Nonlinear planar and non-planar vibrations of viscoelastic fluid-conveying pipes with external and internal resonances

Abstract

In this study, nonlinear vibrations of a viscoelastic fluid-conveying pipe under in-plane and out-of-plane transverse excitations are investigated, with emphasis on external and internal resonances. For this purpose, a three-dimensional pipe model that couples the in-plane and out-of-plane vibrations is developed based on the Euler–Bernoulli beam theory together with the Kelvin–Voigt viscoelasticity. The Galerkin method is performed on the nonlinear governing equations, and the resultant discretized equations are solved via the pseudo-arclength continuation method and a direct integration method. The dynamical responses of the pipe are examined in both sub- and super-critical regions and presented by plotting the frequency–amplitude curves, force–amplitude curves, basins of the attraction, time histories, and phase portraits. A comparison between the simplified two-dimensional model and the present three-dimensional model is presented to showcase the possibility of the in-plane and out-of-plane vibrations in regions of primary and principal parametric resonances, highlighting the modal interaction between the in-plane and out-of-plane transverse modes. In the presence of a two-to-one internal resonance, in addition to typical jumping and hysteresis, the two-peak resonance behavior is observed. The numerical results not only show complex nonlinear dynamics of the fluid–structure interaction system, but also provide hints for safe design and novel application of such structures.