Natural Frequency of Free Transverse Vibrations Due to a Point Load Acting Over a Simply Supported Shaft

          Consider a light shaft AB of length 'l' between supports, carrying a point load W at C which is at a distance of l₁ from A and l₂ from B, as shown in Fig., A little consideration will show that when the shaft is deflected and suddenly released, it will make (or undergo) transverse vibrations. The deflection of the shaft is proportional to the load W which is acting on it and if the shaft is deflected beyond the static equilibrium position under load W, then the load along with the beam will vibrate with simple harmonic motion (S.H.M) (as by helical spring). If 'δ' is the static deflection due to load 'W', then the natural frequency of the transverse vibration is 

ω_n = √g/δ