Experimental evaluation of vibration energy harvest with non-linear oscillators

Transcription

1 Experimental evaluation of vibration energy harvest with non-linear oscillators A. Cammarano 1, S.G. Burrow 2, D.A. Barton 3 1 University of Bristol, Department of Aerospace Engineering, Unversity Walk, BS8 1TR, Bristol,UK andrea.cammarano@bristol.ac.uk 2 University of Bristol, Department of Aerospace Engineering, Unversity Walk, BS8 1TR, Bristol,UK 3 University of Bristol, Department of Engineering Mathematics, Unversity Walk, BS8 1TR, Bristol,UK Abstract In this paper a vibration energy harvester with variable reluctance is presented. The disposition of the magnets and the use of iron in the magnetic circuit creates the variable reluctance which in turns results in a device with highly non linear dynamics. In addition the profile of the reluctance can be changed by increasing or decreasing the air gap in the magnetic circuit used for the transduction. An experimental procedure to characterize this kind of device is presented and some interesting results shown. In particular the work focusses on the effects of air gap variation on the frequency response and on the mechanical to electrical transduction. Some considerations on the elastic characteristics and on the losses will be discussed and further work suggested. 1 Introduction The recent advances in low-power autonomous sensor technology has encouraged the development of energy sources that are able to harvest energy from the environment in which they are present. Mechanical energy sources, such as human body motion or machine vibrations [1], [2], are a prime target for energy harvesting since the energy density available is relatively high compared with other ambient energy sources. The majority of the vibration energy harvesters presented in literature employ a mass-spring-damper type resonating system to enhance the input excitation [3] [4] []. Although this type of resonator produces good results around the natural frequency, these systems typically have a very narrow bandwidth. Using magnets to induce non linear behaviour, has been pointed out as a possible solution to widen the frequency range in which the harvester has high energy vibrations [6]. The system presented in this paper is built around an electromagnetic transducer. It consists of a cantilever beam with a set of magnets mounted on the tip which, when placed on a vibrating object, exhibits relative displacements about a ferromagnetic core mounted on the external frame. A coil wound around the core converts the changing magnetic flux into a potential difference. The non-linearity of the system is a future of the transduction mechanism. This is the biggest difference with other similar works: in [7] and [8] the mechanism with which the non linearity is created is completely independent from the transduction mechanism. The design of the harvester is made to allow changes in the distance between the magnets and the core. The influence that this has on the strength of the non-linearity and the transduction mechanism is investigated. From previous work on a similar harvester by the authors [9] [1], it has been shown that the device can 37

2 376 PROCEEDINGS OF ISMA21 INCLUDING USD21 Air gap Excitation Magnets coil u v Slide y x flux z x Iron Figure 1: Schematic representation of the energy harvester: top view and lateral view exhibit both monostable and bistable behaviours. This work will show the way the device passes from one to the other as the air gap changes and how the response evolves in the process. In the case in which the harvester is operated in a bistable configuration, chaotic oscillations can be induced [11] and [7]. In this work, several example configurations are considered. The purpose is to understand to what extent it is advantageous to use a non-linear oscillator for energy harvesting and in what configuration it is preferable to operate in. Particular attention is paid to the influence of the gap between the magnets and iron core to the extraction and conversion of energy. For each configuration experimental results are presented. The restoring force surfaces method technique [12],[13] is adapted to this particular case. Some variations of the previous technique are suggested to study the dependency of the generated voltage on other parameters as air gap, displacement and velocity. The experimental results are analysed using non-linear system identification techniques in order to estimate the characteristic parameters of the system in each configuration. This will provide a more detailed information about the way non linearities affect the harvesters behaviour and will give some indications for future improvements of the device itself. In addition, this work has a fundamental role to improve the models suggested in [1] and [11]. This will be subject of a following work. 2 Experimental rig The harvester consists of a thin spring steel plate, fully constrained at one end and holding an mass on the other end. The suspended mass forms the armature of the transduction mechanism and is made up of two laminated-iron pieces separated by two aluminium plates (for further details see figure 1). Each laminatediron bit houses two sets of NdFeB permanent magnets.a U-shaped laminated iron-core mounted on a linear slide forms the stator of the transducer. A sliding guide is used to adjust the distance between the stator core and the magnets. The magnets are arranged to induce a magnetic flux: its path forms a magnetic circuit with the stator and the armatures. The direction of the magnetic flux depends on the set of magnets which closes the loop. As the tip of the beam moves in a full oscillation the magnetic circuit in the stator is closed and reopened four times, twice for each set of magnets. The change of flux induces a net voltage in the copper coil winded around the iron core (in this phase, no load has been applied to the harvester and solely the open circuit voltage has been considered). The experimental setup in scheme 2 has been used to acquire the experimental data. The acceleration has been measured on the constraint and at the tip of the seismic mass. A strain gauge installed on the under side of the plate,in the proximity of the constrain, has been used to measure the deflection of the beam. This automatically limits the frequency range for the experiments. In fact, the ratio between the strain and the deflection of the tip is constant only for those frequencies in which the plate deforms following the first modal shape of a cantilever beam (see figure 3). To demonstrate the previous statement in figure 3b a measure of the relative acceleration of the tip taken with the accelerometers is compared to the acceleration obtained differentiating the displacement measured

3 NON-LINEARITIES: IDENTIFICATION AND MODELLING 377 Control/Logging PC 1 Strain Gauge 2 Accelerometer Base 3 Accelerometer Mass 4 Coil Voltage Drive to the shaker Dspace Amplifier Accelerometers SGA* Strain-gauge Shaker *SGA: Strain Gauge Amplifier Figure 2: Test setup. Schematic representation: the shaker is installed on a shaker driven by Dspace (the generated signal is amplified by a voltage amplifier). The green rectangles show the position of the accelerometer. The orange line represents the strain gauge. The signal of the strain gauge must be amplified by a strain gauge amplifier before it can be redirected to the logger (Dspace). Photo with the strain gauge. While the signals are perfectly matching around the first peak, they diverge for higher frequencies. The mismatch visible at low frequency is due to the sensitivity of the accelerometers. The amplitude of base displacement used in the test allows to measure the base acceleration only if the frequency is higher than Hz. In the end, the frequency range we can use for our experiments is [ 6] Hz A sinusoidal base excitation is provided by a shaker. The amplitude of the excitation is controlled with an external loop to maintain the amplitude of the base displacements constant. The sensor used to perform the control is the accelerometer mounted on the constraint of the beam. Real time integration and control have been achieved by a DSpace systems. The input excitation is a step sine in the range 6 Hz it has been swept for each test in both directions. The frequency step is of.2 Hz and the amplitude of the base displacements has been kept at.2 mm. If different values of base excitation have been considered, it will be explicitly stated. 3 System Characterisation A valid tool to investigate the behaviour of the device from the acquired measurements is the restoring force technique [12], [13]. The procedure has been extensively explained in literature so here just a short overview will be provided. First, the oscillator is considered with the iron core removed. In this configuration no magnetic force is reflected back on the seismic mass. In the range of frequency of interest, only the first structural mode can be experienced and the cantilever beam can be thought as a single degree of freedom spring-damper-mass system. The mass of this equivalent system is loaded with three forces: the inertial force F i of the mass, the elastic force F e and the damping force F d. F i + F d + F e =. (1) In equation 1 the first force depends on the acceleration of the mass z measured on an inertial reference frame, the latter forces depend respectively on displacement u and velocity u relative to a frame placed on

4 378 PROCEEDINGS OF ISMA21 INCLUDING USD Transmissibility Trf(u.. ) frequency [Hz] frequency [Hz] Figure 3: Structural characterization (the iron core has been removed from the stator, no magnetic force is reflected on the tip of the beam). Transmissibility. Transfer function of the relative acceleration: differential measurement between the accelerometers (black dots) and evaluation from the strain gauge voltage (gray dots). the stator. The origin of this reference frame is chosen in correspondence of neutral axis of the beam so that u = when the device is not excited. In this particular case equation 1 can be rewritten as z + c m u + k u =, (2) m where m, c, and k are respectively the equivalent mass, damping factor and stiffness of the system. The advantage of writing the equation in this way is that z and u can be directly measured and u can be easily obtained by differentiating the displacement. A plot of these quantities in a 3D graph will result in the desired restoring force surface. Sectioning this surface at u = and u = is possible obtain two straight lines whose slope is exactly k/m and c/m. In presence of the magnetic core, an extra force will load the seismic mass. Since the new force is unknown, the restoring force surface now should be written in a more general form z + c m u + k u + M(u, u, t) = (3) m This new force M(u, u, t) can be decomposed in its conservative F em and its non-conservative component F md. For sake of simplicity, it has been assumed that the conservative forces depend uniquely on the relative displacement and the non conservative forces on the relative velocity. In other terms, the conservative forces are thought to be modelled by an extra spring and the non-conservative as an extra damper of which the characteristics are unknown. In this hypothesis, it is possible to sum together the elastic forces and dissipative forces. The new resultant forces will be addressed to as equivalent conservative force or total elastic force F c and equivalent non conservative forces or total damping force F nc. In the next sections a more detailed description of them will be provided.

5 NON-LINEARITIES: IDENTIFICATION AND MODELLING Displacement [mm] Velocity [m/s] Figure 4: Structural characteristics of the oscillator: elastic force and damping force. The y-scale has been maintained equal to show the difference in magnitude. 3.1 Conservative forces For the assumption done above, a section of the restoring force surface at u = will provide a measure of F c. Since F c = F e + F em, (4) and F e is known from our previous test, it is possible to evaluate F em. This force depends on how distant the magnets are from the iron core. For a fix air gap, the reluctance of the magnetic circuit formed by armaturestator, varies with the displacement of the mass. This results in non linear characteristic for this force as shown in figure 4a. It is now possible to verify if the assumption made, that the total elastic curve does not depend on the velocity is true. If this is the case, sections of the restoring force surface at different values of u should give results in curves having the same shape but translated in the y-axis direction. In figure b several curves are shown, but the translation is so small that actually no shift can be seen. This means that the order of magnitude of the dissipative forces is much smaller compared to the conservative forces. The assumption about the independence of the total elastic force on the velocity can be considered acceptable. Should such a dependence occur its effect is so small that it can be neglected. Figure a) put in evidence the effect of the air gap, that is, the position of the core in respect with the beam. When the gap which separates the core from the magnets is particularly big, the system tends to the linear case in which the iron-core has been removed (red diamond in figure a). As the air gap decreases, the forces that the magnets exert on the iron become greater and greater and the curve of the total elastic forces twists around the stable position. If the gap is small enough ( 1.7 mm), a region of zero stiffness can be observed (red dotted curve in figure a). For air gap bigger than this value, the system presents a hardening stiffness characteristic (curves in gray). For smaller gap, the central position becomes unstable and two new stable positions appear; the system is now bistable, and both hardening and softening behaviour can be experienced. Figure 6 shows the stable (black points) and unstable (grey points) positions for the system as a function of the gap. The action of gravity, possible imperfections in the clamping system and in the magnets setup, induce the bifurcation to be non symmetric. Before the bifurcation occurs the stable position diverges

6 38 PROCEEDINGS OF ISMA21 INCLUDING USD Displacement [mm] Displacement [mm] Figure : Elastic curves. dependency with the air-gap: 6 mm red diamonds, 2. mm gray dots, 1.7 mm red dots, 1. mm black dots. Dependency with the velocity (showing three superimposed curves): forces measured at 3 m/s, m/s and 3 m/s (lines are not distinguishable because of the magnitude of the dissipative forces). from it original position u = : the total elastic force is not zero in correspondence of the structural neutral position. This effect means that the conservative part of M(u, u) is not symmetric in this reference frame. The separation between the two branches of the bifurcation diagram indicates that one position is preferred in respect with the other. This can be easily proved with a static test: be the devices, for a small air gap, forced to the upper branch of the bifurcation curve and be the air gap slowly increased. For values very close to 1.7 mm, the tip abruptly jumps to the other stable position. Once this position is reached, if the gap is reduced and the distance is brought to its original values, the harvester does not reach the original stable position. 3.2 Dissipative forces The same technique used to evaluate the elastic force, can be used to evaluate the dissipative forces, with the only difference that the restoring force surface must be cut with planes u = const. Since, as previously verified, the total elastic force depends entirely on the displacement, sections of the restoring force surface at different u are expected to produce parallel lines.from figure 7a it seems that the total dissipative force characteristic can be considered linear with a good approximation. A better analysis of these forces shows that the losses do not have a constant ratio with the velocity, not even when the beam is not loaded by the magnetic forces. In this phase, it is not particularly clear what what causes the damping force to have this shape. Several test have excluded the possibility that it is generated from imperfection in the clamping system, from measurement errors or from post processing errors. The hypothesis is that some aero-elastic effects are taking place: future test will have the aim to verify this hypothesis. Independently from the way these losses are produced, figure 7b and 7d show that the magnetic forces can influence the total losses in the system. In particular they introduce a dependency with the deflection of the beam (see figure 7d)

7 NON-LINEARITIES: IDENTIFICATION AND MODELLING displacement [mm] Air gap [mm] Figure 6: Bifurcation diagram: stable positions (black dots) and unstable positions (grey dots). 4 Measured behaviour Changing the air gap has two main effects: emphasising the effect of the non linear compliance on tip of the beam altering the transduction between the mechanical and electrical domain In this section both these effects will be shown and some consideration about the 4.1 Frequency response The change in elastic characteristics shown in figure a have incredible effects on the frequency response. Here an overview of the main effect is given. The red dotted line in figure a is considered will be considered as a reference for the following discussion. In fact it identifies two main possible behaviour that here will be addressed as monostable and bistable behaviour, considering the number of stable position possessed by the system in each condition. When the system is monostable (bigger air gap), the oscillator has the typical frequency response of a non-linear oscillator with a hardening spring: increasing the frequency the oscillation suddenly passes from high energy lever to low energy level, decreasing the frequency a sudden jump up to high energy oscillations is experienced. The difference between the jump-up frequency and the jump-down frequency is highly dependent on the excitation level. Higher level of excitation are able to increase the frequency range in which high amplitude oscillations are experienced and hence the width of the hysteretic loop. As the gap decreases, the peak of the response shifts to lower frequencies and its amplitude becomes smaller. This is mainly due to the lower energy content of the excitation being the base displacement kept constant as the frequency changes. When the system is bistable (air gap smaller than 1.7 mm) a second peak appears in the frequency response. This can be seen more clearly with lower input (.1 mm) as shown in figure 8d 4.2 Induced voltage The voltage that the harvester is able to produce, deeply depends on the distance between the magnets and the iron core. For instance, the smaller is the gap, the better is the average transduce ratio θ = voltage/velocity

8 382 PROCEEDINGS OF ISMA21 INCLUDING USD Velocity [m/s] Velocity [m/s] Velocity [m/s] (c) Velocity [m/s] (d) Figure 7: Damping curves. Restoring forces surface for 6 mm and 3 mm of air gap, cut at different values (in the range 1 mm,1 mm ) of displacement. In (c) and (d) the cut at 1 mm (grey squared) and at 1 mm (black dots) of the figures above have been translated to evidence changes in the shape. In (c) the cut at mm (grey triangles) has been superimposed as well. as shown in figure 9a. The dependency between θ and the gap can be considered with a very good approximation linear. A more detailed analysis, reveals that, for a fixed gap, a dependency between θ and the displacement u exists. In particular the transduction mechanism works better in the for small deflection. This can be explained taking in account that, when the beam deflects, the tip of the beam has a total displacement which has a component in the u direction and a component in the v direction. If the component in the v direction is

9 NON-LINEARITIES: IDENTIFICATION AND MODELLING frequency [Hz] frequency [Hz] frequency [Hz] (c) frequency [Hz] (d) Figure 8: Frequency response. Response for air gap of 2. mm (black diamond), 2.2 mm (black circles), 1.81 mm (black crosses). Softening responses can be seen in for air gap values of 1.6 mm (gray diamond), 1. mm (grey triangles) and 1. mm (grey diamonds.). The response of the system run with an air gap of 1.7 mm is shown in (c). In figure (d) the 1. mm air gap response is shown for a base excitation of.1 mm considered, it can be seen that the tip, and consequently the magnets, are further away from the iron core when u is bigger.

10 384 PROCEEDINGS OF ISMA21 INCLUDING USD21 Transduction coefficient [V/(m/s)] Air gap [mm] Voltage [V] Air gap [mm] 2 Displacement [mm] Figure 9: Induced voltage. Dependency of the average transduction coefficient on the air gap. A description of the variation of this coefficient with the displacement is shown in figure Conclusions The experimental results obtained with the procedure shown in this paper, have shown the effect of the distance between magnets and iron core on a non linear energy harvester. The experiments suggest that smaller air gap could have beneficial effects of both widening the frequency response and improving the electromagnetic transduction. Nevertheless, some side effects have to be considered. Small change in the air gap greatly influence the total stiffness of the device. This means that a control of the stiffness, and hence of the frequency response, through changing the gap, could be a challenging task since high accuracy is needed. When the oscillator is highly non linear, even though high deflections can be experienced over broader frequency range, the magnitude of the peak deflection is smaller. This negatively effects the voltage generation. For a softening configuration the period doubling and the appearance of a second peak is a notable characteristic. The strong dependency of the response on the initial conditions and on the input excitation have both positive and negative sides: small changes in the initial conditions can dramatically change the generation of electrical energy even for similar base excitations. On the other hand, this could allow some sort of control. These experiments indicates that further investigations are needed for the characterization of the non conservative forces. With a better understanding of these forces, an analytical model can be attempted to find one or more optimal working conditions. This will be the main target for future work.

11 NON-LINEARITIES: IDENTIFICATION AND MODELLING 38 References [1] R. Tashiro, N. Kabei, K. Katayama, Y. Ishyzyka, F. Tsuboi, and K. Tsuchiya. Development of an electrostatic generator that harnesses the motion of a living body:(use of a resonant phenomenon). JSME international journal. Series C, Mechanical systems, machine elements and manufacturing, 43(4): , 2. [2] T. von Buren, P.D. Mitcheson, T.C. Green, E.M. Yeatman, A.S. Holmes, and G. Troster. Optimization of inertial micropower generators for human walking motion. Sensors Journal, IEEE, 6(1):28 38, Feb. 26. [3] R. Amirtharajah and A.P. Chandrakasan. Self-powered signal processing using vibration-based power generation. Solid-State Circuits, IEEE Journal of, 33():687 69, May [4] SP Beeby, MJ Tudor, and NM White. Energy harvesting vibration sources for microsystems applications. Measurement science and technology, 17(12):17, 26. [] P. Glynne-Jones, MJ Tudor, SP Beeby, and NM White. An electromagnetic, vibration-powered generator for intelligent sensor systems. Sensors & Actuators: A. Physical, 11(1-3): , 24. [6] BP Mann and ND Sims. Energy harvesting from the nonlinear oscillations of magnetic levitation. Journal of Sound and Vibration, 28. [7] S. C. Stanton, C.C. McGhee, and B.P. Mann. Nonlinear dynamics for broadband energy harvesting: investigation of bistable piezoelectric inertial generator. In press. [8] F. Cottone, H. Vocca, and L. Gammaitoni. Nonlinear energy harvesting. Phys. Rev. Lett., 12(8):861, Feb 29. [9] S.G. Burrow and L.R. Clare. A resonant generator with non-linear compliance for energy harvesting in high vibrational environments. 1:71 72, May 27. [1] DAW Barton, SG Burrow, and LR Clare. Energy harvesting from vibrations with a nonlinear oscillator. In Proc. ASME DETC, 29. [11] SG Cammarano, A Burrow and DAW Barton. An energy harvester with bistable compliance characteristics. In Proc. ASME DETC, 21. [12] SF Masri, RK Miller, AF Saud, and TK Caughey. Identification of nonlinear vibrating structures: Part i formulation. Journal of Applied Mechanics, 19(4): , [13] SF Masri, RK Miller, AF Saud, and TK Caughey. Identification of nonlinear vibrating structures: Part ii applications. Journal of Applied Mechanics, 19(4): , 1987.

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