 Mechanical susceptibility

Mechanical susceptibility is the linear frequency response converting applied force into displacement. With the convention $F(t)=\mathrm{Re}(F_{\omega }{e}^{i\omega t})$, the steady response of a forced harmonic oscillator is $x(t)=\mathrm{Re}(\chi (\omega )F_{\omega }{e}^{i\omega t})$, whereSubstitution 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.