Resonance occurs when a system is driven by a repeated force at or near its natural frequency. At this frequency, energy is transferred very efficiently, so the amplitude of the motion can grow much larger than usual. This idea matters in music, engineering, electronics, and everyday vibrations.
It explains why a swing can go higher with well-timed pushes and why some structures must be designed to avoid dangerous vibration buildup.
Every vibrating system has one or more natural frequencies, set by its mass, stiffness, shape, and boundary conditions. When the driving frequency matches a natural frequency, each push adds energy in step with the motion, producing a peak on an amplitude versus driving frequency graph. Damping, such as friction or air resistance, removes energy and lowers the peak amplitude.
Engineers use resonance to amplify useful signals in instruments and circuits, but they also control it in bridges, buildings, and machines to prevent damage.
Key Facts
- Resonance occurs when f_drive = f0.
- Natural frequency for a simple mass spring system: f0 = (1 / 2π) sqrt(k / m).
- Period and frequency are related by f = 1 / T.
- At resonance, amplitude reaches a maximum because energy is added in phase with the motion.
- Damping reduces the resonance peak and spreads it over a wider range of frequencies.
- A higher quality factor means a sharper resonance peak: Q = f0 / Δf.
Vocabulary
- Resonance
- Resonance is the large increase in amplitude that occurs when a system is driven near its natural frequency.
- Natural frequency
- Natural frequency is the frequency at which a system tends to vibrate when disturbed and then left to oscillate freely.
- Driving frequency
- Driving frequency is the frequency of an external force applied repeatedly to a system.
- Amplitude
- Amplitude is the maximum displacement of an oscillating object from its equilibrium position.
- Damping
- Damping is the loss of mechanical energy from an oscillating system due to effects such as friction, air resistance, or internal resistance.
Common Mistakes to Avoid
- Saying resonance happens at the largest driving force, not the matching frequency. Resonance depends mainly on timing, so even a modest force can create a large amplitude if f_drive is near f0.
- Confusing natural frequency with amplitude. Natural frequency is a rate of vibration measured in hertz, while amplitude is the size of the motion.
- Ignoring damping when predicting resonance size. Real systems lose energy, so damping can greatly reduce the peak amplitude and prevent unlimited growth.
- Assuming resonance is always harmful. Resonance can be destructive in bridges and machines, but it is useful in musical instruments, radios, and filters.
Practice Questions
- 1 A mass spring system has spring constant k = 200 N/m and mass m = 2.0 kg. Calculate its natural frequency using f0 = (1 / 2π) sqrt(k / m).
- 2 A child on a swing has a natural period of 2.5 s. What driving frequency should a person use to push the swing for maximum amplitude?
- 3 A bridge begins vibrating strongly when wind gusts arrive at a steady frequency close to one of the bridge's natural frequencies. Explain why the amplitude grows and describe one design change that could reduce the risk.