When a mine hoist is lifting a heavy weight, the natural frequency of the system varies slowly with the changes of mass and stiffness. So the mine hoist system is an oscillatory system with slowly varying parameters if the mass of steel rope is taken into account.The multiple scales method of KuzmakLuke was firstly applied to obtain the periodicity condition of solutions of strongly nonlinear oscillators with generally nonlinear spring, and then Jacobian elliptic functions were used to express the leading order approximate solutions of quadratic and cubic nonlinear oscillators. Secondly, the obtained results were applied respectively to mine hoist systems with slowly varying mass and with quadratic or cubic nonlinear spring. Finally,the theoretical results were used to deal with certain mine hoist systems, and comparisons of asymptotic solutions with numerical solutions of two examples showed the efficiency of the proposed method.