Dynamics · eigenvalues

Mass-spring-damper — predict, then check

The archetype two-state system: ẍ = −k·x − c·ẋ (m = 1), the same skeleton the aircraft's short period wears. Its matrix promises the eigenvalue pair λ = −c/2 ± i·√(k − c²/4) — a decay rate σ and a rotation rate ω, computed before anything moves. Tune k and c and watch the pair move on the s-plane; poke the mass (button, or drag it sideways and let go) and check the promise against the simulation. Then be honest with it: switch on the cubic spring ε (a Duffing term the matrix cannot see), add quadratic damping c₂, or crank the poke to a large amplitude — and watch the measured period and decay walk away from the prediction.

What you're looking at

The amber trace is the true plant integrated with RK4: ẍ = −k·x − ε·x³ − c·ẋ − c₂·ẋ|ẋ|. The cyan dashed trace is the linearisation about x = 0 given the same poke. With ε = c₂ = 0 they coincide exactly — the linearisation is the plant.
The dashed envelope ±x₀·e^σt and the period tick marks every 2π/ω_d are drawn from the eigenvalues alone — from the matrix, not from the flight. The readouts then measure period and decay from the nonlinear response by peak picking, and the agreement line reports the difference. Nothing is faked: when the numbers match, it's because linear theory works.
With ε > 0 (hardening spring) the restoring force grows faster than linear, so bigger pokes oscillate faster — the classic Duffing frequency–amplitude dependence. The eigenvalues, blind to ε, keep promising the small-amplitude period, and the agreement line goes sour as x₀ grows. With c₂ > 0 the decay is amplitude-dependent — fast while the mass moves quickly, slow once it doesn't — so no single σ can describe it and the exponential envelope fails visibly.
The moral for the rest of the course: an aircraft's eigenvalues are the linearisation's promise — valid near the trim point, at small amplitude. That is exactly where flight dynamics lives most of the time, and exactly what stops being true in a spin.