Section 3.4: Resonance
Resonance occurs when a system is driven at its natural frequency, resulting in maximum amplitude. Understanding resonance is crucial in engineering, acoustics, and structural design.
Key Concepts:
- Resonance frequency: \( \omega_r \approx \omega_0 = \sqrt{k/m} \)
- Maximum amplitude occurs at resonance
- Phase difference at resonance is 90° between driving force and displacement
- Excessive resonance can lead to structural failure (e.g., bridges, buildings)
Example 1
A mass-spring system with m = 0.5 kg and k = 200 N/m is driven by a periodic force. Determine the resonance frequency.
Natural frequency: \( \omega_0 = \sqrt{k/m} = \sqrt{200/0.5} = \sqrt{400} = 20 \text{ rad/s} \)
Resonance occurs when driving frequency \( \omega = 20 \text{ rad/s} \)
Practice Problems
- A lightly damped oscillator has m = 1 kg and k = 150 N/m. Find the resonance frequency.
- A forced oscillator with damping coefficient b = 0.1 kg/s is driven at resonance. Sketch amplitude vs frequency.
- A bridge resonates at 2 Hz. If its natural frequency shifts to 2.5 Hz after retrofitting, explain the effect on resonance.
- Compute maximum amplitude for a system with F₀ = 5 N, m = 0.2 kg, at resonance.
- Explain why phase difference between driving force and displacement is 90° at resonance.