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== 2.1#h(0.6em)The Three Damping Regimes


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Objectives]


      - Write down the characteristic equation of a constant-coefficient second-order equation and interpret its roots.
- Distinguish underdamped, critically damped, and overdamped motion by the sign of the discriminant.
- Explain why the real part of a root controls decay and the imaginary part controls oscillation.
]


    A mass on a spring, slowed by friction and pushed by nothing, obeys


    #math.equation(block: true, alt: "x double prime plus 2 a x prime plus open parenthesis a squared plus b squared close parenthesis x equals 0 .")[$x^(″) + 2 a x^(′) + ( a^(2) + b^(2) ) x = 0 .$]


    The coefficients are written in this particular way because of what happens next. Guessing #math.equation(block: false, alt: "x equals e to the power r t")[$x = e^(r t)$] and substituting turns the differential equation into an ordinary algebraic one — the #emph[characteristic equation]


    #math.equation(block: true, alt: "r squared plus 2 a r plus open parenthesis a squared plus b squared close parenthesis equals 0 ,")[$r^(2) + 2 a r + ( a^(2) + b^(2) ) = 0 ,$]


    whose roots are #math.equation(block: false, alt: "r equals minus a ± i b")[$r = − a ± i b$]. The guess pays off because differentiating an exponential returns the same exponential times a constant, so the whole equation collapses to a polynomial. The roots then tell you everything: the general solution is built from #math.equation(block: false, alt: "e to the power minus a t cos b t")[$e^(− a t) cos b t$] and #math.equation(block: false, alt: "e to the power minus a t sin b t")[$e^(− a t) sin b t$], a decaying amplitude #math.equation(block: false, alt: "e to the power minus a t")[$e^(− a t)$] multiplying an oscillation of frequency #math.equation(block: false, alt: "b")[$b$]. The real part #math.equation(block: false, alt: "minus a")[$− a$] is the decay rate; the imaginary part #math.equation(block: false, alt: "b")[$b$] is the ringing.


    One particular solution, with #math.equation(block: false, alt: "x open parenthesis 0 close parenthesis equals 1")[$x ( 0 ) = 1$] and a specific starting velocity, is


    #math.equation(block: true, alt: "x open parenthesis t close parenthesis equals e to the power minus a t open parenthesis cos b t plus the fraction 1 over 2 sin b t close parenthesis ,")[$x ( t ) = e^(− a t) ( cos b t + frac(1, 2) sin b t ) ,$]


    and that is what the figure below plots. Both parameters are live.


    
      {"camera":{"fov":50,"position":\[7.5,0,12\],"projection":"perspective","target":\[7.5,0,0\]},"grid":{"axisColors":{"x":"\#ef4444","y":"\#22c55e","z":"\#3b82f6"},"divisions":16,"gridColor":"\#e5e7eb","showAxes":true,"showLabels":true,"size":16,"visible":true},"id":"7e1089f3-a291-4f99-a725-18a16e1687b2","metadata":{"alt\_text":"A red oscillating curve that starts at height 1 and wiggles along the horizontal time axis inside a shrinking envelope. The slider a controls the damping: at a = 0 the wiggles keep full size forever, and as a grows they die out faster until barely one swing survives. The slider b sets how many oscillations fit in the window.","created\_at":"2026-08-01T00:00:00.000Z","description":"The damped-oscillation family x(t) = exp(-a t)\*(cos(b t) + 0.5 sin(b t)) on t in \[0, 15\], with damping slider a in \[0, 2\] and frequency slider b in \[0.5, 5\]; sweeping a walks the response from sustained oscillation toward the overdamped look.","tags":\["diffeq","second-order","damping","oscillation","slider"\],"updated\_at":"2026-08-01T00:00:00.000Z"},"objects":\[{"id":"ebb987ec-51e0-4355-a179-17df597f754d","kind":"parameter","label":{"text":"a = damping","visible":true},"max":2,"min":0,"name":"a","step":0.05,"value":0.3,"visible":true},{"id":"6fd01723-07bc-42fb-840c-e34ea4005c08","kind":"parameter","label":{"text":"b = frequency","visible":true},"max":5,"min":0.5,"name":"b","step":0.05,"value":3,"visible":true},{"id":"8bfa01ed-4126-48b5-979d-58213fcf897b","kind":"parametric-curve","label":{"text":"x(t) = e^(-at)(cos bt + 0.5 sin bt)","visible":true},"samples":1500,"style":{"color":"\#ef4444","dash":\[\],"lineWidth":4},"tDomain":\[0,15\],"visible":true,"xExpr":"t","yExpr":"exp(-a\*t)\*(cos(b\*t) + 0.5\*sin(b\*t))","zExpr":"0"}\],"title":"Damped vibrations x(t) = e^(-at)(cos bt + 0.5 sin bt)","version":1}
      Damped free vibration #math.equation(block: false, alt: "x open parenthesis t close parenthesis equals e to the power minus a t open parenthesis cos b t plus the fraction 1 over 2 sin b t close parenthesis")[$x ( t ) = e^(− a t) ( cos b t + frac(1, 2) sin b t )$]. The slider #math.equation(block: false, alt: "a")[$a$] is the damping and the slider #math.equation(block: false, alt: "b")[$b$] the oscillation frequency; together they are the real and imaginary parts of the characteristic roots.
    
    
=== Explore


    + #strong[Observe.] Set #math.equation(block: false, alt: "a equals 0")[$a = 0$], #math.equation(block: false, alt: "b equals 3")[$b = 3$]. With no damping the motion repeats forever at constant amplitude. Count how many complete oscillations fit in the window, and predict what setting #math.equation(block: false, alt: "b equals 1.5")[$b = 1.5$] will do to that count before you try it.
+ #strong[Predict.] Keeping #math.equation(block: false, alt: "b equals 3")[$b = 3$], what changes as #math.equation(block: false, alt: "a")[$a$] grows from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "0.5")[$0.5$] — the #emph[height] of the wiggles, the #emph[spacing] of the wiggles, or both? Commit to an answer.
+ #strong[Verify.] Drag #math.equation(block: false, alt: "a")[$a$] to #math.equation(block: false, alt: "0.5")[$0.5$] and check. The curve stays inside an invisible envelope #math.equation(block: false, alt: "± C e to the power minus a t")[$± C e^(− a t)$]; the wiggle spacing is set by #math.equation(block: false, alt: "b")[$b$] alone. Confirm the two sliders act independently.
+ #strong[Observe.] Now push #math.equation(block: false, alt: "a")[$a$] to #math.equation(block: false, alt: "2")[$2$] with #math.equation(block: false, alt: "b")[$b$] still at #math.equation(block: false, alt: "3")[$3$]. How many times does the curve cross zero before it becomes indistinguishable from the axis? Compare with the count at #math.equation(block: false, alt: "a equals 0.3")[$a = 0.3$].
+ #strong[Predict, then verify.] Set #math.equation(block: false, alt: "a equals 2")[$a = 2$] and drag #math.equation(block: false, alt: "b")[$b$] down to #math.equation(block: false, alt: "0.5")[$0.5$]. The curve barely completes a single swing. What would #math.equation(block: false, alt: "b equals 0")[$b = 0$] mean for the characteristic roots, and what shape would the motion have?


    
=== Connect


    Written in the standard form #math.equation(block: false, alt: "x double prime plus p x prime plus q x equals 0")[$x^(″) + p x^(′) + q x = 0$], the roots are #math.equation(block: false, alt: "r equals open parenthesis minus p ± the square root of p squared minus 4 q close parenthesis / 2")[$r = ( − p ± sqrt(p^(2) − 4 q) ) / 2$], and the sign of the discriminant #math.equation(block: false, alt: "p squared minus 4 q")[$p^(2) − 4 q$] splits behavior into three cases. When #math.equation(block: false, alt: "p squared less than 4 q")[$p^(2) < 4 q$] the roots are complex, #math.equation(block: false, alt: "minus a ± i b")[$− a ± i b$] with #math.equation(block: false, alt: "b greater than 0")[$b > 0$]: #emph[underdamped] motion, oscillating inside a decaying envelope, everything the figure shows with #math.equation(block: false, alt: "b")[$b$] positive. When #math.equation(block: false, alt: "p squared greater than 4 q")[$p^(2) > 4 q$] both roots are real and negative: #emph[overdamped] motion, a sum of two decaying exponentials that crosses zero at most once and never oscillates. The boundary case #math.equation(block: false, alt: "p squared equals 4 q")[$p^(2) = 4 q$] gives a repeated real root and is called #emph[critically damped]; its solutions are #math.equation(block: false, alt: "open parenthesis c sub 1 plus c sub 2 t close parenthesis e to the power minus a t")[$( c_(1) + c_(2) t ) e^(− a t)$], and among all damping choices it returns the mass to rest fastest without overshoot.


    That last fact is engineering, not trivia. A screen door closer, a car's shock absorber, and an instrument needle are all deliberately tuned near critical damping — enough resistance to kill the ringing, not so much that the return crawls. In the figure, dragging #math.equation(block: false, alt: "a")[$a$] upward with #math.equation(block: false, alt: "b")[$b$] fixed walks you toward that tuning: the oscillations you have to eliminate are visible right up to the point where the last one disappears.


    #examplebox("Example 1")[Reading the roots][
Classify the motion of #math.equation(block: false, alt: "x double prime plus 4 x prime plus 13 x equals 0")[$x^(″) + 4 x^(′) + 13 x = 0$] and write the general solution. Then state which slider settings in the figure correspond to it.


      The characteristic equation is #math.equation(block: false, alt: "r squared plus 4 r plus 13 equals 0")[$r^(2) + 4 r + 13 = 0$], with discriminant #math.equation(block: false, alt: "16 minus 52 equals minus 36 less than 0")[$16 − 52 = − 36 < 0$], so the roots are complex: #math.equation(block: false, alt: "r equals open parenthesis minus 4 ± 6 i close parenthesis / 2 equals minus 2 ± 3 i")[$r = ( − 4 ± 6 i ) / 2 = − 2 ± 3 i$]. The motion is underdamped, and the general solution is #math.equation(block: false, alt: "x equals e to the power minus 2 t open parenthesis c sub 1 cos 3 t plus c sub 2 sin 3 t close parenthesis")[$x = e^(− 2 t) ( c_(1) cos 3 t + c_(2) sin 3 t )$]. That is the figure with #math.equation(block: false, alt: "a equals 2")[$a = 2$] and #math.equation(block: false, alt: "b equals 3")[$b = 3$] — visibly one or two swings before the amplitude collapses. For contrast, #math.equation(block: false, alt: "x double prime plus 8 x prime plus 13 x equals 0")[$x^(″) + 8 x^(′) + 13 x = 0$] has discriminant #math.equation(block: false, alt: "64 minus 52 equals 12 greater than 0")[$64 − 52 = 12 > 0$] and real roots near #math.equation(block: false, alt: "minus 1.35")[$− 1.35$] and #math.equation(block: false, alt: "minus 6.65")[$− 6.65$]: overdamped, no oscillation at all, a case the figure cannot draw because it has no complex pair to plot.
]