Questions: Normal Modes and Collective Oscillations

Question 1

A coupled two-mass system is initialized in exactly the in-phase normal mode (both masses displaced equally in the same direction). What is the subsequent motion?

• A. The system gradually transfers energy to the out-of-phase mode, producing beats after many oscillations
• B. The system oscillates at the in-phase frequency indefinitely with fixed amplitude ratios, with no energy entering the out-of-phase mode
• C. Both masses oscillate at two frequencies simultaneously, since coupling always activates both modes
• D. The coupling spring causes the motion to become chaotic after many oscillations

Answer: (B)

This is the defining property of a normal mode: when excited purely in a single mode, the system remains in that mode indefinitely, oscillating at that mode's frequency with fixed amplitude ratios. Normal modes are the eigensolutions — they evolve independently. No energy transfers between modes when the initial condition is a pure normal mode. Energy exchange and beats only occur when BOTH modes are simultaneously excited by a general initial condition.

Question 2

What mathematical structure does finding normal modes reduce to, and what do the solutions represent physically?

• A. A system of first-order ODEs; eigenvalues give decay rates, eigenvectors give phase relationships
• B. A generalized eigenvalue problem Kv = ω²Mv; eigenvalues ω² are squared normal mode frequencies, eigenvectors v give amplitude ratios
• C. A Fourier series expansion; eigenvalues are harmonic frequencies, eigenvectors are Fourier coefficients
• D. A linear programming problem; eigenvalues are energy bounds, eigenvectors are stable configurations

Answer: (B)

Assuming oscillatory solutions x(t) = v·e^{iωt}, the equations of motion M**ẍ** = −**K**x become **K**v = ω²**M**v — a generalized eigenvalue problem. Each eigenvalue ω² gives a squared normal mode frequency, and the corresponding eigenvector v gives the amplitude ratios between masses (e.g., equal same-direction motion for the in-phase mode; equal opposite-direction motion for the out-of-phase mode). This is the direct mechanical application of eigenvalue decomposition.

Question 3

Any motion of a coupled oscillator system can be expressed as a superposition of its normal modes, each evolving independently at its own frequency.

• T. True
• F. False

Answer: True

The normal mode eigenvectors form a complete orthogonal basis for the configuration space (in the metric defined by the mass matrix M). Any initial condition can be decomposed into contributions from each mode, and since modes evolve independently, the actual motion is their sum. This is the mechanical analog of Fourier decomposition: just as any periodic function is a sum of sinusoids, any coupled oscillator motion is a sum of independently oscillating normal modes.

Question 4

In a symmetric coupled two-mass system (equal masses, equal outer springs, coupling spring k_c), the out-of-phase normal mode has a lower frequency than the in-phase mode.

• T. True
• F. False

Answer: False

It is the opposite. In the in-phase mode (masses moving together), the coupling spring is unstretched — only the outer springs contribute to the restoring force, giving ω₁ = √(k/m). In the out-of-phase mode (masses moving in opposition), the coupling spring is compressed and stretched, adding to the restoring force: ω₂ = √((k + 2k_c)/m) > ω₁. The coupling spring raises the out-of-phase frequency. The in-phase mode is always the lowest-frequency mode.

Question 5

Explain why decomposing coupled oscillator motion into normal modes is useful, and how it transforms the problem mathematically.

Model answer: The coupled equations M**ẍ** = −**K**x are a system of coupled differential equations — difficult to solve directly. By transforming to normal mode coordinates (projecting the initial conditions onto the eigenvectors), the equations decouple completely: each normal coordinate ξᵢ satisfies the independent equation ξ̈ᵢ = −ωᵢ²ξᵢ, a simple harmonic oscillator with known solution Aᵢcos(ωᵢt + φᵢ). The physical motion is recovered by summing mode contributions: x(t) = Σ ξᵢ(t)·vᵢ. This converts k coupled ODEs into k independent ODEs, each trivially solvable. The eigenvalue problem is solved once; everything else follows by superposition.

The technique generalizes across physics: a vibrating string's normal modes are harmonics; a molecule's normal modes determine its infrared absorption spectrum; a crystal lattice's normal modes are phonons. The mathematical structure — eigenvalue decomposition of the coupled equations — is identical across all these domains, making normal modes one of the most transferable tools in theoretical physics.