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

TL;DR

This paper presents analytical solutions for a complex 3-D dissipative oscillator driven by aperiodic forces, using hypergeometric functions, and explores effects like friction and inclination through advanced mathematical methods.

Contribution

It introduces hypergeometric function solutions to model a 3-D dissipative oscillator with complex driving forces and additional effects like friction and inclination.

Findings

Explicit solutions for the oscillator's motion using hypergeometric functions.

Analysis of transient behaviors in the oscillator dynamics.

Inclusion of friction and inclination effects through advanced mathematical techniques.

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.