Section 3.2: Damped Oscillations
Damped oscillations occur when resistive forces (like friction or air resistance) reduce the amplitude over time. The motion is no longer purely sinusoidal but decays exponentially.
Key Equations:
- Displacement: \( x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi) \)
- Damping coefficient: \( \gamma = \frac{b}{2m} \)
- Damped angular frequency: \( \omega_d = \sqrt{\omega_0^2 - \gamma^2} \)
- Natural frequency: \( \omega_0 = \sqrt{k/m} \)
- Types of damping: Underdamped, critically damped, overdamped
Example 1
A 0.5 kg mass on a spring with k = 200 N/m experiences a damping force with b = 2 kg/s. Find the damping coefficient and damped frequency.
\( \gamma = b/(2m) = 2/(2*0.5) = 2 \text{ s}^{-1} \)
Natural frequency: \( \omega_0 = \sqrt{k/m} = \sqrt{200/0.5} = 20 \text{ rad/s} \)
Damped frequency: \( \omega_d = \sqrt{20^2 - 2^2} = \sqrt{400-4} = \sqrt{396} \approx 19.9 \text{ rad/s} \)
Practice Problems
- A mass-spring system has m = 1 kg, k = 100 N/m, b = 5 kg/s. Find γ and ωd.
- A damped oscillator has amplitude 0.2 m and γ = 0.5 s⁻¹. Find amplitude after 4 s.
- Determine whether a system with m = 0.5 kg, k = 50 N/m, b = 10 kg/s is underdamped, critically damped, or overdamped.
- A mass-spring system is displaced 0.1 m and released. Damping reduces amplitude to 0.05 m in 3 s. Find γ.
- Compute the damped period for a system with ω0 = 15 rad/s and γ = 1 s⁻¹.