1. Introduction
Energy harvesting (EH), supplying freely available energy, arouses a lot of interest and attracts the attention of many constructors and designers, especially in the IoT (Internet of Things) systems industry. The predictions about the global use of energy are alarming. According to the IAE report [
1] and further research [
2], it is modeled that the projected average energy intensity will improve by 2.2% per year by 2030. Therefore, any work with rational and efficient energy consumption is very much desired.
The perspective of collecting energy from the environment seems intriguing. In many applications, it allows the elimination of an external power source, thus simplifying the design of the device and its subsequent operation. Energy harvesting is the result of progress in the field of materials and technologies that enable energy recovery from the background from different sources, yet omitted [
3]. The reason for this was the low efficiency of energy conversion and the high cost of manufacturing the devices necessary for this purpose, namely the harvesters. Decreasing the energy consumption of microsystems is also of key importance, which means that energy sources with a power of milliwatts and even microwatts are of practical importance and allow the elimination of traditional power systems using cable systems, batteries, or accumulators of hazardous environmental effects [
4].
At the current stage of development of this technology, it is possible to collect energy from many sources, such as electromagnetic radiation, light, heat, or vibration motion, and then convert it into electricity [
5].
The area of interest of this manuscript is the kinetic energy of vibrations occurring in the environment of the system, which can be used as a source of electrical energy. Mechanical energy harvesting (MEH) can be obtained from human motion [
6,
7], fluids [
8,
9], rotating machines [
10,
11], and many other sources. The most recent studies on MEH in the transport industry involve road systems: vehicle motion [
12], automotive suspension system [
13], rotating energy [
14,
15] as well as railway systems [
16] and their elements (such as railroads [
17,
18] or railway bridges [
19]) or handling equipment (such as elevators [
20]), harbor electrical cranes [
21], gantry cranes [
22], and conveyors [
23]. The ability to obtain energy is becoming a viable way to increase the energy available in vehicles of various modes of transport.
The main methods for the conversion of the kinetic energy of vibrations are distinguished: electromagnetic, magnetostrictive, electrostatic, triboelectric, and piezoelectric methods [
24].
The electromagnetic method uses the phenomenon of electromagnetic induction and the Villari effect, or the inverse of the magnetostriction phenomenon. This effect consists of changing the magnetic parameters of a ferromagnetic because of its deformation [
25,
26]. As a result, the mechanical energy of vibration is converted into magnetic field energy, which is then converted into electrical energy. Magnetostrictive materials convert magnetic energy into elastic deformation energy [
27]. The electrostatic method [
28] requires the use of electrets, i.e., materials with permanent dipole polarization, which are the electrical equivalent of permanent magnets [
29]. Vibrations are transferred to the surface of the electret, causing the generation of a charge through the triboelectric effect, which is the phenomenon of generating an electric charge through friction or the deformation of materials with electrostatic properties.
Piezoelectric energy harvesting [
30,
31,
32], shown in the research part of this manuscript, uses the piezoelectric effect. Electric charges appear on the surface of some types of materials (crystals, ceramics, composites, polymers, and bioinspired materials) under the influence of mechanical stress. These stresses can be generated by vibrations propagating in the environment, resulting in the direct conversion of mechanical energy into electrical energy. Researchers work on techniques to increase efficiency for piezoelectric energy harvesting through a nonlinear method [
33,
34,
35,
36], piezoelectric frequency upconverting [
37,
38,
39], double pendulum system [
40,
41], circuit management [
42,
43], and others.
The energy recovery systems based on the piezoelectric effect are widely described in the literature [
36,
44,
45,
46]. Most of the proposed solutions are based on a kind of vibrating beam—a resonance system with one or several degrees of freedom, in the form of both linear and nonlinear systems [
47,
48,
49,
50], with particular emphasis on nonlinearity aimed at improving efficiency and applying to a wider spectrum of excitations. Linear systems in the form of a simple vibrating beam are characterized by significant limitations because they have only one resonant frequency, which limits their range of applicability. Most often, nonlinearities aimed at changing these undesirable properties are obtained by adding different configurations of magnetic systems in the form of permanent magnet systems or applied systems based on magnetic induction. The use of the aforementioned magnetic systems causes the system to go into a wider range of resonance frequencies, thus increasing efficiency in a wider frequency spectrum to force the vibration signal. Typically, when applied to self-powered systems such as sensors based on IoT technology, magnetic systems are used that give the mechanical system the character of multistable operation, i.e., bistable and multistable systems, allowing for a wide range of their applicability because of different behaviors in a wide range frequency of extortion. These systems have been extensively described and tested in the literature [
51,
52,
53,
54].
In this study, the dynamics of the nonlinear bistable vibration acquisition system were investigated. This system is characterized by the work with two stable states, which were forced by the specific configuration of the magnetic system in the tested system. Because the analysis of the behavior of such strongly nonlinear systems is complicated, numerical simulations, which are presented in the following chapters, turned out to be necessary. The parameter values should be taken into careful consideration because in a nonlinear system, even a small change in their value can lead to changes in the behavior of the system, which is a positive feature in the case of parameter control and forcing a specific behavior of the vibrating system. To conclude, the article considered a two-well potential system based on a vibrating beam with attached piezoelectric plates in a system with a motion converter in the form of a classic crosshead crank mechanism with the possibility of using it on an overhead crane (bottom pulley). Therefore, the test system can be modeled as an elastic beam that is subjected to an initial elastic deformation by introducing the housing base into motion, utilizing a rotary to reciprocate the motion converter. Two permanent magnets are attached to the housing and one at the end of the vibrating beam, which enables bistable operation. The second model to be made is the crane-hoisting mechanism model to relate the operation of the energy-harvesting system to the movement of the crane load.
The main research goal of this paper is to formulate a mathematical model and conduct simulations for a new energy-harvesting device based on a vibrating beam on a crane-hoisting mechanism with a motion transformation system for powering an independent measurement device allowing energy harvesting.
This manuscript is organized as follows.
Section 1 includes the introduction and review of the scientific literature of the manuscript and outlines previous research on electromechanical energy harvesting and the methods applied.
Section 2 describes the research approach and specifies the methodology. It presents mathematical models focused on the selected energy-harvesting device and its configuration properties in the context of energy-harvesting efficiency and its cooperation with the hoisting mechanism of a crane.
Section 3 outlines the results of computer simulations limited to specified initial conditions for the crane-hoisting mechanism.
Section 4 presents a discussion of the results obtained during the analysis performed. The results are visualized with the RMS voltage diagram induced on piezoelectric elements, a bifurcation diagram, and phase portraits, showing the impact of the individual control parameter configuration on the proposed motion energy–harvesting device on the efficiency of energy harvesting. Finally,
Section 5 is devoted to summary and conclusions.
3. Model Test Results
The model tests were divided into two stages: the first includes simulations of the dynamics of the load-hoisting mechanism, and the second concerns the model tests of the M-EHS. Simulations of the dynamic load-hoisting model were performed for the excitation in the form of a step signal forcing the engine to work (
Figure 4), assuming loose ropes in the start-up phase (zero initial conditions). The simulation results are presented in
Figure 8.
Motor start was performed using the step curve described by relation (b) in Equation (4), which reflects immediate engine start.
Figure 8 shows a series of graphs containing data relevant according to the possibility of obtaining energy in the considered mechanism. As can be seen, the sudden start-up and the lack of initial tension in the wire ropes cause a significant increase in the amplitude of the girder vibrations
concerning the static value; thus, the amplitude of the acceleration of the girder center reaches a value of about
. After passing the transient state, the system stabilizes, and the lifting speed
reaches the nominal value of about
and the vibrations of the bridge and the axis of the lower pulley stop after 3 s. The results obtained are typical for cranes, characterized by intermittent motion. However, the use of this type of mechanical vibration with a relatively rapidly decreasing amplitude may turn out to be difficult and unjustified in terms of the hoisting mechanism. Therefore, it is reasonable to use a vibration recovery system in the form of the system shown in
Figure 6 based on the rotation of the lower pulley. In the case of the vibrations of both the bridge and the axis of the lower pulley, the voltage generated by the system will be short-lived, and the accumulation of energy in this way may turn out to be ineffective. In the next stage, the crane model will be replaced with a system of equations representing the operation of the lower pulley and by modifying its operating parameters, such as speed and the parameters of the rotary motion converter system to reciprocate; by changing the parameters of the arm length
and
, their impact on the efficiency of energy acquisition will be examined using voltage diagrams
.
Taking into account the course of the angular velocity of the lower pulley (
Figure 8i), it was assumed that the energy acquisition would occur in steady motion; therefore, it is reasonable to use the proposed Equation (12) to reflect the operation of the pulley.
4. Model Test Results and Effectiveness of M-EHS for Different Excitation Parameters
Model tests were conducted with the energy acquisition system, for which the energy potential is described by two wells (
Figure 7). It was assumed that the system was affected by excitation
with amplitude
(converter arm) and frequency
(angular velocity of the lower pulley). Model studies were performed with the assumption of the adopted data and phenomenological models included in the article. To determine the dependence influence of the adopted control parameters,
(converter arm) and the frequency
(angular velocity of the lower pulley), many tools were used to reveal the impact of the assumed parameters on the efficiency of the energy acquisition of the M-EHS under examination. Some tools were used to reveal the location of both chaotic and periodic zones. For this purpose, two numerical tools were used: bifurcation diagrams and diagrams of the largest Lyapunov exponent [
63,
64]. Both chaotic zones and zones of periodic solutions are presented in the form of a multicolored map showing the distribution of the maximum Lyapunov exponent (
Figure 9).
The map in
Figure 9 covers a wide range of frequencies, starting at 5 and ending in
, thus presenting the dynamics of the system in a wide spectrum. This approach visually characterizes how motion vibration affects the M-EHS. The two-dimensional map shown in
Figure 9 was determined with zero initial conditions:
.
The control parameters, in addition to the frequency, are also the length of the converter arm
, which has a significant impact on the dynamics of the system under examination. In the system, given the Lyapunov coefficients, in addition to zero initial conditions, a specific difference was assumed between the trajectories in the phase space, and its value was assumed as
[
65]. To generate the map shown in
Figure 9,
simulations had to be performed, which is very time-consuming from the numerical point of view and allows obtaining an image of 500 × 500 points.
The values of the coefficient
presented in
Figure 9 define specific solutions, where positive values of
refer to the chaotic dynamic response of the system and where their negative values,
, mean the system response as periodic with appropriate phase trajectories heading to stable points or periodic orbits, which is presented below. When
approaches zero, we are dealing with the so-called bifurcation points (or quasiperiodic solutions—bifurcation). As presented in
Figure 9, in the low-frequency range (lifting speed multiplied by the
ratio), the chaotic zone occurs almost exclusively for the arm length in the range of
to 0.15 m. The widest chaotic zone is observed in the range of medium and high excitation frequencies in the entire range of arm length
. Because the RMS value of the voltage generated on the piezoelectric electrodes was chosen as the indicator of the efficiency of energy acquisition from the M-EHS, the discussed map of the maximum Lyapunov exponents was compared with the diagrams of the RMS voltage in the entire excitation frequency range. As can be seen in
Figure 10, for selected values of the
(0.07, 0.035, 0.01), both chaotic zones and zones of periodic solutions, and the corresponding RMS voltage values, can be clearly defined.
The determined value of
(RMS: root mean square) of the voltage has been compared with the value of the excitation frequency
. For the value
, the bifurcation diagram (
Figure 10) and the corresponding changes in the RMS value of the voltage are presented. In a range from
to
and
to
(
Figure 9 and
Figure 10), there are two chaotic zones for which the values of the generated RMS voltage reach values in the range of
. The M-EHS is characterized by high efficiency in the range of medium and high-frequency excitation reaching the value of
for
.
Subsequently, for and , a decrease in efficiency is observed, which is related to the specificity of the operation of the motion converter (decrease in speed ). For points and , respectively, there are wide chaotic zones, similarly for the range from point to the end of the range of the tested excitation frequency. In the range , a bifurcation in the form of the multiplication of periodicity can be observed, as it can in the range . The least effective system works for , where for low and medium excitation frequencies, the effective voltage reaches only a few volts . As in the previous case, there are four chaotic zones, but in a narrower frequency band. The discussed chaotic zones and zones of periodic solutions are presented below in the form of time diagrams and in the form of attractors allowed to present chaotic regions. Classically, to obtain the geometric shape of the attractor, visualization in the form of a Poincaré section was used.
To identify attractors, the value of the fractal dimension is widely used; however, the purpose of this study is not to conduct an in-depth analysis of the behavior of the system, only its effectiveness, which is why it was limited only to the presentation of selected cross sections on the two-dimensional phase plane, intending to show both multiperiod and chaotic solutions.
Figure 11 presents an exemplary one
and two-period solutions
for
and the corresponding time series (
Figure 12) of beam end vibrations and the voltage generated on piezoelectric electrodes. In this case, the time series of voltage changes are characterized by high repeatability, which results in significant efficiency for the RMS values.
Figure 13 presents two chaotic solutions that perfectly reflect the Poincaré sections for
and
, i.e., for the excitation value corresponding to the considered hoisting mechanism of the tested crane. Thus, it turns out that for the arm length of
and the speed of
, the system generates an effective voltage in the range of approximately
. Accordingly, for the attractors presented in
Figure 14, the time series of the beam vibrations and voltage, as the well as phase trajectories for chaotic solutions, are listed. Reflecting the least effective solution, i.e., for an arm
,
Figure 15 presents two-period solutions
and four-period solutions for
, which can also be observed in the bifurcation diagram (
Figure 10) for specific values of
. For the selected multiperiod solutions, the voltage-time series and the amplitude of the M-EHS beam end vibrations were also compared (
Figure 16). For
, there are also chaotic solutions presented (
Figure 17), for
and for
. In
Figure 17 and
Figure 18, the corresponding phase trajectories and the time series of the voltage on the piezoelectric electrodes and the amplitude of the beam vibrations are shown.