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A doubly degenerate quadratic model: every direction is a mode, the modal basis is not intrinsic, and coupling splits the double frequency

Physical scope

A worked instance of degeneracy in the coupled quadratic model: two identical uncoupled oscillators, M = m I, K = k I, with the double modal stiffness k/m, the full two-dimensional eigenspace, the four-parameter family of motions, the non-intrinsic status of every modal basis and the splitting of the double modal stiffness into k/m and (k+2c)/m, with corresponding frequencies sqrt(k/m) and sqrt((k+2c)/m), when a coupling c>0 of the two-mass model is switched on. The parameters are model data; the example asserts no measured frequency and no general stability-of-degeneracy theorem.

Example

The algebra below uses the fixed-unit numerical convention of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium, with coordinate units metres, time unit $1 s$ and energy unit $1 J$ . Physical model parameters are given their stated units; in matrix products, initial-vector identities and coefficient formulas their numerical coefficients in these fixed units are used. Modal amplitudes have common unit $a_{*} = (1 J)^{1 / 2} (1 s)$ ; the $j$ -th component of a mass-normalized modal vector has unit $s_{j} / a_{*}$ . Modal oscillatory and hyperbolic coefficients have unit $a_{*}$ , and zero-mode slopes have unit $a_{*} / (1 s)$ . Physical coordinates and energies are reconstructed with the coordinate units and $1 J$ , respectively. In particular a numerical single-mode datum $q (0) = u_{i}$ means a modal amplitude of one in this common amplitude unit.

Global-time, growth and boundedness claims below refer to the unique maximal extension of the autonomous free equation on $R$ ; local motions remain its restrictions. Values at $0$ in coefficient formulas are evaluated on that extension when the original interval omits $0$ .

Consider two uncoupled oscillators of equal mass and stiffness, modelled by 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 $n = 2$ ,

$M = m I_{2}, K = k I_{2}, J \equiv 0, m > 0, k > 0.$

The configuration-vector coefficients $a, b$ in clause 2 are not modal coefficients: their physical components have the coordinate unit $1 m$ ; converting them to modal amplitudes multiplies by $M^{1 / 2}$ in the numerical convention.

Then:

$M$ and $K$ are symmetric positive definite, the matrix $M^{- 1 / 2} K M^{- 1 / 2} = (k / m) I_{2}$ has the double eigenvalue $λ = k / m$ , the modal eigenspace is $E_{λ} = R^{2}$ and $N_{λ} : = R^{2}$ ; every orthonormal basis ${e_{1}, e_{2}}$ of $R^{2}$ is an orthonormal eigenbasis of that matrix, so the decoupling data are not unique;
every motion is $q (t) = \cos (ω t) a + \sin (ω t) b, a, b \in R^{2}, ω = \sqrt{k / m},$ and every nonzero vector $a \in R^{2}$ spans a pure mode: the datum $q (0) = a$ , $q^{'} (0) = 0$ produces the single-frequency oscillation $q (t) = \cos (ω t) a$ ;
the frequency $ω$ and the subspace $N_{λ} = R^{2}$ are intrinsic, while individual modal vectors are not: there is no basis-independent meaning to "the first mode" and no separation of the motions into two distinguished one-parameter families;
splitting by coupling. The two-mass model of Two coupled masses on a line: the in-phase and anti-phase normal modes, their frequencies, mode-energies and the zero-stiffness translation mode with coupling $c > 0$ has stiffness $K_{c} = (\begin{array}{cc} k + c & - c \\ - c & k + c \end{array})$ , which equals $k I_{2}$ at $c = 0$ and coincides with the data of this example when $c = 0$ ; for $c > 0$ its modal stiffnesses are the distinct numbers $k / m$ and $(k + 2 c) / m$ , so an explicit change of the model data separates the double eigenvalue into the unchanged in-phase frequency and the raised relative frequency.

Facts & Assumptions

Given: The data $m, k > 0$ , $M = m I_{2}$ , $K = k I_{2}$ , the frequency $ω = \sqrt{k / m}$ and, in clause 4, the mass metric and stiffness $K_{c}$ of Two coupled masses on a line: the in-phase and anti-phase normal modes, their frequencies, mode-energies and the zero-stiffness translation mode with coupling $c > 0$ .

[F1]

Model dictionary (The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium): the data $(M, K, 0)$ declare a free coupled quadratic model with two degrees of freedom and the modal vocabulary of clause (e).

[F2]

Free-mode theory (The free coupled quadratic model: global existence, the complete modal classification, mode energies, intrinsic degenerate mode subspaces, boundedness and the unstable direction clauses 2, 4 and 5): motions are the modal superpositions; for a modal stiffness $λ$ the pure- $λ$ motions are the solutions of $q^{''} + λ q = 0$ with values in the intrinsic subspace $N_{λ}$ ; a positive definite stiffness gives bounded motions.

[F3]

Coupled two-mass data and modes (Two coupled masses on a line: the in-phase and anti-phase normal modes, their frequencies, mode-energies and the zero-stiffness translation mode clauses 1, 2 and 4): for $M = m I_{2}$ and $K_{c} = (\begin{array}{cc} k + c & - c \\ - c & k + c \end{array})$ with $c > 0$ , the modal stiffnesses are $λ_{+} = k / m$ and $λ_{-} = (k + 2 c) / m$ with the stated modal vectors.

[F4]

Direct linear algebra (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 $R^{n}$ ): $M^{- 1 / 2} = \frac{1}{\sqrt{m}} I_{2}$ , so $M^{- 1 / 2} K M^{- 1 / 2} = \frac{k}{m} I_{2}$ ; a scalar matrix commutes with every orthogonal matrix, so every orthonormal basis diagonalizes it.

[F5]

Scalar oscillator calculus (The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data): for $λ > 0$ the scalar equation $z^{''} + λ z = 0$ has exactly the family $A \cos (ω t) + B \sin (ω t)$ with $ω = \sqrt{λ}$ .

Verification

Proof technique: direct; identify the scalar multiple of the identity, read the double eigenvalue and the flexibility of the modal basis, and compare the data with the coupled two-mass instance.

1.1F1F4algebra

Degeneracy: by [F4] the matrix $M^{- 1 / 2} K M^{- 1 / 2}$ is $(k / m) I_{2}$ , whose only eigenvalue is $k / m$ with eigenspace $R^{2}$ ; any orthonormal basis diagonalizes it because a scalar matrix commutes with every matrix. Hence $λ_{1} = λ_{2} = k / m$ , $E_{λ} = R^{2}$ , $N_{λ} = R^{2}$ and the decoupling data of clause (e) of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium are non-unique. This is clause 1.

2.1F1F2F5step 1.1

Motions: by [F2] and [F5] the motions are the superpositions of the scalar families with the single frequency $ω = \sqrt{k / m}$ in every direction, i.e. the displayed four-parameter family $q (t) = \cos (ω t) a + \sin (ω t) b$ ; the datum $q (0) = a$ , $q^{'} (0) = 0$ corresponds to $b = 0$ and gives $q (t) = \cos (ω t) a$ , a motion whose values lie in the line spanned by $a$ for all $t$ : by clause 4 of [F2] this is a pure- $λ$ motion, and every nonzero $a$ gives one. This is clause 2.

2.2F3step 1.1

Splitting by coupling: the stiffness of [F3] reduces to $k I_{2}$ at $c = 0$ , so the data of this example and of the coupled two-mass model agree at $c = 0$ ; for $c > 0$ , [F3] gives the distinct modal stiffnesses $k / m$ (the in-phase mode, unchanged by the coupling) and $(k + 2 c) / m$ (the relative motion, raised by the coupling). This is clause 4.

3.1F2step 1.1step 2.1

Intrinsic versus non-intrinsic data: the number $λ = k / m$ and the subspace $N_{λ} = R^{2}$ are determined by $M$ and $K$ alone, whereas an individual modal vector is a member of a chosen orthonormal basis and different admissible decouplings assign the same motion different coordinate labels; hence no basis-independent decomposition of the four-parameter family into two distinguished one-parameter families exists here. This is clause 3.

4.1step 1.1step 3.1step 2.2∎

Clauses 1-4 are verified on the explicit data; the splitting statement is a comparison of two explicitly specified models at different parameter values and is not a general assertion that degeneracies are always removed by perturbations.

Remarks

Degeneracy is a property of the data. The double eigenvalue exists because the two oscillators are identical and uncoupled; the identity of the oscillators is a model assumption, and a model with slightly different masses or stiffnesses need not have any degeneracy. Nothing in this example asserts how degeneracies behave under general parameter changes.
What clause 4 does show. In the specific family of the two-mass model, switching on the coupling from $c = 0$ to $c > 0$ splits the double eigenvalue $k / m$ into $k / m$ and $(k + 2 c) / m$ : the in-phase frequency is unchanged and the relative frequency is raised. The two limiting descriptions are consistent, as they must be, since the data agree at $c = 0$ .

A doubly degenerate quadratic model: every direction is a mode, the modal basis is not intrinsic, and coupling splits the double frequency

Example

Facts & Assumptions

Verification

Remarks

Depends on

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