Chatter in milling is still a main obstacle in surface finish,machining accuracy and tool life.Its prediction and avoidance are challenging subjects in the machining field.In this paper,an improved semi-discretization method is proposed to predict variable spindle speed milling with helix angle.Based on tool geometry and machining theory,the cutting region is divided into five different cases to calculate the cutting force.The influences of radial immersion rate and modulation parameters relative to variable spindle speed milling are explored.By comparison with constant spindle speed,the simulation results show that the variable spindle speed scheme can obtain a larger range of stability.In short,the helix angle and variable spindle speed play an important role in the stability of milling process.
In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The P1D controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.
