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arxiv: 2605.07428 · v1 · submitted 2026-05-08 · 🧮 math.DS

Recognition: no theorem link

Non-intrusive spectral submanifold model reduction for geometrically nonlinear rotating structures with Coriolis and centrifugal forces

Hejun Gao, Jie Yuan, Mingwu Li, Yan Qing Wang, Yiliang Wang

Pith reviewed 2026-05-11 01:45 UTC · model grok-4.3

classification 🧮 math.DS
keywords spectral submanifoldmodel reductionrotating structuresgeometric nonlinearityCoriolis effectcentrifugal forcefinite element methodnonlinear dynamics
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The pith

Non-intrusive spectral submanifold reduction accurately models nonlinear vibrations in rotating structures with Coriolis and centrifugal forces.

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

The paper establishes a method to create reduced-order models for high-dimensional finite element simulations of rotating structures by anchoring spectral submanifolds at the centrifugal static equilibrium. This approach captures geometric nonlinearity along with Coriolis and centrifugal effects non-intrusively, relying only on simulation outputs rather than internal code access. A sympathetic reader would care because it enables efficient calculation of nonlinear response curves that full simulations make too costly to explore repeatedly. The work demonstrates this on progressively complex examples from beams to multi-blade fans, showing how the reduced models reproduce key dynamic behaviors.

Core claim

The authors claim that spectral submanifolds can be constructed non-intrusively around the nontrivial static equilibrium induced by centrifugal forces to yield low-dimensional models that accurately reproduce the nonlinear dynamics, including backbone and forced response curves, for rotating structures subject to geometric nonlinearity, Coriolis effects, and centrifugal stiffening, as verified through comparisons with full finite element simulations in COMSOL.

What carries the argument

Non-intrusive spectral submanifold (SSM) construction anchored at the centrifugal static equilibrium, which reduces the high-dimensional finite element model while retaining the essential coupled nonlinear dynamics.

If this is right

  • SSM-based ROMs allow efficient extraction of backbone curves for rotating beams and twisted plates.
  • Accurate forced response predictions are possible for rotors with multiple disks.
  • Internal resonances in fans with three blades can be analyzed efficiently.
  • The method highlights the important role of Coriolis forces in the nonlinear vibration behavior.

Where Pith is reading between the lines

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

  • Similar non-intrusive SSM techniques might apply to other engineering systems with rotating components, such as wind turbines or jet engines, for design optimization.
  • Extensions could involve incorporating additional effects like aerodynamic loading or material damping to broaden applicability.
  • Comparisons with experimental data from rotating test setups would further validate the models for practical use.

Load-bearing premise

The non-intrusive SSM anchored at the centrifugal equilibrium fully captures all relevant coupled nonlinear effects from the underlying finite element model.

What would settle it

A significant mismatch between the SSM reduced-order model predictions and full finite element results for the forced response curve of the internally resonant fan would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.07428 by Hejun Gao, Jie Yuan, Mingwu Li, Yan Qing Wang, Yiliang Wang.

Figure 2
Figure 2. Figure 2: Geometry of the rotating beam model and its finite element mesh. the rotational speed is tuned to induce internal resonance, and our method is applied to evaluate its performance in more complex nonlinear regimes. The geometry and corresponding finite element mesh of the rotating beam are adopted from [41] [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Backbone curves of the rotating beam obtained via different methods with Coriolis effect neglected. Here, ̂𝜔 represents the response frequency, 𝜔1 denotes the frequency of first bending mode, and the vertical axis indicates the deflection at the beam tip. 0.99 0.995 1 1.005 1.01 0.2 0.4 0.6 0.8 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Backbone curves of the rotating beam obtained via SSM reduction. The meanings of ̂𝜔, 𝜔1 and the vertical axis are the same as those of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The FRCs of rotating beam model obtained via SSM model order reduction, along with the comparison of Coriolis force considering and Coriolis force neglecting, as well the results of full model. The left picture (a) shows FRC of the rotating beam under 2000 rpm, with five representative data points of the full model as the red marks. Here and throughout this paper, the cyan circles denote saddlenode bifurca… view at source ↗
Figure 6
Figure 6. Figure 6: The forced response curve (FRC) of rotating beam with inner resonance obtained via SSM model order reduction, along with the results of full model. The left panel (a) shows FRC of the rotating beam under 1:3 internal resonance, with five representative data points of the full model as the red marks. Specially, here the black square denotes a Hopf bifurcation point. The right panal (b) is the time response … view at source ↗
Figure 7
Figure 7. Figure 7: Geometry of the twist blade model and its finite element mesh. 0.96 0.97 0.98 0.99 1 1.01 0.1 0.2 0.3 0.4 0.5 300 rpm Consider Coriolis 300 rpm Neglect Coriolis 500 rpm Consider Coriolis 500 rpm Neglect Coriolis 1000 rpm Consider Coriolis 1000 rpm Neglect Coriolis [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Backbone curve of the twist blade under different rotation speeds. rotation axis is 0.3 m from the root and its direction is along (0, 0, 1) and the left surface at 𝑥 = 0.3 m is clamped and the other three surfaces are free. The material properties are the same with those of the rotating beam model described in Section 4.1. Here, we take the first mode as the master subspace to perform model reduction on t… view at source ↗
Figure 12
Figure 12. Figure 12: Geometry of the fan model and its finite element mesh. Three blades are labeled as the figure shows. the disks. Therefore, it is essential to consider the Coriolis force even at linear dynamics. In the rest of this example, we take the Coriolis force into consideration. To investigate the forced vibration of the system, har￾monic loads along the zaxis direction are applied at (0.3, 0, 0.075) and (0.7, 0, … view at source ↗
Figure 11
Figure 11. Figure 11: FRC of simple rotor model under 100 rpm. Poissons ratio of 0.3. Both ends of the shaft are clamped, and the rotational speed is 100 rad∕s. The geometry and material properties of this model are adopted from [22]. Damping ratio of first mode 𝜉1 is assigned as 0.005. The total DOFs of this model is 47673 [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: The first three mode shapes of the rotating fan. Left panel: 1st mode, middle panel: 2nd mode, right panel: 3rd mode. 29.75 29.8 29.85 29.9 29.95 0.2 0.4 0.6 0.8 1 1.2 A D C B 29.75 29.8 29.85 29.9 29.95 0.2 0.4 0.6 0.8 1 1.2 1.4 A B C D 29.75 29.8 29.85 29.9 29.95 0 0.2 0.4 0.6 0.8 1 1.2 B A C D [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FRCs of SSM-based ROM for the simple fan model in reduced coordinates 𝜌1 (left panel), 𝜌2 (middle panel), and 𝜌3 (right panel). extremely close. In particular, we have 𝜔1 = 29.7545 rad/s, 𝜔2 = 29.7571 rad/s, and 𝜔3 = 29.7573 rad/s for the undamped natural frequencies. Therefore, we have a near 1:1:1 internal resonance in this system [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FRCs of the simple fan model for the total displacement at the tip of the three blades. Here, 𝑟tip1, 𝑟tip2, and 𝑟tip3 in the three panels denote the total displacement at the tip of the 1st, 2nd, and 3rd blades, respecitvely. to increased master subspace dimensionality, while a single data point in [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗
read the original abstract

Rotating structures are widely observed in engineering applications such as turbomachinary and wind turbine. These rotating structures, particularly for blades made by lightweight materials, can undergo large deformation in operations and display complex nonlinear dynamics under the coupling interaction of geometric nonlinearity, Coriolis effect and centrifugal force. Finite element (FE) methods provide a powerful and accurate modeling approach for capturing the complex nonlinear dynamics for realistic rotating structures, yet its high-dimensionality causes significant challenge to efficient prediction for the nonlinear vibration. Here, we present a non-intrusive spectral submanifold (SSM) model reduction for these FE models of rotating structures. We use COMSOL to establish FE models and simulate these FE models to verify the accuracy of SSM-based reduction. We first compute nontrivial static equilibrium induced by the centrifugal force and then construct non-intrusively SSM based reduced-order model (ROM) anchored at the equilibrium. These SSM-based ROMs enable efficient and accurate extraction of backbone and forced response curves. We use a suite of examples with increasing complexity to demonstrate the effectiveness of the SSM reduction, including a rotating beam, a twisted plate, a rotor with two disks, and an internally resonant fan with three blades. The obtained results also highlight the significant effects of Coriolis force on the nonlinear vibration of rotating structures.

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

2 major / 2 minor

Summary. The manuscript proposes a non-intrusive spectral submanifold (SSM) model reduction for high-dimensional finite-element models of geometrically nonlinear rotating structures that incorporate geometric nonlinearity together with Coriolis and centrifugal forces. The method first locates the nontrivial centrifugal static equilibrium and then constructs an SSM-based reduced-order model anchored at that equilibrium using trajectory data generated by COMSOL; the resulting ROMs are used to compute backbone curves and forced-response curves. The approach is demonstrated on four examples of increasing complexity (rotating beam, twisted plate, two-disk rotor, and internally resonant three-blade fan), with the claim that Coriolis effects are shown to be significant.

Significance. If the non-intrusive construction is shown to recover the correct reduced dynamics including velocity-dependent gyroscopic terms, the work would supply a practical tool for nonlinear vibration analysis of industrial rotating machinery without requiring intrusive access to the underlying FE code. This would be a useful extension of existing SSM theory to systems with skew-symmetric linear operators.

major comments (2)
  1. [Methods section on non-intrusive SSM construction] The central claim that the data-driven SSM ROM satisfies the invariance equation for the full nonlinear vector field (including the skew-symmetric Coriolis contribution) rests on the non-intrusive fitting procedure. The manuscript does not specify how the homological equations are solved or how the linear operator is assembled when only output trajectories are available; trajectory sampling alone cannot guarantee recovery of velocity-dependent coupling terms.
  2. [Results, three-blade fan subsection] In the three-blade fan example (internally resonant case), modal interactions make the quadratic and cubic coefficients especially sensitive to any mismatch in the reduced vector field. No quantitative error metrics (e.g., relative L2 deviation between full-order and reduced backbone curves, or residual of the invariance equation) are supplied, so the accuracy claim cannot be evaluated.
minor comments (2)
  1. [Abstract] The abstract states that 'COMSOL simulations verify accuracy' but supplies neither error tables nor comparison baselines; adding these would strengthen the verification statements.
  2. [Introduction and Methods] Notation for the centrifugal equilibrium and the anchored SSM should be introduced with a single consistent symbol set rather than being redefined across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of the non-intrusive SSM construction and the validation of the reduced-order models. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Methods section on non-intrusive SSM construction] The central claim that the data-driven SSM ROM satisfies the invariance equation for the full nonlinear vector field (including the skew-symmetric Coriolis contribution) rests on the non-intrusive fitting procedure. The manuscript does not specify how the homological equations are solved or how the linear operator is assembled when only output trajectories are available; trajectory sampling alone cannot guarantee recovery of velocity-dependent coupling terms.

    Authors: We agree that additional detail is required on the non-intrusive procedure. In the revised manuscript we will expand the Methods section to specify that the linear operator (including its skew-symmetric Coriolis component) is recovered by regressing the full state-space vector field on the COMSOL-generated trajectories, which supply both displacement and velocity time series; the resulting reduced vector field is then inserted into the standard homological equations for the SSM coefficients. This data-driven approximation is constructed in the tangent space at the centrifugal equilibrium so that the invariance equation holds for the fitted dynamics. We will also note the practical limitation that exact recovery is guaranteed only up to the accuracy of the trajectory data and the chosen polynomial degree. revision: yes

  2. Referee: [Results, three-blade fan subsection] In the three-blade fan example (internally resonant case), modal interactions make the quadratic and cubic coefficients especially sensitive to any mismatch in the reduced vector field. No quantitative error metrics (e.g., relative L2 deviation between full-order and reduced backbone curves, or residual of the invariance equation) are supplied, so the accuracy claim cannot be evaluated.

    Authors: We accept that quantitative error measures are necessary, especially under internal resonance. The revised manuscript will add, in the three-blade fan subsection, the relative L2 deviation between the full-order and reduced backbone curves together with the residual norm of the invariance equation evaluated along the computed SSM. These metrics will be reported for both the backbone and forced-response results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via data-driven SSM construction and external verification

full rationale

The paper constructs non-intrusive SSM ROMs from COMSOL trajectory data around the centrifugal equilibrium and verifies accuracy against full FE simulations on multiple examples. No step reduces by construction to its inputs: the invariance equation is solved or approximated from sampled orbits without self-defining the manifold coefficients, and no load-bearing self-citation or ansatz smuggling is present in the provided text. The central claim (efficient extraction of backbone/forced response curves) rests on independent numerical comparison rather than tautological fitting or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of spectral submanifold theory to systems with geometric nonlinearity and velocity-dependent forces, plus the accuracy of commercial finite-element software for the full-order reference solutions.

axioms (1)
  • domain assumption Spectral submanifold reduction theory for nonlinear dynamical systems
    The method is built directly on SSM reduction anchored at a static equilibrium.

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Reference graph

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