Critical (Whirling) Speed of a Rotating Shaft
Critical (Whirling) Speed of a Rotating Shaft
Any rotating shaft carrying a mass — a rotor, a pulley, a flywheel — has a critical speed at which its natural bending frequency coincides with the rotational speed, causing the shaft to whirl with rapidly growing lateral deflection if it is run at or near that speed. Identifying this critical speed is essential for any rotating machinery design so the operating speed range can be kept safely clear of it.Running a shaft through (rather than up to and stopping before) its critical speed is actually standard practice for many high-speed machines, since the resonance is only a problem if the machine dwells there — but any machine expected to operate continuously at or very near a critical speed needs either a stiffer shaft, a different supported mass, or added damping, because sustained operation at resonance can lead to rapid bearing wear or catastrophic shaft failure.
For a simply supported shaft with a central mass, the natural (critical) angular frequency is omega_n = sqrt(48·E·I_s/(m_r·L_s^3)), from classical beam vibration theory. where E is the shaft modulus of elasticity, I_s is the shaft's cross-sectional moment of inertia, m_r is the mass of the central rotor, L_s is the shaft span between supports, omega_n is the resulting natural angular frequency, and f_n is the same frequency expressed in cycles per second.
Inputs
E := 200 GPa=200000 MPa
d_s := 50 mm=50 mm
m_r := 25 kg=25 kg
L_s := 1.2 m=1.2 m
The shaft's resistance to bending depends on its cross-section through the moment of inertia, which for a solid circular shaft is a simple function of diameter to the fourth power.
I_s := pi·d_s^4/64=3.07e-07 m^4
This classical beam-vibration formula combines the shaft's bending stiffness with the central mass it supports and the span between bearings to find the angular frequency at which the shaft naturally wants to whirl.
omega_n := sqrt(48·E·I_s/(m_r·L_s^3))=261.11 Hz
Converting the angular frequency to cycles per second gives the critical speed in the units typically used to specify a machine's operating speed range.
f_n := omega_n/(2·pi)=41.56 Hz
Results
A critical speed of about 41.6 Hz (roughly 2500 rpm) for this shaft means the machine's continuous operating speed should be kept well clear of that value — commonly by a margin of at least 20–25% — to avoid dwelling near resonance during normal running. Because the critical speed depends on the shaft diameter to the fourth power (through I_s) but only on the span cubed in the denominator, a relatively small increase in shaft diameter raises the critical speed far more effectively than shortening the span by the same percentage. If the machine must operate above this critical speed (a "supercritical" design, common in some turbomachinery), it needs to accelerate through the resonance quickly rather than dwelling there, and often needs damping to limit the transient whirl amplitude during that passage.
Assumptions & limitations
- Simply supported shaft with a single central point mass — distributed shaft mass and multiple masses are neglected.
- Shaft assumed massless relative to the central rotor mass (a common simplifying assumption for a first estimate).
- Only the first (fundamental) critical speed is found; higher-order critical speeds exist but are not calculated here.
- Gyroscopic effects and bearing flexibility are not included in this simplified beam-theory model.