= Structural dynamics
{title2=$M\ddot u+C\dot u+Ku=f(t)$}
\b[Structural dynamics] studies the time-dependent motion of structures under applied forces and their response after a disturbance. <Inertia>, <stiffness> and <damping> jointly determine that motion. A common linear small-displacement model is
$$
M\ddot u+C\dot u+Ku=f(t),
$$
where $u$ collects displacement coordinates, $M$ is the <mass matrix>, $C$ the <damping matrix>, $K$ the <stiffness matrix> and $f(t)$ the applied-force vector. Boundary conditions and the chosen coordinates are part of the model; a static force balance omits the velocity and acceleration terms.
A scalar version makes the consequences quantitative. Choose $m=1000\,\mathrm{kg}$, $k=40000\,\mathrm{N\,m^{-1}}$ and $c=400\,\mathrm{N\,s\,m^{-1}}$. Its undamped <natural frequency> is
$$
\omega_0=\sqrt{k/m}=\sqrt{40}\simeq6.325\,\mathrm{rad\,s^{-1}},\qquad
\nu_0=\omega_0/(2\pi)\simeq1.007\,\mathrm{Hz}.
$$
For a constant 100-newton force, static displacement is $100/k=2.5\,\mathrm{mm}$. For a sinusoidal force $100\cos(\omega_0t)$, the steady-state displacement amplitude of this linear <forced harmonic oscillator> is instead
$$
U=\frac{100}{\sqrt{(k-m\omega_0^2)^2+(c\omega_0)^2}}=\frac{100}{c\omega_0}\simeq39.5\,\mathrm{mm}.
$$
These are original model values, not measurements of a building. They show why the force amplitude alone does not determine displacement: forcing frequency and damping matter.
Real structures have multiple <normal modes>. A 2025 study of a viscous <inerter> damper uses a numerical 20-storey building example and shows that inertia and flexibility associated with discarded modes affect the reduced model's tuning. Matching one nominal mode while ignoring those contributions can therefore misrepresent the coupled device–structure behavior.<Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance by Jan Høgsberg>[Abstract, modal interaction, discarded-mode corrections and numerical building example]{ref}
Measured motion can test a chosen model, while its predictions remain limited by assumptions about linearity, boundary conditions and loading. A large predicted response is a reason to examine those assumptions, not proof that an actual structure will follow the linear response indefinitely.
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