Small Oscillations and Normal Modes

Linearization near equilibrium and modal decomposition. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Near a stable equilibrium, almost any mechanical system behaves, to first approximation, as a set of independent harmonic oscillators, even though the original coordinates are completely coupled to each other. This surprising result is the reason the simple harmonic oscillator appears again and again in physics: not because nature is full of ideal springs, but because any stable equilibrium, seen up close, looks like one.

This leaf covers how to find that decomposition: linearizing the equations of motion around equilibrium, solving the generalized eigenvalue problem that gives the frequencies $\omega_n$ and modes $\phi_i^{(n)}$, and reconstructing the general motion as a superposition of those modes. To fix ideas with concrete numbers, this leaf uses throughout the system of two identical blocks of mass $m$, each attached to a wall by a spring of constant $k_w$ and connected to each other by a coupling spring of constant $k_c$.

🟢 Essential level

The essential idea is that, near a stable equilibrium, any system's potential looks like a quadratic surface (a multidimensional paraboloid), and the resulting motion, even though the coordinates may seem coupled in a complicated way, always decomposes into a set of independent modes, each oscillating at its own fixed frequency.

In the concrete two-coupled-block system, this idea is directly visible: there are exactly two independent modes. In the symmetric mode, both blocks move exactly alike, in the same direction and with the same amplitude, and the coupling spring plays no role at all. In the antisymmetric mode, the blocks move in opposite directions, and the coupling spring does stretch, adding effective stiffness and therefore a higher frequency.

Any initial condition of the two blocks, however complicated it may look, can always be written as a combination of these two simple superposed motions; no other kind of motion is possible in the small-oscillation regime.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

The formal level rests on six relations, expressed in terms of the small-displacement vector $\eta_i$ about equilibrium:

Linearized equation of small oscillations

Modal eigenvalue problem

Modal superposition

Modal orthogonality

Symmetric mode's frequency (two coupled oscillators)

Real student questions

"Why doesn't the symmetric mode depend on the coupling spring?" Because in that mode the two blocks move exactly alike, so the distance between them never changes, and the coupling spring neither stretches nor compresses.

"How do I know if two modes are really independent?" One checks modal orthogonality with respect to $M$; if two modes correspond to different frequencies, that orthogonality holds automatically as a mathematical consequence of the eigenvalue problem.

Final synthesis

By the end of this leaf, the student should be able to linearize a system near a stable equilibrium, solve the generalized eigenvalue problem to obtain normal frequencies and modes, and apply the symmetric two-coupled-block case's closed forms as a reference example.

The relevant final check is one of consistency: verifying that, in the no-coupling limit ($k_c$=0), the concrete system's two frequencies exactly match the frequency of a single simple harmonic oscillator with constant $k_w$ and mass $m$.