Natural Frequency of Free Transverse Vibrations of a Shaft Fixed at Both Ends Carrying a Uniformly Distributed Load

          Consider a shaft AB fixed at both ends and carrying a uniformly distributed load 'w' per unit length as shown in Fig.,

    We know that the static deflection at a distance 'x' from A is given by

Integrating the above equation on both sides  with limits from 0 to l, 

Now integrating  y²  within the limits from 0 to l, 

We know that

and natural frequency,

Since the static deflection of a shaft fixed at both ends and carrying a uniformly distributed load is