A cyclist training system having a polymeric pivoting assembly which is adapted to allow a cyclist of a bicycle to nutate about the perpendicular axis when exerting lateral forces on the bicycle. The cyclist training system includes a tubular base support, a tubular bicycle support; and means to securely retain the bicycle within the cyclist training system. The polymeric pivoting assembly couples the tubular bicycle support to the tubular base support and is constructed from a polyurethane elastomer. Various embodiments of the polymeric pivoting assembly are provided having Shore A hardness in the range of 40-90 durameters and Shore D hardness in the range of 45-65 durameters. The polymeric pivoting assembly minimizes the unnatural bounce provided by other cyclist training systems known in the relevant art.