Short Answer:
Natural frequency is the frequency at which a body or system tends to vibrate freely when it is disturbed from its equilibrium position and allowed to vibrate without any external periodic force. Every object capable of vibration has its own natural frequency depending on its mass and stiffness.
In simple words, if you strike a tuning fork or stretch and release a spring, it will vibrate at its natural frequency. This frequency is unique to every system and plays an important role in vibration and resonance studies in mechanical engineering.
Detailed Explanation :
Natural Frequency
In mechanical and structural systems, the term natural frequency refers to the frequency at which the system oscillates when it is disturbed from its position of rest and allowed to move freely. This frequency depends on the physical properties of the system, such as its mass (inertia) and stiffness (elasticity).
When a mechanical system such as a beam, spring, or machine component is displaced and released, it begins to vibrate on its own. The rate at which it vibrates is called its natural frequency. Each system or structure has one or more natural frequencies depending on its shape, material, and mode of vibration.
Understanding natural frequency is very important in engineering design because if a machine or structure is subjected to an external periodic force that matches its natural frequency, resonance occurs, leading to large vibrations and possible failure.
Definition
Natural frequency can be defined as:
“The frequency at which a system tends to vibrate freely when it is displaced from its equilibrium position and allowed to vibrate without any external periodic force is called its natural frequency.”
It is an inherent property of all elastic bodies and represents the system’s tendency to oscillate due to the balance between its inertial and elastic restoring forces.
Physical Explanation
When a body is disturbed from its rest position, a restoring force acts to bring it back to equilibrium. However, due to inertia, the body overshoots its equilibrium position and moves in the opposite direction. This back-and-forth motion continues, resulting in vibration.
The natural frequency is determined by how easily the body can move (its mass) and how strongly it resists deformation (its stiffness).
- If a body is stiffer, it will vibrate faster, resulting in a higher natural frequency.
- If a body is heavier, it will vibrate slower, resulting in a lower natural frequency.
Mathematical Expression
Consider a spring-mass system with mass and stiffness .
When displaced and released, the restoring force in the spring is given by Hooke’s law:
According to Newton’s second law:
Equating both:
This is the equation of motion for free vibration.
The solution of this equation is:
where
is the natural circular frequency (in radians per second), and
is the natural frequency in Hertz (cycles per second).
Thus, the natural frequency depends on:
- Stiffness (k): Directly proportional
- Mass (m): Inversely proportional
Hence, the natural frequency increases with stiffness and decreases with mass.
Factors Affecting Natural Frequency
- Mass of the System (m):
Higher mass reduces natural frequency because heavier bodies require more time to complete one oscillation. - Stiffness of the System (k):
Greater stiffness increases the natural frequency since the restoring force acts more strongly to return the system to equilibrium. - Geometry of the Structure:
The shape and size of a component affect its stiffness and inertia, which in turn affect its natural frequency. - Material Properties:
The modulus of elasticity (E) and density (ρ) of the material play a major role in determining stiffness and mass, thereby influencing natural frequency. - Boundary Conditions:
The way a structure is supported or fixed (e.g., fixed, hinged, free) changes its natural frequency. A fixed beam has a higher natural frequency than a free one. - Damping:
Though damping slightly reduces amplitude, it also affects the natural frequency marginally by dissipating energy from the system.
Examples of Natural Frequency
- Tuning Fork:
When struck, a tuning fork vibrates at its natural frequency and produces a sound of a particular pitch. - Pendulum:
The time period of a pendulum depends on its length and gravity, which determines its natural frequency. - Car Suspension System:
The spring and damper system of a vehicle vibrates at a specific natural frequency that affects ride comfort. - Bridge and Buildings:
Structures have natural frequencies that determine their response to wind or earthquake forces. - Machine Components:
Shafts, beams, and rotating parts vibrate at their own natural frequencies during operation.
Importance of Natural Frequency in Engineering
- Resonance Avoidance:
Engineers design machines and structures to ensure that the operating frequency does not match the natural frequency, preventing resonance and damage. - Vibration Control:
Helps in designing dampers and isolators to minimize unwanted vibrations. - Dynamic Analysis:
Determines how systems respond to varying loads, impacts, and periodic forces. - Structural Safety:
Used in earthquake engineering to analyze how buildings behave under seismic vibrations. - Design Optimization:
Natural frequency analysis helps select materials and shapes for best performance and stability.
Experimental Determination of Natural Frequency
- Free Vibration Test:
The system is displaced and released; the time period of oscillation is measured to calculate natural frequency. - Resonance Test:
The system is subjected to varying external frequencies; resonance occurs when amplitude becomes maximum, indicating the natural frequency. - Modal Analysis:
Performed using sensors and computers to find natural frequencies and vibration modes of complex structures.
Conclusion
Natural frequency is the inherent property of a mechanical or structural system that defines how it vibrates when disturbed and left to move freely. It depends mainly on the mass and stiffness of the system. Understanding natural frequency is essential in mechanical and civil engineering because it helps prevent resonance, ensures stability, and improves machine performance. By designing systems with suitable natural frequencies, engineers can achieve greater efficiency, safety, and durability of machines and structures.