Critical speed induced by imbalance forces is a well-known dynamic behavior of rotating shafts. Such problems are typically found in flexible shafts or rigid shafts with flexible supports when the frequency of rotation reaches the natural frequencies of the shaft. This simple critical speed problem is well understood and formulated in many engineering texts.
However, not all critical speed phenomena are induced by imbalance. A perfectly balanced shaft with certain inertial properties also reaches a critical speed condition at a rotational speed that is not equal to the natural frequency of the shaft. Several variables of the dynamic system play a role on the critical speed condition, which is mainly induced by the unstable gyroscopic moment acting on the shaft.
The unstable gyroscopic moment forces the shaft bearings to deflect causing precession about the undeflected geometric centerline of the shaft, but the rotation and precession speeds remain synchronized at low speeds. Immediately after critical speed, precession and rotation speed bifurcate rendering the system to suddenly become stable due to diminished precession. Synchronization of the precession and rotation speeds would reoccur after the critical speed if the shaft rotation speed were further increased as the unstable gyroscopic moment continues to deflect the rigid shaft on its flexible supports.
The phenomenon has been realized during the development cycle of a mechanical cooling fan of an automotive engine. This paper outlines the measurements made to understand the physics of the problem, it gives some of the key measured dynamic characteristics and finally explains the phenomenon through a simplified dynamic model where the critical speed problem induced by unstable gyroscopic moment is formulated and solved in closed form for a rigid shaft with flexible supports.