Transient tribo-dynamic model for journal bearings during start-up considering 3D thermal characteristic

Journal Pre-proof Transient tribo-dynamic model for journal bearings during start-up considering 3D thermal characteristic Guo Xiang, Yanfeng Han, Tao He, Jiaxu Wang, Ke Xiao, Junyang Li PII:

S0301-679X(19)30637-1

DOI:

https://doi.org/10.1016/j.triboint.2019.106123

Reference:

JTRI 106123

To appear in:

Tribology International

Received Date: 25 August 2019 Revised Date:

14 December 2019

Accepted Date: 15 December 2019

Please cite this article as: Xiang G, Han Y, He T, Wang J, Xiao K, Li J, Transient tribo-dynamic model for journal bearings during start-up considering 3D thermal characteristic, Tribology International (2020), doi: https://doi.org/10.1016/j.triboint.2019.106123. 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 Ideas: Guo Xiang, Yanfeng Han Modelling and programming: Guo Xiang Numerical analysis and parametric study: Guo Xiang, Yanfeng Han, Tao He Writing the paper: Guo Xiang Improving the English grammar: Yanfeng Han, Tao He, Ke Xiao, Junyang Li Critical revision of this article: Guo Xiang, Yanfeng Han, Jiaxu Wang, Tao He Final approval of this article: Jiaxu Wang

Transient tribo-dynamic model for journal bearings during start-up considering 3D thermal characteristic Guo Xianga,b, Yanfeng Hana,b*, Tao Hec, Jiaxu Wanga,b, Ke Xiaoa,b, Junyang Lia,b a

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China b

College of Mechanical Engineering, Chongqing University, Chongqing, 400044, China

c

School of Aeronautics and Astronautics, Sichuan University, Chengdu, 610065, China *email: [email protected]

Abstract The study presents a transient tribo-dynamic model for journal bearings to reveal the mutual effect between the mixed thermoelastohydrodynamic (mixed-TEHD) performance and the journal dynamic behavior during start-up. In the developed model, the bearing shell, oil film, and journal are considered as a heat conduction coupling system, whose transient 3D thermal characteristic is determined by a general transfer equation. The established model is verified by comparing the numerical predictions, including the transient temperature distribution, maximum temperature, axis orbit and contact time, with the published results. The evolution of the mixed-TEHD performance during start-up is presented. And the impacts of the acceleration time, radius clearance and bearing shell thickness on the transient mixed-TEHD performance are evaluated. The simulation results indicate that the dynamic contact load during start-up may be underestimated when the thermal characteristic is absent. The simulation results also indicate that a maximum temperature can be observed before the hydrodynamic pressure is fully formed. Furthermore, the present study demonstrates that although a short acceleration time can reduce the asperity contact pressure, it leads to a relatively large temperature rise. The present study also demonstrates that a smaller radial clearance and a thinner bearing shell tend to generate a larger maximum temperature and thermal expansion, which leads to the increase in the risk of the thermally induced failure during start-up. Key words: Tribo-dynamic model; Journal bearings; Start-up; 3D thermal characteristic

Nomenclature:

kr,kθ,kz

thermal conductivities in r,θ,z direction, W/m.K

C

radial clearance, mm

Φ

heat source

RB

bearing inner radius, mm

CP

specific heat, J/(kg.K)

RJ

journal radius, mm

αJ

shaft thermal expansion

Tshell

bearing shell thickness, mm

L

bearing width, mm

ε

eccentricity ratio

ϕ

attitude angle, rad

dV

control volume, m3

e

eccentricity, mm

dA

control area, m2

p

pressure, MPa

hh

transfer coefficient, W(m2.K)

η

viscosity, Pa.s

ambient temperature,

ρ

density, kg/m3

T∞ Tin

φ

flow/shear/contact factor

Tav

average temperature,

ω

angular velocity, rad/s

Tr

temperature of recirculation

h

average film thickness, µm

t

time, s

Ts

temperature of supply oil,

ph

hydrodynamic pressure, MPa

η0

initial viscosity, Pa.s

σ

surface roughness, µm

β

viscosity-temperature

χ

roughness orientation

h0

geometric film thickness, µm

F

load capacity, N

δ

deformation, µm

f

friction force, N

GE

elastic influence coefficient

µ

friction coefficient

GT

thermal influence coefficient

µc

boundary friction coefficient

∆T

temperature rise,

m

equivalent mass ,kg

δ BT

average thermal expansion of

W

static load, N

bearing shell, µm

ta

acceleration time, s

Ceff

effective radial clearance, mm

Qr

flow rate recirculation oil, m3/s

n

total number of surface nodes

Qs

flow rate of supply oil, m3/s

pc

contact pressure, MPa

U

linear velocity, m/s

Pc

dimensionless contact pressure

Subscripts:

E

elastic modulus, GPa

ν

Poisson ratio

B,J,L

bearing, journal, lubricating oil

γ

asperity aspect ratio

h,c,s

hydrodynamic, contact, shear

coefficient, µm/(m.K)

αB

bearing thermal expansion coefficient, µm/(m.K)

oil inlet temperature,

oil,

coefficient

HY

dimensionless bearing hardness

j,k

circumference, axial node

λx

autocorrelation lengths along

θ,z,r

circumference, axial ,radius

the x direction, mm

direction

E*

equivalent elastic modulus, GPa av

average

v

flow velocity, m/s

κ,ζ

normal, tangential direction

γT

vector of asperity aspect ratio

E,T

elastic and thermal

1. Introduction Journal bearings are important mechanical element used for supporting the rotated shaft under heavy load and high speed conditions. The bearing shell surface and the journal surface can be completely separated by the formed hydrodynamic film when the journal reaches the lift-off speed. However, during start-up, the severe asperity contact is yielded, leading to the excessive friction heat and the sliding wear at the contacting surfaces [1 -3 ,6 ,9 ,13 -14] . Therefore, it is of great significance to study the dynamic behavior of journal bearings during start-up to optimize the tribology performance and improve the serve life. However, to the authors’ best knowledge, the comprehensive theoretical model associated with this topic is still limited. Mokhtar et al. [1] performed an experiment to study the dynamic behavior of journal bearings during rapid starting and stopping operation. In their study, the effects of the external load, rotational speed and radial clearance were investigated experimentally. They observed that the contact time is very short and the hydrodynamic force is rapidly formed during start-up. Later, Mokhtar et al. [2] further experimentally studied the wear behavior of journal bearings operated at numerous starting and stopping cycles. The authors found that the bearing shell is prone to wear during the early stage of the starting operation, and the surface finish of the bearing shell is improved (i.e. smoothing surface effect) during the first 1000 cycles of staring and stopping cycles (after initial wear process). In 2011, the friction torque of journal bearings during start-up under different specific pressures was tested by Bouyer and Fillon [3] . The authors concluded that both the surface roughness of the bearing shell and journal should be considered to attain more precise predictions. Subsequently, Cristea and Bouyer [4] tested the transient pressure and temperature field of the grooved journal bearing during start-up. Sander et al. [5] measured the friction torque of the journal bearings with “new” and “worn” surface under mixed lubrication

condition. In this study, the smoothing surface effect on the dynamic friction behavior of journal bearings during running-in was evaluated. More recently, Sander et al. [6] further investigated, both the numerically and experimentally, the starting and stopping behavior of the journal bearing with the worn surface. The authors found that the contact area yielded by the worn journal bearing is larger than that yielded by the new journal bearing. Although the experiment [1-6] is an efficient method to investigate the dynamic behavior of journal bearings during start-up, it is difficult to capture the time-varying tribology performances due to the response time of the sensors, especially at the early stage of start-up. Therefore, a theoretical model is necessary for understanding and predicting the transient fluid-solid-thermal coupled performance of journal bearings during start-up. The numerical model, which is used to simulate the dynamic behavior of journal bearings during start-up, has been attracted increasing attentions in recent years. Kucinschi et al. [7] performed a transient thermoelastohydrodynamic (TEHD) analysis for plain journal bearings during start-up, and the numerical predictions were validated by the experimental measurement. In their numerical study, the journal position was determined by the balance between the formed hydrodynamic force and the static load. Subsequently, Monmousseau and Fillon [8] successfully predicted the transient thermoelastohydrodynamic performance of titling-pad journal bearings during start-up by a developed numerical model. In the studies above [7 -8] , the transient asperity contact and friction heat were ignored during start-up and the journal bearing was operated at hydrodynamic lubrication regime. Chun and Khonsari [9] presented a numerical model to analyze the wear behavior of the journal bearing subjected to mixed friction. This numerical model can be used to predict the wear volume of journal bearings during repeated starting and stopping cycles. Recently, Sander and Allmaier [10]

proposed a coupled model considering the transient

interaction between the behaviors of the mixed lubrication and sliding wear to calculate the wear volume of journal bearings during start-up. However, in this study, the equivalent temperature collected from experimental data was applied to consider the thermal effect, and the transient thermal expansion due to the temperature rise seems to be ignored. In 2018, Prölß et al. [11] conducted a transient thermal analysis of journal bearings during run-up, in which the asperity friction heat has been considered. Similarly, the journal position was determined by a load equilibrium equation in Prölß’s study. Further, Liu et al. [12] developed a tribo-dynamic model of

crankshaft-main bearing system during start-up, in which the shaft axis orbit was renewed by the journal fully dynamic equation. The authors found that the starting temperature significantly affects the dynamic contact pressure. However, this study introduced the temperature effect by the predetermined value. More recently, Cui et al. [13] performed a pioneered work to study the dynamic contact behavior of the journal bearings during start-up. In their study, the journal dynamic model and the isothermal mixed lubrication model were bridged. Subsequently, Cui et al. [14] further explored the effects of the surface roughness and surface mode on the dynamic contact behavior of journal bearings. Additionally, it is noteworthy that scholars have been demonstrated the important role of the thermal characteristic in simulating the tribology performance of journal bearings

subjected to

severe condition

(Monmousseau and Fillon [8] , Liu et al.[12] , Bouyer and Fillon [15] ). Therefore, it is necessary to develop a theoretical model which bridges the tribol-dynamic model and the thermal model for journal bearings during start-up. The mixed elastohydrodynamic (mixed-EHL) performance of journal bearings is also an area of concern [16 -21] . Wang et al. [16-18] developed a mixed-TEHD model for journal bearings, which has been successfully predicted the mixed lubrication performance and temperature characteristic of journal bearings under serve condition. Further, Han et al. [19] developed a new singularity treatment approach to calculate the mixed lubrication performance of the herringbone microgroove journal bearing by a fast computing technology. In 2015, Han et al. [20] integrated the transient effect into the previously developed mixed lubrication model. More recently, based on the studies conducted by Han et al. [19 -20] , Xiang et al. [21] established a coupling transient mixed lubrication and wear model to predict the time-varying sliding wear performance of journal bearings. However, the journal dynamic behavior and transient thermal characteristic are absent in the studies mentioned above [19 -21] , which plays a crucial role in simulating the transient tribology performance of journal bearings during start-up. As shown in Fig.1, for journal bearings during start-up, the journal dynamic behavior, hydrodynamic pressure, asperity contact, elastic/thermal deformation and thermal characteristic are strongly coupled, which imposes a challenge to perform a numerical study. On the other hand, scholars have been focused on the dynamic behavior of the rotor under hydrodynamic lubrication condition [22 -27] . At present, two main numerical methods were used to calculate the non-linear film force in the

tribo-dynamic model of journal bearings. One is the direct method that the hydrodynamic force is determined by solving the Reynolds equation. The other one is the indirect method that the hydrodynamic force is calculated using eight linear dynamic and damping coefficients. In 2013, Cha et al. [22] compared the journal motion orbit generated by the linear dynamic model (i.e. indirect method) and the non-linear dynamic model (i.e. direct method). They pointed out that the linear model is only suitable for a relatively small shaft unbalance. Yang et al. [23] presented a new nonlinear model to analyze the dynamic characteristic of the rotor system in the hydrodynamic lubrication regime. Ren and Feng [24] discussed the anti-shock characteristic of the water lubricated bearing considering the asperity contact effect. In their study, the non-linear film force was directly solved by the finite element method. More recently, Maharshi et al. [25] , Merelli et al. [26] and Jin et al. [27] investigated the dynamic characteristic of journal bearings by the developed non-linear dynamic coefficient method. Although the insightful findings were presented by the recently publications related to the dynamic characteristic of the lubricated rotor [22 -27] , the transient tribo-dynamic behavior considering the 3D thermal characteristic and mixed friction for journal bearings during start-up has not yet fully understood, which still requires the further investigations. The novelty of the present study is to develop a transient tribo-dynamic model, which bridges the transient mixed lubrication model, time-varying 3D thermal model and elastic/thermal deformation model, for journal bearings during start-up. The axis orbit of the journal is excited by the calculated transient force, including the hydrodynamic and contact force. During transient analysis, the evolution of the mixed-TEHD performance over operating time is presented, and the effects of the acceleration time, radial clearance and bearing shell thickness on the numerical predictions were evaluated. It is expected that the developed model can achieve a better understanding of the transient fluid-solid-thermal performance of journal bearings during start-up.

Fig.1. Coupling relationship of journal bearings during start-up.

2. Mathematic model 2.1. Transient average Reynolds equation A typical journal bearing exposed to the mixed friction is illustrated in Fig.2. One can anticipate that the time-varying lubrication gap occurs due to the dynamic journal during start-up, which leads to the transient hydrodynamic pressure. To consider the roughness effect, the average Reynolds equation derived by Patir and Cheng [28] was applied. Furthermore, the contact factor derived by Wu and Zheng [29] was introduced to consider the asperity contact effect. During lubrication analysis, the lubricant was assumed to be incompressible, laminar and Newtonian. Consequently, the transient hydrodynamic pressure can be governed by ∂ 2 RB ∂θ

 ρ L h3 ∂ph  ∂  ρ L h3 ∂ph  ∂φ  ∂ρ h ω  ∂ρ L h +ρ Lσ s  + L φc  φθ  +  φ z  =  ∂θ ∂θ  ∂t  12η ∂θ  ∂z  12η ∂z  2 RB 

(1)

where φθ and φz [28] are the flow factor in the θ and z direction, respectively;

φs [28] is the shear flow factor, φc [29] is the contact factor, ph is the hydrodynamic pressure, RB and ω are the bearing radius and rotational speed, respectively; η and ρL are the viscosity and density of the lubricant, respectively; σ is the combined surface roughness, t is the operating time.

Fκ

θ

z Journal

θ

Lubricating oil Bearing shell

ϕ

Shaft OB

OB OJ

e

OJ

h min

Asperity contact (Friction heat)

Fζ

h Fluid lubrication

Composite rough surface

(Shearing heat)

Bearing shell

Fig.2. Schematic of the journal bearing operated at mixed lubrication condition.

2.2 Transient lubrication gap In the presented model, the geometric clearance, elastic deformation and thermal deformation were incorporated to calculate the transient lubrication gap, which is given by

h(θ , z, t ) = h0 (θ , z, t ) + δB (θ , z, t, ∆TB , p) + δJ (θ , z, t, ∆TJ , p)

where

(

h0 (θ , z, t )

is

the

time-dependent

)

geometric

clearance,

(2) with

h0 (θ , z, t ) = C 1 + ε ( t ) cos (θ ( t ) − ϕ ) , where ε ( t ) and θ ( t ) are the eccentricity ratio

and attitude angle determined by the journal dynamic behavior, respectively; ∆ TB and ∆ TJ are the temperature rise of the bearing shell and journal, respectively; δB and δJ are the transient deformation, including the elastic and thermal deformation, of the bearing and journal, respectively; which can be calculated by

δ B (θ , z, t , ∆TB , p ) = δ BE (θ , z , t , p ) + δ BT (θ , z , t , ∆TB , p )

δ J (θ , z , t , ∆TJ ) = δ JE (θ , z , t ) − δ JT (θ , z, t , ∆TJ )

(3)

The elastic deformation of the journal can be ignored because whose elastic modulus is much larger than the bearing shell, and the thermal deformation of the journal was treated as linear thermal expansion, which can be calculated by

δ JT (θ , ∆T , t ) = α J ∆TRJ (1 + ε ( t ) cos (θ ( t ) − ϕ ) )

(4)

where αJ is the thermal expansion coefficient of the journal, which was considered to be 11.9 µm/(m.K) in the presented study, RJ is the radius of the journal. In order to calculate the elastic deformation, the influence-function method, developed by Woodward and Paul [33] and applied by Wang and Shi [16 -18] , was used, which is given by

δ BE (θ j , zk , t , p ) = ∑∑ GBE (θ j , zk ,θξ , zη ) ×  ph (θξ , zη , t ) + pc (θξ , zη , t )  ξ

(5)

η

δ BT (θ j , zk , t , ∆T ) = ∑∑∑ GBT (θ j , zk ,θξ , zη , rζ ) × ∆T (θξ , zη , rζ , t ) ξ

η

ζ

(6)

where GBE and GBT are the influence coefficients for the calculation of the elastic and thermal deformation. A calculation procedure based on the finite element method was developed to calculate these coefficients. During start-up, the thermal expansion of the journal and bearing shell can result in the loss of the bearing clearance, which may cause the seizure failure [8 ,30-32] . In this study, the transient effective radial clearance was introduced to evaluate the clearance loss caused by the thermal expansion for both the journal and bearing shell, which is given by

Ceff ( t ) = C − αJ RJ ∆TJ ( t ) − δBT ( t )

(7)

where δ BT is the average thermal expansion of the bearing shell, which can be calculated by

δBT ( t ) =

(

1 ∑∑δBT θ j , zk ,t, ∆TB n j k

)

(8)

where n is the total number of nodes on the bearing shell surface. 2.3 Transient asperity contact model It can be expected that for the journal bearing subjected to a static load, the asperity contact is dominant at the initial period of start-up because the hydrodynamic film has not been fully formed. Lee and Ren [34] developed an elastic-plastic asperity contact model considering the effects of the topography and material hardness, which was used to calculate the contact pressure during start-up, and the transient version can be written as  h ( γ , H Y , Pc (θ j , zk , t ), t ) i  4  = exp ∑ ( γ T [Gi ] H Y ) ( Pc (θ j , zk , t ) )  , Pc < H Y σ   i =0    h ( γ , H Y , Pc (θ j , zk , t ), t ) = 0, Pc ≥ H Y  σ 

(

)

(

(9)

)

where Pc θ j , zk , t = pc θ j , zk , t / Cpr , with Cpr = π E*σ / ( 2λx ) , where E * is the equivalent elastic modulus ( E * = 2 (1 −ν J2 ) / EJ + (1 −ν B2 ) / EB  , where ν B , ν J and −1

EB , EJ are the Poisson ratio and elastic modulus of the bearing shell and journal,

respectively), λx is the autocorrelation lengths along x direction. HY is the dimensionless hardness of the bearing, with H y = 2 H B λx / (π E *σ ) , where H B is the

hardness of the bearing shell. Further, γ is the asperity aspect ratio, which was assumed to be 1 in this study, γ T = 1, γ −1 , γ −2 , γ −3  and the transpose of H Y is H YT = 1, H Y−1 , H Y−2 , H Y−3  . The parametric matrix [Gi] was given in Ref.[34] .

2.4 Transient heat transfer model Based on the Euler method, the bearing shell, lubricant and rotating journal were considered as a heat conduction coupling system, and a general transient energy equation in cylindrical coordinate system was used to calculate the thermal characteristic of this coupling system, which is given by ∂T ∂T ∂T  ∂  ∂T  ∂T + vr + vθ + vz =  kr ∂r r ∂θ ∂z  ∂r  ∂r  ∂t

ρ CP 

∂  +  r ∂θ

 ∂T  kθ  r ∂θ

 ∂  ∂T   +  kz +Φ  ∂z  ∂z 

(10)

where T is the temperature of the coupling system, ρ is the density of the bearing shell( ρ B ), journal( ρ J ) or lubricant( ρ L ), C P is the specific heat of bearing shell ( C PB ), journal( C PJ ) or lubricant ( C PL ), kr ( krB , krJ , krL ) , kθ ( kθ B , kθ J , kθ L ) and

kz ( kzB, kzJ , kzL ) are the thermal conductivities in the r , θ , z direction ,respectively;

vr ,vθ ,vz represent the velocity components in the r,θ, z direction, respectively; Further, Φ is the heat source generated at the journal-bearing interface , in the mixed lubrication regime, which consists of two parts: the asperity friction heat ΦC and the viscous shear heat ΦL, and which can be calculated by Φ dA + Φ L dV Φ= C dV

(11)

where dV and dA are the finite volume and its corresponding control area, respectively. It is reasonable that the gradient of radial velocity was ignored during shear rate calculation [7 -8 ,11 ,16 -17] . Therefore, the heat generated by the asperity friction and fluid shear can be calculated by Φ C = µc pcU  2 2  ∂vθ  2  ∂vz  2   φθ ∂ph  (12) U U φsσ   φ z ∂ph  Φ = + = h − 2 C + + + h − 2 C η ( ) ( )           L h h 2   2η ∂z   ∂r   ∂r    2η RB ∂θ 

where U is the linear velocity of the journal, with U = 2πω RB / 60 , vθ and vz are the flow velocity along the circumferential and axial direction, respectively; which can be calculated by

φsσ C 1 ∂ph C  2 vθ = φθ 2η R ∂θ ( C − Ch ) + h U + h h U  1  ∂ p 1 h v = φ ( C 2 − Ch ) z  z 2η ∂z

(13)

2.5 Transient load capacity and friction force In the mixed friction regime, both the hydrodynamic and contact pressure contribute to the load capacity, the transient force in the ζ and κ direction (as shown in Fig.2) can be determined by the following formulas L 2π   Fhζ ( t ) = ∫ ∫ ph (θ , z , t ) sin θ RB dθ dz  0 0  L 2π F (t ) = − ph (θ , z, t ) cos θ RB dθ dz ∫ ∫  hκ  0 0  L 2π   Fcζ ( t ) = ∫ ∫ pc (θ , z, t ) sin θ RB dθ dz 0 0  L 2π   Fcκ ( t ) = − ∫ ∫ pc (θ , z, t ) cos θ RB dθ dz  0 0

(14)

Subsequently, the transient load capacity can be determined by

F ( t ) = Fζ2 ( t ) + Fκ2 ( t ) =

( Fhζ ( t ) + Fcζ ( t ) ) + ( Fhκ ( t ) + Fcκ ( t ) )2 2

(15)

Similarly, the transient friction force and friction coefficient are dependent on the combined effect between the fluid viscous shearing and the asperity contact, which can be calculated by L 2π L 2π   ηω RB2 h ∂ph (θ , z, t )  +  f ( t ) = ∫ ∫  dθ dz + ∫ ∫ µc pc (θ , z, t ) RB dθ dz h 2 ∂θ   0 0  0 0  f (t )  µ (t ) =  F (t ) 

(16)

where µc is the dry friction coefficient of the bearing shell. Krithivasan and Khonsari [32] reported that the dry friction coefficient typically ranges from 0.1 to 0.3 for boundary-lubricated sliding. According to the studies presented by Wang et al. [16 -18] , it is justified that dry friction coefficient was considered to be 0.1 for the current simulation case. 2.6 Motion equation of the journal In the tribol-dynamic model, the journal axis orbit was excited by the transient force both in the horizontal and the vertical direction. The motion equation was used

to determine the trajectory of the journal during start-up, which is presented by

mζ ′′ = Fhζ ( t ) + Fcζ ( t )  mκ ′′ = Fhκ ( t ) + Fcκ ( t ) − W

(17)

where W is the static load acted at the journal(as shown in Fig.2). It is evident that the calculated journal axis orbit changes the mixed-TEHD performance due to the time-varying lubrication gap during start-up. Similar to the study performed by Cui et al. [13 -14] , a equivalent journal mass according to the static load was adopted in the calculation procedure. 2.7 Boundary conditions (1) Cavitation boundary condition In the presented study, the Reynolds boundary condition was applied to consider the cavitation, which is given by  ph ( θ ,0 ) = ph ( θ , L ) = 0   ph ( θ0 , z ) = 0, ∂ph ( θ0 , z ) / ∂θ = 0

(18)

In which θ 0 represents the circumferential position where the oil film rupture occurs. (2) Elastic deformation boundary condition Figure 3(a) shows the boundary condition related to the elastic deformation of the bearing shell. It can be seen that for calculating the elastic deformation, the outer radius of the bearing shell was considered to be fixed, and the inner surface of the bearing shell can deform freely due to the formed hydrodynamic and contact pressure. (3) Thermal boundary condition Figure 3(b) depicts the boundary condition for the thermal analysis of the heat conduction coupling system, which can be classified as three categories: 1) Internal boundary condition As depicted in Fig.3 (b), the BC1 and BC2 are the bearing-lubricant and journal-lubricant interface, respectively. The bearing-lubricant interface obeys the following boundary condition. TL = TB , krL

∂TL ∂T = krB B ∂r ∂r

(19)

In this study, the temperature of the journal along the circumferential direction was assumed to be average. And the continuity condition at the journal-lubricant interface gives

krJ

2π ∂T ∂TJ 1 L = krL ∫ dθ 0 ∂r 2π ∂r

(20)

2) External boundary condition As shown in Fig.3 (b), BC3 thorough BC7 represent the external boundaries of the thermal coupling system. In the presented study, the convective condition was applied to these external boundaries, which can be described by the following equation k

∂T = − hh (T − T∞ ) ∂n

( r ,θ , z ) ⊂ BC 3 − BC 7

where T∞ is the ambient temperature, which was set to be 40

(21) . In the thermal

model, it was assumed that the initial temperature in the solid domain (bearing shell and journal) equals to the ambient temperature; hh is the heat transfer coefficient. During thermal analysis, the heat transfer coefficient was assumed to be 80 W/ (m2.K) for all free faces of the bearing thermal model. Furthermore, the lubricant was sufficiently supplied by the oil inlet hole located at the top of the journal bearing, as shown in Fig.3 (b). 3) Oil inlet boundary condition As shown in Fig.3 (b), BC8 represents the oil inlet boundary condition. The temperature at BC8 required to be adjusted by the temperature of the recirculation oil and oil supply, which can be calculated by Tin =

where Ts , Tr and

Qs Qr Ts + Tr Qs + Qr Qs + Qr

(22)

Qs , Qr are the temperature and flow rate of the oil supply

and recirculation oil, respectively.

(a)

(b) Fig.3. Geometry of boundary conditions: (a) elastic deformation; (b) thermal model.

In the thermal model, a simplification has made that the lubricant viscosity along the cross-film direction is average. One can anticipate that, in mixed friction regime, the upstream oil-film temperature is almost constant [35] , and the maximum temperature gradient along the radial direction is near the minimum film thickness, where the asperity contact effect may be dominant [16 -17] . Therefore, the average lubricant viscosity along the cross-film direction was used during mixed lubrication analysis, and an exponential expression was applied to evaluate the effect of the temperature on the average viscosity at the node of ( θ, z ) , which is given by β (Tav ( θ , z )−T0 )

η ( θ , z ) = η0e

(23)

where β is a constant depended on viscosity-temperature relationship, which was set to be -0.0298, Tav(θ,z) is the average oil-film temperature at the node of (θ , z ) along the cross-film direction. 2.8 Numerical algorithm The flowchart of the transient tribo-dynamic simulation is depicted in Fig.4. It can be seen that the numerical procedure can be divided into two parts: the first part was used to determine an initial equilibrium position of the loaded journal, in which the static load was completely supported by the asperity contact, by calculating the steady mixed-EHL model under an extremely low speed (such as 0.001r/min); the second part was implemented to handle the strongly coupled relationship between the behavior of the mixed-TEHD and the journal dynamic during start-up. During the numerical calculation, the journal dynamic equation was used to determine the axis position at the current time t, and the calculated axis location renews the lubrication gap in the next time step. Consequently, the mixed-TEHD performance is modified at the time t + ∆t , and subsequently which leads to a new

axis location by the renewed excited force, including the hydrodynamic force, Fh , and the contact force, Fc . In the presented model, the average Reynolds equation was solved by control volume method [36] , the elastic/thermal deformation of the bearing shell and the 3D temperature distribution of the coupling system were calculated by finite element method[16 -18] , and the journal dynamic equation was solved by four order Runge-kutta method. Firstly, the second order ordinary differential equations (ODEs) illustrated in Eq. (18) can be transformed into one order ODEs, which can be expressed as

ζ ′ = f1 ( t , ζ , κ , ζ ′, κ ′ )   ′ ′ Fhζ ( t ) + Fcζ ( t ) = f 2 ( t , ζ , κ , ζ ′, κ ′ ) ( ζ ) = m  κ ′ = f 3 ( t , ζ , κ , ζ ′, κ ′ )  F t + F t −W ( κ ′ )′ = hκ ( ) cκ ( ) = f 4 ( t , ζ , κ , ζ ′, κ ′ )  m

(24)

Secondly, according to the four order Runge-kutta method, the displacement and velocity along ζ and κ direction can be iteratively calculated by

1  ( i +1) i = ζ ( ) + ( k11 + 2k12 + 2k13 + k14 ) ζ 6  ζ ′( i +1) = ζ ′( i ) + 1 ( k + 2k + 2k + k ) 21 22 23 24  6  κ ( i +1) = κ ( i ) + 1 ( k + 2k + 2k + k ) 31 32 33 34  6  1 κ ′( i +1) = κ ′( i ) + ( k41 + 2k42 + 2k43 + k44 ) 6 

(25)

where the coefficients above can be calculated by  k n1 =   k n 2 =  kn3 =   k n 4 =

fn (t, ξ (t ) , κ (t ) , ξ ′ (t ) , κ′ (t ))

f n ( t + 0.5 ∆t , ξ ( t ) + 0.5 ∆tk11 , κ ( t ) + 0.5 ∆tk 21 , ξ ′ ( t ) + 0.5 ∆tk31 , κ ′ ( t ) + 0.5 ∆tk 41 )

f n ( t + 0.5 ∆t , ξ ( t ) + 0.5 ∆tk12 , κ ( t ) + 0.5 ∆tk 22 , ξ ′ ( t ) + 0.5 ∆tk32 , κ ′ ( t ) + 0.5 ∆tk 42 )

(26)

f n ( t + 0.5 ∆t , ξ ( t ) + 0.5 ∆tk13 , κ ( t ) + 0.5 ∆tk 23 , ξ ′ ( t ) + 0.5 ∆tk33 , κ ′ ( t ) + 0.5 ∆tk 43 )

where n = 1, 2, 3, 4 . Consequently, the displacement and velocity of the journal center can be attained through the coupled Eqs. (24)- (26) . In addition, the convergences of the pressure (both for the hydrodynamic and contact pressure) and the temperature can be controlled by the following expressions

n

m

∑∑ p( j =1 k =1 n

new) j ,k

m

∑∑ p j =1 k =1

− p(j ,k

old )

( old )

Tmax

(27)

j ,k

( old ) ( new ) Tmax − Tmax ( old )

≤ 1.0 ×10−6

≤ 1.0 × 10 −6

(28)

1 (TJ max + TB max + TL max ) , with TJ max ,TBmax and TL max are the 3 maximum temperature of the journal, bearing shell and lubricant, respectively. where Tmax =

Film thickness initiation

N

Solution for steady Reynolds equation Eq.(7)

Solution for contact pressure

N Convergence of pressure?

Adjusting the eccentricity ratio

Y N

Convergence of Attitude angle?

Y Convergence of Load?

Y

Solution for motion equation

Eq.(14)

Updating the journal center

Solution for temperature

Y

Solution for elastic deformation Adjusting the attitude angle

N

Convergence of the temperature?

Convergence of pressure?

Reaching the given time?

Solution for transient thermal deformation

Eq.(3)

Solution for transient elastic deformation

Eq.(3)

Solution for transient contact pressure

Eq.(6)

Y Fine

N

t = t + ∆t

N Solution for transient Reynolds equation

Eq.(1)

Y Initial journal center position

Determine initial Journal position

Temperature initiation

Solution for transient tribo-dynamic model

Fig.4. Flowchart for the numerical calculation.

3. Results and discussion 3.1 Model verification 3.1.1 Temperature verification In order to verify the temperature distribution calculated by the developed numerical procedure, the temperature predictions yielded by the presented model were compared with the results measured by Prölß et al.[11] . In Prölß’s experiment, the test parameters, including the geometric and operating condition, were listed in Table 1. And the thermal characteristic parameters of the bearing shell, lubricant and shaft were summarized in Table 2. As illustrated in Fig.5 (a), the numerical results and the measurements show a good agreement at the two cases of operating conditions. Figure 5(b) shows that the maximum temperatures predicted by the developed model are in reasonable agreement with those given by the Prölß et al., including both the

tested results and numerical predictions. To further validate the predicted transient temperature, the evolution of the temperature distributions during start-up predicted by the developed model were compared with the experimental results given by Kucinschi et al. [7] . The parameters of the tested bearing, including the operating condition, geometric, and thermal, were listed in Table 3. Generally, the circumferential temperature distributions at different operating times match well with those tested by Kucinschi et al.[7] .Therefore, these validated results presented in Figs.5-6 demonstrate the developed model is capable of predicting the thermal performance of the journal bearing during start-up. 100

Test results (F=4kN,ω =4000rpm) Test results (F=17.5kN, ω =4000rpm)

90

75.10

1.0

71.21

0.8

Axial direction(z/L)

80

Axial direction(z/L)

Temperature(℃ )

1.0

67.32 63.44

0.6

59.55

0.4

55.66 51.78

0.2

47.89 44.00

0.0 0

1

2 3 4 Circumferential direction(rad)

5

Present results Present results 83.00 78.14

0.8

73.28 68.41

0.6

63.55

0.4

58.69 53.83

0.2

48.96 44.10

0.0

6

0

1

2 3 4 Circumferential direction(rad)

5

6

70

60

50

40 0

40

80

120 160 200 240 280 320 Circumferential direction(Degree)

360

400

3.6

4.0

(a) Test results

80

Simulation results[11] The present results

75 70 65 A xial direction(z/L)

Maximum temperature Tmax(℃ )

85

60

1.0

82.00

0.8

74.52

0.6

67.04

0.4

59.56

52.08

0.2

44.60

0.0 0

55

1

2 3 4 5 Circumferential direction(rad)

6

Increasing load 50 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

Specific pressure pa(MPa)

(b) Fig.5. Verification of the (a) circumferential temperature distribution and the (b) maximum structure temperature of the journal bearing during transient running up.

40

1.0

42.80

t=600s

0.8

40.97

0.6

39.14

0.4

37.31

0.2

35.49 33.66

0.0 0

38

1

2 3 4 Circumferential direction(rad)

5

6

36 1.0 Axial direction(z/L)

Temperature(℃ )

t=30s(Present) t=60s(Present) t=90s(Present) t=120s(Present) t=240s(Present) t=600s(Present)

t=30s (test) t=60s (test) t=90s (test) t=120s(test) t=240s(test) t=600s(test)

42

Axial direction(z/L)

44

34 32

36.00

t=30s

0.8

34.80

0.6 33.60

0.4 32.40

0.2 31.20

0.0 0

30 0

50

100 150 200 250 300 Circumferetial direction(Degree)

1

2 3 4 Circumferential direction(rad)

5

6

350

Fig.6. Verification of the transient circumferential temperature distribution of the journal bearing during start-up. Table 1 Bearing parameters (Prölß et al.[11] ) Bearing

Case A

Shaft diameter Bearing outer diameter Relative radial clearance Bearing Width Supply temperature Supply pressure Speed range

Case B 100mm 130mm

0.16%

0.32% 50mm

30

45 2bar 0-4000rpm

Table 2 Material parameters of the bearing shell, lubricant and journal (Prölß et al.[11] ) Property

Lubricant

Bearing

Journal

Material Density Specific heat capacity Heat conductively Thermal expansion coefficient Elasticity modulus

ISOVG 32 853kg/m3 2090J/(kg.K)-1 0.134W/(m.K) -

CuSnZn4Pb7 8830 kg/m3 380 J/(kg.K)-1 68W/(m.K)

42CrMo4 7720 kg/m3 470 J/(kg.K)-1 42.6 W/(m.K)

18µm/(m.K) 101GPa

11µm/(m.K) 210GPa

Table 3 Bearing parameters (Kucinschi et al.[7] ) Parameters

Value

Parameters

Value

Bearing radius Bearing width Radial clearance Bearing elastic modulus Bearing Poisson ratio Shaft elastic modulus Shaft Poisson ratio Bearing thermal conductivity Shaft thermal conductivity

50mm 80mm 0.123mm 120GPa 0.3 210GPa 0.33 65 W/m.K 50 W/m.K

Bearing specific heat Shaft specific heat Lubricant specific heat Bearing thermal expansion Shaft thermal expansion Supply temperature Lubricant viscosity(30 ) Bearing density Shaft density

380 J/(kg.K) 490 J/(kg.K) 2000 J/(kg.K) 17µm/(m.K) 12µm/(m.K) 30 0.05 Pa.s 8940 kg/m3 7700 kg/m3

Lubricant conductivity Start-up time

0.13 W/m.K 7s

Lubricant density

870 kg/m3

3.1.2 Verification of the axis orbit and contact time In the current section, the axis orbit and contact time of the journal during rapid start-up calculated by the developed numerical model were compared with the tested results reported by Mokhtar et al.[1] and the numerical results calculated by Cui et al.[13] . As shown in Fig.7 (b), it is evident that the predicted axis orbit is in accordance with that tested by Mokhtar et al.[1] . Figure 7(c) is presented to further verify the developed numerical procedure. It can be found that, during the start-up, the contact time calculated by the presented numerical model agrees well with that predicted by Cui et al.[13] and tested by Mokhtar et al.[1] . Therefore, these validated results demonstrated that the developed calculation procedure is capable of predicting the axis orbit and contact time of the journal during start-up. The geometric and operating condition parameters adopted by Mokhtar and Cui were listed in Table 4, in which the speed of the journal with radius of 37.326mm linearly increases from 0rpm to 850 rpm within 0.3s.

(a)

(b) Fig.7. Validation of the axis orbit and contact time of the journal bearing during start-up: (a) journal axis orbit; (b) contact time. Table 4 Parameters of the validated journal bearing adopted by Mokhtar et al. [1] Parameters

value

Parameters

value

Bearing inner radius/RB

37.326mm

Acceleration time/ta

0.3s

Bearing width/L

76.2mm

Journal roughness/σJ

0.12µm

Radial clearance/C

0.12mm

Bearing roughness/σB

1.473µm

Rotation speed/ω

850r/min

Viscosity of the oil/η

0.074Pa.s

3.2 Transient analysis of the journal bearing behaviors during start-up The parameters used in the following numerical analysis were listed in Table 5. To demonstrate the important role of the thermal effect during start-up, the dynamic contact force and axis orbit yielded by the isothermal model and thermal model were compared, as shown in Fig.8. It can be observed that the dynamic contact load may be underestimated when the thermal effect is absent. One reasonable explanation of this observation is that the thermal expansion caused by the temperature rise leads to the increase in the occurrence of the metal-metal contact during start-up. Interestingly, the eccentricity ratio predicted by the thermal model is less than that predicted by the isothermal model, which can also be attributed to the thermal expansion of the journal and bearing shell. Therefore, the results presented in Fig.8 demonstrate the important role of the thermal effect for dynamic contact load predictions. 1.0

Considering thermal effects

250

0.6 255

Contact load(kN)

Not considering thermal effects 0.8

0.4 260 265

270

0.2

0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Time(s)

Fig.8. Comparison of the dynamic contact load and the axis orbit during start-up predicted by the isothermal model and the thermal model.

Table 5 Parameters of the simulated journal bearing Parameters

value

Parameters

value

Bearing inner radius/RB

12mm

Acceleration time/ta

0.1s

Bearing shell thickness/Tshell

2mm

Journal thermal expansion/αJ

11.9µm/(m.K)

Bearing width/L

8mm

Bearing thermal expansion/αB

19µm/(m.K)

Radial clearance/C

0.04mm

Bearing hardness/HY

1.05GPa

Rotation speed/ω

1000r/min

heat transfer coefficient /hh

80W(m2.K)

Bearing elastic modulus/E1

73GPa

Oil thermal conductivity/kL

0.13W/m.K

Journal elastic modulus/ E2

210GPa

Oil specific heat /CPL

2000 J/(kg.K)

Bearing Poisson ratio/υ1

0.37

Bearing thermal conductivity/kB

117 W/m.K

Journal Poisson ratio/υ2

0.3

Bearing specific heat /CPB

963 J/(kg.K)

Bearing density/ρB

2710kg/m3

Journal thermal conductivity/kJ

51.9 W/m.K

Journal density/ρJ

8700 kg/m3

Journal specific heat /CPJ

515 J/(kg.K)

Composite roughness/σ

0.55µm

Inlet oil temperature/Tin

Roughness orientation/χ

1

Boundary friction coefficient/µc

Oil viscosity(40 )/η

0.0599Pa.s

External load/F

1kN

40 0.1

Figure 9(a) shows the evolutions of the maximum temperature, contact load and maximum fluid pressure over operating time. As shown, the maximum temperature first increases rapidly for the time ranged from 0ms to 24ms, and then continuously decreases with the time increased from 24ms to 73ms. Figure 9 (b) illustrates that the minimum film thickness is smaller than 0.5µm at the initial stage of start-up, implying the asperity contact effect is dominant at such a stage, as shown in Fig.9 (a). As a consequence, a rapid increase in the maximum temperature is observed at the initial stage of start-up due to the excessive friction heat. However, with the further increase in operating time (after 24ms,), the hydrodynamic film was rapidly formed to lift the loaded journal, as shown in Fig.9 (a), which reduces the temperature by the convection effects of the oil film and the decreased asperity contact. In the current simulation, the full-film lubrication regime can be achieved when the time is larger than 73ms; the corresponding lift-off speed is 730r/min. The evolutions of the temperature distribution both in the circumferential and axial direction during start-up were illustrated in Fig.9 (c) and (d), respectively; and six time points (as shown in Fig.9 (a)) were chosen to present the corresponding transient temperature distribution both in the circumferential and axial direction, as

shown in Figs.9(c) and (d). It can be seen that the temperature distribution is concentrated at the bottom position of the bearing shell when the time equals 1ms and 6ms (at the initial stage of during start-up). With the further increase in the operating time, the temperature distribution of the contacting surface gradually becomes uniform due to the heat transfer effect, especially at the lift-off time (73ms) when the

1200

0.95

1000

157.5

Contact load Maximum temperature Maximum fluid pressure

800

Contact load(N)

162.5 165.0

0.97

167.5 172.5 170.0

0.98

177.5180.0 175.0

0.99

35

60

28

55 600 50

400

200

1.00

21

14

45

7

40

0

Axis orbit 0

1.01

-200

2.5

5.0 7.5 10.0 Attitude angle(Degree)

12.5

15.0

0.00

0.02

0.04

0.06

0.08

0.10

Time(s)

(a) 2.0

1.0

1.0

26.40

44

Contact pressure(MPa) 0.8

21.12

0.6

15.84

0.4

10.56

0.2

5.28

0.0

Minimum film thickness

0.00

2.8

3.0

3.2 3.4 Circumferential direction(rad)

42

3.6

0.5 Effective radial clearance 0.0

40

-0.5 38 -1.0

Effective radial clearance(µ µ m)

46

1.5 Axial direction(z/L)

0.0

µ m) Minimum film thickness(µ

Eccentricity ratio

160.0

0.96

65

Temperature(℃ )

0.94

Fluid pressure(MPa)

journal is completely separated from the bearing shell.

36 -1.5 -2.0

34 0.00

0.02

0.04

0.06

0.08

0.10

Time(s)

(b)

45.22

54.75

63.60

44.18

51.80

58.88

43.13

48.85

54.16

42.09

45.90

49.44

41.04

42.95

44.72

40.00

40.00

40.00

Journal

45.2

Lubricating oil

54.7

63.4

Bearing shell 54.20

42.64

43.63

51.36

42.11

42.89

48.52

41.58

42.16

45.68

41.06

41.42

42.84

40.53

40.69

40.00

40.00

39.95

54.1

4

3

2

1

42.6

43.6

6

5

(c)

1.0

45.22

0.8

43.13 42.09

0.6

41.04 40.00

0.4

0.2

0.0

45.90 0.6

42.95 40.00

0.4

6

1

2 3 4 5 Circumferential direction(rad)

0.8

42.83 40.00

0.4

0.2

Axial direction(z/L)

0.6

41.05 0.6

40.52 40.00

0.4

1

2 3 4 5 Circumferential direction(rad)

6

6

43.59 42.87

0.8

42.15 41.44 40.72

0.6

40.00 0.4

0.2

0.0

0.0 0

2 3 4 5 Circumferential direction(rad)

1.0

41.57

0.2

0.0

1

42.10

48.49 45.66

40.00

0

42.62

51.32 0.8

44.70

0.4

6

1.0

54.15

49.40 0.6

0.0 0

Axial direction(z/L)

2 3 4 5 Circumferential direction(rad)

54.10

0.2

0.0 1

1.0

Axial direction(z/L)

58.80 0.8

48.85

0.2

0

63.50

51.80

Axial direction(z/L)

Axial direction(z/L)

0.8

1.0

54.75

44.18

Axial direction(z/L)

1.0

0

1

2 3 4 5 Circumferential direction(rad)

6

0

1

2 3 4 5 Circumferential direction(rad)

6

(d) Fig.9. Dynamic behavior of the journal bearing during start-up: (a) axis orbit, contact load and maximum oil film temperature; (b) minimum film thickness and effective radial clearance: Temperature distribution of the coupling system in the (c) circumferential direction and the (d) axial direction.

Figure 10(a) shows that the contact area continuously decreases with the operating time, especially for the time ranged from 36ms to 70ms due to the rapidly formed hydrodynamic pressure. It can also be observed that the contact region moves toward the rotational direction of the journal gradually during start-up, corresponding to the axis orbit (Fig.9 (a)) and the transient thermal distribution (Fig.9 (d)) of the journal bearing. It is noteworthy that the asperity contact starts to concentrate on the both sides of the bearing shell surface after t=48ms. Figure 10(b) is presented to explain this phenomenon. As shown, in the mixed lubrication regime, the lubrication gap is governed by the elastic deformation driven by the formed hydrodynamic pressure, leading to the decrease in the contact pressure at the centre region [20 21] .

21.12

0.4

14.08

0.2

7.04

0.0

0.00

2.2

2.4

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

3.6

3.8

Axial direction(z/L)

0.6 0.4 0.2 0.0 2.0

2.2

2.4

Axial direction(z/L)

1.0

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

3.6

3.8

0.6 0.4 0.2 0.0 2.2

2.4

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

3.6

3.8

6.76 0.00

2.2

2.4

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

3.6

3.8

4.0

1.0

20.50 18.45 16.40 14.35 12.30 10.25 8.20 6.15 4.10 2.05 0.00

Contact pressure(MPa) 0.8

Time=36ms

0.6 0.4 0.2 0.0 2.0

8.86 7.97 7.09 6.20 5.32 4.43 3.54 2.66 1.77 0.89 0.00

Time=48ms

2.0

13.52

0.2

4.0

Contact pressure(MPa) 0.8

20.28

0.4

2.0 30.40 27.36 24.32 21.28 18.24 15.20 12.16 9.12 6.08 3.04 0.00

Contact pressure(MPa) Time=24ms

27.04

Time=12ms

0.6

4.0

1.0

33.80

Contact pressure(MPa) 0.8

0.0

Axial direction(z/L)

2.0

Axial direction(z/L)

28.16

Time=1ms

0.6

0.8

1.0

35.20

Contact pressure(MPa) 0.8

2.2

2.4

1.0 Axial direction(z/L)

Axial direction(z/L)

1.0

0.8

3.6

3.8

4.0 0.08 0.07 0.06 0.05 0.05 0.04 0.03 0.02 0.02 0.01 0.00

Time=70ms

0.6 0.4 0.2 0.0 2.0

4.0

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

Contact pressure(MPa)

2.2

2.4

2.6 2.8 3.0 3.2 3.4 Circumferential direction(rad)

3.6

3.8

4.0

1.4

1.2

1.2

1.0

1.0

0.8 Film thickness

0.8

0.6

Elastic deformation 0.6

0.4

0.4

0.2

0.2

Film thickness(µ µ m)

Elastic deformation(µ µ m)

(a)

0.0 0.0

0.2

0.4 0.6 Axial direction(z/L)

0.8

1.0

(b) Fig.10. (a) Transient contact pressure distribution during start up; (b) oil film thickness and elastic deformation along the axial direction when the operating time equals 48ms.

Figure 11 shows the circumferential distribution of the mixed-TEHD performances, including the thermal expansion, fluid pressure and contact pressure, at the different operating times. Figure 11(a) reveals that the thermal expansion increases with the time increased from 1ms to 24ms, and its maximum of 1.5µm can be identified at 24ms. However, the decreased thermal expansion was observed with the further increase in operating time due to the heat dissipation effect of the hydrodynamic oil-film. Figures 11(b) and (c) show the development of the fluid pressure and contact pressure distribution during start up. It can be observed that the increasing fluid pressure over time generates the decrease in the contact pressure. This is because that the formed fluid pressure reduces the occurrence of the asperity

contact by enlarging the lubrication gap.

(a)

(b)

(c) Fig.11. Distribution of the (a) thermal deformation (b) fluid pressure and (c) contact pressure in the circumferential direction.

3.3 Effects of the acceleration time The current section aims to identify the effect of the acceleration time on the transient tribol-dynamic performance. Figure 12(a) shows that a short acceleration time leads to a relatively large temperature and thermal expansion. The reason for this is because that a shorter acceleration time generates a larger sliding velocity at the same time, which yields a larger friction heat. However, a short acceleration time can also lead to the increases in heat dissipation rate, which may be attributed to the fact that a shorter acceleration time can form the hydrodynamic film quicker, as shown in Fig.12 (b). And it can be seen from Fig.12 (b) that the maximum fluid pressures at the lift-off time for three acceleration times are most the same (around 30MPa). Figure 12(c) indicates that a short acceleration time gives a relatively small lift-off time and a relatively large lift-off speed. Additionally, a shorter acceleration time yields a smaller contact load (or friction coefficient) before the lift-off speed is achieved, as shown in

Fig.12(c), implying that a short acceleration time is potential to reduce the sliding wear during start-up. The results shown in Fig.12 (d) illustrate that a shorter acceleration time can result in a smaller eccentricity ratio during start-up. This is because that a short acceleration time can promote the hydrodynamic effect, as shown in Fig.12(b), which lifts the journal to a higher position along the radial direction. The almost identical sliding distance can be identified among three acceleration times, as shown in Fig.12 (e). Generally, the results presented above demonstrated that, although a larger maximum temperature and thermal expansion are yielded, a short acceleration time can improve the tribol-dynamic performances, in terms of decreasing the asperity contact and increasing the hydrodynamic pressure. Therefore, on the operating condition that the maximum temperature and thermal expansion are in allowable range, a short acceleration time may be recommended. Figure 13 was presented to further evaluate the effect of the acceleration time on the maximum temperature and lift-off speed under different radial clearances. It can be seen from the Fig.13 (a) that a smaller maximum temperature can be observed at a larger acceleration time and a larger radial clearance, and the radial clearance seems has limited effect on the maximum temperature. Moreover, Fig.13 (b) gives that a larger lift-off speed can be identified at a shorter acceleration time and a larger radial clearance

(a)

(b)

(c)

(d)

(e) Fig.12. Comparison of the dynamic behavior under different acceleration times: (a) maximum temperature and thermal deformation;(b) maximum fluid and contact pressure;(c) contact load and friction coefficient;(d) eccentricity ratio and minimum film thickness; (e) axis orbit;

(a)

(b)

Fig.13 Effects of the radial clearance and the acceleration time on the (a) maximum temperature and the (b) lift-off speed.

3.4 Effects of the radial clearance Figure 14 shows that the transient mixed-TEHD behavior of the journal bearing during start-up under different radial clearances. As shown in Fig.14 (a), the maximum temperature and thermal expansion decreases with the increase in the radial clearance, which may be due to that a larger clearance space is more beneficial for heat dissipating. It is evident that, as shown in Fig.14 (b), the maximum fluid and contact pressure are significantly affected by the radial clearance, and a small radial clearance yields a low maximum fluid and contact pressure. Moreover, a shorter lift-off time (or lift-off speed) can be observed at a smaller radial clearance. However, the radial clearance slightly affects the contact load and friction coefficient, as shown in Fig.14(c). Figure 15 is presented to explain this observation. As shown in Fig.15 (a),

although a smaller radial clearance leads to a lower maximum fluid pressure, it gives a larger area for hydrodynamic pressure spreading. Similarly, as shown in Fig.15 (b), the contact load generated by a relatively small radial clearance is comparable with that generated by a relatively large radial clearance. Consequently, a slight difference in the dynamic contact load and the friction coefficient among these three radial clearances can be identified. Figure 14(d) shows that the increasing radial clearance can reduce the minimum film thickness. The axis orbit of the journal at three different radial clearances is illustrated in Fig.14 (e). It is evident that the decrease in the radial clearance generates the increase in sliding distance along the rotational direction, exhibiting a consistent trend with the published results in the literatures [13 14] . Furthermore, in order to evaluate the risk of seizure, the minimum effective radial clearance and maximum temperature at different static loads and radial clearances were compared, as shown in Figs.16 (a)-(b). It can be found that, as shown in Fig.16 (a), the effective radial clearance decreases with the increasing static load. The reason for this is that a larger static load generates a higher temperature, as shown in Fig.16 (b), which leads to a larger thermal expansion. To be specific, for the simulated case of F=1400N, C=0.01mm, the minimum effective radial clearance is only 4.21 µm, implying the seizure may occur if the static load is further increased. As expected, the maximum contact and fluid pressure is sensitive to the radial clearance variation, as shown in Figs.16(c)-(d).

(a)

(b)

(c)

(d)

(e) Fig.14. Comparison of the dynamic behavior under different radial clearances: (a) maximum temperature and thermal deformation;(b) maximum fluid and contact pressure;(c) contact load and friction coefficient;(d) eccentricity ratio and minimum film thickness; (e) axis orbit.

(a)

(b) Fig.15. Distribution of the (a) fluid pressure and the (b) contact pressure under different radial clearances. 100

40

F=600N F=800N F=1000N F=1200N F=1400N

Maximum temperature(℃ )

Effective clearance(µ µ m)

50

30

4.21µ m 20 10

80

F=600N F=800N F=1000N F=1200N F=1400N

60 40 20 0

0 0.01

0.02

0.03 0.04 Radial clearance(mm)

(a)

0.05

0.01

0.02

0.03 0.04 Radial clearance(mm)

(b)

0.05

40

Maximum contact pressure(MPa)

Maximum fluid pressure(µ µ m)

50 F=600N F=800N F=1000N F=1200N F=1400N

30 20 10 0 0.01

0.02

0.03 0.04 Radial clearance(mm)

0.05

50 40

F=600N F=800N F=1000N F=1200N F=1400N

30 20 10 0 0.01

0.02

0.03 0.04 Radial clearance(mm)

0.05

(c) (d) Fig.16. Effects of the radial clearance and static load on the mixed-TEHD performance: (a) effective radial clearance; (b) maximum temperature; (c) maximum fluid pressure; (d) maximum contact pressure.

3.5 Effect of the bearing shell thickness The bearing shell is another parameter affecting the tribo-dynamic performances, especially for the temperature and elastic deformation. As shown in Fig.17 (a), the decreasing bearing shell thickness can increase the maximum temperature and thermal expansion during start-up, which is in line with the results described in Fig.17 (d). Figure 17(c) shows that among three different bearing shell thicknesses, a slight discrepancy in the maximum contact and fluid pressure occurs during start-up, which may be attributed to the difference in the yielded elastic deformation, as demonstrated in Fig.17(c). Figure 18 further evaluate the effect of the bearing shell thickness on the maximum temperature, effective radial clearance, maximum contact and fluid pressure. It can be found that the decreasing thickness of bearing shell leads to the increase in the maximum temperature. Accordingly, the loss of clearance increases with the decreasing thickness of bearing shell, as shown in Fig.18 (b). It can also be found that, as depicted in Figs.18(c)-(d), a thinner bearing shell yields a larger contact and fluid pressure during start-up. Based on these results, the thin bearing shell is not recommended at a relatively small radial clearance in order to prevent the thermally induced seizure during start-up.

70

40

100

2.0

Tshell=2.5mm Maximum temperature

1.0

40

0.5

30

0.0

20 10

-1.0

0

-1.5

-10

-2.0 0.02

0.04

0.06 Time(s)

0.08

20

60

Maximum fluid pressure Tshell=1.5mm

0

Tshell=2.0mm

40

Tshell=2.5mm

-0.5

Maximum thermal deformation

0.00

80

-20

20 Maximum contact pressure

-40

0

-20

0.10

Maximum fluid pressure(MPa)

Maximum temperature(℃ )

Tshell=2.0mm

50

1.5

Thermal deformation(µ µ m) Maximum contact pressure(MPa)

Tshell=1.5mm

60

-60 0.00

0.02

(a)

0.04 0.06 Time(s)

0.08

0.10

(b) 75

Temperature(℃ )

70

Tshell=1.5mm,t=10ms

Tshell=2.5mm,t=10ms

Tshell=1.5mm,t=20ms

Tshell=2.5mm,t=20ms

Tshell=1.5mm,t=30ms

Tshell=2.5mm,t=30ms

Tshell=2mm,t=10ms Tshell=2mm,t=20ms

Maximum temperature

Tshell=2mm,t=30ms

65

60

55 2.6

(c)

2.8

3.0 3.2 3.4 3.6 Circumferential direction(rad)

3.8

4.0

(d)

Fig.17. Comparison of the dynamic behavior at different bearing shell thicknesses: (a) maximum temperature and thermal expansion; (b) maximum contact and fluid pressure; and circumferential distribution of the (c) deformation distribution and the (d) temperature.

(a)

(b)

(c)

(d)

Fig.18. Effects of the bearing shell thickness on the (a) maximum temperature, (b) effective radial

clearance, (c) maximum contact pressure and (d) maximum fluid pressure.

4. Conclusion Herein, a transient tribo-dynamic model was developed to numerically calculate the mixed-TEHD behavior of the journal bearing during start-up, in which the journal dynamic model and the transient mixed-TEHD model were bridged. The numerical predictions, including the temperature distribution, maximum temperature, axis orbit, and contact time of the journal bearing during start-up, are in reasonable agreement with those given by the published results, including both the experimentally and numerically. Furthermore, parametric studies have been performed to evaluate the effect of the acceleration time, radial clearance and bearing shell thickness on the numerical predictions during start-up. The following can be concluded (1) The maximum temperature first rapidly increases to its maximum value due to the severe asperity contact at initial stage of start-up, and then continually decreases because of the formed hydrodynamic film. Consequently, the maximum transient temperature occurs before the journal is completely separated from the bearing. (2) Although a shorter acceleration time can reduce the asperity contact, it also generates the increase in the maximum temperature and thermal expansion. Additionally, a smaller lift-off speed can be identified at a larger start-up time. (3) The decreasing radial clearance significantly reduces the maximum contact pressure, while it leads to the slightly increase in the maximum temperature and thermal expansion. And the journal bearing with a relatively small radial clearance is prone to seizure during start-up, especially for the cases of rapid start-up and heavy load. (4) The increasing maximum temperature and decreasing effective radial clearance can be observed when the bearing shell becomes thinner. Moreover, the maximum contact and fluid pressure are affected by the bearing shell thickness. A thinner bearing shell yields a larger contact pressure at the initial stage of start-up.

Acknowledge The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present study is partially supported by National Natural Science Foundation of China (51605053 and 51975064), the General Projects of Basic Science and Frontier Technology Research of Chongqing (cstc2018jcyjAX0442), and the China Postdoctoral Science Foundation funded

project (2018M631059 and 2019T120805)

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1. A transient tribo-dynamic model for journal bearings during start-up has been developed. 2. The transient 3D thermal characteristic has been integrated into the developed model. 3. The effects of the radius clearance, acceleration time and bearing shell thickness on the dynamic behavior considering the 3D thermal characteristic have been identified.

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.