Machining Dynamics Involving Whirling Part II: Machining Motions Described by Nonlinear and Linearized Models

Abstract

The nonlinear model presented in Part I is linearized and numerically evaluated to investigate the impact of linearization on interpreting turning dynamics. As described in Part I, the equations of motion are so derived that the motion of the workpiece machining surface is in the plane orthogonal to the spindle axis and the motion of the tool is along the axis and coupled with the machining surface. The nonlinear 3D model is linearized about an operating point at (0, 0, Z0), where zt = Z0 is the equilibrium location of the tool associated with the ( x2 = 0 , y2 = 0) position of the workpiece. The (0, 0, Z0 ) operating point is selected to satisfy the ultimate machining goal for always achieving precise workpiece geometry without surface error and waviness. Taylor series expansion about the operating point is applied to linearize the nonlinear equations of motion. Modifications are also made to the nonlinear tool stiffness term and the cutting force to minimize linearization errors. The mass and stiffness of the workpiece remain functions of time after the modifications are performed. Numerical results show that the linearized model underestimates tool vibrations in the time domain and overestimates system behavior in the frequency domain; whereas the nonlinear model agrees with the physical results reported in the literature in describing machining stability and chatter.