COINCIDES with the natural frequency of lateral/transverse vibrations, the shaft tends to bow out with a large amplitude. This speed is termed as critical/whirling speed.Whirling speed or Critical speed of a shaft is defined as the speed at which a rotating shaft will tend to vibrate violently in the transverse direction if the shaft rotates in the horizontal direction.In other words, the whirling or critical speed is the speed at which resonance occurs.Hence we can say that Whirling of the shaft occurs when the natural frequency of transverse vibration MATCHES the frequency of a rotating shaft.It is the speed at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite is known as critical or whirling speed.Deflection of the shaft due to transverse vibration of the shaft.\(y = \frac{e}{{{{\left( {\frac{{{\omega _n}}}{\omega }} \right)}^2} - 1\;}}\)At critical speed,\(\omega = {\omega _n} = \sqrt {\frac{k}{m}} = \sqrt {\frac{g}{\delta }} \)where, ω = Angular velocity of the shaft, k = Stiffness of shaft, e = initial eccentricity of the center of mass of the rotor, m = mass of rotor, y = additional of rotor due to centrifugal force.As the shaft is loaded centrally, it acts as a simply supported beam with a point load at centre:Deflection is this case is given by:\({\delta} = \frac{{P{L^3}}}{{48EI}}\)if the shaft is circular, I = \(\frac{\pi d^4}{64}\)therefore PUTTING values in equation 1, we will COME to know that critical speed depends upon the span and diameter of the shaft.