**Preface to "Vibration Energy Harvesting for Wireless Sensors"**

Mechanical vibrations occur in most technical systems in operation. High level vibrations could indicate an overloaded or damaged technical system; these states and behaviours can be monitored or reported. The ambient vibrations may, in turn, be used as a source of energy. The vibrational energy harvesters used in this frame could be an alternative for the supply of low-power autonomous electronic systems for remote sensing of operations. However, a level of energy harvested by using such an approach is usually very low and the whole concept of vibration energy harvesting system operations (including power management electronics and wireless sensors) must be adapted for factual target applications.

This special issue, through twelve diverse contributions, intends to capture some of the contemporary challenges, solutions and insights around the outlined issues and provides an overview on this rapidly evolving topic. The papers collected in the SI represent the variety of energy harvesting sources, as well as the need to create numerical and experimental evidence bases around them. Qualifying and quantifying their performances in terms of energy harvesting levels, as well as their consistency and potential applications, are all reflected in the papers. The importance of fundamental understanding of the ensuing sensors, along with their possible integration within the respective application areas, including those related to the effective communication of measurement data, are also emphasized through this special issue.

In the SI Mech et. al present magnetomechanical harvesting and related possi-bility of data transfer and subsequently investigate a novel hybrid solution focusing on rapid demagnetization [2]. Hadas et al. [3] investigate an electromagnetic harvester for railway applications, where the energy is transferred from the vibration of the rails during their operational conditions, thereby leading to applications that can encompass monitoring. Litak et al. [4] on the other hand focus on the fundamentals of energy harvesting and investigate bifurcation aspects due to nonlinearities in them, whereby different domains of harvesting exist. In particular, the impact of hysteresis is ana-lysed, which should be considered in more detail by the energy harvesting community. Koszewnik et al. [5] demonstrate a smart beam with Macro Fiber Composites (MFC) and demonstrate numerically and experimentally how energy harvesting can be used for damage detection in this field. Machu et al. [6] further investigate the design of en-ergy harvesting-powered sensors through various configurations, creating experi-mental verifications of analytical models, which creates the possibility of developing robust models with minimized computational effort, in-keeping with fundamental physics. Kunz et al [7] focus their efforts towards novel methods to assess the performance of these harvesters in terms of power flow. Bae and Kim [8], on the other hand, approach sensor performance issues in terms of load resistance optimisation for a bi-stable system. Okosun et al. [9] address one of the core sustainable development goals of availability of drinking water and experimentally demonstrate how energy harvesting patches can be used for pipeline leak detection, creating a respective benchmark. The topic of experimental validation is continued in Chen et al. [10] for impact driven harvesters in a magnetic field; for such harvesters, the source can be important, and a comparison between piezoelectric and triboelectric harvesting of en-ergy is investigated by Thainiramit et al. [11]. Finally, Phan et al. [12] provide a short and impactful investigation of electromagnetic harvesters with linear and nonlinear springs.

The dynamism and breadth of the sector is clearly observed in the contributions to this special issue, as is the variety of approaches that are available. We expect the considered sector to move further in an interdisciplinary manner in the near future, giving rise to new sensors, methods of measurement and impactful applications around a range of sectors, established through scientific rigour, along with numerical and experimental benchmarks.

> **Zdenek Hadas, Saˇsa Zelenika, and Vikram Pakrashi** *Editors*

## *Article* **Use of Magnetomechanical Effect for Energy Harvesting and Data Transfer**

**Rafał Mech \*, Przemysław Wiewiórski and Karol Wachtarczyk**

Faculty of Mechanical Engineering, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland; przemyslaw.wiewiorski@pwr.edu.pl (P.W.); karol.wachtarczyk@pwr.edu.pl (K.W.) **\*** Correspondence: rafal.mech@pwr.edu.pl; Tel.: +48-(71)-3202899

**Abstract:** The presented paper describes a method where, with the use of a dedicated SMART Ultrasonic Resonant Power System (SURPS) developed by the authors, a power and data transfer between two devices can be performed at the same time. The proposed solution allows power to be supplied to the sensor, located in a hardly accessible place, with simultaneous data transfer in a half-duplex way (e.g., "question–response"). The power transmission mechanism is based on the excitation of a construction with a sinusoidal wave, with an actuator transforming this wave into useful, electrical power through a harvester device. Data transfer is achieved with the use of the F2F (Frequency Double Frequency) procedure, which is a kind of frequency modulation. To receive optimized parameters for each construction, an original software is developed, which allows the selection of the proper type of actuator, modulation, and frequency.

**Keywords:** smart materials; magnetostriction; Terfenol-D; wireless sensors; ultrasonic system

#### **1. Introduction**

In the past few decades, the development of wearable and wireless devices has been growing significantly. It became possible to reduce the power which is needed to supply these devices to only tens of milliwatts [1]. At those power levels, traditional batteries are limited to only short-term operation, mainly due to dimension limitations. Additionally, in the case of long-term operation, batteries need to be replaced or recharged while, at the same time, undergoing degradation. Meanwhile, other components behind wearable and portable devices improved rapidly following Moore's law [2]. To overcome the problem of traditional batteries, researchers started to work on energy harvesting. This is a technique that can extract electrical power from ambient sources and might supplement and even replace batteries.

Energy Harvesting (EH), originally known as power harvesting or energy scavenging, is a set of techniques that provide electrical energy through energy conversion from different sources, such as mechanical, thermal, solar, and electromagnetic energy and salinity gradients, etc., e.g., [3]. Generally, the main goal is to use sources that are commonly available in the environment, which, in most cases, are undesirable and suppressed (e.g., noise, impact, and mechanical vibration from equipment and constructions and different sources of heat from friction or combustion or as a result of electric current flow and engine cooling, etc.). Energy harvesting is also based on commonly available energy sources (solar light, wave energy, salinity differences, and biochemical processes, e.g., plants), as well as on energy connected with human biology (motion, body heat, etc.). Nowadays, it is said that EH might be a useful source of "low-cost or cost-free" (excluding installation costs) power supply to low-power electric devices [4–8]. Currently, many types of research are being carried out in relation to vast energy harvester networks which provide a relatively large amount of energy in a short time.

One of the sources of wasted energy is structural vibrations. In many cases, it can be a consistent source of energy, even though its amplitude and frequency can vary significantly,

**Citation:** Mech, R.; Wiewiórski, P.; Wachtarczyk, K. Use of Magnetomechanical Effect for Energy Harvesting and Data Transfer. *Sensors* **2022**, *22*, 3304. https://doi.org/10.3390/s22093304

Academic Editors: Vikram Pakrashi, Zdenek Hadas and Saša Zelenika

Received: 16 March 2022 Accepted: 20 April 2022 Published: 26 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

depending on the location. Vibrations in civil engineering constructions, such as buildings or bridges, have low amplitudes and frequencies (0.1 g and 0.1 Hz); at the same time, various small electrical devices, such as ovens, microwaves, and others, have higher amplitudes and frequencies (0.5 g and about 150 Hz, respectively) [9]. In other constructions, such as cars, planes, or helicopters, vibration amplitudes are relatively high, while amplitudes are varied dependent on operation conditions [10,11]. The above conditions have inspired multiple types of harvester to be developed and described in the literature.

Two types of harvester device can be distinguished, i.e., passive materials and active materials. Those which are passive-type harvesters can be divided into electromagnetic and electrostatic. The electromagnetic devices use Faraday's law, and they are built from a coil and permanent magnet. The relative motion of these two elements generates AC voltage [12,13]. The electrostatic devices are variable capacitors with movable electrodes and a dielectric layer between them. The motion of the layers caused by the vibrations induces AC currents [14–17]. In the case of active harvesters, most devices are based on magnetostrictive or piezoelectric (PZT) materials [18,19]. At this point, it should be noted that piezoelectric harvesters are capacitive sources of energy; therefore, they have a high output impedance. This implies that appropriate energy management circuits must be used to be able to supply electrical devices. On the other hand, there are magnetostrictive harvesters, which are inductive. Thus, they can provide low impedance at frequencies characteristic of most common sources of vibrations (described above).

Of the passive vibration energy harvesters, magnetostrictive harvesters supply higher energy density. What is more, a comparison of magnetostrictive devices with those based on piezoelectric material showed that both can generate similar levels of output energy; however, there is no need for additional special power management circuits in the case of solutions based on magnetostrictive material. Among the magnetostrictive devices, the two most common types are the axial type and bending type, based on the stress state in the material. The axial-type devices are usually mounted in places where a large excitation force is provided [20–28]. Because of these high loads, they can generate relatively high power density levels up to even 10 W/cm−<sup>3</sup> [23]. However, to protect core material from damage, proper mechanisms of protection are needed, especially in the case of brittle Terfenol-D [23,24]. Contrary to axial-type harvesters, the bending type of vibration energy harvesting device can be mounted on any source of vibrations [29–33]. However, the output power density is much lower, and the typical level is 10 mW/cm−<sup>3</sup> [30,32]. Additionally, bending harvesters can be divided into three different types: a single-layer magnetostrictive beam, the output of which is relatively small [29,30]; a double layer, the output of which is greater than for single layer [34] but is still relatively small, mainly because of dominant shear stresses; and a composite magnetostrictive beam [31,32], which provides the highest amount of energy of these three types but requires further investigations. It should be noted that, in the case of magnetostrictive harvesters, their efficiency varies and depends on many factors, such as load, frequency of operation, method of mounting, or the material on which the given device is based. In study [32], it was seen that the efficiency of the energy conversion of the proposed device was 16% at 395 Hz; however, in [31], the maximum conversion efficiency was 35%, and it was achieved at an input frequency of 202 Hz.

In the literature on the subject, it can be seen that the research focused mainly on piezoelectric transducers [5–7,35,36]. However, it turns out that, in some cases, a better solution is the use of magnetostrictive harvesters [37]. Taking into account the previous research conducted in this area, the main goals of this research are:


In this work, the axial-type harvester is presented. Such a harvester was chosen after analysis of already developed harvesters, which can be found in the literature. Additionally, such a solution was also connected with the predicted amount of energy which might be generated with the use of this type of harvester. It can be seen, based on the literature research, that higher amounts of energy can be obtained from axial-type harvesters.

#### **2. Materials and Methods**

The results presented in the paper are related to the magnetomechanical effect. The main material used in the research was Terfenol-D (a material with so-called giant magnetostriction). Additionally, the research used material prepared by the authors, which was produced by the suction casting method.

The prepared material was Fe57Co10B20Si5Nb4V4, which is given using atomic notation. The elements used for the preparation of the alloy were melted into uniform material with the use of a laboratory furnace. High accuracy of chemical composition was obtained by using elements of very high purity and using a scale with high accuracy. The weighted elements were mixed and placed in the furnace to create alloy material. After melting the elements three times in the argon atmosphere, the obtained alloy was again heated to the temperature of the liquid and then rapidly sucked into the copper mold, which was constantly water-cooled. The materials prepared with this method were in form of rods.

To achieve the main goal set by the authors, the design of a multiphase magnetostrictive actuator was developed. This device served as an actuator of a multiphase mechanical vibration regulator, which allowed the positioning of objects that are freely placed on the beam. Positioning was performed using mechanical vibrations. It was assumed that the proposed solution will make it possible to supply energy in a controlled manner and remotely perform mechanical work (e.g., displacement in a given direction, rotation of a mechanism, unscrewing an element by pressing a switch). The described process should be possible to carry out with the use of vibrations up to 30 kHz. To accomplish this task with the use of magnetostrictive actuators, the idea of generating phase-shifted vibrations that have the same frequency was implemented. The implementation of this idea consisted of switching on the actuators, which were placed one after the other, and the signals sent by them were shifted relative to each other by an appropriate angle, which was controlled by the developed algorithm. The whole system used feedback obtained from a vibration sensor, thanks to which the presented method allowed the use of the so-called Structural Stiffness Code (CSS).

#### *2.1. Magnetostrictive Actuators*

To be able to generate mechanical vibrations, first, it is necessary to understand the operation of the different types of actuator. The selection of an actuator for a given task depends on the vibration level (PSD—Power Spectral Density), the operating frequency, and the level of amplitude-phase distortions. When we consider devices without moving elements, i.e., solid-state devices, we limit the choice between two types of transducer, namely, piezoelectric (PZT) and magnetostrictive (usually based on Terfenol-D). In the case of piezoelectric devices, it should be noted that they only work in a strictly defined range of resonance frequencies. This range is related to the design of the transducer. In the case of the operation of such devices outside their scope, the actuator overheats, and the ceramic material is usually destroyed. In the case of actuators based on a magnetostrictive material, the frequency range in which they can operate is much wider compared to PZT actuators. The operating frequency of the magnetostrictive actuator can reach 100 kHz [38] and also includes the resonant frequencies of the actuator itself [39]. Additionally, such actuators generate large forces, which, depending on the design and size of the device, can reach up to several hundred kN. Despite the above advantages, these actuators also have some drawbacks, the most important of which are the non-linear operating characteristics, low vibration amplitude, and the limited operating temperature associated with the use of an induction coil. Additionally, the price of such actuators is relatively high when compared

to devices based on PZT. However, due to the abovementioned advantages, it was decided to use magnetostrictive actuators in the solution presented in this paper.

#### *2.2. Magnetostrictive Harvesters*

In the case of magnetostrictive harvester actuators, the most important element is the magnetostrictive core. Such a core may consist of one or more elements, depending on the size, length, and purpose of the device. The material from which the core is made is also an important issue. Such material must be characterized by gigantic magnetostriction (GMM— Giant Magnetostrictive Materials, e.g., Terfenol-D, nano-ferrite cobalt). An additional element is a system that allows you to adjust the pre-magnetization of the material, which are usually properly selected neodymium magnets. The number of elements that make up the core of the device has a significant impact on the frequency of operation. The smaller the number of elements, the higher the operating frequency can be. To optimize the construction of the actuator, the magnetostrictive material and neodymium magnets are arranged alternately.

In the case of magnetostrictive materials, the parameters of the physical fields that affect the material from the outside, i.e., the magnetic field and also the mechanical field, are of great importance. Therefore, to obtain the best working parameters, permanent magnets and appropriate spring systems are used. The springs are used to create initial stress in the magnetostrictive material, the appropriate level of which makes it possible to increase the amplitude of work. In addition, the main task of the permanent magnets is to shift the starting point of the material's operation, thanks to which such devices can work in the linear range of their characteristics.

The actuators/harvesters presented in this paper have relatively large dimensions: the diameter of the device was 44 mm and its height was 47 mm. The geometry was forced mainly by the necessity of applying the appropriate actuator pressure to the tested structure. Inside the casing was a coil with a resistance Rcoil = 5.5 Ω. The devices worked in a wide frequency range from 10 Hz to 30 kHz to find the resonant frequency of the system within which the system obtained the highest voltage values. Additionally, both harvesters and actuators were pre-stressed with a force of 400 N. This value was determined based on experimental tests, during which the magnetomechanical response of the system depending on the applied load was determined.

The above information is based on the authors' extensive experience in the construction and modernization of magnetostrictive actuators [38]. Subsequent constructions and modifications allow more and more power to be obtained; therefore, these devices began to be used as a source of electricity (Figure 1). The amount of electricity, i.e., the values of current and voltage, must be appropriate for the power supply of the sensor and the builtin processor (with a matched converter) and the communication system. In addition to these parameters, the generator voltage/current conditioning system is equally important. Additionally, to properly design the harvester's electrical circuits, it is necessary to know the characteristics of the receiving device.

**Figure 1.** Harvester scheme.

#### *2.3. Electric Transducer*

Harvesters can be divided into direct current and alternating current. DC harvesters include devices that use the thermoelectric or photovoltaic effect. On the other hand, AC harvesters are devices that generate energy from vibrations (magnetostrictive, piezoelectric, and the Faraday effect). Harvesters that use a mechanical impulse (shock) to generate energy are a special case [38]. In the event of an impact, electricity is produced and available for a very short time. Impact harvesters are characterized by a strong current pulse that appears in the form of an alternating current. At the same time, in the generated signal, one can observe the frequencies related to the magnetic resonance between the core and the coil of the device. Harvesters of this type differ in the amount of magnetostrictive material and their form (solid, composite, powdered) in the magnetic circuit. The power that can be obtained from this type of harvester depends on the type of material, layout, and dimensions of the core.

#### **3. Results**

#### *3.1. Multiphase Actuator*

Mechanical vibrations in a wide frequency range were generated with the use of a multiphase magnetostrictive actuator. The system also had an integrated sensory part. The control of the system was carried out using the HERON Advanced Multiphase software. This software allowed us to supply the actuator with control signals, which kept the whole structure in resonance. The vibration controller used an electronic system containing DSP (Digital Signal Processor) and measurement modules, i.e., input–output modules, DDS (Direct Digital Synthesis) generators, and an ICP (Integrated Electronics Piezoelectric) sensor from PCB Piezotronics. A dedicated CDM-P1 device was responsible for conditioning the signal and changing the vibration amplitude.

To generate the so-called multiphase vibrations, it is necessary to use a head that contains many magnetostrictive actuators. The research presented in the paper was carried out with the use of a head consisting of four actuators equipped with Terfenol-D cores and a nanocrystalline alloy prepared by the authors. Such an arrangement of actuators allowed for their analog control similar to that of a typical stepper motor. In addition, a PCB-type vibration sensor, which was located at the central point of the system, was used to measure the vibration values in the tested object. A diagram of the arrangement of actuators with the PCB sensor is presented in Figure 2.

The prototype head was manufactured according to the design shown in Figure 2. The actuators were arranged symmetrically (Figure 2b). In addition, Figure 2c schematically shows the positioning system that used the generated vibrations for a straight rail. This circuit worked with a feedback loop, which is described below.

Figure 3 shows the vibration control system with the designed head. Importantly, in the case of this system, it was possible to control each actuator separately, thanks to which it was possible to generate even very complex mechanical vibrations. In addition to the designed head, the vibration controller consisted of many advanced electronic systems based on the HERON card with a floating-point DSP and the Texas Instruments C6000 card with expansion modules. Moreover, the system was supplemented with dedicated software using API.

(**c**)

**Figure 2.** The idea of the four magnetostrictive actuators (A1–A4) system: (**a**) actuators operating simultaneously with vibration measurement path; (**b**) symmetrically distributed actuators; (**c**) linear positioning system using vibration.

**Figure 3.** Elements of the control system based on Hunt Engineering DSP.

The Hunt Engineering HERON system was responsible for signal processing in the system. Thanks to the use of this system, it was possible to comprehensively design the experiment, because the software allowed for the acquisition and conditioning of the sensor signal and the generation of phase-shifted control signals. For this purpose, both a DAC (Digital-to-Analog Converter) and a DDS were used.

The PCB sensor received vibrations in the form of electrical signals and then, after being appropriately supplied, they were collected by a module of 16-bit ADCs-HEGD12. They were used as the basis for the determination of the setpoints of the digital vibration controller which released subsequent DDS values in the feedback loop through HEGD4. The role of CDM-1P was only the conditioning of sensor signals and power amplification for the four magnetostrictive actuators.

The HERON Advanced Multiphase software used in the system was applied to maintain the structure in resonance, although the resonant frequencies were changing. To achieve such an effect, the system generated a control signal for actuators with the appropriate frequency of resonant vibration. Determining the appropriate resonant frequency was possible based on the analysis of the signal with a given moment of frequency. In the case of a decrease in the vibration amplitude at a given excitation, a deviation from the resonance state was found. In the next step, the system checked how the system would behave in the case of excitation with a higher and lower frequency. If, in any of the cases, the amplitude of the vibrations increased, then the system defined this frequency as the new resonant frequency. In the next step, the operation of checking the amplitude value was repeated.

#### *3.2. Remote Object Positioning*

One of the tasks that were assigned to the designed system was the positioning of the object on the structure with the use of vibrations. To carry out this task, in the first stage, it was necessary to generate vibrations with a resonance frequency in the structure. Then, a non-magnetic object weighing about 30 g was placed on the vibrating structure (a steel beam 6 m long). The vibrations caused by the beam set the mass in motion—the element was jumping on the beam and hitting it. These impacts were used to determine its position on the object through changes in the phase shifts in the generated signals.

Due to the application of the acceleration sensor in the developed multiphase head, it was possible to register the acoustic events that occurred when the mass separated from the beam. Due to the small mass of the object (20 g), the frequency of the system only slightly changed (0.04%). In addition, the previously described acoustic event related to the separation of the mass from the beam had a much higher frequency than the resonant frequency of the beam itself. The changes in vibrations were recorded as a sinusoidal signal with a frequency of 667 Hz.

#### *3.3. Code of the Structure Stiffness—CSS*

The above-described acoustic event that occurred in the system during the impact of the mass against the beam was characteristic of each structure. In the event of a change of mass or construction change, the response changed. It was related to the different amounts of energy accumulated over time. This response served as a control signal for the vibration controller. In addition, the ability to control the amplitude of vibrations and measure the response of the structure made it possible to detect changes in it, including the appearance of defects or damage.

Based on these tests, it can be concluded that the number and nature of recorded acoustic events are mainly influenced by the energy (in the form of mechanical vibrations) supplied to the tested structure and the temporary state of the structure in which they are located. Hence, the sequence of acoustic events as a modulated binary waveform (via F2F-frequency/dual-frequency modulation or modified MFM frequency) depends on the changing parameters of the medium in which the vibrations propagate, including propagation obstructions. The obtained results were mainly influenced by: the type of

mechanical structure (its stiffness), the frequency of induction characteristic for any of the structural elements, and material defects such as pores or cracks. The resulting sequence of acoustic events was characteristic of each object and depended on the energy supplied to the object. This relationship was referred to as the Structural Stiffness Code (CSS). The use of this relationship and the ability to correctly interpret the received binary signal enabled the operation of the vibration controller in a wide frequency range, thanks to which it was possible to move the object with a specific algorithm.

Figure 4 shows the application of the CSS method. As a result of moving the object along with the structure, it was possible to isolate several acoustic events in the signal of the ICP sensor. The essence of the CSS method is the determination of a binary series based on the duration of acoustic events and the appropriate energy released into the system by a moving object. The main influence on the amount of released energy is the period in which a high level is maintained from the initiation of a given event. In this way, a unique code connected with the state of the structure can be obtained. Work on this code and its use are the subjects of further work.

**Figure 4.** Decoded signals from ICP sensor, where the numbers from 1 to 13 represent code of the structure stiffness (CCS).

The number of acoustic events is characteristic of a specific state of the structure. Thanks to the use of a vibration regulator in a wide frequency range, it is possible to determine a specific structure "code" and to determine the dynamic characteristics of a mechanical structure. To accomplish this task, it is necessary to use a programmable pure sinusoidal excitation.

This issue requires further research, but, even at this stage, according to the authors, it can be stated that it was possible to develop a unique method that allows for the positioning of objects and the assessment of their structure/condition. The measurement technique based on the proposed solution can be an alternative to vibroacoustics and can potentially be used in SHM (Structural Health Monitoring).

#### *3.4. Power and Data Transmission*

In previous research by the authors, it turned out that SMART magnetic materials can be effectively used for wireless transmission of power and information [38]. The achieved results also indicated the high effectiveness of this method. The system that allowed for simultaneous data and power transmission was developed by the authors and is called SURPS (SMART Ultrasonic Resonant Power System). This system provides transmission through various solids, as well as through liquids. An additional advantage of the system is the possibility of using various transmitter–receiver configurations [38].

The results presented in this paper are a continuation of work on the use of the magnetomechanical effect in the case of energy harvesting which was more widely described in [38]. As was shown earlier (Figure 2), four actuators were used instead of one. This was due to the desire to check whether it was possible to transmit energy continuously during data transmission. For this purpose, the Phase Shift Vibration Algorithm (PSVA) was used, which is schematically presented in Figure 5. In the case of this algorithm, the vibrations caused by the actuators were out of phase with each other, while each of the harvesters received a signal with a specific phase.

**Figure 5.** Power transmission through ultrasonic vibration—scheme.

The transmission was carried out by an actuator that transmits mechanical energy in the form of a pure sinusoidal ultrasonic wave and then this wave was picked up by the harvester, which converted this wave into an electric current using the magneto- or electrostrictive material contained therein. This way of transmitting energy also allowed information to be sent. What is more, it was possible to send energy through different materials, and the choice of material depended mainly on the distance the energy needed to be sent.

To transmit the information, the F2F procedure was used, which is a type of frequency modulation. The modulation worked in such a way that the data transmission frequency was one order lower than the resonant frequency of the structure. Figure 6 shows, schematically, how the data were transmitted via a magnetostrictive (AT) actuator and how the signal was received on the harvester with a magnetostrictive core. In the case of using a single harvester, it is visible that there was a break in the collection of energy while receiving the signal, which means that the harvester was not powered at the moment. In the case of using more harvesters, this was transferred to each of them. The use of PSVA allowed the creation of a dedicated circuit in which each harvester received the signal in the appropriate phase so that, when data were sent to one of the harvesters, others were still powered continuously.

**Figure 6.** Data transmission and receiving information for PSVA.

#### *3.5. System Structure*

The SURPS was designed to work with various actuator–harvester configurations. One such configuration is a system where harvesters are connected in series and are located between two parallel beams. Such a system is characterized by a resonance frequency above 20 kHz. The test stand, consisting of two steel rails with magnetostrictive transducers between them, is shown in Figure 7. This system made it possible to simultaneously power the microprocessor on the side of the energy harvesters and transmit data in both directions.


**Figure 7.** Actuator–harvester magnetostrictive system based on two beams.

Based on the above-described solution, supplemented with the current state of knowledge in the field of ultrasonic techniques and wireless power transmission, an original and innovative transmitting–receiving system was developed, equipped with a microcontroller, frequency modulators, and dedicated proprietary software. Figure 8 shows a schematic overview of how the system works. The developed system has several variants, depending on what medium is used for data and energy transmission. More information on the tested system can be found in [3].

**Figure 8.** A block diagram of SURPS structure.

The main applications of SURPS encompass:


The results showing the frequency-amplitude characteristics for the circuit shown in Figure 7 are presented in Figure 9. It is worth noting that the highest voltage value (the highest efficiency) was achieved for the frequency located in the above acoustic band (above 20 kHz). The zone defined as SW in the figure defines the acceptable range of resonance frequencies and is approximately 20 kHz. The dashed line shows the voltage value at the level of 2.5 V, above which loss of the microprocessor system occurred. In addition, point A mark the most advantageous frequency ranges in the case of broadcasting information. As can be seen, several frequency ranges could be distinguished that allow the system to be powered and, depending on the conditions and needs, choices could be made between the required ranges of carrier signals. It should also be mentioned that it was possible to connect more microprocessors to the harvester network, but, in that case, it was necessary to follow a strictly defined sequence of their activation. This solution allows the system to be used in SHM (Structural Health Monitoring) applications with numerous sensors.

**Figure 9.** Frequency response of the dual-beam system.

The system was tested using a Gecko microprocessor chip with a 32-bit Cortex-M3 EFM32TG840 processor from Energy Micro. The solution for energy and information transmission proposed in the paper had to allow for powering the Gecko system, as it is a typical system used in industry. Additionally, the proposed system had to ensure a simultaneous half-duplex transmission. This meant that data were sent in one direction and at a certain time but in a two-way channel. The processor software was developed by the authors of the paper and supplemented with numerous useful functions.

The F2F-AM algorithm was used for data transmission. This algorithm guaranteed the stability of the information flow and allowed the system to achieve the transmission value at the level of 1 kbps, which was a sufficient speed in the analyzed case. To obtain higher bit rates, other frequency modulations had to be used. Figure 10 shows the results achieved for the test stand shown in Figure 7. The signal was transmitted by a magnetostrictive actuator, and the nature of the signal was sinusoidal with slight harmonic distortions. The frequency transmitted by this actuator was modulated in the on–off mode, which means that the actuator was turned on and off, which caused temporary shortages in the power supply of the energy harvester. To ensure continuous operation of the microprocessor on the harvester's side, the system was equipped with a set of capacitors, the capacity of which was sufficient for 0.5 s of microprocessor operation. Thanks to this solution, it was possible to continuously power the microprocessor despite the temporary shortages of power supplied by the harvester and to transmit many bits of data encoded in ASCII.

The results obtained during the research showed that the developed proprietary SURPS enables the transmission of energy over distances up to 6 m without the need for wires and using only various types of mechanical structure. This solution allows the use of various types of harvester in many configurations, while the selection of the appropriate harvester system is influenced by the material and form of the medium through which energy and data are transmitted, as well as the ultrasonic wavelength.

To obtain the highest possible efficiency of both information and energy transmission while maintaining a low level of generated noise, the original software was developed. This software allows for the selection of an appropriate actuator type for a given design, as well as modulation and the recommended frequency band, thanks to which it is possible to precisely determine the values of resonance frequencies. Finally, the software selects the resonance range in which the transmission is most effective.

**Figure 10.** Graphic presentation of the results of power and data transfer.

#### **4. Conclusions**

The paper presents the results of works devoted to the transmission of energy and information through various centers. The results achieved include the following:


The results presented in this paper are current and constitute the basis for further work in the field of energy and data transmission. One of the main scopes for future research is the usage of controlled vibrations made by the designed actuator in NDE diagnostics. The presented method can be considered as an expansion of the methods known as energy harvesting due to the possibility for it to transform various forms of energy supplied to/collected from the system to perform mechanical work, e.g., object positioning.

**Author Contributions:** Conceptualization, R.M. and P.W.; methodology, P.W.; software, P.W. and K.W.; validation, R.M. and P.W.; formal analysis, P.W.; investigation, R.M. and P.W.; resources, P.W.; data curation, K.W.; writing—original draft preparation, R.M. and P.W.; writing—review and editing, R.M. and P.W.; visualization, P.W.; supervision, R.M.; project administration, R.M.; funding acquisition, R.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by The National Centre for Research and Development within the project "Composite magnetostrictive-nanocrystalline materials for use in the field of energy harvesting and transformation", grant number LIDER/21/0082/L-9/17/NCBR/2018.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data supporting reported results can be provided upon request. Currently, these data are collected as part of the ongoing project and only after its completion will the data be made available to the public.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


### *Article* **Rapid Demagnetization of New Hybrid Core for Energy Harvesting**

**Rafał Mech \*, Przemysław Wiewiórski and Karol Wachtarczyk**

Faculty of Mechanical Engineering, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland; przemyslaw.wiewiorski@pwr.edu.pl (P.W.); karol.wachtarczyk@pwr.edu.pl (K.W.) **\*** Correspondence: rafal.mech@pwr.edu.pl; Tel.: +48-71-320-2899

**Abstract:** This paper presents the results obtained using the rapid demagnetization method in the case of an NdFeB magnet and a new hybrid core. The developed core consists of three basic elements: an NdFeB magnet, Terfenol-D, and a specifically developed metallic alloy prepared by means of a suction casting method. The main goal of proposing a new type of core in the event of rapid demagnetization is to partially replace the permanent magnet with another material to reduce the rare-earth material while keeping the amount of generated electricity at a level that makes it possible to power low-power electrical devices. To "capture" the rapid change of magnetic flux, a small number of coils were made around the core. However, the very low voltage level at very high current required the use of specialized electronic transducers capable of delivering a voltage level appropriate for powering a microprocessor system. To overcome this problem, a circuit designed by the authors that enabled voltage processing from low impedance magnetic circuits was used. The obtained results demonstrated the usefulness of the system at resonant frequencies of up to 1 MHz.

**Keywords:** SMART materials; magnetostriction; Terfenol-D; energy harvesting

#### **1. Introduction**

Interest in energy harvesting (EH) is continuously increasing from year to year. The primary impulse for development in the discussed area is related to the shortage of electricity and the search for new possible sources of energy. In this context, the idea of EH would also include the generation of energy from wind, solar plants, and other natural power sources on a large scale. In the literature, the discussion of energy harvesting is limited to small electrical and electronic devices that can operate in a self-sufficient manner [1–3]. On the basis of this limitation, it has become clear that EH cannot be considered as a feasible power source in high-power applications. However, as could be predicted, increasing the number of devices that can be powered from harvested energy would make it possible to relieve other energy sources [4–8]. Obtaining energy for devices from the environment has found possible applications in many different markets, and the number of possible applications will grow continuously [9–13]. Currently, the consumer device market is not large, but it is predicted that it will increase continuously, and will have an influence on the direction of future research. Important features and attributes have had, and still have, an influence on the directions of EH development and potential target markets, including:


Due to their small size and lack of need for complicated powering assemblies, these devices can be used in difficult-to-access environments [14–16], such as in complicated

**Citation:** Mech, R.; Wiewiórski, P.; Wachtarczyk, K. Rapid Demagnetization of New Hybrid Core for Energy Harvesting. *Sensors* **2022**, *22*, 2102. https://doi.org/ 10.3390/s22062102

Academic Editors: Zdenek Hadas, Saša Zelenika and Vikram Pakrashi

Received: 10 January 2022 Accepted: 4 March 2022 Published: 9 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

machines, structures, etc. Research on the use of EH devices in very fragile environments such as the human body was performed at Southampton University Hospital in the UK. A prototype of an electrodynamics harvester was created that was able to generate enough energy from heartbeats to supply power to a pacemaker for the entire life of a patient. The output was 4.3 micro joules per heartbeat; with the use of new and better polymer materials, that value is expected to double [1].

Additionally, self-sufficient devices mean less cable. This applies mainly to sensors, and is a very important advantage, making monitoring constructions, machines, and different types of elements much easier and safer to use, while providing better information for decision-making processes like with respect to the maintenance needs of an object [17]. These advantages were noticed by the American Society of Civil Engineers, which together with the University of Texas (UT), began testing a new wireless bridge monitoring system [18]. Another system for monitoring the condition of railway bridge structures was presented in Ref. [19]. Sensors that do not need a separate powering system make it possible to test bridges more quickly and receive more detailed information in real time. The importance of such kinds of systems is evidenced by the fact that in America in 2008, over 17,000 bridges did not meet the requirements necessary for operation.

The research and development on generating energy is progressing very quickly, opening new markets for a growing range of different devices. Consumers and the military, along with third-world markets, are the main groups, and will create the biggest market for devices by value. Their share accounted for around 88% of all value predicted for such kinds of device in 2014. As projects are run mainly for profit (including both present and future profits, both financial and strategic), it seems to be significant to realize the importance and the full potential of the EH concept and technology. According to the report presented by IDTechEx Ltd. (Cambridge, UK) [20], these potential markets could represent an income of about USD 1.156 trillion per year.

The main idea of the presented research was to determine whether the chosen types of harvesting device and the proposed type of excitation could be useable for mechanical purposes. The main goal of this investigation was to determine whether the new proposed hybrid core structure, when applied in dedicated harvesters, is able to provide sufficient energy to supply the chosen ATMEL microcontroller. The tests were carried out on a dedicated stand, allowing the impulse load to be applied with a fixed value. The results of the impulse investigations are discussed in detail to provide information about the usefulness of the proposed hybrid core in the area of energy harvesting.

#### **2. Materials and Methods**

The study presented in this paper was based on a well-known giant magnetostriction material, Terfenol-D, and on a material prepared by the authors, produced by the suction casting method. In addition, commercially available NdFeB neodymium magnets were also used in the investigation.

#### *2.1. Suction Casting Alloy*

The newly created material in the form of a metallic alloy described in this paper was developed based on literature data describing a patented soft, magnetic, amorphous material known as FINEMET [21,22]. It should be noted that the material with the composition Fe57Co10B20Si5Nb4V4 described in this paper was developed in the opposite direction to what has been the case in many studies showing materials based on FINEMET. The material presented in this work has a lower content of the basic elements, such as Fe and Co. It should also be emphasized that when describing the material, its composition is given in atomic notation, not mass notation. An arc furnace (arc-melter) was used to produce the material. To ensure the highest possible representation of the developed chemical composition of the alloy, very high-purity elements—up to 99.999%—were used for its production. Each of the alloying elements was weighed out to 4 decimal places, and then these components were placed in the arc furnace chamber. Next, the furnace chamber

was sealed, and then the air was removed from it by pumping to a vacuum of 5 × <sup>10</sup>−5, after which the entire chamber was flushed with argon and pumped down again. After several chamber flushing processes, the alloy was prepared by melting all the elements several times to homogenize them in the entire volume. Finally, the liquid alloy was sucked into a specially prepared mold, which made it possible to obtain rods with a diameter of 3 mm and a length of 150 mm. The material prepared in this way was then subjected to mechanical treatment in order to be able to place it in devices for energy harvesting.

#### *2.2. Ferro Magnetic Generator (FMG)*

The tests performed in this work are mainly based on the principles of Ferro Magnetic Generators (FMG). As is generally known, the working principle is based on ferromagnets that are demagnetized either by shock loading or from the impact of high-speed flyers. The same effect might also occur as a result of the detonation of high explosives. The team at the Agency for Defense Development in Korea presented the output characteristics of annulusand cylinder-type explosion-driven devices based on NdFeB magnets [23]. In this paper, the authors decided to use similar principles to FMG to obtain energy that could possibly be used to supply microcontrollers, but without the use of an explosion-driven device, and only with a force impulse. The force impulse causes a change in the magnetization around the material, then this change in the magnetic field is picked up by the coil and converted into electric energy.

In the case of a magnet, during an impact, it will be demagnetized (partially), and with a sufficiently large number of impacts, it could be fully demagnetized. In addition, in the event of an impact with sufficiently high energy, it could even be destroyed, generating a high amount of energy. On the other hand, the results showed that the use of magnets alone as a source of energy may be insufficient, as can be seen from the results shown in this paper.

#### **3. Results**

This section is divided into appropriate subsections, allowing the presentation of the course of the processes and the evolution of subsequent solutions, leading to the achievement of the aims of the study. The first part of this section describes the devices for energy harvesting that were then used in the experimental activities described in the second part.

#### *3.1. Harvesters*

The research object is a unique basic Energy Harvester Device (EHD) model, shown in Figure 1. The model, called TCCM (Top Core Coil Magnet), is a construction consisting of four major elements: the Top, the role of which is to transfer the shock to the core, and the Coil, Magnet, and Core, which process the impact energy passed from the top, transforming it into electricity. For the aforementioned shock, a reference shock with specific repeatable parameters was used, thanks to which it was possible to test the harvester cores under the same working conditions. However, it should be noted that these devices can also operate in the case of various oscillating systems (with vibrations) of appropriate amplitude.


As can be seen from the cross-section (Figure 1), TCCM harvesters can be considered to be one of the simplest models, especially since they do not have a system allowing the introduction of prestress, which makes it a structure with relatively low stiffness, especially when the force impulse is given. It should be noted that due to the simplicity of this structure, the device allows a large number of different materials to be tested relatively quickly for suitability in the field of energy recovery.

**Figure 1.** TCCM harvester scheme.

The construction of a more advanced version of TCCM called Double Top Core Coil Magnet (DTCCM) is shown in Figure 2. This is a modified version of the TCCM harvester in which two magnetoelectric circuits are applied. The magnetic circuit of this device is also based on neodymium magnets, and it has low dispersion of magnetic field on the outside due to the permendur plates.


Additionally, different arrangements of cores were also used.

#### *3.2. Test Stand and Measurements*

Figure 3 shows the scheme of the stand developed to perform impact tests for energy harvesting devices, in particular, TCCM devices. A horizontally mounted PZT sensor was used to measure the impact force. The sensor was placed on a rigid, non-deformable surface of non-magnetic plate. To generate the impulse force in the device, an aluminum rod was used, which was mounted so that it always hit the center of the upper part of the device. The velocity of the rod movement, and thus the value of the impact force, was regulated. A fast MOSFET transistor interacting with a linear motor was used for regulation, which made it possible to set a precisely defined velocity. Additionally, the impact energy can be changed by changing the load applied to the moving element using weights of a known mass. The mass of each weight was determined with an accuracy of four decimal places. A maximum of 2 kg can be placed on the movable element, but it was decided to apply 0.5 kg due to the small size of the part with the aluminum rod. Thanks to the applied solutions, it was possible to obtain repeatable values of impact energy Ek.

**Figure 2.** DTCCM harvester: (**a**) scheme, (**b**) real device.

WRDFTXLVLWLRQDQGFRQWROVXEV\VWHPRI3LFR3RZHU'HYHORSPHQW3ODWIRUPEDVHGRG+XQW+HURQ6\VWHP 706&'63

**Figure 3.** Test stand for TCCM harvester investigations.

The prepared system made it possible to test the cores placed inside the coil of the TCCM device. The impulse force resulting from the movement of the hammer in the form of the aluminum rod is transmitted to the core of the device through its upper part (Top). Figure 4 shows the response of the system during the experiment. Performing the analysis with respect to the experimental process, it is possible to specify individual phases within the experiment. The first phase is the phase just before it hits the device, where the voltage can be observed to rise. This change in voltage is probably caused by the movement of the ram in the magnetic field of the applied neodymium magnet. The next phase is the impact, which causes a shock wave to go through the consecutive elements of the harvester, to the magnet located at the bottom. This changes the magnetic flux, thus inducing a voltage on the used coil.

**Figure 4.** Example of a waveform obtained from TCCM.

In the case of the TCCM harvester, only the top and the core of the system affect the resonance frequency of the system. The shock wave generated by the impact can circulate in the core–magnet system until it is completely suppressed, or exit the system through a magnet that is in contact with another rigid surface. However, it should be noted that this wave is not transferred to the coil itself. It can be observed that the mechanical resonance of the device resulting from the impact is a decisive factor for the signal obtained from the device.

In the case of the energy harvesting device used in the experiment, it was found that the prestress system did not need to be used. It was observed that the prestress effect was obtained during the first moments of impact. This was also the moment at which the maximum response values of the tested system were obtained. It should be noted that the values of the generated voltage were dependent on the applied electromagnetic system, which was in the form of a coil. When applying the same impact energy, the obtained voltage values, apart from the materials used for the core of the device, were also dependent on the number of turns in the coil.

The main goal of this investigation was to show the differences between the materials used to produce the core of the harvester in terms of the current values obtained for each of them, as well as with respect to the reaction of the PZT sensor and TCCM for impact. The final research step was to perform the core selection, which is a significant element of the harvesting device.

The original current signals from the coil were windowed using the Hamming windowing function. All current measurements were performed using a sampling rate of 1 MHz. In line with this, FFT analysis was performed for a spectrum of 500 kHz. Due to the applied windowing, the analysis of the spectrum was narrowed to 5 kHz and 10 kHz when using Terfenol-D as the core material.

As an output of the impact (the applied shock force), the velocity and voltage were determined, and their values are presented in Figure 5, where a correlation between the output signals and the actual test phases is presented.

The first phase of the test encompassed the acceleration of the aluminum ram up to a velocity of 1.1 m/s, Ek = 1.21 J, as shown in Figure 5. The second phase was the moment of impact, upon which the ram rapidly slowed down, and its reflection moment was controlled and damped. In the worst scenario, when the ram was not aligned with the harvester axis, vibration occurred throughout the whole device. On the graphs in Figure 5, the moment of impact is visible. A rapid increase in the current value appears at the moment of impact.

In Figure 6, the results of an experiment with a series of four impacts with the same parameters and a TCCM harvester are presented. Analysis of the signal from the PZT piezo patch revealed the nature of the impact, as well as the reaction of the surface to which the harvester was fixed. However, according to the fact that loading the PZT with an impedance value of 1 MΩ decreases the surface signal one hundred times to the order of a few μV, the practical influence of that signal is negligible. By registering the signal from the PZT, it was possible to control the repeatability of the shock tests, except that, in the future, it will be possible to use that signal as a coupling to change the stiffness of the next generation of harvesters, with the intention of absorbing the maximum amount of energy from the impact with minimal transmission of that energy to the surface. However, the repeatability of the signal in the PZT did not have a place in the current signal of the coil. The reverse magnetostriction effect (Villari's effect), and ensuring that the system has the appropriate operating parameters are very important. Even small changes in the impact conditions, mainly differences in the ram or the core, resulted in significant changes in the current values obtained. The operating parameters of the system are also greatly influenced by the external bias field, the pre-stress, and the impact energy.

**Figure 5.** Impact correlated with graphs of velocity and output voltage.

It was important to assess the influence of the state of aggregation (solid, powder) of the core material on the output values. The main criterion for analyzing this influence was the absolute value (the ideal looseness straightens the AC signal) of the current during the 1 ms following the impact moment.

The fundamental frequencies of mechanical resonance are the second parameter describing the core composition. The coil resistive load value Rc = 125 Ω was determined using the math library package Numerix SIGLIB v6.0, implemented in the Agilent VEE Pro software.

During the impact tests, the couple–core–magnet system reacts rapidly, while the coil's body is not expanded by the core material. When the impact energy becomes critical, Ek max, an irreversible change in system parameters occurs, resulting in the deformation of the coil body. This becomes an important issue when the core is in a powder state, because the powder adapts itself to the shape of the coil.

During the investigation, it could be seen that the core material had a crucial influence on the results of the tests. These results are shown in Table 1, where the differences between four measured parameters can be seen. It can be observed that the highest value of Imax was obtained for the Terefenol-D rod with small pieces of the prepared Fe57Co10B20Si5Nb4V4 alloy used as the core. However, due to the high cost of Terfenol-D and its brittleness, the number of tests was limited. As an alternative for a solid rod made using that material, a powder form that was capable of withstanding impact energies greater than those achievable for steel and NdFeB was used. Additionally, the Terenol-D powder was partially mixed with powdered Fe57Co10B20Si5Nb4V4 alloy. The results of the impact tests for the powder core are shown in Figure 7. The only limitations of the

experiment were the performance of the body and the durability of the coil windings. However, these two parameters were taken under consideration.

**Figure 6.** Reaction on impact: the PZT sensor signal, current of coil for TCCM, and their FFT spectrums.

**Table 1.** Comparison of different harvesting cores with parameters as follows: TotalLoad = 125 Ω, Ek = 5.6 J, Rc = 180 Ω for V = 6.7 m/s, and m = 0.5 kg.


Where:


<sup>μ</sup> <sup>μ</sup>

**Figure 7.** Example of the coil current measured for solid Terfenol-D with Fe57Co10B20Si5Nb4V4 alloy pieces, as well as its FFT.

The test performed using powdered Terfenol-D mixed with Fe57Co10B20Si5Nb4V4 alloy as the harvester core, even without to the application of prestress, showed that with this type of core, it was possible to achieve higher results for the investigated parameters than when using the other listed materials. The lowest results were obtained for the sintered ferrite-type core.

The tests carried out using the TCCM devices made it possible to perform a comparison of the different types of materials used for the harvester core. However, it should be noted that regardless of the material used, the amount of energy obtained from the device was insufficient to power the microcontroller. Therefore, in the next step, it was decided to modify the test stand to carry out tests with the DTCCM harvester type. For this purpose, the PicoPower platform was specifically developed by the authors, based on the Hereon Hunt system. The block diagram for the research stand is shown in Figure 8. The data acquisition system that was used played a significant role in the stand. This system made it possible to obtain data with a very low level of noise in real time from all sources plugged into it.

The main component of data transfer between the Heron DSP chip and the host was one FIFO buffer that was connected to the PCI interface on the motherboard. The HEDG12 module was used to acquire the coil parameters in the form of current and voltage. This module is a 16-bit ADC converter with eight channels. Thanks to the use of this module, it was possible to acquire data on all eight channels using sigma-delta conversion.

The sigma-delta conversion technique used made it possible to eliminate analog antialiasing filters from the measurement path. Simply put, this technique oversamples the waveform that appears at the input eight times. The next step is to digitally filter the over-sampled data. Finally, the resulting samples are reduced eightfold.

The devices presented in this paper are not intended to serve as continuous energy sources for microcontrollers. The main purpose of the developed harvesting devices is to provide a sufficiently large energy pulse to be able to charge a large-capacity capacitor. However, this does not mean that they cannot act as such a source of energy. For the devices shown in the paper, the manner in which they work will depend on the system specification. If the system is intended to send signals every few milliseconds, then the amount of energy supplied may not be large enough to accomplish such tasks. However, if the signal is to be sent every few hours, days, etc., then such a system could be regarded as a continuous source of energy. Figure 9 shows the principle of the system's operation. Additionally, the capacitors would have to act fast enough to capture the current impulse in tens of μm.

**Figure 8.** The PicoPower Development Platform was applied as a system to construct a new type of harvesting power supply.

**Figure 9.** Supply current and voltage measurement scheme for detecting the lifetime of a powered microcontroller.

Figure 10 shows the implementation of the developed energy harvesting device in practice, i.e., with the use of a microcontroller. The voltage drop that can be noticed on the measuring resistor, Rsense = 4.7 (Figure 9), can be read as the current consumed by the capacitor–microcontroller system. Because the capacitors C1 and C2 (Figure 9) are charged in the first phase, a significant increase in current can be noticed. These capacitors represent a large load for the developed harvester. In the moment at which current consumption decreased, the voltage reached a value of Uc = 3 V (Figure 9), making it possible to initiate microcontroller operation. In the first step, the microcontroller performed a reset, and then started to implement the saved algorithm. The microcontroller was supposed to wait until the capacitors were charged to about 5 V before performing the next actions. In the moment at which this value was reached, the microcontroller was to start generating signal sequences, which were then counted and forwarded by the RS232 HEGD3 module.

The lifetime algorithm of the program worked from Umax = 5 V to Umin = 1.8 V. During this voltage drop, the microprocessor was able to send about 50 pulses, giving a microcontroller life of about 3 ms. By selecting various mechanical excitation sources, values of up to 200 pulses were obtained, which extended the lifetime of the microcontroller to 8 ms. Finally, an Energy Harvesting Device platform was developed that was able to supply a popular microcontroller, realizing its code for a lifetime of 3 ms with a low impact

energy of Ek = 0.25 J. The presented device was based on a core made of Terfenol-D powder mixed with Fe57Co10B20Si5Nb4V4 alloy powder.

**Figure 10.** The lifetime of the ATMEGA48V microcontroller circuit powered by the DTCCM harvesting device at a low impact energy of Ek = 0.25 J.

In the case of the proposed system, it seems difficult to estimate its efficiency. It cannot be easily calculated, because the electrical efficiency of the system is dependent on the efficiency of each of its components, and the measurement path is complicated. It is obvious that the higher the efficiency of converting energy from one form to another, the better it will be. However, attention should be paid to the fact that the proposed solution needs to work in places where energy is not processed in any way, but only lost in the form of mechanical vibrations. In such cases, recovering or changing even a small part of the energy into another form of energy (in this case electric) seems to make the most sense, even though the efficiency of this transformation is not at a high level. The device shown in this work is based on the reverse magnetostriction effect, but of course, work is also underway using inverse pizoelectric effects, the Farraday effect, and electrostatic charges. In the case of the presented solution, the goal was to develop a device that would not have any moving parts, thus limiting the possibility of defects to some extent. In further research work, it is planned to modify both the measurement system and the harvester in order to increase the amount of electricity generated.

#### **4. Conclusions**

Two types of simple energy harvesting devices were developed and investigated using dedicated test stands. The first type of harvester device was used to perform basic research on the influence of the core type on the resulting values of electric energy. On the basis of this series of tests, it is possible to conclude that:

• The proper choice of the coil (the number of coils and its impedance) and the core material for the harvesting device is very important, and has a significant influence on the results.


Moreover, the results obtained in the presented work may constitute a base for further application work with the developed materials, particularly in the field of energy harvesting.

The solution proposed in this paper may find application in various areas, including the transport and manufacturing industries. These devices could be placed under or in the vicinity of railway tracks or on railway carriages. Another example would be trucks or loading bases, where devices could be placed in the ground (e.g., around speed bumps). In addition, devices of this type could find application in the manufacturing industry, especially in steel mills and pressing plants, near presses, or hammers. Another interesting application is the use of the proposed device in ballistics equipment. As additional work, cooperation has been initiated with a group of scientists from the Department of Mechanics, Mechanical and Biomedical Engineering, researching ballistic shields. When testing such shields, large amounts of energy are generated and lost at the same time. Work is currently underway on the effective use of the proposed devices to recover at least a small part of the energy supplied to ballistic shields.

**Author Contributions:** Conceptualization, R.M. and P.W.; methodology, P.W.; software, K.W.; validation, R.M. and P.W.; formal analysis, P.W.; investigation, R.M., P.W. and K.W.; resources, P.W.; data curation, K.W.; writing—original draft preparation, R.M. and P.W.; writing—review and editing, R.M.; visualization, P.W.; supervision, R.M.; project administration, R.M.; funding acquisition, R.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by The National Centre for Research and Development within the project: "Composite magnetostrictive-nanocrystalline materials for use in the field of energy harvesting and transformation", grant number LIDER/21/0082/L-9/17/NCBR/2018.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data supporting reported results can be provided upon request. Currently, these data are collected as part of the ongoing project, and only after its completion will the data be made available to the public.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Kinetic Electromagnetic Energy Harvester for Railway Applications—Development and Test with Wireless Sensor**

**Zdenek Hadas \*, Ondrej Rubes, Filip Ksica and Jan Chalupa**

Faculty of Mechanical Engineering, Brno University of Technology, 616 69 Brno, Czech Republic; Ondrej.Rubes@vut.cz (O.R.); Filip.Ksica@vutbr.cz (F.K.); chalupa@fme.vutbr.cz (J.C.)

**\*** Correspondence: hadas@fme.vutbr.cz

**Abstract:** This paper deals with a development and lab testing of energy harvesting technology for autonomous sensing in railway applications. Moving trains are subjected to high levels of vibrations and rail deformations that could be converted via energy harvesting into useful electricity. Modern maintenance solutions of a rail trackside typically consist of a large number of integrated sensing systems, which greatly benefit from autonomous source of energy. Although the amount of energy provided by conventional energy harvesting devices is usually only around several milliwatts, it is sufficient as a source of electrical power for low power sensing devices. The main aim of this paper is to design and test a kinetic electromagnetic energy harvesting system that could use energy from a passing train to deliver sufficient electrical power for sensing nodes. Measured mechanical vibrations of regional and express trains were used in laboratory testing of the developed energy harvesting device with an integrated resistive load and wireless transmission system, and based on these tests the proposed technology shows a high potential for railway applications.

**Keywords:** energy harvesting; train; electromagnetic transducer; model; vibration; test; wireless sensor

**1. Introduction**

Modern railways are required to provide an improved quality of service and high levels of safety. Reliable trackside infrastructure maintained in good condition is important for smooth transportation of goods and passengers. To accomplish that, preventive maintenance and scheduled maintenance techniques are currently being used for trackside infrastructure, which can reveal critical wears, defects or failures. However, continuous condition monitoring and long-time sensing using modern electronics could detect incipient wears, failures and degradation that could affect safe railway operation.

Monitoring of trackside systems is important in order to reveal significant changes in functional parameters (e.g., deformation, vibration and temperature). This type of monitoring and diagnostics is widely known as condition-based maintenance, and its main goal is to provide significant savings in infrastructure operational costs. Predictive maintenance techniques require detailed trackside monitoring and the employment of many sensing systems. Reliable and low-maintenance power supplies are essential prerequisites to reliable predictive maintenance results.

Electrical power for these monitoring systems could be delivered from a catenary that is a part of the trackside infrastructure electrical grid. The catenary, however, could be difficult to access due to tight restrictions set up by the infrastructure management and operation in order to maintain the reliability of the track systems. Furthermore, even in the case of a modern railway network, many electrical systems access points are still remote or quite difficult to access due to poor infrastructure and a lack of foresight in regard to modern wireless sensing system power management. Cables and wires are an expensive part of the infrastructure, are often subject to theft, and are difficult to maintain, especially when the layout of railway tracks is changed. Auxiliary railway systems are

**Citation:** Hadas, Z.; Rubes, O.; Ksica, F.; Chalupa, J. Kinetic Electromagnetic Energy Harvester for Railway Applications—Development and Test with Wireless Sensor. *Sensors* **2022**, *22*, 905. https://doi.org/ 10.3390/s22030905

Academic Editor: Amir H. Alavi

Received: 12 December 2021 Accepted: 23 January 2022 Published: 25 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

ready to accept alternative power sources and achieve economically efficient operation by using alternative and energy harvesting sources to power them. Renewable energy sources, such as solar panels or wind turbines, could be used for remote applications that are quite demanding in terms of their power consumption (e.g., warning and signal lights, track switches, grade crossing signals, point machines, positive train control systems and train positions, communication access points etc.). Other energy harvesting sources are widely discussed for embedded monitoring systems in railways. Energy harvesting has been used for wireless sensor nodes and low-power autonomous systems for over 20 years [1]. In general, energy harvesting is based on the conversion of ambient energy into useful electricity. In trackside environment, the passing train by itself could deliver a wide variety of ambient mechanical energy (e.g., mechanical vibration, rail deformation, the sag of sleepers or rails etc.) that could be utilized for such systems.

Individual trackside energy harvesting technologies are summarized in this paper, and the physical principle of a kinetic energy harvesting solution based on converting track vibration into electricity is proposed. On the basis of a mathematical model, a design for a maintenance-free kinetic energy harvester is developed and described, including experimental results and the testing of a complete system with a sensor node.

#### **2. Energy Harvesting Technologies for Trackside Applications**

Recent developments in wireless technologies have resulted in a significantly smaller size, lower price and decreased energy consumption of these systems. For this reason, in railways applications wired sensors are often being abandoned and replaced by wireless alternatives. Their main advantages, on top of the abovementioned ones, are their easy installation and simplified maintenance. The primary battery source and operation in lowpower mode could assure the reliable operation of these sensors for more than a year [2]. However, such a period is still close to the required maintenance period of the sensor itself. For this reason, energy harvesting technologies are investigated in order to achieve several years of maintenance-free operation of these sensor nodes.

Wireless sensor nodes with autonomous energy harvesters could find their way into various engineering applications, as they could operate autonomously in maintenancefree mode for long periods of time. As an example, these solutions are currently used in heavy industry applications, structural health monitoring systems [3], aerospace [4] and transportation [5]. Current energy harvesting technologies have been investigated as a possible source of power for wireless applications, which would otherwise be difficult to connect to the existing power grid.

Track condition monitoring applications are developed on the basis of acceleration sensors [6] or strain gauges [7], mainly for the condition monitoring of a crossing [8–10]. Bridge monitoring solutions have also been widely discussed in recent publications. Paper [11] discussed the feasibility of a bridge monitoring system in terms of its operational life. It illustrated how the traffic on a bridge over time could accentuate the identification of damage, which was necessary to know the state and health of the structure. A segmental prefabrication and assembly of the bridge on the Guangzhou Metro was presented in [12]. Passing vehicles induced vibrations used for energy harvesting, and using appropriate modelling and dynamic analyses of the bridge system a new type of electromagnetic vibration energy harvester was proposed. This device was designed in a way that could power strain-collection units for a bridge health monitoring system. These typical sensing applications, such as crossings and bridges, provide a measurable dynamic response of the track infrastructure to the passing train. In this case, the measured response from the passing train could serve as a suitable source of energy for monitoring applications.

Many published papers dealing with trackside energy harvesting solutions showed that harvesting energy from passing trains has a great potential for wireless sensing applications in railways. Authors of paper [13] investigated the possibility of establishing a self-powered wireless sensor network by integrating the ZigBee stack protocol together with an energy harvesting power source. This system is used for the condition monitoring

of urban rail transit utilizing localized energy harvesting. Authors of the previous article also present another complex system for the smart monitoring of an underground railway by local energy generation in paper [14].

A study and the results of a portable electromagnetic energy harvesting system are presented in papers [15,16]. Their proposed solution consists of a mechanical transmission and an electrical regulator that converts sags in the rail into electricity, providing a peak voltage of 58 V at 1 Hz with a displacement of 2.5 mm. Authors from Stony Brook presented a preliminary prototype of a mechanical-rectifier-based harvester [17]. This study illustrated that sufficient power can be harvested by the device, which is based on a motion rectifier design. A novel direct-motion-driven harvester was described in publication [18], where the authors describe how an anchorless mounting results in a higher power capacity without the requirement of any special preparation during its installation. An installation and test under a fully loaded freight train running at 64 km/h was also presented in this paper. Paper [19] presented a design, modelling, in-lab experiment and field-test results of a mechanical motion rectifier mechanism which is based on a compact ball-screwbased electromagnetic energy harvester. A theoretical study of a cam mechanism was presented by the University of Nebraska in publication [20], where it was used to exploit the contact between a train wheel and a harvester mechanism to drive an electromagnetic generator. A solution based on a direct load piezoelectric harvesting device was proposed in paper [21], offering a structurally simple solution in the form of a piezoelectric drum device placed under sleepers. Piezoelectric solutions for strain-based energy harvesting have been widely discussed, where piezoceramic patches or piezo stacks transduce deformation into electricity [22].

The abovementioned technology mainly converts a direct train load in form of direct contact, deformation or element strain. These devices have the potential to provide peak output power of several watts; however, they are not suitable for high-speed rail applications due to their necessity for a mechanical contact. In contrast, the subsequently presented drum and patch type piezoelectric element solutions are suitable for high-speed operation at the cost of a lower power output in the range of several microwatts. These piezoelectric elements provide a very high voltage but a low current. This disadvantage could be eliminated by multilayer piezoelectric composites; however, the manufacturing of such materials is a very expensive process and for this reason it is not suitable for cost-effective wireless sensor nodes.

Kinetic energy harvesting solutions capable of transducing kinetic energy from vibrations under the passing train into electricity serve as maintenance-free sources of energy. Authors of publication [23] investigated the possibility of harvesting energy from the vertical vibrations of sleepers generated by passing trains at various speeds. A model combining the track structure and the energy harvesting system was used. Results indicated the generated power was around 100 mW, assuming a 2 mm rail displacement amplitude at a frequency of 6 Hz. The presented track model was validated with UK network experimental data. Testing of a piezoelectric vibration cantilever harvester in publication [24] was focused on energy harvesting at a frequency of 5 to 7 Hz. An output power of 4.9 mW and a peak-to-peak voltage of 22.1 V were achieved on a rail vibrating with amplitudes of 0.2 to 0.4 mm at a frequency of 7 Hz. A design of a resonant electromagnetic harvester was published in papers [13,25]. An approach based on magnetic levitation was capable of energy harvesting at a broadband low-frequency vibration in range of 3 to 7 Hz. This device induced a peak-to-peak voltage 2.32 V and an output power of 119 mW when subjected to vibrations with 1.2 mm amplitudes and with an optimal resistive load of 44.6 Ohm.

An innovative approach was presented in paper [26], where authors deployed piezoelectric energy harvesting devices for monitoring a full-scale bridge structure undergoing forced dynamic testing by passing trains. A similar approach was used in paper [27], where the damage detection and structural health monitoring of a laboratory-scaled bridge was observed using a vibration energy harvesting device, in particular a cantilever-based piezoelectric energy harvesting device. The published approach had an advantage over the

conventional accelerometer-based method in terms of power requirements, because energy storage and data transmission units were the only power-consuming parts of the system.

#### **3. Model Based Design of Electromagnetic Trackside Energy Harvester**

A passing train provides mechanical vibrations in rails and sleepers. The vertical deflection of a sleeper is depicted in Figure 1 using the variable *z*. This sag in a sleeper depends on the passing train's mass, velocity and the quality of the rail subgrade. The proposed energy harvesting system is based on a principle of kinetic energy harvesting which can convert the kinetic energy from sleeper oscillations into useful electricity. A mechanical resonator is used for the transfer of input kinetic energy into the free oscillation of a seismic mass. A design of this kinetic energy harvester is based on a mass *m* which is suspended on two steel cantilevers with a known stiffness *k*<sup>1</sup> and known mechanical damping *dm*. This longitudinal design could be placed on the top of a sleeper, or it could be embedded inside a new generation of innovative sleepers.

**Figure 1.** Physical principle of kinetic energy harvester under passing train vibrations.

On the basis of the previously published analysis, the electromagnetic energy transducer provides an effective harvesting power for this application. The oscillating mass is a part of a magnetic circuit, and its free oscillation *x* against a fixed coil generates useful electricity, which provides electromagnetic damping forces in the form of electrical damping *de*.

#### *3.1. Mathematical Model of One Degree of Freedom System*

The kinetic electromagnetic energy harvesting system could be described by a multidomain model in the form of coupled mechanical and electrical systems, as depicted in Figure 2. The mechanical resonator is excited by ambient mechanical shocks *z* to the oscillating sleeper and this results in the relative movement *x*. The relative movement *x* of the mass *m* in the magnetic circuit consisting of the frame and a fixed coil is inversely proportional to the mechanical damping *dm*. Due to Faraday's law, the relative movement of the magnetic circuit results in a change in the magnetic field of the coil, inducing an electromotive voltage *ui*. A model of an electromagnetic coupling coefficient was used for the description of the interaction between both the mechanical and electrical domains. The induced voltage depends on the design of the electromagnetic transducer (the electromagnetic coupling coefficient *cEH*) and its relative velocity. When a resistive electrical load *RL* is connected to a coil, then a current flows through the coil and electrical power is extracted

from the system. The electrical power extracted from this system provides electromechanical feedback in a form of an electrical damping, which is depicted as a damper *de*. This electrical damping feedback is proportional to the electromagnetic coupling coefficient *cEH*. A derived mathematical model with one degree of freedom was used for predicting the harvested power in a resonance operation.

**Figure 2.** Coupled mechanical and electromagnetic models of the proposed kinetic energy harvester.

A second-order equation according to the mechanical model in Figure 2 describes mechanical oscillations of a seismic mass as a response to the kinetic excitation of the sleeper:

$$m\ddot{\mathbf{x}} + d\_m \dot{\mathbf{x}} + d\_c \dot{\mathbf{x}} + k\mathbf{x} = -m\ddot{\mathbf{z}}\tag{1}$$

where *x* is the relative displacement of the oscillating mass, *z* is the absolute displacement of the vibrating sleeper, *m* is the moving mass, *dm* is the mechanical damping, *de* is the electrical damping, and *k* is the mechanical stiffness.

The mechanical stiffness combines stiffness of both cantilevers. The stiffness of a single beam *k*<sup>1</sup> can be calculated using this equation:

$$k\_1 = \frac{3 \cdot E \cdot f}{I^3} \tag{2}$$

where *E* is Young's modulus of the used material (steel), *J* is a second moment of the area, and *l* is the length of the cantilever. The mechanical stiffness *k* is then simply 2·*k*<sup>1</sup> for this double suspended system.

The mechanical damping *dm* can be calculated using this relation:

$$d\_m = \frac{1}{2Q\_m} 2m\Omega\tag{3}$$

where *Qm* is the mechanical quality factor, either estimated or calculated from an experiment. The natural frequency Ω can be calculated using a commonly known formula for a single degree of freedom system:

$$
\Omega = \sqrt{\frac{k}{m}}\tag{4}
$$

The electrical damping *dm* of the electromechanical system can be calculated using this equation:

$$d\_{\varepsilon} = \frac{\left(BNI\right)^{2}}{R\_{\mathbb{C}} + R\_{L}} = \frac{\left(c\_{EH}\right)^{2}}{R\_{\mathbb{C}} + R\_{L}}\tag{5}$$

where *B* is the magnetic flux density in the coil, *N* is the number of turns, *l* is the active length of one turn, *RC* is the coil resistance, *RL* is the load resistance, and *cEH* is the electromagnetic coupling coefficient of the energy harvester, where *cEH* = *BNl*.

The induced voltage on the coil *ui*, can be calculated using equation:

$$
\mu\_{\dot{l}} = B N l \cdot \dot{\mathbf{x}} = \mathbf{c}\_{EH} \cdot \dot{\mathbf{x}} \tag{6}
$$

The first-order electric differential equation of the electrical circuit in Figure 2 is then:

$$L \cdot \frac{di}{dt} + i \cdot (\mathcal{R}\_{\mathbb{C}} + \mathcal{R}\_{L}) = \mu\_{i} \tag{7}$$

where *L* is the inductance of the coil, and *i* is the electric current. By design, the inductance of the coil in our harvester is very small (for a coil with an air core) and the current change is very slow, therefore the first term is irrelevant and the equation can simplified:

$$\dot{\mathbf{u}} = \frac{\mathcal{L}\_{EH} \cdot \dot{\mathbf{x}}}{\mathcal{R}\_{\mathbb{C}} + \mathcal{R}\_{L}} \tag{8}$$

The coupled mechanical equation can modified, where Equations (5) and (8) provide the electrical damping as a function of the electric current:

$$m\ddot{\mathbf{x}} + d\_m \dot{\mathbf{x}} + c\_{EH} \dot{\mathbf{t}} + k\mathbf{x} = -m\ddot{\mathbf{z}}\tag{9}$$

The fundamental performance of the energy harvester is the equation for the output power:

$$p\_{\rm out} = i^2 \cdot R\_L \tag{10}$$

The displacement amplitude (peak values) could be simply calculated from these equations assuming a resonance operation. The mechanical amplitudes of both the displacement and velocity follow these relations:

$$\dot{x}\_A = z\_A Q\_T = \frac{\ddot{z}\_A}{\Omega^2} Q\_T \to \dot{x}\_A = \frac{\ddot{z}\_A}{\Omega} Q\_T \tag{11}$$

where the term *QT* is the total quality factor of both the mechanical and electrical damping. This quality factor is a compound on the basis of the following relation:

$$Q\_T = \frac{1}{2\frac{d\_m + d\_c}{2m\Omega}} = \frac{m\Omega}{d\_m + d\_c} \tag{12}$$

The calculation of the velocity amplitude can be used for the calculation of the amplitude of the induced voltage: .

$$
\mu\_{i\_A} = \mathfrak{c}\_{EH} \cdot \dot{\mathfrak{x}}\_A \tag{13}
$$

and the amplitude of the output voltage on the resistive load is:

$$
\mu\_{L\_A} = \mu\_{i\_A} \frac{R\_L}{R\_C + R\_L} \tag{14}
$$

The output power amplitude can then be expressed using either voltage or current:

$$p\_{out\_A} = i\_A^2 \cdot R\_L = \frac{u\_{L\_A}^2}{R} \tag{15}$$

#### *3.2. Design of Energy Harvesting Device*

The proposed design of the electromagnetic kinetic energy harvester could be capable of converting sleeper vibrations into useful electricity. Resonance operation is not possible due to the pulse excitation characteristics produced by the passing train. However, the free oscillation response to the passing train provides a relative oscillation of the suspended seismic mass against the fixed base with a coil, which has the potential to generate satisfactory levels of useful electrical power. In the case of the longitudinal design of the device mounted on top of the sleeper, the suspension system consists of a pair of steel cantilevers with dimensions of 400 × <sup>30</sup> × 3 mm3. The mechanical resonator is by design tuned up to have a natural frequency of 12 Hz, which provides a sufficient relative movement. The

long steel cantilever design results in a mechanical resonator with one degree of freedom in the vertical direction, which makes it sensitive to the train induced vibrations. The relative movement amplitude is important for a correct design of the magnetic circuit, which is fixed inside the seismic mass. A concept of a sleeper kinetic energy harvester design for trackside application is shown in Figure 3.

**Figure 3.** Proposed integration of KEH design for railway applications.

The fundamental part of the seismic mass is a magnetic circuit with 16 rare earth FeNdB magnets and ferromagnetic holders. A pair of ferromagnetic holders with permanent magnets moves freely in the air gap of a fixed coil. A planar finite element analysis of this magnetic circuit was conducted in order to calculate the average magnetic flux density in the area of the coil for a given relative movement. The coil was designed to have an air core and wound up around a plastic frame fixed to the base. All active turns of the coil were placed in the air gap of the magnetic circuit. A relative position of the magnetic circuit and coil was set with a minimal air gap to achieve efficient electro-mechanical energy conversion. This analysis was done in an FEMM environment and the calculated magnetic field is shown in Figure 4.

**Figure 4.** Planar FEMM model of magnetic circuit; analysis of magnetic flux density *B*.

The developed and assembled kinetic electromagnetic device for trackside application is shown in Figure 5 and it consists of:


**Figure 5.** Design of proposed kinetic energy harvester.

The model from the previous chapter was used for the design of the individual parameters with respect to the required harvested power. The parameters of the final model and the assembled device (see Figure 5) are summarized in Table 1.



#### **4. Electromagnetic Kinetic Energy Harvester Testing with Resistive Load**

*4.1. Resonance Operation: Model Results and Experiment*

The designed parameters of the model are used to predict the output voltage and power in a resonance operation. The presented electro-mechanical equations in combination with the model of peak voltage and peak power described in Section 3.1 were used for output calculations for a variable resistive load. The experiment was conducted on a laboratory shaker, an RMS SW8142–SWH600APP, connected to its auxiliary measurement instruments and depicted in Figure 6. The harvested voltage was measured on an oscilloscope, a Rigol MSO 5204. Both the model and experiment were excited in a resonance frequency with an acceleration amplitude of 1 ms<sup>−</sup>2.

**Figure 6.** Shaker lab test of kinetic energy harvester.

The calculated output voltage and power are shown in Figure 7, and both outputs are compared with values obtained from the experiment. The correlation between the model and experiment is very good for low values of the resistive load. Based on the harvester model, the maximal power was expected with a resistive load of 3 kΩ. However the experimental results showed that the maximal power was harvested for a resistive load of 2 kΩ, but the harvested power was very similar across a wide range of resistive loads, 2–3 kΩ. The experimentally measured voltage and power for a higher resistance were lower than the theoretical values and it seemed that the real damping was higher compared to the damping model at higher speeds, causing a less pronounced increase in the output voltage due to the voltage being proportional to the speed.

**Figure 7.** Voltage and power responses in resonance operation vs. load resistance—simulation results and measurements with excitation acceleration amplitude of 1 ms<sup>−</sup>2.

#### *4.2. Test of Kinetic Energy Harvester—Harmonic Vibration*

The frequency response is very important for characterizing an energy harvester's performance. A shaker test with sinusoidal vibrations was used to measure the frequency response around resonance. Both sweep up and sweep down tests with a slowly changing input frequency of vibration were used for voltage and output power response measurements in the frequency domain. The input acceleration with an amplitude of 1 ms−<sup>2</sup> and frequency rate of change of 0.1 Hz/s were used in the tests. These sweep tests were realized around the resonance frequency with different load resistance.

The measured peak voltage and peak power are depicted in Figure 8. The lab experiment was made with four different load resistances: 1, 2, 3 and 5 kΩ. The voltage response was proportional to the load resistance. It was caused by an increase in the velocity due to decreasing electrical damping. However, the output power peaks at around 2 kΩ, and below and above that value the power amplitudes decreased. This power maximum was observed in a resonance operation, but outside of the resonance operation the device harvested higher power with the lowest load resistance of 1 kΩ. The kinetic energy harvester with a lower resistance provided higher power outside of a resonance operation. Based on this fact, it is suitable to use a resistive load of 2 kΩ in a resonance operation but use a lower value of 1 kΩ for a non-resonance operation. This fact is discussed further in the next section.

**Figure 8.** Voltage and output power depending on excitation frequency with different load resistance. Video S1.

#### *4.3. Test of Kinetic Energy Harvester—Train-Induced Vibrations*

The typical transient dynamic response of a kinetic energy harvester was provided with real train-induced vibrations as inputs. Real vibrations in the trackside sleeper, where the vibrations were not pure sinusoidal, but rather had the characteristic of a series of mechanical pulses, are shown in Figure 9. The acceleration and displacement of the sleeper was measured on a trackside in the Czech Republic using an inertial accelerometer on the sleeper and a capacitive displacement sensor mounted between the sleeper and a fixed point. This detail is for a single bogie consisting of two train wheels. The movement of the sleeper had general characteristics based on many parameters of the track subgrade. For this particular sleeper movement, the shown input vibrations for whole trains are used for the kinetic energy harvester test. Experiments with different electric loads were made with the aim of finding an optimal resistive load for a maximal energy harvesting potential.

**Figure 9.** Typical vibrations of sleeper under passing train—detail of bogie/two wheels.

Real acceleration and displacement measurements of sleeper movement for two typical trains were used for lab tests of the developed kinetic energy harvesters:


As Figure 10 shows, the maximal average power was measured with a resistive load of 150 Ω, which was the same as the coil resistance. The output power was lower at higher resistive loads due to lower damping and lower energy harvesting from trackside vibrations. The power was lower at lower resistive loads due to higher energy dissipation on the coil resistance rather than the load resistance.

**Figure 10.** Average power depending on load resistance for Train 1 and Train 2.

Different weights of the seismic mass of this device were tested in order to achieve the optimal design of this energy harvesting device. The default weight of the moving mass, 800 g, was increased using an additional external weight. Unfortunately, decreasing the weight below the default value without significant structural modifications was not possible. A relation of this mass value to the generated power was experimentally verified, and the results of his test are shown in Figure 11. It was possible to add an extra mass to the initially calculated seismic mass of 800 g; however, it is evident that the output power would decrease with higher values of the seismic mass, and it was verified that the modelled mass of 800 g provided the most effective energy harvesting operation with the given train vibration data. On the other hand, it was not possible to decrease the mass in the current design due to the nature of the magnetic circuit, which needs a minimal cross-section area of the core to function properly.

**Figure 11.** Average power depending on moving mass for Train 1 and Train 2.

Time-domain measurements of the voltage and power with the excitation acceleration and displacement for both testing trains are shown in Figure 12. These measurements were done with the optimal resistive load of 150 Ω. The average harvested power and total harvested energy for a passing train are presented in Table 2.

**Figure 12.** Lab shaker test with real acceleration data from regional Train 1 (**a**) and from express Train 2 (**b**); measured voltage and power with load resistance of 150 Ω for input mechanical vibrations of the shaker.


**Table 2.** Harvested power and energy with optimal load resistance.

Both the voltage and power strongly depended on the input acceleration during the train's passage. While the average power was 9.1 mW for the regional Train 1, the maximum power during much more pronounced acceleration peaks was up to 300 mW. In the case of the express Train 2, this vibration could generate average power of 29.3 mW and several peaks above 600 mW could be observed. Given all the information it is necessary to mention that the variable nature of the output power must be considered during the design of power management electronics for railway applications.

#### **5. Kinetic Energy Harvester as Source of Energy for Wireless Sensing**

The previous chapter shows that the kinetic energy harvester could generate useful electric power from passing trains. However, a test with power management electronics, a sensing node, and a communication module is required to fully assess the feasibility of this energy harvesting technology. For this reason, operational tests of the kinetic energy harvester utilized as a source of energy for a wireless sensor node were conducted for both types of trains. Our lab successfully tested autonomous wireless sensor nodes with the vibration energy harvester in lab conditions, and these results, including the design of a power management, sensing unit and communication module, were successfully published in our research paper [28].

The electric diagram of an energy harvesting system with a sensor node and communication module is shown in Figure 13. The developed kinetic energy harvester is connected to previously published electronic systems. It consists of a rectifier, a storage capacitor with a capacity of 493 μF, an LTC 3588 power management circuit, an analogue front end for the sensing signal, and a communication module based on the NRF24L01 chip from Nordic Semiconductor.

**Figure 13.** Schematic diagram of energy harvesting system with highlighted electrical parameters measurement.

Measured vibration signals for both types of passing trains (the same as those in the previous section) were used for the test of this autonomous source of energy for the sensing node and communication module. Measurements of electrical signals highlighted in Figure 13 (the voltage induced by the energy harvester, the input voltage and the current into the LTC power management circuit of the wireless sensor node) were used for the harvester performance assessment.

Experimental results of the autonomous wireless sensing node operation are depicted in Figure 14 for both train types. The voltage induced on the developed kinetic energy harvester was very similar to the response with a resistive load. The input voltage into the LTC circuit rose quickly during the first second of the train's passage and then began to fluctuate around a constant value for the regional Train 1, or slowly increase for the express Train 2, which provided more power in general. After a train passed, the voltage decreased slowly and transmission continued. The current was discontinuous due to characteristics of the LTC power electronics, and this pattern reflected the transmission of the measured data.

**Figure 14.** Shaker test with real acceleration data from Train 1 (**a**) and Train 2 (**b**) with connected power electronics; measured voltage and current according to electric schema. Video S2.

During this experiment, the average power consumption of the radio transmission unit was around 4.5 mW, which was less than the average power acquired from each train. The power harvested during a train's passage was stored in the capacitor, and after the passage it was used to continue the data transmission. This energy could also be stored for later if the transmission was no longer active after the respective train passed. However, it would strongly depend on the sensor setup and monitoring requirements for any given application. Nevertheless, experiments with both types of trains confirmed that the proposed system was able to provide enough power for continuous transmission, as is evident in Figure 14.

Measurement with different capacitor values are depicted in Figure 15. The excitation acceleration data were acquired from Train 1, similarly to Figure 14a. For a better understanding of the energy storage capabilities, the graph of the voltage is accompanied by a graph depicting the calculated energy stored in the capacitor. Initially, the lower capacity caused the voltage to increase rapidly; however, after a few seconds of operation the stored energy was lower compared to the case with higher capacity. Furthermore, with lower capacity values the voltage tended to fluctuate significantly.

**Figure 15.** Charging process of capacitor of different value with continuous power consumption of wireless sensor.

#### **6. Potential Applications**

Several manufacturers of railway infrastructure systems (e.g., sleepers/bearers, railway switches, point machines, axle counters etc.) have begun to seriously consider the development of new equipment with energy harvesters for predictive maintenance applications of their products. There are two potential workstreams: retrofitting existing products, and novel design of products with embedded energy harvesting. Both emphasize the main advantages of energy harvesting devices, which would result in a reduction in wiring and cables (in both communication and power), reduction of losses due to cable theft, decreased costs of the energy power supply and battery replacement etc.—and all of these aspects contribute to a significant reduction in maintenance costs. This fact is amplified by the subsequent application of predictive maintenance methods, which replace periodic inspections and, compared to the conventional methods, can reveal potential defects and abnormal degradation before a fatal system failure occurs.

#### *6.1. Smart Railway Monitoring Application*

A new generation of smart sleepers could detect overloaded trains, trains with abnormal wheel wear or problems with suspension systems, all of which contribute to a significantly higher load on the track and a rapidly increasing wear of railways. The railway wear or significant changes in subgrade and ballast properties—mainly the gap under a sleeper—could also be detected by smart monitoring systems embedded in sleepers. The proposed concept of autonomous application is shown in Figure 16. The kinetic energy harvester could be integrated or embedded inside innovative sleepers and provide electricity for autonomous monitoring. The sensing node could be integrated into rails in the form of a piezoelectric layer that generates an active voltage signal and does not consume power. This system could be even more affordable if PVDF piezopolymers for structural monitoring were used.

**Figure 16.** Proposed autonomous railway application of autonomous piezoelectric sensing. Video S3.

This maintenance system could be interesting for infrastructure management and freight train providers interested in detecting critical wear or damage of both the railway and trains. The comprehensive monitoring system could solve the sensitive question of whether freight cars are subjected to excessive wear due to poor track quality, or conversely whether damaged freight cars in operation are causing excessive wear of railways.

#### *6.2. Concept of Smart Turnouts*

Switches and crossings are the parts of a railway track most impacted by the dynamic forces applied by trains. From a maintenance point of view, it is important to recognize faults or degradation processes at an early stage, before any significant limitation in operability occurs. The best way to monitor the conditions of crossings is with condition monitoring systems capable of measuring and evaluating the dynamic impacts on crossings over longer periods of time. The concept of an autonomous sensing node with a vibration energy harvester could represent a suitable solution for this system, and kinetic energy harvesters could provide sufficient energy, especially if piezopolymer materials were utilized as the active piezoelectric sensors (e.g., PVDF), mainly because they do not require an external power source to operate. The concept of an autonomous wireless node depicted in Figure 17 can transmit signals from the turnout structure to a close trackside IoT point. There is also the option to process signals on-site and transmit only the results of embedded data analyses. Energy harvesting could mostly be useful in the case of short-distance wireless communication between the track structure and the IoT point, which would use electricity from the power grid.

**Figure 17.** Proposed concept of smart turnout based on energy harvesting.

#### **7. Conclusions**

The main design goal of all energy harvesting devices should be to deliver sufficient power for the operation of a given system. In the case of railway applications, replacing cables, cutting off railway systems from the power grid, and using energy harvesting sources for every system is not always an optimal solution, and in many cases is nearly impossible to implement. It is important to keep in mind that energy harvesting devices are utilized as power sources for specific tasks and must be developed and designed with respect to the system they are used to power in order to achieve reliable, low-power and maintenance-free operation over long periods of time. Only such an approach would inevitably lead to a long-term deployment of energy harvesting devices with a new generation of smart railway systems and parts for sustainable rail transportation.

The maintenance of cable systems could result in damage to the wires used in wired sensing systems. For this reason, using embedded kinetic energy harvesters as a source of energy for autonomous wireless sensing nodes is an advantageous approach for long-term railway sensing and monitoring. Two potential applications are presented in this paper for smart rail monitoring and a turnout predictive maintenance system.

The main aim of this paper was to present the development of a kinetic energy harvesting device for rail track applications, a device that is able to provide sufficient power for short-distance communication. The developed device was tested under lab conditions with the input vibration signals of two different trains, regional and express. In lab tests, vibrations obtained from a real rail track used as an input provided enough energy for the communication module and transmission of the sensing signal, with the results presented in this paper. The concept it uses, of placing a kinetic energy harvester on the top of sleepers, could generate an average output power in range of 5–35 mW, depending on the train speed. In this case, this technology could be attractive for the retrofitting of the existing railway infrastructure and for innovative products.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/s22030905/s1, Video S1: Resonance, Video S2: Train, Video S3: Communication.

**Author Contributions:** Conceptualization, Z.H.; data curation, O.R., F.K. and J.C.; formal analysis, O.R. and F.K.; funding acquisition, Z.H.; investigation, Z.H., O.R., F.K. and J.C.; methodology, Z.H., O.R., F.K. and J.C.; project administration, Z.H.; resources, Z.H.; software, J.C.; supervision, Z.H.; validation, O.R. and J.C.; visualization, O.R. and F.K.; writing—original draft, Z.H., O.R. and F.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The presented energy harvesting research and development were supported by H2020 projects ETALON S2R-OC-IP2-02-2017 and I2T2 S2R-OC-IP2-02-2020. The sensing part was supported by the Czech Science Foundation project GA19-17457S 'Manufacturing and analysis of flexible piezoelectric layers for smart engineering', Czech Republic.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


## *Article* **On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters**

**Grzegorz Litak 1,\*, Jerzy Margielewicz 2, Damian G ˛aska 2, Andrzej Rysak <sup>1</sup> and Carlo Trigona <sup>3</sup>**


**Abstract:** The piezoelectric energy-harvesting system with double-well characteristics and hysteresis in the restoring force is studied. The proposed system consists of a bistable oscillator based on a cantilever beam structure. The elastic force potential is modified by magnets. The hysteresis is an additional effect of the composite beam considered in this system, and it effects the modal solution with specific mass distribution. Consequently, the modal response is a compromise between two overlapping, competing shapes. The simulation results show evolution in the single potential well solution, and bifurcations into double-well solutions with the hysteretic effect. The maximal Lyapunov exponent indicated the appearance of chaotic solutions. Inclusion of the shape branch overlap parameter reduces the distance between the external potential barriers and leads to a largeamplitude solution and simultaneously higher voltage output with smaller excitation force. The overlap parameter works in the other direction: the larger the overlap value, the smaller the voltage output. Presumably, the successful jump though the potential barrier is accompanied by an additional switch between the corresponding shapes.

**Keywords:** vibration energy-harvesting system; hysteretic effect; bistable oscillator; bifurcation

#### **1. Introduction**

Mechanical vibrations typically induced during machine operation are a disadvantageous phenomenon. In most practical applications, the influence of mechanical vibrations is limited by means of vibration-reduction systems, while the complete elimination of vibration is practically impossible. On the other hand, there are devices in technology in which vibrations are intentionally induced: vibrating conveyors, compactors, pneumatic hammers, etc. Currently, there is growing interest from both scientists and industry in the harvesting of this irretrievably lost energy resulting from vibrations [1–3]. This is possible because of energy harvesters that use, among others, the piezoelectric effect.

Piezoelectric energy harvesting from ambient vibration sources has been widely studied in recent years [1–11]. The applied devices are made up of a vibration resonator formed as a beam-mass resonator structure and a piezoelectric transducer [1,2]. To improve efficiency, a nonlinear oscillator was proposed [3–12], which is characterized by inclinations of resonance curves [13] and additional resonances defined by the multiplied rational and fractions of the main resonance frequencies [11,12,14]. Simultaneously, multiple coexisting solutions are present in such a system and depend on initial conditions [13–15]. Among the main disadvantages of such systems, one can distinguish the difficulty of controlling particular solutions [16]. In recent years, several studies have been conducted to investigate, among others, the hysteretic effects of elastic beam materials and related structures, as well as piezo elements [17–19]. In this study, we continue the investigation of the dynamics of

**Citation:** Litak, G.; Margielewicz, J.; G ˛aska, D.; Rysak, A.; Trigona, C. On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters. *Sensors* **2022**, *22*, 381. https://doi.org/ 10.3390/s22010381

Academic Editor: Chelakara S. Subramanian

Received: 22 November 2021 Accepted: 29 December 2021 Published: 5 January 2022

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a nonlinear bistable energy harvester with hysteresis induced by the snap-through phenomenon of a bistable beam [17,19]. Our study was inspired by a recent work [17], wherein composite bistable beams were used for energy harvesting, and by [20], where bistability was used in combination with an additional magnetic field to model the potential shape. In [17], the authors proposed a piezoelectric energy harvester that stores the potential energy induced by the mutual self-constraint of the subbeams and harvests the large energy released during the rapid shape transition.

Regarding nonlinear systems, the assessment of the impact of individual parameters is difficult to implement without detailed numerical experiments [8,12]. This claim is justified because even a small change in the value of any parameter in a nonlinear system can lead to drastic changes in the dynamics of the system [21]. In this publication, we are considering cutting the potential barrier and shifting the cut parts so that they overlap. This can be modelled by an elastic cantilever beam that is subjected to initial elastic deformation. An example of such an arrangement would be a typical hairpin. Another possibility is to use a shape-memory material.

#### **2. Mathematical Model**

The starting point for the numerical experiments performed is the piezoelectric beam resonator system (Figure 1a) [22], which is the subject of many publications (see [23]). In general, the analyzed system consists of a flexible beam I with piezoelectric transducers II attached to its flat surfaces. The flexible beam I is rigidly fixed in the frame IV, which is bolted to the vibrating subassembly of the mechanical system, from which energy is recovered by means of screws III. In our research, the impact of the hysteresis loop (presumably from the composite beam) on the efficiency of the energy harvested was assessed. For simplicity, we used the initial configuration of the system without hysteresis (Figure 1a) accompanied by the interacting permanent magnets to support the double-well solution (Figure 1b). The hysteresis loop caused by the beam complexity is introduced to the above system potential as an effect of cut and sift the left- and right-hand sides of potential characteristic (Figure 1c) with respect to the central point at the top of the potential barrier by the distance of d (Figure 1c).

**Figure 1.** System and its potential characteristics: (**a**) schematic diagram without indicating the origin of hysteresis, modelling of the hysteretic branches: (**b**) potential without hysteresis loop, (**c**) with hysteresis loop, *d* indicates the overlap shift, *q* is the displacement of the tip point of the beam. *i* is the current in the electrical circuit.

In numerical simulations, it was assumed that the system was influenced by mechanical vibrations of the frame mapped by the harmonic function having an amplitude A and a frequency *ωW*, *f* = *A*sin(*ω<sup>W</sup> t*). The differential equations of motion, taking into account the electromechanical coupling, have a general form of:

$$\begin{cases} m\_1 \frac{d^2 q}{dt^2} + b\_B \left( \frac{dq}{dt} - \frac{df}{dt} \right) + c\_B (q - f) - c\_1 (q - f) + c\_2 (q - f)^3 + k\_P l L\_P = 0, \\\ C\_P \frac{d l I\_P}{dt} + \frac{l I\_P}{K\_O} - k\_P \left( \frac{dq}{dt} - \frac{df}{dt} \right) = 0, \end{cases} \tag{1}$$

where *m*1, *bB*, *cB* are effective (modal) beam parameters corresponding to the mass, damping and stiffness, respectively. *c*<sup>1</sup> and *c*<sup>2</sup> denote the coefficients of the magnetic force. *CP* and *RO* are parameters in the electrical circuit indicating capacity and resistance, respectively. Finally, *kp* is the coupling parameter and *Up* is the voltage output on the resistor [4,12,22].

Considering quantitative and qualitative computer simulations, the system of Equation (1) was transformed into a dimensionless form. A new coordinate *y*, *y* = *q* − *f*, was introduced, defining the difference between the displacement of the free end of the flexible beam *q* and the point of its attachment to the rigid frame IV *f* (see Figure 1a). The differential equations of motion, considering dimensionless time and dimensionless displacement, finally take the form:

$$\begin{cases} \ddot{\mathbf{x}} + 2\delta \dot{\mathbf{x}} - a(\mathbf{x} + \mathbf{a} \cdot \mathbf{d}) \cdot \left[ 1 - (\mathbf{x} + \mathbf{a} \cdot \mathbf{d})^2 \right] + \theta \mathbf{u} = \omega^2 p \sin(\omega \tau), \\\ \dot{\mathbf{u}} + \sigma \mathbf{u} - \theta \dot{\mathbf{x}} = 0. \end{cases} \tag{2}$$

where:

$$\omega\_0^2 = \frac{\varepsilon\_1 - \varepsilon\_B}{m}, \quad \delta = \frac{b\_B}{m\omega\_0}, \quad \alpha = \frac{\varepsilon\_2 \mathbf{x}\_0^\sharp}{\varepsilon\_1 - \varepsilon\_B}, \quad \theta = \frac{k\mathbf{p}}{\omega\_0^2 \mathbf{x}\_0 m}, \quad p = \frac{A}{\omega\_0}, \quad \omega\_0 = \frac{A}{\omega\_0}, \quad \omega\_0 = \frac{A}{\omega\_0 \omega\_0}, \quad \omega\_0 = \frac{A}{\omega\_0}, \quad \omega\_0 = \frac{A}{\omega\_0}$$

In the model tests, *x*<sup>0</sup> represents the scaling parameter equal to the absolute value of the coordinate defining the position of the minimum potential barrier. On the other hand, the variable a occurring in the mathematical model reflects control quantity responsible for the shift of the operating point from one half of the potential barrier to the other. In general terms, the variable a takes the value of 1 or −1 depending on the hysteretic branch. Here, we limit ourselves to listing the numerical values of the mathematical model coefficients, based on which quantitative and qualitative computer simulations were carried out, i.e.,: *δ* = 0.05, *ϑ* = 0.5, *θ* = 0.05, *σ* = 0.05, *α* = 0.5.

#### **3. The Results of Model Tests**

Based on the assumed numerical data characterizing the mathematical model of the analyzed energy-harvesting system, numerical experiments were performed to investigate the impact of the d shift on the efficiency of energy harvesting. In the first stage, the impact of the shift on the location of the zones of chaotic and periodic solutions was assessed. Among the periodic solutions, small and large orbits were found. Areas of chaotic solutions can be identified via different numerical tools, such as bifurcation diagrams [21], 0–1 test [19,24] or by determining the maximal Lyapunov exponent [25]. In our numerical simulations (performed in Mathematica), the areas of chaotic and periodic solutions were visualized in the form of multicolor maps showing the distribution of the maximal Lyapunov exponent (Figure 2). According to the authors, this approach makes it possible to look at system dynamics with a wide range of variability of the parameters characterizing the source of mechanical vibration affecting the energy-harvesting system. The maximal Lyapunov exponent was estimated in a three-dimensional phase space following the numerical procedure proposed by Wolf et al. [26]. The phase space was spanned by the computed coordinates (*x*, . *x*, *u*). All the included two-dimensional, multicolor distribution maps were plotted for the assumed nodal initial conditions: *<sup>x</sup>*(0) = 0, . *x* (0) = 0 and *u*(0) = 0. Furthermore, two control parameters, ω and p, were selected to describe changes in the external excitation source. Technically, the small distance of the initial conditions between the tested trajectory and the

reference trajectory was assumed to be ε(0) = 10<sup>−</sup>5. To obtain a satisfactory resolution, the ranges of the control parameters ω and *p* were divided into 500 subintervals.

**Figure 2.** Influence of the parameter *d* and damping on the distribution of the largest Lyapunov exponent: (**a**) *d* = 0, (**b**) *d* = 0.3 with nodal initial conditions.

It is worth mentioning that the positive values of λ relate to the chaotic dynamic response of the system, otherwise (negative, λ < 0) the system response is regular with the corresponding phase trajectories tending to stable points or periodic orbits. However, when λ approaches values equal to zero, we are dealing with the so-called bifurcation points (or quasiperiodic solutions). In a low frequency range ω, narrow repetitive zones of chaotic solutions are arranged along the curves that can be approximated by functions with a fairly high exponential growth. Regardless of the parameter *d* value, the largest area of chaotic solutions is located in the central parts of the multicolor maps showing the distribution of the maximal Lyapunov exponent. For the case of *d* = 0, this region is located in the band ω [1,2] (see Figure 2a). However, for *d* = 0.3, this zone stretches towards higher values of the dimensionless frequency ω [1, 2.4] (see Figure 2b). An indicator expressed as the effective value of the voltage induced on the piezoelectric electrodes was taken as a measure of energy-harvesting efficiency. To determine the effect of the solution on the efficiency of energy harvesting, the multicolor maps of the maximal Lyapunov exponent distribution were compared with the diagrams of effective voltage values. The results clearly indicate that when the solution changes from periodic to chaotic, the voltage induced on the piezoelectric electrodes is limited. Examples of these landmarks are marked in black. The RMS(*u*) diagrams (RMS—root mean square) were identified in relation to every value of the dimensionless excitation frequency from the range ω [0,4], based on a time sequence of 50 periods of the mechanical vibration signal affecting the energy-harvesting system.

With regard to the zero initial conditions, it can be observed that—irrespective of the displacement value of the "cut" halves of the potential barrier d (Figure 1b,c) and the amplitude of the excitation source p—periodic solutions with a periodicity of 1T (single excitation period T) dominate in the range of low, dimensionless excitation frequencies of *ω* < 0.5. It should be noted that increasing the width of the hysteresis loop reduces the energy-harvesting capability. Consequently, the reduction in the voltage induced on the piezoelectric electrodes is particularly noticeable in the range of higher parameter *p* values. In the range of low amplitudes of forced vibration, *p* < 0.1, particularly in the zone of chaotic solutions, no significant decrease in effective voltage value is observed. As predicted by

RMS(*u*), the voltage value is directly proportional to the amplitude and average kinetic energy of cantilever beam vibration.

#### *3.1. Influence of the Parameter d on the Geometrical Structure of a Chaotic Attractor*

The below graphs (Figure 3) show examples of attractors that occur in the regions of chaotic solutions. It is worth mentioning that in the classic visualization of the Poincaré cross-section, numerical results are presented as a set of points located on the phase plane. Much more information about the geometric structure of chaotic attractors can be obtained by analyzing the density of the distribution of the points of intersection of the trajectory and the control plane. In this way, one obtains information from the areas of the phase plane where the trajectory most often intersects with the control plane. The graphical representations of the Poincaré cross-sections are then plotted against bifurcation diagrams. From a theoretical point of view, bifurcation diagrams can be drawn based on the following: Poincaré cross-section, phase trajectory and time sequence. In our case, these diagrams were plotted based on the local minima (points marked in red) and maxima (points marked in blue) of the analyzed time series (Figure 3).

The plotted Poincaré cross-sections show different geometrical structures of attractors depending on the external excitation frequency value affecting the energy-harvesting system. To quantify the geometric structure of the chaotic attractors, their correlation dimension was identified, and the effective voltage values induced on the piezoelectric electrodes were also given. Based on the model tests, it was found that in the case of the "developed" attractors (a developed attractor is understood as a structure consisting of the Poincaré cross-section points forming an even distribution along the geometric structure), the correlation dimension of the attractor of the system with a displaced characteristic (*ω* = 1.7 Figure 3b) is comparable to the attractor plotted for the classical potential function (*ω* = 1.7 Figure 3a). In conclusion, the parameter *d* does not significantly affect the position of the band of chaotic solutions. Both for *d* = 0 and *d* = 0.3, unpredictable solutions are located in comparable ranges of the variability *ω* [1.5, 1.9]. Similar values of the correlation dimension (DC) also result from the similarity of the geometric structure of the chaotic attractors and the density distribution of the points of intersection between the trajectory and the control plane.

The most important element distinguishing the two discussed cases is the parameter representing the effective value of the voltage induced on the piezoelectric electrodes. In the case of the "split" and shifted characteristic (*ω* = 1.7 in Figure 3b), the value is lower by approximately 0.08. This RMS voltage value is directly related to the lower amplitude of the displacement of the free end of the flexible beam I (Figure 1). The influence of the displacement of the "cut" halves of the potential barrier is best visible in the bifurcation diagrams. In particular, this is very visible in the range of low values of *ω*, where increasing the parameter *d* results in narrowing the zones of chaotic solutions. It is also worth noting that in the range of low values of ω, the zones of unpredictable solutions are additionally shifted, as a result of which it is impossible to directly refer to the Poincaré cross-sections that were plotted for the cases *d* = 0 and *d* = 0.3. The numerical modelling results presented in this section were obtained for zero initial conditions. This approach however does not fully reflect the capability of harvesting energy from vibrating mechanical devices. Therefore, the numerical simulations focused on the assessment of initial conditions from the point of view of coexisting solutions.

**Figure 3.** Influence of the parameter *d* on the geometric structure of the Poincaré cross-section: (**a**) *d* = 0, (**b**) *d* = 0.3. Starting from the top, the horizontal panels correspond to the Poincaré map and the corresponding time series for selected ω and amplitude–frequency spectrum. The bottom panels show the bifurcation diagrams based on the local minima (red) and the local maxima (blue) compared with RMS(u) for *d* = 0.3. To clarify the influence of the parameter *d* on bifurcations in the selected frequency interval, (**c**,**d**) magnify the difference between the cases *d* = 0 and *d* = 0.3, respectively. The simulation results were obtained assuming zero initial conditions.

With reference to the presented Poincaré sections, Fourier amplitude–frequency spectra were plotted. They provide information on the dominant harmonics in the time series, which constitutes the formal basis for depicting Poincaré maps. Based on the spectra, it is possible to conclude that in the case of developed attractors, for which the correlation dimension *DC* > 1.5, harmonic components representing the frequency of the source of excitation dominate in the Fourier spectra. We deal with such geometrical structures of Poincaré cross-sections in the widest zone of chaotic solutions, located in the band *ω* [1.4, 1.9]. However, in the range of low ω values, the correlation dimension of the Poincaré cross-sections *DC* <1.5. At the same time, in the amplitude–frequency spectra, it is represented by the domination of harmonic components that are a multiple of the frequency of mechanical vibrations affecting the energyharvesting system. We deal with such a sequence of dominant harmonic components of the Fourier spectrum when the correlation dimension of Poincaré cross-sections is in the *DC* range of [1.1, 1.5]. At this point, it is worth noting that such spectra occur both in the case of continuous and smooth (*p* = 1, *d* = 0, ω = 0.42) and discontinuous (*p* = 1, *d* = 0.3, *ω* = 0.26) characteristics representing the potential barrier. On the basis of the presented results of numerical experiments, it is also possible to state that for correlation dimensions *DC* < 1.1, in the amplitude–frequency spectra, the dominant harmonic components are the multiples of the combination of the two fundamental frequencies ω and ω1. As in the previous example, this is the case with both continuous and smooth (*p* = 1, *d* = 0, *ω* = 0.21) and discontinuous (*p* = 1, *d* = 0.3, *ω* = 0.48) characteristics representing the barrier potentials.

The diagrams (the bottom panels Figure 3a,b) show a direct comparison of Feigenbaum's steady state bifurcation diagrams with the diagrams of RMS values of the voltage induced on the piezoelectric electrodes. In the wide range of variability of the dimensionless excitation frequency *ω* [0, 4], bifurcation diagrams are dominated by one relatively wide band of chaotic solutions, within the *ω* range of [1.4, 1.9]. On the other hand, in the band *ω* < 1, there are many bands corresponding to unpredictable solutions, separated by pieriodic areas and bifurcation zones. The individual bands characterizing the dynamics of the tested energy-harvesting system were distinguished by colors: the bands of chaotic solutions were depicted with a light cyan color, and the bifurcation zones with a light magenta color. Based on the presented graphs, it is difficult to unambiguously characterize the dynamics of the system, because both in the bands of chaotic solutions and bifurcation zones we deal with an increase and a decrease in the effective voltage induced on piezoelectric electrodes. We are also dealing with both the decrease and increase in the voltage induced on piezoelectric electrodes in the areas of periodic solutions. For example, the voltage drop induced on piezoelectric electrodes in the area of periodic solutions is in the ω band of [0.225, 0.245] (Figure 3c). The same is the case when the halves of the "cut" potential barrier overlap with *ω* in [0.23, 0.255] (Figure 3d), and the range of variation of the dimensionless excitation frequency is similar. In the remaining bands of periodic solutions, for example *ω* [0.337, 0.395] (Figure 3c) and *ω* [0.355, 0.42] (Figure 3d), we deal with the opposite situation, i.e., an increase in the voltage induced on the electrodes is observed. While any detailed inference regarding the correlation of the bifurcation diagram with the diagram of rms voltage values in the bands of chaotic solutions and bifurcation zones is difficult due to the large number of points, in the case of periodic solutions it is possible to formulate a probable diagnosis.

It is possible to formulate the hypothesis that if the mean of the slope coefficients approximating the branches of the bifurcation diagram in the area of periodic solutions is positive, then we are dealing with an increase in the effective value of the voltage induced on the piezoelectric electrodes. On the other hand, when the average value of the slope approximating branches of bifurcation diagrams takes negative values, a decrease in the RMS voltage is observed on the electrodes. If the mean value of the slope of the branch takes values in the vicinity of zero, then the RMS voltage diagram assumes a constant value.

#### *3.2. Identification of Multiple Solutions*

It should be emphasized that the results presented in the previous section pertain to selected cases of system dynamics for zero initial conditions. However, one of the most important properties of nonlinear dynamic systems is related to the coexistence of different solutions depending on the initial conditions. For the defined system and excitation parameters, one has to find the system's evolution with various initial conditions. This problem comes down to the study of phase plane orbit topologies [22], the origins of which are located in different places. In the case of systems with a greater number of degrees of freedom, coexisting periodic and chaotic solutions are identified in a multidimensional phase space. The space dimension is a multiple of the number of degrees of freedom. Considering the research problem formulated in this way, it is possible to conduct complex numerical calculations, the results of which can be illustrated in the form of a diagram of solutions (DS) [27,28]. In this approach, the information about the efficiency of harvesting energy from coexisting solutions is neglected. For this reason, this work contributes to the state of the art by proposing a different approach, the essence of which boils down to multiple plotting of voltage effective value diagrams. Every diagram is plotted for randomly chosen initial conditions of the phase space. Such presentation of computer simulation results provides information about the number of coexisting solutions. However, it does not provide information about their periodicity. For this reason, additional detailed computer simulations were performed to identify the periodicity of individual branches appearing in the diagrams of effective voltage induced on the piezoelectric electrodes (Figure 4). The following convention was adopted to define periodicity: the digit before the letter T indicates the periodicity of the solution, while the number of solutions with a given periodicity is denoted by the right subscript.

It should be noted that both the plotted DS diagrams and the diagrams of effective values plotted for randomly selected RMS(*u*) initial conditions were obtained in a simplified manner. As a result, their accuracy is a compromise between the precision of numerical calculations and the time of computer simulation. Below are some examples of diagrams showing the effective values of the voltage induced on the piezoelectric electrodes. On their basis it was possible to determine the ranges of variability of the dimensionless frequency ω in which energy harvesting is most effective. Examples of numerical results showing the effects of the dimensionless excitation amplitude p and the shift d of the left and right potential halves overlap in the barrier zone are given in the graphs (Figure 4).

Given the identified branches on the plotted diagrams of the RMS values of the voltage induced on the piezoelectric electrodes, additional numerical simulations were performed to identify phase trajectories of the coexisting solutions. The periodicity of individual branches was identified for the frequency *ω* > 0.4. Irrespective of the *d* shift value of the "cut" potential barrier, the highest energy-harvesting efficiency was obtained for 1Tperiodic solutions in the band *ω* < 2. A comparison of the numerical results presented in this section with the RMS(*u*) diagrams (Figure 2) reveals that the highest energy-harvesting efficiency in this band is achieved by assuming zero initial conditions. However, in the zone *ω* > 2, energy-efficient solutions can be obtained too, for example by initiating appropriate initial conditions. It is also worth noting that the parameter *d* does not affect the nature (periodicity) of a given solution. Its influence is mainly visible in the shift of individual branches in relation to the frequency axis ω. However, with increasing the parameter *d* value, this shift is directed towards higher values of the dimensionless excitation frequency ω. Moreover, as the parameter *d* is increased, the efficiency of energy harvesting is reduced. A detailed examination of the single points and branches in the diagrams (Figure 4) shows that they represent transient solutions which finally become attracted to permanent periodic solutions over long enough time spans. This is the case with, e.g., the 4T2 branch that appears in the diagrams identified for *p* = 2.

**Figure 4.** Bifurcation diagram: effect of parameters *d* and *p* on energy recovery efficiency with various initial conditions considered simultaneously for the given system parameters. The parameters are indicated in the figures. For better clarity, the solutions are marked by *nTm* (*n*—the response period, *m*—the number of different solutions with the same response period). The bottom panels are the summery of the upper left and right parametric cases to show the tendencies of the system evolution with *d* parameter changes.

For clarity, the selected cases were studied with the help of phase plane trajectories (Figure 5). The images are shown against the background of the potential barrier. It should be noted that the multiple solutions, which are marked in different colors depending on the solution periodicity, occur with the evolution of the frequency *ω*. The Poincaré points are also identified, which confirms the periodicity of the solutions obtained. The multiple solutions of the same period are plotted using the same color. Usually, they are trapped in the left and right potential wells.

The trajectories, which were depicted against the background of the "cut" and shifted halves of the potential barrier, illustrate the possible cases of coexisting solutions. Figure 6 presents selected examples of coexisting solutions. The examples were selected to illustrate the effect of the superimposition of the "cut" halves of the potential barrier, with respect to large (Figure 6a) and small (Figure 6d) orbits of coexisting solutions and unpredictable responses (Figure 6b). The following convention was adopted during the graphical visualization of the results of computer simulations: the colors of the coexisting solutions were assigned to the individual branches of the diagrams of the RMS values of the voltage induced on the piezoelectric electrodes (Figure 4), while the coexisting solutions are plotted with dashed lines.

**Figure 5.** Influence of the parameters d and p on the efficiency of energy harvesting: 3D orbits with the vertical axis indicate the total mechanical energy. For clarity, the corresponding hysteretic potential is plotted with a division into two-color flaps with overlap. The system parameters are indicated in the figures.

**Figure 6.** Examples showing the influence of parameter *d* on coexisting solutions, plotted for: (**a**) and (**b**) coexisting solutions for *p* = 1, *ω* = 1.6, (**c**) *p* = 1, *ω* = 2.0, (**d**) *p* = 1, ω = 3.4.

Based on the results of computer simulations, it can be concluded that the increase in the value of the parameter *d* does not change the nature of the solution. This behavior of the system is observed both in the case of periodic and chaotic solutions. It is also worth noting that the vibrations of the elastic cantilever beam for periodic solutions do not undergo a phase shift with the increase in the parameter *d* (Figure 6a,c). In the case of large and medium orbits, only a reduction in the amplitude of vibrations is observed. However, amplitude limitations were not observed for solutions whose orbits are located inside the potential well (Figure 6d). If the response of the system is given in the form of a chaotic solution (Figure 6b), the change of parameter *d* does not cause a significant deformation of the geometric structures of the Poincaré sections, and their similarity is preserved. The differences in the plotted Poincaré cross-sections become apparent when the correlation dimensions are identified. With an increase in the value of the parameter *d*, a decrease in the value of the correlation dimension *Dc* is observed. On the other hand, in graphical images of amplitude–frequency spectra, it is manifested by limiting the amplitude of the excited harmonics, with a simultaneous slight shift towards higher values.

#### **4. Conclusions**

The influence of parameter *d* reduces the distance between the external potential barriers, and as a result, the efficiency of energy harvesting changes. On the one hand, the system easier undergoes the potential barrier, on the other, the large orbit size is limited. The small orbit solutions are very similar for any d as they are governed by the linearized equations. In the hysteretic potential case, chaotic solutions may appear easier as a result of the higher competition between unstable small and large orbits. It is worth noting that the hysteretic property of the system can be a side effect of bistable structures such as bistable plates or beams [20,29]. Consequently, the amplitude of resonator oscillations is smaller, and the efficiency of energy harvesting is decreased with respect to the system without hysteresis. In the next step, we would perform experimental verification of the observed tendencies with a suitable metrological characterization. Based on the numerical experiments carried out, it is possible to formulate the following, more detailed conclusions:


The presented graphs clearly indicate that when designing energy-harvesting systems, the impact of hysteresis caused by overlapping potential barriers should be avoided or minimized. From an engineering point of view, such characteristics of potential barriers can be found in mechanical systems that have been pre-deformed, or in the case of shapememory materials.

**Author Contributions:** Conceptualization, G.L., J.M., D.G. and A.R.; methodology, G.L., J.M. and D.G.; calculations, J.M., D.G., A.R.; validation, G.L., C.T.; formal analysis, J.M. and D.G.; original draft preparation, J.M., G.L., review G.L., J.M., D.G., A.R. and C.T.; visualization, J.M.; supervision, G.L. and J.M.; project administration, G.L. and J.M.; funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Ministry of Science and Higher Education in Poland under the project DIALOG 0019/DLG/2019/10 during the years 2019–2021.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

