Section 6.6: Damped Oscillations
In real systems, oscillations gradually decrease in amplitude due to resistive forces like friction or air resistance. This phenomenon is called damping. A damped oscillator loses mechanical energy over time as it is converted into thermal or other forms of energy.
Damped Motion Equation:
\[
x(t) = A e^{-\beta t} \cos(\omega' t + \phi)
\]
- \( \beta = \tfrac{b}{2m} \) is the damping coefficient
- \( \omega' = \sqrt{\omega_0^2 - \beta^2} \) is the damped angular frequency
- \( \omega_0 = \sqrt{\tfrac{k}{m}} \) is the natural angular frequency
Types of Damping:
- Underdamped: Oscillations with gradually decreasing amplitude.
- Critically damped: Returns to equilibrium as fast as possible without oscillating.
- Overdamped: Returns slowly to equilibrium without oscillations.
Example 1
A 0.5 kg mass is attached to a spring (\(k = 100 \, \text{N/m}\)) with damping constant \(b = 1.0 \, \text{kg/s}\). Find the damped angular frequency.
\(\omega_0 = \sqrt{\tfrac{k}{m}} = \sqrt{\tfrac{100}{0.5}} = \sqrt{200} \approx 14.14 \, \text{rad/s}\)
\(\beta = \tfrac{b}{2m} = \tfrac{1}{1.0} = 1.0 \, \text{s}^{-1}\)
\(\omega' = \sqrt{\omega_0^2 - \beta^2} = \sqrt{200 - 1} \approx 14.1 \, \text{rad/s}\)
Practice Problems
- Explain the physical meaning of the damping coefficient \( \beta \).
- A damped oscillator has \(m=2\) kg, \(k=50\) N/m, \(b=0.8\) kg/s. Find \( \omega' \).
- What distinguishes underdamping from overdamping in terms of system response?
- Show that energy in a damped oscillator decreases exponentially over time.
- Why are critically damped systems important in engineering (e.g., car suspension)?