Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve

Mingwu Li; Shobhit Jain; George Haller

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Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve

Mingwu Li, Shobhit Jain, George Haller

arXiv:2106.05162v2Jun 9, 2021math.DS

TL;DR

High-dimensional finite-element mechanical systems make direct integration, collocation, and harmonic balance computationally impractical for extracting forced response curves. The paper constructs reduced-order models on time-periodic spectral submanifolds associated with internally resonant modes and obtains periodic responses as equilibria of the reduced dynamics. SSM reduction obtained the beam’s FRC in approximately 1 hour for a discretization with 29,998 DOF.

Problem
High-dimensional finite-element mechanical systems make direct integration, collocation, and harmonic balance computationally impractical for extracting forced response curves.
Method
The paper constructs reduced-order models on time-periodic spectral submanifolds associated with internally resonant modes and obtains periodic responses as equilibria of the reduced dynamics.
Results
SSM reduction obtained the beam’s FRC in approximately 1 hour for a discretization with 29,998 DOF.
Takeaways & Limitations
The reduced model’s dimension depends on the resonant modes rather than the full system dimension, enabling efficient FRC computation for large finite-element structures.
Takeaways & Limitations
The approach requires solving a 2n-sized linear system at each sampled excitation frequency, and polar-coordinate continuation can terminate near a singular point.

Abstract

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We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the reduced-order models are discussed in this paper and its companion, Part II, respectively. The dimension of a reduced-order model is determined by the number of modes involved in the internal resonance, independently of the dimension of the full system. The periodic responses of the full system are obtained as equilibria of the reduced-order model on spectral submanifolds. The forced response curve of periodic orbits then becomes a manifold of equilibria, which can be easily extracted using parameter continuation. To demonstrate the effectiveness and efficiency of the reduction, we compute the forced response curves of several high-dimensional nonlinear mechanical systems, including the finite-element models of a von K\'arm\'an beam and a plate.

1 Introduction

Internal resonances make nonlinear mechanical responses complex and make standard periodic-orbit computations impractical for high-dimensional finite-element systems. The paper develops SSM-based reduced-order models whose dimension depends on resonant modes and whose equilibria yield the full system’s periodic responses and FRCs.

Internal resonances can produce energy transfer between modes, saturation, localization, and frequency stabilization in mechanical systems.
For high-dimensional finite-element systems, direct integration, collocation, and harmonic balance become impractical because of excessive time, memory, or nonlinear solve costs.
Earlier SSM computations were limited by eigenbasis requirements and two-dimensional manifolds, while newer physical-coordinate methods motivate extending SSM reduction to internal resonances.
The paper derives SSM-based reduced-order models for harmonically excited systems with internal resonances and extracts FRCs up to arbitrary approximation orders.
A resonant SSM produces a 2m-dimensional reduced model independently of the full system dimension, with periodic full-system orbits corresponding to equilibria of the slow-phase dynamics.
The FRC becomes a manifold of equilibria in the reduced vector field, enabling efficient parameter continuation and stability assessment of periodic orbits.

2 System setup

The paper models a weakly forced, damped nonlinear mechanical system in second-order form and transforms it to a first-order system for SSM analysis. The computation uses selected master modes rather than requiring all eigenmodes.

The system is a weakly forced nonlinear mechanical model with mass, damping, stiffness, and nonlinear terms, driven by harmonic excitation.
The second-order equations are transformed into a first-order system, whose coefficient matrices are symmetric when the original mechanical matrices are symmetric.
Linearization yields 2n generalized eigenvalues with corresponding right and left eigenvectors, ordered by their real parts.
The analysis assumes all eigenvalues have strictly negative real parts, making the linearized equilibrium asymptotically stable.
SSM computation in physical coordinates requires only eigenvectors associated with the selected master modal subspace, not the complete spectrum.

3 Non-autonomous SSM for systems with internal resonance

For internally resonant modes, the paper constructs a time-periodic SSM associated with a 2m-dimensional master spectral subspace. The resulting invariant manifold embeds reduced coordinates into the forced full-system phase space.

The master spectral subspace contains internally resonant modes and has dimension 2m, with underdamping assumed for its spectrum.
The formulation allows near inner resonances among eigenvalues associated with the selected resonant modes.
A periodic SSM is defined as a 2m-dimensional invariant manifold with forcing period 2π/Ω that smoothly perturbs from the master spectral subspace.
Under a non-resonance condition and sufficiently small forcing, the theory guarantees existence of a time-periodic SSM.
The SSM is represented as an embedding of reduced coordinates and forcing phase into the full phase space, with reduced dynamics defined on the manifold.
In high-dimensional systems, the non-resonance condition is checked using a computed subset of low-frequency modes because all eigenvalues may be unavailable or expensive to calculate.

4 Computation of SSM

SSM computation separates the autonomous manifold from forcing-dependent corrections and solves their parameterization equations as truncated Taylor expansions. The resulting reduced dynamics provides the basis for continuation of forced responses.

4.2 Non-autonomous part: Forcing-dependent corrections are obtained from the next invariance equation and approximated through Taylor expansions whose coefficients are periodic functions of the forcing phase.
4.1 Autonomous part: The autonomous SSM and its reduced dynamics are first computed from the unforced invariance equation using Taylor expansions in normal coordinates.
4.1 Autonomous part: Balancing monomials at each expansion order produces recursively solvable linear equations for the SSM and reduced-dynamics coefficients.
4.1 Autonomous part: At leading order, the eigenvectors and eigenvalues of the master spectral subspace solve the autonomous invariance equations, enabling higher-order approximation.
4.2 Non-autonomous part: The reduced dynamics on the SSM is arranged in complex-conjugate blocks associated with each master-mode pair, with forcing contributions determined from a linear system.
4.1 Autonomous part: The SSM expansion order is truncated in practice because formulating and solving the coefficient equations becomes costly at large orders.

5 Reduced dynamics on SSM

The paper derives reduced dynamics on periodic spectral submanifolds for internally resonant, harmonically forced systems, using polar or Cartesian coordinates near external resonance. Periodic responses correspond to reduced fixed points, whose continuation forms the forced response curve and preserves stability information.

5 Reduced dynamics on SSM: The reduced model is 2m-dimensional, where m is the number of master mode pairs, and this dimension is independent of the full system size.The formulation selects master modes according to internal and external resonance conditions.
5.1 Main theorems: The leading-order reduced dynamics is organized in a rotating frame as a slow-fast system in polar coordinates under the stated inner and external resonance conditions.The polar transformation uses amplitude and phase variables for each master mode, while Cartesian coordinates provide an alternative representation.
5.1 Main theorems: Periodic responses of the full system correspond to hyperbolic fixed points of the reduced dynamics on the SSM, with matching stability types.This correspondence holds in both polar and Cartesian representations.
5.1 Main theorems: For fixed excitation amplitude, the forced response curve is obtained as a one-dimensional manifold of reduced fixed points parameterized by excitation frequency.The full fixed-point solution manifold is two-dimensional in excitation frequency and amplitude; fixing amplitude yields the FRC.
5.1 Main theorems: The polar reduced dynamics becomes singular when a modal amplitude vanishes, so the Cartesian formulation is needed to determine fixed-point stability correctly.This singularity arises, for example, on branches where one master-mode amplitude vanishes in a 1:1 external resonance setting.
5.2 FRC extraction: Level-set detection can recover isolated FRC branches for a two-dimensional SSM, whereas higher-dimensional internally resonant cases require numerical solution and parameter continuation.The paper uses continuation of fixed points, implemented with the ep toolbox in COCO, for the general case.

6 Examples

The examples use resonant SSMs to compute forced response curves for nonlinear oscillator systems, capturing modal interactions and bifurcations while reducing continuation to a lower-dimensional problem. Cartesian coordinates avoid the polar-coordinate singularity encountered near vanishing modal amplitude.

6.1 A chain of oscillators: The oscillator-chain FRC contains stable and unstable periodic orbits plus saddle-node and Hopf bifurcations caused by modal interactions.
6.1 A chain of oscillators: The SSM formulation reduces periodic-orbit continuation to a lower-dimensional problem, producing a major speed-up over full-system continuation.
6.1 A chain of oscillators: Cartesian SSM coordinates continue successfully across the tested frequency range, whereas polar continuation terminates near Ω≈1.0054 when ρ3≈2.08 × 10−8.
6.1 A chain of oscillators: When m=n, the SSM analysis is equivalent to normal-form analysis without assuming that the nonlinearity is small; later examples use m≪n for reduction.

6.2 A prismatic beam with axial stretching

A 1:3 resonant SSM reduces the forced beam model to four dimensions and is compared with multiple-scales and collocation calculations across resonances of the first and second modes. The SSM results capture forced modal responses that multiple scales misses, including responses in indirectly excited modes.

6.2 A prismatic beam with axial stretching: The first two beam modes satisfy a near 1:3 internal resonance, so a 20-dimensional phase-space model is reduced using a four-dimensional master spectral subspace.
6.2 A prismatic beam with axial stretching: For forcing near ω1=3.8533, the second-mode amplitude remains nonzero but small relative to the first, and SSM results agree excellently with comparison calculations.
6.2 A prismatic beam with axial stretching: With all 10 modes forced, SSM reduction predicts the nontrivial u3 response and changes in ||u2||∞, whereas MMS retains its previous response and predicts u3=0.
6.2 A prismatic beam with axial stretching: The SSM reduced dynamics are four dimensional and computed by an automated recursive numerical procedure, making them less costly than applying MMS to the full system.
6.2 A prismatic beam with axial stretching: For forcing near ω2=12.4927 on the non-vanishing-ρ1 branch, the first mode can dominate even though forcing is applied only to the second mode, with SSM and MMS matching well.
6.2 A prismatic beam with axial stretching: On the branch with ρ1≡0, SSM predicts non-vanishing u1 while MMS predicts u1=0, despite the two methods agreeing on the principal u2 response.

6.3 A viscoelastic beam with gyroscopic force

The viscoelastic axially moving beam example extends SSM reduction to systems with gyroscopic forces and cubic nonlinear damping. A four-dimensional model for the near 1:3 resonance produces an FRC that converges at fifth order and agrees with full-system collocation.

6.3 A viscoelastic beam with gyroscopic force: The model includes gyroscopic forces and cubic nonlinear damping arising from the viscoelastic constitutive law.
6.3 A viscoelastic beam with gyroscopic force: The first two natural frequencies satisfy ω2≈9.5862≈3ω1 with ω1≈3.1954, motivating a four-dimensional SSM reduction based on the first two mode pairs.
6.3 A viscoelastic beam with gyroscopic force: For base-excitation amplitude ǫ=1.5 × 10−4, the forced response curve converges well at expansion order O(5).
6.3 A viscoelastic beam with gyroscopic force: The SSM-reduced forced response agrees with the collocation method applied to the full system.

6.4 A von K´arm´an beam with support spring

The support spring is tuned to create a 1:3 internal resonance, and a four-dimensional SSM reduction efficiently reproduces the beam’s forced response across discretization sizes. Compared with full-system harmonic balance and collocation, the reduction is much faster while capturing modal energy transfer.

Model and resonance: The support stiffness is set to ks = 37 kg/s2, producing ω2 = 3ω1 with ω1 = 33.20 rad/s and ω2 = 99.59 rad/s.The resonance intersection is reported as robust to the discretization size.
Reduction setup: The SSM reduction uses the first two complex-conjugate mode pairs as its master subspace, yielding a reduced model for the 1:3 resonance.The beam is discretized by finite elements with axial and transverse displacements and rotation included.
Computational efficiency: 14 seconds versus 12.5 hours for harmonic balance and 58.5 hours for collocation at 40 elements, demonstrating the SSM reduction’s computational advantage.The 40-element model has 118 DOF and a 236-dimensional phase space.
Validation: SSM, harmonic balance, and collocation FRCs match well for 8, 20, and 40 elements, while SSM results also agree with full-system Newmark-beta integration at larger sizes.Results at 200 elements already converge with respect to further mesh refinement.
Modal interaction: Near the first-mode resonance, the second-mode amplitude peaks while the first-mode amplitude drops, indicating energy transfer caused by the 1:3 internal resonance.The midspan and quarter-span transverse responses qualitatively track the first- and second-mode amplitudes, respectively.

6.5 A Timoshenko beam carrying a lumped mass

A Timoshenko cantilever beam with a lumped mass exhibits near 1:3 internal resonance and very large forced responses under harmonic end moments. SSM reduction converges at higher order for larger deformations, matches collocation results, and substantially reduces computation time.

Reduction setup: The four-dimensional SSM reduction retains the first two complex-conjugate mode pairs to represent the near 1:3 internal resonance in the 42-dimensional phase space.The beam’s first two natural frequencies are ω1 = 2.2562 rad/s and ω2 = 7.2301 rad/s ≈ 3ω1.
Response amplitude and convergence: The peak response reaches 415 mm for M = 0.84 N · m, exceeding ten beam thicknesses, while 283 mm for M = 0.24 N · m converges at lower SSM order.The larger response requires O(9) expansion, whereas the smaller response converges at O(5).
Validation and modal interaction: SSM FRCs match closely with collocation results for the full system, including the large-amplitude response at M = 0.84 N · m.A small bump near Ω ≈ 2.44 is explained by a second peak in the second-mode response.
Computational efficiency: 29 seconds versus 3.8 hours for collocation at M = 0.84 N · m demonstrates a substantial SSM speed-up over full-system computation.The harmonic-balance continuation also reaches its one-day time threshold in this case.

6.6 A simply supported von K´arm´an plate

The simply supported von Kármán plate has a 1:1 internal resonance between its second and third bending modes, modeled using a high-dimensional finite-element discretization. SSM reduction captures its nonlinear FRCs accurately and remains practical where full-system continuation is computationally prohibitive.

Resonance and reduction: The second and third bending modes form a 1:1 internal resonance, motivating a master spectral subspace containing their two complex-conjugate mode pairs.The finite-element model uses 200 triangular elements and 606 DOF.
FRC convergence: The FRC converges at O(5) for a 50-load case with 2.4 mm peak amplitude, whereas doubling the load produces 5.1 mm and requires O(11).Higher-order SSM expansions are needed for the larger response amplitude.
Validation and nonlinearity: SSM results match closely with shooting-based full-system results, while linear analysis deviates substantially when response amplitudes and geometrical nonlinearities become large.Linear and SSM results agree for small amplitudes or frequencies far from the second natural frequency.
Modal interaction: At resonance, a notch in node A’s FRC coincides with a maximum at node B, reflecting energy transfer between the resonant bending modes.The physical-node responses correspond closely to the reduced modal responses because the nodes lie near modal peaks and nodal lines.
Computational efficiency: A 606-DOF plate’s FRC computation takes about one minute with SSM reduction versus about 6 days with shooting continuation.Harmonic balance and collocation achieve only one and four continuation steps, respectively, in ten days for the same system.
Scalability: SSM reduction remains applicable as the plate model grows to 240,006 DOF, with FRC computation times reported for increasingly fine finite-element discretizations.The study considers models from 2,406 to 240,006 DOF.

6.7 A shallow shell structure

The shallow-shell example uses a 1:2 internal resonance induced by the shell curvature and computes its FRC with a 1320-DOF finite-element model. The SSM FRC converges efficiently and agrees well overall with shooting results despite the latter’s high computational cost.

Resonance setup: Choosing curvature w = 0.041 m induces a near 1:2 internal resonance between the first two modes.The shell is excited by a concentrated 10 cosΩt N load near the first-mode frequency.
Computational efficiency: The O(5) SSM FRC converges in about two minutes for the 1320-DOF shallow-shell model.The model contains 400 triangular shell elements.
Validation and cost: SSM and shooting FRCs match well overall, but the shooting computation cannot cover the full FRC within 7.5 days and takes nearly five days even over a restricted frequency range.Increasing integration steps from 100 to 200 substantially reduces the observed discrepancies.

7 Conclusion

The paper reduces forced nonlinear mechanical systems with internal resonance to time-periodic spectral submanifolds, where periodic responses become fixed points of slow-phase reduced dynamics. Parameter continuation then extracts full-system forced response curves, with examples showing substantial computational savings but limitations in locating isolas and bounding reliable forcing amplitudes.

Reduced formulation: Fixed points of the slow-phase reduced dynamics correspond to periodic orbits on the time-varying SSM, whose solution branches map to the full-system FRC.Normal-form coordinates cancel time-dependent harmonic terms, and parameter continuation constructs the FRC from fixed-point branches.
Validation scope: Seven examples demonstrate the accuracy and efficiency of SSM reduction across resonant oscillator, beam, gyroscopic, nonlinear-damping, and finite-element structural systems.The examples include 1:1:1 and 1:3 internal resonances and finite-element models of beams, plates, and shells.
Computational efficiency: A four-dimensional SSM obtains the 1,320-DOF shallow-shell FRC in two minutes, while shooting-based continuation fails to cover the full FRC within a week.The shell FRC is one of the finite-element structural examples used to demonstrate computational efficiency.
Limitations: Parameter continuation depends on the initial solution and can miss isolated branches, while the implementation lacks an estimate of the forcing-amplitude range where reduction results remain reliable.The authors identify singularity-theory or multidimensional-continuation approaches for isolas and note that estimating convergence bounds under internal resonance is nontrivial.

8 Appendix

The appendix develops the reduced dynamics and proves that fixed points of the slow-phase system generate periodic solutions on the SSM. The proof uses resonance conditions, coordinate transformations, scaling, implicit-function arguments, and averaging to establish persistence under higher-order perturbations.

Reduced dynamics: The appendix restricts the derivation to one harmonic and adapts a leading-order approximation to formulate the reduced dynamics in normal-form coordinates.The reduced dynamics is expressed using the master spectral subspace associated with internally resonant modes.
Resonance conditions: Resonance and eigenvector orthogonality conditions resolve singular linear systems when an eigenvalue matches the excitation frequency.The forcing coefficient is obtained by projecting onto the left eigenvector, while near resonance motivates the same derivation to avoid ill-conditioning.
Periodic solutions: After transforming to polar radii and phase differences, fixed points of the slow-phase system correspond to T-periodic reduced solutions and inherit their stability.The period is T = 2π/(r_dΩ), with r_d defined as the largest common divisor of the relevant rational resonance ratios.
Persistence proof: The persistence proof establishes slow dynamics relative to the excitation phase and applies averaging to continue hyperbolic periodic orbits under higher-order perturbations.The slow-fast structure follows when the real parts of the resonant eigenvalues are close to the corresponding excitation frequencies.
Scaling argument: Scaling and the implicit function theorem show that the fixed-point amplitude satisfies x⋆ ∼ O(ε^(1−q)) for sufficiently small forcing parameters.Invertibility of the fixed-point Jacobian provides the smooth dependence needed for the scaling result.
Continuation computation: Continuation implementations produce beam FRCs with different point counts and runtimes, including 175 points in about six and a half hours for atlas-1d and 248 points in about 11 hours for atlas-kd.The atlas-kd run uses a much smaller observed maximum continuation step size of 0.1 versus about 30 for atlas-1d.