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A self-tuning resonator for vibration energy harvesting
Sensors and Actuators A 201 (2013) 328–334
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A self-tuning resonator for vibration energy harvesting Noha A. Aboulfotoh a , Mustafa H. Arafa b,∗ , Said M. Megahed c a
Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Egypt Mechanical Engineering Department, American University in Cairo, Egypt c Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Egypt b
a r t i c l e
i n f o
Article history: Received 16 February 2013 Received in revised form 3 June 2013 Accepted 24 July 2013 Available online xxx Keywords: Vibration energy harvesting Self-tunable harvester Permanent magnets
a b s t r a c t This work is concerned with developing a vibration-based energy harvester with a tunable natural frequency. The harvester is designed to automatically adjust its own natural frequency to match that of the imposed base excitation. The proposed device consists of a cantilever beam carrying a tip mass in the form of a magnet which is placed in close proximity to another magnet with opposite polarity that can move axially thereby adjusting the beam’s natural frequency by mechanical straining. The system is designed to autonomously adjust the gap between the two magnets so as to achieve a harvester whose natural frequency matches that of the excitation. As such, the movable magnet is mounted on a motordriven tray that undergoes linear motion and adjusts its position in accordance with the frequency of the support motion. The base motion frequency is detected by an electromagnetic means, wherein another magnet, fixed to the base, moves past a stationary coil generating an electric signal. The signal is conditioned through a microprocessor to detect its frequency and is then used to determine the favorable gap between the tuning magnets to achieve resonance from a lookup table. Based on the findings of this work, the natural frequency of the harvester is successfully tuned from 4.7 Hz to 9.0 Hz generating voltage per acceleration from 6.3 V/m/s2 to 1.1 V/m/s2 , respectively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The recent years have witnessed a wealth of research on energy harvesting technologies wherein useful energy is extracted from ambient sources that are otherwise neglected. For maximum power output, vibration-based energy harvesters are usually designed to exhibit natural frequencies that match those of the excitation. As resonators normally possess high quality factors, a slight deviation from operation at resonance due to changes in the operating conditions or manufacturing errors leads to a substantial reduction in the output power generated. This has spurred interest into the design of devices that respond to a wide bandwidth to maintain an acceptable level of harvested power. In this context, the review papers by Tang et al. [1], Zhu et al. [2] and Ibrahim and Ali [3] on the techniques to increase the bandwidth of vibration-based energy harvesters are valuable. Insight into the pertinent literature reveals that two main approaches have been adopted to address this issue. The first approach relies on widening the bandwidth of an essentially passive device, i.e. one whose design parameters are fixed. This is normally accomplished by designing harvesters that exhibit
∗ Corresponding author. Tel.: +20 2 2615 3082. E-mail address: [email protected] (M.H. Arafa). 0924-4247/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2013.07.030
favorable dynamics to cope with variable-frequency excitation environments. Examples of these solutions include the use of nonlinearities [4,5], designs having multiple oscillators [6,7] and structures exhibiting multiple resonance frequencies [8–13] to harvest energy from a wide band of frequencies. The second approach relies on actively tuning the resonance frequency of a harvester, in response to the forcing environment, in order to attain resonance constantly. In this context, systems with manually [14] and automatically [15,16] adjustable natural frequencies have been proposed to improve the performance. While attempts to vary the natural frequency of an energy harvester by changing its length [17] or the center of mass of its inertial mass [18] have been reported, the most popular approaches depend on mechanical stiffening [14–16,19–21] wherein external forces are applied by actuators or magnets to strain the harvester, thereby altering its resonance frequency. A key component in self-tunable harvesters is the frequency sensing technique, which is usually employed as part of a control system to automatically adjust the natural frequency. Several approaches have been proposed in the literature, including maximization of the output voltage [15] and detection of the phase between the base acceleration and structural deflection [22]. This paper is concerned with the development of a low-cost self-tuning electromagnetic energy harvester combining both frequency detection and self-actuation. Frequency adjustment is attained through the use of permanent magnets that induce tensile forces on the resonator. Frequency sensing relies on using a
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329
Fig. 1. Schematic illustration of proposed energy harvester.
microcontroller to measure the inverse of the periodic time of the incoming vibrations by an electromagnetic sensor. This information is utilized to give instructions to drive a linear actuator to tune the resonance frequency in an open-loop control algorithm. The remainder of this paper is organized into four sections. Section 2 outlines the proposed harvester design, together with the frequency sensing mechanism and gap calculation. Section 3 presents an experimental validation of the proposed algorithms, whereas Section 4 is dedicated to conclusions and recommendations.
in order to amplify and convert it to a square wave, which in turn is fed to a microcontroller. The microcontroller is responsible for detecting the frequency of the base motion and issuing the commands to move the stepper motor so as to match the harvester’s natural frequency with that of the incoming vibrations. A description of the different subsystems and components constituting the frequency detection methodology is provided in the next section.
2. Harvester design
Under harmonic base excitation, the signal picked up by the frequency detection coil will be a low-amplitude sine wave. The method proposed to sense the excitation frequency relies on measuring the period of the measured time-domain signal. This is conveniently accomplished by converting the resulting sine wave into a square wave and designing a positive-edge-triggered microcontroller circuit to measure the time between two successive cycles, from which the frequency can be determined by calculating the inverse of that time. The code developed for this purpose contains an infinite loop which instructs the device to keep measuring the frequency indefinitely, allowing the harvester to respond to continuously variable frequencies. The microcontroller chosen in this work is of the type PIC16F877A. As the output signal from the frequency detection coil will be in the order of milli-Volts, whereas the interrupt pin of the microcontroller needs a 2.4 V to operate, an operational amplifier of the type 741 is used to perform the dual functions of amplifying the signal and converting it to a square wave, as illustrated in Fig. 2.
Fig. 1 shows a schematic illustration of the proposed energy harvester. A cantilever beam is mounted on a shaker. The beam carries a tip mass in the form of a permanent magnet that is attached to the beam tip. An opposite magnet is attached to a movable tray that is driven by a stepper motor through a rack and pinion mechanism. In this way, variable attraction forces can be applied to the beam to control its natural frequency. The beam is mounted on a rigid frame that is excited harmonically by a shaker. Two magnet-coil units (with fixed coils) are employed in the present design; one is responsible for energy harvesting (magnet attached on vibrating beam), whereas the second is utilized for detecting the frequency of base motion (magnet mounted on moving frame). As the shaker vibrates, the signal picked up by the frequency detection coil is conditioned and analyzed in real time by a micro-processor to identify the incoming frequency of base motion. This data is then used to drive the stepper motor to adjust the beam’s natural frequency to match the excitation frequency, thereby ensuring resonance for a range of excitation frequencies. The flow diagram shown in Fig. 2 illustrates how the stepper motor moves in response to the signal picked up by the frequency detection coil. For sinusoidal excitation, the signal picked up by the frequency detection coil is first fed into an operational amplifier
2.1. Frequency detection
2.2. Effect of magnetic forces on natural frequencies In order to study the effect of the magnetic forces on the cantilever beam’s natural frequencies, a finite element model of the beam is developed. The beam under investigation is modeled as an
Fig. 2. Flow diagram of the frequency detection algorithm.
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Transverse gap,
N
S
P P
N
t
P
S
N
Fy Fy
Axial gap
N
S
S
P
a
Misaligned
Aligned
Fig. 3. Axial and transverse magnetic forces between aligned and misaligned magnets.
⎡ 12EI 6P P 6P P ⎤ 6EI 12EI 6EI + + + − 3 − 3 2 2 5L 10 5L 10 ⎥ L L L ⎢ L ⎢ 6EI P 2PL 4EI 6EI PL ⎥ P 2EI ⎢ ⎥ + − 2 − − ⎥ e ⎢ L2 + 10 L 15 L 10 30 L ⎢ ⎥ K =⎢ 6P P ⎥ 12EI 6EI ⎢ − 12EI − 6P − 6EI − P ⎥ + − 2 − ⎢ L3 5L 10 5L 10 ⎥ L2 L3 L ⎣ ⎦ 2EI PL − L 30
P 6EI − 2 − 10 L
4EI 2PL + L 15
⎡ 13 ⎢ 35 ⎢ ⎢ 11L ⎢ e ⎢ 210 M = AL ⎢ ⎢ 9 ⎢ ⎢ 70 ⎣
−13L 420
11L 210
9 70
L2 105
13L 420
13L 420
13 35
−L2 140
−11L 210
−13L 420
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −11L ⎥ ⎥ 210 ⎥ ⎦ 2 −L2 140
2.5
(2)
L 105
where is the density and A is the cross-sectional area of the beam. As the beam vibrates, the centerlines of the magnets become misaligned. This causes the magnets to provide not only an axial tensile force, P, acting on the beam, but also a transverse force, Fy , acting on the beam tip trying to align the magnets, as illustrated in Fig. 3. The effect of these forces must be incorporated in the finite element model in order to obtain an accurate assessment of the natural frequency, as shown in this section. In order to measure the forces generated between a pair of magnets as a function of their axial and transverse separation, a device was specifically fabricated where one magnet is firmly mounted on an electronic scale, whereas another magnet is placed on an arm that is linearly guided on a movable frame whose displacement is measured by a vernier caliper. In this way the forces and the corresponding gap between the aligned and/or misaligned magnets can be measured. The magnets used in this work are Nickel plated NdFeB ring magnets of Grade 42 having an outer diameter of 0.75 in., an inner diameter of 0.25 in. and thickness of 0.25 in. Fig. 4 shows the variation of the axial force, P, as a function of the axial gap measured experimentally. The result shows an exponentially decaying force as the magnets are brought further apart.
2 1.5 1 0.5 0 -0.5 0
(1)
where P is the axial force (positive if tensile), E is the modulus of elasticity, L is the beam element length and I is the second moment of area. The mass matrix of the beam element is determined from classical finite element formulations as:
Experiment Fitted Curve
3
20
40
60 80 Axial Gap [mm]
100
120
Fig. 4. Experimentally measured axial force versus axial gap between the tip magnets.
The variation of the transverse magnetic force, Fy , with the transverse gap, ıt , for three different values of the axial gap, ıa , as obtained experimentally, is shown in Fig. 5. These variations are in close agreement with the findings of Ribeiro [24]. The plots indicate that as the magnets become misaligned, a transverse force builds up trying to align the magnets. This force reaches a maximum value, after which the axial separation becomes too large for the magnets to interact, resulting in a drop in the magnetic forces.
0.7
Axial Gap 25 mm Axial Gap 30 mm Axial Gap 35 mm
0.6 Transverse Force [N]
P 6EI + 10 L2
3.5
Axial Force [N]
Euler–Bernoulli beam using one-dimensional beam finite elements having two degrees of freedom per node denoting the transverse and angular displacements. To account for the effect of the axial force, the beam stiffness matrix is adjusted to include the stiffening or weakening effects of the axial load on bending stiffness. Using strain energy concepts, the adjusted element stiffness matrix is given after Chajes [23] as:
0.5 0.4 0.3 0.2 0.1 0 0
5
10
15 20 25 Transverse Gap [mm]
30
35
Fig. 5. Experimentally measured transverse force versus transverse gap between the tip magnets for three values of axial gaps.
N.A. Aboulfotoh et al. / Sensors and Actuators A 201 (2013) 328–334 Table 1 Axial gaps, axial forces and estimation of transverse forces.
331
Table 2 Properties of cantilever beam.
Axial Gap, ıa (mm)
Axial force, P (N)
ky
Cy = ky /P (m−1 )
Elastic modulus
190 GPa
25 30 35
1.3 0.7 0.4
60 37 20
46.15 52.86 50
Beam length Location of harvesting magnet (from support) Beam width Beam thickness Density Mass of tuning magnet Mass of harvesting magnet
185 mm 146 mm 25.5 mm 0.64 mm 7950 kg/m3 37.6 g 24.2 g
It is evident from Fig. 5 that for small perturbations about the equilibrium position, the transverse force can fairly be linearized to give a proportional relation with the transverse gap, i.e. Fy = ky ıt
(3)
where ky is a constant of proportionality which evidently depends on the axial gap. For an axial gap of 25 mm, the transverse force is estimated from Fig. 5 as: Fy = 60ıt
(4)
where Fy is in Newtons and ıt is in meters. When the axial gap is increased to 30 mm, the transverse force becomes: Fy = 37ıt
(5)
Finally, for an axial gap of 35 mm, we have: Fy = 20ıt
(6)
Table 1 lists a summary of these results, together with an interpolation from Fig. 4 of the corresponding value of the axial force, P, for each of the three axial gaps. Defining Cy = ky /P reveals a fairly uniform value which suggests that the transverse force may be approximated by: Fy = Cy Pıt
(7)
As an average, the value of Cy is calculated from Table 1 to be 49.67 m−1 . In this way, the transverse tip force is modeled by a spring whose stiffness is given by ky = Cy P, as shown schematically in Fig. 6, in agreement with the results of Manour et al. [14]. This effect can easily be incorporated into the overall (assembled) stiffness matrix by adding the aforementioned spring stiffness to the entry corresponding to the transverse degree of freedom at the tip. An adjustment of the overall mass matrix is also made to account for the masses of the tuning and harvesting magnets, which are treated as point masses. The natural frequencies can easily be obtained by solving the frequency equation:
[K] − ω2 [M] = 0
(8)
a lookup table which hosts a list of gaps and corresponding natural frequencies. Entries in this lookup table are obtained from the finite element model presented in the previous section. This information is then used by the microcontroller to drive a stepper motor through a sequence of steps via a voltage amplifier to adjust the axial gap in accordance with the detected frequency to enable operation at resonance. 3. Experimental work The frequency detection algorithm outlined in Fig. 2 was first verified against a pure sine wave generated by a function generator (HP type 3314A), to ensure the developed circuit is capable of correctly detecting an incoming sine wave. Results of this test revealed that the proposed approach is capable of measuring frequencies in the range 4–20 Hz with a 0.46% error. Upon this verification, the frequency detection algorithm was then implemented in the tunable energy harvesting mechanism illustrated in Fig. 1. Fig. 8 shows a photograph of the experimental setup of the proposed energy harvester. The cantilever beam, along with the tuning and harvesting magnets, are mounted on a frame, which in turn is mounted on a shaker (Brüel & Kjær type 3386-062). The input signal is generated by a function generator (HP type 3314A) and fed to a power amplifier (Brüel & Kjær type 2706). Base accelerations are measured with an accelerometer (Brüel & Kjær type 4507B) that is mounted directly onto the frame. Resistive load is provided by a decade resistance box (Lutron, type RBOX-408). Data is captured on a multi-channel dynamic signal analyzer (LMS Pimento). The beam is then excited harmonically while capturing both the output voltage and base acceleration. A movable tray is driven by a stepper motor (SLO-SYN type M061-LS02) to adjust the gap between the tuning magnets. In order to observe how the tuning magnet moves during frequency adjustment, a Linear
To verify this formulation, the natural frequencies of a beam having the properties listed in Table 2 are numerically predicted and compared with experimental measurements as shown in Fig. 7. The theoretical and experimental results are shown to compare favorably well, which verifies the accuracy of the adopted model.
Once the frequency has been experimentally determined, a corresponding gap between the magnets is calculated with the aid of
185 mm Tuning magnet Harvesng magnet 146 mm
P ky = CyP
Fig. 6. Modeling of the transverse magnetic force as a spring whose stiffness is a function of the axial load being determined by the proximity of the tuning magnets.
Numerical Experimental
11 Natural Frequency [Hz]
2.3. Gap calculation
12
10 9 8 7 6 5 4 0
20
40 60 Axial Gap [mm]
80
Fig. 7. Numerical and experimental natural frequencies.
100
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Fig. 8. Experimental setup.
Variable Differential Transformer (LVDT, type MTN/EUXL025) was attached to the movable tray and its displacement was monitored with time. To illustrate the behavior of the proposed design, consider a harvester that is initially unstrained (tuning magnet is completely retracted), thereby possessing a natural frequency of 4.7 Hz (see Fig. 7). Exciting this system at 8 Hz instructs the movable magnet to move closer to the tip magnet, thereby straining the beam and increasing its natural frequency to the desired exciting frequency. Fig. 9 shows the time history plot of the open-circuit voltage for this scenario, together with the corresponding magnet displacement. It can be observed that upon starting the system and turning on the frequency detection algorithm, the tuning magnet moves steadily until reaching its final destination. No appreciable increase in voltage is observed during its motion, as no state of resonance was reached. However, just before the tray stops at its final destination, the voltage amplitude steadily builds up and the system operates at resonance (after overcoming a few beat cycles), which indicates successful automatic tuning. Fig. 10 shows the behavior of an initially unstrained harvester that is subjected to a base excitation having a frequency of 9.0 Hz, where the tuning magnet again moves in the correct direction to increase the harvester’s natural frequency. It is also shown that if the excitation frequency is changed to 8.0 Hz during operation, a drop in the voltage output occurs because of the mistuning. This only happens for a certain amount of time, until the magnet
Fig. 9. Resonance tuning from 4.7 to 8.0 Hz. (a) Output voltage; (b) magnet displacement.
Fig. 10. Resonance tuning from 4.7 to 9.0 Hz, then back to 8.0 Hz.
is automatically re-adjusted to a new position where resonance occurs again, as illustrated by the second growth in voltage amplitude. Through the course of these experiments, beats were observed to occur when the excitation frequency is very close, but not exactly equal, to the natural frequency. This can be attributed to slight errors in the microcontroller to measure the frequency. In addition, motor positioning errors can cause beats in the response, especially at higher excitation frequencies where small errors in the gap lead to significant changes in the frequency, as observed in Fig. 7. The behavior can be observed in Figs. 9 and 10, as the magnet adjusts its position to achieve resonance. Pronounced beats is also shown in Fig. 11, where a frequency prediction error causes a harvester to possess a natural frequency of 8.2 Hz instead of the applied 8.3 Hz motion. The experimental output voltage normalized by the base acceleration is shown in Fig. 12 as a function of load resistance, for different axial loads. In general, the trends indicate steady output voltage at higher resistance. It is also evident that the use of excessive axial load to increase the tuning range results in a significant drop in the output voltage. This is attributed to the reduced beam displacement for a given base acceleration at higher axial loads, i.e. higher frequencies, as noted in [14].
Fig. 11. Beating is observed when a slight mismatch occurs between the excitation frequency (8.3 Hz) and the resonance frequency (8.2 Hz).
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4. Conclusions
-2
Voltage per Acceleration [V/ms ]
7 6
P=0 P = 1.1 N P = 2.1 N P = 2.7 N
5 4 3 2 1 0 0
1000
2000 3000 Resistance [ ]
4000
5000
Fig. 12. Output voltage per base acceleration versus load resistance.
Fig. 13 shows the variation of the output power, normalized by the base acceleration, versus load resistance. To obtain the duty cycle of the proposed system, an estimate of the power consumed by the actuator and that produced by the generator, as well as their efficiencies, can be made. From Fig. 9 it can be inferred that the actuator speed is 2 mm/s. For an average magnetic axial force of 1.3 N, and assuming an actuator efficiency of 50%, the power consumed by the actuator is estimated to be 5.2 mW. For a tuning time of 10 s, the energy required for one tuning cycle is 52 mJ. At the generator side, the power generated by the energy harvester depends on the incoming base acceleration, as well as the excitation frequency. For a base acceleration of 0.1 m/s2 , and assuming an average output power of 0.08 Ws4 /m2 , the duty cycle of the proposed device can be estimated to be 217 s, assuming a generator efficiency of 30%. This duty cycle presents the approximate time required for the harvester to remain in a dormant mode in order to collect sufficient energy to enable it to re-adjust its frequency. For the device to produce more power than it consumes, longer duty cycles must be used. It must be noted that operation off-resonance due to a frequency mismatch leads to a significant reduction in output power, which leads to a further increase in the duty cycle. Finally, it is noted that these values serve as rough guidelines that are applicable only to the present proof-of-concept design.
0.12 P=0 P = 1.1 N P = 2.7 N
0.1
4
2
Normalized Power [Ws /m ]
333
0.08
0.06
0.04
0.02
0
0
500
1000
1500
2000
Resistance [ ] Fig. 13. Normalized output power versus load resistance.
2500
This paper presented a self-tuning vibration-based electromagnetic energy harvester. The system consists of a microcontroller that measures the periodic time of the incoming base motion by analyzing the output signal from a base-mounted electromagnetic sensor. The frequency is then converted to a desirable gap between the tuning magnets through a lookup table. This information is used to drive a motor-driven tray to autonomously adjust the gap between the tuning magnets so as to achieve a state of resonance. The frequency detection methodology was first verified against simulated sinusoidal signals, and was then implemented in the energy harvester in the presence of a mismatch between the excitation and natural frequencies. The resonance frequency was successfully tuned from 4.7 Hz to 9.0 Hz by varying the gap between the two tuning magnets from 66 mm to 16.7 mm. The proposed device is effective only for single-frequency excitation schemes, and hence is not applicable for multi-component or broadband excitation. Slight positioning errors can lead to a reduction in the output voltage, as well as beats in the response, as observed in Figs. 9–11. This can be alleviated through the use of higher precision mechanical drives. Although the present device consumes more power that it produces, emphasis in this work is placed on presenting a viable methodology for frequency tuning by using a microcontroller that detects the frequency of base excitation in the time domain. A practical implementation of the proposed design should include a study of the duty cycle, to allow the device to harvest enough energy from its environment before making frequency adjustments. Efforts to downsize the components of the device, thereby reducing their power consumption, are worthwhile to achieve a self-sufficient harvester. Finally, issues of robustness and operation in the presence of noise are valuable to investigate.
Acknowledgements The authors wish to thank Dr. M. Galal Rabie for many useful discussions. The authors also acknowledge the support by Eng. Mohamed Shalaby in programming the microcontroller. This work was funded by the American University in Cairo through a research grant on vibration-based energy harvesting.
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Biographies Noha Aboulfotoh is currently an assistant lecturer in the Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Egypt. She received her B.Sc. and M.Sc., both in Mechanical Design and Production, in 2007 and 2012, respectively. Her research interests include mechatronics, mechanical vibrations, automatic control and robotics. Mustafa Arafa received his B.Sc. and M.Sc., both in Mechanical Engineering, from Cairo University in 1994 and 1997, respectively. He obtained his Ph.D. from the University of Maryland in 2002. He served as an assistant professor at Cairo University from 2002 to 2005 before joining the Mechanical Engineering Department of the American University in Cairo, where he is currently an Associate Professor. His research interests lie in the broad fields of design and dynamics, including vibration-based energy harvesting and vibrations of machinery. Said Megahed received his B.Sc. and M.Sc., both in Mechanical Engineering, from Cairo University in 1973 and 1976, respectively. He worked as a Researcher in LAAS-CNRS, France and was awarded his Doctorat d’état (D.Sc. degree) with honors degree in 1984 in Robotics Engineering. At the Faculty of Engineering, Cairo University he served as Assistant Professor (1984–1990), Associate Professor (1990–1995), Professor (1995–2010) and Emeritus Professor (2010-present). Dr. Megahed has been appointed as Industrial Engineering Department FECU/Fayoum Branch Adjoin-Head (1998–2002), CAPSCU Director (2005–2007), MDP Department/FECU Head (2007–2010), and Dean of Institute of Aviation Engineering and Technology/Ministry of Higher Education (2011/2012). He is currently serving as the Vice President for Research at the Egypt-Japan University of Science and Technology, Alexandria, Egypt. During his career, Prof. Megahed authored, co-authored, and edited five books and conference proceedings, and over a hundred research articles, supervised tens of M.Sc. and Ph.D. Theses, and had received many prizes, awards, and certificates of appreciation.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A self-tuning resonator for vibration energy harvesting Noha A. Aboulfotoh a , Mustafa H. Arafa b,∗ , Said M. Megahed c a
Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Egypt Mechanical Engineering Department, American University in Cairo, Egypt c Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Egypt b
a r t i c l e
i n f o
Article history: Received 16 February 2013 Received in revised form 3 June 2013 Accepted 24 July 2013 Available online xxx Keywords: Vibration energy harvesting Self-tunable harvester Permanent magnets
a b s t r a c t This work is concerned with developing a vibration-based energy harvester with a tunable natural frequency. The harvester is designed to automatically adjust its own natural frequency to match that of the imposed base excitation. The proposed device consists of a cantilever beam carrying a tip mass in the form of a magnet which is placed in close proximity to another magnet with opposite polarity that can move axially thereby adjusting the beam’s natural frequency by mechanical straining. The system is designed to autonomously adjust the gap between the two magnets so as to achieve a harvester whose natural frequency matches that of the excitation. As such, the movable magnet is mounted on a motordriven tray that undergoes linear motion and adjusts its position in accordance with the frequency of the support motion. The base motion frequency is detected by an electromagnetic means, wherein another magnet, fixed to the base, moves past a stationary coil generating an electric signal. The signal is conditioned through a microprocessor to detect its frequency and is then used to determine the favorable gap between the tuning magnets to achieve resonance from a lookup table. Based on the findings of this work, the natural frequency of the harvester is successfully tuned from 4.7 Hz to 9.0 Hz generating voltage per acceleration from 6.3 V/m/s2 to 1.1 V/m/s2 , respectively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The recent years have witnessed a wealth of research on energy harvesting technologies wherein useful energy is extracted from ambient sources that are otherwise neglected. For maximum power output, vibration-based energy harvesters are usually designed to exhibit natural frequencies that match those of the excitation. As resonators normally possess high quality factors, a slight deviation from operation at resonance due to changes in the operating conditions or manufacturing errors leads to a substantial reduction in the output power generated. This has spurred interest into the design of devices that respond to a wide bandwidth to maintain an acceptable level of harvested power. In this context, the review papers by Tang et al. [1], Zhu et al. [2] and Ibrahim and Ali [3] on the techniques to increase the bandwidth of vibration-based energy harvesters are valuable. Insight into the pertinent literature reveals that two main approaches have been adopted to address this issue. The first approach relies on widening the bandwidth of an essentially passive device, i.e. one whose design parameters are fixed. This is normally accomplished by designing harvesters that exhibit
∗ Corresponding author. Tel.: +20 2 2615 3082. E-mail address: [email protected] (M.H. Arafa). 0924-4247/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2013.07.030
favorable dynamics to cope with variable-frequency excitation environments. Examples of these solutions include the use of nonlinearities [4,5], designs having multiple oscillators [6,7] and structures exhibiting multiple resonance frequencies [8–13] to harvest energy from a wide band of frequencies. The second approach relies on actively tuning the resonance frequency of a harvester, in response to the forcing environment, in order to attain resonance constantly. In this context, systems with manually [14] and automatically [15,16] adjustable natural frequencies have been proposed to improve the performance. While attempts to vary the natural frequency of an energy harvester by changing its length [17] or the center of mass of its inertial mass [18] have been reported, the most popular approaches depend on mechanical stiffening [14–16,19–21] wherein external forces are applied by actuators or magnets to strain the harvester, thereby altering its resonance frequency. A key component in self-tunable harvesters is the frequency sensing technique, which is usually employed as part of a control system to automatically adjust the natural frequency. Several approaches have been proposed in the literature, including maximization of the output voltage [15] and detection of the phase between the base acceleration and structural deflection [22]. This paper is concerned with the development of a low-cost self-tuning electromagnetic energy harvester combining both frequency detection and self-actuation. Frequency adjustment is attained through the use of permanent magnets that induce tensile forces on the resonator. Frequency sensing relies on using a
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Fig. 1. Schematic illustration of proposed energy harvester.
microcontroller to measure the inverse of the periodic time of the incoming vibrations by an electromagnetic sensor. This information is utilized to give instructions to drive a linear actuator to tune the resonance frequency in an open-loop control algorithm. The remainder of this paper is organized into four sections. Section 2 outlines the proposed harvester design, together with the frequency sensing mechanism and gap calculation. Section 3 presents an experimental validation of the proposed algorithms, whereas Section 4 is dedicated to conclusions and recommendations.
in order to amplify and convert it to a square wave, which in turn is fed to a microcontroller. The microcontroller is responsible for detecting the frequency of the base motion and issuing the commands to move the stepper motor so as to match the harvester’s natural frequency with that of the incoming vibrations. A description of the different subsystems and components constituting the frequency detection methodology is provided in the next section.
2. Harvester design
Under harmonic base excitation, the signal picked up by the frequency detection coil will be a low-amplitude sine wave. The method proposed to sense the excitation frequency relies on measuring the period of the measured time-domain signal. This is conveniently accomplished by converting the resulting sine wave into a square wave and designing a positive-edge-triggered microcontroller circuit to measure the time between two successive cycles, from which the frequency can be determined by calculating the inverse of that time. The code developed for this purpose contains an infinite loop which instructs the device to keep measuring the frequency indefinitely, allowing the harvester to respond to continuously variable frequencies. The microcontroller chosen in this work is of the type PIC16F877A. As the output signal from the frequency detection coil will be in the order of milli-Volts, whereas the interrupt pin of the microcontroller needs a 2.4 V to operate, an operational amplifier of the type 741 is used to perform the dual functions of amplifying the signal and converting it to a square wave, as illustrated in Fig. 2.
Fig. 1 shows a schematic illustration of the proposed energy harvester. A cantilever beam is mounted on a shaker. The beam carries a tip mass in the form of a permanent magnet that is attached to the beam tip. An opposite magnet is attached to a movable tray that is driven by a stepper motor through a rack and pinion mechanism. In this way, variable attraction forces can be applied to the beam to control its natural frequency. The beam is mounted on a rigid frame that is excited harmonically by a shaker. Two magnet-coil units (with fixed coils) are employed in the present design; one is responsible for energy harvesting (magnet attached on vibrating beam), whereas the second is utilized for detecting the frequency of base motion (magnet mounted on moving frame). As the shaker vibrates, the signal picked up by the frequency detection coil is conditioned and analyzed in real time by a micro-processor to identify the incoming frequency of base motion. This data is then used to drive the stepper motor to adjust the beam’s natural frequency to match the excitation frequency, thereby ensuring resonance for a range of excitation frequencies. The flow diagram shown in Fig. 2 illustrates how the stepper motor moves in response to the signal picked up by the frequency detection coil. For sinusoidal excitation, the signal picked up by the frequency detection coil is first fed into an operational amplifier
2.1. Frequency detection
2.2. Effect of magnetic forces on natural frequencies In order to study the effect of the magnetic forces on the cantilever beam’s natural frequencies, a finite element model of the beam is developed. The beam under investigation is modeled as an
Fig. 2. Flow diagram of the frequency detection algorithm.
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Transverse gap,
N
S
P P
N
t
P
S
N
Fy Fy
Axial gap
N
S
S
P
a
Misaligned
Aligned
Fig. 3. Axial and transverse magnetic forces between aligned and misaligned magnets.
⎡ 12EI 6P P 6P P ⎤ 6EI 12EI 6EI + + + − 3 − 3 2 2 5L 10 5L 10 ⎥ L L L ⎢ L ⎢ 6EI P 2PL 4EI 6EI PL ⎥ P 2EI ⎢ ⎥ + − 2 − − ⎥ e ⎢ L2 + 10 L 15 L 10 30 L ⎢ ⎥ K =⎢ 6P P ⎥ 12EI 6EI ⎢ − 12EI − 6P − 6EI − P ⎥ + − 2 − ⎢ L3 5L 10 5L 10 ⎥ L2 L3 L ⎣ ⎦ 2EI PL − L 30
P 6EI − 2 − 10 L
4EI 2PL + L 15
⎡ 13 ⎢ 35 ⎢ ⎢ 11L ⎢ e ⎢ 210 M = AL ⎢ ⎢ 9 ⎢ ⎢ 70 ⎣
−13L 420
11L 210
9 70
L2 105
13L 420
13L 420
13 35
−L2 140
−11L 210
−13L 420
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −11L ⎥ ⎥ 210 ⎥ ⎦ 2 −L2 140
2.5
(2)
L 105
where is the density and A is the cross-sectional area of the beam. As the beam vibrates, the centerlines of the magnets become misaligned. This causes the magnets to provide not only an axial tensile force, P, acting on the beam, but also a transverse force, Fy , acting on the beam tip trying to align the magnets, as illustrated in Fig. 3. The effect of these forces must be incorporated in the finite element model in order to obtain an accurate assessment of the natural frequency, as shown in this section. In order to measure the forces generated between a pair of magnets as a function of their axial and transverse separation, a device was specifically fabricated where one magnet is firmly mounted on an electronic scale, whereas another magnet is placed on an arm that is linearly guided on a movable frame whose displacement is measured by a vernier caliper. In this way the forces and the corresponding gap between the aligned and/or misaligned magnets can be measured. The magnets used in this work are Nickel plated NdFeB ring magnets of Grade 42 having an outer diameter of 0.75 in., an inner diameter of 0.25 in. and thickness of 0.25 in. Fig. 4 shows the variation of the axial force, P, as a function of the axial gap measured experimentally. The result shows an exponentially decaying force as the magnets are brought further apart.
2 1.5 1 0.5 0 -0.5 0
(1)
where P is the axial force (positive if tensile), E is the modulus of elasticity, L is the beam element length and I is the second moment of area. The mass matrix of the beam element is determined from classical finite element formulations as:
Experiment Fitted Curve
3
20
40
60 80 Axial Gap [mm]
100
120
Fig. 4. Experimentally measured axial force versus axial gap between the tip magnets.
The variation of the transverse magnetic force, Fy , with the transverse gap, ıt , for three different values of the axial gap, ıa , as obtained experimentally, is shown in Fig. 5. These variations are in close agreement with the findings of Ribeiro [24]. The plots indicate that as the magnets become misaligned, a transverse force builds up trying to align the magnets. This force reaches a maximum value, after which the axial separation becomes too large for the magnets to interact, resulting in a drop in the magnetic forces.
0.7
Axial Gap 25 mm Axial Gap 30 mm Axial Gap 35 mm
0.6 Transverse Force [N]
P 6EI + 10 L2
3.5
Axial Force [N]
Euler–Bernoulli beam using one-dimensional beam finite elements having two degrees of freedom per node denoting the transverse and angular displacements. To account for the effect of the axial force, the beam stiffness matrix is adjusted to include the stiffening or weakening effects of the axial load on bending stiffness. Using strain energy concepts, the adjusted element stiffness matrix is given after Chajes [23] as:
0.5 0.4 0.3 0.2 0.1 0 0
5
10
15 20 25 Transverse Gap [mm]
30
35
Fig. 5. Experimentally measured transverse force versus transverse gap between the tip magnets for three values of axial gaps.
N.A. Aboulfotoh et al. / Sensors and Actuators A 201 (2013) 328–334 Table 1 Axial gaps, axial forces and estimation of transverse forces.
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Table 2 Properties of cantilever beam.
Axial Gap, ıa (mm)
Axial force, P (N)
ky
Cy = ky /P (m−1 )
Elastic modulus
190 GPa
25 30 35
1.3 0.7 0.4
60 37 20
46.15 52.86 50
Beam length Location of harvesting magnet (from support) Beam width Beam thickness Density Mass of tuning magnet Mass of harvesting magnet
185 mm 146 mm 25.5 mm 0.64 mm 7950 kg/m3 37.6 g 24.2 g
It is evident from Fig. 5 that for small perturbations about the equilibrium position, the transverse force can fairly be linearized to give a proportional relation with the transverse gap, i.e. Fy = ky ıt
(3)
where ky is a constant of proportionality which evidently depends on the axial gap. For an axial gap of 25 mm, the transverse force is estimated from Fig. 5 as: Fy = 60ıt
(4)
where Fy is in Newtons and ıt is in meters. When the axial gap is increased to 30 mm, the transverse force becomes: Fy = 37ıt
(5)
Finally, for an axial gap of 35 mm, we have: Fy = 20ıt
(6)
Table 1 lists a summary of these results, together with an interpolation from Fig. 4 of the corresponding value of the axial force, P, for each of the three axial gaps. Defining Cy = ky /P reveals a fairly uniform value which suggests that the transverse force may be approximated by: Fy = Cy Pıt
(7)
As an average, the value of Cy is calculated from Table 1 to be 49.67 m−1 . In this way, the transverse tip force is modeled by a spring whose stiffness is given by ky = Cy P, as shown schematically in Fig. 6, in agreement with the results of Manour et al. [14]. This effect can easily be incorporated into the overall (assembled) stiffness matrix by adding the aforementioned spring stiffness to the entry corresponding to the transverse degree of freedom at the tip. An adjustment of the overall mass matrix is also made to account for the masses of the tuning and harvesting magnets, which are treated as point masses. The natural frequencies can easily be obtained by solving the frequency equation:
[K] − ω2 [M] = 0
(8)
a lookup table which hosts a list of gaps and corresponding natural frequencies. Entries in this lookup table are obtained from the finite element model presented in the previous section. This information is then used by the microcontroller to drive a stepper motor through a sequence of steps via a voltage amplifier to adjust the axial gap in accordance with the detected frequency to enable operation at resonance. 3. Experimental work The frequency detection algorithm outlined in Fig. 2 was first verified against a pure sine wave generated by a function generator (HP type 3314A), to ensure the developed circuit is capable of correctly detecting an incoming sine wave. Results of this test revealed that the proposed approach is capable of measuring frequencies in the range 4–20 Hz with a 0.46% error. Upon this verification, the frequency detection algorithm was then implemented in the tunable energy harvesting mechanism illustrated in Fig. 1. Fig. 8 shows a photograph of the experimental setup of the proposed energy harvester. The cantilever beam, along with the tuning and harvesting magnets, are mounted on a frame, which in turn is mounted on a shaker (Brüel & Kjær type 3386-062). The input signal is generated by a function generator (HP type 3314A) and fed to a power amplifier (Brüel & Kjær type 2706). Base accelerations are measured with an accelerometer (Brüel & Kjær type 4507B) that is mounted directly onto the frame. Resistive load is provided by a decade resistance box (Lutron, type RBOX-408). Data is captured on a multi-channel dynamic signal analyzer (LMS Pimento). The beam is then excited harmonically while capturing both the output voltage and base acceleration. A movable tray is driven by a stepper motor (SLO-SYN type M061-LS02) to adjust the gap between the tuning magnets. In order to observe how the tuning magnet moves during frequency adjustment, a Linear
To verify this formulation, the natural frequencies of a beam having the properties listed in Table 2 are numerically predicted and compared with experimental measurements as shown in Fig. 7. The theoretical and experimental results are shown to compare favorably well, which verifies the accuracy of the adopted model.
Once the frequency has been experimentally determined, a corresponding gap between the magnets is calculated with the aid of
185 mm Tuning magnet Harvesng magnet 146 mm
P ky = CyP
Fig. 6. Modeling of the transverse magnetic force as a spring whose stiffness is a function of the axial load being determined by the proximity of the tuning magnets.
Numerical Experimental
11 Natural Frequency [Hz]
2.3. Gap calculation
12
10 9 8 7 6 5 4 0
20
40 60 Axial Gap [mm]
80
Fig. 7. Numerical and experimental natural frequencies.
100
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Fig. 8. Experimental setup.
Variable Differential Transformer (LVDT, type MTN/EUXL025) was attached to the movable tray and its displacement was monitored with time. To illustrate the behavior of the proposed design, consider a harvester that is initially unstrained (tuning magnet is completely retracted), thereby possessing a natural frequency of 4.7 Hz (see Fig. 7). Exciting this system at 8 Hz instructs the movable magnet to move closer to the tip magnet, thereby straining the beam and increasing its natural frequency to the desired exciting frequency. Fig. 9 shows the time history plot of the open-circuit voltage for this scenario, together with the corresponding magnet displacement. It can be observed that upon starting the system and turning on the frequency detection algorithm, the tuning magnet moves steadily until reaching its final destination. No appreciable increase in voltage is observed during its motion, as no state of resonance was reached. However, just before the tray stops at its final destination, the voltage amplitude steadily builds up and the system operates at resonance (after overcoming a few beat cycles), which indicates successful automatic tuning. Fig. 10 shows the behavior of an initially unstrained harvester that is subjected to a base excitation having a frequency of 9.0 Hz, where the tuning magnet again moves in the correct direction to increase the harvester’s natural frequency. It is also shown that if the excitation frequency is changed to 8.0 Hz during operation, a drop in the voltage output occurs because of the mistuning. This only happens for a certain amount of time, until the magnet
Fig. 9. Resonance tuning from 4.7 to 8.0 Hz. (a) Output voltage; (b) magnet displacement.
Fig. 10. Resonance tuning from 4.7 to 9.0 Hz, then back to 8.0 Hz.
is automatically re-adjusted to a new position where resonance occurs again, as illustrated by the second growth in voltage amplitude. Through the course of these experiments, beats were observed to occur when the excitation frequency is very close, but not exactly equal, to the natural frequency. This can be attributed to slight errors in the microcontroller to measure the frequency. In addition, motor positioning errors can cause beats in the response, especially at higher excitation frequencies where small errors in the gap lead to significant changes in the frequency, as observed in Fig. 7. The behavior can be observed in Figs. 9 and 10, as the magnet adjusts its position to achieve resonance. Pronounced beats is also shown in Fig. 11, where a frequency prediction error causes a harvester to possess a natural frequency of 8.2 Hz instead of the applied 8.3 Hz motion. The experimental output voltage normalized by the base acceleration is shown in Fig. 12 as a function of load resistance, for different axial loads. In general, the trends indicate steady output voltage at higher resistance. It is also evident that the use of excessive axial load to increase the tuning range results in a significant drop in the output voltage. This is attributed to the reduced beam displacement for a given base acceleration at higher axial loads, i.e. higher frequencies, as noted in [14].
Fig. 11. Beating is observed when a slight mismatch occurs between the excitation frequency (8.3 Hz) and the resonance frequency (8.2 Hz).
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4. Conclusions
-2
Voltage per Acceleration [V/ms ]
7 6
P=0 P = 1.1 N P = 2.1 N P = 2.7 N
5 4 3 2 1 0 0
1000
2000 3000 Resistance [ ]
4000
5000
Fig. 12. Output voltage per base acceleration versus load resistance.
Fig. 13 shows the variation of the output power, normalized by the base acceleration, versus load resistance. To obtain the duty cycle of the proposed system, an estimate of the power consumed by the actuator and that produced by the generator, as well as their efficiencies, can be made. From Fig. 9 it can be inferred that the actuator speed is 2 mm/s. For an average magnetic axial force of 1.3 N, and assuming an actuator efficiency of 50%, the power consumed by the actuator is estimated to be 5.2 mW. For a tuning time of 10 s, the energy required for one tuning cycle is 52 mJ. At the generator side, the power generated by the energy harvester depends on the incoming base acceleration, as well as the excitation frequency. For a base acceleration of 0.1 m/s2 , and assuming an average output power of 0.08 Ws4 /m2 , the duty cycle of the proposed device can be estimated to be 217 s, assuming a generator efficiency of 30%. This duty cycle presents the approximate time required for the harvester to remain in a dormant mode in order to collect sufficient energy to enable it to re-adjust its frequency. For the device to produce more power than it consumes, longer duty cycles must be used. It must be noted that operation off-resonance due to a frequency mismatch leads to a significant reduction in output power, which leads to a further increase in the duty cycle. Finally, it is noted that these values serve as rough guidelines that are applicable only to the present proof-of-concept design.
0.12 P=0 P = 1.1 N P = 2.7 N
0.1
4
2
Normalized Power [Ws /m ]
333
0.08
0.06
0.04
0.02
0
0
500
1000
1500
2000
Resistance [ ] Fig. 13. Normalized output power versus load resistance.
2500
This paper presented a self-tuning vibration-based electromagnetic energy harvester. The system consists of a microcontroller that measures the periodic time of the incoming base motion by analyzing the output signal from a base-mounted electromagnetic sensor. The frequency is then converted to a desirable gap between the tuning magnets through a lookup table. This information is used to drive a motor-driven tray to autonomously adjust the gap between the tuning magnets so as to achieve a state of resonance. The frequency detection methodology was first verified against simulated sinusoidal signals, and was then implemented in the energy harvester in the presence of a mismatch between the excitation and natural frequencies. The resonance frequency was successfully tuned from 4.7 Hz to 9.0 Hz by varying the gap between the two tuning magnets from 66 mm to 16.7 mm. The proposed device is effective only for single-frequency excitation schemes, and hence is not applicable for multi-component or broadband excitation. Slight positioning errors can lead to a reduction in the output voltage, as well as beats in the response, as observed in Figs. 9–11. This can be alleviated through the use of higher precision mechanical drives. Although the present device consumes more power that it produces, emphasis in this work is placed on presenting a viable methodology for frequency tuning by using a microcontroller that detects the frequency of base excitation in the time domain. A practical implementation of the proposed design should include a study of the duty cycle, to allow the device to harvest enough energy from its environment before making frequency adjustments. Efforts to downsize the components of the device, thereby reducing their power consumption, are worthwhile to achieve a self-sufficient harvester. Finally, issues of robustness and operation in the presence of noise are valuable to investigate.
Acknowledgements The authors wish to thank Dr. M. Galal Rabie for many useful discussions. The authors also acknowledge the support by Eng. Mohamed Shalaby in programming the microcontroller. This work was funded by the American University in Cairo through a research grant on vibration-based energy harvesting.
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Biographies Noha Aboulfotoh is currently an assistant lecturer in the Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Egypt. She received her B.Sc. and M.Sc., both in Mechanical Design and Production, in 2007 and 2012, respectively. Her research interests include mechatronics, mechanical vibrations, automatic control and robotics. Mustafa Arafa received his B.Sc. and M.Sc., both in Mechanical Engineering, from Cairo University in 1994 and 1997, respectively. He obtained his Ph.D. from the University of Maryland in 2002. He served as an assistant professor at Cairo University from 2002 to 2005 before joining the Mechanical Engineering Department of the American University in Cairo, where he is currently an Associate Professor. His research interests lie in the broad fields of design and dynamics, including vibration-based energy harvesting and vibrations of machinery. Said Megahed received his B.Sc. and M.Sc., both in Mechanical Engineering, from Cairo University in 1973 and 1976, respectively. He worked as a Researcher in LAAS-CNRS, France and was awarded his Doctorat d’état (D.Sc. degree) with honors degree in 1984 in Robotics Engineering. At the Faculty of Engineering, Cairo University he served as Assistant Professor (1984–1990), Associate Professor (1990–1995), Professor (1995–2010) and Emeritus Professor (2010-present). Dr. Megahed has been appointed as Industrial Engineering Department FECU/Fayoum Branch Adjoin-Head (1998–2002), CAPSCU Director (2005–2007), MDP Department/FECU Head (2007–2010), and Dean of Institute of Aviation Engineering and Technology/Ministry of Higher Education (2011/2012). He is currently serving as the Vice President for Research at the Egypt-Japan University of Science and Technology, Alexandria, Egypt. During his career, Prof. Megahed authored, co-authored, and edited five books and conference proceedings, and over a hundred research articles, supervised tens of M.Sc. and Ph.D. Theses, and had received many prizes, awards, and certificates of appreciation.