Shaft Critical Speed (Rayleigh Method)
The first critical (whirl) speed of a shaft carrying up to three wheels, rotors, or pulleys.
Example
You enter
- Mass 1 weight (lb) 100
- Static deflection at mass 1 (in) 0.002
- Mass 2 weight (lb, optional) 60
- Static deflection at mass 2 (in) 0.0015
- Mass 3 weight (lb, optional) 0
- Static deflection at mass 3 (in) 0
- Operating speed (rpm, blank to skip) 3600
You get
- First critical speed 4,369 rpm
- Speed ratio 0.824
No published worked example is on file for this calculation. It is checked against a hand-worked derivation of its method and an independent reference test.
Details, formula, and sources
ScopeFrom each mass and the shaft's static deflection under it: omega = sqrt(g sum(w y)/sum(w y^2)). One mass is the shop rule 187.7/sqrt(deflection in inches). A 100 lb rotor deflecting 0.002 in and a 60 lb sheave at 0.0015 in give about 4,369 rpm, so a 3,600 rpm motor runs at 82% of critical, inside the band to avoid. A screen; rotor dynamics governs.
Formula or tableomega = sqrt(g sum(w_i y_i) / sum(w_i y_i^2)), g = 386.0886 in/s^2, w in lb and y the static deflection (in) at each mass under all the loads; N_c = 60 omega / (2 pi) rpm; one mass: N_c = 187.7/sqrt(y); ratio = operating / N_c.
Source and editionRayleigh's method for the first critical speed of a shaft (Shigley, Mischke and Budynas, Mechanical Engineering Design, Ch. 7), by name.
Free accessA published energy method; the weights and static deflections are the user's inputs (from a beam calculation or a measurement).
What governsEstimate. AHJ and licensed professional govern.
API field namesweight_1_lb, deflection_1_in, weight_2_lb, deflection_2_in, weight_3_lb, deflection_3_in, operating_rpm, critical_rpm, speed_ratio
- Rayleigh's method lumped masses at the static deflected shapeShigley Ch. 7
- Resonance band within 20% of the first criticalcommon machinery practice