Chinese Journal of Aeronautics, (2015), 28(4): 1296-1304

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

TEHL analysis of high-speed and heavy-load roller bearing with quasi-dynamic characteristics

Shi Xiujiang, Wang Liqin *

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China

Received 28 September 2014; revised 10 December 2014; accepted 12 March 2015 Available online 20 June 2015

KEYWORDS

Aero-engine mainshaft roller

bearing;

Ellipticity;

Quasi-dynamics; Radial clearance; TEHL; Velocity

Abstract In this paper, the aero-engine mainshaft roller bearing D1842926 under typical operating conditions is taken as a case study, a new integrated numerical algorithm of quasi-dynamics and thermal elastohydrodynamic lubrication (TEHL) is put forward, which can complete the bearing lubricated analysis from global dynamic performance to local TEHL state and break out of the traditional analysis way carried out independently in their own field. The 3-D film thickness distributions with different cases are given through integrated numerical algorithm, meanwhile the minimum film thickness of quasi-dynamic analysis, integrated numerical algorithm and testing are compared, which show that integrated numerical results have good agreements with the testing data, so the algorithm is demonstrated available and can judge the lubrication state more accurately. The parameter effects of operating and structure on pv value, cage sliding rate, TEHL film pressure, thickness and temperature are researched, which will provide an important theoretical basis for the structure design and optimization of aero-engine mainshaft roller bearing.

© 2015 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Numerous studies of aeroengine mainshaft bearings have been undertaken due to the special operating conditions—high speed, heavy load and great heat,1-6 which may lead to a sudden and catastrophic failure more easily than the common bearings. But with the development of bearing material, the

main reason on bearing failure hardly comes from fatigue and more from the surface injure related to the lubrication state.7

In engineering practice, the film thickness ratio8 is generally used to judge the lubrication state, thus the calculation accuracy of film thickness has a significant impact on the judgement of the lubrication state, particularly in the aero-engine mainshaft bearing which requires high reliability and precise limit design. But almost all the computations of minimum film thickness are obtained by fitting formulas in bearing dynamic analysis, and the most common one comes from Hamrock-Dowson (H-D)9 whose minimum film thickness in general operating conditions agrees well with numerical results.10,11 However, H-D formula at high speed and under heavy load conditions is limited on account of neglecting thermal effect, which had been proven by the previous researches of Luo12

* Corresponding author. Tel.: +86 451 8640 2012. E-mail addresses: shixiujiang@163.com (X. Shi), lqwanghit@163. com (L. Wang).

Peer review under responsibility of Editorial Committee of CJA.

http://dx.doi.Org/10.1016/j.cja.2015.06.013

1000-9361 © 2015 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

and Sun et al.13 Later, Wilson-Sheu14 (W-S) came up with the modified formula with a modified factor on the basis of H-D's, which have been demonstrated to still have a large error in this paper because of not covering all the effects like bearing operation, structure, material, lubricant performance, etc. Besides, Lubrecht,15 Canzi16 and Zhu et al.17 also pointed out the limitation of the fitting formulas and suggested the algorithm of numeric or testing should be adopted.

The numerical analysis18'19 of thermal elastohydrodynamic lubrication (TEHL) is an effective and accurate algorithm for film thickness computation. But during the last years, the algorithm has been hardly connected with application engineering and has been used only for theoretical calculation of the single rolling element and the race. The elastohydrodynamic lubrication (EHL) analysis and the dynamic analysis are both carried out in their own field for the complicated calculation, which make the bearing numerical analysis become incomplete and inaccurate. So in this study the integrated numerical algorithm of aero-engine mainshaft roller bearing under typical operating conditions will be researched to provide an important theoretical basis for the judgement of lubrication state and structural optimization.

2. Governing equation

2.1. Force and motion analysis

The quasi-dynamic model with high accuracy and applicability is adopted, and the force equilibrium of the bearing and the roller is shown in Fig. 1. In which OXYZ is the fixed coordinate system of the bearing and ObjXayaza is the follow-up coordinate system of the roller; j represents the jth roller, the roller is divided into n wafers, and k means the kth wafer; 1 and 2 represent the outer race and inner race; Q1jk and Q2jk mean the normal contact loads; T1jk and T2jk stand for the traction forces; P1jk and P2jk mean the film pressure; Fcj and fcj are the normal and tangential forces of the cage; Fdj refers to the resistance of gas-oil mixture; Fyj, Fzj and MXj, Myj, Mzj are the initial forces and moments; Fx, Fy, Fz and My, Mz represent the applied forces and moments; Wj (j =1,2,...) is the

azimuth angle of the roller; x2 and wm, are the angular velocities of the inner race, the revolution of the cage; xXj and woj are the rotation and the revolution of the roller. The quasi-dynamic equations' solution of cage and race can be obtained in Ref.20 and the balanced equations are given in Eq. (1).

YsiTjk - Tj + Pjk - Pyk)+ Fcj + Fdj + Fj = 0 k=1

E(Qjk - Qijk)-fcj + Fj = 0

Mj = 0

£(Tj + Tijk)-fc

Y^QjkXk - Qljkxk)+ Myj = 0 k=1 n

J2(P2jkXk - PjkXk)+ Mj = 0

Fig. 2 presents the roller structure and the contact motion diagrams between the roller and the race. In which h0 is the center film thickness of rigid bodies, Rx and Ry are the curvature radii of roller and modified roller radii (0.01 m), I is the total roller length and the straight length is 0.9/, Db is the roller diameter, œbj is the rotation of roller, x, y, z represent the directions of the rolling, the roller length and the film thickness. Assuming that outer race is fixed, the entrainment velocity U1j and U2j (average velocity of roller and outer/inner race), as well as the relative sliding velocity DU1j and AU2j are expressed as Eqs. (2)-(5), which can be solved through quasi-dynamic analysis.20

- C xbj

Uj = D1 [(1 + y>m + c'xbj]

U2j = [(1 - c')(x2 - xm)-

DUj = Dm [(1 + c')®m - y'xbj\ DÜ2J = Dm [(1 - c')(X2 - Xm) y'xbj\

where c' is defined as Db/Dm, and D diameter.

is the bearing pitch

Inner raceway Film

(!/=07360°

(a) Force of bearing (b) Force of roller

Fig. 1 Force equilibrium of bearing and roller.

Fig. 2 Roller structure and contact motion state of roller and races.

2.2. TEHL equations

The difference between thermal and isothermal Reynolds21 equation is the density and viscosity of TEHL varying in the film thickness direction. But the equation format is similar, only q/g and q are replaced by (q/g)e and q , which can be reduced to the following form:

d(q*h) dx

where p and h are the film pressure and thickness, g and q are the film viscosity and density, and the density/viscosity-pres sure-temperature equations come from Ref.22; (q/g)e and q are defined as

( (q/g)e = 12(geqe/ge - qe)

I q* - 2(qe- g eqe)

where ge, ge, qe, qe, q^ can be obtained by

ge= h/ /o ^

ge- h2//0h^

qe = h/û h Pdz

t 1 r h r

qe = h^Jû qJû qe = ^/ûh qSo

,z' dz'

The film thickness equation taking elastic deformation into account is calculated by

h(x,y) = ho + + 2Rx

x2 (|y|— 0.9/)'

p(x ,y )

(y — y' )2

= dx' dy'

where E is the general elastic modulus and the forth term in the equation represents the elastic deformation of a point (X,y) under the pressure p(x',y).

The energy equation of the film can be expressed in the form

' dT qU dx~

qv dy-

'"dz2"

T dq dp q dT dx

where z is the coordinate along film thickness direction, c0 and k0 are the film specific heat and thermal conductivity, u and v are the flow velocities along x and y axis, and q is flow of the lubricant.

3. Numerical analysis method

Fig. 3 presents the basic solving steps. First, the initial conditions are given, which take the H-D's minimum film thickness as the starting film thickness, then the quasi-dynamic equations are solved to get the micro-contact field motion and stress state by Newton-Raphson and steepest descent meth-ods.20 Further, TEHL analysis is conducted to acquire the EHL performance including film thickness, pressure and temperature. In the solving process of TEHL analysis, the lubrication equations are dimensionlessed, dispersed22 and solved by multi-grid method. The grids employed to solve film pressure include 5 layers, which have 128 nodes in x direction and 1024 nodes in y direction on the top layer. And the column-scan method is used to solve thermal field whose grid number is the same as that in the top layer of pressure solution; the nodes in film thickness direction contain each 6 nodes in the bodies and 10 nodes in the film.

4. Results and discussion

4.1. Minimum film thickness comparison with different algorithms

The basic parameters of aeroengine mainshaft roller bearing D1842926 and aero lubricant 4109 are given in Tables 1 and 2.

Fig. 3 Flow diagram of numerical analysis.

Table 1 Basic parameters of aeroengine mainshaft roller bearing D1842926.23

Material parameter Value Geometry parameter Value

Elastic modulus of roller 218 Inner radius (mm) 130

and race (GPa)

Poisson's ratio of roller 0.3 Outer radius (mm) 180

and race

Density of roller and race 7870 Roller diameter 12

(kg/m3) (mm)

Elastic modulus of cage 209 Total roller length 12

(GPa) (mm)

Poisson's ratio of cage 0.3 Straight roller length 10.8

Density of cage (kg/m3) (mm)

7870 Roller number 30

Specific heat (J/(kg °C)) 460 Thermal conductivity (N/ (s-°Q) 15

Table 2 Basic parameters of aero lubricant 4109.

Parameter Value Parameter Value

Environmental 0.033 viscosity (Pa s) Environmental 970 density (kg/m3) Specific heat (J/ 1910 (kg -°C)) Viscosity-temperature 0.032 coefficient (°C_1) Viscosity-pressure 1.85 x 10~8 coefficient (Pa-1) Thermal conductivity 0.0966 (N/(s-°C))

Since the outer race radii is larger than the inner race one, which means the relative velocity of the out race is larger, the film forms easier in the outer race and the minimum film thickness would appear in the micro-contact field of the inner race and the rollers. In quasi-dynamic analysis, the minimum film thickness use to be calculated by fitting formulas. Fig. 4 shows the contact operating conditions (contact load and entrain-ment velocity) in micro-contact field between rollers and inner race which are obtained by quasi-dynamic analysis under 4 cases from Ref.23. The applied radial loads of the 4 cases are 8 kN, 12 kN, 18 kN and 24 kN, and the velocities of the inner race are all 13218 r/min. As seen in Fig. 4, the contact loads and the loaded roller number increase with the increasingly applied radial load, and the entrainment velocities almost keep unchanged because of the same inner race velocities.

The integrated numerical analysis of quasi-dynamics and TEHL is put forward in this paper. The minimum film thicknesses all appear at the 360° azimuth angle of the roller, which is caused by the maximum load at this location, and the 3-D film thickness distributions in 4 cases are provided in Fig. 5, which reveals that the minimum film thicknesses are 0.514 im, 0.490 im, 0.471 im and 0.445 im, and b represents the half width of Hertz contact. As can be seen in Fig. 5, the minimum film thicknesses emerge on both sides of the roller for the stress concentration, which becomes more and more apparent from Case1 to Case 4 due to the increasing load.

At the same time, the minimum film thickness distributions between the inner race and rollers from Case 1 to Case 4 with 4 methods are described in Fig. 6, which indicate that H-D's minimum film thickness has a large error due to neglecting thermal effect, though the error of W-S's decreases, which is still above 40% for not covering all the testing conditions such as bearing operating, structure, material, lubrication parameters, etc. The numerical results have a great agreement with the testing results23 whose error is only within 10%, which prove that the algorithm is available and can judge the lubrication state more accurately.

4.2. Operating condition effects on bearing tribology performance

4.2.1. Radial load effects

Above all, thermal effect has a large influence on the film thickness. In bearing dynamic analysis, the cage sliding ratio defined as the ratio of the difference between the actual cage velocity and the theoretical velocity to the theoretical velocity and the pv value represented as the product of the contact stress and the relative sliding velocity between the rollers and the inner race are often used to characterize heat distribution, and the larger the cage sliding ratio and the pv value, the more the heat.

As the rollers and inner race contact, the distributions of cage sliding ratios and the pv values under different applied radial loads (the velocity are all 13218 r/min) are shown in Fig. 7, which suggests that the cage sliding ratio decreases with the increase of the load. The loaded rollers increase with the increase of the load, which causes the cage velocity to increase and the sliding ratio decreases. However, the pv values increase from 8 kN to 12 kN and decrease from 12 kN to 18 kN. The reasons are that the loaded rollers keep the same from 8 kN to 12 kN firstly, the high load leads to high pressure, so the pv values increase; then, the loaded rollers increase with the

Fig. 4 Contact load and entrainment velocity in micro-contact field between rollers and inner race.2

Fig. 5 3-D film thickness distributions in different cases.

increasing load from 12 kN to 18 kN, the pressure of the single roller decreases, and the pv values decrease.

Meanwhile, Fig. 8 presents the distributions of dimension-less film pressure, thickness and temperature with different applied radial loads (the velocities are all 13218 r/min) in the rolling direction at 360° azimuth angle of the roller through

the integrated numerical algorithm, where p is the dimensionless film pressure, h the dimensionless film thickness, and T the dimensionless film temperature, X the dimensionless coordinate in the half width direction of Hertz contact. As shown in Fig. 8, with increasing stress and friction heat the pressure and the temperature increase, and the film thickness decreases.

Fig. 6 Minimum film thickness distributions with four methods. 4.2.2. Velocity effects

Fig. 9 displays the distributions of cage sliding ratios and the pv values at different velocities (the loads are all 18 kN), which shows that the cage sliding ratios increase with the increase of the velocity. Since the loaded field decreases with the increasing centrifugal force caused by the increase of the velocity, which make the cage velocity decrease and the sliding ratios increase. And the pv values increase with the increase of the velocity, because the relative velocity increases with the increasing velocity, which leads the pv values to increase.

Further, the distributions of dimensionless film pressure, thickness and temperature at different velocities (the load are all 18 kN) are portrayed in Fig. 10, which states that the

pressures are almost the same, the temperature increases because of the increasing relative sliding velocity which increases with the increase of the velocity and the film thickness increases with the increase of fluid hydrodynamic effect.

4.3. Structure effects

4.3.1. Radial clearance effects

Take Case 4 as an example, the distributions of cage sliding ratios and the pv values under different radial clearance (the ellipticity is 0) are depicted in Fig. 11, which shows the cage sliding ratios and the pv values increase with the increase of the radial clearance. Since the loaded roller number decreases with the increase of the radial clearance, which makes the cage sliding ratios increase, the contact stress and the relative velocity increase and the pv values increase.

In the meantime, Fig. 12 gives the distributions of dimen-sionless film pressure, thickness and temperature under different radial clearance (the ellipticity is 0), which shows that the pressure and the temperature increase on account of the increasing relative sliding velocity and the stress with the increase of the radial clearance, and the film thickness decreases.

4.3.2. Ellipticity effects

Fig. 13 displays the cage sliding ratio and the pv value distributions under different ellipticity (the radial clearance is 60 im), which indicates the sliding ratios and the pv values decrease with the increase of ellipticity. As the ellipticity exists, the mainshaft will be forced more times during a rotation period,

Fig. 7 Radial load effects on cage sliding ratio and pv value.

Fig. 8 Radial load effects on film pressure, thickness and temperature.

(a) Velocity effects on cage sliding ratio (b) Velocity effects onpv value

Fig. 9 Velocity effects on cage sliding ratio and pv value.

(b) Film thickness

Fig. 10 Velocity effects on film pressure, thickness and temperature.

Fig. 11 Radial clearance effects on cage sliding ratio and pv value.

(a) Film pressure (b) Film thickness (c) Film temperature

Fig. 12 Radial clearance effects on film pressure, thickness and temperature.

Fig. 13 Ellipticity effects on cage sliding ratio and pv value.

Fig. 14 Ellipticity effects on film pressure, thickness and temperature.

which means the more rollers will be preloaded, the loaded rollers will increase, the cage sliding ratio will decrease, the contact stress and the relative velocity will decrease, which will lead pv values to decrease.

Furthermore, the distributions of dimensionless film pressure, thickness and temperature under different ellipticity24 (the radial clearance is 60 im) defined as the distance of raceway and base circle are shown in Fig. 14, which declares that the pressure and the temperature decrease with the increase of ellipticity and the film thickness increases. This is caused by the decreasing relative sliding velocity and pressure with the increase of ellipticity; the temperature decreases, the contact load decreases and the film thickness increases.

5. Conclusions

(1) The 3-D film thickness distributions with different cases are given through integrated analysis and the minimum film thickness of quasi-dynamic analysis and integrated analysis are compared. Results show that the integrated numerical minimum film thicknesses are closer to the testing results, which prove that the integrated analysis is available and can be used to judge the lubrication state more accurately.

(2) The effects of operating conditions on bearing tribology performance are researched, which indicate that the cage sliding ratios decrease with the increasing loads, the pv values increase firstly then decrease; the film pressure and the temperature increase, and the thickness decreases. The cage sliding ratios and pv values increase

with the increase of the velocity, the film pressure almost keeps the same, and the temperature and the thickness increase.

(3) The influences of structural conditions on bearing tribol-ogy performance are discussed, which claim that the cage sliding ratios and pv values increase with the increase of the radial clearance; the film pressure and the temperature increase; the thickness decreases. The cage sliding ratios and pv values decrease with the increasing ellipticity, the film pressure and the temperature decrease, and the thickness increases.

Acknowledgments

This study was supported by the National Key Basic Research Program of China (No. 2013CB632305) and the National Natural Science Foundation of China (No. 51373186).

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Shi Xiujiang is a Ph.D. student at the School of Mechatronics Engineering, Harbin Institute of Technology. He received the B.S. and M.S. degrees in mechanical engineering from Qingdao Technological University in 2011 and 2014 respectively. His area of research includes rolling bearing dynamics and TEHL.

Wang Liqin is a professor and Ph.D. supervisor at the College of Mechatronic Engineering, Harbin Institute of Technology. His current research interests are aerospace tribology and rolling bearing of high performance.