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Stable islands in the stability chart of milling processes due to unequal tooth pitch

Stable islands in the stability chart of milling processes due to unequal tooth pitch

Categories Zeitschriften/Aufsätze (reviewed)
Year 2011
Authors Sellmeier, V., Denkena, B.:
Published In International Journal of Machine Tools & Manufacture, 51 (2011), S. 152-164.
Description

The use of unequal tooth pitch is a known means to influence and to prevent chatter vibrations in milling. While the process dynamics of equally pitched end mills can be modeled by non-autonomous differential equations with a single constant delay, the dynamics of unequally pitched end mills lead to differential equations with multiple constant delays. In this paper the process stability of an unequally pitched end mill is investigated experimentally and theoretically. The numerical approximation of the stability limit relies on two fundamental methods: Ackermann’s method to control systems with delay and the method of the piecewise constant subsystems. On the basis of these two methods two ways for the theoretical stability analysis are derived. The first way neglects the time dependency of the system by replacing the time varying system matrices by their means. The second way accounts for the time dependency of the system by combining Ackermann’s method to control systems with delay with the method of the piecewise constant subsystems, which results in the semi-discretization method. Besides the exemplary investigation of a specific end mill the two methods are compared for a simple one degree of freedom system for different number of teeth, different alternating and linear tooth pitch variations and different helix angles. It is shown, that unlike equally pitched end mills also at high radial immersions the time dependency of the system leads to significant differences between the stability limits of the unequally pitched end mills, predicted by the two methods. Depending on the time variance of the system and the unequal tooth pitch stable islands can arise, which are largely located within the stable peaks of the stability diagram of the system where the time varying system matrices are replaced by their means. The correctness of the results are backed up for several operating points by time domain simulations, taking into account the trochoidal movement of the cutting edges, the time varying character of the system and teeth jumping out of contact.