On internal resonances in pipes conveying pulsating fluid for beam, stretched-beam, and string models

Abstract

In this paper, we investigate an initial-boundary value problem for a linear Euler-Bernoulli beam equation governing the dynamics of pipes conveying fluid. The fluid flow velocity inside the pipe is assumed to have a small amplitude and to be time-varying, that is, V(t)=ε(V0+V1sin(Ωt)), where a two time-scales perturbation method is applied to construct approximations of the solutions. We explore fluid velocity pulsation frequencies Ω that lead to resonance in the pipe system. Depending on the order of bending stiffness, we study beam, stretched-beam and string models, each displaying distinct resonance behaviour. For special values of the frequency Ω and the bending stiffness, resonance frequencies can coincide, leading to internal resonances that exhibit more complex dynamical behaviour. We investigate both pure and detuned internal resonance scenarios, highlighting how early truncation in the number of oscillation modes leads to erroneous approximations and incorrect stability conclusions.