The forced coupled quadratic model: modal responses, resonance, and boundedness for positive definite stiffness off resonance

Use the fixed coordinate, time and energy units and numerical matrix convention of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium. Every displayed matrix identity and phase norm is in those numerical coordinates; physical positions, velocities, modal amplitudes, frequencies and energies are recovered by the stated fixed scale factors. This does not change the set of physical motions.

Adopt the coupled quadratic model of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium with data $(M,K,J)$ in which $M$ is symmetric positive definite, $K$ is symmetric and non-negative, $I_0\subseteq \mathbb{R}$ is an interval with nonempty interior and $J:I_0\to {\mathbb{R}}^n$ is continuous; let $O,\mathrm{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, ${\omega }_i=\sqrt{{\lambda }_i}$ for ${\lambda }_i>0$ and the modal coordinates $z=O^{\mathsf{T}}{M}^{1\mathrm{/}2}q$ be the modal data of clause (e) of that definition, and call $\psi :=O^{\mathsf{T}}{M}^{-1\mathrm{/}2}J$ the modal force. Its physical unit is $a_{\ast }\mathrm{/}{t}_{\ast }^2$; the modal initial velocity ${\eta }_i$ has unit $a_{\ast }\mathrm{/}{t}_{\ast }$ and ${\xi }_i$ has unit $a_{\ast }$. Every convolution and particular-solution amplitude therefore reconstructs a modal coordinate in unit $a_{\ast }$. Fix $t_0\in I_0$ and a datum $(q_0,v_0)$, and write ${\xi }_i:=z_i(t_0)$, ${\eta }_i:=z_i^{\mathrm{\prime }}(t_0)$ for the modal data of the motion. Then:

1. Unique motion and modal convolution. There is exactly one motion $q:I_0\to {\mathbb{R}}^n$ with that datum; it is $q=M^{-1\mathrm{/}2}Oz$ with

$z_i(t)={\xi }_i\cos ({\omega }_i(t-t_0))+\frac{{\eta }_i}{{\omega }_i}\sin ({\omega }_i(t-t_0))+\frac{1}{{\omega }_i}{\int }_{t_0}^t\sin ({\omega }_i(t-s)){\psi }_i(s)ds({\lambda }_i>0),$

$z_i(t)={\xi }_i+{\eta }_i(t-t_0)+{\int }_{t_0}^t(t-s){\psi }_i(s)ds({\lambda }_i=0)\mathrm{.}$

2. Sinusoidal or constant generalized force. Suppose $I_0=\mathbb{R}$ and $J(t)=J_0\cos (\mathrm{\Omega }t)$ with $J_0\in {\mathbb{R}}^n$ and $\mathrm{\Omega }\ge 0$, so that $\psi (t)={\psi }_0\cos (\mathrm{\Omega }t)$ with ${\psi }_0=O^{\mathsf{T}}{M}^{-1\mathrm{/}2}{J}_0$. Then for each index $i$: (a) if ${\lambda }_i>0$ and ${\mathrm{\Omega }}^2\mathrm{\ne }{\lambda }_i$, then $z_i^{\mathrm{p}}(t):={\psi }_{0,i}\cos (\mathrm{\Omega }t)\mathrm{/}({\lambda }_i-{\mathrm{\Omega }}^2)$ solves $z_i^{\mathrm{\prime }\mathrm{\prime }}+{\lambda }_i{z}_i={\psi }_{0,i}\cos (\mathrm{\Omega }t)$ on $\mathbb{R}$, with amplitude $\mathrm{\mid }{\psi }_{0,i}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_i-{\mathrm{\Omega }}^2\mathrm{\mid }$; (b) if ${\lambda }_i>0$ and ${\mathrm{\Omega }}^2={\lambda }_i$, then $z_i^{\mathrm{p}}(t):={\psi }_{0,i}t\sin ({\omega }_{i}t)\mathrm{/}(2{\omega }_i)$ solves that equation and is unbounded on $[0,\mathrm{\infty })$ when ${\psi }_{0,i}\mathrm{\ne }0$; (c) if ${\lambda }_i=0$ and $\mathrm{\Omega }>0$, then $z_i^{\mathrm{p}}(t):=-{\psi }_{0,i}\cos (\mathrm{\Omega }t)\mathrm{/}{\mathrm{\Omega }}^2$ solves $z_i^{\mathrm{\prime }\mathrm{\prime }}={\psi }_{0,i}\cos (\mathrm{\Omega }t)$ and is bounded; if ${\lambda }_i=0$ and $\mathrm{\Omega }=0$, then $z_i^{\mathrm{p}}(t):={\psi }_{0,i}{t}^2\mathrm{/}2$ solves $z_i^{\mathrm{\prime }\mathrm{\prime }}={\psi }_{0,i}$ and is unbounded on $[0,\mathrm{\infty })$ when ${\psi }_{0,i}\mathrm{\ne }0$.
3. Reduction, resonance and boundedness. In every case of clause 2 the modal particular functions $z_i^{\mathrm{p}}$ combine into the particular motion $q^{\mathrm{p}}:=M^{-1\mathrm{/}2}O{z}^{\mathrm{p}}$ of the model, and every motion of the model on $\mathbb{R}$ is $q=q^{\mathrm{p}}+q^{\mathrm{h}}$ with $q^{\mathrm{h}}$ a motion of the free model of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium. Consequently: (a) resonance. If some index $i$ has ${\psi }_{0,i}\mathrm{\ne }0$ and either ${\lambda }_i>0$ with ${\mathrm{\Omega }}^2={\lambda }_i$, or ${\lambda }_i=0$ with $\mathrm{\Omega }=0$, then every motion of the model is unbounded on $[0,\mathrm{\infty })$; (b) boundedness. If $K$ is positive definite and no index $i$ with ${\psi }_{0,i}\mathrm{\ne }0$ satisfies ${\mathrm{\Omega }}^2={\lambda }_i$, then every motion of the model is bounded on $[0,\mathrm{\infty })$. The resonant drive angular frequencies are exactly the values $\mathrm{\Omega }=\sqrt{{\lambda }_i}$ with ${\lambda }_i>0$ and ${\psi }_{0,i}\mathrm{\ne }0$, together with $\mathrm{\Omega }=0$ when some index with ${\lambda }_i=0$ has ${\psi }_{0,i}\mathrm{\ne }0$.

The generalized force is prescribed external data. For $\mathrm{\Omega }>0$ this is the sinusoidal-drive formulation of The sinusoidally driven oscillator model: a prescribed external force $f_0\cos (\mathrm{\Omega }t)$ added to the damped oscillator; for $\mathrm{\Omega }=0$ it is the constant-force case allowed by clause (b) of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium, whose $J$ is any continuous prescribed generalized force. In both cases the source is not modelled dynamically, and no back-reaction, damping, measured response or empirical value is asserted.

• Degenerate resonance. If a modal stiffness is degenerate, the modal force components ${\psi }_{0,i}$ depend on the choice of diagonalization, but whether the drive resonates with the corresponding pure-$\lambda$ family does not: by clause 4 of The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction the family is the intrinsic subspace $N_{\lambda }$, and the resonance condition is that the mass-metric projection of $M^{-1}{J}_0$ onto $N_{\lambda }$ be nonzero, a basis-independent statement.

• Why the zero mode is special. A zero modal stiffness means the potential of the quadratic model has a flat direction to second order; a constant generalized force component along such a direction produces quadratic growth in time, the degenerate analogue of resonance, and no oscillation at all is present in that direction.