japan Society of Mechanical Engineers) 1989; 55 (517) :2336–2341. [14] Yang S, Sheu G, On the spillover of steady state unbalance response of a rotating shaft under velocity feedback, ASME Journal of Vibration and Acoustics 2006;128:143– 147. [15] Ishida Y, Liu J, Vibration suppression of rotating machinery utilizing discontinuous spring characteristics, ASME Journal of Vibration and Acoustics 2008; 130:143– 147. [16] Santos IF. Design and evaluation of two types of active tilting pad journal bearings. In: Burrows CR, Keogh PS, editors. The active control of vibration. ed.1. London, UK: Mechanical Engineering Publications Ltd; 1994. p. 79–87. [17] Santos IF. On the adjusting of the dynamic coefficients of tilting-pad journal bearings. STLE Tribol Trans 1995;38(3):700–6. [18] Santos IF. Theoretical and experimental identification on the stiffness and damping coefficients of active-tilting pad journal bearings. In: Friswell M, Mottershead J, editors. Identification in engineering systems. Swansea, UK: The Cromwell Press Ltd.; 1996. p. 325–34. [19] Palazzolo AB, Lin RR, Alexander RM, Kascak AF, Montague J. Test and theory for piezoelectric actuator for active vibration control of rotating machinery.ASME J Vib Acoust 1991;113:167–75. [20] Przybylowicz P. Active stabilisation of a rigid rotor by a piezoelectrically controlled mobile journal bearing system. Australian J Mech Eng 2004;1(2):123–8. [21] Tuma J, Simek J, Skuta J, Los J. Active vibrations control of journal bearings with the use of piezoactuators. Mech Syst Signal Process 2013;36:618–29. [22] Tallian T, Gustafsson O. Progress in rolling bearing vibration research and control, ASLE Transactions 1965;8 (3):195–207. [23] Itou T. Sound and vibration of rolling bearing, Mechanics Design 1962;6 (12):30–36.

[24] Kirk R, Gunter E, The effect of support flexibility and damping on the synchronous response of a single-mass flexible rotor, ASME Journal of Engineering for Industry 1972;94 (1):221–232. [25] Ota H, Kanbe Y, Effects of flexible mounting and damping on the synchronous response of a rotor-shaft system, ASME Journal of Applied Mechanics 1976;98 (1) :144– 149. [26] Vance JM, Li, J. Test results of a new damper seal for vibration reduction in turbomachinery, ASME J. Eng. Gas Turbines Power 1996;118:843–846. [27] Andres L San, Lubell D. Imbalance response of a test rotor supported on squeeze film dampers, ASME J. Eng. Gas Turbines Power 1998;120:397–404. [28] El-Shafei A, Hathout JP. Development and control of hsfds for active control of rotor-bearing systems, ASME J. Eng. Gas Turbines Power 1995;117:757–766. [29] Althaus J, Ulbrich H. A fast hydraulic actuator for active vibration control. ImechE 1992;C432/045:141–2. [30] Goodwin MJ, Penny JET, Hooke CJ. Variable impedance bearings for turbogenerator rotors. ImechE 1984;C288/84:535–41.

[31] Kicinski J, Materny P. Non-linear vibrations in multi degree freedom system on the example of turbine 13K215. In: Proceedings of the international conference on vibration and noise; 1995. p. 120–7. [32] Sun L, Krodkiewski JM, Cen Y. Self-tuning adaptive control of forced vibration in rotor system using an active journal bearing. J Sound Vib 1988;213:1–14. [33] Chasalevris A, Dohnal F, A journal bearing with variable geometry for the reduction of the maximum response amplitude during passage through resonance, ASME J. Vib. Acoust. 2012;134:061005. [34] Chasalevris A, Dohnal F, Vibration quenching in a large scale rotor-bearing system using journal bearings with variable geometry, J. Sound Vib. 2014;333:2087–2099. [35] Papadopoulos CA, Nikolakopoulos PG, Gounaris GD. Identification of clearances and stability analysis for a rotor journal-bearing system, Mech. Mach. Theory 2008;43 (4): 411–426. [36] Chasalevris A, Nikolakopoulos PG, Papadopoulos CA. Dynamic effect of bearing wear on rotor rotor-bearing system response, ASME J. Vib. Acoust. 2013;135: 011008.

Appendix. The answers to the reviewers Reviewer #1: 1) The cross-rigidity and damping of the bearings do have a large impact on system stability. The figure below is the four calculated stiffness coefficients. Comparing the four stiffness coefficients, it is found that the cross stiffness is smaller than the main stiffness, but their values are an order of magnitude. So it is not because the cross stiffness is too small to ignore it.

Fig.1 The stiffness coefficient varies with the rotational speed under different bearing clearance conditions.

The research in this paper is consistent with the comments made by the reviewer. Because the research in this paper only needs to use numerical simulation to qualitatively study the influence of bearing clearance on rotor vibration, the cross term can be ignored. 2) Fig. 9 of the manuscript shows other vibration frequencies when the bearing clearance is adjusted to 30% Cr. This phenomenon can be explained by the axis orbit and vibration frequency domain analysis. Fig.2 is the axis orbit of journal under different bearing clearance. Fig.3 is the frequency domain data of journal under different bearing clearance.

Clearance=90% Cr

15 10 5

20

50

Frequency (Hz)

Clearance=90% Cr

100

(33, 17.3)

15 10 5 0

0

Amplitude (μm)

20

0

Clearance=30% Cr

25

(33, 23.5) Amplitude (μm)

Amplitude (μm)

25

Clearance=70% Cr Clearance=50% Cr Fig.2 Axis orbit of journal under different bearing clearance

0

50

100

Frequency (Hz)

Clearance=70% Cr Clearance=50% Cr Fig.3 Frequency domain data of journal under different bearing clearance

Clearance=30% Cr

From the analysis results in Fig. 2, it is found that the axis orbit of journal is elliptical, and the smaller values for the clearance result in a smaller rotor vibration amplitude. The bearing clearance is adjusted from 90%cr to 30%cr, the axis orbit gradually decreases. When the clearance is adjusted to 30%cr, the rotor amplitude response is the smallest, and the amplitude suppression degree can reach more than 60%. In Fig. 2, the axis orbit appears as a small circle inside the large circle when the clearance decreases to 30%cr. The analysis results in Fig. 3 show that when the clearance is adjusted to 30%cr, the system has a new vibration component besides the vibration caused by the mass imbalance, and the frequency is half of

the co-frequency vibration. This vibration component is due to the oil film whirl. Oil film whirl has little effect on the normal operation of the rotor and usually does not destroy the normal lubrication of the bearing, but it is a precursor to the instability of the rotor bearing system. In addition, the phenomenon of oil film whirl appears with the decrease of clearance (see Fig.2), and the clearance continues to decrease the axis orbit continues to decrease. The results show that the rotor vibration can be effectively suppressed, even if oil film whirl occurs. However, it is worth noting that due to the existence of the half-frequency vibration frequency, the rotor bearing system has a great safety hazard. In Fig. 10 and Fig. 11 of the manuscript, after the critical speed, the frequency component of the rotor amplitude consists of two parts, one is the co-frequency vibration component of the rotor and the other is the vibration component of the critical speed. When the two parts of the vibration are superimposed, large fluctuations occur. This phenomenon was studied by the famous professor of the University of Virginia, E. Gunter, and explained this phenomenon. Explain in detail in page 8 (point III) of Experimental Study of the Critical Speed Response, of a Jeffcott Rotor with Acceleration. This phenomenon can be explained by a waterfall diagram. Before the rotor reaches the critical speed, the rotor has only the frequency component of the co-frequency vibration. When the critical speed is reached, the frequency component of the critical speed (non-synchronous vibration component) begins to appear. After crossing the critical speed, the rotor has two frequency components: the co-frequency vibration component and the critical speed frequency component. Their superimposed combination causes amplitudes of fluctuations after critical speed. 3) Creating a design feature for a sleeve bearing that allows for adjusting its diameter is a challenging task. The increment of adjustment for the radial clearance must be at least one order of magnitude smaller than its nominal value.

Fig. 4. Structural design of a radial clearance adjustable bearing and adjustable property

The radial clearance adjustable bearing design is shown in Fig.4. The bearing structure is designed with two bearing pads, and the vertical movement of the lower bearing bushing allows for the adjustment of the bearing clearance. The mechanical structure design utilizes a screw mechanism to convert rotational motion into horizontal sliding. The servo motor can accurately position the angle of the rotary motion, and then vertical motion of the bearing bushing can be amplified by the motion amplification mechanism to accurately adjust the bearing clearance. Since the bearing bushing is moved upward from its original position where it forms a circle, the top clearance of the adjustable bearing is smaller than the side clearance. Thus, the upper and lower pads can form an oil wedge, which can produce better operating stability.

(a) Front view of bearing device (b) Top view of partly device. Fig. 5. View and description of the physical structure of the radial clearance adjustable bearing device.

The adjustment of the bearing radial clearance is achieved by the structural design as shown in Fig.5. The bearing ring is divided into two parts: the upper part is fixed with the housing while the lower part can slide up and down. The inclined iron connection is the lead screw and the bearing’s lower part. The inclined iron is used to convert the rotational motion of the lead screw into a vertical sliding of the lower pad. The slope of the plane in contact with the inclined iron and the bearing lower part is 1:125. The lead screw rotates one revolution to drive the inclined iron to move 1mm horizontally and the bearing’s lower pad to slide 8µm vertically. Through the mechanical structure design, the adjustment of the small clearance of the bearing can be accurately set. 4) Some sections of the paper can be omitted. For example, Appendix A uses the finite difference method (FDM) to evaluate the adjustable bearing lubrication pressure. This section can be omitted. Remove Appendix A according to the reviewer's comments. A simple description of the finite element method to calculate the bearing lubrication pressure is added to the original paragraph. 5) Modifications to phrases and grammar, for example: - "radius clearance" is modified to "radial clearance" - Introduction: "unbalance of the rotating machinery" is modified to "imbalance of the rotating machinery" . - Introduction: "is shifted to…"is modified to "is shifting to…" - Section 2 : "…until the requirements are met." is modified to “If the amplitude of the rotor does not meet the requirements, the bearing clearance can be continuously adjusted through the above process until the rotor vibration is optimal.” - Section 2 : "To create a design feature for a sleeve bearing…" is modified to “Creating a design feature for a sleeve bearing that allows for adjusting its diameter is a challenging task.” - Section 2 : "Finally, the adjustment of the bearing radius clearance is ingeniously achieved…" is modified to “Through the mechanical structure design, the adjustment of the small clearance of the bearing can be accurately set.” - Section 3: "the oil film pressure can be generated in both… " is modified to “Since the radial clearance adjustable bearing is composed of two arc-shaped bearing pads, and the top clearance of the adjustable bearing is smaller than the side clearance, when the journal is operating under the load W, there are both a convergent gap and an open gap between the journal and both the upper and lower pads.” - Section 4: Delete this sentence "This section applies the adjustable bearing to a rotating machine to simulate…"

- Section 5.4: " The bearing design work is for the safe and stable operation of the rotor system…" is modified to “Since the radial clearance adjustable bearing is composed of two arc-shaped bearing pads, and the top clearance of the adjustable bearing is smaller than the side clearance, when the journal is operating under the load W, there are both a convergent gap and an open gap between the journal and both the upper and lower pads.”

Reviewer #2: First, I am very grateful to the reviewer for his comments. The reviewer gave a comprehensive evaluation of my article and pointed out the areas where the article needed improvement. In this study, a radial clearance adjustable bearing structure is designed firstly, and then the influence of bearing clearance on the vibration response of the rotor system is studied under various working conditions. The purpose of these studies is to first determine whether the adjustable bearing can improve the vibration of the rotor system and how to adjust the bearing clearance to improve the vibration of the rotating machine. This research process is an attempt to build a database to find a way to adjust the bearing clearance to improve the vibration of the rotating machinery. For different rotor vibration conditions, find the corresponding bearing clearance adjustment method. Through research, the rule of adjustable bearing suppression rotor vibration is summarized, and then the theoretical basis for further research on bearing active control method is provided. The subject of this paper is the effect of a radial clearance adjustable bearing on the vibration response of the rotor system. Active control of the bearings has not been achieved to suppress rotor vibration. It can be said that the research in this paper is the basic research of bearing active control. In the future, our research team will continue to study the active control method of bearings. I will make a clear statement in the introduction section about the difference between this study and the published article by Chasalevris et al. Conduct a more in-depth literature review to compare the ideas of this study with the literature. One of my research focuses in this article is the design of adjustable bearing structure and its working principle. Although this kind of bearing is also an adjustable bearing, it has different structure and function from the bearing proposed by Chasalevris et al (Fig. 6). The bearings they designed regulate the clearance of the bearings in a passive manner. Only when the fluid film forces exceed a preload of an external spring, the bearing bush moves downward to increase the bearing clearance. This kind of bearing needs to first set the preload of an external spring, and it is only possible to make the clearance adjustment when there are some special cases where the oil film force is extremely increased. The bearing designed in this paper is a semi-active adjustment bearing (Fig. 7), which can increase or decrease the bearing clearance arbitrarily according to the vibration of the rotor. The working principle of the two bearings is completely different. The bearings designed by him can only passively increase the clearance according to the set external preload. The bearings designed by our research team need to study the dynamic characteristics of the rotor and actively adjust the bearing clearance according to the vibration of the rotor.

Fig. 6 Adjustable bearings proposed by Chasalevris. Description in the literature: The proposed concept for vibration reduction during passage through resonance establish a journal bearing that possesses the ability to introduce an additional fluid film zone by displacing its lower semibearing part if, and only if, the fluid film forces exceed a well defined preload of an external spring

Fig. 7 The radius clearance adjustable bearing proposed in this paper. The proposed bearing can not only quantitatively adjust the clearance of the bearing, but also semi-actively increase or decrease the clearance of the bearing according to the vibration condition of the rotor. The fluid film thickness can be adjusted throughout the acceleration process to improve the vibration of the rotor

Bearings designed by Chasalevris et al only increase the bearing clearance and do not control the amount of bearing clearance change. However, my research details the precise adjustment of the bearing clearance. The bearing clearance can not only increase and decrease according to working conditions, but also quantitatively control the size of the bearing clearance. The precise adjustment of the bearing clearance allows a more intuitive study of the effect of bearing clearance on rotor vibration. At each experiment, the bearing clearance can be adjusted quantitatively to record the effects. Designing the mechanical structure to adjust the bearing clearance is key to this research. The bearing radial clearance is very small (it is one thousandth of the bearing diameter), and it is very difficult to adjust it directly. In this study, the bearing clearance is precisely adjusted by the mechanical structure, so the clearance size of the bearing can be accurately set. In future research, it is necessary to continuously optimize the adjustment structure and adjustment method, which will be the development direction of bearing research. Because the bearings designed by me and Chasalevris are different, and the working principles of the two bearings are also different, the following computational research is also focused on different bearings. I and his research draw similar conclusions, which verifies the correctness of our research conclusions, but it is not a repetitive work. In addition, his research work mainly focuses on passively increasing bearing clearance to reduce resonance vibration during resonance. My research not only studies the active increase of bearing clearance to reduce the vibration of the rotor when passing resonance, but also studies the reduction of bearing clearance to reduce the vibration of the rotor when moving away from the resonance speed. It is also studied that when the rotational speed is increased to twice the critical rotational speed, if oil film whirl occurs, the active adjustment of the bearing clearance suppresses the development of oil film whirl. This is a complete study of the entire acceleration process of a rotating machine. My article mainly studies the effect of bearing clearance on rotor vibration. The study of the dynamic behavior of the adjustable bearing rotor system has not been carried out in depth. Thanks to the reviewers for some very good suggestions for me, I will focus on the rotor dynamics behavior in later research, applying the the state of the art regarding theoretical understanding of the dynamic behavior of rotors over plain journal bearings.

Reviewer #3: 1. One of my research focuses in this study is the design of adjustable bearing structure and its working principle. Although this bearing is also an adjustable bearing, its structure and function are different from those proposed in the previous literature. The bearings designed in the literature adjust the clearance of the bearings in a passive manner. When the fluid film forces exceed a preload of an external spring, the bearing bush moves downward to increase the bearing clearance. The bearing designed in this article is a semi-active adjustment bearing which can adjust the bearing clearance arbitrarily according to the vibration of the rotor. Designing the mechanical structure to adjust the bearing clearance is key to this research. The bearing radial clearance is very small (it is one thousandth of the bearing diameter), and it is very difficult to adjust it directly. Bearings designed in the previous literature only increase the bearing clearance and cannot control the amount of bearing clearance. In this study, the bearing clearance is precisely adjusted by the mechanical structure, so the clearance size of the bearing can be accurately set. The precise adjustment of the bearing clearance allows a more intuitive study of the effect of bearing clearance on rotor vibration. In future research, it is necessary to continuously optimize the adjustment structure and adjustment method, which will be the development direction of bearing research. Because the bearings designed by me and Chasalevris are different, and the working principles of the two bearings are also different, the following computational research is also focused on different bearings. I and his research draw similar conclusions, which verifies the correctness of our research conclusions, but it is not a repetitive work. In addition, his research work mainly focuses on passively increasing bearing clearance to reduce resonance vibration during resonance. My research not only studies the active increase of bearing clearance to reduce the vibration of the rotor when passing resonance, but also studies the reduction of bearing clearance to reduce the vibration of the rotor when moving away from the resonance speed. It is also studied that when the rotational speed is increased to twice the critical rotational speed, if oil film whirl occurs, the active adjustment of the bearing clearance suppresses the development of oil film whirl. This is a complete study of the entire acceleration process of a rotating machine. 2. Athanasios's [34]…. is modified to Chasalevris and Dohnal’s [34] latest theoretical and experimental studies have found that… 3. The screw is included in the bearing's housing structure, which functions to convert the rotational motion of the motor into the vertical movement of the bearing lower. The type of lead screw is the ball screw, which has a diameter of 15 mm and a lead of 1 mm. The screw rotates one revolution to drive the inclined iron to move 1mm horizontally. The lead screw only plays a mechanical transmission role in this paper, and its friction loss is beyond the scope of this paper. Moreover, the screw we selected has self-locking property, and even if a load is applied to the screw slider mechanism, it does not move, so no work is done to generate friction loss. 4. The load on the bearing is applied to the inclined iron, and the inclined iron is directly supported by the bearing housing, so the screw is almost unloaded, and the screw can be considered to be rigid inside the bearing housing (Fig. 8). Since the lead screw only plays the role of changing the direction of motion, and it is free from force, it can be easily rotated, so its thread wear due to high loads is not considered in this paper.

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Fig. 8. View and description of the physical structure of the radial clearance adjustable bearing device. The load on the bearing is transmitted to the inclined iron, and the inclined iron transmits the load directly to the bearing housing. The lead screw is not directly subjected to the load.

Vertical displacement amplitude, mm Horizontal displacement amplitude, mm

5. The bearing designed in this study adjusts the bearing clearance through a series of mechanical components, and the load on the bearing is directly applied to the bearing housing. Because the mechanical transmission parts inside the bearing housing do not bear the load, they can be moved quickly to complete the bearing radius adjustment. During the experiment, the adjustment speed of the bearing clearance is set according to the specific rotor vibration variation. For example, in the acceleration process, if there is a half-frequency vibration component (i.e., oil whirl) in the vibration signal of the rotor, which is when the rotor speed continues to increase to twice the critical speed, the half-frequency vibration and the critical speed coincide, resulting in oil whirl resonance(i.e., oil whip). As the oil whirl develops rapidly, it is necessary to complete the adjustment of bearing clearance in a very short time to prevent the oil whip. In this study, the velocity is linearly accelerated from 0 to 3000 r/min with a constant acceleration of 1.5 rad/s2. In this experiment, it takes 6 seconds from the appearance of oil whirl to the development of oil whip (Fig. 9). After many clearance adjustment tests, it was found that the bearing clearance should be adjusted before the third second of the appearance of oil whirl, so as to avoid the appearance of oil whip. Therefore, it is theoretically possible to complete the clearance adjustment within three seconds. This test completed the clearance adjustment within 0.2 seconds to avoid oil film whirl. 150 100 50 0 -50 -100 -150 150 0

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Fig. 9. The journal vibration response of the adjustable bearing position during acceleration. During the experiment, the radial clearance was not adjusted, so the oil whirl occurred at the 20-seconds mark, and the emergency stop was engaged at the 30-second mark. After many clearance adjustment tests, it was found that the bearing clearance should be adjusted

before the third second of the appearance of oil whirl, so as to avoid the appearance of oil whip.

6. The working function of the clearance adjustable bearing is to reduce the vibration response of the rotor system by adjusting the clearance. Fig. 10 is a working flow chart of the radial clearance adjustable bearing. The displacement sensor is mounted on the rotor. If the amplitude of the rotor is too large, the controller issues a command to control the rotation of the servo motor, and the servo motor drives the screw to rotate. The screw then converts the rotary motion into a horizontal motion to achieve a horizontal movement of the inclined iron. Because the inclined iron and the bearing’s lower pad are in inclined contact, the horizontal sliding of the inclined iron can be converted into the vertical sliding of the bearing’s lower pad. The adjustment of the bearing radial clearance is achieved by the series of motion transformations described above. As shown in Fig. 10, it is a closed-loop control process. If the amplitude of the rotor does not meet the requirements, the bearing clearance can be continuously adjusted through the above process until the rotor vibration is optimal. In this study, a radial clearance adjustable bearing structure is designed firstly, and then the influence of bearing clearance on the vibration response of the rotor system is studied under various working conditions. The purpose of these studies is to first determine whether the adjustable bearing can improve the vibration of the rotor system and how to adjust the bearing clearance to improve the vibration of the rotating machine. This research process is an attempt to build a database to find a way to adjust the bearing clearance to improve the vibration of the rotating machinery. For different rotor vibration conditions, find the corresponding bearing clearance adjustment method. Through research, the rule of adjustable bearing suppression rotor vibration is summarized, and then the theoretical basis for further research on bearing active control method is provided. The subject of this paper is the effect of a radial clearance adjustable bearing on the vibration response of the rotor system. Active control of the bearings has not been achieved to suppress rotor vibration. It can be said that the research in this paper is the basic research of bearing active control. In the future, our research team will continue to study the active control method of bearings.

Fig. 10 Work flow chart of the radial clearance adjustable bearings

7. In this study, the theoretical calculation was first carried out, and the influence of the bearing clearance on the rotor vibration was obtained. To verify the simulation calculation, a rotor bearing test bench is designed which is in good agreement with the numerical analysis model. Therefore, the dynamic characteristics of the test bench are the same as the numerical model. The calculation results of the theoretical model are consistent with the experimental results, and the correctness of the theoretical calculation is verified by experiments. 8. The change in linear stiffness and damping coefficient in this study is essentially due to changes in bearing clearance. Although the screw movement causes the bearing clearance to change, the screw does not actually bear the load due to the clever design of the bearing housing (Fig.8). So the screw does not affect the linear stiffness and damping coefficient. The load on the bearing is applied to the inclined iron, and the inclined iron is directly supported by the bearing housing, so the screw is not subjected to the load, and the screw can be considered to be rigid inside the bearing housing. Since the force applied to the bearing directly acts on the bearing housing, the bearing housing is a rigid body whose stiffness is much greater than the stiffness of the bearing oil film. Therefore, the influence of screw motion on stiffness and damping coefficient is not considered in this study. 9. In this experiment, the clearance of the bearing can be measured, and the variation of the vibration amplitude of the rotor under different bearing clearances can be measured. The focus of this study is on the influence of bearing clearance on rotor vibration, so only the bearing clearance and the vibration amplitude of the rotor are measured in the experiment. Then through experimental research to find out the influence of bearing clearance on the vibration characteristics of the rotor system. The measurement of these two sets of experimental data can meet the requirements of this study. Thanks to the reviewer's comments, the measurement of stiffness and damping coefficients in the experiment can help us better understand the dynamic characteristics of the rotor system and the essence of bearing clearance changing rotor vibration. However, this experimental device does not have equipment for measuring stiffness and damping coefficient. In the next experimental study, we will increase the measurement of stiffness and damping coefficient. 10. The bearing proposed in this study is to change the clearance of the bearing through the mechanical transmission system. Its response speed is not as fast as that of the electromagnetic bearing, but the advantage of the bearing is that the bearing capacity and damping are large. This type of bearing relies on the mechanical structure to adjust the journal bearing clearance and enhance the bearing's ability to adapt to changing working conditions. This bearing concept inherits the characteristics of journal bearings and electromagnetic bearings. It has the advantages of large bearing capacity and large damping of the journal bearing, and also has the active control capability. Journal bearings are mainly used in large heavy-duty machines, which have the advantages of high bearing capacity and high stability. The dynamic characteristics of the journal bearings are relatively stable, so there is no need to adjust the bearing clearance frequently and quickly. This paper mainly studies the effect of bearing clearance adjustment on rotor vibration. The time response of the active control system has not been studied in detail. At this stage, research focuses on the influence of the clearance adjustment on the dynamic characteristics of the rotor system. 11. Our research team also conducted other studies on the effects of bearing clearance on the dynamic

characteristics of the bearing. This paper focuses on the effects of bearing clearance on rotor vibration. Beyond to x(t) and y(t) responses, other calculations and experimental data have been published in other articles. 1) Vibration suppression mechanism research of adjustable elliptical journal bearing under synchronous unbalance load, Tribology International Volume 132, April 2019, Pages 185-198. 2) An experimental study on vibration suppression of adjustable elliptical journal bearing-rotor system in various vibration states Mechanical Systems and Signal Processing, Available online 1 November 2019. 12. Cavitation may occur if the bearing clearance is unreasonably adjusted during the course of the experiment. However, the purpose of this experiment is to optimize the adjustment of the bearing clearance and find a reasonable adjustment method to improve the stability of the rotor system. In the course of the experiment, the situation of collecting data confusion began to appear, but after continuously optimizing the experimental steps, the experimental device was debugged, and finally the experiment can be completed according to the expected situation, and the same experimental result can be repeatedly made. 13. The effect of the bearing radial clearance on the system vibration response is the focus of this paper. In order to study the vibration suppression effect of the radial gap more comprehensively, the clearance adjustment range is set as wide as possible. When the bearing’s lower pad moves down to the limit position, the top clearance is the same as the side clearance, and the bearing becomes a conventional circular bearing (100%cr). When the bearing’s lower pad moves upward, in order to have good lubrication between the bearing and the journal, it is necessary to ensure that the top clearance of the bearing is not too narrow, so the minimum radial clearance is set to 30% cr (According to the mechanical design manual of the journal bearing). The adjustable bearing clearance in this study can be adjusted from 30%cr to 90%cr. However, during the experimental study, the rotor dynamics was almost unchanged when the bearing clearance was 90% cr, and the influence of the bearing clearance on the rotor vibration was not observed. Through experimental research, it is found that when the radius gap is 70%cr, the vibration characteristics of the rotor begin to change. The smaller the bearing clearance is adjusted, the more obvious the dynamic characteristics of the rotor system change. However, the bearing clearance is too small to affect the lubrication effect of the bearing, and the bearing stability is also poor. Therefore, the adjustment range of the bearing clearance is selected to be 30%cr to 70% cr.

Highlights of “A radial clearance adjustable bearing reduces the vibration response of the rotor system during acceleration” a) One of my research focuses in this study is the design of adjustable bearing structure and its working principle. Although the bearing is also an adjustable bearing, its structure and function are different from those proposed in the previous literature. The bearings designed in the literature adjust the clearance of the bearings in a passive manner. The bearing designed in this article is a semi-active adjustment bearing which can adjust the bearing clearance arbitrarily according to the vibration of the rotor. b) Designing the mechanical structure to adjust the bearing clearance is key to this research. The bearing radial clearance is very small (it is one thousandth of the bearing diameter), and it is very difficult to adjust it directly. In this study, the bearing clearance is precisely adjusted by the mechanical structure, so the clearance size of the bearing can be accurately set. The precise adjustment of the bearing clearance allows a more intuitive study of the effect of bearing clearance on rotor vibration. c) When the rotational speed does not reach the critical speed, reducing the radial clearance can effectively reduce the vibration response of the system, and the vibration suppression effect can reach 67%. When the rotational speed approaches the critical speed, increasing the radial clearance can significantly reduce the resonance response of the system, and the vibration suppression effect reaches about 37%. d) When the oil whip occurs during the acceleration process, increasing the bearing radial clearance can suppress the half-frequency vibration and control the oil whip. The attenuation of the half-frequency vibration component can significantly reduce the amplitude of the rotor, ensuring that the amplitude response of the system is within safe limits.

e) Radial clearance adjustable bearings have the ability to reduce rotor vibration throughout the acceleration process. Therefore, a database for adjusting the bearing clearance can be established to make quick adjustment responses according to the vibration state of the rotor, which is very helpful for the stability of the rotor system.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: