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Analytical consideration of the radial clearance to reduce cage slip of the turbo engine roller bearing

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Abstract

Roller bearings of aircraft turbo engines are operated under high-speed conditions of over 2 million DN. If contact force between rollers and the raceway is insufficient, then relative slip between the roller and raceways occurs and speed of the cage decreases. This condition may induce skid damage in the raceway. Large operating radial clearance can cause skid damage, but their analytical correlations have rarely been studied. In this study, cage slip ratio was quantified according to clearance changes in various operating conditions and geometric parameters, and causes of cage slip were analyzed using the simplified dynamic model. The design methodology will be proposed to select the proper operating clearance in consideration of cage slip and fatigue life.

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Abbreviations

Δδ Fit :

Clearance change by interference fit

Δδ T :

Clearance change by temperature changes

Δδ Cir :

Clearance change by centrifugal force

Δδ total :

Total clearance change during operation

I w :

Roller width

cr :

Radial clearance

r sh,o :

Radius of outer-race shoulder

r sh,i :

Radius of inner-race shoulder

r w :

Radius of roller

r i :

Radius of inner-raceway

r o :

Radius of outer-raceway

r m :

Radius of bearing pitch

r ci :

Cage inner radius

r co :

Cage outer radius

x i :

Displacement of mass center of the inner-race along X-axis

y i :

Displacement of mass center of the inner-race along Y-axis

x .i :

Velocity of mass center of the inner-race along X-axis

y i :

Velocity of mass center of the inner-race along Y-axis

δ ij :

Contact deformations between inner-race and j-th roller

δ oj :

Contact deformations between outer-race and j-th roller

r .j :

Radial direction movement of j-th roller

r j :

Radial direction velocity of j-th roller

θ j :

Azimuth angle of j-th roller

N ij :

Contact force between inner-race and j-th roller

N oj :

Contact force between outer-race and j-th roller

k j :

Contact stiffness between inner-race and roller

k o :

Contact stiffness between outer-race and roller

x c :

Displacement of mass center of the cage along X-axis

y .c :

Displacement of mass center of the cage along Y-axis

x .c :

Velocity of mass center of the cage along X-axis

y c :

Velocity of mass center of the cage along Y-axis

δ ci :

Contact deformations between inner-race and cage

θ ci :

Contact point angle between inner-race shoulder and cage

δ co :

Contact deformations between outer-race and cage

θ ci :

Contact point angle between outer-race shoulder and cage

N ci :

Contact force between inner-race shoulder and cage

N co :

Contact force between outer-race shoulder and cage

k ci :

Contact stiffness between inner-race shoulder and cage

k co :

Contact stiffness between outer-race shoulder and cage

δ pcf,j :

Contact deformations between front pocket and j-th roller

δ pcr,j :

Contact deformations between rear pocket and j-th roller

r pc :

Radial position of j-th roller from the center of cage

θ c,j :

Azimuth angle of j-th cage pocket

θ pc :

Angular gap of cage pocket

N pcf,j :

Contact force between front pocket and j-th roller

N pcr,j :

Contact force between rear pocket and j-th roller

k pc :

Contact stiffness between cage pocket and roller

\({\underset{..}{m}_i}\) :

Mass of inner-race

\(\matrix{{{x_i}} \cr {..} \cr} \) :

Acceleration of mass center of the inner-race along X-axis

\(\matrix{{{y_i}} \cr {..} \cr} \) :

Acceleration of mass center of the inner-race along Y-axis

\({\underset{..}{x}_c}\) :

Acceleration of mass center of the cage along X-axis

y c :

Acceleration of mass center of the cage along Y-axis

F ci :

Frictional force between inner-race shoulder and cage

F ij :

Frictional force between inner-race and j-th roller

F ig :

Applied radial load on the inner-race

u :

Slip velocity

μ :

Frictional coefficient

m r :

Mass of roller

F pcf,j :

Frictional force between front pocket and j-th roller

F pcr,j :

Frictional force between rear pocket and j-th roller

F cir,j :

Centrifugal force on j-th roller

F g,j :

Weight of j-th roller

φ j :

Orbital angle relative to the initial angle of j-th roller

\({\mathop {\underset{.}{\varphi}}\limits^. _j}\) :

Orbital angular velocity of j-th roller

φ j :

Orbital angular acceleration of j-th roller

F oj :

Frictional force between outer-race and j-th roller

F D,j :

Lubrication oil drag on j-th roller

J r :

Moment of inertia of roller

ω rj :

Rotational velocity of j-th roller

\({\dot \omega _{rj}}\) :

Rotational acceleration of j-th roller

n :

Total number of rollers

C D :

Drag coefficient

ρ :

Density of lubrication oil

V j :

Linear velocity of j-th roller

A :

Cross-sectional area of the roller

Re :

Reynolds number

d :

Diameter of roller

μ vis :

Viscosity of lubrication oil

F cg :

Weight of cage

F cir, c :

Centrifugal force on cage

m c :

Mass of cage

J c :

Moment of inertia of cage

\({\underset{.}{\varphi}_c}\) :

Rotational angle of cage

\({\underset{.}{\varphi}_c}\) :

Rotational speed of cage

\({\underset{.}{\varphi}_c}\) :

Angular acceleration of cage

θ co :

Orbital angular acceleration of cage

t k :

k-th time

Δt :

Calculation time step

ω c, ideal :

Epicyclic (ideal) angular speed of cage

ω i :

Rotational speed of shaft

s :

Cage slip ratio

θ contact :

Angular range of the contact zone

δ i,max :

Maximum amount of deformation of the raceway

LS norm :

Normalized radial load/normalized speed2

C r,limit :

Radial clearance limit in terms of cage slip

C r,f,limit,− :

Negative radial clearance limit in terms of normal contact force

C r,f,limit,+ :

Positive radial clearance limit in terms of normal contact force

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Acknowledgments

This work was supported by the project “Technology of Turbofan Engine System Integration Development” of Defense Acquisition Program Administration and Agency for Defense Development.

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Authors and Affiliations

  1. The 4th R&D Institute, Agency for Defense Development, 34186, Daejeon, Korea

    Sun Je Kim

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  1. Sun Je Kim
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Correspondence to Sun Je Kim.

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Sun Je Kim is a Senior Researcher of the 4th R&D Institute of Agency for Defense Development, Daejeon, Korea. He received his B.S., M.S., and Ph.D. in Mechanical Engineering from Korea Advanced Institute of Science and Technology, Daejeon, Korea. His research interests include dynamic modeling, mechanism design, power-train design and control, and actuator design.

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Kim, S.J. Analytical consideration of the radial clearance to reduce cage slip of the turbo engine roller bearing. J Mech Sci Technol 35, 2827–2839 (2021). https://doi.org/10.1007/s12206-021-0606-0

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  • DOI: https://doi.org/10.1007/s12206-021-0606-0

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