5.3 Forced Oscillations and Resonance

Forced oscillations occur when external forces act on an oscillating system. These forces can be periodic or non-periodic, affecting the system's behavior. Understanding forced oscillations is crucial for analyzing real-world systems like car suspensions and AC circuits.

Resonance happens when the forcing frequency matches the system's natural frequency, causing maximum amplitude. This phenomenon is important in various fields, from music to engineering. Resonance can lead to dramatic effects, like shattering wine glasses or enhancing radio reception.

Forced Oscillations

Forcing Functions and Harmonic Forcing

• A forcing function $f(t)$ is an external force applied to an oscillating system
• Forcing functions can be periodic or non-periodic
• Harmonic forcing involves a sinusoidal forcing function of the form $f(t) = F_0 \cos(\omega t)$
  • $F_0$ represents the amplitude of the forcing function
  • $\omega$ represents the angular frequency of the forcing function
• Example: A child pushing a swing at regular intervals applies a periodic forcing function to the swing system

Damped Forced Oscillations and Amplitude Response

• Damped forced oscillations occur when a damped oscillating system is subjected to an external forcing function
• The equation of motion for a damped forced oscillator is $m\ddot{x} + c\dot{x} + kx = f(t)$
  • $m$ represents the mass of the oscillator
  • $c$ represents the damping coefficient
  • $k$ represents the spring constant
• The steady-state solution for a damped forced oscillator with harmonic forcing is $x(t) = A \cos(\omega t - \phi)$
  • $A$ represents the amplitude of the steady-state oscillation
  • $\phi$ represents the phase shift between the forcing function and the oscillator's response
• The amplitude response $A$ depends on the frequency of the forcing function and the natural frequency of the oscillator
• Example: A car's suspension system can be modeled as a damped forced oscillator, with the road's irregularities acting as the forcing function

Phase Shift in Forced Oscillations

• The phase shift $\phi$ represents the lag between the forcing function and the oscillator's response
• The phase shift depends on the frequency of the forcing function, the natural frequency of the oscillator, and the damping ratio
• For underdamped systems, the phase shift varies from 0 to $\pi$ as the forcing frequency increases
• For overdamped systems, the phase shift varies from 0 to $\pi/2$ as the forcing frequency increases
• Example: In an AC circuit with a resistor and an inductor, the current lags behind the voltage, representing a phase shift between the forcing function (voltage) and the response (current)

Resonance

Resonance and Resonant Frequency

• Resonance occurs when the frequency of the forcing function matches or is close to the natural frequency of the oscillator
• The resonant frequency is the frequency at which resonance occurs, leading to maximum amplitude of oscillation
• At resonance, the amplitude of the steady-state oscillation is maximum, and the phase shift is $\pi/2$ for underdamped systems
• Example: A singer hitting a high note can cause a wine glass to shatter if the note's frequency matches the glass's natural frequency, leading to resonance and large amplitude oscillations

Quality Factor and Bandwidth

• The quality factor $Q$ is a measure of the sharpness of the resonance peak
  • Higher $Q$ values indicate sharper resonance peaks and less damping
  • Lower $Q$ values indicate broader resonance peaks and more damping
• The quality factor is defined as $Q = \frac{\omega_0}{\Delta \omega}$, where $\omega_0$ is the resonant frequency and $\Delta \omega$ is the bandwidth
• The bandwidth is the range of frequencies over which the amplitude of oscillation is at least $1/\sqrt{2}$ times the maximum amplitude
• Example: In a radio receiver circuit, a high-quality factor allows for better selectivity of the desired radio station, while a low-quality factor may lead to interference from adjacent stations

Beating Phenomenon

• Beating occurs when two harmonic forcing functions with slightly different frequencies are applied to an oscillator
• The resulting oscillation exhibits a periodic variation in amplitude, known as beats
• The beat frequency is the difference between the two forcing frequencies
• The amplitude of the beats is determined by the amplitudes of the individual forcing functions
• Example: When two tuning forks with slightly different frequencies are struck simultaneously, a beating sound can be heard, with the volume of the sound varying periodically at the beat frequency