Variable-Length Pendulum-Based Mechatronic Systems for Energy Harvesting: A Review of Dynamic Models

Abstract

The ability to power low-power devices and sensors has drawn a great deal of interest to energy harvesting from ambient vibrations. The application of variable-length pendulum systems in conjunction with piezoelectric or electromagnetic energy-harvesting devices is examined in this thorough analysis. Because of their changeable length, such pendulums may effectively convert mechanical vibrations into electrical energy. This study covers these energy-harvesting systems’ basic theories, design concerns, modeling methods, and performance optimization strategies. This article reviews several studies that look at dynamic models, the effects of damping coefficients, device designs, and excitation parameters on energy output. The advantages and disadvantages of piezoelectric and electromagnetic coupling techniques are demonstrated by comparative research. This review also looks at technical advances and future research prospects in variable-length, pendulum-based energy harvesting. An expanded model for an energy harvester based on a variable-length pendulum derived from the modified, swinging Atwood machine is more specifically presented. This model’s numerical simulations, estimated current and voltage outputs, and produced power from the electromagnetic and piezoelectric devices integrated at various points in a 4-DOF variable-length pendulum model all indicate encouraging results. This necessitates extra study, changes, and optimizations to improve the usefulness of the proposed model. Finally, important dynamic models on developing variable-length, pendulum-based energy harvesters for usage in a range of applications to create sustainable energy are summarized.

1. Introduction

Variable-length pendulums refer to those whose length can be adjusted either manually or automatically [1]. These pendulums find applications across diverse fields, serving as energy harvesters (e.g., in wave energy converters), load-lifting mechanisms (used in cranes), components in robotics, timekeeping devices, experimental tools, and even attractions in amusement parks [2]. Due to their dynamic or manual length adjustment capability, variable-length pendulums have proven versatile in scientific research, education, and practical contexts [3]. Let us enumerate the main types of these physical concepts below.

1.1. Manually Adjustable Pendulum

A manually adjustable pendulum allows for the manual modification of the pendulum rod’s length. Typically, it involves a mechanism—such as a clamp or screw—that can be loosened to alter the length and then tightened to secure the new length in place. These pendulums find common use in educational settings, where they serve to demonstrate the fundamental principles of pendulum motion and the relationship between length and period [1].

1.2. Automated Adjustable Pendulum

Automated adjustable pendulums use mechanical or electronic systems to change the pendulum’s length dynamically. A computer or a feedback mechanism can control these systems to adjust the pendulum length in real time [1]. This pendulum type is used in advanced physics experiments and systems that require the precise control of pendulum dynamics.

1.3. Tuning-Fork Pendulum

In certain horological applications, tuning fork pendulums come into play. These pendulums utilize a tuning fork—a slender, fork-shaped metal component—as their oscillating element. The effective length of the pendulum can be adjusted by manipulating the position of the tuning fork’s weights [4]. This fine-tuning capability allows for precise control over the pendulum’s period, thus contributing to accurate timekeeping.

1.4. Spring-Mounted Pendulum

A spring-mounted pendulum employs a spring mechanism to vary its effective length. By compressing or extending the spring, the pendulum’s length changes, affecting its period. Ancient clocks and experimental setups often utilize this type of pendulum.

1.5. Elastic Cord Pendulum

Unlike a rigid rod, this pendulum employs an elastic cord. The gravitational and Hooke’s potentials drive its oscillations [5,6]. As the cord stretches and contracts, the pendulum’s length varies. This type of pendulum is commonly used in instructional demonstrations and can exhibit intricate oscillatory behavior due to the elasticity of the cord.

In this study, we emphasize the utilization of variable-length pendulum dynamic models in conjunction with piezoelectric or electromagnetic devices for energy harvesting. Our aim is to explore the efficiency and effectiveness of these innovative applications in energy-harvesting systems.

According to a review article [2], the variable-length pendulum system is recognized as highly demanding due to its complex modeling and analysis requirements. However, the authors suggested that a solid understanding of constant-length pendulums can serve as a foundation for creating a more accurate mathematical description of physical processes. Additionally, the review highlighted that numerical investigations have confirmed the feasibility of energy harvesting, particularly noting the improved efficiency associated with shorter, reduced pendulum lengths. This conclusion is supported by both numerical simulations and experimental results [2,7,8].

The exploration of energy harvesting from variable-length pendulums presents an intriguing field of renewable energy research, attracting the interest of mechanical engineers. This relatively novel approach focuses on the kinetic energy generated by the oscillatory motion of pendulums with adjustable lengths. The concept is promising for various applications, from powering low-energy devices to contributing to larger energy grids [9,10,11]. This review examines variable-length pendulums from the perspective of mechanics and mechanisms, as well as their potential applications in energy harvesting.

Using experimental setups and mathematical models, scientists have extensively studied the variable-length pendulum in the field of energy harvesting, aiming to harness energy from the pendulum’s vibrations [12,13,14,15,16,17,18]. Numerical analyses have demonstrated the feasibility of energy harvesting, with increased efficiency observed for shorter, reduced lengths of the pendulum: a finding that is supported by both numerical and experimental investigations [17,19]. By adjusting its length in response to the oscillations it undergoes [20,21,22,23], the pendulum experiences variations in length as it swings, causing a corresponding shift in its frequency [24,25]. This change in frequency can then be utilized to generate electrical energy [26,27].

The variable-length pendulum can be used as a tool for harvesting energy from the surrounding environment [13,17,18]. The basic idea of energy harvesting using a variable-length pendulum is to transform the kinetic energy generated by the oscillations of the pendulum into electrical energy [14,28,29,30]. At the pendulum’s end, the mechanical energy is transformed into electrical power via a generating mechanism [31,32]. These mechanisms include the utilization of piezoelectric materials [33,34,35], electromagnetic induction [36,37], or mechanical-to-electrical transducers [38,39,40,41]. The energy that is produced can subsequently be used to power electronics or could be stored in a battery.

1.6. Piezoelectric Materials

By attaching piezoelectric materials to the pendulum or its pivot point, the mechanical stress from oscillation can be converted into electrical voltage [42]. Piezoelectric materials generate voltage when subjected to the mechanical strain induced by the motion of a pendulum [9,10,11,43,44,45,46].

1.7. Electromagnetic Induction

By placing a coil of wire near a moving magnet on the pendulum, an electric current can be induced in the coil as the magnet moves back and forth [13,47]. This principle is similar to the operation of a generator.

Mechanical-to-electrical transducers, such as capacitive or inductive transducers, convert the mechanical displacement of the pendulum into electric energy [12,48,49].

Energy harvesting techniques from variable-length pendulums offer sustainable and environmentally friendly alternatives to conventional power sources. They have diverse applications, ranging from powering small electronic devices like wearable electronics, wireless sensor networks, and condition monitoring systems [36,43,47,50]. Furthermore, energy harvesting can effectively supply power to remote sensors across various domains, including industrial, commercial, and medical applications.

The effectiveness of the variable-length pendulum relies on several key factors, such as the length of the pendulum, oscillation frequency, and the design of the generator [51]. Optimization of these factors enhances the efficiency of the energy conversion process [28,46,52,53].

2. Piezoelectric Device Energy Harvesting from a Variable-Length Pendulum

Piezoelectric devices attached to a variable-length pendulum can generate voltage by transforming mechanical energy into electric energy [13,17]. Piezoelectric materials possess the ability to produce an electric charge when subjected to mechanical stress [36,42,54]. When the variable-length pendulum swings, mechanical stress is applied to the piezoelectric material. As the pendulum oscillates back and forth due to gravitational force and external excitation, it exerts varying levels of mechanical stress on the attached piezoelectric devices [55,56]. This stress deforms the piezoelectric material, causing the generation of electric charge [34,44].

The electric charge produced by the piezoelectric material leads to the development of a potential difference across the material, resulting in the generation of voltage [54,57,58,59]. This voltage can be utilized in energy harvesting and also to power electronic devices or charge batteries [17]. When using a variable-length pendulum, one can optimize the pendulum’s natural frequency to match the excitation frequency of the ambient environment [17,18,48]. This resonance enhancement maximizes the mechanical energy transferred to the piezoelectric devices, thereby increasing the generated voltage.

The voltage output generated by the piezoelectric devices attached to the variable-length pendulum depends on various factors, including the material properties of the piezoelectric element, the amplitude and frequency of motion of the pendulum, and the efficiency of the energy-harvesting circuitry [17,54,60,61,62]. This setup holds potential applications in harvesting energy from ambient vibrations, such as in structures, machinery, or even human motion [28,54,61,63,64,65,66,67,68]. It provides a renewable and sustainable source of power for low-power electronics or sensor networks in remote or inaccessible locations.

Integrating piezoelectric devices with a variable-length pendulum presents a promising approach for generating voltage from mechanical motion, with implications for various energy harvesting applications. Several methods involve utilizing a piezoelectric device attached to a variable-length pendulum for energy harvesting.

In [17], a variable-length pendulum with two degrees of freedom linked to a piezoelectric device was explored (refer to Figure 1 for more information). The governing equations were described using two dimensionless equations, as expressed in Equation (1), which were solved either numerically or even analytically using the multiple-scale technique.

$$\begin{array}{cc}\hfill & \ddot{z}+c_1\dot{z}+z+3\alpha {\zeta }_{s}z({\zeta }_s+z)+\alpha z^3-r{p}^2(cosp\tau +\theta sinp\tau )+\frac{1}{2}{W}^2{\theta }^2+\mu v-(1+z){\dot{\theta }}^2\hfill \\ & =f_{1}cos{p}_1\tau ,\hfill \\ \hfill & {(1+z)}^2\ddot{\theta }+c_2\dot{\theta }+(1+z)[W^2(\theta -\frac{{\theta }^3}{6})-r{p}^{2}sinp\tau (1-\theta )]+2(1+z)\dot{z}\dot{\theta }=f_{2}cos{p}_2\tau ,\hfill \\ \hfill & \dot{v}+\frac{v}{R_p{c}_p{\omega }_1}=\frac{l_x\beta \dot{l}}{C_p}.\hfill \end{array}$$

(1)

The above non-dimensional equations were obtained by considering the following dimensionless parameters:

$$\begin{array}{cc}\hfill & {\omega }_1^2=\frac{k_1}{m},{\omega }_2^2=\frac{g}{l_x},{\omega }^2=\frac{{\omega }_2^2}{{\omega }_1^2},r=\frac{R}{L},p=\frac{\Omega }{{\omega }_1},\alpha =\frac{k_2{l}_x^2}{{\omega }_1^{2}m},\mu =\frac{\beta }{{\omega }_1^2},\hfill \\ \hfill & c=\frac{C}{{\omega }_{1}m},f_1=\frac{F_0}{l_x{\omega }_1^{2}m},f_2=\frac{M_0}{l_x{\omega }_1^{2}m},\tau ={\omega }_{1}t,\hfill \end{array}$$

(2)

where $z_r+\alpha z_r^3=W^2$, $z_r={\xi }_s{l}_x$, $x=z{l}_x+z_r$, $l_0=l_x-z_r$, $k_1$ and $k_2$—linear and nonlinear stiffness coefficients, R—radius of circular path, $\Omega$—angular velocity with respect to a harmonic torque, m—mass of the pendulum, c—damping coefficient, $l_0$—normal length of the spring, g—acceleration due to gravity, $C_p$—capacitance of the piezoelectric circuit, $l_m$—inductance of the coil, $\beta$—circuit linear coupling coefficients, and $R_p$—resistive load of the piezoelectric circuit.

This study examined how excitation amplitudes, damping coefficients, and different frequencies affect the output of the energy-harvesting device. A comparison of performance between piezoelectric and electromagnetic devices revealed that the piezoelectric device generates significantly more output power than the electromagnetic one. However, it was concluded that increasing the damping coefficient c decreases the maximum voltage, current, and power values. The solution’s time histories, the piezoelectric device’s output voltage, and the generated power were all plotted to validate the proposed model. Figure 1b–d aims to validate the findings discussed in [17], and they align with the authors’ assertions despite the results being transitional.

In the work by Wu et al. [69], a nonlinear harvester was introduced, featuring two cantilevers with masses and magnets attached to their free ends. These cantilevers had identical, permanent magnets that generated a repulsive magnetic force between them. The second mass included a piezoelectric cantilever (see Figure 2). The primary DOF captured the ambient, low-frequency vibrations and excited the secondary DOF to vibrate significantly at its resonant frequency. Since the resonant frequency of the secondary DOF was three times higher than the initial frequency, the harvester achieved frequency up-conversion.

An external acceleration $\Gamma$ was applied to the model, resulting in a relative displacement ${x_i}_{(i=1,2)}$ between the masses and the base frame. After incorporating the magnetic force $F_B$ and the electrical equation, the following equation governing dynamics of the system was found:

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{m}_1\\ m_2\end{array}\right]\Gamma =& \left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right]\left[\begin{array}{c}{\ddot{x}}_1\\ {\ddot{x}}_2\end{array}\right]+\left[\begin{array}{cc}{c}_1& 0\\ 0& c_2\end{array}\right]\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]+\left[\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}-F_B\\ \beta v+F_B\end{array}\right],\hfill \\ \hfill I=& \beta {\dot{x}}_2-C_p\dot{v}.\hfill \end{array}$$

(3)

where $\Gamma$ represents the external acceleration, $\beta$ represents the piezoelectric voltage coefficient, v denotes the piezoelectric output voltage, $k_i,{c_i}_{(i=1,2)}$ are the global equivalent stiffness and damping coefficients, I stands for the output current, and $C_p$ signifies the clamped capacitance of the electric transducers. This study explored simulated performances and experimental validation, thereby extending the understanding of the dynamic model and theoretical power harvester.

The investigation revealed that the resonant frequency can be adjusted to align with the ambient excitation frequency, potentially tripling the output frequency and expanding the operating bandwidth under high-level excitations. Nonetheless, there is a pressing need to enhance the piezoelectric generator’s coupling coefficient and perform in-depth studies on the frequency up-conversion effect related to the coupling parameters within the harvester structure.

A bistable, 2-DOF piezoelectric energy harvester with controllable magnetic coupling was presented in [70]. In this concept, two resonant peaks are produced by a linear parasitic oscillator coupled to a primary energy-harvesting beam. Nonlinear dynamics is induced by magnetic coupling to produce electrical outputs. Figure 3 shows the nonlinear electromechanical model as the system of Equation (4) gives the dynamic model:

$$\begin{array}{cc}\hfill & m_1{\ddot{x}}_1=-c_1\left({\dot{x}}_1-{\dot{x}}_0\right)-k_1\left(x_1-x_0\right)-\beta v+c_2\left({\dot{x}}_2-{\dot{x}}_1\right)-k_2\left(x_2-x_1\right),\hfill \\ \hfill & m_2{\ddot{x}}_2=-c_2\left({\dot{x}}_2-{\dot{x}}_1\right)-k_2\left(x_2-x_1\right),\hfill \\ \hfill & \beta \left({\dot{x}}_1-{\dot{x}}_0\right)+C_P\dot{v}+\frac{v}{R_p}=0.\hfill \end{array}$$

(4)

A comprehensive parametric study explored various parasitic oscillator configurations, and it was complemented by the experimental validation of the theoretical analysis that was also reported in [71]. Their findings revealed that nonlinear responses may manifest at resonant peaks. Notably, the load resistance significantly influenced the peak, exhibiting strong nonlinear responses and resulting in a noticeable peak shift. Their study established that optimal power output coincides with a reduced bandwidth due to the peak adjustment facilitated by a proper tuning of the parasitic oscillator.

In [9], an innovative energy harvester utilizing a piezoelectric spring architecture was presented. This design incorporates a binder clip structure into a spring pendulum, facilitating the conversion of energy of the dynamic mass into electrical energy within the piezoelectric transducer. The coupling mechanism is depicted in Figure 4.

The dynamic model for the piezoelectric spring pendulum harvester (PSPH system) is expressed in the following form:

$$\begin{array}{cc}\hfill & m\left(l_0+l\right)\ddot{\phi }+c_1\left(l_0+l\right)\dot{\phi }+2m\dot{\phi }\dot{l}+mgsin\phi =m{\ddot{x}}_{H}cos\phi -m{\ddot{x}}_{V}sin\phi ,\hfill \\ \hfill & m\ddot{l}+c_2\dot{l}+\left(k_s+k_p\right)l-m\left(l_0+l\right){\dot{\phi }}^2-mgcos\phi +mg+\beta v=m{\ddot{x}}_{H}sin\phi +m{\ddot{x}}_{V}cos\phi ,\hfill \\ \hfill & C_P\dot{v}-\beta \dot{l}+\frac{v}{R_p}=0.\hfill \end{array}$$

(5)

This system of equations is derived using the Lagrange method, where $x_H$ and $x_V$ represent the displacement in the horizontal and vertical directions, respectively. Additionally, $k_s$ and $k_p$ denote the equivalent stiffness of the mechanical spring and piezoelectric elements. During the numerical analysis, it was observed that a maximum power output of $13.29$ mW was achieved with a motion excitation of $2.03$ Hz and $0.26$ g. Remarkably, the experimental results closely aligned with the simulation, thus highlighting the reliability of the findings.

3. Electromagnetic Device Energy Harvesting from Constant- and Variable-Length Pendulums

Electromagnetic devices attached to a variable-length pendulum can also generate voltage by converting the mechanical energy into electrical energy [17,48,72]. Electromagnetic induction occurs when a varying magnetic field generates an electromotive force (EMF) or voltage in a conductor [73,74,75], thus forming the basis for the operation of electromagnetic generators. In the case of a variable-length pendulum, the motion of the pendulum causes a change in the magnetic field relative to a coil or wire within the electromagnetic device [13,76]. This change induces an EMF or voltage in the coil according to Faraday’s law of electromagnetic induction [73,75]. As the pendulum swings back and forth, the magnetic field passing through the coil or wire within the electromagnetic device changes, thereby inducing an electric current in the coil and generating a voltage across its terminals.

Like piezoelectric devices, using a variable-length pendulum allows for adjusting the pendulum’s natural frequency to match external excitation frequencies. This optimization enhances the mechanical energy transferred to the electromagnetic device, thereby increasing the generated voltage [14,31,77]. The voltage output generated by the electromagnetic device attached to the variable-length pendulum depends on factors such as the strength of the magnetic field, the number of wire turns in the coil, the velocity of the pendulum, and the efficiency of the energy conversion process [78,79,80,81]. Electromagnetic devices attached to variable-length pendulums have applications in energy harvesting from various sources of mechanical motion, including vibrations, oscillations, or even human movement, thus yielding a sustainable and even renewable power source for low-power electronics, networks of sensors, or remote monitoring systems.

3.1. Principle of the Electromagnetic Coupling and Pendulum Application

In this section, how electromagnetic coupling proves effective is discussed, as studied in [82] and applied to variable-length pendulums for various engineering applications in [7].

The magneto-mechanical oscillator operates with a cylindrical permanent magnet plunger that is propelled by an external magnetic field generated by a solenoid, as illustrated in Figure 5a. Through utilizing the Biot–Savart law and the difference element method, we can model the magnetic field at any point along the axial direction of the coils. Additionally, employing the charge model introduced in [82,83,84,85,86], enables the calculation of the force acting on the cylindrical magnet bar within the external magnetic field.

It is important to mention that, due to symmetry, no lateral force is exerted when the cylindrical magnet moves along the coil axis [74,87]. However, the frame structure can be configured to induce a slight vertical motion in the event of significant deflection [7,88].

To deduce the magnetic flux density B at each point P along the solenoid axis, an infinitesimal element is considered from the coil loop, which is denoted by $dl$ in Figure 5a. Consequently, the magnetic flux density at the desired location P can be expressed as a vector according to the Biot–Savart law [82]. When considering only the magnetic field along the x-axis, we have the following:

$$\overrightarrow{B_x}={\int }_0^{2\pi }\frac{{\mu }_{0}I{R}^2}{{(x^2+R^2)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}d{\theta }_{\overrightarrow{i}}=\frac{{\mu }_{0}I{R}^2}{2{(x^2+R^2)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\overrightarrow{i},$$

(6)

where I represents the flow of current through the coils; ${\mu }_0$ is the vacuum permeability, which is a constant $4\pi \times {10}^{-7}$ H/m; R is the radius of the coils; and $\overrightarrow{i}$ is the unit vector in the x-axis direction. The number of coils n that turn through a unit area is determined by considering a cross-section of the solenoid along its radius direction

$$n=\frac{N}{\left(\frac{D_2}{2}-\frac{D_2}{2}\right)L}=\frac{2N}{\left(D_2-D_1\right)L}$$

(7)

and it is provided by the current differential element traveling through the unit region

$$dI=I\left(ndxdR\right)=nIdxdR.$$

(8)

When substituting Equation (7) into Equation (8), a variable of the magnetic field at an arbitrary point P along the x-axis is described as follows:

$$\begin{array}{cc}\hfill d{B}_x& =\frac{{\mu }_{0}I{R}^2}{2{(x^2+R^2)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}dI=\frac{{\mu }_{0}NI{R}^2}{\left(\left(D_2-D_1\right)L{(x^2+R^2)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}dxdR=\frac{{\mu }_{0}NI{R}^2}{2\left(D_2-D_1\right)L}\hfill \\ & \left[\left(L-2x\right)ln\frac{D_2+\sqrt{{\left(2x-L\right)}^2+D_2^2}}{D_1+\sqrt{{\left(2x-L\right)}^2+D_1^2}}+\left(L+2x\right)ln\frac{D_2+\sqrt{{\left(2x+L\right)}^2+D_2^2}}{D_1+\sqrt{{\left(2x+L\right)}^2+D_1^2}}\right].\hfill \end{array}$$

(9)

According to the charge model delivered in [83,89], the force of the permanent magnet in an external magnetic field can be proposed:

$$\overrightarrow{F}={\int }_v\rho {\overrightarrow{B}}_{ext}dv+{\oint }_s\sigma {\overrightarrow{B}}_{ext}ds,$$

(10)

where $\rho =-\nabla ·\overrightarrow{M}$ and $\sigma =\overrightarrow{M}·\overrightarrow{N}$ are the equivalent volume and surface charge density, respectively; ${\overrightarrow{B}}_{ext}$ is the external magnetic field, and the permanent magnet’s magnetic moment per unit volume is denoted by the symbol $\overrightarrow{M}$. Additionally, the magnetic moment can be calculated using the formula $\overrightarrow{M}=\frac{B_r}{{\mu }_0}$, where $B_r$ is the permanent magnet’s residual flux density. The cylindrical magnet plunger is polarized along its axis with fixed and uniform magnetization, which is denoted as $\overrightarrow{M}=M_s\overrightarrow{i}$, thus leading to $\rho =-\nabla ·\overrightarrow{M}=0$, and the magnetic force along the x-axis is

$$\overrightarrow{F}={\oint }_s\sigma {\overrightarrow{B}}_{ext}ds.$$

(11)

If the magnetic plunger is thought of as a hollow cylinder, then, to evaluate $\sigma$, the unit surface normal is determined according to the following:

$$\left\{\begin{array}{cc}\hfill -\overrightarrow{i},& x_l=x-\frac{l}{2},\hfill \\ \hfill \overrightarrow{i},& x_r=x+\frac{l}{2},\hfill \\ \hfill -e_{\overrightarrow{r}},& r=\frac{d_1}{2},\hfill \\ \hfill e_{\overrightarrow{r}},& r=\frac{d_2}{2},\hfill \end{array}\right.$$

(12)

where $e_{\overrightarrow{r}}$ is the unit vector in the direction of $\overrightarrow{r}$. If the north and south poles of the permanent magnet are on the left and right sides, respectively, then the charge density in the permanent magnet’s end surface yields

$$\left\{\begin{array}{cc}\hfill M,& x_l=x-\frac{l}{2},\hfill \\ \hfill -M,& x_r=x+\frac{l}{2},\hfill \end{array}\right.$$

(13)

where l, $d_1$, and $d_2$ are the magnet cylinder’s length, inner radii, and outer radii, respectively; and $x_l$ and $x_r$ are the coordinates of the left and right region of the permanent magnet on the x-axis, respectively.

When using Equations (10) and (13), the force acting on the permanent magnet plunger has the following form:

$$\begin{array}{cc}\hfill F_x& ={\oint }_s{\sigma }_m{B}_{x}ds=M\left[B_x\left(x-\frac{l}{2}\right)-B_x\left(x+\frac{l}{2}\right)\right]{\int }_{\frac{d_1}{2}}^{\frac{d_2}{2}}{\int }_0^{2\pi }rdrd\theta \hfill \\ \hfill & =\frac{B_r\pi \left(d_2^2-d_1^2\right)}{4{\mu }_0}\left[B_x\left(x-\frac{l}{2}\right)-B_x\left(x+\frac{l}{2}\right)\right],\hfill \end{array}$$

(14)

where $B_x\left(·\right)$ is the function with respect to the variable x.

When using the parameter values outlined in

Table 1. The parameters utilized in Figure 5 and Figure 6 display the properties of the solenoid actuation forces $D_1$, $D_2$, L, $d_1$, $d_2$, and l in mm.

Set	${\mathit{D}}_1$	${\mathit{D}}_2$	L	${\mathit{d}}_1$	${\mathit{d}}_2$	l	I(A)	N
1	50	60	150	6	20	20	6	540
2	50	60	150	6	22	20	6	540
3	50	60	150	6	26	20	6	540
4	50	60	180	6	26	20	6	540
5	50	60	180	6	26	20	6	540

and considering $B_r=0.005$ T, the magnetic force exerted by the actuator can be visualized, as shown in Figure 5b,c. These plots reveal that the maximum actuation force occurs when the magnet’s center moves relatively close to the coil end in a single-solenoid actuator setup. Figure 5d illustrates the impact of current variation on the electromagnetic forces of a single-solenoid actuator. It demonstrates the effect of increasing the current from 1 A to 5 A. Furthermore, Figure 6 indicates a near-linear relationship between the generated force and the solenoid’s magnet placement.

Based on the computation conducted for the single-solenoid actuator, it was evident that the amplification of the driving force depends on the displacement x, as indicated by the earlier computation for a single-solenoid actuator. A more effective electromagnetic actuator can be achieved for mechatronic applications utilizing two similar and coaxial solenoids with their coils wound in opposite directions and supplied by the same currents. This configuration results in a constant displacement–independent force [82,90,91].

Figure 6a shows the details of a double-identical solenoid actuator. The double-identical solenoid actuator operates by employing two solenoids mounted in parallel with each other [83,84]. The use of two identical solenoids in the actuator provides several benefits. First, it ensures that the actuator produces a balanced force, as the two solenoids are similar in size and strength. Second, it provides redundancy in cases where one of the solenoids fails, as the other solenoid can continue to provide motion [83,84,92,93]. Finally, it allows for enhanced motion control since the two solenoids can be independently controlled [83,94,95,96].

It was assumed that the distance between the centers of the two solenoids ($O^{\prime }$ and $O^{″}$) is denoted by d. We designated the midpoint between $O^{\prime }$ and $O^{″}$ along the axial line as the coordinate origin O for the new system. Importantly, it is worth noting that, along the axial line, the origin O coincides with the position of the magnetic plunger’s center. The electromagnetic force was obtained from Equation (14) in the following way:

$$F=F_x\left(x-\frac{d}{2}\right)+F_x\left(x-\frac{d}{2}\right),$$

(15)

where $F_x\left(·\right)$ is the function of the variable x. To provide further detail and explanation, one can present the following expanded form:

$$F=F_x\left(B_2-B_1\right)-F_x\left(B_2-B_1\right),$$

(16)

where $F_x=\frac{B_r\pi }{4{\mu }_0}\left(d_2^2-d_1^2\right)$, and also

$$\begin{array}{cc}\hfill & B_i=\frac{{\mu }_{0}NI}{2\left(D_2-D_1\right)L}\hfill \\ \hfill & \left[\left(L-2x_i\right)ln\frac{D_2+\sqrt{{\left(2x_i-L\right)}^2+D_2^2}}{D_1+\sqrt{{\left(2x_i-L\right)}^2+D_1^2}}+\left(L+2x_i\right)ln\frac{D_2+\sqrt{{\left(2x_i+L\right)}^2+D_2^2}}{D_1+\sqrt{{\left(2x_i+L\right)}^2+D_1^2}}\right],\hfill \end{array}$$

(17)

where $i=1,2,3,4$.

For $B_1$, $x_1=\left(x-\frac{d}{2}\right)-\frac{l}{2}$; for $B_2$, $x_2=\left(x-\frac{d}{2}\right)+\frac{l}{2}$; for $B_3$, $x_3=\left(x+\frac{d}{2}\right)-\frac{l}{2}$; for $B_4$, $x_4=\left(x+\frac{d}{2}\right)+\frac{l}{2}$.

To examine the characteristics of the magnetic force in the design of two identical solenoid actuators, the data corresponding to the first row in Table 1 were utilized to generate the respective force curves, as illustrated in Figure 6b.

The characteristics of the electromagnetic force generated by a constant direct current, as depicted in Figure 6b, can be used to determine one of the four types of electromagnetic force being produced: quasi-constant force (QCF), oblique linear force (OLF), single peak force (SPF), and double peak force (DPF).

Various force types reflect the intricate relationships between force and displacement that are achievable through adjustments in the electromagnetic actuator’s characteristics. These adjustments may include parameters such as the solenoid’s length, thickness, and radius, as well as the number of turns in the coil and the separation between solenoid pairs. For instance, in Figure 6b, the QCF zone is evident around $x=0$ when $d=83$ mm. As the separation d decreases or increases, the force curves transition into SPFs and DPFs, as observed at $d=75$ mm and 92 mm, respectively.

The OLF region, which is characterized by forces between $-10$ N and 15 N, is notable in Figure 6b. Additionally, the segments demonstrating OLF and SPF behaviors can be observed in Figure 5b,c for the single-solenoid actuator. Notably, an actuator driven by a single solenoid can only produce SPFs and OLFs. Conversely, an actuator powered by a pair of identical solenoids can generate all four force types.

Integrating electromagnetic devices with variable-length pendulums offers a promising approach for generating voltage from mechanical motion, with broad implications for energy harvesting and power generation applications.

Several approaches by different authors have explored using an electromagnetic device attached to a variable-length pendulum for energy harvesting. However, let us start from a basic system that incorporates a pendulum with a constant length, where the swinging motion across the electromagnetic device proves adequate for voltage generation. Nonetheless, because the pendulum’s velocity significantly influences voltage generation, employing a variable-length pendulum becomes crucial for amplifying pendulum oscillation. Consequently, we present the previous research on energy harvesting with both constant-length and variable-length pendulum configurations.

3.2. Energy Harvesting with a Constant-Length Pendulum Using Electromagnetic Devices

Musharraf et al. explored an interesting concept involving a wing-oscillating coil rod pendulum that is designed to generate a low current and voltage [79]. Central to this system lies a copper coil pendulum, which interacts with a flapping wing that is attached to it, oscillating at a low frequency upon wind impact. The kinetic energy from the wind drives the pendulum’s oscillation, converting wind energy into electrical energy. This conversion is facilitated by a semicircle-shaped permanent magnet positioned beneath the coil rod pendulum. A bar magnet, connected to the pendulum’s bob via a nylon fiber or rope over a pulley, ensures smooth movement, thus reducing the friction between the pulley and the fiber. The pendulum oscillates when the wing encounters air thrust, causing the bar magnet to move within the copper coil.

Consequently, as long as the pendulum oscillates, the coil induces EMF. However, the discussion primarily focuses on the system’s construction and electrical design in utilizing the principles of a simple pendulum. Further investigations are necessary to ascertain the output electrical power and to explore the hardware setup and simulation results.

To overcome the constraints of a simple pendulum confined to a plane and its restricted degree of excitation, as shown in Musharraf et al. [79], the dynamics of a spherical pendulum experiencing translational support excitation in three directions under generic forcing conditions were explored in the study by Sommermann et al. [97]. The system is represented using two generalized coordinates to enhance pre-prototypes and to explore the impact of the power take-off on pendulum dynamics. Bifurcation diagrams and Poincaré sections were used to analyze the system’s dynamics and optimize its performance.

The geometrical configuration of the spherical pendulum is illustrated in Figure 7, delineating a local coordinate system ($0,x,y,z$) affixed to the pivot of the pendulum alongside a distinct frame ($0,X,Y,Z$).

The equations of motion derived using Lagrange equations of the second kind are given in Equations (18) and (19) below:

$$\begin{array}{cc}\hfill & \ddot{\phi }\left(t\right)+2{\epsilon }_{\phi }{\omega }_n\dot{\phi }\left(t\right)+\frac{g}{l}sin\left(\phi \left(t\right)\right)-sin\left(\phi \left(t\right)\right)cos\left(\phi \left(t\right)\right){\dot{\phi }\left(t\right)}^2=\hfill \\ \hfill & \frac{U_0{\Omega }_U^2}{l}cos\left(\phi \left(t\right)\right)sin\left(\phi \left(t\right)\right)cos\left({\Omega }_{U}t\right)+\frac{V_0{\Omega }_V^2}{l}cos\left(\phi \left(t\right)\right)cos\left(\varphi \left(t\right)\right)cos\left({\Omega }_{V}t\right)\hfill \\ \hfill & -{\omega }_0{\Omega }_{\omega }^{2}sin\left(\phi \left(t\right)\right)cos\left({\Omega }_{\omega }t\right)-\frac{2T_{\phi }}{\pi m{l}^2}{tan}^{-1}\left(\frac{\dot{\phi }\left(t\right)}{0.01}\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \ddot{\varphi }\left(t\right)+\frac{2{\epsilon }_{\varphi }{\omega }_n\dot{\varphi }\left(t\right)}{{sin\left(\phi \left(t\right)\right)}^2}+\frac{2\dot{\phi }\left(t\right)cos\left(\phi \left(t\right)\right)\varphi \left(t\right)}{sin\left(\phi \left(t\right)\right)}=-\frac{U_0{\Omega }_U^2}{lsin\left(\phi \left(t\right)\right)}cos\left(\varphi \left(t\right)\right)cos\left({\Omega }_{U}t\right)\hfill \\ \hfill & -\frac{V_0{\Omega }_V^2}{lsin\left(\phi \left(t\right)\right)}sin\left(\varphi \left(t\right)\right)cos\left({\Omega }_{V}t\right),\hfill \end{array}$$

(19)

where $\phi$ and $\varphi$ describe the planer oscillation and rotation of the pendulum’s plane of oscillation, respectively; ${\epsilon }_{\phi ,\varphi }$ is the classical linear viscous damping term; $T_{\phi }$ is the power take-off term; $u\left(t\right)$, $v\left(t\right)$ and $\omega \left(t\right)$ are the prescribed excitations operating on the pivot; $U_0$, $V_0$, and $W_0$ correspond to the excitation amplitudes; and ${\Omega }_u$, ${\Omega }_v$, and ${\Omega }_w$ refer to the excitation frequencies.

The dynamic behavior of the energy harvester was scrutinized through the utilization of bifurcation diagrams and Poincaré sections. Generally, the system exhibits a more periodic nature when its power is extracted. Notably, it was observed that increasing the excitation amplitude enhanced the conversion of energy. However, a careful balance is required for the power take-off torque to ensure it is not excessively high. Further experimentation in a real-time setting and an analytical evaluation of the issue through an asymptotic approach of the perturbation method of multiple scales is deemed necessary to unveil the potential internal resonances.

In the work of [55], the finite element method (FEM) was employed to simulate an electromagnetic energy harvester featuring a rotational pendulum. The authors experimented with the entire system utilizing mechanical vibrations to harvest electrical energy. Their study involved developing a simplified mechanical model for the rotational pendulum, which was expressed as follows:

$$\begin{array}{cc}\hfill & I_M\ddot{\phi }+c\ddot{\phi }+m{\omega }^{2}Ahcos\left(\omega t\right)sin\phi +mghsin\phi +F_{mag}sin\left(2\phi +1.5\right)=0,\hfill \end{array}$$

(20)

where $I_M$—the pendulum’s moment of inertia, h—distance from the center of gravity to the pivot axis, g—gravitation, $\omega$—the angular velocity of the pendulum, A—the amplitude of vertical excitation, and $F_{mag}$—the magnetic force amplitude acting on the pendulum.

Equation (20) represents the mechanical model of the energy harvester, specifically one with a rotational pendulum. While the equation excludes the magnetic circuit, it encompasses all the variables influencing the pendulum’s motion. The findings indicate the system’s significant potential for broad-frequency responses. Furthermore, simulations revealed a substantial frequency range of rotational solutions, thus incorporating sub-harmonic solutions, which proves advantageous for energy harvesting, as noted by [71,98,99]. Their study concluded that increasing the number of coils with magnets can generate multiple potential wells when exploring the potential for multi-stable phenomena. However, system optimization remains a prospect through adjustments to its moment of inertia, and the broader orbit requires thorough examination in subsequent studies, including analysis of the basins of attraction.

In the work of [100], an energy harvester was introduced that employs Wiegand wires for a rotating object. The device features an eccentric pendulum mounted on a wheel rim, as illustrated in Figure 8. In this configuration, the $\overline{x}$-axis is delineated as the radial direction extending from the center of the rotating wheel to the pivot joint of the eccentric pendulum, while the $\overline{y}$-axis represents the tangential direction along the wheel rim.

The governing Equation (21) was formulated by employing the Euler–Lagrange equation, and it yielded the following:

$$\begin{array}{cc}\hfill & \left(I_M+m{r}^2\right)\ddot{\phi }+\left(\left(I_M+m{r}^2+mr{R}_{2}cos\phi \right)\ddot{\theta }+gmrcos\left(\phi +\theta \right)+mr{R}_2{\dot{\theta }}^{2}sin\phi \right)=-c_T\dot{\phi },\hfill \end{array}$$

(21)

where r—the distance between the pivot joint and the mass center of the eccentric pendulum, and $R_2$—the distance from the wheel center to the rotational center of the eccentric pendulum.

When the vehicle maintains a constant speed, $\ddot{\theta }=0$, it is assumed that the eccentric pendulum swings within a relatively small angle, where $sin\phi \approx \phi$ and $cos\phi \approx 1$. Consequently, Equation (21) can be simplified to

$$\begin{array}{cc}\hfill & \ddot{\phi }+\frac{1}{I_M+m{r}^2}\left(gmrcos\left(\phi +\theta \right)+mr{R}_2{\dot{\theta }}^{2}sin\phi \right)=-c_T\dot{\phi }.\hfill \end{array}$$

(22)

The natural frequency of the eccentric pendulum is determined as follows: ${\omega }_n=\dot{\theta }\sqrt{R_2/L^{*}}$, where $L^{*}=(r^2+k_h^2)/r$ represents the characteristic length of the system and $k_h^2=I_M/m$.

The studies suggest that expanding the rotation range increases the output power of both Wiegand wires. The discrepancies between the experimental and simulation results, potentially stemming from unpredictable and nonlinear frictional torque, were noted. Moreover, the experimental findings suggested that the proposed energy harvester could be a promising power source for a tire-pressure monitoring system. However, further exploration is required to understand the nonlinear friction torque within the energy harvester.

The work by Kuang et al. [101] presented an analyzed and innovative electromagnetic pendulum energy harvester that featured a mechanical motion rectifier (MMR). The presentation included the design and fabrication details of two prototypes of the pendulum energy harvester, one with MMR and the other without MMR. The MMR pendulum energy harvester was developed using the Lagrangian method, and the equation of motion, specifically when the rotation speed of the bottom-level gear exceeds that of the driving gears, was provided in two decoupled systems. The equations are as follows:

$$\left\{\begin{array}{c}{J}_p\ddot{\phi }+c_{m1}\dot{\phi }+mg{l}_c\phi =mcos\left({\omega }_{f}t\right),\hfill \\ J_c{\ddot{\phi }}_g+\left(c_e+c_{m2}\right){\dot{\phi }}_g=0,\hfill \end{array}\right.$$

(23)

and for the non-MMR pendulum energy harvester under harmonic excitation is as follows:

$$\begin{array}{c}\hfill \left(J_p+J_c\right)\ddot{\phi }+\left(c_e+c_m\right){\dot{\phi }}_g+mg{L}_c\phi =mcos\left({\omega }_{f}t\right),\end{array}$$

(24)

where $J_c$ and $c_e$—the corresponding rotational inertia and equivalent damping of an MMR pendulum energy harvester; $J_p$—the total rotational inertia of the pendulum system with respect to z-axis; $c_{m_1}$—the linearized damping coefficient in view of the mechanical damping in the bevel gears and frame; $c_{m_2}$—the linearized damping coefficient, which considers the mechanical damping in the planetary generator; $L_c$—the distance from the center of mass of the pendulum to the central shaft; and $c_m$—the linearized damping coefficient of the whole system.

The non-MMR system under harmonic excitation represents a classical single-DOF, energy-harvesting system that has been extensively explored in various studies [102,103]. The average output power of a non-MMR pendulum energy harvester was calculated using the following formula:

$$\begin{array}{cc}\hfill & P_{{\left(non-MMR\right)}_{max}}=\frac{\omega m_N^2}{4\sqrt{{\left[1-{\omega }^2\left(1+{\mu }_m\right)\right]}^2+4{\omega }^2{\zeta }_m}+8\omega {\zeta }_m}.\hfill \end{array}$$

(25)

The corresponding optimal electrical damping can be expressed as

$$\begin{array}{c}\hfill {\left({\zeta }_m\right)}_{opt}=\sqrt{{\zeta }_m^2+\frac{{\left[1-{\omega }^2\left(1+{\mu }_m\right)\right]}^2}{4{\omega }^2}},\end{array}$$

(26)

and the corresponding non-dimensional dynamic equations of MMR pendulum energy harvester are as follows:

$$\begin{array}{cc}\hfill & P_{{\left(MMR\right)}_{max}}=\frac{\omega m_N^2}{4X_{\omega \mu c}+8\omega \left(c_{m_1}+c_{m_2}\right)},\hfill \\ \hfill & X_{\omega \mu c}=\sqrt{{\left[1-{\omega }^2\left(1+{\mu }_m\right)\right]}^2+4{\omega }^2\left(c_{m_1}+c_{m_2}\right)}.\hfill \end{array}$$

(27)

The optimal electrical damping at the equivalent mass ${\mu }_m$ is given by

$$\begin{array}{c}\hfill {\zeta }_{{\left(MMR\right)}_{opt}}=\sqrt{c_{m_1}^2+c_{m_2}^2+\frac{{\left[1-{\omega }^2\left(1+{\mu }_m\right)\right]}^2}{4{\omega }^2}}.\end{array}$$

(28)

The above equations for the output power of both the MMR and non-MMR energy harvesters, along with the corresponding optimal electrical damping, numerically solve Equations (25)–(28) (see Figure 9).

The results obtained align with those presented by Kuang et al. in [101]. Notably, the output power of the MMR pendulum energy harvester surpassed that of its non-MMR counterpart at higher frequencies. Additionally, the bandwidth of the MMR pendulum harvester was expanded due to the disengagement of one-way clutches in the MMR system.

The experimental results presented in [101] confirm that a MMR pendulum energy harvester’s bandwidth and high-frequency output power are greater than those of a non-MMR pendulum energy harvester. However, the system stability needs to be studied.

The work of [104] proposed an electromagnetic energy harvester model based on a pendulum mechanism employing anti-phase motion. They assessed simulations and experimental outcomes, including the frequency response, arc angle, output voltage, and power. Their experimental data indicated that the anti-phase motion type yielded a maximum output power of 247 µW at an input frequency of 2 Hz, thus outperforming single-phase motion. Anti-phase motion demonstrated a $21\%$ reduction in induction time and a $37\%$ increase in output power. Additionally, the proposed anti-phase motion phenomenon effectively enhanced the output voltage of the electromagnetic induction-based energy harvesters.

The work of [105] utilized Euler–Lagrange equations to characterize the motion of a portable electromagnetic energy harvester, which employs magnetic spring technology to capture the kinetic energy from human body movement. Figure 10 illustrates the equivalent model of the energy harvester. The nonlinear differential equation representing this model is given in Equation (29):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(M+m\right)\ddot{y}+m\left(R+r_m\right)\left(\ddot{\phi }cos\phi -{\dot{\phi }}^{2}sin\phi \right)+c\dot{y}=F_{1}cos\left(\omega t\right),\hfill \\ \left[m{\left(R+r_m\right)}^2+I\right]\ddot{\phi }+c_m\dot{\phi }+c_e\dot{\phi }=-m\left(R+r_m\right)\ddot{y}cos\phi ,\hfill \end{array}\right.\end{array}$$

(29)

where M—the rest weight of the device, $\rho$—the density of the NdFeB magnet, $c_e$—the damping ratio, $c_m$—the mechanical damping ratio, $R_{coil}$—the coil resistance, $R_{load}$—the load resistance, $L_{coil}$—the coil inductance, $\ddot{y}$—the acceleration of the shell, and $\ddot{\phi }$—the angular acceleration of the moving magnet.

The equation describing the motion of the magnet can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\begin{array}{c}i{L}_{coil}+\left(R_{load}+R_{coil}\right)=Bl\dot{\phi }=v\left(\phi ,\dot{\phi }\right),\hfill \\ \begin{array}{c}\hfill F_L\left(i,\dot{\phi }\right)l\dot{\phi }=iv\left(\phi ,\dot{\phi }\right),\end{array}\hfill \end{array}\hfill \end{array}\right.\end{array}$$

(30)

where $F_L$—the electrical damping force, and $v\left(\phi ,\dot{\phi }\right)$—the induced voltage.

The proposed energy harvester represents a 2-DOF system designed to efficiently capture ambient vibrations. It boasts several advantages, such as a low operating frequency, efficient space utilization, and enhanced reliability. The experimental results revealed that the electromagnetic vibration energy harvester can generate a peak voltage of $2.2$ V. However, optimization efforts are necessary to improve the output power density. The current findings suggest promising applications for the electromagnetic energy harvester in wearable electronic devices.

The work by Graves et al. [106] introduced an electromagnetic pendulum energy harvester that features a counterweight explicitly designed to capture the low-frequency vibrations from ocean waves, and it is intended for use in uncrewed surface vehicles. The device’s design was fabricated and subsequently subjected to experimental testing. The schematic of the counterweight pendulum, as proposed by Graves et al. [106], is depicted in Figure 11a, consisting of two rigid pendulum arms fixed at a ${180}^{\circ }$ angle to one another.

This paper exclusively presents experimental findings. However, in the current research, we illustrate the time histories of the angular position, $\phi \left(t\right)$, as described in Equation (31) and plotted in Figure 11b. This visualization aims to provide insights into the model’s behavior and identify which parameters influence the system most. Our observations reveal that, in addition to the length and damping coefficient, the amplitude of excitation, A, significantly impacts the system model. Therefore, it is crucial to carefully select this parameter to ensure optimal efficiency and reliability for energy harvesting purposes.

The assumption made was that the masses of the arms are small. Consequently, the dynamics of the pendulum subjected to a horizontal excitation of $Acos\left(\omega t\right)$ was modeled in the following manner:

$$\begin{array}{cc}\hfill & \left(m{l}_m^2+M{l}_M^2\right)\ddot{\phi }+c\dot{\phi }+\left(m{l}_m-M{l}_M\right)gsin\phi =\left(m{l}_m-M{l}_M\right)gcos\phi Acos\left(\omega t\right),\hfill \end{array}$$

(31)

where $l_m$—the length of the first pendulum from the point of rotation O, M—the counterweight mass, and $l_M$—the distance from O to the counterweight mass M.

The undamped resonance frequency of the pendulum is given by the following formula:

$$\begin{array}{c}\hfill {\omega }_n=\sqrt{\frac{\left(m{l}_m-M{l}_M\right)g}{m{l}_m^2+M{l}_M^2}},\end{array}$$

(32)

where $l_m$—the length of the first pendulum from the point of rotation O, $l_M$—the distance from O to the counterweight mass M, and A and $\omega$—the amplitude and frequency of the excitation acceleration, respectively. When $\phi$ is small, then the restoring torque stiffness of the pendulum $\left(m{l}_m-M{l}_M\right)g$ and the moment of inertia of the pendulum $m{l}_m^2+M{l}_M^2$ are present.

Equation (32) indicates that the resonance frequency decreases with an increase in either the counterweight’s mass M or the length $l_M$ of the counterweight arm. This effect occurs because the counterweight reduces the pendulum’s restoring torque stiffness while increasing its moment of inertia. The inclusion of adjustable pendulum and counterweight arms facilitate the testing and analysis of various configurations, and this is supported by empirical evidence. The experimental results underscore the counterweight’s ability to adjust the system to different resonant frequencies. The reduction in resonant frequency was found to scale linearly with the ratio of the arm lengths. In conclusion, the energy harvester design effectively lowered the pendulum’s natural frequency without increasing its length, while also maintaining satisfactory power output at these reduced frequencies.

In their study [107], Dotti and Sosa assessed the feasibility of harvesting energy, utilizing a pendulum harvester, from the vibrations of trucks and sleepers as a high-speed train passes. The diagram of the pendulum harvester is illustrated in Figure 12, with the analysis conducted using a Lagrange equation for single DOF non-conservative systems. The second-order differential equation governing the dynamics of the system is given by

$$\begin{array}{c}\hfill m{l}^2\ddot{\phi }+b{l}^2\dot{\phi }+ml\left(\ddot{Y}+g\right)sin\phi +kJsgn\dot{\phi }=0,\end{array}$$

(33)

where $Y=Y\left(t\right)$ is provided by the vibration of the track or sleepers, and b stands for the viscous damping coefficient.

If excitation begins at $t=0$, then energy extraction is initiated at a time $t_0>0$, i.e.,: if $t<t_0$, then $k=0$; if $t\ge t_0$, then $k=1$. Therefore, the instantaneous power extracted from the system is given by $P=kJ\left|\dot{\phi }\right|$, while the average power is $\overline{P}=\frac{1}{t_s-t_0}\left({\int }_{t_0}^{t_s}J\left|\dot{\phi }\right|dt-E_0\right)$, where $t_s$ is the time required by the pendulum to come to a complete stop at its rest position, and $E_0$ denotes the initial energy of the following system:

$$\begin{array}{c}\hfill E_0=\frac{1}{2}m{l}^2{\dot{\phi }}_0^2+mgl\left(1-cos{\phi }_0\right)-\frac{1}{2}b{l}^2{\dot{\phi }}_0^2.\end{array}$$

(34)

Numerical simulations have revealed that a rotating pendulum harvester with sufficiently low damping can generate a practical average power, typically ranging from 5 W to 6 W per unit. This level of power output proves adequate for supplying various rail-side equipment, such as wireless sensors or warning light systems, especially when considering a modular arrangement of devices. However, effective control actions are essential given the challenging task of selecting suitable initial conditions.

The studies by Basheer et al. [50] and Izadgoshasb [68] have suggested that a double pendulum demonstrates more efficient energy conversion than a single pendulum due to its complex dynamics [68,108]. Kumar et al. [19] conducted a parametric study on a base-excited double pendulum configured with two pendulums connected by rigid, massless links. In this setup, a magnet affixed to the lower pendulum’s end interacted with coils placed near the magnet loop’s curvature (refer to Figure 13). The system utilizes small amplitude oscillations to create an irregular magnetic track, thus inducing electricity in the coils, which is further amplified by chaotic magnet motion.

The parameters $M_m$, $l_m$, and $r_m$ represent the cylindrical magnet’s mass, length, and radius, respectively. The equations of motion were derived using the following Euler–Lagrange equation for a progressive system:

$$\begin{array}{cc}\hfill & \left(m_1+m_2\right)l_1^2{\ddot{\phi }}_1+m_2{l}_1{l}_{2}cos\left({\phi }_1-{\phi }_2\right){\ddot{\phi }}_2+m_2{l}_1{l}_{2}sin\left({\phi }_1-{\phi }_2\right){\dot{\phi }}_2^2+\hfill \\ \hfill & \left(m_1+m_2+M_m\right)g{l}_{1}sin{\phi }_1+d_m{\dot{\phi }}_1=-\left(m_1+m_2\right)l_{1}cos{\phi }_1{\ddot{x}}_g,\hfill \\ \hfill & \left(m_2{l}_2^2+I_m\right){\ddot{\phi }}_2+m_2{l}_1{l}_{2}cos\left({\phi }_1-{\phi }_2\right){\ddot{\phi }}_1+m_2{l}_1{l}_{2}sin\left({\phi }_1-{\phi }_2\right){\dot{\phi }}_1^2+\hfill \\ \hfill & \left(m_2+M_m\right)g{l}_{2}sin{\phi }_2+\left(c_m+c_e\right){\dot{\phi }}_1=-m_2{l}_{2}cos{\phi }_2{\ddot{x}}_g,\hfill \end{array}$$

(35)

where $d_m$ and $d_e$ denote the linear mechanical and electrical damping, respectively, and the input function $x_g=Asin\left(\omega t\right)$ is the system support’s transverse motion.

The system dynamics analysis focused solely on the impact of magnet mass while disregarding the magnet’s influence. Remarkably, it was observed that the system could harvest substantial energy in both linear and nonlinear scenarios. The harvested energy demonstrated an upward trend with increasing excitation amplitude and frequency. In conclusion, it was determined that increasing the number of coils around the harvester would lead to a corresponding increase in harvested energy.

Ma et al. [16] investigated the utilization of parametric resonance to enhance vibration-based energy harvesting, particularly at low frequencies around 4 Hz. The device integrates a pendulum component for movement and electromagnetic induction for energy conversion. The proposed architecture of the device is illustrated in Figure 14.

The equation of motion is expressed as follows:

$$\begin{array}{cc}\hfill & \left(ML+\frac{I_m}{L}\right)\ddot{\phi }\left(t\right)+c_{m}L\dot{\phi }\left(t\right)+M\left[g+{\ddot{y}}_0\left(t\right)\right]sin\phi \left(t\right)+F_{em}\frac{{\rho }_w}{L}=0,\hfill \end{array}$$

(36)

where $F_{em}$—the total tangential electromagnetic force generated by the winding; ${\rho }_w$—the effective winding radius; $r_w$—the winding resistance; $L_w$—the leakage inductance of the winding; $R_L$—the load resistance; and $e_A$, $e_B$, and $i_A$, $i_B$—the instantaneous electromotive force and current in the respective winding, respectively.

The performance of the device was thoroughly examined through theoretical, numerical, and experimental approaches, ensuring a comprehensive evaluation.

It was noted that the numerical results closely matched the experimental measurements, albeit with a slight overestimation of the power output in the theoretical analysis. Nevertheless, the proposed energy harvesting method exhibited potential advantages over the linear method in scenarios with low electrical loads. This superiority can be attributed to identifying an optimal damping level that maximizes power output. Moreover, the bandwidth of the proposed device was found to be nearly inversely proportional to the damping level, which is in contrast to the linear method where the bandwidth diminishes significantly at lower damping levels. Further investigation was validated in terms of exploring the impact of complex electrical load circuits and conditions that influence the device behavior, particularly those about low and high-amplitude attractors.

3.3. Energy Harvesting with a Variable-Length Pendulum Using Electromagnetic Devices

The research by Hassan et al. [18] introduced a self-tuning energy harvester using pendulum oscillations. Activated by a slider-crank mechanism, it includes a mechanical rectifier to convert vibrations into electrical energy. The system autonomously adjusts its natural frequency to match base excitation. The governing differential equation, derived using Lagrange equations, is a second-order ordinary differential equation that is expressed as follows:

$$\begin{array}{cc}\hfill & \ddot{\phi }+\frac{c\dot{\phi }}{\left(2m_d+0.5m_r\right)l}+\frac{\left({2m}_d+m_r\right)g}{\left(2m_d+0.5m_r\right)l}sin\phi +\frac{\left({2m}_d+m_r\right)}{\left(2m_d+0.5m_r\right)l}\frac{1}{{\left({b_r}^2-{a_r}^2{\left(sin\left(\omega t\right)\right)}^2\right)}^{1.5}}\hfill \\ \hfill & (a_r{\omega }^2\left(cos\omega t\right){\left({b_r}^2-{a_r}^2{\left(sin\left(\omega t\right)\right)}^2\right)}^{1.5}+{a_r}^3{\left(sin\left(\omega t\right)\right)}^4-a_r{b_r}^2{\left(sin\left(\omega t\right)\right)}^2+a_r{b_r}^2\hfill \\ \hfill & {\left(cos\left(\omega t\right)\right)}^2)cos\phi =0,\hfill \end{array}$$

(37)

where $a_r$ and $b_r$—the lengths of the crank and connecting rods, respectively, and $m_d$ and $m_r$—the pendulum and rod mass, respectively.

The differential equation of dynamics, which encompasses both mechanical and electrical damping, can be expressed as follows

$$\begin{array}{cc}\hfill & \ddot{\phi }+\frac{c_e\dot{l}}{(2m_d+0.5m_r)l}+\frac{c_t\dot{\phi }}{(2m_d+0.5m_r)l}+\frac{(2m_d+m_r)g}{(2m_d+0.5m_r)l}sin\phi +\frac{(2m_d+m_r)}{(2m_d+0.5m_r)l}\hfill \\ \hfill & \left(\frac{1}{{(b^2-a^2{(sin\left(\omega t\right))}^2)}^{1.5}}\right){\left(a{\omega }^2(cos\omega t)\right(b^2-a^2{(sin\left(\omega t\right))}^2)}^{1.5}+a^3{(sin\left(\omega t\right))}^4-a{b}^2\hfill \\ \hfill & {(sin\left(\omega t\right))}^2+a{b}^2{(cos\left(\omega t\right))}^2)cos\phi =0,\hfill \end{array}$$

(38)

where $c_e$ is a constant for the internal loss coefficient in the motor, while the voltage–current relation is found through

$$\begin{array}{c}\hfill I\left(R_e+R_L\right)-C_{e}Z\dot{\phi }+L_e\frac{dI}{dt}=0,\end{array}$$

(39)

where Z—the gear ratio, $R_e$—the armature resistances, and $L_e$—the armature inductance.

The harvested power can be calculated using the formula $P=I^2{R}_L$, where I represents the current from the DC source and $R_L$ denotes the load resistance. The experimental measurements indicated that the device achieves a maximum output power of $4.1$ W, which is observed at a resistance of 3 $\Omega$ when the pendulum length reaches 900 mm. This occurs at a tuned resonance frequency of $0.47$ Hz and an excitation amplitude of 120 mm. Control units significantly broaden the bandwidth, extending it from $0.47$ Hz to $0.68$ Hz.

Zhang et al. [109] introduced a groundbreaking configuration for a pendulum absorber harvester, incorporating magnets to enhance its vibration suppression and energy harvesting capabilities significantly. This innovative design integrates theoretical analysis with experimental evaluations to assess its performance. In this setup, two cylindrical magnets are fixed on either side of the pendulum, as illustrated in Figure 15.

The magnetic force exerted on both sides amplifies the amplitude of the pendulum’s oscillation, facilitating a more efficient transformation of the structure’s vibrational energy into pendulum motion. Consequently, an electromagnetic harvester installed on the pendulum’s pivot can capture and harvest more energy. The model depicted above exemplifies a two-degree-of-freedom system, where $F_{m_{l1}}$ and $F_{m_{l2}}$ represent the lateral magnetic forces, while $F_{m_{a1}}$ and $F_{m_{a2}}$ denote the axial attractive magnetic forces. The exciting force on the primary structure was assumed to be harmonic, and it is given by $F\left(t\right)=f_{0}cos\left(\omega t\right)$. Therefore, employing the Lagrange approach and Kirchhoff’s law yielded differential equations that describe the hoisting structure, the pendulum, and the electromagnetic induction. The resulting differential equations modeling the dynamics are given in the following form:

$$\begin{array}{cc}\hfill & \left(m_1+m_2\right){\ddot{x}}_1+c_1{\dot{x}}_1+k_1{x}_1+m_{2}l\left({\dot{\phi }}^{2}cos\phi +\ddot{\phi }sin\phi \right)=f_{0}cos\left(\omega t\right),\hfill \\ \hfill & m_2{l}^2\ddot{\phi }+c_p\dot{\phi }+\alpha l+m_{2}l\left(g+{\ddot{x}}_2\right)sin\phi +\left(F_{m_{l1}}+F_{m_{l2}}\right)lsin\phi =\left(F_{m_{a1}}-F_{m_{a2}}\right)lcos\phi ,\hfill \\ \hfill & L\dot{I}+\left(R_l+R_c\right)I-\alpha \dot{\phi }=0,\hfill \end{array}$$

(40)

where $\alpha$—the electromechanical coupling coefficient depending on the harvester physical construction, and $\alpha l=m_e$—the electromagnetic moment acting on the pendulum.

The steady-state solution for linear and nonlinear responses is determined when the pendulum is activated. The amplitudes of the hoisting structure and the pendulum are expressed as follows:

$$\begin{array}{cc}\hfill & X_{linear}=\frac{f}{\sqrt{{\left(-1+{\Omega }^2\right)}^2+4{\Omega }^2{\zeta }_1^2}},{\phi }_{linear}=0,\hfill \\ \hfill & X_{nonlinear}=\frac{\sqrt{{\left({\Omega }^2-4x_{st}^2+4q_1\right)}^2+16{\Omega }^2{x}_{st}^2{\zeta }_2^2}}{2{\Omega }^2{x}_{st}^2},{\phi }_{nonlinear}=\sqrt{\frac{R_1+4\Omega \sqrt{R_2{R}_3}}{R_4}}.\hfill \end{array}$$

(41)

In the above equations, $\Omega$—the normalized excitation frequency, f—the amplitude of the normalized harmonic force, $x_{st}$—the normalized natural frequency of the pendulum, ${\zeta }_1$ and ${\zeta }_2$—the damping ratio of the structural mass and the pendulum, respectively, and $q_1$—the dimensionless components of the axial magnetic force. Other parameters are expressed below:

$$\begin{array}{cc}\hfill & {\zeta }_2=\left({\zeta }_p+{\zeta }_e\right),q_1=x_{st}^2-\frac{1}{4},\hfill \\ \hfill & R_1=2\left(-1+{\Omega }^2\right)\left({\Omega }^2-4x_{st}^2+4q_1\right)-16{\Omega }^2{x}_{st}{\zeta }_1{\zeta }_2,\hfill \\ \hfill & R_2=-\left(\Omega +4q_1\right){\zeta }_1+x_{st}^2\left(f\Omega +4{\zeta }_1\right)-2\left(-1+{\Omega }^2\right)x_{st}{\zeta }_2,\hfill \\ \hfill & R_3=x_{st}^2\left(f\Omega -4{\zeta }_1\right)+\left({\Omega }^2+4q_1\right){\zeta }_1+2\left(-1+{\Omega }^2\right)x_{st}{\zeta }_2,R_4={\Omega }^4{x}_{st}^2{\beta }_1.\hfill \end{array}$$

(42)

The simulation results demonstrate that the proposed absorber–harvester model exhibits superior vibration mitigation and energy harvesting capabilities compared to the conventional designs. For real-world applications, it is crucial to select the damping ratio ${\zeta }_1$ meticulously and the dimensionless components of the axial magnetic forces $q_1$ to optimize both vibration suppression and energy harvesting performances, respectively.

He et al. [15] concentrated on vibration reduction and energy harvesting within a dynamic system that featured a spring-pendulum configuration leveraging an independent electromagnetic system. Their aim wsa to optimize the harvester’s energy harvesting and vibration mitigation capabilities. The harvesting mechanism used relied on a magnet within an oscillating coil, as illustrated in Figure 16.

The equations governing the kinematics of the system were derived by applying Lagrange equations (refer to Equation (43)). These equations were subsequently solved asymptotically, employing the multiple-scale method.

$$\begin{array}{cc}\hfill & m_1{\left(l_0+l\right)}^2\ddot{\phi }+c_1\dot{\phi }+2m_1\left(l_0+l\right)\dot{l}\dot{\phi }+m_{1}g\left(l_0+l\right)sin\phi +m_1{\Omega }^2\left(l_0+l\right)\hfill \\ \hfill & \left[asin\phi cos\left(\Omega t\right)-bcos\phi sin\left(\Omega t\right)\right]=M_0\left(t\right)cos\left({\Omega }_{1}t\right),\hfill \\ \hfill & m_1\ddot{l}+c_2\dot{l}+kl+m_{1}g\left(1-cos\phi \right)+m_1\left(l_0+l\right){\dot{\phi }}^2-m_1{\Omega }^2\hfill \\ \hfill & \left[bsin\phi sin\left(\Omega t\right)+acos\phi cos\left(\Omega t\right)\right]+B{L}_{coil}I=F_0\left(t\right)cos\left({\Omega }_{2}t\right).\hfill \end{array}$$

(43)

In the next step, the Kirchhoff voltage law and Faraday’s law of electromagnetic induction for the circuit illustrated in Figure 16b are applied: $L_{ind}\dot{I}+RI-B{l}_{coil}\dot{v}=0$, where R is the summation of $R_L$ and $R_C$, and $l_{coil}$ is the length of the coil. The temporal histories of the modified phase and amplitude of generalized coordinates, the frequency response curves, and the stability and instability areas are depicted to reveal the influence of various parameters on the behavior of the analyzed system. The findings highlight zones of stability and instability, indicating that the system remains stable across a broad spectrum of parameters. Recently, this model has emerged as a critical tool through leveraging control sensors in industrial applications, buildings, infrastructure, automobiles, and transportation sectors.

As discussed in Abohemer’s work [17], the exploration also focused on the configuration of a 2-DOF, variable-length pendulum coupled to an electromagnetic device (refer to Figure 17 for more information). Two dimensionless forms defined the dynamics equations, which are expressed in Equation (1)). The equation governing the energy-harvesting component is $\dot{I}+R_{m}I/\left(l_m{\omega }_1\right)$ = $\left(l\beta \dot{I}\right)/l_m$, where $R_m$ is the resistive load of the electromagnetic circuit. Upon comparing the performance of the two devices, it was observed that the piezoelectric device consistently generates significantly more output power than its electromagnetic counterpart.

Finally, this review thoroughly examined energy-harvesting systems that use variable-length pendulums combined with piezoelectric or electromagnetic devices. We studied these cutting-edge systems’ underlying theories, technological progress, and real-world impacts. However, it is important to note that piezoelectric and electromagnetic devices have their own strengths and weaknesses. Hence, the effectiveness of energy-harvesting devices depends on the specific environment or system in which it is implemented to optimize power generation. Here, we outline some of the advantages and limitations of piezoelectric and electromagnetic devices.

3.3.1. Advantages of Piezoelectric Coupling

Piezoelectric materials offer high efficiency by efficiently converting mechanical energy into electrical energy, thus making them suitable for harvesting energy from a variety of sources such as vibrations, motions, and deformations [54,57,65]. Additionally, these devices are generally compact and lightweight [26,44], facilitating integration into small and portable systems without significantly increasing their size or weight. They operate over a wide frequency range [53], enabling energy harvesting from different types of mechanical vibrations and movements. This wide frequency range also contributes to their versatility, allowing for energy harvesting from various sources of mechanical vibrations [33,43]. Moreover, piezoelectric energy harvesting is environmentally friendly and sustainable [65] as it provides a clean and renewable energy source [68]. Furthermore, due to their solid-state nature, piezoelectric devices typically have fewer moving parts [45,46], resulting in lower maintenance requirements than other energy harvesting technologies.

3.3.2. Limitations of Piezoelectric Coupling

While piezoelectric energy harvesting has numerous advantages, there are also several limitations to consider. Piezoelectric devices typically operate most efficiently within a narrow frequency band [53], which can restrict their effectiveness in environments with broad frequency spectra. Moreover, the power output of piezoelectric energy harvesters is often limited [68], mainly when extracting energy from low-amplitude vibrations or movements. Additionally, piezoelectric materials are sensitive to fluctuations in temperature and humidity [57,65], impacting their performance and reliability over time. Furthermore, some piezoelectric materials can be brittle and susceptible to mechanical fatigue under repetitive stress or strain [54], thus diminishing their efficiency and lifespan and necessitating careful handling and protection in specific applications. Lastly, high-quality piezoelectric materials tend to be expensive [57,65], thus potentially increasing the overall cost of piezoelectric energy-harvesting systems, especially for large-scale implementations.

3.3.3. Advantages of Electromagnetic Coupling

Electromagnetic energy-harvesting systems offer a range of advantages that make them appealing for various applications. They boast high-power output capabilities [62,64,96], making them suitable for tasks with elevated energy demands. Their wide frequency range allows them to harvest energy from diverse mechanical vibrations and movements effectively [36,80,96]. Moreover, these systems are highly scalable, and they are capable of being adjusted to meet specific power requirements [80,87,96], thereby enhancing their adaptability across different scenarios. Unlike piezoelectric devices, electromagnetic systems depend less on specific resonant frequencies for optimal operation [83,96], granting greater flexibility in real-world applications. They are often robust and durable, and they are capable of withstanding harsh environmental conditions and mechanical stress [83]. Additionally, electromagnetic energy harvesting displays temperature stability, showing resilience against some levels of temperature fluctuations in diverse environments [80,87]. Simplified designs with fewer components contribute to easier fabrication and maintenance, further highlighting the appeal of electromagnetic generators in various settings.

3.3.4. Limitations of Electromagnetic Coupling

While electromagnetic energy-harvesting systems offer numerous benefits, they also present several challenges to consider. These systems can be larger and heavier [80] compared to piezoelectric counterparts, thus potentially limiting their integration into smaller and more compact devices. Moreover, the complexity of electromagnetic designs, which involves components such as coils, magnets, and electronic circuitry, often requires a careful tuning of parameters like coil size, magnet strength, and positioning for optimal performance [70,83,89], thereby increasing manufacturing costs and complexity. Although electromagnetic devices can operate across a wide frequency range [36,62], their efficiency may diminish at lower frequencies, thus restricting their effectiveness in specific applications [64]. Additionally, the cost of electromagnetic materials and components, particularly high-quality magnets and coils, can be significant, which potentially impacts the overall cost-effectiveness of energy-harvesting systems. Maintenance is another concern, as electromagnetic systems may require regular upkeep due to their moving parts and electronic components, thus leading to increased operational costs and potential downtime. Furthermore, electromagnetic coupling relies on the strength of the magnetic field [83], and fluctuations in the magnetic field strength may affect the device’s efficiency and power output [89,96], introducing another layer of dependency and potential variability in performance.

Table 2. A comparative assessment of the different pendulum-based energy harvesting devices.

Ref.	Structure	Conversion Method	Applications	Input Condition	Output Power
[13]	Spring-mounted/ Elastic cord	Electromagnetic	Duffing oscillator	12 Hz	$0.25$ µW
[16]	Automated adjustable	Electromagnetic	Parametric resonance	4 Hz	−
[17]	Elastic cord	Piezoelectric	Uniform circular motion	10 Hz	−
[17]	Elastic cord	Electromagnetic	Uniform circular motion	10 Hz	−
[18]	Automated adjustable	Electromagnetic	Rotating systems	$0.68$ Hz	$4.1$ W
[28]	Automated adjustable	Electromagnetic	Freight train-based railway	90 Km/h (train speed)	$5.321$ W
[51]	Manually adjustable	Electromagnetic	Rotating systems	89 rpm	$9.5$ mW
[58]	Automated adjustable	Piezoelectric	Underfloor energy harvester	5 Hz	$1.25$ mW
[68]	Manually adjustable	Piezoelectric	Human motion	2 Hz	$86.12$ µW
[9]	Elastic cord	Piezoelectric	Asymmetric cantilever beams	$2.03$ Hz	$13.29$ mW
[99]	Spring-mounted	Electromagnetic	Bistable energy harvesters	70 Hz	100 µW
[100]	Eccentric and Wiegan wires	Electromagnetic	Rotating systems	660 rpm	$1.24$ µW
[101]	Automated adjustable	Electromagnetic	Rail track vibration	$9.7$ Hz	1 mW
[104]	Manually adjustable	Electromagnetic	Anti-phase motion	2 Hz	247 µW
[106]	Counterweight	Electromagnetic	Marine environment	$0.75$ Hz	$0.997$ W
[107]	Manually adjustable	Electromagnetic	Railroad tracks and sleepers	3 Hz	5–6 W
[109]	Spring-mounted	Electromagnetic	Absorber- harvester	$3.6$ Hz	−

provides an overview of studies on pendulum-based energy harvesters, including both constant and variable-length pendulums. It focuses on the conversion mechanism, applications, operational conditions, and output power.

Through understanding that the efficiency of energy-harvesting devices hinges on the particular environment or system in which they operate, a groundbreaking energy harvester was introduced, as shown in Section 4. This innovative device employs a variable-length pendulum and integrates piezoelectric and electromagnetic components to generate power from two distinct points within the system. This design stems from adapting the swinging Atwood machine, incorporating multiple pendulums within a 4-DOF system.

4. A Novel Variable-Length Pendulum Energy Harvester Utilizing the Modified Swinging Atwood Machine

This section presents the derivation of a mathematical model for the four-degree-of-freedom mechanical system shown in Figure 18, which was used to simulate an extended swinging Atwood’s machine (SAM) based on the equations of motion derived in [2,7].

As a result of the complexity of the mathematical model and practical considerations, two independent approaches were utilized to deduce the equations of motion governing the dynamics of the proposed prototype SAM. The first approach relied solely on Newton’s second law of motion, while the second approach utilized an energy-based method employing Euler–Lagrange equations. Both approaches were expected to yield an accurate mathematical representation of the investigated dynamical system.

However, before proceeding with the derivation of the mathematical model, certain assumptions were taken into account:

• A damped spring model connects the two pendulum masses $m_1$ and $m_2$;
• The mass of the cord (also called rope) connecting all system masses is insignificant and, therefore, neglected;
• The system is non-symmetric, that is, the two pendulum masses $m_1$ and $m_2$ have more inertia as a result of the kinematic excitation that was applied to their point of suspension, i.e., the right-hand side of the pendulum of the two coupled masses $m_1$ and $m_2$ (refer to Figure 18 for more information);
• The drag forces generated by the system’s masses due to air resistance are neglected.

A spring pendulum was incorporated on the opposite side of the counter mass M, as outlined in [2]. The system includes a Maxwell element-based suspension with two masses, $m_1$ and $m_2$, a stiffness parameter k, and a damping coefficient c between the masses. Point $O_1$ is fixed, while $O_2$ can oscillate along the line $(O,X)$, thus allowing variation in the length $l_1\left(t\right)$, the first state variable, and the double pendulum joints. Length $l_2\left(t\right)$ is the state variable measured after a dynamic extension due to the spring connection with stiffness k between the two pendulum masses. The length $l_{20}$ between the two point-focused masses remains constant under a static gravitational force.

Based on the reviews of energy harvesting when using a variable-length pendulum with piezoelectric and electromagnetic devices, this proposal introduces a 4-DOF, modified, and swinging Atwood’s machine (MSAM) for energy harvesting at two locations. An electromagnetic device was employed at point M, where the counterweight mass was located. Moreover, M is a magnetic plunger that moves freely across the electromagnetic device, converting the pendulum’s mechanical energy into electrical energy, as supported by the findings of various authors, which was detailed in the previous section.

Conversely, a piezoelectric device was attached to the spring between masses $m_1$ and $m_2$. Consequently, energy can be generated simultaneously through the piezoelectric coupling.

4.1. Mathematical Modeling of the Modified SAM Energy Harvester

The derivation of the equations of motion for the 4-DOF MSAM model without the energy harvester is detailed in [2,7]. These works used both the Euler–Lagrange equations and Newton’s second law of motion, resulting in Equation (44), which details the scenario when neglecting the low friction in the pulley bearing.

$$\begin{array}{cc}\hfill {\ddot{l}}_1=& \frac{1}{m_1+M}\left(T_{2}cos({\phi }_2-{\phi }_1)-(m_{1}sin{\phi }_1+M){\ddot{X}}_0+m_1(l_1{\dot{\phi }}_1^2+gcos{\phi }_1)-Mg+\beta I\right),\hfill \\ \hfill {\ddot{l}}_2=& \frac{1}{{2m}_1{m}_2(m_1+M)}((M{m}_1{T}_2(cos\left(2({\phi }_2-{\phi }_1)\right))-1)M{m}_1{m}_2\left({\ddot{X}}_0\right(2cos({\phi }_2-{\phi }_1)\hfill \\ \hfill & +sin({\phi }_2-{2\phi }_1)-sin{\phi }_2)+g)+2m_1{m}_2({Ml}_1{\dot{\phi }}_1^{2}cos({\phi }_2-{\phi }_1)+\hfill \\ \hfill & (m_1+M)(l_1+l_{20}){\dot{\phi }}_2^2)-{2m}_1{T}_2(m_1+m_2+M)+\beta v),\hfill \\ \hfill {\ddot{\phi }}_1=& \frac{T_{2}sin\left({\phi }_2-{\phi }_1\right)-m_1\left(2{\dot{l}}_1{\dot{\phi }}_1+{\ddot{X}}_{0}cos{\phi }_1+gsin{\phi }_1\right)}{m_1{l}_1},\hfill \\ \hfill {\ddot{\phi }}_2=& \frac{1}{2(m_1^2+M{m}_1)(l_2+l_{20})}\left(\right(-M{T}_{2}sin\left(2({\phi }_2-{\phi }_1)\right)-Mg{m}_1(2sin({\phi }_2-{\phi }_1))\hfill \\ \hfill & +sin({\phi }_2-{2\phi }_1)+sin{\phi }_2)-M{m}_1{\ddot{X}}_0(2sin({\phi }_2-{\phi }_1)-cos({\phi }_2-{2\phi }_1)+cos{\phi }_2)\hfill \\ \hfill & -2M{m}_1{l}_1{\dot{\phi }}_1^{2}sin({\phi }_2-{\phi }_1)-4m_1{\dot{l}}_2{\dot{\phi }}_2(m_1+M)),\hfill \\ \hfill \dot{I}+& \frac{R_{m}I}{l_m}=\frac{\beta \dot{l_1}}{l_m},\dot{v}+\frac{v}{R_p{c}_p}=\frac{\beta \dot{l_2}}{c_p},\hfill \end{array}$$

(44)

where $T_2=c{\dot{l}}_2+k{l}_2$, $l_1$, $l_2$, ${\phi }_1$, and ${\phi }_2$—t-dependent state variables; ${\ddot{X}}_0=-{\omega }^2{f}_{0}sin\omega t$; $f_0$—the amplitude of the excitation force; $\omega$—the excitation frequency; ${\phi }_1\left(t\right)$ and ${\phi }_2\left(t\right)$—the pendulum rotation angles, and $l_1\left(t\right)$, $l_{20}+l_2\left(t\right)$ (relatively to its predecessor)—the lengths of each pendulum arm (refer to Figure 18 for more information).

The following dimensionless parameters were considered:

$$\begin{array}{cc}\hfill & A=\frac{l_{20}}{l},G=\frac{M}{\left(m_1+M\right)},G_1=\frac{m_1}{\left(m_1+M\right)},{\omega }_2^2=\frac{g}{l},{\omega }_3^2=\frac{k}{m_1},{\omega }_1^2=\frac{k}{m_1+M},\hfill \\ \hfill & {\lambda }^2=\frac{{\omega }_1^2{\omega }_1}{G_1},{\delta }_0=G\lambda ,c_1=\frac{c}{\left(m_1+M\right){\omega }_1},F=\frac{f_0{\omega }^2}{l{\omega }_1^2},c_2=\frac{c}{{\lambda \omega }_1},b_1=\frac{M}{m_1},\hfill \\ \hfill & b_2=\frac{M}{m_2},b_3=\frac{m_1}{m_2},{\omega }_4^2=\frac{{\omega }_2^2}{{\omega }_1^2},{\omega }_5^2=\frac{{\omega }_3^2}{{\lambda \omega }_1^2},c_2=\frac{c}{\lambda {\omega }_1},{\sigma }_1={\omega }^2\lambda +G{\omega }_4^2-\frac{{\omega }^2{\omega }_1^2{\omega }_4^2}{{\lambda }^2},\hfill \\ \hfill & {\sigma }_2=FG,{\sigma }_3=\frac{F{\omega }^2{\omega }_1^2}{{\lambda }^2},{\sigma }_4=\frac{F{\omega }^2{\omega }_1^2}{6{\lambda }^2},{\sigma }_5=\frac{{\omega }^2{\omega }_1^2{\omega }_4^2}{2{\lambda }^2},{\zeta }_4=\frac{1}{\lambda },{\sigma }_6={\omega }^2{\omega }_1,{\sigma }_7=\frac{{\omega }^2{\omega }_1^2}{\lambda },\hfill \\ \hfill & {\sigma }_8=\frac{{2\omega }^2{\omega }_1^2}{\lambda },{\sigma }_9=\frac{{\omega }^2{\omega }_1^2}{{\lambda }^2},{\delta }_1=\frac{G{\lambda }^3}{{\omega }_1^2}+\frac{5G{\omega }_4^2}{2},{\delta }_2=\frac{G{\lambda }^2}{{\omega }_1^2},{\delta }_3=FG{\omega }_0,{\delta }_4=\frac{G{\omega }_4^2}{4},\hfill \\ \hfill & {\delta }_5=\frac{FG}{12},{\delta }_6=\frac{2G\lambda }{{\omega }_1},{\delta }_7=b_2{c}_1,{\delta }_8=b_3{c}_1,{\delta }_9=\frac{2G{\lambda }^2}{{\omega }_1},{\zeta }_1=\frac{F}{3}+\frac{5G{\omega }_4^2}{2},{\zeta }_2=\frac{{\omega }_4^2}{6},\hfill \\ \hfill & {\zeta }_3=\frac{2}{{\omega }_1},{\xi }_1=\frac{G{\lambda }^3}{A{\omega }_1^2}+2G{\omega }_4^2,{\xi }_2=\frac{G{\lambda }^2}{A{\omega }_1^2},{\xi }_3=G{\omega }_5^2,{\xi }_4^2=\frac{G{\lambda }^3}{A{\omega }_1^2}+G{\omega }_4^2,{\xi }_5=\frac{hFG}{4},\hfill \\ \hfill & {\xi }_6=\frac{G{\omega }_4^2}{12},h=1,{\xi }_7=c_{2}G,{\xi }_8=\frac{2G{\lambda }^2}{A{\omega }_1},{\xi }_9=\frac{2G\lambda }{A{\omega }_1},{\xi }_{10}=\frac{G\lambda }{A},{\xi }_{11}=\frac{G}{A},\hfill \\ \hfill & {\xi }_{12}=2G{A}^{-1},{\xi }_{13}=2G_1^{-1},{\xi }_{14}=A^{-1}.\hfill \end{array}$$

(45)

This equation represents the system of four second-order ordinary differential equations that govern the system’s dynamical behavior, which are expressed in terms of the general coordinates $l_1\left(t\right)$, $l_2\left(t\right)$, ${\phi }_1\left(t\right)$, and ${\phi }_2\left(t\right)$. When considering harvesters, the electromagnetic and piezoelectric coupling equations are also included.

Through using Equation (45), the dimensionless form of Equation (44) can be written as follows:

$$\begin{array}{cc}\hfill & {\sigma }_1-sin\left(\overline{\omega }\tau \right)({\sigma }_2+{\sigma }_3{\varphi }_1\left(\tau \right)-{\sigma }_4{\varphi }_1^3\left(\tau \right))-w^2{x}_1\left(\tau \right)-{\omega }_0{x}_2\left(\tau \right)+{\sigma }_5{\varphi }_2^2\left(\tau \right)\hfill \\ \hfill & -c_1{\dot{x}}_2\left(\tau \right)-{\sigma }_6{\varphi }_1\left(\tau \right)-{\dot{\varphi }}_1^2\left(\tau \right)\left({\sigma }_7+{\sigma }_9{x}_1\left(\tau \right)\right)-{\sigma }_8{x}_1\left(\tau \right){\dot{\varphi }}_1\left(\tau \right)+{\beta }_{1}I-{\ddot{x}}_1\left(\tau \right)=0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}& {\delta }_1-{\sigma }_{2}sin\left(\overline{\omega }\tau \right)+{\delta }_2{x}_1\left(\tau \right)-x_2\left(\tau \right)-b_2{x}_2\left(\tau \right)-b_2{x}_2\left(\tau \right)-b_3{x}_2\left(\tau \right)+{\delta }_{3}sin\left(\overline{\omega }\tau \right){\varphi }_1\left(\tau \right)\hfill \\ & -{\delta }_4{\varphi }_2^2\left(\tau \right)-{\delta }_{5}sin\left(\overline{\omega }\tau \right){\varphi }_2^3\left(\tau \right)-c_1{\dot{x}}_2\left(\tau \right)-{\delta }_7{\dot{x}}_2\left(\tau \right)-{\delta }_8{\dot{x}}_2\left(\tau \right)+{\delta }_9{\dot{\varphi }}_1\left(\tau \right)+{\delta }_6{x}_1\left(\tau \right){\dot{\varphi }}_1\left(\tau \right)\hfill \\ & +{\delta }_0{\dot{\varphi }}_1^2\left(\tau \right)+G{x}_1\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right)+{\dot{\varphi }}_2^2\left(\tau \right)\left(\frac{1}{2}AG\left(\tau \right)+{AG}_1\left(\tau \right)+G_1{x}_2\left(\tau \right)\right)+{\beta }_{2}v-{\ddot{x}}_2\left(\tau \right)=0,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & Fsin\left(\overline{\omega }\tau \right)-{\omega }_4^2{\varphi }_1+{\omega }_5^2{x}_2\left(\tau \right){\varphi }_1\left(\tau \right)+{\zeta }_1{\varphi }_1^3\left(\tau \right)-{\zeta }_{1}sin\left(\overline{\omega }\tau \right){\varphi }_1^2\left(\tau \right)-{\omega }_5^2{x}_2\left(\tau \right){\varphi }_2\left(\tau \right)-\hfill \\ \hfill & {\zeta }_3{\dot{x}}_1\left(\tau \right)-c_2{\varphi }_1\left(\tau \right){\dot{x}}_2\left(\tau \right)+c_2{\varphi }_2\left(\tau \right){\dot{x}}_2\left(\tau \right)-2{\zeta }_4{\dot{x}}_1\left(\tau \right){\dot{\varphi }}_1\left(\tau \right)-{\ddot{\varphi }}_1\left(\tau \right)({\zeta }_4{x}_1\left(\tau \right)+1)=0,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & {\varphi }_1\left(\tau \right)({\xi }_1+h{\sigma }_{2}sin\left(\overline{\omega }\tau \right)+{\xi }_2{x}_1\left(\tau \right)+{\xi }_3{x}_1\left(\tau \right)+{\xi }_7{\dot{x}}_2\left(\tau \right)+{\xi }_8{\dot{\varphi }}_2^2\left(\tau \right))-{\xi }_4^2{\varphi }_2\left(\tau \right)\hfill \\ \hfill & -{\sigma }_{2}sin\left(\overline{\omega }\tau \right){\varphi }_2\left(\tau \right)-{\xi }_2{x}_1\left(\tau \right){\varphi }_2\left(\tau \right)-{\xi }_3{x}_2\left(\tau \right){\varphi }_2\left(\tau \right)-{\xi }_{5}sin\left(\overline{\omega }\tau \right){\varphi }_2^2\left(\tau \right)\hfill \\ \hfill & -{\xi }_6{\varphi }_2^3\left(\tau \right)-{\xi }_7{\varphi }_2\left(\tau \right){\dot{x}}_2\left(\tau \right)+{\varphi }_1\left(\tau \right)({\xi }_9{x}_1\left(\tau \right){\dot{\varphi }}_2^2\left(\tau \right)+{\xi }_{10}{\dot{\varphi }}_1^2\left(\tau \right)-{\xi }_{11}{x}_1\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right))\hfill \\ \hfill & -{\xi }_8{\varphi }_2\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right)-{\xi }_9{x}_1\left(\tau \right){\varphi }_2\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right)-{\xi }_{10}{\varphi }_2\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right)-{\xi }_{11}{x}_1\left(\tau \right){\varphi }_2\left(\tau \right){\dot{\varphi }}_1^2\left(\tau \right)\hfill \\ \hfill & +{\xi }_{12}{\dot{x}}_2\left(\tau \right){\dot{\varphi }}_2^2\left(\tau \right)+{\xi }_{13}{\dot{x}}_2\left(\tau \right){\dot{\varphi }}_2^2\left(\tau \right)-{\xi }_{14}{x}_2\left(\tau \right){\ddot{\varphi }}_2\left(\tau \right)-{\ddot{\varphi }}_2\left(\tau \right)=0,\hfill \end{array}$$

(49)

$$\dot{I}+\frac{R_{m}I}{l_m}=\frac{{\beta }_1\dot{x_1}}{l_m{\omega }_1},\dot{v}+\frac{v}{R_p{c}_p{\omega }_1}=\frac{{\beta }_2\dot{x_2}}{c_p}.$$

(50)

4.2. Simulation Results

The modified SAM energy harvester concept simulation results are presented in Figure 19 and Figure 20. The computation was performed with high accuracy using the following set of parameter values:

$$\begin{array}{cc}\hfill & A=0.5,c_1=c_2={\xi }_1={\xi }_3={\delta }_3=0.01,{\sigma }_5={\zeta }_3={10}^{-4},{\xi }_4=1.61,\hfill \\ \hfill & {\sigma }_6={\sigma }_7={\xi }_{10}={\xi }_{11}=2·{10}^{-4},{\omega }_5={\xi }_2=2·{10}^{-3},{\sigma }_8=4·{10}^{-4},{\xi }_{12}=4·{10}^{-5},\hfill \\ \hfill & {\xi }_5=5·{10}^{-3},{\sigma }_9={\xi }_{14}={\zeta }_4=5·{10}^{-5},{\delta }_2={10}^{-3},{\delta }_4=3·{10}^{-3},{\delta }_5=8·{10}^{-5},\hfill \\ \hfill & {\xi }_7=1.2·{10}^{-3},h={\omega }_0=1,{\delta }_8=8·{10}^{-3},{\delta }_9=9·{10}^{-5},{\delta }_0=2·{10}^{-5},\varpi =10,\hfill \\ \hfill & {\xi }_6=8·{10}^{-5},{\xi }_9=2.1·{10}^{-5},{\xi }_{13}=1.5·{10}^{-4},{\sigma }_1=0.15,{\sigma }_2=0.464,b_2=2.11,\hfill \\ \hfill & b_3=1.63,G=0.8,G_1=0.2,F=0.81,{\omega }_4=1.72,{\zeta }_1=0.05,{\sigma }_3=1.15,\hfill \\ \hfill & w=0.25;R_m=R_p=10.0;{\beta }_1={\beta }_2=1.28·{10}^{-2};\hfill \\ \hfill & v=I=0.5;l_m=C_p=5·{10}^{-3}.\hfill \end{array}$$

(51)

This application demonstrated that the chaotic dynamics of MSAM can be harnessed to generate electrical energy from both bobs of the pendulum simultaneously.

Figure 19 shows the time histories of $l_1$, $l_2$, ${\phi }_1$, and ${\phi }_2$, while Figure 20 presents the time histories of the generated current, voltage, and power of the electromagnetic and piezoelectric devices.

The current and voltage outputs, along with the generated power from the MSAM variable-length pendulum energy harvester, displayed oscillatory behavior with varying amplitude and frequency. These variations depend on design parameters such as system damping, oscillation frequency, and electrical load. Only the results with optimal parameter values are shown here, as optimizing these factors is crucial for maximizing energy harvesting efficiency.

The power generated by the piezoelectric device is more promising and practical than that of the electromagnetic device. This is likely due to the denser oscillations of the piezoelectric device. The disparity may stem from the variable-length pendulum’s design and the electromagnetic device’s placement. However, system design adjustments can optimize power generation.

5. Conclusions

This article provides an overview of energy-harvesting systems that use variable-length pendulums coupled with piezoelectric or electromagnetic devices. The theoretical foundations, mathematical advancements, and real applications of these systems were examined.

This research highlights the adaptability and potential of variable-length pendulum energy harvesters in generating electricity from mechanical energy. These systems capture energy from vibrations and movements using piezoelectric and electromagnetic devices.

Our comparative study of the dynamic modeling outlined the strengths and limitations of piezoelectric and electromagnetic solutions. While piezoelectric devices are efficient and reliable, electromagnetic counterparts offer more flexibility in certain applications.

The impact of various parameters on the functionality of these devices, including excitation amplitude, damping coefficients, and frequency, was covered. Numerical simulations and experimental validations provided insights into the power output and efficiency.

Moreover, this review emphasized the importance of variable-length pendulum energy harvesters as very promising ways through which to access ambient energy. Integrating electromagnetic and piezoelectric devices with a modified SAM generates current and voltage from mechanical motion, with potential applications in various fields.

Future research in this field looks interesting, with new technologies, materials, and designs enhancing the scalability and efficacy of these systems. Subsequent work will involve extensive analytical research, dynamical analyses, stability evaluations, and experimental validations to clarify the possibilities and capacities of these integrated systems.