On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers

Abstract

This paper examines the energy efficiency of three variations of the two-degree-of-freedom transverse galloping energy harvester. These variants differ in the number and placement of electromechanical transducers. By utilizing the harmonic balance method, the limit cycles of mathematical models of the devices were determined. Analytical expressions derived from the models were then used to formulate the efficiency of the systems. It was demonstrated that efficiency depends on flow speed and can be comprehensively characterized by the following criteria parameters: peak efficiency, denoting the maximum efficiency of the system, and high-efficiency bandwidth, which describes the range of flow speeds within which the efficiency remains at no less than 90% of peak efficiency. The values of these parameters are heavily reliant on two other parameters: the speed at which the system achieves peak efficiency, referred to as the nominal speed, and also the flow speed at which the system undergoes Hopf bifurcation, namely the critical speed. Comparative analysis revealed that only the device equipped with two electromechanical transducers can potentially outperform a simple one-degree-of-freedom system. For selected parameters, this gain reached nearly 10%.

1. Introduction

The ongoing energy crisis and burgeoning concept of the Internet of Things (IoT) [1] are continually motivating factors for researchers from various scientific fields to develop newer technologies that enable cheaper and more efficient generation of usable energy. Among the most innovative of these technologies are those that allow energy recovery from vibrations induced by flow. A commercial example of this is the Vortex Bladeless power plants [2], whose vibrations are excited by von Karman vortices. According to the manufacturer, the cost of energy production using these turbines is only 70% of the cost associated with traditional wind turbines of comparable size [2]. Moreover, their sleek, bladeless design allows for closer placement than standard turbines, facilitating more efficient use of wind farm space.

The discussed advantages of vortex-induced vibration energy harvesters may be partially overshadowed by a significant limitation. Due to their reliance on resonance with the vortex frequency, these devices can generate energy only within a narrow flow speed range. An alternative type of power plant without this limitation is the galloping energy harvester (GEH). Transverse galloping [3] is a phenomenon where vibrations are induced due to the flow separation from a bluff body with an appropriate shape, leading to negative, speed-dependent aerodynamic (hydrodynamic) damping. When the flow speed exceeds a critical value, the negative aerodynamic (hydrodynamic) damping surpasses structural damping, resulting in the overall damping in the system becoming negative. In this situation, the damping will supply energy to the system rather than dissipate it, causing the system to become unstable.

The key advantage of using transverse galloping as an excitation mechanism for energy harvesters is that, beyond the critical speed, the amplitude of oscillation increases indefinitely with the flow speed and remains independent of the frequency or the presence of von Karman vortices.

Early studies [4] explore its potential for energy harvesting, emphasizing the crucial role of aerodynamic coefficients in its efficiency. Subsequent research [5] delves into analytical, numerical, and experimental analyses of GEHs, indicating the detrimental impact of structural damping on the performance of the device.

Due to the critical importance of the aerodynamics of the flowing body, many subsequent works have been devoted to examining the efficiency provided by GEHs of various shapes. Building on this trend, it has been demonstrated, for instance in [6], that rhomboidal shapes exhibit a significant propensity for galloping. The article [7] is dedicated to the study of the properties of a device equipped with a bluff body in the shape of various variants of an isosceles triangle. Furthermore, research [8,9] shows that adding a stream splitter favors power generation. The studies [10,11,12] have also demonstrated the benefits of employing non-typical shapes. According to the reports presented in [13,14,15,16], the efficiency of a GEH can also be increased by giving its surface an appropriate geometric metastructure.

Another extensively investigated area in the development of GEHs is the examination of the impact of nonlinear elasticity on its performance. Studies [17,18] propose a method for analyzing the effect of the nonlinear elasticity coefficient on the characteristics of Hopf bifurcation. The study [19] focuses on improving the energy harvesting performance of galloping systems through the use of magnetic coupling. Conversely, the work [20] presents modeling and experimental investigations of asymmetric distances using magnetic coupling based on a galloping piezoelectric energy harvester, aiming to enhance the stability of the generated power. In the articles [21,22], it has been demonstrated that incorporating nonlinear stiffness into the system can significantly increase its efficiency. Research [23] demonstrates that a GEH composed of two oscillators coupled by a nonlinear magnetic interaction can exhibit a lower critical speed. Furthermore, an appropriate selection of the distance between the magnetic segments can contribute to an increase in the system’s efficiency. A similar coupling, but in reference to a single-degree-of-freedom system, was discussed in [24]. This approach resulted in a tristable system, which exhibited an efficiency over 35% greater than that of a standard GEH.

According to the literature, enhancing the energy extraction efficiency of a GEH can be also achieved by extending its mechanical structure with an additional degree of freedom. In [25], two variants of GEHs with two degrees of freedom (GEH2Ds), differing in the location of the applied aerodynamic force, were compared to a single-degree-of-freedom device. Experimental comparisons of the electric voltage generated by the variants were presented in [26] and extended to numerical studies of systems with up to three additional masses in [27]. Comprehensive analytical studies of the voltage generated by a GEH2D, considering the possibility of two different vibration modes, were presented in [28]. A two-mass system, whose segments are magnetically coupled, was discussed in [29]. It was demonstrated that such a system can exhibit both a reduced critical speed and enhanced efficiency.

The literature review clearly indicates that extending the mechanical structure to include an additional degree of freedom is seen as a potential way to increase the efficiency of a GEH. However, the literature does not discuss the fact that within the family of two-mass systems, three device variants can be distinguished, differing in the placement of the electromechanical transducer: it can be located between the stationary base and the first mass, between the masses, or in both of these places.

The aim of this work is therefore to analytically examine the efficiency of all three GEH2D variants, compare them, and assess whether any of them can exhibit greater efficiency than the device of basic design.

2. Efficiency of Reference Variant

While the primary focus of this study lies in analyzing the efficiency of systems with two degrees of freedom, understanding the characteristics of this class of devices can be facilitated by comparing them with a system possessing well-established properties—a linear device with one degree of freedom, hereafter referred to as the reference device (Figure 1). Therefore, it is justified to conduct an analysis of the efficiency of this system, particularly as we aim to highlight a certain intriguing feature of this variant, which has not been previously addressed in the literature. According to articles [3,4], the mathematical model of a GEH takes the form shown in Equation (1) with parameters detailed in

Table 1. List of GEH parameters.

Symbol	$\mathbf{P}\mathbf{a}\mathbf{r}\mathbf{a}\mathbf{m}\mathbf{e}\mathbf{t}\mathbf{e}\mathbf{r}$	$\mathbf{U}\mathbf{n}\mathbf{i}\mathbf{t}$
$\widehat{m}$	Mass	$\mathrm{k}\mathrm{g}$
$\widehat{k}$	Stiffness coefficient	$\mathrm{N}/\mathrm{m}$
$\widehat{c}$	Damping coefficient	$\mathrm{k}\mathrm{g}/\mathrm{s}$
$\widehat{u}$	Flow speed	$\mathrm{m}/\mathrm{s}$
$\widehat{\rho }$	Planar fluid density	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^2$
$\widehat{h}$	Characteristic dimension	$\mathrm{m}$
$\widehat{\theta }$	Piezoelectric coefficient	$\mathrm{N}/\mathrm{V}$
$\widehat{C_p}$	Equivalent capacity	$\mathrm{F}$
$\widehat{R}$	Circuit electrical resistance	$\mathsf{\Omega }$
$\tau$	Time	$\mathrm{s}$
$z=z\left(\tau \right)$	Vibration vs. time function	$\mathrm{m}$
$\xi =\xi \left(\tau \right)$	Voltage vs. time function	$\mathrm{V}$

.

$$\widehat{m}\ddot{z}+\widehat{c}\dot{z}+\widehat{k}z-\theta \xi =\widehat{F_L}\left(\widehat{u}\right)=-\frac{1}{2}\widehat{\rho }{\widehat{u}}^2\widehat{h}\left(a_1\frac{\dot{z}}{\widehat{u}}+a_3{\left(\frac{\dot{z}}{\widehat{u}}\right)}^3\right),$$

(1a)

$$\widehat{C_p}\dot{\xi }+\frac{\xi }{\widehat{R}}+\widehat{\theta }\dot{x}=0.$$

(1b)

The efficiency $\eta$ of such a system is defined as the ratio of the average power generated by the device $\widehat{P_g}$ over a time period equal to the period $\widehat{T}$ of the voltage function ξ(τ) to the surface flow power density $\widehat{P_f}$:

$${\eta }^L=\frac{\widehat{P_g}}{\widehat{P_f}}=\frac{\frac{1}{\widehat{T}}{\int }_0^{\widehat{T}}\frac{{\mathsf{\xi }}^2\left(\tau \right)}{\widehat{R}}d\tau }{\frac{1}{2}\widehat{\rho }\widehat{h}{\widehat{u}}^3},$$

(2)

By introducing the following dimensionless quantities:

$$\begin{gathered} y=\frac{z}{\widehat{h}},v=\xi \frac{\widehat{\theta }}{\widehat{m}\widehat{h}{{\widehat{\omega }}_n}^2},{\widehat{\omega }}_n=\sqrt{\frac{\widehat{k}}{\widehat{m}}},c=\frac{\widehat{c}}{\widehat{m}{\widehat{\omega }}_n},u=\frac{\widehat{u}}{\widehat{h}{\widehat{\omega }}_n},\rho =\widehat{\rho }\frac{{\widehat{h}}^2}{2\widehat{m}},r=\widehat{C_p}\widehat{R}{\widehat{\omega }}_n, \\ \kappa =\frac{{\widehat{\theta }}^2}{\widehat{C_p}\widehat{m}{{\widehat{\omega }}_n}^2},t=\tau {\widehat{\omega }}_n, \end{gathered}$$

(3)

the device dimensionless mathematical model can be written as

$$\ddot{y}+c\dot{y}+y-v=\rho \left(a_{1}u\dot{y}+a_3\frac{{\dot{y}}^3}{u}\right),$$

(4a)

$$\dot{v}+\frac{v}{r}+\kappa \dot{y}=0.$$

(4b)

Assuming that $T_c$ is the period of the function $v\left(t\right)$, efficiency in terms of dimensionless parameters can be expressed as

$${\eta }^L=\frac{\frac{1}{T_c}{\int }_0^{T_c}\frac{v^2\left(t\right)}{\kappa r}dt}{\rho u^3}.$$

(5)

To characterize the efficiency ${\eta }^L$ of the system, it is essential to derive the voltage function $v\left(t\right)$ generated by it, which can be accomplished using the harmonic balance method. Consequently, it was assumed that the limit cycle of the system would be described by a set of solutions in the following form:

$$y=A_y\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right),$$

(6a)

$$v=A_v\cos \left(\omega t+\phi \right),$$

(6b)

where the four unknown quantities $A_y$, $A_v$, $\omega =\frac{\widehat{\omega }}{{\widehat{\omega }}_n}$, and $\phi$ represent the dimensionless vibration amplitude, dimensionless voltage amplitude, dimensionless vibration frequency, and the phase shift between the oscillator vibrations and voltage oscillations, respectively. The parameter $\widehat{\omega }$ denotes the unknown dimensional frequency of the system’s vibrations. Substituting solutions of the form Equation (6) into the model Equation (1) leads to the transformation of the system of differential equations into a system of algebraic equations:

$$A_y\left(1-{\omega }^2\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+A_y^3\frac{\rho {\omega }^3{a}_3}{u}{\mathrm{s}\mathrm{i}\mathrm{n}}^3\left(\omega t\right)+A_y\omega \left(u\rho a_1-c\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)-A_v\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t+\phi \right)=0,$$

(7a)

$$A_v\left(\frac{\cos \left(\omega t+\phi \right)}{r}-\omega \sin \left(\omega t+\phi \right)\right)-A_y\kappa \omega \mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)=0.$$

(7b)

This condition will be satisfied for every moment of time $t$ if and only if the sum of the coefficients with corresponding time functions equals zero. From Equation (7b), the following was deduced:

$$\frac{A_v\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\phi \right)-r\omega \mathrm{s}\mathrm{i}\mathrm{n}\left(\phi \right)\right)}{r}=0,$$

(8a)

$$\frac{A_v\left(r\omega \mathrm{c}\mathrm{o}\mathrm{s}\left(\phi \right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi \right)\right)+A_y\kappa r\omega }{r}=0.$$

(8b)

Equation (8a) shows that

$$\mathrm{t}\mathrm{g}\left(\phi \right)=\frac{1}{\omega r},\sin \left(\phi \right)=\frac{1}{\sqrt{{\left(\omega r\right)}^2+1}},\cos \left(\phi \right)=\frac{\omega r}{\sqrt{{\left(\omega r\right)}^2+1}}.$$

(9)

Based on Equation (8b) and Equation (9), the expression describing the relationship between the vibration amplitude and the voltage amplitude can be derived:

$$A_v={-A}_y\frac{r\kappa \omega }{\sqrt{1+r^2{\omega }^2}}.$$

(10)

By balancing the harmonics of Equation (7a), the remaining two algebraic equations necessary to determine an approximate solution of the device model are obtained. After the previously derived relations Equations (9) and (10) are taken into account, they take the form

$$A_y{k}_e^L\omega -A_y\left({\omega }^2-1\right)=0,$$

(11a)

$$A_{y}u\rho \omega a_1+A_y^3\frac{3\rho {\omega }^3{a}_3}{4u}-A_{y}c\omega -A_y{e}^L\omega =0,$$

(11b)

where $k_e^L=\kappa \frac{r^2{\omega }^2}{1+r^2{\omega }^2}$ and $e^L=\kappa \frac{r}{1+r^2{\omega }^2}$ are the piezoelectric stiffness and electrical damping of the linear system. Based on the above system of equations, it can be shown that

$$\begin{array}{c}{\omega }_1^2\\ {\omega }_2^2\end{array}=\frac{r^2\left(1+\kappa \right)-1\pm \sqrt{4r^2+{\left(r^2\left(1+\kappa \right)-1\right)}^2}}{2r^2}$$

(12a)

$${A_y}^2=\frac{4u\left(c+e^L-u\rho a_1\right)}{3\rho {\omega }^2{a}_3}.$$

(12b)

Equation (12a) shows that ${\omega }_2^2<0$, regardless of the system parameters. Therefore, in the following part of the work, the notation ${\omega }_1=\omega$ has been adopted. Returning now to the general definition of efficiency Equation (5) and substituting the expressions Equations (6b), (10), and (12) into it, we obtain

$${\eta }^L=\frac{2e^L\left(c+e^L-u\rho a_1\right)}{3u^2{\rho }^2{a}_3}.$$

(13)

Figure 2 illustrates the efficiency characteristics of the system. The efficiency is depicted as the ratio ${\eta }^L/{\eta }_p^L$, and this representation is maintained throughout the study. The figure also includes the analogous characteristic obtained numerically for initial conditions $y\left(0\right)=0.1$, $\dot{y}\left(0\right)=0$, $n\left(0\right)=0$, utilizing the fourth-order Runge–Kutta method.

In this figure, it can be observed that the system undergoes Hopf bifurcation at a certain speed, from now on referred to as the critical flow speed $u_{cr}$. Its value results directly from Equation (13):

$${\eta }^L=\frac{2e^L\left(c+e^L-u\rho a_1\right)}{3u^2{\rho }^2{a}_3}=0\Rightarrow u=u_{cr}=\frac{c+e^L}{\rho a_1},$$

(14)

Another noteworthy observation from the same set of graphs is that irrespective of the parameter set, there exists a specific nominal flow speed $u=u_p^L$ for which the efficiency ${\eta }^L$ attains a maximum value, denoted as the peak efficiency ${\eta }_p^L$. These quantities are given by the expressions

$$\frac{\delta {\eta }^L}{\delta u}=0\Rightarrow u=u_p^L=2\frac{c+e^L}{\rho a_1}=2u_{cr},$$

(15)

$${\eta }_p^L={\eta }^L\left(u_p^L\right)=\frac{-a_1^2{e}^L}{6a_3\left(c+e^L\right)}.$$

(16)

The aforementioned identities were previously derived in [3,4]. However, the remainder of the article discusses entirely original content. An unexplored property of GEHs, which can be inferred from Figure 2 or deduced from the identity $u_p^L=2u_{cr}$, is noteworthy. A system with a low critical speed $u_{cr}$ will experience a more pronounced decline in efficiency due to the deviation of the flow speed $u$ at which it operates from the nominal speed $u_p^L$. Let the measure of this phenomenon be the flow speed bandwidth in which the system efficiency ${\eta }^L$ does not fall below 90% of the maximum efficiency ${\eta }_p^L$, hereafter referred to as the high-efficiency band $B^L$. According to the definition, $B^L$ is given as

$${\eta }^L=0.9{\eta }_p^L\Rightarrow \begin{array}{c}{u}_2\\ u_1\end{array}=\frac{20\pm 4\sqrt{10}}{9}\frac{\left(c+e^L\right)}{\rho a_1},$$

(17)

$$B^L=u_2-u_1=\frac{4\sqrt{10}}{9}\frac{\left(c+e^L\right)}{\rho a_1}\approx 1.4u_{cr}\approx 0.7u_p^L.$$

(18)

The quantities critical speed $u_{cr}$, nominal speed $u_p^L$, peak efficiency ${\eta }_p^L$, and high-efficiency bandwidth $B^L$ (Figure 2) will be further referred to as the criterion parameters.

3. Efficiency of Two-Degree-of-Freedom System

According to the information provided in Section 1, the presence of additional mass in systems with two degrees of freedom implies the necessity to consider the placement of the electromechanical transducer. It may be positioned between the main mass and the stationary base (Figure 3a), between the masses (Figure 3b), or in both locations (Figure 3c).

To evaluate the influence of piezoelectric placement on efficiency, the efficiency characteristics of the variant with both transducers will be derived. Subsequently, the consequence of the absence of one of them will be examined.

The dynamics of devices within the discussed category can be described by a general dimensionless mathematical model of the form Equation (20). In addition to the identities in Equation (3), the following also holds:

$$\begin{gathered} y_i=\frac{z_i}{\widehat{h}},v_i=\xi \frac{\widehat{{\theta }_i}}{\widehat{m_i}\widehat{h}{{\widehat{\omega }}_n}^2},c_i=\frac{{\widehat{c}}_i}{{\widehat{m}}_1{\widehat{\omega }}_n},k_2=\frac{{\widehat{k}}_2}{{\widehat{k}}_1},\kappa =\frac{{\widehat{{\theta }_i}}^2}{\widehat{C_{pi}}\widehat{m_1}{{\widehat{\omega }}_n}^2}, \\ \theta =\frac{{\widehat{\theta }}_1}{{\widehat{\theta }}_2},M=\frac{{\widehat{m}}_2}{{\widehat{m}}_1},i=\mathrm{1,2} \end{gathered}$$

(19)

$${\ddot{y}}_1+y-k_2\left(y_2-y_1\right)+c_1{\dot{y}}_1-c_2\left({\dot{y}}_2-{\dot{y}}_1\right)-v_1+v_2=\rho \left(a_{1}u{\dot{y}}_1+a_3\frac{{\dot{y}}_1^3}{u}\right),$$

(20a)

$${\kappa }_1{\dot{v}}_1+\frac{v_1}{r_1}+{\dot{y}}_1,$$

(20b)

$${M\ddot{y}}_2+k_2\left(y_2-y_1\right)+c_2\left({\dot{y}}_2-{\dot{y}}_1\right)-v_2=0,$$

(20c)

$${\kappa }_2{\dot{v}}_2+\frac{v_2}{r_2}+{\dot{y}}_2-{\dot{y}}_1=0.$$

(20d)

To derive the solutions of the GEH2D mathematical model, the procedure outlined in [28] was adapted. It was assumed that the approximate solution of the model Equation (20) will have the following form:

$$y_1=A_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)+B_1\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)+G_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)+H_1\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right),$$

(21a)

$$v_1=n_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)+n_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)+n_3\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)+n_4\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right),$$

(21b)

$$y_2=A_2\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)+B_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)+G_2\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)+H_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right),$$

(21c)

$$v_2={\vartheta }_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)+{\vartheta }_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)+{\vartheta }_3\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)+{\vartheta }_4\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right),$$

(21d)

Substituting the above identities into Equations (20b) and (20d) and then balancing the harmonics $\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)$, $\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)$, $\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)$, $\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right)$ allows for the derivation of the relationships between the amplitudes of voltage and the amplitudes of vibration.

$$\begin{gathered} n_1=B_1{\omega }_1{\epsilon }_{D11}-A_1{\delta }_{D11},n_2=-A_1{\omega }_1{\epsilon }_{D11}-B_1{\delta }_{D11}, \\ {n}_3=H_1{\omega }_2{\epsilon }_{D12}-G_1{\delta }_{D12},n_4=-G_1{\omega }_2{\epsilon }_{D12}-H_1{\delta }_{D12}, \end{gathered}$$

(22a)

$$\begin{gathered} {\vartheta }_1=\theta \left({\epsilon }_{D21}\left(\theta B_2-B_1\right){\omega }_1+{\delta }_{D21}\left(A_1-\theta A_2\right)\right), \\ {\vartheta }_2=\theta \left({\epsilon }_{D21}\left(A_1-\theta A_2\right){\omega }_1+{\delta }_{D21}\left(B_1-\theta B_2\right)\right), \\ {\vartheta }_3=\theta \left({\epsilon }_{D22}\left(\theta H_2-H_1\right){\omega }_2+{\delta }_{D22}\left(G_1-\theta G_2\right)\right), \\ {\vartheta }_4=\theta \left({\epsilon }_{D22}\left(G_1-\theta G_2\right){\omega }_2+{\delta }_{D22}\left(H_1-\theta H_2\right)\right). \end{gathered}$$

(22b)

where ${\delta }_{Dij}={\kappa }_i\frac{r^2{\omega }_{\mathrm{j}}^2}{1+r^2{\omega }_{\mathrm{j}}^2}$ and ${\epsilon }_{Dij}={\kappa }_i\frac{r}{1+r^2{\omega }_{\mathrm{j}}^2}$ represent the piezoelectric stiffness and electric damping of the $i$-th piezoelectric for vibration with the $j$-th frequency.

Harmonic balancing of $\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)$ of the algebraic equation obtained by substituting the solutions of Equation (21) into Equation (20c) and using the identities Equation (22) leads to the following system of equations:

$$A_2=B_1{p}_1+A_1{q}_1,$$

(23a)

$$B_2=B_1{q}_1-A_1{p}_1,$$

(23b)

where

$$p_1=\frac{(k_2+\theta {\delta }_{D21})(c_2+{\theta }^2{\epsilon }_{D21}){\omega }_1-(c_2+\theta {\epsilon }_{D21})(k_2+{\theta }^2{\delta }_{D21}-M{\omega }_1^2){\omega }_1}{{(c_2+{\theta }^2{\epsilon }_{D21})}^2{\omega }_1^2+{(k_2+{\theta }^2{\delta }_{D21}-M{\omega }_1^2)}^2},$$

(23c)

$$q_1=\frac{(c_2+\theta {\epsilon }_{D21})(c_2+{\theta }^2{\epsilon }_{D21}){\omega }_1^2+(k_2+\theta {\delta }_{D21})(k_2+{\theta }^2{\delta }_{T21}-M{\omega }_1^2)}{{(c_2+{\theta }^2{\epsilon }_{D21})}^2{\omega }_1^2+{(k_2+{\theta }^2{\delta }_{D21}-M{\omega }_1^2)}^2}.$$

(23d)

By substituting the above identities into the harmonic balance equations for $\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{1}t\right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{1}t\right)$ of expression (20a), the following identities can be formulated:

$$A_1{\alpha }_1+B_1{\beta }_1-A_1\frac{3\rho a_3{\omega }_1\left(\right(A_2^2+B_2^2){\omega }_1^2+2(G_2^2+H_2^2\left){\omega }_2^2\right)}{4u}=0,$$

(24a)

$$A_1{\beta }_1-B_1{\alpha }_1+B_1\frac{3\rho a_3{\omega }_1\left(\left(A_2^2+B_2^2\right){\omega }_1^2+2\left(G_2^2+H_2^2\right){\omega }_2^2\right)}{4u}=0,$$

(24b)

where

$${\alpha }_1=\left(p_1\right(k_2+{\theta }^2{\delta }_{D21})+(c_1+{\epsilon }_{T11}+\left(c_2+{\epsilon }_{D21}\right)(1-\theta q_1)){\omega }_1-u\rho {\omega }_1{a}_1),$$

(25a)

$${\beta }_1=1+{\delta }_{T11}+\left({\delta }_{D21}\theta +k_2\right)\left(1-\theta q_1\right)-{\omega }_1\left(p_1\left(c_2+{\theta }^2{\epsilon }_{D21}\right)+{\omega }_1\right).$$

(25b)

Adapting the above procedure to balance the harmonics $\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{2}t\right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{2}t\right)$ of Equations (20c) and (20a) allows for the derivation of the following identities:

$$G_2=H_1{p}_2+G_1{q}_2,$$

(26a)

$$H_2=H_1{q}_2-G_1{p}_2,$$

(26b)

$$\begin{array}{c}{G}_1{\alpha }_2+H_1{\beta }_2-G_1\frac{3\rho a_3{\omega }_2\left(2\right(A_1^2+B_1^2){\omega }_1^2+(G_1^2+H_1^2\left){\omega }_2^2\right)}{4u}=0\end{array}$$

(26c)

$$G_1{\beta }_2-H_1{\alpha }_2+H_1\frac{3\rho a_3{\omega }_2\left(2\right(A_1^2+B_1^2){\omega }_2^2+(G_1^2+H_1^2\left){\omega }_2^2\right)}{4u}=0$$

(26d)

where

$$p_2=\frac{(k_2+\theta {\delta }_{D22})(c_2+{\theta }^2{\epsilon }_{D22}){\omega }_2-(c_2+\theta {\epsilon }_{D22})(k_2+{\theta }^2{\delta }_{D22}-M{\omega }_2^2){\omega }_2}{{(c_2+{\theta }^2{\epsilon }_{D22})}^2{\omega }_2^2+{(k_2+{\theta }^2{\delta }_{D22}-M{\omega }_2^2)}^2},$$

(27a)

$$q_2=\frac{(c_2+\theta {\epsilon }_{T22})(c_2+{\theta }^2{\epsilon }_{D22}){\omega }_2^2+(k_2+\theta {\delta }_{D22})(k_2+{\theta }^2{\delta }_{D22}-M{\omega }_2^2)}{{(c_2+{\theta }^2{\epsilon }_{D22})}^2{\omega }_2^2+{(k_2+{\theta }^2{\delta }_{D22}-M{\omega }_2^2)}^2},$$

(27b)

$${\alpha }_2=\left(p_2\right(k_2+{\theta }^2{\delta }_{D22})+(c_1+{\epsilon }_{T12}+\left(c_2+{\epsilon }_{T22}\right)(1-\theta q_2)){\omega }_2-u\rho {\omega }_2{a}_1),$$

(27c)

$${\beta }_2=1+{\delta }_{D12}+\left({\delta }_{D22}\theta +k_2\right)\left(1-\theta q_2\right)-{\omega }_2\left(p_2\left(c_2+{\theta }^2{\epsilon }_{D22}\right)+{\omega }_2\right).$$

(27d)

By adding Equation (24a) multiplied by $B_1$ to Equation (24b) multiplied by $A_1$, an expression was obtained that allows the frequency ${\omega }_1$ to be explicitly formulated:

$$1+k_2+{\delta }_{D11}-k_2{q}_1+{\delta }_{D21}\theta \left(1-\theta q_1\right)-p_1\left(c_2+{\theta }^2{\epsilon }_{D21}\right){\omega }_1-{\omega }_1^2=0.$$

(28)

Similarly, by adding Equation (26c) multiplied by $H_1$ to Equation (26d) multiplied by $G_1$, an equation was obtained from which the frequency ${\omega }_2$ can be derived:

$$1+k_2+{\delta }_{D12}-k_2{q}_2+{\delta }_{D22}\theta \left(1-\theta q_2\right)-p_2\left(c_2+{\theta }^2{\epsilon }_{D22}\right){\omega }_2-{\omega }_2^2=0.$$

(29)

Note that in each pair of parameters ($p_1$, $p_2$), ($q_1$, $q_2$), (${\epsilon }_{D21},{\epsilon }_{D22}$), (${\delta }_{D11},{\delta }_{D12}$), (${\delta }_{D21},{\delta }_{D22}$), the only difference is the frequency in their definition—${\omega }_1$ or ${\omega }_2$. Considering the similarity between Equations (28) and (29), one can conclude that the frequencies ${\omega }_1$ and ${\omega }_2$ must be equal, thereby excluding the possibility of polymodal vibrations in the system. The expression describing the vibration frequency ${\omega }_2={\omega }_1$ can therefore be derived by solving only one of the above equations.

The relationship between the vibration amplitude and the system frequency remains unknown. The first of the equations necessary to determine this relationship was obtained by subtracting Equation (24b) multiplied by $B_1$ from Equation (24a) multiplied by $A_1$. The second one is the difference of Equation (26c) multiplied by $G_1$ and Equation (26d) multiplied by $H_1$:

$$p_1\left(k_2+{\theta }^2{\delta }_{D21}\right)+\left(c_1+c_2\left(1-q_1\right)+{\epsilon }_{D11}+{\epsilon }_{D21}\theta \left(1-\theta q_1\right)\right){\omega }_1-u\rho {\omega }_1{a}_1-\frac{3\rho {\omega }_1\left(A_y^2{\omega }_1^2+2G_y^2{\omega }_2^2\right)a_3}{4u}=0,$$

(30a)

$$p_2\left(k_2+{\theta }^2{\delta }_{D22}\right)+\left(c_1+c_2\left(1-q_1\right)+{\epsilon }_{D12}+{\epsilon }_{D22}\theta \left(1-\theta q_2\right)\right){\omega }_2-u\rho {\omega }_2{a}_1-\frac{3\rho {\omega }_2\left(2A_y^2{\omega }_1^2+G_y^2{\omega }_2^2\right)a_3}{4u}=0,$$

(30b)

where $A_y^2=A_1^2+B_1^2$ and $G_y^2=G_1^2+H_1^2$. The set of Equation (30) has three non-trivial solutions in terms of $A_y^2$ and $G_y^2$, which, after recalling the ${\omega }_2={\omega }_1$ identity, take the following form:

$$\begin{gathered} A_y^2=\frac{4u\left(p_1\left(k_2+{\theta }^2{\delta }_{D21}\right)+\left(c_1+c_2\left(1-q_1\right)+{\epsilon }_{D11}+{\epsilon }_{D21}\theta \left(1-\theta q_1\right)\right)-u\rho a_1\right)}{3\rho {{\omega }_1}^3{a}_3}, \\ {G}_y^2=0 \end{gathered}$$

(31a)

$$\begin{gathered} A_y^2=0, \\ {A}_y^2=0, \\ {G}_y^2=\frac{4u\left(p_1\left(k_2+{\theta }^2{\delta }_{D21}\right){\omega }_1+\left(c_1+c_2\left(1-q_1\right)+{\epsilon }_{D11}+{\epsilon }_{D21}\theta \left(1-\theta q_1\right)\right)-u\rho a_1\right)}{3\rho {{\omega }_1}^3{a}_3}, \end{gathered}$$

(31b)

$$\begin{gathered} A_y^2=\frac{4u\left(3p_1\left(k_2+{\theta }^2{\delta }_{D21}\right){\omega }_1+c_1+c_2\left(1-q_1\right)+{\epsilon }_{D11}+{\epsilon }_{D21}\theta \left(1-\theta q_1\right)\right)-u\rho a_1}{9\rho {{\omega }_1}^2{a}_3}, \\ {G}_y^2=\frac{4u\left(-3p_1\left(k_2+{\theta }^2{\delta }_{D21}\right){\omega }_1+c_1+c_2\left(1-q_1\right)+{\epsilon }_{T11}+{\epsilon }_{D21}\theta \left(1-\theta q_1\right)\right)-u\rho a_1}{9\rho {{\omega }_1}^2{a}_3}. \end{gathered}$$

(31c)

These expressions, along with the previously derived identities, enable the explicit formulation of solutions for the system Equation (20) in the form Equation (21). It should be noted that the identities given by Equations (31a) and (31b) correspond to the same solution in the form of Equation (21); thus, only one of them, namely Equation (31a), will be further analyzed. Moreover, no set of system parameters and initial conditions has been found that would lead to the excitation of vibrations with amplitude Equation (31c). Therefore, this solution was considered unstable, and the efficiency has been derived based only on Equation (31a):

$${\eta }_D=\frac{\frac{1}{T_c}{\int }_0^{T_c}\left(\frac{{v_1}^2\left(t\right)}{r_1{\kappa }_1}+\frac{{v_2}^2\left(t\right)}{r_2{\kappa }_2}\right)dt}{\rho u^3}=\frac{2\lambda (\mu -u\rho a_1)}{3u^2{\rho }^2{a}_3},$$

(32)

where

$$\lambda =({\epsilon }_{D11}+{\epsilon }_{D12}{\theta }^2(1-2\theta q_1+{\theta }^2({p_1}^2+{q_1}^2)\left)\right),$$

(33a)

$$\mu =\left(c_1+\left(c_2+\theta {\epsilon }_{D11}\right)\left(1-\theta q_1\right)\right)+\frac{p_1\left(k_2+{\theta }^2{\delta }_{TD21}\right)}{{\mathsf{\omega }}_1}.$$

(33b)

The critical speed $u_{cr}^D$ and nominal speed $u_p^D$ of the GEH2D are given by the following expressions:

$${\eta }_D={\mathsf{\gamma }}_1\frac{2\lambda (\mu -u\rho a_1)}{3u^2{\rho }^2{a}_3}=0\Rightarrow u=u_{cr}^D=\frac{\mu }{\rho a_1},$$

(34)

$$\frac{\delta {\eta }_D}{\delta u}=0\Rightarrow u=u_p^D=2\frac{\mu }{\rho a_1}=2u_{cr}^D.$$

(35)

Now it is possible to derive the peak efficiency of the system:

$${\eta }_p^D={\eta }^D\left(u_p^D\right)=\frac{-\lambda a_1^2}{6{\mathsf{\gamma }}_2{a}_3}.$$

(36)

To fully determine the features of the analyzed variant, it is necessary to define its high-efficiency bandwidth $B^D$. It is

$${\eta }^D=0.9{\eta }_p^D\Rightarrow \begin{array}{c}{u}_2\\ u_1\end{array}=\frac{\frac{2}{0.9}\rho a_1\mu \pm \frac{\sqrt{0.4}}{0.9}\rho a_1\mu }{{\left(\rho a_1\right)}^2},$$

(37)

$$B^D=u_2-u_1=\sqrt{\frac{160}{81}}\frac{\mu }{\rho a_1}\approx 1.4u_{cr}^D\approx 0.7u_p^D.$$

(38)

Figure 4 depicts the relationship between efficiency ${\eta }^D$ and flow speed $u$, represented by the function Equation (32), compared with an analogous relationship obtained through numerical integration of the model Equation (20) for the following initial conditions: $y_1\left(0\right)=0.1$, ${\dot{y}}_1\left(0\right)=0$, $n_1\left(0\right)=0$, $y_2\left(0\right)=0$, ${\dot{y}}_2\left(0\right)=0$, $n_2\left(0\right)=0$. Moreover, the figure shows the values characterizing the efficiency of the variant—$u_{cr}^D$, $u_p^D$, ${\eta }_p^D$, and $B^I$.

4. Comparison and Discussion

As suggested by Equations (34), (35), and (38), critical speeds, nominal speed, and high-efficiency bandwidths are interrelated in the same manner in both analyzed GEH variants. Consequently, comparing the efficiency of these systems can only be achieved by comparing their peak efficiencies. Since the efficiency of both devices depends on their nominal speeds, it is necessary to compare systems with the same nominal speeds. The electrical damping of the linear system $e^L$ will therefore be selected in such a way that this condition is satisfied. In the following derivation, it is assumed that the structural damping of the linear system c is equal to the structural damping $c_1$ of the damper connecting the lower mass of the GEH2D with the base, i.e., $c=c_1$. The shape of the flowing body, represented by the coefficients $a_1$ and $a_3$, and the density of the fluid remain the same for all variants.

$$u_p^I=u_p^L,$$

(39a)

$$2\frac{\mu +{\mathsf{\gamma }}_2}{\rho a_1}=2\frac{e^L+c_1}{\rho a_1},$$

(39b)

$$e^L={\mathsf{\gamma }}_2+\mu -c_1.$$

(39c)

According to Equations (16) and (39c), the efficiency of a system with one degree of freedom with a nominal speed equal to the nominal speed of the GEH2D is equal to

$${\eta }^L\left(u_p^D\right)=-\frac{\left(\mu -c_1\right)a_1^2}{6\mu a_3}.$$

(40)

The ratio of peak efficiencies of compared systems with the same nominal speed is therefore given by

$$\frac{{\eta }^L\left(u_p^D\right)}{{\eta }^D\left(u_p^D\right)}=\frac{s_1+s_2}{s_1+s_3},$$

(41)

where

$$s_1={\epsilon }_{T11}\left(\right(k_2+{\theta }^2{\delta }_{D21}\left)\right(k_2+{\theta }^2{\delta }_{T21}-2M{{\omega }_1}^2)+{(c_2+{\theta }^2{\epsilon }_{D21})}^2{{\omega }_1}^2+M^2{{\omega }_1}^4),$$

(42a)

$$s_2=M{{\omega }_1}^2\left((k_2{\epsilon }_{D21}-c_2{\delta }_{D21})(\theta -1)\theta +M{{\omega }_1}^2(c_2+\theta {\epsilon }_{D21})\right),$$

(42b)

$$s_3={\theta }^2{\epsilon }_{D21}\left(M{{\omega }_1}^2\left(2k_2\right(\theta -1)+M{{\omega }_1}^2)+({c_2}^2{{\omega }_1}^2+{k_2}^2){(\theta -1)}^2\right).$$

(42c)

Clearly, according to Equation (41), the discussed system with two degrees of freedom will have greater efficiency than the reference system if and only if $s_3>s_2$. For variant with only one lower transducer (Figure 3a), where ${\epsilon }_{D21}={\delta }_{D21}=0$, this inequality takes the form

$$0\ngtr c_2{M}^2{{\omega }_1}^4,$$

(43)

which implies that such a device cannot be more efficient than the reference system, regardless of its parameters. This conclusion contradicts the results presented in [25]. The reason for this discrepancy lies in the fact that in the cited work, the operating conditions of the compared systems were not standardized—the devices had different critical speeds and, consequently, different nominal speeds. Despite the unquestionable value of this article, the conclusion stated therein can be subject to questioning. In the case of the variant with only the upper transducer (Figure 3b), where ${\epsilon }_{T11}={\delta }_{T11}=0$, the condition for efficiency improvement takes the following form:

$${\epsilon }_{D21}\ngtr {\epsilon }_{D21}+c_2,$$

(44)

which indicates that this variant does not offer efficiency enhancement. Similarly, an identical and impossible-to-satisfy requirement is associated with the special case of the third variant, where there are two identical transducers, meaning ${\epsilon }_{D11}={\epsilon }_{T21}$, ${\delta }_{D11}={\delta }_{D21}$, and $\theta =1$. However, the inequality $s_3>s_2$ can be satisfied for the most general variant—the one equipped with two different transducers. In this case, it can be reduced to the condition

$$\left(\theta -1\right)\theta \left({\epsilon }_{D21}{k_2}^2\left(\theta -1\right)\theta +c_2{\delta }_{D21}M{{\omega }_1}^2+{c_2}^2{\epsilon }_{D21}{{\omega }_1}^2\left(\theta -1\right)\theta +{\epsilon }_{D21}{k}_{2}M{{\omega }_1}^2\left(2\theta -1\right)\right)>M^2{{\omega }_1}^4(c_2-{\epsilon }_{D21}(\theta -1\left)\theta \right),$$

(45)

which, after substituting the solution of Equation (27), can be solved numerically for a chosen parameter. For the parameters presented in

Table 2. GEH2D system parameter values.

Parameter	Value
$M$	0.5
$k_2$	0.3
$c_1$	0.1
$c_2$	0.25
${\kappa }_1$	0.9
${\kappa }_2$	1.8
$r_1$	3
$r_2$	2
$\theta$	1.3
$a_1$	2.3
$a_3$	−18
$\rho$	0.02

, solving the above inequality for $\theta$ results in obtaining a threshold value of approximately $\theta \approx 1.13$. An example of the efficiency characteristics of the GEH with $e^L$ given by Equation (39c) and GEH2D is shown in Figure 5, where condition Equation (45) has been met by adopting parameters according to Table 2, with $\theta =1.3>1.13$. It should be noted that parametric analysis does not allow one to indicate the optimal gain that the discussed system can provide. Determining this fact can be considered as a potential direction for further research.

5. Conclusions

The aim of the study was to investigate the efficiency of a galloping energy harvester with two degrees of freedom, considering its three variants, which differ in the number and location of electromechanical transducers. The realization of this objective commenced with the analysis of the reference variant with one degree of freedom. Utilizing the harmonic balance method, an approximate solution of the mathematical model of the system was derived, followed by the formulation of an expression describing the efficiency of the variant. Based on this, key criteria parameters were defined, providing comprehensive information about the variant’s efficiency: peak efficiency, high-efficiency bandwidth, critical speed, and nominal speed.

Subsequent sections of the study delineated the multitude of configurations that a system with two degrees of freedom can adopt. Three different subvariants of the device were characterized, differing from each other by the location and number of the electromechanical transducers. Parameters characterizing the efficiency of all subvariants were derived, demonstrating that the critical speed, nominal speed, and high-efficiency bandwidth are related to each other in the same manner as in the case of the reference system. It was then demonstrated that among the three indicated subvariants of the two-degree-of-freedom system, only the one with two transducers can be more efficient than the reference system.