Geometrical Modeling of Surface Generation Through Wrapping

A number of methods exist for determining profiles of cutting tools that work by wrapping. Most of these methods are based on the envelope theory and almost inevitably require cumbersome analytical formulations not always easy to resolve. This work presents a new method for studying conjugated surfaces associated with rolling axodes. Originally devised in an analytical form in a previous work of the first author, a purely numerical method is developed here based on a theorem which we call “the theorem of the minimal distance.” The advantage is twofold: first, geometrical modeling of tool profile calculation is possible even for profiles which cannot be described analytically; second, a very useful tool is provided for the inverse problem—starting from the measured cutting edges profiles, one can calculate the effectively generated surface on the workpiece. Several examples are shown for rack, shaper, and rotational cutters.