Abstract
Wind energy harvesters are emerging as a viable alternative to standard, large horizontal-axis wind turbines. This study continues a recent investigation on the operational features of a torsional-flutter-based apparatus, proposed by the author to extract wind energy. The apparatus is composed of a non-deformable, flapping blade-airfoil. A nonlinear torsional spring mechanism, either simulated by a Duffing model or a hybrid Duffing–van der Pol model, installed at equally spaced supports, enables limit-cycle, post-critical vibration. To enhance the output power, stochastic resonance principles are invoked through a novel, negative stiffness mechanism that is coupled to the eddy current device for energy conversion. The output power is explored by numerically solving the stochastic differential equation of the model, accounting for incoming flow turbulence. Three main harvester types with variable configuration are examined; the chord length of the blade-airfoil, used for energy harvesting, varies between 0.5 and 1 m; the spanwise-length-to-chord aspect ratio is four. The flapping frequency varies between 0.10 and 0.25 Hz. The study demonstrates that exploitation of negative stiffness mechanism can improve the performance of the harvester.
1 Introduction
Several harvesting devices [1] have been proposed to exploit wind energy in the low wind speed range and for small-scale applications. These wind energy technologies are triggered by aeroelastic instability [2,3]. Most existing harvesters are designed as micro-mechanical units and used for recharging miniature sensors [4]. By contrast, “meso-scale” harvesting technologies are still an active research field. For example, a hybrid device has been installed on a highway bridge and utilizes two different energy sources (natural wind flow and structural vibrations) [5]. Furthermore, vortex-induced vibrations of multi-unit circular cylinders in water flows [6] have been considered. Harvesting devices, triggered by water axial-flow instabilities [7], or by flutter of an inverted and flexible flag [8], or by vortex-induced vibration of cylinders inside ventilation ducts [9,10], have been studied. The exploitation of coupled, bending-torsion vibration of streamlined bodies, i.e., the “flutter mill,” has been proposed for energy harvesting in wind flows [2,3,6,9]; several flapping foil mechanisms have been derived from this concept [11]. All of the above share the common task of scavenging the kinetic energy of the flow at low wind speeds.
Along this line of research, this paper continues recent investigations on the use of a novel wind energy technology, which considers a purely torsional-flutter harvester [12]. Contrary to the previously mentioned devices, the apparatus has a simple operational mechanism that is ideal for implementation; it is also designed to supplement energy to one or few residential housing units.
The proposed apparatus is inspired by seminal works [13,14]. It is composed of a rigid, flapping blade-airfoil that rotates about a pivot axis, positioned upwind with respect to the airfoil’s aerodynamic center. The blade-airfoil has a chord length equal to about 1 m and a spanwise length between 4 and 10 times the chord (i.e., aspect ratio (AR)). Both a nonlinear torsional spring mechanism (Duffing model) and a hybrid one (Duffing and van-del-Pol model), installed at equally spaced spanwise supports close to the pivot position, have been considered in previous studies to convert the kinetic energy of the flow to electrical energy by eddy currents.
Stochastic modeling and analysis of the harvester are used to evaluate the output power [12]. The state vector of the model includes physical states (flapping angle and angular velocity) and aeroelastic torque unsteady flow states, corrected for three-dimensional flow effects. Turbulence is also included in the stochastic dynamic equation. A wide range of stationary turbulent winds can be simulated, typical of complex urban flow environments.
In this study, an enhancement is explored by considering the principles of stochastic resonance, i.e., the aptitude of an external noise (turbulence) to amplify the unstable flapping. This condition is achieved by adding a negative stiffness mechanism, embedded within the transmission shaft that is connected to the eddy current power system. The negative stiffness destabilizes the torsional restoring torque provided by a positive, cubic nonlinear Duffing model. The conceptual idea is also borrowed from the bi-modal Duffing oscillators excited by stationary white noise, for which several investigations and solutions are available [15].
The negative stiffness mechanism is installed at equally spaced supports; it is used to enhance the amplitude of transverse, post-critical vibration of a permanent magnet within the electro-magnetic coil of the power system. Conversion to electrical power is in fact warranted by an eddy current power system with multi-loop magnetic coil and a translating permanent magnet [16]. Energy is stored in a battery for future use.
In this initial exploration, example configurations are considered with an adjustable negative stiffness. The rotation, pivot axis position is placed at the windward edge of the blade-airfoil. Stationary wind flows are exclusively examined. Turbulence intensity equal to 10% is used to investigate realistic types of wind flow. Stability is studied by solving the stochastic differential equations in time domain, by ensemble averaging and Monte-Carlo sampling.
2 Theoretical Derivations
The apparatus (Fig. 1) is composed of a flapping blade-airfoil with dimensions (half-chord width) and (spanwise, longitudinal length). The main degree-of-freedom (DOF) of the model is the torsional rotation about pivot “O.” Aeroelastic torque is modeled by standard aeroelastic formulation for small angles of attack, corrected for three-dimensional flow effects that depend on the aspect ratio of the blade.
The load model also replicates the main features of synoptic, stationary turbulent winds and combines mean outflow wind speed with along-wind stationary turbulence . Since the blade longitudinal plane is oriented on a vertical plane (), gravity is not relevant to energy conversion.
2.1 Non-Rigid Transmission to Electrical Power Circuit
2.1.1 Foreword.
Contrary to previous studies, a non-rigid transmission system is employed to transfer the flow energy to electrical energy. A schematic diagram of the energy conversion system and secondary power circuit is depicted in Fig. 2. The foremost leeward edge of the flapping airfoil (node A) undergoes rotation about the windward-edge pivot “O” (Fig. 1). A rigid linkage AB is connected by a hinge at B to a massless shaft BC. The BC shaft is equipped with a permanent magnet at the upper end (toward node C in Fig. 2) that slides inside a multi-loop coil. This mechanism is used to convert the rotation to transverse translation along the axis. The conversion is based on eddy current, induced by the permanent magnet that slides inside the multi-loop coil. A frictionless collar is used to guide the magnet along a trajectory that, for small flapping amplitudes , is approximately directed along the y axis in the figure. This type of multi-loop coil system is inspired by an electro-magnetic harvester, triggered by random vibration sources [16].
2.1.2 Dynamic Equilibrium at Node “C”.
The equilibrium equation at node C of the transmission shaft BC is examined in detail (Fig. 2). In Fig. 2, is the absolute translation at the massless linkage AB; is the relative displacement at “C.” A non-rigid connector is inserted between nodes “B” and “C” of the transmission shaft that connects the linkage AB to the electro-magnetic induction converter, i.e., the magnet that slides through a frictionless collar into the multi-loop coil. The moving magnet introduces magnetic induction and interacts with the “BC” shaft, translating in the direction inside the winding coil (Fig. 2). The quantity is the eddy-induced output current of the power system and is the electro-mechanical coupling coefficient in units of Newton/Ampère [12].
If the mass is negligible, the previous equation can be simplified as , which coincides with the internal constraint force transferred to the primary system through linkage AB in the direction. For small flapping angles about O, we note that .
2.1.3 Electrical Power System Equation.
For a rigid shaft, it is noted that and .
2.2 Dynamic Equilibrium Equation of the Mechanical System.
In Eq. (7), the external aeroelastic torque is about the pivot O, and is the total polar mass moment of inertia about O. Structural damping is simulated through a linear term in Eq. (7) with damping ratio ; is a function of and that encloses the nonlinear restoring and damping torque mechanisms, and that enables power conversion and limit-cycle oscillation [12,18].
In Eq. (8), is the dimensionless current, is the reference current intensity, and is the electro-mechanical coupling coefficient, as derived in Sec. 2.1.3.
Duffing model: , i.e., the nonlinear restoring force effect is simulated by the term , with being a suitable dimensionless constant.
Hybrid Duffing–van Der Pol model: The equilibrium incorporates both Duffing and van der Pol models [19] to improve energy conversion. The modified nonlinear restoring torque function reads . Besides parameter , there is a nonlinear damping coefficient , which models the self-limiting feature of the negative damping mechanism proportionally to the linear damping term .
3 Stochastic Differential Equations
3.1 Random Turbulence Wind Flows.
The aeroelastic torque is proportional to the mean flow speed , contaminated by along-wind stationary turbulence . The flapping foil of the apparatus has a reference diagonal dimension with being the chord length, the spanwise width, and the geometric aspect ratio. Therefore, the main transitory feature of the flow that influences the stability is the instantaneous magnitude of the flow speed.
In Eq. (9), turbulence is random, stationary, and normalized as . The properties of the stationary process are represented, without any loss of generality, as a Gaussian white noise process, simulated in Sec. 3.4 by a Wiener process [20] of independent Gaussian increments with standard deviation .
Finally, since the diagonal reference dimension of the blade-airfoil is small compared to the integral turbulence length scales, the gust load is fully coherent across the apparatus (Fig. 1).
3.2 Aeroelastic Torque With Random Perturbations.
Furthermore, random load perturbation is introduced by replacing the deterministic quantity with a random, time-dependent ; and is a zero-mean, Gaussian perturbation; also accounts for load measurement error and modeling simplifications. It is assumed that the unsteady torque features are not affected either by turbulence or any other perturbation.
3.3 Dynamic Equilibrium With Leading Edge Flapping Pivot.
3.4 State-Space Vector Equation.
in Eq. (12) is the state vector; includes both physical, aeroelastic states and dimensionless output current .
In Eq. (12), Wiener noise separately addresses turbulence perturbation from , used for load perturbation. Quantity is a nonlinear vector-function; is a nonlinear turbulence diffusion vector-function that depends on turbulence intensity . is a constant, diffusion matrix that controls the load perturbation and depends on the standard deviation of , . Both and are nonlinear; they are two vector-functions.The non-zero elements of the matrix are two only. The Wong-Zakai [25] correction terms are introduced in both and . Derivation of , , and is omitted for the sake of brevity. A detailed description of these equations may be found in previous works [12,18].
3.5 Mean-Square Stability.
As explained by Xie [27,30] is a measure of the slow dynamics of a nonlinear system; it can be interpreted as the role of damping in the linear deterministic case. Specifically, indicates that the variance of the 2-norm of zero-mean vector in Eq. (13) vanishes as time tends to infinity, i.e., the system is asymptotically stable in the mean squares.
From a practical perspective, in Eq. (13) is approximated by arithmetic averages after collecting a sample of 200 repeated, numerical solutions of Eq. (12). Furthermore, Eq. (13) is numerically found and examined at , i.e., after about 47 equivalent periods of aeroelastic flapping and, if the is noted, the harvester is labeled as stable, i.e., not engaged in energy conversion.
3.6 Output Power Estimation.
4 Description of the Modeled Units and Flow Properties
Simulations examine various configurations of the apparatus, with rotation axis pivot at the apex of the blade-airfoil (). Three “types” are selected from Refs. [12,18], with the main properties described in Table 1; in this table, quantities such as angular frequency and damping ratio must be interpreted as the properties of the linearized dynamic equation of the harvester.
Main harvester and flow properties
AR | |||||||
---|---|---|---|---|---|---|---|
Type | (m) | (Hz) | (%) | (–) | (%) | (–) | |
0 | 0.25 | 20 | 0.25 | 0.25 | 4 | 10 | 0.07 |
1 | 0.25 | 40 | 0.25 | 0.30 | 4 | 10 | 0.07 |
2 | 0.50 | 300 | 0.10 | 0.30 | 4 | 10 | 0.07 |
AR | |||||||
---|---|---|---|---|---|---|---|
Type | (m) | (Hz) | (%) | (–) | (%) | (–) | |
0 | 0.25 | 20 | 0.25 | 0.25 | 4 | 10 | 0.07 |
1 | 0.25 | 40 | 0.25 | 0.30 | 4 | 10 | 0.07 |
2 | 0.50 | 300 | 0.10 | 0.30 | 4 | 10 | 0.07 |
The nonlinear stiffness parameter in the Duffing model is constant and set to , irrespective of the type. The nonlinear damping parameter is used in the hybrid Duffing–van der Pol model. Coupling with the power circuit is achieved by setting and [12].
Initial conditions are imposed on the random vector at by considering an initial flapping angular motion at that triggers the instability [12]. The random initial amplitude of is set to zero mean and standard deviation equal to 2 deg, coincident with small, realistic angular deviations from the static equilibrium.
In all the simulations, the wind flow is turbulent with intensity and load perturbation with standard deviation .
5 Numerical Results
Numerical simulations are executed in the range only, since the efficacy of the harvester at low wind speeds is desirable.
In the next four figures, the legend labels “Ty.0,” “Ty.1,” and “Ty.2,” respectively, refer to type-0, type-1, and type-2 apparatuses, described in Table 1.
5.1 Operational Regimes.
Figure 3 illustrates the reference simulation result for the linear model (, found for turbulent wind flows of intensity 10% and moderate load perturbation. The flexible transmission link has a moderate negative stiffness with .
Examining the linear dynamic model is useful to evaluate the incipient instability threshold, in a linear system, divergent vibrations are only possible and no limit cycle exists. The figure demonstrates that both type-2 and type-0 apparatuses, with transmission shaft equipped with negative stiffness mechanism, tend to be unstable at low wind speeds, e.g., .
By contrast, Fig. 4 examines the Duffing case with nonlinear stiffness coefficient and negative stiffness apparatus. The system is stable at for all the wind speeds examined. The type-2 case diverges due to numerical integration instability [18]. Type-2 apparatus is unstable and therefore operational at . Stability, however, may still correspond to operational wind conditions, since limit-cycle vibration entails a periodic response (zero damping).
Finally, Fig. 5 examines the hybrid model at with both nonlinear damping and nonlinear stiffness in the primary system (, ; the negative stiffness effect through the passive connector is again set to . Turbulence of intensity and load perturbation are the same as before.
Again, an apparently stable behavior is observed at high wind speeds , suggesting that the apparatus may require re-designing the transmission mechanism between flapping blade tip and permanent magnet (Fig. 2), if the hybrid Duffing–van der Pol apparatus is employed.
5.2 Output Power.
Figure 6 presents preliminary output power results at various mean flow speeds for an apparatus equipped with nonlinear Duffing restoring torque mechanism and negative stiffness transmission shaft using a soft coupling parameter . This figure transfers the information, originally included in Fig. 4 and discussed in the previous sub-section, to energy conversion. In the two figure panels, the mean output power is estimated numerically by Eq. (14) in the time interval . The power estimation for in the panel of Fig. 6(b) is interrupted at , since numerical integration issues arise (also refer to Fig. 4(b)).
![Mean output power E[Pout(τ)]versus time τ for the nonlinear Duffing model and apparatus at mean flow speeds: (a)U=12.0m/s and (b)U=14.8m/s](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/lettersdynsys/5/2/10.1115_1.4067394/1/m_aldsc_5_2_021006_f006.png?Expires=1748032016&Signature=L9bB4-4d-3pgcfaD68FEUqT47AkEwh6L7cL~ECf48EghekoQJvzx44er27spGkXLj4T29z8YfnuAmSLqEgN~NlN3cQO2J4uRHr4imUFeC0kxiAJ1fjNA-R1q1ej2QJHVRUxALYvAof~IWDChqG4vTZaxBAv55SM4sO6FPskCNaYlrzu75CXTredUMo5Nq95ApEV1e7Tp5D2UE~xYzG9lq0qJLMvmuJ~Ted3wFZ0nzFS5nmHvJteSTGHOfnq~iQv9yXcsmwqHITg4h19EJm0s62QVVGz2qRFVOdh~wVdrYLSgh2VDZt9GRfeSvdq~bO6R9vqF17qqYz20bkBQ9syQrA__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Mean output power versus time for the nonlinear Duffing model and apparatus at mean flow speeds: and
This figure demonstrates that type-2 apparatus is active and produces energy that might be stored in a battery for future re-use [12]. Nevertheless, the available power is rather small and possibly unsuitable to practical design and implementation. Additional investigations are necessary to optimize the negative stiffness transmission shaft with soft coupling parameter.
6 Concluding Remarks
An initial inspection of the numerical results suggested that the use of a hybrid model with a nonlinear torsional stiffness term and a nonlinear damping term may produce a stable system at all wind speeds. Therefore, any non-linearity, installed in the primary mechanism that enables the airfoil flapping, may neutralize the torsional flutter and, consequently, be detrimental to the energy conversion.
To enhance the harvester vibrations, a flexible link was proposed, exploiting the properties of a negative stiffness mechanism connected to the eddy current power device. The study demonstrates that the output power can be increased, approximately by a factor of two for a type-2 apparatus, and that the performance can be remarkably improved.
In spite of the promising outcome, future studies are still necessary to evaluate the output power at other flow regimes, induced by the flapping blade-airfoil and enhanced by the negative stiffness mechanism. Energy conversion may also require further improvements. For instance, since the added mass of the secondary system connected to the negative spring mechanism has been neglected, enhancements should be possible by specifically including this quantity. Consequently, the ratio between mass moments of inertia of the flapping foil and the secondary power device system could possibly be adjusted to achieve a better performance.
Footnote
This work was completed while the author was on sabbatical leave as a visiting professor (January–April 2024) in the Department of Civil, Environmental and Mechanical Engineering, University of Trento (Italy), and a visiting professor (June 2024) in the Department of Civil and Environmental Engineering, University of Perugia (Italy).
Acknowledgment
This work was completed while L. Caracoglia was on sabbatical leave in early 2024; L. Caracoglia gratefully acknowledges the support of the Universities of Trento and Perugia (Italy) during his sabbatical leave in 2024. This material is based in part upon work supported by the National Science Foundation (NSF) of the United States of America, Award CMMI-2020063. Any opinions, findings, and conclusions or recommendations are those of the author and do not necessarily reflect the views of the NSF.
Conflict of Interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data Availability Statement
Data and models that support the findings of this study are available from the corresponding author upon reasonable request.
Nomenclature
- =
dimensionless position of the pivot “O”
- =
half-chord width of the blade-airfoil
- =
spanwise length of the blade-airfoil
- =
absolute transverse displacement of DOF B
- =
time (s)
- =
along-wind stationary turbulence ()
- =
vertical axis coordinate
- =
normalized along-wind stationary turbulence
- =
output current of the power system (A)
- =
mean outflow wind speed ()
- =
parameters of the function ()
- =
mean value of random exponent parameter
- =
parameters of function ()
- =
dimensional axial stiffness, flexible BC shaft
- =
reduced frequency, one-DOF flapping foil
- =
lumped mass of the transmission shaft (kg)
- =
relative transverse displacement BC (transmission shaft)
- =
nonlinear drift vector-function
- =
nonlinear turbulence diffusion function
- =
scalar Wiener noise for turbulence
- =
scalar Wiener noise for load perturbation
- =
internal constraint force of linkage AB
- =
polar mass moment of inertia of flapping foil about “O” ()
- =
impedance of output power circuit (Henries)
- =
aeroelastic torque (N m)
- =
electromotive torque (N m)
- =
resistance of output power circuit (Ohms)
- =
nonlinear restoring torque function
- =
diffusion matrix of load perturbation
- =
random state vector
- =
aspect ratio of the blade-airfoil
- =
flapping angle of the blade-airfoil, about “O”
- =
nonlinear dimensionless damping (van der Pol)
- =
soft coupling coefficient, Eq. (4)
- =
random perturbation to parameter
- =
normalized inertia parameter
- =
structural damping ratio of the flapping foil
- =
three-dimensional flow effect parameter
- =
dimensionless induced current, power circuit
- =
dimensionless cubic torsional stiffness
- =
second moment Lyapunov exponent
- =
generalized impedance of the power circuit
- =
aeroelastic state ()
- =
aeroelastic state ()
- =
standard deviation of
- =
standard deviation of
- =
dimensionless time
- =
discrete time instant used to find MLE
- =
sub-vector of random vector
- =
unsteady aeroelastic forcing function
- =
unsteady aeroelastic forcing function at
- =
dimensional electro-mechanical coupling coefficient (N/A)
- =
dimensionless stiffness of the flexible shaft
- =
electro-mechanical coupling coefficient
- =
angular vibration frequency ()
- =
pulsation of the one-DOF flapping foil ()