Dynamic Modeling and Investigation of a Tunable Vortex Bladeless Wind Turbine

Abstract

This paper investigates the dynamics of an electromagnetic vortex bladeless wind turbine (VBWT) with a tunable mechanism. The tunable mechanism comprises a progressive-rate spring that is attached to an oscillating magnet inside an electromagnetic coil. The spring stiffness is progressively adjusted as the wind speed changes to tune the turbine fundamental frequency to match the shedding frequency of the vortex-induced vibration (VIV) due to the wind flow crossing over the oscillating mast. Coupled nonlinear equations of motion of the tunable turbine are developed using the lumped-mass representation and Lagrange formulation. Numerical results show that the tunable turbine performs effectively beyond a threshold wind speed. An analytical expression of the threshold speed is derived based on the mechanical fundamental frequency of the turbine. The analytical results are in reasonable agreement with the numerical evaluations. At a given wind speed past the threshold, the tunable turbine has an optimum spring stiffness at which the output power is maximum. Numerical studies also show that the output power of the 2 m long tunable turbine is tens of times larger in comparison to a conventional bladeless turbine. For example, at a wind speed of 4.22 m/s, the output rms power of the tunable turbine is around 1105 mW versus 17 mW of the conventional VBWT. The power can be further maximized at an optimum external load. This research work demonstrated the feasibility and merits of the proposed tunable mechanism to enhance the overall performance of the bladeless wind turbine.

1. Introduction

Vortex bladeless wind turbines (VBWTs) are vibration-based energy harvesters that have been successfully demonstrated to scavenge wind energy [1]. As the name implies, one of the main features of this type of turbine is that they do not have rotating blades or moving parts. As such, they are considered an environmentally friendly, economical, and efficient alternative to traditional propeller-type turbines [2]. In comparison to horizontal and vertical axis wind turbines, VBWTs can generate power at lower wind speeds, which, in turn, expands their operating range and improves their power density.

The turbine is excited to oscillate due to the vortex-shedding force induced on the body mast. Once the frequency of the shedding force is close to the fundamental frequency of the turbine, the lock-in phenomenon is initiated [3]. At this particular point, the mast will oscillate at the shedding frequency of the flow vortices, which is directly proportional to the wind speed. This phenomenon allows expanding the turbine bandwidth by sustaining the oscillation between the turbine structure and the shedding vortices unaltered beyond the fundamental frequency of the turbine. In general, the maximum amount of energy is extracted from the wind within the lock-in region [4]. However, the upper bound at which the lock-in region collapses is determined by the mass ratio of the mast to the added mass by the airflow. The bandwidth of the lock-in region is narrowed down as the ratio increases (i.e., added mass increases with the increasing speed of the wind) [3].

The lock-in region is also narrow-banded because the turbine structure is often unchanged and damped. There have been many research attempts to improve and optimize the output power, bandwidth, and efficiency of the wind turbine. Yazdi [5,6] proposed a tuning mechanism with a linear actuator and an optimal control strategy to slide a mass inside the hollow mast of the piezoelectric-based turbine; as such, the structure resonance could be regulated close to the shedding frequency at a different airflow speed. At a wind speed of 1.6 m/s, the numerical results suggested an increase in the 6 m long turbine bandwidth and power of 190% and 294%, respectively. This method, however, is complex and requires external power to operate the actuator and the controller. Meliga et al. [7] presented a feedback control approach that utilizes an actuator located in the cylindrical mast of the turbine. The actuator ejects a controlled stream of flow at an optimized velocity, position, and angle to suppress the vibration of the cylinder to remain within the limit cycle orbits of the shedding vortices of the flow. This control approach leads to a 3.5% increase in the harvested energy. Using this approach, the system has more robustness against small inaccuracies of the structural parameters.

Optimizing the shape of the body mast is another method to increase the harnessed power. Ding et al. [8] numerically investigated several cross-sectional shapes of the body mast. A quasi-trapezoid shape (i.e., a shape between square and triangle) generates more power than the other designs, including triangular prism, square cylinder, and circular cylinder. However, for VBWTs, the cylindrical body mast remains preferred over the other designs due to its less sensitivity to air flow direction. Square cylinders are best used for energy harvesters utilizing the galloping effect of the flow. On the other hand, Villarreal et al. [9,10] introduced an arrangement of two pairs of annular magnets to the structure of the VBWT. The magnets have opposite polarities to repel each other to form a repulsive restoring force. This nonlinear restoring force tunes the overall stiffness of the structure. As the wind speed increases, the structure becomes stiffer, and, thus, the resonance frequency increases to match the shedding frequency of the flow at higher wind speeds. This method, in turn, widens the lock-in region of the turbine.

In this research work, a tunable mechanism based on a progressive-rate spring is presented. The spring stiffness varies as the wind speed changes. The spring is attached to a permanent magnet, allowing it to oscillate vertically inside the electromagnetic coil. The key feature of this mechanism is to keep the fundamental frequency of the turbine locked with the shedding frequency of the flow. As such, the lock-in phenomenon is established at a wider range of wind speeds. In conventional VBWTs, the magnet or the coil usually moves in an arched profile relative to each other. In this proposed configuration, an extra degree of freedom is added. The magnet is allowed to oscillate in the longitudinal direction of the mast body. The principal motive of this research work is to explore the feasibility of utilizing a passive tunable progressive-rate spring. To quantify the harvested power characteristics of the turbine, a mathematical model describing the nonlinear coupled dynamics of the turbine is presented in Section 3. The proposed design structure is introduced next.

2. Turbine Design and Structure

In general, bladeless wind turbines are based on piezoelectric [5] and/or electromagnetic [9] harvesting units. Salvador et al. [11] experimentally investigated two configurations of magnets in a harvester, namely disc-shaped and arc-shaped. In this design, the magnets were attached to the base of the mast body that was pivoted to swing as a pendulum. Results show that the arc-shaped magnet yields a higher voltage output. This might be attributed to the larger size of the arc-shaped magnet tested in the study compared with the disc-shaped magnet. Villarreal [12] proposed a set of magnets arranged around the circular circumference of the turbine base. Each magnet is placed between two coils, which are attached to the mast body. As the mast oscillates, the coils move relatively to the magnets to generate an induced electrical power. Sets of magnets and coils are stacked above each other to generate more power. Gautam et al. [13] compared two configurations of magnet field arrangements. The first is to arrange the magnets in a circular pattern, and the other is to arrange them radially. It was observed that the magnetic flux change is enhanced in the radial configuration for small vibrations. This research also investigated stacking the magnets in layers. It was concluded that two opposite radial magnets per layer stacked vertically in three layers would further improve the flux change rate. The layers were oriented 120° in-plane relative to each other. Moreover, five different coil configurations were proposed in [14] to improve the field gradient for small vibrations. Each design showed a maximum field density at a different location from the mean or center position of the oscillating mast body. The coil configurations were mainly made of four Golay coils that are generally used to create a magnetic field gradient perpendicular to the main magnetic field. A flux density variation of 900 mT was achieved by the optimum design of the coil configuration.

The structure of the proposed vortex bladeless wind turbine (VBWT) is presented in Figure 1. The turbine comprises three main components: a cantilever beam, a cylindrical mast body, and a harvesting unit. The underlying concept of the VBWT is to capture the vortex-induced vibration of the body mast attached to a vertical flexible cantilever beam using a permanent magnet (PM) oscillating inside a multi-layer coil. Among others, this method has been proven to provide high electromechanical energy conversion efficiency, energy density, and high voltage [13]. The electromagnetic harvesting unit is hosted inside the turbine to harness the mechanical energy of the vibrating mast into electrical energy based on Faraday’s law of induction. The mechanical vibration of the mast is harnessed to electrical energy due to the relative motion between the magnet and the coil. In order to provide an efficient harvesting mechanism, the coil is fixed to the stationary body of the turbine, and the magnet is attached to the vibrating body mast. Using this configuration, the permanent magnet will have a circular arc (2D) motion relative to the coil. Moreover, in this unique design, the magnet is connected to a spring, which allows it to slide in the longitudinal direction of the body mast and the coil. A rigid guide rod is used to confine the lateral and rotary motion of the magnet not to introduce instability during the operation of the turbine.

The tuning mechanism that changes the fundamental frequency of the turbine as the wind speed changes can be achieved by altering the intrinsic stiffness of the spring attached to the sliding permanent magnet of the harvester. It will be shown later in the following sections that the fundamental frequency of the turbine is a function of the spring stiffness. As such, the lock-in phenomenon can be initiated theoretically at any wind speed by tuning the fundamental frequency of the turbine (i.e., by changing the spring stiffness) to match the shedding frequency of the flow at a given wind speed. Progressive-rate springs such as Belleville [15,16], wave [17], or variable-pitch [18] springs can be selectively engineered to offer the displacement–force characteristics needed for the tunable mechanism of the turbine. The dynamic model of this turbine is presented in the next section.

3. Dynamic Modeling

In this section, the nonlinear dynamic modeling of the proposed design of the tunable bladeless turbine is developed. Assuming the fundamental frequency of the turbine dominates the dynamics during the operation, the turbine can be schematically presented by the lumped-mass configuration depicted in Figure 2. The permanent magnet (PM) of the harvester, which translates and swings simultaneously, is considered a point mass ($m_2$) oscillating on an inverted pendulum mounted on a cart. The cart with mass ($m_1$), on the other hand, represents the equivalent oscillation motion of the free end of the cantilever beam. $k_1$ and $k_2$ symbolize the equivalent stiffness of the beam and the tunable spring stiffness, respectively. The turbine is driven by a vortex-induced force that is represented by $F$. An electromagnetic coil is introduced to harness the PM motion to electrical power. The resistor ($R_L$) connected to the coil imitates the equivalent external load or circuit attached to the turbine.

Without loss of generality, the mast body is assumed to be made of a lightweight material, so its weight and rotary inertia are negligible. The magnet is confined to slide over a guide rod to maintain its motion in the longitudinal direction of the mast body while it is oscillating. As such, the rotational angle ($\theta$) of the oscillating magnet is equal to the rotational angle of the mast body which, in turn, is identical to the free-end angle of the beam. Moreover, the free-end angle ($\theta$) of the beam is related to its free-end deflection ($x$). The relationship is derived following the assumptions of the linear transverse vibrations of the undamped Euler–Bernoulli beam. The governing equation of motion of the cantilever beam carrying a tip mass is obtained as

$$EI\frac{{\partial }^{4}w}{\partial y^4}+m\frac{{\partial }^{2}w}{\partial t^2}=0$$

(1)

where $w\left(y,t\right)$ is the transverse displacement of the beam at position y along the cantilever beam. $EI$ and $m$ are the bending stiffness and the mass per unit length of the beam, respectively. The boundary condition of forces at the free end of the beam carrying a tip mass of $m_2$ (the permanent magnet) is expressed as

$$EI\frac{{\partial }^{3}w\left(L,t\right)}{\partial y^3}-m_2\frac{{\partial }^{2}w\left(L,t\right)}{\partial t^2}=0$$

(2)

Using the Galerkin procedure [19] to discretize Equation (2), the transverse displacement ($w$) can be approximated by introducing spatial and temporal functions as

$$w\left(y,t\right)={\displaystyle \sum }_{i=1}^n{\varphi }_i\left(y\right){\eta }_i\left(t\right)$$

(3)

where ${\varphi }_i\left(y\right)$ ($i$ = 1, 2, 3 …, n) are the mode shapes of the clamped-free cantilever beam. Following our assumption that the fundamental frequency is dominant, then only the first mode can be considered in this analysis. The first mode is expressed as [20]

$${\varphi }_1\left(y\right)=A_1\left[\cos \frac{{\beta }_1}{L_b}y-\cosh \frac{{\beta }_1}{L_b}y+{\gamma }_1\left(\sin \frac{{\beta }_1}{L_b}y-\sinh \frac{{\beta }_1}{L_b}y\right)\right]$$

(4)

in which, $L_b$ is the length of the beam and

$${\gamma }_1=\frac{\sin {\beta }_1-\sinh {\beta }_1+{\beta }_1\frac{m_2}{m{L}_b}\left(\cos {\beta }_1-\cosh {\beta }_1\right)}{\cos {\beta }_1+\cosh {\beta }_1-{\beta }_1\frac{m_2}{m{L}_b}\left(\sin {\beta }_1-\sinh {\beta }_1\right)}$$

(5)

Assuming the rotary inertia of the tip mass is negligible, the eigenvalue of the first vibration mode denoted by ${\beta }_1$ is calculated from the characteristic equation

$$1+\cos {\beta }_1\cosh {\beta }_1+{\beta }_1\frac{m_2}{m{L}_b}\left(\cos {\beta }_1\sinh {\beta }_1-\sin {\beta }_1\cosh {\beta }_1\right)=0$$

(6)

For a lumped-mass representation of the turbine, the free-end displacement ($x$) and rotation ($\theta$) of the beam are expressed as $w\left(L_b,t\right)$ and $w^{\prime }\left(L_b,t\right)$, respectively. Defining the parameter $r$ as the ratio of the rotation ($\theta$) over the displacement ($x$) results in

$$r=\frac{\theta }{x}=\frac{w^{\prime }\left(L_b,t\right)}{w\left(L_b,t\right)}=\frac{{\varphi }_1^{\prime }\left(L_b\right)}{{\varphi }_1\left(L_b\right)}=-\frac{{\beta }_1\left[\sin {\beta }_1+\sinh {\beta }_1-{\gamma }_1\left(\cos {\beta }_1+\cosh {\beta }_1\right)\right]}{L_b\left[\cos {\beta }_1-\cosh {\beta }_1+{\gamma }_1\left(\sin {\beta }_1-\sinh {\beta }_1\right)\right]}$$

(7)

The superscript ($\prime$) represents the derivative with respect to $y$. As it is clearly seen, the parameter $r$ can be determined at a given magnet mass ($m_2$), beam length ($L_b$), and mass per unit length of the beam ($m$) by first calculating the eigenvalue (${\beta }_1$) from Equation (6) and then substituting it into Equation (5) to calculate the modal parameter (${\gamma }_1$). Once ${\beta }_1$ and ${\gamma }_1$ are known, Equation (7) is then used to determine the ratio $r$. It is worth noting that using the guide rod imposes restrictions on the rotational motion of the magnet that is now prescribed or constrained by the motion of free-end displacement of the cantilever beam, or vice versa, as described by Equation (7). The constraint equation that restricts the rotational motion of the magnet is given by

$$C\left(x,\theta ,t\right)=\theta \left(t\right)-rx\left(t\right)=0$$

(8)

The equations of motion of the turbine are developed using the Lagrange formulation. Lagrange’s multiplier ($\lambda$) [21,22] is introduced to impose the constraint rotational motion of the magnet. The turbine equations of motion are formulated by implementing the Lagrange given equation [22]

$$\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial {\dot{q}}_i}-\frac{\partial \mathcal{L}}{\partial q_i}+\frac{\partial D}{\partial {\dot{q}}_i}=Q_i+\lambda \frac{\partial C}{\partial q_i}$$

(9)

where $\mathcal{L}=T-U$ denotes the Lagrangian that is the difference between the kinetic ($T$) and potential ($U$) energies of the system. For $i=1, 2, \cdots n$, $q_i$ and $Q_i$ are the generalized coordinates and forces of the system, respectively. $D$ represents Rayleigh’s dissipation function of the turbine. The last RHS term of Equation (9) is the constraint force applied to the rotational motion of the magnet. By defining four generalized coordinates, which are beam deflection ($q_1=x$), mast rotation angle ($q_2=\theta$), magnet sliding motion ($q_3=s$) relative to the mast body, and output charge ($q_4=q_c$) of the electromagnetic coil, the kinetic, potential, and dissipative energies of the turbine have the form:

Kinetic Energy:

$$T=\frac{1}{2}{m}_1{\dot{x}}^2+\frac{1}{2}{m}_2\left[{\left(\dot{x}+\dot{s}\sin \theta +s\dot{\theta }\cos \theta \right)}^2+{\left(s\dot{\theta }\sin \theta -\dot{s}\cos \theta \right)}^2\right]+\frac{1}{2}{L}_c{\dot{q}}_c^2+{\dot{q}}_c\left({\alpha }_{x}x+{\alpha }_{s}s\right)$$

(10)

Potential Energy:

$$U=\frac{1}{2}{k}_1{x}^2+\frac{1}{2}{k}_2{\left(s-l_0\right)}^2+m_{2}gs\cos \theta$$

(11)

Dissipation Function:

$$D=\frac{1}{2}{c}_1{\dot{x}}^2+\frac{1}{2}\left(R_c+R_L\right){\dot{q}}_c^2$$

(12)

Here, $L_c$ and $R_c$ are the electromagnetic coil inductance and resistance, respectively. $l_0$ is the initial length of the tunable spring ($k_2$). $g$ is the acceleration due to gravity (9.81 $m/s^2$). The last term in Equation (10) represents the coupling energy between the mechanical and electrical domains [23]. The electromagnetic coupling factors in the $x$- and $s$-directions are denoted by ${\alpha }_x$ and ${\alpha }_s$, respectively.

In addition to the above, the generalized forces ($Q_i$) are represented only by the lift force in the $x$-direction (i.e., $Q_1=F$). The lift force due to the vortex shedding acting on the body mast of the turbine given by [24]

$$F=\frac{1}{2}{\rho }_a{U}_f^2{D}_m{L}_m{C}_L\sin \left(2\pi f_{v}t\right)$$

(13)

where, ${\rho }_a$ is the density of the air. $D_m$ and $L_m$ are the diameter and length of the mast body. $C_L$ is the lifting coefficient that on average equals 0.50 for Reynold’s numbers between 10³ and 10⁵ [25]. $U_f$ denotes the wind speed, and $f_v$ is the shedding frequency in Hz that is analytically presented with the following expression [26]

$$f_v=\frac{S_t{U}_f}{D_m}$$

(14)

where $S_t$ is the Strouhal number (~0.20) [26] that represents a measure of the ratio of the inertial forces of the flow to the inertial forces of the convective acceleration. In this preliminary study, increasing the output power of the turbine is the primary interest, so the dynamics within the lock-in region that is associated with the bandwidth of the turbine are not considered.

Substituting Equations (10)–(13) into Lagrange’s Equation (9), the equations of motion are readily obtained. However, the equations of motion can be further simplified. For small angles, the trigonometric functions are approximated as $\sin \theta \approx \theta$ and $\cos \theta \approx 1$. Further, for an infinitesimal oscillation of the beam, the centrifugal force term ($m_{2}s{\dot{\theta }}^2$) is neglectable. By doing this, the system equations of motion become

The $x$-direction equation:

$$\left(m_1+m_2\right)\ddot{x}+k_{1}x+c_1\dot{x}+m_2\ddot{s}\theta +2m_2\dot{s}\dot{\theta }+m_{2}s\ddot{\theta }-{\alpha }_x{\dot{q}}_c-r\lambda =F_o\sin \left(2\pi f_{v}t\right)$$

(15)

The $\theta$-direction equation:

$$m_{2}s\ddot{\theta }+2m_2\dot{s}\dot{\theta }+m_2\ddot{x}-m_{2}g\theta +\lambda =0$$

(16)

The $s$-direction equation:

$$m_2\ddot{s}+m_2\ddot{x}\theta +m_{2}g+k_2\left(s-l_0\right)-{\alpha }_s{\dot{q}}_c=0$$

(17)

The electromagnetic circuit equation:

$$L_c{\ddot{q}}_c^2+\left(R_c+R_L\right){\dot{q}}_c+{\alpha }_x\dot{x}+{\alpha }_s\dot{s}=0$$

(18)

Moreover, Equation (8) yields the fifth EOM that is needed to describe the full dynamics of the turbine, including the four generalized coordinates and Lagrange’s multiplier. In general, if the constraint equation ($C$) is explicitly a function of time and the generalized coordinates vector ($\mathit{q}={\left[q_1, q_2, q_3, q_4\right]}^T$) such that $C=C\left(\mathit{q}, t\right)$, then taking the second time derivative of the constraint equation yields [27]

$${\mathit{C}}_{\mathit{q}}\ddot{\mathit{q}}+2{\mathit{C}}_{\mathit{t}\mathit{q}}\dot{\mathit{q}}+{\mathit{C}}_{\mathit{q}\mathit{q}}\dot{\mathit{q}}+C_{tt}=0$$

(19)

where ${\mathit{C}}_{\mathit{q}}$ is the Jacobian matrix of the constraint equation associated with the generalized coordinates that is

$${\mathit{C}}_{\mathit{q}}=\left[\frac{\partial C}{\partial q_1}, \frac{\partial C}{\partial q_2},\frac{\partial C}{\partial q_3},\frac{\partial C}{\partial q_4}\right]$$

${\mathit{C}}_{\mathit{t}\mathit{q}}$ is the vector of the time-partial derivative of the Jacobian matrix such that

$${\mathit{C}}_{\mathit{t}\mathit{q}}=\left[\frac{{\partial }^{2}C}{\partial t\partial q_1}, \frac{{\partial }^{2}C}{\partial t\partial q_2},\frac{{\partial }^{2}C}{\partial t\partial q_3},\frac{{\partial }^{2}C}{\partial t\partial q_4}\right]$$

${\mathit{C}}_{\mathit{q}\mathit{q}}$ is the vector of partial derivative of the term ${\mathit{C}}_{\mathit{q}}\dot{\mathit{q}}$ calculated as

$${\mathit{C}}_{\mathit{q}\mathit{q}}=\left[\frac{\partial {\mathit{C}}_{\mathit{q}}\dot{\mathit{q}}}{\partial q_1}, \frac{\partial {\mathit{C}}_{\mathit{q}}\dot{\mathit{q}}}{\partial q_2},\frac{\partial {\mathit{C}}_{\mathit{q}}\dot{\mathit{q}}}{\partial q_3},\frac{\partial {\mathit{C}}_{\mathit{q}}\dot{\mathit{q}}}{\partial q_4}\right]$$

and $C_{tt}$ is the second time-partial derivative of the constraint equation. In the preceding equation and for our system under the investigation, only the first term remains nonzero. Thus, Equation (19) reduces to

$$\ddot{\theta }-r\ddot{x}=0$$

(20)

Next, the electromagnetic circuit model parameters, including the coil inductance, resistance, and coupling factors, can be estimated using closed-form approximations. For instance, the coil inductance ($L_c$) in Henry is calculated using Wheeler’s spiral equation [28]

$$L_c=\frac{7.87\times {10}^{-6}{d}_c^2{N}_c^2}{3d_c+9h_c+10w_c}$$

(21)

in which, $N_c$ is the number of turns of the coil. $h_c$, $w_c$, and $d_c$ are the height, width, and mean diameter of the multilayer coil. The equation is accurate for $w_c>0.2d_c$. The coil resistance ($R_c$) in Ohms is estimated using [29]:

$$R_c=\frac{4{ϵ}_e{N}_c{d}_c}{d_w^2}$$

(22)

${ϵ}_e$ is the copper wire resistivity (i.e., 1.68 × 10⁻⁸ Ωm [30]), and $d_w$ is the wire diameter of the coil. On the other hand, the coupling factors ${\alpha }_x$ and ${\alpha }_s$ are calculated based on Faraday’s law of induction [XX15]. The induced electromotive force ($f_{em}$) is described by

$$f_{em}=-\frac{d\mathsf{\Phi }}{dt}=-\left(\frac{\partial \mathsf{\Phi }}{\partial x}\dot{x}+\frac{\partial \mathsf{\Phi }}{\partial s}\dot{s}\right)=-\left({\alpha }_x\dot{x}+{\alpha }_s\dot{s}\right)$$

(23)

where $\mathsf{\Phi }$ is the total flux of the coil. In our analysis, the coupling factors are constant and evaluated at the maximum flux point. The coupling factor in $s$-direction is given by [23]

$${\alpha }_s=\frac{N_c{B}_r{V}_m}{2h_c{w}_c}$$

(24)

$B_r$ is the residual magnetic flux density of the magnet. $V_m$ is the volume of the magnet. It can be shown from the analysis conducted by Behtouei et. al. [31] that the coupling factor in the $s$-direction is approximately twice that of the $x$-direction. For a sinusoidal output current at steady state, the output rms electrical power ($P_T$) of the turbine is calculated as

$$P_T=\frac{1}{2}{R}_L{\dot{q}}_{c,max}^2$$

(25)

To perform the dynamic analysis of the turbine, the coupled equations of motion are numerically solved in the next section using the MATLAB^® ode45 solver [32].

4. Results and Discussion

In this section, the dynamic investigation of the turbine is discussed. Based on the global distribution of the winds, the mean annual wind speed is estimated to be less than 6.9 m/s at an elevation of 80 m ± 20 m above the ground [33]. Thus, in this work, the equations of motion are solved numerically for a wind speed less than 10 m/s. The nominal parameters used in our simulations are listed in

Table 1. Nominal parameters of the turbine used in this study.

Description	Parameter	Value	Unit
Equivalent mass of the beam	$m_1$	0.133	kg
Equivalent stiffness of the beam	$k_1$	306.3	N/m
Equivalent damping of the beam	$c_1$	0.383	N/m/s2
Total length of the beam	$L_b$	1.00	m
Magnet mass	$m_2$	0.250	kg
Total length of the mast body	$L_m$	2.00	m
Mast body outer diameter	$D_m$	0.200	m
Unstretched length of the spring	$l_o$	0.050	$\mathrm{m}$
Coil inductance	$L_c$	35.00	H
Coil resistance	$R_c$	2.00	$\mathrm{k}\mathsf{\Omega }$
External resistive load	$R_L$	5.00	$\mathrm{k}\mathsf{\Omega }$
$s$-direction coupling factor	${\alpha }_s$	2200	N/A
Ratio defined by Equation (7)	$r$	2.473	rad/m
Lift coefficient	$C_L$	0.50	-
Wind speed	$U_f$	0.0 to 10.0	m/s

. The parameters typically represent a 2 m long vortex bladeless turbine. The primary objective of this research work is to investigate the effect of interposing a tunable spring between the magnet and the mast body on the overall performance of the turbine in terms of the output power and bandwidth. In the following analysis, the equations of motion were simulated numerically using ode45 in MATLAB^®.

4.1. Effect of Coupling Factors on the Turbine Dynamics

The equations of motion (Equations (15)–(18) and (20)) represent a 3-DOF system, namely $x$, $s$, and $q_c$. It is important to note that the rotational motion ($\theta$) of the mast body is constrained to the motion of the free-end beam following Equation (8). At a spring stiffness ($k_2$) of 2 kN/m, Figure 3 illustrates the charge frequency spectrum at different levels of coupling factors. As expected, at low coupling factors, the system has three distinct resonances associated with its three degrees of freedom. However, as the coupling factor increases, the system tends to have only one resonance. This can be attributed to the increasing coupling among the modeling parameters. If the effect of the inductance of the coil is assumed negligible, substituting ${\dot{q}}_c$ from Equation (18) into Equations (15) and (17), the effective damping of the system in the $x$- and $s$-directions becomes proportional to (${\alpha }_s^2$). In particular, the damping due to the coupling factor dominates the overall dynamics of the system.

It is also observed from the figure that the overall electrical energy harvested from the wind increases as the coupling factor increases at a wider bandwidth in comparison to the case of a small coupling factor. With a negligible coil inductance, the output rms power is proportional to (${\alpha }_s^2$) and the velocity amplitudes of the modal parameters as shown

$$P_T=\frac{1}{2}\frac{{\alpha }_s^2{R}_L}{{\left(R_c+R_L\right)}^2}{\left(0.5{\dot{x}}_{max}+{\dot{s}}_{max}\right)}^2$$

(26)

However, it is very crucial to note that the power is not always maximum at the maximum coupling factor. A large coupling factor will damp out the system and, thus, reduce the velocity amplitudes of the modal parameters. Therefore, modal parameters should be selected optimally to maximize the output power of the turbine.

4.2. Effect of Tunable Spring Stiffness on the Output Power

Using the nominal parameters, the frequency spectrum of the output charge of the turbine at different values of the spring stiffness is shown in Figure 4. Obviously, the fundamental frequency of the turbine can be tuned by adjusting the spring stiffness. The stiffer the spring, the smaller the harvested power. This is due to the fact that as the spring stiffness is high, the magnet displacement in the $s$-direction will be small. At an extremely high stiffness, the spring is considered a rigid link. In this case, the system presents a conventional bladeless wind turbine.

Figure 5 depicts the time history of the four generalized coordinates of the turbine at a wind speed ($U_f$) of 4.22 m/s and a spring stiffness of 2034 N/m. The vortex-shedding frequency (see Equation (14)) at this wind speed coincides with the fundamental frequency of the turbine at its nominal parameters and the simulated spring stiffness. As expected, the four generalized coordinates are oscillating at the same resonance frequency. Moreover, the ratio $r$ given by Equation (7) is approximately 2.473 rad/m or 141.72 deg/m. Examining the figure, this ratio and, hence, the constraint equation, remains valid between the free-end displacement of the beam ($x$) and the rotational angle ($\theta$) of the body mast.

In the conventional VBWT, the magnet is rigidly fixed to the mast body such that it cannot oscillate in the $s$-direction. In this analysis, the spring with a stiffness of 30 kN/m is assumed a rigid link. The output power of the tunable turbine is compared with that of the conventional turbine, as shown in Figure 6. The results simulate the case of a turbine with and without a tuning spring at the same wind speed. As clearly seen, the tunable turbine outperforms the conventional turbine. The output power of the tunable turbine is about 65 times of the conventional turbine at a wind speed of 4.22 m/s. The output rms power of the tunable turbine is around 1105 mW versus 17 mW of the conventional turbine. As mentioned earlier, the tunable turbine is at resonance, which explains the large amount of power generated at this wind speed. Nevertheless, it is necessary to recall that the main objective of using the tunable spring is to maintain the turbine at resonance at any given wind speed.

To further investigate the dynamics of the tunable turbine, the system is simulated at a wind speed of 7.0 m/s using two different cases. The first case represents a resonating tunable turbine with a spring stiffness of 8081 N/m, and the other is a nonresonating case with a spring stiffness of 2034 N/m. Figure 7 shows the output power of the two cases against the output power of the conventional turbine. The rms power of the resonating turbine is 2028 mW, which is 100 times higher than the case of nonresonating one. On the other hand, the output rms power of the conventional turbine is around 133 mW, which is 12 times larger than that of the nonresonating turbine. As such, it is very important the spring stiffness be tuned precisely to achieve the resonance of the turbine at a given wind speed. Otherwise, the turbine performance may be degraded in comparison to the conventional one. However, with an accurate tuning mechanism, the simulated results illustrate the effectiveness of using a tunable spring to improve the output power of the turbine.

4.3. Optimum Spring Stiffness

This subsection explores the optimum spring stiffness at a given wind speed. Figure 8 shows the output power of the turbine under various wind speeds as the tuning spring stiffness ($k_2$) varies. The output rms power is normalized with respect to the maximum power associated with each case. It could be observed that, at high wind speeds, there is an optimum spring stiffness at which the output power of the turbine is maximum. On the contrary, at relatively low wind speeds, no optimum spring stiffness exists. Thus, the conventional bladeless turbine is desired at low wind speeds. This suggests that the tunable spring might be feasible within a range of wind speeds.

To find the threshold wind speed beyond which the tunable spring is effective, Figure 9 presents the optimum spring stiffness of wind speeds within the range of 1 m/s to 10 m/s. In this analysis, the spring is assumed rigid if its stiffness is more than 30 kN/m. As predicted, the tunable turbine operates immediately right after a threshold wind speed (denoted by $U_t$) is passed. For the nominal parameters listed in Table 1, the threshold wind speed is approximately 2.875 m/s. As the wind speed increases beyond the threshold speed, the required optimum spring stiffness increases quadratically. On the other hand, at low wind speeds less than the threshold speed, the conventional turbine is more effective. The output rms power of the tunable and conventional wind turbines as a function of the wind speed is depicted in Figure 10. The magnitude of harnessed power depends on the wind speed. The output power of the tunable turbine surpasses that of the conventional turbine past the threshold speed. For instance, at wind speeds of 5 m/s and 8 m/s, the output power of the tunable turbine is 50 and 11 times larger in comparison to the conventional turbines, respectively. The output power of the tunable turbine is 2120 mW at a wind speed of 8.0 m/s. However, the actual harvested power is expected to be less if the weight of the mast body and the rotary inertia of the magnet are included in the analysis. As a concluding remark, the power at high wind speeds is less sensitive to the variation of the spring stiffness around its optimum value as illustrated in Figure 8.

4.4. Threshold Wind Speed

Next, the threshold wind speed ($U_t$) is investigated in detail in this subsection. In order to achieve an intuitive understanding of the dynamics around this speed, the effect of the cantilever beam on the threshold speed is studied. Figure 11 presents the optimum spring stiffness at various cantilever beam diameters of the tunable turbine. As a first insight, the threshold wind speed generally increases with increasing beam diameter. At a large beam diameter, the turbine becomes stiffer to oscillate at low wind speeds. Consequently, the wind speed needed to operate the turbine is high, which, in turn, reduces the usable range of the turbine. The optimum spring stiffness is quadratically proportional to the wind speed for all diameters beyond the threshold speed. The equivalent stiffness of the cantilever beam ($k_1$) can be determined by [34]

$$k_1=\frac{3\pi }{64}\frac{E_b{d}^4}{L_b^3}$$

(27)

where $E_b$ is the Young’s modulus of the beam material, and $d$ is the beam diameter. Research studies have suggested using glass fiber material for its durability at large amplitude of oscillations [35].

The threshold wind speed can be estimated analytically from the dynamic model of the turbine. At low wind speeds, the turbine oscillations are small, and the nonlinearities introduced by the Coriolis, centripetal, and centrifugal forces are likely negligible. Hence, the dynamic model given by Equations (15)–(18) and (20) is reduced to:

$$\left(m_1+m_2\right)\ddot{x}+k_{1}x+c_1\dot{x}-{\alpha }_x{\dot{q}}_c-r\lambda =F_o\sin \left(2\pi f_{v}t\right)$$

(28)

$$m_{2}s\ddot{\theta }+m_2\ddot{x}-m_{2}g\theta +\lambda =0$$

(29)

$$m_2\ddot{s}+m_{2}g+k_2\left(s-l_0\right)-{\alpha }_s{\dot{q}}_c=0$$

(30)

$$\left(R_c+R_L\right){\dot{q}}_c+{\alpha }_x\dot{x}+{\alpha }_s\dot{s}=0$$

(31)

Equation (30) expresses the dynamic of the magnet at low wind speeds and oscillations. The magnet motion is not driven sufficiently by the lateral motion of the turbine (e.g., compare Equations (17) and (30), $m_2\ddot{x}\theta \approx 0$). As such, the magnet motion cannot be initiated at low wind speeds, and it remains stationary as if it is fixed rigidly to the mast body of the turbine. In this case, the tunable turbine acts as a conventional one. Similar dynamics were also observed in [36] for a pendulum tuned mass damper (TMD) system that has a limited effect at small vibrations.

In order to increase the turbine oscillations, it should be driven at resonance. As the wind speed continues to increase, the speed at which the turbine starts to resonate is the threshold wind speed. At that speed, the system will violently oscillate and, therefore, lead to exciting the motion of the magnet in the $s$-direction. Nonlinear terms of the system play an important role in giving the turbine its unique dynamic characteristics beyond the threshold wind speed. At small oscillations, the electromagnetic coupling factor (${\alpha }_s$) does not dampen the system motion. The threshold wind speed can be estimated by the undamped fundamental frequency of the turbine at low wind speeds. Using Equations (20), (28) and (29), the fundamental frequency of the turbine is systematically obtained as

$$f_n=\frac{1}{2\pi }\sqrt{\frac{k_1+r^2{m}_{2}g}{m_1+\left(1+r+l_o{r}^2\right)m_2}}$$

(32)

The turbine resonates when the shedding frequency of the wind flow matches its fundamental frequency, i.e., $f_n=f_v$. Thus, equating Equations (14) and (32), the threshold wind speed ($U_t$) is analytically given by

$$U_t=\frac{f_n{D}_m}{S_t}$$

(33)

The threshold wind speeds at different beam diameters shown in Figure 11 are used to validate Equation (33). Using the nominal parameters,

Table 2. Comparison study between the analytical and numerical estimations of the threshold wind speed of the turbine.

Diameter (d) (mm)	Analytical ($U_t$) (m/s)	Numerical ($U_t$) (m/s)
15	1.6539	1.6207
20	2.7476	2.8367
25	4.1141	4.1034
30	5.6855	5.7755

presents a comparative analysis between the analytical and numerical threshold wind speeds. The values of the analytical threshold speeds computed using Equation (33) are in reasonable agreement with the numerical results at an average difference of 1.8%. Equation (33) presents an important design parameter of the tunable turbine. It can be used to define the usable wind range of the turbine.

4.5. Design Optimization

The optimization procedure can be conducted to determine the optimum design parameters of the turbine to increase its output power and operational range. The output rms power given by Equation (26) depends heavily on the value of the external load of the turbine. To find the optimal external load of the turbine, an illustrative numerical example is examined at a wind speed of 7.0 m/s. The output power of the turbine is simulated over a range of external loads and spring stiffnesses, as shown in Figure 12. The optimum external load and spring stiffness of the harvester are clearly determined to be 2246 $\mathsf{\Omega }$ and 8081 N/m, respectively. The maximum harvested rms power at these optimum parameters is 2412 mW. To this end, the turbine can be optimized further by considering the other design parameters such as the beam stiffness, the mass of the magnet, and the coupling factors. However, the lock-in phenomena in a tunable turbine should be first investigated in future analysis. This will provide more insights into the dynamics of the turbine within the lock-in region and the way the tunable spring affects the region’s characteristics, including its power and bandwidth.

5. Conclusions

The aim of the present research is to numerically examine the dynamics of a 3-DOF vortex bladeless wind turbine with a tunable mechanism. The mechanism is based on a progressive-rate spring placed between the oscillating permanent magnet and the body mast of the turbine. The cardinal idea of this work is to tune the fundamental frequency of the turbine to match the vortex-shedding frequency within a range of wind speeds by adjusting the spring stiffness. A mathematical model was set up to describe the nonlinear dynamics of the tunable turbine. It was observed that the electromechanical coupling factors alter the dynamics of the turbine greatly to have only one fundamental frequency at high wind speeds. The results of this investigation show that the turbine can only be operated effectively beyond a threshold wind speed. An analytical expression was developed to calculate the threshold speed. The analytical results are in good agreement with the numerical analysis. The research has also shown that, at a wind speed larger than the threshold speed, there exists an optimum value of a spring stiffness at which the output power of the turbine is maximized. As the wind speed increases, the required optimum spring stiffness increases quadratically. The tunable turbine outperforms the conventional VBWT. The power realized by a tunable turbine is several tens of times larger than that of the conventional one. For instance, using the nominal parameters at a wind speed of 8.0 m/s, the output rms power of a 2 m long tunable turbine can reach 2120 mW with an optimum spring stiffness of 11,210 N/m. The output power can be maximized at an optimum external load.

In general, the tunable turbine performs low at wind speeds below the threshold speed, but this limitation can be overcome by proper design optimization. The threshold wind speed is a function of the modal parameters that can be designed to keep the threshold as small as possible. Further studies need to be carried out to include the lock-in phenomenon dynamics in the developed mathematical model of the turbine. A wide band of power can be realized with the lock-in region. In conclusion, this research work demonstrated the feasibility of using a tunable turbine at high wind speeds.