A new approach for uncoupling the equations of motion typical for rotordynamical systems is presented. The method does not neglect the speed dependent effects, like gyroscopic effects, and can be particularly valuable in the controller design of actively controlled rotors. In the presence of hysteretic type of damping, the resulting uncoupled gyroscopic systems come with an equivalent viscous damping, equivalent in a sense of the same natural frequency and decay rate. The approach is illustrated through the example of technical interest. The generated results demonstrate that the developed approach is correct and straightforward.