The damped driven oscillator model: second-order equation, phase-plane system and characteristic data

Let $m>0$, $c\ge 0$ and $k\in \mathbb{R}$ be given real numbers, let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to \mathbb{R}$ be continuous (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). The damped driven oscillator model with data $(m,c,k,f)$ is the second-order equation

$m{x}^{\mathrm{\prime }\mathrm{\prime }}(t)+c{x}^{\mathrm{\prime }}(t)+kx(t)=f(t),t\in I\mathrm{.}$

A solution of the model on an interval $J\subseteq I$ with nonempty interior is a twice differentiable $x:J\to \mathbb{R}$ such that the identity holds for every $t\in J$, derivatives being ordinary real derivatives (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). The model is called homogeneous when $f\equiv 0$, undamped when $c=0$, and strictly damped when $c>0$.

The phase-plane equation of the model has right-hand side

$G_f:I\times {\mathbb{R}}^2⟶{\mathbb{R}}^2,G_f(t,x,v):=(v,\frac{f(t)-cv-kx}{m})\mathrm{.}$

Here ${\mathbb{R}}^2$ is the phase plane, with coordinates $(x,v)$, where $x$ is the displacement and $v$ the velocity. For an interval $J\subseteq I$ with nonempty interior, a differentiable curve $z=(x,v):J\to {\mathbb{R}}^2$ solves the phase-plane equation when $z^{\mathrm{\prime }}(t)=G_f(t,z(t))$ for every $t\in J$, using the domain-relative derivative at any included endpoint (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). If $x$ is a solution of the model on $J$, then $t\mapsto (x(t),x^{\mathrm{\prime }}(t))$ solves this equation; conversely, if $t\mapsto (x(t),v(t))$ solves it, then $v=x^{\mathrm{\prime }}$ and $x$ is a solution of the model on $J$. When $J$ is open, this is the first-order system of First-order systems, initial value problems, and solutions on intervals on the open domain $J\times {\mathbb{R}}^2$. For the homogeneous model the right-hand side is written $G_0(x,v)=(v,-(cv+kx)\mathrm{/}m)$.

The characteristic polynomial of the model is

$p(\lambda ):=m{\lambda }^2+c\lambda +k,$

with discriminant $\mathrm{\Delta }:=c^2-4mk$. When $k>0$ define the undamped frequency ${\omega }_0:=\sqrt{k\mathrm{/}m}$, the damping rate $\gamma :=c\mathrm{/}(2m)$ and, when $4mk>c^2$, the damped frequency $\omega :=\sqrt{k\mathrm{/}m-{\gamma }^2}=\sqrt{4mk-c^2}\mathrm{/}(2m)$; the corresponding square roots are the nonnegative real ones.

• Equilibrium of the homogeneous phase system. A point $(x,v)\in {\mathbb{R}}^2$ satisfies $G_0(x,v)=(0,0)$ if and only if $v=0$ and $kx=0$; hence the zero set of $G_0$ is $\{(0,0)\}$ when $k\mathrm{\ne }0$ and the whole line $\{(x,0):x\in \mathbb{R}\}$ when $k=0$. For the forced model, $G_f(t,x,v)=(0,0)$ holds if and only if $v=0$ and $kx=f(t)$.

• Interpretation. This item is a purely mathematical model: $m,c,k$ are real parameters, $f$ an arbitrary continuous real function, and no physical units, force law, constitutive assumption or empirical claim is made. The classical-mechanical reading of $m,c,k,f$ and any identification with measured data belong to the consuming physics item, not to this definition.

• Regularity. $G_f$ is continuous relative to $I\times {\mathbb{R}}^2$ and, for each $t$, Lipschitz in $(x,v)$ with constant $L:=\max (1,(c+\mathrm{\mid }k\mathrm{\mid })\mathrm{/}m)\sqrt{2}$, since $\mathrm{\mid }-c\mathrm{\Delta }v-k\mathrm{\Delta }x\mathrm{\mid }\le c\mathrm{\mid }\mathrm{\Delta }v\mathrm{\mid }+\mathrm{\mid }k\mathrm{\mid }\mathrm{\mid }\mathrm{\Delta }x\mathrm{\mid }$ and $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }\le \sqrt{2}\sqrt{a^2+b^2}$. At every interior time $t\in I^{\circ }$, its restriction to a sufficiently small open time interval about $t$ satisfies the Picard-Lindelof hypotheses of Picard-Lindelöf local existence and uniqueness for first-order systems. If $I$ includes an endpoint, this item does not assert existence across that endpoint unless $f$ is also defined on an open time neighborhood there.