Supersymmetric partner chirping of Newtonian free damping

We connect the classical free damping cases by means of Rosner's construction in supersymmetric quantum mechanics. Starting with the critical damping, one can obtain in the underdamping case a chirping of instantaneous physical frequency \omega ^{2}(t) \propto \omega_{u}^{2}sech^2(\omega_{u}t), whereas in the overdamped case the "chirping" is of the (unphysical) type \omega ^{2}(t)\propto\omega_{o}^{2}sec^{2}(\omega_{o}t), where \omega_{u}$ and $\omega_{o} are the underdamped and overdamped frequency parameters, respectively