Tolerance analysis of the volumetric error of heavy-duty machine tool based on interval uncertainty

Abstract

Tolerance design is one of the main stages in the robust design process of machine tool. For heavy-duty machine tools, traditional tolerance analyses based on probabilistic methods may not be suitable because it is generally difficult and costly to obtain the accurate probability distribution of error by the limited geometric error measurements in large dimension. In this paper, a new tolerance analysis method for heavy-duty machine tools is proposed based on the interval uncertainty. Considering independent geometric error components have certain bounds but accurate distributions are unknown, the interval theory is introduced to the kinematic modeling of the static volumetric error of heavy-duty machine tools based on the multi body system (MBS) and homogeneous transformation matrix (HTM). Geometric error components are described in forms of interval numbers to extend the error model to the interval number system. The interval extension of the volumetric error function is further modified so that the variation interval of the volumetric error is thus accurately evaluated by the interval arithmetic. Comparative studies on tolerance analyses of different typical probabilistic/non-probabilistic methods are conducted through numerical simulations. It is found that under the given computational condition, the tolerance analysis based on the interval uncertainty can accurately evaluate the variation range of the volumetric error.

Appendix

According to the MBS in Fig. 2, the detailed expressions of all the actual HTMs of position and orientation in Eq. (6) are shown as follows:

$$ {\mathrm{T}}_{12}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{2x}\\ {}0& 1& 0& {p}_{2y}\\ {}0& 0& 1& {p}_{2z}\\ {}0& 0& 0& 1\end{array}\right]{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4} $$

(19)

$$ {\displaystyle \begin{array}{c}{\mathrm{T}}_{23}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{3x}\\ {}0& 1& 0& {p}_{3y}\\ {}0& 0& 1& {p}_{3z}\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& {S}_{xw}& 0\\ {}0& 1& -{S}_{yw}& 0\\ {}-{S}_{xw}& {S}_{yw}& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\\ {}\left[\begin{array}{cccc}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}0& 0& 1& w\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{\varepsilon}_z(w)& {\varepsilon}_y(w)& {\delta}_x(w)\\ {}{\varepsilon}_z(w)& 1& -{\varepsilon}_x(w)& {\delta}_y(w)\\ {}-{\varepsilon}_y(w)& {\varepsilon}_x(w)& 1& {\delta}_z(w)\\ {}0& 0& 0& 1\end{array}\right]\end{array}} $$

(20)

$$ {\displaystyle \begin{array}{c}{\mathrm{T}}_{34}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{4x}\\ {}0& 1& 0& {p}_{4y}\\ {}0& 0& 1& {p}_{4z}\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{S}_{xy}& 0& 0\\ {}{S}_{xy}& 1& 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\\ {}\left[\begin{array}{cccc}1& 0& 0& x\\ {}0& 1& 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{\varepsilon}_z(x)& {\varepsilon}_y(x)& {\delta}_x(x)\\ {}{\varepsilon}_z(x)& 1& -{\varepsilon}_x(x)& {\delta}_y(x)\\ {}-{\varepsilon}_y(x)& {\varepsilon}_x(x)& 1& {\delta}_z(x)\\ {}0& 0& 0& 1\end{array}\right]\end{array}} $$

(21)

$$ {\displaystyle \begin{array}{c}{\mathrm{T}}_{45}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{5x}\\ {}0& 1& 0& {p}_{5y}\\ {}0& 0& 1& {p}_{5z}\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& {S}_{xz}& 0\\ {}0& 1& -{S}_{yz}& 0\\ {}-{S}_{xz}& {S}_{yz}& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\\ {}\left[\begin{array}{cccc}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}0& 0& 1& z\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{\varepsilon}_z(z)& {\varepsilon}_y(z)& {\delta}_x(z)\\ {}{\varepsilon}_z(z)& 1& -{\varepsilon}_x(z)& {\delta}_y(z)\\ {}-{\varepsilon}_y(z)& {\varepsilon}_x(z)& 1& {\delta}_z(z)\\ {}0& 0& 0& 1\end{array}\right]\end{array}} $$

(22)

$$ {\mathrm{T}}_{56}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{6x}\\ {}0& 1& 0& {p}_{6y}\\ {}0& 0& 1& {p}_{6z}\\ {}0& 0& 0& 1\end{array}\right]{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4} $$

(23)

$$ {\mathrm{T}}_{17}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{7x}\\ {}0& 1& 0& {p}_{7y}\\ {}0& 0& 1& {p}_{7z}\\ {}0& 0& 0& 1\end{array}\right]{\mathrm{I}}_{4\times 4}\left[\begin{array}{cccc}1& 0& 0& 0\\ {}0& 1& 0& y\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{\varepsilon}_z(y)& {\varepsilon}_y(y)& {\delta}_x(y)\\ {}{\varepsilon}_z(y)& 1& -{\varepsilon}_x(y)& {\delta}_y(y)\\ {}-{\varepsilon}_y(y)& {\varepsilon}_x(y)& 1& {\delta}_z(y)\\ {}0& 0& 0& 1\end{array}\right] $$

(24)

$$ {\displaystyle \begin{array}{c}{\mathrm{T}}_{78}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{8x}\\ {}0& 1& 0& {p}_{8y}\\ {}0& 0& 1& {p}_{8z}\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& {S}_{xc}& 0\\ {}0& 1& -{S}_{yc}& 0\\ {}-{S}_{xc}& {S}_{yc}& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\\ {}\left[\begin{array}{cccc}\cos \gamma & -\sin \gamma & 0& 0\\ {}\sin \gamma & \cos \gamma & 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& -{\varepsilon}_z\left(\gamma \right)& {\varepsilon}_y\left(\gamma \right)& {\delta}_x\left(\gamma \right)\\ {}{\varepsilon}_z\left(\gamma \right)& 1& -{\varepsilon}_x\left(\gamma \right)& {\delta}_y\left(\gamma \right)\\ {}-{\varepsilon}_y\left(\gamma \right)& {\varepsilon}_x\left(\gamma \right)& 1& {\delta}_z\left(\gamma \right)\\ {}0& 0& 0& 1\end{array}\right]\end{array}} $$

(25)

$$ {\mathrm{T}}_{89}^{\prime }=\left[\begin{array}{cccc}1& 0& 0& {p}_{9x}\\ {}0& 1& 0& {p}_{9y}\\ {}0& 0& 1& {p}_{9z}\\ {}0& 0& 0& 1\end{array}\right]{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4}{\mathrm{I}}_{4\times 4} $$

(26)

where p_ix, p_iy, and p_iz (j=2, 3, …, 9) are X, Y, and Z position coordinates in the T_ijs of each body.