Recovering an oscillator from its ring-down peaks

A mass attached to a linear spring and viscous damper obeys \[ m\ddot x+b\dot x+kx=0, \qquad m,b,k>0. \] The motion is underdamped. Two successive positive displacement maxima have magnitudes \(12.0\,\mathrm{mm}\) and \(8.50\,\mathrm{mm}\), and the elapsed time between them is \(0.800\,\mathrm{s}\). The mass is \(m=0.500\,\mathrm{kg}\). Starting from the characteristic roots, derive the decaying sinusoidal response and prove that the logarithmic decrement between successive same-sense peaks is \(\delta=\beta T_d\), where \(\beta=b/(2m)\) and \(T_d=2\pi/\omega_d\). Use the data to determine \(\beta,\omega_d,\omega_0=\sqrt{k/m},b,k\), and \(Q=\omega_0/(2\beta)\). Compute the ratio of mechanical energies at the two peaks and the time required for the peak envelope to fall to two percent of its initial value. State why absolute peak magnitudes must be used if alternating maxima are recorded, identify the underdamping check, and explain which inferred quantity would be wrong if the measured peak interval were mistakenly treated as half a damped period.

Write the real solution as \(C e^{-\beta t}\cos(\omega_dt-\phi)\); the phase does not affect the ratio of same-sense peak envelopes.