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Compressible aerodynamics and rigid-body support motion in post-flutter piezoelectric energy harvesting from a pitch-plunge-flap aerofoil
Nikolaos D. Tantaroudas, Ilias Karachalios, Andrew J. McCracken
TL;DR
This paper asks how compressible aerodynamics and a freely translating support combine in a piezoelectric aerofoil harvester, after each change was studied separately. It models both together using a flow solver that needs no additional mesh motion. The corrections multiply for harvested power but do not generally compose for the flutter boundary, where the outcome depends on the transducer regime.
Problem
Compressible aerodynamics and a freely translating support each alter transducer-induced flutter-boundary shifts and harvested power, but their combined effect had not been established.
Source: §1 · p. 2, §4.2 · p. 7Method
The paper combines compressible Euler aerodynamics and a free-translating fuselage in a pitch-plunge-flap aerofoil harvester, using a motion map so support translation requires no new flow-solver mesh motion.
Source: §1 · p. 2, §2.2 · p. 3, §7 · p. 12Results
The corrections multiply for harvested power, yielding 82.5% against the predicted 82.8%, but on the flutter boundary they more than compound and can reverse the shift at strong coupling.
Source: §7 · p. 11, §8 · p. 13Takeaways & Limitations
Boundary corrections compose to within seven parts in a thousand above the electrical-pole crossing, but below it their product can be wrong by up to fourteen and by a sign on the pitch mounting.
Source: §8 · p. 13The study fixes Mach number at M = 0.10, maps composition at χ = 0.50 using four resistances, and compares harvested power at one operating point.
Source: §7 · p. 12
Abstract
from arXiv · showhide
A piezoelectric transducer on a pitching and plunging aerofoil with a trailing edge flap turns the limit cycle that follows flutter into electrical power, and two modelling changes to that harvester have been studied separately. Replacing incompressible strip theory by the compressible Euler equations reduces the stabilising effect of a transducer mounted on the pitch, where the electrical time constant meets the flutter frequency, and raises the harvested power, and letting the section react against a fuselage free to translate, instead of rigid ground, does the same two things by about the same amounts. Here both changes are put in one model, which needs no new mesh motion in the flow solver, since the support's translation displaces the wall exactly as the plunge does. On the harvested power the two published corrections simply multiply. On the flutter boundary they more than compound, and at strong enough coupling they reverse the sign of the transducer's effect at the load resistance where both companion papers harvest, although neither change alone reverses it. The free support widens the band of resistance over which the two aerodynamic models disagree in sign, carrying its upper edge out by about a factor of three. Above the resistance where the electrical pole meets the flutter frequency, where the transducer acts as an added stiffness, the product of the published corrections predicts the shift to within seven parts in a thousand. Below it, where the transducer acts as a damper whose effect depends on phase, the product is wrong by a factor of up to fourteen, and on the pitch mounting by a sign. The same transducer lowers the boundary while flattening the growth of the instability above it. Whether corrections found one change at a time may be multiplied therefore depends on the quantity and, for the flutter boundary, on the regime of the transducer.
1 Introduction
Flutter-driven limit cycles can be both a fatigue problem and a source of electrical power, but transducer coupling also shifts the flutter boundary. This study combines compressible aerodynamics and a translating free support, two changes previously examined separately.
1 Introduction
- Flutter-driven limit cycles convert flow energy into structural motion, which an embedded piezoelectric element turns partly into electricity, while transduction itself shifts the flutter boundary.Because a linear system at its flutter boundary has no bounded amplitude, harvesting operates above the boundary on a nonlinearity-fixed cycle.Source: §1 · p. 2
- The paper combines the two previously separate modelling changes—compressible aerodynamics and a free-translating support—to test how they act together on the same harvester.The harvester is a pitch-plunge aerofoil with a finite-mass trailing-edge flap and a piezoelectric transducer.Source: §1 · p. 2
- The two prior changes each roughly halve the pitch mounting’s stabilising boundary shift where the electrical time constant meets the flutter frequency, leaving their combined effect unresolved.The prior studies could not determine whether the effects compound or partly describe the same mechanism.Source: §1 · p. 2
- Free-free coupling links a wing’s elastic motion with a body’s rigid-body motion, a feature of aeroelastic-flight-dynamic models of flexible aircraft [10].Reduced-order models in that field have been used to identify worst-case gusts rapidly [11].Source: §1 · p. 2
- A free support does not require a four-degree-of-freedom mesh motion: support translation moves the wall like plunge, removing an apparent obstacle to high-fidelity flow modelling.The paper explains this equivalence in Section 2.Source: §1 · p. 2
2 Model
The model couples a pitch-plunge-flap aerofoil, a translating fuselage, a piezoelectric transducer, and compressible flow. A motion map lets the flow solver retain three aerodynamic coordinates despite the model’s fourth structural coordinate.
2.1 The section, the support and the transducer
- The four structural coordinates are relative wing plunge, pitch, flap deflection, and fuselage translation; the wing’s absolute plunge is the sum of relative plunge and fuselage translation.The support mass enters through MR = m/Mf; MR = 0 recovers rigid support, and the transducer is mounted on plunge, pitch, or flap with coupling χ and resistance r.Source: §2.1 · pp. 2–3
2.2 Why a flow solver needs no fourth mesh motion
- The aerodynamic model uses three coordinates—absolute plunge, pitch, and flap—and the fourth structural coordinate reaches it through a motion map, with generalized forces returned through R.The fuselage row of R is scaled by MR.Source: §2.2 · p. 3
- Fuselage translation produces the same wall displacement as plunge, so the flow solver keeps its existing mesh motion and the fourth coordinate is handled by the motion map.The fuselage aerodynamic force is exactly MR times the plunge force; rigid translation leaves the flow state unchanged in mesh coordinates, making the aerodynamic block of the free-free system’s null vector zero.Source: §2.2 · p. 3
2.3 The compressible aerodynamics
- The compressible model marches each cell’s conservative variables with the structure, using rigid mesh motion for pitch and plunge and radial-basis-function deformation for the flap.Wall forces come from the residual-applied wall-face flux; there is no apparent-mass term, and the Mach number is an input while reduced-time aerodynamics do not depend on reduced velocity.Source: §2.3 · p. 3
3 Implementation and validation
The implementation carries the free-free motion maps into both aerodynamic models and validates the compressible solver against companion results. Additional checks assess reproducibility and whether time-step and mesh choices affect the comparisons.
3.1 What was carried over and what was written
- The shared motion maps connect the compressible flow provider to the third companion paper’s solver, leaving the solver below the interface unchanged.The implementation also supplies the flow model’s rigid-body null vector directly and uses bisection or Illinois-rule flutter searches depending on spectrum cost.Source: §3.1 · p. 4
3.2 Validation
- At M = 0.10 with MR = 0, the baseline boundary is U*F = 6.288873 versus 6.288873 in the companion result, with a difference of 2.9×10^-7.The flow-model frequency is 0.08352 versus 0.0835 in the companion paper.Source: §3.2 · p. 4
- All eight recorded rigid-support boundaries at χ = 0.50 reproduce the companion paper’s values to 4 × 10^-11 across four load resistances on each of two mountings.Table 1 covers the comparison of companion-paper boundaries with the four-degree-of-freedom solver’s frozen-support results.Source: §3.2 · p. 4, Table 1 · p. 4
- The two repeated computations agree to 1.9×10^-12, and changing the strip-theory time step alters mean power by only 1.5×10^-4 of itself.The 1.5×10^-4 change is three orders below reported differences; mesh refinement gives a 0.160% boundary spread, with the companion-paper mesh within 0.077% of the finest.Source: §3.2 · p. 5
4 Results
With the pitch-mounted transducer at χ = 0.50 and r = 1, combining compressible flow and a free support reverses the positive boundary shift found when either change is applied alone. Across load resistance, the two corrections compose in the added-stiffness regime but not in the phase-dependent damping regime.
4 Results
- At χ = 0.50 and r = 1, the pitch-mounted shift changes from +4.06% for rigid-support strip theory to −0.533% with both changes, beyond the +1.951% and +1.94% single-change shifts.The combined case is the previously uncomputed cell in Table 3.Source: §4 · pp. 5–6, Table 3 · p. 6
4.1 The same result without the root-finder
- At U∗ = 6.15, the transducer raises the leading real part from −2.664 × 10−3 to −2.053 × 10−4, while its branch crosses zero earlier and rises more slowly.The slopes are 0.0084 versus 0.0453 per unit reduced velocity; at U∗ = 6.25, the transducer-on system is more stable.Source: §4.1 · p. 6
4.2 Load Resistance
- For the pitch mounting in compressible flow, the model shifts are negative at r = 0.1, 0.3, and 1 on a free support, but positive by r = 3; strip theory stays positive at all four resistances.The free support moves the sign-change interval from between r = 0.3 and r = 1 to between r = 1 and r = 3, where the electrical pole meets the flutter frequency at r = 1.89.Source: §4.2 · p. 7
4.3 Where the two corrections do multiply
- Above r = 1.89, multiplying the two single-change corrections predicts the shift within seven parts in a thousand; below it, the product fails on both mountings and can predict the wrong sign for pitch.The transducer acts as added stiffness above the crossover and as a phase-dependent damper below it, where changes in flutter mode and frequency make the corrections non-independent.Source: §4.3 · p. 8, Table 4 · p. 8
4.4 How strong the coupling must be
- The combined shift changes sign between χ = 0.28 and 0.35, at χ = 0.315 by linear interpolation; the corrections under-predict the loss of stabilisation at both tested couplings.At χ = 0.20 the combined shift is +0.2672%, versus a predicted +0.829%; at χ = 0.50 it is −0.5326%, versus a predicted +0.931%.Source: §4.4 · pp. 8–9
4.5 The support mass on its own, in a compressible flow
- Without the transducer, increasing support mass lowers the boundary by 1.276% in compressible flow and 1.168% in strip theory; both changes therefore destabilise the baseline.At r = 1, the transducer's phase-dependent damping is sensitive to the flutter mode and frequency, which both modelling changes alter.Source: §4.5 · p. 9, Figure 3 · p. 9
5 The limit cycle and the harvested power
At the anchor operating point, compressibility and a free support each raise harvested power, and their gains nearly multiply. Both also increase cycle motion, especially plunge, while a free support introduces measurable fuselage motion.
5 The limit cycle and the harvested power
- At χ = 0.20 and r = 1, compressibility raises mean power by 35.1% and a free support by 35.3% over the rigid-support strip-theory baseline.The respective mean powers are 2.431 × 10−5 and 2.434×10−5, versus 1.799 × 10−5 for the baseline.Source: §5 · p. 9
- 82.5% is the combined harvested-power gain over rigidly supported strip theory, nearly matching the 82.8% expected if the two gains multiplied.The combined mean power is 3.283 × 10−5, against 1.799 × 10−5 for the baseline.Source: §5 · pp. 9–10
- The power gains compound because both changes raise cycle amplitude by making the path between the air and transducer more compliant.The combined gain is small in excess of the product of the separate gains; it is not claimed to be exactly zero.Source: §5 · p. 10
- Across measured cycle pairs, compressible-flow plunge is 17.1%–20.1% larger than strip-theory plunge, while pitch is only 1.9%–3.1% larger.The plunge mounting outperforms the pitch mounting by about a factor of four in both models.Source: §5 · p. 10
- On a free support, compressible flow produces a larger fuselage-motion amplitude than strip theory, including 0.0211 versus 0.0176 at MR = 1 with the transducer.The passage characterizes the difference as between a fifth and a quarter across the four measured pairs with support motion.Source: §5 · p. 10
6 The compressible section with and without a free support
Adding a free support to the compressible section changes the flutter boundary and increases motion and harvested power. The combined model also exposes effects that either single-change model cannot show.
6 The compressible section with and without a free support
- The pitch-mounted transducer’s boundary shift changes from +1.951% on a rigid support to −0.533% on a free support, while harvested power rises by 35.1%.With the fourth degree of freedom, the flutter frequency rises, the cycle grows in every coordinate, and its period shortens.Source: Table 6 · p. 11, §6 · p. 11
- Table 6 compares the compressible section on a rigid support with the same section on a free support, isolating the fourth degree of freedom.The aerodynamics are held fixed in this comparison.Source: Table 6 · p. 11, §6 · p. 11
- Strip theory under-predicts harvested power for compressible flow by 35.1% on a rigid support and 34.9% on a free support.The two gains agree within two tenths of a percentage point; the plunge mounting retains its advantage across models and supports.Source: §6 · p. 11
- Only the model combining compressible flow and a free support shows a sign reversal at the anchor resistance and a measurable super-compounding boundary shift.Neither the compressible rigid-support section nor the incompressible free-support section alone reveals these combined effects.Source: §6 · p. 11
7 Discussion
Whether the separate modelling corrections compose depends on the measured quantity and, for the flutter boundary, on transducer regime. The discussion explains the contrasting outcomes, describes a low-cost solver extension, and identifies limits on the study’s scope.
7 Discussion
- The power corrections multiply to within 0.3 percentage points, but multiplying the boundary corrections predicts a small positive shift where the combined result is negative.The latter discrepancy is confined to a load-resistance regime; above r = 1.89, the corrections compose again.Source: §7 · pp. 11–12
- When the electrical pole is above the flutter frequency, the corrections multiply to within seven parts in a thousand on both mountings.The failure to compose belongs to the damping regime, which the companion papers used.Source: §7 · p. 12
- Power gains compound because both changes increase cycle amplitude, whereas boundary shifts reflect phase-dependent damping affected by changes to the flutter mode and its frequency.Above r = 1.89, where the transducer stops acting as a damper, the corrections compose again.Source: §7 · p. 12
- A free-support flow-solver extension needs no new mesh motion because support translation displaces the wall as plunge does.The implementation uses two motion maps at the provider interface rather than a new solver.Source: §7 · p. 12
- The study fixes Mach number at M = 0.10, compares power at one operating point, and models the fuselage as a point mass translating in plunge alone.Mach dependence and shifts of the coupling crossing with load resistance or support mass remain unresolved.Source: §7 · p. 12
8 Conclusions
The combined model changes require no new flow-mesh motion, and their effects on harvested power and the flutter boundary differ: power corrections multiply, while boundary corrections can more than compound. Boundary composability depends on the transducer regime, with the combined effects also changing stability growth above flutter.
8 Conclusions
- The free-support flow model retains three solver coordinates because support translation displaces the wall identically to plunge.A motion map and force map carry the fourth coordinate at the interface, leaving mesh motion, residual, and loads unchanged.Source: §8 · p. 12
- In the heavy-fuselage limit, the implementation reproduces the companion compressible paper’s flutter boundary, pitch-mounted shift, and measured full-limit-cycle quantities.This checks the coupled model against the rigid-support case.Source: §8 · p. 12
- At both tested couplings, compressible flow and a free support together reduce the pitch-mounted transducer’s stabilising boundary shift by more than the product of their separate effects.At coupling 0.50, the shift changes sign at the anchor load resistance; at 0.20, it falls to a tenth of its rigid-support strip-theory value but remains positive.Source: §8 · pp. 12–13
- The free support widens the resistance band where the aerodynamic models disagree in sign, moving its upper edge from between r = 0.3 and r = 1 to between r = 1 and r = 3.The widened band includes the harvesting resistance; the plunge mounting does not reverse sign in the tested sweep.Source: §8 · p. 13
- Above the resistance where the electrical pole meets flutter frequency, multiplying the published corrections predicts boundary shifts within seven parts in a thousand on both mountings.Below that resistance, where the transducer acts as a phase-dependent damper, the product errs by a factor of 1.5 to 14 and predicts the wrong sign for the pitch mounting at the harvesting resistance.Source: §8 · p. 13
- The transducer lowers the flutter boundary while flattening the instability growth-rate curve above it.Its effect is destabilising for cycle onset but stabilising for growth rate.Source: §8 · p. 13
- 82.5% harvested power compares with the 82.8% predicted by multiplying the two separate corrections.The corrections compose for power, unlike the flutter-boundary shifts.Source: §8 · p. 13
Declaration of generative AI use
Claude (Anthropic) assisted with drafting, editing, proofreading, and checking analysis scripts and consistency with computed data; the authors reviewed the content and take responsibility.
Declaration of generative AI use
- Claude (Anthropic) assisted with text editing and analysis-script checks, while the authors reviewed the content and take responsibility.The assistance included proofreading and checking consistency with computed data.Source: Declaration of generative AI use · p. 13