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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 ω²(t) ∝ ωᵤ²sech^2(ωᵤt), whereas in the overdamped case the "chirping" is of the (unphysical) type ω²(t)∝ωₒ²sec²(ωₒt), where ωᵤ$ and $ωₒ are the underdamped and overdamped frequency parameters, respectively
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Rosu et al. (1997) studied this question.