# Damped Harmonic Oscillators :config: { :override-date: 2025-01-05 } :: :author: { :name: Dr. Henry Jekyll :affiliation: Department of Oscillatory Studies, Pemberley Physics Institute } :: :author: { :name: Elizabeth Bennet :affiliation: Department of Oscillatory Studies, Pemberley Physics Institute } :: :abstract: { :keywords: {damped oscillator, differential equations, physics} } The damped harmonic oscillator extends the simple harmonic oscillator by incorporating energy dissipation. This paper examines the three characteristic regimes—underdamped, critically damped, and overdamped—and demonstrates how the damping coefficient fundamentally determines system behavior. An interactive simulation enables exploration of parameter space, building intuition for this fundamental physical system. :paragraph: {:class: {press-note, small}} *Example scroll made with RSM Studio for Scroll Press.* :: ## Equation of Motion Consider a mass $m$ on a spring (constant $k$) subject to damping proportional to velocity: $$ m\frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt} $$ Rearranging with $\omega_0 = \sqrt{k/m}$ and $\gamma = b/(2m)$, $$ \frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = 0 $$ The general solution depends on the discriminant $\gamma^2 - \omega_0^2$, $$ x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi) \quad \text{(underdamped, } \gamma < \omega_0\text{)} $$ $$ x(t) = (A + Bt)e^{-\gamma t} \quad \text{(critically damped, } \gamma = \omega_0\text{)} $$ $$ x(t) = A_1 e^{r_1 t} + A_2 e^{r_2 t} \quad \text{(overdamped, } \gamma > \omega_0\text{)}, $$ where $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ is the damped frequency and $r_{1,2}$ are negative real roots. ## Energy Dissipation :theorem: {:label: energy-dissipation} For a damped harmonic oscillator with $\gamma > 0$, the total mechanical energy $$ E(t) = \frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}kx^2 $$ is strictly decreasing over time. :: :proof: :step: Define the total mechanical energy as $$ E(t) = \frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}kx^2. $$ :: :step: Differentiate with respect to time: $${:label:eqn} \frac{dE}{dt} = m\frac{dx}{dt}\frac{d^2x}{dt^2} + kx\frac{dx}{dt}. $$ :: :step: Substitute the equation of motion $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = 0$ in :ref:eqn::. $$ \frac{dE}{dt} = m\frac{dx}{dt}\left(-2\gamma\frac{dx}{dt} - \omega_0^2 x\right) + kx\frac{dx}{dt}. $$ :: :step: Since $k = m\omega_0^2$, the terms involving $x$ cancel, leaving $$ \frac{dE}{dt} = -2m\gamma\left(\frac{dx}{dt}\right)^2 \leq 0, $$ with equality only when $\frac{dx}{dt} = 0$. :qed: :: :: :corollary: {:label: stability} The equilibrium position $(x=0, v=0)$ is asymptotically stable for any positive damping coefficient. :: ## Three Damping Regimes The underdamped regime (${:label:under, :class: hl}\gamma < \omega_0$) produces oscillations with exponentially decaying amplitude $Ae^{-\gamma t}$. Energy decays as $E(t) \propto e^{-2\gamma t}$. This behavior is common in mechanical systems where oscillation before settling is acceptable. Critical damping (${:label:crit, :class: hl}\gamma = \omega_0$) represents the boundary case, returning to equilibrium in minimum time without overshooting. This regime is optimal for shock absorbers and door closers. The overdamped regime (${:label:over, :class: hl}\gamma > \omega_0$) causes the system to slowly creep to equilibrium without oscillating. Return is slower than critical damping. This behavior is common in highly viscous environments. ## Interactive Simulation The visualization below demonstrates all three regimes. Adjustable parameters include the natural frequency $\omega_0$ (oscillation rate without damping), damping coefficient $\gamma$ (energy dissipation strength), and initial conditions $x_0$ and $v_0$ (starting position and velocity). As predicted by :ref:energy-dissipation,Theorem 1::, energy continuously decreases for any positive damping. Watch the system transition from :ref:under,underdamped:: (oscillatory) through :ref:crit, critically damped:: (optimal) to :ref:over,overdamped:: (slow decay) as $\gamma$ increases past $\omega_0$. The simulation shows position vs. time with exponential envelope (underdamped case) and phase space trajectories converging to the origin, illustrating :ref:stability,the asymptotic stability:: established in Corollary 1. :html: { :path: damped_oscillator_widget.html } :caption: Interactive damped oscillator simulation showing position vs. time and phase space trajectories for all three damping regimes. :: ## Applications In mechanical engineering, shock absorbers are designed as underdamped ($\zeta \approx 0.3-0.4$) to allow one oscillation after impact, providing comfort while maintaining tire contact. Buildings use underdamped dampers and tuned mass dampers for seismic protection, dissipating earthquake energy. RLC circuits provide an electrical analog where inductance $L \leftrightarrow m$, capacitance $C \leftrightarrow 1/k$, and resistance $R \leftrightarrow b$. Door closers are designed as critically or slightly overdamped to close smoothly without bouncing. ## Conclusion The damping coefficient $\gamma$ relative to natural frequency $\omega_0$ determines qualitative behavior. :ref:crit,Critical:: damping provides fastest non-oscillatory return, while :ref:under,underdamping:: and :ref:over,overdamping:: represent the oscillatory and slow-decay regimes. Understanding these fundamental dynamics is essential for applications from mechanical design to signal processing. :references: @book{landau1976, title={Mechanics}, author={Landau, L. D. and Lifshitz, E. M.}, year={1976}, publisher={Butterworth-Heinemann}, doi={10.1016/C2009-0-25569-3} } @book{benacquista2018, title={Classical Mechanics}, author={Benacquista, Matthew J. and Romano, Joseph D.}, year={2018}, publisher={Springer International Publishing}, doi={10.1007/978-3-319-68780-3} } :: :paragraph: {:class: {press-note, small}} *Example scroll made with RSM Studio for Scroll Press.*

Damped Harmonic Oscillators

January 05, 2025

Dr. Henry Jekyll

Department of Oscillatory Studies, Pemberley Physics Institute

Elizabeth Bennet

Department of Oscillatory Studies, Pemberley Physics Institute

Abstract

The damped harmonic oscillator extends the simple harmonic oscillator by incorporating energy dissipation. This paper examines the three characteristic regimes—underdamped, critically damped, and overdamped—and demonstrates how the damping coefficient fundamentally determines system behavior. An interactive simulation enables exploration of parameter space, building intuition for this fundamental physical system.

Example scroll made with RSM Studio for Scroll Press.

Keywords: damped oscillator, differential equations, physics

1. Equation of Motion

Consider a mass $m$ on a spring (constant $k$) subject to damping proportional to velocity:

$$ m\frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt} $$

(1.1)

Rearranging with $\omega_0 = \sqrt{k/m}$ and $\gamma = b/(2m)$,

$$ \frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = 0 $$

(1.2)

The general solution depends on the discriminant $\gamma^2 - \omega_0^2$,

$$ x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi) \quad \text{(underdamped, } \gamma < \omega_0\text{)} $$

(1.3)

$$ x(t) = (A + Bt)e^{-\gamma t} \quad \text{(critically damped, } \gamma = \omega_0\text{)} $$

(1.4)

$$ x(t) = A_1 e^{r_1 t} + A_2 e^{r_2 t} \quad \text{(overdamped, } \gamma > \omega_0\text{)}, $$

(1.5)

where $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ is the damped frequency and $r_{1,2}$ are negative real roots.

2. Energy Dissipation

Theorem 2.1.

For a damped harmonic oscillator with $\gamma > 0$, the total mechanical energy

$$ E(t) = \frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}kx^2 $$

(2.1)

is strictly decreasing over time.

Proof.

Define the total mechanical energy as

$$ E(t) = \frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}kx^2. $$

(2.2)

⟨1⟩

Differentiate with respect to time:

$$ \frac{dE}{dt} = m\frac{dx}{dt}\frac{d^2x}{dt^2} + kx\frac{dx}{dt}. $$

(2.3)

⟨2⟩

Substitute the equation of motion $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = 0$ in (2.3).

$$ \frac{dE}{dt} = m\frac{dx}{dt}\left(-2\gamma\frac{dx}{dt} - \omega_0^2 x\right) + kx\frac{dx}{dt}. $$

(2.4)

⟨3⟩

Since $k = m\omega_0^2$, the terms involving $x$ cancel, leaving

$$ \frac{dE}{dt} = -2m\gamma\left(\frac{dx}{dt}\right)^2 \leq 0, $$

(2.5)

with equality only when $\frac{dx}{dt} = 0$. QED

⟨4⟩

Corollary 2.2.

The equilibrium position $(x=0, v=0)$ is asymptotically stable for any positive damping coefficient.

3. Three Damping Regimes

The underdamped regime ($\gamma < \omega_0$) produces oscillations with exponentially decaying amplitude $Ae^{-\gamma t}$. Energy decays as $E(t) \propto e^{-2\gamma t}$. This behavior is common in mechanical systems where oscillation before settling is acceptable.

Critical damping ($\gamma = \omega_0$) represents the boundary case, returning to equilibrium in minimum time without overshooting. This regime is optimal for shock absorbers and door closers.

The overdamped regime ($\gamma > \omega_0$) causes the system to slowly creep to equilibrium without oscillating. Return is slower than critical damping. This behavior is common in highly viscous environments.

4. Interactive Simulation

The visualization below demonstrates all three regimes. Adjustable parameters include the natural frequency $\omega_0$ (oscillation rate without damping), damping coefficient $\gamma$ (energy dissipation strength), and initial conditions $x_0$ and $v_0$ (starting position and velocity). As predicted by Theorem 1, energy continuously decreases for any positive damping. Watch the system transition from underdamped (oscillatory) through critically damped (optimal) to overdamped (slow decay) as $\gamma$ increases past $\omega_0$. The simulation shows position vs. time with exponential envelope (underdamped case) and phase space trajectories converging to the origin, illustrating the asymptotic stability established in Corollary 1.

Damped Harmonic Oscillators: Three Characteristic Regimes. Dr. Henry Jekyll, Elizabeth Bennet.