Theory

Machine components at a standstill may behave very differently when they are moving, even at relatively low speeds. A solid shaft able to support a hundred times its own weight plus the weight of the components mounted on it may, when rotating at certain speeds, bend and vibrate. The speeds are called ‘critical speeds and the bending and the vibration is known as ‘whirling’. If this ‘critical speed of whirling’ is maintained then the resulting amplitude becomes sufficient to cause buckling and failure. However if the speed is rapidly increased before such deleterious effects occur then the shaft is seen to re-stabilize and run true again until at another specific speed a double bow is produced.

Whirling is usually associated with fast-rotating shafts. When a shaft rotates it is subjected to radial or centrifugal forces, which cause the shaft to deflect from its rest position. These centrifugal forces are unavoidable, since material inhomogeneities and assembly difficulties ensure that the center of gravity of the shaft or its attached masses cannot coincide with the axis of rotation. Dunkerley first investigated the centrifugal forces involved and determined that the only restabilizing or restoring force was that due to the elastic properties or stiffness of the shaft. Hence, he was able to deduce the speed at which the shaft would suffer an infinite deflection due to whirling.

When the speed of rotation is increased the centrifugal force also increases and so does the restoring force. Below the critical speeds, the restoring forces increase with increasing shaft deflection faster than the centrifugal forces, so the deflection is held in check. At the critical speeds, the restoring forces increase at the same rate as the unbalance forces, so they cancel each other out. Shaft deflection is unchecked and the shaft behaves as though it is very flexible. Above the critical speeds the unbalance forces hold sway, and the shaft rotates about the center of mass of the assembly (which is very close to the center of the shaft).