On Periodic Solutions of Delay Differential Equations with Impulses

The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces R ( [ − r , 0 ] , R n ) . The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schäffer fixed point theorem in regulated space R ( [ − r , 0 ] , R n ) .

The Effect of Helix Angle Variation on Milling Stability

Helical milling tools of nonuniform helix angles are widely used in manufacturing industry. While the milling tools with these special cutting edges are already available in the market, their cutting dynamics has not been fully explored. Also, there have been several attempts to introduce complex harmonically varied helix tools, but the manufacturing of harmonic edges is extremely difficult, and their effect on cutting dynamics is not clear either. In this study, a general mechanical model is introduced to predict the linear stability of these special cutters with optional continuous variation of the helix angle. It is shown that these milling tools cause distribution in regeneration. The corresponding time-periodic distributed delay differential equations are investigated by semi-discretization. This work points out how the nonuniform and harmonically varied helix cutters behave in case of high and low cutting speed applications.

The Effect of Harmonic Helix Angle Variation on Milling Stability

The linearly varied helix tool is widely used in manufacturing industry and milling tools are available in the market with these special cutting edges. There were several attempts to introduce complex harmonically varied helix tools, but the manufacturing of sinusoid edges is extremely difficult and its effect on cutting dynamics is not clear yet. In this study a mechanical model is introduced to predict the linear stability of these special cutters. It is shown that these milling tools cause distribution in regeneration and the corresponding time periodic distributed delay differential equations are investigated by semi-discretization. This work points out how the harmonically varied helix cutters behave in case of high and low cutting speed applications.