Friday, 23 December 2016

Natural Frequency of Free Transverse Vibrations

          Consider a shaft of negligable mass whose one end is fixed and the other end carries a body of weoght W, as shown in fig. 

               Let           s = Stiffness of shaft,
                            δ = Static deflection due to weight of the body,
                                 x = Displacement of body from mean position after time t.
                                m = Mass of body = W/g As discussed in the previous article, 

Restoring force  =     – s.x          . . .                       (i)
and accelerating force  =   m   d2 x               . . . (ii)
Equating equations (iand (ii), the equation of motion becomes

Hence, the time period and the natural frequntly of the transverse vibrations are same as that of longitudinal vibrations. Therefore 

Note : 

The shape of the curve, into which the vibrating shaft deflects, is identical with the static deflection curve of a cantiliver beam loaded at the end. It has been proved in the text book on strength of Materials that the static deflection of a cantiliver beam loaded at the free end is 

                           δ  =      Wl3                    (in metres)


Where        W = Load at the free end, in newtons,
                     l = Length of the shaft or beam in metres,
                  E = Young’s modulus for the material of the shaft or beam in N/m2, and
                    I = Moment of inertia of the shaft or beam in m4.

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