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

Abstract

• Complete case of the Coriolis theorem under viscous drag. • ODE solved via the hypergeometric functions. • Solution analysis for different parameters. • Polar trajectory. Analytic approximate solution to the ODE with dry friction. • Fourier–Bessel expansion to evaluate integrals in the Lagrange variational method. 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) 1 F 1 . 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.