= Mechanical dynamic stiffness
{title2=$K_{\rm dyn}(s)=F(s)/X(s)$}
<Mechanical dynamic stiffness> is the complex <force> required per unit displacement in a specified zero-initial-state <frequency response>. If <mechanical receptance> is $H(s)=X(s)/F(s)$, its scalar inverse is $K_{\rm dyn}(s)=1/H(s)$, wherever that inverse exists. Units are $\mathrm{N/m}$, but this dynamic quantity includes inertia and <damping> as well as the static <spring> stiffness. The measured coordinate, applied-force direction and boundary conditions are part of the definition.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[Printed pp.14–16, complex force/response functions and conversion between displacement, velocity and acceleration]{ref}
For $m\ddot x+c\dot x+kx=F$, zero <initial conditions> give
$$
K_{\rm dyn}(s)=ms^2+cs+k,\qquad K_{\rm dyn}(i\omega)=k-m\omega^2+i\omega c.
$$
Thus displacement and <force> generally differ in phase. With $m=2\,\mathrm{kg}$, $c=4\,\mathrm{N\,s/m}$ and $k=200\,\mathrm{N/m}$, at <angular frequency> $\omega=10\,\mathrm{rad/s}$ the result is $40i\,\mathrm{N/m}$. A real displacement <complex amplitude> $X=0.020\,\mathrm m$ requires $F=0.80i\,\mathrm N$, rather than the static estimate $kX=4\,\mathrm N$. Elastic and inertial contributions cancel at this frequency; <damping> still requires an applied <force>.
For several coupled coordinates, a <mechanical dynamic-stiffness matrix> maps displacement to <force>:
$$
\mathbf F=\mathbf D\mathbf X,\qquad \mathbf D=s^2\mathbf M+s\mathbf C+\mathbf K,\qquad \mathbf H=\mathbf D^{-1}.
$$
A <matrix inverse> cannot be replaced by reciprocating individual entries. For example, two masses connected by a <spring> of stiffness $k_c$, each also grounded by stiffness $k_0$, have $D_{11}=D_{22}=a=ms^2+k_0+k_c$ and $D_{12}=D_{21}=-k_c$. If coordinate2 is free of external <force>, $H_{11}=a/(a^2-k_c^2)$, so the driving-point dynamic stiffness is $a-k_c^2/a$ where defined. If coordinate2 is clamped, the required ratio is instead $a$. The measurement constraints determine which inverse is physically relevant.
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