Think of a pendulum or a spring that keeps moving back and forth. The motion repeats over time – that’s an oscillation 🎢.
Every system has a “preferred” speed of oscillation, called the natural frequency:
\$\omega_0 = \sqrt{\frac{k}{m}}\$
Where k is the stiffness (spring constant) and m is the mass. Imagine a guitar string – pluck it and it vibrates at its natural frequency 🎸.
Real systems lose energy (to friction, air resistance, etc.). Damping forces oppose motion:
\$c\dot{x}\$
The equation of motion becomes:
\$m\ddot{x} + c\dot{x} + kx = 0\$
Damping reduces amplitude over time, like a swing that eventually comes to rest if you stop pushing it 🚲.
If we apply an external periodic force, the system responds:
\$m\ddot{x} + c\dot{x} + kx = F_0\cos(\omega t)\$
Here, F₀ is the force amplitude and ω is the driving frequency. Think of pushing a child on a swing at a steady rhythm – if you push at just the right time, the swing goes higher! 🎠.
Resonance happens when the driving frequency matches the natural frequency:
\$\omega = \omega_0\$
At this point, the amplitude reaches a maximum (limited by damping). It’s like a tuning fork that rings louder when struck at its own pitch 🎶.
The steady‑state amplitude is:
\$A(\omega)=\frac{F_0}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}}\$
Notice how the denominator is smallest when ω ≈ ω₀, giving the largest A.
Q tells us how “sharp” the resonance peak is:
\$Q = \frac{m\omega_0}{c}\$
High Q = low damping → narrow, high peak. Low Q = high damping → broad, low peak. Think of a drum – a thin drumhead (high Q) rings for a long time, while a thick one (low Q) stops quickly.
| ω (rad/s) | Amplitude A (m) |
|---|---|
| 0.5 ω₀ | Low |
| ω₀ | Maximum (Resonance) |
| 1.5 ω₀ | Low |
The curve is symmetric around ω₀ and its width depends on damping.