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*__d__^{2}*x*. . .**(***ii***)***dt*

^{2}

Equating equations

**(***i***)**and**(***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 =

__Wl____(in metres)__^{3}
3

*EI*
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/m

^{2}, and
I = Moment of inertia of the shaft or beam in m

^{4}.
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