Short Answer:

The natural frequency of a cantilever beam is the frequency at which the beam vibrates freely after being displaced and released, without any external force acting on it. It depends on the beam’s length, material properties, cross-sectional dimensions, and mass distribution.

In simple terms, when a cantilever beam (fixed at one end and free at the other) is bent and released, it starts to oscillate up and down. The rate or frequency of this oscillation is its natural frequency. The stiffer and shorter the beam, the higher will be its natural frequency, while longer and flexible beams have lower frequencies.

Detailed Explanation :

Natural Frequency of a Cantilever Beam

A cantilever beam is a structural member that is fixed at one end and free at the other. When this beam is subjected to a dynamic load or deflected and then released, it vibrates freely about its equilibrium position. The frequency with which it vibrates naturally, without any external periodic force, is called its natural frequency.

The natural frequency of a cantilever beam is one of the most important parameters in mechanical and structural engineering. It determines how the beam will respond to vibrations and external loads. If the frequency of an external force matches the beam’s natural frequency, resonance occurs, which can lead to excessive vibrations and even failure. Hence, it is essential to determine and control the natural frequency while designing beams and machine components.

1. Concept of Natural Frequency

When a cantilever beam is deflected due to an applied force, it stores strain energy in the material due to bending. Once the force is removed, the beam tries to return to its original position because of its elastic restoring force.

However, due to inertia, the beam overshoots its mean position and continues to oscillate about it. This repetitive motion continues at a particular frequency known as the natural frequency of vibration. The beam vibrates freely until the energy is gradually dissipated through damping.

The natural frequency depends mainly on:

• Elastic properties of the material (Young’s modulus, ),
• Mass per unit length (),
• Moment of inertia of the cross-section (),
• Length of the beam ().

2. Mathematical Expression for Natural Frequency

The general differential equation for the vibration of a beam according to Euler-Bernoulli beam theory is:

Where,

•  = Young’s modulus of elasticity of the beam material,
•  = Moment of inertia of the beam’s cross-section,
•  = Transverse deflection at distance ,
•  = Mass per unit length of the beam,
•  = Time.

The solution of this equation gives the natural frequency for a beam, depending on the type of support.

For a cantilever beam, the first mode (fundamental) natural frequency is given by:

Where,

•  = Natural frequency in Hz (cycles per second),
•  = Length of the beam,
•  = Young’s modulus,
•  = Moment of inertia,
•  = Mass per unit length.

3. Explanation of the Formula

• The term  represents the bending stiffness of the beam. A stiffer beam (high ) will vibrate at a higher frequency.
• The term  represents the mass per unit length. Heavier beams have more inertia, reducing the frequency.
• The length () affects the frequency most significantly — since it appears with , even small increases in length greatly reduce the natural frequency.
• The constant  corresponds to the first mode shape of vibration for a cantilever beam. For higher modes, different constants (4.694, 7.855, etc.) are used.

4. Factors Affecting Natural Frequency

1. Length of the Beam (L):
   The natural frequency decreases rapidly with an increase in length because longer beams are more flexible.
2. Material Property (E):
   Beams made from stiffer materials (such as steel) have higher natural frequencies compared to softer materials (such as aluminum or wood).
3. Moment of Inertia (I):
   A beam with a larger cross-sectional area or greater moment of inertia resists bending more and thus vibrates at a higher frequency.
4. Mass per Unit Length (m):
   A heavier beam with greater mass vibrates more slowly because more energy is required for motion.
5. Boundary Conditions:
   The type of support influences natural frequency. A cantilever beam has a lower frequency compared to a beam fixed at both ends because it is more flexible.

5. Mode Shapes of Cantilever Beam

A cantilever beam can vibrate in multiple modes or shapes depending on the excitation frequency:

• First Mode (Fundamental Frequency):
  The beam bends in a simple curve with maximum deflection at the free end.
• Second Mode:
  There is one node (a point that remains stationary) along the length.
• Higher Modes:
  More nodes appear as the mode number increases, but each has a higher natural frequency.

In most engineering applications, the first mode is of primary interest, as it has the largest amplitude and greatest influence on system stability.

6. Applications

The concept of natural frequency in cantilever beams is used in:

• Bridge design to avoid resonance due to moving loads.
• Aircraft wings and turbine blades to ensure structural stability under aerodynamic forces.
• Machine tool arms and robotic arms for precision operations.
• Sensors and micro-electromechanical systems (MEMS) that use cantilever beams to detect small forces or vibrations.

7. Importance of Knowing Natural Frequency

• Helps to avoid resonance, which can cause large vibrations and failure.
• Ensures structural safety by designing components that operate away from natural frequency.
• Improves performance and accuracy in machines where vibration must be minimized.
• Allows for predictive maintenance, as changes in natural frequency indicate damage or cracks in the structure.

Conclusion:

The natural frequency of a cantilever beam is the rate at which it freely oscillates when disturbed. It depends on the material’s stiffness, beam geometry, and mass distribution. The fundamental natural frequency is given by

Understanding and controlling the natural frequency is crucial to prevent resonance, ensure safety, and maintain the efficient operation of mechanical and structural systems. Proper design of cantilever beams helps to achieve stability and longevity in various engineering applications.