Dunkerley's empirical formula to find the natural frequency of transverse vibration for a shaft carrying a number of point loads and uniformly distributed load (UDL) is given by [Where, fn = Natural frequency of transverse vibration of the shaft carrying point load and uniformly distributed load; fn1, fn2 ------ = Natural frequency of transverse vibration of each point load; fns = Natural frequency of transverse vibration of UDL (or due to mass of shaft]

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BHEL Engineer Trainee Mechanical 24 Aug 2023 Official Paper

\( \frac{1}{f_n^2}=\frac{1}{f_{n 1}^2}+\frac{1}{f_{n 2}^2}+\frac{1}{f_{n 3}^2}+\cdots \ldots \ldots \frac{1}{f_{n s}^2} \)
\(f_n=f_{n 1}+f_{n 2}+f_{n 3}+\cdots \ldots \ldots f_{n s}\)
\(\frac{1}{f_n^3}=\frac{1}{f_{n 1}^3}+\frac{1}{f_{n 2}^3}+\frac{1}{f_{n 3}^3}+\cdots \ldots \ldots \frac{1}{f_{n s}^3}\)
\(f_n^2=f_{n 1}^2+f_{n 2}{ }^2+f_{n 3}{ }^2+\cdots \ldots \ldots f_{n s}{ }^2\)

Answer 1

Explanation:

Dunkerley's Empirical Formula for Natural Frequency of Transverse Vibration:

Dunkerley's empirical formula is a crucial method used to determine the natural frequency of transverse vibration for a shaft carrying a number of point loads and uniformly distributed load (UDL). This formula is particularly useful in mechanical and structural engineering applications where predicting the vibration characteristics of shafts is essential for ensuring stability and preventing resonance.
The natural frequency of a system is the frequency at which it tends to oscillate in the absence of any driving or damping force. Dunkerley's formula provides a simplified approach to estimate the natural frequency of a complex system by considering the contributions of individual components separately.

Given:

Let \( f_n \) = Natural frequency of the complete system

\( f_{n1}, f_{n2}, f_{n3}, \dots, f_{ns} \) = Natural frequencies due to individual point loads and UDL

Calculation:

According to Dunkerley's empirical formula, the natural frequency is given by:

\( \frac{1}{f_n^2} = \frac{1}{f_{n1}^2} + \frac{1}{f_{n2}^2} + \frac{1}{f_{n3}^2} + \cdots + \frac{1}{f_{ns}^2} \)