The Damped Harmonic Oscillator

Derivation

The Damped Harmonic Oscillator

D-059 Home PU-103 Threads force · energy Depends on Simple Harmonic Motion from a Linear Restoring Force, second-order-linear-odes

Statement

For a mass \(m\) subject to a linear restoring force \(-kx\) and a linear viscous drag \(-c\dot{x}\), the equation of motion \(m\ddot{x}+c\dot{x}+kx=0\) is a homogeneous second-order linear ODE. The exponential trial solution \(x=e^{\lambda t}\) reduces it to the characteristic equation \(m\lambda^{2}+c\lambda+k=0\), with roots \(\lambda_{\pm}=-\gamma\pm\sqrt{\gamma^{2}-\omega_0^{2}}\) where \(\gamma\equiv c/2m\) and \(\omega_0^{2}\equiv k/m\). The sign of the discriminant \(\Delta=c^{2}-4mk\) (equivalently \(\gamma^{2}-\omega_0^{2}\)) partitions the motion into exactly three regimes: under-damped (\(\Delta<0\), oscillatory decay), critically damped (\(\Delta=0\), fastest non-oscillatory return), and over-damped (\(\Delta>0\), slow non-oscillatory return), with the boundary at the critical damping \(c_{c}=2\sqrt{mk}=2m\omega_0\).

Why it matters

Every real oscillator loses energy. The undamped result \(x=A\cos(\omega_0 t+\phi)\) is an idealisation; adding the simplest physically admissible loss — one linear in velocity — is the minimal correction that keeps the equation linear and closed-form solvable, and it already reproduces the qualitative ringdown of pendulums in air, \(RLC\) circuits, and galvanometer needles.

The three-way classification by a single dimensionless ratio, the damping ratio \(\zeta=c/c_c\), is the template for all of linear systems engineering. Shock absorbers, control-loop stability margins, and seismometer response are all tuned by pushing \(\zeta\) toward the critical value, where a system returns to rest as fast as possible without overshoot. Reading a qualitative threshold off a discriminant, rather than off a plotted curve, is the transferable skill.

Assumptions

The drag is linear in velocity, \(F_{\text{drag}}=-c\dot{x}\) with constant \(c>0\).If drag is quadratic (\(\propto\dot{x}^2\), high-speed motion in air) or dry Coulomb friction (\(\propto\operatorname{sgn}\dot{x}\)), the equation becomes nonlinear or piecewise, the exponential ansatz fails, and the discriminant classification is lost.

The restoring force is Hookean, \(-kx\), with constant \(k>0\).If \(k\) varies with amplitude (anharmonic potential) the frequency becomes amplitude-dependent, superposition of the two modes fails, and normal-mode analysis no longer applies.

The coefficients \(m,c,k\) are constant in time (linear, time-invariant system).If they drift, the coefficients become time-dependent, the characteristic equation loses meaning, and one needs Floquet, Mathieu, or WKB methods — parametric resonance can pump energy in despite \(c>0\).

One degree of freedom, no external drive (homogeneous transient only).An applied force \(F(t)\) adds a particular solution; classifying by \(\Delta\) alone then describes only the transient, not the steady state.

Mass strictly positive and dissipation non-negative (\(m>0,\ c\ge0\)).Negative effective damping (an active/feedback element) sends \(\operatorname{Re}\lambda\) positive, turning the "decaying" solution into exponential growth — instability, not settling.

Derivation

\[ m\ddot{x} = -kx - c\dot{x} \]

Newton's second law with the two body forces: Hookean restoring force and linear drag opposing velocity. A

\[ m\ddot{x} + c\dot{x} + kx = 0 \]

Collect all terms on one side; homogeneous because there is no external drive. This is the canonical second-order linear ODE. A

\[ \ddot{x} + 2\gamma\,\dot{x} + \omega_0^{2}\,x = 0, \qquad \gamma \equiv \frac{c}{2m}, \quad \omega_0^{2} \equiv \frac{k}{m} \]

Divide by \(m\neq0\) and name the two combinations that carry physical meaning: the decay rate \(\gamma\) and the undamped natural frequency \(\omega_0\). The factor \(2\) is chosen so later formulae come out clean. A

\[ x(t) = e^{\lambda t} \;\Rightarrow\; \dot{x}=\lambda e^{\lambda t}, \;\; \ddot{x}=\lambda^{2} e^{\lambda t} \]

For a constant-coefficient linear ODE, exponentials are the eigenfunctions of \(d/dt\), which acts on \(e^{\lambda t}\) by multiplication; this ansatz spans the solution space (prior result: second-order-linear-odes). B

\[ \left(\lambda^{2} + 2\gamma\lambda + \omega_0^{2}\right) e^{\lambda t} = 0 \]

Substitute the ansatz; each time-derivative brings down a factor \(\lambda\), and \(e^{\lambda t}\) is common to every term. A

\[ \lambda^{2} + 2\gamma\lambda + \omega_0^{2} = 0 \]

Since \(e^{\lambda t}\neq 0\) for all finite \(t\), it may be cancelled. What remains is the algebraic characteristic (auxiliary) equation. A

\[ \lambda_{\pm} = -\gamma \pm \sqrt{\gamma^{2} - \omega_0^{2}} \]

Quadratic formula. The single quantity under the root, \(\gamma^2-\omega_0^2\), decides whether the roots are complex, real-repeated, or real-distinct. A

\[ \operatorname{sgn}\!\left(\gamma^{2}-\omega_0^{2}\right) = \operatorname{sgn}\!\left(\frac{c^{2}}{4m^{2}}-\frac{k}{m}\right) = \operatorname{sgn}\!\left(c^{2}-4mk\right) = \operatorname{sgn}\Delta \]

Multiply the discriminant of step 7 by \(4m^2>0\) to recover the discriminant \(\Delta=c^2-4mk\) of the un-normalised polynomial \(m\lambda^2+c\lambda+k\); the sign is invariant, so the classification does not depend on how the equation was scaled. B

\[ \Delta<0:\quad \lambda_{\pm}=-\gamma \pm i\omega_d,\qquad \omega_d\equiv\sqrt{\omega_0^{2}-\gamma^{2}} \]

\[ x(t)=e^{-\gamma t}\left(A\cos\omega_d t + B\sin\omega_d t\right) \]

Under-damped. Complex-conjugate roots give a real solution via Euler's formula; the real part \(-\gamma\) is the envelope decay, the imaginary part \(\omega_d<\omega_0\) the ringing frequency. C

\[ \Delta=0:\quad \lambda=-\gamma \ (\text{repeated}) \;\Rightarrow\; x(t)=(A+Bt)\,e^{-\gamma t} \]

Critically damped. A repeated root gives only one exponential; the second independent solution \(t\,e^{-\gamma t}\) comes from reduction of order (or the \(\omega_d\to0\) limit of 9a). This is the fastest non-oscillatory return. C

\[ \Delta>0:\quad \lambda_{\pm}=-\gamma \pm \beta,\quad \beta\equiv\sqrt{\gamma^{2}-\omega_0^{2}}<\gamma \;\Rightarrow\; x(t)=A\,e^{\lambda_{+} t}+B\,e^{\lambda_{-} t} \]

Over-damped. Two distinct real roots, both negative since \(\beta<\gamma\); pure decay on two time scales, the slower root \(\lambda_{+}\) dominating the long-time approach. No oscillation. C

Result

\[ m\lambda^{2}+c\lambda+k=0,\qquad \lambda_{\pm}=-\gamma\pm\sqrt{\gamma^{2}-\omega_0^{2}},\qquad \Delta=c^{2}-4mk \]

\[ \boxed{\;\Delta<0\ \text{under-damped}\quad \Delta=0\ \text{critical}\quad \Delta>0\ \text{over-damped}\;}\qquad \zeta=\frac{c}{2\sqrt{mk}}=\frac{\gamma}{\omega_0} \]

Reading. Every root has real part \(-\gamma\), so all free motion decays with envelope time constant \(1/\gamma=2m/c\), set by drag and mass alone — independent of the spring. What the spring (through \(\omega_0\)) decides is only the character of the decay: whether the system rings on the way down (\(\zeta<1\)), does the fastest non-oscillatory return (\(\zeta=1\)), or crawls (\(\zeta>1\)). The under-damped ring frequency \(\omega_d=\omega_0\sqrt{1-\zeta^2}\) is always below \(\omega_0\): friction slows the wobble. All roots have negative real part for \(c>0\), so equilibrium is always asymptotically stable.

Units check. \([\gamma]=[c/2m]=(\text{kg s}^{-1})/\text{kg}=\text{s}^{-1}\) and \([\omega_0]=\sqrt{[k/m]}=\sqrt{(\text{N m}^{-1})/\text{kg}}=\sqrt{\text{s}^{-2}}=\text{s}^{-1}\); both are rates, so \(\lambda\) has units \(\text{s}^{-1}\) and the exponent \(\lambda t\) is dimensionless as required. \([\Delta]=[c^2]=[mk]=\text{kg}^2\text{s}^{-2}\) (consistent), and \(\zeta=c/2\sqrt{mk}\) is dimensionless.

Limiting cases

\(c\to0\) (\(\zeta\to0\)): \(\gamma\to0\), \(\lambda_\pm=\pm i\omega_0\), recovering undamped SHM \(x=A\cos(\omega_0 t+\phi)\) (prior result: shm-from-linear-restoring-force).
\(\zeta\ll1\) (light damping): \(\omega_d\approx\omega_0\left(1-\tfrac12\zeta^2\right)\); the frequency shift is second order, so weak damping barely changes the period but steadily bleeds amplitude.
\(\zeta=1\) exactly: the two roots merge at \(-\omega_0\) and the under-damped modes coalesce into \((A+Bt)e^{-\omega_0 t}\); fastest settling.
\(\zeta\gg1\) (heavy damping): \(\lambda_{+}\approx -\omega_0^{2}/(2\gamma)=-k/c\) (slow, spring-limited) and \(\lambda_{-}\approx-2\gamma=-c/m\) (fast, drag-limited); the two time scales separate widely and inertia becomes irrelevant to the slow mode.
\(k\to0\): \(\omega_0\to0\), roots become \(\lambda=0\) and \(\lambda=-c/m\); a free particle coasting to rest under drag, no restoring force.

Breaks when

Nonlinear drag. At high speed in a fluid, drag scales as \(\dot{x}^2\) (quadratic), or for dry contact as constant Coulomb friction. The equation is no longer linear, superposition fails, and there is no discriminant to classify regimes; Coulomb decay is linear-in-time and stops dead at a nonzero displacement.
Anharmonic potential. For large amplitude the restoring force departs from \(-kx\) (e.g. a real pendulum, \(-mg\sin\theta\)). The characteristic-equation method collapses: frequency becomes amplitude-dependent and the modes are no longer exponentials.
Negative or zero damping (\(c\le0\)). If \(c<0\) (energy-injecting medium) the real part of \(\lambda\) turns positive and the "decaying" solution grows without bound — the labels survive but the settling interpretation is void.
Time-varying or memory-laden media. If \(c=c(t)\), \(\omega_0=\omega_0(t)\), or the drag depends on the whole velocity history (viscoelastic, retarded fluid), the constant-coefficient ansatz \(e^{\lambda t}\) is not a solution and one needs Laplace/convolution or Floquet theory.

Failure modes

Using \(\omega_0\) for the ring frequency. The observed oscillation is at \(\omega_d=\omega_0\sqrt{1-\zeta^2}\), always smaller; only in the \(\zeta\to0\) limit do they coincide.
Dropping the \(t\,e^{-\gamma t}\) term at critical damping. Writing \(x=Ae^{-\gamma t}\) alone gives a one-parameter family that cannot satisfy two independent initial conditions \((x_0,\dot{x}_0)\).
Forgetting the factor of \(2\) in \(\gamma=c/2m\). Setting \(\gamma=c/m\) misplaces the critical condition by a factor of \(2\) and corrupts every downstream number.
Thinking over-damped returns fastest. Beyond \(\zeta=1\) the slow root \(-k/c\) approaches zero — the system gets slower, not faster. Critical is the optimum.
Treating \(\zeta\) as having units. Comparing \(c\) directly to \(1\) without the \(2\sqrt{mk}\) normalisation; only the dimensionless \(\zeta\) sorts regimes.
Confusing amplitude and energy decay rates. The amplitude envelope decays as \(e^{-\gamma t}\), so the energy \(\propto A^2\) decays as \(e^{-2\gamma t}\) — twice the rate.

Discussion

The whole story lives in the complex \(\lambda\)-plane. Each root is a point whose real part is a decay rate and imaginary part an oscillation frequency. As \(c\) increases from zero, the conjugate pair starts on the imaginary axis at \(\pm i\omega_0\) (pure SHM), slides left along a circle of radius \(\omega_0\) while staying conjugate (under-damped), collides on the negative real axis at \(-\omega_0\) (critical), then splits into two real roots that walk apart — one racing toward \(-\infty\), the other creeping toward the origin (over-damped). Critical damping is precisely the moment of collision, which is why it is a genuine boundary and not merely "a lot of damping."

Energetically, the drag does negative work at rate \(P=-c\dot{x}^2\le0\), so the mechanical energy \(E=\tfrac12 m\dot{x}^2+\tfrac12 kx^2\) is monotonically non-increasing: \(\dot{E}=-c\dot{x}^2\). This is the physical content behind every root having \(\operatorname{Re}\lambda<0\). The quality factor \(Q=\omega_0/2\gamma=\sqrt{mk}/c=1/2\zeta\) repackages the same physics: \(Q\) counts, up to \(2\pi\), the oscillations before the energy falls by \(e^{-1}\). Under-damped is \(Q>\tfrac12\), critical exactly \(Q=\tfrac12\), over-damped \(Q<\tfrac12\); a wine glass has \(Q\sim10^3\), a quartz crystal \(\sim10^6\).

Term for term, the same characteristic equation is the series \(RLC\) circuit \(L\ddot{q}+R\dot{q}+q/C=0\), under the map \(m\to L,\ c\to R,\ k\to1/C\): inductance is inertia, resistance is drag, inverse capacitance is stiffness. Under-, critically, and over-damped ringdown of a circuit are the electrical translation of this mechanical result, which is why one differential-equations toolkit serves mechanics, electronics, and acoustics alike.

Deeper still, the roots \(\lambda_\pm\) are the eigenvalues of the companion matrix of the first-order system \(\dot{x}=v,\ \dot{v}=-\omega_0^2 x-2\gamma v\), and the three regimes are the phase-plane classification of a fixed point: stable spiral (\(\Delta<0\)), degenerate/improper node (\(\Delta=0\)), stable node (\(\Delta>0\)). In the Laplace domain the same \(\lambda_\pm\) are the poles of the transfer function \(H(s)=1/(ms^2+cs+k)\); \(\Delta<0\) places them off the real axis at \(-\gamma\pm i\omega_d\) in the left half-plane, their distance from the imaginary axis being the decay rate and their height the ringing frequency. The complex-to-real transition at \(\Delta=0\) is a codimension-one event whose generalisation decides the linear stability of equilibria in every autonomous dynamical system.

Common misconceptions. "More damping always settles faster" is false past critical; "damped frequency equals natural frequency" ignores the \(-\gamma^2\) shift; and "over-damped means no motion" is wrong — the system still moves, it simply crawls back monotonically without crossing zero.

Worked examples

Classify a suspension and find its ring frequency: effective mass \(m=250\ \text{kg}\), spring \(k=9.0\times10^{4}\ \text{N m}^{-1}\), damper \(c=2.4\times10^{3}\ \text{N s m}^{-1}\).

\[ \omega_0=\sqrt{\frac{k}{m}}=\sqrt{\frac{9.0\times10^{4}}{250}}=\sqrt{360}=18.97\ \text{s}^{-1},\qquad \gamma=\frac{c}{2m}=\frac{2.4\times10^{3}}{500}=4.80\ \text{s}^{-1} \]

\[ \zeta=\frac{\gamma}{\omega_0}=\frac{4.80}{18.97}=0.253<1,\qquad \Delta=c^2-4mk=5.76\times10^{6}-9.0\times10^{7}<0 \;\Rightarrow\; \text{under-damped} \]

\[ \omega_d=\sqrt{\omega_0^{2}-\gamma^{2}}=\sqrt{360-23.04}=\sqrt{336.96}=18.36\ \text{s}^{-1} \]

Set up the two rates symbolically, form \(\zeta\) and check the discriminant, then compute the ring frequency. B

\[ \text{Under-damped};\quad \omega_d=18.4\ \text{s}^{-1}\ (f_d\approx2.92\ \text{Hz}),\quad \text{envelope}\ e^{-4.80\,t}\ (\tau=0.21\ \text{s}) \]

Reading. The wheel oscillates about three times a second, each swing \(1/e\) smaller after \(0.21\ \text{s}\) — noticeable float. Critical damping would need \(c_c=2m\omega_0=2(250)(18.97)\approx9.5\times10^{3}\ \text{N s m}^{-1}\), about four times the fitted value.

Units check. \(\gamma,\omega_0,\omega_d\) all in \(\text{s}^{-1}\); \(\zeta\) dimensionless; envelope exponent \(\gamma t=(\text{s}^{-1})(\text{s})\) dimensionless.

Tune for critical damping, then find settling: \(m=0.50\ \text{kg}\), \(k=200\ \text{N m}^{-1}\). What \(c\) gives \(\zeta=1\), and how long to fall to \(1\%\) of an initial displacement \(x_0\) released from rest?

\[ c_{c}=2\sqrt{mk}=2\sqrt{(0.50)(200)}=2\sqrt{100}=20\ \text{N s m}^{-1},\qquad \gamma=\frac{c_c}{2m}=\omega_0=\sqrt{\frac{200}{0.50}}=20\ \text{s}^{-1} \]

\[ x(t)=(A+Bt)e^{-\gamma t}:\quad x(0)=x_0\Rightarrow A=x_0;\quad \dot{x}(0)=B-\gamma A=0\Rightarrow B=\gamma x_0 \]

\[ x(t)=x_0\,(1+\gamma t)\,e^{-\gamma t},\qquad \frac{x}{x_0}=0.01:\ (1+20t)e^{-20t}=0.01 \;\Rightarrow\; 20t\approx6.64 \;\Rightarrow\; t\approx0.33\ \text{s} \]

Set \(c=c_c\), apply both initial conditions to the degenerate solution, then solve the transcendental \(1\%\) condition (dominated by the exponential, corrected by the \((1+20t)\) prefactor). C

\[ \boxed{\,c_c=20\ \text{N s m}^{-1},\qquad t_{1\%}\approx0.33\ \text{s}\,} \]

Reading. Critical damping requires exactly \(20\ \text{N s m}^{-1}\). The mass returns to within \(1\%\) of equilibrium in about a third of a second without ever overshooting; any larger \(c\) would take longer.

Units check. \([c_c]=2\sqrt{\text{kg}\cdot\text{N m}^{-1}}=2\sqrt{\text{kg}\cdot\text{kg s}^{-2}}=\text{kg s}^{-1}=\text{N s m}^{-1}\); the exponent \(\gamma t\) is dimensionless and \(t\) emerges in seconds.

Problems

A mass \(m=2.0\ \text{kg}\) on a spring \(k=50\ \text{N m}^{-1}\) has damping \(c=30\ \text{N s m}^{-1}\). Classify the regime and give the roots.
Solution
\(\Delta=c^2-4mk=900-4(2.0)(50)=900-400=+500>0\Rightarrow\) over-damped. Check: \(\omega_0=\sqrt{50/2.0}=5.0\ \text{s}^{-1}\), \(\gamma=30/(2\cdot2.0)=7.5\ \text{s}^{-1}\), \(\zeta=7.5/5.0=1.5>1\). Roots \(\lambda_\pm=-7.5\pm\sqrt{56.25-25}=-7.5\pm5.59\), i.e. \(\lambda_+=-1.91\ \text{s}^{-1}\) and \(\lambda_-=-13.1\ \text{s}^{-1}\); both real and negative, decay dominated by the slow \(-1.91\ \text{s}^{-1}\) mode.
A system has \(m=0.20\ \text{kg}\), \(k=5.0\ \text{N m}^{-1}\), \(c=0.40\ \text{N s m}^{-1}\). Classify it, give \(\omega_d\), and find the \(c\) that would make it critical.
Solution
\(\omega_0=\sqrt{5.0/0.20}=\sqrt{25}=5.0\ \text{s}^{-1}\), \(\gamma=0.40/(2\cdot0.20)=1.0\ \text{s}^{-1}\). Since \(\gamma=1.0<\omega_0=5.0\Rightarrow\) under-damped. \(\omega_d=\sqrt{25-1}=\sqrt{24}=4.90\ \text{s}^{-1}\) (about \(2\%\) below \(\omega_0\)). Critical: \(c_c=2\sqrt{mk}=2\sqrt{(0.20)(5.0)}=2\sqrt{1.0}=2.0\ \text{N s m}^{-1}\); the fitted \(c\) is one-fifth of critical.
Compare settling speeds. For fixed \(\omega_0=10\ \text{s}^{-1}\), find the dominant (slowest) decay rate at \(\zeta=1\) and at \(\zeta=2\), and say which returns to rest faster.
Solution
At \(\zeta=1\): repeated root \(\lambda=-\gamma=-\omega_0=-10\ \text{s}^{-1}\); envelope \(\sim e^{-10t}\) (times a slowly rising \(t\)). At \(\zeta=2\): \(\gamma=2\omega_0=20\), \(\beta=\sqrt{\gamma^2-\omega_0^2}=\sqrt{400-100}=17.3\); slow root \(\lambda_+=-20+17.3=-2.68\ \text{s}^{-1}\). The over-damped case is governed by the slow \(-2.68\ \text{s}^{-1}\) mode, far slower than the \(-10\ \text{s}^{-1}\) of critical damping. Critical wins — increasing \(c\) past \(c_c\) slows the return.
An \(RLC\) series circuit has \(L=0.10\ \text{H}\), \(R=200\ \Omega\), \(C=1.0\ \mu\text{F}\). Using \(m\to L,\ c\to R,\ k\to1/C\), classify the charge oscillation and find any ring frequency.
Solution
\(\Delta=R^2-4L/C=200^2-4(0.10)/(1.0\times10^{-6})=4.0\times10^{4}-4.0\times10^{5}<0\Rightarrow\) under-damped. \(\omega_0=1/\sqrt{LC}=1/\sqrt{(0.10)(1.0\times10^{-6})}=1/\sqrt{1.0\times10^{-7}}=3.16\times10^{3}\ \text{s}^{-1}\). \(\gamma=R/(2L)=200/0.20=1.0\times10^{3}\ \text{s}^{-1}\). \(\omega_d=\sqrt{\omega_0^2-\gamma^2}=\sqrt{1.0\times10^{7}-1.0\times10^{6}}=\sqrt{9.0\times10^{6}}=3.0\times10^{3}\ \text{s}^{-1}\). The charge rings at \(\approx3.0\times10^{3}\ \text{rad s}^{-1}\) with envelope \(e^{-1000t}\).
A critically-damped system (\(\gamma=\omega_0=2.0\ \text{s}^{-1}\)) starts from rest at \(x(0)=0.10\ \text{m}\). Using \(x=(A+Bt)e^{-\gamma t}\), find \(A,B\) and the time of maximum speed.
Solution
Position: \(x(0)=A=0.10\ \text{m}\). Velocity \(\dot{x}=(B-\gamma(A+Bt))e^{-\gamma t}\); at rest \(\dot{x}(0)=B-\gamma A=0\Rightarrow B=\gamma A=2.0(0.10)=0.20\ \text{m s}^{-1}\). So \(x(t)=(0.10+0.20t)e^{-2t}\). Speed is extremal where \(\ddot{x}=0\): \(\ddot{x}=\gamma(-2B+\gamma(A+Bt))e^{-\gamma t}\). Setting the bracket to zero: \(-2(0.20)+2.0(0.10)+2.0(0.20)t=0\Rightarrow-0.40+0.20+0.40t=0\Rightarrow t=0.50\ \text{s}\).