Consider a shaft AB of negligible mass loaded with point loads W_{1} , W_{2}, W_{3 }and_{ }W_{4}_{ }etc., as shown in Fig (a)., Let m_{1 }, m_{2 }, m_{3} and m_{4 }etc., be the corresponding masses in Kg. The natural frequency of such a shaft may be found out by the following two methods:

**1. Energy (or Rayleigh’s) method**

Let y_{1 }, y_{2 }, y_{3} and y_{4 }etc. be total deflection under loads W_{1} , W_{2}, W _{3 }and_{ }W_{4}_{ }etc., as shown in Fig (a).,

We know that maximum potencial energy

and maximum kinetic energy

Equating the maximum potencial energy and maximum kinetic energy we have,

Natural frequency of transverse vibration

**2. Dunkerley’s method**

The natural frequency of transverse vibration for a shaft carrying a number of point loads and uniformly distributed from Dunkerley's empirical formula. According to this

Where f_{n} = Natural frequency o transverse vibration of the shaft carrying point loads and uniformly distributed load.

f_{n1 }, f_{n2 }, f_{n3 }etc., = Natural frequency of transverse vibration of each point load.

f_{ns }= Natural frequency of transverse vibration off the uniformly distributed load (or due to the mass of the shaft).

Now consider a shaft AB loaded as shown in Fig (b).,

Let δ_{1} , δ_{2 }, δ3 etc., = Static deflection due to the load W_{1} , W_{2}, W _{3 }etc., when considered seperately

δS = Static deflection due to the uniformly distributed load or due to the mass of the shaft.

We know that natural frequency of transverse vibration due to the load W_{1} ,

Similarly, natural frequency of transverse vibration due to the load W_{2} ,

and, natural frequency of transverse vibration due to the load W_{3} ,

Also natural frequency of transverse vibration due to uniformly distributed load or weight of the shaft,

Therefore, according to Dunkerly's empirical formula, the natural frequency of the whole system,

**Note:** 1. When there is no uniformly distributed load or mass of the shaft is negligible, then δS = 0.

2. The value of etc., for a simply supported shaft may be obtained from the relation
Where δ= Static deflection due to load W,

a and b = Distance of the load from the ends,

E = Young's modulus for the material of the shaft,

I = Moment of inertia of the shaft, and

l = Total length of the shaft.