Harvesting energy from Faraday waves

In 1831, in the Appendix of one of his seminal papers titled “On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles upon Elastic Surfaces,”¹ Faraday described an interesting phenomenon concerning the waves developing on the free surface of a liquid resting upon a vertically vibrating support. He noted that “For each fro and to motion of the plane, or one complete vibration one of the sets appeared, so that in two complete vibrations the cycles of changes was complete.” To put this statement in today's scientific language, Faraday reported that when the fluid was excited parallel to its height, the fluid surface completed one cycle of oscillations for every two cycles of excitation.

Twenty eight years later, this phenomenon was explored further by Melde² using a vertically excited pendulum and then in the famous paper by Rayleigh on the crispation of fluid resting upon a vibrating support.³ It was concluded that large-amplitude motions can arise when an oscillator is excited near twice its fundamental frequency in a phenomenon known today as the principal parametric resonance.⁴ 

Faraday's waves were studied extensively in the open literature mainly for the purpose of estimating the hydrodynamic forces resulting from the nonlinear sloshing of fluids in partially filled containers.^5,6 On the one hand, such forces can have a detrimental influence on the stability and performance of the system. For instance, if not carefully accounted for, hydrodynamic forces can have a destabilizing effect on the carrier vehicle during the transportation of fluids.^7,8 On the other hand, forces culminating from the sloshing of a liquid inside a container can be effectively utilized in the design of vibration absorbers to mitigate large-amplitude oscillations in civil structures.^9–13 

In the present work, we explore a novel and possibly very useful application of Faraday waves. In particular, we exploit the motion of the free surface of a magnetic fluid contained in a rigid container for the purpose of harvesting energy from environmental vibrations. Vibratory energy harvesting has recently emerged as a major thrust area in the field of micro-power generation. In particular, a scalable electro-mechanical transduction mechanism is used to transform mechanical vibrations into electricity to power and maintain low-power consumption sensors such as those used for ecological studies and for structural health monitoring.^14–16 

Generally, vibratory energy harvesters incorporate solid materials for energy transduction. For instance, as shown in Fig. 1, a typical electromagnetic energy harvester consists of a solid magnet suspended by a spring and allowed to move with respect to a stationary coil. The harvester is placed in a vibratory environment, and the stiffness of the spring is chosen such that the system's natural frequency matches the dominant frequency component of the environmental vibrations. This sets the magnet into large-amplitude motion which induces an electric current in the coil as per Faraday's law of induction.¹⁷ Nevertheless, the solid nature of the transduction element can place limitations on the conformability of the energy harvester to different shapes and its sensitivity to external excitations. Such issues have lead the authors, among a few other researchers, to explore the use of liquid-state materials to transform mechanical energy into electricity.^18,19

In a series of previous papers, the authors investigated the utilization of ferro-fluids as the transduction element in electromagnetic energy harvesters.^13,19,20 Ferro-fluids are stable ferrous nanoparticles colloidally suspended in a carrier fluid.^21,22 As shown in Fig. 2, each ferrous particle, which forms a nanoscale permanent magnetic dipole, is coated with a surfactant such that, in the absence of an external magnetic field, the dipoles are randomly oriented in the carrier fluid. When an external magnetic field is applied, the average direction of the fluid magnetization becomes parallel to the magnetic field lines, thereby creating a net magnetic dipole in the fluid.

When the (magnetized) fluid is subjected to external vibratory input, large-amplitude surface waves can be excited especially when the excitation is coupled through resonance to the fluid. The sloshing motion rotates the net magnetic dipole of the fluid creating a time-varying magnetic flux which can then be used to induce an electromotive force in a coil located adjacent to the container. As per Faraday's induction law, this process generates an electric current.

A key difference between this work and the previous research of the authors lies in the direction of excitation and, as a result, the fundamental mechanism by which energy is transferred from the environment to the fluid.^13,19,20 In their previous work, the authors excited the fluid perpendicular to its height. As such, the surface was directly excited resulting in the typical resonance behavior whereby large-amplitude surface waves arise when the excitation frequency matches one of the infinitely many modal frequencies of the fluid column. Here, however, and for the first time in the context of energy harvesting, the container is excited parallel to its height resulting in the development of large Faraday surface waves when the excitation frequency is twice the modal frequency of the fluid column. These waves develop due to the activation of the principal parametric resonance of one of the infinitely many fluid sloshing modes. A key advantage of parametric resonance as compared to the traditional direct primary resonance lies in the fact that, when the level of input excitation exceeds a certain threshold, the growth of motion amplitude associated with parametric pumping is not limited by the total linear damping present in the system but is only limited by the effective nonlinearity of the surface waves.²³ Nonetheless, even in the absence of this key advantage, vibratory environmental excitations can occur in different directions. Thus, it is equally important to study the performance of the proposed harvester under parametric excitations.

This paper is organized as follows: Sec. II discusses some initial experiments used to provide a proof-of-concept for the proposed idea. Section III investigates the frequency and force response of the harvester near the principal parametric resonance of the second sloshing mode. Section IV discusses the frequency and force response of the harvester near the principal parametric resonance of the first sloshing mode. Section V derives a mathematical model of the harvester. The model is used to shed more light onto the process of energy harvesting via Faraday waves and to provide a platform for future optimization studies. Section VI provides an experimental validation of the proposed model. Finally, Sec. VII offers the concluding remarks.

To evaluate the proposed idea, the setup shown in Fig. 3 was designed and constructed. The setup consists of a plastic rectangular container of height 15.24 cm, width 7.62 cm, and depth 15.24 cm. For the purpose of energy generation, the container is partially filled with the ferrofluid and mounted on top of an electrodynamic shaker as shown in Fig. 3. The shaker can be used to generate vibration inputs similar to those encountered in a typical environment. To induce a net magnetization in the fluid, two static magnets are mounted near either side of the container. The location of the magnets can be changed vertically and horizontally to permit changing the magnetic field distribution. The choice of permanent magnets eliminates the need for external energy to generate the magnetic field. The current generated as a result of the change in magnetic flux due to the motion of the fluid is collected using a brass coil (gauge 12) wound around the container as shown in Fig. 3.

The magnitude and frequency of the parametric base acceleration provided by the electrodynamic shaker are controlled via a closed-loop controller which measures the acceleration using an accelerometer mounted on the shaker base and minimizes the error between the measured and desired value.

To obtain an overview of the response behavior of the harvester, the container is first subjected to parametric excitations having a fixed acceleration magnitude. The frequency of acceleration is varied linearly using a chirp signal between 2 and 15 Hz at a rate of 0.0025 Hz/s. At each frequency step, the output voltage is recorded across a resistive load, R_L = 137 Ω specifically chosen to match the impedance of the coil, thereby creating favorable conditions for energy flow from the source to the load. The fluid height is initially fixed at 46 mm, and the center of the static magnets is placed at a distance of 2 cm from the side of the container and at 2 cm above the fluid surface. The magnets are initially oriented to have opposite polarities resulting in an even, almost quadratic, variation of the magnetic field with the container width as shown in Fig. 4.

For this magnetic field distribution, which is also denoted as EMFD in the figure, a voltage-frequency sweep results in several pronounced voltage peaks as shown in Fig. 5(a). These peaks occur near twice one of the modal frequencies of the fluid column. Except for the first mode, the peaks are more pronounced near twice the modal frequencies of the even sloshing modes (modes 2, 4, and 6). The peak voltages observed during the frequency sweep are approximately 0.01 V (3 μW/g) near the first mode, 0.012 V (5.15 μW/g) near the second mode, and 0.004 V (0.57 μW/g) near the third mode.

The experimentally observed peaks occur at frequencies that are slightly lower than the theoretical values, $ωn0$⁠, associated with the horizontal waves along the width of the container which can be obtained using the relationship²⁴ 

where g is the gravitational acceleration, L is the width of the container, h is the height of the fluid, and n refers to the sloshing mode number. This shows that the applied field serves the purpose of reducing the effective stiffness of the fluid. At first glance, the reduction in modal frequencies can be counter-intuitive since the magnetic field should increase the stiffness of the fluid. Nonetheless, this perplexing behavior can be explained by noting that, when the magnets are brought very close to the surface, the fluid is pushed towards the side walls creating a convex surface. One can think of this curvature as surface tension but in the opposite direction. We therefore theorize that, when compared to surface tension, this curvature has an opposite effect on the natural frequencies. Note that typical surface tension in which the surface is concave increases the modal frequencies of the surface waves.

When one of the magnets is reversed such that the two magnets have similar polarity, the magnetic field, referred to as OMFD, becomes an odd function of the container width as shown in Fig. 4. In such a case, peaks in the frequency response curve become only pronounced near twice one of the odd Faraday frequencies. As described in greater detail in Section IV, this behavior is related to the orthogonality between the sloshing modes shown in Fig. 6 and the magnetic field distribution.

In this section, the response near the parametric resonance of the second sloshing mode is analyzed. Figure 7 investigates the voltage output at different acceleration levels using the EMFD. At a base acceleration of 1.5 m/s², the response exhibits a softening-type nonlinear behavior such that large-amplitude responses occur at frequencies that are lower than twice the modal frequency of the second mode. Due to the hysteretic nature of the nonlinear response, the harvester exhibits a larger bandwidth in the backward sweep (dots), as compared to the forward sweep (rectangles). The maximum recorded voltage is 0.0105 Volts across a resistive load of 137 Ω.

As the base acceleration is increased to 2 m/s², the effective bandwidth of the harvester increases in both directions of the frequency sweep. In the backward sweep, the bandwidth increases to about 0.6 Hz, and, in the forward sweep, it increases to about 0.5 Hz. The maximum recorded voltage does not increase appreciably and is around 0.011 Volts. The reason behind the very small voltage increase is not yet clear. However, in the authors' opinion, it could have stemmed from the saturation of the voltage taking place when the amplitude of the surface waves increases. This is indicated by the abrupt change in the slope of the voltage-frequency curve as shown in Fig. 7(b).

As the base acceleration is further increased to 3 m/s², the effective bandwidth of the harvester increases regardless of the frequency sweep direction. In the backward sweep, the bandwidth increases to about 0.7 Hz, while, in the forward sweep, it increases to approximately 0.6 Hz, which are both almost twice the bandwidth at a base acceleration of 1.5 m/s². Furthermore, due to the previously speculated voltage saturation, the maximum voltage amplitude increases only slightly to reach approximately 0.012 Volts. The abrupt change in the slope of the frequency response curve is more pronounced when compared to the 2 m/s² base acceleration case. For this relatively high acceleration level, visual inspection of the surface waves during the experiments revealed that, at the point where the slope of the voltage-response curve changes, the surface waves become so large that they almost reach the top of the container further reinforcing the voltage saturation argument.

The influence of the magnetic field strength on the output voltage is investigated in Fig. 8. To this end, two magnets of opposite polarity and equal strength were placed at either side of the container's walls such that their centers are at a distance of 2 cm above the surface of the fluid. The horizontal distance between the walls and the center of the magnets was then changed. The resulting magnetic field distribution at the surface of the fluid was measured using a Gauss meter and recorded in Fig. 9 for three different horizontal positions from the container walls, namely, 1 cm (MFD 1), 2 cm (MFD 2), and 3 cm (MFD 3). For the three cases, the magnetic field exhibits dependence on the distance, with MFD 1 producing a higher magnetic field over the whole container width.

The results shown in Fig. 8 demonstrate that the output voltage increases with the magnetic field. Therefore, when the magnets are placed closer to the container walls, larger output voltages are obtained. It can also be noted that the bandwidth of parametric resonance shifts towards lower frequencies when the magnetic field is increased. This points to the validity of the proposed argument that the magnetic field acts to soften the motion of the surface waves.

In addition to the frequency response curves, the forced response of the harvester was also investigated for the three different magnetic field distributions shown in Fig. 9. In each case, the base acceleration is increased from 0 to 4 m/s² and the output voltage is recorded at a single frequency corresponding to twice the second modal frequency of the fluid column. As shown in Fig. 10(a), for the lowest magnetic field strength (MFD 3), the output voltage remains negligible up to a threshold base acceleration of 1.1 m/s² beyond which the harvester starts to produce measurable output. The output increases monotonically with the acceleration level until it saturates near 0.006 Volts. The transition of the output voltage from zero to large values occurs smoothly without hysteresis.

For the case involving the medium magnetic field, Fig. 10(b), the threshold base acceleration required to initiate measurable responses increases to 1.4 m/s² followed by a sudden jump to large amplitude voltages. The maximum voltage level achieved in this case is approximately 0.01 Volts which is much larger than that achieved for the lower magnetic field. A clear hysteretic behavior due to a sub-critical bifurcation can also be seen near 1.4 m/s². Due to the nonlinear hysteresis behavior, the jump to large-amplitude voltages in the forward sweep occurs at a higher value when compared to the jump from the large amplitude voltages in the backward sweep.

For the case involving the highest magnetic field values shown in Fig. 10(c), the threshold acceleration level is not very clear since there is no clear transition from zero voltage output to measurable values. Nonetheless, the harvester can now produce higher voltage levels around 0.014 Volts at the higher end of the base acceleration.

The voltage output of the harvester near the principal parametric resonance of the first mode is investigated for the two different magnetic field distributions shown in Fig. 4. The response associated with EMFD is depicted in Fig. 11(a) while that associated with OMFD is shown in Fig. 11(b). It can be clearly seen that the output voltage associated with OMFD is almost four times that obtained using the EMFD. These results seem to point to a direct relationship between the output voltage of the harvester and a weighted orthogonality between the mode shape of the response and the magnetic field distribution. Note that the first mode shape is odd in nature.

To further confirm this observation, we repeated the experiment for excitation frequencies near twice the second even mode as shown in Fig. 12. As speculated, when the magnetic field is an odd function of the container width, which is orthogonal to the second sloshing mode, the harvester does not produce any measurable voltage levels as shown in Fig. 12(b). On the other hand, when both the magnetic field and modal shape are even, the harvester produces large voltage outputs as depicted in Fig. 12(a).

To shed light onto the intriguing energy harvesting phenomenology that exploits Faraday waves and to provide a platform for future optimization studies, a mathematical model is constructed to adequately describe the dynamics of the ferrofluid harvester. To this end, the two-dimensional finite amplitude sloshing dynamics of an irrotational, incompressible fluid in a rectangular container of width L is considered. The fluid of mass density ρ is assumed to be of height h as shown in Fig. 13. As the fluid starts to move due to external perturbations, surface waves of height $η¯(x¯,t¯)$ are excited. The equations and associated boundary conditions governing the planar motion of the fluid can be written as

Here, the subscripts denote partial differentiation with respect to the independent variables. Equation (2a) states the irrationality of the velocity field by expressing the velocity field denoted by u as the gradient of the scalar potential function $ϕ¯$⁠. Equation (2b) is a consequence of the incompressibility assumption for which the continuity equation requires the Laplacian of the velocity potential to vanish. Equation (2c) states that the velocity normal to the sidewalls and bottom wall vanishes. Equation (2d) is the kinematic boundary condition at the surface which states that the velocity of a fluid particle on the surface must be equal to the velocity of the surface itself. Finally, Eq. (2e) represents the dynamic boundary condition at the surface obtained by enforcing the unsteady Bernoulli equation.

In Eq. (2e), the first term accounts for the unsteadiness in the velocity field; the second term represents the kinetic energy of the fluid; the third term accounts for the potential energy; and the fifth term represents forces exerted on the surface due to a harmonic base acceleration in the $z¯$ direction. Here, Z₀ and ω denote the amplitude and frequency of the parametric excitation, respectively. Finally, the sixth term represents the magnetic moment exerted on the surface, where $M¯$ and $H¯$ are the magnetization and magnetic field, respectively; μ₀ represents the permeability of vacuum, and $C¯$ is a constant.

In this study, the applied magnetic field will be static and will vary along the $x¯$-axis only, i.e., $H¯(x¯,y¯,z¯,t¯)=H¯(x¯)$⁠. The magnetic field is further assumed to have a linear relationship with the magnetization, $M¯(x¯)=χmH¯(x¯)$⁠, that is,

Equation (2) can be further nondimensionalized by introducing the following nondimensional quantities:

where $ω10=πgLδ1$ is the first modal frequency of the fluid column in the absence of surface tension and magnetic field, $δ1=tanh(πh/L)$⁠, and $H0=Lω102ρμ0$⁠. Enforcing the stated nondimensionalization yields the following non-dimensional equations:

Equation (5) can be solved numerically using finite-difference or finite-element techniques. Nevertheless, when available at all, analytical solutions provide a more insightful and less computationally expensive approach. Since Eqs. (5c) and (5d) are nonlinear, an exact solution cannot be easily found. To overcome this issue, we obtain an approximate analytical solution of the equations using the method of multiple scales. To this end, we expand the time dependence in the equation into multiple time scales in the form²³ 

where ε is a bookkeeping parameter. By using Eq. (6), the time derivatives can be expressed as

where $Dk=∂∂Tk$⁠. Furthermore, we expand $ϕ$⁠, η, and C in the following series of ε:

To express the nearness of the excitation frequency to twice the modal frequency of a given sloshing mode, we let $Ω=2ωk+ε2σ$⁠, where σ is a small detuning parameter. Furthermore, we treat only soft excitations for which z₀ is very small and can be scaled as order $ε2$⁠.

Since the dynamic and kinematic boundary conditions are evaluated at the surface $η(x,t)$⁠, which is still unknown, we expand the dependence of $ϕ$ on η in a Taylor series around η = 0. In other words, we let $ϕ(η)≈ϕ(0)+εϕz(0)η+ε2/2ϕzz(0)η2+O(ε3)$⁠. Note that this assumption is accurate as long as the surface waves are finite but sufficiently small.

Upon substituting Eqs. (6)–(8) into Eqs. (5c) and (5d) and by collecting terms of like powers of ε, we obtain the following cascade of linear partial differential equations:

• $O(ε1):$

  $D0ϕ1+1πδ1(η1−βη1xx)=0,$

  (9a)

  $D0η1−ϕ1z=0,$

  (9b)
• $O(ε2):$

  $D0ϕ2+1πδ1(η2−βη2xx)=−12(ϕ1z2+ϕ1x2)−D1ϕ1+η1D0ϕ1z+C2,$

  (10a)

  $D0η2−ϕ2z=−D1η1+η1ϕ1zz−η1xϕ1x,$

  (10b)
• $O(ε3):$

  $D0ϕ3+1πδ1(η3−βη3xx)=−3β2πδ1η1x2η1xx−ϕ1zϕ2z−η1ϕ1zϕ1zz−ϕ1xϕ2x−η1ϕ1xϕ1xz−D2ϕ1−D1ϕ2−η2D1ϕ1z+η1D0ϕ2z−12η12D0ϕ1zz+4z0ωk2η1 sin (2ωkT0)+C3,$

  (11a)

  $D0η3−ϕ3z=−D2η1−D1η2+η2ϕ1zz+η1ϕ2zz+12η12ϕ1zzz−η2xϕ1x−η1xϕ2x−η1η1xϕ1xz.$

  (11b)

When a given sloshing mode is being excited parametrically near twice one of its modal frequencies and that mode is not in an internal resonance with any of the other modes, a single-mode response is sufficient to describe the dynamics of Eqs. (9a) and (9b). Thus, the solution of the first-order problem can be written as

By substituting Eq. (12) into the second-order problem, Eqs. (10a) and (10b), and by enforcing the solvability conditions, wherein the right-hand side of Eq. (10a) is forced to be orthogonal to every solution of the adjoint homogeneous problem, i.e., orthogonal to Eq. (12), we obtain $D1Ak=D1A¯k=0$⁠. Upon substituting the solvability conditions into the second order problem, the particular solution can be expressed as

where $C2k(x)=cos (2kπx), S2k(x)=sin (2kπx), Ch2k(z)=cosh(2kπ(z+h/L))cosh(2kπ(h/L))$⁠,

Here, $ω2k=2kδ2k/δ1$ and the coefficient C₂ is determined by forcing $∫−1/21/2η2dx$ to vanish to preserve continuity.

Equations (12) and (13) are then substituted into Eq. (11a). The secular terms, i.e., terms which render the particular solution of the resulting equation non-uniform, are eliminated by enforcing the solvability conditions. In other words, the right-hand side of Eq. (11a) is forced to be orthogonal to every solution of the adjoint homogeneous problem, i.e., orthogonal to Eq. (12). This yields

where $Fk=πz0ωk3δ1$⁠, and N_eff is the effective nonlinearity coefficient. This coefficient represents an average nonlinear stiffness expression, which captures the nature of the nonlinear response of the finite-amplitude sloshing modes, and is given by

The solution of Eq. (15) is obtained by expressing the unknown complex-valued function A_k in the polar form $Ak=1/2akeiβk, A¯k=1/2ake−iβk$⁠. Upon substituting the polar transformation into Eq. (15) and separating the real and imaginary parts, we obtain the following equations which govern the slow modulation of the waves amplitude and phase:

where the prime represents a derivative with respect to T₂, $γk=σT2−2βk$ is the phase, and $μ1k$ and $μ2k$ are the linear and nonlinear modal damping coefficients added to capture the viscous damping of the fluid, respectively.

Equation (17a), also known as the amplitude equation, describes the slow modulation of the response amplitude, while Eq. (17b), known as the phase equation, governs the slow modulation of the response phase.

To obtain the long time behavior of the system, we set the time derivatives in Eqs. (17a) and (17b) to zero, then square, and add the outcome to obtain

where $a0k$ is the steady-state modal amplitude. Equation (18), known as the nonlinear parametric frequency-response equation, exhibits six different roots, two of which are always zero indicating that zero steady-state amplitude is always a possible response of the harvester. The four other solutions are non-trivial but can sometimes take negative or complex values which do not represent physically meaningful solutions.

The stability of the resulting steady-state solutions can be assessed by finding the eigenvalues of the Jacobian matrix associated with Eqs. (17a) and (17b) and evaluating it at the steady-state roots. If at least one of the eigenvalues of the Jacobian matrix has a positive real part, then the associated steady state solution is unstable and hence physically unrealizable.

Upon obtaining the steady-state roots $a0k$⁠, the steady-state potential field and surface-wave amplitude can be written as

where

is the steady-state phase of the response.

To obtain the voltage induced in one coil, V_o, Faraday's Law is applied. For a rectangular tank with coils wound in the vertical direction and magnetic field applied along the $x¯$-axis, the voltage output per unit width of the harvester can be written as

where $dA$ is an element on the moving surface $Σ(t¯)$⁠, B is the vector of magnetic flux density, and $(·)$ indicates the dot product. For the problem at hand, the previous equation can be written as

where 2 b is the depth of the container. The magnetic flux density, B, can be further related to the applied field via $B(x¯)=μ0(1+χm)H¯(x¯)$ which, upon substitution into Eq. (22) and carrying out the integration, yields

To determine the average output voltage generated in the total number of coils, N, we multiply Eq. (24) by the number of coils, N, and average the results over the width of the container, L. This yields

where $αc=−2NbhLμo(1+χm)$⁠. It is worth noting that when $H¯(x¯)$ is orthogonal to $dη¯dt¯$ along the $x¯$-axis, the voltage V_o approaches zero.

To determine the current in the induced coil, we apply Kirchhoff's Law and obtain

where $i¯$ is the induced current, R_L is the load resistance, and R_c and L_c are the resistance and inductance of the collecting coil, respectively.

In this section, we evaluate the fidelity of the model developed in Sec. V in describing the qualitative behavior of the harvester under different conditions. We start by investigating the response near the principle parametric resonance of the second mode using the magnetic field EMFD and two different acceleration levels as shown in Fig. 14.

For the lower acceleration level of 1.5 m/s², there is a good quantitative agreement between the theory and the experiment for most of the frequency range. The bandwidth and magnitude of the voltage are both in good agreement with the experimental data. For the higher acceleration level of 2 m/s², there still is good quantitative agreement between the theory and the experiment especially in terms of the harvester's bandwidth. However, the theoretical model predicts a response larger than the experimental response especially near the lower end of frequencies. As described earlier, this can be attributed to the saturation of the experimental voltage when the surface waves become very large.

As shown in Fig. 15, when the magnetic field strength is increased to MFD1, the theoretical model still predicts the experimental findings in terms of bandwidth and magnitude with relatively good accuracy.

The theoretical forced response was also compared to experimental findings near the second mode as depicted in Fig. 16. Two cases were considered using the distributions MFD 2 and MFD 3. For the case involving the lower magnetic field MFD 3 shown in Fig. 16(a), the theoretical model correctly predicts lower voltage magnitudes and nonhysteretic transition between the trivial and non-trivial solutions. For the higher distribution MFD 2, the theoretical model also correctly predicts the qualitative nature of the response. Specifically, it predicts the presence of the cyclic-fold and trans-critical bifurcations observed in the experiments. However, due to the saturation effect of the voltage, the theoretical model over-estimates the experimental results at large base accelerations.

The output power of the harvester is also accurately predicted as shown in Fig. 17. Both the theory and the experiments agree in predicting that the peak voltage occurs near a load resistance, R_L = 137  Ω, which is equal to the measured coil resistance. However, due to model uncertainties, the theory over-predicts the experimental data over the whole range of electric loads considered. Keep in mind that any discrepancies between the theoretical and experimental voltage measurements propagate in the power calculation since the power is proportional to the square of the voltage.

Finally, we evaluate the ability of the model to predict the influence of the magnetic field distribution on the response near the principle parametric resonance of the first sloshing mode. As previously noted in Sec. IV and further highlighted in the theoretical model, the harvester produces much lower power levels when the mode shape of the Faraday surface wave is nearly orthogonal to the magnetic field distribution, more specifically, when $∫−L/2L/2H¯(x¯)dη¯dt¯dx¯≈0$⁠. Theoretical results obtained using the derived model and shown in Figs. 18(a) and 18(b) confirm this finding. The results clearly show that the harvester produces much lower voltages when excited near twice the first odd mode while using an even magnetic field distribution, EMFD.

Through a theoretical and experimental investigation, this article demonstrated the feasibility of using Faraday waves for vibration energy harvesting. In particular, it has been shown that standing waves which form on the surface of a parametrically excited magnetized ferrofluid generate change in the surrounding magnetic flux that is sufficient to induce an electric current in an adjacent coil. The influence of the magnetic field strength and distribution as well as the level of excitation on the performance of the harvester was investigated for the principle parametric resonance of the first and second sloshing modes. It was observed that the output voltage and bandwidth of the harvester increase as the levels of excitation and magnetic field strength are increased. It was also shown that the output power of the harvester depends on a weighted orthogonality between the mode shape of the surface response and the magnetic field distribution along the width of the container. In specific, very little power can be generated when the mode shape of the surface wave is orthogonal to the magnetic field distribution.

We gratefully acknowledge support by the National Science Foundation under Grant CMMI-1335049.