If the shaft is therefore to revolve stably this force must be balanced at any instant by the elastic resistanc of the shaft to deflexion.
The elastic resistance evoked by this deflexion is proportional to the deflexion, so that if c is a constant depending upon the form,, material and, method of support of the shaft, the following equality must hold if the shaft is to rotate stably at the stated speed- -~ (y+e)ai=cy, from which y=Wafe/(gcWa2).
The corresponding value of a, namely ~ gc/w, is called the critical velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins.
How would you define stably? Add your definition here.