An underdamped oscillator keeps oscillating while it fades. The oscillation has its own frequency, the pseudo-frequency ω_d, which is lower than the natural frequency ω₀, and it sits inside a pair of exponential curves that shrink at the rate c/(2m). Successive peaks are one pseudo-period apart, and each is smaller than the last by one fixed factor.
Core ideas
Three ideas: the new frequency, the curve that bounds the motion, and the fixed factor between peaks.
The roots are α ± ω_d·i with α = −c/(2m), and the quadratic formula gives ω_d = √(4mk − c²)/(2m). Squared, ω_d² = k/m − c²/(4m²) = ω₀² − (c/(2m))². The damper subtracts a square from ω₀², so ω_d < ω₀ for every c > 0: damping slows the oscillation as well as shrinking it.
Write the solution as e^(αt)·(C₁cos(ω_d t) + C₂sin(ω_d t)). The bracket is a swing of constant amplitude A, so the whole is at most A·e^(αt) in size. The curves ±A·e^(αt) are the envelope, with decay rate c/(2m). The lab draws them with the motion.
The bracket repeats every pseudo-period T_d = 2π/ω_d, and the decay multiplies by e^(α·T_d) over that time. So every peak is e^(−c·T_d/(2m)) times the one before it, whatever the start. The factor is irrational and the lab prints it with ≈.
The material
The roots again, the pseudo-frequency, the envelope, the fixed ratio of peaks, and three presets with a different damping each.
The previous lesson ended with the underdamped roots. This lesson reads them as numbers. For m·r² + c·r + k = 0 with c² − 4mk < 0 the roots are a complex pair, and writing out the real and imaginary parts gives the three quantities that describe the motion.
In the underdamped solution x = e^(αt)·(C₁cos(ω_d t) + C₂sin(ω_d t)) the number ω_d is the pseudo-frequency and T_d = 2π/ω_d the pseudo-period. The motion is not periodic, since it keeps shrinking, which is why the prefix.
The curves ±A·e^(αt) are the envelope; the rate c/(2m) = −α says how fast it falls.
In an underdamped motion, the value of x one pseudo-period later is e^(α·T_d) times its value now, at every t.
The bracket C₁cos(ω_d t) + C₂sin(ω_d t) has period T_d = 2π/ω_d, so replacing t by t + T_d leaves it unchanged.
The factor e^(αt) becomes e^(αt)·e^(α·T_d). So x(t + T_d) = e^(α·T_d)·x(t) for every t, and differentiating both sides gives the same for the rate: x′(t + T_d) = e^(α·T_d)·x′(t). If x has a peak at t₁, where x′(t₁) = 0, then x′(t₁ + T_d) = 0 as well, so the next peak is exactly one pseudo-period later and is e^(α·T_d) = e^(−c·T_d/(2m)) times the first. The peaks of x are not where the bracket peaks: each comes a little earlier, where the bracket's rate balances the decay. They are spaced by T_d all the same.
□
With m = 1: ω₀² = 5, c/(2m) = 1, and ω_d² = 5 − 1 = 4, so ω_d = 2 and T_d = 2π/2 = π. The envelope falls like e^(−t), and each peak is e^(−π) times the one before, which the lab prints as about 0.0432139 and rounded.
That is a very fast fade, and the lab marks it rounded. Compare the undamped period of this spring, 2π/√5, about 2.80993, with the pseudo-period π, about 3.14159. The damped oscillation is slower.
Take m = 1, c = 1/2, k = 4. Now c/(2m) = 1/4, and ω_d² = 4 − 1/16 = 63/16. The pseudo-frequency ω_d = √63/4 is a surd, barely below ω₀ = 2.
A light damper changes the frequency by very little and the peaks by a modest factor: the lab prints the ratio as about 0.453116, rounded. Damping shows first in the envelope and only second in the frequency.
Take m = 2, c = 2, k = 5. The rate is c/(2m) = 1/2, and ω_d² = 5/2 − 1/4 = 9/4, so ω_d = 3/2 exactly. The pseudo-period is 4π/3.
Compared with the first preset, doubling the mass halves the envelope rate for the same damper (c/(2m) has m in the denominator), so the heavy mass fades more slowly: the lab's ratio of successive peaks is about 0.123145, rounded, against 0.0432139.
The same three quantities are all that describe a ringing bell, a plucked string heard as a decaying tone, or a car's suspension after a bump. The frequency tells you the note, the envelope rate tells you how long it rings, and the ratio of peaks is how much quieter it is each time round.
Interactive
Free, damped and forced motion, with every exact quantity
Method
Five moves. The first is the check that the case is underdamped at all.
If it is not, there is no oscillation and there is no pseudo-frequency; go back to the previous lesson's classification.
A fraction. For m = 1, c = 2 it is 1; for m = 2, c = 2 it is 1/2.
ω_d² = k/m − (c/(2m))². This is the formula the lab prints. Take a root only if you need the frequency itself, and keep it as a surd if the root is not rational.
T_d = 2π/ω_d, a symbol with π; round it only at the end, with ≈.
e^(−(c/(2m))·T_d), rounded and marked ≈. For the first preset the exponent is −1·π, and the ratio is e^(−π).
The first preset: m = 1, c = 2, k = 5, started at x(0) = 2 with x′(0) = 0.
The pseudo-frequency 2 is below the natural frequency √5, as it must be, since ω_d² = 5 − 1. The constants for the start (2, 0) are C₁ = 2 and C₂ = 1, from x′(0) = −C₁ + 2·C₂ = 0; the lab does not fit them, but the envelope it draws is built from them.
You should be able to compute ω_d² = k/m − c²/(4m²) exactly, the envelope rate c/(2m), and the pseudo-period 2π/ω_d, give the ratio of successive peaks as a rounded number marked ≈, and say why ω_d is below ω₀.
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