A Two-Part Analytical Solution for the Shock Response Spectrum with Coulomb Damping

The present work primarily focuses on arriving at an analytical solution for the response of a Coulomb damped harmonic oscillator when subjected to aperiodic excitation.

The signum function used in modelling the Coulomb friction model is reduced to a first harmonic term. This is carried out by assuming that vibratory motion exists for at least one full cycle, by means of harmonic balance (HB) method. The equation of motion thus reduced is solved to obtain an analytical solution for half-sine, rectangular, and triangular shock inputs. The solution thus obtained is combined with another analytical solution derived by the phase space coordinate averaging method. The total solution is now expressed as a two-part analytical solution, which comprises the response during the action of the input pulse and after the pulse ends.

The analytical results are compared with numerical ones to establish validity and accuracy. A comparison is also made with the experimental readings, and a reasonably good agreement is observed for the response of the oscillator with Coulomb damping. The study is extended further to plot the shock response spectrum (SRS) for the three shock inputs over a wide range of damping values ( \(\alpha \) $\alpha$ ) and frequency ratios ( \(\Omega \) $\mathrm{\Omega }$ ). The \(1/2\Omega \) $1/2\mathrm{\Omega }$ range over which the \(n^\textrm{th}\) $n^{\text{th}}$ peak of the system response is dispersed in the SRS increases with a corresponding increase in \(\alpha \) $\alpha$ .

The two-part analytical solution proposed here predicts the SRS for the first dominant peak in the vibratory response to a good degree of accuracy and is quantified by directly comparing it with experiment(s).