Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments

Abstract

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

State-Dependent, Non-Smooth Model of Chatter Vibrations in Turning

“…Now ifL(t) :=L(t, 0), then the substitution of (26) to (15) and (16) gives the corresponding cases in (10). Therefore, for constant spindle speed and under the neglection of vibrations in direction x, (23) gives (9), that is, at angular position φ = 0 the extended PDE-ODE model gives the DDAE model.…”

“…The term fly-over refers to the tool's motion for the case when it is out of cut. During fly-over the tool vibrates freely and the delayed terms disappear from the governing equations which results the non-smoothness of the mathematical model [10].…”

Modeling of Abradable Coating Removal in Aircraft Engines Through Delay Differential Equations

“…It usually involves one or two degree-of-freedom (dof) models of the cutter and/or the workpiece. The stability analysis of these equations is used to maximize material removal rates without reaching instability zones [13].…”

“…These types of equations are used in many fields such as species dynamics [10], chemistry [11] and torque control within combustion engines [12] to name a few. In manufacturing, DDE have proved to be very suitable for the modeling of the cutter-workpiece interaction phenomena occurring in various manufacturing processes such as milling [13] and turning [14]. In these research areas, it is highly desirable to avoid self-excited vibrations between the cutter and the workpiece, also known as chatter effect, in order to obtain Laboratoire de Dynamique des Structures et Vibrations 3 smooth surfaces without major wear of the tool.…”

Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation

“…The center manifold algorithms in [1, 2,4,5] cannot be directly applied to the time-periodic case without additional complexities associated with redefining the adjoint equation. Nevertheless a center manifold computation for the case of flip bifurcation in single DOF low immersion milling was carried out in [15,16] by assuming that the system can be accurately modeled using a 2-dimensional map of the cutting tool's position and velocity from one instantaneous cut to the next. For such a system, the coefficients are essentially periodically-spaced Dirac delta functions.…”