Experimental and numerical assessment of a high-performance electromagnetic vibration energy harvester in a double-deck railway tunnel

Experimental and numerical assessment of a high-performance electromagnetic vibration energy harvester in a double-deck railway tunnel Victor Ordo ˜ nez1, Hassan Liravi1, Robert Arcos1,2, Jordi Romeu1and Behshad Noori3 Abstract This paper evaluates the performance of an electromagnetic vibration energy harvester for powering wireless sensor networks in railway tunnels. A field test was conducted in a double-deck railway tunnel, measuring vibrations as well as the mechanical and electrical responses of a prototype of the harvester during 139 passing trains. Numerical simulations have been carried out to estimate the harvester’s output performance from which a good agreement has been reached when comparing the experimental and simulated induced voltages. Using the validated model and measured vibration data, the harvester’s performance adopting a fully frequency-tuned resonant system or different implementation locations, such as the tunnel wall and the rail, has been simulated. Cumulative electrical energy generated by the harvester for different locations is presented for the total period of railway-induced vibration measurements. Maximum cumulative electrical energies of approximately 1805 mJ, 0.52 mJ, and 7838 mJ have been estimated to be generated by the harvester when placed on the interior floor, the wall, and the rail, respectively, in about nine hours of operation. Keywords Electromagnetic vibration energy harvester, Railway-induced vibration, Structural health monitoring, Vibration-based generator, Ballastless track tracks, specially at the rail (Zhang et al. 2024;Gao et al. 2017;Yang et al. 2021;Gao et al. 2016). Nonetheless, the high amplitudes of vibration that are typically induced in these systems, particularly in the rail where displacements and accelerations of vibration can reach 12 mm (Sadeghi et al. 2017;Pan et al. 2019) and 30 g (Gao et al. 2017), respectively, may put at risk the structural integrity of the mechanical subsystem of the harvester or reduce its reliability and lifespan (Kim et al. 2017). The present paper is focused on the application of VEHs in other subsystems of the railway system, such as slabs, sleepers, and the tunnel structure. Vibration levels in these locations are usually significantly lower than those at the rail. Consequently, producing enough electrical energy on their basis can result in a challenging task for most vibrationbased generators. Only a few studies have proposed VEHs to be installed on these other subsystems of the railway system. For instance, Hou and his colleagues developed an 1Acoustical and Mechanical Engineering Laboratory (LEAM), Universitat Polit` ecnica de Catalunya (UPC), c/Colom, 11, 08222 Terrassa (Barcelona), Spain. 2Serra H´ unter Fellow, Universitat Polit` ecnica de Catalunya (UPC). 3AV Ingenieros, c/Joan XXIII, 23, 08173 St. Cugat del Vall` es (Barcelona), Spain. Corresponding author: Victor Ordo˜ nez, Acoustical and Mechanical Engineering Laboratory (LEAM), Universitat Polit` ecnica de Catalunya (UPC), c/Colom, 11, 08222 Terrassa (Barcelona), Spain. Email: victor.hugo[email protected] Introduction Nowadays, there is a substantial and increasing demand for using railway transportation systems for mobilizing people in their daily activities and transferring goods for all types of industrial purposes due to their safety, efficiency, and punctuality among other transportation systems. Railway systems are playing, thus, a vital role in the connectivity and economic development of modern societies. Hence, it is of great importance to maintain railway network systems in optimum working conditions. For this purpose, onboard and track-side electrical devices such as ultra-low-power sensors (Perez et al. 2020; Wang et al. 2021; Zuo et al. 2021) for structural health monitoring (SHM) of railroad vehicles and railway tracks are typically employed. When access to the electricity grid is complex or unavailable, these sensors are normally powered by batteries. Batteries provide a continuous and stable power supply, but, unfortunately, they are not a cost-effective solution (Lin et al. 2015) that also introduces new conflicts for maintenance and logistics personnel. Since kinetic energy associated with vibrations induced by railway traffic i s a bundant, t here i s a high potential for vibration energy harvesters (VEHs) to scavenge that energy for powering wireless sensors (Zuo et al. 2023). In the last few years, several designs of VEHs have been proposed by different researchers for onboard railroad vehicle purposes, mainly installed on the bogie system of the railway vehicle (Cao et al. 2023; Perez et al. 2020; Ortiz et al. 2014; Bradai et al. 2018; Park and Kim 2016; De Pasquale et al. 2012), and also to be installed in railway 2 and described in the next section has been experimentally tested in the mentioned railway system under its normal operating conditions. Various numerical simulations have also been conducted to validate the simulation methodology in a real scenario and to determine the output performance of the EMVEH model in its actual condition as well as for the case when its natural frequency is tuned to optimize the electrical power generation. Moreover, to enrich this investigation, a simulation of the performance of the fabricated prototype model virtually applied to the wall and rail of the underground tunnel under evaluation are also exposed, based on experimental vibrations measured at these two locations. Finally, the cumulative electrical energy generated by the fabricated prototype in each case study and location is presented for the total period of railway-induced vibration measurements to have a general perspective of its applicability for powering WSNs in an underground railway environment. Due to the high-performance capabilities of the proposed EMVEH provided by its ring-shaped architecture and Halbach configuration, the results presented in this paper on the harvested energy from the slab and even the tunnel wall potentially demonstrate the applicability of the proposed device in this context. The present article can also serve as a survey of vibration levels occurring in a double-deck railway tunnel that could be employed, for example, to test other vibration energy harvesting devices for applications in this kind of underground railway systems. Methodology Description of the EMVEH The employed EMVEH (Ordo˜ nez et al. 2022) is mainly consisting of a “Magnet in-line coil” electromagnetic transducer mechanism based on three ring magnets with a linear Halbach array configuration that concentrates their magnetic field in the inner space of the mechanism where a single vertically-centered concentric coil has been located. Specifically, it consists of one radial ring magnet located in between two axial ring magnets with repelling forces. This particular structure allows for increasing the resonant mass within fixed dimensions of the transducer and to reduce the coil resistance for the same number of turns, enhancing the power generation capability of this device. Moreover, two helical compression springs have been selected as the elastic elements of the oscillating system, which is also composed of the three ring magnets and a magnet holder. The natural frequency of the harvester (61.7 Hz) is designed to be within the range of railway-induced vibrations in tunnels, which are usually between 30 Hz and 100 Hz for all components except the rails (Balastegui et al. 2013;Gupta et al. 2008). Still, this does not mean that it is expected to be exactly tuned to one of the most dominant frequencies of the system response. The overall dimensions of the fabricated prototype (including its external case) are 2.55 cm of radius and 4.7 cm of height, giving a total volume of 96 cm3. The rest of geometrical characteristics and properties of the employed device are thoroughly presented in the mentioned previously published work of the authors (Ordo˜ nez et al. 2022). electromagnetic vibration energy harvester (EMVEH) to be applied in the slab track of Guangzhou Metro (Hou et al. 2018). The numerical simulation results of the performance of the proposed device estimated a peak power density value of 176.5 µW/cm3, which is also reported to be much higher than that of most existing similar devices. Later on, the same authors proposed in Hou et al. (2021) a multilayer piezoelectric vibration energy harvester (PVEH) meant to be applied at the edge of the slab in the floating slab track system (FSTS) of the railway bridge system under study. Different numerical simulations were carried out to determine that by installing 144 of these devices, arranged on 36 units of the FSTS, the amount of electrical energy produced in 17 hours of train operation would reach 31.4 kJ. As a result, they have far exceeded the 12.85 mJ required in the working cycle of each wireless sensor network (WSN) node used in their monitoring systems. Gatti et al. (2016) also presented a numerical investigation of how much energy can be harvested with a single-degree-of-freedom (SDOF) linear VEH from the vertical vibration of a sleeper of the Great Western Main Line in the United Kingdom as an Intercity 125 train passed by at a speed of 195 km/h. Results indicate that the maximum energy that could be harvested in this system is about 0.25 J per unit (kg) of energy harvester mass at a frequency of about 17 Hz. Cahill et al. (2014) investigated the feasibility of piezoelectric-based energy harvesting on a single-span steel-concrete composite railway bridge in Sweden for different types of passing trains. Experimental results determined that the proposed VEHs can produce an electrical output up to 588 µW during the passage of a passenger train. The previous literature review discussed vibration energy harvesting devices that directly transform the kinetic energy of the linear vibration to electrical power. However, alternative energy transformation technologies have been applied to design and develop other types of vibrationbased generators for the railway sector. Among them, a popular category comprises devices that convert the linear displacement of the rail into rotational motion to drive electrical generators. Some examples of these types of VEHs can be found in (Zhang et al. 2016; Pan et al. 2019; Gao et al. 2020; Wang et al. 2012; Lin et al. 2018b,a). However, it is worth noting that these VEHs tend to be bulky and that their application is restricted to the rail itself, becoming a less suitable alternative to power WSNs. Nowadays, concrete slabs are common components of modern railway systems. Mounting VEHs on tunnel concrete slabs presents several advantages compared to other locations within the tunnel. Essentially, they are flat and extended surfaces that facilitate the installation, the mechanism of fixation, a nd t he u pscaling o f t he device (if ever required). Also, when an EMVEH is installed on a concrete slab instead of a rail system, there is a lack of electromagnetic interactions that could otherwise affect the dynamic response of the device (Kuang et al. 2021). In this context, this paper attempts to assess a Halbach-based EMVEH for harvesting the kinetic energy of the vibration induced by underground railway traffic on a concrete slab that performs as the interior floor o f a double-deck tunnel of Metro Barcelona. For this purpose, the fabricated prototype developed in (Ordo˜nez et al. 2022) METHODOLOGY 3 Study of the influence of the vibration amplitude on the performance of the EMVEH prototype In the referenced study, the authors present an experimental characterization of the fabricated EMVEH prototype using sinusoidal base excitation applied by an electrodynamic shaker, with a constant amplitude of 0.03 g. However, the vibration levels induced by train traffic can vary significantly from this value. To characterize the prototype performance at different amplitudes, the laboratory tests reported in Ordo˜ nez et al. (2022) have been further complemented for the present work with results at two other excitation amplitudes: 0.006 g and 0.15 g for input RMS acceleration. The corresponding results are depicted in Figs. 1and 2, illustrating the transmissibility and the induced voltage transfer function (TF) of the transducer. These frequency response functions describe the relationships between the base acceleration, measured using two accelerometers, and both the acceleration of vibration of the resonant mass and the induced voltage of the harvester. Results indicate that 60 61 62 63 64 Frequency (Hz) 101 102 103 Transmissibility Input acc. = 0.006 g Input acc. = 0.03 g Input acc. = 0.15 g Figure 1. Transmissibility of the fabricated prototype considering various values for the input RMS acceleration. 60 61 62 63 64 Frequency (Hz) 101 102 103 Induced Voltage TF (V/g) Input acc. = 0.006 g Input acc. = 0.03 g Input acc. = 0.15 g the damping of the fabricated prototype is highly dependent on the input amplitude, with a significant reduction in performance as the amplitude increases. This can be better observed from the normalized power density (NPD) (refer to Appendix A for the formulation) obtained for the three excitation amplitudes, 0.006 g, 0.03 g, and 0.15 g, which are 62.2 mW cm−3g−2, 16.1 mW cm−3g−2, and 1.2 mW cm−3g−2, respectively. These NPD values are computed considering the technical specification and details presented in Ordo˜ nez et al. (2022). However, vibration levels induced by railway traffic in a tunnel system, particularly those near the resonant frequency of the harvester, fall within the lower range of the RMS values considered in these laboratory tests. Therefore, it can be concluded that the fabricated prototype is expected to achieve high performance when harvesting energy in a railway tunnel environment. Experimental field test Description of the test site: With the aim of evaluating the performance and applicability of the previously mentioned EMVEH on railway tunnels, with a special focus on concrete slab components, an experimental field test was carried out on a tunnel section of Metro Barcelona. More specifically, the field test was conducted in a section of the double-deck railway tunnel of lines L9 and L10 of Metro Barcelona near Torrassa station, located at L’Hospitalet de Llobregat in Barcelona, Spain. The general dimensions of this doubledeck tunnel correspond to an inner diameter of 10.9 m and a thickness of 0.336 m, whereas the width and height of the interior floor that supports train passages in the upper deck are 10 m and 0.3 m, respectively. As indicated in Fig. 3(a), two railway tracks exist on each tunnel deck, all based on UIC54 rails with standard gauge. The fabricated prototype was installed on the middle point of the interior floor, located between railway lines 1 and 2, as observed in Fig. 3(b). Moreover, experimental vibration measurements on the wall and the rail were conducted in parallel to tests on the interior floor location. It is worth mentioning that subway trains in lines L9 and L10 typically pass through railway line 2 on the upper deck in one direction and on the lower deck for the opposite one, while railway line 1 is dedicated (in both cases) to special situations. Moreover, subway trains for both lines are scheduled to arrive at Torrassa station at the same time with a periodicity of approximately 480 seconds. However, since the measurement sites were not at the station but deeper inside the tunnel, there is a passing difference of approximately 120 seconds between trains that reach Torrassa station at the same time from opposite directions. Consequently, the railway-induced vibration measurements coming from the trains on each deck can be easily identified in the vibration measurements at the interior floor, wall, and rail locations. Field test setup: Fig. 4shows the experimental setup adopted to conduct the proposed field test. On the one hand, the physical prototype of the EMVEH was wax-fixed to a steel plate that was first glue-fixed onto the interior floor. Two PCB Piezotronics accelerometers 352C65 (Acc1 & Acc2) with a frequency range of 0.5 Hz to 10000 Hz and a sensitivity of approximately 100 mV/g were magnetically Figure 2. Induced voltatge transfer function (TF) of the fabricated prototype considering various values for the input RMS acceleration. 4 1.53 m D E 1.53 m 1.0 m (09(+ 5DLOVOLQH 5DLOVOLQH Figure 3. The field test environment. (a) Schematic of the tunnel and location of the EMVEH. (b) General view of the physical prototype mounted on the interior floor of the double-deck tunnel site. a) b) Acc3 Prototype Acc1 & Acc2 LMS SCADAS $FFHOHURPHWHURQUDLO1.5 m $FFHOHURPHWHURQZDOOF) Figure 4. Experimental setup adopted in the field test. (a) Mounting configuration of the three accelerometers and the prototype. (b) Schematic of the double-deck tunnel and the mounting locations of the accelerometers on the wall and rail. (c) Connection to the data acquisition system and computer. RESULTS AND DISCUSSION 5 physical components of the system. Figs. 5and 6show diagrams of the model implementation for the open circuit operation case (without external load resistance) and for the closed circuit operation case (with external load resistance), respectively. Moreover, there are two main inputs in these numerical algorithms: The first input used in all simulation cases is the external base excitation, corresponding to the railwayinduced vibration measurements at each location. The second input comprises all mechanical, electromagnetic, and coil parameters of the fabricated prototype, which have been established and/or experimentally estimated in the design process and experimental validation of the fabricated prototype in Ordo˜ nez et al. (2022). However, it is important to note that the natural frequency of the harvester will be modified in these simulations (from one case or location to another) as they are considering frequency-untuned and frequency-tuned conditions. It is also important to note that the damping coefficient employed in the simulations is based on laboratory tests for an input RMS acceleration of 0.03 g, as this value was found to provide results most closely matching the experimental data. This damping coefficient was applied in all simulations, not just those related to the fabricated prototype, as it is expected that an improved version of the proposed EMVEH design will exhibit more stable damping behaviour. Results and discussion Experimental results Interior floor location: The railway-induced vibration and harvester response signals measured in the previously described experimental campaign allow for determining the dominant frequencies and acceleration levels at the interior floor location selected in this investigation, as well as analyzing the mechanical behavior and electrical performance of the fabricated prototype. In this regard, Figs. 7and 8show the acceleration spectral density (ASD) and acceleration spectrum in one-third octave bands of the vibration signals induced by the passage of trains on the lower and upper decks, respectively, while Fig. 9illustrates the acceleration and induced voltage spectra in one-third octave bands of the EMVEH response to all 139 passing trains. The acceleration responses and induced voltage of the one-third octave band results are presented in terms of dB with a reference of 10−6and 1, respectively. The onethird octave bands are normalized using the average time that takes the first and the last wheel of the train to pass from the same point. It is relevant to mention that Figs. 7and 8 denote the average of measured accelerations in Acc1 and Acc2, from which dominant frequencies are found to occur at about 50 Hz for both the lower and upper deck passing trains. However, as expected and observed in Fig. 9(a), the dominant frequency of the mechanical subsystem is around 63 Hz, which is consistent with the natural frequency of the fabricated EMVEH. Consequently, the maximum induced voltage also happens at the one-third octave band of 63 Hz, as can be observed in Fig. 9(b). This information allows for determining that the fabricated prototype is not performing at its best since the natural frequency of the harvester is not matching the most dominant frequencies of the vibration fixed t o t he s teel p late t o m easure t he i nput railwayinduced vibration, while one PCB Piezotronics miniature accelerometer 352B10 (Acc3) with a frequency range of 2 Hz to 10000 Hz and a sensitivity of approximately 10 mV/g was wax-fixed to the resonant mass of the prototype to measure its response, as observed in Fig. 4(a). On the other hand, two accelerometers have been employed in this experimental setup, one fixed t o t he t unnel wall and the other to the rail, to measure their railway-induced vibration response. Fig. 4(b) shows a schematic of the location of these two accelerometers within the tunnel. For the sake of reliability, each accelerometer was previously calibrated with an IMI 699A02 handheld shaker. Based on this experimental test setup, the vibration measured from those five accelerometers and the electrical response of the harvester were acquired for nine consecutive hours using the LMS SCADAS data acquisition system (Fig. 4(c)), considering a sampling frequency of 512 Hz, from which a total of 139 passing trains in both directions were recorded. More specifically, t his c orresponds t o 7 0 a nd 6 9 passing trains running through the upper and lower decks of the tunnel, respectively. EMVEH numerical simulation Different numerical simulation cases have been conducted to estimate the electrical output performance of the proposed EMVEH applied to the interior floor o f t he double-deck tunnel section under study. First, the frequency-untuned device, which corresponds to the fabricated prototype model, was simulated. This simulation aimed to compare the induced voltage (under open circuit operation conditions) to that obtained experimentally, assessing the accuracy of the proposed numerical algorithm for the specific measured input signals. Second, the load voltage and electrical output power (closed circuit operation condition) of the frequency-untuned device have been estimated. Third, the overall output performance of the fabricated prototype model has been simulated for the case of a frequency-tuned system. Furthermore, simulations of the electrical output performance of a frequency-tuned EMVEH model applied to the wall and the rail of the double-deck tunnel section under study have been also carried out. The aim of including these simulations is to demonstrate the capabilities of the proposed harvester in underground railway applications from a global perspective. The experimentally measured railway-induced vibration signals (at the interior floor, w all, a nd r ail) correspond to random vibrations mainly generated by the wheel-rail contact of passing trains. Therefore, the electrical response of the harvester to these vibrations also corresponds to a transient random induced voltage. To simulate the response of the EMVEH to these transient random signals, a numerical time-domain algorithm implemented in Simulink has been developed based on the governing equations and modeling approach presented in Ordo˜nez et al. (2022); Ordo˜nez et al. (2021); Ordonez et al. (2021). A thorough review of the theory and governing equations underlying the employed simulation methodology is presented in Appendix A for the reader’s convenience. In this proposed implementation, the physical modeling blocks of the Simscape extension of Simulink have been employed to represent some of the actual 6 Figure 5. Diagram of the numerical algorithm implemented in Simulink for computing the transient induced voltage of the fabricated harvester model for an open circuit operation case and input random vibrations. Figure 6. Diagram of the numerical algorithm implemented in Simulink for computing the transient load voltage and output power of the fabricated harvester model for a closed circuit operation case and input random vibrations. passages numbers 8 and 98), and the remaining one is related to a lower deck train passing (train passage 49). The railway-induced vibrations and EMVEH responses generated by the mentioned trains are illustrated in Figs. 10 and 11, respectively. The time histories are presented in a concatenated view for comparison purposes. As appreciated in Fig. 10(a), the vertical railway-induced vibration can reach peak amplitudes of up to 0.7 g, while the ASD of each of the chosen passing trains, depicted in Fig. 10(b), allows for determining that the most dominant frequencies of the interior floor response are 49.2 Hz, 36.9 Hz, and 54.3 Hz, respectively. Besides, Fig. 11(a) shows that responses of the source. Nevertheless, the natural frequency of this device has been designed without knowing these particular values, but modifying it for this or any other application (once the dominant frequencies are identified) i s a s imple t ask that mainly involves redesigning the helical compression spring parameters. Even though a relatively consistent pattern was observed when analyzing the 139 railway-induced vibration responses, the vibration signals induced by three non-consecutive train passages with clearly diverse amplitudes and dominant frequencies were selected. Two of these chosen events are associated with trains passing by the upper deck (train RESULTS AND DISCUSSION 7 8 10 12.5 16 20 25 31.5 40 50 63 80 Frequency (Hz) Acceleration (dB ref. 10-6 g) 0 20 40 60 80 Frequency (Hz) $6' (g/Hz) ED 40 60 80 100 120 0 0.2 0.4 0.6 0.8 Figure 7. Railway-induced vibration response (average acceleration recorded by Acc1 and Acc2) of the 69 passing trains on the lower deck in terms of (a) acceleration spectral density and (b) one-third octave bands. Grey lines represent the response to each passing train, while black lines correspond to their average. 8 10 12.5 16 20 25 31.5 40 50 63 80 Frequency (Hz) Acceleration (dB ref. 10-6 g) 0 20 40 60 80 Frequency (Hz) $6' (g/Hz) ED 40 60 80 100 120 0 0.2 0.4 0.6 0.8 Figure 8. Railway-induced vibration response (average acceleration recorded by Acc1 and Acc2) of the 70 passing trains on the upper deck in terms of (a) acceleration spectral density and (b) one-third octave bands. Grey lines represent the response to each passing train, while black lines correspond to their average. 8 10 12.5 16 20 25 31.5 40 50 63 80 Frequency (Hz) 60 80 100 120 140 Acceleration (dB ref. 10-6 g) 8 10 12.5 16 20 25 31.5 40 50 63 80 Frequency (Hz) -100 -80 -60 -40 -20 0 20 Induced Voltage (dB ref. 1 V) DE Figure 9. The one-third octave band spectra of the mechanical and electrical responses of the EMVEH prototype for the 139 passing trains in terms of (a) the acceleration recorded by Acc3 and (b) the induced voltage. Grey lines represent the response to each passing train, while black lines correspond to their average. It is interesting to notice how similar the acceleration and induced voltage curves of the EMVEH are, showing the linearity of the transformation process from kinetic to electrical energy of the prototype, as well as its high sensitivity. Also, as observed in Fig. 11, the experimental results of the mechanical and electrical responses of the oscillating mass can reach peak amplitudes of up to 5.6 g, whereas Fig. 11(b) illustrates the experimentally measured induced voltage of the fabricated EMVEH for an open circuit operation case, with a maximum peak amplitude of almost 4 V. 8 0 20 40 60 Time (s) -1 -0.5 0 0.5 1 Acceleration (g) a) WKWrain WKWrain 9WKWrain 0 20 40 60 80 Frequency (Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 A6' (g/Hz) b) I n  49.2 Hz I n  54.3 Hz I n  36.9 Hz Figure 10. Railway-induced vibration response at the EMVEH prototype location (average acceleration recorded by Acc1 and Acc2). (a) Time histories of the selected samples. (b) Acceleration spectral density of the vibration induced by each passing train sample. 0 20 40 60 Time (s) -6 -2 -4 0 2 4 6 Acceleration (g) a) 0 20 40 60 Time (s) -5 0 5 Induced Voltage (V) b) Figure 11. Time histories of the mechanical and electrical responses of the fabricated prototype for the three train samples under study. (a) The experimentally measured acceleration recorded by Acc3. (b) The experimentally measured induced voltage. RESULTS AND DISCUSSION 9 Simulation and validation of the frequency-untuned EMVEH on the interior floor To validate the proposed simulation models presented in the previous section, the induced voltage responses of the fabricated prototype have been simulated, and the results have been compared with the experimental measurements carried out. Hence, a natural frequency of 61.7 Hz has been employed in these simulations, corresponding to the exact natural frequency of the prototype. The experimental and simulated induced voltage results of the proposed harvester on the interior floor location for the three train passages selected are presented in Fig. 14. This figure allows for determining that a good approximation has been reached by the simulation in terms of amplitude and shape. This can be appreciated in more detail in the moving RMS (mRMS) results, which are shown for both simulated and experimental signals in the same figure. The mRMS has been calculated considering an integration time of one second. Also, the RMS induced voltage of each passing train has been computed and presented in Table 1for both the experimental and simulated results as a reference value to estimate the relative error of the simulation. In this manner, the proposed numerical algorithm, implemented in Simulink, is validated for estimating the generated voltage of a SDOF EMVEH subjected to random vibration with a maximum deviation of about 13.58%. It is found that one of the main reasons for these discrepancies between the experimental and simulation results is the observed slight changes in the mechanical damping of the system each time the harvester prototype is reassembled. Moreover, Fig. 15 illustrates the results of the simulation for the load voltage and electrical output power. The load resistance value utilized in this simulation corresponds to 10910 Ωsince it was previously determined as the optimal load resistance of this device (Ordo˜ nez et al. 2022). For these particular samples, a maximum peak amplitude of approximately 3.45 V and 1.10 mW are observed for the load voltage and electrical output power, respectively, while the electrical energy generated by the EMVEH for each sample of passing trains numbers 8, 49, and 98 has been estimated to be 0.98 mJ, 0.34 mJ, and 0.43 mJ, respectively. Nevertheless, it is relevant to remind the reader at this point that the cumulative energy harvested by the EMVEH in response to the 139 passing trains for each case study and location will be presented at the end of this section. Simulations of the frequency-tuned EMVEH Interior floor location: For simulations of the frequencytuned case, the natural frequency of the fabricated prototype model is now set to 49.2 Hz to match the most dominant frequency when analyzing the average ASD of all 139 passing trains, which coincides with the most dominant frequency of the three evaluated excitation samples. The results of these simulations are depicted in Fig. 16, from which it can be observed that the load voltage and output power induced by the vibrations generated by the passage of trains numbers 8 and 98 are dramatically higher than those generated by train number 49. The explanation for this behavior reduces to the dominant frequencies of the vibration signal of train 49, previously found to be occurring around harvester mainly amplify the signal corresponding to train number 8 when its most dominant frequency (49.2 Hz) is farther from the natural frequency of the harvester (61.7 Hz) than that of train number 98 (54.3 Hz). The explanation for this behavior is related to the higher peaks of train number 8 in the range of 60 Hz to 63 Hz than those corresponding to train number 98. Wall location: Following the same criteria and procedure as with the interior floor location, the 139 railway-induced vibration measurements on the wall have been divided as they correspond to trains passing through the upper and lower decks of the tunnel. The vibration signals induced by three non-consecutive train passages were chosen for their particular study, and time histories are presented in a concatenated view for comparison purposes (the same procedure is also employed for the rail location exhibited in the following subsection). Fig. 12 illustrates the experimental radial railway-induced vibrations on the tunnel wall for the selected samples consisting of the same three passing trains previously analyzed in the case of the harvester placed on the interior floor o f t he d ouble-deck t unnel. A s observed in Fig. 12(a), low vibration levels are occurring at the tunnel wall, with a maximum peak amplitude of about 0.035 g, corresponding to almost 20 times less amplitude in comparison to the maximum peak acceleration amplitude of the three train samples at the interior floor. Besides, the ASD of each passing train sample, shown in Fig. 12(b), allows for determining that their most dominant frequencies are 67.7 Hz, 67.6 Hz, and 67.7 Hz, respectively. Rail location: The experimentally measured vertical railway-induced vibrations on the rail are depicted in Fig. 13. In contrast to the samples employed in the interior floor and tunnel wall evaluations (trains numbers 8, 49, and 98), the samples used for the rail location correspond to trains going only through the upper deck, meaning that train number 49 has been replaced by train number 50. The reason for this decision is that the vibration responses on the upper deck rail due to trains passing through the lower deck are drastically lower (Clot et al. 2016) compared to those going through the upper deck, as one could expect, and illustrating these signals in the same plot offers no visual information. Nonetheless, lower deck trains will be considered in the final cumulative electrical energy estimation. As appreciated in Fig. 13(a), accelerations at the rail can reach maximum peak amplitudes of around 30 g, far exceeding the peak accelerations at the wall and interior floor. Yet, t hese very high amplitudes seem to be an extraordinary response occurring only for the vibration measurements of train number 8, which can occur due to the variability in wheel roughness, axle weights, and even slight variations in the train speed, braking, or acceleration patterns passing through the site under study. However, very high peak amplitudes of approximately 10 g are also observed in response to the passage of trains 50 and 98. Additionally, the most dominant frequencies of the samples under study are found to be 58.4 Hz, 58.5 Hz, and 70.8 Hz for the passing trains numbers 8, 50, and 98, respectively. 16 (Spreemann and Manoli 2012). Fig. 20 depicts the circuit representation of the electromagnetic subsystem connected in series to a load resistance Rl(Beeby and Kazmierski 2011). The governing equation of this circuit, then, can be expressed as Lc di(t) dt + (Rc+Rl)i(t) = ε(t),(3) in which Lcis the coil inductance, i(t)is the current flowing through the coil, and Rcis the coil resistance. By neglecting the coil inductance under the assumption of low frequencies of vibration, the current can be found to be i(t) = kt˙z(t) Rc+Rl ,(4) which leads to that the voltage Vlacross the load resistance and the corresponding electrical output power Pout can be estimated as Vl=Rl Rl+Rc kt˙z(t),(5) and Pout =V2 l Rl .(6) Figure 20. Circuit diagram representation of the electromagnetic subsystem of the energy harvester. According to Lenz’s law, a feedback electromagnetic force Feis generated due to the current flowing through the coil, and it can be written as (Williams et al. 2001) Fe=kti(t) = k2 t Rc+Rl ˙z(t).(7) The electromagnetic force can also be expressed as a product of the electromagnetic damping coefficient and the relative velocity of the oscilating mass. Consequently, the electromagnetic damping coefficient induced by the electromagnetic subsystem of the transduction mechanism can be found to be (Saha 2011;Thein et al. 2019) ce=k2 t Rc+Rl .(8) The maximum power transfer theorem would lead one to set Rl=Rcto obtain maximum electrical output power. Still, that approach does not consider the effect of the electromagnetic damping on the mechanical behavior. In turn, the optimal load resistance that maximizes the output power of an EMVEH is given by (M¨ osch and Fischerauer 2019) Rl=Rc+k2 t cm .(9) Energy harvesting performance To quantify the performance of a VEH, Beeby et al. (2007) have proposed a formula for NPD. 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