This page provides natural frequency formulas for beams and rotating shafts, including cantilever, simply supported and fixed end conditions. These equations are widely used in vibration analysis to determine resonance, critical speed and whirling behaviour in mechanical systems.
The information below relates to natural frequency of traverse vibration. The units for the various
parameters must be consistent. i.e for metric calculations length is m, force = N, mass = kg. etc.
It can be shown that the critical whirling speed for a shaft is equal to the fundamental frequency of transverse vibration.
Background notes showing how the natural frequencies are derived and the relationship to the shaft whirling velcities are found at Derivation of Natural Frequencies
For members of excelcalcs.com I have created a set of spreadsheet calcs based on the equation below ref. ExcelCalcs.com calculation Shafts -Natural Frequencies / Whirling speeds
The units below relate to application of the equations using SI metric values
|
L = Length (m) m = mass/Unit length (kg/m) M = Mass of Disk (kg) E = Youngs Modulus (N/m)2 f = frequency of vibration /whirling frequency (Hz= cycles/s) g= acceleration due to gravity (m/s 2) I = Area Moment of Inertia (m4) G = Torsional Modulus (N.m-2) s = Shaft torsional stiffness = GJa/L (Nm.rad) J = Shaft polar moment of inertia (m4) JM1, JM2,JM3...= polar moment of inertia of disc mass = MX.k2 (X= 1,2,3...)(kg.m2) k = radius of gyration of disc (m) y = deflection (m) |
It can be shown that the critical whirling speed for a shaft is equal to the fundamental frequency of transverse vibration.
g = acceleration due to gravity (9.81 m/s2 ). y = static deflection at mass
The values for natural frequencies relate to cycle/unit time. The higher harmonic modes are not listed. To obtain values for wn in radians/s multiply values by 2.p.
| Mode | 1 | 2 | 3 | 4 |
| K | 3,52 | 22,0 | 61,7 | 121 |
Note: if equation used for whirling speed assume rotating member is supported using "a long bearing /bearing proving substantial angular support e.g. needle bearings"
| Mode | 1 | 2 | 3 | 4 |
| K | 9,87 | 39,5 | 88,8 | 158 |
Note: if equation used for whirling speed assume rotating member is supported using "short bearings providing little angular restraint e.g a spherical ball bearing"
| Mode | 1 | 2 | 3 | 4 |
| K | 22,4 | 61,7 | 121,0 | 200 |
Note: if equation used for whirling speed assume rotating member is supported using "a long bearing /bearing proving substantial angular support e.g. needle bearings"
| Mode | 1 | 2 | 3 | 4 |
| K | 15,4 | 50 | 104 | 178 |
Note: if equation used for whirling speed assume rotating member is supported using "a long bearing /bearing proving substantial angular support e.g. needle bearings" and a short bearing providing little angular support e.g a spherical ball bearing
Note: if equation used for whirling speed assume rotating member is supported using "a long bearing /bearing proving substantial angular support e.g. needle bearings"
Note: if equation used for whirling speed assume rotating member is supported using "short bearings providing little angular restraint e.g a spherical ball bearing"
Note: if equation used for whirling speed assume rotating member is supported using "short bearings providing little angular restraint e.g a spherical ball bearing"
Note: if equation used for whirling speed assume rotating member is supported using "long bearings providing full angular restraint e.g needle bearings"
Note: if equation used for whirling speed assume rotating member is supported using "short bearings providing little angular restraint e.g a spherical ball bearing"
The following two examples identify the natural torsional frequency of shafts with heavy disks on the ends.
Note: Assume shaft of negligible mass
When a shaft rotates at a speed equal to the natural frequency of traverse vibration, it results in vibratons at this frequency which becomes large and this is identified as shaft whirling. This also occurs at multiples of the resonant frequency......It must be noted that a Perfectly balanced shaft will not whirl at any speed
Additional notes on this topic are found at Natural Frequencies to Traverse Vibrations- Whirling