Normal Modes of a System of Masses

Live Script exploring normal modes of four masses densely connected by springs moving in one dimension.

The normal modes of oscillation of N masses connected by springs is a standard topic in an intermediate-level course in mechanics. Normal modes are harmonic solutions to the equations of motion with frequencies determined by finding the roots of a characteristic polynomial equation of order Nand, for each root, the associated eigenvector. A general solution may be constructed as a linear superposition of normal mode solutions.

This educational Live Script illustrates normal modes of a closed system of four masses moving along a line with springs connecting all mass pairs. The masses and spring constants are arbitrary and may be chosen to study lower dimensional coupled oscillator problems.

The eigenvalue problem determining the normal mode frequencies is set up symbolically and the zero frequency mode corresponding to uniform motion of the closed system is identified. For the user's choice of masses and spring constants, the eigenvalue problem is solved numerically, and each normal mode motion is illustrated through an animation. Then, for the user's choice of initial conditions, subject to a constraint of zero net momentum, the exact solution is computed using the normal mode expansion. That motion is visualized in an animation and the frequency spectrum associated with the motion of each mass is found.

This script may interest students and instructors in physics and related fields. 'Try this' suggestions, coding 'Challenges', hyperlink, and references are provided for further exploration. Additional educational Live Scripts by the author are here.