Natural Frequency of Free Transverse Vibrations For a Shaft Subjected to a Number of Point Loads

           Consider a shaft AB of negligible mass loaded with point loads W₁ , W₂,  W₃ and W₄ etc., as shown in Fig (a)., Let m₁ , m₂ , m₃ and m₄ 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 y₁ , y₂ , y₃ and y₄ etc. be total deflection under loads W₁ , W₂, W ₃ and W₄ 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 δ₁ , δ₂ , δ3 etc., = Static deflection due to the load W₁ , W₂, 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 W₁ ,

Similarly, natural frequency of transverse vibration due to the load W₂ ,

and, natural frequency of transverse vibration due to the load W₃ ,

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.