Real-time thermal modelling approach of a machine tool spindle based on bond graph method


The thermal characteristics of spindles have a significant influence on the workpiece quality. This paper proposes a novel real-time thermal characteristic modelling approach for spindles based on the bond graph method, with simplified thermal structure and brief estimation and calibration of thermal parameters. First, the thermal characteristics of this spindle are analysed and simply divided into different thermal components. Then, a network of thermal capacitances and thermal resistances is established based on the mechanism analysis of heat generation and heat transfer, and consequently, a thermal characteristics model is developed based on the bond graph method. Parameters of thermal conditions of the spindle are briefly estimated using theoretical analysis and empirical formulas. An experiment is designed to calibrate these thermal parameters of the model, followed by verification experiments of accuracy and robustness, the results of which indicate stable good prediction performance, with the maximum error of 0.7355 °C and the average error of 0.1989 °C. Finally, this model is applied in the quantitative investigation of the influence of working conditions on the thermal characteristics of the spindle and the real-time prediction of spindle’s thermal deformation with 5.26 μm maximum error and 1.45 μm average error. The results indicate that this approach has considerable advantages in the real-time prediction of thermal behaviours of spindles and can be used in industrial applications.

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Data availability

All data generated or analyzed during this study are included in this published article and its supplementary information files.


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This work was supported by the National Key R&D Program of China (Grant number 2018YFB1701204) and the National Natural Science Foundation of China (Grant numbers 51975372).

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Zhengchun Du was in charge of the whole trial; Yun Yang wrote the manuscript; Xiaobing Feng assisted with sampling and laboratory analyses; Jianguo Yang supported the equipment and experimental guidance.

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Correspondence to Zhengchun Du.

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All custom code during this study are included in the supplementary information files

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Yang, Y., Du, Z., Feng, X. et al. Real-time thermal modelling approach of a machine tool spindle based on bond graph method. Int J Adv Manuf Technol 113, 99–115 (2021).

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  • Real-time thermal model
  • Bond graph method
  • Machine tool spindle
  • Calibration
  • Verification