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Natural Frequencies to Traverse Vibrations



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.

Typical Units

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.

Fundamental Natural Frequencies of Beams (bending) and Shafts (Torsion)



Cantilevered Beam


Mode 1 2 3 4
K3,52 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
K9,87 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
K22,4 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
K15,4 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

Nat Freq 9

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

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Last Updated 10/08/2009