Equilibrium Behavior of a Tethered Autogyro: Application in Extended Flight and Power Generation

An unpowered rotorcraft producing steady lift from self-rotating blades is in a state of autorotation. Harnessing significant prevailing winds found consistently at high altitudes [1–5], and it is possible for an autorotating rotorcraft, or autogyro, to generate power. This ability can be exploited for the purpose of extended, efficient flight of rotorcraft, and can provide a platform for accessing the largely untapped energy source of high-altitude jet streams. This document presents a numerical model for describing a tethered, power-generating autogyro and analyzes the feasible flight region to verify practicality for both a larger, high-altitude configuration focusing on power generation, and a smaller, lower altitude configuration focused on energy efficient flight for surveillance applications.

Many efforts have been made to explore the power generation capabilities of airborne wind energy (AWE) systems in the form of tethered airfoils/kites, [6–14], as well as lighter-than-air (LTA) tethered wind turbines [15–17]. Airfoil/kite AWEs largely rely on mechanical stimulation of the tether through crosswind maneuvers. Thus, such a system becomes impractical at higher altitudes where wind energy density is prominent, as increased uncertainty in tether dynamics can decrease the efficiency of mechanical transmission to the base. LTA windmills may be effective at higher altitudes, but are quite lacking in maneuverability, as suggested by Ref. [18], which can be essential for effective surveillance. The use of tethered autorotation combines the benefits of on-board power generation seen in LTAs with the heightened maneuverability of smaller, and more aerodynamic designs are seen in airfoil/kite AWEs.

A review of the timeline and challenges of autogyro development is presented in Refs. [19,20]. Following the creation of autogyros for piloted flight in the early 1920s, Glauert [21] developed a compact initial model for describing the steady-state behavior of an autogyro utilizing the blade element momentum approach [22]. Many works followed to relax the assumptions of Ref. [21] and expand upon their model [23–27]; perhaps most notable of which is the work of Wheatley [25], who not only presented a more detailed model but also provided experimental validation through testing on a commercially available autogyro of the time. This analysis expanded upon Ref. [21] to include a linear variation in pitch. Further work in Ref. [28] expanded the analysis to explore the effects of periodic blade twist. Many works have focused on autorotation in the context of the descent of a helicopter [29]. Analysis of autogyro performance analogous to the flight modes of helicopters and fixed-wing vehicles is also present in the literature [30], with some works testing and modeling the performance of hybrid rotor/fixed-wing crafts in the conditions of autorotation [31–33]. In Ref. [34,35], a simple model was tuned through small-scale experimentation to explore instability in autorotation. In this article, we take a different approach considering the context of a stationary tethered rotorcraft autorotating in a prevailing wind field. It is unpowered when autorotating, and if excess wind power is available at its operating altitude, it can transfer the power to ground via the tether.

Improving upon Ref. [25], initial efforts have been made in Refs. [36–39] to model and explore the effects of regenerative braking on the steady-state behavior of an autogyro, as well as the practicality of tethered autorotation as a source of high-altitude power generation. The following document provides an updated and refined numerical method for solving aerodynamic equilibrium behavior. An additional numerical method is presented to couple the response of a static tether to the aerodynamic forces provided by an autogyro in a realistic wind profile. The coupled numerical process has been thoroughly simulated to visualize the steady-state behavior of a tethered autogyro over a dense range of inputs, not only for the case of high-altitude power generation but also lower altitude surveillance. The contributions of this work are as follows:

1. The aerodynamics of autorotation is generalized by incorporating regenerative braking, lower rotor speeds, and wind velocity and atmospheric density variations with altitude.

2. Numerical methods are proposed to solve for the equilibrium conditions.

3. The generalizations yield a larger space of equilibrium conditions of the system.

4. Tether mechanics are incorporated and equilibria are determined for the tether-autogyro assembly.

5. The possibility of power generation is quantitatively explored through this model-based approach.

The rest of this article is organized as follows: In Sec. 2, we present the aerodynamic model that incorporates regenerative braking and lower rotor speeds. A numerical approach to solve for equilibrium conditions and model validation results is provided. Section 3 presents the model of the catenary. In Sec. 4, we discuss the numerical approach to solve for the equilibria of the combined tether-autogyro system in the presence of wind speeds and air density, which vary with altitude. Simulation results are provided in Sec. 5. This is followed by concluding remarks, acknowledgments, and references.

The model presented here expands upon the work of Wheatley, [25], to explore the equilibrium aerodynamic behavior of an autogyro under regenerative braking and low rotor speeds. With this as the focus, we skip the details of model development, for which the reader is suggested to read [25,36]. Here, we concisely present the governing equations for steady autorotation. Consider a tethered autogyro under equilibrium conditions, as shown in Fig. 1(a). In describing the aerodynamics, this model employs the use of two dimensionless ratios shown in Eq. (1).

In this section, we present an algorithm that can be used to determine the various autorotational equilibria. The proposed algorithm considers three variables, namely μ, Q_e, and V, to form the input space of the autorotational equilibria. A total of ten unknown variables are determined from the set of nonlinear equations composed of Eqs. (1)–(12) and Eq. (14) (with the adjustments of Eqs. (15) and (16)). Due to the coupling of equations introduced by the inclusion of the final term in Eq. (3) and the addition of regenerative braking in Eq. (10), an iterative solving method is necessary to converge on an equilibrium state. A suggested numerical process is presented below:

1. Provide a selected μ, Q_e, and V.

2. Initialize λ and α with reasonable guesses.

3. Calculate Ω with the tip speed relation of Eq. (1).

4. Calculate a₀, a₁, b₁, a₂, and b₂ with Eqs. (3)–(7).

5. Calculate λ with Eq. (10).

6. Calculate C_T with Eq. (11).

7. Calculate λ₁ with Eq. (14).

8. Adjust b₁, a₂, and b₂ with Eq. (15).

9. Recalculate λ by appending Eq. (16) to Eq. (10).

10. Calculate α with Eq. (12).

11. Adjust guesses for λ and α in direction of calculated values.

12. Repeat steps 3–11 until the absolute change of λ and α is below a desired threshold suggesting convergence (chosen as 10⁻⁵ in this study).

Figure 2 displays simulated profiles for the flapping Fourier coefficients of Eq. (2). The trends denoted by x’s represent the presented model with the blade weight moment term of Eq. (3) neglected. The profiles resulting from this assumption are independent of wind velocity, V. With the blade weight moment term of Eq. (3) included in the analysis, consideration of V becomes important. This is apparent by observing the solid contours representing constant V across the span of μ. The dash-dot profile is the simulated results with the blade weight moment term of Eq. (3) incorporated while selecting input V values analogous to that of the experimental data, as outlined in the previous paragraph. The experimental data collected by Wheatley [25] is visualized with circular points. As the assumption of a nonnegligible effect of blade weight moment directly pertains to Eq. (3), it is reasonable that a₀ would be the most significantly affected coefficient. The other four coefficients are negligibly affected by the inclusion of the blade weight moment term of Eq. (3), especially in the region μ < 0.5. It was suggested by Glauert [21] and substantiated by experiments in Ref. [25] that μ < 0.5 is the range of validity for blade element momentum calculations. This is based on the condition that the relative wind velocity be positive over the outer half of the retreating blade. Thus, it is expected that error will occur beyond this point. The adjusted magnitude of a₀ directly relates to the average of the flapping angle, β. It can be observed that for high wind velocities, V, the assumption of a negligible effect of blade weight moment becomes more and more accurate. It follows from Eq. (1) that, for a given μ and uniform α, a larger V will result in higher angular velocity Ω. Therefore, by providing sufficiently large V and sustaining a nonzero value of cosα, the final term of Eq. (3) will be dominated by its denominator and thus be rendered negligible.

In Figs. 2–4, note the trends labeled “Prescribed Wing Loading.” These trends are obtained under the specific flight conditions used to report results in the study by Wheatley [25]. We include these specific trends to put our results in context. The wing loading refers to the weight per unit area supported by the fixed wing of the PCA-2 Autogyro, whose experimental results are reported in Ref. [25]. The PCA-2 Autogyro has both a fixed wing and a rotor. The experimental setup and data are detailed in Ref. [31]. The prescribed wing loading has a specific one-to-one correspondence with the air-speed of the PCA-2 Autogyro, which is equivalent to the wind velocity, V, in our work. It also has a similar dependence with the tip-speed ratio μ of the rotor. Thus, the data in Ref. [31] effectively considers V as a function of μ. In Ref. [25], this relationship is not explicitly given. However, it can be determined from the Ω versus μ and α versus μ data, together with the definition of μ. The calculated dependence matches closely with the experimentally obtained relation reported in Ref. [31]. For example, from Ref. [31], the wing aloading at μ = 0.25 is $≈20%$⁠, which corresponds to an effective air speed of ≈ 56 mph (equivalent to ≈ 82 ft/s). It can be observed in Fig. 4 that the “Prescribed Wing Loading” trend as well as the data cross the velocity level curves at ≈ 82 ft/s.

In Fig. 3, it can be seen that the coefficient of thrust C_T is largely independent of V below μ = 0.5, where as angle of incidence, α, displays only negligible changes with respect to V for the entire range of μ. While the model results show deviation from the experimental data of C_T, this deviation is particularly small (within ≈ 5%) in the range 0.15 < μ < 0.35, a region where feasible autorotational equilibria are predominantly concentrated. It is noted that for μ < 0.1, α assumes large angles where the momentum theory is not valid and hence model predictions for μ < 0.1 may not be reliable. The angular velocity Ω shown in Fig. 4 varies significantly with differing V. It can be seen from the dash-dot line in Fig. 4 that the prescribed-wing-loading-based V generates nearly constant Ω over the range μ > 0.1. This behavior as well as the magnitude agrees with the experimental behavior over the range of 0.1 < μ < 0.5 reported by Wheatley [25]. The trends of Fig. 4 are representative of a nonnegligible effect of blade weight moment, but it was observed that exclusion of this term does not significantly affect the value of Ω. Comparison to the results of Ref. [25] was shown for a case of no regenerative braking, i.e., for Q_e = 0.

Addition of regenerative braking significantly affects the performance of an autogyro, and thus, the visuals presented in Figs. 5–7 serve to provide insight into this behavior. The solid trends represent a constant input of torque with consideration to nonnegligible blade weigh moment effects; the dash-dot trends show the same torque applied with blade weight moment neglected. These trends were simulated at a constant wind velocity of 100 ft/s. The behavior of a₀, a₁, a₂, b₁, and b₂ with differing torque input is displayed in Fig. 5. The increase in magnitude with increasing torque relationship seen for a₀ and b₁ is also true for the remaining coefficients, and thus, the same constant torque inputs from 0 to 1000 lb · ft should be associated with the a₁, a₂, and b₂ trends from bottom to top, respectively. This increase in flapping coefficient magnitude suggests an overall increase to the average flapping angle as well as the variation of the flapping angle over a full rotation of the rotor. A similar trend is shown in Fig. 6(a) with regards to the coefficient of thrust, C_T. Once again, C_T does not appear to be significantly affected by the inclusion of the effects of the blade weight moment term of Eq. (3); however, an increase in regenerative braking does appear to change the performance, suggesting the ability to tune the thrust output of an autogyro with regenerative braking. The parameters α and Ω remain largely unaffected by a change in input torque.

It should be noted that the plots shown are in relation to a span of μ. The ratio μ is defined in Eq. (1) to be a function of V, α, and Ω. With μ and V being inputs, it follows from this observation that the weak relationship between Q_e and α causes only a small change in Ω with different input Q_e. Logic and experience tells us that applying a braking torque to a rotating shaft will decrease its angular speed; however, by selecting a value for μ as input, the presented model neglects causality. As this work is an exploration of steady-state behavior, it need not be upheld to the same causal demands of dynamic analysis. Nevertheless, a practical interpretation of Fig. 4 can be as follows: a decrease in angular speed for constant V and uniform α would in fact increase the tip speed ratio. Therefore, for a scenario where a torque is applied to an autogyro at a slow enough rate to be considered quasi-static, the analysis suggests that the angular speed would move along a trend analogous to the 100 ft/s trend shown in Fig. 7.

The aforementioned tether model is valid for finding equilibrium positioning of a tethered autogyro system in a uniform wind velocity field. With this assumption, the aerodynamics of the autogyro and the static equilibrium of the tether can be solved independently. However, relaxing this assumption to a more realistic wind field, where V increases linearly as a function of altitude, y, couples the solving of the autogyro aerodynamics and tether profile, as the equilibrium altitude, y_e, solved for by the catenary model may not coincide with the value of V selected as input to the aerodynamic calculations. Instead, it is necessary to implement an iterative method to converge on an altitude where the input V matches the wind speed expected at y_e. It should be noted that for many cases, this equilibrium altitude will not exist, suggesting the given conditions are not suitable for sustainable steady-state tethered autorotation. An equilibrium altitude can be found through the following process:

1. Select a value for μ, Q_e, and l_t.

2. Initialize V as the velocity at an altitude of l_t.

3. Use the above aerodynamic model to solve for T and α.

4. Calculate the altitude at x_e using Eqs. (21)–(26).

5. Compare the velocity at this altitude, and adjust the guess in the direction of the new velocity.

6. Repeat steps 3–5 until the absolute change of V is below a desired threshold (chosen as 10⁻³), or until the calculated altitude has dropped below a selected minimum threshold suggesting failure.

Utilizing the aforementioned numerical process, one can explore the feasible equilibrium space for a selected autogyro design and visualize the optimal flight conditions for power generation. The range of equilibria was explored for a wind field with wind velocity and air density varying with altitude. These altitude variations are shown in Fig. 9. Standard atmospheric density variation data are obtained from Ref. [41]. Wind velocity distribution with altitude is based on Refs. [2,42]. Linear velocity distributions were approximated. We present two sets of simulation results. The first concerns high altitudes and a large and heavy autogyro, see Sec. 5.1. In this scenario, we explore regenerative power generation as a feature. The second simulation considers lower altitude deployment and a smaller autogyro, Sec. 5.2. Here we only explore equilibrium autorotation, without power generation. For the two scenarios, the linear wind velocity distributions have slightly different approximations, as shown in Fig. 9.

Selected physical parameters for the autogyro and tether used in this simulation are provided in Eq. (17). For blade physical and aerodynamic properties, refer to Sec. 2.2 and the study by Wheatley [25]. The tether is considered to have a weight per unit length of 0.00 lb/ft. This mass density is based on mechanical properties of commercial Kevlar rope found online, for example, Ref. [43]. Specifically, tensile strength was considered and compared against simulation results (presented in Fig. 11). The autogyro weight is chosen at 23.3 slugs, i.e., ≈750 lb. A main component of the weight is anticipated to come from the on-board motor/generator. For the possible power generation determined from simulations (to be discussed using Fig. 11), such motors can weigh as low as ≈50 lb, [44]. The 750 lb assumption can therefore be considered conservative. The results of tethered autogyro simulation are displayed for a range of μ and Q_e values in Figs. 10 and 11. Figure 10 presents the converged equilibrium altitudes for various tether lengths. The flight conditions that did not allow convergence have been filtered to an altitude value of zero to emphasize the successful equilibrium states. It should be noted that converging states along the border of these successful regions will likely have other undesirable traits such as tether profiles that drop below zero altitude between the base and the autogyro. For optimization purposes, it would be necessary to select a higher threshold altitude to avoid subterranean tether profiles; however, for the purposes of general equilibrium analysis, this filtering was deemed unnecessary. It was observed for the selected autogyro/tether dimensions and weights, see Eq. (17), that successful equilibrium altitude convergence was not achievable for the entire range of μ and Q_e input values for tether lengths below approximately 20,000 ft (≈6 km). From Fig. 10(c), it is evident that higher values of Q_e result in equilibria at lower altitudes for a given tether length. This is expected, since higher Q_e effectively reduces the thrust T. It is also evident that μ ≈ 0.2 gives higher equilibrium altitude over the range of feasible Q_e, and hence for a given tether length yields a lower horizontal drift.

Figures 11(a)–11(c) display the lateral drift of the autogyro from the base, the power generated, and the maximum static tension of the tether, respectively, for a 32,000 ft (approximately 10 km) tether. The lateral drift, power, and tension are important considerations in assessing the characteristics of the equilibria. A high lateral drift may be undesirable as it implies a significant deviation from the launch point. On the other hand, high power is desirable if power generation is the primary goal. We note from the regions enclosed by dashed lines in Figs. 11(a) and 11(b) that these two goals are conflicting. If flying at a higher altitude is desired, then the equilibria enclosed by the dashed line in Fig. 11(a) is preferred since it equivalently corresponds to high-altitude flight, as shown in Fig. 10(c). Tether tension, Fig. 11(c), is an important practical consideration since it relates to the mechanical integrity of the system. The tether length 32,000 ft was the maximum length considered for this study as the assumption of V varying linearly with y loses justifiablity after this altitude. From Figs. 10(c) and 11(a), it can be seen that the lateral drift of the autogyro has an inverse relationship to that of the equilibrium altitude. This is expected, as a lower flying craft would have more available tether to drift from the base. From Figs. 11(a) and 11(b), we note that generated power is positively correlated with rotor torque Q_e. This is a sensible result, as assuming Ω does not significantly vary, power is directly related to the regenerative braking, Q_e, applied. It is also apparent that power output favors lower values of μ. As μ is defined by an inverse relationship with Ω, it follows that this relation will propagate through to the power output. This inverse relationship between μ and power appears to break down; however, as the autogyro approaches the border of the successful equilibrium space. Given that this region is represented by low altitude flight, it follows that low altitude wind speeds are insufficient for producing the rotational speeds for optimal power output.

Figure 12 plots a combined altitude and power-based fitness for the case of 32,000 ft tether. The fitness function is of the form,

In this section, we explore the autorotational equilibria of lighter, smaller autogyros at lower altitudes. Here, owing to lesser wind power available at lower altitudes, emphasis is given only on equilibrium hovering conditions and power generation is de-emphasized. Such a setup is deemed suitable for surveillance purposes and for energy efficient flight using autorotation. A lightweight motor, e.g. [45], can be installed on-board to lift the autogyro to the desired altitude range and as a supplement to compensate for variable wind speeds. We consider a autogyro of 35 lb weight and a tether of ≈ 0.005 lb/ft weight per unit length. This leads to a total weight of ≈ 50 lb if a 1 km long tether is used. The blade length R was consider to be 10 ft. The blade chord c was proportionately reduced. The parameters I₁, M_W, γ, and σ were accordingly scaled.

Figure 13 shows the equilibrium profiles of the system in a 26 ft/s wind field at different angles of incidence α. All simulations were conducted with Q_e = 0. We notice that as α reduces, the equilibrium value of μ increases, but the equilibrium altitude reaches a maximum at an intermediate value of μ. This is consistent with the trends observed in Figs. 10(b) and 10(c), where maximum altitude consistently occurred at μ ≈ 0.2. From Fig. 13, it is evident that the equilibria at high α, e.g., cases (1) and (2), have a dominant drag force. This causes high drift and taut tether at equilibrium. Lowering α causes the equilibria to be progressively driven by the lift force, e.g., cases (4) and (5), and hence achieve high elevations. Further lowering of α reduces the magnitude of lift force and the resulting equilibria are at low altitudes, e.g., cases (7) and (8). In Fig. 14, we extend the results of the light autogyro to three tether lengths, namely, 3000, 7000, and 10,000 ft and for Q_e = 0. The plots, along with the results of Sec. 5.2, confirm that μ ≈ 0.2 is an optimal value with respect to maximizing the flying altitude, lowering the horizontal drift and achieving low tether tension.

This article details a model-based study of the equilibrium characteristics of a tethered autogyro. The aerodynamic model, the tether mechanics, and the proposed numerical method together facilitate the consideration of a wider range of operating conditions than reported in prior works on modeling autorotation. Specifically, a range of aerodynamic equilibria are obtained as a function of the operating variables, namely, μ, Q_e, and V. It is shown that the results of the presented model accurately match the experimental data of the study by Wheatley [25] when its prescribed wing loading profile is replicated in our model, thereby demonstrating our model’s efficacy. The model is simulated to visualize the steady-state behavior of an autogyro and to explore the effects of differing wind velocities and regenerative braking. The aerodynamics is coupled with a static catenary model to approximate the behavior of a tethered autogryo. This coupled model is numerically solved for a wind velocity profile varying with altitude.

The investigation shows the feasibility of performing energy efficient and long duration hovering through autorotation, which can be useful for surveillance of a confined area. In such applications, autorotation could be intermittent or sustained, depending on the consistency of wind speed. In the context of possible power generation, it is observed that sustainable tethered autorotation with simultaneous power generation is likely achievable at significantly high altitudes of a few to several kilometers. Furthermore, it is observed that maximizing the power output for a given autogyro configuration is accompanied by significant drift from the base. It is therefore reasonable to assume that a compromise must be made between the area required by the tethered autogyro mechanism and the power output of the rotor; however, it is possible that an array of smaller autogyros flown at a lower altitude would be able to produce an analogous power output in a more compact space. Testing of this claim would require analysis into the optimal geometric design of a tethered autogyro, which is a topic of future research. The aforementioned analysis is at steady state and thus gives lesser insight into the dynamic behavior of tethered autogyros. Efforts to explore this dynamic behavior and controlling the operating altitude, the drift or the angle of incidence, through active regenerative braking or active tether actuation are currently underway.

This work was supported by the National Science Foundation CMMI Grant #1762986, USA.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

slope of lift curve

b =

number of blades

c =

blade chord

g =

acceleration due to gravity

v =

induced axial velocity

x =

autogyro horizontal position

y =

autogyro altitude

B =

blade radius fraction less tip losses

K =

amplitude of flow variation

H =

longitudinal force

R =

blade radius

T =

thrust force

V =

wind velocity

l_t =

length of tether

m_a =

mass of autogyro

m_t =

mass of tether

x_e =

autogyro horizontal equilibrium position

y_e =

autogyro vertical equilibrium position

C_T =

rotor thrust coefficient

F₀ =

tether tension at base

F₁ =

tether tension at autogyro

F_t,d =

total drag force of tether

I₁ =

blade moment of inertia about flapping hinge

M_W =

flapping moment from blade weight

M_T =

flapping moment from thrust

Q_e =

regenerative Braking Torque

a₀, a₁, b₁, a₂, b₂ =

Fourier series parameters of flapping

α =

angle of incidence

β =

flapping angle

γ =

nondimensional mass constant

δ =

average drag coefficient

η₀ =

tether angle at base

η₁ =

tether angle at autogyro

θ₀ =

blade pitch angle at hub

θ₁ =

blade pitch slope

λ =

axial flow ratio

μ =

tip speed ratio

ρ =

density of air

σ =

blade disc solidity

ψ =

angular position of blade

Ω =

rotor angular velocity