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

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 an assigned angular speed ω(t). The bead’s z-vertical time law is obvious, whilst its x-motion along the horizontal arm is ruled by a linear differential equation we solve through the Hermite functions and the Kummer (1837) [1] confluent Hypergeometric Function (CHF) 1F1. After the rotation θ(t) has been computed, we know completely the m-motion in a cylindrical frame of reference so that some transients have then been analyzed. 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: we resort to Lagrange method of variation of constants, helped by a Fourier–Bessel expansion, in order to manage the relevant intractable integrations.