How damping kills an oscillation
Pluck a string, push a swing, hit a bell. It vibrates, then stops. Damping is the reason it stops — and it changes the shape of the motion, not just how long it lasts.
How damping kills an oscillation — the interactive part
Displacement from rest
Seconds
- Displacement
= exp(-damping * x) * cos(frequency * x) - Envelope
= exp(-damping * x) - Envelope (lower)
= -exp(-damping * x)
This chart as text
- Displacement = exp(-damping * x) * cos(frequency * x)
- Envelope = exp(-damping * x)
- Envelope (lower) = -exp(-damping * x)
- Plotted for x from 0 to 20, y from -1.15 to 1.15.
- one full swing is marked at x = 3.927.
Amplitude halves after is 3.85 seconds. One swing takes is 3.93 seconds.
These curves are drawn from expressions written by AI and checked numerically — sampled across the whole x range and every control for undefined results and sudden jumps. That makes them well-behaved, not necessarily right, which is why each one is shown next to the maths that draws it.
Two things happening at once
The motion is a product, not a sum: a cosine that swings back and forth, multiplied by an exponential that shrinks. The cosine sets the rhythm and the exponential sets the ceiling. Neither one knows about the other, which is why changing the frequency does not change how fast the motion dies away.
Why the envelope is the useful part
Ask how long a bell rings and nobody means the individual vibrations, which are far too fast to count. They mean the envelope. That is why damping is usually quoted as a half-life or a decay time: it is the one number that says when the motion stops mattering, independent of how fast it was oscillating.
Check your understanding
You double the damping. What happens to the time between successive peaks?
Pick one to check yourself.