§ Paper 01 · Derivation · MER 2.0

A Three-Mass Coupled Pendulum Students Can Verify Without a Teacher

Anderson (2018) uses a two-mass spring-coupled pendulum to make eigenvalues physical. MER 2.0 extends the system to three masses, where the added mode gives us a dimensionless frequency ratio that a student can check against prediction without knowing any material constants.

This page is a working derivation — the math a Foothill Math 2B student should be able to follow with an eigenvalue-decomposition sheet and Newton's second law.

Status: Working draft · not yet mentor-reviewed
Author: Henry Fan (student of Prof. Jeff Anderson)
For: Paper 01 of The Modeling Bench — MER 2.0
Feedback: via mentor update, following the 8-step modeling structure

§1Physical setup

Three identical pendula of mass $m$ and effective length $L$ hang from a shared horizontal rail. Two identical springs of stiffness $k_c$ couple pendulum 1 to pendulum 2, and pendulum 2 to pendulum 3. There is no spring between pendulum 1 and pendulum 3 directly. The only restoring force on each pendulum individually is gravity.

Figure 1. Three identical pendula coupled by two identical springs. The displacements $x_1$, $x_2$, $x_3$ are measured horizontally from each pendulum's equilibrium position. The derivation below uses the small-angle approximation so the restoring force from gravity is linear: each pendulum alone has effective stiffness $k_p = mg/L$.

§2Warm-up: the two-mass case

Before adding the third pendulum, review the two-mass system Anderson (2018) uses in the MER project. Let $x_1$ and $x_2$ be the horizontal displacements of two coupled pendula, with small-angle restoring stiffness $k_p = mg/L$ each and a coupling spring of stiffness $k_c$ between them.

Newton's second law on each mass gives, in the small-angle limit:

$$ \begin{aligned} m\ddot{x}_1 &= -k_p\,x_1 - k_c(x_1 - x_2) = -(k_p + k_c)\,x_1 + k_c\,x_2 \\ m\ddot{x}_2 &= -k_p\,x_2 - k_c(x_2 - x_1) = -(k_p + k_c)\,x_2 + k_c\,x_1 \end{aligned} $$

In matrix form with $\mathbf{x} = (x_1, x_2)^\top$:

$$ m\,\ddot{\mathbf{x}} = -K_2\,\mathbf{x},\qquad K_2 = \begin{pmatrix} k_p + k_c & -k_c \\ -k_c & k_p + k_c \end{pmatrix}. $$

Try the standing-wave ansatz $\mathbf{x}(t) = \mathbf{v}\,e^{i\omega t}$. Substituting and cancelling the time factor:

$$ \bigl(K_2 - m\omega^2 I\bigr)\,\mathbf{v} = \mathbf{0}. $$

So $\omega^2$ and $\mathbf{v}$ are an eigenvalue and eigenvector of $\frac{1}{m}K_2$. For the symmetric $2\times 2$ case above, the eigenvalues come out by inspection: the matrix $K_2$ splits as $k_p\,I + k_c\,A_2$ where

$$ A_2 = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}. $$

$A_2$ has eigenvalues $\mu = 0$ (eigenvector $(1, 1)^\top$) and $\mu = 2$ (eigenvector $(1, -1)^\top$), so the two-pendulum system has natural frequencies

$$ \omega_1^2 = \frac{k_p}{m},\qquad \omega_2^2 = \frac{k_p + 2 k_c}{m}. $$

The $\omega_1$ mode is both pendula swinging in phase — the coupling spring stays at its natural length and contributes nothing, so the system oscillates at the bare pendulum frequency. The $\omega_2$ mode is the two pendula swinging exactly out of phase, stretching and compressing the spring maximally.

Why Anderson (2018) stops here. Two modes are enough to teach the idea that eigenvectors are physical things and eigenvalues are measurable frequencies. But two modes don't give the student a way to check their model against their data without assuming a value for $k_p$ or $k_c$. The three-mass case below fixes that.

§3Three-mass stiffness matrix

With three pendula and two coupling springs (Fig. 1), Newton's second law on each mass gives:

$$ \begin{aligned} m\ddot{x}_1 &= -k_p x_1 - k_c(x_1 - x_2) \\ m\ddot{x}_2 &= -k_p x_2 - k_c(x_2 - x_1) - k_c(x_2 - x_3) \\ m\ddot{x}_3 &= -k_p x_3 - k_c(x_3 - x_2) \end{aligned} $$

Collecting terms, $m\,\ddot{\mathbf{x}} = -K_3\,\mathbf{x}$ with stiffness matrix

$$ K_3 = \begin{pmatrix} k_p + k_c & -k_c & 0 \\ -k_c & k_p + 2 k_c & -k_c \\ 0 & -k_c & k_p + k_c \end{pmatrix}. $$

Split as before: $K_3 = k_p\,I + k_c\,A_3$, where

$$ A_3 = \begin{pmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{pmatrix}. $$

The eigenvalue problem reduces to diagonalizing $A_3$. Expanding the characteristic polynomial:

$$ \det(A_3 - \mu I) = (1-\mu)\bigl[(2-\mu)(1-\mu) - 1\bigr] - (1-\mu). $$

Factor out $(1-\mu)$:

$$ = (1-\mu)\bigl[(2-\mu)(1-\mu) - 2\bigr] = (1-\mu)\bigl[\mu^2 - 3\mu\bigr] = \mu\,(1-\mu)\,(\mu - 3). $$

The eigenvalues of $A_3$ are therefore $\mu = 0,\ 1,\ 3$, and the natural frequencies of the three-mass system are

$$ \omega_1^2 = \frac{k_p}{m},\qquad \omega_2^2 = \frac{k_p + k_c}{m},\qquad \omega_3^2 = \frac{k_p + 3 k_c}{m}. $$

The three eigenvectors (normal modes)

Solving $(A_3 - \mu I)\,\mathbf{v} = \mathbf{0}$ at each eigenvalue gives three orthogonal modes:

Mode	$\mu$	$\omega^2$	Eigenvector $\mathbf{v}$	Physical picture
$\omega_1$ — In-phase	$0$	$k_p / m$	$(1,\ 1,\ 1)^\top / \sqrt{3}$	All three pendula swing together. Neither spring ever compresses or stretches — the bare pendulum frequency.
$\omega_2$ — Antisymmetric (middle still)	$1$	$(k_p + k_c) / m$	$(1,\ 0,\ -1)^\top / \sqrt{2}$	Outer pendula move exactly out of phase; the middle pendulum stays at rest. Each spring feels only one moving mass.
$\omega_3$ — Breathing	$3$	$(k_p + 3 k_c) / m$	$(1,\ -2,\ 1)^\top / \sqrt{6}$	Middle pendulum swings against the two outer pendula. Both springs stretch and compress together at maximum amplitude — highest frequency.

A student can check orthogonality directly: the three eigenvectors above have zero pairwise dot products, which is the experimental guarantee that any initial condition can be decomposed into a sum of these three modes that evolve independently.

§4The teacher-free verification test

The clean pedagogical payoff of the three-mass case is a dimensionless prediction that depends on no measured constants. Subtract $\omega_1^2$ from $\omega_2^2$ and $\omega_3^2$:

$$ \omega_2^2 - \omega_1^2 = \frac{k_c}{m},\qquad \omega_3^2 - \omega_1^2 = \frac{3 k_c}{m}. $$

Divide one by the other: everything that depends on $k_c$, $k_p$, $m$, and $L$ cancels.

Prediction — MER 2.0 central test

For three identical pendula coupled by two identical springs, the three measured natural frequencies must satisfy:

$$\boxed{\;\dfrac{\omega_3^2 - \omega_1^2}{\omega_2^2 - \omega_1^2} = 3.\;}$$

What the student measures: three natural frequencies from their phone-camera footage. What they compute: one dimensionless ratio. What they compare to: the integer 3.

No knowledge of $k_p$, $k_c$, $m$, or $L$ is required. A student at their kitchen table can verify — or falsify — the eigenvalue prediction without a teacher present. That is the verifiability floor the paper is about.

What a good pilot looks like

If the student's measured ratio lands in the range $[2.7,\ 3.3]$ (a 10% tolerance chosen to absorb spring nonlinearity, finite-amplitude corrections, and phone-video timing jitter), Paper 01 treats the pilot as a success. Systematic deviations — say, a measured ratio of $2.4$ across every group — would indicate a real physical effect we don't yet understand and would become the focus of the "Analyze" and "Verify" steps in Anderson's eight-step modeling process.

§5Measurement protocol

What the student actually does with the kit, distilled:

Mount the three pendula on a shared rail with a printed coordinate scale behind them.
Record the pendula for 30 seconds at 60 frames per second with a phone on a tripod. Two runs: one excited in the in-phase mode, one excited in a mixed initial condition so that all three modes are present.
Track the three bob positions through the MER 2.0 OpenCV pipeline to produce a CSV of $(t,\, x_1,\, x_2,\, x_3)$.
FFT each displacement signal and locate the three peaks. (A student who has not yet seen Fourier transforms can use a simpler peak-spacing protocol: read the period off a few consecutive zero crossings.)
Compute the ratio $(\omega_3^2 - \omega_1^2)/(\omega_2^2 - \omega_1^2)$ and compare to $3$.

The entire measurement protocol uses no physics beyond "the bob goes back and forth at a frequency I can read off a timestamped camera," and no math beyond eigenvalue decomposition. That matches the Math 2B prerequisite exactly.

§6Open questions

Things this derivation does not yet handle; items to discuss with Jeff in the Monday call:

Spring nonlinearity at finite amplitude. Real coil springs have a cubic term that shifts $\omega_3$ more than $\omega_1$. Does the $\pm 10\%$ tolerance swallow this, or do we need to derive a first-order correction?
Asymmetric pendula. If the three pendulum lengths $L_1, L_2, L_3$ are not identical, the clean $3$ prediction collapses and we get a messier formula. The paper should report the measurable sensitivity and say how carefully the student needs to match pendulum lengths — mm? cm?
Damping. The ansatz $\mathbf{x} = \mathbf{v}\,e^{i\omega t}$ assumes no damping. Air and pivot friction make the signal decay. For a 30-second video the decay is usually small enough that FFT still resolves the three peaks cleanly, but this needs empirical confirmation.
The "find" step. Anderson's eight-step modeling process begins with find — something in the world that matters. The three-pendulum system is pedagogically clean but physically contrived. Is there a real coupled-oscillator story (a building on a shake table? a bridge? a diatomic molecule?) that makes the $\omega_3^2 - \omega_1^2 = 3(\omega_2^2 - \omega_1^2)$ prediction feel like it answers a question a student already had?