Dunkerley Lower Bound Approximation

Let W₁, W₂ ,….W_n be the concentrated loads on the shaft due to masses m₁, m₂,….m_n and D₁, D₂,…… D₃ are the static deflections of the shaft under each load. Also let the shaft carry a uniformly distributed mass of m per unit length over its whole span and static deflection at the mid span due to the load of this mass be Δs. Also

Let
 
ω_n = Frequency of transverse vibration of the whole system.
 
ω_rε = Frequency with distributed load acting alone
 
ω_n1, ω_n2..... = Frequency of transverse vibration when each of W₁, W₂ ,….W_n ....act alone.

According to Dunkerley's empirical formula

 
Dunkerley's method gives lower bound approximation.

For a simply supported Euler Bernoulli's beam

   
for simply supported beam with uniformly distributed load, maximum deflection occur at midpoint.


 

 
Similarly for a fixed-fixed beam with loading the maximum deflection can be given by

 
In this case for the first mode
 

 
For Cantilever Beam

 
In this case for the first mode

 
So,

 
In case of concentrated loading the natural frequencies can be determined from the relation,

where Δ is the deflection under that load. One may note for the commonly used cases.

 
The Rayleigh-Ritz method

This is considered as an extension of Rayleigh's method. A closer approximation to the natural mode can be obtained by superposing a number of assumed functions than using by a single assume functions as in Rayleigh's method.

It gives the more accurate result than the previous method.

In the case of transverse vibration of beams, if n functions are chosen for approximating the deflection W(x), can be written as

 
where, are linear independent functions of the spatial coordinate x which satisfy the boundary condition of the problem, and c₁, c₂ ...., c_n  are the coefficient to be found.

As the Rayleigh quotients have stationary value near the natural mode by differentient by differentiating the

Rayleigh quotient with respect to these coefficients will yield a set of homogeneous algebraic equations, which can be solved to obtain the frequencies.

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