Dynamic responses of the 2DOF electromagnetic vibration energy harvester through different electrical coil connections

Highlights

• Friction present in the system will reduce the isolation frequency.

• Electromagnetic damping on series is highest/sum of those on individual coil.

• Individual 2 connection has highest normalized power density of 14.464 ${\mathrm{W}\mathrm{m}}^{-1}{\mathrm{s}}^{-5}$.

• Series and individual 2 outperform other mode and two SDOF in bandwidth and power.

• Operational impedance divides to high/low on series/individual 2 modes respectively.

Abstract

This work investigated the dynamic responses and evaluated the performance of a two-degree-of-freedom (2DOF) coupled coil-magnet electromagnetic vibration harvester excited at low frequencies. Three different connection modes with individual, in-series, and in-parallel configurations of the transduction coils were considered. Such approach was taken to identify the most suitable mode for usage because different sensor nodes have different power, current and impedance ratings and the optimum load associated with each connection mode differs. The above reasons also lead to the need for impedance matching of the sensor and the harvesters optimum load, as otherwise, the power deliverable to the sensor would be significantly lower than those achievable under a matched impedance condition. The mathematical model was established and verified experimentally, and a 97.8 % agreement is shown across the connection types at resonance of 9.69 Hz and 0.30 g excitation level. At optimum loads, the individual and series connections output the highest power at 497.50 mW and 410.40 mW, respectively. The optimum power peaked at densities of 223.13 Wm³, 1216.36 Wm³, 1009.03 Wm³, and 658.38 Wm³ for coils 1 and 2 (individual), series and parallel connections, respectively. Overall analysis concludes that the optimum harvestable power using in-series and individual coil 2 connection is higher by 47.41 % and 36.25 %, respectively, over the in-parallel connection and 81.77 % and 77.90 % over individual 1 connection. A comparison of the 2DOF harvester with an equivalent two SDOF harvesters at same resonance, excitation level and mechanical damping coefficient shows that that 2DOF series and individual coil 2 connections perform consistently well in preference to the equivalent two SDOF systems in terms of the power harvested, bandwidth, and normalized power density while associated optimum load resistance dichotomized the practical operational impedance into the high and low on the series and individual coil 2, respectively in the operational bandwidth corresponding to each design.

Introduction

With the recent need to ensure a reduction of the carbon footprint in compliance with the sustainable development goals, novel green, low-cost and eco-friendly methods for energy generation have attracted growing research interests. The concept of scavenging energy from free ambient energy such as that associated with vibration, thermal gradient, force/pressure gradients, tidal and solar radiations, has been reportedly utilized and with appropriate transduction mechanism converted into useful electrical energy to power sensor nodes, microelectronic and even some household electronic gadgets.

A two degree of freedom (2DOF) system has been analyzed and presented in literatures as a system having two independent directions/axes of response to external vibration. Engineering applications of a 2DOF system among others include a quarter-vehicle model which has found practical applications in the modelling and analysis of suspensions (shock absorbers) in automobiles, helicopter cockpits, vibration isolators/absorbers, etc. One unique feature of a 2DOF system is that the system has two normal vibration modes corresponding to two natural frequencies such that if an arbitrary initial excitation is imposed on the system, the resulting free vibration will be a superposition of the two normal modes of vibration corresponding to both natural frequencies.

Most of the harvester type reported in literature is a resonant single-DOF (SDOF) system such that the performance of such an SDOF system is optimum when the external excitation coincides with the predefined resonance. While noting that the concept of resonance is likewise applicable to 2DOF system, readily available sources of vibrations include, among others, those induced from train motion, vehicle suspensions, air conditioning systems, vibrations in cockpits and wingspans of flying vehicles etc. Real life applications of energy harvesters tap from this vibration sources to either monitor the structural health or power up sensors and micro gadgets operational on the vehicles. A review, analysis and comparison of methods for harvesting train induced vibration was investigated using electromagnetic, piezoelectric, triboelectric and hydraulic transduction methods by harvesting considerable amount out of the total 14 % energy loss associated with vibration, traction and heat during train motion to achieve a power supply for the structural health monitoring sensors on the track line and the vehicle while mentioning limitations such as stability, durability and economy as problems that are yet to be fully resolved in current research literatures [1]. An efficient and approximate method for computation of the harvested energy for train induced bridge vibrations was presented [2] as equivalent to the analytical modal solutions of the simply-supported Euler- Bernoulli beam transverse response under moving loads, while conceptualizing the train as a moving load it concludes that the harvested energy is strongly dependent on the rail traffic and a clear succinct mathematical model for the energy harvested found the optimum amount of energy harvested per unit mass is proportional to the product of the square of the input base acceleration amplitude and the square of the input duration [3]. Similarly, a conceptualization and optimization of a real life electromagnetic harvester using rail was demonstrated [4] and validated using the multi-physics model, and an average electrical power of $6.5mW$ was obtained experimentally. Likewise, various morphologies and designs of energy harvester; mostly electromagnetic based was reported to have been incorporated into various parts of automobiles like the suspension [5], [6], [7], rack and gear transmission [8] etc., and tested to give satisfactory performance to have harvest considerable vibration energy useable by payload and other micro gadgets/sensors on the automobiles.

Most of the harvester design reported in literature are resonant energy harvesters such that once they are factory tuned to a resonant, they cannot self-adjust to the dominant frequency of its environment in the event of spurious and stochastic vibration discharge. This situation constitute the major limitations to the efficient and autonomous operation of energy harvesting devices since such resonant harvesters are characterized by a narrow bandwidth, however, recent advances undertaken to overcome the demerits of such resonant systems [9], [10]. In an attempt to ensure a consistent, high, broadband and autonomous power supply is available to remote sensors, authors have reported on various methodologies of achieving such a novel ambition by varying the harvester’s design parameters to achieve higher response or frequency tuning/up conversion using pair of magnets [11], [12], stoppers [10], [13], [14] spring [15], parametric pendulums [16], antiphase motion [17] etc. One recent approach to achieving a better response culminating in harvesting a higher power at a lower resonant frequency in using a multi-DOF energy harvester. A novel example of such approach is the wave energy converter (WEC) that uses a spherical submerged bodies to increases the average captured power by 26 % for the WECs going from 2DOF to 4DOF while a 19 % decrease going from 2DOF to 5DOF was observed in the resonant frequency. These results translate to capturing power more efficiently at a lower resonant frequency [18] while another approach resulted in an increased bandwidth [19]. The use of a six degree of freedom (6 DOF) triboelectric nanogenerator (TENGs) designed to mimic the petal of a flower floating on ocean to harvest the chaotically stochastic ocean waves along the six possible DOFs was demonstrated to capture blue (ocean/sea) energy. The impressive device triggered by a water wave frequency of about 1.3 Hz and a wave height of about 8 cm charged a capacitor of $220\mathrm{\mu }\mathrm{F}$ to a voltage of $1.3\mathrm{V}$ in 1 minute while the harvested power was used to power a watch, a calculator, and a hygrometer thereby showing promising applications in developing self-powered smart marine sensors and distributed power systems in oceans [20]. Likewise a novel approach to harvest vibrational energy in the freight cable was introduced in [21]. The systems were demonstrated to be highly efficient, portable, reliable but plagued by two main challenges. These challenges as reported are capturing arbitrary random/stochastic vibrational energy efficiently and increasing the output power so that the system is suitable for cableway equipment that requires high power.

A parametrically excited harvester was demonstrated to have resulted in a large amplitude response and a potential buildup of harvested power because unlike the conventional harvesting technology that depends on the direct activation of the fundamental modes of resonance. The parametric excitation introduces a paradigm shift distinct from the normal resonant excitation because at least one of the system parameters is modulated to be time dependent. However, such feats come at potentially expensive limiting factors of requiring the excitation amplitude to exceed a certain initiation threshold prior to onset of the parametric resonant regime. A novel mechanism and design to reduce the short-comings of a parametrically excited vibration energy harvester (PEVEH) for practical realization are investigated [22]. As stated earlier, [23] it was iterated in Ref. [22] that the wideband performance of a parametric harvester was limited by the nature of the ambient excitation whose amplitudes are often not high enough to initiate a parametric excitation. An attempt to reduce the potential barrier that initiates a parametric excitation includes a wideband two-element piezoelectric energy harvester with both bi-stability and parametric resonance characteristics employing magnetic coupling effects between a parametrically excited beam with another directly excited beam [23]. The use of nonlinear stress-strain curves to achieve desired nonlinearities through field-induced striction by magnetostriction or electrostriction different from existing approaches, where external fields are harvested using strictive effects, the authors reported employing external fields that manipulate the effective Young’s modulus to achieve parametric excitations [24]. Using a pendulum-based harvester-absorber that allows for an improved vibration suppression and harvesting simultaneously by fixing the same poled cylindrical magnet to the sides of the pendulum hinged on a rotor and stator mechanism to initiate electromagnetic transduction, while other sets of same poled magnets are fitted to the primary structure in a position such that their field could interact with those on the pendulum was reported [25]. Thus, additional vibration energy of the primary structure can be transferred to the motion of the pendulum if properly tuned hence energy will be harvested by the electromagnetic harvester mounted on the pendulum’s pivot [25].

At this point it becomes necessary to state that some harvester system prototypes, including the one presented in this work, are fully dependent on friction, since friction is introduced into the system due to the gliding/relative movement of the free parts. Friction is usually an unwanted feature in a mechanical system since it is mostly the cause of energy waste manifested as wear and tear, unnecessary noise, and heat. However, in energy harvesting systems, they could be beneficial to achieve an enhanced performance in term of response or bandwidth if properly tuned. For the majority of the work reported, it was found that the Coulomb damping model was able to produce the closest match to the experimental data although the LuGre model proved more suitable in one case having a relatively high level of friction [26].

It was earlier stated that frictional force is a major but unwanted part of the harvested system whose disadvantage could be exploited to achieve a better performance and or enhanced bandwidth, several attempts that describe a correct model to characterize the friction forces were explored and presented by authors. The smooth Coulomb friction was adequately modelled in [27], [27] and was found to suppress the vibration response and effectively dissipate vibration response. Works reported in [26], [27], [28] investigated the response of mechanical system under different friction models. Here, the Coulomb friction models were observed to give a result with the closest match to the experiment. Hence, such a model can be adopted to characterize the nature of the friction in our design, while noting that the design approach adopted in this work ensured that this predatory friction type is reduced to the possible minimum in the moving parts as it has the capacity to limit/reduce the systems response.

In this paper, the focus will center on the design, modeling, verification, and experimentation with a 2DOF electromagnetic vibration energy harvester. The analysis of the harvester’s performance was investigated under three different coil connection configurations: individual, series, and parallel connections to ascertain which configuration type is the most appropriate to ensure a suitable impedance matching between the sensor and the harvester. A detailed analysis of the 2DOF system presented here opens a new potential for a performance trade-off in the power harvested, power density and operational bandwidth of vibration energy harvester.

The analysis of the harvester’s performance was investigated under three different coil connection configurations: individual, series, and parallel connections to ascertain which configuration type is the most appropriate to ensure a suitable impedance matching between the sensor and the harvester. The system design reported in this work demonstrates practical applicability such as harvesting vibration energy on automobile body and suspension during motion to power sensors used for monitoring the structural health and working condition of the vehicle, as well to power the Light Emitting Diode (LED) lightning systems since recent car designers have opted for LEDs as lightning to ensure energy optimization [6]. The proposed design, its geometrical and electromagnetic damping are yet to be optimized, noting that the optimization will further reduce the overall mass and enhance the systems performance. Optimization offers prospect for expanding the scope of usability of the harvester design to cover a wide spectrum of applications, including from powering lighting system in automobiles and train/rail track to onboard sensor nodes in automobiles, trains, and air/spacecrafts.

The paper presented here is organized as follows. Section 2 introduces the governing equation of forced coulomb-damped 2DOF system where the analytical solution for the steady state responses and the associated phase of each degree of freedom was obtained. The steady state response analysis of each mass was investigated to assert the extent and nature of the response while the dynamic nature of the coulomb friction on the response was imposed as a tool for response tuning, hence categorizing the response as either continuous, stick or stick-slip in nature. Derivation of the electromagnetic damping ratio, voltage, and power equation for different connections configurations were introduced in Section 3. In Section 4, a five-stage experimentation on the determination of the spring’s stiffness, mechanical damping, optimum load resistance, system response and the harvested voltage/power was undertaken and compared with the theoretical results introduced in Section 2. The harvested power and power densities using optimum load resistance was considered and normalized with respective mass accelerations and frequencies and compared with those reported in recent literatures in Section 5. A brief comparison of the harvested voltage/power and bandwidth of the proposed 2DOF design with an equivalent SDOF was reported in section 6. Finally, the work was concluded in Section 7.