For ball screw feed system on a lathe, the randomness of geometric parameters deriving from the inevitable manufacturing and assembly errors significantly affected the system dynamic characteristics and turning accuracy. In this study, a novel lumped dynamic model accounting for the analytical piecewise restoring force function involving the overall axial deformation is proposed. The equivalent mechanical model of bearing joints including the change of nominal contact angle is made a reasonable simplification to facilitate acquisition of the system analytical equation of motion. Additionally, the approximate analytical solution of the proposed equation is obtained by the modified Lindstedt–Poincaré method (MLP). The advanced first order second moment method (AFOSM) is employed to evaluate the reliability of dynamic response error. Furthermore, the numerical simulation aims to demonstrate the capabilities of the MLP and AFOSM method in the practical applications. Eventually, the effects of external excitation, sliding platform position, and nominal contact angles on the dynamic response are discussed and the reliability and sensitivity analysis in accordance with actual turning accuracy level is presented.