Short Answer:

At resonance, the amplitude of vibration becomes very large because the frequency of the external periodic force matches the natural frequency of the system. The system absorbs maximum energy from the external force, resulting in a sharp rise in vibration amplitude.

In simple words, when resonance occurs, the vibration amplitude increases rapidly and can reach dangerous levels if there is no damping. The amplitude at resonance depends on the amount of damping present — smaller damping leads to higher amplitude, while greater damping reduces the amplitude at resonance.

Detailed Explanation :

Amplitude at Resonance

When a mechanical or structural system is subjected to a periodic external force, it vibrates with an amplitude that depends on the relationship between the forcing frequency (ω) and the natural frequency (ωₙ) of the system. The condition when these two frequencies become equal is known as resonance.

At this condition, the system absorbs maximum energy from the external force, leading to maximum amplitude of vibration. The phenomenon is called resonance because the external excitation “resonates” with the natural tendency of the system to vibrate.

In practical terms, this means the system vibrates with very high amplitude at resonance. If the damping in the system is small, the amplitude can become dangerously large and may result in failure or damage to mechanical parts or structures.

Mathematical Explanation of Amplitude at Resonance

The steady-state amplitude of a forced damped vibration system is given by:

Where:

•  = amplitude of vibration
•  = amplitude of applied force
•  = stiffness of the system
•  = frequency ratio
•  = damping ratio

At resonance, the forcing frequency (ω) equals the natural frequency (ωₙ), so .
Substituting this in the equation:

This equation clearly shows that the amplitude at resonance () is inversely proportional to the damping ratio ().
Thus:

• If damping is very small, amplitude becomes very large.
• If damping is large, amplitude remains low even at resonance.

This relationship demonstrates that damping plays a crucial role in controlling the amplitude at resonance.

Behavior of Amplitude at Resonance

1. In an Undamped System (ξ = 0):
   • There is no damping to absorb or dissipate energy.
   • The amplitude at resonance theoretically becomes infinite.
   • Practically, such systems do not exist because some damping is always present in real materials and systems.
2. In a Lightly Damped System (ξ is small):
   • The amplitude at resonance is very high but finite.
   • The resonance curve is sharp and narrow.
   • This condition is common in mechanical systems such as engines, rotating machinery, or suspension systems.
3. In a Heavily Damped System (ξ is large):
   • The amplitude at resonance is much smaller.
   • The resonance curve becomes broad and flat.
   • Energy absorption and vibration transmission are reduced, making the system stable.

Thus, as damping increases, the amplitude at resonance decreases, and the sharpness of the resonance peak reduces.

Energy Transfer at Resonance

At resonance, the energy supplied by the external force in each vibration cycle matches exactly with the energy lost by damping in the same cycle. As a result, the system continues to absorb and store energy, causing a significant increase in amplitude.

If there is no damping, the energy input per cycle is not dissipated, and it accumulates continuously. This leads to a steady rise in amplitude, theoretically reaching infinity. In real systems, damping limits this growth, but even a small damping value can make resonance extremely powerful.

Therefore, the amplitude at resonance represents the condition of maximum energy transfer between the external force and the system.

Graphical Explanation (Concept)

If we plot the amplitude ratio (X/Xst) versus the frequency ratio (r = ω/ωₙ), the graph shows:

• At : The amplitude gradually increases.
• At : The amplitude reaches a maximum (resonance).
• At : The amplitude decreases again.

The height of the resonance peak (maximum amplitude) depends on damping:

• Lower damping → higher and sharper peak.
• Higher damping → lower and broader peak.

This graphical behavior helps in identifying the effect of damping on resonance amplitude.

Practical Examples of Amplitude at Resonance

1. Machinery and Rotating Equipment:
   When machines like compressors, turbines, or engines operate at speeds close to their natural frequency, the amplitude of vibration increases sharply, leading to excessive noise and possible failure.
2. Bridges and Structures:
   Resonance due to wind, traffic, or earthquakes can cause large oscillations in buildings or bridges. The Tacoma Narrows Bridge collapse (1940) is a famous example.
3. Vehicles:
   In vehicle suspension systems, resonance can cause excessive bouncing at certain speeds if damping is low.
4. Musical Instruments:
   In controlled cases, resonance increases sound amplitude and quality, such as in guitars, drums, or tuning forks.
5. Electrical Systems:
   In alternating current (AC) circuits, resonance causes maximum current amplitude when inductive and capacitive reactances are equal.

Effect of Damping on Amplitude Control

To prevent excessive amplitude at resonance, engineers use damping mechanisms such as:

• Viscous dampers (oil-based).
• Rubber mounts in vehicles or machines.
• Vibration isolators in buildings and precision equipment.

Proper damping ensures that the system does not experience damaging amplitudes and maintains safe operation even near resonance.

Key Observations about Amplitude at Resonance

• Amplitude is maximum when forcing and natural frequencies are equal.
• The amplitude is inversely proportional to the damping ratio.
• Damping reduces the amplitude and broadens the resonance curve.
• Resonance is both useful (when controlled) and dangerous (when uncontrolled).

Conclusion

In conclusion, at resonance, the amplitude of vibration becomes maximum because the system’s natural frequency matches the frequency of the external force. This causes the system to absorb maximum energy, resulting in large oscillations. The amplitude at resonance depends on the damping present in the system — less damping means higher amplitude, while more damping reduces it. Controlling the amplitude at resonance through proper design and damping is essential to ensure the safety, stability, and reliability of mechanical and structural systems.