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The science
Shear-building model
Each floor is a lumped mass m connected to the one below by a storey spring k (the stiffness of the columns). With relative displacements u measured from the moving ground, the equations of motion are M u'' + C u' + K u = -M 1 ag(t). K is tridiagonal, so each time step is a cheap tridiagonal solve. Time integration is the Newmark average-acceleration method (beta 1/4, gamma 1/2), which is unconditionally stable and conserves energy for an undamped frame.
Natural frequencies
For N equal floors fixed at the base the frequencies are ωq = 2 sqrt(k/m) sin((2q-1)π / (2(2N+1))) with mode shapes φq(i) = sin(i(2q-1)π/(2N+1)). A taller frame (larger N) or softer storeys (smaller k) lowers f1; the first period of a real tall building is roughly 0.1 s per floor. Damping is Rayleigh type, C = αM + βK, fitted so that modes 1 and 3 have the chosen damping ratio.
Resonance
Harmonic ground motion at ω drives a mode with the dynamic amplification 1 / sqrt((1-r²)² + (2ζr)²), r = ω/ωq. At r = 1 the response is 1/(2ζ) times the static one, which is why a 0.1 g shake at f1 can do more harm than a 0.5 g shake at a far-off frequency. The frequency-response graph is computed by solving (K - ω²M + iωC) x = -M 1 directly.
Tuned mass damper
A damper mass μN m on the roof with spring and dashpot tuned by Den Hartog's rules, ft = f1/(1+μ) and ζt = sqrt(3μ/(8(1+μ)³)), splits the single resonance peak into two lower peaks and removes much of the response (switch on the preset "Resonance + tuned mass damper" and compare the roof displacement graph).
Soft storey
If one storey is much weaker than the rest, the drift concentrates there: the drift ratio plot shows a spike at that storey even though the roof moves less.
Model checks (run in the page)
1. Free vibration of a 10-storey uniform frame must reproduce the closed-form period above. 2. The steady-state roof amplitude at resonance from the time-domain integration must equal the frequency-domain complex solve (two independent methods). 3. Energy balance: input work equals kinetic + strain + dissipated energy.
Honest simplifications
The frame is linear elastic: it never yields, cracks or collapses, so "damage" is an indicator (peak drift ratio versus your chosen limit), not a prediction. Floors are rigid, motion is horizontal in one direction, there is no soil-structure interaction, torsion or P-delta effect. Ground motions are synthetic records scaled to the chosen peak ground acceleration. Real design uses recorded motions, nonlinear analysis and local building codes; this is a teaching model of inanimate structures only.