Internal Resonance Energy Harvesting

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Jan 29, 2015 - x1 ہ k2ًx2 ہ x1ق m1g ¼ 0 k2ًx2 ہ x1ق m2g ¼ 0. (2) which can be numerically solved for given parameters. The stabil- ity of the equilibrium can ...
Li-Qun Chen1 Department of Mechanics, Shanghai University, Shanghai 200444, China; Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China; Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China e-mail: [email protected]

Wen-An Jiang Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China e-mail: [email protected]

Internal Resonance Energy Harvesting Internal resonance is explored as a possible mechanism to enhance vibration-based energy harvesting. An electromagnetic device with snap-through nonlinearity is proposed as an archetype of an internal resonance energy harvester. Based on the equations governing the vibration measured from a stable equilibrium position, the method of multiple scales is applied to derive the amplitude–frequency response relationships of the displacement and the power in the first primary resonances with the two-to-one internal resonance. The amplitude–frequency response curves have two peaks bending to the left and the right, respectively. The numerical simulations support the analytical results. Then the averaged power is calculated under the Gaussian white noise, the narrow-band noise, the colored noise defined by a second-order filter, and the exponentially correlated noise. The results demonstrate numerically that the internal resonance design produces more power than other designs under the Gaussian white noise and the exponentially correlated noise. Besides, the internal resonance energy harvester can outperform the linear energy harvesters with the same natural frequencies and in the same size under Gaussian white noise. [DOI: 10.1115/1.4029606] Keywords: nonlinearity, energy harvesting, internal resonance, double-jumping, randomness

1

Introduction

Harvesting ambient waste kinetic energy to run low-powered electronics has emerged as a prominent research area and continues to grow at the rapid pace. Therefore, vibration-based energy harvesting has been extensively investigated in recent years. A vibration-based energy harvester is essentially a resonator working in a limited frequency range. To increase the working frequency range is a challenging problem [1,2]. A promising solution approach is to introduce intentionally nonlinearities into the design of energy harvesters [3], for example, adding magnetic restoring forces [4,5]. Multiple stable equilibriums resulted from nonlinearity may enhance vibration-based energy harvesting [6,7]. In addition to multistabilities, another typical nonlinear behavior, the bending of amplitude–frequency response curves can be employed to broaden the harvester working frequency bands. In contrast to linear forced vibration with symmetrical resonance band in both increasing and decreasing directions of the forcing frequency, resonance band in nonlinear forced vibration may bend to the forcing frequency increasing direction (hardening nonlinearity) or the decreasing direction (softening nonlinearity). Many scientists explored the applications of nonlinearity in vibration-based energy harvesting. Mann and Sims modeled an energy harvester using magnetic levitation as the Duffing oscillator, applied the method of multiple scales to predict the bending of hardening nonlinearity type, and validated the prediction by experimental tests [8]. Barton et al. used magnets to generate the nonlinearity in an electromagnetic energy harvester, revealed experimentally the hardening nonlinearity bending, and compared the experimental results with numerical simulations of the mathematical model, a Duffing equation [9]. Quinn et al. introduced the nonlinearity due to the geometrical configuration into a piezoelectric energy harvester, sought analytical solutions via the method of average, and found increased performance of the nonlinear harvesting system in terms of the mean power harvested under impulsive excitation [10]. Masana and Daqaq modeled mathematically an axially loaded, tunable, and clamped–clamped energy harvesting beam with the 1 Corresponding author. Manuscript received August 16, 2014; final manuscript received January 12, 2015; published online January 29, 2015. Assoc. Editor: Alexander F. Vakakis.

Journal of Applied Mechanics

account for geometrical nonlinearities, determined the steady-state amplitude via the method of multiple scale after one-term Galerkin truncation, and validated experimentally the hardening nonlinearity bending of the amplitude–frequency response curves [11]. Wu et al. studied a synthesis of a coupled linear oscillator and bistable energy harvester, applied the method of harmonic balance to predict the interwell vibration, and validated numerically and experimentally the key analytical predictions [12]. Xie and Du designed a frequency-tunable nonlinear electromagnetic energy harvester, predicted analytically the hardening nonlinearity bending of the amplitude–frequency response curves, and validated the analytical prediction by the experimental data [13]. Although the bending of amplitude–frequency response curves creates larger resonance frequency ranges, such enlargements are only toward higher or lower frequency direction due to the jumping. In addition to bending and multistability, internal resonance is a typical nonlinear vibration phenomenon [14]. In the presence of internal resonance, amplitude–frequency response curves will bend to both increasing and decreasing frequency directions. Such double-jumping or double-bending will preserve even in close but not exact internal resonance [15]. Therefore, internal resonance may be a possible mechanism to enhance energy harvesting. To authors’ best knowledge, internal resonance has not been applied in energy harvesting. The present work is motivated to explore the possibility of broadening vibration-based harvesters working frequency ranges by introducing a subsystem designed via internal resonance. It should be remarked that there are experimental and numerical works to demonstrate the outperformance of nonlinear two-degree-of-freedom energy harvesters [16,17]. Consisting of a piezoelectric beam attracted by a movable magnet [16] or an outer beam with an inner piezoelectric beam repelled by a magnet [17], the first two resonant response peaks can be tuned to be close enough and both with adequate amplitude under appropriate parameters. However, internal resonance will increase the width of external resonant bandwidth at the fundamental frequency. Nonlinearity is a necessary condition to tune internal resonance. In the present investigation, nonlinearity is implemented through snap-through mechanism consisting of two inclined linear springs connected to a mass. Each spring is linear, but the geometrical setting results in a nonlinear restoring force. It is a bistable system. Thompson and Hunt first studied the snap-through mechanism as

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a simplified model of buckling [18]. McInnes et al. used the snapthrough mechanism to explore the enhancement of the kinetic energy harvesting via stochastic resonance and demonstrated numerically the significant increase of the energy extraction under random excitations [19]. Ramlan et al. examined the positive effects of the snap-through mechanism, as a nonlinear stiffness, on kinetic energy harvesting and found numerically the benefits for low frequency excitations [20]. The investigations [19,20] focused on harvesting kinetic energy rather than electrical energy. Li and Xiong used the snap-through mechanism to harvest electrical energy via an electromagnetic generator and analyzed numerically its nonlinear behavior under periodic excitations [21]. Jiang and Chen employed the snap-through mechanism to harvest electricity from random vibration through piezoelectricity and revealed numerically that the snap-through energy harvester can outperform the linear energy harvester in the similar size under Gaussian white noise [22]. All the mentioned devices based on the snap-through mechanism are with single-degree-of-freedom. A two-degree-of-freedom electromagnetic energy harvester will be designed at internal resonance so that intrawell vibration possesses a double-bending amplitude–frequency response curve. An amplitude–frequency response curve reflects an intrinsic characteristic of a vibratory system. Such a characteristic will affect the system response to random excitations despite the deterministic derivation of the responses curve. Randomness inherent in real-world circumstances may significantly change the behaviors of energy harvesters, and there are many investigations on the effects of nonlinearity on enhancing energy harvesting, for example [4,9,23–27]. As commented in Ref. [3], hardening-type nonlinearities cannot be used to improve performance under a white noise, but softening-type nonlinearities resulting in an asymmetric potential function may enhance performance under such excitations. To examine the performance of the internal resonance energy harvesting, the averaged power will be calculated under different types of random excitations to compare the internal resonance energy harvester with various energy harvesters without internal resonance. The paper is organized as follows. Section 2 proposes the design of an electromagnetic energy harvester based on internal resonance and establishes its mathematical model. Section 3 derives the amplitude–frequency curves in the first primary resonances and two-to-one internal resonance from the method of multiple scales and validates the approximately analytical outcomes by numerical integrations. Section 4 compares the internal resonance design with linear designs and other nonlinear designs by computing their averaged power under the Gaussian white noise, the narrow-band noise, the colored noise defined by a second-order filter, and the exponentially correlated noise. Section 5 ends the paper with concluding remarks.

Fig. 1 Schematics of a snap-through electromagnetic energy harvester with an additional oscillator

In the following, the design of the additional mass-spring system will be explored. Suppose the base excitation is prescribed by displacement xb. Then the dynamical equations can been derived from Newton’s second law and Kirchhoff’s second law. L ffiÞx1 m1 x€1 þ c1 x_1 þ c2 ðx_ 1  x_ 2 Þ þ 2k1 ð1  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x1 þ l2  k2 ðx2  x1 Þ þ m1 g þ BILcoil ¼ m1 x€b m2 x€2 þ c2 ðx_2  x_1 Þ þ k2 ðx2  x1 Þ þ m2 g ¼ m2 x€b Lind I_ þ RI  BLcoil x_1 ¼ 0

where viscous damping terms are added to account for the inevitable energy dissipation. The static equilibrium positions of the system satisfy the algebraic equations

2k1

! L ffi x1  k2 ðx2  x1 Þ þ m1 g ¼ 0 1  pffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ l2

k2 ðx2  x1 Þ þ m2 g ¼ 0

2

(1)

(2)

Formalizations

Figure 1 depicts schematically an electromagnetic energy harvester based on internal resonance. The nonlinear restoring force is introduced by the snap-through mechanism. Mass m1 is connected to the vibrating framework by two inclined spring with stiffness k1 and original length L, l is the distance between the center and the edge of the frame and h is the inclination of the spring with respect to the horizontal. Mass m2 is connected to mass m1 by a spring with stiffness k2. The motion of the two masses are described by displacements x1 and x2 measured from the line across the two connect points on the framework. Connected to mass m1 by a massless rigid bar, coils move in the magnetic fields, with magnetic flux intensity B, produced by permanent magnets which are fixed to the enclosure. Resulting in electric current output I, the electromagnetic part is characterized by electromagnetic leakage resistance R, inductance Lind, and effective coil length Lcoil. The system is essentially a conventional electromagnetic energy harvester subjected to nonlinear spring force and attached to an additional linear mass-spring oscillator.

which can be numerically solved for given parameters. The stability of the equilibrium can be determined by the local linearization. Usually, there are two stable equilibriums separated by an unstable equilibrium. Under small excitations, the snap-through electromagnetic system undergoes an intrawell motion around an equilibrium point. In this case, the nonlinear term in Eq. (1) can be expanded into the Taylor series at the stable equilibrium ðx10 ; x20 Þ. Omitting higher order terms in the resulting expanding expression and shifting the origin of the coordinate by introducing the new variable y1 ¼ x1 x10 ; y2 ¼ x2  x20 yield m1 y€1 þ ðc1 þ c2 Þy_1  c2 y_2 þ d1 y1 þ d2 y21 þ d3 y31  k2 ðy2  y1 Þ þ BILcoil ¼ m1 x€b m2 y€2  c2 y_1 þ c2 y_2 þ k2 ðy2  y1 Þ ¼ m2 x€b Lind I_ þ RI  BLcoil y_1 ¼ 0

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where

!

L 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x10 þ l2

d1 ¼ 2k1 d2 ¼ d3 ¼

6k1 Lx10 þ l2 Þ3=2

ðx210

6k1 L



 ðx210 þ l2 Þ3=2

þ

2k1 Lx210 ðx210 þ l2 Þ3=2

6k1 Lx310 ðx210 þ l2 Þ5=2 36k1 Lx210 ðx210 þ l2 Þ5=2

ypj $ eypj ; I $ eI; fjk $ efjk ;

(11) (4)

30k1 Lx410 ðx210 þ l2 Þ7=2

Equation (3) governs small-amplitude motion measured from the stable equilibrium position.

3

Amplitude–Frequency Response

3.1 Multiscale Analysis. The present investigation focuses on the internal resonance in the mechanical part. Therefore, consider undamped free vibration without the electric coupling m1 y€1 þ ðd1 þ k2 Þy1  k2 y2 ¼ 0 m2 y€2  k2 y1 þ k2 y2 ¼ 0

1 f½ðm1 þ m2 Þk2 þ m2 d1  2m1 m2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ½ðm1 þ m2 Þk2 þ m2 d1 2 4m1 m2 d1 k2

Then Eq. (9) becomes  d2 y€P1 þ x2P1 yP1 ¼ e f11 y_P1  f12 y_P2  ðyP1 þ yP2 Þ2 M1  d3 BLcoil m1 þp1 m2 2 e ðyP1 þ yP2 Þ3  Iþ f X cos Xt M1 M1 M1  d2 ðyP1 þ yP2 Þ2 y€P2 þ x2P2 yP2 ¼ e f21 y_P1  f22 y_P2  M2  d3 BLcoil m1 þp2 m2 2 e ðyP1 þ yP2 Þ3  Iþ f X cos Xt M2 M2 M2 Lind I_ þ RI  BLcoil ðy_P1 þ y_P2 Þ ¼ 0

x21;2 ¼

(6)

Equation (12) defines a nonlinear oscillator with quadratic term. There may be the internal resonance if x2 ¼ 2x1. The amplitude– frequency response will be determined in the first primary resonance in the presence of 2:1 internal resonance. Introduce the detuning parameters r0 and r1 to describe the nearness of X to x1 and x2 to 2x1, respectively. Thus, the frequency relations for the first primary resonance and the 2:1 internal resonance are described by X ¼ x1 þ er0 ; x2 ¼ 2x1 þ er1

ðj ¼ 1; 2Þ

(7)

y2 ¼ p1 yP1 þ p2 yP2

yP1 ðt; eÞ ¼ yP10 ðT0 ; T1 Þ þ eyP11 ðT0 ; T1 Þ þ    yP2 ðt; eÞ ¼ yP20 ðT0 ; T1 Þ þ eyP21 ðT0 ; T1 Þ þ    Iðt; eÞ ¼ I0 ðT0 ; T1 Þ þ eI1 ðT0 ; T1 Þ þ   

(14)

(8)

Based on the frequencies and the modes, the linear mechanical part in Eq. (3) can be decoupled in the modal coordinates

The derivatives with respect to time t are expansions in terms of the partial derivatives with respect to new time scales T1 and T2 d d2 ¼ D0 þ eD1 þ    ; 2 ¼ D20 þ 2eD0 D1 þ    dt dt

x2P1 yP1

y€P1 þ f11 y_P1 þ f12 y_P2 þ d2 d3 ¼ ðyP1 þ yP2 Þ2  ðyP1 þ yP2 Þ3 M1 M1 BLcoil m1 þp1 m2  I x€b M1 M1 2 y€P2 þ f21 y_P1 þ f22 y_P2 þ xP2 yP2 d2 d3 ¼ ðyP1 þ yP2 Þ2  ðyP1 þ yP2 Þ3 M2 M2 BLcoil m1 þp2 m2  I x€b M2 M2 Lind I_ þ RI  BLcoil ðy_P1 þ y_P2 Þ ¼ 0

(13)

Define the fast and the slow time scales T0 ¼ t and T1 ¼ et. The solution to Eq. (12) is assumed to be

Introduce modal coordinates y1 ¼ yP1 þ yP2 ;

(12)

(5)

One can solve the natural frequencies

and the corresponding modes 0 1   1 1 2 ¼ @ d1 þ k2  m1 xj A pj k2

ðj; k ¼ 1; 2Þ

;

; þ

BLcoil BLcoil $e Mj Mj

(15)

where Dj (j ¼ 1, 2) stands for the partial differential operator @=@Tn . Substituting Eqs. (14) and (15) into Eq. (12) and equating coefficients of e0 and e1 in the resulting equations lead to (9) D20 yP10 þ x2P1 yP10 ¼ 0 D20 yP20 þ x2P2 yP20 ¼ 0 Lind D0 I0 þ RI0 ¼ BLcoil ðD0 yP10 þ D0 yP20 Þ

(16)

and where Mj ¼ m1 þ

p2j m2 ;

c1 þ c2  pj c2 þ pk ðpj  1Þc2 fjk ¼ Mj

ðj; k ¼ 1; 2Þ

(10)

Assume that the base excitation is harmonic and small, i.e., xb ¼ e2 f cos Xt, where small parameter e is a book keeping device in the subsequent multiscale analysis. To account for the smallness of the displacements, the current, the damping, and the coupling between the mechanical and the electric parts, rescale the parameters, and the coefficients as

D20 yP11 þx2P1 yP11 ¼ 2D0 D1 yP10 f11 D0 yP10 f12 D0 yP20 BLcoil d2  2 m1 þp1 m2 2  y þ2yP10 yP20 þy2P20  I0 þ X f cosXt MP1 P10 MP1 MP1 D20 yP21 þx2P2 yP21 ¼ 2D0 D1 yP20 f21 D0 yP10 f22 D0 yP20 BLcoil d2  2 m1 þp2 m2 2  y þ2yP10 yP20 þy2P20  I0 þ X f cosXt MP2 P10 MP2 MP2 Lind D0 I1 þRI1 ¼ D1 I0 þBLcoil ðD0 yP11 þD1 yP10 þD0 yP21 þD1 yP20 Þ

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The solution of linear equation (16) can be expressed as yP10 ¼ AP1 ðT1 Þ expðixP1 T0 Þ þ cc; yP20 ¼ AP2 ðT1 Þ expðixP2 T0 Þ þ cc; I0 ¼

  ixP1 BLcoil AP1 ðT1 Þ ixP2 BLcoil AP2 ðT1 Þ R expðixP1 T0 Þ þ expðixP2 T0 Þ þ EðT1 Þ exp  T0 þ cc R þ ixP1 Lind R þ ixP2 Lind Lind

(18)

where APj(T1) (j ¼ 1, 2) are undetermined functions and cc stands for the complex conjugate of the preceding terms. Substituting Eqs. (14) and (18) into Eq. (17) and eliminating the secular terms yield

D1 AP1 ¼ C11 AP1 þ C12 AP1 AP2 expðir1 T1 Þ þ f C13 expðir0 T1 Þ

(19)

D1 AP2 ¼ C21 AP2 þ C22 A2P1 expðir1 T1 Þ where f11 B2 L2coil  ; 2 2MP1 ðR þ ixP1 Lind Þ f B2 L2coil ¼  22  ; 2 2MP2 ðR þ ixP2 Lind Þ

d2 ; xP1 MP1 i d2 ¼ 2MP2 xP2 i

C11 ¼ 

C12 ¼ 

C21

C22

C13 ¼

ðm1 þ p1 m2 ÞX2 ; 4xP1 MP1 i

Functions APj(T1) (j ¼ 1,2) can be expressed in the polar form 1 APj ðT1 Þ ¼ aPj ðT1 Þ exp½ihPj ðT1 Þ 2

(20)

(21)

where aPj and hPj are real functions with respect to T1. Substitution of Eq. (21) into Eq. (19) and separation of the real and the imaginary parts in the resulting equations give 1 D1 aP1 ¼ ReðC11 ÞaP1 þ aP1 aP2 ½ReðC12 Þ cos c1  ImðC12 Þ sin c1  þ 2f ½ReðC13 Þ cos c2  ImðC13 Þ sin c2  2 a2 D1 aP2 ¼ ReðC21 ÞaP2 þ P1 ½ReðC22 Þ cos c1 þ ImðC22 Þ sin c1  2 D1 c1 ¼ r1 þ ImðC21 Þ  2ImðC11 Þ  aP2 ½ReðC12 Þ sin c1 þ ImðC12 Þ cos c1  (22) 4f a2P1 ½ReðC13 Þ sin c2 þ ImðC13 Þ cos c2   ½ReðC22 Þ sin c1  ImðC22 Þ cos c1   aP1 2aP2 1 2f ½ReðC13 Þ sin c2 þ ImðC13 Þ cos c2  D1 c2 ¼ r0  ImðC11 Þ  aP2 ½ReðC12 Þ sin c1 þ ImðC12 Þ cos c1   2 aP1 where c1 ¼ hP2  2hP1 þ r1 T1 ; c2 ¼ hP1 þ r0 T1

(23)

For the steady-state response, aPj and cPj should be independent of T1. Therefore 1 0 ¼ ReðC11 ÞaP1 þ aP1 aP2 ½ReðC12 Þ cos c1  ImðC12 Þ sin c1  þ 2f ½ReðC13 Þ cos c2  ImðC13 Þ sin c2  2 a2 0 ¼ ReðC21 ÞaP2 þ P1 ½ReðC22 Þ cos c1 þ ImðC22 Þ sin c1  2 0 ¼ r1 þ ImðC21 Þ  2ImðC11 Þ  aP2 ½ReðC12 Þ sin c1 þ ImðC12 Þ cos c1 

(24)

4f a2  ½ReðC13 Þ sin c2 þ ImðC13 Þ cos c2   P1 ½ReðC22 Þ sin c1  ImðC22 Þ cos c1  aP1 2aP2 1 2f ½ReðC13 Þ sin c2 þ ImðC13 Þ cos c2  0 ¼ r0  ImðC11 Þ  aP2 ½ReðC12 Þ sin c1 þ ImðC12 Þ cos c1   2 aP1 Eliminating c1 , c1 , and a2 from Eq. (24), one can obtain the amplitude–frequency response relationship in the first mode K1 a6P1 þ K2 a4P1 þ K3 a2P1 þ K4 ¼ 0 031004-4 / Vol. 82, MARCH 2015

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where n o K1 ¼ Im2 ðC22 ÞIm2 ðC12 Þ Re2 ðC21 Þ þ ½2r0  r1  ImðC21 Þ2 ; n o K2 ¼ 8ImðC22 ÞImðC12 Þ Re2 ðC21 Þ þ ½2r0  r1  ImðC21 Þ2 fReðC11 ÞReðC21 Þ ½r0  ImðC11 Þ½2r0  r1  ImðC21 Þg; n on o2 K3 ¼ 16 Re2 ðC11 Þ þ ½r0  ImðC11 Þ2 Re2 ðC21 Þ þ ½2r0  r1  ImðC21 Þ2 ; n o2 K4 ¼ 64f 2 Im2 ðC13 Þ ½2r0  r1  ImðC21 Þ2 þ Re2 ðC21 Þ

Then the amplitude–frequency response relationship in the second mode is jImðC22 Þj aP2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2P1 2 2 Re ðC21 Þ þ ½r1  2r0 þ ImðC21 Þ2

(27)

The stability of the steady-state responses can be determined via the Lyapunov theory. The disturbance equation of Eq. (22) is ð D1 DaP1

D1 DaP2

¼ Jð DaP1

D1 Dc1

DaP2

Dc1

D1 Dc2 ÞT Dc2 ÞT

(28)

where superscript T denotes transpose and J is the Jacobian matrix calculated at the fix points defined by Eq. (24). If all the eigenvalues of J are with negative real parts, the corresponding steadystate response is stable. If J has at least an eigenvalue with a positive real part, then the corresponding steady-state response is unstable. 3.2 Results and Discussion. The parameters of the energy harvester listed in Table 1. In this case, there is the 1:2 internal resonance with x1 ¼ 65.8469 s1 and x2 ¼ 131.682 s1. Thus, the corresponding internal detuning parameter r1 is zero. A stable static equilibrium position is solved from Eq. (2) as x10 ¼ 0.0674 and x20 ¼ 0.0665. Equation (4) yields d1 ¼ 73.4430, d2 ¼ 436.9840, and d3 ¼ 5829.19. The modes given by Eq. (7) are p1 ¼ 1.7346 and p2 ¼ 1.4412. The modal parameters given by Eq. (10) are MP1 ¼ 0.0220, MP2 ¼ 0.0183, n11 ¼ 0.0114, n12 ¼ 0.00094, n21 ¼ 0.00110, and n22 ¼ 0.0434. Book-keeping parameter e is set to 0.01. The default value of external excitation amplitude f is set to 0.005 m. In the above mentioned exact internal resonance case, the amplitude–frequency response relationships in the two modes leads to those of the displacement of mass m1 and the electric current output I. Equations (18) and (21) yield y1 ¼ aP1 cosðxP1 T0 þ hP1 Þ þ aP2 cosðxP2 T0 þ hP2 Þ

(29)

BLcoil I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½xP1 aP1 cosðxP1 T0 þ hP1 þ b1 Þ 2 R þ x2P1 L2ind þ xP2 aP2 cosðxP2 T0 þ hP2 þ b2 Þ

(30)

where tan bj ¼ R=ðxPj Lind Þ. Then the harvested power is given by PðtÞ ¼ I 2 ðtÞR

(31)

Based on Eqs. (29)–(31), the variations of the amplitudes of y1 and P can be derived from the amplitude–frequency response relationships defined by Eq. (24). The amplitude–frequency response

(26)

relationships will be illustrated in the following for different parameters. In those figures, solid lines represent stable responses and dotted lines represent the unstable responses. Figure 2 shows the displacement and the power amplitude– frequency response curves for different external excitation amplitudes (ƒ ¼ 0.001, 0.005, and 0.008). There are typical double-jumpings in the curves for large enough f. The curves have two separated multivalued ranges so that there are two peaks bending to the left and the right, respectively. The doublejumping leads to broadband vibration-based energy harvesting. In addition, Fig. 2 reveals the effects of the external excitation amplitude on the amplitude–frequency response curves. Both the response amplitude and the resonance range increase with the excitation amplitude. In energy harvesting, the resonance range is desired operating bandwidth. Figure 3 plots the amplitude–frequency response curves for different damping coefficients (c1, c2 ¼ 0.0002, 0.008, and 0.02). Both the displacement and the power frequency response curves show that the effect of nonlinearity becomes more dramatic with smaller damping coefficients. In addition to the increase of the peak response amplitudes, the bandwidth of energy harvesting also increases as the damping coefficient decreases. Figure 4 depicts the amplitude–frequency response curves for different resistance (R ¼ 2, 10, and 16). The results indicate the dissimilar effects on the displacement and the power. The displacement amplitude is insensitive to the resistance large enough, while the power amplitude peak decreases with the decreasing resistance. The resonance ranges of both the displacement and the power increase with the resistance. Figure 5 illustrates the amplitude–frequency response curves for different magnetic flux densities (B ¼ 0.05, 0.06, and 0.08). For small enough magnetic flux density, there are doublejumpings in the curves, while the curves are with two peaks without jumping for large magnetic flux density. Both the response amplitude and the resonance range decrease with the magnetic flux density in the displacement response curves. However, the

Table 1 Values for parameters used in simulation Prameters

Values

Mass m1 (kg) Mass m2 (kg) Spring stiffness k1 (N/m) Spring stiffness k2 (N/m) Damping c1 (N/(m/s)) Damping c2 (N/(m/s)) Spring length L (m) Distance l (m) Magnetic flux B (T) Resistance R (X) Inductance Lind (H) Coil length Lcoil (m)

0.01 0.004 50 40.95 0.0002 0.0001 0.08 0.04 0.05 10 0.005 10

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Fig. 2 Amplitude–frequency response curves for different excitation amplitudes. (a) Displacements and (b) power.

response amplitude increases and the resonance range decreases with the magnetic flux density in the power response curves. Physically, larger magnetic flux tends to lead to higher induced damping in the system. Actually, the magnetic flux has the same effects on the system as the damping shown in Figs. 3(b) and 3(d). Figure 6 demonstrates the amplitude–frequency response curves for different coil length (Lcoil ¼ 10, 12, and 15). For large enough coil length, the double-jumpings disappear and emerge two peaks without jumping, and for small enough coil length, there are double-jumpings. With the increase of the coil length, the displacement amplitude decreases, the power increases, and both the displacement and the power resonance ranges decrease. It should be remarked that there are two different types of electrical circuit equations used in electromagnetic energy harvesting. Mann and Sims [8], Daqaq [23,28], Green et al. [26], and Leadenham and Erturk [29] proposed a uncoupled electromechanical equation for lumped-parameter energy harvesters in which the

inductance was neglected. Mann and Owens [30] and Li and Xiong [21] presented a coupled electromechanical equation for lumped-parameter energy harvesters which accounted for the inductance. The approximate analysis here indicated that the inductance in Eq. (1) has a limited effect on the amplitude–frequency response curve. 3.3 Numerical Validations. To verify the approximately analytical results, Eq. (3) is numerically integrated via the fourthorder Runge–Kutta algorithm. In addition, to examine the errors of omitting high power terms, Eq. (1) is also numerically integrated via the same algorithm. For specific excitation frequencies, the amplitudes of the displacement and the current steady-state responses can be obtained from the numerical solutions. The numerical amplitude–frequency response curves (actually only its stable portions) are compared with the analytical amplitude– frequency response curves, shown in Fig. 7, in which the solid

Fig. 3 Amplitude–frequency response curves for different damping coefficients. (a) Displacements for varying c1, (b) power for varying c1, (c) displacements for varying c2, and (d) power for varying c2.

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Fig. 4 Amplitude–frequency response curves for different resistances. (a) Displacements and (b) power.

Fig. 5 Amplitude–frequency response curves for different magnetic flux densities. (a) Displacements and (b) power.

and the dotted lines is the stable and the unstable analytical results, the solid triangles are numerical results based on Eq. (1), and the hollow circles are numerical results based on Eq. (3). The amplitude–frequency response curves have double-jumpings, which qualitatively agrees with the analytical ones. Quantitatively, the numerical resonance ranges are small than the analytical ones, and so are the peak values. Numerical solutions of Eqs. (1) and (3) are demonstrated to be in good agreement. Besides, numerical integrations reveal that the unstable portions of the analytical amplitude–frequency response curves correspond to quasi-periodic motions.

3.4 Contrast to Linear Harvesters. In this subsection, the energy harvesting performances of the proposed design, as shown in Fig. 1, and its linear counterparts are contrasted. The counterparts are two linear elcetromagnetic energy harvesters: one is with the same natural frequencies and the other is in the same static equilibrium, which implies the same size of the harvester. The linear elcetromagnetic energy harvester is schematically depicted in Fig. 8. Linear model 1 is with k1 ¼ 73.4430 N/m so that its natural frequencies are x1 ¼ 65.8469 s1 and x2 ¼ 131.682 s1, which are the same as the nonlinear model under investigation. Linear model 2 is with k1 ¼ 70 N/m so that its static equilibrium is the

Fig. 6 Amplitude–frequency response curves for different coil length levels. (a) Displacements and (b) power.

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Fig. 7 Comparisons of analytical and numerical results. (a) Displacements and (b) power.

Table 2

Fig. 8 Schematics of two linear electromagnetic energy harvesters

same as the nonlinear model. In this case, its natural frequencies are x1 ¼ 64.6320 s1 and x2 ¼ 130.9771 s1. The governing equations of the linear elcetromagnetic energy harvester are m1 x€1 þ ðc1 þ c2 Þx_1  c2 x_ 2 þ ðk1 þ k2 Þx1  k2 x2 þ BILcoil ¼ m1 x€b m2 x€2  c2 x_ 1 þ c2 x_2  k2 x1 þ k2 x2 ¼ m2 x€b Lind I_ þ RI  BLcoil x_ 1 ¼ 0

k2 (N/m)

x1 (s1)

x2 (s1)

x1:x2

40.95 21.05 211.8 0 0

65.8469 58.5149 71.3525 85.6988 72.4287

131.682 106.2440 276.3749

1:2 pffiffiffi 1:pffiffiffiffiffi 3 1: 15

and k2 are the spring stiffness, B is magnetic flux intensity, I is output electric current, R is resistance, Lind is the inductance, Lcoil is effective coil length, and xb is the base displacement. Figure 9 shows the displacement and the power amplitude–frequency response curves for the snap-through energy harvester and the corresponding two-degrees-of-freedom linear energy harvesters. There are typical double-jumpings in the curves for snap-through nonlinearity energy harvester with the two-to-one internal resonance. The curves have two separated multivalued ranges so that there are two peaks bending to the left and the right, respectively. The double-jumping leads to broadband vibrationbased energy harvesting than the corresponding linear energy harvesters. However, the height of the peaks is almost the same.

4 Harvesting Performance Comparisons Under Random Excitations (32)

where m1 and m2 are the masses, x1 and x1 are the displacements of masses m1 and m2, c1 and c2 are the damping coefficients, k1

Fig. 9

Values of parameters for five different design models

To explore the merits of the internal resonance energy harvesting, the averaged power E½PðtÞ ¼ E½I 2 ðtÞR is computed for different designs under various types of random excitations. In Fig. 1, there are five designs under investigation, three designs with an additional mass-spring system, and two designs without, which have the parameters listed in Table 2. In the designs with

Amplitude–frequency response curves of nonlinear and linear energy harvesters

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Fig. 10 The averaged power versus Gaussian white noise excitation intensity

Fig. 11 Comparisons harvesters

of

nonlinear

and

linear

energy

an additional mass-spring system with m2 ¼ 0.004 kg, termed twodegrees-of-freedom systems, the 1:2 internal resonance occurs for k2 ¼ 40.95 N/m with x1 ¼ 65.8469 s1, x2 ¼ 131.682 s1. In other 1 two designs, one choice is k2 ¼ 21.05 N/m pffiffiffi with x1 ¼ 58.5149 s , 1 x2 ¼ 106.2440 s and x1 : x2  1 : 3, and the other choice is

Fig.12 The averaged power versus the center frequency of a narrow-band colored noise

Fig. 13 The averaged power versus the intensity of narrowband colored noise

k2 ¼ 211.8 N/m with x1 ¼ 71.3525 s1, x2 ¼ 276.3749 s1 and pffiffiffiffiffi x1 : x2  1 : 15. These ratios are chosen such that x1:x2 are smaller and larger than 1:2 and far away from any simple fractions. For the designs without an additional mass-spring system, termed one-degree-of-freedom system, one design is to remove the additional mass-spring system and the other is to remove the additional spring k2 and to put m2 on m1. That is, one is simply with m1 ¼ 0.01 kg and m2 ¼ 0, k2 ¼ 0 (undamped linearized frequency x1 ¼ 85.6988 s1) and the other is with m1 ¼ 0.014 kg and m2 ¼ 0, k2 ¼ 0 (undamped linearized frequency x1 ¼ 72.4287 s1). The latter case is actually the attachment of m2 to mass m1 (mathematically set k2 ! 1). Equation (3) is numerically integrated with initial conditions set at a stable equilibrium position, namely, yi ð0Þ ¼ 0; y_i ð0Þ ¼ 0; Ið0Þ ¼ 0 (i ¼ 1, 2). First, the random base excitation zðtÞ ¼ € xb is assumed to be the Gaussian white noise [4,23,24,26]. Figure 10 shows the averaged power varying with the excitation intensity D. For all excitation intensity calculated, the numerical results demonstrate that the internal resonance leads to the largest output power and the differences increase with the increasing excitation intensity. Figure 11 plots the averaged power of the snap-through nonlinearity energy harvester and two linear energy harvesters (shown as Fig. 8 in Sec. 3.4) varying with the excitation intensity D, and the

Fig. 14 The averaged power versus the center frequency of colored noise

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Fig. 15 The averaged power versus the intensity of colored noise Fig. 17 The averaged power versus the bandwidth of exponentially correlated noise

where X is the center frequency, b is the bandwidth, n(t) is the Gaussian white noise with zero mean and unit variance. As a leading factor, the center frequency changes significantly the averaged power, as shown in Fig. 14 with excitation intensity D ¼ 0.002 ms5/2 and b ¼ 0.001 s1. The internal resonance design has the advantages but they are not substantial. In a frequency favorable to the internal resonance design with X ¼ 65.85 s1 and b ¼ 0.001 s1, Fig. 15 shows the output power for the changing intensity of the colored noise, measured by the intensity of excitation nðtÞ in Eq. (34). Last, random excitation z(t) is the exponentially correlated noise defined by the Ornstein–Uhlenbeck equation [28] z_ þ cz ¼ nðtÞ

Fig. 16 The averaged power versus the intensity of exponentially correlated noise

results indicate that the internal resonance produces larger power than the corresponding linear energy harvesters. Second, random excitation z(t) is the narrow-band noise defined by [31] zðtÞ ¼ A cosðxt þ qW ðtÞÞ

(33)

where A > 0 is a deterministic amplitude, x is the center frequency, q is the intensity of random excitation, W(t) is a standard _ Wiener process satisfying WðtÞ ¼ nðtÞ, and n(t) is the Gaussian white noise with zero mean and a unit variance. Figure 12 highlights the effect of center frequency x on the averaged power for five different designs with q ¼ 0.05 and A ¼ 0.02 m/s2. Each design with its natural frequency has a different resonance range, and the internal resonance design has the highest peak and the seemingly widest resonance bandwidth. Anyway, the advantages of the internal resonance design are not considerable. The center frequency is the key issue, and the excitation intensity usually cannot change the tendency. For example, in a frequency favorable to the internal resonance design with x ¼ 65.85 s1 and A ¼ 0.02 m/s2, the averaged power varying with the excitation intensity q for five different designs is shown in Fig. 13. Third, random excitation z(t) is the colored noise defined by a second-order filter [23] €z þ bz_ þ X2 z ¼ X

pffiffiffi bnðtÞ

(34)

(35)

where c is the noise bandwidth of the, and n(t) is Gaussian white noise with zero mean and unit correlation function. The effect of excitation intensity D on averaged power for five different designs is shown in Fig. 16 with c ¼ 0.01 s1. The internal resonance design produces the largest power. Fixed D ¼ 0.002 m/s3, the internal resonance design still produces the largest power for varying bandwidth c, as shown in Fig. 17.

5

Conclusions

The work explores the application of internal resonance in energy harvesting. An archetype of an electromagnetic energy harvester is conceptually designed to utilize two-to-one internal resonance. Based on the governing equations, the method of multiple scales is employed to establish the amplitude–frequency response relationship in the first primary resonance in the presence of 1:2 internal resonance. The governing equations are numerically integrated under a harmonic excitation and four types of random excitations, namely, the Gaussian white noise, the narrow-band noise, the colored noise defined by a second-order filter, and the exponentially correlated noise. The investigation yields the following conclusions: (1) The amplitude–frequency response relationships are analytically predicted to have double-bending, two peaks bending to the left and the right, respectively. (2) The analytical prediction is supported by the numerical simulations. (3) The amplitude of the displacement steady-state response increases with the excitation amplitude and the resistance, but decreases with the magnetic flux density and the coil length. The displacement resonance range increases with

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the excitation amplitude, the resistance, and the coil length, while decreases with the magnetic flux density. (4) The amplitude of the power steady-state response increases with the excitation amplitude, the magnetic flux density, and the coil length, but decreases with the resistance. The power resonance range increases with the excitation amplitude, the resistance, and the coil length, while decreases with the magnetic flux density. (5) Under the Gaussian white noise, the internal resonance design is numerically demonstrated to outperform the same frequencies and the same size linear energy harvesters. (6) The superiority of the internal resonance design is numerically demonstrated over other designs under the Gaussian white noise and the exponentially correlated noise.

Acknowledgment This work was supported by the State Key Program of National Natural Science of China (No. 11232009), Shanghai Leading Academic Discipline Project (No. S30106) and China Scholarship Council.

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