3-Degree-of-freedom electromagnetic vibration energy harvester with serially connected leaf hinge joints

Abstract

This article presents a new design method of a planar 3-degree-of-freedom serial manipulator-type electromagnetic vibration energy harvester in which any desired ratio of power peaks and three target resonant frequencies can be specified arbitrarily. The design of the harvester aims to achieve minimum difference between the power peaks generated at target frequencies. The geometrical positions of three normal modes are first determined and the corresponding stiffness matrix of the harvester is found. Second, the stiffness matrix can be synthesized by three serially connected torsional springs. Third, the leaf hinge joints corresponding to torsional springs are designed using the newly developed design equations. Finally, the array and the locations of the magnets are found using the sequential quadratic programming (SQP) algorithm. The experiments are conducted to verify the design method. Three resonant frequencies are measured at 23.4, 29.2, and 34.8 Hz comparing to the target frequencies of 25, 30, and 35 Hz. The peak powers of 1.28, 0.89, and 1.32 mW are obtained across the optimal load resistor of 1.01 kΩ under the condition of the constant acceleration of 1.5 m/s².

Keywords

Vibration resonance 3-degree-of-freedom serial device energy harvester electromagnetic transducer stiffness synthesis leaf hinge design method

Introduction

As the power consumption required to drive sensors and electronics rapidly decreases, the interests in harvesters in their use as a power supply to these devices increase (Bhatnagar and Owende, 2015). An energy harvester based on vibration is a device that converts mechanical energy into electric energy using ambient vibration as an energy source. The vibration energy harvesters generate peak powers at resonant frequencies through diverse transducers such as piezoelectric, electromagnetic, electrostatic, and magnetoelectric transducers (Beeby et al., 2006; Castagnetti and Dallari, 2017; Cook-Chennault et al., 2008; Lefeuvre et al., 2005; Naifar et al., 2017). The frequencies of ambient vibration sources are variable in the low frequency range within 100 Hz (Khan and Ahmad, 2016; Roundy et al., 2003). Many researchers have developed some interesting designs of the structures of harvesters to locate a number of target resonant frequencies in the low frequency range (Castagnetti, 2012, 2015b; Li et al., 2013; Wang and Tang, 2017; Zhou et al., 2012).

Cantilevers harvesting vibrational energy with piezoelectric materials have been widely used by virtue of a simple geometrical structure. In general, a cantilever harvester captures vibration energy only when the harvester resonates with the vibration source. Otherwise, the harvesting efficiency becomes very low. A simple solution to this problem may be to consider using multiple cantilevers. An effort to implement multi-resonant frequencies was to present the design of a set of independent beam–mass systems with various beam lengths and masses without an experimental validation (Shahruz, 2006). A system of two masses placed on one cantilever was suggested to obtain two resonant frequencies (Ou et al., 2010); however, the second resonant frequency was considerably distant from the first one. The design of a two-mass system harvester with one mass suspended by the main cantilever and the other mass by the inner secondary reverted cantilever was proposed to locate the first two resonant frequencies closely (Seo et al., 2012; Wu et al., 2012). Another 2-degree-of-freedom (DOF) cantilever-type harvester used a primary beam as an impact-enhanced dynamic magnifier with a stopper to obtain the broad working frequency range from the reverted secondary beam (Halim and Park, 2015, 2016). Multi-DOF harvesters greater than 2-DOF were developed using stacked cantilevers or side-mounted cantilevers to obtain multi-resonant peaks within 100 Hz (Qi et al., 2010; Xiong and Oyadiji, 2014). Some other harvester designs with different geometrical shapes such as the V-shape, the zigzag, and the spiral beam were also presented to demonstrate the implementation of the multi-DOF harvesters (Bai et al., 2014; Castagnetti, 2013, 2015a; Dhote et al., 2015; El-Hebeary et al., 2013; Karami and Inman, 2011).

Although much efforts have been made to develop the multi-DOF harvesters, the limitations on performance associated with large differences in the magnitudes of power peaks and narrow useful bandwidths were shown. To overcome these limitations, the design method that enables specifying any desired ratios of the peak magnitudes at target resonant frequencies was introduced (Kim et al., 2015). Using this design method, the 3-DOF serial linkage-type harvester with piezoelectric transducers (Lee et al., 2014) and the parallel-type harvester with electromagnetic transducers (Park et al., 2017) were developed. The study on the 3-DOF serial linkage-type harvester showed the distinct possibility of designing the identical power peaks generated at equally spaced three target resonant frequencies using the polyvinylidene fluoride (PVDF) films (Kim et al., 2017).

In this line of research, this article presents a novel design of a 3-DOF serial manipulator-type harvest utilizing electromagnetic transducers that can produce the equal power peaks at three resonant frequencies. The harvester is composed of a proof mass with magnets supported by serially connected torsional springs which are realized in the form of leaf hinge joints. A new design method of a leaf hinge joint is derived from a lumped element model of a cantilever to find hinge parameters such as a pivot point and a torsional stiffness. The power is obtained from the fixed coils located near the magnets oscillating with the proof mass. The different design method (from the work presented in Kim et al., 2017) involves the arrangement of an array of magnets and the realization of torsional springs by means of leaf hinge joints.

The basic design theory is summarized in the next section which includes the geometrical determination methods for the normal modes, the stiffness matrix, the locations of magnets, and the leaf hinge joints. Detailed description of the design of a prototype harvester is followed. Finally, the experiments are carried out to demonstrate the validation of the design and the results are discussed.

Equations of motion for base excitation

We consider a rigid body that is supported by serially connected springs and dampers as shown in Figure 1. The conceptual design of 3-DOF serial device of Figure 1(a) is to be realized by a harvester shown in Figure 1(b). Resonances at three target frequencies of the harvester will be brought about by properly located torsional springs. Thus, the first step in the design is to find the locations and stiffness constants of torsional springs of the harvester, which is described in the next section.

If the rigid body is subject to the base excitation $X_{0}$ , the equation of the motion of the rigid body can be expressed by

M \overset{\cdot\cdot}{Z} + D \overset{\cdot}{Z} + K Z = - M {\overset{\cdot\cdot}{X}}_{0}

(1)

and

Z = X - X_{0} = \hat{Z} e^{j ω t}

(2)

where $M, D,$ and $K$ are the $3 \times 3$ mass, damping, and stiffness matrices in a plane, respectively. $ω$ is the frequency of the base excitation. $Z$ is the relative displacement of the rigid body against to the base and written as

Z = \hat{Z} e^{j ω t} = {[\begin{matrix} δ x & δ y & δ γ \end{matrix}]}^{T} e^{j ω t} = δ γ \hat{S} e^{j ω t}

(3)

where $δ x$ and $δ y$ are the x - and y -components of small translational displacement of the rigid body, and $δ γ$ is the small rotational angular displacement. The ordered triple numbers of the normalized line vector $\hat{S} = [\begin{matrix} y & - x & 1 \end{matrix}]^{T}$ in equation (3) are Plücker’s axis coordinates (Duffy, 1996) and represent the line perpendicular to the xy -plane passing through the point $(x, y)$ . The essential part of the design theory lies in the understanding of the geometrical meaning and relationship of three normal modes of a planar vibration system with a single rigid body. The line vectors ${\hat{S}}_{i}^{'} s$ are interpreted as the instant centers of vibration, or shortly, vibration centers, in geometrical terms (Blanchet, 1998). Thus, the solution $\hat{Z}$ of equation (3) implies that the rigid body oscillates about the vibration center $\hat{S}$ with the angular displacement $δ γ$ , as shown in Figure 2.

The time-independent form of equation (1) can be written as

(- ω^{2} M + j ω D + K) \hat{Z} = ω^{2} M {\hat{X}}_{0} (= {\hat{F}}_{ext})

(4)

The right side of equation (4) corresponds to an external force ${\hat{F}}_{ext}$ of the base excitation. It is noted here that in order to harvest electrical energy, we chose the electromagnetic transducer composed of an array of magnets which acts like a damper. The power absorbed by the damper can be estimated from

P = {\overset{\cdot}{\hat{Z}}}^{T} D \overset{\cdot}{\hat{Z}} = ω^{2} {\hat{Z}}^{T} D \hat{Z}

(5)

Design method of planar vibration system

Clearly, a general planar vibration system of a single rigid body has three normal modes. We can construct a triangle defined by three normal modes (or, vibration centers) ${\hat{S}}_{i}$ and it is referred to as a modal triangle. It is noted that the orthocenter of the modal triangle is located at the mass center (Blanchet, 1998). Such geometrical characteristics of the modal triangle are utilized to find the positions of normal modes when any corresponding natural frequencies are specified (Kim et al., 2015). For a given mass matrix, the objective here is to find the stiffness matrix that eventually yields any desired three normal modes for specified natural frequencies. In the following two sections, the design equations that relate the positions of three normal modes of a modal triangle to the absorbed power by the transducer are derived. Once the positions of three normal modes are determined and the corresponding stiffness matrix is found, the next step is to design the array of the magnets of the electromagnetic transducer that is used in this study.

Determination of normal modes

If the mass center is positioned at the origin of the coordinate system, the mass matrix can be denoted by

M = [\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I \end{matrix}]

(6)

where $m$ and $I$ are the mass and the moment of inertia, respectively. The ith normal mode ${\hat{S}}_{i}$ positioned at $(x_{i}, y_{i})$ can be written as

{\hat{S}}_{i} = [\begin{matrix} y_{i} & - x_{i} & 1 \end{matrix}]

(7)

Now, the coordinate axes are placed in such a way that the origin is located at the mass center and the mode ${\hat{S}}_{1}$ is positioned on the x -axis while the other two modes are positioned on the vertical line parallel to the y -axis as shown in Figure 3. Thus, three normal modes can be expressed by

\begin{matrix} {\hat{S}}_{1} = {[\begin{matrix} 0 & - x_{1} & 1 \end{matrix}]}^{T}, {\hat{S}}_{2} = {[\begin{matrix} y_{2} & - x_{2} & 1 \end{matrix}]}^{T}, \\ and {\hat{S}}_{3} = {[\begin{matrix} y_{3} & - x_{2} & 1 \end{matrix}]}^{T} \end{matrix}

(8)

The normal modes are orthogonal to each other with respect to the mass and the stiffness matrices, which can be expressed by

S^{T} MS = [\begin{matrix} {\tilde{m}}_{1} & 0 & 0 \\ 0 & {\tilde{m}}_{2} & 0 \\ 0 & 0 & {\tilde{m}}_{3} \end{matrix}] and S^{T} KS = [\begin{matrix} {\tilde{k}}_{1} & 0 & 0 \\ 0 & {\tilde{k}}_{2} & 0 \\ 0 & 0 & {\tilde{k}}_{3} \end{matrix}]

(9)

where $S = [\begin{matrix} {\hat{S}}_{1} & {\hat{S}}_{2} & {\hat{S}}_{3} \end{matrix}]^{T}$ , ${\tilde{m}}_{i} = {\hat{S}}_{i}^{T} M {\hat{S}}_{i}$ , and ${\tilde{k}}_{i} = {\hat{S}}_{i}^{T} K {\hat{S}}_{i}$ for $i = 1, 2, 3$ . The generalized mass ${\tilde{m}}_{i}$ is given by

{\tilde{m}}_{i} = m (x_{i}^{2} + y_{i}^{2} + μ^{2})

(10)

where $μ = \sqrt{I / m}$ denotes the radius of gyration. We define two parameters of $a$ and $b$ that represent the horizontal and the vertical ratio of the x - and y -coordinates of the vertices of a modal triangle, respectively

a = - \frac{x_{2}}{x_{1}} and b = - \frac{y_{3}}{y_{2}}

(11)

From equations (9) and (11), the four coordinates $x_{1}$ , $x_{2}$ , $y_{2}$ , and $y_{3}$ can be expressed in terms of a and b by

\begin{matrix} x_{1} = - μ \sqrt{\frac{1}{a}}, x_{2} = μ \sqrt{a}, y_{2} = μ \sqrt{\frac{(1 + a)}{b}}, \\ and y_{3} = - μ \sqrt{(1 + a) b} \end{matrix}

(12)

If the base excitation force ${\hat{F}}_{ext}$ in equation (4) is applied in the y -direction, it can be written as

{\hat{F}}_{ext} = ω^{2} M {\hat{X}}_{0} = ω^{2} md {[\begin{matrix} 0 & 1 & 0 \end{matrix}]}^{T}

(13)

where $d$ is the magnitude of the base displacement. The relative displacement $\hat{Z}$ in equation (4) can be written in terms of the ith normal mode ${\hat{S}}_{i}$ and the angular displacement $ν_{i}$ as

{\hat{Z}}_{i} = ν_{i} {\hat{S}}_{i} = ν_{i} {[\begin{matrix} y_{i} & - x_{i} & 1 \end{matrix}]}^{T}

(14)

Equation (14) is substituted into equation (4) and both sides of equation (4) are premultiplied by ${\hat{S}}_{i}^{T}$ to utilize the orthogonality given in equation (9). Then after some algebra (described in Kim et al., 2015), the angular displacement $ν_{i}$ can be expressed by

ν_{i} = \frac{d x_{i}}{2 j (x_{i}^{2} + y_{i}^{2} + μ^{2}) ζ_{i}}

(15)

where $ζ_{i}$ is the ith modal damping ratio. As mentioned in the previous section, electrical power can be converted from absorbed power by the transducer, equation (5) is used to find the position parameters of the normal modes instead of using the work done by base excitation on the rigid body in (Kim et al., 2015). Substituting equations (8), (14), and (15) into equation (5), the power absorbed by the damper can be determined as

P_{i} = \frac{m d^{2} Ω_{i}^{3} x_{i}^{2}}{2 ζ_{i} (x_{i}^{2} + y_{i}^{2} + μ^{2})}

(16)

and

\begin{matrix} P_{1} = \frac{m d^{2} Ω_{1}^{3} x_{1}^{2}}{2 ζ_{1} (x_{1}^{2} + μ^{2})}, P_{2} = \frac{m d^{2} Ω_{2}^{3} x_{2}^{2}}{2 ζ_{2} (x_{2}^{2} + y_{2}^{2} + μ^{2})}, \\ and P_{3} = \frac{m d^{2} Ω_{3}^{3} x_{2}^{2}}{2 ζ_{3} (x_{2}^{2} + y_{3}^{2} + μ^{2})} \end{matrix}

(17)

From equation (12), equation (17) can be rewritten as

\begin{matrix} P_{1} = \frac{m d^{2} Ω_{1}^{3}}{2 ζ_{1} (a + 1)}, P_{2} = \frac{m d^{2} Ω_{2}^{3} ab}{2 ζ_{2} (1 + a) (1 + b)}, \\ and P_{3} = \frac{m d^{2} Ω_{3}^{3} a}{2 ζ_{3} (1 + a) (1 + b)} \end{matrix}

(18)

Now, if any desired ratio of the powers is specified such that $P_{1} : P_{2} : P_{3} = 1 : γ_{1} : γ_{2}$ , the position parameters $a$ and $b$ can be found as

a = \frac{Ω_{1}^{3} (Ω_{2}^{3} γ_{2} ζ_{3} + Ω_{3}^{3} γ_{1} ζ_{2})}{ζ_{1} Ω_{2}^{3} Ω_{3}^{3}} and b = \frac{γ_{1} Ω_{3}^{3} ζ_{2}}{γ_{2} Ω_{2}^{3} ζ_{3}}

(19)

Consequently, the normal modes ${\hat{S}}_{i}$ are obtained from equations (8) and (12).

Determination of the stiffness matrix

From equation (4), the equation of motion for undamped free vibration becomes

(- ω^{2} M + K) \hat{Z} = 0

(20)

For the given resonant frequencies $Ω_{i}$ and the mass matrix $M$ , once the normal modes of the modal matrix $S = [\begin{matrix} {\hat{S}}_{1} & {\hat{S}}_{2} & {\hat{S}}_{3} \end{matrix}]^{T}$ are determined, the corresponding stiffness matrix $K$ can be determined from

K = MS [Ω Ω^{2}] S^{- 1}

(21)

where $[Ω^{2}] = diag (Ω_{1}^{2}, Ω_{2}^{2}, Ω_{3}^{2})$

Determination of electrical damping matrix

Referring to El-hami et al. (2001), the value of the electrical damping $D_{e}$ of the electromagnetic transducer has been determined from

D_{e} = \frac{ϕ^{2}}{R_{L} + R_{c} + j ω L_{c}}

(22)

where $R_{L}$ , $R_{c}$ , and $L_{c}$ are the load resistance, the coil resistance, and the coil inductance, respectively. The transformation factor $ϕ$ is given by

ϕ = NBl

(23)

where $N$ , $B$ , and $l$ are the number of turns of a coil, the average flux density of a magnet, and the effective length of each coil, respectively. In general, the inductive impedance is much lower than the resistive impedance. Thus, the electrically generated damping ratio at the resonant frequency $Ω_{i}$ can be written as

ζ_{ei} = \frac{D_{e}}{2 m Ω_{i}} = \frac{ϕ^{2}}{2 m Ω_{i} (R_{L} + R_{c})}

(24)

Now, the electrical damping matrix $D_{e}$ is determined from

D_{e} = 2 MS [Ω Ω] [ζ_{e}] S^{- 1}

(25)

where $[ζ_{e}] = d i a g (ζ_{e 1}, ζ_{e 2}, ζ_{e 3})$ and $[Ω] = d i a g (Ω_{1}, Ω_{2}, Ω_{3})$ . The power generated by the electromagnetic transducer can be computed from

P_{e} = {\overset{\cdot}{\hat{Z}}}^{T} D_{e} \overset{\cdot}{\hat{Z}} = ω^{2} {\hat{Z}}^{T} D_{e} \hat{Z}

(26)

Determination of the array of magnets

The moving directions of the proof mass are determined according to the normal modes ${\hat{S}}_{i}$ . One magnet moving across the fixed coil cannot generate the equal power at all resonant frequencies. For this reason, it is needed to use multiple magnets, and three magnets of the same number of modes are used in this study. The locations of three magnets satisfying the following two conditions: (1) the mass center of the total mass of magnets is located at the origin of the coordinate system and (2) the circumcenter of the triangle formed by center points of three magnets is matched the origin of the coordinate system, consequently formed the equilateral triangle. The position vector $m_{k}$ to the kth magnet located at the point ( $\begin{matrix} x_{k}, & y_{k}, & 0 \end{matrix}$ ) in the xyz coordinate system can be written as

m_{k} = {[\begin{matrix} x_{k} & y_{k} & 0 \end{matrix}]}^{T}

(27)

When the kth magnet oscillates about the vibration center of the normal mode ${\hat{S}}_{i}$ with the angular displacement $ν_{i}$ of equation (15) at the ith resonant frequency $Ω_{i}$ , the maximum velocity vector $v_{ik}$ of the kth magnet can be computed from

v_{ik} = Ω_{i} ν_{i} (r_{ik} \times {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T})

(28)

where $r_{ik} = s_{i} - m_{k}$ and $s_{i} = [\begin{matrix} x_{i} & y_{i} & 0 \end{matrix}]^{T}$ denote the position vector of the vibration center of the normal mode ${\hat{S}}_{i}$ . The output voltage of the coil $V_{o}$ can be found by summing the voltages $V_{k}$ induced by oscillating magnets

V_{io} = \sum V_{ik} = \sum {(b \times v_{ik}) \cdot t_{k} Nl}

(29)

where $b = [\begin{matrix} 0 & 0 & B \end{matrix}]^{T}$ and $t_{k} = [\begin{matrix} y_{k} & - x_{k} & 0 \end{matrix}]^{T} / \sqrt{(x_{k}^{2} + y_{k}^{2})}$ . Referring to Figure 4, $b$ and $t_{k}$ are the magnetic flux density vector passing through the area of $m_{k}$ and the unit tangent vector of the coil at the kth magnet center, respectively.

The maximum voltages generated at the resonant frequencies depend on the magnet array $M_{arr} (M_{1}, M_{2}, M_{3})$ and the angle of the equilateral triangle $φ$ . To find the optimum values for these parameters, the following objective function is defined

\begin{matrix} Minimize L (φ) = 1 / \\ {{(V_{median} - V_{\max})}^{2} + {(V_{median} - V_{\min})}^{2}} \end{matrix}

(30)

Subject to 0 \leq φ \leq 2 π (rad)

(31)

where $V_{\max}$ , $V_{median}$ , and $V_{\min}$ are the maximum, the median, and the minimum value of the output voltage of the coil generated at the resonant frequencies.

Synthesis method of the stiffness matrix

The stiffness matrix $K$ can be synthesized by a set of torsional springs (Hong and Choi, 2012). In the previous study presented in Kim et al. (2017), the matrix $K$ was realized by first synthesizing the parallel connections of three line springs and then converting it into three serially connected torsional springs. Here we present a new method for the direct determination of three serially connected torsional springs, as shown in Figure 5.

The rank 3 compliance matrix $C$ which is defined as the inverse matrix of $K$ may be written as

\begin{matrix} C = J [c_{\emptyset}] J^{T} = [\begin{matrix} {\hat{J}}_{1} & {\hat{J}}_{2} & {\hat{J}}_{3} \end{matrix}] [\begin{matrix} c_{\emptyset 1} & 0 & 0 \\ 0 & c_{\emptyset 2} & 0 \\ 0 & 0 & c_{\emptyset 3} \end{matrix}] \\ {[\begin{matrix} {\hat{J}}_{1} & {\hat{J}}_{2} & {\hat{J}}_{3} \end{matrix}]}^{T} \end{matrix}

(32)

C = \sum_{k}^{3} c_{\emptyset k} {\hat{J}}_{k} {\hat{J}}_{k}^{T}

(33)

where ${\hat{J}}_{k}$ and $c_{\emptyset k}$ are respectively the positions and the compliance constants of torsional springs. ${\hat{J}}_{k}$ is expressed in the normalized form $[\begin{matrix} y_{k} & - x_{k} & 1 \end{matrix}]^{T}$ of Plücker’s axis coordinates, which represents the rotation axis of the torsional spring positioned at $(x_{i}, y_{i})$ .

The stiffness matrix $K$ can be synthesized by finding the compliance matrix $C$ instead, which means that it is needed to determine six values for the positions of three rotation axes $(x_{i}, y_{i})$ of the torsional springs together with three torsional compliance constants $c_{\emptyset k}$ . Since C is symmetric, six equations can be obtained from three diagonal and three off-diagonal elements of C in equation (32). Therefore, there must be three free choices in synthesis. First, at least one rotation axis can be taken arbitrarily. We let ${\hat{J}}_{1}$ be the position of the torsional spring to be first selected. From equation (33), the compliance matrix $C_{2}$ is defined and determined as

C_{2} = C - c_{\emptyset 1} {\hat{J}}_{1} {\hat{J}}_{1}^{T} = c_{\emptyset 2} {\hat{J}}_{2} {\hat{J}}_{2}^{T} + c_{\emptyset 3} {\hat{J}}_{3} {\hat{J}}_{3}^{T}

(34)

Since the rank of $C_{2}$ is 2, the determinant of $C_{2}$ becomes 0

\begin{matrix} \det (C_{2}) = \det (C - c_{\emptyset 1} {\hat{J}}_{1} {\hat{J}}_{1}^{T}) \\ = \det (C) (1 - c_{\emptyset 1} {\hat{J}}_{1}^{T} C^{- 1} {\hat{J}}_{1}^{T}) = 0 \end{matrix}

(35)

Now, from equation (35), $c_{\emptyset k}$ can be found as

c_{\emptyset k} = 1 / {\hat{J}}_{k}^{T} C^{- 1} {\hat{J}}_{k} = 1 / {\hat{J}}_{k}^{T} K {\hat{J}}_{k}

(36)

If we consider two lines perpendicular to and passing through the xy -plane that represent two normalized bases of $C_{2}$ , then two pass-through points can be identified. Any point lying on the line joining two pass-through points can be selected for the position of the second torsional spring ${\hat{J}}_{2}$ . The corresponding second torsional compliance constant $c_{\emptyset 2}$ can also be determined from equation (36). Consequently, the rank 1 compliance matrix $C_{1}$ is obtained from

C_{1} = C_{2} - c_{\emptyset 2} {\hat{J}}_{2} {\hat{J}}_{2}^{T}

(37)

The position of the third torsional spring ${\hat{J}}_{3}$ is determined from the normalized form of a single basis of $C_{1}$ and again, the corresponding constant $c_{\emptyset 3}$ is computed using equation (36). Finally, the torsional stiffness constants $k_{\emptyset k}$ are determined from

k_{\emptyset k} = \frac{1}{c_{\emptyset k}}

(38)

Now, the torsional stiffness $k_{\emptyset k}$ at ${\hat{J}}_{k}$ can be realized by means of a hinge joint. The harvester can be designed in the form of a 3-DOF serial manipulator-type structure consisting of three hinge joints located at ${\hat{J}}_{k}^{'} s$ and a proof mass attached to its end as shown in Figure 6.

Design method of a leaf hinge joint

In this section, the lumped element model of a cantilever is first viewed as the serial manipulator connected by torsional springs and the design equations of the corresponding leaf joints are derived. If a cantilever has two different rectangular cross sections in series, the thinner one corresponds to a hinge part and the wider one to a rigid part. The orientation of the rigid part can be expressed by the pivot point and the angular displacement. In order to use the hinge joint as a torsional spring, the pivot point of the hinge should be located at ${\hat{J}}_{k}$ with the spring constant $k_{\emptyset k}$ . However, the pivot point can be easily moved according to the position of the applied force. Although the specific hinge types such as the elliptic and the circular hinges as shown in Figure 7 have the advantage of localizing the pivot point by concentrating the deflection of the hinge on the thinnest section, the design equations are complex and the fabrication is rather difficult (Smith, 2000; Tseytlin, 2002). A leaf hinge joint can be a useful solution to this problem if both the pivot point and the angular displacement can be designed properly. Referring to the analysis of a simple cantilever given in Selig and Ding (2001), a new design method of a leaf hinge joint is described in the following section.

Lumped element model of a cantilever

We consider a cantilever beam of the length L divided into n discrete elements of the same length. The origin of the reference coordinate system is placed at the fixed end of the beam as shown in Figure 8. The motion of the beam is assumed to remain in the xy -plane. The centerline of the beam is parallel to the x -axis. Referring to Figure 8, each of the beam elements is considered to be linked to the adjacent element by a torsional spring. It is necessary to establish the local coordinate systems whose origins are located at the linking points (see Figure 8). Thus, the cantilever beam can be assumed as the serial manipulator consisting of n-joints that can rotate about the axes perpendicular to the xy -plane passing through each of the origins of the local coordinate systems.

If the vertical force F is applied at the free end in the negative y -direction, the force can be expressed in terms of Plücker’s ray coordinates as $\hat{F} = F [\begin{matrix} 0 & - 1 & L \end{matrix}]^{T}$ , where $(0, - 1)$ and $L$ denote respectively the unit direction vector of the force in xy -plane and the shortest distance from the origin to the applied force. The force ${}^{i}{\hat{F}}$ caused by $\hat{F}$ at the ith element can be expressed in the ith local coordinate system as

{}^{i}{\hat{F}} = F {[\begin{matrix} 0 & - 1 & (L - x) \end{matrix}]}^{T}

(39)

Considering a cantilever beam as a serial manipulator connected by torsional springs, the position of the free end of the beam is determined by only the angular displacements of the torsional springs. In general, the angular displacement $δ \emptyset_{x}$ at the position x is given by $δ \emptyset_{x} = F (L - x) c_{\emptyset}$ . The rotational compliance constant of the beam $c_{\emptyset}$ can be written as $c_{\emptyset} = 1 / E I_{b}$ , where $E$ and $I_{b}$ are Young’s modulus $E$ and the second moment of cross-sectional area of the beam. Thus, the compliance matrix of the ith element ${}^{i}C$ can be expressed by

{}^{i}C = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{E I_{b}} \end{matrix}]

(40)

The displacement vector of the ith element ${}^{i}{\hat{T}}$ can be obtained as

{}^{i}{\hat{T}} = {}^{i}C {}^{i}{\hat{F}} = {[\begin{matrix} 0 & 0 & \frac{F}{E I_{b}} (L - x) \end{matrix}]}^{T}

(41)

This means that the ith element is rotated about the origin of the ith local coordinate system with the angular displacement $F / E I_{b} (L - x)$ . The transformation matrix $E_{i}$ can be used to transform a line vector expressed in the ith coordinate system into the one in the global coordinate system

E_{i} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - x \\ 0 & 0 & 1 \end{matrix}]

(42)

The displacement vector of the ith element can be expressed in the global coordinate system as

\begin{matrix} {\hat{T}}_{i} = E_{i} {}^{i}{\hat{T}} & = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - x \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 0 \\ 0 \\ \frac{F}{E I_{b}} (L - x) \end{matrix}] \\ = [\begin{matrix} 0 \\ - \frac{F}{E I_{b}} x (L - x) \\ \frac{F}{E I_{b}} (L - x) \end{matrix}] \end{matrix}

(43)

The total displacement of the free end of the beam $\hat{T}$ can be obtained by integrating the displacements of the elements ${\hat{T}}_{i}$ over the interval between the origin and the position x

\begin{matrix} \hat{T} = \int_{0}^{x} {\hat{T}}_{i} dx = [\begin{matrix} 0 \\ \frac{F}{E I_{b}} (\frac{1}{3} x^{3} - \frac{1}{2} x^{2} L) \\ \frac{F}{E I_{b}} (- \frac{1}{2} x^{2} + Lx) \end{matrix}] \\ = \frac{F}{E I_{b}} (- \frac{1}{2} x^{2} + Lx) [\begin{matrix} 0 \\ \frac{2 x^{2} - 3 xL}{- 3 x + 6 L} \\ 1 \end{matrix}] \end{matrix}

(44)

Equation (44) means that the free end of the beam is rotated about the pivot point $L_{c} (x, y)$ with the angular displacement $\emptyset$ which are given by

\emptyset = \frac{F}{E I_{b}} (- \frac{1}{2} x^{2} + Lx)

(45)

and

L_{c} (x, y) = (\frac{2 x^{2} - 3 xL}{3 x - 6 L}, 0)

(46)

Finally, the rotational stiffness constant $k_{\emptyset}$ can be written as

k_{\emptyset} = \frac{F L_{F}}{\emptyset}

(47)

where $L_{F} (= L - L_{c})$ denotes the distance between the applied force $F$ and the pivot point $L_{c}$ .

Design of a leaf hinge

The length of the cantilever $L$ is made up of two parts, the hinge part of $L_{H}$ and the rigid part of $ℓ$ . The origin of the reference coordinate system is located at the fixed end of the hinge part. The load condition is considered such that the force is applied at the free end of the rigid part in the negative y direction as shown in Figure 9.

If we define the ratio $α = L_{H} / ℓ$ , $ℓ$ may be written as

ℓ = α L_{H}

(48)

Using equation (48), the angular displacement $\emptyset$ in equation (45) and the rotation center $L_{c}$ in equation (46) can be rewritten as

\emptyset = \frac{F}{E I_{b}} (- \frac{1}{2} {L_{H}}^{2} + (L_{H} + ℓ) L_{H}) = \frac{F}{E I_{b}} (\frac{1}{2} + α) {L_{H}}^{2}

(49)

and

L_{c} = \frac{2 {L_{H}}^{2} - 3 L_{H} (L_{H} + ℓ)}{3 L_{H} - 6 (L_{H} + ℓ)} = \frac{1 + 3 α}{3 + 6 α} L_{H}

(50)

The distance between the applied force $F$ and the pivot point $L_{c}$ is given by

\begin{matrix} L_{F} = L - L_{C} = α L_{H} + L_{H} - \frac{1 + 3 α}{3 + 6 α} L_{H} \\ = \frac{2 + 6 α + 6 α^{2}}{3 + 6 α} L_{H} \end{matrix}

(51)

Thus, the torsional stiffness $k_{\emptyset}$ can be determined from equation (47)

k_{\emptyset} = \frac{F L_{F}}{\emptyset} = \frac{F \frac{2 + 6 α + 6 α^{2}}{3 + 6 α} L_{H}}{\frac{F}{E I_{b}} (\frac{1}{2} + α) {L_{H}}^{2}} = E I_{b} \frac{4 (1 + 3 α + 3 α^{2})}{3 {(1 + 2 α)}^{2} L_{H}}

(52)

Consequently, the moment of inertia of cross-sectional area of the hinge part $I_{b}$ can be found as

I_{b} = \frac{k_{\emptyset}}{E β}

(53)

where $β = 4 (1 + 3 α + 3 α^{2}) / 3 (1 + 2 α)^{2} L_{H}$ . Table 1 shows the design parameters of the leaf hinge in terms of $L_{H}$ .

Table 1.

Parameters of leaf hinge according to $ℓ$ in terms of $L_{H} s$ .

$ℓ$	$0$	$L_{H}$	$2 L_{H}$	$3 L_{H}$
$L$	$L_{H}$	$2 L_{H}$	$3 L_{H}$	$4 L_{H}$
$\emptyset$	$F {L_{H}}^{2} / 2 E I_{b}$	$3 F {L_{H}}^{2} / 2 E I_{b}$	$5 F {L_{H}}^{2} / 2 E I_{b}$	$7 F {L_{H}}^{2} / 2 E I_{b}$
$L_{c}$	$L_{H} / 3$	$4 L_{H} / 9$	$7 L_{H} / 15$	$10 L_{H} / 21$
$L_{F}$	$2 L_{H} / 3$	$14 L_{H} / 9$	$38 L_{H} / 15$	$74 L_{H} / 21$
$k_{\emptyset}$	$4 E I_{b} / 3 L_{H}$	$28 E I_{b} / 27 L_{H}$	$76 E I_{b} / 75 L_{H}$	$148 E I_{b} / 147 L_{H}$

In what follows, an algorithm describing the design procedure of a harvester which satisfies the specified ratio of the absorbed powers at three target resonant frequencies is presented:

Read $m$ and $I$ of a proof mass in equation (6) and target resonant frequencies $Ω_{1}, Ω_{2}$ , and $Ω_{3}$ ;

Specify the ratio of absorbed powers $P_{1} : P_{2} : P_{3} = 1 : γ_{1} : γ_{2}$ and damping ratios $ζ_{i}$ ;

Compute $a = Ω_{1}^{3} (Ω_{2}^{3} γ_{2} ζ_{3} + Ω_{3}^{3} γ_{1} ζ_{2}) / ζ_{1} Ω_{2}^{3} Ω_{3}^{3}$ and $b = γ_{1} Ω_{3}^{3} ζ_{2} / γ_{2} Ω_{2}^{3} ζ_{3}$ of equation (19);

Compute $x_{1} = - μ \sqrt{1 / a}, x_{2} = μ \sqrt{a}$ , $y_{2} = μ \sqrt{(1 + a) / b}$ , and $y_{3} = - μ \sqrt{(1 + a) b}$ of equation (12);

Obtain ${\hat{S}}_{1} = [\begin{matrix} 0 & - x_{1} & 1 \end{matrix}]^{T}$ , ${\hat{S}}_{2} = [\begin{matrix} y_{2} & - x_{2} & 1 \end{matrix}]^{T}$ , and ${\hat{S}}_{3} = [\begin{matrix} y_{3} & - x_{2} & 1 \end{matrix}]^{T}$ of equation (8);

Compute $K = MS [Ω^{2}] S^{- 1}$ from equation (21);

Compute $C = K^{- 1}$ ;

Select ${\hat{J}}_{1} = [\begin{matrix} y_{1} & - x_{1} & 1 \end{matrix}]$ arbitrarily;

Compute $c_{\emptyset 1} = 1 / {\hat{J}}_{1}^{T} K {\hat{J}}_{1}$ from equation (36);

Compute $C_{2} = C - c_{\emptyset 1} {\hat{J}}_{1} {\hat{J}}_{1}^{T}$ from equation (34);

Select ${\hat{J}}_{2} = [\begin{matrix} y_{2} & - x_{2} & 1 \end{matrix}]$ located on the line passing through two normalized bases of $C_{2}$ ;

Compute $c_{\emptyset 2} = 1 / {\hat{J}}_{2}^{T} K {\hat{J}}_{2}$ from equation (36);

Compute $C_{1} = C_{2} - c_{\emptyset 2} {\hat{J}}_{2} {\hat{J}}_{2}^{T}$ from equation (37);

Obtain ${\hat{J}}_{3} = [\begin{matrix} y_{2} & - x_{2} & 1 \end{matrix}]$ that span in eigenvector space of $C_{1}$ ;

Compute $c_{\emptyset 3} = 1 / {\hat{J}}_{3}^{T} K {\hat{J}}_{3}$ from equation (36);

Compute $k_{\emptyset k} = 1 / c_{\emptyset k}$ from equation (38);

Obtain $L_{Fk}$ from distance between origin and ${\hat{J}}_{k}$ ;

Compute $L_{ck}$ and $L_{Hk}$ from equations (50) and (51);

Compute $I_{bk}$ from equation (53);

Determine magnet array from equation (29).

Figure 10 illustrates the design process of the harvester.

Design of a prototype harvester

The design of a prototype harvester involves the following three steps: (1) the determination of three normal modes and the stiffness matrix $K$ , (2) the synthesis of the stiffness matrix $K$ , and (3) the determination of the parameters of leaf hinge joints.

Design of planar vibration system

In this design of a prototype harvester, the mass and the moment of inertia of the proof mass are first given as 50 g and 1.0e–5 kg m². From equation (6), the mass matrix becomes

M = [\begin{matrix} 0.050 kg & 0 & 0 \\ 0 & 0.050 kg & 0 \\ 0 & 0 & 1.0 e - 5 kg m^{2} \end{matrix}]

Three equally spaced target resonant frequencies are specified at 25, 30, and 35 Hz. The ratio of the absorbed powers is given as $P_{1} : P_{2} : P_{3} = 1 : γ_{1} : γ_{2} = 1 : 1 : 1$ , which means that $γ_{1} = γ_{2} = 1$ under the constant base acceleration $a_{cc}$ . From the relation of $a_{cc} = d_{i} Ω_{i}^{2} = constant$ , the powers absorbed by the dampers given by equation (18) can be rewritten as

\begin{matrix} P_{1} = \frac{m a_{cc}}{2 ζ_{1} (a + 1) Ω_{1}}, P_{2} = \frac{m a_{cc} ab}{2 ζ_{2} (1 + a) (1 + b) Ω_{2}}, \\ and P_{3} = \frac{m a_{cc} a}{2 ζ_{3} (1 + a) (1 + b) Ω_{3}} \end{matrix}

(54)

Two parameters $a$ and $b$ in equation (19) can be determined from

a = \frac{Ω_{2} Ω_{3} (γ_{2} ζ_{3} / Ω_{2} + γ_{1} ζ_{2} / Ω_{3})}{ζ_{1} Ω_{1}} and b = \frac{γ_{1} Ω_{2} ζ_{2}}{γ_{2} Ω_{3} ζ_{3}}

(55)

If all the same values are used for the damping ratio $ζ_{i}$ , the horizontal and the vertical ratios of the vertices of a modal triangle become

a = 2.6000 and b = 0.8571

The four coordinates $x_{1}$ , $x_{2}$ , $y_{2}$ , and $y_{3}$ of the vertices are obtained from equation (12) as

\begin{matrix} x_{1} = - 0.0088, x_{2} = 0.0228, y_{2} = 0.0290, and \\ y_{3} = - 0.0248 \end{matrix}

The normal modes ${\hat{S}}_{i}$ are now determined from equation (8)

\begin{matrix} {\hat{S}}_{1} = {[\begin{matrix} 0 & 0.0088 & 1.000 \end{matrix}]}^{T}, \\ {\hat{S}}_{2} = {[\begin{matrix} 0.0290 & - 0.0228 & 1.000 \end{matrix}]}^{T}, and \\ {\hat{S}}_{3} = {[\begin{matrix} - 0.0248 & - 0.0228 & 1.000 \end{matrix}]}^{T} \end{matrix}

Therefore, the stiffness matrix $K$ can be found using equation (21)

K = 10^{3} [\begin{matrix} 2.0726 N / m & 0.2718 N / m & - 0.0024 N \\ 0.2718 N / m & 1.8752 N / m & - 0.0056 N \\ - 0.0024 N & - 0.0056 N & 0.0003 N m \end{matrix}]

Figure 11 shows the plot of the absorbed power versus damping ratio for the design specification of $P_{1} : P_{2} : P_{3} = 1 : 1 : 1$ . The maximum absorbed powers of 10.0, 5.1, and 3.5 mW are calculated for the damping ratios 0.01, 0.02, and 0.03, respectively.

Synthesis of the stiffness matrix

The positions of the torsional springs ${\hat{J}}_{k}$ and the torsional stiffness constants $k_{\emptyset k}$ are obtained from equations (32) to (38)

\begin{matrix} {\hat{J}}_{1} = {[\begin{matrix} - 0.0154 & - 0.0088 & 1 \end{matrix}]}^{T}, \\ {\hat{J}}_{2} = {[\begin{matrix} 0.0152 & - 0.0067 & 1 \end{matrix}]}^{T}, and \\ {\hat{J}}_{3} = {[- \begin{matrix} 0.0020 & 0.0173 & 1 \end{matrix}]}^{T} \end{matrix}

k_{\emptyset 1} = 1.1788, k_{\emptyset 2} = 0.8065, and k_{\emptyset 3} = 0.6627 N m / rad

These positions of the torsional springs and the modal triangle are shown in Figure 12.

Design of leaf hinge joints

Equation (4) implies that the base excitation can be regarded as the external force acting on the center of the proof mass. Referring to Figure 12, the distances between the applied force and the locations of torsional springs $L_{Fk}$ are

L_{F 1} = 17.7 mm, L_{F 2} = 17.4 mm, and L_{F 3} = 16.6 mm

In this design, we chose the ratio of the hinge to rigid part to be 1.5, that is, $α = L_{H} / ℓ = 1.5$ ; the hinge lengths $L_{Hk}$ are obtained from equation (51) as

L_{H 1} = 8.7 mm, L_{H 2} = 8.5 mm, and L_{H 3} = 8.1 mm

From equation (50), the locations of hinge joints $L_{ck}$ are determined as

L_{C 1} = 4.0 mm, L_{C 2} = 3.9 mm, and L_{C 3} = 3.7 mm

Acrylonitrile-butadiene-styrene (ABS) plastic has been selected for the material of the device. Young’s modulus $E$ and the density of the ABS plastic are 2.3 GPa and $1040 kg / m^{3}$ , respectively. The rectangular section of the leaf hinge has the moment of inertia given by $I_{b} = t w^{3} / 12$ (see Figure 13). For any given width $w$ of the hinge joint, the thickness $t$ is determined by $t = 12 k_{\emptyset} / E β w^{3}$ from equation (53). The dimensions of leaf hinge joints are computed as shown in Table 2.

Table 2.

Dimensions of leaf hinge joints of the prototype.

Parameters	Dimensions	Units
Spring constant $k_{\emptyset 1}$	$1.1788$	N m/rad
Spring constant $k_{\emptyset 2}$	0.6627	N m/rad
Spring constant $k_{\emptyset 3}$	0.8065	N m/rad
Hinge1 ( $L_{H}, L_{C}, t, w$ )	(8.7, 4.0, 1.5, 16)	mm
Hinge2 ( $L_{H}, L_{C}, t, w$ )	(8.5, 3.9, 1.2, 16)	mm
Hinge3 ( $L_{H}, L_{C}, t, w$ )	(8.1, 3.7, 1.4, 12)	mm

The computed values of the parameters of the leaf hinge joints are confirmed by the finite element method (FEM) analysis using ANSYS software. The origin of the FEM model is located at the pivot point and the y-directional force of 1 N is applied to the end of the cantilever which is positioned at the distance of $L_{Fk}$ from the origin as shown in Figure 14. Poisson’s ratio of 0.35 is used as the material property. The result of the FEM strain analysis in Figure 14 shows the deformations of the regions adjacent to the leaf hinge which may affect the stiffness of the leaf hinge and cause any discrepancy. The vertical displacement at the end of the cantilever is measured and compared to the theoretical results. The errors of the displacements are 1.1%, 4.4%, and 6.1%, respectively, as shown in Table 3.

Table 3.

Comparison of the displacements at the end of the cantilever.

Parameters	Theoretical computation (mm)	FEM analysis (mm)	Error (%)
Hinge1	0.267	0.270	1.1
Hinge2	0.459	0.479	4.4
Hinge3	0.342	0.363	6.1

Design of prototype

The same ABS plastic as the leaf hinges is used for the main body of the prototype harvester. Including plastic connectors, the proof mass weighs 50 g and its moment of inertia is 1.0e–5 kg m². The proof mass is mounted on the free end of the serial manipulator-type harvester. The mass effect of the harvester on the resonant frequencies is neglected to obtain the simplified theoretical model. This makes it possible to regard the harvester as a proof mass supported by three serially connected torsional springs which are located at the hinge pivot points.

To produce voltage in the fixed coil, three pairs of round magnets are attached to the proof mass that moves across the fixed coil. A pair of concentric coils is connected serially and positioned on sides of the magnets as shown in Figure 15. The values of the electromagnetic design parameters are given in Table 4.

Table 4.

Design parameters of electromagnetic transducers.

Parameters	Values	Units
Dimensions of the prototype	$77.5 l \times 47.4 w \times 56.0 t$	mm
Average flux density $B$	1.1	T
Distance between coil and magnet	1	$mm$
Height of the magnet	15	$mm$
Diameter of the magnet	8	mm
Mass of the magnet (total)	5.6 (33.6)	g
Number of magnets	6	EA
Number of coils	2	EA
Turns of the coil $N$ (total)	1400 (2800)	turns
Inner radius of the coil	7	mm
Outer radius of the coil	15	mm
Thickness of the coil (total)	4 (8)	mm
Wire diameter of the coil	0.1	mm
Effective length of the coil $l$	6.3	mm

The arrangements of three pairs of round magnets have been found by performing the optimization process defined by equations (30) and (31) in which the sequential quadratic programming (SQP) algorithm in the MATLAB function Fmincon is used. The optimizations are carried out for four different cases of the magnet arrays $M_{arr} (M_{1}, M_{2}, M_{3})$ such as (S, S, S), (S, N, S), (S, S, N), and (S, N, N), where N and S denote the polarities. The magnetic flux densities of the magnets are measured as 1.1 T. The determined parameter values are shown in Table 5. When the magnet array is (S, N, N) and the angle of the equilateral triangle $φ$ becomes 90°, the uniform voltage of 2.57 V is obtained at all the resonant frequencies. Thus, the magnets are placed at $M_{1}$ (0, –11.0), $M_{2}$ (–9.53, 5.50), and $M_{3}$ (9.53, 5.50) mm.

Table 5.

Peak voltages according to array of magnets.

Freq. (Hz)	$ν_{i}$ (rad)	(S, S, S) (V)	(S, N, S) (V)	(S, S, N) (V)	(S, N, N) (V)
25	0.0481	1e–4	1.23	1.38	2.57
30	0.0154	1e–4	4.10	1.38	2.57
35	0.0132	1e–4	1.23	3.73	2.57
$φ$ (°)		90	91.48	92.44	90

Experimental setup

For the validation of the presented design method, an experimental prototype has been fabricated. The main body and the proof masses of the prototype have been produced by 3D printing using the Dimension SST1200ES 3D printer with ABSplus. Despite additive manufacturing of 3D printing with different printing angles, in view of vibration, the printed material is considered to be isotropic in elastic stage in which the harvester performs (Zou et al., 2016). The experimental setup consists of the prototype harvester, a function generator, an electrodynamic shaker (Brüel & Kjær type 4808), an accelerometer and a DAQ adapter (NI PXLE-based system), as shown in Figure 16. The function generator is used to control the frequency and to adjust the amplitude of the shaker together with the power amplifier (Brüel & Kjær type 2712). The accelerometer is mounted on the table of the shaker to measure the input acceleration. The output of the harvester is connected to load resistance and the output voltages across a load resistor are measured using the NI DAQ board. The shaker supplies a sinusoidal base excitation to the prototype in a frequency sweep test maintaining a constant acceleration.

Results and discussion

The resonant frequencies of the prototype are measured as 23.4, 29.2, and 34.8 Hz that are close to the target frequencies of 25, 30, and 35 Hz. Figure 17 shows how the optimal load resistance is determined through the experiments in which the load resistance is increased from 363 Ω to 1.34 kΩ at the second resonant frequency of 29.2 Hz. The optimal load resistance of 1.01 kΩ is found at and the corresponding maximum peak power of 0.39 mW is measured under the acceleration of 1 m/s².

Experiments are carried out in the frequency range from 15 to 45 Hz while the acceleration is kept constant at 1.5 m/s². The frequencies are spaced at intervals of 0.2 Hz. During the frequency sweep, the peak-to-peak voltages $V_{p - p}$ of 2.28, 1.90, and 2.31 V across the optimal load resistance of 1.01 kΩ are measured at the respective resonant frequencies of 23.4, 29.2, and 34.8 Hz as shown in Figure 18. The maximum peak-to-peak voltage is obtained at the third resonant frequency. The maximum discrepancy in resonant frequencies occurred at the first resonant frequency with 6.4% difference for the target value of 25 Hz. It is understood that the fabrication error is a major cause of the discrepancies in resonant frequencies and peak-to-peak voltages. Using the measured values of peak-to-peak voltages, the peak powers are obtained as 1.28, 0.89, and 1.32 mW at three resonant frequencies, as shown in Figure 19, from $P_{peak} = V_{p - p}^{2} / 4 R_{load}^{2}$ , where $R_{load}$ is the optimal load resistance. Utilizing the half-power bandwidth method, three electrical modal damping ratios $ζ_{ei}$ are found to be 0.009, 0.007, and 0.009. In general, the ith total modal damping ratio $ζ_{ti}$ is given by the sum of $ζ_{ei}$ and the mechanical to the electrical power converted from the absorbed power by the damper (see Figure 11) that is determined according to the total modal damping ratio $ζ_{ti}$ . However, it is difficult to estimate the absorbed power by the damper exactly by experiments. For this reason, the measurements of the power are compared with the analytically estimated absorbed power assuming that $ζ_{ti} = ζ_{ei}$ . Consequently, when $ζ_{ti} = 0.009$ , at least 11.5%, 8.0%, and 11.9% of the estimated absorbed power peak of 11.1 mW can be considered as the amount of the available powers at three resonant frequencies.

The experimental results are compared to the previously published ones and summarized in Table 6. It shows that three power peaks with the least difference can be obtained at equally spaced resonant frequencies. The output powers are compared in view of the normalized power density (NPD; Beeby et al., 2007), which can be obtained from

NPD = P_{peak} / (a_{cc}^{2} V_{vol})

(56)

Table 6.

Comparisons of frequency and power peak ratios at first three resonant frequencies in some published works.

References	Types	Number of resonant frequencies	Frequency ratios $((Ω_{3} / Ω_{2}) / (Ω_{2} / Ω_{1}))$	Power Peak ratios $(P_{\max} / P_{\min})$	NPD $(kg s m^{- 3})$
Bai et al. (2014)	ME	5	0.98	4.71	1.32
Castagnetti (2013)	PE	3	2.01	33	0.02
Dhote et al. (2015)	PE	3	1.16	–	–
El-Hebeary et al. (2013)	EM	3	1.10	4.10	2.00
Kim et al. (2017)	PE	3	0.90	1.40	0.07
Park et al. (2017)	EM	3	0.98	31.38	0.27
Qi et al. (2010)	PE	5	0.86	–	–
Seo et al. (2012)	PE	3	0.19	1.31	–
Xiong and Oyadiji (2014)	–	3	1.00	–	–
This article	EM	3	0.96	1.48	2.85

NPD: normalized power density; ME: magnetoelectric transducer; PE: piezoelectric transducer; EM: electromagnetic transducer.

where $V_{vol}$ is the volume of the energy harvester.

Conclusion

A novel design of a 3-DOF vibration energy harvester with electromagnetic transducers is presented. An emphasis in the design has been placed on the realization of the required stiffness by means of three serially connected leaf hinge joints. The design method is aimed at achieving minimum difference between the power peaks generated at three equally spaced target frequencies. For specified three target resonant frequencies, the power peaks absorbed by the damper are estimated and the corresponding normal modes are found to determine the required stiffness matrix. The stiffness matrix is synthesized by three leaf hinge joints. A new design method of a leaf hinge joint is developed to determine both specified pivot points and torsional stiffness constants. The optimal positions of the magnets of the electromagnetic transducer are found, thereby fabricating an experimental prototype. The experiments are conducted to verify the design method. Three resonant frequencies are measured at 23.4, 29.2, and 34.8 Hz comparing to target resonant frequencies of 25, 30, and 35 Hz. The maximum peak powers of 1.28, 0.89, and 1.32 mW are measured across the optimal load resistor of 1.01 kΩ at three resonant frequencies.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Hyun Soo Kim

Yong Je Choi

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