A general framework for the design of efficient passive pitch systems

Unsteady load management is crucial across many engineering disciplines, including the operation of small unmanned aerial vehicles (UAVs), which are significantly affected by environmental factors,^1–5 and in the mitigation of vibration and noise in aircraft and rotorcraft,^6–11 as well as in enhancing the efficiency and durability of wind and hydrokinetic turbines.^12–14 In comparison with larger aircraft, small UAVs face unique challenges in terms of control, flight stability, efficiency, and resilience, due to their lower inertia and thus increased vulnerability to disturbances such as gusts and atmospheric turbulence.^15–20 The use of UAVs in urban areas and under extreme weather conditions underscores the necessity for developing efficient systems to handle unsteady load fluctuations.

Actively controlled lift surfaces, such as trailing edge flaps, are standard,²¹ while passive load alleviation mechanisms are less common, although they are gaining attention due to their inherent reliability and zero energy demand.^22–25 The natural world offers exemplary models of passive musculoskeletal adjustments, as seen in birds and insects, whose flight stability in turbulent conditions far surpasses that of human-made flyers. These creatures employ a mix of active and passive mechanisms to maintain position and velocity and to counteract the effects of wind gusts and lulls.^26–31 The kestrel's (Falco tinnunculus) ability to stabilize its gaze relative to the earth while hovering has intrigued both scientists and bird enthusiasts alike.^32–35 Distinguishing between active and passive responses to unsteady loads remains challenging, yet recent findings indicate a widespread reliance on passive adaptations for this purpose.^36–40 The degree to which engineering systems can emulate these biological responses to mitigate load fluctuations, given the complexity of factors involved, including diverse wing shapes and gust profiles, is an ongoing area of investigation.

By analyzing a simplified two-dimensional foil model, free to adjust passively to changes in flow velocity and direction, we can isolate the essential physics of passive load mitigation. This analysis sheds light on the limits and potential of passive mechanisms for load alleviation, guides the interpretation of biological adaptations in nature, and informs the design of more effective and reliable engineering solutions for unsteady load challenges. This paper deliberately abstains from delving into specific application details, focusing instead on findings with broad implications across various design scenarios.

The manuscript is structured to begin with a formal definition of the passive pitch system (Sec. II) and a quasi-steady analysis of the initial and final state of the system for an arbitrary change in the free-stream speed and/or direction (Sec. III). It then progresses to the description of the dynamic low-order model (Sec. IV) and its application to streamwise gusts to investigate the optimal kinematics and pivot point location (Sec. V). It concludes by summarizing key outcomes and drawing conclusions (Sec. VI).

Consider a two-dimensional rigid foil in an incompressible flow that rotates in response to a change in the magnitude and direction of the free-stream velocity (Fig. 1). The initial flow velocity is uniform and constant and exerts a constant fluid force on the foil, which is a function of the foil angle of attack. Then, the velocity varies over a transitory period to a different uniform velocity, which is kept constant until the foil position and fluid force converge to a new constant value.

For a constant free-stream velocity, the foil is held in position by an externally applied torque around the center of rotation, point P on Fig. 1, hereafter the pitching axis. The foil response is entirely passive: the torque is not adjusted based on the state of the system. In a biological system, the torque represents the musculoskeletal tension exerted on a joint. In an engineering system, the torque could be provided by a torsional spring. For example, Gambuzza et al.⁴¹ considered a tidal turbine blade free to pitch around its spanwise axis, with a torsional spring around the axis. The large preload of the spring resulted in approximately constant torque, and the system allowed mitigation of the thrust and power fluctuations experienced by the turbine.

A bird wing is likely to both translate upward and backward and rotate around, for instance, its spanwise axis in response to a gust. At every instant, the combination of a rotation and a translation can be defined as a single rotation around a center of rotation. Therefore, while we consider some cases with centers of rotation that are far from the wing, a real mechanism could be made more compact by combining a rotation about a closer pitching axis with an additional translation to generate the same overall motion.

In this paper, we limit our study to fixed centers of rotation over the entire dynamic response period. Hence, for example, the foil cannot first translate and then rotate. Furthermore, we note that any in-plane translation of a rigid body is equivalent to a pure rotation around a center infinitely far from the body. For example, a horizontal translation is equivalent to a pure rotation around a center at an infinite distance above or below the foil. Similarly, a vertical translation of the foil is equivalent to a pure rotation around a center at an infinite distance left or right of the foil. However, a pure translation is not a physical solution because a rotation is necessary for the foil to change its angle of attack and reach a final equilibrium position.

Consider a foil subject to an instantaneous freestream velocity having absolute value $u ̂$ at an angle α with the foil chord (Fig. 1). The foil is hinged around a pivot P at position $x p = ( x p , y p )$ so that the only motion that is allowed is a rigid rotation around P.

All dimensional quantities are overlined with a caret denoted as $· ̂$ and are made non-dimensional by using the fluid density $ρ ̂$⁠, the foil chord $c ̂$⁠, and the freestream speed at the initial state $u ̂ 0$⁠, as characteristic density, length, and velocity, respectively.

The foil chord forms an angle β with an arbitrary direction in the plane (Fig. 1), which is positive if a counterclockwise rotation is necessary to move from the arbitrary direction to the chord. We define, for this foil, a frame of reference such that x is parallel to the chord of the foil directed from the leading edge to the trailing edge, y is perpendicular to x and directed from the pressure side to the suction side, and the origin is located on the quarter-chord.

In this frame of reference, the pivot P has coordinates $x P , y P$⁠. The flow generates a system of forces on the foil that can be decomposed into two in-plane forces and one moment normal to the plane, which are considered here to be applied at the quarter-chord of the foil.

Let us denote the initial equilibrium condition with the subscript 0, so that the initial free-stream speed is $u ̂ 0$⁠, the initial angle of attack is α₀, and the initial angular position of the foil is β₀. Let us furthermore assume that the freestream velocity changes in magnitude and direction, so that the final absolute value is $u ̂ 1$ and the angle with respect to the original location of the chord is $α 0 + α ′$ (Fig. 1). The foil pitches around the pivot P to accommodate for the changes in inflow; as this is a solid rotation around P, the coordinates of P do not change in the foil frame of reference. At the new steady state, the new angle of attack is α₁, and the new forces and moments are $F ̂ x ( α 1 ) , F ̂ y ( α 1 )$⁠, and $M ̂ ( α 1 )$⁠, respectively, while the opposing torque is $T ̂ ( β 1 )$⁠.

With prior knowledge of α₀, and if the position of the pivot $( x P , y P )$ is known, then this equation can be solved to obtain the final angle of attack α₁. This equation is, however, non-linear in α₁ as the forces depend on C_L, C_D, and C_M, which are themselves not necessarily linear with α, and thus, the solution of Eq. (13) requires numerical root-finding.

These results demonstrate that, for any arbitrary change in the onset flow speed and direction, pivots placed along the same line defined by Eq. (14) result in the same final angular position β₁ of the foil.

By the inspection of the coefficients of lift, drag, and moment of a generic foil, it is trivial to show that, in general, there is no rotation or translation of the foil that would allow the same resultant force for both the initial and final flow velocity (see the  Appendix for details). As the free-stream velocity changes, the foil passively rotates up to a position where the final moment equals the initial moment, but the initial and final resultant forces differ. We conclude that for a generic flow velocity fluctuation, a complete unsteady load mitigation is impossible by a passive rotation or translation.

However, we observe that for most natural flyers and engineering technologies, there is a direction along which it is more important to keep the force constant. For example, for a bird gliding near and parallel to the sea surface, maintaining a constant altitude and, therefore, upward force is critical, while accelerations in the direction parallel to the sea surface are not such a concern. We thus wonder whether complete mitigation of the load fluctuations in a selected direction is possible by a passive rotation or translation.

As described in the discussion following Eq. (15), Eq. (14) describes a line where both α₁ and β₁ are constant. As α₁ is constant, so are $C x ( α 1 )$ and $C y ( α 1 )$ as these only depend on α, and so are $F x 1$ and $F y 1$ following Eq. (7), and following that, the ratios $F x 1 / F x 0$ and $F y 1 / F y 0$ are also constant. This implies that for any generic direction n with respect to an earth-fixed frame, the ratio between the final and the initial force component $F n 1 / F n 0$ is also constant along the line. For example, consider a bird flying near the sea surface experiencing a streamwise gust. If the wings can passively pitch around a point P, the relative change of the force in the vertical direction depends only on the line on which P lies and not on the position of P along the line [Eq. (14)]. Furthermore, there is a line for which the latter relative change is zero, and thus, the force along the vertical direction remains constant.

As an example, Fig. 2 shows the lines of Eq. (14) for a set of forces computed with the coupled Euler-boundary layer solver Xfoil⁴² for a NACA 0012 foil. The initial Reynolds number is $R e 0 ≡ u 0 c / ν = 10 3$⁠, where c is the chord length and ν is the kinematic viscosity of the fluid; the initial angle of attack is $α 0 = 5 °$⁠; the torque T is constant; and the gust ratio is $u 1 / u 0 = 2$⁠. In this example, n is the lift direction. Here, we assume $u 0$ to be the mean flow velocity, and $u 1$ the perturbation, and thus, we define lift the force in the direction orthogonal to $u 0$ also after the velocity has changed from $u 0$ to $u 1$⁠. The colors show the ratio between the final and initial force along n, i.e., the lift ratio $L 1 / L 0$⁠. Therefore, by passively pitching around an axis that intersects the line $L 1 / L 0 = 1$ in Fig. 2, the lift remains constant despite the flow speed doubling.

These results reveal that the quasi-steady variations of an arbitrary chosen force component due to a gust ratio can be entirely canceled by passively pitching the foil around a specific pitching axis. Note that an optimal center of rotation at $x P < 0$ and $y P ≈ 0$ as suggested in Fig. 2, is equivalent to an upward and backward translation, and a nose-down rotation around the center of mass, which is consistent with the expected gust response of natural flyers.^27,30

We now consider a free-stream velocity that varies in direction but not in magnitude. As we noted before, for a given flow speed, there is only one angle of incidence at which the fluid torque is in equilibrium with the externally applied constant torque T. Therefore, for any change $α ′$ of the angle of attack, the foil will rotate exactly by an angle $α ′$⁠, resulting in the same initial and final resultant force vector with respect to a frame fixed with the foil. The projection of the resultant force vector F along any earth-fixed direction n would change by $cos α ′$⁠.

This reveals that, for a generic variation of the angle of attack, the force on the foil in a specific direction cannot be kept constant by a passive rotation or translation. However, for moderate angle of attack variations (⁠ $α ′$⁠), the force in any generic direction n changes only by a small amount (1- $cos α ′$⁠), which depends only on the amplitude of the angle of attack variation and not on, for example, the pitching axis location. For example, variations of $10 °$ and $20 °$ result in F_n changes of 1.5% and 6%, respectively.

Overall, these results reveal that (1) the fluctuations in an arbitrary chosen force component due to due to quasi-steady variations in flow speed can be entirely canceled by passive pitch; (2) the optimum pitching axis is at any arbitrary position along a line that depends only on the geometry of the foil [via its $C l ( α ) , C d ( α )$⁠, and $C m ( α )$ curves], the initial and final Reynolds number; and (3) force fluctuations due to quasi-steady changes in the flow direction cannot be canceled, but substantial mitigation will be provided by pitching about any pitching axis.

Given the previous findings, one can determine where to place the pitching axis with respect to the foil so as to keep one component of the force generated by the foil constant before and after a given gust.

To avoid overloading the notation, we assume that the arbitrary direction, along which we desire to keep the forces constant, is the same as that which has been used to define the angles β in Fig. 1; as this direction is arbitrary, this choice does not affect the generality of our conclusions.

Once α₁ is found, this can be used in Eq. (14) and following to determine where the pivot should lie.

As all solutions of this system require that α₁ be constant, the set of values of $( x P , y P )$ for which this is satisfied define an iso-line of α₁ in the $x P − y P$ space and therefore represent a straight line.

While in Fig. 2 we showed an example where T is constant, T can be any function of β (Sec. III). It is interesting to note that any pivot position results in stable equilibrium if $d T / d β$ is sufficiently small. Finally, it is noted that the modeled foil has only one degree of freedom in pitch, and thus, flutter is inhibited. Real wings, however, have multiple degrees of freedom and flutter instabilities might arise.

To investigate the transient from the initial to the final equilibrium position, we employ a low-order model based on Kirchhoff's equations.⁴³ We consider the quasi-steady circulatory forces and added mass forces, but neglect the unsteady circulatory forces, which depend on the history of the flow field. This methodology has been validated against several experiments of free-falling bodies at $R e = 10 3$ to 10⁴.^44–47 

Figure 4 shows the vectors corresponding to the forces and moments considered in Eq. (25). The fluid force is modeled as the sum of the quasi-steady circulatory force $F QS$ applied at the quarter-chord point Q with position vector, $x Q$ from the center of mass O, and the added-mass force $F AM$ acting at the mid-chord M with position vector $x M$⁠. The quasi-steady fluid moment is $M QS$⁠, and the externally applied torque is T. The foil's angular velocity is $ω$⁠. Added mass forces are computed as follows.

Apart from the added-mass force, the remainder of the lift, drag, and moment during the gust encounter are modeled in a quasi-steady manner based on tabulated CFD data. These data are generated by solving the two-dimensional Navier–Stokes equations for incompressible flows and Newtonian fluids with OpenFOAM v2106. The equations are solved in an inertial frame where the X and Y coordinates are parallel and orthogonal to $u 0$⁠, respectively. The computational domain consists of a rectangle 34c long (in the X direction) and 20c wide (in the Y direction), discretized by a structured grid (Fig. 5) generated using ICEM-CFD. Uniform velocity and pressure boundary conditions are Dirichlet conditions at the domain's upstream and downstream boundaries, respectively, while a slip condition is applied at the side boundaries.

Simulations are performed for $R e = 10 3$ and $R e = 2 × 10 3$⁠, for a range of angles of attack from 0° to 12°. The computed $C L S , C D S$⁠, and $C M S$ are shown in Fig. 6. Here, we set the Reynolds number of $O ( 10 3 )$ first because we are interested in the gust response of natural flyers (insects and small birds), nano air vehicles, and micro flyers. Second, Re = 1000 is sufficiently high to ensure the vorticity diffusion is negligible compared to advection, and sufficiently low to suppress lift fluctuations for a constant onset flow and angle of attack (e.g., due to wake instabilities and vortex shedding). For a higher Reynolds number, say $R e > O ( 10 4 )$⁠, one can use XFOIL,⁴² which provides sufficiently accurate prediction of forces.

It is important to note that this change in the flow speed occurs over a short time frame compared to the advection timescale of the flow, and the kinematics of the pitching foil would have changed marginally if the speed had increased over an even shorter time frame. In fact, as the gust period t_G over which the velocity changes from u₀ to 2u₀ vanishes, the added mass force tends to infinity, while its duration tends to zero, and thus, the resultant added mass impulse would plateau to a finite value. The effect of the added mass is to generate vorticity (with net zero circulation) on the surface of the foil that is proportional to the added mass impulse. Hence, as $t G → 0$⁠, the amount of added mass vorticity generated on the foil tends to a finite value. Therefore, the vorticity field at the start of the kinematics would not change significantly if $t G ≪ c / u 0$⁠.

The low order model described in Sec. IV is used to investigate the effect of the pitching axis location on the unsteady load mitigation, and specifically on the maximum amplitude of the lift fluctuation during the transient phase. Consider a change in the onset flow speed as detailed in Eq. (32). The initial condition is $α 0 = 5 °$⁠, and the Reynolds number increases from $R e 0 = 10 3$ to $R e 1 = 2 × 10 3$ in one convective time. The initial angle of attack $α 0 = 5 °$ is chosen because it is an intermediate angle between the zero-lift condition for a symmetric foil and the stall angle, which is typically of the order of 10°. The time history of the lift L(t) non-dimensionalized by the initial lift L₀ is shown in Fig. 8, for an example case with a pitching location at $x P = ( − 1.5 , 0.25 )$⁠. The first sharp peak at $1 < t u 0 / c < 2$ is due to added mass and, for the fixed foil, is proportional to the onset flow acceleration [i.e., the time derivative of Eq. (32)]. This first peak is substantially mitigated by the pitching foil, as it passively pitches in response to the load change. However, the pitching foil experiences a damped oscillation before reaching its final equilibrium position. Therefore, unlike for the fixed foil, the minimum lift is not necessarily the initial lift L₀: for some pitching locations, there is a lower minimum reached during the oscillation. In the example shown in Fig. 8, the minimum lift is achieved after the end of the gust at $t ≈ 2 c / u 0 .$

The map of $ε DY$ (Fig. 9) suggests that the maximum unsteady load alleviation is achieved for a pitching axis located along the extended chord line (indicated with a white dotted line in Fig. 9). This is consistent with the results shown for a quasi-steady analysis in Fig. 2. Furthermore, the sensitivity of $ε DY$ to the exact location decreases as the pitching axis approaches the extended chord line. Hence, these results suggest that the optimum location of the pitching axis is along the extended chord line. While the quasi-steady analysis showed that the unsteady load alleviation depends on the line on which the pitching axis lies and not on its distance from the foil (see Fig. 2), the dynamic analysis, in contrast, suggests that the position along the extended chord line is important. We will discuss the optimal position along the extended chord line in Sec. V B.

To further understand the underlying physics resulting in the optimum pitching axis, we study the effect of pitching axis on the maximum and minimum lift peaks separately. The maximum lift $C L max$ is only marginally affected by pitching axis location [Fig. 10(a)], and the maximum alleviation of $C L max$ is achieved with a pitching axis at approximately $X P = ( − 1 , 0.5 )$⁠. On the other hand, the minimum lift $C L min$⁠, which is due to the oscillations of the foil as it approaches the new equilibrium position, is highly dependent on the pitching axis location [Fig. 10(a)], and thus, the optimum pitching position is governed by $C L min$ [cf. Figs. 9(a) and 10(b)]. Specifically, the maximum alleviation of the negative lift peak is achieved for pitching axes along the extended chord line of the foil. Moreover, similarly to $ε DY , C L min$ decreases with increasing X/c.

We further dissect the contribution of the added mass and quasi-steady lifts to the maximum and minimum lift. While the position of the pitching axis minimizing the positive lift peak $C L max$ is approximately at $X P = ( − 1 , 0.5 )$⁠, the pitching axis would need to be further away from the foil to minimize the sole added mass lift peak, and closer to the foil to minimize the quasi-steady lift peak [Figs. 11(a) and 11(b)]. This result reveals that the optimal distance of the pitching axis to mitigate the maximum lift is a compromise between added mass and quasi-steady force, even if the maximum lift plays a minor part in the overall lift fluctuation, which, instead, is governed by the minimum lift. Conversely, placing the pitching axis along the extended chord line contributes to increasing both the minimum lift due to added mass and quasi-steady force [Figs. 11(c) and 11(d)].

To understand what determines the optimal distance of the pitching axis and the balance between added mass and quasi-steady forces, we consider the pitching axis on the extended chord line (i.e., on the white dotted line in Fig. 9), and we plot the trend of $ε DY$ vs the X coordinates of the pitching axis (Fig. 12). We observe that $ε DY$ is minimum at $X / c = − 0.83$⁠, which is also where the maximum lift has the lowest value [Fig. 12(b)]. Around $X p / c = − 0.73 , C L min$ increases with decreasing distance from the foil, and at $X p / c = − 0.73$⁠, it has the same value as the initial lift value, $C L 0 = 0.248$ [Fig. 12(c)].

We will show now that this optimum pitching axis coordinate, $X p / c = − 0.73$⁠, depends on the inertia of the system. For an increased distance of the pitching axis from the foil, the inertia increases, and the overshoot generating the minimum lift decreases [Fig. 12(c)]. Therefore, $ε DY$ initially decreases with the distance of the pitching axis from the foil. However, for a further increased inertia, the foil is slow to accelerate at the beginning of the gust, and thus, the maximum lift increases [Fig. 12(b)]. Therefore, there is an optimal system inertia such that the minimum lift peak due to the overshoot is not lower than the initial lift. Specifically, the higher the inertia, the closer the optimal pitching axis will be to the wing.

To demonstrate this, we consider the time history of lift C_L for three selected cases [identified as case 1, 2, and 3 in Fig. 12(a)]. Case 1 has the pitching axis at $X p / c = − 0.5$⁠, case 2 at $X p / c = − 0.73$ (the optimal pitching axis location), and case 3 at $X p / c = − 722$⁠. In case 1 [Fig. 13(a)], the minimum lift is the initial lift, $C L min = C L 0$ and the lift after the peak is always higher than $C L 0$⁠. In case 2 [Fig. 13(b)], which is the maximum lift alleviation case, the lift after the peak becomes as low as the initial lift at $t u 0 / c ≈ 2$⁠. Finally, in case 3 [Fig. 13(c)], the lift after the peak is smaller than the initial lift $C L 0$⁠. These results explain why the minimum $ε DY$ is minimum for the same $X p$ for which $C L , max$ is minimum, even if before (Fig. 10) we concluded that the optimum pitching axis location is governed by $C L , min$ and not $C L , max$⁠.

The kinematics of the three cases (Fig. 14) reveal the underlying physics for the time series of the lift observed in Fig. 13. The angle of attack decreases in response to the change in the free-stream speed to reach the new equilibrium position [Fig. 13(a)]. The higher the inertia, the lower the magnitude of the rate of change of the angle of attack [Fig. 13(b)]. Consistently, the acceleration of the angle of attack shows a negative peak corresponding to the foil angular velocity becoming negative, and a positive peak corresponding to the vanishing angular velocity as the foil approaches the new equilibrium position.

Noting that $b ≪ c$⁠, the dominant added mass force component is proportional to the acceleration $u ̇ y$ [Eq. (29)], which is correlated with the acceleration of the angle of attack through $r mc α ̈ = u ̇ x 2 + u ̇ y 2$⁠, where $r mc$ is the distance between the pivot and the center of mass (|PO|). For a pitching axis along the extended chord line and small initial angle of attack, the rotation along the pivot corresponds to a chord normal velocity, i.e., $r mc α ̈ ≈ u ̇ y 2$⁠. The velocity $r mc α ̇$ and the acceleration $r mc α ̈$ are shown in nondimensional form in Fig. 15. As one would expect, the higher the inertia (from cases 1 to 3), the later the maximum velocity is reached [Fig. 15(a)] and the slower the velocity decreases after the peak. Therefore, while the positive acceleration peak increases with inertia (case 3), the magnitude of the deceleration peak decreases with inertia. These results reveal why the maximum added mass lift peak is better mitigated by pitching axes far from the foil [Fig. 11(a)], while the minimum added mass lift peak is better mitigated by pitching axes closer to the foil [Fig. 11(c)].

The quasi-steady force, instead, depends on the effective angle of attack $α eff$ [Fig. 16(a)], which is the angle between the chord and the relative velocity experienced by the foil $u ∞ + u M$⁠. The time histories of $α eff$ are similar for the three cases [Fig. 16(a)]. However, when these are scaled with the square of the instantaneous free-stream velocity [Fig. 16(b)], the amplitude of both the maximum and the minimum peak increases with increasing inertia. This reveals why both the maximum [Fig. 11(b)] and minimum [Fig. 11(d)] quasi-steady lift are better mitigated by pitching closer to the foil.

In this paper, a general approach for the design of efficient passive pitch systems has been developed. Their capabilities and limitations in mitigating unsteady loads have been investigated. We considered a foil in a uniform free stream, which is free to rotate around a pitching axis and is held in position by an externally applied torque. The system is passive because the applied torque is either constant or varies linearly with the angular position of the foil. Such a system can be achieved, for example, by a linear torsional spring. We developed a mathematical framework to compute how a generic force component varies when the onset flow velocity changes, in a quasi-steady manner, to a new uniform speed and direction. We showed that any force component can be kept constant if the locus of the pivot is a line that depends on the initial and final flow velocity and the aerodynamic polars (lift, drag, and moment coefficients) of the foil.

Furthermore, we developed a dynamic model that includes analytical formulations of the added mass forces and tabulated quasi-steady aerodynamic polars. The model enables estimation of the dynamic response of the foil in the transient between the initial and final equilibrium state. We applied the model for the lift generated by a foil at a small angle of incidence (⁠ $5 °$⁠), experiencing a rapid twofold increase in freestream velocity. More than 2/3 of the lift fluctuation can be canceled by placing the pivot point along the extended chord line at a distance from the foil that increases with decreasing inertia. The foil first experiences a lift increase and thus pitches reducing the angle of incidence, and then, it oscillates until it reaches the final equilibrium position. The optimum pivot position is such that the minimum lift during the transient is equal to the initial lift. While the optimal positioning of the pitching axis may vary with gust parameters, the efficacy of load mitigation shows minimal sensitivity to the precise location within the vicinity of the optimal locus.

The present study focuses on the passive pitch system at relatively low Reynolds numbers, $R e = O ( 10 3 )$⁠, but yet the pressure forces dominate the viscous forces. As long as this remains true, such as at higher Reynolds numbers, the present results are expected to stand. For situations where the viscous forces dominate, such as very low Reynolds numbers or non-Newtonian flows, the results should be reviewed.

In the present work, the pivot point is fixed during the transient. Therefore, this work may potentially underestimate the efficacy of advanced systems such as adjustable musculoskeletal tension, dynamic pitching axes or joints, and temporary modifications in foil positioning through dihedral angle adjustments, among others. Additionally, the passive mechanism examined here is inadequate for handling stall conditions, which necessitate active stabilization methods.

Overall, this work provides a new methodology for studying different gust types and designing optimal passive pitch systems. It also provides new insights into the underlying physics of gust mitigation through passive pitching. As such, it may also enrich our understanding of complex biological passive systems observed in nature.

This work was supported by the UK Engineering and Physical Sciences Research Council through the grant “Morphing-Blades: New-Concept Turbine Blades for Unsteady Load Mitigation” (No. EP/V009443/1).

The authors have no conflicts to disclose.

Shūji Ōtomo: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead). Stefano Gambuzza: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead). Yabin Liu: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Anna M. Young: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Riccardo Broglia: Funding acquisition (supporting); Supervision (equal); Writing – review & editing (equal). Edward D. McCarthy: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Ignazio Maria Viola: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Methodology (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Figure 17 reports the polars for four different foils at four different values of chord-based Reynolds number Re, spanning from 10³ to 10⁶. These are the NACA 0012 presented in this work; a flat plate operating at $R e = 1.1 × 10 4$ representative of a dragonfly wing;⁴⁹ a NACA 2414 operating at $R e = 10 5$ representative of a small horizontal-axis wind turbine;⁵⁰ and a NACA 63-618 for use in larger-scale wind turbines.⁵¹ The NACA 0012 data are computed with direct numerical simulations, while all other polars come from experimental measurements in wind tunnels. It can be seen that the polar curves do not self-intersect in any case, demonstrating that, in general, there is at most one value of α for each pair of C_L and C_D.