Hypergeometric solutions to a three dimensional dissipative oscillator driven by aperiodic forces

Abstract

We model the dynamical behavior of a three dimensional (3-D) dissipative oscillator consisting of a $m$-block whose vertical fall occurs against a spring and which can also slide horizontally on a rigid truss rotating at a known angular speed law $\omega(t)$. The $z$-vertical time law is obvious, whilst its $x$-motion along the horizontal arm is ruled by a linear differential equation to be solved through the Hermite functions and the Confluent Hypergeometric Function (CHF) $_{1}F_{1}$ (Kummer). After the rotation time law $\theta(t)$ has been computed, we know completely the mass motion in a cylindrical coordinate reference: some transients have then been discussed. Finally, further effects as an inclined slide and a contact dry friction have been added to the problem, so that the motion differential equation becomes inhomogeneous and we resort to Lagrange method of variation of constants, helped by a Fourier-Bessel expansion, in order to manage the relevant intractable integrations.