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Optimal tolerance allocation for five-axis machine tools in consideration of deformation caused by gravity

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Abstract

Due to the effect of gravity on machine tools, the small deformations inevitably exist. The existing tolerance allocation methods are based on the rigid body assumption, which ignore the small deformations. It will make optimization results inaccurate and increase manufacturing cost. Therefore, a new optimal tolerance allocation method, which integrates the small deformations, is presented in this paper. The establishment of a geometric error model based on tolerance is involved at first. Based on this model and multi-body system theory, the mapping relationship between tolerance and volumetric error of the five-axis machine tool (FAMT) is formulated. Secondly, the small deformations of the FAMT are obtained based on finite element analysis. Then, the optimal tolerance allocation model is established by integrating the small deformations into the constraint conditions. Thirdly, simulation analysis is carried out with this model by using a genetic algorithm. Then, the optimal tolerance allocation scheme is obtained, and the total manufacturing cost after optimization is reduced by approximately 11.5%. Finally, the volumetric errors of the FAMT are calculated based on the two tolerance allocation schemes. The results show that the volumetric errors are within the permitted ranges. Therefore, the proposed method in consideration of the small deformation is feasible and effective.

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Funding

This work is financially supported by the National Natural Science Foundation of China (No. 51775010 and 51705011) and Science and Technology Major Projects of High-end CNC Machine Tools and Basic Manufacturing Equipment of China (No. 2016ZX04003001).

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Authors and Affiliations

  1. Beijing Key Laboratory of Advanced Manufacturing Technology, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Jinwei Fan, Haohao Tao, Ri Pan & Dongju Chen

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  1. Jinwei Fan
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  2. Haohao Tao
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  3. Ri Pan
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  4. Dongju Chen
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Correspondence to Ri Pan.

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Appendices

Appendix 1

$$ \left\{\begin{array}{c}{\delta}_x(x)={t}_1\sin \left(\frac{2\pi x}{\lambda}\right)+ ax\\ {}{\delta}_y(x)=\frac{t_2\left(\sin \left(\frac{2\pi {x}_{i+1}}{\lambda}\right)+\sin \left(\frac{2\pi {x}_{i-1}}{\lambda}\right)\right)}{2}\\ {}{\delta}_z(x)=\frac{t_3\left(\sin \left(\frac{2\pi {x}_{i+1}}{\lambda}\right)+\sin \left(\frac{2\pi {x}_{i-1}}{\lambda}\right)\right)}{2}\\ {}{\varepsilon}_x(x)=\frac{t_4\left(\sin \left(\frac{2\pi {x}_{i+1}}{\lambda}\right)+\sin \left(\frac{2\pi {x}_{i-1}}{\lambda}\right)\right)}{2L}\\ {}{\varepsilon}_y(x)=\frac{t_3\left(\sin \left(\frac{2\pi {x}_{i+1}}{\lambda}\right)-\sin \left(\frac{2\pi {x}_{i-1}}{\lambda}\right)\right)}{B}\\ {}{\varepsilon}_z(x)=\frac{t_2\left(\sin \left(\frac{2\pi {x}_{i+1}}{\lambda}\right)-\sin \left(\frac{2\pi {x}_{i-1}}{\lambda}\right)\right)}{B}\end{array}\right. $$
(10)
$$ \left\{\begin{array}{c}{\delta}_x(c)={t}_{13}{\sin}^2\left({\theta}_{ci}\right)\\ {}{\delta}_y(c)={t}_{13}\sin \left({\theta}_{ci}\right)\cos \left({\theta}_{ci}\right)\\ {}{\delta}_z(c)={t}_{14}\sin \left({\theta}_{ci}\right)\\ {}{\varepsilon}_x(c)=\frac{t_{13}\sin \left({\theta}_{ci}\right)\cos \left({\theta}_{ci}\right)}{L_c}\\ {}{\varepsilon}_y(c)=\frac{t_{13}{\sin}^2\left({\theta}_{ci}\right)}{L_c}\\ {}{\varepsilon}_z(c)={t}_{14}\sin \left({\theta}_{ci}\right)\end{array}\right. $$
(11)

Appendix 2

$$ \left\{\begin{array}{c}\begin{array}{c}{E}_x={\delta}_{\mathrm{x}}(x)+{\delta}_{\mathrm{x}}(y)+{\delta}_{\mathrm{x}}(z)-{\varepsilon}_{\mathrm{z}}(x)y+{\varepsilon}_{\mathrm{y}}(x)z+{\varepsilon}_{\mathrm{y}}(y)z-{\varepsilon}_{\mathrm{x}\mathrm{y}}y+{\varepsilon}_{\mathrm{x}\mathrm{z}}z+{\delta}_{\mathrm{x}}(c)\cos (c)-{\varepsilon}_{\mathrm{y}}(b)\sin (c)-{\delta}_{\mathrm{y}}(c)\sin (c)+{\varepsilon}_{\mathrm{y}}(x){q}_{3\mathrm{z}}\\ {}+{q}_{4\mathrm{z}}\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)\right)+{q}_{5\mathrm{z}}\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{z}}\right)+{q}_{6\mathrm{z}}\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{c}}+{\varepsilon}_{\mathrm{x}\mathrm{z}}+{\varepsilon}_{\mathrm{y}}(c)\cos (c)+{\varepsilon}_{\mathrm{x}}(c)\mathit{\sin}(c)\right)\\ {}-{\varepsilon}_{\mathrm{z}}(x){q}_{3\mathrm{y}}-{q}_{4\mathrm{y}}\left({\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)-{q}_{5\mathrm{y}}\left({\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)+{\delta}_{\mathrm{x}}(b)\cos (b)\cos (c)+\left({\delta}_{\mathrm{z}}(b)-l\right)\cos (c)\sin (b)\\ {}-l\cos (b)\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{c}}+{\varepsilon}_{\mathrm{x}\mathrm{z}}\right)-\left({\varepsilon}_{\mathrm{z}}(c)+{\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)\left({q}_{6\mathrm{y}}\cos (c)+{q}_{6\mathrm{x}}\sin (c)\right)-{\delta}_{\mathrm{x}}(b){\varepsilon}_{\mathrm{y}}(x)\sin (b)\\ {}-{\varepsilon}_{\mathrm{x}}(b)l\sin (c)-l\cos (b)\cos (c)\left({\varepsilon}_{\mathrm{y}}(b)+{\varepsilon}_{\mathrm{y}}(c)+{\varepsilon}_{\mathrm{x}}(c)\right)-{\varepsilon}_{\mathrm{bz}}l\cos (b)\sin (c)+\left({\varepsilon}_{\mathrm{z}}(c)+{\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{b}}+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)l\sin (b)\sin (c)\end{array}\\ {}\begin{array}{c}{E}_y={\delta}_{\mathrm{y}}(x)+{\delta}_{\mathrm{y}}(y)+{\delta}_{\mathrm{y}}(z)-{\varepsilon}_{\mathrm{x}}(x)z-{\varepsilon}_{\mathrm{x}}(y)z-{\varepsilon}_{\mathrm{y}\mathrm{z}}z+{\delta}_{\mathrm{y}}(b)\cos (c)+{\delta}_{\mathrm{y}}(c)\cos (c)+{\delta}_{\mathrm{x}}(c)\sin (c)-{\varepsilon}_{\mathrm{x}}(x){q}_{3\mathrm{z}}-{q}_{4\mathrm{z}}\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)\right)\\ {}-{q}_{5\mathrm{z}}\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{y}\mathrm{z}}\right)-{q}_{6\mathrm{z}}\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{y}\mathrm{c}}+{\varepsilon}_{\mathrm{y}\mathrm{z}}+{\varepsilon}_{\mathrm{x}}(c)\cos (c)-{\varepsilon}_{\mathrm{y}}(c)\sin (c)\right)+{\varepsilon}_{\mathrm{z}}(x){q}_{3\mathrm{x}}+l{\varepsilon}_{\mathrm{x}}(b)\cos (c)\\ {}+{q}_{5\mathrm{x}}\left({\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{x}\mathrm{y}}+{\varepsilon}_{\mathrm{z}}(z)\right)+{\delta}_{\mathrm{z}}(b)\sin (b)\sin (c)-l\cos (b)\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{y}\mathrm{c}}+{\varepsilon}_{\mathrm{y}\mathrm{z}}\right)+\left({\varepsilon}_{\mathrm{bz}}-{\varepsilon}_{\mathrm{x}}(c)\right)l\cos (b)\cos (c)\\ {}+{q}_{6\mathrm{x}}\cos (c)\left({\varepsilon}_{\mathrm{z}}(c)+{\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)-{q}_{6\mathrm{y}}\sin (c)\left({\varepsilon}_{\mathrm{z}}(c)+{\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)+{q}_{4\mathrm{x}}\left({\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)\\ {}+\cos (b)\sin (c)\left({\delta}_{\mathrm{x}}(b)+{\varepsilon}_{\mathrm{y}}(c)-{\varepsilon}_{\mathrm{y}}(b)\right)-l\cos (c)\sin (b)\left({\varepsilon}_{\mathrm{z}}(c)+{\varepsilon}_{\mathrm{z}}(x)+{\varepsilon}_{\mathrm{z}}(y)+{\varepsilon}_{\mathrm{z}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{b}}+{\varepsilon}_{\mathrm{x}\mathrm{y}}\right)\end{array}\\ {}\begin{array}{c}{E}_z={\delta}_{\mathrm{z}}(c)+{\delta}_{\mathrm{z}}(x)+{\delta}_{\mathrm{z}}(y)+{\delta}_{\mathrm{z}}(z)-{q}_{4\mathrm{x}}\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)\right)+\sin (b)\left(\left({\varepsilon}_{\mathrm{y}}(b)l+{\varepsilon}_{\mathrm{y}}(c)l-{\delta}_{\mathrm{x}}(b)\right)-l\sin (c)\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{y}\mathrm{c}}+{\varepsilon}_{\mathrm{y}\mathrm{z}}\right)\right)\\ {}+{q}_{6\mathrm{y}}\left({\varepsilon}_{\mathrm{x}}(c)+{\varepsilon}_{\mathrm{y}\mathrm{c}}\cos (c)+\sin (c)\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{c}}+{\varepsilon}_{\mathrm{x}\mathrm{z}}\right)\right)+{\varepsilon}_{\mathrm{x}}(x){q}_{3\mathrm{y}}+{q}_{4\mathrm{y}}\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)\right)+{\varepsilon}_{\mathrm{x}}(b){\varepsilon}_{\mathrm{y}}(z){\varepsilon}_{\mathrm{x}\mathrm{b}}l\cos (c)\\ {}-{q}_{5\mathrm{x}}\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{z}}\right)+{q}_{6\mathrm{x}}\left(\sin (c)\left({\varepsilon}_{\mathrm{x}}(x)+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{y}\mathrm{c}}+{\varepsilon}_{\mathrm{y}\mathrm{z}}\right)-\cos (c)\left({\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{c}}+{\varepsilon}_{\mathrm{x}\mathrm{z}}+{\varepsilon}_{\mathrm{y}}(x)\right)-{\varepsilon}_{\mathrm{y}}(c)\right)\\ {}+{q}_{5\mathrm{y}}\left({\varepsilon}_{\mathrm{y}\mathrm{z}}+{\varepsilon}_{\mathrm{x}}(y)+{\varepsilon}_{\mathrm{x}}(z)+{\varepsilon}_{\mathrm{x}}(x)\right)+l\cos (c)\sin (b)\left({\varepsilon}_{\mathrm{y}}(x)+{\varepsilon}_{\mathrm{y}}(y)+{\varepsilon}_{\mathrm{y}}(z)+{\varepsilon}_{\mathrm{x}\mathrm{c}}+{\varepsilon}_{\mathrm{x}\mathrm{z}}\right)+{\varepsilon}_{\mathrm{x}}(x)y+{\delta}_{\mathrm{z}}(b)\cos (b)-{\varepsilon}_{\mathrm{y}}(x){q}_{3\mathrm{x}}\end{array}\end{array}\right. $$
(12)

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Fan, J., Tao, H., Pan, R. et al. Optimal tolerance allocation for five-axis machine tools in consideration of deformation caused by gravity. Int J Adv Manuf Technol 111, 13–24 (2020). https://doi.org/10.1007/s00170-020-06096-x

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