A closed-form numerical algorithm (CFNA) is analyzed in detail. CFNA is widely used in mechanical dynamics for periodic solution of second-order original differential equations (SODE) with periodic time-variant coefficients. The principle of the algorithm is to discretize the motion period into many short time intervals, so the coefficient matrices of the equation set are regarded as constant in a time interval. Defects are found in the original algorithm in treating the modal coordinates at the two end-nodes and important modifications to the defects is made for the algorithm. The modified algorithm is finally used to solve the dynamic problem of a three-ring planetary gear transmission. A closed-form numerical algorithm (CFNA) is analyzed in detail. CFNA is widely used in mechanical dynamics for periodic solution of second-order original differential equations (SODE) with periodic time-variant coefficients. The principle of the algorithm is to discretize the motion period into many short time intervals, so the coefficient matrices of the equation set are considered as constant in a time interval. Defects are found in the original algorithm in treating the modal coordinates at the two end-nodes and important modifications to the defects is made for the algorithm. The modified algorithm is finally used to solve the dynamic problem of a three-ring planetary gear transmission.