Physlib.ClassicalMechanics.DampedHarmonicOscillator.Basic

For a damped harmonic oscillator with spring constant $k$, the spring constant is non-zero, i.e., $k \neq 0$.

For a damped harmonic oscillator with mass $m$, the mass is non-zero, i.e., $m \neq 0$.

For a damped harmonic oscillator with mass $m$ and spring constant $k$, the natural (undamped) angular frequency $\omega_0$ is defined as the square root of the ratio of the spring constant to the mass: $$\omega_0 = \sqrt{\frac{k}{m}}$$

For a damped harmonic oscillator, the natural angular frequency $\omega_0$ is strictly positive, i.e., $\omega_0 > 0$.

For a damped harmonic oscillator with mass $m$ and spring constant $k$, the square of the natural angular frequency $\omega_0$ is equal to the ratio of the spring constant to the mass: $$\omega_0^2 = \frac{k}{m}$$

For a damped harmonic oscillator with mass $m$, damping coefficient $\gamma$, and spring constant $k$, a position function $x : \text{Time} \to \mathbb{R}$ satisfies the equation of motion if for all times $t \in \text{Time}$, the following differential equation holds: $$m \ddot{x}(t) + \gamma \dot{x}(t) + k x(t) = 0$$ where $\dot{x}(t)$ and $\ddot{x}(t)$ represent the first and second time derivatives of $x$ at time $t$, respectively.

For a damped harmonic oscillator with mass $m$, given a position function $x: \text{Time} \to \mathbb{R}$, its kinetic energy is the function mapping each time $t$ to $$\frac{1}{2} m (\dot{x}(t))^2$$ where $\dot{x}(t)$ is the time derivative of the position $x$ at time $t$.

For a damped harmonic oscillator with spring constant $k$, given a position function $x: \text{Time} \to \mathbb{R}$, its potential energy is the function $t \mapsto \frac{1}{2} k (x(t))^2$.

For a damped harmonic oscillator with mass $m$ and spring constant $k$, given a position function $x: \text{Time} \to \mathbb{R}$, its mechanical energy is the function $t \mapsto E(t)$ defined as the sum of its kinetic and potential energies: $$E(t) = \frac{1}{2} m (\dot{x}(t))^2 + \frac{1}{2} k (x(t))^2$$ where $\dot{x}(t)$ is the time derivative of the position $x$ at time $t$.

For a damped harmonic oscillator with mass $m$, and for any position function $x: \text{Time} \to \mathbb{R}$, the kinetic energy is given by the function $t \mapsto \frac{1}{2} m (\dot{x}(t))^2$, where $\dot{x}(t)$ is the time derivative of $x$ at time $t$.

For a damped harmonic oscillator $S$ with spring constant $k$, and for any position function $x: \text{Time} \to \mathbb{R}$, the potential energy is given by the function $t \mapsto \frac{1}{2} k (x(t))^2$.

For a damped harmonic oscillator with damping coefficient $\gamma$, and for any position function $x : \text{Time} \to \mathbb{R}$, the energy dissipation rate is defined as the function $t \mapsto -\gamma (\dot{x}(t))^2$, where $\dot{x}(t)$ is the time derivative of $x$ at time $t$.

For a damped harmonic oscillator with damping coefficient $\gamma$, let $x : \text{Time} \to \mathbb{R}$ be a position function of class $C^\infty$ that satisfies the equation of motion $m \ddot{x} + \gamma \dot{x} + kx = 0$. Then the time derivative of the mechanical energy $E(t)$ at any time $t$ is given by $$\frac{dE}{dt}(t) = -\gamma (\dot{x}(t))^2,$$ where $\dot{x}(t)$ denotes the time derivative of $x$ at time $t$.

Consider a damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $\gamma > 0$. Let $x : \text{Time} \to \mathbb{R}$ be a position function of class $C^\infty$ that satisfies the equation of motion $m \ddot{x} + \gamma \dot{x} + kx = 0$. If the velocity at time $t$ is non-zero, $\dot{x}(t) \neq 0$, then the time derivative of the mechanical energy $E(t) = \frac{1}{2} m (\dot{x}(t))^2 + \frac{1}{2} k (x(t))^2$ is strictly negative: $$\frac{dE}{dt}(t) < 0.$$

The discriminant of a damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $\gamma$ is defined as $\gamma^2 - 4mk$. This value determines the damping regime (underdamped, critically damped, or overdamped) of the system.

A damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $\gamma$ is said to be **underdamped** if its discriminant $\gamma^2 - 4mk$ is strictly less than zero.

A damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $\gamma$ is said to be **critically damped** if its discriminant $\gamma^2 - 4mk$ is equal to zero.

A damped harmonic oscillator with mass $m$, spring constant $k$, and damping coefficient $\gamma$ is said to be **overdamped** if its discriminant $\gamma^2 - 4mk$ is strictly greater than zero.