Section 3.3: Forced Oscillations
Forced oscillations occur when an external periodic force drives the system. The amplitude depends on the driving frequency and can lead to resonance if the frequency matches the system's natural frequency.
Key Concepts:
- Equation of motion: \( m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t) \)
- Steady-state amplitude: \( A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}} \)
- Resonance: Maximum amplitude occurs at \( \omega \approx \omega_0 \)
- Phase difference between driving force and displacement varies with frequency
Example 1
A 1 kg mass-spring system with k = 100 N/m is driven by a force \( F = 10 \cos(5t) \) N. The damping is negligible. Find the amplitude of oscillation.
Natural frequency: \( \omega_0 = \sqrt{k/m} = \sqrt{100/1} = 10 \text{ rad/s} \)
Amplitude: \( A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2}} = \frac{10/1}{\sqrt{(10^2-5^2)^2}} = \frac{10}{\sqrt{(100-25)^2}} = \frac{10}{75} \approx 0.133 \text{ m} \)
Practice Problems
- A damped oscillator with m = 2 kg, k = 50 N/m, b = 0.5 kg/s is driven by \( F = 5\cos(3t) \) N. Find steady-state amplitude.
- Find the resonance frequency of a mass-spring system with k = 200 N/m and m = 0.5 kg.
- For a driven system with ω = 8 rad/s and ω₀ = 10 rad/s, calculate phase difference.
- A system with negligible damping is driven at ω = ω₀. Determine amplitude if F₀ = 15 N and m = 1 kg.
- Sketch amplitude vs frequency curve for a lightly damped system.