A mathematical theory of nonlinear chatter is developed. In this, the structure is represented by an equivalent single degree of freedom system with nonlinear stiffness characteristics and the cutting force by a third degree polynomial of the chip thickness. This model leads to a second order differential equation with nonlinear stiffness and nonlinear time delay terms from which the conditions of steady state chatter are derived. These are then discussed by applying them to an equivalent system derived from experimental data pertaining to a face milling process. The theory provides an explanation for the stages in which chatter develops and also for the “finite amplitude instability” phenomenon.
A graphical method for the investigation of regenerative machine tool chatter is presented. The method is based on the harmonic response locus of the machine tool structure and allows the determination of the stable and unstable cutting speed ranges. The chip thickness variation effect as well as the penetration rate effect are taken into consideration. The method is illustrated by a number of examples relating to drilling or spot facing chatter arising on a radial drilling machine. The effects of mode interaction and of the penetration rate on the stability and on the variation of the chatter frequency are discussed. A critical assessment of the method is presented, in comparison with other methods available.
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