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Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems
Journal of Sound and Vibration 330 (2011) 5583–5597
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Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems M. Amin Karami a,n, Daniel J. Inman b a b
Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, United States Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, United States
a r t i c l e i n f o
abstract
Article history: Received 23 June 2010 Received in revised form 3 May 2011 Accepted 27 June 2011 Handling Editor: K. Worden Available online 30 July 2011
A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Energy harvesting refers to scavenging small amounts of power from the ambient energy in the environment. This paper focuses on energy harvesting from vibrations. Such ambient energy can come from bridge vibrations, tire motion or the human heart beating. The energy can power up sensor nodes and therefore reduce the wiring complications or eliminate the frequent need of changing batteries. For more information on general energy harvesting the reader may refer to Refs. [1–6]. During the past two years nonlinear energy harvesting has received substantial attention. The main advantage of nonlinear energy harvesters over their linear counterparts is that the nonlinear harvesters scavenge energy over a broader frequency range of vibrations. Vibrational energy harvesters typically have low damping ratios to maximize the harvested power at resonance. The low damping ratio however, makes the harvesting device highly frequency sensitive (Fig. 1a). As depicted in Fig. 1b the presence of nonlinearity makes the peak of frequency response function lean toward higher or lower frequencies (this depends on the type of nonlinearity). The nonlinear energy harvester therefore acts over a wider
n
Corresponding author. E-mail addresses: [email protected] (M.A. Karami), [email protected] (D.J. Inman).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.06.021
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
legend is base acc. ms-2
1
0.1 0.4 0.7 1
end def.
0.8 0.6
legend is base acc. ms-2
1
0.1 0.4 0.7 1
0.8 end def.
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0.4 0.2
0.6 0.4 0.2
0
0 -1
0
1 2 Frequency (Hz)
3
-1
0
1 2 Frequency (Hz)
3
Fig. 1. Frequency response: (a) linear and (b) nonlinear.
range of base excitation frequencies. Fig. 1b shows the behavior of softly nonlinear, mono-stable Duffing oscillator and is only one example of possible nonlinear scenarios. The nonlinearity can be natural (for example the nonlinear material properties of the piezoelectric substance [7] ) or can be synthetic. If in addition to the lateral direction the beam is excited longitudinally, the governing equation of the system includes some nonlinear expression in the form of parametric excitation [8]. The most common mechanism of making the beam nonlinear is by placement of permanent magnets [9–17]. After modeling their systems and deriving the nonlinear governing equations most of these researchers have used numerical or experimental methods to solve the governing equations. Among the mentioned literature on magnetically nonlinear harvesters only Ref. [15] uses analytical perturbation methods, where they simplify the electro-mechanical interactions by considering an electro-mechanical damping. They next analyze the nonlinear vibrations of the single-degree-of-freedom mechanical system. The nonlinear vibrations problems are complicated in nature. Since the principle of superposition does not hold for nonlinear systems the concept of transfer function is invalid. In addition the response of nonlinear oscillators is sensitive to initial conditions, especially when the system is close to bifurcation points. If the system undergoes bifurcations, a slight change in parameters of the system or a slight alteration of the level or frequency of excitations could change the entire nature of the resulting oscillations. In nonlinear energy harvesting problems the situation is even more complicated. The nonlinear mechanical system is coupled with one or more electrical subsystems and together they form a system of nonlinear equations. The entire system has to be solved to account for the interactions between mechanical and electrical subsystems. In some situations the coupling can significantly lengthen the investigation of the nonlinear energy scavengers. Our experience shows that, for example, extracting a chaotic strange attractor of a coupled hybrid nonlinear energy harvester is ten time more numerically intensive than finding the strange attractor of its mechanical subsystem. This is in spite the fact that the interesting phenomena and the complicated behavior of nonlinear energy harvesting systems are rooted in their mechanical subsystems. The presented reduced-order modeling approach can fundamentally simplify analysis of energy harvesting systems. It allows breaking down the electro-mechanical system of equations and facilitates solving the portions individually. In the first step we investigate the backwards coupling phenomenon (the effect of electro-mechanical coupling on mechanical vibrations). We show that the backward coupling effect can be interpreted as some extra damping together with variations of the frequency base excitations. Before solving the system we project the backward coupling effect by altering the damping and frequency of the excitations. We only solve the equivalent mechanical system and evaluate the mechanical vibrations. Due to coupling mechanisms, the mechanical vibrations excite the electrical systems. In the second step we use the calculated mechanical response and solve the electrical equations to evaluate the power output. The proposed approximation method is accurate and at the same time significantly reduces the computation time. The paper is organized as following. To arrive at governing equations we investigate a nonlinear hybrid energy harvester. The configuration is novel at the same time since it is a hybrid system it includes the equations corresponding to piezoelectric mechanism as well as the formula associated with the electromagnetic harvesting. We use the novel hybrid design for the case studies further down the paper. Next we look into mono-stable harvesters which comprise linear vibrational harvester and softly nonlinear energy scavengers. The perturbation methods are valid for these configurations and we use them to derive the approximation methods. A case study on linear harvesting and another on softly nonlinear systems showcase the use of approximation method. The bi-stable systems are considered next. The vibrations of such systems can be chaotic or exhibit Limit Cycle Oscillations (LCO) and thus have a complicated form. Before deriving the approximation technique, we first characterize the complex oscillations by the underlying periodic orbits of the system. Since the amplitude of motion in bi-stable systems can be large, the perturbation techniques are not valid. The approximation method for bi-stable systems is guessed based on the perturbation analysis and then validated against numerical solutions. The advantage of approximation technique is that it is a unified method for linear, softly nonlinear and bi-stable harvesting systems.
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2. Governing equations of the hybrid case study and common energy harvesting systems To illustrate the general form of governing differential equations of energy harvesting systems, we present our proposed nonlinear hybrid energy harvesters. The hybrid nature of the nonlinear harvesting device proposed here is illustrated in Fig. 2. We use magnetic forces in our system to induce nonlinear behavior. The magnetic force between the tip and base magnets is repulsive and therefore counteracts the elastic restoring force. The existence of nonlinear forces acting on the beam introduces nonlinear hardening terms, which are explained in Section 2. The piezoelectric element, bounded to the beam, harvests energy from beam deflection. As a novel approach we have placed electromagnetic coils in the system. When the beam vibrates the magnetic tip mass passes by the coils and generates additional electricity. The system is a hybrid energy harvester in the sense that, it uses two different methods (piezoelectric and electromagnetic transduction) for power harvesting. We use the energy methods to model the dynamics of the system. The harvester is made up of three coupled systems; the cantilever beam which is characterized by the deflection of beam, the circuit connected to the piezoelectric element and the circuit connected to the coils. The electrical circuit for harvesting is simplified to be only a resistor in order to focus on the transduction. The value of the resistive load in the piezoelectric circuit is R1 and the value of the resistive load in the electromagnetic circuit is R2. The energy in various components of the entire electro-mechanical system is: the elastic and magnetic potential energies stored in the beam and the magnetic field (V), the electrostatic energy stored in the piezoelectric patch (We), the kinetic energy stored in the beam and the tip mass (T), the magnetic potential energy between the tip and base magnets (G), the electromagnetic energy stored in the coils (Wm) and the energies dissipated by the resistors and damping of the beam. The displacement of the beam, the flux linkage across the piezoelectric element and the charge through the coils are the coordinates used for identifying the system. Following the approach in Ref. [18], the Lagrangian of the system is Lg ¼ TV þ We þ Wm
(1)
The continuous vibrations of the beam are simplified by only considering the dynamics of its first mode. The deflection at each point and at a certain time relative to the base is w(x,t)¼ f(x)u(t). The approach is a common practice in study of nonlinear vibrations of bi-stable structures [9] and is justified by center manifold reduction [19]. The assumed mode shape is the first analytically driven mode shape of a composite beam with varying cross-section and a tip mass. The following integrals are defined to facilitate abbreviation of formulas: Z L Z L1 Z L Z L1 FL20 ¼ fðxÞ2 dx, FL201 ¼ fðxÞ2 dx, FL22 ¼ f00 ðxÞ2 dx, FL221 ¼ f00 ðxÞ2 dx (2a) 0
0
FL10 ¼
Z 0
L
fðxÞ dx, FL101 ¼
0
0
Z
L1 0
fðxÞ dx, FL121 ¼
Z
L1
f00 ðxÞ dx
(2b)
0
Each of the terms in Lagrangian are related to the states as follows: ZZZ 1 1 We V ¼ cij Si Sj þ eij Ei Sj þ eij Ei Ej Gðwend ðtÞÞ 2 2 vol
(3)
where cij are the stiffness coefficients, eij are the piezoelectric constants [20], Ei are the electric field components and G(wend(t)) is the magnetic force potential. The Magnetic force is experimentally measured and is characterized as
Fig. 2. Schematic view of the harvester, R1 and R2 represent piezoelectric and electromagnetic loads accordingly.
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f ¼ awend þ bw3end . The magnetic force potential is therefore: Gðwend Þ ¼ ða=2Þw2end þ ðb=4Þw4end . We let Ys denote Young’s modulus of the steel substrate, Yp Young’s modulus of the piezoelectric patch, Is the area moment of inertia of the steel beam about its geometric center and Ip stand for the area moment of inertial of the cross-section of each piezoelectric patch about the center line of the steel substructure. Eq. (3) is simplified to ( ) 1 2Ap zp FL22 e13 _ e AL bfðLÞ4 afðLÞ2 1 L1 L 4 uðtÞ þ Ys Is F22 þ Yp Ip F22 uðtÞ2 þ l 1 ðtÞuðtÞ þ 33 2p 1 l_ 1 ðtÞ2 (4) We V ¼ 2 4 2 hp hp In Eq. (4), l1(t) is the flux linkage across the piezoelectric patch, Ap is the cross-sectional area and zp is the z-coordinate of the centroid of the patch. The z and x coordinates have been defined in Fig. 2. The base motion, characterized by r(t), should be taken into account when calculating the kinetic energy. The kinetic energy is evaluated as h i 1 1 1 _ r_ ðtÞ þ Mtip þ Ms þ 2Mp r_ ðtÞ2 _ 2 þ rs As FL10 þ 2rp Ap FL101 þ Mtip fðLÞ uðtÞ T¼ rs As FL20 þ rp Ap FL201 þ Mtip fðLÞ2 uðtÞ (5) 2 2 2 The densities of the steel substrate and the piezoelectric patch are rs and rp, respectively. The total mass of the substrate and each of the piezoelectric patches are Ms and Mp. The cross-sectional area of the substrate is rs and Mtip stands for the mass of the tip magnet. When the tip magnet passes by the coils some electromagnetic energy conversion occurs. The electromagnetic coupling can be characterized by the coupling coefficient, Tm. _ end , a force of magnitude Tmi2 impedes the motion of tip When the tip magnet passes by the coils with the velocity w magnet. The current in the coils is i2. At the same time a potential difference is generated across the coil which equals _ end . The charge passing through the coils is noted by q2 and the overall inductance of the coils is l. The following two Tm w terms in the Lagrangian represent the electro-mechanical energy in the coils: n V ¼ Wm
1 lq_ 2 þ Tm q_ 2 fðLÞu 2 2
(6)
The Euler–Lagrange equations for our three-degrees-of-freedom system is 8 @L d @Lg > @ug ¼ cf u_ > > dt @u_ > > < @L @L _ g d @lg1 ¼ lR11 dt @l_ 1 > > > d @Lg @L > > : dt @q_ 2 @qg2 ¼ R2 q_ 2
(7)
The damping coefficient of the mechanical spring is denoted by cf. Performing the derivatives in Eq. (7), dividing by the modal mass and grouping the terms results 8 ~ 3 ¼ du ~ V þ g~ q_ m _ c ^ r€ ðtÞ u€ þku þ bu > 2 1 1 > < c~ 2 _ V1 _ (8) V 1 þ R1 C0 ¼ C0 u > > : di2 R2 ~ _ þ l i2 ¼ g u dt The coefficients in Eq. (8) are ~ ¼ rs As FL20 þ 2rp Ap FL201 þ Mtip fðLÞ2 , m bfðLÞ4 b~ ¼ , ~ m
d¼
cf , ~ m
^ ¼ m
rs As FL10 þ 2rp Ap FL101 þ Mtip fðLÞ
c~ 1 ¼
^ m 1 2Ap zp FL12 e13 , ~ p mh
~ , ~c c~ 2 ¼ m 1
1 afðLÞ2 þ Ys Is FL22 þ 2Yp IP FL22 , ~ m 2e33 Ap L1 C0 ¼ , g~ ¼ Tm fðLÞ h2p
,
k¼
(9)
The second and third terms on the right-hand side of Eq. (8a) represent the backward coupling terms introduced by the piezoelectric patch and the electromagnetic coils. The energy transferred to the electric circuits reduces the mechanical energy of the beam and therefore suppresses its oscillations. The sign of the linear restoring coefficient, k, can be positive or negative. The familiar positive coefficient corresponds to low magnetic forces. In this situation the zero deflection equilibrium is stable and the system is a ‘‘nonlinear mono-stable oscillator’’ coupled to the piezoelectric and electromagnetic circuits. If the tip magnet is close to the base the repelling force between the magnets, which forces the tip away from the zero deflection, becomes significant. The u ¼0 equilibrium will be unstable but there will be two stable equilibriums on the left and right side of zero deflection (u¼ 7u*). In this situation the system is ‘‘nonlinear bi-stable oscillator’’ coupled to the piezoelectric and electromagnetic circuits. The nonlinear vibrations of the nonlinear bi-stable oscillator is discussed in Section 6. The form of Eq. (8) is typical of governing equations of energy harvesting systems. Typically the temporal parameters of the vibration modes are governed by second-order differential equations. The second-order equation includes base excitation terms that derive each mode, the nonlinear terms, and electro-mechanical coupling terms that relate the mechanical subsystem and each electrical subsystem. From this point on we address the governing equations in their general form (Eq. (8)) and references to the physical parameters are only made in the case studies. The piezoelectric and electromagnetic energy harvesters are both special cases of the hybrid harvesting system. For piezoelectric harvesters the
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~ coupling coefficient g~ vanishes and Eq. (8c) becomes irrelevant. In electromagnetic harvesters the coupling coefficient c 1 is zero and Eq. (8b) should be excluded. 3. Mono-stable oscillators The key feature of the governing equations of the nonlinear mono-stable oscillator is that the coefficient k in Eq. (8) is positive. If the nonlinearities pand ffiffiffi the couplings were not present, the spring would act as a simple harmonic oscillator and with natural frequency at on ¼ k. The frequency can be used in nondimensionalizing Eq. (8). The dimensionless time is t ¼ ont. ^ r€ ðrÞ ¼ 2F cosðotÞ and the equations of motion become: If the base excitations are harmonic the forcing term in Eq. (8) becomes m 8 2 d u du 3 > 2 þ e2z dt þ u ¼ ebu ec1 V1 þ eg1 i2 þ e2f cosðOtÞ > > < dt c2 du dV1 V1 þ R1 C0 on ¼ C0 dt (10) dt > > > : di2 þ R2 i2 ¼ g2 du dt lon l dt The following relates the coefficients in Eq. (10) to the coefficients in Eq. (8):
z¼
d , 2on
b¼
b~
on
,
c1 ¼
c~ 1
o2n
,
c2 ¼ c~ 2 , g1 ¼
g~ F , g2 ¼ g~ , f ¼ 2 o2n on
and
O¼
o on
The e is a small bookkeeping parameter. It has been assumed that the parameters z, b, c1 , g, f , c2 and R2 are small (of the order of 0.1 or less) and R1 is large (in the order of 10 or more). As a mid-step in deriving Eq. (10) from Eq. (8) the mentioned coefficients have been replaced by the following parameters to facilitate sorting small parameters from ordinary parameters: ez, eb, ec1 , eg, ef , ec2 , eR2 and R1 =e. All the terms in the second and third lines of Eq. (8) included an e which was crossed out. The method of multiple scales [21,22] is used to analyze the system in Eq. (10). The solution of Eq. (10) is expanded in terms of powers of e u ¼ u0 þ eu1 þ . . .,
V1 ¼ V10 þ eV11 þ. . .,
i2 ¼ i20 þ ei21 þ . . .
(11)
The solutions u0 ðtÞ, V10 ðtÞ and i20(t) are the solutions of Eq. (10) when e is set to zero. The change of displacement, voltage and current occur over different time scales, namely T0 ¼ t, T1 ¼ et,. . .. The time scale T1 is slower compared to the time scale T0 since the time is multiplied by the small parameter e and it takes more time for T1 to become notable. Each of the variables in Eq. (11) is assumed to be a function of the scales T0 and T1 rather than a function of t and e. We define D0 ¼ @=@T0 and D1 ¼ @=@T1 . The chain rule is used to re-evaluate the time derivatives in Eq. (10). This yields 8 > ðD2 þ 2eD0 D1 Þðu0 þ eu1 Þ þ 2ezD0 u0 þ u0 þ eu1 ¼ ebu30 ec1 V10 þ eg1 i20 þ e2f cosðOT0 Þ > > 0 < ðD0 þ eD1 ÞðV10 þ eV11 Þ þ VR101 Cþ0eoVn11 ¼ cC02 ðD0 þ eD1 Þðu0 þ eu1 Þ > > > : ðD þ eD Þði þ ei Þ þ R2 ði þ ei Þ ¼ g2 ðD þ eD Þðu þ eu Þ 0
1
20
21
lon
20
21
l
0
1
0
1
The above equations must hold for all values of e, therefore the coefficients of the like powers of e must satisfy the equations. This results two sets of relations corresponding to zeroth and first power of e. 8 > D20 u0 þu0 ¼ 0 > > < c2 V10 0 (12) Oðe Þ D0 V10 þ R1 C0 on ¼ C0 D0 u0 > > g > D i þ R2 i20 ¼ 2 D u : 0 20
lon
l
0 0
8 > D2 u þ u1 ¼ 2D0 D1 u0 2zD0 u0 bu30 c1 V10 þ g1 i20 þ 2f cosðOT0 Þ > > 0 1 < c2 V11 Oðe1 Þ D0 V11 þ R1 C0 on ¼ D1 V10 þ C0 ðD0 u1 þD1 u0 Þ > > > D i þ R2 i21 ¼ D i g2 ðD u þ D u Þ : 0 21
lon
1 20
l
0 1
(13)
1 0
The general solutions of the equations for zero power of e (Eq. (12)) are 8 > u ¼ AðT1 ÞejT0 þ AðT1 ÞejT0 > > 0 < c2 c2 V10 ¼ jC0 þ ð1=ðR AðT1 ÞjejT0 jC0 þ ð1=ðR AðT1 ÞjejT0 1 on ÞÞ 1 on ÞÞ > > g2 >i ¼ : AðT1 ÞjejT0 jl þðRg22=on Þ AðT1 ÞjejT0 Þ 20 jl þ ðR2 =on Þ
(14)
The coefficients of AðT1 Þ in Eq. (14) are independent of T0 but vary as T1 changes. Since the T1 is the slowly varying scale A(T1) change slowly in time. The complex conjugate of A is represented by A. The value of the A-coefficients is derived by eliminating the secular terms. The solution in Eq. (14) is replaced in the right-hand side of Eq. (14-1). The secular terms are those which are proportional to eiT0 , and they must sum up to zero otherwise the assumptions of small vibrations is violated [21] dA c1 c2 g1 g2 þ Aj þ f ejðOT0 T0 Þ ¼ 0 2zAj3bA2 A (15) 2j DT1 Co j þð1=ðR1 on ÞÞ lj þðR2 =on Þ
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The base oscillations are considered in the vicinity of the natural frequency. A detuning parameter (s) is defined which nondimentionalizes the difference between the excitation frequency and the linear natural frequency: O ¼1þ se. The complex number A can be written in polar coordinates: A ¼ ða=2Þejn . Both amplitude (a) and phase (n) are functions of T1. Since Eq. (15) is a complex equation it has two equations embedded which give the values for the two unknowns a and n. We define 1 c1 c2 g1 g2 þ ¼ G þ Zj 2 Co j þ ð1=ðR1 on ÞÞ ljþ ðR2 =on Þ In other words G represents the real part of the coupling expression and Z is its imaginary part. Also to have autonomous equations we define d ¼ sT1 n. Therefore Eq. (15) gives the following two equations corresponding to its imaginary and real parts: 8 da < dT þ ðz þ GÞa ¼ f sinðdÞ 1 (16) d d : a dT aðs þ ZÞ þ 38 ba3 ¼ f cosðdÞ 1 We are mostly interested in the steady-state response of the beam. For steady-state situation Eq. (16) becomes 8 2 < a20 ðz þ GÞ2 þ a0 ðs þ ZÞ 38 ba30 ¼ f 2 zþG : tan d ¼ s þ Zð3=8Þba 2
(17)
0
Note that to the first approximation the steady-state solution is 8 u0 ðtÞ ¼ a0 cosðotdÞ > > > > c2 > o a sinðotd þR c o Þ < V10 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 0 n C02 þ ð1=ðR21 o2n ÞÞ on > > g > 2 n ffi o a0 sin otd þ lo > > i20 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 : l2 þ ðR2 =o2 Þ on 2
(18)
n
Eq. (17) identifies a0 (the steady-state amplitude of oscillations) and d (the phase shift between the excitations and response). This is a useful expression for understanding the effect of the harvesting on the vibrations of the spring. Eq. (16) suggests that energy harvesting effects are equivalent to a change in the damping ratio of the structure and a change in the excitation frequency. The damping ratio changes by
G¼
1 c1 c2 R2 g1 g2 þ 2R1 on ð1=ðR21 o2n ÞÞC02 2on ðR22 =o2n Þl2
and the excitation frequency changes from o to
o þ Zon ¼ o þ on
C0 c1 c2 l g1 g2 þ 2 ð1=ðR21 o2n ÞÞC02 2 ðR22 =o2n Þl2
!
To better illustrate the implications of the approximation method, we perform the following two case studies. 4. Linear piezoelectric energy harvesting Linear piezoelectric energy harvesting is a special case of nonlinear harvesting and our solution in the previous section is also valid for linear harvesting. For linear piezoelectric systems, the coefficients b and g1 in Eq. (10) are zero and Eq. (10c) become irrelevant. The exact solution for linear piezoelectric energy harvesting exists in the literature [23]. In this section we compare the proposed the approximate solution with the exact to investigate the accuracy of the approximate method. At the same time we can see if the approximation facilitates either derivation or interpretation of the closed form solution for linear piezoelectric systems. Since the nonlinear coefficient, b, is zero, Eq. (17) simplifies to a0 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 ðz þ GÞ þ ðs þ ZÞ2 The equivalent exact formula for a single mode harvester is [23] a0 c1 ððjoc2 o2n Þ=ðo2 þ j2zoon þ o2n ÞÞ 2o2n ¼ 1 f ð1=R1 Þ þjoC0 þððjoc1 c2 o2n Þ=ðo2 þ j2zoon þ o2n ÞÞ o2 þj2zoon þ o2n
(19)
(20)
Comparison of the perturbation and exact solution has been illustrated in Fig. 3. The different lines correspond to different resistances. By changing the resistive load the resonance frequency gradually changes form the short circuit to open circuit value. The perturbation solution is almost indistinguishable from the exact solution. This is in spite of the fact that the approximate formula, Eq. (19), is much simpler and significantly more intuitive than the exact solution, Eq. (20).
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
90
5589
102
80
a/f
a/f
70 60 50 Analytical Perturbation
40
Analytical Perturbation
30
101 47
47.5
48 ω
48.5
49
42
44
46
48 ω
50
52
54
Fig. 3. Perturbation vs. exact solution: (a) about resonance and (b) big picture.
Table 1 Prototype specification. L (mm) hs (mm) b (mm) Substrate material Piezoelectric patches Length of the piezoelectric out of the clamp (mm) Inductance of the coils (mH) The electromagnetic coupling coefficient (Tm) Tip mass (g)
127 0.254 25.4 Spring steel Mide QP10n quick pack 38.1 347 0.081 48
Table 2 Parameters used for harmonic case study. b C0
2 108 93 nF
g1 g2
0.0013 6.5
c1 c2
4.7 10 8 0.077
z
on
0.01 70.6
5. Case study of nonlinear mono-stable harvester This section serves as an example for the analytical results derived in Section 3. The mechanical, piezoelectric and electromagnetic parameters of the system considered here correspond to the experimental setup discussed in Ref. [24]. Table 1 lists the design specifications of the prototype. As a parametric study we are curious to see how the system behaves for different base excitations and at different R1 and R2 resistances. The gap between the tip and base magnets are set such that the beam is still mono-stable but there is some hardening nonlinearity acting on the beam. Eq. (10) coefficients used for the case study have been summarized in Table 2. These coefficients are based on the prototype in Ref. [25] when the magnets distance is 50 mm. Fig. 4a–c illustrates that at small base excitations the system acts like a linear oscillator but for moderate and large base excitations, the nonlinear effects become dominant. For those levels of excitation the tip of the displacement vs. base excitation curves, lean to the right side (towards higher frequencies). This behavior is typical of nonlinear hardening springs [22]. As a result the oscillator vibrates over a broader range of frequency. As can be seen in Fig. 4b and c the power generation is less sensitive to the base excitation frequency. Like nonlinear oscillators there are two stable values for the amplitude of oscillation if the base excitation is larger than the natural frequency. At such frequencies, the behavior of the system depends on the initial conditions and a sudden force can make the system jump between its two stable equilibriums. Fig. 4d illustrates that changing R1 acts similar to changing the damping coefficient and changing the excitation frequency of the effective system. It shows that the optimal resistance corresponds to the smallest amplitude of response or equivalently largest electro-mechanical damping. For the system examined here, the optimal resistance is close to 200 kO. A similar argument is valid for the electromagnetic power generation, but as Fig. 4g–i proposes the optimal value of the resistance is too low. In fact, since the optimal resistance is lower that the coil wiring resistance (5 O), it cannot be experimentally achieved. The optimal electromagnetic shunt resistance (R2) is equal to the equivalent resistance of the coils. 6. Nonlinear bi-stable energy harvester When the repelling magnetic force is sufficiently large and makes the beam’s zero deflection position unstable, the energy harvester is called bi-stable. The vibrations of bi-stable harvesters can take complicated forms. In this section we
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
2.5 2 1.5 1 0.5 0
1
0.5
10
11 12 13 Frequency (Hz)
14
x 10-3
3
piezo power watt
4 end def. (m)
1.5
3 13 50 200 794 3162
2 1 0 10 11 12 13 14 Base Excit. Freq. (Hz)
x 10-3
11 12 13 Frequency (Hz)
piezo power watt
2 1.5 1 0.5 0 10 11 12 13 Base Excit. Freq. (Hz)
3 2 1 9
3.5
10
11 12 13 Frequency (Hz)
14
x 10-5 3 13 50 200 794 3162
3 2.5 2 1.5 1 0.5 0
10
x 10-3
11 12 13 14 Frequency (Hz)
15
9
10
legend is R2 (Ω)
11 12 13 14 Frequency (Hz)
15
legend is R2 (Ω) -6 7 x 10 2 3.5 5 6.5 8
1
2 3.5 5 6.5 8
6 5 4 3
0.5
2 1
0 9
4
legend is R1 (kΩ)
3 13 50 200 794 3162
9
1.5 2 3.5 5 6.5 8
0.1 0.4 0.7 1
5
14
0.5
15
6
legend is R1(kΩ)
x 10-3
1
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0 9
10 11 12 13 Base Excit. Freq. (Hz)
14
9
10 11 12 13 Base Excit. Freq. (Hz)
14
Fig. 4. Mono-stable parameter study: (a) mechanical vibration in relation to base acceleration; (b) piezo-power in relation to base acceleration; (c) electromagnetic power in relation to base acceleration; (d) mechanical vibration in relation to R1; (e) piezo-power in relation to R1; (f) electromagnetic power in relation to R1; (g) mechanical vibration in relation to R2; (h) piezo-power in relation to R2 and (i) electromagnetic power in relation to R2.
first tailor the governing equations for the bi-stable case. We next look into stable and unstable periodic orbits of the harvester to characterize its complicated vibrations. The first step in analyzing the bi-stable oscillator is nondimensionalizing the equations of motion. The k coefficient in Eq. (8) pffiffiffiffiffiffiffiffiffiffi is negative, we define o0 ¼ 2k and use the following change of time and deflection variables [19] to nondimensionalize qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ r€ ðtÞ ¼ F cosðotÞ: Eq. (8): t ¼ o0t and u ¼ y 2k=b~ . If the base excitations are harmonic the forcing term in Eq. (8) becomes m 8 2 d y > 1 y þ y3 ¼ 2z dy fV1 þ gi2 þ f cosðOtÞ > > dt < dt2 2 dy dV1 þ a1 V1 ¼ b1 dt dt > > > : di2 þ a2 i2 ¼ b dy
(21)
2 dt
dt
The following relates the coefficients in Eq. (21) to the coefficients in Eq. (8):
z¼
d , 2o0
f¼
c~ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
g¼
g~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
f¼
F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
a1 ¼
1 RC0 o0
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
b1 ¼
c~ 2 C0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ , ðð2kÞ=bÞ
a2 ¼
R2 lo0
and
b2 ¼
g~ l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ : ðð2kÞ=bÞ
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(22)
The system in Eq. (21) is the coupled single mode model of the nonlinear oscillator. The nonlinear bi-stable coupled oscillator shows three behaviors based on the level of excitations. If the excitation is too low or too large the response of the system is periodic. The mid-range excitations result in chaotic motion which is unpredictable. We can undoubtedly characterize the periodic motions of the oscillator by its periodic orbits. Later in this section we show even when the vibrations are chaotic, the unstable periodic orbits are correlated with the shape of the strange attractors. We therefore use the periodic orbits to characterize the vibrations of the bi-stable system in all vibration regimes. The shooting method is a numerical method for finding the periodic solutions of autonomous and non-autonomous systems [26]. To find the limit cycles of the non-autonomous system _ ¼ FðX,tÞ X
(23)
we start with guessing a point on the phase space (Z0) and see if that point is on a periodic orbit. In Eq. (23) X is the state vector containing the position, velocity, voltage and current values. Since the system is force driven the period of the cyclic motion is equal to the period of the forcing or its multiples [26]. If the period is noted by t0, we numerically integrate Eq. (23) over one period. If Xðt0 , g0 Þ ¼ g0 that means g0 has indeed been on a periodic orbit. Otherwise we correct the initial guess using Newton–Raphson method. The correction in initial guess (dg) is calculated from @X ðt0 , g0 ÞI dg ¼ g0 Xðt0 , g0 Þ (24) @Z To be able to use Eq. (24), we need to find @X=@Zby numerically integrating the following system over period t0: d @X @X @X , ð0Þ ¼ I ¼ Dx F dt @g @g @g
(25)
Numerically integrating Eq. (25) would give the monodromy matrix as a by-product
U¼
@X ðt 0 Þ @g
(26)
The stability of each periodic orbit is determined by the eigenvalues of the monodromy matrix. The Floquet multipliers of the system are the eigenvalues of the monodromy matrix [26]. If any of the eigenvalues of U is outside the unit circle on complex plane, the periodic orbit is unstable. In contrast if all the eigenvalues of U are within the unit circle, the periodic orbit is stable. For the nonlinear harvesting system (Eq. (21)) the X and F vectors are defined as 2 3 2 3 y X2 6 dy 7 6 X1 X3 2zX fX þ gX þ f cosðOtÞ 7 2 3 4 6 dt 7 62 7 1 7 6 7 (27) X¼6 6 V1 7 and F ¼ 6 7 a X þ b X 1 3 2 1 4 5 4 5 a2 X4 b2 X2 i2 To solve Eq. (21) we use many initial conditions spread over the phase space to identify all the stable and unstable periodic orbits.
7. Periodic orbits as land marks in Poincare’ maps In this section we look at the Poincare map of the vibrations in addition to the periodic orbits. The dynamic system (Eq. (21)) is non-autonomous and the period of the forcing function is t0 ¼ 2p=O. To get the Poincare map we read the X-vector, defined in Eq. (27), every t0. If the vibrations are periodic the Poincare map will be a single point in phase space. The Poincare map of a chaotic vibration will be its strange attractor [26]. We use both the shooting method and the Poincare map to analyze the vibrations. The shooting method gives the stable and unstable limit cycles, which give structure to the dynamic behavior of the system. The existence of a stable limit cycle however, does not mean that all the initial conditions evolve into a periodic motion. The Poincare map identifies what the dynamic behavior is based on the initial conditions. In plotting the Poincare maps we start at different initial conditions. If the trajectory ends in a periodic orbit, its Poincare section will be a point in phase space. We present this trajectory by a thin line connecting the initial condition to the Poincare point. Otherwise, if the initial condition gives rise to a quasiperiodic or chaotic trajectory, we show the entire Poincare map as a collection of points. In this way the periodic and nonperiodic trajectories can be visually distinguished and the basin of attractions can be approximately illustrated. The same prototype discussed in Section 5 can demonstrate bi-stable motion. For this purpose the gap between the tip and base magnets is reduced to 25 mm. The shorter gap strengthens the repulsive magnetic force and therefore makes the no deflection position unstable. The parameters of Eq. (21) in the case study, estimated from the prototype, have been summarized in Table 3.
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Table 3 Parameters used for bi-stable case study.
a1 a2
9.6 2.6 103
6.7 10–4 0.024
g
3.1 0.32
b1 b2
f
z k
0.03 62.8
The oversimplified version of the energy harvester where the backward couplings have been neglected is the forced, damped Duffing oscillator: 2
d y 1 dy y þy3 ¼ 2z þ f cosðOtÞ dt dt2 2
(28)
The vibration of the simplified system (illustrated in Fig. 5a) is purely mechanical. Fig. 5b shows the bifurcation of the coupled system identified by Eq. (21). The phase space of this system is four dimensional but we have only plotted y and y_ for illustration purposes. It can be seen from Fig. 5 that the vibrations of the system changes from oscillations about each of the static equilibriums to chaotic motion and then to a large limit cycle circling both static equilibriums. They also show that multiple types of oscillations can exist for a single base acceleration. For example for the base acceleration of 0.5 m s 2 the forced damped Duffing system can vibrate about one of the static equilibriums, it can be chaotic, or it can perform large Limit Cycle Oscillations. Comparison of the responses of forced damped Duffing system and the coupled harvesting system shows that energy harvesting notably affects the mechanical response. The different types of vibrations were also demonstrated. We choose the periodic orbits (calculated using shooting method) as the characteristic of the vibrations. The periodic orbits perfectly resemble the steady-state motion in small amplitude vibrations and limit cycle oscillations. In chaotic vibrations, although the periodic orbits are unstable, they are the back bones of the strange attractor and can be used as a simple method to quantify the oscillations. 8. Approximate solution for oscillations of the bi-stable system Before we proceed we clarify our terminology. The electro-mechanical energy harvesting system in its exact form (Eq. (21)) is a ‘‘coupled system’’. Neglecting the coupling terms ( fV1 þ gi2) in the mechanical system results in the purely mechanical ‘‘forced damped Duffing system’’ Eq. (28). The response of the forced damped Duffing system is different from the mechanical response of the coupled system due to the role of backward coupling. We breakdown the coupled system and first solve the ‘‘equivalent mechanical system’’, which is the mechanical subsystem modified to account for backwards coupling. The response of the equivalent mechanical system should resemble the mechanical response of the coupled system. In the second step we derive the power outputs of each harvesting circuit based on the response of equivalent mechanical system. We first start with small amplitude oscillations of the bi-stable nonlinear harvesters. Since the amplitude of vibrations are small we can use perturbation methods to derive the approximate system. pffiffiffi First we shift the origin of our system to thepstatic equilibrium: x ¼ yð1= 2Þ. By scaling the variables as x-ex, z-e2 z, ffiffiffi 2 2 2 f-e f, g-e g, f -e f and by defining y2 ¼ 3 2=2, y3 ¼1 we obtain 8 2 d x > þ x þ ey2 x2 þ e2 y3 x3 ¼ e2 2z ddxt e2 fV1 þ e2 gi2 þ e2 f cosðOtÞ > > < d t2 dV1 þ a1 V1 ¼ b1 ddxt (29) dt > > > di : 2 þ a i ¼ b dx dt
2 2
2 dt
The general form of Eq. (29) is similar to the form of Eq. (10). The main difference between the two systems is the presence of quadratic nonlinearities in the former. Due to the second-order nonlinearity we use ‘‘second-order’’ method of multiple scales [21,22]. We expand the variables as x ¼ x0 þ ex1 þ e2 x2 þ . . .,
V1 ¼ V10 þ eV11 þ e2 V12 þ . . ., T0 ¼ t,
T1 ¼ et,
i2 ¼ i20 þ ei21 þ e2 i22 þ. . .
T2 ¼ e2 t,. . .
After performing the chain rule (similar to Section 3) the terms with the same power of e are grouped 8 2 > < D0 x0 þ x0 ¼ 0 Oðe0 Þ D0 V10 þ a1 V0 ¼ b1 D0 x0 > : D i þ a i ¼ b D x 0 20
2 0
2
(30) (31)
(32)
0 0
OðeÞ : D20 x1 þx1 ¼ 2D0 D1 x0 y2 x20
(33)
Oðe2 Þ : D20 x2 þ x2 ¼ 2D0 D1 x1 D21 x0 2D0 D2 x0 y2 2x0 x1 2zD0 x0 y3 x30 fV10 þ gi20 þ f cosðOT0 Þ
(34)
We first solve Eq. (32) for x0, V10 and i20. Then, we substitute the solution in Eq. (33), eliminate the secular terms and find its homogeneous solution. Next, the results of the previous two steps are substituted in Eq. (34) and the secular terms are
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forced damped Duffing system
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1 0.5 dy/dτ
0.5 dy/dτ
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0
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-1 -2
-1
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1
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2 1
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-1
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1
2
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1
coupled system
1.5
dy/dτ
-0.5
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2
-1
0 y
1
2
Fig. 5. Bifurcations of uncoupled (a) and coupled (b) systems subject to (1) 0.1, (2) 0.5, (3) 1, (4) 1.5 and (5) 2 m s 2 base acceleration.
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set to zero. After introducing the detuning parameter (s ¼ O 1) , putting the secular terms equation in polar coordinates and making the system autonomous by defining d ¼ sT2 b, we get the following expression for x(t): a2 a2 (35) xðtÞ ¼ a0 cosðOtdÞey2 0 cosð2Ot2dÞ þ 0 6 2 The steady-state amplitude factor (a) and phase shift (d) are calculated from 8 < 0 ¼ ðz þ GÞa0 þ 2f sinðdÞ 2 : 0 ¼ 3y3 ð10=3Þy2 a30 þ a ðs þ ZÞ þ f cosðdÞ 0 2 8
(36)
The coefficients G and Z are defined similar to Section 3 as 1 b1 f b g G þ jZ ¼ þ 2 2 j þ a1 j þ a2
(37)
The steady-state solution, Eq. (35) contains harmonic oscillations as well as double frequency and constant terms. The latter two terms are due to the existence of second-order nonlinearity. Although the vibrations about the static equilibrium of bi-stable oscillator are different from the oscillations of mono-stable Duffing system, the effects of couplings are similar for both cases. The mechanical vibrations of the hybrid energy harvesting system is equivalent to the mechanical vibrations of forced damped Duffing spring with modified damping and excitation frequency. The damping of the equivalent mechanical system is that of the coupled harvester plus G. The excitation frequency of the equivalent mechanical system is the excitation frequency of the electro-mechanical device plus Z. This correspondence is the same so far for linear, softly nonlinear (mono-stable) and small amplitude bi-stable energy harvesters. The method of multiple scales (MMS) is an approximate method and is not accurate for large vibrations of the harvesting system (Fig. 6). Therefore, for the majority of bi-stable oscillations, although the perturbation solution suggests the approximation of the coupled system by the equivalent system, it cannot prove the exact correspondence. We, therefore, consider the suggested equivalent system as a ‘‘hypothesis’’. We check the validity of this hypothesis in the next section using numerical methods. We return to the case study investigated in Section 7. We have previously compared the coupled system and the oversimplified forced damped Duffing system and we saw that they are notably different. In this case study we compare the derived equivalent mechanical system with the other two and check if the approximate system resembles the exact coupled system. The results are summarized in Fig. 7. The periodic orbits of the equivalent mechanical system are almost indistinguishable from the periodic orbits of the coupled system. This is valid for small amplitude oscillations, chaotic vibrations, and limit cycle oscillations. The case study clearly validates the accuracy of the approximation method for all types of vibrations of the bi-stable energy harvesting system. 9. Case study of bi-stable harvester Thus far the analysis of the bi-stable energy harvesting system was focused on its mechanical part. In this section we investigate how the bifurcations affect the power harvesting. As a case study we consider the system described in Table 3. The shunt resistances are R1 ¼ 200 kO and R2 ¼10 O. The piezoelectric and electromagnetic harvested powers have been
0.6 0.4
dy/dτ
0.2 0 -0.2 -0.4 -0.6 0
0.2
0.4
0.6
0.8
1
y Fig. 6. Comparison of MMS results (rigid line) with numerical solution (dashed). The periodic orbits correspond to base acceleration of 0.1, 0.45, 0.8, 1.15 and 1.5 m s 2. The larger the base acceleration the larger the limit cycle is.
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
0.2
1.5
Froced Damped System
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Froced Damped System
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0.5 dy/dτ
0 -0.5
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0 -0.5
0 -0.5
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-1
Coupled System and Approximate
-1
-1.5
-1.5 -2
-1
0 y
1
2
-2
-1
0 y
1.5
1.5 1
1
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Froced Damped System
0.5 dy/dτ
0.5 dy/dτ
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0 -0.5
-0.5 Coupled System and Approximate
-1 -1.5 -2
-1
0 y
Coupled System and Approximate
-1 -1.5 1
2
-2
-1
0 y
Fig. 7. Limit cycles, difference between coupled and uncoupled systems and difference between coupled and equivalent systems. The limit cycles correspond to base acceleration of (a) 0.1, (b) 0.6, (c) 1, (d) 1.5, (e) 2 and (f) 3 m s 2. The thin lines represent the unstable periodic orbits.
summarized in Figs. 8 and 9. The powers correspond to the limit cycles calculated by the shooting method. The lower curves correspond to small limit cycles about each of the equilibriums, which are stable for small base excitations but become unstable for large excitations. The upper power curves correspond to the large limit cycle circling both static equilibriums. This limit cycle is only stable for large base accelerations. The figures show that the power harvested from the large limit cycle is more than an order of magnitude larger than the power from the smaller limit cycles. Figs. 8 and 9 only illustrate periodic solutions and do not show the chaotic behavior of the harvester. In reality the harvester can show chaotic behaviors if the acceleration level is between 0.7 and 1.7 m s 2.Generally the power produced in the chaotic regime is larger than the power produced from small limit cycles but is short of the power generated by the large limit cycles. The power produced by the small LCO increases rapidly with the base acceleration but the power from large LCO is less sensitive to the level of excitations. The power harvested from electromagnetic coils is less than but comparable to the power from piezoelectric elements. The main obstacle in optimization of the coils is their resistance which is by itself higher than the optimal resistance for electromagnetic mechanism. 10. Conclusions It was shown that the electro-mechanical backward coupling effect on the mechanical vibrations of energy harvesters can be approximated by a change in their damping coefficient and excitation frequency. Although this approximation solution was derived separately for linear, softly nonlinear, and bi-stable energy scavengers, the result was the same for all cases. The approximation was derived for hybrid piezoelectric-electromagnetic energy harvesters and can easily be applied to piezoelectric or electromagnetic systems. The first utility of this approximation method is in modeling of energy harvesters. Instead of solving the coupled system of governing equations, one can derive the mechanical vibrations of the
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PZT power (watt)
10-2
10-3
10-4
10-5 0
0.5
1
1.5
2
2.5
3
base acc ms-2 Fig. 8. Piezoelectric power, solid line: stable, dashed line:unstable.
10-4
power EM (watt)
10-5
10-6
10-7
10-8
0
0.5
1
1.5
2
2.5
3
base acc ms-2 Fig. 9. Electromagnetic power, solid line: stable, dashed line:unstable.
harvester first and solve the first-order electrical equations in the second step for power generation. The other outcome of this approximation is in interpreting and understanding the response of energy scavengers. In the energy harvesting literature people observed that when shunt resistance changes from small to large values the resonance of the piezoelectric systems shifts from short circuit natural frequency to open circuit natural frequency. It was also observed that the amplitude of mechanical vibrations is the smallest when an optimal shunt resistance is implemented. Both of this phenomena can be easily explained with the developed approximation method. The accuracy of the approximation method was verified through analysis of a novel hybrid energy harvester. The nonlinear energy harvester is novel since it utilizes both piezoelectric and electromagnetic transduction mechanisms. The hybrid harvester was modeled through the use of Hamilton’s principle in electro-mechanical systems. Individual case studies illustrated the accuracy of the approximation method and verified its use for linear, softly nonlinear and bi-stable harvesters. The approximation method reduces the computation efforts for analysis of some bi-stable harvesting systems by an order of magnitude. It was also shown that by designing the energy harvester to perform limit cycle oscillations one can enhance the output power by more than an order of magnitude. Acknowledgment This work was performed under the support of the US Department of Commerce, National Institute of Standards and Technology, Technology Innovation Program, Cooperative Agreement Number ‘‘70NANB9H9007’’. This work was supported in part by the Institute for Critical Technology and Applied Science (ICTAS).
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The first author would like to thank Prof. Ali H. Nayfeh for his teachings and help on the subject. References [1] S.P. Beeby, M.J. Tudor, N.M. White, Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology 17 (2006) 175. [2] S. Anton, H. Sodano, A review of power harvesting using piezoelectric materials (2003–2006), Smart Materials and Structures 16 (2007) 1. [3] D. Arnold, Review of microscale magnetic power generation, IEEE Transactions on Magnetics 43 (2007) 3940–3951. [4] K. Cook-Chennault, N. Thambi, A. Sastry, Powering MEMS portable devices—a review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems, Smart Materials and Structures 17 (2008) 043001. [5] S. Priya, Advances in energy harvesting using low profile piezoelectric transducers, Journal of Electroceramics 19 (2007) 167–184. [6] S. Priya, D. Inman, Energy Harvesting Technologies, Springer, 2008. [7] A. Triplett, D. Quinn, The effect of non-linear piezoelectric coupling on vibration-based energy harvesting, Journal of Intelligent Material Systems and Structures 20 (2009) 1959. [8] M. Daqaq, C. Stabler, Y. Qaroush, T. Seuaciuc-Osorio, Investigation of power harvesting via parametric excitations, Journal of Intelligent Material Systems and Structures 20 (2009) 545. [9] S. Stanton, C. McGehee, B. Mann, Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator, Physica D: Nonlinear Phenomena (2010). [10] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Physical Review Letters 102 (2009) 80601. [11] S. Shahruz, Increasing the efficiency of energy scavengers by magnets, Journal of Computational and Nonlinear Dynamics 3 (2008) 041001. [12] A. Erturk, J. Hoffmann, D. Inman, A piezomagnetoelastic structure for broadband vibration energy harvesting, Applied Physics Letters 94 (2009) 254102. [13] B. Mann, Energy criterion for potential well escapes in a bistable magnetic pendulum, Journal of Sound and Vibration 323 (2009) 864–876. [14] B. Mann, B. Owens, Investigations of a nonlinear energy harvester with a bistable potential well, Journal of Sound and Vibration (2009). [15] B. Mann, N. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration 319 (2009) 515–530. [16] S. Stanton, C. McGehee, B. Mann, Reversible hysteresis for broadband magnetopiezoelastic energy harvesting, Applied Physics Letters 95 (2009) 174103. [17] D.A.W. Barton, S.G. Burrow, L.R. Clare, Energy harvesting from vibrations with a nonlinear oscillator, Proceedings of the ASME-IDETC, San Diego, CA, 2009. [18] A. Preumont, Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems, Kluwer Academic Publishers, 2006. [19] F. Moon, P. Holmes, A magnetoelastic strange attractor, Journal of Sound Vibration 65 (1979) 275–296. [20] D.J. Leo, Engineering Analysis of Smart Material Systems, Wiley, 2007. [21] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981. [22] A. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley-VCH, 1995. [23] A. Erturk, D.J. Inman, Piezoelectric Energy Harvesting, John Wiley & Sons, Ltd., 2011. [24] M.A. Karami, D.J. Inman, Nonlinear Hybrid Energy Harvesting utilizing a piezo-magneto-elastic spring, Presented at the 17th Proceedings of the SPIE Annual International Symposium on Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring, San Diego, CA, 2010. [25] M.A. Karami, P.S. Varoto, D.J. Inman, Experimental study of the of the nonlinear hybrid energy harvesting system, Proceedings of the SEM International Modal Analysis Conference, IMAC-XXIIIX, Jacksonville, FL, 2011. [26] A. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, New York, 1995.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems M. Amin Karami a,n, Daniel J. Inman b a b
Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, United States Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, United States
a r t i c l e i n f o
abstract
Article history: Received 23 June 2010 Received in revised form 3 May 2011 Accepted 27 June 2011 Handling Editor: K. Worden Available online 30 July 2011
A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Energy harvesting refers to scavenging small amounts of power from the ambient energy in the environment. This paper focuses on energy harvesting from vibrations. Such ambient energy can come from bridge vibrations, tire motion or the human heart beating. The energy can power up sensor nodes and therefore reduce the wiring complications or eliminate the frequent need of changing batteries. For more information on general energy harvesting the reader may refer to Refs. [1–6]. During the past two years nonlinear energy harvesting has received substantial attention. The main advantage of nonlinear energy harvesters over their linear counterparts is that the nonlinear harvesters scavenge energy over a broader frequency range of vibrations. Vibrational energy harvesters typically have low damping ratios to maximize the harvested power at resonance. The low damping ratio however, makes the harvesting device highly frequency sensitive (Fig. 1a). As depicted in Fig. 1b the presence of nonlinearity makes the peak of frequency response function lean toward higher or lower frequencies (this depends on the type of nonlinearity). The nonlinear energy harvester therefore acts over a wider
n
Corresponding author. E-mail addresses: [email protected] (M.A. Karami), [email protected] (D.J. Inman).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.06.021
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legend is base acc. ms-2
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3
Fig. 1. Frequency response: (a) linear and (b) nonlinear.
range of base excitation frequencies. Fig. 1b shows the behavior of softly nonlinear, mono-stable Duffing oscillator and is only one example of possible nonlinear scenarios. The nonlinearity can be natural (for example the nonlinear material properties of the piezoelectric substance [7] ) or can be synthetic. If in addition to the lateral direction the beam is excited longitudinally, the governing equation of the system includes some nonlinear expression in the form of parametric excitation [8]. The most common mechanism of making the beam nonlinear is by placement of permanent magnets [9–17]. After modeling their systems and deriving the nonlinear governing equations most of these researchers have used numerical or experimental methods to solve the governing equations. Among the mentioned literature on magnetically nonlinear harvesters only Ref. [15] uses analytical perturbation methods, where they simplify the electro-mechanical interactions by considering an electro-mechanical damping. They next analyze the nonlinear vibrations of the single-degree-of-freedom mechanical system. The nonlinear vibrations problems are complicated in nature. Since the principle of superposition does not hold for nonlinear systems the concept of transfer function is invalid. In addition the response of nonlinear oscillators is sensitive to initial conditions, especially when the system is close to bifurcation points. If the system undergoes bifurcations, a slight change in parameters of the system or a slight alteration of the level or frequency of excitations could change the entire nature of the resulting oscillations. In nonlinear energy harvesting problems the situation is even more complicated. The nonlinear mechanical system is coupled with one or more electrical subsystems and together they form a system of nonlinear equations. The entire system has to be solved to account for the interactions between mechanical and electrical subsystems. In some situations the coupling can significantly lengthen the investigation of the nonlinear energy scavengers. Our experience shows that, for example, extracting a chaotic strange attractor of a coupled hybrid nonlinear energy harvester is ten time more numerically intensive than finding the strange attractor of its mechanical subsystem. This is in spite the fact that the interesting phenomena and the complicated behavior of nonlinear energy harvesting systems are rooted in their mechanical subsystems. The presented reduced-order modeling approach can fundamentally simplify analysis of energy harvesting systems. It allows breaking down the electro-mechanical system of equations and facilitates solving the portions individually. In the first step we investigate the backwards coupling phenomenon (the effect of electro-mechanical coupling on mechanical vibrations). We show that the backward coupling effect can be interpreted as some extra damping together with variations of the frequency base excitations. Before solving the system we project the backward coupling effect by altering the damping and frequency of the excitations. We only solve the equivalent mechanical system and evaluate the mechanical vibrations. Due to coupling mechanisms, the mechanical vibrations excite the electrical systems. In the second step we use the calculated mechanical response and solve the electrical equations to evaluate the power output. The proposed approximation method is accurate and at the same time significantly reduces the computation time. The paper is organized as following. To arrive at governing equations we investigate a nonlinear hybrid energy harvester. The configuration is novel at the same time since it is a hybrid system it includes the equations corresponding to piezoelectric mechanism as well as the formula associated with the electromagnetic harvesting. We use the novel hybrid design for the case studies further down the paper. Next we look into mono-stable harvesters which comprise linear vibrational harvester and softly nonlinear energy scavengers. The perturbation methods are valid for these configurations and we use them to derive the approximation methods. A case study on linear harvesting and another on softly nonlinear systems showcase the use of approximation method. The bi-stable systems are considered next. The vibrations of such systems can be chaotic or exhibit Limit Cycle Oscillations (LCO) and thus have a complicated form. Before deriving the approximation technique, we first characterize the complex oscillations by the underlying periodic orbits of the system. Since the amplitude of motion in bi-stable systems can be large, the perturbation techniques are not valid. The approximation method for bi-stable systems is guessed based on the perturbation analysis and then validated against numerical solutions. The advantage of approximation technique is that it is a unified method for linear, softly nonlinear and bi-stable harvesting systems.
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2. Governing equations of the hybrid case study and common energy harvesting systems To illustrate the general form of governing differential equations of energy harvesting systems, we present our proposed nonlinear hybrid energy harvesters. The hybrid nature of the nonlinear harvesting device proposed here is illustrated in Fig. 2. We use magnetic forces in our system to induce nonlinear behavior. The magnetic force between the tip and base magnets is repulsive and therefore counteracts the elastic restoring force. The existence of nonlinear forces acting on the beam introduces nonlinear hardening terms, which are explained in Section 2. The piezoelectric element, bounded to the beam, harvests energy from beam deflection. As a novel approach we have placed electromagnetic coils in the system. When the beam vibrates the magnetic tip mass passes by the coils and generates additional electricity. The system is a hybrid energy harvester in the sense that, it uses two different methods (piezoelectric and electromagnetic transduction) for power harvesting. We use the energy methods to model the dynamics of the system. The harvester is made up of three coupled systems; the cantilever beam which is characterized by the deflection of beam, the circuit connected to the piezoelectric element and the circuit connected to the coils. The electrical circuit for harvesting is simplified to be only a resistor in order to focus on the transduction. The value of the resistive load in the piezoelectric circuit is R1 and the value of the resistive load in the electromagnetic circuit is R2. The energy in various components of the entire electro-mechanical system is: the elastic and magnetic potential energies stored in the beam and the magnetic field (V), the electrostatic energy stored in the piezoelectric patch (We), the kinetic energy stored in the beam and the tip mass (T), the magnetic potential energy between the tip and base magnets (G), the electromagnetic energy stored in the coils (Wm) and the energies dissipated by the resistors and damping of the beam. The displacement of the beam, the flux linkage across the piezoelectric element and the charge through the coils are the coordinates used for identifying the system. Following the approach in Ref. [18], the Lagrangian of the system is Lg ¼ TV þ We þ Wm
(1)
The continuous vibrations of the beam are simplified by only considering the dynamics of its first mode. The deflection at each point and at a certain time relative to the base is w(x,t)¼ f(x)u(t). The approach is a common practice in study of nonlinear vibrations of bi-stable structures [9] and is justified by center manifold reduction [19]. The assumed mode shape is the first analytically driven mode shape of a composite beam with varying cross-section and a tip mass. The following integrals are defined to facilitate abbreviation of formulas: Z L Z L1 Z L Z L1 FL20 ¼ fðxÞ2 dx, FL201 ¼ fðxÞ2 dx, FL22 ¼ f00 ðxÞ2 dx, FL221 ¼ f00 ðxÞ2 dx (2a) 0
0
FL10 ¼
Z 0
L
fðxÞ dx, FL101 ¼
0
0
Z
L1 0
fðxÞ dx, FL121 ¼
Z
L1
f00 ðxÞ dx
(2b)
0
Each of the terms in Lagrangian are related to the states as follows: ZZZ 1 1 We V ¼ cij Si Sj þ eij Ei Sj þ eij Ei Ej Gðwend ðtÞÞ 2 2 vol
(3)
where cij are the stiffness coefficients, eij are the piezoelectric constants [20], Ei are the electric field components and G(wend(t)) is the magnetic force potential. The Magnetic force is experimentally measured and is characterized as
Fig. 2. Schematic view of the harvester, R1 and R2 represent piezoelectric and electromagnetic loads accordingly.
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f ¼ awend þ bw3end . The magnetic force potential is therefore: Gðwend Þ ¼ ða=2Þw2end þ ðb=4Þw4end . We let Ys denote Young’s modulus of the steel substrate, Yp Young’s modulus of the piezoelectric patch, Is the area moment of inertia of the steel beam about its geometric center and Ip stand for the area moment of inertial of the cross-section of each piezoelectric patch about the center line of the steel substructure. Eq. (3) is simplified to ( ) 1 2Ap zp FL22 e13 _ e AL bfðLÞ4 afðLÞ2 1 L1 L 4 uðtÞ þ Ys Is F22 þ Yp Ip F22 uðtÞ2 þ l 1 ðtÞuðtÞ þ 33 2p 1 l_ 1 ðtÞ2 (4) We V ¼ 2 4 2 hp hp In Eq. (4), l1(t) is the flux linkage across the piezoelectric patch, Ap is the cross-sectional area and zp is the z-coordinate of the centroid of the patch. The z and x coordinates have been defined in Fig. 2. The base motion, characterized by r(t), should be taken into account when calculating the kinetic energy. The kinetic energy is evaluated as h i 1 1 1 _ r_ ðtÞ þ Mtip þ Ms þ 2Mp r_ ðtÞ2 _ 2 þ rs As FL10 þ 2rp Ap FL101 þ Mtip fðLÞ uðtÞ T¼ rs As FL20 þ rp Ap FL201 þ Mtip fðLÞ2 uðtÞ (5) 2 2 2 The densities of the steel substrate and the piezoelectric patch are rs and rp, respectively. The total mass of the substrate and each of the piezoelectric patches are Ms and Mp. The cross-sectional area of the substrate is rs and Mtip stands for the mass of the tip magnet. When the tip magnet passes by the coils some electromagnetic energy conversion occurs. The electromagnetic coupling can be characterized by the coupling coefficient, Tm. _ end , a force of magnitude Tmi2 impedes the motion of tip When the tip magnet passes by the coils with the velocity w magnet. The current in the coils is i2. At the same time a potential difference is generated across the coil which equals _ end . The charge passing through the coils is noted by q2 and the overall inductance of the coils is l. The following two Tm w terms in the Lagrangian represent the electro-mechanical energy in the coils: n V ¼ Wm
1 lq_ 2 þ Tm q_ 2 fðLÞu 2 2
(6)
The Euler–Lagrange equations for our three-degrees-of-freedom system is 8 @L d @Lg > @ug ¼ cf u_ > > dt @u_ > > < @L @L _ g d @lg1 ¼ lR11 dt @l_ 1 > > > d @Lg @L > > : dt @q_ 2 @qg2 ¼ R2 q_ 2
(7)
The damping coefficient of the mechanical spring is denoted by cf. Performing the derivatives in Eq. (7), dividing by the modal mass and grouping the terms results 8 ~ 3 ¼ du ~ V þ g~ q_ m _ c ^ r€ ðtÞ u€ þku þ bu > 2 1 1 > < c~ 2 _ V1 _ (8) V 1 þ R1 C0 ¼ C0 u > > : di2 R2 ~ _ þ l i2 ¼ g u dt The coefficients in Eq. (8) are ~ ¼ rs As FL20 þ 2rp Ap FL201 þ Mtip fðLÞ2 , m bfðLÞ4 b~ ¼ , ~ m
d¼
cf , ~ m
^ ¼ m
rs As FL10 þ 2rp Ap FL101 þ Mtip fðLÞ
c~ 1 ¼
^ m 1 2Ap zp FL12 e13 , ~ p mh
~ , ~c c~ 2 ¼ m 1
1 afðLÞ2 þ Ys Is FL22 þ 2Yp IP FL22 , ~ m 2e33 Ap L1 C0 ¼ , g~ ¼ Tm fðLÞ h2p
,
k¼
(9)
The second and third terms on the right-hand side of Eq. (8a) represent the backward coupling terms introduced by the piezoelectric patch and the electromagnetic coils. The energy transferred to the electric circuits reduces the mechanical energy of the beam and therefore suppresses its oscillations. The sign of the linear restoring coefficient, k, can be positive or negative. The familiar positive coefficient corresponds to low magnetic forces. In this situation the zero deflection equilibrium is stable and the system is a ‘‘nonlinear mono-stable oscillator’’ coupled to the piezoelectric and electromagnetic circuits. If the tip magnet is close to the base the repelling force between the magnets, which forces the tip away from the zero deflection, becomes significant. The u ¼0 equilibrium will be unstable but there will be two stable equilibriums on the left and right side of zero deflection (u¼ 7u*). In this situation the system is ‘‘nonlinear bi-stable oscillator’’ coupled to the piezoelectric and electromagnetic circuits. The nonlinear vibrations of the nonlinear bi-stable oscillator is discussed in Section 6. The form of Eq. (8) is typical of governing equations of energy harvesting systems. Typically the temporal parameters of the vibration modes are governed by second-order differential equations. The second-order equation includes base excitation terms that derive each mode, the nonlinear terms, and electro-mechanical coupling terms that relate the mechanical subsystem and each electrical subsystem. From this point on we address the governing equations in their general form (Eq. (8)) and references to the physical parameters are only made in the case studies. The piezoelectric and electromagnetic energy harvesters are both special cases of the hybrid harvesting system. For piezoelectric harvesters the
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~ coupling coefficient g~ vanishes and Eq. (8c) becomes irrelevant. In electromagnetic harvesters the coupling coefficient c 1 is zero and Eq. (8b) should be excluded. 3. Mono-stable oscillators The key feature of the governing equations of the nonlinear mono-stable oscillator is that the coefficient k in Eq. (8) is positive. If the nonlinearities pand ffiffiffi the couplings were not present, the spring would act as a simple harmonic oscillator and with natural frequency at on ¼ k. The frequency can be used in nondimensionalizing Eq. (8). The dimensionless time is t ¼ ont. ^ r€ ðrÞ ¼ 2F cosðotÞ and the equations of motion become: If the base excitations are harmonic the forcing term in Eq. (8) becomes m 8 2 d u du 3 > 2 þ e2z dt þ u ¼ ebu ec1 V1 þ eg1 i2 þ e2f cosðOtÞ > > < dt c2 du dV1 V1 þ R1 C0 on ¼ C0 dt (10) dt > > > : di2 þ R2 i2 ¼ g2 du dt lon l dt The following relates the coefficients in Eq. (10) to the coefficients in Eq. (8):
z¼
d , 2on
b¼
b~
on
,
c1 ¼
c~ 1
o2n
,
c2 ¼ c~ 2 , g1 ¼
g~ F , g2 ¼ g~ , f ¼ 2 o2n on
and
O¼
o on
The e is a small bookkeeping parameter. It has been assumed that the parameters z, b, c1 , g, f , c2 and R2 are small (of the order of 0.1 or less) and R1 is large (in the order of 10 or more). As a mid-step in deriving Eq. (10) from Eq. (8) the mentioned coefficients have been replaced by the following parameters to facilitate sorting small parameters from ordinary parameters: ez, eb, ec1 , eg, ef , ec2 , eR2 and R1 =e. All the terms in the second and third lines of Eq. (8) included an e which was crossed out. The method of multiple scales [21,22] is used to analyze the system in Eq. (10). The solution of Eq. (10) is expanded in terms of powers of e u ¼ u0 þ eu1 þ . . .,
V1 ¼ V10 þ eV11 þ. . .,
i2 ¼ i20 þ ei21 þ . . .
(11)
The solutions u0 ðtÞ, V10 ðtÞ and i20(t) are the solutions of Eq. (10) when e is set to zero. The change of displacement, voltage and current occur over different time scales, namely T0 ¼ t, T1 ¼ et,. . .. The time scale T1 is slower compared to the time scale T0 since the time is multiplied by the small parameter e and it takes more time for T1 to become notable. Each of the variables in Eq. (11) is assumed to be a function of the scales T0 and T1 rather than a function of t and e. We define D0 ¼ @=@T0 and D1 ¼ @=@T1 . The chain rule is used to re-evaluate the time derivatives in Eq. (10). This yields 8 > ðD2 þ 2eD0 D1 Þðu0 þ eu1 Þ þ 2ezD0 u0 þ u0 þ eu1 ¼ ebu30 ec1 V10 þ eg1 i20 þ e2f cosðOT0 Þ > > 0 < ðD0 þ eD1 ÞðV10 þ eV11 Þ þ VR101 Cþ0eoVn11 ¼ cC02 ðD0 þ eD1 Þðu0 þ eu1 Þ > > > : ðD þ eD Þði þ ei Þ þ R2 ði þ ei Þ ¼ g2 ðD þ eD Þðu þ eu Þ 0
1
20
21
lon
20
21
l
0
1
0
1
The above equations must hold for all values of e, therefore the coefficients of the like powers of e must satisfy the equations. This results two sets of relations corresponding to zeroth and first power of e. 8 > D20 u0 þu0 ¼ 0 > > < c2 V10 0 (12) Oðe Þ D0 V10 þ R1 C0 on ¼ C0 D0 u0 > > g > D i þ R2 i20 ¼ 2 D u : 0 20
lon
l
0 0
8 > D2 u þ u1 ¼ 2D0 D1 u0 2zD0 u0 bu30 c1 V10 þ g1 i20 þ 2f cosðOT0 Þ > > 0 1 < c2 V11 Oðe1 Þ D0 V11 þ R1 C0 on ¼ D1 V10 þ C0 ðD0 u1 þD1 u0 Þ > > > D i þ R2 i21 ¼ D i g2 ðD u þ D u Þ : 0 21
lon
1 20
l
0 1
(13)
1 0
The general solutions of the equations for zero power of e (Eq. (12)) are 8 > u ¼ AðT1 ÞejT0 þ AðT1 ÞejT0 > > 0 < c2 c2 V10 ¼ jC0 þ ð1=ðR AðT1 ÞjejT0 jC0 þ ð1=ðR AðT1 ÞjejT0 1 on ÞÞ 1 on ÞÞ > > g2 >i ¼ : AðT1 ÞjejT0 jl þðRg22=on Þ AðT1 ÞjejT0 Þ 20 jl þ ðR2 =on Þ
(14)
The coefficients of AðT1 Þ in Eq. (14) are independent of T0 but vary as T1 changes. Since the T1 is the slowly varying scale A(T1) change slowly in time. The complex conjugate of A is represented by A. The value of the A-coefficients is derived by eliminating the secular terms. The solution in Eq. (14) is replaced in the right-hand side of Eq. (14-1). The secular terms are those which are proportional to eiT0 , and they must sum up to zero otherwise the assumptions of small vibrations is violated [21] dA c1 c2 g1 g2 þ Aj þ f ejðOT0 T0 Þ ¼ 0 2zAj3bA2 A (15) 2j DT1 Co j þð1=ðR1 on ÞÞ lj þðR2 =on Þ
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The base oscillations are considered in the vicinity of the natural frequency. A detuning parameter (s) is defined which nondimentionalizes the difference between the excitation frequency and the linear natural frequency: O ¼1þ se. The complex number A can be written in polar coordinates: A ¼ ða=2Þejn . Both amplitude (a) and phase (n) are functions of T1. Since Eq. (15) is a complex equation it has two equations embedded which give the values for the two unknowns a and n. We define 1 c1 c2 g1 g2 þ ¼ G þ Zj 2 Co j þ ð1=ðR1 on ÞÞ ljþ ðR2 =on Þ In other words G represents the real part of the coupling expression and Z is its imaginary part. Also to have autonomous equations we define d ¼ sT1 n. Therefore Eq. (15) gives the following two equations corresponding to its imaginary and real parts: 8 da < dT þ ðz þ GÞa ¼ f sinðdÞ 1 (16) d d : a dT aðs þ ZÞ þ 38 ba3 ¼ f cosðdÞ 1 We are mostly interested in the steady-state response of the beam. For steady-state situation Eq. (16) becomes 8 2 < a20 ðz þ GÞ2 þ a0 ðs þ ZÞ 38 ba30 ¼ f 2 zþG : tan d ¼ s þ Zð3=8Þba 2
(17)
0
Note that to the first approximation the steady-state solution is 8 u0 ðtÞ ¼ a0 cosðotdÞ > > > > c2 > o a sinðotd þR c o Þ < V10 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 0 n C02 þ ð1=ðR21 o2n ÞÞ on > > g > 2 n ffi o a0 sin otd þ lo > > i20 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 : l2 þ ðR2 =o2 Þ on 2
(18)
n
Eq. (17) identifies a0 (the steady-state amplitude of oscillations) and d (the phase shift between the excitations and response). This is a useful expression for understanding the effect of the harvesting on the vibrations of the spring. Eq. (16) suggests that energy harvesting effects are equivalent to a change in the damping ratio of the structure and a change in the excitation frequency. The damping ratio changes by
G¼
1 c1 c2 R2 g1 g2 þ 2R1 on ð1=ðR21 o2n ÞÞC02 2on ðR22 =o2n Þl2
and the excitation frequency changes from o to
o þ Zon ¼ o þ on
C0 c1 c2 l g1 g2 þ 2 ð1=ðR21 o2n ÞÞC02 2 ðR22 =o2n Þl2
!
To better illustrate the implications of the approximation method, we perform the following two case studies. 4. Linear piezoelectric energy harvesting Linear piezoelectric energy harvesting is a special case of nonlinear harvesting and our solution in the previous section is also valid for linear harvesting. For linear piezoelectric systems, the coefficients b and g1 in Eq. (10) are zero and Eq. (10c) become irrelevant. The exact solution for linear piezoelectric energy harvesting exists in the literature [23]. In this section we compare the proposed the approximate solution with the exact to investigate the accuracy of the approximate method. At the same time we can see if the approximation facilitates either derivation or interpretation of the closed form solution for linear piezoelectric systems. Since the nonlinear coefficient, b, is zero, Eq. (17) simplifies to a0 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 ðz þ GÞ þ ðs þ ZÞ2 The equivalent exact formula for a single mode harvester is [23] a0 c1 ððjoc2 o2n Þ=ðo2 þ j2zoon þ o2n ÞÞ 2o2n ¼ 1 f ð1=R1 Þ þjoC0 þððjoc1 c2 o2n Þ=ðo2 þ j2zoon þ o2n ÞÞ o2 þj2zoon þ o2n
(19)
(20)
Comparison of the perturbation and exact solution has been illustrated in Fig. 3. The different lines correspond to different resistances. By changing the resistive load the resonance frequency gradually changes form the short circuit to open circuit value. The perturbation solution is almost indistinguishable from the exact solution. This is in spite of the fact that the approximate formula, Eq. (19), is much simpler and significantly more intuitive than the exact solution, Eq. (20).
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
90
5589
102
80
a/f
a/f
70 60 50 Analytical Perturbation
40
Analytical Perturbation
30
101 47
47.5
48 ω
48.5
49
42
44
46
48 ω
50
52
54
Fig. 3. Perturbation vs. exact solution: (a) about resonance and (b) big picture.
Table 1 Prototype specification. L (mm) hs (mm) b (mm) Substrate material Piezoelectric patches Length of the piezoelectric out of the clamp (mm) Inductance of the coils (mH) The electromagnetic coupling coefficient (Tm) Tip mass (g)
127 0.254 25.4 Spring steel Mide QP10n quick pack 38.1 347 0.081 48
Table 2 Parameters used for harmonic case study. b C0
2 108 93 nF
g1 g2
0.0013 6.5
c1 c2
4.7 10 8 0.077
z
on
0.01 70.6
5. Case study of nonlinear mono-stable harvester This section serves as an example for the analytical results derived in Section 3. The mechanical, piezoelectric and electromagnetic parameters of the system considered here correspond to the experimental setup discussed in Ref. [24]. Table 1 lists the design specifications of the prototype. As a parametric study we are curious to see how the system behaves for different base excitations and at different R1 and R2 resistances. The gap between the tip and base magnets are set such that the beam is still mono-stable but there is some hardening nonlinearity acting on the beam. Eq. (10) coefficients used for the case study have been summarized in Table 2. These coefficients are based on the prototype in Ref. [25] when the magnets distance is 50 mm. Fig. 4a–c illustrates that at small base excitations the system acts like a linear oscillator but for moderate and large base excitations, the nonlinear effects become dominant. For those levels of excitation the tip of the displacement vs. base excitation curves, lean to the right side (towards higher frequencies). This behavior is typical of nonlinear hardening springs [22]. As a result the oscillator vibrates over a broader range of frequency. As can be seen in Fig. 4b and c the power generation is less sensitive to the base excitation frequency. Like nonlinear oscillators there are two stable values for the amplitude of oscillation if the base excitation is larger than the natural frequency. At such frequencies, the behavior of the system depends on the initial conditions and a sudden force can make the system jump between its two stable equilibriums. Fig. 4d illustrates that changing R1 acts similar to changing the damping coefficient and changing the excitation frequency of the effective system. It shows that the optimal resistance corresponds to the smallest amplitude of response or equivalently largest electro-mechanical damping. For the system examined here, the optimal resistance is close to 200 kO. A similar argument is valid for the electromagnetic power generation, but as Fig. 4g–i proposes the optimal value of the resistance is too low. In fact, since the optimal resistance is lower that the coil wiring resistance (5 O), it cannot be experimentally achieved. The optimal electromagnetic shunt resistance (R2) is equal to the equivalent resistance of the coils. 6. Nonlinear bi-stable energy harvester When the repelling magnetic force is sufficiently large and makes the beam’s zero deflection position unstable, the energy harvester is called bi-stable. The vibrations of bi-stable harvesters can take complicated forms. In this section we
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
2.5 2 1.5 1 0.5 0
1
0.5
10
11 12 13 Frequency (Hz)
14
x 10-3
3
piezo power watt
4 end def. (m)
1.5
3 13 50 200 794 3162
2 1 0 10 11 12 13 14 Base Excit. Freq. (Hz)
x 10-3
11 12 13 Frequency (Hz)
piezo power watt
2 1.5 1 0.5 0 10 11 12 13 Base Excit. Freq. (Hz)
3 2 1 9
3.5
10
11 12 13 Frequency (Hz)
14
x 10-5 3 13 50 200 794 3162
3 2.5 2 1.5 1 0.5 0
10
x 10-3
11 12 13 14 Frequency (Hz)
15
9
10
legend is R2 (Ω)
11 12 13 14 Frequency (Hz)
15
legend is R2 (Ω) -6 7 x 10 2 3.5 5 6.5 8
1
2 3.5 5 6.5 8
6 5 4 3
0.5
2 1
0 9
4
legend is R1 (kΩ)
3 13 50 200 794 3162
9
1.5 2 3.5 5 6.5 8
0.1 0.4 0.7 1
5
14
0.5
15
6
legend is R1(kΩ)
x 10-3
1
legend is R2 (Ω)
2.5 end def. (m)
10
0 9
legend is base acc. (ms-2) x 10-6
0 9
legend is R1 (kΩ)
3
7
0.1 0.4 0.7 1
0 9
5
legend is base acc. (ms-2) x 10-3
elec. mag. power watt
end def. (m)
1.5
0.1 0.4 0.7 1
piezo power (W)
3
legend is base acc. (ms-2) x 10-3
elec. mag. power (W)
5590
14
0 9
10 11 12 13 Base Excit. Freq. (Hz)
14
9
10 11 12 13 Base Excit. Freq. (Hz)
14
Fig. 4. Mono-stable parameter study: (a) mechanical vibration in relation to base acceleration; (b) piezo-power in relation to base acceleration; (c) electromagnetic power in relation to base acceleration; (d) mechanical vibration in relation to R1; (e) piezo-power in relation to R1; (f) electromagnetic power in relation to R1; (g) mechanical vibration in relation to R2; (h) piezo-power in relation to R2 and (i) electromagnetic power in relation to R2.
first tailor the governing equations for the bi-stable case. We next look into stable and unstable periodic orbits of the harvester to characterize its complicated vibrations. The first step in analyzing the bi-stable oscillator is nondimensionalizing the equations of motion. The k coefficient in Eq. (8) pffiffiffiffiffiffiffiffiffiffi is negative, we define o0 ¼ 2k and use the following change of time and deflection variables [19] to nondimensionalize qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ r€ ðtÞ ¼ F cosðotÞ: Eq. (8): t ¼ o0t and u ¼ y 2k=b~ . If the base excitations are harmonic the forcing term in Eq. (8) becomes m 8 2 d y > 1 y þ y3 ¼ 2z dy fV1 þ gi2 þ f cosðOtÞ > > dt < dt2 2 dy dV1 þ a1 V1 ¼ b1 dt dt > > > : di2 þ a2 i2 ¼ b dy
(21)
2 dt
dt
The following relates the coefficients in Eq. (21) to the coefficients in Eq. (8):
z¼
d , 2o0
f¼
c~ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
g¼
g~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
f¼
F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ~ o2 ðð2kÞ=bÞ 0
a1 ¼
1 RC0 o0
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
b1 ¼
c~ 2 C0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ , ðð2kÞ=bÞ
a2 ¼
R2 lo0
and
b2 ¼
g~ l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ : ðð2kÞ=bÞ
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(22)
The system in Eq. (21) is the coupled single mode model of the nonlinear oscillator. The nonlinear bi-stable coupled oscillator shows three behaviors based on the level of excitations. If the excitation is too low or too large the response of the system is periodic. The mid-range excitations result in chaotic motion which is unpredictable. We can undoubtedly characterize the periodic motions of the oscillator by its periodic orbits. Later in this section we show even when the vibrations are chaotic, the unstable periodic orbits are correlated with the shape of the strange attractors. We therefore use the periodic orbits to characterize the vibrations of the bi-stable system in all vibration regimes. The shooting method is a numerical method for finding the periodic solutions of autonomous and non-autonomous systems [26]. To find the limit cycles of the non-autonomous system _ ¼ FðX,tÞ X
(23)
we start with guessing a point on the phase space (Z0) and see if that point is on a periodic orbit. In Eq. (23) X is the state vector containing the position, velocity, voltage and current values. Since the system is force driven the period of the cyclic motion is equal to the period of the forcing or its multiples [26]. If the period is noted by t0, we numerically integrate Eq. (23) over one period. If Xðt0 , g0 Þ ¼ g0 that means g0 has indeed been on a periodic orbit. Otherwise we correct the initial guess using Newton–Raphson method. The correction in initial guess (dg) is calculated from @X ðt0 , g0 ÞI dg ¼ g0 Xðt0 , g0 Þ (24) @Z To be able to use Eq. (24), we need to find @X=@Zby numerically integrating the following system over period t0: d @X @X @X , ð0Þ ¼ I ¼ Dx F dt @g @g @g
(25)
Numerically integrating Eq. (25) would give the monodromy matrix as a by-product
U¼
@X ðt 0 Þ @g
(26)
The stability of each periodic orbit is determined by the eigenvalues of the monodromy matrix. The Floquet multipliers of the system are the eigenvalues of the monodromy matrix [26]. If any of the eigenvalues of U is outside the unit circle on complex plane, the periodic orbit is unstable. In contrast if all the eigenvalues of U are within the unit circle, the periodic orbit is stable. For the nonlinear harvesting system (Eq. (21)) the X and F vectors are defined as 2 3 2 3 y X2 6 dy 7 6 X1 X3 2zX fX þ gX þ f cosðOtÞ 7 2 3 4 6 dt 7 62 7 1 7 6 7 (27) X¼6 6 V1 7 and F ¼ 6 7 a X þ b X 1 3 2 1 4 5 4 5 a2 X4 b2 X2 i2 To solve Eq. (21) we use many initial conditions spread over the phase space to identify all the stable and unstable periodic orbits.
7. Periodic orbits as land marks in Poincare’ maps In this section we look at the Poincare map of the vibrations in addition to the periodic orbits. The dynamic system (Eq. (21)) is non-autonomous and the period of the forcing function is t0 ¼ 2p=O. To get the Poincare map we read the X-vector, defined in Eq. (27), every t0. If the vibrations are periodic the Poincare map will be a single point in phase space. The Poincare map of a chaotic vibration will be its strange attractor [26]. We use both the shooting method and the Poincare map to analyze the vibrations. The shooting method gives the stable and unstable limit cycles, which give structure to the dynamic behavior of the system. The existence of a stable limit cycle however, does not mean that all the initial conditions evolve into a periodic motion. The Poincare map identifies what the dynamic behavior is based on the initial conditions. In plotting the Poincare maps we start at different initial conditions. If the trajectory ends in a periodic orbit, its Poincare section will be a point in phase space. We present this trajectory by a thin line connecting the initial condition to the Poincare point. Otherwise, if the initial condition gives rise to a quasiperiodic or chaotic trajectory, we show the entire Poincare map as a collection of points. In this way the periodic and nonperiodic trajectories can be visually distinguished and the basin of attractions can be approximately illustrated. The same prototype discussed in Section 5 can demonstrate bi-stable motion. For this purpose the gap between the tip and base magnets is reduced to 25 mm. The shorter gap strengthens the repulsive magnetic force and therefore makes the no deflection position unstable. The parameters of Eq. (21) in the case study, estimated from the prototype, have been summarized in Table 3.
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Table 3 Parameters used for bi-stable case study.
a1 a2
9.6 2.6 103
6.7 10–4 0.024
g
3.1 0.32
b1 b2
f
z k
0.03 62.8
The oversimplified version of the energy harvester where the backward couplings have been neglected is the forced, damped Duffing oscillator: 2
d y 1 dy y þy3 ¼ 2z þ f cosðOtÞ dt dt2 2
(28)
The vibration of the simplified system (illustrated in Fig. 5a) is purely mechanical. Fig. 5b shows the bifurcation of the coupled system identified by Eq. (21). The phase space of this system is four dimensional but we have only plotted y and y_ for illustration purposes. It can be seen from Fig. 5 that the vibrations of the system changes from oscillations about each of the static equilibriums to chaotic motion and then to a large limit cycle circling both static equilibriums. They also show that multiple types of oscillations can exist for a single base acceleration. For example for the base acceleration of 0.5 m s 2 the forced damped Duffing system can vibrate about one of the static equilibriums, it can be chaotic, or it can perform large Limit Cycle Oscillations. Comparison of the responses of forced damped Duffing system and the coupled harvesting system shows that energy harvesting notably affects the mechanical response. The different types of vibrations were also demonstrated. We choose the periodic orbits (calculated using shooting method) as the characteristic of the vibrations. The periodic orbits perfectly resemble the steady-state motion in small amplitude vibrations and limit cycle oscillations. In chaotic vibrations, although the periodic orbits are unstable, they are the back bones of the strange attractor and can be used as a simple method to quantify the oscillations. 8. Approximate solution for oscillations of the bi-stable system Before we proceed we clarify our terminology. The electro-mechanical energy harvesting system in its exact form (Eq. (21)) is a ‘‘coupled system’’. Neglecting the coupling terms ( fV1 þ gi2) in the mechanical system results in the purely mechanical ‘‘forced damped Duffing system’’ Eq. (28). The response of the forced damped Duffing system is different from the mechanical response of the coupled system due to the role of backward coupling. We breakdown the coupled system and first solve the ‘‘equivalent mechanical system’’, which is the mechanical subsystem modified to account for backwards coupling. The response of the equivalent mechanical system should resemble the mechanical response of the coupled system. In the second step we derive the power outputs of each harvesting circuit based on the response of equivalent mechanical system. We first start with small amplitude oscillations of the bi-stable nonlinear harvesters. Since the amplitude of vibrations are small we can use perturbation methods to derive the approximate system. pffiffiffi First we shift the origin of our system to thepstatic equilibrium: x ¼ yð1= 2Þ. By scaling the variables as x-ex, z-e2 z, ffiffiffi 2 2 2 f-e f, g-e g, f -e f and by defining y2 ¼ 3 2=2, y3 ¼1 we obtain 8 2 d x > þ x þ ey2 x2 þ e2 y3 x3 ¼ e2 2z ddxt e2 fV1 þ e2 gi2 þ e2 f cosðOtÞ > > < d t2 dV1 þ a1 V1 ¼ b1 ddxt (29) dt > > > di : 2 þ a i ¼ b dx dt
2 2
2 dt
The general form of Eq. (29) is similar to the form of Eq. (10). The main difference between the two systems is the presence of quadratic nonlinearities in the former. Due to the second-order nonlinearity we use ‘‘second-order’’ method of multiple scales [21,22]. We expand the variables as x ¼ x0 þ ex1 þ e2 x2 þ . . .,
V1 ¼ V10 þ eV11 þ e2 V12 þ . . ., T0 ¼ t,
T1 ¼ et,
i2 ¼ i20 þ ei21 þ e2 i22 þ. . .
T2 ¼ e2 t,. . .
After performing the chain rule (similar to Section 3) the terms with the same power of e are grouped 8 2 > < D0 x0 þ x0 ¼ 0 Oðe0 Þ D0 V10 þ a1 V0 ¼ b1 D0 x0 > : D i þ a i ¼ b D x 0 20
2 0
2
(30) (31)
(32)
0 0
OðeÞ : D20 x1 þx1 ¼ 2D0 D1 x0 y2 x20
(33)
Oðe2 Þ : D20 x2 þ x2 ¼ 2D0 D1 x1 D21 x0 2D0 D2 x0 y2 2x0 x1 2zD0 x0 y3 x30 fV10 þ gi20 þ f cosðOT0 Þ
(34)
We first solve Eq. (32) for x0, V10 and i20. Then, we substitute the solution in Eq. (33), eliminate the secular terms and find its homogeneous solution. Next, the results of the previous two steps are substituted in Eq. (34) and the secular terms are
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1.5
dy/dτ
-0.5
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2
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Fig. 5. Bifurcations of uncoupled (a) and coupled (b) systems subject to (1) 0.1, (2) 0.5, (3) 1, (4) 1.5 and (5) 2 m s 2 base acceleration.
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set to zero. After introducing the detuning parameter (s ¼ O 1) , putting the secular terms equation in polar coordinates and making the system autonomous by defining d ¼ sT2 b, we get the following expression for x(t): a2 a2 (35) xðtÞ ¼ a0 cosðOtdÞey2 0 cosð2Ot2dÞ þ 0 6 2 The steady-state amplitude factor (a) and phase shift (d) are calculated from 8 < 0 ¼ ðz þ GÞa0 þ 2f sinðdÞ 2 : 0 ¼ 3y3 ð10=3Þy2 a30 þ a ðs þ ZÞ þ f cosðdÞ 0 2 8
(36)
The coefficients G and Z are defined similar to Section 3 as 1 b1 f b g G þ jZ ¼ þ 2 2 j þ a1 j þ a2
(37)
The steady-state solution, Eq. (35) contains harmonic oscillations as well as double frequency and constant terms. The latter two terms are due to the existence of second-order nonlinearity. Although the vibrations about the static equilibrium of bi-stable oscillator are different from the oscillations of mono-stable Duffing system, the effects of couplings are similar for both cases. The mechanical vibrations of the hybrid energy harvesting system is equivalent to the mechanical vibrations of forced damped Duffing spring with modified damping and excitation frequency. The damping of the equivalent mechanical system is that of the coupled harvester plus G. The excitation frequency of the equivalent mechanical system is the excitation frequency of the electro-mechanical device plus Z. This correspondence is the same so far for linear, softly nonlinear (mono-stable) and small amplitude bi-stable energy harvesters. The method of multiple scales (MMS) is an approximate method and is not accurate for large vibrations of the harvesting system (Fig. 6). Therefore, for the majority of bi-stable oscillations, although the perturbation solution suggests the approximation of the coupled system by the equivalent system, it cannot prove the exact correspondence. We, therefore, consider the suggested equivalent system as a ‘‘hypothesis’’. We check the validity of this hypothesis in the next section using numerical methods. We return to the case study investigated in Section 7. We have previously compared the coupled system and the oversimplified forced damped Duffing system and we saw that they are notably different. In this case study we compare the derived equivalent mechanical system with the other two and check if the approximate system resembles the exact coupled system. The results are summarized in Fig. 7. The periodic orbits of the equivalent mechanical system are almost indistinguishable from the periodic orbits of the coupled system. This is valid for small amplitude oscillations, chaotic vibrations, and limit cycle oscillations. The case study clearly validates the accuracy of the approximation method for all types of vibrations of the bi-stable energy harvesting system. 9. Case study of bi-stable harvester Thus far the analysis of the bi-stable energy harvesting system was focused on its mechanical part. In this section we investigate how the bifurcations affect the power harvesting. As a case study we consider the system described in Table 3. The shunt resistances are R1 ¼ 200 kO and R2 ¼10 O. The piezoelectric and electromagnetic harvested powers have been
0.6 0.4
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y Fig. 6. Comparison of MMS results (rigid line) with numerical solution (dashed). The periodic orbits correspond to base acceleration of 0.1, 0.45, 0.8, 1.15 and 1.5 m s 2. The larger the base acceleration the larger the limit cycle is.
M. Amin Karami, D.J. Inman / Journal of Sound and Vibration 330 (2011) 5583–5597
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2
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Fig. 7. Limit cycles, difference between coupled and uncoupled systems and difference between coupled and equivalent systems. The limit cycles correspond to base acceleration of (a) 0.1, (b) 0.6, (c) 1, (d) 1.5, (e) 2 and (f) 3 m s 2. The thin lines represent the unstable periodic orbits.
summarized in Figs. 8 and 9. The powers correspond to the limit cycles calculated by the shooting method. The lower curves correspond to small limit cycles about each of the equilibriums, which are stable for small base excitations but become unstable for large excitations. The upper power curves correspond to the large limit cycle circling both static equilibriums. This limit cycle is only stable for large base accelerations. The figures show that the power harvested from the large limit cycle is more than an order of magnitude larger than the power from the smaller limit cycles. Figs. 8 and 9 only illustrate periodic solutions and do not show the chaotic behavior of the harvester. In reality the harvester can show chaotic behaviors if the acceleration level is between 0.7 and 1.7 m s 2.Generally the power produced in the chaotic regime is larger than the power produced from small limit cycles but is short of the power generated by the large limit cycles. The power produced by the small LCO increases rapidly with the base acceleration but the power from large LCO is less sensitive to the level of excitations. The power harvested from electromagnetic coils is less than but comparable to the power from piezoelectric elements. The main obstacle in optimization of the coils is their resistance which is by itself higher than the optimal resistance for electromagnetic mechanism. 10. Conclusions It was shown that the electro-mechanical backward coupling effect on the mechanical vibrations of energy harvesters can be approximated by a change in their damping coefficient and excitation frequency. Although this approximation solution was derived separately for linear, softly nonlinear, and bi-stable energy scavengers, the result was the same for all cases. The approximation was derived for hybrid piezoelectric-electromagnetic energy harvesters and can easily be applied to piezoelectric or electromagnetic systems. The first utility of this approximation method is in modeling of energy harvesters. Instead of solving the coupled system of governing equations, one can derive the mechanical vibrations of the
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PZT power (watt)
10-2
10-3
10-4
10-5 0
0.5
1
1.5
2
2.5
3
base acc ms-2 Fig. 8. Piezoelectric power, solid line: stable, dashed line:unstable.
10-4
power EM (watt)
10-5
10-6
10-7
10-8
0
0.5
1
1.5
2
2.5
3
base acc ms-2 Fig. 9. Electromagnetic power, solid line: stable, dashed line:unstable.
harvester first and solve the first-order electrical equations in the second step for power generation. The other outcome of this approximation is in interpreting and understanding the response of energy scavengers. In the energy harvesting literature people observed that when shunt resistance changes from small to large values the resonance of the piezoelectric systems shifts from short circuit natural frequency to open circuit natural frequency. It was also observed that the amplitude of mechanical vibrations is the smallest when an optimal shunt resistance is implemented. Both of this phenomena can be easily explained with the developed approximation method. The accuracy of the approximation method was verified through analysis of a novel hybrid energy harvester. The nonlinear energy harvester is novel since it utilizes both piezoelectric and electromagnetic transduction mechanisms. The hybrid harvester was modeled through the use of Hamilton’s principle in electro-mechanical systems. Individual case studies illustrated the accuracy of the approximation method and verified its use for linear, softly nonlinear and bi-stable harvesters. The approximation method reduces the computation efforts for analysis of some bi-stable harvesting systems by an order of magnitude. It was also shown that by designing the energy harvester to perform limit cycle oscillations one can enhance the output power by more than an order of magnitude. Acknowledgment This work was performed under the support of the US Department of Commerce, National Institute of Standards and Technology, Technology Innovation Program, Cooperative Agreement Number ‘‘70NANB9H9007’’. This work was supported in part by the Institute for Critical Technology and Applied Science (ICTAS).
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The first author would like to thank Prof. Ali H. Nayfeh for his teachings and help on the subject. References [1] S.P. Beeby, M.J. Tudor, N.M. White, Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology 17 (2006) 175. [2] S. Anton, H. Sodano, A review of power harvesting using piezoelectric materials (2003–2006), Smart Materials and Structures 16 (2007) 1. [3] D. Arnold, Review of microscale magnetic power generation, IEEE Transactions on Magnetics 43 (2007) 3940–3951. [4] K. Cook-Chennault, N. Thambi, A. Sastry, Powering MEMS portable devices—a review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems, Smart Materials and Structures 17 (2008) 043001. [5] S. Priya, Advances in energy harvesting using low profile piezoelectric transducers, Journal of Electroceramics 19 (2007) 167–184. [6] S. Priya, D. Inman, Energy Harvesting Technologies, Springer, 2008. [7] A. Triplett, D. Quinn, The effect of non-linear piezoelectric coupling on vibration-based energy harvesting, Journal of Intelligent Material Systems and Structures 20 (2009) 1959. [8] M. Daqaq, C. Stabler, Y. Qaroush, T. Seuaciuc-Osorio, Investigation of power harvesting via parametric excitations, Journal of Intelligent Material Systems and Structures 20 (2009) 545. [9] S. Stanton, C. McGehee, B. Mann, Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator, Physica D: Nonlinear Phenomena (2010). [10] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Physical Review Letters 102 (2009) 80601. [11] S. Shahruz, Increasing the efficiency of energy scavengers by magnets, Journal of Computational and Nonlinear Dynamics 3 (2008) 041001. [12] A. Erturk, J. Hoffmann, D. Inman, A piezomagnetoelastic structure for broadband vibration energy harvesting, Applied Physics Letters 94 (2009) 254102. [13] B. Mann, Energy criterion for potential well escapes in a bistable magnetic pendulum, Journal of Sound and Vibration 323 (2009) 864–876. [14] B. Mann, B. Owens, Investigations of a nonlinear energy harvester with a bistable potential well, Journal of Sound and Vibration (2009). [15] B. Mann, N. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration 319 (2009) 515–530. [16] S. Stanton, C. McGehee, B. Mann, Reversible hysteresis for broadband magnetopiezoelastic energy harvesting, Applied Physics Letters 95 (2009) 174103. [17] D.A.W. Barton, S.G. Burrow, L.R. Clare, Energy harvesting from vibrations with a nonlinear oscillator, Proceedings of the ASME-IDETC, San Diego, CA, 2009. [18] A. Preumont, Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems, Kluwer Academic Publishers, 2006. [19] F. Moon, P. Holmes, A magnetoelastic strange attractor, Journal of Sound Vibration 65 (1979) 275–296. [20] D.J. Leo, Engineering Analysis of Smart Material Systems, Wiley, 2007. [21] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981. [22] A. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley-VCH, 1995. [23] A. Erturk, D.J. Inman, Piezoelectric Energy Harvesting, John Wiley & Sons, Ltd., 2011. [24] M.A. Karami, D.J. Inman, Nonlinear Hybrid Energy Harvesting utilizing a piezo-magneto-elastic spring, Presented at the 17th Proceedings of the SPIE Annual International Symposium on Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring, San Diego, CA, 2010. [25] M.A. Karami, P.S. Varoto, D.J. Inman, Experimental study of the of the nonlinear hybrid energy harvesting system, Proceedings of the SEM International Modal Analysis Conference, IMAC-XXIIIX, Jacksonville, FL, 2011. [26] A. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, New York, 1995.