Abstract: Based on cesaro method of differential geometry, successive and dispersed mapping models are established. These models have good effect on the investigation of the rolling character, ratio function and “polygonal” effect. Instantaneous centers, pitch curves and cycloid are discussed in detail. The study shows, though the pitch curves of the sprockets 1, 2 purely rolling on the pitch line of the chain, however their instantaneous arc-length does not in general equal. In order to shed a light on “polygonal” effect to ratio function, numerical examples are given.

Key words: Cycloid, Noncircular sprocket (band wheel), Pitch curve, Ratio function