THE CHAOTIC OSCILLATIONS OF HIGHSPEED MILLING
G. Stépán*, R. Szalai*, S. J. Hogan** * Department of Applied Mechanics, Budapest University of Technology and Economics Budapest H-1521, Hungary Phone: +36 1 463 1369 Fax: +36 1 463 3471 e-mail:
[email protected],
[email protected] **Department of Engineering Mathematics, University of Bristol Bristol BS8 1LN, England Phone: +44(0)117 928 7754 Fax: +44(0)117 925 1154 e-mail:
[email protected]
Abstract:
In case of highly interrupted machining, 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 vibration or flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the classical selfexcited vibrations or secondary Hopf bifurcations. The present work investigates the nonlinear vibrations in case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. Also, the appearance of chaotic oscillation ‘outside’ the unstable period-two oscillation is proved for low-immersion high-speed milling processes.
Key words:
flip bifurcation, high-speed milling, time delay
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G. Stépán*, R. Szalai*, S. J. Hogan**
INTRODUCTION
High-speed milling is one of the most efficient cutting processes nowadays. In the process of optimizing this technology, it is a challenging task to explore its special dynamical properties, including the stability conditions of the cutting process, and the nonlinear vibrations that may occur near to the stability boundaries. These dynamical properties are mainly related to the underlying regenerative effect in the same way as it is in case of the classical turning process having complicated but well-studied and understood stability properties. Still, some new phenomena appear for lowimmersion milling as predicted by Davies et al. (2000, 2002) and Bayly et al. (2001). Insperger and Stépán (1999, 2000a,b) also described these phenomena in case of milling independently from the immersion or speed characteristic of the milling processes. High-speed milling has specific properties like small tool diameter, low number of milling teeth (2, 3 or 4), and high cutting speed. Together, all these lead to the so-called highly interrupted cutting. This means that, most of the time, none of the tool cutting edges is in contact with the work-piece, while cutting occurs during those short time-intervals only when one of the teeth hits the workpiece. Actually, the time spent cutting to not cutting may be less than 10%, so it can often be considered as a small parameter. For the case of highly interrupted cutting, the short contact periods between the tool and the workpiece can be described as kind of impacts, where the linear impulse coming from the cutting force contains, again, a past-effect, i.e., the regenerative effect still has a central role, even in highspeed milling. The corresponding mathematical model is similar to that of an impact oscillator, but it also involves time delay. The governing equations of the tool free-flights and the subsequent impacts can be solved analytically, and a closed form nonlinear Poincare mapping can be constructed. In the subsequent sections, the simplest possible, but still nonlinear highly interrupted cutting model is described. The resulting discrete mathematical model is two-dimensional and nonlinear. The linear stability analysis is presented in the same form as it appears in the specialist literature. The bifurcation analysis is carried out along the stability limit related to period doubling bifurcation. This requires center manifold reduction and normal form transformation. The tedious algebraic work can be carried out in closed form and it leads to a phenomenon similar to the one experienced in the case of the Hopf bifurcation in the turning process. It is suspected that chaotic oscillations may occur for those technological parameters of turning where the stationary cutting is unstable, or unstable periodic orbits arise ‘around’ stable stationary cutting (Stépán, 2000). In a similar way, chaotic oscillations are experienced also for high-speed milling. The existence and the structure of these chaotic oscillations are proved and explained for the introduced nonlinear discrete model of high-speed milling
The chaotic oscillations of high-speed milling
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in those parameter domains where the stationary cutting loses its stability via period doubling bifurcation.
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MECHANICAL MODEL
The simplest possible one degree-of-freedom (DOF) model of highly interrupted cutting is presented in Figure 1 (for a 2 DOF model see Altintas and Budak, 1995). Here, the number of the cutting edges is one only, and it is in contact with the workpiece material periodically with time period . The time it spends in contact is where
