Large post-critical dynamics of an inextensible spinning fluid-conveying pipe with pinned-roller supports: high-order Galerkin and a modified Hencky bar-chain framework

Abstract

This paper investigates the stability and large post-critical dynamics of an inextensible spinning fluid-conveying pipe with pinned-roller supports. Replacing the pinned-pinned support of the extensible counterpart with a sliding support removes the axial-stretching restoring mechanism and fundamentally changes the governing equations of motion. Derived here for this configuration, these equations contain a different set of nonlinear terms -- arising from the inextensibility constraint and the bending curvatures rather than the single axial-stretching term -- that drives a post-critical regime with large deflections. The regime is analysed with two complementary methods. The first is a Galerkin discretisation in which the bending curvatures are Taylor-expanded to ninth order, shown to be the lowest order resolving the post-critical amplitude; the standard cubic truncation overestimates the deflection significantly by missing the geometric stiffening from inextensibility. The second is a modified Hencky bar-chain model with a global angular description: a closed, $n$-independent matrix framework with exact trigonometric kinematics, directly implementable in any standard programming environment with matrix routines and adaptable to both extensible and inextensible configurations through a single boundary-condition reduction. The linearised dynamics give an ellipse-like stability boundary in the flow-velocity--rotational-speed plane with semi-axes $U=\pi$ and $\Omega=\pi^{2}$; three damping regimes are identified, including a high-rotation instability driven by rotating damping. Close agreement between the two methods across linear-stability, bifurcation, and time-history comparisons confirms the ninth-order Galerkin truncation and establishes the modified Hencky bar-chain as a reliable general-purpose discrete framework for spinning fluid-conveying pipes.