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Rigid-body support motion and post-flutter piezoelectric energy harvesting from a pitch-plunge-flap aerofoil

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

arXiv:2610.00211v1cs.CE

TL;DR

The paper asks whether piezoelectric harvesting design findings from a rigidly grounded aerofoil hold when its support is a freely translating fuselage. It models the free-free section and finds that support motion affects mountings unequally, while load matching remains unchanged and power comparisons depend on how operating speed is defined.

  • Problem

    Earlier aerofoil-harvesting design findings use rigid-ground support, leaving open whether mounting rankings, load matching, and optimum power persist when the fuselage can move.

    Source: §1 · p. 2

  • Method

    The study adds a vertically translating fuselage mass to the companion pitch-plunge-flap aerofoil and transducer, creating a four-degree-of-freedom free-free model.

    Source: §2.1 · p. 3

  • Results

    The support amplifies plunge destabilisation, reduces pitch stabilisation in the time-constant-matching regime, and leaves flap stability effects untouched; fixed-margin power gains persist for plunge but not pitch.

    Source: §8 · p. 17

  • Takeaways & Limitations

    The coupling and load-resistance optimum stays in the same design cell, while harvested-power results must specify whether flow speed or margin above flutter is held fixed.

    Source: §5.2 · p. 12, §8 · p. 17

    The aerodynamics use incompressible strip theory, and the combined effect of that model’s bias with support-mass effects remains unresolved.

    Source: §7 · p. 16

Abstract

from arXiv · show

A piezoelectric transducer on a pitch-plunge aerofoil with a finite-mass trailing-edge flap turns the limit cycle that follows flutter into electrical power. The design findings for such a harvester have so far been obtained with the section's suspension reacting against rigid ground, which is what a wind-tunnel mounting provides. Here a 3-dof aerofoil with a transducer is carried by a fuselage mass that is itself free to translate, so that the coupled system is free-free and the wing's plunge is measured relative to a support that moves. The support degree of freedom is destabilising on its own, softening the flutter mode by taking a share of it in antiphase with the wing's plunge, and it acts on the three transducer mountings unequally, amplifying the plunge mounting's destabilisation, leaving the largest pitch-mounted stabilisation untouched while halving it where the electrical time constant meets the flutter frequency, and leaving the flap mounting alone. The optimum coupling and load resistance do not move, so the matching rules of the rigidly supported section carry over unchanged. The harvested power rises with the support mass at a fixed flow speed for every mounting, and that rise is shown to be an artefact of the fixed-speed convention rather than a property of the cycle. Measured instead at a fixed margin above each system's own flutter boundary, which divides out the boundary's motion, the plunge mounting keeps a real gain while the pitch mounting loses power, so the apparent benefit of a soft support to a pitch-mounted harvester is entirely the boundary coming down to meet the flow speed. A harvester result quoted at a fixed speed and read as a property of the limit cycle is therefore wrong by the whole of that difference.

1 Introduction

The paper asks whether piezoelectric harvesting design findings from a rigidly supported aerofoil survive when the support is a freely translating fuselage. It extends the companion section to a free-free system and examines how support motion affects stability, mounting choice, load matching, and power.

1 Introduction

  • Post-flutter limit cycles provide a deterministic source of motion for piezoelectric harvesting, unlike broadband random excitation.Earlier work includes a built and tested flutter-driven harvester.Source: §1 · pp. 1–2
  • A freely translating fuselage makes the wing’s plunge relative to its support, so rigid-ground results may not transfer directly to a flying vehicle.The paper tests whether mounting rankings, electrical-load matching, and optimum harvested power are properties of the section or its mounting.Source: §1 · p. 2
  • The paper positions the free-free case against prior rigid-ground studies and asks which of their design conclusions survive support motion.The section’s finite-mass trailing-edge flap and unsteady strip-theory aerodynamics match the companion study [1].Source: §1 · p. 2
  • The study retains the companion paper’s section, aerodynamics, transducer, and test cases, adding a vertically translating fuselage mass parameterized by the wing-to-fuselage mass ratio.The resulting structure has four degrees of freedom.Source: §1 · p. 2

2 Electro-aeroelastic model

The model adds a freely translating fuselage to a pitch-plunge section with a finite-mass flap, while the wing’s absolute plunge drives the aerodynamics and its relative plunge describes motion against the support. A piezoelectric transducer couples to one of three structural mountings, and the free support adds a neutral rigid-body mode that affects stability analysis.

2.1 Structure

  • The wing’s relative plunge is ξ and its absolute plunge is ξ + y, where y is the freely translating fuselage displacement.The plunge spring and transducer act between wing and fuselage; MR is the wing-section-to-fuselage mass ratio, with MR = 0 representing rigid support.Source: §2.1 · p. 3, Figure 1 · p. 3

2.2 Unsteady aerodynamics

  • The aerodynamic model uses indicial strip theory, with lag states driven by pitch, absolute plunge, and flap motion; gust input is zero in the reported results.Driving the plunge lag states by ξ + y makes aerodynamic loading depend on motion through the air, not laboratory position.Source: §2.2 · p. 4

2.3 Piezoelectric transduction

  • The transducer is a resistively shunted lumped element driven by the velocity of one structural degree of freedom, with plunge, pitch, and flap as the three possible mountings.No fourth mounting exists because the free fuselage has no spring to react against.Source: §2.3 · p. 5

2.4 What a free support does to the spectrum

  • A free-free system has an exact zero eigenvalue because its position is indifferent, so stability cannot be assessed by taking the spectrum’s largest real part at face value.The fuselage-velocity mode is damped by the air, while the neutral mode represents rigid translation.Source: §2.4 · p. 5

3 Implementation and validation

The four-degree-of-freedom model is mapped from the companion solver and checked against an independent implementation and several distinct validation tests. These checks support its limiting behavior, flutter and limit-cycle predictions, while exposing numerical hazards created by the free-free rigid-body mode.

3.1 Against the solver this one came from

  • The independent 2013 solver agrees with the flap-deleted model’s trajectories to 1.0 × 10^-14 relative and eigenvalues to 10^-15, and reproduces its flutter boundary to 10^-6.The boundary comparison is U*F = 2.180076 at frequency 0.4341.Source: §3.1 · p. 6

3.2 Five independent checks

  • The MR = 0 flutter boundary and frequency match the companion paper, and the boundary shifts smoothly by −1.168% at MR = 1.The boundary is 6.3245919 versus 6.324592 in the companion paper, with frequency 0.084552 in both.Source: §3.2 · p. 6
  • Above flutter, pitch amplitude follows the square root of excess velocity within 1.2% over a ten per cent range at MR = 0, 0.3, and 1.The nonlinear march decays to the origin at 0.998 U*F.Source: §3.2 · p. 6
  • The linear and nonlinear marches agree below the boundary to 2.4 × 10^-4, but above it the linear response diverges while the nonlinear cycle settles at 0.0517.The nonlinear cycle period is 75.69 versus the linear value 84.51.Source: §3.2 · pp. 6–7
  • Across five cycles, air-work and load dissipation differ by 7.1 × 10^-4 to 2.9 × 10^-3 of the dissipated work, with the gap shrinking as the cycle is further converged.The energy balance requires undoing the equation normalizations before comparing work and power.Source: §3.2 · p. 7

3.3 What the free-free mode cost

  • A max-Re-λ bisection reports the free-free rigid-body mode at zero frequency rather than the flutter boundary, and modulus tolerances cannot safely separate it from a genuine electrical pole.The rigid-body eigenvector is a pure translation and is the reliable solver test; the transducer pole can be of comparable size to numerical noise.Source: §3.3 · p. 7
  • A growing limit-cycle transient gives the neutral mode a net impulse, so the cycle can sit on an offset and drift until the damped velocity mode decays.The fuselage amplitude is measured about its record mean, and the centre of mass is checked cycle by cycle.Source: §3.3 · p. 7

4 The flutter boundary

A translating fuselage mildly lowers the flutter boundary and softens the mode. Its effects on transducer-induced boundary shifts differ by mounting: plunge destabilization is amplified, pitch stabilization changes near tuning, and the flap effect remains unchanged.

4.1 The support mass on its own

  • The fuselage carries 10.5% of the flutter mode at MR = 1 and moves antiphase with wing plunge, reducing suspension compression and softening the mode.The limit-cycle plunge-to-pitch ratio rises from 2.66 to 2.97 over the same support-mass range.Source: §4.1 · p. 8
  • Figure 3 and Table 2 show the boundary falling from 6.3246 at MR = 0 to 6.2507 at MR = 1 (−1.168%), while flutter frequency rises from 0.084552 to 0.086315.The support mass produces a first-order boundary shift on the same scale as a plunge transducer at r = 10 with χ = 0.17.Source: Figure 3 · p. 8, Table 2 · p. 8, §4.1 · p. 8

4.2 The transducer on a free support

  • Table 3 and Figure 4 report transducer shifts relative to each support mass’s own boundary, separating transducer effects from support-induced shifts.The MR = 0 block reproduces the companion paper; MR > 0 blocks provide the new results.Source: Table 3 · p. 9, §4.2 · p. 9, Figure 4 · p. 10
  • At χ = 0.50 open circuit, plunge-mounted destabilization grows from −3.57% at MR = 0 to −4.83% at MR = 1 as the support and transducer softenings compound.The shift is −3.90% at MR = 0.3.Source: §4.2 · p. 10
  • Pitch-mounted stabilization at r = 1 falls from +4.06% at MR = 0 to +1.94% at MR = 1, while its largest stabilizing shift barely changes.At χ = 0.50, the optimum load resistance moves from r = 14.1 to r = 17.8; the open-circuit stiffness effect is untouched, unlike finite-resistance phase-dependent damping.Source: §4.2 · p. 10
  • The flap mounting shifts the boundary by −0.20% at χ = 0.50 open circuit for every support mass tested.Its effect is an order of magnitude weaker than the other mountings on a rigid support, and support mass does not change that.Source: §4.2 · p. 10

5 The limit cycle and the harvested power

The section examines the limit cycle and harvested power as support mass changes, distinguishing fixed-velocity power from power measured at a fixed margin above each system’s flutter boundary. The fixed-margin comparison shows that the pitch mounting’s apparent gain does not reflect a better cycle, while the plunge mounting retains a gain.

5.1 The cycle

  • At U∗ = 7.0 without a transducer, pitch amplitude rises from 0.0516 to 0.0539 rad and plunge from 0.1372 to 0.1601 over MR = 0 to 1, while period falls from 76.15 to 75.23.At MR = 1, the fuselage carries 9.0% of the orbit.Source: §5.1 · p. 10, Table 4 · p. 11

5.2 The coupling and load-resistance plane

  • At MR = 0, the plunge optimum is χ = 1.00, r = 1.74 with mean power 9.335×10−5, while the pitch optimum is χ = 1.00, r = 0.55 with mean power 2.080 × 10−5.The maps show mean harvested power over the coupling and load-resistance plane; grey cells have no converged limit cycle.Source: §5.2 · p. 11, Table 5 · p. 12, Figure 5 · p. 12

5.3 Power against the support mass, measured two ways

  • +29.9% for the plunge mounting and +51.9% for the pitch mounting at MR = 1 when power is compared at fixed reduced velocity.Figure 6 presents the support-mass trends using both fixed reduced velocity and fixed margin above each system’s flutter boundary.Source: §5.3 · pp. 12–13, Figure 6 · p. 13
  • +11.8% for the plunge mounting and −9.3% for the pitch mounting at a fixed margin above each system’s flutter boundary, unlike their fixed-velocity gains.This comparison divides out the support-mass-induced boundary shift.Source: §5.3 · pp. 12–13, Figure 6 · p. 13
  • As reduced velocity is rescaled by each system’s flutter boundary, the power curves for the three support masses nearly collapse.Against reduced velocity itself, the curves separate.Source: §5.3 · p. 13, Figure 7 · p. 13

6 The reduced model

The reduced model is tested on a free support, where rigid-body modes sit outside the flutter-band basis and may affect predicted motion and harvested power. The two-pair model reproduces pitch closely but can substantially misstate power; retaining rigid-body modes helps only when the slow modes coalesce.

6 The reduced model

  • The flutter-band basis misses the rigid-body and slow support motion, which can leave the fuselage amplitude misrepresented by roughly a quarter.The two-pair basis error falls from +28.2% at MR = 0.1 to +21.4% at MR = 1.Source: §6 · pp. 14–15
  • The reduction projects the residual onto a real basis formed from retained right and left eigenvectors, comparing structural-pair and rigid-body-mode combinations.The reduced linear operator is evaluated at the running velocity on the frozen basis, and the cubic spring is projected directly.Source: §6 · p. 14
  • The two-pair basis keeps pitch error between 0.1% and 0.9% but power error grows from −27.9% at MR = 0 to −48.2% at MR = 1.Voltage error grows from −21.9% to −35.3% over the same support-mass range.Source: Table 6 · p. 14, §6 · p. 15
  • At MR = 1, the six-state model has 9.5% power error versus 48.2% for the four-state model, after the two slow modes coalesce above MR = 0.3519.The six-state model’s pitch, fuselage, voltage, and period errors are 0.5%, 5.1%, 8.5%, and 0.4%, respectively.Source: Table 6 · p. 14, §6 · p. 15, Figure 8 · p. 15
  • Replacing the second structural pair with the flutter pair and rigid-body modes worsens every quantity at every support mass, including pitch by more than ten per cent and power by 40% to 200%.The rigid-body modes should be added to, not substituted for, the companion paper’s basis.Source: §6 · p. 15
  • At MR = 0, rigid-body modes cannot be retained because the support coordinate is frozen and decoupled, making biorthonormalisation singular.The repeated zero eigenvalues belong to an eigenspace the coupled system does not enter.Source: §6 · p. 15

7 Discussion

The discussion separates findings that carry over from the rigidly supported section from those changed by a free support. It also bounds the support-mass effect and identifies unresolved interaction between aerodynamic and support-modeling effects.

7 Discussion

  • The plunge mounting still harvests about four and a half times the pitch mounting’s power, the flap mounting’s stability effect remains an order of magnitude smaller, and the mounting effects retain their signs.The optimum coupling and load-resistance position and the supercritical Hopf bifurcation with a square-root amplitude law also persist.Source: Table 7 · p. 16, §7 · p. 16
  • At MR = 1, plunge destabilisation grows by a third, while intermediate-load pitch stabilisation is more than halved and its optimum resistance moves outward by a quarter.A pitch transducer tuned for stabilisation on a rigid rig is therefore mistuned on a free support.Source: §7 · p. 16
  • A fuselage as heavy as the wing section moves the flutter boundary by just over one per cent; for a much heavier real-aircraft fuselage, the effect is a fraction of a per cent.Support mass matters most as MR approaches unity, such as for a small unmanned vehicle or a free-body harvester.Source: §7 · p. 16
  • The joint effect of compressible-flow modeling and a free support on pitch stabilisation remains unresolved, and this model represents a free-free section rather than a free-free aircraft.The fuselage is a point mass in plunge alone, with no pitch freedom or second wing.Source: §7 · pp. 16–17

8 Conclusions

The conclusions show that a free support changes the flutter mode and affects transducer mountings unequally, while rigid-support matching rules remain valid. Power comparisons depend on whether flow speed or distance above flutter is held fixed, and reduced models can misrepresent the free-free system.

8 Conclusions

  • The free-fuselage model applies aerodynamic forces to absolute plunge and springs to relative plunge, and its heavy-fuselage limit reproduces the companion paper’s flutter table and power map to their rounding.The system is assembled from the rigidly supported section using two maps.Source: §8 · p. 17
  • The free support mildly destabilises flutter, amplifying plunge-mounted destabilisation by a third, reducing pitch-mounted stabilisation by more than half at the time-constant match, and leaving flap mounting unchanged.A fuselage as heavy as the wing section produces a shift comparable to a moderately coupled plunge-mounted transducer.Source: §8 · p. 17
  • Rigid-support optimum coupling and load resistance remain optimal with a free support, and the plunge mounting retains its advantage over the pitch mounting.The matching conclusion is stated for the studied section and test cases.Source: §8 · p. 17
  • At a fixed margin above each system’s flutter boundary, the plunge mounting keeps a power gain while the pitch mounting loses power; fixed-speed comparisons can misattribute boundary motion to the limit cycle.The authors say harvesting results quoted at fixed speed should specify that convention.Source: §8 · p. 17
  • The companion-style four-state reduced model predicts pitch within one per cent but can misstate harvested power by up to a factor of two and fuselage motion by about a quarter.Retaining rigid-body modes fixes these errors only above the support mass where the two slow modes coalesce; below it, enlarging the basis can worsen fuselage-motion predictions.Source: §8 · p. 17
  • For this free-free system, the largest spectral real part and a modulus-tolerance exclusion are unsafe stability tests; the closed-form rigid-body eigenvector is the reliable test.An almost open-circuited transducer can have a genuine pole similar in size to the rigid-body mode.Source: §8 · p. 17

Declaration of generative AI use

The authors disclose using Claude to assist with drafting, editing, and analysis checks, while retaining responsibility for the paper.

Declaration of generative AI use

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