According to the dynamic characteristics of a rigid rotor system with flexible supports, the dynamic equations for transient responses were deduced by Lagrangian approach, where the assumption of constant speed was removed, and the rotating and nonrotating damping were taken into account, respectively. Based on the proposed model,both the amplitude and phase characteristics of the transient response were investigated theoretically by use of the precision integration method. Moreover,the effects of the acceleration,rotating and nonrotating damping coefficients were also addressed for the transient analysis. The results show that the timevarying rotating speed directly brings out coupling stiffness coefficients and additional forces, and the maximum amplitude greatly decreases with the increases in passing speed and damping coefficients. A phase angle less than 90 degrees, rather than the typical 90-degree phase angle, appears at the critical speed, when the system passes through resonance. Additionally, such a phase angle is independent of the acceleration and rotating damping ratio, but can be increased by enhancing the non-rotating damping ratio.