Theory of finite amplitude machine tool instability

Abstract A non-linear theory of machine tool chatter is presented. It is shown that even when the machine tool structure is linear or only slightly non-linear (as appears to be the case), large non-linearity is introduced by two causes. Firstly, by the chatter amplitudes exceeding a certain value, dependent on the mean chip thickness and the vibrating tool leaving the workpiece. Secondly, by a non-linearity of the cutting force characteristics. By combining these non-linear effects with the linear operative receptance of the machine tool structure, non-linear stability conditions are derived which describe a surface in the three-dimensional space defined by rotational speed, depth of cut and chatter amplitude. Sections through this surface provide the traditional (linear) stability chart relating depth of cut and rotational speed, as well as non-linear charts relating chatter amplitude with depth of cut. The latter are used for discussing the mechanism of stabilisation of chatter amplitudes and the phenomenon of finite amplitude instability. It is shown that the physical cause of finite amplitude instability lies in the non-linearity of the cutting force characteristic. The theory presented leads to conclusions of practical significance. The most important of these is that when the cutting force characteristic is non-linear the cutting process may be stabilised by a large increase of the mean chip thickness. Theoretical work is supported by experiments and good correlation between these is noted.