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Abstract

In this paper, a theoretical mathematical model in conjunction with an electrical generation model is examined. Using a simulated algorithm, the optimal design of a two-mass energy harvester that finds the maximal electrical power will be assessed. Before the optimal design is performed, the influence of the electrical power with respect to design parameters such as the magnet’s height, the diameter, the stiffness of the lower springs, the stiffness of the upper springs, the revolution of the lower coil, the revolution of the upper coil, the diameter of the coil’s wire, and the electrical resistance of the loading will be analyzed. Results reveal that the design parameters play essential roles in maximizing electrical power. The two mode shapes of the two-mass energy harvester also occur at the targeted forcing frequencies. The electrical power is optimally extracted at the two primary forcing frequencies, i.e. 12 and 30 Hz. Moreover, it is obvious that the induced electrical power of the two-mass energy harvester will be superior to that of the one-mass energy harvester.

Keywords

Two-mass vibration harvester

Introduction

Because portable devices are unusually popular, durable electrical power for these devices is a concern. However, there is the problem of electrical exhaust during prolonged use. So as to provide sufficient and continuous electrical power, interest in portable energy harvesters with a spring–mass system has surfaced. In 1997, Williams et al.,¹ doing research with small generators, invented a wireless sensing device that was triggered by vibrational energy using a sensor that was fixed onto a bridge to detect structural status. Now, there are many energy resources including mechanical energy, thermal energy, and potential energy. Conventional work in the vibration control is to eliminate the vibrational energy using various active vibration control methods and vibration absorbers.^2–5 As a view point of energy saving, it is an energy loss for the vibrational equipment system; therefore, along with these, there are numerous energy extraction methods such as the piezoelectric, electromagnetic, magnetostrictive, and electrostatic transducers.^6,7 The electrical application of the above electrical generation devices that do not need continuous amounts of electricity is still sufficient even though the electrical power created from the vibration-based generator is small. However, because of progress in the semiconductor field, MEMS (Micro-Electro-Mechanical Systems)⁸ along with various integrated circuit (IC) devices with low electrical power was developed. An example of the MEMS would be the application of vibration-based ICs used for the diagnosis of equipment using a vibrational signal via a remote monitoring system.⁹

In order to extract more energy from environmental vibrational energy, many energy harvesters have been proposed. Also, various piezoelectric energy harvesters with low electrical power have been created. However, they are fragile and therefore risky when used at lower frequencies to extract energy of higher vibrational amplitude. So, to extract energy from a lower frequency vibrational source, an electromagnetic energy harvester was presented. In previous studies, a portable one-mass vibration-based energy harvester used in extracting vibrational energy from equipment has been developed. Sapinski¹⁰ developed a generator for a linear magnetorheological damper using permanent magnets and coil with foil winding. However, electrical power with one-tone resonation has been limited and is insufficient.¹¹ Later, Chiu et al.¹² assessed three kinds of portable one-mass pendulum-arm energy harvesters using experimental tests. But, because only a resonating vibration will be induced for a one-mass vibrating system, the energy extracted from the one-mass vibrational system was limited to vibration with one forcing frequency. Therefore, in order to extract more vibrational energy from equipment having multitone forcing vibration, the development of a multimass energy harvester used to extract the vibrational energy at the multitone resonating frequencies is essential. There are many related works found in literatures.¹³

In this paper, in order to expand the electrical power of various frequencies (12 and 30 Hz), a portable vibration-based two-mass energy harvester is proposed. To maximize the extracted energy, two resonant frequencies of the two-mass vibrational system will be similarly tuned as two external vibrating frequencies emitted from the vibrational source. Eight kinds of design parameters—the magnet’s height (H_m), the diameter (D_m), the stiffness of the lower springs (k₁), the stiffness of the upper springs (k₂), the revolution of the lower coil (N₁), the revolution of the upper coil (N₂), the diameter of the coil’s wire (d_w), and the electrical resistance of the loading (R_load)—used in tuning the system’s natural frequencies are adopted. The optimization of a two-mass electromagnetic energy harvester is performed by using the simulated annealing (SA) method. Consequently, a comparison of the outputted electrical power between the one-mass energy harvester and the two-mass energy harvester will also be assessed and discussed.

Mathematical models

Motion of the two-mass vibration system

The two-dimensional vibration-based electromagnetic energy harvester is depicted in Figures 1 and 2. In case of a base excitation is inspired simultaneously by $y_{f_{1}} (t) = Y_{f_{1}} e^{i ω_{1} t}$ and $y_{f_{2}} (t) = Y_{f_{2}} e^{i ω_{2} t}$ , the related displacement responses with respect to magnet 1 and magnet 2 are assumed as

x_{1} (t) = X_{1 - f_{1}} (e^{i ω_{1} t}) + X_{1 - f_{2}} (e^{i ω_{2} t})

(1a)

x_{2} (t) = X_{2 - f_{1}} (e^{i ω_{1} t}) + X_{2 - f_{2}} (e^{i ω_{2} t})

(1b)

Figure 1.

Two-dimensional base excitation with two supporting springs and two coils. (a) A mechanism of two-mass energy harvester and (b) experimental equipment for a vibration-based electromagnetic energy harvester.

Figure 2.

Kinetic diagram for a two-mass base-excitation energy harvester.

The motion equation for a two-mass harvester using differential operation is

\begin{array}{l} m_{1} {\ddot{x}}_{1} = - (x_{1} - y) k_{1} - ({\dot{x}}_{1} - \dot{y}) c_{1} + (x_{2} - x_{1}) k_{2} + ({\dot{x}}_{2} - {\dot{x}}_{1}) c_{2} \\ m_{2} {\ddot{x}}_{2} = - (x_{2} - x_{1}) k_{2} - ({\dot{x}}_{2} - {\dot{x}}_{1}) c_{2} \end{array}

(2)

Plugging equation (1) into equation (2) and performing Laplace transform yields

{\begin{array}{l} \begin{array}{l} s^{2} m_{1} [X_{1 - f_{1}} + X_{1 - f_{2}}] = - (X_{1 - f_{1}} - Y_{f_{1}}) k_{1} - (X_{1 - f_{2}} - Y_{f_{2}}) k_{1} - (X_{1 - f_{1}} - Y_{f_{1}}) s c_{1} - (X_{1 - f_{2}} - Y_{f_{2}}) s c_{1} \\ + (X_{2 - f_{1}} - X_{1 - f_{1}}) k_{2} + (X_{2 - f_{2}} - X_{1 - f_{2}}) k_{2} + (X_{2 - f_{1}} - X_{1 - f_{1}}) s c_{2} + (X_{2 - f_{2}} - X_{1 - f_{2}}) s c_{2} \end{array} \\ \begin{array}{l} \times s^{2} m_{2} [X_{2 - f_{1}} + X_{2 - f_{2}}] = - (X_{2 - f_{1}} - X_{1 - f_{1}}) k_{2} - (X_{2 - f_{2}} - X_{1 - f_{2}}) \\ - (X_{2 - f_{1}} - X_{1 - f_{1}}) s c_{2} - (X_{2 - f_{2}} - X_{1 - f_{2}}) s c_{2} \end{array} \end{array}

(3)

The matrix form of equation (3) is

[\begin{array}{c} m_{1} s^{2} + (c_{1} + c_{2}) s + k_{1} + k_{2} & - (k_{2} + c_{2} s) \\ - (k_{2} + c_{2} s) & m_{2} s^{2} + k_{2} + c_{2} s \end{array}] [\begin{matrix} X_{1 - f_{1}} + X_{1 - f_{2}} \\ X_{2 - f_{1}} + X_{2 - f_{2}} \end{matrix}] = [\begin{array}{c} k_{1} + c_{1} s \\ 0 \end{array}] (Y_{f_{1}} + Y_{f_{2}})

(4)

Using the Cramer’s rule in solving $X_{1 - f_{1}, f_{2}}$ and $X_{2 - f_{1}, f_{2}}$ yields

\begin{array}{l} X_{1 - f_{1}, f_{2}} = Y_{f_{1}, f_{2}} \frac{(k_{1} + c_{1} s) (m_{2} s^{2} + c_{2} s + k_{2})}{{(k_{2} + c_{2} s)}^{2} - (m_{2} s^{2} + c_{2} s + k_{2}) [m_{1} s^{2} + (c_{1} + c_{2}) s + k_{1} + k_{2}]} \\ = A_{1 - f_{1}, f_{2}} + j B_{1 - f_{1}, f_{2}} \end{array}

(5a)

\begin{array}{l} X_{2 - f_{1}, f_{2}} = - Y_{f_{1}, f_{2}} \frac{(k_{1} + c_{1} s) (c_{2} s + k_{2})}{{(k_{2} + c_{2} s)}^{2} - (m_{2} s^{2} + c_{2} s + k_{2}) [m_{1} s^{2} + (c_{1} + c_{2}) s + k_{1} + k_{2}]} \\ = A_{2 - f_{1}, f_{2}} + j B_{2 - f_{1}, f_{2}} \end{array}

(5b)

The resulting solution of $x_{1} (t)$ yields

\begin{array}{l} x_{1 - f_{1}, f_{2}} (t) = \sum_{i = 1}^{2} X_{1 - f_{i}} \cdot e^{s_{i} t} = \sum_{i = 1}^{2} \sqrt{A_{1 - f_{i}}^{2} + B_{1 - f_{i}}^{2}} e^{j (ω_{i} t + α_{i})} \end{array}

(6a)

where

α_{i} = \tan^{- 1} (B_{1 - f_{i}} / A_{1 - f_{i}})

(6b)

Similarly, the resulting solution of $x_{2} (t)$ is

\begin{array}{l} x_{2 - f_{1}, f_{2}} (t) = \sum_{i = 1}^{2} X_{2 - f_{i}} \cdot e^{s_{i} t} = \sum_{i = 1}^{2} \sqrt{A_{2 - f_{i}}^{2} + B_{2 - f_{i}}^{2}} e^{j (ω_{i} t + β_{i})} \end{array}

(7a)

where

β_{i} = \tan^{- 1} (B_{2 - f_{i}} / A_{2 - f_{i}})

(7b)

The relative displacements of the lower magnet $z_{1} (t)$ and the upper magnet $z_{2} (t)$ are

\begin{array}{l} z_{1 - f_{1}, f_{2}} (t) = \sum_{i = 1}^{2} x_{1 - f_{i}} (t) - y_{f_{i}} (t) = \sum_{i = 1}^{2} [\sqrt{A_{1}^{2} + B_{1}^{2}} e^{j (ω_{i} t + α)} - Y_{f_{i}} (e^{ω_{i} t})] \end{array}

(8a)

\begin{array}{l} z_{2 - f_{1}, f_{2}} (t) = \sum_{i = 1}^{2} x_{2 - f_{i}} (t) - y_{f_{i}} (t) = \sum_{i = 1}^{2} [\sqrt{A_{2}^{2} + B_{2}^{2}} e^{j (ω_{i} t + β)} - Y_{f_{i}} (e^{ω_{i} t})] \end{array}

(8b)

In case of an excitation $y (t) = Y_{f_{1}} \sin ω_{1} t + Y_{f_{2}} \sin ω_{2} t$ , the relative displacement of the lower magnet and the upper magnet is

\begin{array}{l} z_{1} (t) = Im {z_{1 - f_{1}, f_{2}} (t)} = Im {\sum_{i = 1}^{2} [\sqrt{A_{1}^{2} + B_{1}^{2}} e^{j (ω_{i} t + α)} - Y_{f_{i}} (e^{ω_{i} t})]} \end{array}

(9a)

\begin{array}{l} z_{2} (t) = Im {z_{2 - f_{1}, f_{2}} (t)} = Im {\sum_{i = 1}^{2} [\sqrt{A_{2}^{2} + B_{2}^{2}} e^{j (ω_{i} t + β)} - Y_{f_{i}} (e^{ω_{i} t})]} \end{array}

(9b)

To simplify the simulation, the diameter and resistance of the coil’s wires around the upper magnet and the lower magnet are similarly designed. Therefore, for a single-layer coil, the relative damping coefficients (c₁ and c₂) are expressed as

\begin{array}{l} c_{1} = \frac{K_{1}^{2}}{R_{i 1} + R_{l o a d}} = \frac{{(N_{1}^{2} B d_{c})}^{2}}{ρ_{c} \frac{4 N_{1} D_{c}}{d_{c}^{2}} + R_{l o a d}} \end{array}

(10a)

\begin{array}{l} c_{1} = \frac{K_{2}^{2}}{R_{i 2} + R_{l o a d}} \\ = \frac{{(N_{2}^{2} B d_{c})}^{2}}{ρ_{c} \frac{4 N_{2} D_{c}}{d_{c}^{2}} + R_{l o a d}} \end{array}

(10b)

where

K_{1} = N_{1} B l_{c 1}, K_{2} = N_{2} B l_{c 2}, R_{i 1} = ρ_{c} \frac{l_{c 1}}{A_{c 1}}, and R_{i 2} = ρ_{c} \frac{l_{c 2}}{A_{c 2}}

(10c)

For n-layer coil, the related damping coefficient yields

c_{1} = \frac{{(n N_{1}^{2} B d_{c})}^{2}}{R_{i 1} + R_{l o a d}}

(11a)

c_{2} = \frac{{(n N_{2}^{2} B d_{c})}^{2}}{R_{i 2} + R_{l o a d}}

(11b)

where

R_{i 1} = \sum_{n = 1}^{L_{1}} ρ_{c} \frac{8 N_{1} [\frac{D_{c}}{2} + (n - 1) d_{c}]}{d_{c}^{2}}

(11c)

R_{i 2} = \sum_{n = 1}^{L_{2}} ρ_{c} \frac{8 N_{2} [\frac{D_{c}}{2} + (n - 1) d_{c}]}{d_{c}^{2}}

(11d)

The magnetic flux for a permanent magnet

For a cylindrical permanent magnet, it is assumed that the inner magnetization is uniform. The coupled currents method is then applied for calculating the magnetic field of the permanent magnet.¹⁴ The effective current distribution is the same as an ideal solenoid. Since the magnetic field is along the z-axis only, integration of the magnetic field along the z-axis is obtained and shown in equation (12a) using the Biot–Savart law. The distribution flux density through the center of cylindrical axis is expressed in equation (13a)¹⁵

\int d {\overset{⇀}{B}}_{z} = \int_{- \frac{H_{m}}{2}}^{\frac{H_{m}}{2}} \int_{0}^{2 π} \frac{μ_{0} I}{4 π r_{1}^{2}} (\frac{\sin θ_{1} - \sin θ_{2}}{r_{1}^{2} - r_{2}^{2}}) R d θ d z ({\hat{u}}_{ϕ} \times \hat{r})

(12a)

\begin{matrix} where R = \frac{D_{m}}{2}, r_{1} = \sqrt{{(z + \frac{H_{m}}{2})}^{2} + {(\frac{D_{m}}{2})}^{2}}, \\ r_{2} = \sqrt{{(z - \frac{H_{m}}{2})}^{2} + {(\frac{D_{m}}{2})}^{2}} \end{matrix}

(12b)

B_{z} (z) = B_{r} [\frac{z + \frac{D_{m}}{2}}{\sqrt{4 {(z + \frac{H_{m}}{2})}^{2} + D_{m}^{2}}} - \frac{z - \frac{D_{m}}{2}}{\sqrt{4 {(z - \frac{H_{m}}{2})}^{2} + D_{m}^{2}}} - C]

(13a)

where

{\begin{cases} C = 1 (\frac{- H_{m}}{2} < z < \frac{H_{m}}{2}) \\ C = 0 (\frac{- H_{m}}{2} > z, z > \frac{H_{m}}{2}) \end{cases}

(13b)

The electromagnetic energy output

According to Wang’s analysis¹⁶ and experimental work, a best allocation for the coil can be obtained. Results reveal that the magnetic voltage will increase if the layer of the coil increases. In order to simplify the analysis, only a one-layer solenoid/winding coil is considered. In addition, the magnet will be allocated at the center of the coil. According to Faraday’s law, the individual induced voltage with respect to each circular coil yields

ε = - \frac{d ϕ}{d t} = - A_{c} \frac{{d B}_{z}}{d z} \cdot \frac{d z}{d t}

(14)

For a single-layer coil, the cross area can be expressed as

A_{c} = π {(\frac{D_{c}}{2})}^{2}

(15)

For an n-layer coil, the related cross area of each layer yields

A_{a} = π {[\frac{D_{c}}{2} + (a - 1) d_{c}]}^{2}, {a \in N : a > 0}

(16)

In the case of a single-layer coil, the relative displacement $d_{n}$ between a moving magnet and each turn of a fixed coil with $N$ coil revolutions can be expressed as equation (17)

d_{n} = Z - d_{c} [(\frac{N + 1}{2}) - n]

(17)

where n is the index number of each revolution from the top to the bottom.

Subsequently, the induced voltage of each turn is

ε_{n} = - A_{c} B_{r} [\frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} + \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}} - \frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} - \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}}] \frac{d z}{d t}

(18)

Moreover, the total induced voltage yields

\begin{array}{l} V_{t o t a l} = \sum_{n = 1}^{N} ε_{n} \\ = \sum_{n = 1}^{N} - A_{c} B_{r} [\frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} + \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}} - \frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} - \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}}] \frac{d z}{d t} \end{array}

(19)

Likewise, in the case of an n-layer coil, M is the number of coil layers and N is the number of revolutions of each coil layer. The total induced voltage yields

\begin{array}{l} V_{t o t a l} = \sum_{a = 1}^{M} \sum_{n = 1}^{N} ε_{n} \\ = \sum_{a = 1}^{M} \sum_{n = 1}^{N} - A_{a} B_{r} [\frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} + \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}} - \frac{D_{m}^{2}}{\sqrt{{[4 {(d_{n} - \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{2}}}] \frac{d z}{d t} \end{array}

(20)

According to Faraday’s law, the individual induced voltage with respect to each circular coil yields

ε = - \frac{d ϕ}{d t} = - \frac{π D_{c}^{2}}{4} \frac{{d B}_{z}}{d z} \cdot \frac{d z}{d t}

(21a)

where

\frac{d B_{z} (z)}{d z} = B_{r} [\frac{D_{m}^{2}}{\sqrt{{[4 {(z + \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{3}}} - \frac{D_{m}^{2}}{\sqrt{{[4 {(z - \frac{H_{m}}{2})}^{2} + D_{m}^{2}]}^{3}}}]

(21b)

Consequently, the total voltage induced by a moving magnet will be obtained by summing up all the coil’s induced electrical voltages

V_{t o t a l} (D_{m}, H_{m}, k_{1}, k_{2}, N_{1}, N_{2}, d_{c}, R_{l o a d}) = \sum_{n = 0}^{N_{c} - 1} ε_{n}

(22)

Objective function

The induced system’s electrical voltage will be maximized if the system’s two natural frequencies ( $ω_{d 1}$ and $ω_{d 2}$ ) are the same as the two external inputted frequencies ( $ω_{e x 1}$ and $ω_{e x 2}$ ). Here, $ω_{e x 1}$ is smaller than $ω_{e x 2}$ . In order to reach the targeted natural frequencies, an objective function of the summation of the frequency’s squared deviation is established

\begin{array}{l} O B J (D_{m}, H_{m}, k_{1}, k_{2}, N_{1}, N_{2}, d_{c}, R_{l o a d}) \\ = {(ω_{e x 1} - ω_{d 1})}^{2} + {(ω_{e x 2} - ω_{d 2})}^{2} \end{array}

(23)

The optimal design data where the system’s two natural frequencies are very close to the two targeted frequencies ( $ω_{e x 1}$ and $ω_{e x 2}$ ) will be obtained by adjusting eight parameters (D_m, H_m, k₁, k₂, N₁, N₂, d_c, R_load) during the optimization process.

Model check

To verify the accuracy of the mathematical model and experimental data, the experimental work of a two-mass vibration-based electromagnetic energy harvester in conjunction with a vibrating source shown in Figure 1 has been established. As indicated in Figure 1, two NdFeB-made cylindrical permanent magnets serve as a vibrator. The vibrator that has a 12 mm diameter and is 16 mm in height is compressed by two springs (Stainless A313) at both ends. The selected spring is about 71 mm in l_s (free length of the spring), 0.5 mm in d_s (diameter of the spring’s wire), and 11.57 mm in D_s. Before the theoretical formula is calculated, some physical conditions will be preset and experimentally measured. The physical conditions are given and shown in Table 1. As indicated in Figure 1, to obtain the spring’s stiffness and damping ratio, experimental work using the logarithmic decrement method in conjunction with a piezoelectric pressure sensor is carried out. The damping coefficient can be obtained by using an experimental measurement without the coil. The damping ratio relationships are shown as below

c_{s y s} = 2 ς_{s y s} \sqrt{k m} = c_{m} + c_{e} = 2 ς_{m} \sqrt{k m} + c_{e}

(24a)

c_{e} = 2 ς_{s y s} \sqrt{k m} - 2 ς_{m} \sqrt{k m} = 2 \sqrt{k m} (ς_{s y s} - ς_{m})

(24b)

where damping ratio

(ς_{m})

for a mechanical motion is 0.01184.

Table 1.

Given data of the vibrating system and the two-mass vibration-based electromagnetic energy harvester.

Item	Properties	Unit
Upper mass	$m_{1} = 16.02$	g
Lower mass	$m_{2} = 14.91$	g
Permanent magnet	Diameter $D_{m} = 12$	mm
Permanent magnet	Height $H_{m} = 16$	mm
Coil	Number of layers $L_{n} = 4$ Turns of each layers $N = 34$ Height of each layer $h_{c} = 73$	layersturnsmm
Supporting spring	k = 88.2528	N/m
Loading resistor	$R_{l o a d} = 200$	Ω
Excited amplitude	$Y = 500$	µ $m$

The logarithmic, δ, can be calculated using the logarithmic decrement method

δ = \frac{1}{n} \ln (\frac{V_{o}}{V_{n}})

(25)

where

V_{o}

is the first peak amplitudes and

V_{n}

is the nth period peak amplitude. Therefore, the related damping ratio is

ς_{s y s} = \frac{1}{\sqrt{1 + {(\frac{2 π}{δ})}^{2}}}

(26)

Using the above data in the theoretical calculation, the profile of the theoretically induced electrical voltage and the experimental voltage with respect to frequency is obtained and shown in Figure 3. The lower spring stiffness will decrease and the upper spring stiffness will increase due to the magnetic effect on the springs. In addition, because the center axis of the helical spring, the center axis of cylindrical magnet, and the center axis of the vibration source may not be set up on the same line, some vibrational energy might be dissipated in the uniaxial direction. Therefore, the practical energy detected in the experimental work will be smaller than that of the theoretical prediction. As depicted in Figure 3, the performance curve of the theoretical and experimental data is in agreement. Moreover, the maximum electromagnetic voltage is tuned to the desired frequencies of 7 and 18.4 Hz. Therefore, the mathematical model is acceptable. Consequently, the model linked by the SA numerical method is used for optimizing the unit-induced electrical voltage of a two-mass vibration-based electromagnetic energy harvester in the following section.

Figure 3.

Performance of a two-mass vibration-based electromagnetic energy harvester when comparing the theoretical simulation to the experimental work. (a) The response of the lower magnet and (b) the response of the upper magnet.

Case study

There are two kinds of base vibrating sources (excited frequencies of 12 and 30 Hz) with the same displacement amplitude of 0.1 mm occurring in the equipment. To optimally extract these vibrations, a two-mass electromagnetic energy harvester is adopted. To simplify the simulation, the dimensions of two magnets and the diameter of two coal wires are similarly designed.

Eight kinds of design parameters—the magnet’s height (H_m), the diameter (D_m), the stiffness of the lower springs (k₁), the stiffness of the upper springs (k₂), the revolution of the lower coil (N₁), the revolution of the upper coil (N₂), the diameter of the coil’s wire (d_w), and the electrical resistance of the loading (R_load)—used in tuning the system’s natural frequencies are adopted. The optimization of a two-mass electromagnetic energy harvester is performed using the SA method described in the following sections. The related ranges of the design parameters are shown in Table 2. Moreover, the data for the vibrating system and the vibration-based electromagnetic energy harvester are shown in Table 3.

Table 2.

The ranges of the design parameters.

Optimization parameter defined	Optimized range	Unit
$D_{m}$	0.01–0.03	m
$H_{m}$	0.01–0.03	m
$N_{1}$	$⌊ \frac{H_{m}}{d_{c}} ⌋ \times 0.2 \sim ⌊ \frac{H_{m}}{d_{c}} ⌋ \times 2$
$N_{2}$	$⌊ \frac{H_{m}}{d_{c}} ⌋ \times 0.2 \sim ⌊ \frac{H_{m}}{d_{c}} ⌋ \times 2$
$k_{1}$	$0.5 ω_{e x 1} - 2 ω_{e x 2}$	N/m
$k_{2}$	$0.5 ω_{e x 1} - 2 ω_{e x 2}$	N/m
$d_{c}$	0.0001–0.0001	m
$R_{l o a d}$	100–10,000	Ω

Table 3.

The related physical property of the magnet and the coil.

Neodymium permanent magnet
Symbol	Description
$ρ_{m}$	Neodymium magnet density	$7.4 \times 10^{3}$	${kg/m}^{3}$
$B_{r}$	Residual magnetic flux density	1.2	T
Wounding coil

Optimization process

The OBJ function is linked to the SA method. The concept behind SA was first introduced by Metropolis et al.¹⁷ and later developed by Kirkpatrick et al.¹⁸ The optimization flow diagram using the SA method is shown in Figure 4.

Figure 4.

Flow diagram of a SA optimization.

For the SA optimization process, a new random solution (X′) will be chosen from the neighborhood of the current solution (X). If the change in the objective function is negative (i.e. ΔF ≤ 0), a new solution will be acknowledged as the new current solution with the transition property pb(X′) of 1; if it is not negative (i.e. ΔF > 0), then a new transition property (pb(X′)) varying from 0 to 1 will be calculated using the Boltzmann factor (pb(X′) =exp(ΔF/CT)) as shown in equation (27)

p b (X') = {\begin{matrix} 1, Δ F \leq 0 \\ \exp (\frac{- Δ F}{C T}), Δ F > 0 \end{matrix}

(27a)

Δ F = O B J (X_{n}^{'}) - O B J (X_{n})

(27b)

where C and T are the Boltzmann constant and the current temperature. Simultaneously, a new random probability of rand(0, 1) will be given to compare with the above new transition property (pb(X′)). If the transition property (pb(X′)) is greater than the random number of rand(0, 1), the new uphill solution, which results in a higher energy condition, will then be accepted; if not, it will be rejected. Here, the uphill solution with an inferior solution will then have a better possibility of escaping from the local optimum. The algorithm reiterates the perturbation of the current solution and the measurement of change in the objective function. Each successful swap of the new current solution will point to the decay of the current temperature as

T_{n e w} = k k \cdot T_{o l d}

(28)

where kk is the cooling rate. The process is reiterated until the predetermined number (iter_max) of the outer loop is reached.

Results and discussion

For a base-vibrating system excited by two frequencies ( $f_{d 1}$ = 12 Hz, $f_{d 2}$ = 30 Hz), the optimal results by varying the SA’s control parameters (kk, iter_max) are shown in Table 4. As indicated in Table 4, the optimal design data can be obtained when the SA parameters at (kk, iter_max) = (0.96, 10,000) are applied. As indicated in Table 4, the optimal design data occurring at the 20th set are ( $D_{m} = 0.0 22$ , $H_{m} = 0.0 247$ , $d_{c} = 0.000 9$ , $k_{1} = 1054 .00 2$ , $k_{2} = 897.155$ , $N_{1} = 28$ , $N_{2} = 13$ , $n_{1} = 30$ , $n_{2} = 3$ , $R_{l o a d} = 5909$ ). The related system’s natural frequencies of f_ex1 and f_ex2 are 12.048 and 30 Hz, respectively. The designed tones (f_ex1 and f_ex2) for two magnets at x₁ and x₂ are plotted in Figure 5. The response of the displacements of two magnets with respect to time at the external excitation frequency of 12 Hz is shown in Figure 6. Additionally, the response of the displacements of two magnets with respect to time at the external excitation frequency of 30 Hz is shown in Figure 7. Inputting the design data into the electrical power’s calculation, the resulting electrical power of the magnets with respect to time is plotted in Figures 8 and 9.

Table 4.

Optimal OBJ for the vibration-based electromagnetic energy harvester at various kk and iter_max (at targeted tones of 12 and 30 Hz with an amplitude of 0.01 m).

SA controlparameter		Design parameter										Result
kk	iter _max	D _m	H _m	d _c	k ₁	k ₂	N ₁	N ₂	n ₁	n ₂	R_load	OBJ
		m	m	m	N/m	N/m					Ω
100	0.9	0.023	0.0233	0.0004	672.749	1066.768	38	18	35	25	8028	54.964
100	0.91	0.0294	0.0105	0.0007	873.416	673.695	5	23	3	2	3458	96.435
100	0.92	0.0135	0.0261	0.0009	1150.708	1099.709	45	23	26	16	9707	10.664
100	0.93	0.0229	0.0167	0.0006	685.782	691.118	41	24	23	7	4235	11.186
100	0.94	0.0269	0.0153	0.0009	709.65	943.682	8	4	47	11	5817	49.227
100	0.95	0.0244	0.0151	0.0008	728.773	730.911	17	28	39	49	791	69.609
100	0.96	0.0277	0.0128	0.0003	843.845	745.468	23	35	17	23	2070	10.017
100	0.97	0.0136	0.0231	0.0005	1114.461	843.545	44	57	33	27	1452	41.089
100	0.98	0.0241	0.0165	0.0004	1066.093	616.93	47	59	44	12	6749	33.668
100	0.99	0.022	0.0247	0.0009	1054.002	897.155	28	13	30	3	5909	12.185
1000	0.96	0.023	0.0233	0.0004	672.749	1066.768	38	18	35	2	8028	7.885
1000	0.96	0.0294	0.0105	0.0007	873.416	673.695	5	23	3	2	3458	1.659
2000	0.96	0.0135	0.0261	0.0009	1150.708	1099.709	45	23	26	16	9707	0.098
3000	0.96	0.0258	0.0155	0.0002	635.718	879.902	97	54	29	8	8038	0.638
4000	0.96	0.0269	0.0153	0.0009	709.65	943.682	8	4	47	11	5817	0.154
5000	0.96	0.0244	0.0151	0.0008	728.773	730.911	17	28	39	49	791	0.399
6000	0.96	0.0277	0.0128	0.0003	843.845	745.468	23	35	17	23	2070	0.767
7000	0.96	0.0136	0.0231	0.0005	1114.461	843.545	44	57	33	27	1452	0.458
8000	0.96	0.0241	0.0165	0.0004	1066.093	616.93	47	59	44	12	6749	0.069
9000	0.96	0.022	0.0247	0.0009	1054.002	897.155	28	13	30	3	5909	0.029

OBJ: objective function; SA: simulated annealing.

Figure 5.

Comparison of the frequency response between the target tones ( $f_{d 1}$ and $f_{d 2}$ ) and the designed tones (f_ex1 and f_ex2).

Figure 6.

The displacements of two magnets with respect to time at the external excitation frequency of 12 Hz.

Figure 7.

The displacements of two magnets with respect to time at the external excitation frequency of 30 Hz.

Figure 8.

The induced voltage of two coils with respect to time at the external excitation frequency of 12 Hz during one period of time.

Figure 9.

The induced voltage of two coils with respect to time at the external excitation frequency of 30 Hz during one period of time.

Moreover, considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two-mass energy harvester, the electrical power of a two-magnet harvester with respect to time is shown in Figure 10.

Figure 10.

The induced voltage with respect to time of two-mass vibration system excited by two frequencies of 12 and 30 Hz.

As indicated in Table 4, the optimal design data can be obtained when the iteration number (iter_max) increases and the SA’s cooling rate (kk) is 0.96. In addition, Figure 5 indicates that the system’s optimal natural frequencies (f_ex1 and f_ex2) are very close to the external forcing frequencies ( $f_{d 1}$ =12 Hz, $f_{d 2}$ =30 Hz). Obviously, the system’s motion will resonate at the external forcing frequencies ( $f_{d 1}$ , $f_{d 2}$ ) of 12 and 30 Hz. The induced resonant motion will result in a large reciprocating motion between the magnets and the related coils. With this, the induced electrical power will be maximized. As indicated in Figure 6, the amplitude of the upper magnet (x₂) is larger than that of the lower magnet (x₁) when the external forcing frequency is at 12 Hz. The phase angle difference between x₁ and x₂ is 0°. Moreover, as indicated Figure 7, the amplitude of the lower magnet (x₁) is larger than that of the upper magnet (x₂) when the external forcing frequency is at 30 Hz. The phase angle difference between x₁ and x₂ is 180°. Obviously, the phase angle difference of 0° shown in Figure 6 represents the first mode shape that occurs during the reciprocating motion of the two-mass vibrational system. Additionally, the phase angle difference of 180° shown in Figure 7 represents the second mode shape that occurs during the reciprocating motion of the two-mass vibrational system. For a two-mass vibrational system, the physical phenomena of two mode shapes during the resonant motion are thus verified.

As can be seen in Figures 8 and 9, the immediate electrical power for two magnets varies periodically when two kinds of external base excitation forcing frequencies (12 and 30 Hz) are added individually. Considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two-mass energy harvester, the electrical power of two magnets with respect to time is shown in Figure 10.

To appreciate the electrical efficiency of one-mass and two-mass harvesters in extracting the vibrational energy from a piece of vibrating equipment with two excitation frequencies of 12 and 30 Hz, the electrical power of a one-magnet harvester with respect to time is optimally assessed and presented in Table 5 and Figure 11. As indicated in Figures 10 and 11, it is obvious that the electrical efficiency of the two-mass harvester is superior to that of the one-mass harvester.

Table 5.

Optimal OBJ for the one-mass vibration-based electromagnetic energy harvester (at targeted tones of 12 and 30 Hz with an amplitude of 0.01 m).

m₁ (kg)	d_c (m)	k₁ (N/m)	c₁ (N m/s)	N₁ (turns)	L₁ (layers)
0.0134	0.0002	76.319	0.10073	18	11

OBJ: objective function.

Figure 11.

The induced voltage with respect to time of one-mass vibration system excited by two frequencies of 12 and 30 Hz.

To realize the influence of parameters (D_m, H_m, k₁, k₂, d_c, N₁, N₂, R_load) with respect to the induced voltage has been assessed and shown in Figure 12. As indicated in Figure 12(a) and (b), there are two peaks of voltage occurred at two specified values of D_m (the magnet’s diameter) and H_m (the magnet’s height) due to a two-mode resonating system. And, as illustrated in Figure 12(c) and (d), a peak of voltage occurred at a specified k₁ and k₂. Also, result in Figure 12(e) shows that the induced electrical voltage (V) will increase if d_c (coil’s wire diameter) decreases. As illustrated in Figure 12(f) and (g), the peak of voltage will occur at a compromised number of coil turns (N₁ and N₂). It is because that the retarding force to the magnetic motion will increase when number of coil turns (N₁ and N₂). Therefore, to reach maximal electrical voltage, the coil turns must be compromised. Consequently, as indicated in Figure 12(h), the electrical voltage will remain the same when R_load (the electrical resistance) is beyond a specified value.

Figure 12.

The induced voltage with respect to various design parameters (D_m, H_m, k₁, k₁, d_c, N₁, N₂, R_load). (a) The induced voltage with respect to D_m while other parameters are fixed, (b) the induced voltage with respect to H_m while other parameters are fixed, (c) the induced voltage with respect to k₁ while other parameters are fixed, (d) the induced voltage with respect to k₂ while other parameters are fixed, (e) the induced voltage with respect to d_c while other parameters are fixed, (f) the induced voltage with respect to N₁ while other parameters are fixed, (g) the induced voltage with respect to N₂ while other parameters are fixed, and (h) the induced voltage with respect to R_load while other parameters are fixed.

Conclusion

It has been shown that SA can be used in the optimization of a two-mass vibration-based electromagnetic energy harvester. The SA parameters of the kk (cooling rate) and the iter_max (maximum iteration number) are essential during the SA optimization. The higher iter_max will result in a better solution.

Concerning the geometric allocation, eight parameters (H_m: the magnet’s height; D_m: the magnet’s diameter; k₁: the stiffness of the lower springs; k₂: the stiffness of the upper springs; N₁: the revolution of the lower coil; N₂: the revolution of the upper coil; d_w: the diameter of the coil’s wire; R_load: the electrical resistance of the loading) which are tightly related to the induced electrical voltage are adopted for maximization of electrical power. To increase the electrical power extracted from vibrating equipment with two primary forcing frequencies ( $f_{d 1}$ =12 Hz, $f_{d 2}$ =30 Hz), a two-mass energy harvester induced to resonate via two natural frequencies (f_ex1 and f_ex2) is built and optimized using an SA optimizer. The resulting motions of two magnets at two individual forcing frequencies are obtained and shown in Figures 6 and 7. The figures indicate that the two mode shapes of the two-mass energy harvester occur at the targeted forcing frequencies ( $f_{d 1}$ =12 Hz, $f_{d 2}$ =30 Hz). The frequency response in Figure 5 also indicates that the two-mass energy harvester is optimally tuned at the two primary forcing frequencies of the vibrating equipment. Considering two external excitation frequencies of 12 and 30 Hz simultaneously acting on the two-mass energy harvester, the electrical power of two magnets with respect to time is shown in Figure 10. Consequently, as indicated in Figures 10 and 11, the electrical efficiency of the two-mass harvesters is superior to that of the one-mass harvester.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support of the National Science Council (NSC 100-2221-E-036-019, ROC).

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Optimization of a two-mass vibration-based electromagnetic energy harvester using simulated annealing method

Abstract

Keywords

Introduction

Mathematical models

Motion of the two-mass vibration system

The magnetic flux for a permanent magnet

The electromagnetic energy output

Objective function

Model check

Case study

Optimization process

Results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Appendix

References