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Transducer Placement and the Limits of a Four-State Reduced Model in Post-Flutter Piezoelectric Energy Harvesting from a Pitch-Plunge-Flap Aerofoil

Nikolaos D. Tantaroudas, Ilias Karachalios, Andrew J. McCracken

arXiv:2609.12788v1cs.CEcs.SYeess.SYmath-phmath.MP

TL;DR

This paper studies how transducer placement and reduced-model construction affect post-flutter piezoelectric energy harvesting, where nonlinear time marches make design sweeps costly and a structural basis may miss the electrical state carrying power. It compares reduced predictions with a full-order system and finds that re-projecting the exact Jacobian restores structural response without enlarging the basis, while harvested voltage remains inaccurate.

  • Problem

    Harvester design sweeps require a nonlinear time march to a converged limit cycle at every parameter point, and a structural-mode basis may miss the electrical state carrying harvested power.

    Source: §1 · p. 2

  • Method

    The study compares a reduced-order model with the full system and tests operator updates and retained eigenmode bases for a fifteen-state aerofoil harvester.

    Source: §3 · p. 5, §4 · p. 6

  • Results

    Exact Jacobian re-projection reduces peak pitch-amplitude error from 46% to 3.4%, while voltage remains 20 to 23% low.

    Source: §7 · p. 14

  • Takeaways & Limitations

    Harvester ROM accuracy should be assessed on the harvested quantity itself, because structural response accuracy does not ensure voltage accuracy.

    Source: §7 · p. 14

    The attached, incompressible, two-dimensional aerodynamic model holds the limit cycle below five degrees and does not support claims about stall-dominated LCOs.

    Source: §6 · p. 14

Abstract

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Aeroelastic ?utter is normally a failure mode to be designed against, yet the limit-cycle oscillations (LCOs) that follow it convert flow energy into sustained structural motion that a piezoelectric transducer can turn into electrical power. A transducer is embedded in a three-degree-of-freedom pitch-plunge aerofoil with a finite-mass trailing-edge flap and unsteady strip-theory aerodynamics, giving a fifteen-state electro-aeroelastic system with a cubic hardening pitch spring, and the system is reduced by biorthonormal projection of the Taylor-expanded residual onto eigenvectors of the coupled Jacobian. The testbed is deliberately low order, so that every reduced prediction can be checked against the full-order system it replaces. The degree of freedom that carries the transducer is a first-order design variable, since it sets both the sign of the shift in the flutter boundary and the magnitude of the harvested power. The error of the reduced model is dominated not by the size of the retained basis but by how the reduced operator is made to depend on low speed. Expanding the retained eigenvalues leaves the reduced operator block-diagonal, with no coupling between the retained modes at any order of the expansion, whereas re-projecting the exact Jacobian onto the same frozen basis supplies that coupling and restores the structural response without enlarging the basis. It is found that the error that remains lies in the harvested voltage, which the equilibrium eigenvectors carry in a fixed proportion to the motion, and this is not removed by adding the electrical modes to the basis, whereas the flap detection, which the same basis also misses, is recovered by retaining the flap modes. The accuracy of a harvester reduced-order model must therefore be reported on the harvested quantity itself.

1 Introduction

Post-flutter limit-cycle oscillations can power piezoelectric harvesters, but designing them requires costly nonlinear sweeps, and a structural-mode basis may miss the electrical state carrying harvested power. This work examines transducer placement and reduced-model errors against a deliberately low-order full system.

1 Introduction

  • Post-critical operation matters because a linear system at flutter has no bounded amplitude, whereas practical harvesters operate on nonlinear, bounded limit cycles.Prior work reported 10.7 mW at linear flutter and noted that transduction shifts the flutter boundary.Source: §1 · p. 2
  • A design sweep requires a nonlinear time march to a converged limit cycle at every point, making sweeps prohibitive for high-fidelity models.Harvesting also requires a basis that captures the electrical state carrying power, not only the dominant structural mode.Source: §1 · p. 2
  • Transducer placement changes both the sign of the flutter-boundary shift and harvested power by a factor of 4.5; the testbed permits every reduced prediction to be checked against the full-order model.The authors report that the remaining reduced-model error lies in the harvested quantity, not the pitch and plunge response.Source: §1 · p. 2
  • The study extends the authors’ nonlinear reduced-order formulation from aeroelastic energy dissipation to harvesting, using the pitch-plunge aerofoil with finite-mass trailing-edge flap characterized in.The formulation’s earlier applications include gust-load analysis and control applications.Source: §1 · p. 2

2 Electro-aeroelastic model

The model couples plunge, pitch, and flap motion with unsteady aerodynamic lag states and a resistively shunted piezoelectric transducer. Its nondimensional formulation produces a fifteen-state autonomous electro-aeroelastic system, with cubic pitch stiffness bounding the limit cycle.

2.1 Structure and unsteady aerodynamics

  • The structural model has three coordinates—plunge, pitch, and finite-mass trailing-edge flap—with diagonal stiffness and zero structural damping.The elastic axis is at the quarter chord, where circulatory lift exerts no moment.Source: §2.1 · pp. 2–3
  • Only the pitch spring is nonlinear, and its cubic stiffness βα = 100 keeps the limit cycle below five degrees of pitch.Because the restoring forces have no quadratic term, the reduction’s quadratic operator vanishes identically.Source: §2.1 · p. 3
  • Unsteady aerodynamics use strip theory in indicial form with two-exponential Wagner and Küssner fits, represented by lag states for each coordinate.The gust velocity is zero throughout; the gust states are retained for the solver’s fifteen-state layout and are inert in reported results.Source: §2.1 · pp. 3–4

2.2 Piezoelectric transduction

  • The transducer is a lumped element on one structural degree of freedom, with linear electromechanical coupling and capacitance, shunted by a resistive load.Scaled voltage follows a first-order charge-balance equation driven by the mounted coordinate’s velocity.Source: §2.2 · p. 4
  • Adding the voltage state to the structural and aerodynamic states yields a fifteen-state autonomous system, x′ = F(x; U*) = A(U*)x + f(x; U*).The linear part A(U*) is the Jacobian at the origin.Source: §2.2 · p. 4

3 Nonlinear reduced-order model

The nonlinear reduced model projects a Taylor-expanded residual onto a biorthonormal eigenvector basis. The retained basis has four real states, and the paper compares ways of updating its reduced operator as flow speed changes.

3 Nonlinear reduced-order model

  • The reduction evaluates second- and third-order Fréchet derivatives matrix-free along retained basis vectors, avoiding assembly of high-order full-space tensors.Biorthonormal right and left eigenvectors define the real basis, with WᵀV = I.Source: §3 · p. 5
  • Two complex-conjugate pairs are retained, giving four real states: the flutter pair and a second pitch-plunge-dominated pair retained because it affects limit-cycle amplitude.The flap pair is omitted in the principal basis and tested by adding it later.Source: §3 · p. 5
  • The retained eigenvalues are expanded in flow-speed change with K = 3 in reported results, while the corresponding operator is reassembled from those eigenvalues.The table describes the fifteen-state Jacobian spectrum at U* = 7.0 for plunge mounting, χ = 0.20, and r = 1.0.Source: §3 · p. 5, Table 1 · p. 5
  • The alternative velocity treatment re-projects the exact Jacobian onto the same frozen basis, at one Jacobian evaluation and two matrix products per velocity.The Taylor-expanded residual construction remains unchanged.Source: §3 · p. 5
  • The eigenvalue-expansion operator remains block-diagonal at every expansion order, so it cannot couple retained modes; re-projecting the exact Jacobian instead yields a general matrix.This structural difference governs the models’ accuracy.Source: §3 · p. 6

4 Time integration and measurement of the limit cycle

Full-order and reduced responses are marched from rest using the same fixed-step Heun scheme, without gust forcing. Limit-cycle convergence, decay, and measurement are assessed using successive long time blocks and complete cycles.

4 Time integration and measurement of the limit cycle

  • Both models use Heun’s explicit second-order Runge–Kutta scheme with fixed step 0.05 in τ, about 1,500 steps per cycle, starting from a small pitch perturbation.The reduced model starts from the projection of the full initial condition onto its basis; no gust is applied.Source: §4 · p. 6
  • A limit cycle is declared when the largest pitch amplitude changes by less than 0.1% between successive 20,000-step blocks; cases reaching the 40-block cap are unconverged.Responses are classified as decayed when the equilibrium is linearly stable and amplitude is falling, or when amplitude falls below 10^-9.Source: §4 · p. 6
  • Amplitudes, period, and mean power are measured over whole cycles in the last block, bounded by the first and last upward voltage zero crossings.This avoids biasing mean power with a partial cycle.Source: §4 · p. 6

5 Results

Transducer placement changes both the flutter boundary and harvested power, while reduced-model accuracy away from its construction speed depends strongly on how the operator is continued. Re-projecting the Jacobian restores structural-response accuracy without enlarging the basis, but basis-related voltage error remains.

5.1 Verification

  • Limiting-case checks recover the baseline fourteen-state system and, with zero coupling, its eigenvalues plus the electrical pole and power relation.The reported checks cover 5 ≤ U* ≤ 10 and r = 0.1, 1, and 10.Source: §5.1 · p. 6

5.2 Effect of the transducer on the flutter boundary

  • Plunge mounting lowers the flutter boundary to 6.1002 (−3.5%), whereas pitch mounting raises it to 8.0058 (+26.6%); both shifts vanish in the short-circuit limit.The baseline boundary is U*F = 6.3246; an open-circuited transducer acts as added stiffness despite delivering no power.Source: §5.2 · p. 6

5.3 Limit-cycle oscillation and harvested power

  • At U* = 7.0, plunge mounting yields 9.35 × 10−5 mean power versus 2.09 × 10−5 for pitch mounting, a factor of 4.5 on the common reference.The reported optima occur at χ = 1.00, with r = 1.78 for plunge and r = 0.56 for pitch.Source: §5.3 · p. 7, Figure 3 · p. 8

5.4 Accuracy of the reduced model

  • At the construction point, four states reproduce pitch and plunge amplitudes to +0.1% and +0.8%, but voltage is 21.9% low; adding flap modes reduces flap error to −3.9%.The voltage deficit stems from the fixed voltage-to-motion proportion carried by equilibrium eigenvectors; adding electrical modes only partly improves it, while retaining flap modes recovers flap motion.Source: §5.4 · pp. 8–10
  • Re-projecting the exact Jacobian onto the frozen basis reduces pitch-amplitude errors to −3.4% to +1.3% over 6.40 ≤ U* ≤ 8.00 and places the Hopf point at 6.3130 (+0.10%).The eigenvalue-expansion treatment instead gives pitch errors from +46.4% to −16.9% and a spurious limit cycle below the full-order Hopf point.Source: §5.4 · p. 9, Table 3 · p. 9

5.5 Cost of the reduction

  • At this model scale and aerodynamic fidelity, reduction yields no significant time-integration advantage; it begins to pay only at about one hundred states.The residual is a small share of measured cost, while the projected cubic operator is dense and can cost more than the fifteen-state residual.Source: §5.5 · p. 11

5.6 The linear measure does not reveal the error

  • Linear growth-rate errors are similar for the two operator treatments, yet their nonlinear amplitude errors differ by more than an order of magnitude near Hopf.Near Hopf, limit-cycle amplitude scales with the square root of excess growth rate, amplifying small linear errors.Source: §5.6 · pp. 11–12

5.7 Why the velocity expansion fails

  • The eigenvalue-expansion operator omits inter-mode coupling at every order; the omitted coupling reaches 8.6 × 10−2, comparable to retained eigenvalues, and higher order does not restore it.Re-projecting the Jacobian supplies the missing coupling; the voltage deficit remains a basis property unaffected by this remedy.Source: §5.7 · pp. 12–13, Table 6 · p. 13

6 Discussion

The four-state model’s broad pitch error is chiefly a velocity-treatment error, while its remaining errors are concentrated in harvested voltage and power. Mounting choice also trades flutter margin against energy capture, within a limited aerodynamic and circuit scope.

6 Discussion

  • Plunge mounting produces 4.5 times the power of pitch mounting, but the mountings shift the flutter boundary in opposite directions.A stabilising mounting delays harvesting, whereas the higher-harvesting destabilising mounting erodes the flutter margin.Source: §6 · p. 13
  • The NROM pitch error spans +46% to −17%, although the same four-state basis is accurate at the construction point; exact re-projection removes the spurious limit cycle below Hopf.At the construction point, pitch and plunge errors are 0.1% and 0.8%; the large errors away from it arise because the expanded operator has zero inter-mode coupling.Source: §6 · p. 13
  • Voltage remains 20 to 23% low and mean power 28 to 34% low, despite pitch and plunge being accurate to a few per cent.The flap error of 58 to 83% is a truncation error that is removed by retaining the flap pair at six states.Source: §6 · p. 14
  • The scope excludes stall-dominated LCOs and real-circuit losses: the flow is attached, incompressible, and two-dimensional, and the load is purely resistive.The limit cycle is held below five degrees; reported powers are upper bounds because rectification, storage, and impedance matching are omitted.Source: §6 · p. 14

7 Conclusions

The study applies a four-state reduction to a fifteen-state pitch-plunge-flap harvester and finds that mounting choice affects both flutter shift and power. Exact Jacobian re-projection improves structural predictions without enlarging the basis, but harvested-voltage error remains.

7 Conclusions

  • Plunge mounting yields 4.5 times the power of pitch mounting, while the flutter-boundary shift reaches −3.5% for plunge and +26.6% for pitch.The transducer is embedded in a three-degree-of-freedom aerofoil with a finite-mass flap and unsteady strip-theory aerodynamics.Source: §7 · p. 14
  • Exact re-projection onto the same frozen basis reduces peak pitch-amplitude error from 46% to 3.4% at negligible cost.The error is dominated by how the reduced operator depends on flow speed, rather than by basis size.Source: §7 · p. 14
  • Eigenvalue expansion leaves inter-mode coupling identically zero at every order, reproducing the linear spectrum but not the limit cycle.Adding expansion terms does not repair this structural failure.Source: §7 · p. 14
  • Voltage remains 20 to 23% low even when pitch and plunge are accurate to a few per cent; retaining the flap pair recovers flap deflection to within 4%.Equilibrium eigenvectors fix the voltage-to-displacement ratio at their own eigenvalue, below the gain on the periodic cycle.Source: §7 · p. 14
  • At fifteen states, reduction provides no wall-clock savings because both marches are limited by array-operation dispatch rather than arithmetic.The cost argument is intended for full-order models an order of magnitude larger.Source: §7 · p. 14

Declaration of generative AI use

Claude assisted with drafting and editing the text, proofreading, and checking analysis scripts and consistency with computed data; the authors reviewed the content and take responsibility for it.

Declaration of generative AI use

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