damped free vibration explained
The last lesson kept an ideal vibration alive forever, because its energy never left. Real motion does not work that way. A plucked guitar string fades, a struck bell goes quiet, and a car that clears a bump rocks a few times and then stops. Energy is leaking out of the motion. Friction, air drag, and the losses inside the material all sit under one name, damping. Picture a damper on the system, a viscous-friction device. It pushes back against the motion and, on every cycle, turns a little of the energy into heat. That is why each round trip reaches a bit less far, until the mass finally rests. How heavy that damping is can be written as one number, the damping ratio ζ. When the damping grows large enough, the oscillation never starts. The mass does not cross to the other side. It simply sags back to where it began. Heat that has left the motion does not come back as motion.
The upper and lower systems share the same spring and the same mass. Only one thing differs, and the lower one carries a damper. Release both from the same width. The upper mass, with no damper, keeps that width, while the lower mass, with a damper, reaches a little less on every trip. Inside the damper a piston drives through viscous fluid and turns kinetic energy into heat. The force always faces against the velocity, so it can take energy out and cannot put energy in. The ½kx²+½mv² = constant of the last lesson therefore stops holding, and the sum leaks. A shrinking reach is the visible sign that energy has left. Energy follows the square of the width (E∝A²), so once the width is cut in half only 1 part in 4 of the stored energy remains, a quarter. The number underneath reports what is left.
Light damping still leaves an oscillation, but the motion is trapped inside an exponential envelope that keeps narrowing. The displacement is written x(t) = A·e−ζωₙt·cos(ωd t). The orange dashed curve is the envelope e−ζωₙt, and the blue curve is the displacement moving inside it. The width does not lose a fixed length each cycle. It loses a fixed fraction. Follow the gold peaks and each one is the same multiple r of the peak before it. If the first peak falls from 10 to 8, the next falls from 8 to 6.4, and the one after that from 6.4 to 5.12, every time times 0.8. The logarithm of that steady ratio is the logarithmic decrement. Two peaks are enough to read how damped the system is, and raising ζ makes the envelope steeper. A steeper envelope leaves less width after the same time, and the ratio is the cleaner reading.
The number that measures damping is the damping ratio ζ = c(2√(km)). Here c is the viscous damping coefficient, how strong the damper is, and k and m are the stiffness and the mass already in hand. ζ ties the three into one dimensionless dial, so a large system and a small one die away in the same shape when ζ is the same. Raising ζ does two things together. The decay quickens, and the wobble itself slows a little. With damping the system does not swing at its natural frequency ωₙ. It swings at the lower damped frequency ωd = ωₙ√(1−ζ²). Under light damping √(1−ζ²) is nearly 1, so ωd ≈ ωₙ, but as ζ approaches 1 the ωd falls toward 0. That is the moment just before the wobble disappears. One dial sets the shape of the decay no matter how large the system is.
The moment ζ passes 1, the kind of motion changes. Give the mass a displacement and let it go from rest, and the return curve splits cleanly. Underdamped (ζ<1), it crosses 0, travels to the other side, and comes back oscillating. Critically damped (ζ=1), it never crosses 0 even once and returns home as fast as a return can, with no overshoot (the gold dashed curve). Overdamped (ζ>1) also refuses to cross, but the motion is so sticky that it creeps home. Slide ζ from 0 to 2 and the live curve moves among these three shapes. Critical damping is special because ζ=1 is exactly the edge that stops soonest without overshooting. It suits a design that must settle quickly and must not ring. The fastest return that never rings is what critical means, and the slider makes that edge visible as the gold curve takes over.
These three regimes are design choices an engineer makes on purpose, not a piece of mathematics left on the page. A car shock absorber is set a little underdamped (ζ≈0.3). After a bump it swings softly once or twice and settles, because too stiff a ride is harsh and too soft a ride keeps wallowing. A door closer sits near critical (ζ≈0.9). It must not slam shut and it must not bounce open. It has to close smoothly and simply stop. An instrument needle, and some hydraulic gear, is overdamped (ζ≈1.6). Slow is acceptable. What matters is that it comes to rest on one value with no wobble, so the mark can be read exactly. The buttons swap the three machines, and the same ζ story answers differently in each one.