Damped and Driven Oscillations
The same equation governs a mass on a spring and a series circuit. Damping decides how the free motion dies away, and driving near the natural frequency produces resonance.
What a learner can do afterwards
- Classify motion as underdamped, critically damped or overdamped from the roots
- Identify the transient and steady-state parts of a driven solution
- Explain resonance in terms of the driving frequency and the natural frequency
1 · Read
The same equation governs a mass on a spring and a series circuit, so learn it once and spend it twice. Without damping or driving, the mass traces a pure sine wave with a fixed natural frequency set by stiffness over mass. Real oscillators lose energy to friction, which wraps the swing in a decaying exponential. The characteristic roots classify the case: a complex pair means underdamped oscillation, a repeated real root means critically damped, and distinct real roots mean overdamped.
Read two root reports. Roots minus 2 and minus 3 are real, distinct, and negative, so the motion is overdamped: a slow return with no oscillation, like a heavy door creeping shut. Roots minus 1 plus 2 i and minus 1 minus 2 i form a complex pair with negative real part, so the motion is underdamped: it swings with slowly shrinking height. Car suspensions aim near critical, which returns to rest fastest without overshooting.
Driving adds a persistent forced wave on top. The full driven answer always splits into a transient that dies with the damping plus a steady state that persists at the driving frequency. After a while only the steady hum remains, beating at the rhythm of the driver rather than the natural song. Combining a sine and a cosine into one wave gives that steady part its amplitude and phase.
Push a swing at its own rhythm and tiny pushes build a huge arc: that is resonance. Near the natural frequency the steady amplitude spikes, limited only by damping. Engineers design to stay clear of resonance, or to exploit it, but never to ignore it.
Roots name the damping, the transient fades, the steady state follows the driver, and matched rhythms resonate.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.