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Two coupled masses on a line: the in-phase and anti-phase normal modes, their frequencies, mode-energies and the zero-stiffness translation mode

Physical scope

A worked instance of the coupled quadratic model: two equal masses m on a line joined by a coupling spring c with outer springs k to fixed points, the explicit normal modes, frequencies sqrt(k/m) and sqrt((k+2c)/m), the single-mode initial data, the conserved mode energies, and the zero-stiffness case k=0 in which the pair's translation is an affine zero mode and only the relative motion oscillates. The masses and spring constants are model parameters with declared units; no measured value, no realized apparatus and no empirical claim is present.

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 particle labels of equal mass $m > 0$ constrained to a line, with coordinates $x_{1}, x_{2}$ in metres read in a fixed inertial frame (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Each label is attached to a fixed point by a spring of stiffness $k > 0$ and the labels are coupled to each other by a spring of stiffness $c > 0$ ; both constants are read in kilograms per second squared. The model is 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 $n = 2$ , coordinates $q = (x_{1}, x_{2})$ , mass metric matrix $M = m I_{2}$ and stiffness

$K = (\begin{array}{cc} k + c & - c \\ - c & k + c \end{array}),$

so that the equation $M q^{''} + K q = 0$ is the linear chain $m x_{1}^{''} = - (k + c) x_{1} + c x_{2}$ , $m x_{2}^{''} = c x_{1} - (k + c) x_{2}$ . Then:

the vectors $u_{\pm} : = (1, \pm 1) / \sqrt{2 m}$ satisfy $u_{+}^{T} M u_{+} = u_{-}^{T} M u_{-} = 1$ , $u_{+}^{T} M u_{-} = 0$ and $K u_{\pm} = λ_{\pm} M u_{\pm}$ with $λ_{+} = \frac{k}{m}, λ_{-} = \frac{k + 2 c}{m};$
the modal coordinates are $z_{\pm} = e_{\pm}^{T} M^{1 / 2} q$ with $e_{\pm} = (1, \pm 1) / \sqrt{2}$ , that is $z_{+} = \sqrt{m / 2} (x_{1} + x_{2})$ and $z_{-} = \sqrt{m / 2} (x_{1} - x_{2})$ , and every motion is $q (t) = (A \cos (ω_{+} t) + B \sin (ω_{+} t)) u_{+} + (C \cos (ω_{-} t) + D \sin (ω_{-} t)) u_{-}$ with $ω_{+} = \sqrt{k / m}$ and $ω_{-} = \sqrt{(k + 2 c) / m}$ ;
the initial data $q (0) = u_{+}$ , $q^{'} (0) = 0$ excite exactly the in-phase mode $q (t) = \cos (ω_{+} t) u_{+}$ , whose mechanical energy is the constant $E = k / (2 m)$ ; the data $q (0) = u_{-}$ , $q^{'} (0) = 0$ excite exactly the anti-phase mode $q (t) = \cos (ω_{-} t) u_{-}$ , with $E = (k + 2 c) / (2 m)$ ;
the in-phase frequency $ω_{+}$ is independent of the coupling $c$ and the anti-phase frequency $ω_{-}$ is strictly increasing in $c$ ;
if $k = 0$ while $c > 0$ , then $λ_{+} = 0$ and $u_{+}$ spans a zero mode: the general motion is $q (t) = (A + B t) u_{+} + (C \cos (ω_{-} t) + D \sin (ω_{-} t)) u_{-}$ , the pair translates freely as a whole and only the relative coordinate oscillates, at frequency $\sqrt{2 c / m}$ .

Facts & Assumptions

Given: The masses $m > 0$ , the stiffnesses $k > 0$ (and in clause 5, $k = 0$ ), $c > 0$ , the matrices $M = m I_{2}$ and $K$ of the Example, and the vectors $u_{\pm}, e_{\pm}$ .

[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, kinetic energy $\frac{1}{2} m (x_{1}^{' 2} + x_{2}^{' 2})$ , potential energy $\frac{1}{2} q^{T} K q$ and phase system $z^{'} = A z$ ; the modal vocabulary is that of clause (e) with mass metric $M$ and stiffness $K$ .

[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, 5 and 6): motions of the free model are the modal superpositions of that theorem, with oscillatory families for $λ_{i} > 0$ , affine families for $λ_{i} = 0$ and exponential families for $λ_{i} < 0$ ; these are complete; a positive definite stiffness gives bounded motions.

[F3]

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 $R^{n}$ ): matrix-vector products and inner products in $R^{2}$ are computed componentwise; $M^{1 / 2} = \sqrt{m} I_{2}$ , so $u_{\pm} = M^{- 1 / 2} e_{\pm}$ and $z = O^{T} M^{1 / 2} q$ with $O = [e_{+} e_{-}]$ ; the displayed products below are direct computations.

[F4]

Scalar oscillator calculus (The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps and diffeomorphisms): the scalar equations $z_{\pm}^{''} + λ_{\pm} z_{\pm} = 0$ with $λ_{\pm} > 0$ have exactly the families $A \cos (ω_{\pm} t) + B \sin (ω_{\pm} t)$ with $ω_{\pm} = \sqrt{λ_{\pm}}$ , and the displayed curves are differentiated by the chain rule.

Verification

Proof technique: direct; verify the modal identities by explicit matrix multiplication and invoke the free-mode theorem for the motion and energy statements.

1.1F1F3algebra

Modal identities: $M$ is symmetric positive definite and $K$ is symmetric with $q^{T} K q = (k + c) (x_{1}^{2} + x_{2}^{2}) - 2 c x_{1} x_{2} = k (x_{1}^{2} + x_{2}^{2}) + c (x_{1} - x_{2})^{2} \geq 0$ for $k > 0, c > 0$ (indeed positive definite), so the model is an oscillatory coupled quadratic model of [F1]. Multiplying, $K u_{+} = (k, k) / \sqrt{2 m} = k (1, 1) / \sqrt{2 m} = \frac{k}{m} M u_{+}$ and $K u_{-} = (k + 2 c, - (k + 2 c)) / \sqrt{2 m} = \frac{k + 2 c}{m} M u_{-}$ ; and $u_{\pm}^{T} M u_{\pm} = m ∣ u_{\pm} ∣_{2}^{2} = m \cdot \frac{2}{2 m} = 1$ , $u_{+}^{T} M u_{-} = m (1 - 1) / (2 m) = 0$ . This proves clause 1 with $λ_{\pm}$ as displayed.

2.1F1F3step 1.1algebra

Modal coordinates: with $e_{\pm} = (1, \pm 1) / \sqrt{2}$ the matrix $O = [e_{+} e_{-}]$ is orthogonal and $M^{- 1 / 2} K M^{- 1 / 2} = \frac{1}{m} K$ has the orthonormal eigenbasis $e_{\pm}$ with eigenvalues $λ_{\pm}$ by clause 1; hence $z = O^{T} M^{1 / 2} q$ has $z_{\pm} = e_{\pm}^{T} \sqrt{m} q = \sqrt{m / 2} (x_{1} \pm x_{2})$ and $u_{\pm} = M^{- 1 / 2} e_{\pm}$ , so the modal 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 exactly those displayed.

2.2step 1.1algebra

Frequency comparison: $ω_{+} = \sqrt{k / m}$ contains no $c$ , while $c \mapsto ω_{-} = \sqrt{(k + 2 c) / m}$ is strictly increasing on $c > 0$ with $ω_{-} > ω_{+}$ : the in-phase oscillation is softer than the anti-phase one. This is clause 4.

2.3F2step 1.1algebra

Zero stiffness: if $k = 0$ then $K = c (\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ , $K u_{+} = 0$ and $K u_{-} = \frac{2 c}{m} M u_{-}$ , so $λ_{+} = 0$ and $λ_{-} = 2 c / m$ ; by [F2] the general motion is the affine-plus-oscillatory family displayed in clause 5, in which the pair translates as a whole (arbitrary constant velocity $B$ ) and only the relative coordinate $x_{1} - x_{2}$ oscillates, with $ω_{-} = \sqrt{2 c / m}$ . This is clause 5.

3.1F2F4step 1.1step 2.1

Motions: by [F2] and [F4], every motion is the superposition of clause 2 with $λ_{\pm} > 0$ , i.e. the displayed two-frequency family, and the coefficients $(A, B, C, D)$ are bijective onto the data $(q (0), q^{'} (0)) = (A u_{+} + C u_{-}, B ω_{+} u_{+} + D ω_{-} u_{-})$ . This is clause 2.

4.1F2step 1.1step 3.1algebra

Single-mode data: for $q (0) = u_{+}$ , $q^{'} (0) = 0$ the coefficient system gives $A = 1$ , $B = C = D = 0$ , so $q (t) = \cos (ω_{+} t) u_{+}$ ; its energy is $E = \frac{1}{2} q^{' T} M q^{'} + \frac{1}{2} q^{T} K q = \frac{1}{2} ω_{+}^{2} \sin^{2} (ω_{+} t) + \frac{1}{2} \frac{k}{m} \cos^{2} (ω_{+} t) = \frac{k}{2 m}$ , constant because $ω_{+}^{2} = k / m$ ; the same computation with $λ_{-} = (k + 2 c) / m$ gives $E = (k + 2 c) / (2 m)$ for the anti-phase datum. This is clause 3.

5.1step 1.1step 3.1step 4.1step 2.2step 2.3∎

Clauses 1-5 are verified on the explicit data; all parameters are model data with declared units, and no measured frequency, damping or apparatus behaviour is asserted.

Remarks

The coupling acts only on the relative coordinate. The change of coordinates $(x_{1}, x_{2}) \mapsto (x_{1} + x_{2}, x_{1} - x_{2})$ decouples the chain: the sum is governed by the outer springs alone and the difference by the outer springs plus twice the coupling. This is the physical reading of clauses 1-4 and the reason the in-phase frequency is independent of $c$ .
The zero mode is a translation, not an oscillation. Clause 5 is the two-mass instance of the zero-mode warning recorded in Scope of the coupled quadratic model: zero modes, degenerate eigenspaces, unstable quadratic directions and the limits of the approximation statements: the mode of zero frequency is the uniform motion of the pair, its family is affine, and it is unbounded whenever the pair has nonzero total momentum.

Two coupled masses on a line: the in-phase and anti-phase normal modes, their frequencies, mode-energies and the zero-stiffness translation mode

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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