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arxiv: 2606.27552 · v1 · pith:GXDNRKN5new · submitted 2026-06-25 · 🌊 nlin.CD

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

Ali Fasihi , Grzegorz Kudra , Maryam GhandchiTehrani , Jan Awrejcewicz This is my paper

Pith reviewed 2026-06-29 00:39 UTC · model grok-4.3

classification 🌊 nlin.CD
keywords inextensible pipefluid-conveying pipespinning pipepost-critical dynamicsGalerkin truncationHencky bar-chainpinned-roller supportsnonlinear stability
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The pith

For an inextensible spinning fluid-conveying pipe with pinned-roller supports, ninth-order Galerkin truncation is required to resolve the post-critical amplitude because cubic terms miss the geometric stiffening.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the stability and large-amplitude post-critical behavior of a spinning pipe carrying fluid, supported by pinned-roller ends that allow sliding and remove axial stretching resistance. The inextensibility constraint introduces nonlinear terms from the bending curvatures that govern the large deflections after the stability limit is crossed. A Galerkin projection with bending curvatures expanded to ninth order is shown to be the minimal truncation that correctly captures the amplitude, while the common cubic approximation significantly overestimates the deflection. A modified Hencky bar-chain discretization with exact trigonometric kinematics provides an alternative discrete model that agrees closely with the continuous approach across linear and nonlinear regimes.

Core claim

The governing equations for this pinned-roller configuration feature nonlinear bending-curvature terms instead of axial stretching, leading to an elliptical stability boundary with semi-axes at flow velocity U = π and rotational speed Ω = π². The post-critical regime exhibits large deflections whose amplitude is only correctly predicted when the bending curvatures are Taylor-expanded to ninth order; lower orders, including the cubic truncation, miss the stiffening effect and overestimate deflections. The modified Hencky bar-chain model, formulated with a global angular description and n-independent matrices, reproduces the same linear stability, bifurcation diagrams, and time histories, conf

What carries the argument

Ninth-order Taylor expansion of the bending curvatures within the Galerkin discretization, which incorporates the geometric stiffening arising from the inextensibility constraint.

If this is right

  • The linear stability boundary forms an ellipse-like region in the flow-velocity versus rotational-speed plane with semi-axes U=π and Ω=π².
  • Three distinct damping regimes appear, one of which is a high-rotation instability driven by rotating damping.
  • The modified Hencky bar-chain model with global angular description works as a closed matrix framework adaptable to both extensible and inextensible cases via boundary conditions.
  • Agreement between the high-order Galerkin and the discrete model holds for linear stability, bifurcation points, and time-history responses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may extend to other constrained beam or pipe systems where inextensibility produces similar geometric nonlinearities that demand high-order expansions.
  • The discrete Hencky framework could enable efficient parameter studies or control design for spinning fluid-conveying pipes in engineering applications.
  • Similar support changes might alter post-critical dynamics in related problems like rotating shafts or flexible risers.

Load-bearing premise

That the post-critical amplitude cannot be resolved without carrying the Taylor expansion of the bending curvatures to ninth order, as lower truncations miss essential stiffening terms.

What would settle it

A high-fidelity numerical simulation or physical experiment measuring the steady post-critical deflection amplitude at a point above the critical flow speed and rotation rate that matches the ninth-order prediction but deviates from the cubic truncation result.

Figures

Figures reproduced from arXiv: 2606.27552 by Ali Fasihi, Grzegorz Kudra, Jan Awrejcewicz, Maryam GhandchiTehrani.

Figure 1
Figure 1. Figure 1: Spinning fluid-conveying pipe with pinned–roller supports. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Linear stability boundary in the (𝑈, Ω) plane and its parametric dependence, reproduced from [23]: (a) three damping regimes – 𝛼 = 0; 𝛼 > 0 at low Ω; 𝛼 > 0 at high Ω; (b) dependence on the mass ratio 𝛽; (c) dependence on the flow-profile modification factor 𝛾. A. Fasihi et al.: Preprint submitted to Elsevier Page 9 of 32 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Midpoint deflection |𝑟(0.5, 𝑡)| at 𝑈 = 4, Ω = 4 for the full model and for the four cases with one nonlinear group (NMT, NGT, NCT, NST) switched off. Parameters: 𝛼 = 0.023, √ 𝛽 = 0.536925 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Steady-state midpoint amplitude |𝑟(0.5)| for the full model and for each case with one group switched off, with the percentage shift relative to the full model annotated. The velocity-dependent groups NMT, NGT leave the equilibrium amplitude unchanged; NCT and NST shift it, with NST dominant. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the steady-state midpoint deflection with the truncation order of the bending curvatures [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Inertial-frame post-critical response at [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Rotating-frame counterpart of Fig [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Three-dimensional bifurcation surface: steady-state midspan deflection [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Steady-state midpoint deflection versus flow velocity for the extensible pinned–pinned pipe (left axis) and [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Modified Hencky bar-chain model of the spinning fluid-conveying pipe: (a) rigid-link chain with flexural [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Wall-clock computation time for a fixed simulation as a function of the number of links [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Stability boundary in the 𝑈–Ω plane from the frequency analysis of the modified Hencky model for 𝑛 = 3, 7, 11, 15, 30, compared with the continuum ellipse-like reference. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bifurcation diagram of |𝑟(0.5)| versus flow velocity 𝑈 at Ω = 0: (a) modified Hencky model at several values of 𝑛 vs the Galerkin reference; (b) magnified view of the onset, including 𝑛 = 30. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Bifurcation diagram of |𝑟(0.5)| versus rotational speed Ω at 𝑈 = 0, modified Hencky model (𝑛 = 15) vs the Galerkin reference. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Bifurcation diagram of |𝑟(0.5)| versus flow velocity 𝑈 at non-zero rotational speed: (a) Ω = 5; (b) Ω = 8. Modified Hencky model (𝑛 = 15) vs the Galerkin reference. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Time history of the midpoint response under combined flow and rotation: (a) rotating-frame [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Bifurcation diagram of |𝑟(0.5)| versus flow velocity 𝑈 at Ω = 0 for the pinned–roller pipe; modified Hencky model at 𝑛 = 5, 7, 11, 15. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Steady-state midpoint deflection |𝑟(0.5)| versus flow velocity at Ω = 8 for the pinned–roller pipe: modified Hencky model (𝑛 = 15) vs the Galerkin (𝜀 9 ) reference. Parameters as in [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Time history of the midpoint response under combined flow and rotation for the pinned–roller pipe, modified [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the stability and large post-critical dynamics of an inextensible spinning fluid-conveying pipe with pinned-roller supports. Replacing pinned-pinned supports removes axial stretching, yielding new nonlinear terms from the inextensibility constraint and bending curvatures. The equations are discretized via a ninth-order Galerkin expansion of curvatures (claimed minimal for accurate post-critical amplitudes, as cubic overestimates due to missing geometric stiffening) and a modified Hencky bar-chain model with exact trigonometric kinematics and n-independent matrix form. Linearised dynamics produce an ellipse-like stability boundary (semi-axes U=π, Ω=π²) in the flow-velocity–rotational-speed plane; three damping regimes are identified, including high-rotation instability. Close agreement is reported between the two methods on linear stability, bifurcations, and time histories.

Significance. If the results hold, the work establishes the necessity of high-order curvature expansions for post-critical regimes in inextensible pipes and introduces an adaptable, matrix-based discrete framework usable for both extensible and inextensible cases. The explicit cross-validation between Galerkin and Hencky discretizations across multiple diagnostics is a methodological strength that supports the identified damping regimes and instability mechanisms.

minor comments (3)
  1. [Abstract and Galerkin section] Abstract and Galerkin section: the claim that ninth-order is the lowest truncation resolving post-critical amplitude (with cubic overestimating due to absent geometric stiffening) should be supported by a dedicated convergence plot or table quantifying amplitude error versus truncation order.
  2. [Linearised dynamics] Linearised dynamics: the ellipse-like stability boundary with semi-axes U=π and Ω=π² is presented as an outcome of the linearised equations; the characteristic equation or explicit reduction steps should be shown to allow direct verification.
  3. [Hencky model] Hencky model: the reduction yielding the n-independent matrix framework after imposing pinned-roller boundary conditions should include a brief algorithmic outline or pseudocode for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no individual points requiring rebuttal or clarification at this stage. We will make any minor adjustments needed for the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the governing PDEs from the inextensibility constraint and pinned-roller boundary conditions, obtains the elliptical stability boundary (semi-axes U=π, Ω=π²) directly from linearization, and demonstrates the necessity of ninth-order curvature truncation via explicit convergence comparisons and cross-validation against an independent modified Hencky bar-chain model using exact trigonometric kinematics. No load-bearing step reduces by construction to a fitted input, self-citation chain, or renamed ansatz; the two discretization frameworks are mutually independent and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard beam-theory assumptions plus the modeling choice of inextensibility; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The pipe is inextensible
    Explicitly stated as the property that removes axial-stretching restoring force and changes the nonlinear terms.
  • ad hoc to paper Bending curvatures require ninth-order Taylor expansion to resolve post-critical amplitude
    Claimed as the minimal order that captures geometric stiffening; lower orders are said to overestimate deflection.

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Reference graph

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