The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium

(a) Data of a coupled quadratic model. A coupled quadratic model (a quadratic oscillator model) is declared by the following data: an integer $n\ge 1$, the number of degrees of freedom; a mass metric matrix $M\in M_n(\mathbb{R})$, symmetric and positive definite, that is $M=M^{\mathsf{T}}$ and $v^{\mathsf{T}}Mv>0$ for every nonzero $v\in {\mathbb{R}}^n$ (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); a stiffness matrix $K\in M_n(\mathbb{R})$, symmetric, $K=K^{\mathsf{T}}$; and an interval $I_0\subseteq \mathbb{R}$ with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) together with a continuous generalized force $J:I_0\to {\mathbb{R}}^n$ (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions). When no force or time interval is prescribed, take $I_0:=\mathbb{R}$ and $J(t):=0$ for every $t\in \mathbb{R}$; in particular this is the convention for the small-oscillation model in clause (d). The coordinates $q_1,\dots ,q_n$ of ${\mathbb{R}}^n$ are read in the chart in which the model is declared and carry declared dimensions ${\gamma }_1,\dots ,{\gamma }_n$ and units, and time is read in seconds and mass in kilograms (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The data carry the dimensions

$[M_{ij}]=\mu {\mathrm{\ell }}^2{\gamma }_i^{-1}{\gamma }_j^{-1},[K_{ij}]=\mu {\mathrm{\ell }}^2{\gamma }_i^{-1}{\gamma }_j^{-1}{\tau }^{-2},[J_i]=\mu {\mathrm{\ell }}^2{\tau }^{-2}{\gamma }_i^{-1},$

so that $\frac{1}{2}{v}^{\mathsf{T}}Mv$, $\frac{1}{2}{q}^{\mathsf{T}}Kq$ and the pairing $J\cdot \delta q$ of the generalized force with a coordinate variation all have the energy dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$; if the coordinate $q_i$ is length-valued, ${\gamma }_i=\mathrm{\ell }$, the entry $J_i$ has the force dimension $\mu \mathrm{\ell }{\tau }^{-2}$. This is the dimension bookkeeping of a generalized coordinate description in the sense of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, whose kinetic metric and generalized forces are the framework objects the data present; the matrices $M$ and $K$ are model parameters, not measured quantities.

Fixed-unit numerical convention. Choose a positive coordinate unit $s_i$ of dimension ${\gamma }_i$ for each direction, a time unit $t_{\ast }>0$ and an energy unit $e_{\ast }>0$, held fixed throughout a model calculation. Matrix products, square roots, eigenvectors, derivatives and Euclidean norms below act on numerical coefficients in these units, not on arrays of unlike physical quantities. Explicitly, if $Q_i$, ${\mathsf{M}}_{ij}$, ${\mathsf{K}}_{ij}$ and ${\mathsf{J}}_i$ denote the dimensional data, the symbols in the algebra are $q_i=Q_i\mathrm{/}{s}_i,t=t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{t}_{\ast },M_{ij}={\mathsf{M}}_{ij}{s}_i{s}_j\mathrm{/}(e_{\ast }{t}_{\ast }^2),K_{ij}={\mathsf{K}}_{ij}{s}_i{s}_j\mathrm{/}{e}_{\ast },J_i={\mathsf{J}}_i{s}_i\mathrm{/}{e}_{\ast }\mathrm{.}$ Then $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=J$ is exactly the physical equation in these coefficients, $Q_i=s_i{q}_i$, ${\dot{Q}}_i=(s_i\mathrm{/}{t}_{\ast })q_i^{\mathrm{\prime }}$, and the physical energy is $e_{\ast }E$. This is an invertible choice of units, imposing no restriction on the dimensions ${\gamma }_i$. For a variable metric and potential the same convention means $U(q)=U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(s_1{q}_1,\dots ,s_n{q}_n)\mathrm{/}{e}_{\ast }$ and $M_{ij}(q)={\mathsf{M}}_{ij}(s_1{q}_1,\dots ,s_n{q}_n)s_i{s}_j\mathrm{/}(e_{\ast }{t}_{\ast }^2)$. Congruence by the positive diagonal coordinate-scale matrix and multiplication by positive constants preserve symmetry, definiteness and critical-point type.

(b) Model equation, motions and phase system. On any interval with nonempty interior, $C^r$ for $r=1,2$ means that the first $r$ derivatives exist and are continuous on that interval, using the domain-relative derivative of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral at included endpoints; on open intervals this agrees with $C^k$ Euclidean maps and diffeomorphisms. The equation of the model with data $(M,K,J)$ is

$M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)+Kq(t)=J(t)(t\in I_0)\mathrm{.}$

A motion of the model is a curve $q:J\to {\mathbb{R}}^n$ of class $C^2$ on an interval $J\subseteq I_0$ with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) satisfying the equation for every $t\in J$. A datum at $t_0\in I_0$ is a pair $(q_0,v_0)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$. The phase system of the model is the first-order linear system $z^{\mathrm{\prime }}=Az+b(t)$ on $I_0\times {\mathbb{R}}^{2n}$ with $z=(q,v)$ and

$A=\left(\begin{array}{cc}0& I_n\\ -M^{-1}K& 0\end{array}\right),b(t)=\left(\begin{array}{c}0\\ M^{-1}J(t)\end{array}\right),$

where $M^{-1}$ is the inverse of $M$. By the product rule and invertibility of $M$, a $C^1$ curve $t\mapsto (q(t),v(t))$ solves the phase system if and only if $q$ is a motion of the model and $v=q^{\mathrm{\prime }}$. The model is free when $J\equiv 0$ and forced, with generalized force $J$, otherwise; the read of $J$ as a prescribed external generalized force is the modelling status of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets for the external contributions of a force model.

(c) Quadratic energies and the mass metric. The kinetic energy, the potential energy of the stiffness and the mechanical energy of the model are

$T(v):=\frac{1}{2}{v}^{\mathsf{T}}Mv,U_K(q):=\frac{1}{2}{q}^{\mathsf{T}}Kq,E(q,v):=T(v)+U_K(q),$

whose physical readings are $e_{\ast }$ times the displayed numerical quantities, all of energy dimension. $U_K$ is defined for every symmetric $K$; the name records that $K$ is the matrix through which the model enters, and when $K$ is indefinite $U_K$ is not non-negative, the model is not presented by a potential with a local minimum at the origin, and no stability vocabulary is attached to it by this item. The quadratic form $v\mapsto v^{\mathsf{T}}Mv$ is positive definite and is called the mass metric of the model; the word "metric" records only this positive-definiteness, and no distance on configuration space is defined here.

(d) Small-oscillation model at an equilibrium. Let a mechanical system in a chart be given by a $C^2$ field $q\mapsto G_{\mathrm{k}\mathrm{i}\mathrm{n}}(q)$ of symmetric positive definite matrices on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and a $C^2$ potential $U:\mathrm{\Omega }\to \mathbb{R}$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart), and let $q_0\in \mathrm{\Omega }$ be a mechanical equilibrium, $\mathrm{\nabla }U(q_0)=0$. Its small-oscillation model at $q_0$ is the coupled quadratic model with degrees of freedom $n$, mass metric matrix $M:=G_{\mathrm{k}\mathrm{i}\mathrm{n}}(q_0)$, stiffness matrix $K_0:=D^{2}U(q_0)$ — the Hessian of the potential (The Hessian matrix and critical points of a scalar field, symmetric by The Hessian of a $C^2$ scalar field is symmetric) — and zero generalized force, read in the displacement coordinates $\xi :=q-q_0$. When the chart arises from the generalized-coordinate dictionary of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, the matrix $G_{\mathrm{k}\mathrm{i}\mathrm{n}}(q_0)=DX(q_0{)}^{\mathsf{T}}\mathsf{M}DX(q_0)$ is the value at $q_0$ of that dictionary's kinetic metric. This clause assigns model data to the equilibrium; the identification of the assigned model with the derivative of the mechanical field, and every statement about its motions, are separate results and are not asserted here.

(e) Modal vocabulary of the decoupling data. All the following matrix operations use the fixed-unit numerical convention. By clauses 1-3 of Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity, $M^{1\mathrm{/}2}$ and its inverse $M^{-1\mathrm{/}2}$ exist and are symmetric positive definite. For every symmetric $K$, without a sign restriction, $S:=M^{-1\mathrm{/}2}K{M}^{-1\mathrm{/}2}$ is symmetric, since $S^{\mathsf{T}}=M^{-1\mathrm{/}2}{K}^{\mathsf{T}}{M}^{-1\mathrm{/}2}=S$. The Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis therefore supplies an orthogonal $O$ and real diagonal $\mathrm{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$ with $S=O\mathrm{\Lambda }{O}^{\mathsf{T}}$. This construction includes negative eigenvalues; the restricted non-negative-$K$ result Decoupling of the constant-coefficient quadratic system $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=0$ into normal modes, with the zero-mode and energy identities is used only in that restricted case. Define modal coordinates $z:=O^{\mathsf{T}}{M}^{1\mathrm{/}2}q$ and modal vectors $u_i:=M^{-1\mathrm{/}2}O{e}_i$, the columns of the inverse transform $q=M^{-1\mathrm{/}2}Oz$. The modal stiffnesses are the real numbers ${\lambda }_i$. Direct multiplication gives $K{u}_i={\lambda }_{i}M{u}_i$ and $u_i^{\mathsf{T}}M{u}_j={\delta }_{ij}$. For the modal units here and in items using this convention, extend the dimension group of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (a) from ${\mathbb{Z}}^3$ to $D_{1\mathrm{/}2}:=(\frac{1}{2}\mathbb{Z}{)}^3$, with the same componentwise addition and negation for products and inverses. The inclusion ${\mathbb{Z}}^3\hookrightarrow D_{1\mathrm{/}2}$ preserves every earlier dimension assignment. For $d=(a,b,c)\in D_{1\mathrm{/}2}$ write its reference unit as $u_d:={\mathrm{m}}^a{\mathrm{s}}^b{\mathrm{k}\mathrm{g}}^c$; a quantity of dimension $d$ is a real multiple $x{u}_d$, and unit multiplication is $u_d{u}_{d^{\mathrm{\prime }}}=u_{d+d^{\mathrm{\prime }}}$. Products, quotients, sums of equal dimension and derivatives retain the rules of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (b). For a positive quantity $X=x{u}_d$ with $x>0$ and $d\in {\mathbb{Z}}^3$, define $\sqrt{X}:=\sqrt{x}{u}_{d\mathrm{/}2}$, using the positive real square root; this construction has dimension $d\mathrm{/}2$ and square $X$. Only integer-dimensional positive quantities are square-rooted by this convention.

In physical units put $a_{\ast }:=\sqrt{e_{\ast }}{t}_{\ast }$, of dimension ${\mu }^{1\mathrm{/}2}\mathrm{\ell }$ and SI unit ${\mathrm{k}\mathrm{g}}^{1\mathrm{/}2}\mathrm{m}$. A modal coordinate has unit $a_{\ast }$; the $j$-th component of a mass-normalized modal vector has unit $s_j\mathrm{/}{a}_{\ast }$, and the physical coordinate is the sum of the physical modal-vector components times their modal amplitudes. The matrix entries of $O$ and its orthonormal eigenvectors are dimensionless numerical coefficients. The physical modal stiffness is ${\lambda }_i\mathrm{/}{t}_{\ast }^2$; an oscillatory or hyperbolic coefficient has unit $a_{\ast }$, while the slope coefficient in a zero-mode family $a_i+b_{i}t$ has unit $a_{\ast }\mathrm{/}{t}_{\ast }$. The physical frequency is $\sqrt{{\lambda }_i}\mathrm{/}{t}_{\ast }$ when ${\lambda }_i>0$.

An index is oscillatory when ${\lambda }_i>0$, with numerical frequency ${\omega }_i:=\sqrt{{\lambda }_i}$; it is a zero mode when ${\lambda }_i=0$, and an unstable (exponential) direction when ${\lambda }_i<0$. Distinct indices are degenerate when their stiffnesses agree. For real $\lambda$ define the modal eigenspace $E_{\lambda }:=\ker (K-\lambda M)$. Its dimension is the multiplicity of $\lambda$, because $q\mapsto M^{1\mathrm{/}2}q$ bijects it with the $\lambda$-eigenspace of $S$. The model is oscillatory when $K$ is non-negative. Indeed $x^{\mathsf{T}}Sx=(M^{-1\mathrm{/}2}x{)}^{\mathsf{T}}K(M^{-1\mathrm{/}2}x)$, so $K\ge 0$ exactly when every ${\lambda }_i\ge 0$; positive definiteness likewise corresponds to all ${\lambda }_i>0$. The zero-mode multiplicity is $\dim \ker K$. Frequencies are read in radians per unit time under Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

(f) Status. This item introduces objects, vocabulary, domains, regularity and dimensions for the restricted model class of this page and assigns its small-oscillation model to a mechanical equilibrium. It contains no existence, uniqueness, conservation, stability, resonance, degeneracy or approximation assertion; those are proved on this page as separate items. It adopts no force law of nature and records no measured value. Every later statement of this page names the data $(M,K,J)$ or the mechanical data $(G_{\mathrm{k}\mathrm{i}\mathrm{n}},U,q_0)$ it uses, and the one-dimensional instance is recorded in the remarks.

• The one-dimensional instance. For $n=1$ the data are $M=m>0$, $K=k\in \mathbb{R}$ and $J=f$, and the equation is $m{x}^{\mathrm{\prime }\mathrm{\prime }}+kx=f(t)$, which is exactly the forced scalar model $m{x}^{\mathrm{\prime }\mathrm{\prime }}+0\cdot x^{\mathrm{\prime }}+kx=f$ of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with $c=0$. To identify this with the physical scalar class of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria clause (f), require a length-valued position or displacement coordinate, ${\gamma }_1=\mathrm{\ell }$, read in a fixed inertial frame, take configuration domain $Q=\mathbb{R}$, and restrict to an open time interval. Reconstructing dimensional data from clause (a), the scalar $\mathsf{M}$ is then a mass in kilograms, $\mathsf{K}$ a stiffness in newtons per metre, and $\mathsf{J}$ a force in newtons; the numerical SI coefficients correspond to that definition's $(m,0,k,f)$. In particular this includes $k>0$ and constant, vanishing or sinusoidal force. With this same length-coordinate identification, at a scalar strict minimum the small-oscillation stiffness is $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })\ge 0$; the free model is the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$ when $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })>0$, and the free zero-stiffness model when $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })=0$. For an angular coordinate the dimensional $\mathsf{M}$ and $\mathsf{J}$ instead have the units of rotational inertia and torque; the mathematical scalar equation still applies, but the physical mass-and-force identification requires a separately declared change to a length coordinate. An isolated strict minimum can have zero Hessian (for example $U(x)=x^4$); the accuracy of that approximation in time and amplitude is not part of the present item.

• Why "mass metric" and not "mass matrix". The matrix $M$ is the value of the chart's kinetic metric at the equilibrium, and the quadratic form it defines is the velocity dependence of the kinetic energy. In the Cartesian-particle case $G_{\mathrm{k}\mathrm{i}\mathrm{n}}=\mathsf{M}$ is the constant mass operator of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator and the name reduces to the familiar one. With arbitrary coordinate dimensions the entries of $M$ are not masses in kilograms; only the pairing $v^{\mathsf{T}}Mv$ and the products with coordinate variations have the stated invariant dimensions.

• The dimension of the generalized force. A generalized force is defined by its pairing with coordinate variations, $J\cdot \delta q$, which must carry the energy dimension; hence its entries carry energy per unit coordinate, $\mu {\mathrm{\ell }}^2{\tau }^{-2}{\gamma }_i^{-1}$ as recorded in clause (a). For a length-valued coordinate this is the force dimension $\mu \mathrm{\ell }{\tau }^{-2}$, while for a dimensionless angle coordinate it is the torque dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$. The generalized force of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces is the same framework object, with the energy-per-coordinate dimension recorded in clause (c) of that definition.

• Prescribed generalized force. In clause (b) the generalized force is declared as a function of time alone, in the same modelling status as the external drive of The sinusoidally driven oscillator model: a prescribed external force $f_0\cos (\mathrm{\Omega }t)$ added to the damped oscillator: a source that reacts to the motion, or whose own dynamics matters, is not represented by the model. The forcing theorems of this page are statements about such prescribed data.