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A thermal model for real-time temperature forecast of rolling linear guide considering loading working conditions

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Abstract

Thermal behaviors significantly affect the contact stiffness and position accuracy of the rolling linear guide, which worsen the performance of the precision machinery. In this paper, based on the modified lumped capacitance method (MLCM) and finite element method (FEM), a thermal model of rolling linear guides has been formulated in terms of the geometric parameters, friction, gyroscopic moment, and loading conditions. Experimental results show that the thermal model can well predict the real-time temperature variation under different working conditions. Further analysis shows that, the applied vertical load, carriage velocity, and toque load significantly influence the temperature variation of linear guides, while the influence of the torque load on temperature variation decreases with increasing vertical load levels. Besides, the reasons of the steady-state temperature variation considering the load distribution model are also discussed.

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Abbreviations

i :

Designator of ball row number (1, 2, 3, and 4)

Q fi :

Frictional heat from corresponding raceway

\( {Q}_c^{"} \) :

Convection heat from the front surfaces of the carriage

c :

Specific heat

T :

Real-time temperature of the measuring point

t :

Time

\( {\beta}_c^{\prime } \) :

Compensation factor for the heat loss from the top and side surfaces of the carriage

H Fij :

Frictional heat due to the motion in the direction of rolling

\( {V}_{ij}^{\prime } \) :

Sliding linear velocity of the carriage raceway over the ball in the direction of rolling

\( {F}_{dij\left({x}_{ti}\right)} \), \( {F}_{dij\left({y}_{ti}\right)} \) :

Frictional force in the direction of xti-axis and yti-axis, respectively

ϵij :

Slid-roll ratio at the contact point

O-X, Y, Z :

Global coordinate system

o-x, y, z :

Coordinate system fixed on the carriage

β :

Angle between the U-axis and the \( {x}_i^{\prime }-{y}_i^{\prime } \) plane

U, r, ∅:

Three components of Polar coordinate system

\( {T}_{\left({O}^{\prime }-O\right)\_ ij} \) :

Mapping matrix

w hij :

Distance from the ball’s center, (\( {o}_i^{\prime } \)), to origin, (O), in the direction parallel to the Y-axis

L bcij :

Distance from the ball’s center, (\( {o}_i^{\prime } \)), to the front surface of the carriage in the direction parallel to the Y-axis

F V :

Vertical load acting on the carriage

ζ i :

Vertical displacement of the carriage

α 0 :

Initial contact angle

ori, oci :

Initial raceway groove curvature centers of the rail and carriage, respectively

fr, fc :

Ratio of the rail and carriage raceway groove radius to a ball diameter, respectively

k H :

Parameter of the Hertz contact area

M x :

Pitch torque applied on the carriage

\( {Q}_{ij}^{\prime } \) :

Additional normal force under rolling torque

S H :

Center distance between the no. 1 and no. 3 rail raceway

\( {Q}_{ij}^{"} \) :

Normal force

\( {\overline{p}}_{ij} \) :

Hydrodynamic pressure

η p :

Absolute viscosity of the lubricant

α P :

Pressure-viscosity coefficient

ω Ri :

Rotational speed of the ball center

R :

Equivalent curvature radius of the elliptic contact surface

l c :

Characteristic length of the convective surface

R e :

Reynolds number

u ∞ :

Airflow velocity

c a :

Specific heat capacitance of air

q ij :

Heat fluxes set on the elliptical region elements in the carriage raceway grooves

h 2 :

Convection coefficient of the front surfaces of the carriage

j :

Designator of ball number (−n,…0…,n)

\( {Q}_c^{\prime } \) :

Convection heat from the top and side surfaces of the carriage

ρ :

Material density

V :

Volume of the carriage

T 0 :

Environment temperature

α13, α24 :

Impact degree factors for the frictional heat generated from the upper and lower raceways, respectively

\( {\beta}_c^{"} \) :

Compensation factor for the heat loss from the front surfaces of the carriage

H Gij :

Frictional heat due to the motion in the direction perpendicular to rolling

\( {V}_{ij}^{"} \) :

Sliding velocity in the direction perpendicular to rolling

\( {M}_{y_i^{\prime }} \) :

Running torques about the \( {y}_i^{\prime } \)-axis

D b :

Diameter of a ball

\( {o}_i^{\prime } \)-\( {x}_i^{\prime },{y}_i^{\prime },{z}_i^{\prime } \) :

Moving coordinate system

\( {o}_i^{\prime } \)-U, V, W :

Rotating coordinate system

β′:

Angle between the projection of U-axis in the \( {x}_i^{\prime }-{y}_i^{\prime } \) plane and the \( {x}_i^{\prime } \)-axis

xti, yti, zti :

Three components of the contact coordinate system

V C :

Linear velocity of the carriage

w vi :

Distance from the ball’s center, (\( {o}_i^{\prime } \)), to origin, (O), in the direction parallel to the Z-axis

δ 0 :

Oversize of the ball

P ij :

Normal force at contact point under vertical load

ε i :

Outward deformation of the carriage in the direction of X-axis

γ i :

Actual contact angle under load

s 0 :

Distance between ori and oci

\( {o}_{ci}^{\prime } \) :

Actual raceway groove curvature center of the carriage under load

∆ε i :

Initial outward deformation without vertical load

M y :

Rolling torque applied on the carriage

L T :

Moment arm of the rolling torque My

S V :

Center distance between the No.3 and No.4 rail raceway

θ x :

Pitch angle of the carriage

aij, bij :

Semi-major and semi-minor radius of contact ellipse, respectively

h c :

The Til film thickness of the contact area center

η dm _ i :

Acceleration of an element of mass dm

\( {M}_{z_i^{\prime }} \) :

Running torques about the \( {z}_i^{\prime } \)-axis

k a :

Thermal conductivity of the air

\( {N}_{u{l}_c} \) :

Nusselt number

P r :

Prandtl number

υ a :

Kinematic viscosity of air

μ f :

Dynamic viscosity of air

h 1 :

Convection coefficient of the top and side surfaces of the carriage

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Funding

This project is financially supported by the National Science and Technology Major Projects of China (Grant No. 2019ZX04010001) and National Natural Science Foundation of China (Grant No. 51905274).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

    Xiao-Yi Wang, Hu-Tian Feng, Chang-Guang Zhou & Kun-Qi Ye

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  1. Xiao-Yi Wang
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  2. Hu-Tian Feng
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  3. Chang-Guang Zhou
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  4. Kun-Qi Ye
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Correspondence to Chang-Guang Zhou.

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Appendix

Appendix

Table 5 Calculated results of heat flux q_ij considering different torque loads under light load (0.796 kN, 4%C).
Full size table
Table 6 Calculated results of heat flux qij considering different torque loads under heavy load (5.970 kN, 30%C)
Full size table

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Wang, XY., Feng, HT., Zhou, CG. et al. A thermal model for real-time temperature forecast of rolling linear guide considering loading working conditions. Int J Adv Manuf Technol 109, 2249–2271 (2020). https://doi.org/10.1007/s00170-020-05723-x

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  • DOI: https://doi.org/10.1007/s00170-020-05723-x

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