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Stability analysis of Euler's elastica ring using harmonic balance

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Pith's one-line read This paper claims that harmonic balance gives accurate stability ranges for the pressurized Euler elastica ring, yielding smooth stable-pressure intervals for symmetric modes up to n = 12.

desk verdict Table 1's stability ranges are not derived in the paper; the 'stability' criterion is geometric self-intersection, so the central claim is unverifiable as written—though the zeroth-order algebra is fine. read the letter →

arxiv 2508.16704 v1 pith:FQCFCWXN submitted 2025-08-22 physics.class-ph

Muhammad Sami Siddiqui This is my paper · ORCID
classification physics.class-ph
keywords Euler'selasticaringharmonicbalancemethodstabilityanalysispost-bucklingbifurcationself-intersectionuniformpressuresymmetricloading
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the harmonic balance method gives reliable stability ranges for a thin elastic ring compressed by uniform pressure, for symmetric buckling modes up to n = 12. The central result is Table 1, which lists the interval of the loading parameter p in which stable equilibrium shapes exist for each mode: for example, (3, 10.5) for n = 2 and (143, 1191) for n = 12. The lower end is the classical threshold n^2 − 1; the upper end is where the computed shape self-intersects and the branch becomes unstable. The paper checks this against a linearization/bifurcation analysis and against numerical shooting, finding agreement, and presents stability diagrams showing the closed-to-contact-to-over-contact-to-irregular progression. These ranges matter because they predict when a pressurized ring will collapse or self-intersect, which is relevant for biological membranes, soft actuators, and structural elements.

What carries the argument

The load-bearing mechanism is the harmonic balance expansion of the curvature difference ν(s) as a cosine series in cos(nks), solved order by order; the paper uses the third-order version from its earlier work. This turns the nonlinear differential equation into algebraic equations for the Fourier coefficients, giving the loading parameters µ, β, and hence p = µ + β − 1, as functions of the amplitude A. Varying A traces the equilibrium branch p versus ν(0); the upper endpoint of the branch, where the computed ring self-intersects, defines the claimed stability boundary. A second, independent check is the linearization/bifurcation analysis, which marks the same unstable parameter region.

What would settle it

Recompute the equilibrium branch for, say, n = 2 and n = 5 by shooting or finite differences with fine arc-length continuation, and check whether a closed, non-self-intersecting ring still exists just above the upper bounds in Table 1 (p = 10.5 and p = 170). If a stable non-self-intersecting solution exists beyond the listed endpoint, the claimed stability boundary is not the true one; if continuation also terminates at exactly those pressures, the table is confirmed.

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Extended reading notes

Core claim

The paper expands the curvature difference ν(s) as an n-fold cosine series and finds its coefficients by harmonic balance. Using the third-order approximation from the author's earlier work, it traces the equilibrium branch for each mode n as the pressure parameter p grows. The branch begins at p = n^2 − 1, passes through closed, contact, and over-contact shapes, and ends where the computed curve self-intersects; that endpoint is taken as the loss of stability. The endpoints form Table 1's stable-pressure intervals. Linearization about the branch and numerical shooting recover the same intervals. The claimed result: harmonic balance gives smooth, accurate stability ranges for all symmetric m

Load-bearing premise

The stability intervals in Table 1 are correct only if the third-order harmonic balance approximation from the author's earlier paper reproduces the true equilibrium shapes and if the pressure at which the computed shape self-intersects really is the point of instability.

Editorial extensions

If this is right

  • For each symmetric mode n, the stable window (n^2 − 1, p_max) gives the full pressure range over which a ring keeps a non-self-intersecting equilibrium shape.
  • The two independent routes — harmonic balance and linearization — agree on the unstable regions, so either can be used to cross-check buckling predictions in related problems.
  • The harmonic balance stability data extend to higher modes such as n = 10 and n = 12, a range not tabulated by earlier elliptic-function solutions.
  • The closed-to-contact-to-over-contact-to-irregular sequence gives a geometric reading of how a ring approaches instability, visible directly in the computed shapes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A pattern in Table 1's upper endpoints (close to 8n^2 for n = 8, 9, 10, 12) suggests p_max may follow a simple quadratic law; fitting this relation and testing it for more modes would give a closed-form design rule the paper does not state.
  • The identification of self-intersection with loss of stability is made numerically; proving that the self-intersection value of p coincides with the first unstable eigenmode of the linearized problem would turn the table into a theorem.
  • Since the harmonic balance ansatz keeps only symmetric cosine modes, the stability statement concerns symmetric perturbations; checking non-symmetric or dynamic perturbations would test whether the branch is stable in full function space.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The manuscript studies the planar Euler elastica ring under uniform, n-fold symmetric pressure p. The author solves the nonlinear curvature-difference equation (Eq. 1) by harmonic balance: a zeroth-order approximation ν0(s) = A cos(ns) is derived in Section 3, first-order corrections are sketched (with poorly formatted formulas), and the third-order solution is taken from the author's prior work [15]. Section 4 presents equilibrium shapes and discusses contact and self-intersection transitions. Section 5 presents stability diagrams (p vs ν(0)) and Table 1, which lists claimed stable p-intervals for loading modes n = 2 through 12, asserted to begin at p = n² − 1 and to end at a 'break-even point' identified with the onset of self-intersecting shapes. The abstract claims that the harmonic balance method gives accurate stability calculations and agrees with the bifurcation theory of linearization.

Significance. If the stability ranges were rigorously derived, the paper would provide a useful systematic map of stable p-intervals for all symmetric buckling modes of the pressurized elastica ring, complementing the classical p > n² − 1 condition of Tadjbakhsh–Odeh and the elliptic-function descriptions of Djondjorov et al. The zeroth-order harmonic balance balance (Section 3, Eq. (4)) is algebraically consistent and worth acknowledging. However, the manuscript ships no code, no machine-checked proof, and reproduces none of the equations that determine Table 1; the third-order formulation of [15] is invoked but not given, and no independent numerical or analytical verification of the table is presented. The claimed agreement with the bifurcation theory of linearization is stated only in words. Consequently, the paper's central quantitative claim is not checkable from the text, and the significance is contingent on material the manuscript does not provide.

major comments (3)
  1. [§5, Table 1; end of §4] The central result—the stable p-intervals in Table 1—is not derived in the manuscript. The only stability criterion stated is geometric: 'if we exceed the external loading, we would get intersecting shape… break-even point, where the stable shapes becomes unstable' (end of Sec. 4; Sec. 5). Self-intersection of the centerline is not a stability boundary: an equilibrium with positive second variation of the elastica energy remains stable even when the curve self-intersects, and contact changes the mechanical system entirely. The paper gives no second-variation calculation, no linearized perturbation eigenvalue problem, and no Jacobi/Floquet condition. The endpoints p = 10.5 (n = 2), 70 (n = 3), 1191 (n = 12) are therefore asserted, not computed, and the abstract's claim of 'accurate stability calculations' is unsupported by the text.
  2. [§3, first-order expressions after Eq. (5)] The displayed expressions for ν1, μ1, β1 are not well-formed: the symbols 'nn', 'M', and 'An' are never defined, and factors such as '8Ann − 32An2' and 'nn − 5n2 − 3' are illegible as mathematics. The relation of ν1 to ν0 is unexplained, and the iterative scheme (5) is not carried beyond the first order, although Section 5 states that the stability results come from the third-order formulation of [15], which is not reproduced. The only harmonic balance derivation actually present in this paper, the zeroth-order balance of Eq. (4), does not determine the quantities reported in Table 1.
  3. [§5, opening sentence] The stability analysis explicitly delegates the solution to the author's own prior paper: 'In our previous work [15], we used the third-order harmonic balance formulation… Therefore, we employ the same formulation to compute stability diagrams.' Since neither that formulation nor the stability boundary is reproduced, and the agreement with 'bifurcation theory of linearization' is claimed without equations, Table 1 rests on an unreproduced, self-referential basis. The statement that 'NSolve gives the same results' is not accompanied by any numerical values or comparison plots, and the offered Mathematica notebooks (Sec. 5) are not shipped. A reader cannot reproduce or verify the paper's predictions from the text.
minor comments (8)
  1. [§1, Introduction] The sentence 'Troger and Steindl ref and Chaskalovic and Naili ref' contains unfilled placeholder 'ref' markers and the corresponding references are missing from the bibliography.
  2. [§4, paragraph 'As external loading move to thepcp < psi'] This paragraph is garbled: pcp and psi are undefined, and the condition 'x−2 = x+1' is illegible as typeset. The contact/self-intersection condition should be formulated explicitly with clear notation.
  3. [§1, last paragraph] The paper says 'Section 5 concludes the paper', but Section 5 is 'Stability Analysis' and the conclusion is actually Section 6.
  4. [Table 1] The row for n = 11 is omitted without explanation, and the notation '(3, 10.5)' does not state whether endpoints are included or how the endpoints were extracted from the diagrams.
  5. [§4, Figure 2 caption] The caption lists p-values (5.247, 21.65, 51.844, 97.834) that are not connected to Table 1 or to the known contact pressures from the elliptic-function or numerical literature [6, 8]; no comparison with those values is given.
  6. [§5, penultimate paragraph] The claim that 'Previous researcher have not computed the stability range and bifurcation diagrams … using elliptic functions' is an unsupported literature assertion; a comparison with at least one existing stability or eigenvalue analysis of elastic rings would be needed to substantiate it.
  7. [Eqs. (1) and (4)] The inline fractions are ambiguous: '3/2ν(s)2 + 1/2ν(s)3' should be typeset as (3/2)ν(s)² + (1/2)ν(s)³ to avoid misinterpretation.
  8. [Figures 4–6] The stability diagrams are never compared quantitatively with the numerical bifurcation diagram (Fig. 6), which is shown only for n = 5; the claimed agreement between HBM and lin- earization bifurcation analysis cannot be assessed from the figures.

Circularity Check

2 steps flagged · score 8.0 of 10

Table 1's 'stable ranges' reduce to the author's own prior third-order HBM model [15] plus a geometric self-intersection criterion; no independent stability analysis is supplied.

  1. self citation load bearing [Section 5, Stability Analysis]
    "In our previous work [15], we used the third-order harmonic balance formulation to calculate the equilibrium shape of the Euler elastica. This formulation provided improved precision for all symmetric loading conditions. Therefore, we employ the same formulation to compute stability diagrams under the loading parameter p."

    The paper's headline numerical output (Table 1's stable p-ranges) is computed from a 'third-order harmonic balance formulation' taken from the author's own prior paper [15]. The present text gives only zeroth- and first-order HBM coefficients; the third-order model, the stability criterion, and the mapping from equilibrium shapes to p-intervals are never reproduced. Thus the central predictive claim is inherited from a self-citation and cannot be checked within the paper.

  2. self definitional [Section 4, Solution: Equilibrium Shapes]
    "If we exceed the external loading, we would get intersecting shape because imposing the two-dimensional condition, and then this will reach to break-even point,where the stable shapes becomes unstable."

    This sentence effectively defines 'unstable' as 'self-intersecting shape' (the break-even point). Consequently the 'stable range' in Table 1 is by construction the set of p-values for which the [15]-computed HBM shape does not self-intersect. No second variation of the elastica energy, no linearized eigenvalue problem, and no Floquet/Jacobi stability analysis is given. The stability classification is therefore a relabeling of the self-intersection threshold rather than an independent stability prediction.

full rationale

The central claim of the paper—accurate stability ranges for Euler's elastica ring—is not derived from an independent mechanical stability analysis. Section 3 derives only the zeroth- and first-order harmonic-balance coefficients; the third-order formulation that actually produces Figures 2-6 and Table 1 is imported from the author's own prior paper [15] ('we employ the same formulation'). The stability boundary in Section 4 is identified with geometric self-intersection ('break-even point, where the stable shapes becomes unstable'), and no energy second variation, linearized eigenvalue, or Floquet criterion is presented. Thus the headline result reduces to (a) the unreproduced third-order HBM equilibrium shapes of the author's own [15] and (b) a definition of 'stable' as 'non-self-intersecting.' The alleged agreement with 'bifurcation theory of linearization' is asserted without equations, so it does not break the self-reference. This is a self-citation chain plus a redefinition, not a self-contained stability derivation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central result rests on a prior self-cited model and an unstated mapping to stability boundaries. The only free parameter is the HBM amplitude A; the stability range numbers are essentially outputs of that model, not derived here.

free parameters (1)
  • A (harmonic balance amplitude)
    The amplitude of the first harmonic in the HBM solution. It parametrizes the solution branch but is not fitted to data; it is a standard variable in HBM. However, the stability boundary values in Table 1 are not derived from A explicitly in this paper.
assumptions (4)
  • domain assumption The Euler elastica ring is governed by Eq. (1): ν'' + μν - β + (3/2)ν^2 + (1/2)ν^3 = 0.
    This equation is taken from Tadjbakhsh and Odeh [21] as the standard force-moment balance model for an inextensible uniform ring under uniform pressure.
  • domain assumption The ring has n-fold symmetry and closure conditions (2) and (3).
    These are standard symmetry and closure conditions for the n-th buckling mode of a closed ring.
  • ad hoc to paper The third-order harmonic balance formulation from the author's prior work [15] is correct and applicable to stability analysis.
    The paper does not reproduce the third-order equations; it directly uses them from [15]. This is the load-bearing assumption for Table 1.
  • domain assumption The stability boundary coincides with the onset of self-intersection or irregular shapes.
    The text in Section 4 states that exceeding a certain loading gives intersecting shapes and 'where the stable shapes becomes unstable', but no precise criterion is given.

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Cite this review

Pith. "Pith review of Stability analysis of Euler's elastica ring using harmonic balance." pith.science (2026). https://pith.science/paper/FQCFCWXN

@misc{pith2026250816704,
  author       = {Pith},
  title        = {Pith review of: Stability analysis of Euler's elastica ring using harmonic balance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQCFCWXN}},
  note         = {Machine review of arXiv:2508.16704}
}
read the original abstract

The stability analysis of elastic rings subjected to various loading conditions is examined, focusing on stable and unstable configurations. The harmonic balance method is employed to investigate the stability range under different loading conditions. This method provides improved accuracy in results and yields a smooth, stable range of values for all modes of symmetry. Additionally, the bifurcation analysis technique of linearization is utilized to identify unstable regions, and it demonstrates good agreement with the results obtained using the harmonic balance method.

Figures

Figures reproduced from arXiv: 2508.16704 by the authors.

Figure 1
Figure 1. Elastic ring model with geometric description We consider a two-dimensional, thin, inextensible, uniform circular elastic ring in its equilibrium configuration. The ring deforms under the application of a uniform loading param￾eter p, which represents a constant inward pressure or com￾pression. The curve Λ describing the ring is parametrized by the arc length s, and its position vector is denoted as r(s) = (x1(s), x… view at source ↗
Figure 2
Figure 2. Equilibrium ring shapes corresponding to (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Plot of v 2 (s) at p = 10.34, v 3 (s) at p = 70.34 and v 6 (s) at p = 600 If we exceed the external loading, we would get intersecting shape because imposing the two￾dimensional condition, and then this will reach to break-even point,where the stable shapes becomes unstable. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Stability diagram between p vs. ν(0) at (a) n = 2 and (b) n = 3, with some stable and unstable elastic ring shapes at specific values of p (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Stability diagram between p vs. ν(0) at (a) n = 4 and (b) n = 5, with some stable and unstable elastic ring shapes at specific values of p Moreover, elastic ring shapes are computed over the stability diagram, which expresses the stable to unstable shape transition. Th…
Figure 6
Figure 6. Figure 6: Bifurcation diagram computed numerically at [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [15]

    Self-contact of a flexible loop under uniform hydrostatic pressure

    Majid A., and Siddiqui S. Self-contact of a flexible loop under uniform hydrostatic pressure. European Journal of Mechanics - A/Solids 84:2020, 104082

    work page 2020

  2. [1]

    Bell, J. F. Volume I: The Experimental Foundations of Solid Mechanics. Springer, Berlin, 1973

    work page 1973

  3. [2]

    PhD dissertation, Gaottin- gen, 1906

    Born, M., Staabilitat der elastischen Linie in Ebene und Raum. PhD dissertation, Gaottin- gen, 1906

    work page 1906

  4. [3]

    The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell

    Canham, P.B. The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell. J. Theoret. Biol 1970:26:61-81

    work page 1970

  5. [4]

    Carrier, G. F. On the buckling of the elastic rings. Journal of Mathematical Physics . 1974;26:94-103

    work page 1974

  6. [5]

    F, John F

    Deborah K. F, John F. Marko, Albert L. Mechanics of Microtubule-Based Membrane Ex- tension. Physics Review Letter 1997;79:22

    work page 1997

  7. [6]

    M., Mladenov, I.M

    Djondjorov, P.A., Vassilev, V. M., Mladenov, I.M. Analytic description and explicit parametrisation of the equilibrium shapes of elastic rings and tubes under uniform hy- drostatic pressure. International Journal of Mechanical Sciences . 2011;53:335-364

    work page 2011

  8. [7]

    Methodus inveniendi lines curves maximi minimive proprtetate gaudents

    Euler, L. Methodus inveniendi lines curves maximi minimive proprtetate gaudents. Lau- sanne 1744

    work page

Show all 24 references
  1. [8]

    and Rubinow, S.I., Post buckling behavior of elastic tubes and rings with opposite sides in contact.SIAM J

    Flaherty, J.E., Keller, J.B. and Rubinow, S.I., Post buckling behavior of elastic tubes and rings with opposite sides in contact.SIAM J. APPL. MATH . 1972;23

    1972

  2. [9]

    Goss, Victor Geoffrey Alan., The history of the planar elastica: insights into mechanics and scientific method, Science & Education 18.8 (2009): 1057-1082

    2009

  3. [10]

    On the buckling of elastic rings by external confinement.Phill

    Hazel Al, Mullin T. On the buckling of elastic rings by external confinement.Phill. Trans. R. Soc. A375:20160227

  4. [11]

    Ouevres de Lagrange 2, pp 125-137, 1770-1773

    Lagrange, J.L., Sur la figure des colones. Ouevres de Lagrange 2, pp 125-137, 1770-1773

  5. [12]

    and He, X., Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators

    Lai, S.K., Lim, C.W., Wu, B.S., Wang, C., Zeng, Q.C. and He, X., Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Mathematical Modelling 33 (2009) 852-866

    2009

  6. [13]

    Levy. M. Mémoire sur un nouveau cas intégrable du probléme de l’élastique et l’une de ces applications. J. Math (Liouville) 1884;3:7

  7. [14]

    Simulation of a soap film spanning a flexible loop.International Journal of Non-Linear Mechanics

    Abdul Majid. Simulation of a soap film spanning a flexible loop.International Journal of Non-Linear Mechanics. 124 (2020) 103497. 7

    2020

  8. [16]

    and Herisanu, N., Nonlinear Dynamical Systems in Engineering: Some Ap- proximate Approaches .Springer

    Marinca, V. and Herisanu, N., Nonlinear Dynamical Systems in Engineering: Some Ap- proximate Approaches .Springer. 2011

    2011

  9. [17]

    Mickens, R.E., A generaliztion of the method of harmonic balance.Journal of Sound and Vibration 515-518 111(3) (1986)

    1986

  10. [18]

    Elastica and Computer Vision

    Mumford, D. Elastica and Computer Vision. Algebraic Geometry and its Applications 1994;491-506

    1994

  11. [19]

    and Hadzhilazova, M., The Many Faces of Elastica

    Mladenov, I.M. and Hadzhilazova, M., The Many Faces of Elastica. Springer 2013

    2013

  12. [20]

    Strogatz, S.H., Nonlinear dynamics and chaos: with applications to physics, biology, chem- istry, and engineering, CRC press, 2018

    2018

  13. [21]

    1997;18:59-74

    Tadjbakhs I, Odeh.Equilibrium state of elastic rings.J Math Anal Appl . 1997;18:59-74

    1997

  14. [22]

    Chan, Sung Ha Kang, Jianhong Shen .Euler’s Elastica and Curvature-Based In- painting

    Tony F. Chan, Sung Ha Kang, Jianhong Shen .Euler’s Elastica and Curvature-Based In- painting. SIAM J. APPL. MATH.2002;63:564-592

    2002

  15. [23]

    Mathematica 11.3.Wolfram Research, Inc 2018

    Wolfram Research, Inc. Mathematica 11.3.Wolfram Research, Inc 2018

    2018

  16. [24]

    International Journal of Mechanical Sciences 2007;49:661-668

    Wu, B., Yu, Y., Li, Z., Analytic approximation to large pos-buckling deformation of elastic rings under uniform hydrostatic pressure. International Journal of Mechanical Sciences 2007;49:661-668. 8

    2007

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