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Joint nonlinearity in a stiffened aluminium wingbox panel and what it requires of a reduced basis

Nikolaos D. Tantaroudas, Evangelos Papatheou

arXiv:2609.10088v1cs.CEcs.NAcs.SYeess.SYmath-phmath.MPmath.NA

TL;DR

Fastened joints can be nearly invisible in low-level modal tests yet matter at operating amplitudes, creating a reduced-basis challenge for jointed structures. The study builds and calibrates a fastener-resolved shell model, evaluates nonlinear joint behavior, and applies projection-based nonlinear reduction. Softening and slip substantially shift resonances, while the reduction requires joint-to-joint receptance that global eigenvectors poorly capture.

  • Problem

    Joint microslip can be nearly invisible in low-level modal tests yet matter at operating amplitudes, and localized joint nonlinearity challenges global-mode reduced bases.

    Source: §1 · p. 1, §6.3 · p. 14

  • Method

    The study builds and calibrates a fastener-resolved shell model, compares nonlinear solution paths, and applies projection-based reduction while keeping residual flexibility outside the basis.

    Source: §3.2 · p. 5, §6.1 · p. 12, §6.4 · p. 14

  • Results

    Softening or slip shifts the first three resonances by −2.1, −4.9, and −9.2%, while hardening is almost invisible; independent solution paths agree to 5.5 × 10−4.

    Source: §7 · p. 18

  • Takeaways & Limitations

    Jointed-structure reduced bases must represent or condense joint-to-joint receptance, rather than relying on linear-response accuracy alone.

    Source: §7 · p. 19

    At the three resonances, 126 eigenvectors reproduce only 2.1, 2.2, and 2.4% of the joint-receptance diagonal, leaving local fastener-region flexibility unresolved.

    Source: §6.3 · p. 14

Abstract

from arXiv · show

A shell finite-element model of a stiffened aluminium wingbox panel, a laboratory structure shared by several experimental studies, is built with its plate-to-stiffener joints at the $78$ physical fastener positions and calibrated against measured modal data, a six-parameter sensitivity update of the substructure moduli within $\pm20\%$ reproducing the first five measured modes to $1.1$--$4.3\%$ with modal-assurance values of $0.88$--$1.00$. Making the fasteners nonlinear, at equal mesh, mass and damping, shows that joint nonlinearity enters the response strongly asymmetrically. Hardening is almost invisible, at most $+1\%$ in frequency, whereas softening or slipping the joints moves the first three resonances by $-2.1$, $-4.9$ and $-9.2\%$, and friction removes up to $80\%$ of the resonant peak before it recovers. Newmark integration of all $18{,}804$ degrees of freedom and a harmonic balance condensed exactly onto the joints and continued in arclength agree to $5.5\times10^{-4}$. A projection-based nonlinear model order reduction then locates the criterion that a jointed structure imposes on a reduced basis. The binding quantity is not the linear response, which eleven vectors reproduce to better than $0.01\%$, but the receptance of the structure between the joints, of which $126$ global eigenvectors carry about $2\%$, and without which the reduced model overpredicts the hardening shift of the fundamental by a factor of thirty and returns a value beyond the rigid-joint limit of the panel.

1 Introduction

Fastened joints can be nearly invisible in low-level modal tests yet produce amplitude-dependent behavior that complicates reduced-order modelling. This study uses a shared stiffened-panel case to examine how localized joint nonlinearity challenges global-mode bases and how to address it.

1 Introduction

  • A fastened interface transmits shear by friction until the interface force exceeds friction capacity, after which the parts slip and response becomes amplitude-dependent.The amplitude dependence may be weak in low-level modal tests but matter at operating levels.Source: §1 · p. 1
  • The same Sheffield stiffened aluminium panel underlies studies of novelty detection, transfer learning, and reduced-basis adaptation.Its shared geometry, instrumentation, and acquisition chain make it a common reference case.Source: §1 · p. 2
  • Built-up structures use nonlinear joint models to represent microslip and damping, while sensitivity updating fixes linear parameters without selecting a nonlinear joint law.The paper contrasts these approaches with reductions developed for distributed, smooth geometric nonlinearities.Source: §1 · p. 2
  • Localized joint nonlinearity interacts with global bases in ways a linear convergence study cannot reveal, motivating a basis criterion specific to jointed structures.The paper builds a physical-fastener shell model, calibrates it, bounds nonlinear resonance effects, and identifies a basis-sizing quantity with a reformulation that avoids enlarging the basis.Source: §1 · p. 2

2 The panel, the measurements and the shell model

The panel model combines a shell representation of its eleven substructures with discrete fasteners at physical joint positions, and its element and assembly choices are checked against folded-structure benchmarks. The section also documents the measured modes used later for calibration.

2.1 Structure and data

  • The free-free modal test uses a shaker and 14 accelerometers; five tracked modes at 24.70, 69.99, 104.31, 117.56, and 124.00 Hz are retained for updating.The fundamental is a torsion mode, while the 104.31 Hz mode is the first bending mode about the short axis.Source: §2.1 · p. 2

2.2 Weak form and kinematics

  • The free-vibration model treats eleven Reissner–Mindlin shell substructures with individual thickness, density, and effective Young’s modulus multipliers.The kinematics use translations and physical right-hand shell-normal rotations; the rotation triplet transforms between local and global frames as a vector.Source: §2.2 · pp. 2–3

2.3 The mixed-interpolation shell element

  • The four-node shell element uses mid-edge tying points to reconstruct transverse shear and avoid shear locking while retaining the Reissner–Mindlin response for thick plates.Its membrane and bending parts use full 2 × 2 Gaussian integration, with drilling stabilization on the normal rotation.Source: §2.3 · p. 3

2.4 Fastener elements and assembly

  • At the 12.5 mm reference mesh, the model has 18,804 degrees of freedom and 78 discrete fasteners connecting plate and stiffener contact-point images.The panel has two rib rows of 16 rivets each and two stringer-leg rows of 23 bolts each, at 30 mm spacing.Source: §2.4 · p. 4, Figure 1 · p. 4

2.5 Verification

  • The folded-assembly checks expose rotation-transformation errors that flat-plate tests cannot detect; the equal-leg angle mode is 184.7 Hz versus 185.2 Hz from beam theory.The free-free box-beam bending frequency is 2.2% below Euler–Bernoulli theory, with the residual attributed to shear deformation.Source: §2.5 · pp. 4–5, Table 1 · p. 5

3 Model updating

A sensitivity-based update calibrates the panel’s substructure moduli against measured modes, then checks the fit’s mesh and fastener-stiffness sensitivity. The calibration improves frequency agreement, while near-coincident modes complicate modal assurance at finer meshes.

3.1 Formulation

  • The sensitivity update uses six substructure Young’s-modulus multipliers, frequency and mode-shape residuals, and a box-constrained Levenberg–Marquardt iteration.Best-MAC pairing is used because finite-element and measured mode order differs for measured modes 4 and 5; the order convention serves as a check.Source: §3.1 · p. 5

3.2 The calibrated model

  • Calibration yields frequency errors of 1.1–4.3%, an effective plate modulus of 81.5 GPa, and all six parameters interior to the ±20% box; models use rigid fasteners at h = 12.5 mm.Source: §3.2 · p. 5, Table 2 · p. 6
  • The residual remains systematic: modes 1 and 5 are low, modes 2–4 slightly high, and no physically bounded modulus combination removes it.Finite-element modes 4 and 5 are within 1 Hz, compared with 6.4 Hz for the measured pair, contributing to their lowest modal-assurance values.Source: §3.2 · p. 6

3.3 Mesh refinement and the fastener stiffness

  • Less than 1% separates the two finest meshes, while measured mode 4’s modal-assurance value dips as near-coincident finite-element modes 4 and 5 hybridise.Across h = 12.5 to 3.5 mm, the frequency errors change by 1 to 2.6 percentage points at the calibration mesh; recalibration at h = 6.25 mm returns moduli within 7% of the reference.Source: §3.3 · p. 6, Figure 3 · p. 7
  • 15% higher torsion-fundamental frequency with rotational coupling than with translation-only fasteners; plausible fastener stiffnesses are already in the saturated regime.The stiffness sweep supports using the rigid-link model as a reference and explains why calibration with compliant fasteners leaves stiffnesses unchanged.Source: §3.3 · p. 7

4 The nonlinear joint model and how it is solved

The nonlinear model changes only the fastener constitutive law and compares four joint laws using full-model time integration and exact harmonic-balance condensation onto joint deformations. Independent solution paths agree closely, supporting the computed nonlinear responses and branch continuation.

4.1 Equation of motion and joint laws

  • The model keeps mesh, mass, and damping fixed while replacing the fastener law; the equation adds the nonlinear joint-force correction to the calibrated linear system.A linear law reproduces the linear model identically, providing a verification test.Source: §4.1 · pp. 7–8
  • Four laws share initial stiffness and knee deformation: cubic hardening, smooth saturation, bilinear softening, and Jenkins friction with retained fraction ϱ = 0.05.The retained fraction represents bearing through the shank after slip; setting it to zero introduces twelve zero-energy mechanisms beyond rigid-body modes.Source: §4.1 · p. 8, Figure 5 · p. 8

4.2 Two independent solution paths

  • The solvers use Newmark integration on all 18,804 degrees of freedom and harmonic balance condensed exactly onto 312 joint deformations.The harmonic-balance receptance is that of the full model between joint degrees of freedom; its modal approximation needs a remainder correction because 150 modes alone are 94% in error.Source: §4.2 · p. 9
  • Softening bends resonance branches into multivalued responses, so pseudo-arclength continuation passes folds and recovers the unstable middle branch.At a strongly nonlinear operating point, the time-integrated and harmonic-balance waveforms and harmonic content agree; distortion is at most 5.1% of the fundamental at resonance peaks.Source: §4.2 · p. 9, §4.3 · p. 9

4.3 Verification

  • 5.5 × 10^-4 relative deviation in fundamental amplitude and 7.0 × 10^-4 in energy dissipated per cycle between the two full-model solution paths.The linear joint law also reproduces the linear model, verifying the nonlinear pipeline against the Section 3 model.Source: §4.3 · p. 9
  • 1:3 superharmonic resonances create narrow spikes and the study’s largest third-harmonic content, making harmonic-set checks important.At those points, including the seventh harmonic changes the fundamental more than anywhere else in the study.Source: §4.3 · p. 9

5 What joint nonlinearity does to the panel

Joint nonlinearity affects the panel asymmetrically: hardening barely shifts resonances, while softening and slipping produce larger frequency shifts and friction-dependent peak loss. The response also changes with excitation level as more joints pass their slip knee.

5 What joint nonlinearity does to the panel

  • Cubic hardening shifts the peak by less than +1%, even when the most-loaded element reaches six times its knee.The joints are already nearly rigid relative to the sheet, so strain energy remains in the plate.Source: §5 · p. 9
  • The saturating and bilinear laws shift the first three peaks by −2.1, −4.9 and −9.2%, while friction removes up to 80% of the resonant peak before recovery.The frequency shifts approach the fully saturated bound to within 0.3%.Source: §5 · p. 10, Table 4 · p. 11, Figure 9 · p. 12
  • These are numerical experiments on a calibrated model, not measurements of the panel; the knee and retained fraction were chosen because low-level modal data do not identify amplitude dependence.A stepped-sine test at several levels is proposed to assess whether the panel's joints behave this way.Source: §5 · p. 10
  • As excitation rises, the fraction of joint elements past their knee grows from a few loaded rows to almost the whole population, and peaks fall before recovering at 30 N.At high level, joints slip through most of the cycle and the structure spends most of its period in the fully saturated configuration.Source: §5 · p. 11, Figure 8 · p. 11

6 Nonlinear model order reduction

The paper applies projection-based nonlinear model order reduction to a jointed panel, where localized joint nonlinearity differs from the smooth distributed geometric nonlinearity for which the approach was developed. It writes the full-order system in descriptor form and projects its Taylor-expanded restoring force onto modal coordinates.

6 Nonlinear model order reduction

  • The reduction projects the response onto a small basis and represents the nonlinear restoring force with a Taylor expansion about equilibrium.The approach was developed for flexible-aircraft systems with distributed geometric nonlinearity, unlike the panel's localized joint nonlinearity.Source: §6.1 · p. 12
  • The singular consistent mass matrix, which has no inertia on drilling degrees of freedom, makes the first-order full-order model a descriptor system rather than an ordinary differential equation.The system is written in first-order form without forming an inverse of the mass matrix.Source: §6.1 · p. 12
  • The panel is unloaded, so the residual expansion is taken about zero equilibrium, with the nonlinear correction represented by second- and third-order operators.The projected state is expressed using the retained eigenvectors and their left eigenvectors.Source: §6.1 · pp. 12–13

6.2 Matrix-free coefficients and the step size

Higher-order operators are evaluated matrix-free through finite differences of the full-order residual along basis directions. Their step size must balance truncation and round-off, with the third-order stencil especially sensitive to round-off.

6.2 Matrix-free coefficients and the step size

  • Only the action of the higher-order operators on basis vectors is needed, obtained by matrix-free central differences of the full-order residual.Mixed arguments follow from polarization identities, and complex arguments are split into real and imaginary parts.Source: §6.2 · p. 13
  • For the cubic law, the third-Jacobian stencil is exact at every step size, leaving only round-off; for the saturating law, truncation error falls as ϵ^4.The third-order stencil is sensitive because it divides by ϵ^3.Source: §6.2 · p. 13, Figure 10 · p. 13
  • At ϵ3 = 10^-3, the two cross and the best attainable error is 1.3 × 10^-9; inherited ϵ3 = 10^-1 from the aeroelastic application has 9.6% error on this normalisation.Source: §6.2 · p. 13

6.3 What sets the size of the basis

The reduced basis must capture joint-to-joint receptance as well as linear response and joint deformation. Global eigenvectors reproduce little of this local flexibility, causing an eigenvector-only model to give an unphysical hardening prediction; residual attachment modes recover it.

6.3 What sets the size of the basis

  • Joint deformation converges about an order of magnitude more slowly than sensor receptance, reaching 1.9% at 126 vectors; the joint force depends on this deformation.Sensor receptance errors are 1.2% at 23.8 Hz and 3.8% at 71.4 Hz with eleven vectors.Source: §6.3 · pp. 13–14
  • At the three resonances, 126 eigenvectors reproduce only 2.1, 2.2 and 2.4% of diagonal joint-to-joint receptance, with force-pattern errors of 91, 28 and 42%.Adding more global modes does not supply the missing local static flexibility of the fastener region.Source: §6.3 · p. 14, Figure 11 · p. 14
  • With joint relaxation missing, the reduced model overpredicts the fundamental hardening shift by a factor of thirty and exceeds the panel's rigid-joint limit, despite passing the linear check.The linear response does not test the receptance through which a joint's nonlinear force relaxes its deformation.Source: §6.3 · p. 14
  • Residual attachment modes reduce the receptance errors from 91, 28 and 42% to 1.8, 0.02 and 0.10% with eleven modes and thirty-five attachment vectors.The repair makes the basis three to four times larger, and above 10 N its per-frequency cost returns to that of the condensed full-order solution.Source: §6.3 · p. 14

6.4 Keeping the residual flexibility out of the basis

The reduction keeps residual joint flexibility outside the modal basis and constrains joint deformation instead. Because truncation errors are amplified as the joints slip, accurately representing the residual operator is essential in that regime.

6.4 Keeping the residual flexibility out of the basis

  • The response is split into retained modal motion and a static remainder, so residual flexibility constrains joint deformation without enlarging the basis.The joint equation ties d to modal amplitudes through Rres and the nonlinear law.Source: §6.4 · p. 14
  • Unlike sensor receptance and joint deformation, joint-to-joint receptance does not converge at any practical basis size, despite governing the nonlinear model.Figure 13 compares errors from k-nearest-neighbour condensation against dense Newton on the complete residual operator.Source: Figure 12 · p. 15, Figure 13 · p. 16
  • The residual operator is nearly local: its 2×2 diagonal blocks carry 99.7% of its Frobenius norm, enabling 156 independent two-variable solves.A vectorised Newton converges in two or three iterations, and the dynamic state remains the m modal amplitudes.Source: §6.4 · p. 15
  • With eleven vectors, joint-receptance errors at the three resonances fall from 99.96, 99.86 and 98.79% to 0.085, 0.146 and 0.289%.The condensed approach reaches these errors with a basis a quarter the size of the one using thirty-five attachment vectors.Source: §6.4 · p. 15

6.5 How accurately the discarded operator has to be represented

  • Discarding 8.3% of the residual-operator norm costs 0.06% in joint deformation below the slip knee but 39% when the joints slip together.The error follows conditioning: at the solution, it is amplified by (1 −ρ)^−1 as ρ grows from 0.02 to 0.94.Source: §6.5 · p. 15
  • Block-preconditioned iteration fails when ρ reaches or exceeds one, whereas an exact banded Newton solve converges through ρ = 14.2.The retained pattern uses k nearest joint elements and reverse Cuthill–McKee ordering; the exact Newton converges in at most ten iterations.Source: §6.5 · pp. 15–16

6.6 Results and cost

Restoring residual coupling brings the reduced nonlinear model close to the full-order response, but accuracy and cost depend on the retained neighbour pattern. Even with coupling restored, the modal basis itself limits some results.

6.6 Results and cost

  • The accurate enriched-basis model costs 399 ms per frequency, compared with 1000 ms for a direct full-model solve, while the third-order model costs 46 ms but is unusable.The linear reduced model costs 1.6 ms; Table 6 separates per-frequency response time from setup costs paid once.Source: §6.6 · p. 16, Table 6 · p. 17
  • The eleven-vector block-diagonal model costs 118 ms per frequency; eight neighbours cost 595 ms, while sixteen cost 1109 ms.Four neighbours cost 288 ms and recover about two thirds of the accuracy at parity of cost; the neighbour count is an accuracy dial, not a speed-up.Source: §6.6 · p. 16
  • On the fundamental at 30 N, increasing k from 0 to 32 moves the predicted shift from −0.90 to −2.10%, bracketing the full-order −2.06%.For mode 2, the sequence passes the full-order −4.87% as k increases; the remaining discrepancy is attributed to the eleven-vector joint-receptance representation.Source: §6.6 · pp. 16–17
  • The restored-coupling model reproduces full-order shifts to 0.01 percentage points for hardening and 0.13 percentage points for the fully slipped fundamental under softening.Table 5 compares peak-frequency shifts on eleven vectors against the nonlinear full-order model at 30 N.Source: §6.6 · p. 16, Table 5 · p. 17, Figure 14 · p. 17

6.7 Verification of the reduction

The reduction is checked against independent references at the basis, Taylor-coefficient, solver and condensation levels. Its Taylor expansion is valid across the studied range only for the cubic law; the local condensation covers more laws, but not Jenkins friction.

6.7 Verification of the reduction

  • The matrix-free third Jacobian matches an analytic tensor, the cubic Taylor model agrees with exact-law Galerkin to 2.7 × 10−11, and banded assembly matches the dense Jacobian to machine precision.The cubic series is exact; the banded condensation exactly solves the truncated operator.Source: §6.7 · pp. 17–18, Table 7 · p. 18

6.8 Where the expansion is valid

  • Only the cubic law lies within the Taylor model’s assumptions over the full study range; the saturating, bilinear and friction laws fall outside them for distinct reasons.The saturating-law series is unbounded below after third-order truncation past the knee, the bilinear law is nondifferentiable there, and friction carries an internal state.Source: §6.8 · p. 18
  • The local condensation evaluates the law itself and reaches saturating and bilinear cases, but Jenkins friction remains outside both formulations because it carries an internal state.A friction-capable reduced model would need a stateful nonlinearity or an operating-point-adapted basis, as in.Source: §6.8 · p. 18

7 Conclusions

The calibrated panel model shows strongly asymmetric effects from nonlinear joints, and independent solution routes agree closely. The reduction identifies joint-to-joint receptance—not linear-response convergence—as the binding basis requirement and motivates condensing localized flexibility.

7 Conclusions

  • The calibrated shell model places joints at 78 physical fastener positions and reproduces the first five measured modes within 1.1 to 4.3% in frequency, with modal-assurance values of 0.88 or better.A six-parameter sensitivity update was constrained to ±20%.Source: §7 · p. 18
  • Softening and slip move the first three resonances by −2.1, −4.9 and −9.2%, while hardening is almost invisible and friction removes up to 80% of the resonant peak before recovery.The time-domain full-order and joint-condensed frequency-domain solutions agree to 5.5 × 10−4.Source: §7 · pp. 18–19
  • A basis of 126 global eigenvectors carries about 2% of joint-to-joint receptance, the quantity governing the nonlinear model, despite eleven vectors reproducing linear response to better than 0.01%.Condensing that operator avoids the accuracy–cost collision of basis enrichment.Source: §7 · p. 19
  • For localized nonlinearities, the unrepresentable operator is localized too, so condensing it is cheaper than spanning it when condensation accuracy is judged by joint slip rather than discarded-operator size.Source: §7 · p. 19
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