# Thermal Characteristic Analysis and Experimental Study of a Spindle-Bearing System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermo-Mechanical Coupling Model of the Spindle-Bearing System

#### 2.1. Deflection of High-Speed Ball Bearing under Applied Load

^{0}is the initial contact angle when there is no load, δ

_{a}and θ are the relative axial and angular displacement of the inner and outer rings, R

_{i}is the radius of the locus of the inner raceway groove curvature center, δ

_{r}is the relative radial displacement of the inner ring center, and ψ is the azimuth angle of the rolling element.

_{1}and X

_{2}are introduced to help the following analysis. Then the inner and outer contact angles for any position of the rolling element can be expressed as:

_{i}

_{(o)}is the normal contact deformation between the rolling element and the inner (outer) raceway groove, D is the diameter of the rolling element, and f

_{o}is the ratio of the curvature radius of the raceway groove to the diameter of the rolling element.

#### 2.2. Frictional Heat Generation of High-Speed Ball Bearing

_{x′n}and F

_{y′n}. In the x′z′ plane, friction force F

_{x′n}, normal load Q

_{n}, and centrifugal force F

_{c}achieve force equilibrium. In the x′y′ plane, friction force F

_{y′n}, and viscous friction force F

_{v}achieve force equilibrium. The moments due to the surface friction shear stresses are M

_{x′n}, M

_{y′n}, and M

_{z′n}. Moments M

_{y′n}and M

_{z′n}achieve equilibrium at gyroscopic moments M

_{y′n}and M

_{z′n}, respectively.

_{i}

_{(o)}is the normal load between the rolling element and the inner (outer) raceway groove; F

_{x’i}

_{(o)}and F

_{y’i}

_{(o)}are the frictional force of the contact area between the rolling element and the inner (outer) raceway groove in the x’ and y’ direction, respectively; F

_{c}is the centrifugal force; F

_{v}is the viscous friction of the rolling element; F

_{a}is the axial force; a

_{i}

_{(o)}and b

_{i}

_{(o)}are the semi-major axis and the semi-minor axis of the contact area between the rolling element and the projection of the inner (outer) raceway groove, respectively; θ

_{i}

_{(o)}is the included angle of the normal center line at any point in the contact area between the rolling element and the projection of the inner (outer) raceway groove; τ

_{x’i(o}

_{)}and τ

_{y’i(o)}are the frictional shear stress between the rolling element and the inner (outer) raceway groove in x’ and y’ direction, respectively; and M

_{gy}

_{’}and M

_{gz’}are the gyroscopic moments of the rolling element in x’ and y’ direction, respectively.

_{m}is the pitch diameter of the bearing, ω

_{m}is the revolution speed of the rolling element, and F

_{v}is the viscous friction force of the lubricant.

## 3. Heat Flow and Thermal Expansion Model

#### 3.1. Heat Transfer Model

#### 3.1.1. Thermal Contact Resistance between Rolling Element and Raceway Groove

#### 3.1.2. Heat Transfer Coefficient of Lubricating Grease

_{m}is the relative velocity between the grease and the rolling element, x is the pitch diameter of the bearing, ν is the kinematic viscosity of the grease, and P

_{r}is the Prandtl number of the grease.

#### 3.1.3. Convective Heat Transfer Coefficient of Cooling Oil

_{gap}is the equivalent diameter, R

_{e}is the Reynolds number, and P

_{r}is the Prandtl number.

^{2}·K).

#### 3.2. Temperature Distribution

_{9-in}, the outlet temperature is T

_{9-out}and the oil film temperature is T

_{9}. Its neighboring temperature nodes are the inner ring 2 of the bearing, the contact area 3 of the rolling element with the inner raceway groove, and the rolling element 4.

_{f}is the friction heat generation, H

_{c}

_{,3i-9}is the heat transferred by thermal conduction between the contact surface of the rolling element and the lubricant grease film, H

_{c}

_{,3b-9}is the heat transferred by thermal conduction between the contact surface of the inner raceway groove and the lubricant grease film, ω is the rotational speed, c is the specific heat capacity of the lubricating grease, and m is the quality of the lubricating grease.

_{ib}is the thermal contact resistance of the rolling element surface, R

_{bi}is the thermal contact resistance of the inner raceway groove surface, T

_{ib}is the contact surface temperatures of the rolling element, and T

_{bi}is the inner raceway groove.

#### 3.3. Thermal Deformation

_{o}is the outer radius of ring, r

_{i}is the inner radius of ring, and Г is the thermal expansion coefficient.

_{z}= 0.

_{r}is a coefficient for the radial elastic contact of the bearing.

## 4. Result Analysis and Discussion

#### 4.1. Experimental Verification

#### 4.2. Numerical Analysis

#### 4.2.1. Effect of Rotational Speed and Preload

#### 4.2.2. Effect of Lubricating Grease Temperature

^{2}/s at 40 °C and 4.7 mm

^{2}/s at 100 °C, respectively. The density of grease is 0.99 g/cm

^{3}. The relationship between kinematic viscosity, temperature, and pressure is expressed by Roelands as follows [20]:

#### 4.2.3. Effect of Cooling System

^{2}·K). When the convective heat transfer coefficient of the cooling system was 520.89 W/(m

^{2}·K), the corresponding radial temperature distribution of the spindle system could be obtained as shown in Figure 18. Compared with Figure 17, although the temperature of the inner raceway groove in the bearing increases by about 3 °C, the radial temperature difference of the spindle system can be controlled under 30 °C even at the highest rotational speed. Moreover, the whole spindle system can reach a thermal balance state after 2.5 h, which meets the temperature control requirement of the spindle system. Meanwhile, the analysis of the cooling system highlights that the convective heat transfer coefficient is not “the bigger the better” and the temperature of the cooling oil is not “the lower the better.” Carmichael and Davies [8] reported an excessively high thermally-induced preload inside the roller bearing for the case of water cooling of the housing, which can lead to the thermal failure of the roller bearing. Conversely, Chun [8] showed that the cooling of the shaft can considerably reduce the thermally-induced preload in the bearing. This shows that if the analysis is not accurate, it can also lead to the thermal failure of the spindle system [19].

#### 4.2.4. Analysis of Thermal Failure

## 5. Conclusions

- (1)
- The thermo-mechanical coupling model, the heat transfer model, and the numerical calculation of the temperature prediction model can be used to analyze the transient and steady state thermal characteristics of the spindle-bearing system owing to the lumped assumption of the spindle and the finite number of temperature nodes of the entire system. The main factors of models such as applied force, preload, lubricating state, surface morphology, and rotational speed are numerically analyzed.
- (2)
- A heat transfer model can be used to estimate critical parameters such as the thermal contact resistance between the rolling element and the raceway, the convective heat transfer coefficient of the cooling system, and the grease. The accuracy of the temperature distribution calculation depends on the selection of the boundary conditions and the initial temperature.
- (3)
- Various experimental schemes are designed and a number of experimental real-time measuring are conducted for comparative analysis. Not only are the effectiveness, accuracy, and practicability of the mathematical models verified, but also a comprehensive understanding about the thermal characteristic of the spindle system at the transient and steady state can be obtained. These experiments can make the spindle avoid the appearance of the instantaneous temperature peak and the unnecessary thermal failure in actual conditions.
- (4)
- Analysis of the spindle rotational speed, the preload of the spindle bearing, the grease temperature, and the cooling system are carried out. The significant effect of the high rotational speeds, preload oil viscosity, and heat transfer coefficients on the temperature or thermal failure of the bearing has been revealed, and schemes to improve the R&D of the spindle-bearing system are provided.

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | distance between raceway groove curvature centers |

f | r/D |

B | f_{i} + f_{o} − 1, total curvature |

d_{m} | diameter of pitch circle |

r | raceway groove curvature radius |

D | ball diameter |

a | semi-major axis of the contact area |

b | semi-minor axis of the contact area |

α^{0} | free contact angle |

α | mounted contact angle |

F_{p} | preload |

E | Young’s modulus |

ξ | Poisson’s ratio |

δ | displacement of the bearing |

θ | angular displacement of the bearing |

Z | number of balls per bearing |

ω | rotational speed |

σ | normal contact stress |

F(ρ) | curvature difference |

Σ(ρ) | curvature sum |

κ | eccentricity |

β | ball pitch angle |

β′ | ball yaw angle |

M_{gy′} | gyroscopic movement in y′ direction |

M_{gz′} | gyroscopic movement in z′ direction |

F_{c} | centrifugal force |

F_{a} | axial force |

R_{i(o)} | the radius of the locus of the raceway groove curvature center |

ψ | azimuth angle of the rolling element |

τ | frictional shear stress |

$\overline{U}$ | speed parameter |

${\overline{Q}}_{Z}$ | load parameter |

G | material parameter |

Q | ball-raceway normal load |

F | frictional force |

F_{v} | viscous friction force |

H | heat generation rate |

## Appendix A. Surface Friction Shear Stress between Rolling Element and Raceway Groove

_{0}is the oil film thickness of the contact center, and R

_{x}and R

_{y}are the equivalent curvature radii along the x, y direction.

_{c}/A

_{0}is

_{SUM}is the area density of summits, and d is the distance between the summit height and the surface mean plane. F

_{1}(t) is the integral

_{0}, m

_{2}, and m

_{4}are known as the zeroth, second, and fourth spectral moments of a profile, respectively. They are equivalent to the mean square height, slope, and second derivative of a profile in an arbitrary direction.

## Appendix B. Heat Transfer System of the Temperature Nodes

**Table B1.**Heat transfer system of the temperature nodes (C represents thermal conduction, V represents thermal convection, F represents heat generation).

Node | A | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | B |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | - | - | C | - | - | - | - | - | - | - | - | C | C | - | - | - | - | - | - | - |

2 | - | C | - | C | - | - | - | - | - | V | - | - | - | C | C | - | - | - | - | - |

3 | - | - | C | - | C | - | - | - | - | C | - | - | - | - | - | - | - | - | - | - |

4 | - | - | - | C | - | C | - | - | - | V | - | - | - | - | - | - | - | - | - | - |

5 | - | - | - | - | C | - | C | - | - | C | - | - | - | - | - | - | - | - | - | - |

6 | - | - | - | - | - | C | - | C | - | V | - | - | - | - | - | C | C | - | - | - |

7 | - | - | - | - | - | - | C | - | V | - | - | - | - | - | - | - | - | C | C | - |

8 | - | - | - | - | - | - | - | V | - | - | - | - | - | - | - | - | - | V | V | - |

9 | - | - | V | C | V | C | V | - | - | - | - | - | - | V | V | V | V | - | - | - |

10 | - | V | - | - | - | - | - | - | - | - | - | V | V | - | - | - | - | - | - | - |

11 | C | C | - | - | - | - | - | - | - | - | V | - | - | C | - | - | - | - | - | - |

12 | - | C | - | - | - | - | - | - | - | - | - | - | - | - | C | - | - | - | - | C |

13 | C | - | C | - | - | - | - | - | - | V | - | C | - | - | - | - | - | - | - | - |

14 | - | - | C | - | - | - | - | - | - | V | - | - | C | - | - | - | - | - | - | C |

15 | C | - | - | - | - | - | C | - | - | V | - | - | - | - | - | - | - | C | - | - |

16 | - | - | - | - | - | - | C | - | - | V | - | - | - | - | - | - | - | - | C | C |

17 | C | - | - | - | - | - | - | C | V | - | - | - | - | - | - | C | - | - | - | - |

18 | - | - | - | - | - | - | - | C | V | - | - | - | - | - | - | - | C | - | - | C |

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**Figure 5.**Temperature and thermal resistance model along the radial direction of the spindle system.

**Figure 9.**Comparison of the predicted housing temperature to the measured value by sensors for the test conditions of 3000 rpm and 1750 N preload.

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**MDPI and ACS Style**

Wu, L.; Tan, Q.
Thermal Characteristic Analysis and Experimental Study of a Spindle-Bearing System. *Entropy* **2016**, *18*, 271.
https://doi.org/10.3390/e18070271

**AMA Style**

Wu L, Tan Q.
Thermal Characteristic Analysis and Experimental Study of a Spindle-Bearing System. *Entropy*. 2016; 18(7):271.
https://doi.org/10.3390/e18070271

**Chicago/Turabian Style**

Wu, Li, and Qingchang Tan.
2016. "Thermal Characteristic Analysis and Experimental Study of a Spindle-Bearing System" *Entropy* 18, no. 7: 271.
https://doi.org/10.3390/e18070271