How Does Energy Harvesting from a Fluttering Foil Influence Its Nonlinear Dynamics?

Abstract

This study investigates the nonlinear aeroelastic behavior and energy harvesting performance of a two-degrees-of-freedom NACA 0012 airfoil under varying reduced velocities and electrical load resistances. The system exhibits a range of dynamic responses, including periodic and chaotic states, governed by strong fluid–structure interactions. Nonlinear oscillations first appear near the critical reduced velocity $U_r^{*}=6$, with large-amplitude limit-cycle oscillations emerging around $U_r^{*}=8$ in the absence of the electrical loading. As the load resistance increases, this transition shifts to higher $U_r^{*}$, reflecting the damping effect of the electrical load. Fourier spectra reveal the presence of odd and even superharmonics in the lift coefficient, indicating nonlinearities induced by fluid–structure coupling, which diminishes at higher resistances. Phase portraits and Poincaré maps capture transitions across dynamical regimes, from periodic to chaotic behavior, particularly at a low resistance. The voltage output correlates with variations in the lift force, reaching its maximum at an intermediate resistance before declining due to a suppressing nonlinearity. Flow visualizations identify various vortex shedding patterns, including single (S), paired (P), triplet (T), multiple-pair (mP) and pair with single (P + S) that weaken at higher resistances and reduced velocities. The results demonstrate that nonlinearity plays a critical role in efficient voltage generation but remains effective only within specific parameter ranges.

1. Introduction

Driven by the global need for sustainable energy, researchers increasingly focus on innovative technologies that can efficiently capture and convert ambient energy in the atmosphere into electrical power. Among these emerging technologies, aeroelastic energy harvesters that utilize the inherent dynamics of fluttering airfoils garnered substantial attention. Unlike traditional rigid systems, fluttering-foil-based mechanisms capitalize on fluid-induced oscillations to generate power continuously. This unique capability to convert oscillatory motion into electrical energy using piezoelectric positions such systems as a promising solution for energy harvesting to be used in diverse applications. Pioneering studies explored the optimization of these systems, examining various configurations and flow conditions to enhance their performance and efficiency [1,2,3,4]. These investigations collectively contributed to explaining the governing physics and improving the energy conversion capabilities of flapping foil-based harvesters, highlighting their potential as a reliable alternative to traditional wind and hydroelectric turbines. Unlike traditional wind turbines that rely on rotational motion, flutter-based energy harvesters exploit the self-sustained oscillatory behavior of airfoils in fluid flow. This approach offered unique advantages, including operation in low-speed flows, scalability, and enhanced power density [5,6,7].

Energy harvesting through passively flapping airfoils, a phenomenon commonly associated with flutter, is fundamentally governed by the principles of fluid–structure interaction (FSI). Flutter refers to a dynamic aeroelastic instability that arises in flexible structures, such as wings of airplanes or blades of turbines, when aerodynamic forces induced by fluid flow couple with the structural modes of the system, leading to self-excited oscillations. These oscillations can grow in amplitude over time, potentially resulting in sustained motion or structural failure. In the context of energy harvesting, this dynamic interaction enables the continuous extraction of energy from the flow through the oscillatory motion of the airfoil [8,9]. This fluid–structure coupling induces limit cycle oscillations (LCOs), allowing for the continuous extraction of energy from the surrounding flow. These oscillatory motions were extensively studied with respect to mechanical efficiency, emphasizing the impact of key design parameters, such as the airfoil geometry [10], mass ratio [11], moment of inertia [11], and the positioning of the pitching axis [11]. The pitching and heaving kinematics significantly influence the overall efficiency of the system for energy conversion, dictated strongly by the vortex dynamics around the foils. Furthermore, the energy harvesting efficiency is affected by Reynolds number ($\mathrm{R}\mathrm{e}$) and reduced velocity ($U_r^{*}$) [10,11,12]. Additionally, some previous studies revealed substantial changes in energy harvesting capabilities of fluttering foils, highlighting the critical role of fluid dynamics in optimizing their aeroelastic behavior [13]. Different electromechanical transducers are integrated with flutter-based energy harvesting systems, which enable the direct conversion of mechanical vibrations into electrical energy. These include piezoelectric [14,15,16], piezoresistive [17], electromagnetic, and electrostatic transduction mechanisms [18] for sustainable micro-power generation. The development of micro-machining techniques and advancements in low-power electronic devices further contributed to stimulating interest in micro-power generators (MPGs) [19,20].

A critical aspect of flutter-based energy harvesters lies in the nature of the nonlinear dynamics that govern their performance. While many traditional systems rely on structural nonlinearities to induce oscillations, recent advancements have demonstrated that fluid-related nonlinearities alone could drive substantial energy harvesting [21,22]. Specifically for linear structural configurations, the nonlinearity introduced by unsteady aerodynamic forces, vortex shedding, and wake interactions play a dominant role in dictating oscillation amplitude, frequency, and stability [23]. This nonlinearity, originating in fluid flows, introduces rich dynamical behaviors, including bifurcations, LCOs, and chaotic oscillations, which directly influences the efficiency and robustness of the energy harvesting process [24,25,26]. Moreover, the mechanical efficiency of such systems is highly sensitive to structural parameters, as well as flow conditions, making their optimization a crucial area of study [2,27].

Despite substantial advancements in the field of aeroelastic energy harvesting, the majority of existing research predominantly emphasized systems with inherent structural nonlinearities, often overlooking the significant role of nonlinear fluid dynamics. To bridge this critical gap, the present study examines the impact of purely fluid-driven nonlinearities on the efficiency of energy harvesting in fluttering airfoils. A strongly coupled electro-aeroelastic model is employed to analyze the dynamics of flutter oscillations in a structurally linear system, wherein nonlinearity is exclusively attributed to fluidic interactions. The influence of these fluid-induced nonlinearities on energy harvesting is explored in terms of output power, system stability, and bifurcation behavior as the reduced velocity increases. Specifically, our current work investigates how the onset of nonlinearity affects power output, potentially triggering dynamic stall and altering the nature of oscillations. The occurrence of subharmonic and superharmonic components is observed in the system, revealing insights into symmetric or asymmetric behaviors as well as vortex shedding patterns that cause large-amplitude oscillations, regardless of the specific type of nonlinearity present. Furthermore, this study discusses the implications of a system’s stability on energy conversion efficiency and provide insights about the conditions under which flow-induced nonlinearities enhance or suppress energy extraction. This approach facilitates a deeper understanding of aerodynamic instabilities and their contributions to energy harvesting performance, thereby offering pathways for optimizing airfoil-based harvesters in highly-unsteady low-speed flow environments [28,29,30,31]. While setting up the primary novelty of our present work here, the findings from this research are expected to bridge the gap in the current literature by providing new insights into the nonlinear fluid dynamics of fluttering airfoils and their implications for energy harvesting. Furthermore, the developed model offers a robust framework for optimizing flutter-based systems, aiming to enhance power output and reliability in real-world applications. This work primarily explores efficiency in terms of electrical output voltage, providing a comprehensive understanding of the design parameters that maximize energy extraction from fluid-induced oscillations.

2. Computational Methodology

To model the coupled dynamics of fluid flows and elastically mounted structures, we employ an Arbitrary Lagrangian–Eulerian (ALE) model [20,32,33]. The equations for the conservation of mass and momentum for incompressible flows in the ALE framework are expressed as follows.

$${\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{x}_j}}=0$$

(1)

$${\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }t}}+{\tilde{u}}_j{\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }{x}_i}}+{\displaystyle \frac{1}{Re}}{\displaystyle \frac{{\mathsf{\partial }}^2{u}_i}{\mathsf{\partial }{x}_j\mathsf{\partial }{x}_j}}$$

(2)

where $u_i=(u,v)$ are the Cartesian components of the velocity of the fluid, $(i,j)=(1,2)$ represent two Cartesian directions, $x_i$ represents a specific direction, $x_j$ represents the spatial coordinates, p is the pressure, and $\rho$ denotes the density of the fluid. We define the Reynolds number as $Re={\displaystyle \frac{\rho c{U}_{\infty }}{\mu }}$, with c as the chord length of the NACA-0012 airfoil, $U_{\infty }$ as the free-stream velocity, and $\mu$ as the dynamic viscosity of the fluid.

In the following expression, ${\tilde{u}}_j=(\tilde{u},\tilde{v})$ denotes the flow velocity relative to the grid, where $u_{gj}$ represents the grid velocity.

$${\tilde{u}}_j=u_j-u_{gj}$$

(3)

We use a non-staggered, body-fitted O-grid, as shown in Figure 1, which is 20C to accurately capture the boundary layer dynamics. In this configuration, all state variables, including the components of the flow velocity and pressure, are defined at the cell centers, while the fluxes are evaluated at the centers of the cell faces.

In the ALE framework, computational grids around moving structures are designed to adjust and deform dynamically in response to a structural body’s motion. This adaptive meshing strategy enables interior grid nodes to move in an arbitrary manner, optimizing the configuration of grid cells while maintaining mesh quality during deformation. This capability is particularly critical for simulations involving large-amplitude oscillations of both rigid and flexible bodies, where significant mesh distortions are expected. To achieve robust grid deformation, a re-meshing algorithm is employed alongside the flow solver. For the current simulations, we adopt the approach formulated by [34], which leverages the radial basis function (RBF) interpolation technique. This method efficiently propagates boundary displacements through the mesh, preserving quality of the grid and preventing inversions in the elements.

The RBF interpolation function $f\left({\mathbf{x}}_{\mathbf{j}}\right)$ for a set of boundary nodes on the structural surface is expressed as:

$$f\left({\mathbf{x}}_{\mathbf{j}}\right)=\sum _{i=1}^N{\alpha }_i\varphi (\parallel {\mathbf{x}}_{\mathbf{j}}-{\mathbf{x}}_{\mathbf{i}}\parallel )+p_1\left({\mathbf{x}}_{\mathbf{j}}\right),$$

(4)

where $p_1\left(x_j\right)$ is a minimal ploynominal of degree one, and ${\alpha }_i$ are the coefficients of the basis function. This interpolation strategy effectively maps the structural displacements from the boundary to the entire computational domain, ensuring smooth mesh deformation without compromising the geometric integrity of the cells. The interpolation function defined through the RBF method relies on a set of control points distributed along the boundary of the structure. The total number of these control points is represented by $N_c$, and the RBF function $\varphi (\parallel ·\parallel )$ is evaluated based on the Euclidean norm. For the current implementation, we employ the global support thin-plate spline (TPS) RBF, which is expressed as:

$$\varphi \left(x\right)=x^{2}log\left(x\right).$$

(5)

Additionally, $p_1\left({\mathbf{x}}_{\mathbf{j}}\right)$, satisfying the following condition:

$$\sum _{i=1}^{N_c}{\alpha }_i{p}_1\left({\mathbf{x}}_{\mathbf{cj}}\left(\mathbf{i}\right)\right)=0,$$

(6)

These coefficients are determined by ensuring that the interpolation function $f\left({\mathbf{x}}_{\mathbf{j}}\right)$ matches the prescribed displacements at the control points, given by:

$$f\left({\mathbf{x}}_{\mathbf{cj}}\right)=\Delta {\mathbf{x}}_{\mathbf{cj}},$$

(7)

where $\Delta {\mathbf{x}}_{\mathbf{cj}}$ represents the assigned displacement for each control point ${\mathbf{x}}_{\mathbf{cj}}$. The selection of these control points is performed through a greedy algorithm, as proposed by [35]. Once the interpolation function is established, the displacement of all interior nodes is computed accordingly. The next stage involves calculating the grid velocities by dividing the displacement of each node by the discrete time-step $\Delta t$. The velocity of a node ${\mathbf{x}}_{\mathbf{j}}$ is computed as:

$${\mathbf{u}}_{\mathbf{gj}}={\displaystyle \frac{f\left({\mathbf{x}}_{\mathbf{j}}\right)}{\Delta t}}.$$

(8)

These grid velocities are then used to evaluate the flux terms ($\tilde{U}$ and $\tilde{V}$) required for the flow solver. One of the primary challenges of employing the RBF-based remeshing technique is its computational cost, as it involves solving a dense linear system whose size scales with the number of control points $N_c$. Furthermore, the system becomes increasingly ill-conditioned as the number of control points increases, making iterative solvers impractical. Consequently, a direct linear solver, the Gaussian elimination method with partial pivoting, is employed to efficiently handle factorization of the coefficient matrix during the preprocessing stage. For an optimal performance, we adopt an explicit version of the RBF method combined with the greedy algorithm for the precomputation of the coefficient matrix. This strategy significantly reduces the computational load during runtime. Successive forward–backward substitutions are performed at each time-step to resolve the unknowns in the interpolation techniques, enhancing the overall efficiency of the simulation.

Despite the computational intensity, the RBF method demonstrates remarkable preservation of the quality of mesh during large translational and rotational displacements, offering superior robustness compared to traditional re-meshing techniques by [36]. Our implementation, which is fully parallelized, is capable of handling three-dimensional simulations for bodies with infinite spans. For the present study, involving flows over a two-dimensional foil, low Reynolds numbers are maintained to prevent spanwise flow instabilities. The impact of various design parameters on the structural response and wake dynamics of oscillating systems provides crucial details about designing alternate energy harnessing resources.

We solve the governing model for fluid flows using the fractional-step method. In this approach, the momentum equations are first solved independently for the velocity components, neglecting the continuity constraint during this stage. The pressure-gradient terms may either be omitted or considered as a known quantity. This results in the computation of an intermediate velocity field, denoted as ${\mathbf{u}}^{*}$. Subsequently, the continuity constraint is enforced by solving a pressure Poisson equation derived from the divergence-free condition of the velocity field. This step updates the pressure field and corrects the predicted velocities to ensure an incompressibility constraint.

To approximate the continuous differential operators, finite difference schemes are employed. Spatial derivatives, except for the convective terms, are discretized using a central difference scheme. For the convective terms, the Quadratic Upwind Interpolation for Convective Kinematics (QUICK) scheme [37] is used. This method utilizes a generic stencil formulation based on the direction of the flux, expressed as:

$${\tilde{U}}_j^{+}={\displaystyle \frac{{\tilde{U}}_j+\left|{\tilde{U}}_j\right|}{2}},{\tilde{U}}_j^{-}={\displaystyle \frac{{\tilde{U}}_j-\left|{\tilde{U}}_j\right|}{2}}.$$

(9)

The temporal term is discretized using a semi-implicit scheme, the diagonal diffusion terms are integrated using the implicit Crank–Nicolson method for its stability, while the remaining terms are updated through the two-step explicit Adams–Bashforth method. For additional details regarding these discretization schemes and boundary conditions, and the parallelization strategy, readers are referred to Refs. [11,20,29].

The resulting linear system of equations is solved using the successive over-relaxation (SOR) method, which is efficient for structured grid layouts. The stability of the proposed numerical strategy is governed by the Courant–Friedrichs–Lewy (CFL) condition, which requires that the maximum CFL number within the computational domain remains below unity. The local CFL number is defined as:

$$\mathrm{CFL}=\left({\displaystyle \frac{\left|u\right|}{\Delta x}}+{\displaystyle \frac{\left|v\right|}{\Delta y}}\right)=\Delta t{J}^{-1}\left(\right|\tilde{U}|+|\tilde{V}\left|\right),$$

(10)

where u and v are the components of flow velocity in the x and y directions, respectively, and J denotes the Jacobian.

2.1. Aerodynamic Forces and Moments

The interactions between a fluid and a solid body immersed in it result in forces predominantly attributed to pressure and shear stress distributions over the surface of the body. The aerodynamic or hydrodynamic performance of such systems is typically quantified through three key nondimensional metrics: the lift coefficient ($C_L$), the drag coefficient ($C_D$), and the moment coefficient ($C_M$). These coefficients are mathematically defined as follows:

$$C_L={\displaystyle \frac{F_y\left(t\right)}{\frac{1}{2}\rho U_{\infty }^{2}L}},C_D={\displaystyle \frac{F_x\left(t\right)}{\frac{1}{2}\rho U_{\infty }^{2}L}},C_M={\displaystyle \frac{M_{\theta }\left(t\right)}{\frac{1}{2}\rho U_{\infty }^2{L}^2}}$$

(11)

Here, $F_x\left(t\right)$ and $F_y\left(t\right)$ represent the instantaneous drag and lift forces exerted on the structure, respectively, while $M_{\theta }\left(t\right)$ denotes the moment about the axis located at $x_p$ associated with the elastic supports. Here, c, being the chord length of the foil, shows the characteristic length of the structure. These nondimensional metrics serve as the important indicators for evaluating the performance and dynamic response of the systems undergoing fluid–structure interactions.

2.2. Electromechanical Model

The electro-mechanical coupling in the energy harvesting system illustrated in Figure 2 primarily relies on the structural motion of the airfoil to generate electrical power via piezoelectric transducers. The governing equations of the coupled aero-elastic system, expressed in a non-dimensional form, characterize the relationship between the structural displacement and the induced voltage. Specifically, the non-dimensional plunging displacement is represented by Equations (12) and (13) referenced from [33].

$${\ddot{h}}^{*}+2{\zeta }_h\left({\displaystyle \frac{2\pi }{U_r^{*}}}\right){\dot{h}}^{*}+{\left({\displaystyle \frac{2\pi }{U_r^{*}}}\right)}^2{h}^{*}-S^{*}\left(cos\theta ·\ddot{\theta }-sin\theta ·{\dot{\theta }}^2\right)-{\left({\displaystyle \frac{1}{U_r^{*}}}\right)}^2{V}^{*}={\displaystyle \frac{2C_L}{\pi m^{*}}}$$

(12)

$$\ddot{\theta }+2{\zeta }_{\theta }\overline{f}\left({\displaystyle \frac{2\pi }{U_r^{*}}}\right)\dot{\theta }+{\left({\displaystyle \frac{\overline{f}2\pi }{U_r^{*}}}\right)}^2\theta -{\displaystyle \frac{S^{*}}{r_{\theta }^2}}cos\theta ·\ddot{h}={\displaystyle \frac{2C_m}{\pi r_{\theta }^2{m}^{*}}}$$

(13)

$${\dot{V}}^{*}+{\sigma }_1{\dot{h}}^{*}+{\displaystyle \frac{{\sigma }_2}{U_r^{*}}}{V}^{*}=0$$

(14)

where $h^{*}=h/c$. In this context, h represents the heaving displacement, whereas c is considered to be unity. The pitching displacement is denoted as $\theta$. Additionally, the radius of gyration, which characterizes the distribution of the airfoil’s mass relative to its axis of rotation, is defined as $r_{\theta }=\sqrt{{\displaystyle \frac{I_{\theta }}{mh{L}^2}}}$. The ratio of natural frequencies is expressed as $\overline{f}={\displaystyle \frac{f_{\theta }}{f_h}}$. For simplification, we set $f_h$ and $f_{\theta }$ such that $\overline{f}=1$, ensuring synchronized pitching and heaving motion. The spring constants are constrained by the relationship ${\displaystyle \frac{k_{\theta 1}}{k_{h1}}}={\displaystyle \frac{mh}{I_{\theta }}}$, as presented in Ref. [12]. Finally, the static imbalance, which reflects the ratio of the distance between the location of the elastic axis and the center of mass to the chord length, is represented as $S^{*}={\displaystyle \frac{b}{c}}$, and is maintained at a fixed value of $-0.04$ in this work.

The natural structural frequency of the airfoil is expressed as:

$$f_h={\displaystyle \frac{\sqrt{k_{h1}/m_h}}{2\pi }}$$

(15)

where $k_{h1}$ represents the linear structural stiffness, and $m_h$ is the mass per unit length of the structure. To model flow-induced forces acting on the airfoil, a forcing term is included in the governing equations for a direct representation of the FSI phenomenon. This approach captures the complex aerodynamic loads experienced by the airfoil during oscillations.

The system’s linear structural damping ratio associated with the heaving motion is described by:

$${\zeta }_h={\displaystyle \frac{c_{h1}}{c_{\mathrm{crit}}}}={\displaystyle \frac{c_{h1}}{2\sqrt{k_{h1}{m}_h}}}$$

(16)

The mass ratio is defined as:

$$m^{*}={\displaystyle \frac{m_h}{m_f}}$$

(17)

where $m_f=\rho \pi L^2/4$ represents the mass of the fluid displaced and $m^{*}=10$. In the context of energy harvesting, the introduction of piezoelectric coupling results in additional non-dimensional parameters. The non dimensional voltage is defined as $V^{*}={\displaystyle \frac{V}{V_0}}$, where the reference voltage $V_0$ is expressed as $V_0={\displaystyle \frac{m_h{f}_h^{2}L}{{\theta }_L}}$, and V represents the generated voltage. Here, ${\theta }_L$ represents the electromechanical coupling coefficient, which has a value of $1.55\times {10}^{-3}\mathrm{N}/\mathrm{V}$ [32]. Parameters ${\sigma }_1={\displaystyle \frac{{\theta }_L^2}{m_h{C}_p{f}_h^2}}$ and ${\sigma }_2={\displaystyle \frac{1}{R{C}_p{f}_h}}$ are used to characterize the piezoelectric response, where $C_p$ is the capacitance, $1.2\times {10}^{-7}$ measured in nanofarads (nF), and R is the load resistance. The first-order ODE expresses the rate of change of voltage in the first term, the piezoelectric coupling strength in the second term, and the energy dissipation is shown by the third term. In addition, $I_{\theta }$ denotes the mass moment of inertia, and r is defined as the radius of gyration, which is $0.548c$ from the axis of rotation.

For numerical simulations of this coupled electro-aeroelastic system, the fluidic loads are computed simultaneously with the structural response, as they are strongly coupled. A Hamming fourth-order predictor–corrector method is employed for time integration, ensuring stability and accuracy. A consistent time-step of $8\times {10}^{-4}$ s is utilized by both the fluid and structural solvers to achieve convergence towards steady-state periodic oscillations.

2.3. Validation and Verification

To ensure mesh-independent results, we perform simulations using three different structured grid sizes: $196\times 256$ (coarse), $304\times 400$ (medium), and $464\times 600$ (fine). The computational setup involved a NACA0012 airfoil undergoing combined pitching and heaving motions at a Reynolds number $Re=1100$. The reduced velocity is defined as $U_r^{*}={\displaystyle \frac{U_{\infty }}{f·c}}$, where $U_{\infty }=1$, and f is the oscillation frequency. We analyze the unsteady aerodynamic response in terms of $C_L$ over non-dimensional time. As shown in Figure 3, the medium and fine grids produced nearly overlapping $C_L$ profiles, indicating negligible variations with the increase in grid size. To quantitatively assess convergence, we compute the root-mean-square (RMS) values of $C_L$ at the statistically steady state, which are found to be $0.800165$, $0.878834$, and $0.884199$, as shown in

Table 1. Grid resolution and corresponding RMS values of $C_L$.

Grid	${\mathit{N}}_{\mathit{x}}$	${\mathit{N}}_{\mathit{y}}$	${\mathit{C}}_{\mathit{L}}$ (RMS)
G1	196	256	0.800165
G2	304	400	0.878834
G3	464	600	0.884199

, for the coarse, medium, and fine grids, respectively. The relative error in the RMS value for $C_L$ is approximately $9.52\%$ for the coarse mesh vs. medium, and 0.61% for the medium mesh vs. fine mesh. Based on this analysis, the medium grid is selected for all subsequent simulations, providing an optimal balance between computational cost and solution accuracy.

Using the medium grid size, we also conducted a convergence study to determine an appropriate time-step size for the present simulations, and we present its results in Figure 4. Three different time-steps were tested and calculated based on 5000, 8000, and 10,000 time-steps per oscillation cycle of the airfoil. The time period $\tau$ was chosen based on the results presented by Farooq et al. [32]. The flow parameters used here are consistent with those in the grid-independence study. The computed RMS values of $C_L$ for the three cases were 0.890442, 0.892498, and 0.894194, as listed in

Table 2. Convergence study for different time-step resolutions and their impact on $C_L$ (RMS).

Time-Steps	Grid	${\mathit{C}}_{\mathit{L}}$ (RMS)
5000	Medium	0.890422
8000	Medium	0.892498
10,000	Medium	0.894194

. The relative error between the results from the cases with 5000 and 8000 time-steps in one cycle is 0.4196%, whereas the error between the results for 8000 and 10,000 time-steps per cycle further reduces to 0.1897%. These results indicate that the solutions become increasingly insensitive to further refinement in the time-step size. Please note that 8000 time-steps per oscillation cycle correspond to $\Delta t=0.000125\mathrm{s}$. Please note that further details on validation of the results obtained through our in-house solver are available in Refs. [32,33].

3. Results and Discussion

We simulate flows over a two-degrees-of-freedom ($2\mathrm{D}\mathrm{o}\mathrm{F}$) NACA0012 airfoil at a Reynolds number of 1100, with a translational damping ratio of $0.008$ (${\zeta }_h$) and rotational damping ratio of $0.050$ (${\zeta }_{\theta }$), selected based on $m^{*}$ and moment of inertia. The simulations were conducted over a range of $U_r^{*}$ from 1 to 12, considering it as the primary varying parameter that influences both effective damping and stiffness. For each case, we computed the RMS values of the heaving displacement ($h^{*}$), pitch angle ($\theta$), $C_L$, and $C_D$, and plot them as functions of $U_r^{*}$ under different values of R, as shown in Figure 5.

From these plots, it is evident that for $U_r^{*}\in [1,5]$, the airfoil remains in a stable regime with no observable oscillatory response, irrespective of the value of the resistance. At approximately $U_r^{*}=6$, a small amplitude emerges, indicating the onset of self-excited oscillations. As $U_r^{*}$ increases to 7, sustained oscillations develop with amplitude progressively increasing up to a certain reduced velocity. This behavior is consistently reflected in the variations in $h^{*}$, $\theta$, $C_L$, and $C_D$. In Figure 5a, which presents the evolution of $h^{*}$, the amplitude increases progressively beyond $U_r^{*}=6$ for $R=0\mathrm{k}\Omega$, reaching a maximum at $U_r^{*}=11$, after which it decreases. For $R>0$, oscillations are initiated when $U_r^{*}$ approaches 6. The peak amplitude occurs at $U_r^{*}=11$ for both $R=250\mathrm{k}\Omega$ and $500\mathrm{k}\Omega$, whereas for $R=750\mathrm{k}\Omega$, these peak values get shifted to $U_r^{*}=10$. This trend indicates that the peak heaving amplitudes are shifted toward lower reduced velocities with an increasing electrical resistance. In Figure 5b, the amplitude of the pitching angle reaches its maximum at $U_r^{*}=8$ for $R=0$, whereas for $R>0$, the maximum shifts to $U_r^{*}=10$. Furthermore, Figure 5c,d exhibit similar trends for $C_L$ and $C_D$, respectively, with maximum amplitudes occurring at $U_r^{*}=8$ for $R=0$ and at $U_r^{*}=10$ for $R>0$. Afterwards, the amplitude in $\theta$, $C_L$, and $C_D$ start decreasing again, which shows that the system may be turning back to the non-oscillatory state due to the load-induced damping. These trends in Figure 5a deviate from this trend for $R=0$, indicating that the heaving amplitude demonstrates a delay in showing the same pattern as the other parameters. Overall, the plots show that the oscillation amplitudes decrease with an increasing resistance, and the bifurcation point is shifted toward higher reduced velocities. Beyond $U_r^{*}=10$, the amplitudes are reduced, implying that the damping effect introduced by electrical resistance dominates the system dynamics, thereby attenuating oscillations and promoting a return to non-oscillatory behavior, as explained in detail in the phase portraits discussion.

From Figure 5, representative cases are selected based on the highest-amplitude responses observed in $C_L$, $C_D$, and $\theta$. Specifically, the case with $R=0\mathrm{k}\Omega$ at $U_r^{*}=8$ is selected, while for nonzero resistance ($R>0$), $U_r^{*}=10$ is chosen for further analyses. Temporal histories of $C_L$ are presented in Figure 6 after the system reaches its steady-state dynamic response. The lift coefficient is of particular importance here, because it plays a critical role in quantifying the underlying fluid–structure interactions as the key indicator of the vortex-shedding frequency. Therefore, the understanding of this parameter is directly linked to the energy harvesting potential of the system. Figure 6a presents the temporal histories of $C_L$, corresponding to $U_r^{*}=8$ and $R=0\Omega$. Here, $C_L$ exhibits large amplitudes and periodic limit cycle oscillations with consistent peak-to-peak variations. The waveform appears non-smooth and skewed, indicating asymmetry and highlighting the presence of nonlinearities in the system. These profiles serve as the indicators of how nonlinearities might kick in at different control parameters to determine the nature of the response of this nonlinear aeroelastic energy harvesting system.

As the load resistance increases to $R=250\mathrm{k}\Omega$ in Figure 6b, the waveform becomes slightly smoother and less distorted, accompanied by a noticeable reduction in the amplitude. This behavior reflects the damping effect introduced by the electrical load. A similar trend is observed in Figure 6c at $R=500\mathrm{k}\Omega$, where the signal’s profile becomes smoother, and the amplitude further decreases, suggesting a stronger damping. At $R=750\mathrm{k}\Omega$ in Figure 6d, the waveform exhibits minimal asymmetry and skewness, indicating that the nonlinear characteristics are largely suppressed. The increased load-induced damping potentially reduces the lift and shifts the system towards near-stable state by extracting energy from the system. These observations highlight the critical role of the aerodynamic response through resistive damping, which suppresses the nonlinearity, as reflected in the progressive changes in the shape and amplitude of the waveforms.

To further investigate the presence of nonlinearities in the system, the frequency spectra of the lift and drag coefficients are analyzed by performing a Fast Fourier Transform (FFT). The analysis focuses on cases at $R=0\mathrm{k}\Omega$ for reduced velocities $U_r^{*}\in [7,9]$, which capture the transition and growth of nonlinear behavior, as reflected by the evolution of spectral peaks. A similar trend is observed for $R>0$, where cases corresponding to $U_r^{*}\in [10-12]$ are considered. These represent the progression from maximum $C_L$ amplitudes toward lower oscillation levels and suppression of the nonlinearity as R increases, which causes damping of the system and affects the nonlinear dynamics.

Next, we perform the Fast Fourier Transform (FFT)-based analysis of the $C_L$ for all cases across the full range of reduced velocities considered here. However, only the representative cases discussed above are presented in Figure 7 and Figure 8. For $U_r^{*}\in [1,5]$, the system does not exhibit significant oscillatory behavior. At $R=0\mathrm{k}\Omega$ and $U_r^{*}=6$, where bifurcation occurs, the frequency spectrum displays a single dominant peak with a low amplitude, indicating a predominantly linear response. As $U_r^{*}$ increases from 7 to 12, the spectra reveal multiple frequencies, signifying the emergence and development of nonlinearity. In the presence of a non-zero low load resistance, a similar multi-frequency response is observed starting from $U_r^{*}=7$ to 12, and these multi-frequencies and a broadband spectrum can give an idea about the chaotic state of the system. Nevertheless, we observe this response at up to a certain higher $U_r^{*}$ as the load increases. The fundamental frequency is the dominant frequency observed in the system’s dynamic response. It corresponds to the primary mode of vibration activated under the given excitation conditions. The natural frequency, on the other hand, is an inherent property of the structure. It is the frequency at which the structure vibrates freely when displaced without external forces or damping. In our case, the fundamental frequency stays close to the natural frequency but does not exactly match it, which can be seen in

Table 3. Fundamental and natural structural frequencies for $R=0$–$750\mathrm{k}\Omega$ at various reduced velocities.

${\mathit{U}}_{\mathit{r}}^{*}$	Fundamental Frequency (Hz)	Natural Structural Frequency (Hz)
6	0.16999	0.16666
7	0.14500	0.14285
8	0.13000	0.12500
9	0.11500	0.11111
10	0.10500	0.10000
11	0.09499	0.09090
12	0.08999	0.08333

, where this slight difference may be a numerical artifact. Consistent dominant frequency appears across all values of the resistance, including ($0\mathrm{k}\Omega$, $250\mathrm{k}\Omega$, $500\mathrm{k}\Omega$, and $750\mathrm{k}\Omega$). The corresponding frequency values are also presented in Table 3. It is important to notice that the fundamental frequency starts differing from the natural structural frequency at higher values of $U_r^{*}$.

The FFT spectra of $C_L$ reveals significant contributions from both odd and even harmonics of the dominant frequency ($f_{\circ }$), indicating the presence of cubic, as well as quadratic, nonlinearities in the system, as illustrated in Figure 7 and Figure 8. None of these spectra exhibit subharmonics that indicate an absence of period doubling and also incommensurate frequencies. The observed superharmonic peaks correspond to nonlinear effects arising from quadratic and cubic terms in the system dynamics. In particular, the quadratic nonlinearities contribute to the asymmetry of the response, and the distribution of energy across the superharmonics as integer multiples of the fundamental frequency (i.e., $2f_{\circ }$, $3f_{\circ }$, and $4f_{\circ }$, etc.) [24]. We make similar observations for the spectra of $C_L$ at higher $U_r^{*}$ from Figure 7 and Figure 8.

The superharmonics and nonlinearities sensitive to variations in load resistance. At $R=0\mathrm{k}\Omega$, both odd and even harmonics appear distinctly. As $U_r^{*}$ and R increase, the amplitudes of these harmonics progressively decrease. In the spectra of $C_L$, the decay of harmonic amplitudes follows a characteristic sequence, where even harmonics diminish first, followed by the odd harmonics. This asymmetric decay pattern in the frequency spectrum was also previously discussed by Hammond et al. [38]. In our work, this trend is evident in Figure 7a–f and Figure 8g–l. At $U_r^{*}=12$ and $R=750\mathrm{k}\Omega$, the FFT spectrum shows a single dominant frequency with very small superharmonics with minimal spectral energy spread over other frequencies, which indicates the system begins to exhibit quasi-nonlinear behavior, characterized by the presence of weak nonlinearity and evidenced by low-amplitude oscillations in the lift coefficient $C_L$. This low amplitude trend also appears in the corresponding time histories in Figure 6.

In order to further examine the absence of any subharmonics in the system’s response, we also plot the spectra of $C_D$ in Figure 9. However, these FFT spectra indicate the presence of only superharmonics, with the second harmonic (at twice the fundamental frequency) being dominant. Even harmonics exhibit higher amplitudes than odd ones, suggesting that even harmonics contribute more significantly to the signal energy and are more prominent in the spectral content, as shown in Figure 9 across different values of load resistance. Since these spectra do not provide any indication of more frequencies appearing in the system with the increasing values of the control parameters, it does not help much in detecting a possible transition to a chaotic response of the system.

To address this limitation, phase portraits of $C_L$ versus $C_D$ are examined. These visualizations offer qualitative insight into the system’s phase-space dynamics, revealing important characteristics about how the system evolves and behaves over time and give insights about the underlying nonlinearities. We construct the phase maps using $C_L$ and $C_D$ to examine the temporal evolution of the system by visualizing its trajectories, as shown in Figure 10 and Figure 11. The phase portraits display increasing skewness, particularly along the $C_D$ towards positive values on the x-axis, with the shape spread more toward the right side compared to the left, where $C_L$ is increasing till $U_r^{*}=8$. Afterwards, it is narrowed and less skewed as $C_L$ is reduced (except the one in Figure 10a, which shows the quasi-nonlinearity on a very small scale). Figure 10c exhibits the most pronounced asymmetry, which is an indicator of strong quadratically nonlinear interactions. As the load resistance increases, the degree of skewness and distortion in the phase portraits progressively decreases, suggesting a suppression of quadratically nonlinear effects due to the enhanced electrical damping. Furthermore, phase portraits are analyzed while varying the $U_r^{*}$. At $R=0\mathrm{k}\Omega$ and $U_r^{*}=6$, the airfoil exhibits low-amplitude oscillations dominated by a single frequency. The corresponding phase portrait in Figure 10a displays a thick, closed loop trajectory with smooth evolution, indicating a near-periodic response driven by fluid–structure interactions.

As $U_r^{*}$ increases to 7 and beyond, the amplitude of $C_L$ progressively increases. At $U_r^{*}=8$, the trajectory forms a thinner and well-defined closed loop, as shown in Figure 10c, signifying the occurrence of a stable periodic limit-cycle oscillation (LCO) [23,24]. This double-looped map between $C_D$ and $C_L$ also indicates that $C_D$ is double the frequency of $C_L$. A complete loop appears for $C_D$, while a half-closed loop is observed for $C_L$, suggesting that $C_D$ varies at twice the frequency of $C_L$, as shown in Figure 10. The airfoil sustains high amplitudes of $C_L$ and $C_D$ through $U_r^{*}=8$ in Figure 10c. As $U_r^{*}$ increases to 12, the phase portraits exhibit non-uniform trajectory, with distorted loop and irregularity, indicating a transition toward chaotic behavior, as illustrated in Figure 10e–g.

To examine the system’s behavior beyond this state, a simulation at a higher $U_r^{*}$ is conducted. The resulting phase portrait displays a non-uniform open-loop-shaped trajectory in Figure 10h,i, which is a characteristic of chaotic dynamics. In this case, both $C_L$ and $C_D$ values are increasing more aggressively, which shows that the aerodynamic forces are more dominant on the structure and greater and dominant nonlinearities coming from the fluid rather than the structure itself.

When the electrical load resistance is activated, the transition path changes. Initially, the trajectory becomes thicker, and $C_L$ and $C_D$ increase with $U_r^{*}$. The system enters a limit-cycle oscillatory regime around $U_r^{*}=10$, as shown in Figure 11, with maximum amplitudes and a wider, closed-loop trajectory. At a higher $U_r^{*}$ and increased resistance, the oscillation becomes damped, and the airfoil’s trajectory begins to form a smooth, closed-loop trajectory. However, the time required to achieve LCO increases with R and $U_r^{*}$, showing an increase in damping and a shifting system towards quasi-nonlinearity, as shown in Figure 8. It shows peaks for the dominant frequency with very small superharmonics, also reflected by the very small closed loop in Figure 11i. Please note that we do not consider this specific response as strictly linear due to the presence of very weak superharmonics in the system’s response.

To further characterize the underlying dynamics, Poincaré maps were constructed. This technique provides insight into the system’s qualitative behavior and allows differentiation of periodic, quasiperiodic, and chaotic regimes based on the spatial structure and distributions of points on the map. The Poincaré maps are plotted for the same cases at different values of R, where higher values of $C_L$ are observed. For $R=0\mathrm{k}\Omega$ maps are presented at $U_r^{*}=8$ and 13. The Poincaré sections are constructed for three characteristic conditions with $C_L$ versus $C_D$, when the heaving amplitude is at its maximum, zero, and minimum values.

For $R=0\mathrm{k}\Omega$ at $U_r^{*}=8$, Figure 12a–c show that the points initially appear far from each other but gradually converge or overlap as time progresses. This behavior indicates that the system is going from the quasi-nonlinear to the periodic nonlinear state, where the system continues coming to the same point in the Poincaré section and does not change path over time at the steady state. This can be seen in Figure 12 a–c, and the same behavior from this system is seen up to $U_r^{*}=12$.

As $U_r^{*}$ increases further, the Poincaré points in Figure 12d–f initially appear clustered and subsequently diverge, forming scattered distributions. This behavior indicates that the system transitions from a periodic LCO state towards chaos via a crisis route, as evidenced by the phase portrait in Figure 10h,i. The trajectory is no longer periodic and appears highly distorted, forming an open loop with irregular bursts as $U_r^{*}$ crosses a critical value. The corresponding FFT spectra show no evidence of subharmonics, thereby excluding period-doubling, period-n, and no signs of the incommensurate frequencies indicative of a quasi-periodic state. The increase in $U_r^{*}$ enhances the dominance of aerodynamic forces more strongly on the structure. An increase in $U_r^{*}$ lowers the damping and spring stiffness, causing the system to follow a non-uniform path that does not revisit the same location in the Poincaré section. This behavior is shown in the phase portraits in Figure 10h,i, which illustrate the onset of chaotic dynamics. Contrarily for higher load resistances, e.g., $R=250$ kΩ (Figure 12g–i), $R=500$ kΩ (Figure 13a–c), and $R=750$ kΩ (Figure 13d–f), the points converge more slowly. As R increases, the Poincaré points are more closely packed, indicating that the system takes significantly longer time to transition from a transient regime to a periodic steady state. It also highlights the damping contributed by the load resistance in the system, which does not let the system go towards the chaotic state. Additionally, the phase portraits and Poincaré maps indicate that the system exhibits periodic behavior when load resistance is applied, whereas in the absence of load resistance, the system transitions to chaotic behavior beyond a $U_r^{*}=12$ through the crisis route.

Next, the phase map is plotted to capture the system’s behavior across a range of reduced velocities from $U_r^{*}=6$ to $U_r^{*}=13$, as introduced earlier. To further examine the system dynamics at $U_r^{*}=13$ at $R=0$, with significance in revealing chaotic behavior, as previously discussed through phase portraits and the Poincaré maps, we examine the crisis route. Now, we present a map for nonlinear solutions on a $U_r^{*}-\mathrm{R}$ plane in Figure 14. For $U_r^{*}=6$, the triangular pattern observed in the phase map indicates quasi-nonlinear behavior. As $U_r^{*}$ increases, the system transitions into a periodic nonlinear regime up to $U_r^{*}=12$. Beyond this point, the response becomes chaotic, as identified in the plot by a circle for the case with no resistance. However, for a resistance of $R=250\mathrm{k}\Omega$, the system exhibits quasi-nonlinear behavior initially, transitions to periodic nonlinear behavior at $U_r^{*}=13$. As the resistance increases to $R=500\mathrm{k}\Omega$, the system demonstrates quasi-nonlinear behavior up to $U_r^{*}=7$, transitions into a periodic nonlinear regime, and then reverts to quasi-linear behavior at $U_r^{*}=13$ again. With a further increase in resistance to $R=750\mathrm{k}\Omega$, the system demonstrates quasi-linear behavior up to $U_r^{*}=7$, becomes periodic nonlinear up to $U_r^{*}=11$, and again returns to quasi-linear dynamics. The phase map reveals that an increasing resistance progressively constrains the range of nonlinear behavior with respect to the reduced velocity, highlighting a resistance-induced modulation of the system’s nonlinear characteristics that potentially provides additional damping characteristics to the system.

It is insightful to characterize the vortex patterns associated with distinct nonlinear states of the system described earlier. Now, we provide the phase map to explain the vortex shedding or flow regimes corresponding to each half cycle of the oscillating foil in Figure 15. We observe various flow regimes emerging across different combinations of resistance R and reduced velocity $U_r^{*}$. The filled contours indicate that a shear layer remains attached to the foil’s surface, corresponding to low-amplitude responses. This behavior is consistent with the low-amplitude lift coefficients observed in Figure 5 and is also associated with the quasi-nonlinear regime indicated in Figure 14. In Regime 2, the vortex shedding pattern transitions to an S (single) pattern, highlighted by triangular symbols in the phase map. This triangular pattern corresponds to the nonlinear regime identified in Figure 14. As the flow and design conditions are further varied, the wake evolves to exhibit multiple vortex shedding patterns namely, single vortices (S), vortex pairs (P), and vortex triplets (T, three vortices shed at the same time). This combination of S, P, and T structures named as multiple pair (mP) reflects an increase in flow complexity and is indicative of enhanced nonlinear behavior. To explain it further, The S pattern involves the shedding of a single vortex in each half oscillation cycle, whereas the P pattern involves shedding of a pair of counter-rotating vortices in each half undulation cycle. These multi-structure vortex patterns are attributed to the periodic nonlinear behavior explained in Figure 14. For the chaotic regime, the system exhibits a more disordered shedding pattern, denoted by circular marker in both Figure 14 and Figure 15. At this stage, the dominant shedding structure follows an mP (multiple pattern), along with an emergent hybrid pattern $\mathrm{P}+\mathrm{S}$ highlighted as Regime 4, demonstrating complex nonlinear dynamics.

As we have discussed, the vortex shedding patten correlates with the non-linearity present in the system and how it changes with varying $U_r^{*}$ and R, we describe the vortex and wake dynamics corresponding to the selected cases previously analyzed. For $U_r^{*}$ ranging from 1 to 6, the shear layer remains attached to the airfoil’s surface, which can also be seen in the phase map in Figure 14, and no significant oscillations are observed in this case. However, once the system reaches the critical point, as shown in Figure 5, vortex shedding initiates at a $U_r^{*}=7$ at lower R. The airfoil starts undergoing synchronous heaving and pitching motions with the forming S pattern. As R increases, that S pattern diminishes as well because of the load-induced damping. Next, the shedding of vortices continues with the oscillations, which promotes boundary-layer separation due to the presence of a strong adverse pressure gradient over the airfoil. This separation further increases with $U_r^{*}$, which leads to the formation of flow features in each half-cycle that deviates from the classical von Kármán vortex street or other typical configurations like $2\mathrm{S}$ or $1\mathrm{P}$. The number of shed vortices increases as the foil achieves larger oscillation amplitudes potentially due to a less damping. The spectra of $C_L$ also experiences multiple frequencies that could be attributed to more aggressive shedding of vortices. In this work, the observed vortex shedding patterns include formation of a variety of distinct flow structures, such as S, $\mathrm{m}\mathrm{P}$ which has (S,P,T) and $\mathrm{m}\mathrm{P},\mathrm{P}+\mathrm{S}$, which was defined earlier, and also reported by Wang et al. [12].

Here, we discuss the selected cases that represent the pattern of vortex shedding. At low R and $U_r^{*}$, vortex shedding initiates around $U_r^{*}=7$ with an S-type pattern. However, at $U_r^{*}=8$, the shedding becomes more aggressive as the airfoil attains higher oscillation amplitudes, fully developing nonlinear characteristics. Also, an unsteady aerodynamic phenomenon occurs and an airfoil experience rapidly changes in the angle-of-attack, leading to delayed flow separation, the formation and shedding of large leading-edge vortices (LEVs), and sudden variations in lift and moment. This phenomenon, observed in Figure 16 and Figure 17 under cases, which attain maximum amplitude, indicates the occurrence of dynamic stall. For this phenomenon, the foil experiences an instantaneous angle-of-attack greater than the static stall angle for the Reynolds number considered here [39]. In Figure 16a, the foil moves in the upstroke direction, in this instant, a low-pressure region develops near the leading edge on the top surface, while the bottom surface experiences a relatively higher pressure. This pressure difference induces strong suction near the leading edge and leads to the formation of a coherent leading edge vortex (LEV). As the foil continues to rise, both lift and effective angle-of-attack increase, causing the LEV to grow in size and form coherent structure. When the foil reaches the peak of its upstroke, i.e., the point where the angle-of-attack is maximum (see Figure 18), the boundary layer on the upper surface cannot remain attached and begins to detach from around the mid chord due to the fast reversal of the foil. The LEV subsequently rolls up and detaches from the surface and goes away from the foil, leading to a sudden drop in lift. At a later stage, the LEV convects toward the trailing edge and is recaptured by the foil, where it creates a low-pressure region, contributing to a gain in lift. The foil experiences dynamics stall here, which can be seen in the temporal profiles as well (see Figure 6). We observe that this recapturing of vortices causes a noticeable delay in vortex shedding as well, as illustrated in Figure 16 and Figure 17. At the instant of peak lift, the flow sheds an S-type vortex, subsequently followed by paired (P) and triplet (T) vortices, indicating a progressive increase in flow instability.

During the upstroke, the airfoil sheds counter-clockwise (CCW) rotating vortices, while clockwise (CW) vortices are shed during the downstroke, maintaining symmetry with respect to the stroke direction. The S-vortex consistently appears as the dominant structure in terms of size and circulation. As previously discussed, the LEV rolls up, detaches, and occasionally reattaches downstream near the trailing edge, where it interacts constructively with a trailing-edge vortex of the same orientation. Additional vortices form and shed from the trailing edge during the downstroke as the effective angle-of-attack increases again. This shedding behavior persists across different R. Comparative phase plots reveal that high amplitudes of $C_L$ correspond to the periodic nonlinear behavior, which is often associated with the $mP$ vortex shedding pattern. However, as R increases along with $U_r^{*}$, the system gradually transitions toward a quasi-linear regime. This transition is also evident in Figure 16 and Figure 17, as the vortex activity remains high at lower R, and this activity gradually reduces with increasing R and $U_r^{*}$.

However, the vortex pattern observed for the case with $R=0$ and $U_r^{*}=13$, which shows a chaotic behavior, is presented in Figure 19. The oscillation amplitude becomes significantly high, with the LEV causing huge loss in lift subsequently due to its detachment from the foil’s surface. In this case, a new vortex shedding mode also emerges, characterized by a combined $P+S$ pattern with multiple paired vortices ($mP$). In Figure 19a–f, the LEV grows progressively in size, indicating an increase in the force exerted by the fluid on the structure. This sustained growth of the LEV and the enhanced interaction between the vortices are indicative of a transition toward a chaotic state, driven by the increasing unsteady aerodynamic loads over time.

As mentioned above, Figure 5 illustrates the variation of $C_L$ with $U_r^{*}$, where it is evident that the range of the bifurcation region narrows as the electrical resistance R increases. This trend indicates that the onset and growth range of oscillations becomes progressively limited with higher resistive loading due to enhanced damping effects.

The further investigation is based on the voltage output during the times when the airfoil start to oscillate and attain max amplitude and, further, we discussed that the voltage generates at different $U_r^{*}$ and R. Furthermore, the investigation extends to the electrical response of the system, where the non-dimensional output voltage is computed by the solver at each time-step. Once the system reaches a steady oscillatory state, the root-mean-square (RMS) value of the non-dimensional voltage is evaluated using the formulation explained in Section 2.2. Now, Figure 20 presents a plot of $V_{rms}^{*}$ as a function of the reduced velocity. The voltage profile follows the bifurcation pattern evident in the dynamic response of the system (see in Figure 20 and $C_L$ vs. $U_R^{*}$ in Figure 5). As the airfoil undergoes self-sustained oscillations, the resulting mechanical vibrations induce voltage through the piezoelectric coupling mechanism. The trend of changes in voltage closely follows that of $C_L$, where larger oscillation amplitudes of $C_L$ correspond to elevated voltage output. The plot also reveals how voltage generation varies across $U_r^{*}$ and R. A comparison with the phase portraits (see Figure 11a,d,g) and vortex shedding patterns (see Figure 16 and Figure 17) indicates that, at higher amplitudes, the foil sheds a greater number of vortices. These instances coincide with the nonlinear periodic behavior in the structural response and elevated electrical output. Conversely, as the system transitions toward a quasi-linear state, both the vortex shedding activity and the voltage amplitude decrease, indicating a reduction in energy transfer from the fluid to the structure. From the profiles of voltage, it is found that more voltage is generated at higher value of R and at the $U_r^{*}$, where the foil has the maximum $C_L$.

4. Conclusions

This study investigates the nonlinear aeroelastic dynamics and energy harvesting performance of a two-degrees-of-freedom NACA 0012 airfoil under varying reduced velocities and electrical resistances. The system shows complex responses driven by strong fluid–structure interactions, including quasi-nonlinear, periodic non-linear, and chaotic behaviors. Oscillations begin near $U_r^{*}=6$, and large amplitude limit cycle oscillations appear around $U_r^{*}=8$ in the absence of electrical loading. As resistance increases, the onset of nonlinearity shifts to higher velocities due to added load-induced damping coming from the R. FFT analysis reveals both odd and even harmonics, indicating energy content, which gradually diminish with higher resistive loads. Phase portraits and Poincaré maps capture transitions between different dynamic states, showing that reduced velocity and resistances govern nonlinear characteristics. Voltage output follows the lift coefficient trends, peaking at an optimal resistance, beyond which it saturates or declines due to suppressed nonlinearities. Vortex shedding patterns such as single (S), paired (P), triplet (T), and multiple pair (mP) and P+S (in chaotic state) formations correlate with aeroelastic bifurcations and reduce in number as damping increases. The study highlights how reduced velocity and resistance control the onset, amplitude, and nature of oscillations, and shows that nonlinearity enhances energy harvesting but only within specific parameter ranges. The scope of the current study is limited to computational analyses of fluid–structure–electrical interactive energy harvesting systems at a low Reynolds number. Our future studies will aim to investigate the multi-physics system’s performance and its governing mechanisms for high-speed flows. Additionally, out current work provides important guidelines for the choice of different physical parameters for experimental studies and real designs of fluttering wing-based energy harvesters.