Short Answer:

The static deflection method is a simple and practical technique used to find the natural frequency of a vibrating system such as a shaft, beam, or spring-mass system. It is based on measuring the static deflection produced by a known weight when the system is at rest.

In this method, the system is loaded with a known force, and the resulting deflection is measured. The natural frequency is then calculated using the relationship between stiffness and deflection. This method is especially useful in mechanical systems like rotating shafts and beams, where direct frequency measurement is difficult.

Detailed Explanation :

Static Deflection Method

The static deflection method is an experimental and analytical approach used to determine the natural frequency of vibration of a mechanical system. It works on the principle that the stiffness of a system can be determined from the static deflection caused by a known static load. Once the stiffness is known, the natural frequency can be easily calculated using simple mathematical relationships.

This method is particularly useful in the analysis of beams, shafts, and simple mechanical systems, where determining the natural frequency directly from dynamic measurements may not be practical. It gives a reliable estimation of the fundamental frequency of vibration, which helps in predicting resonance and ensuring safe operation of machines.

1. Principle of Static Deflection Method

The static deflection method is based on the relationship between stiffness, load, and deflection.
When a system is subjected to a static load, the force causes a deflection proportional to the stiffness of the system. The stiffness  of the system can be expressed as:

Where,

•  = stiffness of the system (N/m),
•  = static load applied (N),
•  = static deflection (m).

Once the stiffness is known, the natural frequency (fₙ) can be calculated using the formula:

Substituting  and , we get:

Where,

•  = acceleration due to gravity (9.81 m/s²),
•  = static deflection due to the load (m).

This shows that the natural frequency depends only on the static deflection and not directly on the mass or stiffness individually.

2. Explanation of the Method

The static deflection method involves the following simple steps:

1. Apply a Known Static Load:
   A known weight or force is applied at a specific point on the system, such as the free end of a cantilever beam or the center of a simply supported beam.
2. Measure the Static Deflection:
   The vertical deflection (δ) of the point due to the applied weight is measured accurately using a dial gauge or similar measuring instrument.
3. Calculate Natural Frequency:
   Using the measured value of δ, the natural frequency of the system is calculated by using the equation:

This gives the natural frequency of the first (fundamental) mode of vibration.

1. Interpretation:
   The larger the static deflection, the smaller the natural frequency, and vice versa. This relationship helps engineers estimate how design changes affect vibration characteristics.

3. Application Example

Example:
Consider a cantilever beam with a known static deflection of 4 mm at its free end due to an applied load.

Given:

Using the formula:

So, the natural frequency of the beam is approximately 7.88 Hz.

This method gives a quick and effective way to estimate the frequency without complex analysis.

4. Advantages of Static Deflection Method

1. Simple and Direct:
   The method requires only a static load and a deflection measurement, making it very easy to perform.
2. No Need for Complex Equipment:
   It avoids dynamic testing instruments like accelerometers or vibration sensors.
3. Quick Estimation:
   The natural frequency can be calculated quickly using simple formulas.
4. Useful for Beams and Shafts:
   Especially suitable for long or slender beams and rotating shafts where measuring vibrations directly is difficult.
5. Cost-Effective:
   Requires minimal setup and low-cost measuring tools.

5. Limitations of Static Deflection Method

1. Applicable Only for Small Deflections:
   The method assumes linear behavior (Hooke’s law). For large deflections, errors occur.
2. Limited to First Natural Frequency:
   It provides only the fundamental frequency, not higher modes of vibration.
3. Neglects Damping and Dynamic Effects:
   It ignores damping and inertia effects, which may influence actual behavior.
4. Accuracy Depends on Measurement:
   Small errors in deflection measurement can lead to significant frequency calculation errors.
5. Assumes Uniform Material and Geometry:
   In cases where the material or shape varies, the results may not be accurate.

6. Practical Applications

The static deflection method is widely used in engineering for:

• Shaft and rotor analysis to find the critical or whirling speed.
• Beam vibration testing to estimate the first mode frequency.
• Design of springs and machine components where vibration control is essential.
• Preliminary vibration analysis before performing complex experimental or numerical studies.

This method is commonly used in laboratories and industries because it is quick and gives reasonable accuracy for flexible systems.

Conclusion:

The static deflection method is a simple and effective way to determine the natural frequency of a mechanical or structural system using static load and deflection measurements. It is based on the principle that the natural frequency depends on the stiffness and static deflection of the system. Though it is an approximate method and neglects damping effects, it provides valuable insights for preliminary vibration analysis. Its simplicity, low cost, and practical usefulness make it a common choice in engineering vibration studies.