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MathidiaMathematics Academy
Analytical mechanics
Lesson 58 of 75
Standard practice

Coupled oscillations

Objective. Find normal modes of linear systems.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Coupled oscillations occur when two or more oscillators interact through springs or forces, producing combined motions called normal modes, each oscillating at a single shared frequency. Writing Newton's equations as a matrix system Mẍ = −Kx, normal modes solve x = a·e^(iωt) giving the eigenvalue problem (K−ω²M)a=0, whose solutions ω² are the mode frequencies. Example: two equal masses m connected by springs of stiffness k (outer) and k (middle) have modes where masses move in phase (ω²=k/m) and out of phase (ω²=3k/m). Any motion of the coupled system is a superposition of these normal modes.

Worked example

Two equal masses m connected by three identical springs of stiffness k (wall–mass, mass–mass, mass–wall) undergo coupled oscillations. Find the two normal mode frequencies squared, ω².

Diagram of a square
  1. 1Write equations: mẍ₁ = −kx₁ − k(x₁−x₂), mẍ₂ = −kx₂ − k(x₂−x₁)
  2. 2Try symmetric mode x₁=x₂: equations reduce to mẍ = −kx, giving ω² = k/m
  3. 3Try antisymmetric mode x₁=−x₂: equations reduce to mẍ = −3kx, giving ω² = 3k/m
  4. 4The two normal mode frequencies squared are k/m and 3k/m

Answer: ω² = k/m and ω² = 3k/m

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Common mistake. Students treat each mass's motion as an independent single oscillator, missing that true natural frequencies come only from solving det(K−ω²M)=0 for the whole coupled system, not from isolated spring-mass pairs.

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