The sinusoidally driven oscillator model: a prescribed external force $f_0\cos (\mathrm{\Omega }t)$ added to the damped oscillator

(i) Setting. Work in the one-dimensional model class of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with time window $I_0=\mathbb{R}$, configuration domain $Q=\mathbb{R}$, mass $m>0$, spring constant $k>0$ (The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$) and drag coefficient $c\ge 0$ (The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $-c\dot{x}$, $c\ge 0$). Let a real drive amplitude $f_0\in \mathbb{R}$ of dimension $\mu \mathrm{\ell }{\tau }^{-2}$ (newtons) and an angular drive frequency $\mathrm{\Omega }>0$ of dimension ${\tau }^{-1}$ (radians per second; the radian is dimensionless in the framework's angle convention, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework) be declared. Adopt, through the additive composition of force contributions of clause (v) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and the external-contribution reading of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, the force law $F(t,x,v):=f_0\cos (\mathrm{\Omega }t)-kx-cv\mathrm{.}$ The sinusoidally driven oscillator model is the one-dimensional model with this force law; its equation of motion is $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=f_0\cos (\mathrm{\Omega }t),$ and its data $(m,c,k,f)$ with $f(t)=f_0\cos (\mathrm{\Omega }t)$ are oscillator-type data in the sense of clause (f) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria. The term $f_0\cos (\mathrm{\Omega }t)$ is the drive or driving force; the model is undamped when $c=0$ and strictly damped when $c>0$.

(ii) The drive is prescribed external data. The driving force is a prescribed function of time alone: it does not depend on $x$ or on $x^{\mathrm{\prime }}$, and the object producing it is not one of the model's labels and is not modelled dynamically. This is the same modelling status as the medium of The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient: the drive is an external contribution declared as part of the model. A real apparatus, background or force generator that reacts to the motion is not represented by this model; representing such a back-reaction requires a larger model with additional states.

(iii) Potential presentation with residual force. The model presents its force law through the harmonic potential $U(x)=\frac{1}{2}k{x}^2$ with residual force $d(t,x,v)=f_0\cos (\mathrm{\Omega }t)-cv$, in the sense of clause (d) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria: $F(t,x,v)=-U^{\mathrm{\prime }}(x)+d(t,x,v)$. The mechanical energy of the model is $E(x,v)=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2$ of that clause, defined along every motion; the model is driven and not a global potential model whenever $f_0\mathrm{\ne }0$.

(iv) Regularity. Let $X:=x_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m})$, $V:=v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m}\mathrm{/}\mathrm{s})$, and $T:=t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{s})$ be the dimensionless numerical SI coordinates (the real inputs of the force map). Write $K:=k_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{N}\mathrm{/}\mathrm{m})$, $C:=c_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s})$, ${\widehat{f}}_0:=f_0\mathrm{/}(1\mathrm{N})$, and $\widehat{\mathrm{\Omega }}:=\mathrm{\Omega }(1\mathrm{s})$. The numerical force map is $\widehat{F}(T,X,V)={\widehat{f}}_0\cos (\widehat{\mathrm{\Omega }}T)-KX-CV,$ so it is continuous and globally Lipschitz in the state variable with numerical-coordinate constant $K+C$, uniformly in time. This is a bound on dimensionless SI coordinates and force values, not a sum of the dimensionful $k$ and $c$ (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). It is an admissible force law of clause (a) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, and its equation is the forced model $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=f$ of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with the continuous function $f(t)=f_0\cos (\mathrm{\Omega }t)$.

(v) Physical status and regime. The model is a restricted idealization adopted rather than derived: exactly sinusoidal, with constant amplitude and frequency and no dependence on the motion. The drive amplitude and frequency are model parameters; when their values come from an apparatus calibration, that use is an empirical premise carrying the apparatus conditions and the measurement uncertainty, and this item asserts no such value. The sinusoidal form is the elementary component used by the recorded treatments; the response to a non-sinusoidal periodic drive is a superposition statement belonging to the mathematical theory, not an additional claim here. The model's undamped case $c=0$ is included as a parameter value, and it is where the resonant unboundedness of the following theorem occurs.

(vi) Status. Adopted assumption of the framework when a model invokes it; it selects no law of nature on its own, and no measured response, no resonance curve of any apparatus and no empirical resonance phenomenon is asserted by the postulate.

• Why a postulate and not a theorem. The equation of motion determines no force law; the drive is additional model input, exactly as the drag of The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $-c\dot{x}$, $c\ge 0$. The linearity of the resulting equation, which makes the explicit response formulas available, is a property of the model's declaration, not a derivation from mechanics.

• Two readings of $\mathrm{\Omega }$. $\mathrm{\Omega }>0$ is an angular frequency in radians per second; the frequency in cycles per second is $\mathrm{\Omega }\mathrm{/}(2\pi )$, and the period of the drive is $2\pi \mathrm{/}\mathrm{\Omega }$. The framework's angle convention makes the radian dimensionless, so $[\mathrm{\Omega }]={\tau }^{-1}$.