Introduction
The information below relates to natrural 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 fundemental frequency of transverse vibration.
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/m2 f = frequency of vibration /whirling frequenct (Hz= cycles/s) g= accelaration 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) |
It can be shown that the critical whirling speed for a shaft is equal to the fundemental frequency of transverse vibration.
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.
Cantilevered Beam
| 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"
Simply Supported Beam
| 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"
Beam with fixed ends
| 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"
Beam with one end fixed and one end Simply Supported
| 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
Mass on Cantilever beam
Note: Assume beam of negligible mass
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"
Central mass on Simply supported beam
Note: Assume beam of negligible mass
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"
Mass on Simply supported beam. Off-Centre
Note: Assume beam of negligible mass
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"
Multiple Mass on beam
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"
Formula for Combined Loading
Mass on Cantilevered Shaft In Torsion
The following two examples identify the natural torsional frequency of shafts with heavy disks on the ends.
Note: Assume shaft of negligible mass
Shaft with mass at both ends
Note: Assume shaft of negligible mass
