People working in workshop have observed that there exist magic spindle speeds for any milling tool-workpiece combination at which optimal depth of cuts are allowed. Some of them have tried to unravel these magic spindle speeds experimentally. This is a very time consuming, costly and inaccurate optimization procedure thus the purpose of this work is to outline an analytical solution to not only the problem of finding the magic spindle speeds but also problem of separating the whole of stable operations from the unstable ones. Slotting three tooth end miller was studied using the method of time finite element analysis (TFEA) and validated using the result of time domain stability analysis stemming from MATLAB dde23 solution of the milling equation. It is seen for the studied system that though accuracy improves with increase in number of time elements used, only marginal gain in accuracy could be achieved by increase in number of elements above ten finite time elements. In other words close agreement exists between the two approaches (TFEA and MATLAB dde23) when the number of time elements is high enough. Fourteen time elements were used to generate a working chart for the studied system. The stability chart is seen to exhibit milling stability characteristics of magic spindle speeds already known by workshop practitioners. The major implication of this study becomes that time loss that is occasioned by experimental trial and error method of determining the productive spindle speeds is avoided.
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