A vibration-based energy harvester

Original Article
Journal of Intelligent Material Systems and Structures 2016, Vol. 27(5) 642–665 ó The Author(s) 2015 Reprints and permissions: https://www.sodocs.net/doc/3d16101116.html,/journalsPermissions.nav DOI: 10.1177/1045389X15575088 https://www.sodocs.net/doc/3d16101116.html,
A vibration-based energy harvester suitable for low-frequency, highamplitude environments: Theoretical and experimental investigations
Reza Ramezanpour, Hassan Nahvi and Saeed Ziaei-Rad
Abstract This article presents theoretical and experimental investigations of a vibration-based energy harvester which is suitable to extract energy from low-frequency, high-amplitude environments. The proposed device consists of a rotating proof mass and eight piezoelectric bimorph beams. A magnet, mounted on the rotating pendulum, actuates the tips of the piezoelectric beams due to magnetic interaction. The free oscillations of the piezoelectric beam generate energy each time the pendulum passes over the beams. Based on energy principles, the nonlinear ordinary differential equations governing the electromechanical behavior of the system are derived and solved numerically. Using the proposed model, the effects of angular velocity of the rotating mass on the generated voltage are investigated. It is shown that the harvester with a relatively high angular velocity generates more voltage than the one with a low angular velocity. The performance of the proposed device in the attractive and the repulsive cases is compared to each other and it is concluded that the generated voltage of the attractive case is more than that of the repulsive case. The overall generated power of the harvester under harmonic external excitations is investigated for various amplitudes and frequencies. Using theoretical model, it is shown that the proposed device can be used for harvesting the out-of-plane vibrational energy. A model of the system has been devised and tested. The obtained experimental results show good qualitative agreements with the theoretical ones. Keywords Energy harvesting, rotating proof mass, high-amplitude low-frequency environments, frequency up-converting mechanism
Introduction
The field of energy harvesting has attracted explosive attention during the last decade. Energy harvesters are designed to extract energy from ambient and transform it into electrical energy which can be used by many critical electronic devices such as wireless sensors (Arms et al., 2005; Roundy et al., 2003; Roundy and Wright, 2004), health monitoring sensors (Du Plessis et al., 2005; Inman and Grisso, 2006), and pacemakers (Sanders and Lee, 1996). The generation of electrical energy from available sources such as mechanical vibrations requires a mechanical coupling between a moving body and a physical device capable of generating electricity (Lhermet et al., 2008). For the case of mechanical vibrations, the ambient energy can be transformed to the electrical one via electrostatic (Roundy et al., 2003), electromagnetic (El-hami et al., 2001), piezoelectric (PZT) (Anton and Sodano, 2007), and magnetostrictive (Lei and Yuan, 2008) mechanisms.
Regardless of the conversion mechanism, the resonant inertial energy harvesters are capable of extracting noticeable amount of energy when they are excited at their resonance frequency. However, the majority of environmental energy sources have spectral content distributed over a broader range of frequencies, and therefore, many methods have been proposed to broaden the usable bandwidth of linear harvesters. In this way, the concept of improving energy harvester performance using the intentional nonlinearities has become the main aim of many researches. Among the first who focused on this concept were Cottone et al. (2009) and
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran Corresponding author: Hassan Nahvi, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran. Email: hnahvi@cc.iut.ac.ir
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Ramezanpour et al. Erturk et al. (2009) in which the effects of deliberately inducing nonlinearities on the frequency bandwidth and the generated voltage of the energy harvesters were investigated. Beside the mentioned efforts on both increasing the generated power and broadening the frequency bandwidth, there are a number of researches that dealt with harvesting energy from low-frequency, high-amplitude excitations such as Michael et al. (2009). In their article, in order to determine the parameters influencing the device performance in terms of energy harvesting, an alternative design based on the impact of a moving mass on PZT bending structures is proposed and analyzed. An inertial power generator for scavenging lowfrequency nonperiodic vibrations called the parametric frequency-increased generator (PFIG) is presented by Galchev et al. (2011). Their device operates more effective than resonant generators, especially when the ambient vibration amplitude is higher than the internal displacement limit of the device. An energy harvesting device in which a low-frequency resonator impacts a high-frequency resonator was investigated theoretically and experimentally in the work of Gu and Livermore (2011). It is shown that the efficiency of transferred electrical power is significantly improved with the coupled vibration approach. A harvester consists of a hung mass and two stiff PZT cantilever beams has been proposed by Ye and Cai (2012). In their model, a series of impacts between the mass and cantilever beams occurr during vibration of the mass which triggers high-frequency vibrations. In the harvester presented by Pillatsch et al. (2012), a cylindrical proof mass actuates an array of PZT bimorph beams through magnetic attraction. After initial excitation, the beams are left to vibrate at their fundamental frequencies. The same authors, later, presented the experimental results for a rotational PZT harvester which works based on the beam plucking mechanism (Pillatsch et al., 2014). In their device, magnetic coupling between two permanent magnets mounted on a piezoelectric beam and an eccentric proof mass were utilized. Considering the device proposed by Pillatsch et al. (2014), it seems that the rotational devices which may accept linear and rotational excitations are more suitable for low-frequency, highamplitude excitations. An investigation on the use of gyroscopic proof masses for energy harvesting can also be found in the work of Yeatman (2008). The device in Pillatsch et al. (2012) works well on an oscillating host structure, but if the device is tilted by 90 °, no voltage can be extracted. Later, this problem has been solved in Pillatsch et al. (2014), but unlike their previous device, only one PZT beam has been used. In this work, a configuration is presented that uses eight PZT beams in a circular configuration. Due to extraction of energy from the full rotational course of
643 the proof mass, using a higher number of PZT beam can increase the overall generated power of the device. This, itself, can decrease the sensitivity of the device to its positioning relative to the direction of excitations. At this point, it is noteworthy that the multi-PZT transducers have been used in some researches for enhancing the overall generated power of the harvester especially those which focused on the wind energy harvesting (Bressers et al., 2011) and (Karami et al., 2013). However, unlike this study, in those devices, the motion of the rotating magnet(s) is caused by the direct external excitation (i.e. the wind stream) while in this study the excitation is exerted to the whole proposed device. In this article, the electromechanical behavior of a coupled energy harvester is investigated. The energy harvester comprises eight discontinuously laminated PZT beams and a rotating proof mass with a mounted magnet on its tip. Mathematical model of the harvester is derived using energy principles. The electromechanical responses of the system under harmonic external excitations are obtained applying numerical simulations. Moreover, the performance of the proposed harvester under excitations with different amplitudes and frequencies is examined. The accuracy of the modeling is qualitatively examined by the experimental results.
Theoretical description of the model
The PZT energy harvester considered in this study is shown in Figure 1. This system consists of eight elastic PZT cantilever beams with nonlinear magnetic interaction between the neodymium magnet mounted on their free ends and a magnet which is attached on the tip of a rotating cylindrical mass. The material of the rotating cylinder is not sensitive to the magnetic fields. Nine identical magnets are used to provide either attractive or repulsive force in between. The beams are equally positioned around the center of a circular plate. The horizontal distance between the tip magnets and the center point of the circular plate is equal to the length of the rotating cylinder. This guarantees that there is no contact between the tip of the moving mass and the beams’ tip during vibration. In repulsive case, while the cylindrical mass is rotating, the nonlinear magnetic interaction between the moving magnet and beam tip magnets enforces the beams to deflect downward. As rotation continues, the beams are left to vibrate at their natural frequencies. The system (shown in Figure 1) is excited by a harmonic acceleration in the form a(t) = A cos (Ot) with various amplitudes and frequencies. The two partially PZT laminates located above and below the neutral axis of the cantilever beam (known as bimorph configuration) are assumed to be perfectly bounded and embedded on either side of the beam. It is also assumed
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Figure 1. Schematic diagram of the proposed nonlinear piezoelectric energy harvester.
that the first mode of vibration is dominant. Other assumptions are as follows:  Deformations of the clamped-free Euler– Bernoulli beams which are considered to be inextensible are assumed to be small. The shear deformation and the effects of rotary inertia are neglected based on thin beam assumptions. Magnetic interaction between the two magnets is based on the dipole assumption. The force acting on the beam due to air particles (air damping) is neglected. Series connection between the two PZT laminates (PZT-5A) is considered. Magnetic interactions between PZT beams are assumed to be small. All beams are identical and the only difference is in their locations on the circular plate. The friction between the rotating mass and the central bar which is fixed at the center of the circular plate is assumed to be negligible. It is assumed that the nonlinear interaction between the rotating magnet and the beam tip magnets has negligible effect on the dynamics of the rotating mass. In practice, this can be achieved using a rotating cylinder with a noticeable amount of mass.
to the electric field generated within a layer as (according to Institute of Electrical and Electronics Engineers (IEEE) standard for piezoelectricity)
Tx = cE xx Sx à ezx Ez Dz = ezx Sx + eS zz Ez e 1T

     
E where eS zz , ezx , and cxx are the PZT material permittivity constant at zero strain, the electromechanical coupling coefficient, and the elastic stiffness, respectively.
Energy description of the system
The kinetic energy of a PZT cantilever system shown in Figure 2 can be written as
L L e ep 1 2 _ (x, t)? dx + rp Ap ?w _ (x, t)?2 dx T = r s A s ?w 2 0 0
e 2T
+
1 _ (L, t)?2 m A ?w 2

PZT modeling
PZT energy harvesting employs active materials that generate charges when stressed mechanically. The relationship between strain Sx , stress Tx , electric field Ez , and displacement Dz of a bimorph cantilever beam is governed by equations that relate the mechanical stress
where Lp , L, and w(x, t) are the length of PZT layers, the length of substrate and transversal deflection of the PZT beam, respectively, and eá T indicates ?=?t. Moreover, A, r, and mA are the cross-sectional area, the material density (s: substrate and p: PZT), and mass of the magnet A, respectively. The total kinetic energy of the system, Tt , is the summation of kinetic energies of all PZT cantilever beams. In equation (2), the first and second terms show the kinetic energy within each layer of the PZT and substrate and the third term shows the kinetic energy of the end mass (magnet A). Using Hook’s law, the potential energy expression for a PZT beam can be written as
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Figure 2. Parameters used for modeling nonlinear interaction due to permanent magnets.
L e 1 2 Us = Es Is ?w00 (x, t)? dx 2 0
e 3T
MA = MA VA sin ni cos uo i + MA VA sinni sin uo j + MA VA cosni k MB = à M B V B k
e 6T e 7T
where Es and Is are the Young’s modulus and the area moment of inertia for the substrate, respectively, and double prime denotes ?2 =?x2 . The total potential energy of the system, Ust , is the summation of potential energies for all PZT cantilever beams. Considering one PZT beam, the bending enthalpy for electrically active layers is (Stanton et al., 2010)
1 2 _ (t) Hp = Ep Ip ?w00 (x, t)? dx à ezx bp (hp + hs )w0 (Lp , t)l 2
0 L ep
where the angles ni and uo are shown in Figure 2 and the index i indicates the ith beam. At any moment, the the vector from the center of magnet B to the center of magnet A is given by
r = ?eLR + a(2 à cos ni )Tcos uo à (LR à b)cos g?i + ?eLR + a(2 à cos ni )Tsin uo à (LR à b)sin g?j + ?wi (L, t) + a sin ni à ZR ?k e 8T
à
1 eS zz bp Lp _ l(t)2 4 hp
e 4T
where Ep is the Young’s modulus of the PZT material _ (t) are the laminate thickness, width, and hp , bp , and l and flux linkage (which its time derivative has units of Volts), respectively. In equation (4), the PZT area moment of inertia is given by 2 + 6 h h + 3 h ) = 12 . Ip = bp hp (4h2 p s p s The total enthalpy, Hpt , is the summation of enthalpies of all PZT beams. The potential energy in the magnetic field generated by two dipoles (magnets A and B) is given by (Yung et al., 1998)
Um =   mo MA á r r 3 á MB r 4p   m M A á MB (MA á r)(MB á r) = o à 3 r5 4p r3
e 5T
where LR and ZR are the length of the rotating cylinder and the height of the rotating magnet, respectively, relative to an imaginary plane passing through the PZT beams at their initial positions. In equation (8), uo is the angular position at which the PZT beams are fixed. In this study, uo = p(i à 1)=4, i = 1, 2, 3, . . . , 8 for the ith cantilever beam. Considering a PZT beam, the nonlinear magnetic potential between the tip magnet and the moving magnet can be obtained by substituting equations (6)–(8) into equation (5). To calculate the total nonlinear magnetic potential between the tip magnets and the magnet on the rotating cylinder, Umt , the mentioned procedure should P be repeated for all beams. In other words, Umt = 8 i = 1 Umi for the ith cantilever beam. Using this potential function, the presented mathematical model is capable to consider the interactions between the rotating magnet and all tip magnets, simultaneously. The Lagrangian functional, L, the difference between the kinetic and the potential energy, is
L = Tt à Ust à Hpt à Umt e 9T
where mo is the vacuum permeability, r is the vector from the center of magnet A to the center of magnet B and r is its two-norm. MA and MB are the magnetic moment vectors oriented as shown in Figure 2 and, according to Stanton et al. (2010), may be considered as a multiplication of the magnet volume V and the magnetization vector M. For permanent magnets, the magnitude of the magnetization vector, M, can be estimated using magnets residual flux density, Br , as M = Br =mo . Hence, according to Figure 2, the magnetic moment vector for the two magnets can be written as
In the next subsection, this functional will be used to obtain electromechanical equations of the system.
Discretized model of the energy harvester
After plucking by the moving magnet, the cantilever beams of the proposed device vibrate at their resonance frequency. In this way, a single eigenfunction expansion of the deflection for the mode of interest is assumed.
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646 As such, we presume wi (x, t) = fi (x)ri (t) where r(t) is the time-varying function and f(x) is the mode shape of oscillation. Moreover, ni in equations (6) and (7) can be written as ni = f0i (L)ri (t). As mentioned, before, the length of the PZT laminate is not equal to the length of the beam; therefore, computing f(x) requires an extra step. A framework for computing non-uniform modes for continuous systems with abrupt material property or geometric changes is provided by Koplow et al. (2006). Their results are used later for modeling of a nonlinear bistable energy harvester by Stanton et al. (2010). The total beam was considered as two separate Euler– Bernoulli beams satisfying continuity and compatibility conditions at their intersection. By incorporating wi (x, t) = fi (x)ri (t) into equation (9), the Lagrangian functional can be discretized as
L=
8  à X 1 i=1
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Figure 3. Top view of the cylindrical mass.
2
á 2 _ _ i (t)2 à v2 r i ri (t) + ui ri (t)li (t)  e10T
1 _ i (t)2 à Um eri (t), gT + Cpi l 4
where vi is the first natural frequency of the ith beam and Cp and u are the equivalent capacitance through one layer and the electromechanical coupling term, respectively; given for one PZT cantilever as
Cp = u= eS zz bp Lp hp
rotating cylindrical mass. In the following, the rotating mass differential equation of motion which is coupled with the rest 16 equations is derived. The coupling is related to the parameter g that denotes the angular position of the rotating mass. In Figure 3, top view of the cylindrical mass is shown. In this figure, G and LG represent the gravitational center point of the cylindrical mass and the distance between this point and the central axis (point o) of the central bar, respectively. Denoting the acceleration vector of point o as ao , the acceleration vector of point G, aG , can be written as
_ 2 er + LG g € eu aG = ao + aG=o = a(t)ey à LG g e13T
1 ezx (hp + hs )bp f0 (Lp ) 2
The governing ordinary differential equations can be found by applying the Euler–Lagrange equations given as
  d ?L ?L = Qi (t), à _i dt ?r ? ri d ?L _i dt ?l   à ?L = Ii (t) e11T ? li
where aG=o is the acceleration of point G relative to point o. Therefore, the acceleration vector of point G can be obtained as
à á _ 2 er + eLG g € à a(t)cos gTeu e14T aG = àa(t) sin g à LG g
In equation (11), Q(t) is a generalized force and I (t) is a generalized current which, in electrical dynamics, is represented by an equivalent resistive load, that is, _ (t)=RL (Stanton et al., 2010). Substituting I (t ) = à l equation (10) into equation (11) leads to
_ € _ i (t ) + v 2 r i ( t ) + 2z i v i r i ri (t) à ui li (t) + ?U m = 0 e12aT ?ri e12bT
Using Newton’s second law in the direction perpendicular to the cylinder, the equation of motion for the cylindrical mass can be found as
€ à a(t)cos g = à g sin g LG g e15T
1 € 1 _ _ i (t ) = 0 Cpi li (t) + li (t) + ui r 2 RL
where z is the modal damping ratio and RL is the resistive load that is assumed to be 150 kO. The last term of equation (12a) indicates the repelling force between two magnets. In general, the theoretical model of the proposed system consists of 17 coupled ordinary differential equations for eight PZT cantilever beams and the
where g is the acceleration of gravity. Obviously, if the term a(t) that represents the horizontal acceleration of the device is set to 0, equation (15) reduces to a nonlinear equation describing the oscillations of a simple pendulum. Table 1 lists the parameters used for calculating the potential energy in this section and the forced response of the proposed energy harvester which is investigated in the next section. The symbolic manipulation software Maple was used to derive the potential field. The simulations in the next section are carried out using MATLAB Simulink environment. Using the parameters listed in Table 1, the magnetic potential, U, caused by two identical permanent
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Table 1. System parameters. PZT cantilever beam properties (all PZT beams are the same) Substrate Parameter Length Width Thickness Density Young’s modulus Coupling coefficient Laminate permittivity Permittivity of free space Damping ratio Magnets’ properties Parameter Mass Length Width Height Residual flux density
PZT: piezoelectric.
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PZT laminate Value 55:01 mm 6:2 mm 0:23 mm 1290 kg=m3 2:41 GPa — — — 0:01 Symbol Lp bp hp rp Ep d31 eS 33 e0 — Value 26:10 mm 3:8 mm 0:254 mm 7700 kg=m3 63 GPa à285310à12 C=N 3200e0 8:854310à12 F=m —
Symbol L b hb rs Es — — — z
Symbol mA i and mB 2a and 2b dA i and dB hA i and hB Br
Value 5 :3 g 9:52 mm 6:01 mm 2:2 mm 1:48 T
magnets are calculated for different angular positions (from àp to p) of the cylindrical mass and different beam tip displacements. The results are shown in Figure 4(a) to (d) for beam numbers 1–4. For beam numbers 5–8, the trend of the magnetic potential, U, is similar to the trend of Figure 4(a) to (d), and therefore they are not shown here. Figure 4 clearly shows that whenever the angular position of the cylindrical mass, g, is equal to the angular position of a PZT beam, the (repulsive) magnetic potential rapidly increases and reaches its maximum value. The maximum of magnetic potential for the first beam occurs at g = 0 (Figure 4(a)) and for the second, third, and fourth beams occurs at g = p=4, g = p=2, and g = 3p=4, respectively (Figure 4(b) to (d)). Figure 4 shows that any increase in the distance between the magnets located at the tips of the moving mass and a particular PZT beam decreases the potential in between. In other words, from Figure 4(a) to (d), it is observed that if the position of the magnet located at the tip of a PZT beam, wi (L), is positive (negative), the magnetic interaction increases (decreases).
time step option and ode45 solver (Runge–Kutta method) are used in the simulations. Without losing the generality of the problem, all simulations are done for the following three cases: Case I. In this case, as shown in Figure 5, the energy harvester consists of one PZT beam. The effect of angular velocity of the rotating cylinder on the generated voltage is investigated and some remarks about the damping ratio of the PZT beam are expressed. Case II. The electromechanical responses of the proposed energy harvester with eight PZT beams are simulated under a predetermined rotational motion of the rotating cylinder. The overall effects of the rotating cylinder angular velocity on the generated voltage are investigated in this case. Moreover, the performance of the harvester for two repulsive and attractive cases is compared to each other. Case III. In this case, the electromechanical responses of the energy harvester under harmonic external excitations, exerted in y direction, are simulated. The effect of gravity on the performance of the harvester is investigated. Finally, the performance of the device in low-frequency, highamplitude environments is examined.
Time-domain numerical simulations of the electromechanical response
Due to strong nonlinearity of equation (12) caused by magnetic interactions between the two magnets, numerical simulation is used to obtain the system timedomain responses. In this way, the electromechanical equations are built in MATLAB Simulink environment considering the parameters listed in Table 1. Variable
Case I: the performance of the energy harvester with a PZT beam under a predetermined motion
The simulations in this section are performed for the repulsive configuration and time period of 6 s. A Ramp
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Figure 4. Static potential energy between two magnets attached to the tip of PZT beam and cylindrical mass, LR = 60 mm, and ZR = 35 mm: (a) first beam, (b) second beam, (c) third beam, and (d) fourth beam.
Figure 5. Schematic of the harvester with one PZT beam.
block is used in Simulink environment to simulate a predetermined rotational motion with a constant angular velocity denoted by g t which implies that the angular position of the rotating cylinder can be written as g (t) = g o + gt t (g o is the initial position). It is assumed that at the initial time, the cylindrical mass is located at g 0 = à p=2 (Figure 5). When the rotating proof mass reaches the position g = 0, due to interaction between the magnets, the PZT beam is actuated. Figure 6 shows the tip displacement and the generated voltage of an individual beam for three angular velocities of the cylindrical mass; g t = 1, 3, and 5 rad/s. According to this figure, if the angular velocity increases, the number of actuations in the PZT cantilever beam increases. This results in an increase in the maximum tip displacement and the maximum generated
voltage. It can be explained that when the rotating mass reaches the PZT beam with a low angular velocity, it enforces the beam to deflect slowly, whereas, at a high angular velocity, the magnetic force exerts rapidly and, consequently, increases the initial velocity of the PZT tip in its downward motion. On the other hand, at a high angular velocity, the rotating mass leaves the beam faster, results in a lower repulsive force, and therefore, the PZT beam oscillates with a higher amplitude. This can be seen in Figure 6(a) and (b). Figure 7 shows the generated voltage across the resistive load for two damping ratios 0.01 and 0.05. It can be seen that for smaller values of damping ratio, the oscillations of an individual beam do not completely die off before the next actuation begins, while higher values of damping ratio shorten the duration of after-actuation oscillations as well as maximum generated voltage. At this point, it should be noted that the general scheme of Figure 7, which is drawn using theoretical simulations, is qualitatively in good agreement with the experimental results in Pillatsch et al. (2012, 2014) and Gu and Livermore (2011).
Case II: the performance of the proposed energy harvester under a predetermined motion
The same as previous section, a Ramp block is used in Simulink environment to simulate the predetermined motions in cylindrical mass but, here, the analyses are performed for the attractive and repulsive configurations. The electromechanical behavior of the harvester with eight PZT beams is governed by 16 coupled
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(a) 3 γ =1 rad/sec 2 1
t t t
γ =3 rad/sec γ =5 rad/sec
W (L, t) (mm)
0 -1 -2 -3 -4 -5 -6 0
1
1
2
3
4
5
6
Time (sec)
(b) 2 1.5 1 0.5 γ =1 rad/sec
t t
γ =3 rad/sec =5 rad/sec
Figure 7. Effects of damping ratio on the generated voltage across the resistive load; ZR = 0:02 m and g t = 7 rad=s.
V (volt)
0 -0.5 -1 -1.5 -2 -2.5 -3 0 1 2 3 4 5 6
Time (sec)
Figure 6. Effects of different angular velocities on the tip displacement (a) and the generated voltage (b) of an individual beam, ZR = 0:02 m and z = 0:01; continuous line for gt = 1 rad=s, dash-dotted line for gt = 3 rad=s, and dashed line for gt = 5 rad=s.
nonlinear ordinary differential equations. The solution of these equations is obtained numerically. The results for the repulsive configuration are depicted in Figure 8(a) and (b), which show the tip displacements and the generated voltages, respectively, for the angular velocities 1 and 5 rad/s. The initial value of the angular position is g 0 = à p=2 rad, implying that at the initial time the rotating cylindrical mass is located over the beam number 7 which leads to an initial tip displacement for this beam (Figure 8(a7) and (b7)). After passing over the beam number 7, as is evident in Figure 8(a8) and (b8), the rotating mass reaches the position of beam number 8 and actuates it. In fact, in this case, the first plucking occurs in beam number 8. The plucking mechanism is repeated for beam numbers 1, 2, 3, and so on. For higher angular velocities, more plucking mechanisms are observed which lead to higher overall generated voltages. The sequence of plucks in Figure 8, especially those drawn for gt = 5 rad=s, can be seen in Pillatsch et al. (2012) and Gu and Livermore (2011). A comparative study on the tip displacements and the generated voltages of the harvester in the repulsive and attractive cases is conducted and the results are shown in Figure 9. The repulsive and attractive magnetic interactions can be introduced into the model by
changing the direction of magnetic moment vector of the moving magnet. From Figure 9, it can be observed that the tip displacements and the generated voltages in the case of attractive configuration are significantly more than those in the repulsive configuration. This can be justified by considering the magnetic force between the tip magnets and the moving magnet. In the attractive case, when a PZT tip magnet is pulled up to the moving magnet, the distance between the magnets decreases which leads to an increase in the attractive force and, subsequently, to a higher tip displacement. On the other hand, in the repulsive case, the distance between the tip magnets increases which leads to a lower force. Hence, as shown in Figure 9(b), higher harvested voltages are expected for the attractive configurations rather than the repulsive configurations.
1
Case III: the performance of the proposed energy harvester under a harmonic external excitation
A Sine block is used in MATLAB Simulink environment to produce harmonic external excitations. The responses of 17 coupled nonlinear differential equations are obtained for the attractive configuration and time period of 100 s while the initial angular position, g 0 , is set to be 0 (at t = 0, the cylindrical mass is supposed to be located at the top of the first PZT beam). Furthermore, it is assumed that LR = 60 mm and ZR = 30 mm. The horizontal and vertical coordinates attached to the device are depicted in Figure 10. For ease of discussion, it is assumed that the external excitation is exerted along the horizontal coordinate. When the device is horizontally positioned (Figure 10(b)), the effect of gravity on the dynamics of the cylindrical mass is assumed to be negligible. Although, always, the vibrations of the PZT beams will be under the influence of gravity, it is assumed that the effect of gravity on the vibrations of PZT beams is also negligible. This
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W 4(L, t) (mm) W 3(L, t) (mm) W 2(L, t) (mm) W 1(L, t) (mm)
5 0 -5 5 -1 -10 10 0 1 2 3 (a2) 5 0 -5 5 -10 -1 10 0 1 2 3 (a3) 2 0 -2 2 -4 4 6 -6 0 2 0 -2 2 -4 4 -6 6 0 1 2 3 4 5 6 1 2 3 (a4) 4 5 6 4 5 6 4 5 6
W 6(L, t) (mm) W 5(L, t) (mm)
(a1)
(a5) 5 0 -5 5 -1 -10 10 0 1 2 3 (a6) 5 0 -5 5 -10 -1 10 0 1 2 3 (a7) 10 0 -10 -10 4 5 6 4 5 6
W 8(L, t) (mm) W 7(L, t) (mm)
0
1
2
3 (a8)
4
5
6
2 0 -2 2 -4 4 -6 6 0 1 2 3 4 5 6
Time (sec)
(b1) 2 2
Time (sec)
(b5)
V 1 (volt)
0 -2 2 -4 0 2 1 2 3 (b2) 4 5 6
V 5 (volt)
0 -2 2 -4 0 2 1 2 3 (b6) 4 5 6
V 2 (volt)
0 -2 2 -4 0 2 1 2 3 (b3) 4 5 6
V 6 (volt)
0 -2 2 -4 0 10 1 2 3 (b7) 4 5 6
V 3 (volt)
0 -2 2 -4 0 2 1 2 3 (b4) 4 5 6
V 7 (volt)
0 0 -10
0
1
2
3 (b8)
4
5
6
2
V 4 (volt)
0 -2 2 -4 0 1 2 3 4 5 6
V 8 (volt)
0 -2 2 -4 0 1 2 3 4 5 6
Time (sec)
Time (sec)
Figure 8. Effects of different angular velocities on (a) tip displacements and (b) generated voltages of the proposed harvester in repulsive configuration, ZR = 0:02 m; continuous line for gt = 1 rad=s and dashed line for g t = 5 rad=s.
assumption is acceptable, especially when the mass of the beams (including tip magnets) is small enough and/ or the magnetic force between the moving magnet and the tip magnets is much more than the gravitational force. As an example, the electromechanical behavior of the device under the influence of a harmonic external excitation with amplitude of 10 m/s2 and a frequency of 5 Hz is presented in Figure 11. It is assumed that the device is vertically positioned, that is, the dynamics of the rotating mass is affected by the gravitational force. Figure 11(a) shows the angular position of the rotating cylinder. The horizontal and vertical coordinates of
its tip are shown Figure 11(b). At the beginning of excitation which is in the positive horizontal direction ( + y), the rotating mass goes to the negative angular positions or to the negative horizontal coordinates (Figure 11(a)). When the direction of the external excitation reverses, the rotating magnet moves toward the positive directions. During the excitation, this oscillation is repeated several times and the path curve of the cylinder tip is obtained as shown in Figure 11(b). Hence, the rotating mass oscillates around its initial position while its tip magnet traces the curve as shown in Figure 11(b). Higher external excitations may lead to a complete circular motion of the rotating mass. In
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W (L, t) (mm) W (L, t) (mm) W (L, t) (mm) W 1(L, t) (mm) 4 3 2
20 10 0 -10 -10 0 1 2 3 (a2) 20 10 0 -1 0 -10 0 1 2 3 (a3) 20 10 0 -1 -10 0 0 1 2 3 (a4) 20 10 0 -10 -10 0 1 2 3 4 5 6 4 5 6 4 5 6 4 5 6
W (L, t) (mm) W (L, t) (mm) 6 5
(a1)
(a5) 20 10 0 -1 -10 0 0 1 2 3 (a6) 20 10 0 -1 0 -10 0 1 2 3 (a7) 20 10 0 -10 -10 0 1 2 3 (a8) 20 10 0 -10 -10 0 1 2 3 4 5 6 4 5 6 4 5 6 4 5 6
Time (sec)
(b1) 10
W 8(L, t) (mm) W (L, t) (mm) 7
Time (sec)
(b5) 10
V 1 (volt)
0 -10 0
V 5 (volt)
1 2 3 (b2) 4 5 6
0 -10 0
0
0
1
2
3 (b6)
4
5
6
10
10
V 2 (volt)
0 -10 0
V 6 (volt)
1 2 3 (b3) 4 5 6
0 -10 0
0
0
1
2
3 (b7)
4
5
6
10 0 -10 0
20
V 7 (volt)
1 2 3 (b4) 4 5 6
V 3 (volt)
0 -20 0
0
0
1
2
3 (b8)
4
5
6
10
10
V 4 (volt)
0 -10 0
V (volt)
1 2 3 4 5 6
0 -10 0
0
8
0
1
2
3
4
5
6
Time (sec)
Time (sec)
Figure 9. (a) Tip displacements and (b) generated voltages of the proposed harvester in repulsive (dashed line) and attractive (continues line) configurations, ZR = 0:02 m.
other words, for the amplitude of 10 m/s2 and frequency of 5 Hz, the rotating cylinder is unable to reach beam numbers 3–7 as is evident in Figure 11(b). For this excitation, the rotating mass oscillates near the tip of the beam numbers 8, 1, and 2, and therefore, it is expected that the tip displacements and the generated voltages of these beams are considerably higher than those of the other beams (Figure 12). According to this figure, while beam numbers 8, 1, and 2 oscillate with maximum amplitude of approximately 8 mm, the amplitudes of oscillations for other beams are less than
0.2 mm (Figure 12 (a1), (a2) and (a8)). For a time interval of 10 s, the beam number 5 (Figure 12 (a5)) has the minimum amplitude which is due to the relatively high distance between its tip magnet and the rotating magnet. Plucking mechanism can be observed as consecutive jumps in Figure 12(a) and (b). In order to investigate the performance of the device in the horizontal and vertical situations, the electromechanical equations of the system are solved for different values of the amplitude and frequency of external excitations. The considered values for amplitudes are 0.1g,
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Figure 10. Vertical and horizontal coordinates in two situations; device in the (a) vertical and (b) horizontal situations.
Figure 11. (a) Angular position and (b) tip path curve of the rotating mass when the device is vertically positioned; LR = 60 mm and ZR = 30 mm.
0.25g, 0.5g, 1g, and 2 g where g = 9.81 m/s2. The values of frequencies are 1, 2,..., 5, 7, 9,..., 15 Hz. All solutions are obtained for 100 s. To eliminate the effects of transient responses on the results, only second half of the responses is included, that is, between t = 50 s and t = 100 s. At first, the root mean square (RMS) of the generated voltage of each beam is calcu2 lated. Then, considering Prms = Vrms =RL , the RMS of the generated power of each beam is calculated. Finally, the overall generated power of the device which
is the sum of the generated power of all beams is determined. Hence, the performance of the harvester under a particular external excitation can be expressed by a single quantity. The overall generated power of a horizontally positioned harvester for different amplitudes, A, and frequencies of external excitations is shown in Figure 13. As can be seen, for a certain value of the amplitude of excitation, generally, it is expected that increasing the excitation frequency to values more than 5 Hz decreases
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Figure 12. (a) Tip displacements and (b) generated voltages across the resistive load when the device is vertically positioned; LR = 60 mm and ZR = 30 mm.
the overall generated power. In other words, when the device is horizontally positioned, more power can be harvested in low frequencies rather than in high frequencies. Figure 14 shows the overall generated power of a horizontally positioned harvester for different frequencies and amplitudes of external excitations. According to this figure, in general, the generated power of the device at frequencies lower than 5 Hz is considerably higher than those at higher frequencies. For a certain excitation frequency, it is also expected that increasing
the excitation amplitude increases the overall generated power. The overall generated power of a vertically positioned harvester for different amplitudes (A) and frequencies of external excitations is shown in Figure 15. Unlike Figure 13, in this position, the harvester can produce noticeable amount of power at high frequencies such as 13 Hz (Figure 15(d)). However, the device is still capable of harvesting energy at low frequencies. Figure 16 shows the overall generated power of a vertically positioned harvester for different frequencies
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Journal of Intelligent Material Systems and Structures 27(5)
(a) 0.07 0.06 0.05
A = 0.1g 0.15
(b)
A = 0.25g 0.4
(c)
A = 0.5g 0.35 0.3
(d)
A = 1g 0.4 0.35 .35 0.3 0.25 .25 0.2 0.15 .15 0.1
(e)
A = 2g
0.3 0.1 0.2
0.25 .25 0.2 0.15 .15
Power (mW)
0.04 0.0 04 0.03 0.0 03 0.02 0 0 0.01 0 0 5 10 Frequency (Hz) 15
0.05 0.1
0.1 0.05 .05
0 0
5 10 Frequency (Hz)
15
0 0
5 10 Frequency (Hz)
15
0 0
5 10 Frequency (Hz)
15
0.05 .05
0
5 10 Frequency (Hz)
15
Figure 13. Overall generated power of the horizontally positioned harvester for different amplitudes of external excitations, A, with the range of frequencies from 1 to 15 Hz; LR = 60 mm and ZR = 30 mm: (a) A = 0.1 g, (b) A = 0.25 g, (c) A = 0.5 g, (d) A = 1 g, and (e) A = 2 g.
Figure 14. Overall generated power of the horizontally positioned harvester for different frequencies of external excitations over a range of amplitudes from 0.1 to 2 g; LR = 60 mm and ZR = 30 mm: (a) frequency = 1 Hz, (b) frequency = 2 Hz, (c) frequency = 3 Hz, (d) frequency = 4 Hz, (e) frequency = 5 Hz, (f) frequency = 7 Hz, (g) frequency = 9 Hz, (h) frequency = 11 Hz, (i) frequency = 13 Hz, and (j) frequency = 15 Hz.
and amplitudes of external excitations. The same as Figure 14, when the frequency of excitation is constant, generally, it is expected that increasing the excitation amplitude increases the overall generated power. Considering Figures 13 to 16, it can be concluded that the proposed configuration can be used for harvesting vibration energy at low-frequency and highamplitude excitations and its capability is not restricted to the mentioned types of excitations, especially in vertically positioned situations. It seems that the generated
power is highly influenced by the amplitude of excitation rather than its frequency. As mentioned in section ‘‘Introduction,’’ in comparison to the device presented by Pillatsch et al. (2014), seven more PZT beams have been used in the proposed harvester to extract vibrational energy even when the rotating proof mass is rotating at angles far from the angular position of the first PZT beam. To demonstrate the effect of using a higher number of generators on the overall generate power, in Figures 17 and 18 the
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Figure 15. Overall generated power of the vertically positioned harvester for different amplitudes of external excitations, A, with the range of frequencies from 1 to 15 Hz; LR = 60 mm and ZR = 30 mm: (a) A = 0.1 g, (b) A = 0.25 g, (c) A = 0.5 g, (d) A = 1 g, and (e) A = 2 g.
Figure 16. Overall generated power of the vertically positioned harvester for different frequencies of external excitations over a range of amplitudes from 0.1 to 2 g; LR = 60 mm and ZR = 30 mm: (a) frequency = 1 Hz, (b) frequency = 2 Hz, (c) frequency = 3 Hz, (d) frequency = 4 Hz, (e) frequency = 5 Hz, (f) frequency = 7 Hz, (g) frequency = 9 Hz, (h) frequency = 11 Hz, (i) frequency = 13 Hz, and (j) frequency = 15 Hz.
overall generated power of the device when it has only one PZT beam is compared to that with eight PZT beams. The excitation characteristics are the same as those in Figures 14 and 16. The initial conditions are _ = 0 which implies that whether the still g = 0 and g device consists of one or eight PZT beams, the rotating proof mass is initially located at the top of the first PZT beam. It should be noted that when the device is considered with only one PZT beam, the obtained configuration is still different from the model proposed by Pillatsch et
al. (2014). In their model, the vibrations of PZT beam and the motion of rotating proof mass occur at the same plane, while in our model, the PZT beam oscillates in a plane perpendicular to the plane of rotating mass. The following comparisons show the effect of using higher number of PZT beams on the performance of the harvesting system. The overall generated power of the horizontally and vertically positioned harvesters for different frequencies of external excitations over a range of amplitudes is shown in Figures 17 and 18, respectively.
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Figure 17. Overall generated power of the horizontally positioned harvester for different frequencies of external excitations over a range of amplitudes from 0.1 to 2 g; continuous line: the device with eight PZT beams; dashed line: the device with one PZT beam; LR = 60 mm and ZR = 30 mm: (a) frequency = 1 Hz, (b) frequency = 2 Hz, (c) frequency = 3 Hz, (d) frequency = 4 Hz, (e) frequency = 5 Hz, (f) frequency = 7 Hz, (g) frequency = 9 Hz, (h) frequency = 11 Hz, (i) frequency = 13 Hz, and (j) frequency = 15 Hz.
Figure 18. Overall generated power of the vertically positioned harvester for different frequencies of external excitations over a range of amplitudes from 0.1 to 2 g; continuous line: the device with eight PZT beams; dashed line: the device with one PZT beam; LR = 60 mm and ZR = 30 mm: (a) frequency = 1 Hz, (b) frequency = 2 Hz, (c) frequency = 3 Hz, (d) frequency = 4 Hz, (e) frequency = 5 Hz, (f) frequency = 7 Hz, (g) frequency = 9 Hz, (h) frequency = 11 Hz, (i) frequency = 13 Hz, and (j) frequency = 15 Hz.
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Ramezanpour et al. According to Figures 17 and 18, regardless of the positioning, the harvester with eight PZT beams can generate more power than the harvester with one PZT beam. It can be seen that in general, regardless of the excitation frequency, there is a small difference between the generated powers of these two configurations in low-amplitude excitations. According to Figure 17, for a specific excitation frequency, when the amplitude of excitation increases, the difference between the amplitude of generated powers of these two models becomes noticeable. However, as can be seen in Figure 18, when the excitation frequency increases to values higher than 9 Hz, the generated power of two configurations becomes close to each other. This can be explained by considering the dynamics of the proof mass. When the harvester is vertically positioned, for these values of excitation frequencies and amplitudes, the proof mass oscillates at the top of beam number 1 while the other beams are not actuated. Therefore, for these excitations, there is no considerable difference between the generated powers of the harvester with eight PZT beams and the one with only one PZT beam. It should be mentioned that using a large number of PZT beams increases the number of magnets in the device which itself can increase the possibility of the internal magnetic coupling among the magnets.
657 During excitation, whether the harvester is positioned horizontally or vertically, almost always the beams are not actuated in the same way. Therefore, due to the difference in the actuations, it is expected that the optimum resistance of a particular beam can be different from that of the other beam. Two cases are considered at this point: (a) it is supposed that each of beams is connected to the same load resistance, RL , and objective is to find RL so that the overall RMS of the generated power is maximized and (b) it is assumed that each beam has its own resistive load, and the objective is to find eight values of RL for eight beams so that the overall generated power is maximized. Expectedly, the second case should lead to higher values of power rather than the first one because in later case all beams generate their maximum powers. All results in this section are obtained for the excitation frequency and amplitude of 3 Hz and 9.81 m/s2, respectively.
Optimum load resistance
In order to maximize the power output of the system, it is necessary to determine the best load resistance for the harvester. The simulation model is used to calculate the RMS power for a large range of resistor values.
Case a: eight PZT beams with the same resistive loads. In this section, the resistive loads attached to the beams are the same. Using the simulation model, the overall RMS of the generated power of the harvester is calculated for a range of load resistance from 0.1 to 4 MO. Figure 19(a) and (b) show the results of horizontally and vertically positioned device, respectively. According to Figure 19, the maximum power delivered to the resistive load occurs at the load resistances of 2 and 1.9 MO for the horizontal and vertical positioning and the optimum generated powers for these positions are obtained as 0.93 and 0.86 mW, respectively.
Figure 19. Overall output power versus load resistance at frequency of 3 Hz and amplitude of 1g: (a) horizontally and (b) vertically positioned harvester, ZR = 30 mm.
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Figure 20. Output power of each beam versus its load resistance at frequency of 3 Hz and amplitude of 1g for horizontally positioned device, ZR = 30 mm.
Case b: eight PZT beams with different resistive loads. In this case, eight resistive load values should be found so that the overall RMS of power of each beam and, consequently, of the harvester is maximized. In Figures 20 and 21, the generated power of each beam is drawn versus the load resistance for the horizontal and vertical positions, respectively. By comparing Figures 19(a) and 20, it can be seen that the optimum resistive load for the harvester is not necessarily equal to the optimum resistive load of any of the eight beams. Furthermore, the summations of the maximum generated powers in Figures 20 and 21 are 0.94 and 0.87 mW, respectively, which are slightly bigger than 0.93 and 0.86 mW obtained from Figure 19. As expected, the generated power of the second case is higher than that of the first case, but the difference between the obtained power values is not significant for this excitation characteristic.
Theoretical description of the model for out-of-plane vibrations
If the clamp point of the PZT beam shown in Figure 2 is subjected to a vertical excitation, its kinetic energy can be written as
L L e ep 1 2 _ (x, t) + z _ (x, t) T = r s A s ?w _ (t)? dx + rp Ap ?w 2 0 0
e16T
+z _ (t)?2 dx +
1 _ (L, t) + z m A ?w _ (t)?2 2
In equation (16), if the term z _ (t) is assumed to be 0, this equation will be identical to equation (2). Using equation (16), the discretized form of the Lagrangian functional is obtained as
L=
8 X á 1à 2 _ i (t ) _ i (t)2 à v2 _ (t)2 + ui ri (t)l r i ri (t) + mz 2 i=1 ! 1 2 _ _ i (t )z + Cpi li (t) + G i r _ (t) à Um eri (t), g T e17T 4
Out-of-plane vibration of the harvester
As mentioned before, the oscillation of PZT beams occurs in a plane perpendicular to the motion plane of pendulum. Therefore, it is claimed that the presented configuration can be used for harvesting the out-ofplane vibrational energy. In this section, the ability of this device to harvest the out-of-plane vibrational energy is demonstrated through an example. In this way, the previously obtained electromechanical equations of each PZT beam (equation (12)) are modified to simulate the out-of-plane excitations (denoted by € z(t)).
where the total mass of the PZT beam (including tip mass), m, and the coefficient G are defined as follows
m = rs As L + 2rp Ap Lp + mA
L ep L e
G = (rs As + 2rp Ap )
0
f(x)dx + rs As
Lp
f(x)dx + mA f(L)
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Figure 21. Output power of each beam versus its load resistance at frequency of 3 Hz and amplitude of 1g for vertically positioned device, ZR = 30 mm.
Applying Euler–Lagrange equations to the functional, L, leads to
?Um _ € _ i (t ) + v 2 = à G i€ z( t ) ri (t) + 2zi vi r i ri (t) à ui li (t) + ? ri e18aT 1 € 1 _ _ i (t ) = 0 C p li ( t ) + li (t) + ui r 2 i RL e18bT
The last term in the left-hand side of equation (18a) represents the effect of rotating magnet on the tip magnets and the term in the right-hand side of equation (18a) denotes the effect of out-of-plane vibrations.
Time-domain response
Again, MATLAB Simulink environment is used to obtain the time-domain response of the electromechanical equations. It is supposed that the horizontally positioned harvester is under the influence of an out-of-plane excitation and also the magnetic interactions due to the pendulum movement. For simplicity, it is also assumed that the out-of-plane excitation is a pulse function with constant amplitude as shown in Figure 22(a). To produce this pulse in Simulink, a Pulse Generator block is used with the following parameters: amplitude of 3 g, period of 5 s, and pulse width (% of period) of 0.005. At first, it is assumed that the rotating pendulum is removed and the system is only under the influence of
the out-of-plane excitation. The generated voltage of first PZT beam is drawn versus time in Figure 22(b). It should be mentioned that the generated voltage of other PZT beams is exactly the same as that of the first PZT beam and therefore is not shown here for the sake of brevity. According to Figure 22(b), expectedly, whenever a pulse is exerted to the system, a jump appears in the generated voltage. Due to the fact that the excitation is applied in the +Z direction (refer to Figure 2), once the excitation occurs, the tip mass of PZT beam goes downward, and therefore, the initial values of the generated voltage are negative. Next, it is presumed that the pendulum rotates with a constant speed of p rad/s, while the out-of-plane excitation is applied. The effects of out-of-plane vibrations on the generated voltage of each PZT beam, in the presence of rotating magnet, are shown in Figure 22(c). To eliminate the transient response, the first 15 s of the signals are excluded. The highamplitude jumps in the figure (e.g. at t = 17 s or at t = 19 s for the fourth beam) are due to the magnetic interaction between the rotating magnet and the tip magnet of PZTs, while the low-amplitude ones (e.g. at t = 20 s or at t = 25 s for the first beam) are due to the external pulses. According to Figure 22(c), although the out-of-plane excitations can lead to a higher value of generated voltages (e.g. at t = 20 s or at t = 25 s for the first beam), but sometimes it can suppress the amplitude of
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(a)
(b)
(c)
Figure 22. (a) Out-of-plane excitation versus time, (b) the generated voltage of the first PZT beam in the absence of the rotating pendulum, and (c) the generated voltage of the system in the presence of the rotating pendulum.
generated voltages (e.g. at t = 25 s for the third beam or at t = 20 s for the seventh beam). At this point, it is demonstrated that the presented configuration is capable to respond to the external excitations which are applied alongside the central bar. Considering the capabilities of the proposed device for harvesting energy, it is worth mentioning that the harvested power can be doubled by adding eight more PZT beams to the device as shown in Figure 23.
Experimental apparatus and instrumentation Excitation
This section describes the experimental equipment and tests performed to examine the electromechanical behavior of the device. All tests are performed in the horizontal position. A conventional mechanism that is able to produce harmonic motion is the Scotch yoke mechanism.
Figure 23. An alternative design of the proposed energy harvester for enhancing the generated power.
Figure 24 shows a graphical layout of this mechanism in which the rotational motion of the wheel (motor) turns into the linear motion in the yoke part. According
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Figure 24. (a) Graphical layout of Scotch yoke mechanism and (b) schematic design of the mechanism.
to Figure 24(a), the displacement and acceleration of yoke are x = e cos(vt) and a = à ev2 cos(vt), respectively. In these two relationships, v is the angular frequency of the wheel (motor) and e indicates the eccentric distance of the pin. The Scotch yoke mechanism is used to produce external excitations with various amplitudes and frequencies. In this way, as shown in Figure 24(b), a plate moves freely over a pair of rails using four concave rollers that prevent its motion perpendicular to the rail direction. In this work, this plate is considered as the yoke part of the mentioned mechanism and, at the same time, as a base plate (main tray) for the harvester (shown in Figure 25). Two kinds of harmonic excitations can be generated by means of the Scotch yoke mechanism: constant frequency excitations and constant amplitude ones. Constant frequency excitations are easily obtained when the angular frequency is kept constant and the eccentric distance varies. In order to produce constant amplitude excitation, the multiplication of ev2 should be kept constant. In this manner, when v varies from v1 to v2 , the eccentric distance should vary from e1 to 2 e2 where e2 is equal to e1 v2 1 =v2 . Therefore, the amplitude remains constant whereas the frequency is varied. A servo motor produces required rotational motions for the Scotch yoke mechanism (Figure 24(b). Its rotational speed, which can be controlled by a PC, varies in the range of 3–33 Hz.
Figure 25. Proposed harvester.
Experimental device
As can be seen in Figure 25, eight clamp supports are attached to the main tray which is an aluminum plate with the dimensions of 0:6 cm 3 38 cm 3 39 cm. The harvester consists of eight bimorph PZT beams ′ Corporation). The effective length of (V22BL; Mide each PZT beams is 52 mm. Using a double-sided adhesive tape, a neodymium magnet is attached to the tip of each beam. A similar magnet is attached to the tip of
the rotating proof mass. The magnets are cylindrical shape with the diameter of 5 mm and height of 2 mm. Residual flux density of each magnet is 0.6 T. The rotating proof mass is made of brass because this material is non-ferrous and therefore does not affect the magnetic field of the attached magnet. The experimental setup is shown in Figure 26. The proof mass is initially positioned at the top of beam number 1 (Figure 10(b)), and when the data logger is recording, the motor starts its rotational motion. To stabilize the plate on the rails, especially when the motor is rotating in relatively high speeds, four added weights are fastened to the main tray (Figure 26). However, despite these weights, when maximum eccentricity is used, we failed to reach the angular speeds higher than approximately 100 rev/min. Once the motor is started, the main tray starts its motion on the rails in an oscillating manner. At the same time, the signal obtained from the accelerometer which is attached to the main tray is sent to the data logger while the signals generated by PZT beams are sent to the resistor box through eight connectors. The resistive load in this
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一生励志的正能量短句子大全

一生励志的正能量短句子大全 一生励志的正能量短句子摘抄 1. 不经历风雨，长不成大树，不受百炼，难以成钢。 2. 耐心和恒心总会得到报酬的。 3. 宝剑锋从磨砺出，梅花香自苦寒来。 4. 表示惊讶，只需一分钟;要做出惊人的事业，却要许多年。 5. 不放弃!决不放弃!永不放弃! ——邱吉尔 6. 不积跬步，无以至千里;不积小流，无以成江海。——荀子 7. 苟有恒，何必三更起五更眠;最无益，只怕一日曝十日寒。——毛泽东 8. 成功最终属于耐心等待得人。 9. 凡是新的事情在起头总是这样一来的，起初热心的人很多，而不久就冷淡下去，撒手不做了，因为他已经明白，不经过一番苦工是做不成的，而只有想做的人，才忍得过这番痛苦。——陀思妥耶夫斯基 10. 放弃时间的人，时间也会放弃他。——莎士比亚 11. 斧头虽小，但经历多次劈砍，终能将一棵最坚硬的

橡树砍刀。 12. 告诉你使我达到目标的奥秘吧，我惟一的力量就是我的坚持精神——巴斯德 13. 一个人最痛苦的时候不是吃不上饭的时候，而是想努力奋斗没有机会。 14. 与其做一个有价钱的人，不如做一个有价值的人;与其做一个忙碌的人，不如做一个有效率的人。 15. 没有目标的人，永远为有目标的人打工。 16. 智者创造机会，强者把握机会，弱者坐等机会。 17. 说出的苦不叫苦，说不出的苦才叫苦。 18. 人若把自己框在一定的范围内，就容易限制了自己的思维和格局。 19. 人往往年轻时用健康换财富，老时再用财富换健康。发达国家的人们是透支金钱，储存健康;我们国家的人是透支健康，储存金钱。 20. 人因为有理想、梦想而变得伟大，而真正伟大就是不断努力实现理想、梦想。 一生励志的正能量短句子精选 1. 一件事被所有人都认为是机会的时候，其实它已不是机会了。 2. 天上最美的是星星，人间最美的是真情。 3. 活鱼会逆流而上，死鱼才随波逐流。 4. 怕苦的人苦一辈子，不怕苦的人苦一阵子。

工作励志正能量句子

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给了时间。其实这个城市什么都不缺少，就是缺少时间。我们来不及恋爱，来不及等待，来不及等你拼搏，也来不及陪你走到未来。我们总是想在极短的时间里找到某种安全感——那种你明知不是最好，却又不得不妥协的安全感。 十四、奋斗者在汗水汇集的江河里，将事业之舟驶到了理想的彼岸。 十五、愿你在平凡的岗位上，创造出不平凡的业绩来，直到实现远大的理想。 人生正能量励志句子【二】一、岂能尽人如意，但求无愧于心！ 二、生命对某些人来说是美丽的，这些人的一生都为某个奋斗。 三、人生最大的失败，就是放弃。 四、生命本身就是奇迹，大自然的奇迹，宇宙间存在的奇迹。每一个人从诞生到成长，整个过程，生命的本身就值得我们去欣赏。也许，每一个人个体，就是一处风景，不同的风景。 五、在成功的道路上，激情是需要的，志向是可贵的，但更重要的是那毫无情趣的近乎平常的坚守的毅力和勇气。 六、生命如自助餐厅，要吃什么菜自己选择。 七、幸福和幸运是需要代价的，天下没有免费的午餐！ 八、勤奋者总感到夜长日短，安逸者总认为夜短日长。 九、你被拒绝的越多，你就成长得越快；你学的越多，就越能成功。

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正能量励志句子经典100句

正能量励志句子经典100句 1、忘掉昨天的烦恼。新的一天。用新的态度。来面对! 2、多一分心力去注意别人，就少一分心力反省自己，你懂吗? 3、老板只能给一个位置，不能给一个未来。舞台再大，人走茶凉。 4、我想要的并不多，可我想证明我能够得到的比任何人都多。 5、放下已经走远的人与事，放下早已尘封的是与非。 6、意外和明天不知道哪个先来。没有危机是最大的危机，满足现状是最大的陷阱。 7、灰白色的天空，不是天要下雨的预兆，而是晴空的安宁。 8、没有伞的孩子，必须努力奔跑! 9、每天肩上新增的不是痛苦，是沉稳的素养。 10、感谢黑夜的来临，我知道今天无论有多失败，全新的明天仍然等待着我去证明自己。 11、人生，没有永远的伤痛，再深的痛，在切之时，伤口总会痊愈。 12、毁灭人只要一句话，培植一个人却要千句话，请你多口下留情。 13、你希望掌握永恒，那你必须控制现在。 14、世间成事，不求其绝对圆满，留一份不足，可得无限美好。 15、没有不会干的事，只有不去干的人。

16、我们怀着美好的希望，勇敢的走着，跌倒了再爬起，失败了就再努力，永远相信明天会更好! 17、也许今天我们一事无成，但难保明天一统天下。 18、知足是富人，平常是高人，无心是圣人。 19、贫穷并不可怕，可怕的是缺少自强自立的精神。 20、仇恨永远不能化解仇恨，只有宽容才能化解仇恨，这是永恒的至理。 21、是非和得失，要到最后的结果，才能评定。 22、无论如何选择，只要是自己的选择，就不存在对错后悔。 23、遭遇挫败痛苦时，告诉自己：不过是归零了，不过是从头再来! 24、抬头看清属于自己的那一片天空，炫耀别人没有的快乐。 25、爱家人，爱朋友，爱伴侣，爱孩子，然而，要真正为自己活。 26、怕苦的人苦一辈子，不怕苦的人苦一阵子。 27、每个人都有潜在的能量，只是很容易被习惯所掩盖，被时间所迷离，被惰性所消磨。 28、这个世界太多的幻想，但幻想却始终不能变成现实。 29、永远对生活充满希望，对于困境与磨难，微笑面对。 30、人生只有出走的美丽，而没有等出来的辉煌。 31、浮夸的语言，疲倦的笑容和迷离的眼神。 32、没有不进步的人生，只有不进取的人! 33、与其等到以后再来缅怀、追忆，不如好好地珍惜现在。 34、有勇气并不表示恐惧不存在，而是敢面对恐惧、克服恐