Forced Oscillation: Amplitude Resonance

Forced Oscillation

Forced oscillation occurs when a system is subjected to an external force or input, causing it to oscillate at a frequency different from its natural frequency. The equation of motion for a forced harmonic oscillator is given by:

\[ m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F_0 \cos(\omega t) \]

where:

• \( m \) is the mass of the system,
• \( c \) is the damping coefficient,
• \( k \) is the spring constant,
• \( x \) is the displacement of the oscillator from its equilibrium position,
• \( F_0 \) is the amplitude of the external force,
• \( \omega \) is the angular frequency of the external force, and
• \( t \) is time.

Amplitude Resonance

Amplitude resonance occurs when the frequency of the external force matches the natural frequency of the system. At resonance, the amplitude of the oscillation becomes significantly larger. The resonant frequency (\( \omega_0 \)) is the natural frequency of the system when there is no external force.

Expression for Resonant Frequency

The resonant frequency (\( \omega_0 \)) can be determined by setting the coefficient of \( x \) to zero in the equation of motion. For a damped harmonic oscillator, the resonant frequency is given by:

\[ \omega_0 = \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2} \]

This expression takes into account both the stiffness of the system (\( k \)) and the damping (\( c \)). When the damping is small compared to the stiffness (\( c \ll \sqrt{4mk} \)), the expression simplifies to:

\[ \omega_0 \approx \sqrt{\frac{k}{m}} \]

This is the resonant frequency for a lightly damped system. For a heavily damped system, the resonant frequency is shifted lower due to the damping effect.

Summary

1. Forced oscillation involves an external force driving a system.
2. Amplitude resonance occurs when the external force frequency matches the natural frequency of the system.
3. The resonant frequency (\( \omega_0 \)) is the natural frequency of the system when subjected to external forces.
4. The expression for resonant frequency depends on the mass (\( m \)), spring constant (\( k \)), and damping coefficient (\( c \)) of the system.