Two Pendulums Coupled by a Spring: Normal Modes, Beating, and Normal Coordinates

Two Pendulums Coupled by a Spring

The MIT OpenCourseWare video presents a clear, hands‑on investigation of two identical pendulums connected by a spring. Substituting for Professor Lee, Bulle Weisloach walks through turning a physical two‑mass–two‑pendulum system into a fixed mathematical form, deriving the horizontal equations of motion under small‑angle approximation, and packaging them in a 2×2 matrix. She then explains how the coupled equations admit two normal modes with fixed frequencies, corresponding to the pendulums moving in the same direction or in opposite directions. The frequencies, the mode shapes, and the concept of normal coordinates are developed, followed by an exploration of beating and energy transfer between the masses, and a demonstration of how changing initial conditions or external gravity modifies the observed motion. The session also previews extensions to more oscillators and driven systems.

Introduction to Coupled Oscillators

The video focuses on a simple physical model: two equal pendulums hanging from a fixed support and connected by a spring. The instructor emphasizes converting this realistic setup into a standard mathematical form suitable for analyzing any coupled oscillator system. With the small angle approximation, gravity and the spring provide the restoring forces, while the horizontal motion of each mass couples through the spring. The problem is parameterized by two coordinates X1 and X2, representing displacements from equilibrium. The goal is to obtain equations of motion in a fixed notation that can be generalized to chains of many oscillators.

Matrix Formulation and Normal Modes

The next step is to assemble the dynamics into matrices. The two mass coordinates yield a 2×2 force matrix, containing gravitational and spring contributions, and a mass matrix with equal masses on the diagonal. The equations of motion are written as X¨ = −M^{-1}K X, where K encodes the fixed system parameters and M is the mass matrix. To find normal modes, the standard ansatz X = A e^{i ω t} is used, leading to a linear eigenvalue problem det(M^{-1}K − ω^2 I) = 0. Solving gives two normal frequencies: ω1^2 = g/L and ω2^2 = g/L + 2K/M. The corresponding mode shapes are found from the eigenvectors. The first mode has both masses moving in unison, the second in opposition. The physics is that the spring does not affect the in‑phase mode, while it stiffens the out‑of‑phase mode, increasing its frequency.

Normal Modes, Mode Shapes, and Superposition

The analysis shows that any motion can be expressed as a linear superposition of the two normal modes. The general solution combines two cosine terms, one for each mode, scaled by amplitudes and phased as determined by initial conditions. In this framework, X1 and X2 trace out the two characteristic patterns: mode 1 where X1 ≈ X2, and mode 2 where X1 ≈ −X2. The instructor illustrates how specific initial displacements and velocities lead to different blends of the two modes, producing a variety of visually rich motions even in this simple system.

Beating, Energy Exchange, and Normal Coordinates

A key phenomenon explored is beating when the two frequencies are close. Writing the displacements as sums of cosines highlights a fast carrier frequency set by the average of the two ωs and a slow envelope set by their difference. The energy slowly shifts between the pendulums in this beat pattern, a hallmark of coupled oscillators. The lecture also introduces normal coordinates U1 = X1 + X2 and U2 = X1 − X2, which decouple the equations, yielding two independent one‑dimensional harmonic oscillators with frequencies ω1 and ω2. This transformation clarifies the dynamics and provides a powerful technique for larger, more complex systems. The talk closes with demonstrations and discussions of how changing conditions, such as gravity, can modify the frequencies and beat behavior, and hints at extending the model to driven oscillators and chains of many masses.

Takeaways and Extensions

Throughout the session, the emphasis remains on building a fixed, systematic approach to any coupled oscillator: identify coordinates, derive the matrix form, solve for normal modes and frequencies, deduce mode shapes, and then reconstruct general motion via superposition. The concepts of normal modes, normal coordinates, and beating are presented as universal tools for understanding complex, even chaotic, behavior in simple symmetric systems. The instructor also teases later topics, including forced oscillations and driving, to broaden the scope from two masses to infinite chains.