27.4 The Coupled Pendula Problem

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The Coupled Pendula Problem

Suppose we have a pair of pendula, each composed of a mass attached to a rod of length $ℓ$ , and that the two masses are coupled together by an extension spring with spring constant $k$ .
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If we attach a ruler directly under the masses and set at least one mass in motion, our challenge is to predict the location of the center of each mass along the ruler at any time.

Remark. This problem is actually 27.1 The Standard Eigenvalue Problem in disguise. When solving for our eigenvalues, we search for a few modes first.

First natural frequency (regular pendulum swing AKA cosine curve)

A {\vec{x}}_{1} = λ_{1} {\vec{x}}_{1}

Second natural frequency (opposite pendulum swing AKA opposite cosine curves)

A {\vec{x}}_{2} = λ_{2} {\vec{x}}_{2}

Claim

The complex functions that arise when starting the system with different position masses can be found by taking 6.1 Linear Combination of Vectors of modes 1 and 2.

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Steps to Mathematize Coupled Pendula Problem

In order to mathematize the coupled pendula problem, we will:

Study the motion and dynamics of a single pendulum
Formulate a modeling framework for our coupled system of masses by introducing proper variables and notation.
Analyze system forces on each mass using Free Body Diagrams (FBD)
Derive a set of coupled Differential Equations to model the motion of masses in the system.
Transform the coupled ordinary different equations from step 4 into a standard eigenvalue problem (SEP) by creative use of matrices.