Short Answer:

Natural frequencies are the specific frequencies at which a mechanical system tends to vibrate freely when disturbed from its equilibrium position without any external force acting on it. Every vibrating system, such as a spring, beam, or machine part, has one or more natural frequencies depending on its structure and material properties.

In simple words, natural frequency is the rate at which a body oscillates naturally due to its own mass and stiffness. When an external force acts on the system at this frequency, resonance occurs, which may cause very large vibrations or even structural failure.

Detailed Explanation :

Natural Frequencies

Natural frequencies are one of the most important concepts in mechanical vibration analysis. They are the frequencies at which a system tends to vibrate when it is displaced and allowed to oscillate freely without any continuous external force. The vibration at these frequencies is called natural vibration, and the system’s response is determined by its physical properties like mass, stiffness, and boundary conditions.

Every object capable of vibration, from a small spring to a large bridge, has its own natural frequencies. A system can have one or multiple natural frequencies depending on how many degrees of freedom it has. When the system is disturbed and released, it will vibrate at one or more of these frequencies.

If an external periodic force is applied to the system at one of its natural frequencies, the amplitude of vibration increases rapidly. This phenomenon is known as resonance, which can cause serious damage if not controlled. Therefore, engineers always design machines and structures to operate safely away from their natural frequencies.

Mathematical Expression of Natural Frequency

Consider a simple spring–mass system in which a mass is attached to a spring of stiffness . When the system is displaced by a small distance and released, it vibrates freely. The restoring force on the mass due to spring stiffness is given by Hooke’s Law:

According to Newton’s second law of motion:

Equating the two forces:

This is the differential equation of free vibration.

The solution of this equation is:

where,

= amplitude of vibration
= phase angle
= natural angular frequency (in radians per second)

Now,

The natural frequency (f) in cycles per second (Hz) is given by:

From the above relation, it is clear that:

The natural frequency increases when stiffness increases.
The natural frequency decreases when mass increases.

This relationship forms the basis of vibration analysis in mechanical engineering.

Factors Affecting Natural Frequencies

Mass of the System (m):
The greater the mass of the body, the lower the natural frequency. Heavier systems vibrate more slowly.
Stiffness of the System (k):
Higher stiffness leads to higher restoring forces, resulting in higher natural frequencies.
Damping:
Damping slightly reduces the natural frequency and causes the vibration to die out over time.
Boundary Conditions:
The way a structure is supported or fixed (such as clamped, free, or simply supported) affects its natural frequencies.
Shape and Material:
The geometry and elastic modulus of materials influence stiffness and thus affect the natural frequencies.
Number of Degrees of Freedom:
Systems with multiple masses and springs have several natural frequencies, one for each independent motion.

Examples of Natural Frequencies

Spring–Mass System:
A mass attached to a spring vibrates at its natural frequency when disturbed.
Pendulum:
A pendulum has a natural frequency depending on its length and gravity :

Tuning Fork:
When struck, the prongs of a tuning fork vibrate at a specific frequency, which is its natural frequency, producing a pure musical note.
Machine Components:
Shafts, beams, and connecting rods have natural frequencies determined by their material and geometry.
Bridges and Buildings:
Structures like bridges and towers have natural frequencies depending on their height, stiffness, and weight distribution.

Importance of Natural Frequencies in Mechanical Engineering

Resonance Avoidance:
When the frequency of external excitation equals a system’s natural frequency, resonance occurs, resulting in large amplitude vibrations. Understanding natural frequencies helps prevent such conditions.
Design Optimization:
Machine components are designed so that their operating frequencies are far from their natural frequencies to ensure stability and longevity.
Vibration Control:
Engineers use vibration isolators and dampers to control unwanted vibrations by modifying the natural frequency of systems.
Safety of Structures:
Knowing the natural frequencies of bridges, towers, and aircraft helps prevent resonance with external forces like wind or engine vibrations.
Testing and Analysis:
Experimental methods such as modal testing and frequency response analysis are used to determine natural frequencies and verify theoretical designs.

Multiple Natural Frequencies in Complex Systems

A simple spring–mass system has only one degree of freedom, and thus one natural frequency. However, complex systems with multiple degrees of freedom, such as beams or engine crankshafts, have multiple natural frequencies.

Each frequency corresponds to a different mode shape — the pattern in which the system vibrates. For example, a beam may have one mode where it bends once and another where it bends twice. Engineers analyze all these modes to ensure safety and efficiency in design.

Practical Example of Natural Frequency

In vehicles, the suspension system has its own natural frequency determined by the mass of the vehicle and the stiffness of the springs. If the road irregularities or engine vibrations match this frequency, resonance can occur, leading to discomfort and possible damage. Hence, designers adjust the stiffness or damping so that the system’s natural frequency stays outside the range of expected excitation frequencies.

Conclusion

Natural frequencies are the inherent frequencies at which mechanical systems tend to vibrate freely when disturbed. They depend on the system’s mass, stiffness, and structural conditions. Every object has at least one natural frequency, and operating near this frequency can lead to resonance, which may cause excessive vibrations and failure. Understanding and controlling natural frequencies is vital for the safe design and operation of machines, structures, and mechanical components.