We propose an imaginary planar rack cutter with variable diametral pitch to manufacture the spherical gears of a wrist mechanism. The concept of the imaginary planar rack cutter is derived from the imaginary rack cutter. In general, the diametral pitch of the imaginary rack cutter is constant. In using the imaginary planar rack cutter with constant diametral pitch to cut a spherical blank material, the teeth generated will interfere with each other. Therefore, two matched imaginary planar rack cutters with a variable diametral pitch are proposed. One of the imaginary planar rack cutters consists of convex teeth and is utilized to generate a spherical blank material with concave teeth. The other consists of a concave one and is used to generate a spherical blank material with convex teeth. Through the two-parameter envelope theory and a software package, mathematical models of the spherical gear pairing were determined. Using a developed mathematical model, an investigation on the undercutting analysis of the proposed imaginary planar rack cutter is performed by tooth contact analysis. Finally, a numerical example is used to demonstrate the geometric model of a spherical gear generated by the imaginary planar rack cutter.
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