Short Answer:

The natural frequency of a simply supported beam is the frequency at which the beam freely vibrates when it is displaced from its equilibrium position and released. It represents the beam’s inherent vibration rate without any external force acting on it.

This frequency depends on the beam’s material properties, dimensions, boundary conditions, and mass. For a simply supported beam, both ends are supported but free to rotate, which allows it to vibrate in several modes. The fundamental or first natural frequency is the lowest and most important for preventing resonance and ensuring the beam’s safe design.

Detailed Explanation :

Natural Frequency of a Simply Supported Beam

A simply supported beam is one that is supported at both ends, with each end able to rotate freely but not translate vertically. When such a beam is subjected to a disturbance (like an impact or load) and then released, it begins to vibrate up and down. The frequency at which it vibrates naturally without any external periodic force is known as its natural frequency.

This frequency depends mainly on the stiffness of the beam, its mass per unit length, and the type of support. In practical systems, understanding the natural frequency helps engineers avoid resonance — a dangerous condition that occurs when the external force frequency matches the natural frequency of the beam, causing large amplitude vibrations that can lead to failure.

1. Concept of Natural Frequency

When a beam is deflected, it stores strain energy due to bending. Upon release, this stored energy is converted into kinetic energy as the beam tries to regain its original position. However, because of inertia, it overshoots and vibrates around its mean position. This repetitive motion continues at a specific rate called the natural frequency.

The natural frequency of a beam depends on:

• Material property (E, Young’s modulus),
• Beam geometry (moment of inertia I),
• Beam length (L),
• Mass per unit length (m), and
• Boundary conditions (support type).

A simply supported beam is less stiff compared to a fixed beam but stiffer than a cantilever beam, so its natural frequency lies between the two.

2. Mathematical Expression for Natural Frequency

For a simply supported beam, the governing differential equation of vibration based on Euler-Bernoulli beam theory is:

Where:

•  = Young’s modulus of the material (N/m²),
•  = Moment of inertia of the cross-section (m⁴),
•  = Lateral displacement,
•  = Mass per unit length (kg/m),
•  = Time (s).

By solving this differential equation with the appropriate boundary conditions for a simply supported beam (deflection  and bending moment  at both ends), the natural frequencies can be determined.

3. Natural Frequency Formula

The nth natural frequency of a simply supported beam is given by:

Simplifying,

Where,

•  = Natural frequency of the nth mode (Hz),
•  = Mode number (1, 2, 3, …),
•  = Young’s modulus,
•  = Moment of inertia,
•  = Length of the beam,
•  = Mass per unit length.

For the fundamental (first) mode, , so:

This is the lowest and most significant natural frequency because it is usually the first mode excited during vibration.

4. Explanation of Terms

• EI (Flexural Rigidity):
  The product of Young’s modulus (E) and moment of inertia (I) represents the beam’s stiffness or resistance to bending. A higher EI means higher stiffness and higher natural frequency.
• m (Mass per Unit Length):
  The mass distribution along the beam affects inertia. A beam with greater mass vibrates at a lower frequency.
• L (Length of Beam):
  The natural frequency is inversely proportional to the square of the length. Hence, doubling the beam length decreases its frequency by four times.
• Mode Number (n):
  The beam can vibrate in different shapes or modes, each with a higher frequency. The first mode has one half-wave, the second has two, and so on.

5. Mode Shapes of a Simply Supported Beam

• First Mode (Fundamental):
  The beam bends in one single curve with maximum deflection at the center and zero deflection at the ends.
• Second Mode:
  There are two half-waves along the beam length with one node (point of zero deflection) at the center.
• Third Mode and Higher:
  More nodes appear along the length, each corresponding to higher frequencies.

The first mode is the most critical in design because it has the largest amplitude and is more likely to cause resonance.

6. Factors Affecting Natural Frequency

1. Length of Beam (L): Longer beams vibrate at lower frequencies.
2. Cross-Section Shape (I): Beams with higher moment of inertia are stiffer and have higher frequencies.
3. Material Property (E): Stiffer materials like steel produce higher natural frequencies.
4. Mass Distribution (m): Higher mass reduces the frequency.
5. Support Conditions: Fixed beams have higher frequencies than simply supported or cantilever beams.

7. Practical Applications

• Bridges and Flyovers:
  The natural frequency helps ensure that the bridge doesn’t resonate with traffic or wind forces.
• Machine Foundations:
  Machine supports are designed so that their natural frequencies do not coincide with operating machine frequencies.
• Aircraft Wings and Turbine Blades:
  The frequency must be analyzed to avoid flutter or resonance.
• Structural Design:
  Helps engineers ensure structural stability and avoid fatigue due to cyclic loads.

Conclusion:

The natural frequency of a simply supported beam is the rate at which it vibrates freely after being disturbed. It depends on the beam’s length, stiffness, mass, and boundary conditions. The fundamental frequency is the most important because it dominates the vibration behavior and determines the safety and stability of the structure. By designing beams with appropriate stiffness and material, engineers can ensure that the operating frequencies stay away from the natural frequency to avoid resonance and possible failure.