= Mechanical susceptibility
{title2=$\chi(\omega)$}
Mechanical susceptibility is the linear frequency response converting applied <force> into <displacement>. With the convention $F(t)=\operatorname{Re}(F_\omega e^{i\omega t})$, the steady response of a <forced harmonic oscillator> is $x(t)=\operatorname{Re}(\chi(\omega)F_\omega e^{i\omega t})$, where
$$
\chi(\omega)=\frac1{k-m\omega^2+ib\omega}.
$$
Substitution into $m\ddot x+b\dot x+kx=F$ verifies the formula. The opposite Fourier time convention changes the sign of the imaginary term, so the convention must accompany the expression.
Its magnitude is $|\chi|=[(k-m\omega^2)^2+b^2\omega^2]^{-1/2}$, and its argument gives the phase shift relative to the drive. At zero frequency, $\chi(0)=1/k$ is the static compliance. At high frequency, $\chi\sim-1/(m\omega^2)$ reflects inertia. Measuring response amplitude and phase while varying the drive <frequency> therefore separates stiffness, mass and <viscous damping>.
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