Piezoelectric Energy Harvesting from an L-Shaped Beam-Mass ...

4 downloads 1314 Views 521KB Size Report
Corresponding author, email: [email protected], phone: 1(540)231-2910. Active and ... The saturation phenomenon has already found an application as a nonlinear.
Piezoelectric Energy Harvesting from an L-Shaped Beam-Mass Structure Alper Erturka , Jamil M. Rennob and Daniel J. Inmanb Center for Intelligent Material Systems and Structures a Department of Engineering Science and Mechanics b Department of Mechanical Engineering Virginia Polytechnic Institute and State University ABSTRACT Cantilevered piezoelectric harvesters have been extensively considered in the energy harvesting literature. Mostly, a traditional cantilevered beam with one or more piezoceramic layers is located on a vibrating host structure. Motion of the host structure results in vibrations of the harvester beam and that yields an alternating voltage output. As an alternative to classical cantilevered beams, this paper presents a novel harvesting device; a flexible L-shaped beam-mass structure that can be tuned to have a two-to-one internal resonance to a primary resonance ω2 ∼ = 2ω1 (which is not possible for classical cantilevers). The L-shaped structure has been well investigated in the literature of nonlinear dynamics since the two-to-one internal resonance, along with the consideration of quadratic nonlinearities, may yield modal energy exchange (for excitation frequency ω ∼ = ω1 ) or the so-called saturation phenomenon (for ω ∼ = ω2 ). As a part of our ongoing research on piezoelectric energy harvesting, we are investigating the possibility of improving the electrical outputs in energy harvesting by employing these features of the L-shaped structure. This paper aims to introduce the idea, describes the important features of the L-shaped harvester configuration and develops a linear distributed parameter model for predicting the electromechanically coupled response. In addition, this work proposes a direct application of the L-shaped piezoelectric energy harvester configuration for use as landing gears in unmanned air vehicle applications. Keywords: L-shaped beam-mass structure, piezoelectricity, energy harvesting, unmanned air vehicles.

1. INTRODUCTION Vibration-based energy harvesting has been investigated by numerous researchers starting with the early work of Williams and Yates 1 , where the possible vibration-to-electric energy conversion mechanisms were described as piezoelectric, electromagnetic and electrostatic. These three mechanisms have been studied by researchers extensively within the last decade. Theoretical and experimental papers are available on modeling and applications of piezoelectric 2 , electromagnetic 3 and electrostatic 4 energy harvesters. Among these three alternatives for vibration-to-electric energy conversion, piezoelectric transduction appears to have received the highest attention as its literature has already been summarized in three review articles 2;5;6 . As can be found in the aforementioned review articles, a commonly used piezoelectric energy harvester configuration is a cantilevered beam with one or two piezoceramic layers (hence it is a unimorph or a bimorph with the historical definitions, respectively). Typically, a cantilevered unimorph or bimorph harvester is located on a vibrating host structure and the dynamic-bending strain induced in the piezoceramic layer(s) results in an alternating electric potential difference between the electrodes covering the piezoceramic layer(s). Practical applications 2;5;6 and mathematical modeling 7–10 of cantilevered piezoelectric energy harvesters have been investigated by numerous researchers in the last five years. Practical applications are limited to low-power systems (such as small sensors) 2;5;6 and the respective electromechanical models include single-degree-of-freedom 7;9 , approximate distributed parameter 8;9 and closed form distributed parameter 10 solutions for the coupled system dynamics of these harvesters. Usually, cantilevered harvesters are designed to have a proof mass, which can be Corresponding author, email: [email protected], phone: 1(540)231-2910.

Active and Passive Smart Structures and Integrated Systems 2008, edited by Mehdi Ahmadian Proc. of SPIE Vol. 6928, 69280I, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.776211

Proc. of SPIE Vol. 6928 69280I-1 2008 SPIE Digital Library -- Subscriber Archive Copy

tuned to have the fundamental natural frequency of the harvester beam close to a dominant excitation frequency available in the ambient vibration energy. Although a cantilevered beam is a simple structure that is not very prone to aggressive improvements, the literature includes a considerable effort for improving the electrical outputs of this configuration. Baker et al. 11 examined the effect of geometry of cantilevered piezoelectric benders on power density to find better alternatives to rectangular beam shape. Erturk et al. 12 discussed how to arrange the electrodes of a cantilevered bender to avoid cancellation in energy harvesting from vibration modes higher than the fundamental mode in the absence and presence of a proof mass. Hu et al. 13 introduced an axial preload to the classical cantilevered bimorph configuration to adjust its natural frequency in case of varying-frequency excitations. Hence, the classical cantilevered beam configuration as a piezoelectric energy harvester has been studied extensively in the literature and a considerable effort has been made for optimizing this simple structure for improved electrical outputs. The aim of this paper is to introduce and analyze a novel harvester configuration; an L-shaped beam-mass structure. This structure has been investigated extensively in the past two decades 14;15 . An L-shaped beam-mass structure can be tuned to have a two-to-one internal resonance to a primary resonance (ω2 ∼ = 2ω1 ), which is not possible for classical cantilevers. In the presence of quadratic nonlinearities, two-to-one internal resonance may result in saturation of the primary resonance (for excitation frequency ω ∼ = ω2 ) or energy exchange between the first two modes (for ω ∼ = ω1 ) 16 . The saturation phenomenon has already found an application as a nonlinear vibration absorber 17 . As a prelude to studying energy harvesting from an L-shaped landing gear of an unmanned air vehicle (UAV), this paper presents a linear electromechanical model of an L-shaped unimorph under base excitation. The remaining of this work is organized as follows. Section 2 discusses the features of an L-shaped structure and its merits as a harvester. In Section 3, we present a linear electromechanical model of the structure. Section 4 presents a case study, and an application for the L-shaped harvester is suggested in Section 5. Section 6 draws conclusions for this work.

2. FEATURES OF THE L-SHAPED STRUCTURE In the following, we discuss some unique features of the L-shaped beam-mass structure. Although the coupled model presented in this paper does not include geometric nonlinearities, the authors are currently investigating the use of quadratic nonlinearities to possibly enhance piezoelectric energy harvesting from base excitation (particularly due to two-to-one internal resonance). We also discuss an advantage of the L-shaped beam-mass structure as a broadband energy harvester.

2.1 Nonlinear Interactions due to Two-To-One Internal Resonance The L-shaped beam-mass structure has been investigated in the literature of nonlinear dynamics by Haddow et al. 14 and Balachandran and Nayfeh 15 . As mentioned by Nayfeh and Mook 16 , multi-degree-of-freedom (MDOF) systems having two or more of their natural frequencies commensurable or nearly so (i.e., there exist integers m1 , · · · , mn such that m1 ω1 + m2 ω2 + · · · + mn ωn ∼ = 0 where ω1 , · · · , ωn are the natural frequencies) may posses internal resonances. The simplest possible MDOF system is a 2-DOF system and the condition of having ω2 ∼ = 2ω1 is a simple case for realizing an internal resonance and this particular case is called the two-to-one internal resonance. Although basic structures like cantilevered beams cannot be designed to obtain ω2 ∼ = 2ω1 , it was shown in the literature 14 that an L-shaped beam-mass structure can have this internal resonance condition (with the third and higher modes well separated from the first two modes). A structure that has a two-to-one internal resonance and quadratic nonlinearities may exhibit energy exchange between the first two modes or the so-called saturation phenomenon in case of sinusoidal excitation near a primary resonance (for ω ∼ = ω1 and ω∼ = ω2 , respectively) 16 . Modal energy exchange is characterized by a continuous back and forth energy exchange between the first two modes. Mode saturation occurs after the excitation amplitude exceeds a certain value. Then, the amplitude of the vibration mode that is directly excited becomes independent of the level of excitation (see Chapter 6 in Nayfeh and Mook 16 ). As a part of our ongoing research, we aim to investigate the possible use of these unique features of the L-shaped harvester structure. This work is a prelude to a detailed investigation on using an L-shaped beam-mass structure with a two-to-one internal resonance as a piezoelectric energy harvester.

Proc. of SPIE Vol. 6928 69280I-2

In order to tune the L-shaped beam-mass structure to have a two-to-one internal resonance, Balachandran and Nayfeh varied the location of M2 experimentally 15 . However, to tune the structure in this work, we model the system as a six degree of freedom (DOF) system (with 3 translational and 3 rotational DOFs at x1 = L1 , x2 = L2 and x3 = L3 ). By neglecting the rotary inertias of the lumped masses, the system reduces to a 3-DOF system 18 . Then, Guyan’s method (static condensation) 19 is used to further reduce the system to a 2-DOF system, since the effective mass at x3 = L3 is negligible. We then force the system to have a frequency ratio of two (i.e. ω2 ∼ = 2ω1 ) by finding appropriate values for L2 .

B

(a) Schematic of the L-shaped unimorph harvester.

(b) Coordinate systems and displacements of the L-shaped unimorph harvester.

Figure 1: Schematic of the L-shaped unimorph harvester, coordinate systems and displacements used in the analysis.

2.2 L-Shaped Structure as a Broadband Energy Harvester Besides the nonlinear interaction between the vibrations modes, the structure is advantageous also as a broadband energy harvester. In most research on piezoelectric energy harvesting, the cantilevered harvester beam has been assumed to be excited at or around its first (fundamental) natural frequency. In other words, the first natural frequency of the harvester is tuned to a frequency that is dominant in the ambient vibration energy. This tuning process is usually realized by means of a proof mass. In reality, however, most of the ambient vibration sources display random behavior in time (see, for instance, the random acceleration history of an automobile compressor measured by Sodano et al. 20 or the sample frequency spectra by Roundy et al. 7 ). Hence, in general, ambient vibration cannot be represented by a single harmonic function. As a consequence, vibration energy available in the ambient excites higher vibration modes of the harvester structure, as well. The nearest vibration mode to the fundamental mode is simply the second mode and this mode gains some importance in case of random vibrations with frequency content higher than the first natural frequency. If the dominant frequency content of the ambient vibration energy is randomly distributed in a frequency band around the first natural frequency ω1 of the harvester, it is preferable to have its second natural frequency ω2 close to the first natural frequency so that the harmonics ω > ω1 of the ambient vibration result in a stronger modal contribution from the second mode. As far as the classical cantilevered harvester configuration is concerned, it is straightforward to show that the second natural frequency is more than 6 times the first natural frequency in the absence of a proof mass (ω2 ∼ = 6.27ω1 ). This is simply obtained by taking the ratio of the squares of the dimensionless frequency parameters (eigenvalues) obtained from the respective eigensolution for a clamped-free beam without a proof mass. The presence of a proof mass increases the spacing of ω1 and ω2 of the harvester on the frequency axis even more. Figure 2a displays the variation of the dimensionless natural frequencies with nondimensional proof mass-to-beam mass ratio for the first two vibration modes. Taking the ratio of the curves in Fig. 2a yields ω2 /ω1 , which is a direct measure of the spacing of the first two vibration modes on the frequency axis and the variation of this ratio with proof mass-to-beam mass ratio is plotted in Fig. 2b. As proof mass-

Proc. of SPIE Vol. 6928 69280I-3

to-beam mass ratio (Mproof /Mbeam ) changes from 0 to 10, the ratio between the first two natural frequencies (ω2 /ω1 ) increases from 6.27 to 28.7. According to Fig. 2b, the spacing between the first two vibration modes increases monotonically with increasing proof mass and the minimum amount of spacing corresponds to the no proof mass case (with ω2 ∼ = 6.27ω1 ). 25

30

20

25

15

20

10

15

5

10

0 0

2

4

6

8

10

(a) First and second dimensionless natural frequencies.

5 0

2

4

6

8

10

(b) Ratio of the dimensionless second natural frequency to the first.

Figure 2: Behavior of the first two dimensionless natural frequencies of a cantilevered beam with a proof mass in transverse vibrations.

Now it is possible to relate this simple nondimensional analysis for a classical cantilevered harvester with a proof mass to our discussion regarding the random frequency behavior of the ambient energy. As a common practice, the designer tunes the first natural frequency of the cantilevered harvester to a frequency (ω1 ) that is dominant in the ambient vibration spectrum. This practice automatically assigns a certain value (ω2 ) to the second natural frequency and we know from Fig. 2a and Fig. 2b that the second natural frequency will not be very close to the first one (at least ω2 ∼ = 6.27ω1 ). As a consequence, the response of the harvester to the harmonics in the ambient vibration lying in a very wide frequency range between ω1 and ω2 will be considerably weak. With this consideration, the L-shaped harvester configuration proposed in this work has an important advantage over the classical cantilevered harvesters (with or without proof mass). While selecting a dominant frequency (ω1 ) in the ambient vibration spectrum and setting the first natural frequency of the L-shaped harvester to this value, the geometric parameters of the harvester can be tuned to have a second natural frequency (ω2 ) that is not very far away from the first natural frequency. For instance, as mentioned before and as will be shown in the case study, ω2 ∼ = 2ω1 is a perfectly realizable case for the L-shaped configuration whereas this case is impossible to realize for the classical cantilevered beams where the lowest spacing (between the first two modes) is for ω2 ∼ = 6.27ω1 . Hence, the total response of the L-shaped harvester to a set of ambient vibration harmonics in a frequency band around the first natural frequency can be expected to be stronger than that of a classical cantilevered harvester with the same first natural frequency. This is simply due to the fact that the L-shaped configuration shows two resonances in a narrow frequency range around ω1 . For the same reason (i.e., because the first two modes are much more close to each other), the L-shaped configuration is less sensitive to variations in the dominant excitation frequency when compared to the classical configuration.

Proc. of SPIE Vol. 6928 69280I-4

3. ELECTROMECHANICAL MODELING In this section, we summarize the electromechanical modeling of the L-shaped unimorph harvester shown in Fig. 1a. The electromechanically coupled modeling approach is based on the analytical cantilevered harvester model proposed by Erturk and Inman 10 . As depicted in Fig. 1a, this structure is a combination of one horizontal and one vertical thin beams with two lumped masses, M1 , and M2 . The structure is excited by the vertical acceleration, aB (t), of its base. The metallic substructure and piezoelectric layers are geometrically uniform along their longitudinal directions. The lumped masses and the location of the second lumped mass, M2 , on the vertical beam are important tuning parameters for the system which make it possible to obtain the two-to-one internal resonance for nonlinear oscillations 15 , as mentioned in the previous section. The vibratory motion of the harvester is examined in three segments, Rk , equiped with the reference frame (xk , yk ), Fig. 1b, such that (1) Rk = {xk |0 ≤ xk ≤ Lk } , where k = 1, 2 and 3 for the remaining of this work. Each segment is modeled for the general case of different lengths Lk , masses per unit length mk , and flexural stiffness Y Ik . Therefore, the metallic substructure and piezoelectric layers as well as their geometric properties can be taken to be different for every segment Rk . The piezoelectric layers are covered with conductive electrode pairs. We assume that the piezoelectric layer and the substructure are perfectly bonded. Regions R1 , R2 and R3 have separate electrode pairs whose leads are connected to a single resistive load in series. Although the configuration is taken as a unimorph in the following analysis, one might consider the bimorph case with a similar procedure. The bimorph configuration allows combining the electrical outputs of different segments in parallel or in series depending on the voltage or current requirements 21 . Note that (xk , yk ) are relative frames of reference for the respective segments Rk . In the next subsection, we present the modal analysis for free vibrations of the structure.

3.1 Modal Analysis for Free Vibrations Since the aspect ratios of typical harvesters allow neglecting the effects of shear deformation and rotary inertia, the following modeling procedure is based on the Euler-Bernoulli beam assumptions. Reasonably, we are interested in bending vibrations of the harvester and therefore the longitudinal vibrations of the beam segments are ignored by assuming the segments to be axially rigid. We consider geometrically linear oscillations along with the assumption of linear elastic material behavior. Equations of motion for undamped free vibrations of each segment in its lateral direction can be written as mk

∂ 2 wk (xk , t) ∂ 2 wk (xk , t) ∂ 2 (Mb )k (xk , t) + + δ M g = 0 xk ∈ Rk , 2k 2 ∂t2 ∂x2k ∂x2k

(2)

where mk is the linear mass density, (Mb )k is the bending moment, and wk (xk , t) is the transverse vibratory motion of segment Rk , and δrs is the Kronecker delta (defined as being equal to unity for r = s and equal to zero for r = s). It is important to note that the piezoelectric effect is included in the bending moment (Mb )k , which can be expanded into a term related to the flexural stiffness Y Ik and a term related to the voltage vk (t) across the electrodes of segment Rk 10 . The dissipative effects due to internal and external damping mechanisms will be introduced as modal damping later. Also the weight M2 g of the second lumped mass acts as an axial load for segment R2 . The boundary conditions, compatibility and continuity conditions, are stated next. In the sequel, the over-dot (˙) will indicate temporal differentiation, whereas the prime ( ) indicates spatial differentiation with respect to the corresponding spatial variable. The essential boundary conditions at the clamped end x1 = 0 are w1 (0, t) = 0

and

∂ w1 (0, t) = 0 . ∂x1

(3)a

The linear/angular displacement and force/moment equilibrium relations at the locations of the lumped masses (x1 = L1 and x2 = L2 ), ∂ w1 ∂ w2 w2 (0, t) = 0 , (L1 , t) = (0, t) , (3)b ∂x1 ∂x2

Proc. of SPIE Vol. 6928 69280I-5

Y I1

∂ 3 w1 (L1 , t) = [M1 + M2 + m2 L2 + m3 L3 ]w ¨1 (L1 , t) , ∂x31

(3)c

∂ 2 w1 ∂w ¨1 ∂ 2 w2 (L , t) + J (L , t) = Y I (0, t) , 1 1 1 2 ∂x21 ∂x1 ∂x22

(3)d

∂ w2 ∂ w3 (L2 , t) = (0, t) , ∂x2 ∂x3

(3)e

Y I1

w3 (0, t) = w2 (L2 , t) M2 g

,

∂ 3 w2 ∂ 3 w3 ∂ w2 (L2 , t) + Y I2 (L , t) = Y I (0, t) + M2 w ¨2 (L2 , t) , 2 3 ∂x2 ∂x32 ∂x33

(3)f

∂ 2 w2 ∂w ¨2 ∂ 2 w3 (L , t) + J (L , t) = Y I (0, t) , 2 2 2 3 ∂x22 ∂x2 ∂x23

(3)g

Y I2

where J1 and J2 are the rotary inertias of the lumped masses M1 and M2 respectively. Finally, the natural boundary conditions at the free end of the harvester can be stated as Y I3

∂ 3 w3 (L3 , t) = 0 ∂x33

, Y I3

∂ 2 w3 (L3 , t) = 0 . ∂x23

(3)h

Based on the expansion theorem, the vibratory motion of beam segment Rk can be represented by an absolutely and uniformly convergent series of the eigenfunctions as wk (xk , t) =

∞ 

φkr (xk )ηr (t) , xk ∈ Rk ,

(4)

r=1

where the system is forced to have a single modal response ηr (t) and the piecewise defined eigenfunctions of the structure are, φ1r (x1 ) = A1r sin(αr x1 ) + B1r cos(αr x1 ) + C1r sinh(αr x1 ) + D1r cosh(αr x1 ) ,

(5)a

φ2r (x2 ) = A2r sin(βr x2 ) + B2r cos(βr x2 ) + C2r sinh(γr x2 ) + D2r cosh(γr x2 ) ,

(5)b

φ3r (x3 ) = A3r sin(µr x3 ) + B3r cos(µr x3 ) + C3r sinh(µr x3 ) + D3r cosh(µr x3 ) .

(5)c

The above eigenfunctions are obtained through the separation of variables solution of the respective partial differential equations, Eq. (2). This formulation relies on the reasonable assumption that the standard forms of the eigenfunctions (based on the Euler-Bernoulli beam model) are not affected by piezoelectric coupling. For harmonic oscillations in time domain, the relations between the frequency parameters of different segments of the structure, natural frequencies and the structural parameters can be obtained as 22 ,   κ2 m κ2 m2 κ κ M2 g 2 4 2 m1 4 2 m3 2 2 αr = ωr + ωr2 + ωr2 , µr = ωr , βr = + , γr = − + , κ= , (6) Y I1 Y I3 2 4 Y I2 2 4 Y I2 Y I2 where ωr is the undamped natural frequency of the r-th vibration mode ∗ . As a common practice, substituting Eq. (4) in Eqs. (3) results in an eigenvalue problem. With the well-known procedure, forcing the resulting (12×12) coefficient matrix to be singular, one can obtain the natural frequencies of the structure from the characteristic equation, which can then be used in Eqs. (5) and Eq. (6) to find the eigenfunctions of each segment. In Eqs. (5), {Akr , Bkr , Ckr , Dkr } are the modal coefficients of the eigenfunction defined over segment Rk , respectively. Therefore, the modal contributions of the eigenfunctions in the series expansions depend on these ∗ As far as the electromechanically coupled structure is considered with light mechanical damping, ωr corresponds to ωrsc , which is approximately the short circuit resonance frequency of the harvester (for Rl → 0). For the coupled system, the short circuit resonance frequency (ωrsc ) shifts to the open circuit resonance frequency (ωroc ) as the resistive load, Rl → ∞.

Proc. of SPIE Vol. 6928 69280I-6

sets of modalcoefficients. Next, it is required to normalize the piecewise defined eigenfunctions over the entire  domain {R1 R2 R3 }. In order to be consistent with the formulation proposed by Erturk and Inman 10 for a classical cantilevered beam, the piecewise defined eigenfunctions of the structure are mass normalized according to the following orthogonality conditions ⎡L ⎤ k 3  ⎣ mk φkr (xk )φks (xk )dxk ⎦ + (M1 + M2 + m2 L2 + m3 L3 )φ1r (L1 )φ1s (L1 ) k=1

0

+M2 φ2r (L2 )φ2s (L2 ) + J1 φ1r (L1 )φ1s (L1 ) + J2 φ2r (L2 )φ2s (L2 ) = δrs , and

(7)

⎡L ⎤ L k 2 3  ⎣ Y Ik φkr (xk )φks (xk )dxk ⎦ − M2 gφ2r (x2 )φ2s (x2 )dx2 − Y I2 φ1r (L1 )φ2s (0) − Y I3 φ2r (L2 )φ3s (0) k=1

+Y

0

I3 φ2r (L2 )φ 3s (0)

+Y

0   I2 φ2r (0)φ2s (0)

  2 − Y I3 φ3r (0)φ 3s (0) + Y I3 φ3r (0)φ3s (0) = δrs ωr .

(8)

3.2 Electromechanical Equations for General Base Excitation In order to derive the electromechanical equations of the harvester, we should consider the effect of mechanical forcing in the system. Since the base excitation is in the vertical direction in the physical coordinates (which is the y1 −direction in Fig. 1b), the direct excitation of the structure is due to its own inertia in the same direction. Therefore, as we are interested in bending vibrations of the harvester, it is straightforward to see from Fig. 1a that the forcing will affect Eq. (2) (k = 1) in the physical coordinates. However, the entire structure will be vibrating due to the formulation given in the previous section. After substituting Eq. (4) in Eqs. (2) and applying the orthogonality conditions, the forced equation of motion can be written in the modal coordinates as, η¨r (t) + 2ζr ωr η˙ r (t) + ωr2 ηr (t) +

3 

χkr vk (t) = Nr (t) ,

(9)

k=1

where ζr is the viscous modal damping ratio of the r-th vibration mode † , Nr (t) is the modal forcing function, vk (t) is the voltage across the electrodes in segment Rk and χkr is the modal electromechanical coupling term such that ⎡ ⎤ L1 χkr = ϑk [φkr (Lk ) − φkr (0)] , Nr (t) = − ⎣m1 φ1r (x1 )dx1 + (M1 + M2 + m2 L2 + m3 L3 )φ1r (L1 )⎦ aB (t) . (10) 0

Here, ϑk is the coupling term in the physical coordinates and it is a function of the cross-sectional geometric parameters, Young’s modulus of the piezoelectric layer and its piezoelectric constant in segment Rk 10 . The modal forcing function given by Eq. (10) is due to the lateral inertia of the distributed mass in R1 as well as the lumped mass at the boundary x1 = L1 which includes the masses of the vertical beam segments, and the lumped masses M1 and M2 . As far as the lateral vibrations of the beam segment R1 is concerned, the effective lumped mass at x1 = L1 is not just M1 and M2 since the total mass of segments R2 and R3 also contribute to the lateral inertia of R1 at x1 = L1 , yielding the modal forcing expression of Eq. (10). Note that, in the modal forcing function, the forcing term due to external damping is neglected 23 . Figure 3 displays the electrical circuit of the harvester where the electrical outputs of the piezoelectric elements of segments R1 , R2 and R3 are combined series and connected to a resistive load R. Each piezoelectric layer is shown as a current source along with its internal capacitance connected in parallel. The electric current ik (t) produced in each segment Rk is given by ik (t) =

∞ 

ψkr η˙ r (t)

where

ψkr = −(d31 )k (Yp )k bk (hpc )k [φkr (Lk ) − φkr (0)] .

(11)

r=1 †

For a detailed discussion on how to relate the modal damping ratio to internal (strain rate) and external (air) damping effects based on the proportional damping assumption, the reader is referred to Ref. 23 .

Proc. of SPIE Vol. 6928 69280I-7

i1 t

i2 t

i3 t

v1 t

v2 t

v3 t

Cp

1

vt

Cp

2

Cp

3

Rl Figure 3: Electrical circuit of the L-shaped harvester (the piezoelectric outputs of the three segments are combined in series and connected to a resistive load).

In Eq. (11), (d31 )k is the piezoelectric constant, (Yp )k is Young’s modulus, bk is the width of the piezoelectric layer and (hpc )k is the center of the piezoelectric layer from the neutral axis of the uniform cross-section in Rk . Note that, the above form of Eq. (11) assumes the electrodes of R1 , R2 and R3 are insulated from each other, i.e., they are discontinuous at x1 = L1 and x2 = L2 . As can be seen from Eq. (11), the amount of current produced in each segment is proportional to the bending slope difference at the electrode boundaries. This is due to the fact that the electric current is the time rate of change of the electric displacement integrated over the electrode area (where the electric displacement is proportional to the curvature of the beam) 10 . Therefore, one should be careful about the mode shapes of the structure when combining the currents in the circuit. Otherwise, cancellations of the current outputs are possible as demonstrated in Section 4. The internal capacitance (Cp )k of the piezoelectric layer in segment Rk can be obtained from (Cp )k =

(S33 )k bk Lk , (hp )k

(12)

where (S33 )k is the permittivity of the piezoelectric layer at constant strain and (hp )k is the thickness of the piezoelectric layer. Again, the form of Eq. (12) assumes that the entire length of the piezoelectric in segment Rk is covered by continuous electrodes. Applying Kirchhoff Voltage Law to the circuit shown in Fig. 3 and employing Eq. (11) result in the following three equations for v1 (t), v2 (t) and v3 (t), ∞  1 1 1 v1 (t) + (Cp )1 v˙ 1 (t) + v2 (t) + v3 (t) = ψ1r η˙ r (t) , Rl Rl Rl r=1 ∞  1 1 1 v1 (t) + v2 (t) + (Cp )2 v˙ 2 (t) + v3 (t) = ψ2r η˙ r (t) , Rl Rl Rl r=1

∞  1 1 1 v1 (t) + v2 (t) + v3 (t) + (Cp )3 v˙ 3 (t) = ψ3r η˙ r (t) . Rl Rl Rl r=1

(13)

Equation (9) and Eqs. (13) constitute four ordinary differential equations for the four unknowns v1 (t), v2 (t), v3 (t) and ηr (t). These equations are the electromechanical equations of the harvester. The voltage response across the resistive load is simply 3  v(t) = vk (t) , (14) k=1

and the coupled mechanical response of the harvester in the desired segment can be obtained by using ηr (t) in Eq. (4) along with the eigenfunctions which are normalized according to Eq. (7) and Eq. (8).

Proc. of SPIE Vol. 6928 69280I-8

3.3 Coupled Voltage Response and Structural Response for Harmonic Base Excitation As a common practice in the energy harvesting literature, we assume the base acceleration to be harmonic of the form aB (t) = AB ejωt where AB is the amplitude of the base acceleration, ω is the excitation frequency and j is the unit imaginary number. For linear oscillations, the steady state expressions for the modal mechanical response of the harvester ηr (t) and the voltage response vk (t) can be written as ηr (t) = Hr ejωt

and vk (t) = Vk ejωt ,

(15)

where Hr and Vk are complex valued. Using Eq. (9) and Eq. (15) results in the following relationship λ r AB − Hr =

3 k=1



χkr Vk

ωr2 − ω 2 + 2jζr ωr ω

where

λr = − ⎣m1

L1

⎤ φ1r (x1 )dx1 + (M1 + M2 + m2 L2 + m3 L3 )φ1r (L1 )⎦ . (16)

0

Eliminating the modal mechanical response term in Eq. (13) results in three equations for V1 , V2 and V3 which can be written in the matrix form as 3 

Qmn Vn = Pm

where

m = 1, 2, 3 .

∞  jωψmr χnr 1 + jω(Cp )m δmn + 2 Rl ω − ω 2 + 2jζr ωr ω r=1 r

and Pm =

(17)

n=1

Here, Qmn =

∞ 

ω2 r=1 r

jωψmr λr AB , − ω 2 + j2ζr ωr ω

(18)

are complex valued. Closed-form solution of the complex voltage amplitude can be obtained from Eq. (17), which can be used in Eq. (15) to obtain the steady state voltage response expressions across the electrodes of the individual piezoelectric elements. Then, the complex voltage amplitude across the resistive load is V=

3 

Vk .

(19)

k=1

The complex amplitudes of the electric current passing through the resistive load and the power are obtained 2 through I = RVl and P = VRl .

4. CASE STUDY In this section, an L-shaped unimorph harvester under harmonic base excitation is analyzed. Rather than specifying a certain frequency for the base excitation, the coupled response characteristics of the harvester structure (mechanical and electrical) are investigated with frequency response functions (FRFs). The material, geometric and electromechanical properties of the harvester are given in Table 1 and Table 2. The metallic substructure is made of steel and the piezoelectric layer is PZT-5A. In addition to the numerical data provided in the tables, the structure has two lumped masses M1 = 0.025 kg and M2 = 0.015 kg at x1 = L1 and x2 = L2 , respectively, and these masses have the rotary inertias J1 = 1.5 × 10−6 kg-m2 and J2 = 1 × 10−6 kg-m2 , respectively. Although the following is a linear analysis based on the model derived in Section 3, the unimorph harvester of this case study is designed to have ω2sc ∼ = 2ω1sc where ω1sc ∼ = 22.8 Hz and ω2sc ∼ = 45.7 Hz under ‡ short circuit conditions (i.e., when the resistive load Rl → 0) . It is observed that the first two open circuit natural frequencies of the harvester are ω1oc ∼ = 23.8 Hz and ω2oc ∼ = 46.5 Hz, respectively, for Rl → ∞. The shift in the natural frequencies with changing resistive load is an expected trend as previously observed for classical cantilevered harvesters 10 . ‡

The 2-DOF model of the structure mentioned in Section 2 is used for realizing ω2 ∼ = 2ω1 with minimum effort.

Proc. of SPIE Vol. 6928 69280I-9

Table 1: Material and electromechanical parameters of the L-shaped unimorph harvester. Substructure Piezoceramic Property Young’s modulus [GPa] 200 66 3 Mass density [kg/m ] 7800 7800 Piezocelectric constant (d31 ) [m/V] −190 × 10−12 Permitivity [F/m] 1.327 × 10−8 Table 2: Geometry of the L-shaped unimorph harvester. Substructure Piezoceramic Segment Property Length [cm] 5 5 R1 Thickness [mm] 0.5 0.5 Width [cm] 1 1 Length [cm] 3.96 3.96 R2 Thickness [mm] 0.3 0.3 Width [cm] 0.8 0.8 Length [cm] 2.04 2.04 R3 Thickness [mm] 0.3 0.3 Width [cm] 0.8 0.8 Before discussing the trends in the resulting FRFs, an important issue regarding the mode shape dependence of electrical outputs must be addressed. It can be recalled from Fig. 3 that the individual piezoelectric segments are modeled as current sources in parallel to their internal capacitances. Equation (11) shows that the current source ik (t) of region Rk is a function of the modal velocity response η˙ r (t) and the mode shape dependent parameter ψkr . In case of excitation around a vibration mode (ω ∼ = ωr ), the summation in Eq. (11) reduces to a single dominating term. Then, from the expression given for ψkr , the amplitude and phase of the current source depend on the bending slope difference φkr (Lk )−φkr (0) at the electrode boundaries in region Rk . To improve the charge collected in region Rk , one should make this slope difference as high as possible as discussed theoretically 10 and demonstrated experimentally 12 in the literature. Strong cancellations may occur in energy harvesting from a certain vibration mode if continuous electrodes cover a region where the curvature (and therefore the bending strain) changes sign for that vibration mode. Figure 4a and Fig. 4b display the piecewise defined mode shapes of the L-shaped harvester for modes 1 and 2, respectively. It is clear from Fig. 4a that the slope difference in all three regions has the same sign for mode 1 excitations. Consequently, the current source terms i1 (t), i2 (t) and i3 (t) of regions R1 , R2 and R3 in Fig. 3 are in-phase and they do not cancel one another for direct combination of the electrode leads in mode 1 excitation. However, as can be seen from Fig. 4b, this is not thecase for the second mode shape since the slope difference at the electrode boundaries in R1 and that in R2 R3 have the opposite sign, which means that the i1 (t) is now 180 degrees out-of-phase when compared to i2 (t) and i3 (t). As a result, the previously mentioned combination of the electrode leads causes cancellation around mode 2 excitations. In order to avoid cancellation for excitations around mode 2, one should simply connect the leads coming from R1 in the reverse manner. Expectedly, this modification that we use in order to avoid cancellation for excitations around mode 2 results in cancellation for excitations around mode 1. This discussion is demonstrated in Fig. 4c, where the voltage FRF (per base acceleration in g) is plotted against a frequency band that includes the first two natural frequencies (and the FRF is for a fixed resistive load Rl = 50 kohms). The solid line corresponds to the first case that is favorable for mode 1 excitation but it results in cancellation in mode 2 excitations. The dashed line belongs to the case where the cancellation in mode 2 excitation is avoided but it yields cancellation for mode 1 excitation. In practice, it is possible to avoid cancellation of the electrical outputs of R1 , R2 and R3 for both vibration modes by employing full-bridge rectifiers 12 . It is important to note that the cancellation phenomenon is an electrical issue that depends on the electrode locations and the major trend in the mechanical FRF is not affected that much. This is depicted in Fig. 4d, which is the FRF that gives the transverse displacement at the free end per base acceleration (in g) for the same resistive load (Rl = 50 kohms). The current and power FRFs can be obtained from the voltage FRF and they are not shown here. A useful practice is to plot the variation of the voltage, current and power amplitudes with load resistance for excitation at a certain natural

Proc. of SPIE Vol. 6928 69280I-10

12

6

10

4

8

2

6

0

4

2

2

4

0

6

1

10

0

10

1

2 0

0.01

0.02

0.03

0.04

0.05

0.06

8 0

(a) Normalized mode 1.

10

2

10

3

0.01

0.02

0.03

0.04

0.05

0.06

(b) Normalized mode 2.

10

0

20

40

60

80

100

(c) Voltage FRF (Rl = 50 kohms).

4

2

10

10

3

10

2

10

4

10

0

10

5

10

0

20

40

60

80

100

(d) Displacement FRF at x3 = L3 (Rl = 50 kohms).

4

10

6

10

8

10

(e) Voltage, current and power magnitudes with varying values of Rl at ω = ω1sc .

Figure 4: Results from the case study.

frequency. Figure 4e shows the variation of the voltage, current and power amplitudes (per base acceleration in g) for excitation at the first natural frequency (ω1sc = 22.8 Hz). The voltage output increases monotonically whereas the current output decreases monotonically with increasing load resistance. Hence, as a consequence of these expected trends in voltage and current, power has a peak value for an optimum resistive load. The highest current output is obtained in short circuits conditions (Rl → 0) as 93 µAmp/g and the highest voltage outputs is obtained in open circuit conditions (Rl → ∞) as 203 volts/g. An optimum power output of 10 mWatts/g is obtained for a resistive load of 2.18 Mohms.

5. AN APPLICATION: L-SHAPED HARVESTER AS A UAV LANDING GEAR Unmanned air vehicles (UAVs) have been investigated by several research groups in the literature 24 . Moreover, in the last decade, a new class of UAVs, the so-called micro air vehicles (MAVs), has emerged and several research programs were initiated, in particular, for use in military-based missions. The main difference between UAVs and MAVs is due to their dimensional definitions. Although wing spans of typical UAVs can exceed 1 m, the definition of MAVs limits their maximum length dimension to 6 inches 25 . From the practical point of view, our brief discussion in this section applies both for UAVs and MAVs, although, for convenience, we will stick to the UAV type throughout this section. The motivating reasons for harvesting energy in a UAV application include increasing the flight time for a prescribed mission and powering the UAV’s sensors or global positioning system units. The literature includes research on powering UAVs with thermoelectric energy generation 26 as well

Proc. of SPIE Vol. 6928 69280I-11

(a) Front view of UAV showing two L-shaped landing gears.

(b) Side view of front L-shaped landing gear.

Figure 5: Schematic of UAV with L-shaped unimorph landing gears. as detailed discussions on the implementation of solar, wind, electromagnetic and autophagous (self-consuming) structure-battery applications for UAVs 27 . Piezoelectric energy harvesting for UAVs was discussed by Erturk et al. 28 and Anton et al. 29 where AFC (active fiber composite) benders were located inside the fuselage (in the cantilevered configuration) and also attached on the wing spars (as structural patches) of a UAV with a wing span of 1.8 m for voltage generation during flight. Recently, Magoteaux 30 studied solar and piezoelectric energy harvesting techniques for implementing into a small UAV. For integrating piezoelectric benders to a UAV, Magoteaux proposed 30 to replace the landing gears by cantilevered uniform and curved beams as two possible configurations. Even though uniform and curved cantilevered benders are easy to find commercially, these classical configurations may not be the most appropriate ones to use as landing gears. Having described the L-shaped piezoelectric energy harvester and its features in this work, we suggest using it as a landing gear for UAV applications. In addition to geometric compatibility of the L-shaped structure as a landing gear (Fig. 5), its parameters can be tuned to obtain the two-to-one internal resonance to a prescribed primary resonance as described under Section 2. Reasonably, it is required to be careful with the size and weight limitations of the UAV while designing the L-shaped harvester. The primary resonance of interest is the dominant excitation frequency and it might be the operating frequency of a vibrating base on which the UAV lands during a mission (Fig. 5). As a concrete example, one can consider an MAV which lands on a vibrating air conditioner unit during the mission to charge its batteries. Typical frequency contents of such devices have been provided in the literature of energy harvesting 7 , hence, one can select a dominant primary resonance to obtain the best electrical results in piezoelectric energy harvesting. Considering Fig. 5, it is worthwhile to add that the L-shaped harvester as a landing gear has somewhat different boundary conditions and mechanical forcing when compared to what has been investigated in this work based on Fig. 1a. One can easily modify the boundary conditions given by Eqs. (3) as well as the mechanical forcing term to represent the excitation coming from the wheels. Note that the wheel of each landing gear can take the place of the second lumped mass between regions R2 and R3 . Alternatively, the given formulation can be used directly by equating the length of R3 to zero and the partial lumped mass of the UAV to M2 (Fig. 1a). Then, the clamped end is the location of the wheel with the ground acceleration aB (t). The mass of the UAV can be split and lumped on each landing gear to analyze the electrical outputs for a given vibratory motion at each wheel.

6. CONCLUSIONS In this work, an L-shaped beam-mass structure is proposed as a novel piezoelectric energy harvester configuration and its electromechanical behavior is analyzed in a certain detail. The structure has been investigated in the literature of nonlinear dynamics in the last two decades due to its tunability for obtaining the two-to-one internal resonance to a primary resonance, which may result in saturation of the primary one (for ω ∼ = ω2 ) or modal energy exchange (for ω ∼ = ω1 ) when quadratic nonlinearities are considered. The use of the saturation phenomenon with the two-to-one internal resonance was previously discussed for consideration as a nonlinear vibration absorber. Here, in this introductory study, the L-shaped beam-mass structure is employed for piezoelectric energy harvesting as an alternative to commonly employed classical cantilevered beams. Besides

Proc. of SPIE Vol. 6928 69280I-12

the nonlinear interactions, it is shown that the L-shaped harvester can be tuned to have the first two natural frequencies much close to each other (on the frequency axis) when compared to the cantilevered beam case. Considering the random frequency content of typical ambient vibration sources (which makes it impossible to excite the harvester exactly at a tuned first natural frequency only), the relatively low spacing between the first two natural frequencies makes the L-shaped harvester preferable. The lowest spacing between the first two modes for a classical cantilevered beam occurs with ω2 ∼ = 6.27ω1 whereas an L-shaped harvester can be tuned to have, for instance, ω2 ∼ = 2ω1 (which corresponds to the two-to-one internal resonance case). Reasonably, having the first two natural frequencies close to each other is a favorable feature in the sense of broadband energy harvesting and also for handling uncertainties in the excitation frequency. An electromechanical model is presented for a detailed analysis where the L-shaped piezoelectric energy harvester is investigated as a generator with three thin beam segments. The coupled model is based on a recent distributed parameter cantilevered harvester model presented by the authors. The electrical outputs of the three piezoelectric segments are combined in series and connected to a single resistive load. The frequency response functions of the voltage drop across the resistive load as well as the coupled mechanical response of the harvester are investigated (per base acceleration). Variation of the voltage, current and power are also examined and the well known qualitative trends are observed (such as the existence of an optimum resistive load for the maximum electrical power and the short circuit and open circuit resonance frequencies). In addition, how to combine the electrodes of piezoelectric elements is highlighted to avoid the possible modes-shape dependent cancellations of the electrical outputs. Finally, a direct application is discussed where the L-shaped harvesters are suggested for use as landing gears of a UAV. Improving the flight time of UAVs during the mission and powering their sensors and global positioning units is an important research topic. The literature includes thermoelectric, solar, wind, electromagnetic and selfconsuming battery applications to generate power for UAVs/MAVs. However, implementation of piezoelectric type of energy generation is rather new for UAVs. The use of piezoelectric cantilevers inside the fuselage and piezoelectric patches on the wing spars for energy generation during the flight was discussed recently. More recently, it was suggested to use classical piezoelectric benders as landing gears, which may not be the best configuration as a landing gear. Here, we suggest using L-shaped harvesters as landing gears, which can be excited to generate electricity for charging the UAV’s batteries when it lands on a vibrating surface during the mission. The L-shaped landing gears with piezoceramic layers must be designed according to the frequency content of ambient sources available for landing in a prescribed mission.

ACKNOWLEDGMENTS Funding for this work is administered by the Center of Intelligent Material Systems and Structures, on behalf of the Air Force Office of Scientific Research MURI under grant FA9550-06-1-0143“Energy Harvesting and Storage Systems for Future Air Force Vehicles” monitored by Dr. B.L. Lee.

References [1] C. Williams and R. Yates, “Analysis of a micro-electric generator for microsystems,” Sensors and Actuators A 52, pp. 8–11, 1996. [2] H. Sodano, D. Inman, and G. Park, “A review of power harvesting from vibration using piezoelectric materials,” The Shock and Vibration Digest 36, pp. 197–205, 2004. [3] D. Arnold, “Review of microscale magnetic power generation,” IEEE Transactions on Magnetics 43, pp. 3940–3951, 2007. [4] P. Mitcheson, P. Miao, B. Start, E. Yeatman, A. Holmes, and T. Green, “MEMS electrostatic micro-power generator for low frequency operation,” Sensors and Actuators A 115, pp. 523–529, 2004. [5] S. Anton and H. Sodano, “A review of power harvesting using piezoelectric materials (2003)-(2006),” Smart Materials and Structures 16, pp. R1–R21, 2007.

Proc. of SPIE Vol. 6928 69280I-13

[6] S. Priya, “Advances in energy harvesting using low profile piezoelectric transducers,” Journal of Electroceramics 19, pp. 167–184, 2007. [7] S. Roundy, P. Wright, and J. Rabaey, “A study of low level vibrations as a power source for wireless sensor nodes,” Computer Communications 26, pp. 1131–1144, 2003. [8] H. Sodano, G. Park, and D. Inman, “Estimation of electric charge output for piezoelectric energy harvesting,” Strain 40, pp. 49–58, 2004. [9] N. duToit, B. Wardle, and S. Kim, “Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters,” Journal of Integrated Ferroelectrics 71, pp. 121–160, 2005. [10] A. Erturk and D. Inman, “A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters,” Journal of Vibration and Acoustics, Transactions of ASME, in press , 2008. [11] J. Baker, S. Roundy, and P. Wright, “Alternative geometries for increasing power density in vibration energy scavenging for wireless sensor networks,” in Proceedings of 3rd International Energy Conversion Engineering Conference, (San Francisco, CA), August 2005. [12] A. Erturk, J. Farmer, and D. Inman, “Effect of strain nodes and electrode configuration on piezoelectric energy harvesting from cantilevered beams,” Journal of Vibration and Acoustics, Transactions of ASME, in review , 2008. [13] Y. Hu, H. Xue, and H. Hu, “A piezoelectric power harvester with adjustable frequency through axial preloads,” Smart Materials and Structures 16, pp. 1961–1966, 2007. [14] A. Haddow, A. Barr, and D. Mook, “Theoretical and experimental study of modal interaction in a twodegree-of-freedom structure,” Journal of Sound and Vibration 97(3), pp. 451–473, 1984. [15] B. Balachandran and A. Nayfeh, “Nonlinear motions of beam-mass structure,” Nonlinear Dynamics 1, pp. 39–61, January 1990. [16] A. Nayfeh and D. Mook, Nonlinear Oscillations, Pure & Applied Mathematics, Wiley, 1979. [17] S. Oueini, A. Nayfeh, and J. Pratt, “A nonlinear vibration absorber for flexible structures,” Nonlinear Dynamics 15, pp. 259–282, 1998. [18] A. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, Pearson Prentice Hall, 3 ed., 2007. [19] R. Guyan, “Reduction of stiffness and mass matrices,” AIAA Journal 3(2), p. 380, 1965. [20] H. Sodano, D. Inman, and G. Park, “Generation and storage of electricity from power harvesting devices,” Journal of Intelligent Material Systems and Structures 16, pp. 67–75, 2005. [21] A. Erturk and D. Inman, “Analytical modeling of cantilevered piezoelectric harvesters for transverese and longitudinal base motions,” in Proceedings of 16th AIAA/ASME/AHS Adaptive Structures Conference, (Schaumburg, IL), April 2008. [22] S. Rao, Vibration of Continuous Systems, Wiley, February 2007. [23] A. Erturk and D. Inman, “On mechanical modeling of cantilevered piezoelectric vibration energy harvesters,” Journal of Intelligent Material Systems and Structures, in press , 2008. [24] R. Gallington, H. Merman, J. Entzminger, M. Francis, P. Palmore, and J. Stratakes, “Unmanned aerial vehicles,” in Future Aeronautical and Space Systems, A. Noor, ed., 72, ch. 6, AIAA, January 1996. [25] D. Pines and F. Bohorquez, “Challenges facing future micro-air-vehicle development,” AIAA Journal of Aircraft 43, pp. 290–305, 2006.

Proc. of SPIE Vol. 6928 69280I-14

[26] J. Fleming, W. Ng, and S. Ghamaty, “Thermoelectric-based power system fir UAV/MAV applications,” in Proceedings of 1st Technical Conference and Workshop in Unmanned Aerospace Vehicles, (Portsmouth, VA), May 2002. [27] M. Qidawi, J. Thomas, and J. Kellogg, “Expanding mission capabilities of unmanned systems through the collection of energy in the field,” in Proceedings of the AIAA 3rd International Energy Conversion Engineering Conference, (San Francisco, CA), August 2005. [28] A. Erturk, S. Anton, and D. Inman, “Energy harvesting from rigid body motions,” in Proceedings of the 18th International Conference of Adaptive Structures and Technologies, (Ottawa, Ontario, Canada), October 2007. [29] S. Anton, A. Erturk, and D. Inman, “Energy harvesting from small unmanned air vehicles,” in Proceedings of the 17th International Symposium on Application of Ferroelectrics, 3rd Annual Energy Harvesting Workshop, (Santa Fe, NM), February 2008. [30] K. Magoteaux, “Investigation of an energy harvesting small unmanned air vehicle,” Master’s thesis, University of Dayton, September 2007.

Proc. of SPIE Vol. 6928 69280I-15