Understanding Dunkerley's Empirical Equation

Dunkerley's empirical equation is a method used to determine the fundamental natural frequency of transverse vibration for a shaft carrying several concentrated loads. This equation provides an approximate value for the critical speed of the shaft, which is directly related to the natural frequency. The equation is called 'empirical' because it is based on experimental observations and results rather than being derived purely from theoretical principles. It is a simplified approach for situations where calculating the exact natural frequency becomes complex due to multiple loads.

Applying Dunkerley's Equation for Transverse Vibration

According to Dunkerley's empirical equation, the reciprocal of the square of the natural frequency of the entire system is equal to the sum of the reciprocals of the squares of the natural frequencies calculated for each load acting alone on the shaft. Let:

• \(f_n\) be the natural frequency of the shaft with all loads attached.
• \(f_{n_1}\) be the natural frequency of the shaft with only the first load acting alone.
• \(f_{n_2}\) be the natural frequency of the shaft with only the second load acting alone.
• \(f_{n_3}\) be the natural frequency of the shaft with only the third load acting alone.
• ...
• \(f_{n_s}\) be the natural frequency of the shaft with only the s-th load acting alone.

Dunkerley's empirical equation for the frequency of the transverse vibration of the system of several loads attached to the same shaft is given by the formula:

\(\frac{{1}}{{f_n^2}}=\frac{{1}}{{f_{n_1}^2}}~+~\frac{{1}}{{f_{n_2}^2}}~+~\frac{{1}}{{f_{n_3}^2}}~+~...+\;\frac{{1}}{{f_{n_s}^2}}\)

This formula effectively combines the influence of individual loads on the overall natural frequency. It's important to remember that this is an approximation, but it is widely used in engineering practice for its simplicity and reasonable accuracy in many cases. Let's look at the options provided:

• Option 1 is \(\frac{{1}}{{f_n}}=\frac{{1}}{{f_{n_1}}}~+~\frac{{1}}{{f_{n_2}}}~+~\frac{{1}}{{f_{n_3}}}~+~...+\;\frac{{1}}{{f_{n_s}}}\). This form relates the reciprocals of frequencies directly, which is not Dunkerley's equation.
• Option 2 is \(\frac{{1}}{{f_n^2}}=\frac{{1}}{{f_{n_1}^2}}~+~\frac{{1}}{{f_{n_2}^2}}~+~\frac{{1}}{{f_{n_3}^2}}~+~...+\;\frac{{1}}{{f_{n_s}^2}}\). This exactly matches Dunkerley's empirical equation.
• Option 3 is \(f_{n} =f_{n_1}+f_{n_2}+f_{n_3}+.....+f_{n_s}\). This suggests a simple summation of frequencies, which is not correct for vibrations in series or combined systems like this.
• Option 4 states None of the above, which is incorrect as Option 2 matches the equation.

Therefore, the correct representation of Dunkerley's empirical equation for the transverse vibration frequency of a shaft with multiple loads is given by the sum of the reciprocals of the squares of the individual frequencies.