Controller design for a magnetically suspended milling spindle based on chatter stability analysis

Abstract

The chatter stability of a rigid milling spindle levitated by five-axis active magnetic bearings (AMBs) is studied for its chatter-free cutting, as the control gains of AMBs vary. The characteristic equation for regenerative chatter loop with a delay element is described by a linear differential-difference equation, accounting for the dynamics of the AMB controllers, the uncut chip thickness equation and the cutting process as well as the rigid spindle dynamics itself. For a quick, yet efficient stability analysis, the time-varying cutting force coefficients are expanded as Fourier series, so that they can be approximated as the time invariant but immersion dependent cutting force coefficients. An efficient chatter stability analysis method is then proposed to predict the stability lobes and chatter frequencies in milling. It features that the delay element, which is an nonrational function, is approximated by a rational function, (2,2) Pade approximant, so that the linear rational expression of the characteristic equation can be readily derived. In this analysis method, a generalized eigenvalue problem is used to obtain the characteristic equation of the chatter loop. The analytically predicted stability lobes are found to be in good agreement with the lobes generated by other methods available in the literature. Using the proposed method, parametric study is also performed to investigate the influences of the damping and stiffness coefficients of AMBs on the chatter free cutting conditions, as they are allowed to vary within the stable region formed by the AMB control gains. The cutting tests with a milling spindle levitated by digitally controlled AMBs are carried out in the laboratory in order to validate the chatter stability analysis, including the chatter free axial depth of cuts, spindle rotational speeds and chatter frequencies. The analysis method is applied to milling operation of a rigid AMB milling spindle with one flute cemented carbide milling cu...