A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects

Abstract

In this investigation, a magnetically coupled double-spring design is presented for harvesting low-level non-stationary random vibrational energy. The sensor relies on multimodal coupling between the translation and rotation of a two-spring magnet and coil system to widen the harvesting bandwidth. Energy methods are used to develop a model to characterize the electromechanical response of the system, the solution of which is obtained using stochastic techniques based on a particle swarm algorithm. This approach provides an efficient method to estimate system parameters that otherwise are difficult or impossible to determine with independent measurements. The experimental results demonstrate agreement with the theoretical predictions over a limited bandwidth. The sensor can effectively harvest non-stationary vibration energy down to 10⁻⁴ g within a limited bandwidth of 130–150 $\mathrm{Hz}$. The sensor prototype has an operational volume of 2.6 $\mathrm{c}{\mathrm{m}}^3$ with a calculated power density of 0.2 $\mathrm{W}/\mathrm{c}{\mathrm{m}}^3$. The sensor’s small size results in a coupling efficiency of approximately 6% across the tested bandwidth.

1. Introduction

Self-powered sensors and networks have become a tangible reality, thanks to the ability to harness ambient energy from the surroundings to power electronic devices. This advancement has a profound impact on the realization of intelligent and adaptable sensor networks. As microelectronics and sensing technologies continue to advance, the power requirements for these devices have reduced to the microwatt regime. At this level, the available ambient energy in the environment becomes an attractive resource to power these systems. Efficient energy harvesting technologies will greatly diminish the reliance on short-lived, battery-powered approaches and eliminate the need for direct power supply to the devices [1,2,3]. This paves the way for the development of truly autonomous wireless sensor networks that can operate indefinitely, independent of routine maintenance and infrastructure requirements. These networks will have diverse applications in structural health monitoring, security perimeter tamper detection, autonomous systems, and seamless integration with existing Internet of Things (IoT) technologies [4,5].

A key challenge for persistent sensing is power supply. Even when power requirements are in the microwatt range, they need to be continuous. Many research efforts are focused on developing sensor technologies that leverage ambient energy sources. For systems operating on machinery or structures with well-defined and repeatable dynamic characteristics (e.g., rotating machinery), the energy harvesting mechanism is designed to target specific vibrational frequencies [6]. This approach achieves optimal performance when the harvesting design resonates at or near one of the fundamental operating frequencies of the monitored system. Even small levels of vibrational energy can contribute significantly to harvested power when the harvesting system operates near a resonance. However, linear systems face performance degradation as the operational frequency deviates from the harvester’s resonance design point [7,8,9]. Single-resonance VEH devices operating in a linear regime typically have an FWHM resonance that is 2–3 $\mathrm{Hz}$. Nonlinear effects have shown an improvement by a factor 5, or 10–15 $\mathrm{Hz}$ [7,8,9].

One approach to introduce nonlinearities is through opposing magnetic fields [10,11]. Muskat provides a thorough review of many electromagnetic vibrational energy harvesting (VEH) designs [12]. In particular, designs utilizing multiple masses and magnets to generate favorable harvesting performance have been the focus of numerous investigations [4,13,14,15]. These designs are based on modifying the electromagnetic coupling of the magnet and the coil with the introduction of additional magnets to generate nonlinear forcing functions that lead to local minima in the potential energy function. The local minima generate a multi-stable vibrational response that leads to improved bandwidth and conversion capabilities. The approach is not without challenges—large potential wells can “trap” the system, reducing the high-energy interwell oscillations. To mitigate these effects, several investigators have developed tri-stable approaches resulting in lower potential interwell boundaries, resulting in lower excitation energies to achieve the interwell boost for improved energy harvesting [16,17].

In the following sections, we explore the use of nonlinear coupled multi-modal behavior to enhance bandwidth and increase energy conversion efficiency. Our focus is on developing simplified ordinary differential equation (ODE) approximations to capture the electromechanical behavior. Experimental measurements and comparisons to the ODE model are used to validate the effectiveness of the sensor design. The nonlinear ODE system is also used to explore parameter values for analyzing performance and “tuning” of the device. Here, model parameters can be fit to match empirical data.

The device is described in Section 2. The state-space ODE theory, derived in Section 3, is based upon a Lagrangian formulation. Experience has shown that when fitting the model parameters to match its output to data, the cost function has multiple local minima. Thus, a particle swarm optimizer was used to fit the model parameters. This is described in Section 4. Section 5 describes the experimental design and testing method. The results and harvesting performance are presented in Section 6. The results are discussed in Section 7 with conclusions presented in Section 8.

2. Device Description

The vibrational energy harvesting (VEH) device is pictured in Figure 1. An idealized diagram of the sensor is shown in Figure 2. The prototype sensor uses dual cylindrical magnets each attached to its own planar spring. Threaded rods are utilized to adjust the separation distance of the magnets and their angular orientation. One magnet is at the center of a cylindrically wrapped copper coil. The magnets are oriented with similar poles counterposed. The repulsive magnetic coupling introduces an additional magnetic force in addition to the back electromotive force (EMF) resulting from the coil/magnet interaction. The effect introduces additional nonlinear force behavior to the response of the VEH. Both magnets experience the same base excitation. Another characteristic of the design which results in rotational motion can be attributed to the intentional asymmetries caused by the threaded rods adjustments and unintentional asymmetries due to device imperfections [17].

A result of the asymmetrical repulsive force is to induce rotational motion between the upper magnet and the lower magnet-coil assembly, which adds an additional degree of freedom to the system. It was observed that the planar springs do not operate in a simple vertical (up/down) motion but rather experience out-of-plane tilting as the opposing magnetic forces push the magnets away from their vertical orientations. Consider holding two bar magnets, one in each hand, arranged with identical poles facing each other. As the magnets are moved closer to each other, the opposing force will not only push them apart along their axes but also deflect them with a force felt in the hands. Thus, the planar springs are modeled as bars with springs on their ends as opposed to a single vertically restoring spring. The center of rotation is not necessarily the center of the bar.

In deriving the theory, attention must be paid to the sign of the tilt angles, ${\theta }_1$ and ${\theta }_2$. Following the diagram of Figure 2, ${\theta }_1$ is negative. Thus, $sin{\theta }_1$ is negative and $-sin{\theta }_1$ is positive, making $x_1-l_{11}sin{\theta }_1$ a positive displacement and $x_1+l_{12}sin{\theta }_1$ a negative displacement. The diagram shows ${\theta }_1$ is positive. Thus, $sin{\theta }_2$ is positive and $-sin{\theta }_2$ is negative, making $x_2-l_{21}sin{\theta }_2$ a negative displacement and $x_2+l_{22}sin{\theta }_2$ a positive displacement.

Referring to Figure 2, the planar spring radii are 10 $\mathrm{m}\mathrm{m}$. Thus, the bar lengths are 10 $\mathrm{m}\mathrm{m}$. The center of rotation is located at $l_{11}$ from the edge of spring 1 and $l_{21}$ from the edge of spring 2. By definition, $l_{12}=10-l_{11}$ and $l_{22}=10-l_{21}$.

The measured linear planar spring constants are listed in

Table 1. Physicalparameters used in the simulation which approximately match those of the experiment.

Parameter	Value	Unit
Known
Magnet masses, $m_n, n=1,2$	1.52, 0.453	$\mathrm{g}$
Linear spring constants, $k_{s,n}, n=1,2$	649, 160	$\mathrm{k}\mathrm{g} {\mathrm{s}}^{-2}$
Planar spring radius, l	10	$\mathrm{m}\mathrm{m}$
Magnetic flux density peak amplitude, B	1	$\mathrm{T}$
Damping, $c_n, n=1,2$	5 × 10−2, 5 × 10−2	$\mathrm{k}\mathrm{g} {\mathrm{s}}^{-1}$
Number of coil turns, N	650	Unitless
Coil area, A	25$\pi$	$\mathrm{m}{\mathrm{m}}^2$
Coil resistance, $R_c$	60	$\Omega$
Coil inductance, $L_c$	1.3	$\mathrm{H}$
Load resistance, $R_l$	50	$\Omega$
Guesses
Magnet radii, $\rho$	1	$\mathrm{m}\mathrm{m}$
Magnet height, h	5	$\mathrm{m}\mathrm{m}$
Magnet volume, $V_n$	$\pi {\rho }^{2}h=15.7$	$\mathrm{m}{\mathrm{m}}^3$
Magnetizations, $M_{nx},M_{ny},M_{nz},n=1,2$	5 × 10−4, 10−4, 10−4	$\mathrm{A} {\mathrm{m}}^{-1}$
Spring radii, $l_{11},l_{12},l_{21},l_{22}$	2, 8, 3, 7	$\mathrm{m}\mathrm{m}$
Damping, $c_1,c_2$	0.05, 0.05	$\mathrm{N} \mathrm{s} {\mathrm{m}}^{-1}$
Particle Swarm Optimized Fit Values
Linear spring constants, $k_{11},k_{12}$	559, 461	$\mathrm{k}\mathrm{g} {\mathrm{s}}^{-2}$
Linear spring constants, $k_{21},k_{22}$	130, 161	$\mathrm{k}\mathrm{g} {\mathrm{s}}^{-2}$
Moments of inertia, $J_1,J_2$	2.2 × 10−2, 4.8 × 10−2	$\mathrm{k}\mathrm{g} {\mathrm{m}}^{-2}$
Unconstrained Derivative-Free Optimized Static Values
$x_1\left(0\right),x_2\left(0\right)$	0.05, −24.14	$\mathsf{\mu }\mathrm{m}$
${\theta }_1\left(0\right),{\theta }_2\left(0\right)$	0.23, 0.08	degrees

. However, the spring constants, $k_{11},k_{12},k_{21}$ and $k_{22}$, which permit both vertical and tilt motion, are unknown. Additionally, the center of rotation as given by $l_{11},l_{12},l_{21}$ and $l_{22}$, the damping coefficients, $c_1$ and $c_2$, and the moments of inertia, $J_1$ and $J_2$, are also unknown.

The unknown parameters are identified by fitting the empirical output voltage spectrum to that of a lumped parameter nonlinear ordinary differential equation (ODE) model of the device. The following section develops the ODE model. Section 4 describes how the unknown parameters are fit.

3. Magnetically Coupled Double-Spring Design

Applying an energy approach [18,19,20], a Lagrangian is defined as

$$\begin{array}{ccc}\hfill \mathcal{L}\left(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2,{\theta }_1,{\theta }_2,{\dot{\theta }}_2,{\dot{\theta }}_1,q,\dot{q}\right)& =& \underset{\mathrm{Mechanical}\mathrm{energy}}{\underbrace{\mathrm{K}.\mathrm{E}.-\mathrm{P}.\mathrm{E}.}}+\underset{\mathrm{Magnetic}\mathrm{energy}}{\underbrace{{\mathrm{W}}_{\mathrm{m}}}},\hfill \end{array}$$

(1)

where the kinetic energy is,

$$\begin{array}{ccc}\hfill \mathrm{K}.\mathrm{E}.& =& {\displaystyle \frac{1}{2}}{m}_1{\dot{x}}_1^2+{\displaystyle \frac{1}{2}}{m}_2{\dot{x}}_2^2+{\displaystyle \frac{1}{2}}{J}_1{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}}{J}_2{\dot{\theta }}_2^2,\hfill \end{array}$$

(2)

the potential energy is,

$$\begin{array}{ccc}\hfill \mathrm{P}.\mathrm{E}.& =& {\displaystyle \frac{1}{2}}{k}_{11}{\left(x_1-l_{11}sin{\theta }_1\right)}^2+{\displaystyle \frac{1}{2}}{k}_{12}{\left(x_1+l_{12}sin{\theta }_1\right)}^2+\hfill \\ & & {\displaystyle \frac{1}{2}}{k}_{21}{\left(x_2-l_{21}sin{\theta }_2\right)}^2+{\displaystyle \frac{1}{2}}{k}_{22}{\left(x_2+l_{22}sin{\theta }_2\right)}^2+\hfill \\ & & U_{12}\left({\mathbf{r}}_{12}\right),\hfill \end{array}$$

(3)

where ${\theta }_n$, $n=1,2$ is positive when the right-hand side of the bar rotates counter-clockwise.

The magnetic energy is [21]

$$\begin{array}{ccc}\hfill {\mathrm{W}}_{\mathrm{m}}& =& N{B}_{12}\left({\mathbf{r}}_{12}\right)l\left(x_2-x_1\right)\dot{q}+{\displaystyle \frac{1}{2}}{L}_c{\dot{q}}^2.\hfill \end{array}$$

(4)

The magnetic potential energy at a magnet located at ${\mathbf{r}}_2$ due to a magnet at ${\mathbf{r}}_1$ is [10]

$$U_{12}\left({\mathbf{r}}_{12}\right)={\mathbf{m}}_2·{\mathbf{B}}_{12},$$

(5)

where the magnetic flux density at point ${\mathbf{r}}_2$ due to a source at ${\mathbf{r}}_1$ is

$$\begin{array}{ccc}\hfill {\mathbf{B}}_{12}\left({\mathbf{r}}_{12}\right)& =& -\frac{{\mu }_0}{4\pi }{\nabla }_2\frac{{\mathbf{m}}_1·{\mathbf{r}}_{12}}{{\left|{\mathbf{r}}_{12}\right|}^3}\hfill \end{array}$$

(6)

and the gradient, ${\nabla }_2$, is with respect to ${\mathbf{r}}_2$ and ${\mathbf{r}}_{12}\equiv {\mathbf{r}}_2-{\mathbf{r}}_1$, and ${\mu }_0\equiv 4\times {10}^{-7}\pi \mathrm{H}/\mathrm{m}$ is the permeability of free space. The physical parameters are as follows:

$m_n$	are the masses of the magnets for $n=1,2$,
$x_n$	are the vertical displacements of the magnets for $n=1,2$,
$k_{n1},k_{n2}$	are the linear spring constants for $n=1,2$,
$l_{n1},l_{n2}$	are the spring lengths about the center of rotation for $n=1,2$,
$J_n$	are the moments of inertia for $n=1,2$,
${\mathbf{m}}_n={\mathbf{M}}_n{V}_n$	are the magnetic moments of the magnets for $n=1,2$,
${\mathbf{M}}_n$	are the magnetizations of the magnets for $n=1,2$,
$V_n$	are the volumes of the magnets for $n=1,2$. Note: $V_1=V_2\equiv V$,
q	is the induced charge,
$L_c$	is the coil inductance,
N	is the number of coil turns,
A	is the coil area,
l	is the coil width,
${\mathbf{B}}_{12}\left({\mathbf{r}}_{12}\right)$	is the magnetic flux density at ${\mathbf{r}}_2$ from a magnet at ${\mathbf{r}}_1$,
$B_{12}\left({\mathbf{r}}_{12}\right)$	is ${\mathbf{B}}_{12}$, and
$U_{12}{(}_{12})$	is the magnetic potential energy between the two magnets.

Refer to Appendix A for the physical units.

Substituting Equations (2)–(4) into Equation (1), the full Lagrangian is obtained,

$$\begin{array}{ccc}\hfill \mathcal{L}\left(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2,{\theta }_1,{\theta }_2,{\dot{\theta }}_1,{\dot{\theta }}_2,q,\dot{q}\right)& =& {\displaystyle \frac{1}{2}}{m}_1{\dot{x}}_1^2+{\displaystyle \frac{1}{2}}{m}_2{\dot{x}}_2^2+{\displaystyle \frac{1}{2}}{J}_1{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}}{J}_2{\dot{\theta }}_2^2-\hfill \\ & & {\displaystyle \frac{1}{2}}{k}_{11}{\left(x_1-l_{11}sin{\theta }_1\right)}^2-{\displaystyle \frac{1}{2}}{k}_{12}{\left(x_1+l_{12}sin{\theta }_1\right)}^2-\hfill \\ & & {\displaystyle \frac{1}{2}}{k}_{21}{\left(x_2-l_{21}sin{\theta }_2\right)}^2-{\displaystyle \frac{1}{2}}{k}_{22}{\left(x_2+l_{22}sin{\theta }_2\right)}^2-\hfill \\ & & U_{12}\left({\mathbf{r}}_{12}\right)+N{B}_{12}\left({\mathbf{r}}_{12}\right)l\left(x_2-x_1\right)\dot{q}+{\displaystyle \frac{1}{2}}{L}_c{\dot{q}}^2.\hfill \end{array}$$

(7)

The electromechanical dissipation is

$$\begin{array}{ccc}\hfill \mathcal{D}\left(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2,q,\dot{q}\right)& =& {\displaystyle \frac{1}{2}}{c}_1{\dot{x}}_1^2+{\displaystyle \frac{1}{2}}{c}_2{\dot{x}}_2^2+{\displaystyle \frac{1}{2}}\left(R_l+R_c\right){\dot{q}}^2,\hfill \end{array}$$

(8)

where

$c_n$	are the damping constants for $n=1,2$,
$R_c$	is the coil resistance, and
$R_l$	is the load resistance.

The mechanical Lagrangian equations are

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\partial \mathcal{L}}{\partial {\dot{x}}_n}}\right)+{\displaystyle \frac{\partial \mathcal{D}}{\partial {\dot{x}}_n}}-{\displaystyle \frac{\partial \mathcal{L}}{\partial x_n}}& =& m_n\ddot{y},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\partial \mathcal{L}}{\partial {\dot{\theta }}_n}}\right)+{\displaystyle \frac{\partial \mathcal{D}}{\partial {\dot{\theta }}_n}}-{\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_n}}& =& 0,\hfill \end{array}$$

(10)

where $n=1,2$ and $\ddot{y}$ is the input acceleration. The electrical Lagrangian equation is

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{q}}}\right)+{\displaystyle \frac{\partial \mathcal{D}}{\partial \dot{q}}}-{\displaystyle \frac{\partial \mathcal{L}}{\partial q}}& =& \underset{\mathrm{No}\mathrm{external}\mathrm{voltage}}{\underbrace{{\overline{)E}}^0}}.\hfill \end{array}$$

(11)

Solving Equations (9)–(11) for the highest-order derivatives in each physical value and noting the current is the time derivative of the charge,

$$\begin{array}{ccc}\hfill i& =& \dot{q},\hfill \end{array}$$

(12)

yields

$$\begin{array}{ccc}\hfill m_1{\ddot{x}}_1& =& -\left(k_{11}+k_{12}\right)x_1-c_1{\dot{x}}_1+\left(k_{11}{l}_{11}-k_{12}{l}_{12}\right)sin{\theta }_1+\hfill \\ & & {\displaystyle \frac{3{\mu }_0{V}^2\left(2M_x^2-M_y^2\right)}{4\pi {|x_1-x_2|}^5}}\left(x_1-x_2\right)-{\displaystyle \frac{lN{B}_{12}}{2\pi }}i+m_1\ddot{y}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill J_1{\ddot{\theta }}_1& =& \left(k_{11}{l}_{11}-k_{12}{l}_{12}\right)x_{1}cos{\theta }_1-\left(k_{11}{l}_{11}^2+k_{12}{l}_{12}^2\right)cos{\theta }_{1}sin{\theta }_1\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill m_2{\ddot{x}}_2& =& -\left(k_{21}+k_{22}\right)x_2-c_2{\dot{x}}_2+\left(k_{21}{l}_{21}-k_{22}{l}_{22}\right)sin{\theta }_2-\hfill \\ & & {\displaystyle \frac{3{\mu }_0{V}^2\left(2M_x^2-M_y^2\right)}{4\pi {|x_1-x_2|}^5}}\left(x_1-x_2\right)+{\displaystyle \frac{lN{B}_{12}}{2\pi }}i+m_2\ddot{y}\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill J_2{\ddot{\theta }}_2& =& \left(k_{21}{l}_{21}-k_{22}{l}_{22}\right)x_{2}cos{\theta }_2-\left(k_{21}{l}_{21}^2+k_{22}{l}_{22}^2\right)cos{\theta }_{2}sin{\theta }_2\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \dot{i}& =& {\displaystyle \frac{1}{L_c}}\left[2lN{B}_{12}\left({\dot{x}}_2-{\dot{x}}_1\right)-\left(R_c+R_l\right)i\right].\hfill \end{array}$$

(17)

Note, the magnet volumes are assumed identical; thus, $V\equiv V_1=V_2$. The derivation details are presented in Appendix B.

Define the state vector as,

$$\begin{array}{ccc}\hfill \overrightarrow{x}\left(t\right)& =& \left[\begin{array}{c}{x}_1\left(t\right)\\ {\dot{x}}_1\left(t\right)\\ x_2\left(t\right)\\ {\dot{x}}_2\left(t\right)\\ {\theta }_1\left(t\right)\\ {\dot{\theta }}_1\left(t\right)\\ {\theta }_2\left(t\right)\\ {\dot{\theta }}_2\left(t\right)\\ i\left(t\right)\end{array}\right],\hfill \end{array}$$

(18)

and express Equation (13) through (17) as a state-space system of ODEs,

$${\displaystyle \frac{d}{dt}}\overrightarrow{x}\left(t\right)=A_{\mathrm{linear}}\overrightarrow{x}\left(t\right)+A_{\mathrm{nonlinear}}\left(\overrightarrow{x}\left(t\right)\right)+\overrightarrow{u}\left(t\right),$$

(19)

where

$$A_{\mathrm{linear}}=\left[\begin{array}{ccccccccc}0& 1& 0& 0& 0& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{m_1}}(k_{11}+k_{12})& -{\displaystyle \frac{c_1}{m_1}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& -{\displaystyle \frac{1}{m_2}}(k_{21}+k_{22})& -{\displaystyle \frac{c_2}{m_2}}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{R_l+R_c}{L_c}}\end{array}\right],$$

(20)

$$A_{\mathrm{nonlinear}}=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{m_1}}\left[\left(k_{11}{l}_{11}-k_{12}{l}_{12}\right)sin{\theta }_1+{\displaystyle \frac{3{\mu }_0{V}^2\left(2M_x^2-M_y^2\right)}{4\pi {x_1-x_2}^5}}\left(x_1-x_2\right)-{\displaystyle \frac{lN{B}_{12}}{2\pi }}i\right]\\ 0\\ {\displaystyle \frac{1}{m_2}}\left[\left(k_{21}{l}_{21}-k_{22}{l}_{22}\right)sin{\theta }_2-{\displaystyle \frac{3{\mu }_0{V}^2\left(2M_x^2-M_y^2\right)}{4\pi {x_1-x_2}^5}}\left(x_1-x_2\right)+{\displaystyle \frac{lN{B}_{12}}{2\pi }}i\right]\\ 0\\ {\displaystyle \frac{1}{J_1}}\left[\left(k_{11}{l}_{11}-k_{12}{l}_{12}\right)x_{1}cos{\theta }_1-\left(k_{11}{l}_{11}^2+k_{12}{l}_{12}^2\right)cos{\theta }_{1}sin{\theta }_1\right]\\ 0\\ {\displaystyle \frac{1}{J_2}}\left[\left(k_{21}{l}_{21}-k_{22}{l}_{22}\right)x_{2}cos{\theta }_2-\left(k_{21}{l}_{21}^2+k_{22}{l}_{22}^2\right)cos{\theta }_{2}sin{\theta }_2\right]\\ {\displaystyle \frac{2N{B}_{12}l}{L_c}}\left({\dot{x}}_2-{\dot{x}}_1\right)\end{array}\right]$$

(21)

and

$$\overrightarrow{u}\left(t\right)=\left[\begin{array}{c}0\\ 1\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]\ddot{y}\left(t\right).$$

(22)

Note, as derived in Appendix B,

$$\begin{array}{ccc}\hfill B_{12}& =& {\displaystyle \frac{{\mu }_{0}V}{4\pi }}\sqrt{{\displaystyle \frac{4M_x^2+M_y^2}{{\left(x_1-x_2\right)}^6}}},\hfill \end{array}$$

(23)

Following Hooke’s law,

$$F=-kx,$$

(24)

one may identify the linear spring constant and the nonlinear spring function within Equations (20) and (21),

$$F_1=-\underset{\mathrm{Linear} \mathrm{spring} \mathrm{constant}}{\underbrace{\left(k_{11}+k_{12}\right)}}{x}_1+\underset{\mathrm{Nonlinear}\mathrm{spring}\mathrm{function}}{\underbrace{{\displaystyle \frac{3{\mu }_0{V}^2\left(2M_x^2-M_y^2\right)}{4\pi {|x_1-x_2|}^5}}}}\left(x_1-x_2\right),$$

(25)

with a similar expression for $x_2$. The highly nonlinear nature of the spring function results in a system which is sensitive to driving functions which are orders of magnitude less than the acceleration due to gravity, g. The driving acceleration used in the current model is of the order ${10}^{-2}$ g [22].

To understand the nonlinear nature of the system, consider an order of magnitude analysis of Equation (25):

$$\begin{array}{ccc}\hfill x_n& \sim & {10}^{-5} \mathrm{m},\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill k_{n,1}+k_{n,2}& \sim & {10}^2 \mathrm{N} {\mathrm{m}}^{-1},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill {\mu }_0& \sim & {10}^{-6} \mathrm{H} {\mathrm{m}}^{-1},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill V& \sim & {10}^{-8}{ }^{{\mathrm{m}}^3},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \left(2M_x^2-M_y^2\right)& \sim & {10}^9 \mathrm{A} {\mathrm{m}}^{-1},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill x_1-x_2& \sim & {10}^{-5} \mathrm{m}.\hfill \end{array}$$

(31)

So, the force order of magnitude is

$$\begin{array}{ccc}\hfill F& \sim & \left(\underset{\mathrm{Linear}}{\underbrace{{10}^{-3}}} \pm \underset{\mathrm{Nonlinear}}{\underbrace{{10}^7}}\right)\mathrm{N}.\hfill \end{array}$$

(32)

The conclusion is the nonlinearity is ten times stronger than the linear components.

Static Condition

The system of equations described by Equation (19) through (22) represent a boundary value problem which is defined for all $t>0$. However, the initial conditions are based on an equilibrium condition between the static forces from the magnetic fields and the forces due to the static deflection of the supporting springs. Hence, when there is zero input, there must be zero output. Thus, the state-space model of Equation (19) must be identically zero for all time when $\left(t\right)\equiv 0$:

$$A_{\mathrm{linear}}\overrightarrow{x}\left(t\right)+A_{\mathrm{nonlinear}}\left(\overrightarrow{x}\left(t\right)\right)=0.$$

(33)

This is the static condition. Consider holding two magnets, one in each hand, orienting them with identical poles facing each other and bringing one’s hands together. The force of the same pole arrangement will push the hands apart in distance and angle. The same occurs in the static case: The springs will displace vertically and rotate to avoid alignment of the magnets.

It is not possible to solve analytically for the static physical values of $\left(0\right)$, particularly the initial displacements and angles, because of the nonlinear nature of the system. Rather, the static values are determined by minimizing,

$$\begin{array}{ccc}\hfill f\left(\overrightarrow{x}\left(0\right)\right)& =& \parallel A_{\mathrm{linear}}\overrightarrow{x}\left(0\right)+A_{\mathrm{nonlinear}}\left(\overrightarrow{x}\left(0\right)\right)\parallel \hfill \end{array}$$

(34)

over $\overrightarrow{x}\left(0\right)$, where the linear and rotational velocities were constrained to be zero. The static values are listed in Table 1 and Figure 3.

It is suspected there are multiple local minima in the static condition objective function. Certain combinations of the known and guessed parameters of Table 1 resulted in static angles in excess of tens of degrees. These static solutions were rejected and the parameters changed before rerunning the static optimization. The initial guesses were not altered before rerunning the optimization.

Some “reasonable” and “acceptable” static values resulted in simulations with “collisions” that are ODE solutions where the magnets crossed paths. These cases were rejected. Again, the initial guesses were not altered before rerunning the static solver and the ODE.

4. Estimation of the Effective Model Parameters

Table 1 lists the model physical parameters divided into four categories,

• Known Those parameters whose values may be empirically measured.
• Guesses Unknown parameters whose values were guessed based upon expertise and approximate empirical measurement.
• Particle Swarm Optimized Fit Values Parameters whose values were determined by a particle swarm optimization which fit the model spectrum to the empirical spectrum, that minimized the difference between the two spectra. A particle swarm optimizer was selected because it was hypothesized that the nonlinear nature of the problem resulted in an error measure with multiple local minima.
• Unconstrained Derivative-Free Optimized Static Values The device static parameters which cannot be solved analytically due to the nonlinear nature of the problem and had to be solved via an optimization method.

Identifying the unknown parameters and verifying and validating the lumped parameter nonlinear ordinary differential equation (ODE) model involved utilizing experimental measurements obtained from the vibration harvesting device prototype. The measurements served as a basis to fine-tune and validate the predictions made by the model. By comparing the experimental results with the numerical simulations, adjustments and optimizations can be made to improve the accuracy of the model’s predictions.

To accomplish the verification, a particle swarm optimization (PSO) algorithm [23,24] was employed. The PSO algorithm was utilized to optimize and fit the parameters of the theoretical model to align with the experimental results. Through an iterative process, the algorithm searches for the best parameter values that minimize the discrepancy between the model predictions and the experimental data. The cost function was the root mean square error between the empirical and modeled output voltage spectra:

$$f\left(\mathbf{p}\right)=\sqrt{\frac{1}{N}\sum _{n=1}^N{\left(V_e\left(n\right)-\alpha V_m(n;\mathbf{p})\right)}^2},$$

(35)

where $V_e\left(n\right)$ is the empirical spectrum, $V_m(n;)$ is the model spectrum, N is the number of sample points and is the vector of parameters to be optimized:

$$\mathbf{p}=\left[\begin{array}{c}{k}_{11}\\ k_{12}\\ k_{21}\\ k_{22}\\ J_1\\ J_2\\ \alpha \end{array}\right].$$

(36)

$\alpha$ is a positive amplitude used to account for unknown scaling between the empirical and model spectra.

The effectiveness of this calibration approach hinges upon the assumption that the theoretical model adequately captures all the relevant physics associated with the system under investigation. It is crucial for the model to approximate the behavior of the system with an acceptable degree of accuracy. By validating the model against experimental data, an assessment of whether this assumption holds true and determination of the reliability of the calibrated model can be made. The results obtained from the PSO optimization indicate that the fitted parameters of the model are not significantly different from the independently measured physical parameters, such as the inertial masses and spring stiffness. This suggests that the calibrated model successfully captures the essential characteristics of the system. The agreement between the model parameters and the independently measured values further reinforces the confidence in the accuracy and reliability of the numerical model.

5. Experimental Design and Testing

A computer controlled and calibrated experimental vibration testbed was developed for analyzing the prototype sensor designs. Figure 4 depicts the testbed and its fundamental components. The input excitation for the sensor is generated by an electrodynamic shaker, designed to operate within a bandwidth ranging from 1 $\mathrm{Hz}$ to 5 $\mathrm{k}\mathrm{Hz}$. The computer (not shown for clarity) interfaces with the lock-in amplifier, digital oscilloscope and a laser vibrometer. The lock-in amplifier provides a sinusoidal drive signal to the shaker which is controlled by a MATLAB-GPIB [25] interface on the computer. The generated drive signal can be swept across the experimental bandwidth with very fine resolution. The drive signal from the lock-in amplifier can be amplified 40 dB as needed to boost the output acceleration of the shaker and drive the VEH sensor into a nonlinear operational regime. A 20 $\mathrm{c}\mathrm{m}$ aluminum angle bar is bolted to the shaker on one end. The VEH sensor and the reference accelerometer are mounted on the free end of the aluminum bar. The primary need for the aluminum bar is to minimize the effects of the strong magnetic field from the electrodynamic shaker on the VEH sensor prototypes. Measurements with a laser vibrometer on the free and clamped ends of the stiff aluminum beam over the frequency range (1 $\mathrm{Hz}$ through 200 $\mathrm{Hz}$) indicate very little variation for modest acceleration levels ($\ddot{y}\left(t\right)<0.5$ g). The experimental measurement setup incorporates three independent methods for characterizing the sensor response. The first method utilizes an optical laser vibrometer which can be focused on the inertial mass, the sensor case of the VEH sensor or on the aluminum beam. This non-contact measurement enables precise velocity measurement of the inertial mass without influencing the dynamic response of the sensor. Output from the laser vibrometer is recorded onto the scope and saved via internal storage to a thumb drive. The nature of this sensor design makes this measurement challenging as the inertial mass is designed to wobble, which reduces the signal to noise of the vibrometer and ultimately limits its utility for this situation. The second measurement is performed by an accelerometer mounted near the VEH sensor location on the rigid aluminum beam. This accelerometer captures the input acceleration, $\ddot{y}\left(t\right)$, experienced by the sensor base. The accelerometer provides an accurate standard reference measurement of the magnitude and transient behavior (e.g., frequency content) of the input signal $\ddot{y}\left(t\right)$ to the VEH sensor. Also, since the accelerometer is calibrated, it is also utilized to determine the relative g-force on the sensor for different input voltages to the shaker. Finally, the third measurement is the actual output of the VEH sensor. Here, either a transient response can be recorded on the oscilloscope or a forced response function can be measured using the in-phase and quadrature capabilities of the lock-in amplifier [26]. The lock-in amplifier provides an alternative approach to measuring the response of a system by not relying upon calculating the instantaneous spectrum in a short time window, as is the case for spectrum analyzers. The nature of the phased locked loop in the lock-in amplifier allows for very accurate narrow-band measurements of the order of a few Hertz to be recorded over the bandwidth of the sweep. In addition, the time constants and filter settings associated with the dwell time can be adjusted to improve the sensitivity.

The current output from the VEH sensor is also directed into an original equipment manufacturer (OEM) energy harvesting circuit, EH300 [27]. EH300 harvesting modules are designed for storing power from weak and intermittent voltage signals generated by the VEH sensors. The modules provide solutions for long-term power storage applications for powering diagnostic sensors and communication requirements.

The characteristic input for this investigation is a small-amplitude, long-duration random noise signal. A signal of this type is what might be experienced by the sensor attached to a structure responding to non-correlated ambient vibrations. Here, the emphasis is to test the VEH sensor’s bandwidth and performance to low-level input stimuli in a controlled and repeatable manner. Harvesting performance of the current VEH sensor prototype is investigated in the next section.

6. Empirical and Model Analysis of the VEH Sensor

In this section, the performance of the VEH sensor is investigated using spectral techniques over a limited bandwidth. The goal is to determine an estimate of the harvester’s sensitivity ($\mathrm{V}/\mathrm{g}$) and power conversion efficiency to low-level vibration input. The first approach analyzes the response spectrum of the sensor utilizing the narrowband capabilities of a lock-in amplifier and a swept sinusoidal drive signal. The second approach utilizes a random noise signal to measure the harvesting performance when the sensor is exposed to stochastic non-stationary vibration input. Here, the noise signal is defined to cover the same bandwidth as determined for the harvesting sensitivity. A calibrated accelerometer is used as a reference to maintain similar input characteristics for the two methods.

6.1. Swept Sine Measurements of VEH Sensor and Optimization of Model

A range of sinusoidal input voltages are generated using the internal frequency generator of the lock-in amplifier (Stanford Research, SRS856, Stanford Research Systems, Sunnyvale, CA, USA). The signal is swept across the estimated operational bandwidth of the prototype. The drive voltage to the shaker is adjusted using a reference accelerometer (PCB Piezotronics, model 352A24, Depew, NY, USA). The accelerometer is mounted on the end of the beam and provides a measurement of the input acceleration experienced by the harvesting sensor. Using the calibrated accelerometer, the drive voltage from the power amplifier was adjusted to create an approximate input acceleration from the shaker of $\ddot{y}\left(t\right)={10}^{-2}$ g across the range of 1 $\mathrm{Hz}$ to 200 $\mathrm{Hz}$. The controlled response of the shaker (BK vibration exciter 4809, Marlborough, MA, USA) across this frequency range and drive level is flat. A laser vibrometer is used to verify the shaker response. Additional vibrometer measurements detected no variation in acceleration magnitude across the length of the aluminum bar either, thus verifying the shaker and the attached aluminum bar are reproducing the drive signal characteristics from either the lock-in amplifier or the signal generator.

The goal is to determine an estimate of the harvester’s coupling sensitivity ($\mathrm{V}/\mathrm{g}$) and power conversion efficiency for low-level acceleration input. Figure 5 shows the empirical voltage spectrum from the VEH prototype measured by the SRS 856. The load resistance is 50 $\Omega$ and the input drive acceleration from the shaker is 8 × 10⁻³ g. The spectrum shows one large peak at 138 $\mathrm{Hz}$ and a secondary smaller one at near 140 $\mathrm{Hz}$. The larger peak at 138 $\mathrm{Hz}$ has a usable bandwidth of approximately 20 $\mathrm{Hz}$. Outside this range, the sensor coupling sensitivity of the prototype is insufficient for harvesting; this will be elaborated upon in the next section. This measurement can now be utilized to improve the analytical model. Since the input acceleration is constant, the root mean square (rms) sensitivity of the sensor is estimated as (rms(Vout)/10⁻² g); so, between 130 $\mathrm{Hz}$ and 150 $\mathrm{Hz}$, the rms coupling sensitivity is 0.58 $\mathrm{m}\mathrm{V}/\mathrm{g}$, with a peak sensitivity of 0.87 $\mathrm{m}\mathrm{V}/\mathrm{g}$ at 138 $\mathrm{Hz}$.

The optimization of the model requires tuning the parameters of the model to fit a given measured output. The parameters utilized in the model described in Equations (19)–(22) are listed in Table 1. Some of these parameters can be easily measured while others cannot. The initial fabrication data of the components (i.e., mass, magnetic strength, dimensions, estimated spring stiffness, etc.) are measured using standard laboratory techniques. The spring stiffness is estimated using a measurement of the first resonances of each of the magnet spring sub-assemblies using the laser vibrometer. The magnetic field strength of each magnet was measured using a EXTECH AC/DC Magnetic Meter, Nashua, NH, USA. These techniques will introduce errors when coupled with the complexity of the equations of motion. It is found through trial and error that the stiffness estimates for the spherical springs and the rotational inertial terms have the strongest effect on the response of the model for the low-level input acceleration. The optimization step is utilized in conjunction with the parameters that have the least uncertainty to estimate the parameters that have higher uncertainty. With this approach, the model is “tuned” to fit the empirical results. For each optimization iteration, the model of Equation (19) is run for 30 $\mathrm{s}$ of model time using MATLAB’s ODE45 [25] solver (this is also used to solve the static case of Equation (34)). The model output is the state vector of Equation (18). The time series of interest is the current, $i\left(t\right)$, which when multiplied by the 50 $\Omega$ load resistance yields the harvested voltage, $v\left(t\right)$. The model voltage spectrum, $V_m(n;)$, is estimated using the Welch [28] method with a window of 8.2 $\mathrm{s}$ with a 50% overlap. Here, the empirical spectrum, $V_e\left(n\right)$, is taken from Figure 5.

Figure 6 shows the optimized model output using the parameters of Table 1 along with the harvested voltage spectrum from the VEH prototype over the frequency range 20−200 $\mathrm{Hz}$. The main spectral peak matches the empirical spectrum and follows the trend on the left-hand side of the peak. The 4 degree-of-freedom lumped parameter model (two linear displacements and two rotations from each magnet) follows the empirical spectrum well after parameter fitting. The root mean square error is $1.4\times {10}^{-7}$.

As a measure of similarity, the normalized root mean squared error was chosen. It is defined as

$$s=100\left(1-{\displaystyle \frac{\parallel V_e-V_m\parallel }{\parallel V_e-{\overline{V}}_e\parallel }}\right),$$

(37)

where ${\overline{V}}_e$ is the mean of the empirical spectrum. The similarity is 81.7% (an identical match is 100%).

The similarity measure provides confidence the model parameters have been tuned such that the model is accurately predicting the response of the VEH prototype. The empirical data and the fitted model show that the sensor has the highest potential harvesting performance over a 20 $\mathrm{Hz}$ bandwidth centered at approximately 140 $\mathrm{Hz}$. In the next section, the response of the sensor to low-level random noise is investigated for harvesting applications.

6.2. Harvesting from Low-Level Non-Stationary Signals

Long-term storage and monitoring application require successful harvesting of potentially low-level and infrequent sources of vibrational energy. In this second analysis, the energy harvesting capability of the VEH sensor is investigated utilizing a random acceleration spectrum generated from the white noise feature of the Keysight 3352B arbitrary waveform generator, Santa Rosa, CA, USA (AWG). For these tests, the AWG drives the shaker directly and the accelerometer is used to set the input acceleration magnitude of ${\ddot{y}}_{\mathrm{RMS}}\sim {10}^{-2}$ g, as determined from the reference accelerometer. The targeted noise spectrum is assumed white, centered at 130 $\mathrm{Hz}$, with a 75 $\mathrm{Hz}$ bandwidth.

Transient accelerometer data and the voltage output from the VEH harvester are recorded over a 7 $\min$ time window using a digital oscilloscope. The sampling rate for the oscilloscope is set at 0.2 $\mathrm{m}\mathrm{s}$, which resulted in a data record of approximately 20 $\mathrm{k}\mathrm{samples}$. The frequency spectra are calculated from the transient data using Fourier methods [28]. Figure 7 shows the acceleration input at the base of the harvesting sensor and the output spectra of the VEH harvester measured with a 50 $\Omega$ load termination. The black line is the input acceleration spectrum measured by the reference accelerometer and converted to $\mathrm{g} \mathrm{m} {\mathrm{s}}^{-2}$. The red line is the VEH sensor voltage output spectrum corresponding to an input random acceleration of ${\ddot{y}}_{\mathrm{RMS}}\sim {10}^{-2}$ g for comparison.

Initially, it was thought that the random acceleration input spectrum would be flat across the bandwidth and the VEH sensor would be driven by a stochastically uniform acceleration input. The calculated input acceleration spectrum is found to be biased towards a peak rms amplitude around 75 $\mathrm{Hz}$, with an approximate $-2$ dB/decade on either side. Unfortunately, a large portion of the energy in this spectrum falls outside the targeted range of the VEH; however, it still provides a test of the sensor’s off nominal harvesting capability.

The highlighted region of the graph is the useful harvesting bandwidth ($130\mathrm{Hz}<\mathrm{VEH}\mathrm{frequency}<150\mathrm{Hz}$) of the sensor with the EH300 harvesting circuit. Essentially, this region represents an estimate based on the EH300 circuit’s power harvesting capabilities defined as 4 $\mathrm{V}$ at 200 $\mathrm{n}\mathrm{A}$ or 0.8 $\mathsf{\mu }\mathrm{W}$ [27]. Across the 50 $\Omega$ load resistance, this translates into a minimum power of 0.04 $\mathsf{\mu }\mathrm{W}$ across the 20 $\mathrm{Hz}$ operational bandwidth of the highlighted region and is marked by the red dashed line relative to the voltage axis. A smaller region exists for the second peak; however, this is not considered for this analysis.

An estimate of the harvesting efficiency of the VEH sensor based on the random acceleration is shown in Figure 8. The harvesting efficiency is defined as $\eta =P/P_{\mathrm{in}}$, where P is the average output power from the harvesting circuit, calculated as $P={V_{\mathrm{out}}}^2/\left(R_l\Delta f\right)$, and $P_{\mathrm{in}}$ is the mechanical input power, calculated as $P_{\mathrm{in}}={\ddot{Y}}^{2}m\Delta f$. Here, m are the magnet masses and Δf is the frequency band [29].

The ratio between the output voltage and the input acceleration spectrum are plotted as a function of frequency across the harvesting bandwidth, essentially normalizing the VEH sensor output sensitivity by dividing out the noise signal response. The similarity between the Figure 5 and Figure 8 is evident in the bandwidth (130−150 $\mathrm{Hz}$). In this region, the sensor has an average rms harvesting efficiency of 6.3%, with a peak value of 9.8% at 138 $\mathrm{Hz}$. Here, the impedance match between the coil resistance, 60 $\Omega$, and the load resistance, 50 $\Omega$, supports that the approximation is close to optimal in terms of impedance matching with the load resistance. In a third set of measurements, the harvesting performance of the VEH prototype is recorded while connected to the EH300 harvesting circuit. The harvested voltage across the 50 $\Omega$ load resistor is connected to the input buffering circuit of the EH300. The oscilloscope is now used to record the voltage gain across the circuit’s storage capacitor, and as before, the sensor is exposed to the same band limited random acceleration signal described previously. The total harvested voltage from the VEH sensor in the EH300 is recorded for approximately 7 min. The sampling rate for the oscilloscope was set at 0.2 $\mathrm{m}\mathrm{s}$, which resulted in a data record of approximately 20 $\mathrm{k}\mathrm{samples}$. This is the largest window that could be held in the oscilloscope’s window before the first samples are lost. The sampling rate ensures a Nyquist limit of 5 $\mathrm{k}\mathrm{Hz}$, which is a factor of 25 times larger than the highest frequency associated with the noise spectrum. Figure 9 shows a linear fit of the data results in a voltage harvesting rate of approximately 10 $\mathrm{m}\mathrm{V}/\mathrm{h}$ for random acceleration noise ${\ddot{y}}_{\mathrm{rms}}\sim {10}^{-2}$ g. The spikes in the temporal voltage signal may be generated in part by electronic noise present in the unshielded circuit. Another consideration is that the random acceleration magnitudes may be introducing the fluctuations in the voltage in the storage capacitor. Finally, it is possible to estimate the VEH sensor’s power density by normalizing the output power with the volume, ${\mathrm{Vol}}_{\mathrm{mag}}$, of the effective moving elements (i.e., the volume of the two coil and magnet assemblies [29]). This can be calculated as $P={V_{\mathrm{out}}}^2/\left(R_l\Delta f\right)/{\mathrm{Vol}}_{\mathrm{mag}}$, with a resulting power density of 0.2 $\mathsf{\mu }\mathrm{W}/\mathrm{c}{\mathrm{m}}^3$. With this information, the prototype sensor can be compared to existing results published in the literature [30].

7. Discussion

A prototype of a nonlinear broadband vibrational energy harvester (VEH) was modeled, designed, and built. The experimental performance was investigated over an input range of 1 $\mathrm{Hz}$ to 200 $\mathrm{Hz}$ with low-level random vibrations $\ddot{y}\left(t\right)\ll g$. Over this range, the useful harvesting bandwidth is determined to be due to the sensitivity of the harvesting sensor and the minimum detectable voltage level from the EH300 harvesting circuit. The tested prototype sensor demonstrated the capability of harvesting usable energy from random ambient vibrations of the order of ${10}^{-3}$g over the operating range of $130\mathrm{Hz}<f<150\mathrm{Hz}$. This underscores the device’s potential to harness energy from minute mechanical disturbances. While the bandwidth is limited to 20 $\mathrm{Hz}$, further testing may improve the performance. The small size of the inertial masses plays a large role in the device’s low efficiency; however, this might be improved with further optimization. In the context of longer duration exposure, the sensor does generate energy which can slowly be stored. The EH300 harvesting circuit is responsible for storing the accumulated energy over time. It is observed that the VEH had a charging performance of 10 $\mathrm{m}\mathrm{V} {\mathrm{h}}^{-1}$, based on the measured voltage across the storage capacitor (1000 $\mathrm{n}\mathrm{F}$) in the EH300 circuit. Based on this performance estimate, the current prototype would take approximately 40 $\mathrm{h}$ to fully cycle the EH300 to an operational state suitable to support typical IoT microprocessor applications. This long charging window is due to the misalignment of the input acceleration spectrum and the VEH sensor’s peak sensitivity. An order of magnitude improvement would be realized if the sensor’s design point matched the input acceleration spectrum. The EH300 provided a convenient method for energy capture, with similar approaches given in the literature [31]; however, an investigation of other potential harvesting circuits was not conducted. The EH300 harvesting circuit can also charge different external capacitors with different capacitances, providing the designer additional flexibility and capabilities [32].

Equations of motion for the coupled electro-mechanical system (ODE) model were developed and used to predict the harvesting performance. With the fitted parameters, the model output voltage spectrum shows good correlation with the model over the predicted resonances, with the overall trend between 20 $\mathrm{Hz}$ and 200 $\mathrm{Hz}$ showing a similarity measure of 81.7%. The similarity in the region of interest, 130 $\mathrm{Hz}$ to 150 $\mathrm{Hz}$, is 73.8%. The decrease in similarity over the 130−150 $\mathrm{Hz}$ range is an artifact of fitting the model to the data over the entire spectrum, 20−200 $\mathrm{Hz}$. Because of the fitting process, the model is biased towards the entire measured spectra, so it is not surprising that the fitted model deviates from the experiment in this reduced range. The similarity would improve by refitting the entire model over the smaller bandwidth; however, this was not done in this investigation. The comparison between the model’s performance and the experimental results provides interesting insights into the behavior of the sensor design. Notably, the presence of asymmetries linked to magnetic fields and height disparities within the mounting hardware introduces a rotational motion component into the sensor response, a factor that the experiment reveals but the model only partially captures. These findings highlight the complexities of the system and emphasize the necessity of refining the model to encompass the nuanced 3D effects and accurately represent the future experimental comparisons. The design also possesses the capability to be mechanically adjusted (tuned) to accommodate different tilt angles and magnetic separation distances. The adjusting process is simple but a bit tedious. With the sensor on the test stand, the hex nuts can be loosened and moved up or down on each threaded rod using a small non-magnetic wrench. The adjustable angular travel is ∼$\pm 3$ degrees and the vertical travel is ∼$\pm 4\mathrm{m}\mathrm{m}$. The repelling forces of the two magnets also contribute to the angular tilt and the overall separation between the coil and the upper magnet. Adjustments can be used in conjunction with the model parameters to improve harvesting and modify performance; however, this was not done in this investigation.

8. Conclusions

Overall, the current sensor was shown to be sensitive at capturing low-level random noise centered near its design spectrum. Further tuning by adjusting the tilt and separation of the magnets may improve the performance. Vibration harvesting sensors targeted to harvest low-level ambient vibration energy face many design challenges. Targeting the correct operational spectrum such that the sensor dynamics take full advantage of the ambient vibrational field can be aided by models that are tuned to the target spectrum. Once tuned, the model then indicates which parameters are more effective at controlling the performance. Future investigations will focus on developing sensor designs and operational parameters that optimize the harvesting process.