Short Answer:

Dunkerley’s method is an approximate analytical method used to calculate the fundamental natural frequency of a system with several components, such as multiple masses or springs. It is particularly useful for systems with distributed or combined stiffness, like beams carrying several loads.

The method is based on the principle that the reciprocal of the square of the overall natural frequency of the system is equal to the sum of the reciprocals of the squares of the natural frequencies of each component acting separately. It provides a simple and safe lower-bound estimate of the natural frequency.

Detailed Explanation :

Dunkerley’s Method

Dunkerley’s method is an empirical and approximate method used in vibration analysis to determine the fundamental natural frequency of a mechanical system that contains multiple components, such as several masses connected by springs or supported on beams. This method was developed by S. Dunkerley, a British engineer, and is especially useful when the system is too complex to solve exactly using differential equations or energy methods.

The key idea of Dunkerley’s method is that each mass or component in the system contributes independently to the overall flexibility of the structure. Therefore, the combined effect of all the components can be obtained by combining their individual frequencies through a reciprocal relation.

Dunkerley’s method is mainly used for beams with multiple loads, shafts carrying several discs, and multi-mass systems where stiffness and mass distribution make exact analysis complicated.

Basic Principle

The fundamental principle of Dunkerley’s method is that the total flexibility (or inverse of stiffness) of a system is approximately equal to the sum of the flexibilities of all the components when considered separately. Since the natural frequency depends on stiffness and mass, the overall frequency of a combined system can be estimated from the natural frequencies of individual components.

Mathematically, Dunkerley proposed that the square of the reciprocal of the combined natural frequency is approximately equal to the sum of the squares of the reciprocals of individual natural frequencies.

This is expressed as:

Where:

•  = overall (combined) natural frequency of the system,
•  = natural frequencies of each individual mass or component acting separately.

The result gives a lower value for the true natural frequency, meaning it is a safe estimate that avoids overestimating the resonance frequency in design calculations.

For practical purposes, it is often expressed in terms of frequency (f) as:

Application in Multi-Mass Systems

Consider a shaft carrying multiple discs (masses) supported by bearings or springs. Each disc can vibrate independently when others are not considered. The natural frequency for each disc (acting alone) can be calculated using simple vibration formulas like:

where  is stiffness and  is mass for the  component.

Using Dunkerley’s formula, the combined natural frequency of the entire system is:

This gives a conservative (safe) value of the system’s overall natural frequency.

Steps in Dunkerley’s Method

1. Identify all Masses and Supports:
   Recognize the masses (loads, discs, or particles) and their positions on the beam, shaft, or spring system.
2. Calculate Individual Frequencies:
   Compute the natural frequency for each mass acting independently while considering the stiffness of the system.
3. Apply Dunkerley’s Equation:
   Substitute the individual frequencies into Dunkerley’s relation to obtain the combined frequency of the complete system.
4. Evaluate Results:
   The smallest value of frequency obtained from the equation represents the fundamental natural frequency of the system.

Example of Dunkerley’s Method

Let a simply supported beam carry two concentrated loads  and  at different points. The natural frequencies for each load acting alone are  and .

Then, the combined fundamental frequency is obtained using:

For instance, if
and , then

 

Thus, the fundamental frequency of the combined system is 12 Hz, which is lower than both  and , as expected.

Advantages of Dunkerley’s Method

1. Simple to Use:
   Does not require solving complex differential equations.
2. Useful for Complex Systems:
   Effective for beams or shafts carrying multiple masses.
3. Safe Design Estimate:
   Provides a lower (conservative) value of natural frequency, preventing resonance in practical applications.
4. Saves Time:
   Quick method for preliminary vibration analysis.

Limitations of Dunkerley’s Method

1. Approximate Method:
   It provides only an approximate value, not an exact solution.
2. Fundamental Frequency Only:
   The method cannot determine higher mode frequencies.
3. Accuracy Depends on System Type:
   Works well for lightly coupled systems; accuracy decreases for strongly coupled or continuous systems.
4. Neglects Mode Shape Coupling:
   Interaction between different masses and modes is not considered.

Applications of Dunkerley’s Method

• Calculation of natural frequencies for rotating shafts carrying several discs.
• Beams with multiple point loads in mechanical or civil structures.
• Estimation of vibration frequencies in machine tools and engine components.
• Analysis of aircraft wings, turbine rotors, and automotive driveshafts.
• Used in preliminary design stages for avoiding resonance conditions.

Conclusion

Dunkerley’s method is an approximate analytical technique used to determine the fundamental natural frequency of systems with several masses or loads. It is based on the reciprocal relationship between the combined and individual frequencies of each component. The method is simple, reliable, and provides a conservative estimate, making it ideal for preliminary vibration analysis of beams, shafts, and multi-mass systems. Although it does not provide exact results or higher mode frequencies, its simplicity and safety margin make it a valuable engineering tool for practical design and analysis.