A multi-stable rotational energy harvester for arbitrary bi-directional horizontal excitation at ultra-low frequencies for self-powered sensing

The schematic of the proposed MEH is presented in figures 1(a)–(c). A sphere magnet is placed in a torus-shaped non-magnetic casing, which acts as a mechanism to convert linear vibration input or limb swing motion into rotational motion. The circumference of the rolling path in the torus casing is n times larger than that of the sphere magnet, where n is a positive integer. This allows the sphere magnet to complete n number of full revolutions during one complete cycle along the rolling path. A set of 2n tethering magnets are evenly distributed along this path as shown on figure 1(b). These magnets are of alternating polarity and at a distance of $\pi r_{m}$ relative to each other, where r_m is the radius of the rolling magnet. Parallel to the tethering magnet array, same number of series-connected coils are placed above the rolling magnet's path, ensuring that the coils are centrally aligned with the tethering magnets.

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The tethering magnets creates attractive forces in the rolling path of the sphere magnet, enabling multi-stability in the system. When the proposed design is exposed to input vibrations, the sphere magnet moves with rotational motion along the path, and current is induced through the coils by electromagnetic induction. Having the tethering magnets aligned with the coils ensures that the sphere magnet can start oscillating from near its vicinity when excitation is induced from its resting position. Additionally, having the tethering magnets placed at a distance of half the circumference of the sphere magnet with alternating polarity enables the sphere magnet to complete exactly a half rotation as it transitions between the potential wells created by the tethering magnets, as demonstrated in figure 1(b). This means that the sphere magnet can maintain rolling motion without slipping, and change its magnetization direction as it travels from coil to coil, in effect maximizing the rate of change of magnetic flux to enhance the energy harvesting performance. The potential applications for this proposed device, such as wind turbine tower condition monitoring or human health monitoring are given in figure 1(d). Self-powered sensor nodes using this proposed energy harvester can be used in an IoT WSN setup, as demonstrated in figure 1(e), where the energy harnessed can be used to power a sensor and a wireless transmission unit, and useful information such as acceleration and temperature can be conveyed remotely for structural health monitoring. With these applications in mind, the size of the device is limited to 100 mm span and a sphere magnet of 10 mm radius is employed as the inertial generator. As a result, n value is derived as 4.

In summary, the operation of the sphere magnet from one stable position to the next occurs as follows: The motion of the sphere magnet originates from one of the potential wells as the initial position when it is subjected to external forcing. At this stage, the magnetic flux is the highest in the vicinity of the coil placed above this potential well, since one of its poles are facing the coil where magnetic flux density is the highest. The motion of the sphere magnet is dependent on the relationship between the kinetic energy gained from the input vibrations and the potential energy from the tethering magnet causing it to remain in the well. If the energy absorbed is greater than that of the attractive potential well, it can conquer the well and continue travelling towards the following well with opposite polarity. As the sphere starts moving, vibration is converted into rotational motion and magnetic flux change occurs in the air gap. As it completes a half rotation, it reaches the following potential well created by the attractive forces from the next tethering magnet with reversed polarity. The opposite polarity between wells ensures the rolling without slipping motion of the sphere magnet, reducing losses due to friction and maximising the rate of change of magnetic flux. As the sphere magnet reaches the next potential well, its polarity is reversed as a result and this phenomenon continues along the looped rolling path.

Low frequency energy harvesting using rolling magnet mechanism has been explored in the past using looped path and ferromagnetic rail [52] or cylindrical magnets with diametrical magnetization [53], but low power output due to weak electromechanical coupling or energy losses due to mechanical limiters and damping constraints are challenges that need to be addressed. The toroidal path introduced in this design allows the inertial mass to achieve continuous unrestricted motion since the path is looped and without any physical barriers and the electromechanical coupling is enhanced by introducing top-mounted coils and alternating tethering magnets in the rolling path. The ratio of circumference of the rolling magnet and that of the effective rolling path in the casing allows the magnet to complete 4 full rotations during each cycle, allowing it to effectively utilise the area available.

By considering the rolling motion of the sphere magnet on the circular path, the equation of motion can be derived using Newton's second law of motion as follows:

$$\begin{align} m\ddot{\theta}r_{m} + \frac{J\ddot{\theta}}{r_{m}} + \hat{\Theta}I + c\dot{\theta}r_{m} + r_{m}\sum^{n}_{i = 1}F^{i}_\textrm{mag} = -m\omega^{2}y\cos{\left(\omega t\right)},\end{align} \tag{ 1 }$$

where the mass and radius of the rolling magnet are m and r_m respectively, c is the mechanical damping coefficient, θ is the angular displacement of the magnet, $F^{i}_\textrm{mag}$ is the sum of attractive forces from the tethering magnets ($i = 1,2,3, {\ldots} 8$), $\hat{\Theta}$ is the electromagnetic coupling factor, I is the current through the coils, ω is the excitation frequency, y is the excitation amplitude and the rotational inertia of the magnet is expressed as $J = \frac{2}{5}mr_{m}^{2}$. The MEH is considered as a voltage source and is connected to a load resistance $R_\textrm{l}$, therefore, the following equation is derived using Kirchhoff's voltage law:

$$\begin{align} L\dot{I}+\left(R_\textrm{c}+R_\textrm{l}\right)I-\hat{\Theta}\dot{\theta}r_m = 0,\end{align} \tag{ 2 }$$

where L and $R_\textrm{c}$ are the coil's total internal inductance and resistance respectively. Equations (1) and (2) are coupled using $\hat{\Theta}$, which represents the rate of change of magnetic flux through the coils with respect to the sphere magnet's displacement. To derive this, the vertical component of magnetic flux density B_z present in the vicinity of the coil's midplane z is first modelled using cylindrical polar coordinates as follows:

$$\begin{align} B_{z} = \frac{\mu_{0}\left|\hat{m}\right|}{4\pi}\cdot\frac{3z\sin\left(\frac{s}{r_\textrm{c}}\right)\left(\phi R - s\right) - \cos\left(\frac{s}{r_\textrm{c}}\right)\left(\left(s-\phi R\right)^2 -2z^2\right)}{\left(\left(s-\phi R\right)^2+z^2\right)^\frac{5}{2}},\end{align} \tag{ 3 }$$

where µ₀ is the permeability of free space, s is the distance the sphere magnet has rolled, φ and R are the angular position and radius of the rolling path. From equation (3), the coupling factor can be calculated as a derivative of the magnetic flux Φ through a coil with N turns as follows:

$$\begin{align} \hat{\Theta} = \frac{\textrm{d}\Phi}{\textrm{d} \theta r_{m}} = \frac{\textrm{d}}{\textrm{d}\theta r_{m}}\cdot\frac{N}{a_{2}-a_{1}}\int^{a_{2}}_{a_{1}}\int^{a}_{0}\int^{2\pi}_{0}B_{z}\left(R,\phi,z\right)\textrm{d}\phi \textrm{d}\rho \textrm{d}a,\end{align} \tag{ 4 }$$

where a₁ and a₂ are the inner and outer radius of the coil. Finally, the attractive magnetic forces can be expressed by considering the dipole moment interaction between the sphere magnet the tethering magnets:

$$\begin{align} F^{i}_\textrm{mag} &= \frac{{ 3}\mu_0}{4\pi r^5}\left[\vphantom{\frac{5\left(m_\textrm{t}\cdot r_i\right)\left(m_\textrm{r}\cdot r_i\right)}{r_i^2}}\left(m_\textrm{r}\cdot r_i\right)m_\textrm{t}+\left(m_\textrm{t}\cdot r_i\right)m_\textrm{r}+\left(m_\textrm{r}\cdot r_i\right)r_i \right. \nonumber\\ & \left. \quad -\frac{5\left(m_\textrm{t}\cdot r_i\right)\left(m_\textrm{r}\cdot r_i\right)}{r_i^2}r_i\right],\end{align} \tag{ 5 }$$

where r_i is the distance between the centre of the $i_\textrm{th}$ tethering magnet and the sphere magnet and $m_\textrm{r}$ and $m_\textrm{t}$ are the dipole moments of the rolling and tethering magnets designed based on [54]. The distance r_i can be expressed as:

$$\begin{align} r_{i} = \left[R \sin\left(\frac{\left(\theta + i - 1\right)r_{m}}{R}\right), R \cos\left(\frac{\left(\theta + i - 1\right)r_{m}}{R}\right), h_\textrm{rt}\right],\end{align} \tag{ 6 }$$

where $h_\textrm{rt}$ is the vertical gap between the rolling and tethering magnets.

The electromechanical properties and dynamics of the MEH are studied by solving equations (1)–(6) numerically using MATLAB. The combined potential energy profile due to the addition of tethering magnets in the rolling path is illustrated in figure 2, showing the stable equilibrium positions and the attractive and repulsive regions. From figure 2(b), 8 stable and 8 unstable equilibria can be seen where the rolling magnet faces attractive and repulsive forces. The strength of the potential barriers can be varied by changing the vertical gap between the rolling and tethering magnets. While the unstable equilibrium positions ensure that the rolling magnet does not initiate its motion from these regions, the potential wells in the negative regions offer 8 stable positions in the rolling path and as a result the operational modes of the rolling magnet can change depending on the excitation conditions. The operation modes can vary based on the relationship between the kinetic energy of the sphere magnet and the potential energy from the tethering magnets.

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To investigate the energy dissipated in the coils as a result of the sphere magnet's motion, the magnetic field is calculated. Figure 3 shows the combined magnetic field distribution in the vicinity of the coil region due to rolling motion of the sphere magnet in the provided path. The 8 tethering magnets are placed apart by a distance of half the perimeter of the rolling magnet, as shown by the peaks are visible at $\pi r_{m}$ displacement in the potential energy profile. By placing the coils centrally aligned to the point where peak amplitudes are shown, maximum voltage can be induced in the coils for harvester's enhanced generation capabilities. At zero displacement position, the vertical magnetic field is negative and maximum, illustrated by the blue tip in figure 3(b). As the sphere magnet travels a distance of $\pi r_{m}$ and is positioned at $\pi /4$ radians in the circular path, the polarity reverses and a positive maximum peak with the same amplitude as that of the previous position is achieved as shown in the yellow tip. Thus, rate of change of magnetic flux is optimised during operation as the sphere magnet passes each coil.

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The dynamic characteristics of the MEH under harmonic excitation is investigated to establish the operation modes as the input is varied. Figure 4 presents the voltage and displacement profiles with 2 Hz excitation frequency, generated using the numerical model developed in section 2.2. The results presented in this section have been solved numerically using MATLAB ode45 ordinary differential equation solver, based on explicit Runge–Kutta formulae of order 4 and 5. The step size is determined by the solver, but is generally fixed to 1 µs for the analysis. The duration for each simulation is set up to 1000 s to ensure steady state response is obtained.

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When the MEH is excited with low energy, the kinetic energy gained by the inertial mass is weaker than the potential energy barrier, causing it to oscillate near its initial resting position. This type of motion is labelled as intra-well motion. Exposing the device to higher energy excitation input causes the sphere magnet to conquer one or more potential wells and travel with oscillatory motion, allowing more energy to be extracted through the transducers mounted above. This behaviour is labelled as cross-well motion. Increasing the excitation further causes the sphere to overcome all the potential wells as it accelerates until reaching a periodic regime, and it is characterised as continuous rotation. With a low amplitude excitation of 0.1 g, the dynamics in figures 4(a)–(c) shows that the sphere oscillates in intra-well motion. At this stage, the output voltage is quite low and does not exceed 22 mV, since the inertial mass oscillates near one stable equilibrium position. Cross-well motion is documented in figures 4(d)–(f) when 0.4 g excitation is given to the system, where output voltage of up to 1.2 V peak can be achieved across a fixed load of 400 Ω. In this mode, the system travels through potential wells and continues oscillatory motion, and higher energy is released compared to intra-well motion as the hopping phenomenon occurs. However, the energy provided is not high enough to sustain continuous rotational motion. As the energy increases slightly with a higher amplitude of 0.7 g in figures 4(g)–(i), continuous rotation is achieved and periodic AC voltage with 1.4 V peak is induced. Continuous rotational motion is the desired mode of operation for the proposed design, since energy from external excitation is absorbed effectively and the output voltage is adequate for power conditioning and storage.

An experimental prototype of the proposed energy harvester was fabricated to study the performance of the designed system. In figures 5(a)–(c), the three parts of the prototype are shown. The tethering magnets were attached to a disk, ensuring that they were alternating polarity relative to each other. The toroidal casing was fabricated in parts and then assembled to fit the sphere magnet inside as shown in figure 5(b) and the copper coils were placed on another circular disk. The parts were then assembled as shown in figure 5(d) ensuring that the tethering magnets and the coil centres were aligned. The casings were printed using 3D printing and the magnets were chosen as commercially available N42 grade NdFeB varieties. The parameters associated with the energy harvester prototype are presented in table 1. The completed prototype was attached to a slider controlled by a variable frequency drive, capable of generating base excitation up to 2.5 Hz and 1 g excitation amplitudes. The electric characteristic of the device was recorded using an oscilloscope as shown in figure 5(d) and the motion of the exposed device was recorded using a camera setup as presented in figure 5(e).

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Table 1. Design parameters and material properties of the multi-stable rotational energy harvester.

Symbol	Description	Value
m	Rolling magnet mass	29.4 g
ζ	Mechanical damping coefficient	0.0291 kg s−1
rm	Rolling magnet radius	10 mm
$r_\textrm{t}$	Tethering magnets' radii	1.5 mm
$l_\textrm{t}$	Tethering magnet height	2 mm
$h_\textrm{rt}$	Vertical gap between sphere and tethering magnet	21 mm
$h_\textrm{rc}$	Vertical gap between sphere magnet and coil	14 mm
a1	Coil inner width	4 mm
a2	Coil outer width	14 mm
a3	Coil outer length	18 mm
R	Radius of rolling path	39 mm
$B_\textrm{r}$	Residual flux density	1.3 T
L	Total Coil inductance	82.1 mH
$R_\textrm{c}$	Total Coil resistance	405.7 Ω

Figure 6(a) depicts the open circuit AC voltage from the prototype when exposed to varying range of base excitation amplitudes at 1.4 Hz. From the profile, it is apparent that the output voltage jumps from 1.1 V to 2 V peak value when the amplitude is increased from 0.2 g to 0.3 g. The relationship will be explored further in the following section. To determine the maximum power achievable, resistive loads between 100 Ω–800 Ω were placed across the harvester while excited at 1.4 Hz and 0.8 g vibrations. As highlighted in figure 6(b), a maximum peak power of 5.78 mW was extracted when external impedance matching the internal resistance of the coils was connected to the device.

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The results in figure 7 are obtained with a 405 Ω load connected across the harvester, matching the internal resistance. In figure 7(a), a clear transition between intra-well to full revolution can be seen as excitation amplitude raised from 0.3 g to 0.4 g at 1.4 Hz, but at higher frequencies the voltage response is more chaotic. In figure 7(b), at 0.3 g the sphere magnet starts to operate in the cross-well mode and at 0.4 g the motion changes between the three operation modes defined in section 3.1. The effect on output voltage due to different operation modes can be analysed using figures 8(a)–(c), where the output voltage is derived due to the harvester being excited with 1.9 Hz and 0.1 g, 0.4 g and 0.8 g acceleration amplitudes respectively. Figure 8(a) shows steady state intra-well response due to 1.9 Hz 0.1 g input excitation as 0.18 V low voltage peaks are generated from the harvester. The energy received by the sphere magnet is not enough to conquer the potential energy well it originates from, causing it to oscillate near one stable equilibrium. Increasing the amplitude to 0.4 g reveals a more chaotic response, as the output voltage profile changes abruptly without reaching steady state. The regions marked A, B and C can be used to describe the characteristics of this mode of operation. From A to C, the voltage profile shows a transition from a few cycles of low amplitude peaks to an envelope of high amplitude output in B before transitioning back to lower amplitude profile in C. This behaviour continues during its operation with a 0.4 g amplitude. This transition indicates that the energy absorbed by the sphere magnet is enough to occasionally escape an initial potential well and hop across a few corresponding wells, but not sufficient to sustain continuous rotational motion. Finally, at 0.8 g, the energy gained by the sphere magnet is enough to continue conquering the potential wells present due to the tethering magnets and sustain steady state rotational motion, as apparent in figure 8(c).

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Using the camera setup shown in figure 5(e), the motion of the sphere magnet during operation was recorded as shown in figure 9. To gain visual access to the motion, the coil plate was removed to partially expose the roof of the MEH to record the effect of potential barriers (videos showing three operating modes in figure 9 are available in the supplementary files). Figures 9(a)–(c) can provide insight for the output behaviour in figures 8(a)–(c) respectively, which was extracted in cartesian coordinates from recorded videos using Tracker software. For each of these experiments, the device is excited from zero initial conditions ensuring the sphere magnet starts from the same potential well. The sphere oscillates near one tethered position when excitation amplitude of 0.2 g is provided in figure 8(a), since the energy gained by the sphere magnet is not enough to conquer the potential barrier depth created by the tethering magnets. This kind of motion corresponds to output voltage shown in figure 8(a) or region A in figure 8(b). Chaotic cross-well motion is present in figure 9(b) where the energy gained is enough to occasionally travel across potential wells and rarely maintain revolutionary motion. Finally, at 0.8 g excitation the sphere magnet can take advantage of the full range of travelling path available as shown in figure 9(c), where periodic and continuous rotation is achieved. This motion is desirable since maximum change in magnetic flux occurs in the coils to provide optimized output power. Based on the intended application, the potential energy from the tethering magnets can be weakened by increasing the gap h_rt to enable full range of motion while still continuing half-rotation of the sphere magnet between wells.

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The damping in the rolling path due to friction from the 3D printed casing is also measured using this camera setup. In figure 10(a), a frame of the Tracker software analysing the damped harmonic motion of the sphere magnet is shown and the extracted data are presented in figure 10(b). The peak values were used to calculate the damping coefficient as shown in equation (7):

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$$\begin{align} c = \frac{\frac{1}{n}\textrm{log}\left(\frac{x(t_{0})}{x(t_{n})}\right)}{\sqrt{4\pi^{2}+\left.\left(\frac{1}{n}\textrm{log}\left(\frac{x(t_{0})}{x(t_{n})}\right)\right)\right)^{2}}} = 0.0291~\text{kg/s},\end{align} \tag{ 7 }$$

where n is the number of velocity peaks in figure 10(b), $x(t_{0})$ and $x(t_{n})$ are the initial and final peak velocities and c is the mechanical damping coefficient.

In order to study the output performance of the proposed system across a broadband low frequency range, the theoretical model is used to predict the RMS output voltage and power achievable and to determine the boundary where the system can transition from a less desirable intra-well to high energy motion. A comparison between numerical model and experimental results is conducted to validate the model before exploring this output behaviour.

Figure 11 presents a comparison of the mean values between the experimental output and the calculated values of the numerical model in (a) and (b). The plot from 1.4 Hz excitations in figure 11(a) shows a very close match at 0.1 g and between 0.3 g–0.8 g amplitudes, though the theoretical model achieves continuous rotational motion at slightly smaller amplitude at 0.2 g compared to 0.3 g with its physical counterpart. The difference is caused due to a small mismatch in the excitation conditions in the experiment. The slider used to provide sinusoidal vibrations tend to contain more noise during low amplitude operation as shown in figure 11(c) relative to higher amplitude motion in figure 11(f). Figure 11(b) shows the comparison for 1.9 Hz excitation, where a close match can also be observed for intra-well and continuous rotational motion at 0.1 g–0.2 g and 0.5 g–0.8 g respectively. The difference at 0.3 g and 0.4 g is caused by the difference in excitation amplitude as shown in figure 11(f), along with other variations including uneven friction in the physical rolling path and the poloidal motion of the sphere magnet that contributes to the deviation from the model during the cross-well mode of operation. To compare the time domain behaviour of the system, voltage time histories at 0.2 g and 0.8 g amplitudes are presented for 1.9 Hz in figures 11(d)–(e) and (g)–(h) respectively. Intra-well mode occurs both in the theoretical (figure 11(d)) and experimental model (figures 11(e)), as 0.25 V peak AC voltage output with similar pattern is obtained due to the sphere magnet oscillating near one of the coils. During a higher excitation amplitude of 0.8 g, the sphere magnet rotates continuously in the looped circular path, and the large-amplitude AC voltage with comparable envelopes occur in figures 11(g) and (h) as the magnet travels past each coil. Hence, an overall agreement between experimental and numerical results can be concluded and the theoretical model is capable of describing the system dynamics.

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In order to examine the ideal operation conditions, a parametric investigation is conducted to define the critical limits for obtaining high output from the MEH. Figure 12(a) presents the RMS output voltage relationship with frequency and amplitude ranged between 0.5–5 Hz and 0.1–3 g respectively. The critical boundary is made apparent from the projected map in figure 12(b), where the minimum excitation amplitude for each frequency is shown for the system to transition from low-energy intra-well mode to desirable continuous rotation mode. The boundary also shows that for higher frequencies, it takes larger amplitudes to make this transition but the output power achieved in the desired region is consequently higher. Figures 12(c)–(e) present the phase portraits to gain further insights in the behaviour of the system at the blue region, near the boundary and the desired region respectively. The response is more chaotic near the boundary, as shown by the poincare map in figure 12(d) for 2 Hz and 0.45 g excitation, where 2.3 mW RMS power can be extracted. While stable response is achievable at the two regions, with intra-well response at 0.06 g 1.5 Hz to extract 0.3 mW in figure 12(c) and continuous rotation at 2 g 4 Hz excitation to get 4.1 mW as shown in figure 12(e).

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