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Modeling of the motorized spindle temperature field considering the thermos-mechanical coupling on constant pressure preloaded bearings

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Abstract

The thermal characteristics of the motorized spindle significantly affect the machining accuracy and efficiency, and many thermal models have been developed to investigate the factors that affect the spindle thermal characteristics. However, the thermomechanical coupling of the bearings with constant pressure preload is rarely considered in the present works. Thus, this paper developed a transient temperature model of motorized spindle to study the influence of the radial thermal stress on the heat generation of the constant pressure preloaded bearings. In this research, an analytical thermal stress model was established first by simplifying the components of the bearings into a rotating ring geometry to calculate the thermal stress loaded on the bearings. Meanwhile, a transient temperature model of the motorized spindle was established based on the finite element method (FEM). Then, the analytical model was integrated into the spindle transient thermal model, so that the heat generated by bearings and the motorized spindle temperature can be revised constantly,  through the iterative calculation between these two models. Finally, verification experiments with different work conditions clarify that the proposed transient thermal characteristic model of the motorized spindle is valid, and the study shows that it is necessary to consider the bearing heat generation induced by the radial thermal stress when the spindle runs at a high speed.

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Abbreviations

A :

Distance between raceway groove curvature centers (mm)

a i/a o :

Semi-major axis of contact ellipse between the ball and inner/outer raceway (mm)

B :

Magnetic flux density (T)

C :

Electrical material constant

c p :

Specific heat capacity (J/(kg·°C))

D i/D o :

Bearing inner/outer nominal diameter (mm)

d i/d o :

Bearing inner/outer raceway diameter (mm)

d m :

Bearing pitch diameter (mm)

D w :

Rolling elements diameter (mm)

E :

Modulus of elasticity (MPa)

f :

Magnetization frequency (Hz)

f 0/f 1 :

Factor related to bearing type and lubrication method/applied force load

F a/F r :

Force in axial/radial direction loaded on bearing (N)

F cj :

Centrifugal force (N)

f i/f o :

Ratio of bearing inner/outer raceway radius to rolling element’s diameter

F tem :

Equivalent radial force loaded on rolling elements (N)

F β :

Force loaded on the bearing (N)

h :

Convective coefficient (W/(m2·℃))

H ij/H oj :

Bearing heating in the inner/outer raceway (W)

H si :

Thickness of silicon steel sheet (mm)

I :

Motor winding currents (A)

I cm :

Excitation current (A)

k equ :

Equivalent conductivity of ring (W/(m·℃))

k fluid :

Fluid thermal conductivity (W/(m·℃))

K ij/K oj :

Contact stiffness between roller and outer/inner groove of bearing (Pa)

k L/k b :

Grease/rolling element conductivity (W/(m·℃))

L :

Hydraulic diameter

L rotor :

Length of motor rotor (m)

M :

Applied moment (N mm)

M 1 :

Bearing friction torque due to load (N mm)

M gj :

Gyroscopic moment (N mm)

M ij/M oj :

Bearing friction torque in the inner/outer ring of bearing (N mm)

M v :

Bearing friction torque due to lubrication (N mm)

n :

Rotational speed (r/min)

N i :

Shape function

Nu:

Nusselt number

P cu/P Fe :

Copper/iron loss (W)

P n, P m :

Mechanical and magnetic loss (W)

Pr:

Prandtl number

P stator/P rotor :

The motor stator/rotor heating power (W)

q :

Heat flux (W/m2)

Q i/Q o :

Rolling element-inner/outer raceway contact normal load (N)

r :

The radius of any point inside the ring (m)

R :

Winding resistance (Ω)

R cm :

Excitation resistance (Ω)

Ra:

Rayleigh number

R all :

Total resistance between bearing inner ring and outer ring (℃/W)

R b :

Rolling element thermal resistance (℃/W)

R ci/R co :

Grease thermal resistance (℃/W)

Re:

Reynolds number

T a :

Environmental temperature (℃)

T i :

Node temperature (℃)

u :

The radial displacement of the ring (m)

W :

Bearing width (mm)

Z :

Numbers of rolling elements

ΔT :

Temperature rise (℃)

Σ i/Σ o :

Compete elliptic integral of the second kind in the contact area between rolling element and inner/outer raceway

\({\sigma }_{{r}_{i}\_\Delta T}^{{\text{II}}}\)/\({\sigma }_{{r}_{o}\_\Delta T}^{{\text{IV}}}\) :

Thermal stress on bearing inner/outer ring (Pa)

\({\sigma }_{{r}_{i}\_0}^{{\text{II}}}\)/\({\sigma }_{{r}_{o}\_0}^{{\text{IV}}}\) :

Initial assembly stress on bearing inner/outer ring (Pa)

\({\sigma }_{{r}_{i}}^{{\text{II}}}\)/\({\sigma }_{{r}_{o}}^{{\text{IV}}}\) :

Total stress on bearing inner/outer ring (Pa)

ℜ i :

Radius of the locus of inner raceway groove curvature centers (mm)

α :

Coefficient of thermal expansion

β :

Ball attitude angle (°, rad)

Γq, Γh :

The second/third boundary

δ a/δ r :

Bearing deflection deformation in axial/radial direction (mm)

δ ij/δ oj :

Normal contact deformation between rolling element and inner/outer raceway (mm)

δ stator -rotor :

Gap between stator and rotor (m)

Δψ :

Angular distance between rolling elements (°, rad)

ε r/ε ϕ :

Radial/circumferential strain of the ring

θ :

Bearing misalignment or angular deflection (°, rad)

μ :

Poisson’s ratio

μ si, μ so :

Friction coefficient between the rolling element and the inner/outer raceway

μ air :

Friction coefficient between rotor and air

ν 0 :

Kinematic viscosity of grease (Pas)

ρ :

Density (kg/m3)

ρ air :

Density of air (kg/m3)

ρ ball :

Density of rolling elements (kg/m3)

ρ m :

Rotor core density (kg/m3)

ρ r :

Core resistivity (Ω m−1)

σ r/σ ϕ :

Radial/circumferential stress of the ring (Pa)

ψ j :

Angle of rolling elements in yz plane (°, rad)

ω :

Rotational speed (rad/s)

ω b :

Speed of rolling element about its own axis (rad/s)

ω c :

Orbital speed of rolling elements (rad/s)

ω si/ω so :

Spinning speed of the inner/outer raceway relative to the rolling elements (rad/s)

[C] :

Heat capacity matrix

[K] :

Heat transfer matrix

{f} :

Thermal load vector

[B] :

Geometry matrix

[D] :

Property matrix

L :

Partial differential operator

[K] dis :

Stiffness matrix

{f} dis :

Load vector

g :

Volumetric force vector

q b :

Load Vector

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Acknowledgements

The first author would like to thank the China Scholarship Council (CSC) for financial support and the scholarship award.

Funding

This work was financially supported by the Key-Area Research and Development Program of Guangdong Province (Grant No. 2020B090927002).

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Authors and Affiliations

  1. State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710054, People’s Republic of China

    Dongxu Su, Yang Li, Wanhua Zhao & Zhe Nie

  2. Collaborative Innovation Center of High-End Manufacturing Equipment in Xi’an Jiaotong University, Xi’an, Shaanxi, 710054, People’s Republic of China

    Huijie Zhang

  3. Institute of Computer Application CAEP, Mianyang, Sichuan, 621000, People’s Republic of China

    Dongxu Su

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Contributions

DS: methodology, data curation, formal analysis, writing – original draft, review and editing; YL: conceptualization, supervision, funding acquisition, writing – review and editing; WZ: conceptualization, supervision, writing – review and editing; HZ, ZN: data curation, validation, writing – review and editing.

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Correspondence to Wanhua Zhao.

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Su, D., Li, Y., Zhao, W. et al. Modeling of the motorized spindle temperature field considering the thermos-mechanical coupling on constant pressure preloaded bearings. Int J Adv Manuf Technol 132, 1969–1988 (2024). https://doi.org/10.1007/s00170-024-13306-3

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