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For $-u^{\prime \prime }+u=f$ with homogeneous Dirichlet boundary conditions, the conforming piecewise-linear hat functions on a uniform grid give a diffusion stiffness matrix $S$ and a mass matrix $M$. Their nonzero entries are $S_{ii}=2/h$, $S_{i,i+1}=-1/h$, $M_{ii}=2h/3$, and $M_{i,i+1}=h/6$, with symmetric counterparts. The sum is a positive-definite matrix because its coefficient quadratic form is $\int ((u_h^{\prime }{)}^2+u_h^2)$. For forcing $f(x)=x$, the load at $x_i$ is $h{x}_i$.

The spatial Galerkin method for the heat equation gives a mass matrix $M$ and a diffusion stiffness matrix $S$. With zero forcing, $\frac{d}{dt}(c^{T}Mc)=-2c^{T}Sc\le 0$. The quadratic form $c^{T}Mc$ is the squared L2 norm of the represented finite-element function, proving an energy contraction. For nonzero forcing, the corresponding identity has the additional term $2c^{T}F$, which can be estimated using the Cauchy-Schwarz inequality.

Consider a uniform thin hoop of radius $a$ and mass $m$, suspended from a point on its rim, with a bead of equal mass sliding frictionlessly around it. Let $\theta$ be the angle of the line from the pivot to the hoop's centre, and let $\varphi$ be the angle from the hoop's centre to the bead, both measured from the downward vertical. The stable equilibrium is $\theta =\varphi =0$.

The hoop's pivot moment of inertia is $2m{a}^2$. Adding its kinetic energy to the bead's gives the quadratic small oscillation LagrangianThus the mass matrix and stiffness matrix are $m{a}^2\left(\begin{array}{cc}3& 1\\ 1& 1\end{array}\right)$ and $mga\left(\begin{array}{cc}2& 0\\ 0& 1\end{array}\right)$. Their generalized eigenvalue problem gives $2c^2-5c+2=0$ for $c=a{\omega }^2/g$. The angular frequencies are therefore $\sqrt{g/(2a)}$ and $\sqrt{2g/a}$. The slow normal mode has $\varphi =\theta$; the fast normal mode has $\varphi =-2\theta$. Specifying the two absolute angles explains the off-diagonal coupling and restores the physical scale hidden by dimensionless matrices.

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 iswhere $u$ 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 $m=1000\mathrm{kg}$, $k=40000\mathrm{N}{\mathrm{m}}^{-1}$ and $c=400\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$. Its undamped natural frequency isFor a constant 100-newton force, static displacement is $100/k=2.5\mathrm{mm}$. For a sinusoidal force $100\cos ({\omega }_{0}t)$, the steady-state displacement amplitude of this linear forced harmonic oscillator is insteadThese 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.^[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.