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REVIEW 4 major objections 4 minor 35 references

Torsional vibration of a coupled cylinder

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Explicit displacement formulas for a vibrated two-section hollow cylinder under torsion, for ideal, soft, and rigid coupling, are derived in closed form via a finite Hankel transform and a Green's function representation, and the paper…

desk verdict Solid ideal-contact solution and a useful damage proxy, but the soft/rigid branches inherit unverified thin-layer asymptotics at the paper's own parameters. read the letter →

arxiv 2506.11582 v3 pith:4CNFNY7A submitted 2025-06-13 physics.class-ph

Igor Istenes , Daniel Peck , Yuriy Protserov , Natalya Vaysfeld , Zinaida Zhuravlova This is my paper · ORCID
classification physics.class-ph MSC 74H4574H1544A1533C10 PACS 46.40.-f62.20.mm43.40.At
keywords torsionalvibrationcoupledcylinderidealcontactsoftrigidfiniteHankeltransformGreen'sfunctionresonancefrequency
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

The paper seeks to establish that the steady-state torsional displacement field of a hollow cylinder made of two vertically stacked sections, joined by an ideal, soft, or rigid interface, can be written explicitly as a series involving a one-dimensional Green's function, with no need for discretization. If correct, these formulas give the displacement and tangential stress at any point for arbitrary axisymmetric loading on the upper face, and they locate resonance frequencies as poles of the denominators. The authors also argue that a weak-interface version of the model can approximate the effect of a hidden ring crack in a single-material cylinder, by choosing the interface shear modulus through a linear relation to a measured mean displacement ratio. The practical payoff would be a fast, parameter-free way to map how coupling type, geometry, material contrast, and vibration rate affect dynamic torsion response, and a simple screening tool for damage detection.

What carries the argument

The load-bearing object is the modified finite Hankel transform with kernel $K(\lambda_n, r)=J_2(\lambda_n a)Y_1(\lambda_n r)-Y_2(\lambda_n a)J_1(\lambda_n r)$, where the eigenvalues $\lambda_n$ are roots of $J_2(\lambda a)Y_2(\lambda R)-Y_2(\lambda a)J_2(\lambda R)=0$, together with the associated Green's function $G_{kn}(z,\eta)$ of the transformed one-dimensional problem. The transform eliminates the radial coordinate and exploits the free-side boundary conditions so that the radial derivatives reduce to $\lambda_n^2 u_{kn}(z)$, while the Green's function carries the vertical behavior and its prescribed jumps at the interface encode the coupling. The explicit displacements are then reassembled by the inverse transform, with the denominators $\Delta^I_{2n}$, $\Delta^S_{2n}$, $\Delta^R_{2n}$ whose zeros give resonance frequencies.

What would settle it

Compute the same steady-state torsional displacement with a high-resolution finite-element or boundary-element simulation for the cuprum--aluminium cylinder of Table 1 with a thin soft layer of height $h_0=0.01$ m and shear modulus $\mu_{\text{soft}}=3.285\times 10^9$ Pa, and compare the displacement and stress profiles at $z/H=0.25,0.5,0.75$ against the series (33); a mismatch beyond the claimed near-linear accuracy of the thin-layer model would falsify the soft-contact solution.

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Extended reading notes

Core claim

The central claim is that equations (27), (33), and (39) provide the complete steady-state torsional displacement field of the coupled hollow cylinder for ideal, soft, and rigid contact, respectively, under arbitrary axisymmetric loading on the upper face. The derivation reduces the governing equation via a finite Hankel transform whose kernel is built from Bessel functions, turning the problem into a one-dimensional boundary value problem that is solved with a Green's function; the unknown interface jumps are then fixed algebraically by each contact condition. The authors report that soft contact produces markedly larger displacements than ideal or rigid contact, while ideal and rigid contact give nearly identical fields except near the base and at high aspect ratios, and they show that the resonance frequencies, obtained by phase-shift scanning of the closed-form solutions, differ substantially between the three couplings, with no shared eigenfrequencies in the studied range. They further claim that a weak interface can approximate a damaged region, with the mean displacement ratio following an essentially linear law in the shear-modulus ratio.

Load-bearing premise

The soft and rigid contact solutions rest on asymptotic thin-layer transmission conditions taken from Mishuris (2004a), and the paper does not verify these effective conditions against a direct numerical solution for the chosen material and geometry ratios; if those conditions are inaccurate at, say, $\mu_{\text{soft}}/\mu_k \approx 0.05$ and $h_0/H=0.1$, the corresponding central formulas would be wrong.

Editorial extensions

If this is right

  • Engineers can compute torsional displacement and stress at an arbitrary point of a coupled cylinder directly from the series, without meshing, and can scan frequencies rapidly to find resonance locations.
  • The three closed-form solutions allow a direct quantitative comparison of ideal, soft, and rigid coupling, showing, per the paper, that rigid coupling approximates ideal contact well while soft coupling departs strongly in displacement magnitude.
  • The linear relation between the mean displacement ratio and the interface shear modulus gives a calibration rule: a measured mean displacement deficit at the top face pins down the effective interface stiffness.
  • The radial distribution of the displacement ratio at the top face can serve as a non-destructive discriminator between uniform interfacial damage and a localized ring crack.
  • Because setting $\omega=0$ reduces the governing equation to the Laplacian, the same formulas describe steady heat flux in a coupled cylinder, extending the model beyond mechanics.

Reading between the lines

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

  • The Green's-function denominators suggest that resonance frequencies of the coupled cylinder could be studied as roots of the characteristic equations $\Delta^I_{2n}=0$, $\Delta^S_{2n}=0$, and $\Delta^R_{2n}=0$ directly, rather than by the scanning algorithm the paper uses; a root-finding analysis would test the robustness of the reported frequencies.
  • The claim that a weak interface approximates a ring crack rests on comparing a smeared, uniform bond defect against a localized crack; a natural testable extension is to check whether the displacement ratio over $r$ can also recover the crack's angular extent or depth, not just its presence.
  • The near-linearity of the mean displacement ratio in $\mu_1/\mu_0$ suggests the approximation may hold for a family of loading profiles $p(r)$, not just the quadratic loading used; verifying this would broaden the non-destructive testing recipe.
  • Since the soft-interface law is an asymptotic thin-layer model, the paper's quantitative conclusions about soft contact inherit its accuracy limitations; one could construct a boundary-layer comparison against direct finite-element simulations at the chosen parameter ratios.
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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

4 major / 4 minor

Summary. The manuscript studies steady-state torsional vibrations of an axisymmetric hollow cylinder made of two vertical sections with possibly different elastic materials, with a fixed base and an arbitrary torsional traction on the top face. Three interface models are considered: ideal bonding, a soft thin layer, and a rigid thin layer. The authors apply a finite Hankel transform and a one-dimensional Green's function to derive explicit series formulas for the displacement, given in Eqs. (27), (33), and (39) for ideal, soft, and rigid contact, respectively. Numerical results explore the displacement and stress fields, the influence of geometry and material parameters, resonance frequencies, and a damage-approximation method that replaces a damaged region by a weak interfacial layer. The paper claims complete explicit solutions for arbitrary axisymmetric loading, valid for different material properties in the two sections.

Significance. The topic is of practical interest, and the explicit formulas, if correct, would provide a fast semi-analytical tool for parametric studies of coupled cylinders and for non-destructive testing. The paper has several strengths: it gives closed-form inverse-transform representations, includes the useful limit checks that K_1=0 and K_2=0 recover ideal contact, and presents a broad numerical survey with tables and figures. However, the central derivation contains internal inconsistencies that affect the general case of different material wave speeds and the soft-contact formula, so the numerical conclusions and the damage-calibration method are not reliable as they stand. The significance of the paper therefore hinges on a substantial correction of the Green's function representation and of the printed final formulas.

major comments (4)
  1. [§3.3-§4] The representation (20)–(21) uses the same forcing term p_n sinh(γ_{2n} z)/(µ_2 γ_{2n} cosh(γ_{2n}H)) for both k=1 and k=2. For the lower section, however, the transformed equation (12) has coefficient γ_{1n}, and this forcing term is not a solution of u''_{1n} − γ_{1n}² u_{1n}=0 when c_1≠c_2. For n=0, substituting (23) into (13) gives u''_{10} + β_1² u_{10} = p_0(β_2²−β_1²) sin(β_2 z)/(µ_2 β_2 cos β_2 H), which is not zero unless β_1=β_2. The ideal-contact continuity condition [[u]]=0 at z=h is also violated: from (23) the jump equals −(µ_2−µ_1)/µ_1 u'_{20}(h+0)[G_{10}(h,h)−G_{20}(h,h)], which generally does not vanish. Since the paper explicitly treats different materials, as in the cuprum–aluminium example of Table 1, Eqs. (27), (33), and (39) are not valid in the paper's main setting, and the resonance frequencies in Fig. 4 are consequently unreliable. The solution should be re-derived with piecewise fundamental solutions in each section or with a correct bimaterial Green's function.
  2. [§4.2] The final soft-contact displacement (33) contains a sign error in the n-sum. The transformed-domain expression (31) has u_{kn}(z) = p_n sinh(γ_{2n}z)/(µ_2 γ_{2n} cosh γ_{2n}H) − [K_1 ∂G_{kn}/∂η + ((µ_2−µ_1)/µ_1)G_{kn}] p_n cosh(γ_{2n}h)/(µ_2 Δ^S_{2n}), whereas in (33) the same bracket enters with a plus sign before the n-sum. As a result, for K_1>0 the n>0 contributions in (33) have the opposite sign to those in (31), so (33) is not the inverse transform of (31). This sign error affects all soft-contact results in Section 5 and the damage approximation in Section 6, and it must be corrected before any quantitative conclusions are drawn.
  3. [§6] Equation (41) is not consistent with the preceding formulas. The denominator contains R^4 − a^2, while the corresponding ideal-contact displacement (27) has (R^4 − a^4)/4 in the n=0 term; the two expressions have different dimensions, so the printed denominator is dimensionally wrong. In addition, using the correct signs from Eqs. (31)–(33), the n-sum contribution in the numerator should be positive, not negative as printed. Consequently the least-squares fit ¯u = 1.000 + 0.02482 µ_1/µ_0 cannot be reproduced from the equations in the manuscript, and the proposed non-destructive-testing calibration is not verifiable as stated.
  4. [§2.2.2] The soft and rigid contact conditions (5) and (7) are leading-order thin-layer asymptotics from Mishuris (2004a). The paper does not state the asymptotic error order and uses parameters h_0/H=0.1, µ_soft/µ_2≈0.13, and µ_rigid/µ_2≈130 in Table 1, for which the first corrections to the reduced conditions are of order h_0/H and of the modulus contrast. Without an error estimate or a comparison against a full three-layer solution or a finite-element computation, the physical accuracy of the soft and rigid displacement fields at these parameters is unverified. This matters for Tables 2–3 and for the damage calibration in Section 6, which relies on the soft-contact model.
minor comments (4)
  1. [§3.1] Equation (10) has a sign error: the identity should read ∫ [∂/∂r(r ∂u/∂r) − (1/r)u] K dr = −λ_n² u_{kn}, since the kernel satisfies L[K] = −λ_n² K. The subsequent transformed equation (12) uses the correct sign, so the error appears to be typographical, but as printed Eq. (10) is inconsistent with Eq. (12).
  2. [§5.3.2] The conclusion that there are no shared eigenfrequencies between the three coupling cases is based on a finite frequency scan and should be qualified as a numerical observation, not a proven property.
  3. [§1] The paper claims to provide the first description of a vibrating hollow cylinder under torsion with 'differing loading on the upper and lower surfaces', but the problem statement has a fixed lower surface and loading only on the upper surface; this phrasing should be corrected.
  4. [§7] The concluding claim that the resonance method can achieve 'arbitrarily high levels of accuracy' is overstated, since the method relies on a phase-shift search and the authors themselves note that boundary-condition accuracy deteriorates near resonance.

Circularity Check

2 steps flagged · score 4.0 of 10

Main displacement derivation is self-contained; circularity is confined to Section 6, where the damage approximation leans on a self-cited prior paper and a least-squares fit to the model's own output.

  1. self citation load bearing [Section 6, 'Using an interfacial layer to approximate damage within a cylinder', final paragraph (p. 29), and the comparison with Zhuravlova et al. (2024).]
    "To demonstrate that this is possible, we perform a comparison with Zhuravlova et al. (2024) which considered a ring crack within a single-material hollow cylinder."

    The validation that a weak interfacial layer approximates a ring crack is anchored to the authors' own previous paper (Zhuravlova, Istenes, Peck, Protserov, Vaysfeld 2024), which has the same author set as the present manuscript. No independent experiment, finite-element benchmark, or re-derivation is supplied here; the non-destructive-testing inference about the 'type of damage present' is justified by 'see Zhuravlova et al. (2024), Fig. 4'. Thus a highlighted application claim is supported by a self-citation rather than by new independent evidence.

  2. fitted input called prediction [Section 6, immediately after Eq. (41) and Fig. 6, p. 28.]
    "This is confirmed in Fig. 6, with least-squares analysis revealing that for the given material parameters and cylinder geometry ¯u can be accurately described by (to 4 significant figures) ¯u = 1.000 + 0.02482 µ1/µ0."

    The slope 0.02482 and intercept 1.000 are obtained by least-squares fitting to values of ¯u computed from the paper's own formula (41). The paper then inverts this fitted line to 'determine' the interface shear modulus µ0 from an observed displacement ratio (Section 6 and Section 7). Because the fit is made to the model's own output and the reported 0.01% relative error is measured against those same computed points, the calibration is internal to the model: the inferred µ0 is the value that makes the model reproduce itself, not an independently validated material parameter. This is a fitted relationship presented as a predictive engineering tool.

full rationale

The central derivation of the displacement formulas (27), (33), and (39) is not circular. The finite Hankel transform, Green's function representation, and the jump conditions are all solved self-consistently from the stated boundary value problem, with classical external references (Pathak and Singh 1982; Popov et al. 1999) for the transform and Green's function identities. The soft and rigid interface conditions are taken from Mishuris (2004a), an external source, not from the authors' own prior work, so no self-citation is load-bearing in the main derivation. The resonance-frequency search is a numerical post-processing of the derived solution. The circularity that exists is confined to Section 6: (i) the damage-approximation validation uses the authors' own 2024 ring-crack paper as the benchmark, making the non-destructive-testing claim rely on a self-citation; and (ii) the linear relation ¯u = 1.000 + 0.02482 µ1/µ0 is a least-squares fit to the model's own computed output, and is then inverted to infer the interface modulus. These issues affect a highlighted application rather than the core elastodynamic solution, so a moderate score is appropriate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central derivation rests on standard elastodynamics, the finite Hankel transform, and thin-layer asymptotic interface conditions from Mishuris (2004a). The paper introduces one modeling construct, the damage-representing weak layer, with no independent validation. Free parameters are mostly hand-chosen simulation inputs plus one least-squares fitted slope.

free parameters (3)
  • slope of linear damage approximation = 0.02482
    Least-squares fit to the model's own computed values of ubar vs mu1/mu0 in Section 6; presented as a result but is a fitted constant, not derived from the governing equations.
  • soft interface simulation parameters = mu0 = 3.285e9 Pa, h0 = 0.01 m, rho0 = 5800 kg/m3
    Chosen by hand in Table 1 (mu_soft = (mu1+mu2)/200) to represent a soft interface; these chosen values affect all numerical results in Section 5.
  • rigid interface simulation parameters = mu0 = 3.285e12 Pa, h0 = 0.01 m, rho0 = 5800 kg/m3
    Chosen by hand in Table 1 (mu_rigid = 100(mu1+mu2)/2) to represent a rigid interface; these chosen values affect all numerical results in Section 5.
assumptions (6)
  • standard math The displacement satisfies the classical linear elastodynamic equation for torsional motion, Eq. (3).
    This is the governing equation of linear elasticity for axisymmetric torsion, stated as the starting point.
  • standard math The finite Hankel transform with kernel K(lambda_n, r) and eigenvalues lambda_n diagonalizes the radial torsion operator, Eq. (10).
    Known result (Pathak and Singh 1982, Korenev 2002), though the printed Eq. (10) has a sign error inconsistent with the kernel's Bessel equation; the subsequent Eq. (12) uses the correct sign.
  • domain assumption The soft interface conditions [[u]] = K1 du2/dz and [[tau]] = 0, with K1 = mu2 mu0^{-1} h0, are valid asymptotics for a thin weak layer.
    Imported from Mishuris (2004a) without derivation in this paper; validity requires h0/H much less than 1 and mu0 much smaller than mu_k.
  • domain assumption The rigid interface conditions, including the equation of motion (7) for the interface displacement v(r), are valid asymptotics for a thin stiff layer.
    Imported from Mishuris (2004a); the derivation of Eq. (7) is not given, and the parameter regime mu0 much larger than mu_k must hold.
  • domain assumption A steady-state harmonic time dependence e^{i omega t} is assumed, with the base fixed and lateral surfaces traction-free.
    This defines the boundary value problem (1)-(3) and is standard for linear vibration analysis.
  • ad hoc to paper A ring crack in a single cylinder can be approximated by a homogeneous soft interfacial layer of height h0 and shear modulus mu0.
    This is the core modeling assumption of Section 6; the paper compares qualitatively with Zhuravlova et al. (2024) but under different loading, and does not validate the mapping quantitatively.
invented entities (1)
  • Weak interfacial layer as a proxy for a ring crack or damage region
    purpose: To model a hidden ring crack in a single-material cylinder as a soft interface, allowing use of the coupled-cylinder solution (33) to estimate displacement and stress.
    The paper does not provide a falsifiable handle outside the model: the mapping from crack geometry to (h0, mu0) is not calibrated against experiments or an independent crack solution under the same loading. The comparison with Zhuravlova et al. (2024) is qualitative and uses different boundary conditions.

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Cite this review

Pith. "Pith review of Torsional vibration of a coupled cylinder." pith.science (2026). https://pith.science/paper/4CNFNY7A

@misc{pith2026250611582,
  author       = {Pith},
  title        = {Pith review of: Torsional vibration of a coupled cylinder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CNFNY7A}},
  note         = {Machine review of arXiv:2506.11582}
}
read the original abstract

The torsion loading of a coupled cylinder, comprising distinct upper and lower cylindrical sections potentially made of different materials, is considered. The bottom of the cylinder is fixed in place, and induces the cylinder vibration. The torsion is applied via an arbitrary loading on the upper face. Three forms of coupling condition between the upper and lower cylinders are outlined: ideal, soft (weak), and rigid (hard/ stiff) contact. The resulting displacements and tangential stresses are obtained using the finite Hankel transform, and a Green's function representation of the displacement. Numerical results are provided, and the impact of the differing coupling conditions investigated for a range of cylinder geometries, material properties and vibration rates. The resonance frequencies of the coupled cylinder are determined. A method for using the coupled cylinder model to approximate the displacement of a cylinder containing a damaged region via a weak interfacial layer is outlined. The properties of the weak interface layer needed for this approximation are determined, and the advantages of its use in non-destructive testing are discussed.

Figures

Figures reproduced from arXiv: 2506.11582 by the authors.

Figure 1
Figure 1. Geometry and coordinate system of two-layered hollow cylinder. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The change of (a), (c), (e) the scaled displacement and (b), (d), (f) the stress [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. The change of (a) the scaled displacement and (b) the stress in a coupled [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The resonance frequencies computed from the scaled displacement [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: It can be seen that the interface position and shear moduli ratio both [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 5
Figure 5. Figure 5: The displacement at the top of the cylinder [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: The arithmetic mean over r of the displacement ratio usoft/uideal (41) at the interface z = h, alongside a linear approximation produced using the least-square method. Results are for a steel cylinder with fixed interface relative height δ = h/H = 0.5. The weak interfa…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.