4 theorems on whirl transform of rotor vibration

The motion of a rotor can be decomposed into forward whirls and backward whirls. This approach is referred to as whirl analysis or whirl transform. It reveals the relationship between motion of a rotor and forces applied more evidently, therefore this method is widely applied to diagnosing problems in rotating machinery. 4 theorems about rotation vibration were established. Theorem 1 states that, an elliptical orbit of a rotor can always be decomposed into a forward whirl and a backward whirl. The area of the elliptical orbit is equal to the absolute value of the difference between the area of the forward whirl and that of backward whirl. Theorem 2 is associated with the circumference of an elliptical orbit and its forward whirl and backward whirl. Theorem 3 states that, the forces applied to a rotor can also be decomposed into forward whirl force and backward whirl force. The forward whirl force can only perform work on the forward whirl but not on backward whirl; vice versa, the backward whirl force can only perform work on the backward whirl but does not on the forward whirl. Theorem 4 states that the work performed by forward whirl force is equal to the dot product of forward whirl force and the displacement of rotor in the forward whirl, similarly the work performed by backward whirl force is equal to the dot product of backward whirl force and the displacement of rotor in backward whirl. The theorems are applied to determine the work performed by unbalance, viscous damping and restoring forces due to skew symmetric stiffness. The results are identical with those obtained from traditional dynamics. However it is found that, when ω_y < Omega; < ω_x, as |r_-| > |r₊|, the skew symmetric stiffness will act as a stabilizing force suppressing backward whirl of a rotor.