A Numerical Investigation of the Aeroelastic Interaction between Transonic Buffet and Structural Nonlinearity

Michael Candon; Oleg Levinski; Pier Marzocca; Vincenzo Muscarello

REVIEW 2 major objections 5 minor 38 references

A Numerical Investigation of the Aeroelastic Interaction between Transonic Buffet and Structural Nonlinearity

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Pitch freeplay can make transonic shock buffet lock onto heave superharmonics, producing 2:1 and 3:1 subharmonic resonances with heave limit-cycle amplitudes up to about 37 times the no-freeplay baseline.

desk verdict Plausible new result on buffet-freeplay superharmonic lock-in, but the flutter exclusion rests on an unvalidated first-order flow approximation that needs a cross-check before the LCOs can be attributed to buffet rather than flutter. read the letter →

arxiv 2505.02412 v1 pith:VDLBS64B submitted 2025-05-05 physics.flu-dyn physics.data-an

Michael Candon , Vincenzo Muscarello , Pier Marzocca , Oleg Levinski This is my paper · ORCID

classification physics.flu-dynphysics.data-an

keywords transonicshockbuffetfreeplaynonlinearitylimitcycleoscillationaerodynamiclock-insuperharmonicresonancesubharmonicURANSaeroelasticity

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 asks what happens when two nonlinearities meet: the self-sustained shock oscillations of transonic buffet and the bilinear stiffness of a hinge with freeplay. It finds that, for a two-degree-of-freedom heave-pitch airfoil in URANS flow, a sufficiently large pitch freeplay redistributes vibrational energy into superharmonics of the heave natural frequency, and the unsteady aerodynamics lock onto those superharmonics. The result is 2:1 and 3:1 subharmonic resonances at heave-to-buffet frequency ratios below 1, with heave limit-cycle amplitudes roughly 16–24 times (2:1) and about 37 times (3:1) the linear baseline. These are exactly the load conditions relevant to fatigue on control surfaces and empennages, and the regime identified is one that linear-structure studies would miss.

What carries the argument

The central object is a two-degree-of-freedom heave-pitch NACA 0012 section with pitch freeplay modeled as a bilinear spring (zero restoring moment inside $\pm\alpha_s$), coupled to URANS aerodynamics through an embedded equation of motion. The load-bearing mechanism is superharmonic lock-in: each impact with the freeplay deadzone boundary injects energy into harmonics $p\omega_h$; when the buffet frequency $k_{sb}$ sits close to $p\hat{k}_h$, the aerodynamic force locks onto that superharmonic, while the structure responds at the subharmonic $\omega_h$. Shock-location tracking and the equivalent angle of attack $\alpha_e=\alpha_0+\alpha-\dot h/U_\infty$ are used to show how the shock and structure synchronize and why the shock dwells downstream in the 3:1 case.

What would settle it

A wind-tunnel test of a NACA 0012 section with pitch freeplay $\alpha_s=0.5^\circ$ at $M_\infty=0.72$, $\alpha_0=6^\circ$, and $\hat{k}_h=0.45$ should show a stable heave LCO about 37 times the no-freeplay amplitude with the lift spectrum locked onto $3\hat{k}_h$; observing only small forced oscillations, or finding the large LCO only when the system is close to the flutter boundary, would settle against the claim.

Watch

Extended reading notes

Core claim

The paper claims that structural nonlinearity alone can move a buffet-driven airfoil from forced-harmonic-oscillator behavior into aerodynamic lock-in at superharmonics of the heave mode. With $\alpha_s=0.25^\circ$ freeplay and $\hat{k}_h\gtrsim0.515$, the lift locks onto $2\hat{k}_h$ while the heave responds at $\hat{k}_h$ (2:1 resonance), giving heave LCO amplification of 16 times, and 24 times with $\alpha_s=0.5^\circ$; with the larger freeplay and $\hat{k}_h<0.49$, a 3:1 resonance locks onto $3\hat{k}_h$, with heave amplification of about 37 times and pitch amplification of about 2 times. The lock-in band sits at $\hat{k}_h=0.4$–$0.555$ with $\hat{k}_\alpha=0.755$, well below the previously known lock-in range near unity. The paper also reports that shock motion synchronizes with structural motion (phase about $8.6^\circ$ with heave rate in the 2:1 case) and that during the 3:1 cycle the large heave rate pushes the equivalent angle of attack below the buffet onset angle, pinning the shock downstream. Sensitivity runs show that 2% structural damping or a doubled mass ratio suppresses the resonance, while 0.5% damping already reduces it substantially.

Load-bearing premise

The flutter check assumes that artificially increasing numerical diffusion in the flow solver removes the shock-buffet oscillations while leaving the linear flutter boundary believable; if that is wrong, the large limit cycles could be flutter rather than buffet-freeplay lock-in.

Editorial extensions

If this is right

Heave LCO amplitudes of 16–24 times (2:1) and about 37 times (3:1) the linear baseline occur with pitch freeplay at heave frequency ratios $\hat{k}_h=0.4$–$0.555$, far below the canonical lock-in range near $\hat{k}_h\approx1$.
The onset of 2:1 lock-in requires only small freeplay ($0.1^\circ<\alpha_s<0.25^\circ$), while 3:1 lock-in needs larger freeplay ($0.25^\circ<\alpha_s<0.5^\circ$), so modest hinge wear can abruptly change the response.
Structural damping of 2% completely suppresses the 3:1 lock-in, and doubling the mass ratio also suppresses it, making the resonance far more sensitive to these parameters than linear-model lock-in.
During 3:1 lock-in the aerodynamic forces respond at $3\omega_h$ while the structure responds at $\omega_h$, a subharmonic resonance, with the shock pinned downstream when the equivalent angle of attack drops below the buffet onset angle.
Flutter screening indicates the test points lie below the linear flutter boundary, with the system at 96% of flutter speed at $\hat{k}_h=0.4$, so the large LCOs are attributed to buffet-freeplay interaction rather than post-flutter motion.

Reading between the lines

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

The same superharmonic lock-in mechanism should arise with other stiffness nonlinearities, such as cubic hardening or bilinear damping, because the energy-redistribution step does not depend on the deadzone specifically; substituting the force law in the structural equation would test this directly.
The equivalent-angle-of-attack pinning mechanism predicts that moving the freestream angle of attack closer to the buffet onset angle should narrow or shift the 3:1 lock-in band, since the large heave rate would no longer push $\alpha_e$ below onset; this is testable with the same solver.
If these subharmonic resonances survive on a finite wing, certification fatigue spectra for tail surfaces and control-surface hinges with freeplay should include buffet-induced subharmonic loads even when the structural mode frequency lies well below the buffet frequency.
The 2-D URANS setting leaves open whether spanwise shock cells on a finite wing broaden or suppress the lock-in band; a half-span wing simulation would indicate whether the mechanism survives three-dimensional buffet.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this paper. First, it is genuinely new: the coupling of transonic shock buffet with a pitch-freeplay structural nonlinearity, and the finding that large freeplay can induce 2:1 and 3:1 superharmonic lock-in at frequency ratios below 1, is not in the cited literature. Prior work combined buffet with linear structure, or freeplay without the global buffet instability. Second, the paper's central claim is plausible but not fully established, because the authors rule out flutter using a flow with artificially increased numerical diffusion that has a different mean shock location, and they never check that approximation against the actual buffet flow.

The paper does several things well. The rigid-airfoil validation is solid: buffet frequency within 1.3% of experiment, grid convergence for ΔCL demonstrated. The parameter sweeps are informative, and the distinction between αs=0.25° (2:1 only) and αs=0.5° (3:1 also) is clean. The kinetic-energy explanation in Figure 24 is post hoc, but it is not circular and it matches the observed mass-ratio sensitivity. The authors do not hide their assumption; they explicitly call the flutter estimate "approximate" and recommend experimental verification.

The soft spots are proportionate to the evidence. The biggest one is Section 4.1. The first-order momentum equation changes the steady shock location to the maximum downstream position of the buffeting flow, as their own Figure 4 shows. That means the linear flutter boundary is computed about a mean flow that is not representative of the second-order buffet condition. At the lowest heave frequency ratio the system operates at 96% of this approximate flutter speed—a thin margin. The time-marching check in Figure 6 uses the same first-order flow, so it cannot validate the boundary for the actual buffet flow. If the true flutter speed is lower, the large heave LCOs could be freeplay-induced flutter rather than buffet lock-in, and the superharmonic peaks would be a symptom rather than a mechanism. I think the lock-in interpretation is quite plausible given the spectra and the distinct freeplay threshold, but the authors have not closed this door.

Minor weaknesses: no aeroelastic validation (the rigid-airfoil validation does not cover the moving-mesh coupling), no mesh or time-step sensitivity for the aeroelastic cases, and no uncertainty quantification. Code and data are not released, so reproduction would require contacting the authors. These are normal for a URANS study but they do limit how definitive the claims can be.

Who is this for? Researchers in transonic aeroelasticity, especially those working on buffet and freeplay, will find it valuable as a new data point and a clear hypothesis to test experimentally. It deserves a serious referee. I would send it to peer review with a request for major revision, and I would insist that the authors either cross-check the first-order flutter boundary against the second-order flow (for instance, by damping buffet through a different mechanism, or by running a few time-marching cases at the boundary with the full model) or substantially soften the claim that the system is below flutter.

For what it's worth, I would cite this paper if I were writing about buffet-freeplay interactions, but I would not rely on its flutter exclusion without more evidence.

Referee Report

2 major / 5 minor

Summary. The paper presents URANS simulations of a NACA0012 airfoil at M∞=0.72 and α0=6°, coupled to a two-degree-of-freedom heave-pitch structural model with pitch freeplay. The authors first validate the rigid-airfoil buffet prediction against the McDevitt-Okuno experiment, then sweep the heave frequency ratio (0.4–0.555) and freeplay angle (0°, 0.25°, 0.5°). They report that with sufficiently large freeplay, the aerodynamic force spectrum locks onto superharmonics 2kh or 3kh of the heave natural frequency while the structure oscillates at kh, producing 2:1 and 3:1 subharmonic resonances with heave limit-cycle amplitudes 16–37 times the linear baseline. The sensitivity of these phenomena to structural-to-fluid mass ratio and structural damping is also examined. The paper claims this is the first demonstration of buffet-freeplay lock-in.

Significance. If the reported phenomenon is confirmed, this is a novel aeroelastic interaction: freeplay-induced superharmonic excitation of the shock-buffet flow at frequency ratios well below the previously reported lock-in range, with potentially significant implications for fatigue and flight safety. The paper's strengths include the careful rigid-airfoil validation (buffet frequency within 1.29% of experiment), mesh refinement for the rigid configuration, and the use of time-frequency analysis, shock-trajectory tracking, and Lissajous curves to characterize the mechanisms. The kinetic-energy interpretation of the mass-ratio effect is a plausible physical explanation. The principal caveat is that the flutter exclusion, which is essential to the interpretation of the LCOs as buffet-driven rather than post-flutter, rests on an ad-hoc first-order spatial discretization that is not cross-validated against the buffeting flow.

major comments (2)

[§4.1, Figs. 4–6] The flutter exclusion relies entirely on reducing the momentum equation to first-order spatial accuracy to suppress the buffet instability, and the time-marching check in Fig. 6 uses the same first-order flow. Figure 4 shows that the first-order solution has a different mean shock location (at the maximum downstream position of the buffeting flow) than the second-order buffet flow, so the linear flutter boundary computed from this altered flow may not represent the actual buffet flow. Because the system is predicted to be at 96% of the flutter speed at the lowest heave frequency (kh=0.4) and the large-amplitude LCOs occur at or near this condition, the possibility that these LCOs are freeplay-triggered post-flutter motions rather than buffet lock-in is not eliminated. Please provide a cross-check using a buffet-stable flow that retains second-order accuracy (e.g., a flow-stabilization method) or a time-marching aeroelastic simulation with the second-order scheme at a lower velocity index to confirm that all reported LCOs are sub-flutter.
[§§4.2–4.5] The coupled aeroelastic simulations are not subject to mesh or time-step sensitivity studies. The validation in §3 is for the rigid airfoil only; the dynamic-mesh, moving-airfoil, and freeplay-impact processes may have different resolution requirements. Since the reported amplification factors (16×, 24×, ~37×) and the abrupt transition between 3:1 and 2:1 lock-in near kh≈0.49 are central quantitative results, a grid and Δτ convergence study for at least one 2:1 case (e.g., kh=0.555, αs=0.5°) and one 3:1 case (e.g., kh=0.45, αs=0.5°) is needed to demonstrate that these observations are not numerical artifacts.

minor comments (5)

[§4.2 and Fig. 13] There are several typographical errors: in §4.2, 'the structure oscillating oscillating at kh' should read 'oscillating at kh'; in the Fig. 13 caption, 'aligns wirg' should read 'aligns with'; in §5, 'signification interest' should read 'significant interest'; in §4.2, 'the the increase' should read 'the increase'.
[Eq. (2) and §4] The text uses αs to denote the half freeplay gap, but the phrase 'freeplay αs=0.25°' may be ambiguous; please state explicitly that αs is the half-gap rather than the full gap, and confirm that the baseline aeroelastic results are computed with zero structural damping in both modes.
[Fig. 11] The Short-Time Fourier Transform in Fig. 11 is used to support the beating transient discussion, but the window function, window length, and overlap are not specified; a brief note on these parameters would improve reproducibility.
[Abstract and Introduction] The claim of 'for the first time' is strong, especially in light of the authors' own related work reported in reference [38]. Please calibrate the novelty statement to distinguish the specific new mechanisms reported here from the earlier conference paper.
[§2.1] The computational cost of the 1×10^6 time-step simulations is not reported; a statement of typical wall-clock time and core count would help readers assess the practical feasibility of the study.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central lock-in result emerges from coupled CFD simulation rather than from fitted inputs or self-citation chains.

full rationale

The paper is a numerical parameter study in which the load-bearing lock-in and resonance observations emerge from coupled URANS/structural simulations, not from an analytical derivation that identifies its conclusion with its premise. No model constant is fitted to the observed LCO amplitudes or frequencies: the structural parameters (kh, kalpha, mu, zeta, alpha_s) are prescribed inputs, and the 2:1/3:1 classifications are assigned post hoc from PSD and time-domain data. The validation against the McDevitt-Okuno experiment in Section III provides external support for the flow solver. Self-citations [33] (RFA details), [37] (s-DOF flutter caveat), and [38] (alpha_s=0.1 threshold) are not load-bearing: [33] supplements a standard external method (Roger RFA), [37] is corroborated by an external reference, and [38] sets a secondary onset threshold that is not required for the main 2:1/3:1 lock-in demonstration at alpha_s = 0.25 and 0.5 degrees. The concern raised about the first-order momentum flutter screening in Section 4.1 is a modeling-approximation and robustness issue, not circularity, because the flutter boundary is computed from a separately stabilized flow and is not used as a fitted input to produce the observed LCOs. Frequency-ratio sweeps prescribe the ratios at which resonances could be sought, but the actual lock-in is evidenced by synchronized shock motion, phase data, and PSD content rather than being enforced by construction. No circular step can be exhibited; the derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on simulation fidelity and the flutter-margin approximation. No free parameters are fitted to target results, but the URANS closure, the first-order diffusion flutter screen, and the 2D section model are imported assumptions that the paper does not independently validate for the coupled aeroelastic case.

assumptions (5)

domain assumption URANS with Spalart-Allmaras and curvature correction accurately represents transonic shock buffet and its interaction with structural motion.
Validation in Section 3 is limited to rigid-airfoil mean pressure, buffet frequency, and Delta CL; no validation of the coupled aeroelastic LCO is provided.
ad hoc to paper First-order momentum artificial diffusion provides a trustworthy linear flutter baseline.
Section 4.1 introduces this method and deems it sufficient without cross-checking against a stabilized nonlinear solution; if wrong, observed LCOs could be post-flutter.
domain assumption The two-degree-of-freedom uncoupled heave-pitch model with pitch freeplay is representative of a wing section.
Section 2.2 uses a 2D section model that neglects spanwise effects, structural coupling, and control-surface detail.
domain assumption The frequency-domain rational function approximation flutter solution accurately extrapolates to the buffet condition.
Figure 5 relies on a least-squares RFA from indicial responses; no validation against experimental flutter data at this condition is shown.
domain assumption Dynamic mesh diffusive smoothing does not contaminate the aerodynamic forces.
Section 2.2 states diffusive smoothing is used, but no mesh-motion verification is presented beyond the rigid-airfoil buffet tests.

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Cite this review

Pith. "Pith review of A Numerical Investigation of the Aeroelastic Interaction between Transonic Buffet and Structural Nonlinearity." pith.science (2026). https://pith.science/paper/VDLBS64B

@misc{pith2026250502412,
  author       = {Pith},
  title        = {Pith review of: A Numerical Investigation of the Aeroelastic Interaction between Transonic Buffet and Structural Nonlinearity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VDLBS64B}},
  note         = {Machine review of arXiv:2505.02412}
}

read the original abstract

Transonic shock buffet is a nonlinear, unsteady aerodynamic phenomenon characterized by self-sustained, periodic shock oscillations that can critically affect aircraft structural integrity. While the aerodynamic aspects of shock buffet have been widely studied, its interaction with nonlinear structural dynamics remains largely unexplored. This paper presents, for the first time, a numerical investigation of aeroelastic interactions arising from the coupling of shock buffet with a nonlinear structural model featuring pitch freeplay. Using unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations coupled with a two-degree-of-freedom heave-pitch airfoil model, the study reveals that structural nonlinearity can induce aerodynamic lock-in to superharmonics of the heave natural frequency, resulting in 2:1 and 3:1 resonance mechanisms and large-amplitude heave limit cycles. These newly identified resonance behaviors expand the current understanding of transonic aeroelastic instabilities. The influence of key parameters such as structural-to-fluid mass ratio and structural damping on these phenomena is also systematically examined. This work introduces a novel class of aeroelastic lock-in mechanisms with significant implications for transonic flight dynamics and aircraft design.

Figures

Figures reproduced from arXiv: 2505.02412 by the authors.

**Figure 1.** Computational grid c 2.2 Nonlinear Aeroelastic Equations-of-Motion As per the work of Raveh and Dowell [14], this research considers the uncoupled 2-DOF aeroelastic equations-of-motion with the addition of freeplay, given by: m(h¨ + 2ζhωhh˙ + ωh 2h) = L, (1a) Iα( ¨α + f(α, α˙)) = Mc/4 (1b) 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Grid refinement for pressure coefficient distribution at the pre-buffet condition [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Lift coefficient comparing first-order and second-order discretization of the momentum equation at [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (24 more)

**Figure 4.** Figure 4: Pressure coefficient contours comparing first-order and second-order discretization of the momentum [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Flutter speed ratio at the nominal operating condition for varying heave natural frequencies. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Time-domain aeroelastic response with first-order discretization of the momentum equation. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 10.** Figure 10: The PSD is normalized to 1 (by the respective maximum power) for visualization purposes. [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

**Figure 7.** Figure 7: Dominant response frequencies with µ = 75, kˆh = 0.4 − 0.555, kˆα = 0.755 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: Response RMS with µ = 75, kˆh = 0.4 − 0.555, kˆα = 0.755 The responses with freeplay αs = 0.25◦ is characterized by significant amplification of the LCO in the heave mode (16× relative to the case without freeplay), and moderate amplification in the pitch mode (1.75× r…

**Figure 9.** Figure 9: Time responses with µ = 75, kˆh = 0.555, kˆα = 0.755 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: Power spectral density of the stable LCO with [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: Short-Time Fourier Transform with µ = 75, kˆh = 0.555, kˆα = 0.755 and αs = 0.25◦ [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: Aerodynamic forces and moments with µ = 75, kˆh = 0.555 and kˆα = 0.755 heave rate, and moment as a function of pitch rotation, are both characterized by two closed sections as a result of the 2:1 relationship between the forcing frequency at 2kh and the pitch respons…

**Figure 13.** Figure 13: Time responses depicting shock location for one lock-in cycle with [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

**Figure 14.** Figure 14: Lissajous curves depicting shock location for one lock-in cycle with [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

**Figure 15.** Figure 15: Time responses with µ = 75, kˆh = 0.45, kˆα = 0.755 [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]

**Figure 16.** Figure 16: Power spectral density of the stable LCO with [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

**Figure 17.** Figure 17: Aerodynamic forces and moments with µ = 75, kˆh = 0.45 and kˆα = 0.755 [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]

**Figure 18.** Figure 18: Time responses depicting shock location for one lock-in cycle with [PITH_FULL_IMAGE:figures/full_fig_p013_18.png]

**Figure 19.** Figure 19: Equivalent angle-of-attack and shock location with [PITH_FULL_IMAGE:figures/full_fig_p014_19.png]

**Figure 20.** Figure 20: Shock location as a function of structural response for [PITH_FULL_IMAGE:figures/full_fig_p015_20.png]

**Figure 21.** Figure 21: Lissajous curves for µ = 75, kˆh = 0.45, kˆα = 0.755 and αs = 0.5 ◦ 4.5 Sensitivity to Mass Ratio and Structural Damping The final analysis is to assess the influence of mass ratio and structural damping on the lock-in mechanism, noting that previous authors [31, 6] h…

**Figure 24.** Figure 24: Notably, the first few impacts (τ < 100) occur with a similar KE, then the first impact after 15 [PITH_FULL_IMAGE:figures/full_fig_p015_24.png]

**Figure 22.** Figure 22: Time responses at kˆh = 0.45, kˆα = 0.755 and αs = 0.5 ◦ with various mass ratios [PITH_FULL_IMAGE:figures/full_fig_p016_22.png]

**Figure 23.** Figure 23: Time responses at kˆh = 0.555, kˆα = 0.755 and αs = 0.5 ◦ with various mass ratios τ = 100 is with a nearly 4× increase in KE (with respect to the previous maximum) for µ = 75, and a 2× increase for µ = 100, while no notable increase can be observed for µ = 150. Moreo…

**Figure 24.** Figure 24: Total kinetic energy at kˆh = 0.45, kˆα = 0.755 and αs = 0.5 ◦ with various mass ratios The influence of structural damping on the 3:1 lock-in mechanism is presented in [PITH_FULL_IMAGE:figures/full_fig_p017_24.png]

**Figure 25.** Figure 25: Time responses at µ = 75, kˆh = 0.45, kˆα = 0.755 and αs = 0.5 ◦ with structural damping 5 Conclusions Transonic shock buffet is a complex unsteady aerodynamic phenomenon that plagues modern aircraft, characterized by large amplitude self-sustaining shock oscillation…

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