Stability Analysis of Cutting under Varying Spindle Speed

The stability of cutting system with periodically varying spindle speed is analyzed by means of functional analysis. The dynamic characteristics of both the machine tool structure and cutting process are considered as the linear operators in Hilbert space and the stability problem of the cutting system is solved as an Eigen value problem of these operators. It is proved that two kinds of beat type vibration exist under the stability limit of the system. The amplitude of these beats changes by the same and twice period as that of the spindle speed variation, respectively. The analytical results show that the maximum amplitude of both types of vibration decrease when the frequency of spindle speed variation increases or when the amplitude of spindle speed variation decreases. The analytical results also show that the stability limit of the system rises up when the amplitude of spindle speed variation increases but it is independent of the frequency of spindle speed variation. Finally the stability of the cutting system containing external random vibration is analyzed by using these results and the results show good agreement with the experimental results.