Short Answer:

Natural frequency is the frequency at which a system or object tends to vibrate when it is disturbed from its rest position and then allowed to vibrate freely without any external force. Every object or structure has its own natural frequency, which depends on its mass and stiffness.

When a system vibrates at its natural frequency, even a small external vibration can cause large amplitude oscillations, known as resonance. Understanding natural frequency is important in engineering to design machines and structures that avoid unwanted vibrations and failures.

Detailed Explanation :

Natural Frequency

Natural frequency is one of the most important concepts in the study of vibrations. It is defined as the frequency at which a system vibrates freely after being disturbed from its equilibrium position and no external force acts on it. In simpler words, if you pull or push an object slightly and then release it, the rate at which it oscillates naturally is called its natural frequency.

For example, when you pluck a guitar string, it vibrates freely at its natural frequency to produce a sound. Similarly, when you hit a tuning fork, it vibrates at its natural frequency and produces a specific musical note. The value of the natural frequency depends on two main factors — the mass of the system and its stiffness.

If a system has high stiffness and low mass, its natural frequency will be high. On the other hand, if the system has high mass and low stiffness, its natural frequency will be low. This relationship can be expressed mathematically for a simple spring–mass system as:

Where:

•  = natural frequency (in Hz)
•  = stiffness of the spring (in N/m)
•  = mass of the object (in kg)

This equation shows that the natural frequency increases when stiffness increases and decreases when mass increases.

Importance of Natural Frequency in Engineering

In mechanical and structural engineering, it is essential to calculate and understand the natural frequency of systems to prevent resonance. Resonance occurs when an external force has a frequency equal or close to the natural frequency of a system, which can cause large and dangerous vibrations. This may lead to structural failure or damage to machinery.

For example, bridges, engines, turbines, and buildings are designed carefully so that their natural frequencies do not match with the frequency of external forces such as wind, engine vibrations, or earthquakes. Engineers use damping materials and structural modifications to control vibrations and ensure safety.

In rotating machinery like motors or fans, the natural frequency helps determine the critical speed, which is the speed at which the machine will resonate. Operating at or near this speed can cause severe vibration, noise, and even mechanical breakdown. Therefore, machines are designed to run either below or above this critical speed safely.

Factors Affecting Natural Frequency

1. Mass of the system:
   Increasing mass lowers the natural frequency because heavier objects resist motion more strongly.
2. Stiffness of the system:
   Increasing stiffness raises the natural frequency because the system resists deformation more quickly.
3. Shape and size:
   The geometry of an object affects how it vibrates. For example, longer beams or rods have lower natural frequencies compared to shorter ones.
4. Material properties:
   Materials with higher modulus of elasticity (like steel) have higher natural frequencies compared to softer materials (like rubber).
5. Boundary conditions:
   The way an object is supported or fixed affects its vibration. A beam fixed at both ends will have a higher natural frequency than one that is simply supported or free.

Applications of Natural Frequency

• Structural design: Engineers analyze natural frequencies to ensure buildings and bridges can withstand vibrations due to wind or earthquakes.
• Machine design: Used to avoid resonance in rotating and reciprocating machines.
• Aerospace and automotive engineering: To minimize vibrations in vehicles, aircraft wings, and engines for better performance and comfort.
• Acoustics: Helps in designing musical instruments, speakers, and soundproofing systems.
• Testing and diagnostics: Used in vibration analysis and condition monitoring to detect faults in machines or structures.

Conclusion

Natural frequency is the inherent vibration frequency of a system that depends mainly on its mass and stiffness. Understanding it is vital in mechanical and structural engineering because it helps in avoiding resonance and preventing vibration-related damage. By controlling factors such as material, design, and support conditions, engineers can ensure that systems operate safely and efficiently without unwanted oscillations.