Delay, parametric excitation, and high-speed milling

Gabor Stepan, Department of Applied Mechanics, Budapest University of Technology and Economics

Machine tool vibrations have negative effects on the quality of machined surfaces. Its physical basis is often a time delay that arises naturally in the cutting process, where the delay is inversely proportional to the cutting speed. Referring to the infinite dimensional nature of the dynamics of cutting, its nonlinear vibrations are often compared to the problem of turbulence in fluid mechanics.

Milling is a kind of cutting where loss of contact between the teeth of the tool and the work-piece occurs typically in a periodic way.
This leads to non-autonomous governing equations similar to the
delayed Mathieu equation.

The parametric excitation in the delay-differential equation yields
secondary Hopf (or Neimark-Sacker) bifurcations, and also period
doubling (or flip) bifurcations. The subcritical nature of both bifurcations are shown, and the development of chaotic oscillations are also explained. The results show how nonlinear dynamics can contribute to the increase of efficiency in material processing, and how these methods generalised to act-and-wait control or periodic flow control.

http://www.mm.bme.hu/~stepan