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Nonlinear thermal behaviors of the inter-shaft bearing in a dual-rotor system subjected to the dynamic load

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Abstract

This paper proposes a new theoretical method to investigate the thermal behaviors of the inter-shaft bearing considering the nonlinear dynamic characteristics of a dual-rotor system by combining heat transfer and nonlinear dynamics. The nonlinearities of the inter-shaft bearing, including the Hertzian contact and the radial clearance, are considered during the dynamic modeling for the system. The dynamic load of the inter-shaft bearing is defined according to the nonlinear dynamic responses of the system. Therefore, some fundamental nonlinear phenomena, i.e., jump and bi-stable phenomena happen to the dynamic load. It makes the dynamic load more appropriate to describe the actual load of the inter-shaft bearing than the static load. Furthermore, a steady-state heat transfer model for the inter-shaft bearing subjected to the dynamic load can be set up with the help of Palmgren’s empirical formula. The variation of temperatures with the rotation speed is obtained by using the Gauss–Seidel iteration. Temperatures of the inter-shaft bearing also show nonlinear thermal behaviors, i.e., jump and bi-stable phenomena. It implies the nonlinear dynamic behaviors of the system have a great impact on the thermal behaviors of the inter-shaft bearing. Moreover, an exhaustive parametric analysis for temperatures and nonlinear thermal behaviors of the inter-shaft bearing affected by dynamic parameters (including the rotation speed ratio, unbalances of rotors, the radial clearance, the stiffness and the roller number of the inter-shaft bearing) and thermal parameters (including the lubricant viscosity and the ambient temperature) is carried out. The results show that the rotation speed ratio has a significant influence on both temperatures and nonlinear thermal behaviors, other dynamic parameters mainly affect nonlinear thermal behaviors, while thermal parameters only affect temperatures. This unique discovery indicates the thermal behaviors of the inter-shaft bearing could be much more complex because of the nonlinear dynamic characteristics of the dual-rotor system. The obtained results will contribute to a better understanding of the nonlinear thermal behaviors of bearings and profoundly reveal the mechanism of the nonlinear thermal behaviors of bearings.

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Abbreviations

M :

Total friction torque

M l :

Friction torque due to the load

M ν :

Friction torque due to the viscosity

Q :

Total FH

Q l :

Load FH

Q ν :

Viscosity FH

Q r :

FH distributed to rollers

Q i :

FH distributed to inner race

Q o :

FH distributed to outer race

f l :

A coefficient depends on the type of roller bearing

f ν :

A coefficient depends on the type of roller bearing and the type of lubrication

r LP :

Inner radius of LP rotor

d :

Nominal bore

r i :

Radius of inner race

D m :

Pitch diameter

r o :

Radius of outer race

D :

Nominal outside diameter

r HP :

Outside radius of HP outer

d r :

Roller diameter

a r :

Roller length

K b :

Stiffness of the inter-shaft bearing

B :

Width of the inter-shaft bearing

A :

Area

∑ ρ i :

Curvature sum of rollers-inner race contact pair

∑ ρ o :

Curvature sum of rollers-outer race contact pair

e 1 :

LP rotor’s unbalance

h :

Convective heat transfer coefficient

ε m :

Aspect ratio

V :

Line speed

k steel :

Thermal conductivity of steel

ν :

Kinematic viscosity of the lubricant

α :

Thermal diffusivity

α steel :

Thermal diffusivity of steel

Adown :

“Jump point”

Bdown :

“Jump point”

\( \omega_{{{\text{A}}_{\text{down}} }} \) :

“Frequency of jump point”

\( \omega_{{{\text{B}}_{\text{down}} }} \) :

“Frequency of jump point”

\( \Delta T_{{{\text{A}}_{\text{down}} }} \) :

“Jump amplitude”

\( \Delta T_{{{\text{B}}_{\text{down}} }} \) :

“Jump amplitude”

\( \Delta \omega_{\text{A}} \) :

“Bi-stable interval”

T :

Common temperature

T L :

Temperature of lubricant

T r :

Temperature of rollers

T i :

Temperature of inner race

T o :

Temperature of outer race

T LP :

Temperature of the portion of LP rotor contact inner race

T HP :

Temperature of the portion of HP rotor contact outer race

T ∞ :

Ambient temperature

R ri :

Thermal resistance of rollers-inner race

R ro :

Thermal resistance of rollers-outer race

R Lr :

Thermal resistance of lubricant rollers

R Li :

Thermal resistance of lubricant-inner race

R Lo :

Thermal resistance of lubricant-outer race

R i :

Thermal resistance of inner race-LP rotor

R o :

Thermal resistance of outer race-HP rotor

R LP :

Thermal resistance of LP rotor-ambient

R HP :

Thermal resistance of HP rotor-ambient

F b :

Dynamic load of the inter-shaft bearing

F n :

Normal force between roller and races

2δ0 :

Radial clearance of the inter-shaft bearing

N b :

Roller number of the inter-shaft bearing

n b :

Stressed roller number

ω 1 :

Rotation speed of LP rotor

ω 2 :

Rotation speed of HP rotor

λ :

Rotation speed ratio

e 2 :

HP rotor’s unbalance

Nu:

Nusselt number

Re:

Reynolds number

Pr:

Prandtl number

Ta:

Taylor number

Bi:

Biot number

Pe:

Peclet number

Pe* :

Modified Peclet number

Aup :

“Jump point”

Bup :

“Jump point”

\( \omega_{{{\text{A}}_{\text{up}} }} \) :

“Frequency of jump point”

\( \omega_{{{\text{B}}_{\text{up}} }} \) :

“Frequency of jump point”

\( \Delta T_{{{\text{A}}_{\text{up}} }} \) :

“Jump amplitude”

\( \Delta T_{{{\text{B}}_{\text{up}} }} \) :

“Jump amplitude”

\( \Delta \omega_{\text{B}} \) :

“Bi-stable interval”

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Acknowledgements

The authors are very grateful for the financial supports from the National Major Science and Technology Projects of China (Grant No. 2017-IV-0008-0045), the National Basic Research Program of China (973 Program) (Grant No. 2015CB057400) and the National Natural Science Foundation of China (Grant Nos. 11972129 and 11602070).

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  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Peng Gao, Yushu Chen & Lei Hou

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Appendix

Appendix

The coefficient matrix A is a symmetric matrix, which is shown as follows:

$$A= \left[ {\begin{array}{*{20}c} { - \frac{1}{{R_{\text{ri}} }} - \frac{1}{{R_{\text{ro}} }} - \frac{1}{{R_{\text{Lr}} }}} & {\frac{1}{{R_{\text{ri}} }}} & {\frac{1}{{R_{\text{ro}} }}} & {} & {} & {\frac{1}{{R_{\text{Lr}} }}} \\ {\frac{1}{{R_{\text{ri}} }}} & { - \frac{1}{{R_{\text{ri}} }} - \frac{1}{{R_{\text{i}} }} - \frac{1}{{R_{\text{Li}} }}} & {} & {\frac{1}{{R_{\text{i}} }}} & {} & {\frac{1}{{R_{\text{Li}} }}} \\ {\frac{1}{{R_{\text{ro}} }}} & {} & { - \frac{1}{{R_{\text{ro}} }} - \frac{1}{{R_{\text{o}} }} - \frac{1}{{R_{\text{Lo}} }}} & {} & {\frac{1}{{R_{\text{o}} }}} & {\frac{1}{{R_{\text{Lo}} }}} \\ {} & {\frac{1}{{R_{\text{i}} }}} & {} & { - \frac{1}{{R_{\text{i}} }} - \frac{1}{{R_{\text{LP}} }}} & {} & {} \\ {} & {} & {\frac{1}{{R_{\text{o}} }}} & {} & { - \frac{1}{{R_{\text{o}} }} - \frac{1}{{R_{\text{HP}} }}} & {} \\ {\frac{1}{{R_{\text{Lr}} }}} & {\frac{1}{{R_{\text{Li}} }}} & {\frac{1}{{R_{\text{Lo}} }}} & {} & {} & { - \frac{1}{{R_{\text{Lr}} }} - \frac{1}{{R_{\text{Li}} }} - \frac{1}{{R_{\text{Lo}} }}} \\ \end{array} } \right] $$

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Gao, P., Chen, Y. & Hou, L. Nonlinear thermal behaviors of the inter-shaft bearing in a dual-rotor system subjected to the dynamic load. Nonlinear Dyn 101, 191–209 (2020). https://doi.org/10.1007/s11071-020-05753-w

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