Section 8.3: Damped and Forced Oscillations

Real oscillating systems experience damping due to resistive forces such as friction or air resistance. In addition, a system can be driven by an external periodic force, resulting in forced oscillations.

\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + k x = F_0 \cos(\omega t) \] where:
\( m \) = mass (kg)
\( b \) = damping coefficient (kg/s)
\( k \) = spring constant (N/m)
\( F_0 \) = amplitude of driving force (N)
\( \omega \) = angular frequency of driving force (rad/s)
\( x \) = displacement (m)

Key points:

Underdamped: Oscillations decay gradually.
Critically damped: Returns to equilibrium as fast as possible without oscillating.
Overdamped: Returns to equilibrium slowly without oscillating.
Resonance: Maximum amplitude occurs when driving frequency matches natural frequency.

Example: Damped Oscillator

A mass-spring system has \( m = 0.5\,\text{kg} \), \( k = 200\,\text{N/m} \), and damping coefficient \( b = 2\,\text{kg/s} \). Determine the type of damping.

Damping ratio: \( \zeta = \frac{b}{2\sqrt{mk}} = \frac{2}{2\sqrt{0.5 \cdot 200}} = \frac{2}{20} = 0.1 \)
Since \( \zeta < 1 \), the system is underdamped.

Practice Problems

A mass-spring system with \( m = 1\,\text{kg} \) and \( k = 100\,\text{N/m} \) has a damping coefficient \( b = 5\,\text{kg/s} \). Determine if it is underdamped, overdamped, or critically damped.
A forced oscillator with natural frequency 10 Hz is driven at 10 Hz. Explain what happens to the amplitude.
A 0.3 kg mass attached to a spring (\( k = 50\,\text{N/m} \)) is lightly damped. Calculate the damping ratio if \( b = 1\,\text{kg/s} \).
Explain the difference between free damped oscillations and forced oscillations.
A mass-spring system is underdamped with a damping ratio of 0.2. Sketch qualitatively how the amplitude changes over time.