Abstract: The differential equation of motion of single pendulum without damping is a kind of nonlinear ordinary differential equation with the physical background and the study of exact solutions in nonlinear science is an important work. In order to lay the foundation for studying nonlinear behavior of these issues, four kinds of exact solutions are obtained by using the arctangent fractional transformation method of sine function and introducing Riccati auxiliary equation based on the F-expansion method. The arctangent fractional transformation method applied in such equation is a universal conclusion.

Key words: single pendulum without damping, nonlinear equation, F-expansion method, arctangent fractional transformation, Riccati equation, exact solution