Abstract: A mathematical model of torsional vibration of the shafting system which has quadratic nonlinearities and harmonic excitation is established. By using of the improved averaging method for nonlinear oscillation, under the conditions ω 3≈2ω 2, ω 2≈2ω 1,and  Ω ≈ω 2, the first approximation solution and its corresponding steady-state solution of the triple resonances are obtained. The steady-state response curves are obtained and the singularity analysis is carried out. It is pointed out that the tr...