Exploiting double jumping phenomenon for broadening bandwidth of an energy harvesting device

Mechanical Systems and Signal Processing 139 (2020) 106614

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Exploiting double jumping phenomenon for broadening bandwidth of an energy harvesting device H. Karimpour a,⇑, M. Eftekhari b a b

Department of Mechanical Engineering, University of Isfahan, Isfahan, Iran Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 9 October 2019 Received in revised form 26 December 2019 Accepted 30 December 2019

Keywords: Internal resonance Double jumping Energy harvester Ambient vibrations

a b s t r a c t The steady state response of a particular harvesting system is investigated under the condition of external and internal resonance, with emphasis on the double jump phenomenon. Following the trend of recent researches on exploring nonlinearity aspects for broadening the frequency resonance bandwidth, the problem of effective energy harvesting from a broadband source is dealt with. The proposed harvester can be embedded within vibrational structures and can dually act as a vibration absorber. The governing non-linear partial differential equations are truncated into a set of perturbed equations via Galerkin method. The method of multiple scales is applied to derive the modulated amplitude versus frequency at the vicinity of flapwise and chordwise primary resonances, as well as around other internal resonance frequencies. The amplitude-frequency response plot reveals resonance peaks bending to the left and right, i.e., splitting into two different tongues in contrast to conventional jumps which lean only toward higher or lower frequency directions. Numerical results demonstrate that this internal resonance-based harvesting design can produce sufficient power for the consumption of typical MEMS devices. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction The trend toward development of wireless sensor networks, together with low power consumption technology has conducted research activities on ambient energy harvesting to deal with enhancing power conversion efficiency [1]. The vibration based energy harvester (VBEH) is a device which is used to convert ambient vibration energy to electrical power. The VBEH device is usually attached to the structure and is operated in the vicinity of resonance conditions to harvest the electrical power. The amount of power swept from the source depends on the design of the VBEH device and the frequency spectrum of vibrations. One main limitation of harvesting techniques is the acute dependency of their performance to the resonant frequencies available in ambient vibrations, which are often random and broadband. By properly tuning, VBEH devices can only be functional over a specific frequency range. To resolve this problem, researches concentrated on adopting nonlinear traits of the harvesting system to broaden the effective frequency bandwidth. Different methods for taking advantage of nonlinear damping and stiffness characteristics to improve the vibration energy harvesting performance have been reported in the literature [2–5]. Among others, multi-stable equilibrium systems which results from the superposition of nonlinear sources such as magnetic forces are also suitable alternatives for such purposes [6,7]. Dai et al. designed an electromagnetic generator mounted on a cantilevered beam [4]. The effects of some parameters such as magnet position, wind ⇑ Corresponding author. E-mail address: [email protected] (H. Karimpour). https://doi.org/10.1016/j.ymssp.2019.106614 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

2

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Nomenclature xyz ngf rðtÞ m Jn ; Jg ; Jf E1 ; E2 ; E3 G12 ; G13 ; G23 ma !

Inertial coordinate system Principle axes coordinate system at local position s Position of moving mass from the clamped end Mass of the beam per unit length l Principal mass moments of inertia Elastic and shear modulus

Moving mass Velocity vector of the moving mass ‘ðs; tÞ Lagrangian density re Equilibrium position of the moving mass uðs; tÞ; v ðs; tÞ; wðs; tÞ Neutral axis deflection along x, y and z axes ci ; i ¼ u; v ; w; / Damping coefficients wðs; tÞ; hðs; tÞ; /ðs; tÞ Beam neutral axis Euler rotation angles ^ei ; i ¼ x; y; z; n; g; f Unit vector along the i axis D11 ; D22 ; D33 ; D13 Bending and stiffness rigidities ! Applied force vector to the moving mass Fc Q i ; i ¼ u; v ; w; / External forces ! Position vector of the moving mass rm ks Nondimensional spring stiffness

vm

speed and etc. were explored on the performance of the harvester. Wei and Jing proposed three different vibration-toelectricity conversion mechanisms and discussed about their efficiency [8]. Tran et al. studied some appropriate techniques based on nonlinearity attributes such as internal resonance, parametric excitations,. . . for enhancing energy exchange from ambient stochastic vibrations [9]. Zheng et al. investigated three nonlinear energy dissipative devices for controlling structural vibrations in engineering applications [10]. Three different categories of dampers have been discussed in their work: 1) dampers with nonlinear stiffness 2) dampers with nonlinear damping and 3) dampers with both nonlinear stiffness and damping. Sharvari et al. proposed a piezoelectric vibration-based energy harvester with a wide bandwidth and high output voltage [11]. Numerical and imperical results demonstrate the enhancement realized by juxtaposing magnetic force in parallel to spring elements to change the harvester stiffness. Lu et al. designed a novel piezoelectric energy harvester using a unimorph or bimorph cantilever beam [12]. The energy harvester efficiency improved by employing multi-layer piezoelectric designs compared to traditional ones. Internal resonance is a nonlinear phenomenon recently used in VBEH devices to enhance performance [13,14]. In the presence of internal resonance, the frequency response will either bend toward the increasing or the decreasing direction [14]. Under certain conditions, this phenomenon can occur bidirectionally, referred as double- jumping or doublebending. It is worthy to mention that similar to internal resonance, parametric resonance may also show similar characteristics and unexpected coupling properties. In an attempt based on this phenomenon for broadening harvesting efficiency, Chen and Jiang proposed a snap-through electromagnetic energy harvester combined to an additional oscillator. This archetype was employed to establish the amplitude–frequency response relationship in the first primary resonance in the presence of 1:2 internal resonance [15]. Yang and Towfigian investigated a piezoelectric cantilever beam carrying a moving magnet [16]. The nonlinear energy harvester combined bi-stability with internal resonance, demonstrating that the frequency bandwidth of energy conversion increased. Some other recent works in the energy harvesting area can be found in Refs. [17–23]. Inspired by the fact that internal resonance may increase the operating bandwidth of VBEH devices, a system exploiting this characteristic is designed here. It is constituted of a typically cantilever harvester with a sliding mass-magnet free to reciprocate along it. The power source is provided by lateral mechanical vibrations at the root of the beam. The longitudinal motion of the magnetic mass is induced through the internal resonance condition established between its translational frequency and the flapwise and chordwise natural frequencies of the cantilever beam. This dynamic coupling will permit to convert ambient vibrations into electromagnetic energy via the magnet that reciprocates within a surrounding coil. Numerical results show that in the vicinity of resonance, both working frequency bandwidth and output voltage increase, improving the harvestering performance in comparison to other conventional systems. Another particularity of the present design is its compactness, as it can be incorporated inside the vibration structure in case of space limitation. This subsystem can be attached to a parent host, acting dually as a vibration absorber or energy harvester. It is essential to mention that the weak nonlinearity inherent in the present system will permit an effective coupling between flapping and axial modes of vibrations,

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

3

despite the apparent disengagement between them in the linear regime. The basic working principle will be explained in more details subsequently. 2. Derivation of equations This work explores the application of internal resonance in energy harvesting. An archetype of an electromagnetic energy harvester is conceptually designed for this purpose. Consider a composite beam [24] along which a moving magnet translates, as shown schematically in Fig. 1. The flapwise, chordwise and torsional deformation variables along the beam arclength s are denoted as v ðs; t Þ, wðs; t Þ and /ðs; tÞ, respectively. The transducer status which is constrained to translate along the beam is determined by coordinate rðtÞ and electrical current I ¼ dq=dt induced through the coil (Fig. 2). For better clarification, the cantilever together with its internal oscillator consist of a mechanical resonator, the moving magnet reciprocating into the coil is intended to play the role of an electromagnetic transducer and the consuming element is modeled as a resistive load. The equations of motion for the system can be obtained by applying the extended Hamilton’s principle,

Zt2 dI ¼

ðdL þ dwe Þdt ¼ 0:

ð1Þ

t1

where dL and dwe present the variation of the Lagrangian and virtual work of non-conservative forces, respectively. The total Lagrangian is constituted of three parts corresponding to the composite beam, the moving mass and the coil, as follows:

dLðt Þ ¼ dLbeam ðtÞ þ dLmass ðt Þ þ dLcoil ðtÞ; Rl dLbeam ðt Þ ¼ d‘b ðs; tÞds; 0 8  9    > _ 2 þ J n x2n þ J g x2g þ J f x2f  eT ½ke > < m u_ 2 þ m_ 2 þ w = h i ; ‘b ðs; tÞ ¼ 12 > > : ; þk 1  ð1 þ u0 Þ2 þ m02 þ w02

Fig. 1. Schematic of the electromagnetic cantilever harvesting system.

Fig. 2. An electromechanical model description of the magnet-coil interaction.

ð2Þ

ð3Þ

4

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Zl dLmass ðtÞ ¼

d

  1 ! ! ma v m  v m dðs  r Þds; 2

ð4Þ

0

Lcoil ðtÞ ¼

1 2 _  r e Þ; LH q_ þ aqðr 2

_ q_ þ ardq_ þ aqdr; _ dLcoil ðtÞ ¼ LH qd

ð5Þ

where the constraint of beam length inextensibility is enforced into the formulation through the introduction of a Lagrange’s multiplier kðs; tÞ. This assumption will permit to reduce the problem dimension. The first term in Eq. (5) represents the magnetic field coenergy stored in the inductor (coil) and the second corresponds to the magnetic field moving across the coil [25], where a ¼ 2npRc B and B designs the magnetic field intensity, n the number of coil turns, LH the magnetic induction of the coil and Rc the coil radius. The velocity vector of the moving mass is expressed in the inertial coordinate by using the moving frame attached to the beam at current position:

!

_ tÞ þ r_ cosðhÞcosðwÞÞs¼rðtÞ ^ex þ ðv_ ðs; tÞ þ r_ cosðhÞsinðwÞÞs¼rðtÞ ^ey þ ðwðs; _ tÞ  r_ sinðhÞÞs¼rðtÞ ^ez : v m ¼ ðuðs;

ð6Þ

where s ¼ rðtÞ is its current arc length position along the deflected beam. Euler rotation angles h; w are related to beam deflections through kinematics. The dirac delta function in Eq. (4), not to be confused with the variational operator, is employed to represent this term within an integral cast. The virtual-work expression of the non-conservative forces is obtained as

Rl ! ! _ _ _ _  ðRint þ Rload Þqdq þ ðQ v  cv v_ Þdv þ ðQ w  cw wÞdwÞ þ ðQ /  c/ /Þd/Þds dwe ¼ ð F c :d r m dðs  rÞ þ ððQ u  cu uÞdu 0

ð7Þ ! where F c is the external force vector applied to the moving mass. Parameters cu ; cv ; cw ; c/ are damping coefficients and Q u ; Q v ; Q w ; Q / are external forces applied to the beam. Rint ; Rload are the internal resistance of the coil and the resistance of ! ! the external load. The vector r m is the position vector of moving mass in the inertial coordinate system. The vectors F c ! and r m can be expressed as follows,

! F c ¼ f c fðcosðhÞcosðwÞÞs¼rðtÞ ^ex þ ðcosðhÞsinðwÞÞs¼rðtÞ ^ey  ðsinðhÞÞs¼rðtÞ ^ez g;

ð8Þ

! r m ¼ ðrðtÞ þ uðs; tÞÞs¼rðtÞ ^ex þ ðv ðs; tÞÞs¼rðtÞ ^ey þ ðwðs; tÞÞs¼rðtÞ ^ez ;

ð9Þ

where f c ¼ ka rðtÞ is the restitutive spring force magnitude and ka is the spring constant. Substituting Eqs. (2) and (7) into Eq. (1) and setting each of the coefficients duðs; tÞ; dv ðs; tÞ; dwðs; tÞ; d/ðs; tÞ and drðtÞ equal to zero, the governing equations of motion and boundary conditions are obtained. The Lagrange multiplier is then eliminated through algebraic manipulation. It is noteworthy that in the derivation of above equations, the variational process for variables which are function of the  _ v_ ; w, _ keeping in mind that it has to mass position is observed according to relation dDðs; tÞ ¼ dD þ @@sD drs¼rðtÞ ; D ¼ u; v ; w; u; reflect the dependence of s to rðtÞ. The nondimensional equations governing this electromechanical system are finally derived, which details are omitted here for the sake of conciseness.

v€ þ ecv v_ þ b33 v iv þ b13 /000  Jf v€ 00 þ ms v€ dðs  rÞ ¼ efF NV g

ð10Þ

€ þ ecw w _ þ b22 wiv  J g w € 00 þ ms wdðs € w  rÞ ¼ efF NW g

ð11Þ

€ þ ecu u _  b11 /00  b13 v 000 ¼ efF N/ g; Jn u

ð12Þ

ms€r þ ks r ¼ efF Nr g;

ð13Þ

kq I_ þ I ¼ eke r_ ;

ð14Þ

where e << 1 is a small dimensionless parameter used as a bookkeeping parameter. The right hand-sides of Eqs. (10)–(13), F NV ; F NW ; F Nu ; F Nr , that include the nonlinear coupling terms are assumed of order OðeÞ and are defined in the Appendix for the @ and @t@ , respectively. The underlined terms in above equations sake of conciseness. Prime and overdot are shorthands for @s denote what is supplemented to the governing equations due to the translating mass in comparison to the case of a simple beam obtained in [24]. It can be noticed that the nonlinearity is mainly due to the direction change of applied forces due to beam deflection. Furthermore, it is worthy to mention that Eq. (13) which governs the reciprocating motion of the mass is uncoupled to the first order of approximation from other degrees of freedom. Hence the corresponding linear model will fail to predict any energy harvesting potential of the proposed system and nonlinear characteristics should absolutely be exploited.

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

5

In harvesting applications, the external loading is mainly due to the acceleration of the base and can be characterized as an inertial load which depends on the base excitation frequency, X. To conform with the literature, a generic load resistance is selected as the external harvesting circuit. The following non-dimensional parameters are used in the equations: 2 3 J s cv l D11 l ; J n ¼ n2 ; Q v ¼ Q ; ; cv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; b11 ¼ l D33 D33 v m D33 ml

s ¼

v

¼

w ¼

v

2 3 Jg cw l D22 l ; J g ¼ ; Q w ¼ Q ; ; cw ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; b22 ¼ 2 l D33 D33 w mD33 ml

2 J w c/ D13 l ; c/ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; b13 ¼ ; J f ¼ f 2 ; Q / ¼ Q ; l D33 D33 / mD33 ml

qffiffiffiffiffiffiffi 3  ; ks ¼ kDa33l ; t  ¼ Dml334 t; b33 ¼ 1; Q u ¼ qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi    2npðRc BÞ2 l3 D33 D33 LH 1 kt ¼ 2np; kq ¼ R þR 4 ; ke ¼ R D þR 33 ml ml4

r  ¼ rl ; ms ¼

ma ml

int

I ¼

int

load

l3 D33

Q u;

load

Rc BL2 I; D33

ð15Þ

Notice that the superscript (*) is dropped from the equations for the sake of clarity. The corresponding nondimensional boundary conditions are

m ¼ 0; m0 ¼ 0; w ¼ 0;

w0 ¼ 0

b33 m00 þ b13 /0 ¼ eð Bv 1 ðtÞÞ 0

0

at s ¼ 1;

b11 /0 þ b13 m00 ¼ eðB/1 ðt ÞÞ rð0Þ ¼ r 0 ;

at s ¼ 1;

at s ¼ 1;

€ ¼ eðBw2 ðt ÞÞ b22 w000  J g w

/ ¼ 0

at s ¼ 1;

€ ¼ eð Bm2 ðtÞÞ b33 m000 þ b13 /00  J f m b22 w00 ¼ eðBw1 ðt ÞÞ

and

at s ¼ 1;

 dr  ¼ 0; dt t¼0

at

s ¼ 0;

ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ ð21Þ ð22Þ

where functions Bv 1 ðtÞ; Bv 2 ðtÞ; Bw1 ðtÞ; Bw2 ðtÞ; B/ 1 ðtÞ are defined in Ref. [24]. The Galerkin method is applied to separate the spatial and time dependency of the equations through the following expansions:

v ðs; tÞ ¼ /v ðsÞ VðtÞ;

ð23Þ

wðs; tÞ ¼ /w ðsÞ WðtÞ;

ð24Þ

/ðs; tÞ ¼ // ðsÞ EðtÞ;

ð25Þ

where /v ; /w ; // are the shape functions of the cantilever beam, excluding the moving mass. By substituting Eqs. (23)–(25) into Eqs. (10)–(14) and multiplying the equations by the shape functions /v ; /w ; /u then integrating by parts twice, the following equations are obtained for the time-dependent variables:

€ þ cv 2 V_ þ cv 3 V þ cv 4 E ¼ ef cv 1 V V

ð26Þ

€ þ cw2 W _ þ cw3 W ¼ ef ; cw1 W W

ð27Þ

€ þ c/ E_ þ c/ E þ c/ V ¼ ef ; c/1 E 2 3 4 /

ð28Þ

ms€r þ ks r ¼ ef r ;

ð29Þ

kq I_ þ I þ ke r_ ¼ 0;

ð30Þ

6

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Coefficients cv i ði ¼ 1::4Þ; cwi ði ¼ 1::3Þ and c/i ði ¼ 1::4Þ in above equations are evaluated at the equilibrium position, s ¼ re . Functions f V ; f W ; f u ; f r are defined in the Appendix. 3. Multiple scale method The following asymptotic expansion are utilized for solving the Eqs. (26)–(30):

VðtÞ ¼ V 0 ðT 0 ; T 1 Þ þ eV 1 ðT 0 ; T 1 Þ;

ð31Þ

WðtÞ ¼ W 0 ðT 0 ; T 1 Þ þ eW 1 ðT 0 ; T 1 Þ;

ð32Þ

EðtÞ ¼ E0 ðT 0 ; T 1 Þ þ eE1 ðT 0 ; T 1 Þ;

ð33Þ

rðtÞ ¼ r 0 ðT 0 ; T 1 Þ þ er 1 ðT 0 ; T 1 Þ;

ð34Þ

IðtÞ ¼ I0 ðT 0 ; T 1 Þ þ eI1 ðT 0 ; T 1 Þ;

ð35Þ

Eqs. (31)–(35) are substituted into Eqs. (26)–(30), then coefficients of zero and first order are set to zero as follows. Notice that the zero-order equations of the oscillator are decoupled from the dynamics of the beam itself. Order e0

ðcv 1 ÞD20 V 0 þ ðcv 3 ÞV 0 þ ðcv 4 ÞE0 ¼ 0;

ð36Þ

ðcw1 ÞD20 W 0 þ ðcw3 ÞW 0 ¼ 0

ð37Þ

ðc/1 ÞD20 E0 þ ðc/3 ÞE0 þ ðc/4 ÞV 0 ¼ 0;

ð38Þ

ms D20 r 0 þ ks r 0 ¼ 0;

ð39Þ

kq D0 I0 þ I0 ¼ ke D0 r 0 ;

ð40Þ

Order

e1

cv 1 D20 V 1 þ cv 3 V 1 þ cv 4E1 ¼ fV 1 ;

ð41Þ

cw1 D20 W 1 þ cw3 W 1 ¼ fW 1 ;

ð42Þ

cu1 D20 E1 þ cu3 E1 þ cu4V 1 ¼ f u1 ;

ð43Þ

ms D20 r 1 þ ks r 11 ¼ f r1 ;

ð44Þ

kq D0 I1 þ I1 ¼ kq D1 I0  ke D1 r0  ke D0 r 1 ;

ð45Þ

Functions fV 1 ; fW 1 ; f u1 ; fr 1 are brought to Appendix. Solutions of Eqs. (36)–(40) are obtained as follows

V 0 ¼ AðT 1 Þ eixT 0 þ AðT 1 Þ eixT 0 ;

ð46Þ

E0 ¼ BðT 1 Þ eixT 0 þ BðT 1 Þ eixT 0 ;

ð47Þ

W 0 ¼ CðT 1 Þ eiqT 0 þ CðT 1 Þ eiqT 0 ;

ð48Þ

r0 ¼ DðT 1 Þ ðeixr T 0 Þ þ DðT 1 Þ ðeixr T 0 Þ;

ð49Þ

I0 ¼ FðT 1 Þeð1=kq ÞT 0 þ ð

ke ðixr ÞDðT 1 Þ ixr T 0 þ cc; Þe ixr kq þ 1

ð50Þ

where AðT 1 Þ; AðT 1 Þ; BðT 1 Þ; BðT 1 Þ and CðT 1 Þ; CðT 1 Þ; DðT 1 Þ; DðT 1 Þ; FðT 1 Þ; FðT 1 Þ denote general complex variable functions of higher time-scales and their complex conjugates. The parameter x is the natural frequency of the flapwise-torsional mode, q is the natural frequency of the chordwise mode and xr is the natural frequency of the oscillator. q and xr are obtained from the linearized or zero-order equations as follows

7

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

sffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi cw3 ks q¼ ; xr ¼ cw1 ms

ð51Þ

x is evaluated from the existence condition for a non-trivial solution by the following relationship: #    ¼ 0; 2 c/3  x c/1 

"  c v  x2 c v  3 1   c/4

cv 4

ð52Þ

Substituting Eqs. (46)–(50) into the right-hand sides of Eqs. (41)–(45) and separating the secular terms, the following equations can be obtained:

^ v ðT 1 ÞeixT 0 þ cc þ NST; ðcv 1 ÞD20 V 1 þ ðcv 3 ÞV 1 þ ðcv 4 ÞE1 ¼ H

ð53Þ

^ w ðT 1 ÞeiqT 0 þ cc þ NST; ðcw1 ÞD20 W 1 þ ðcw3 ÞW 1 ¼ H

ð54Þ

^ / ðT 1 ÞeixT 0 þ cc þ NST; ðc/1 ÞD20 E1 þ ðc/3 ÞE1 þ ðc/4 ÞV 1 ¼ H

ð55Þ

^ r ðT 1 Þeixr T 0 þ cc þ NST; ms D20 r 1 þ ks r1 ¼ H

ð56Þ

^v; H ^ w; H ^/ where NST and cc stand for the non-secular and complex conjugate of the secular terms, respectively. Functions H ^ and Hr are the coefficients of secular terms which have to be set to zero as follows:

^ v ¼ 0; H ^ w ¼ 0; H ^ / ¼ 0; H ^ r ¼ 0: H

ð57Þ

For the sake of conciseness, the details of these terms have been appended in the Appendix. 4. Modulation equations Modulation equations governing the dynamics of interacting modes are obtained from Eq. (57) for a composite beam with h i lay-up 1006 =4504 =9005 for which the properties are tabulated in Table 1 of Ref. [24]. The modulation equations are derived s

for both cases of flapwise and chordwise excitations as follows. 4.1. Flapwise and chordwise excitation The concept beneath this energy harvester consists of exploring the application of internal resonance. Based on the governing equations, the frequency response at the first primary resonance in the presence of 2:1 or 1:1 internal resonance is investigated in order to maximize the power transfer toward the oscillating mass. The beam is excited harmonically at its clamped end along the v-direction (i.e. flapwise excitation) and also along its w-direction (i.e. chordwise excitation). The detuning parameters r and d are defined for a parametric study in the vicinity of the two-to-one and one-to-one internal resonances with the inner oscillator, for both flapwise and chordwise excitation conditions:

Flapwise excitation Q v ¼ f v X2v cosðXv T 0 Þ; Q u ¼ 0; Q / ¼ 0; Q w ¼ 0; Chordwise excitation Q w ¼ f w X2w cosðXw T 0 Þ; Q u ¼ 0; Q / ¼ 0; Q v ¼ 0; case 1 Xv ¼ xð1 þ erÞ; case 2 Xw ¼ qð1 þ erÞ;

xr ¼ 2xð1 þ e dÞ; xr ¼ 2qð1 þ e dÞ;

ð58Þ

ð59Þ

Table 1 Beam properties. Length Width Thickness Lamina thickness Damping coefficients

l = 0.4572 m b = 0.00953 m h = 0.00381 m hk ¼ 0:000127 m cv ¼ 0:07; cw ¼ 0:13; c/ ¼ 0:0003

Bending and torsional rigidities

D11 ¼ 1:251 N:m2 D22 ¼ 15:864 N:m2 D33 ¼ 4:382 N:m2 D13 ¼ 0:720 N:m2

Density Poisson’s ratios

q0 ¼ 1539:37 kg=m3 t12 ¼ t13 ¼ 0:24 t23 ¼ 0:49

Nondimensional mass moments of inertia

J 1 ¼ 5:78704  106 Jg ¼ 3:62192  105 J n ¼ 4:20062  105 E1 ¼ 132:379 Gpa E2 ¼ E3 ¼ 10:755 Gpa G23 ¼ 3:6059 Gpa D13 ¼ 5:6537 Gpa

Elastic and shear modulus

8

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

case 3 Xv ¼ xð1 þ erÞ; case 4 Xw ¼ qð1 þ erÞ;

xr ¼ xð1 þ e dÞ; xr ¼ qð1 þ e dÞ;

ð60Þ

Substituting Eqs. (58)–(60) into Eq. (57) and doing some algebraic manipulations, one can obtain

2ixðC1 ÞA0 ¼ 2ixC2 A  2C3 ADD  2C4 ACC  3C5 A2 A ðC6 A D2 e2ixdT 1 l1 þ C7 A D e2ixdT 1 l2 þ C8 eixrT 1 ÞlF ; 2iqðK1 ÞC 0 ¼ 2iqK2 C  2K3 CDD  2K4 AAC  3K5 C 2 C ðK6 D2 Ce2iqdT 1 l1 þ K7 CDe2iqdT 1 l2 þ K8 eiqrT 1 ÞlC ; 2ixr X 1 D0 ¼ 2X 2 AAD  2X 3 DCC  ðX 4 A2 De2ixdT 1 l1 þ X 5 A2 e2ixdT 1 l2 ÞlF  ðX 6 DC 2 e2iqdT 1 l1 þ X 7 C 2 e2iqdT 1 l2 ÞlC þ ðiX 8 þ X 9 ÞD;

ð61Þ

ð62Þ

ð63Þ

where l1 and l2 are two tracers identifying the terms associated with the one-to-one internal resonance and the two-toone internal resonance, respectively. l1 = 1 implies l2 = 0 and vice-versa, l2 = 1 requires that l1 = 0. In case l1 = l2 = 0, no internal resonance shall occur. Also, lF and lC are two tracers associated with the flapwise (lF ¼ 1; lC ¼ 0) and chordwise (lF ¼ 0; lC ¼ 1) excitation cases. The coefficients Ci ði ¼ 1::8Þ; Ki ði ¼ 1::8Þ; X i ði ¼ 1::9Þ are defined in the Appendix. Complex variables A; C; D in Eqs. (61)–(63) are converted to polar form using the following change of variables,

AðT 1 Þ ¼

1 1 1 a1 ðT 1 Þeib1 ðT 1 Þ ; CðT 1 Þ ¼ a2 ðT 1 Þeib2 ðT 1 Þ ; DðT 1 Þ ¼ a3 ðT 1 Þeib3 ðT 1 Þ ; 2 2 2

ð64Þ

Substituting Eq. (55) in Eqs. (52)–(54) and after some simplifications, we obtain

a01 ¼

C2 C6 C7 C8 a f a1 a23 sinðt1f Þl1 þ a1 a3 sinðt2f Þl2 þ sinðtf ÞglF C1 1 8xC1 4 x C1 xC1

ða1 t0f  a1 xrÞlF ¼  4xC3C1 a1 a23  4xC4C1 a1 a22  83xCC51 a31  f8xC6C1 a1 a23 cosðt1f Þl1 þ 4xC7C1 a1 a3 cosðt2f Þl2 þ xCC81 cosðtf ÞglF a02 ¼

K2 K6 K7 K8 a2  f a2 a23 sinðt1c Þl1 þ a2 a3 sinðt2c Þl2 þ sinðtc Þglc K1 8qK1 4qK1 qK1

ða2 t0c  qrÞlc ¼  4qKK3 1 a2 a23  4qKK4 1 a21 a2  83qKK51 a32  f8qKK6 1 a2 a23 cosðt1c Þl1 þ 4qKK7 1 a2 a3 cosðt2c Þl2 þ qKK81 cosðtc Þglc a03 ¼ f8xXr4X 1 a3 a21 sinðt1f Þl1 þ 4xXr5X 1 a21 sinðt2f Þl2 glF þ f8xXr6X1 a3 a22 sinðt1c Þl1 þ 4xXr7X 1 a22 sinðt2c Þl2 glc þ X28 a3

ð65Þ

ð66Þ

ð67Þ

ð68Þ

ð69Þ

t0

a3 fð 21f  t0f þ xðr  dÞl1 þ ðt02f  2t0f þ 2xðr  dÞl2 ÞglF t0

X3 2 2 a3 fð 21c  t0c þ qðr  dÞl1 þ ðt02c  2t0c þ 2qðr  dÞl2 Þglc ¼ 4X xr X 1 a1 a3  4xr X 1 a2 a3

f8xXr4X1 a21 a3 cosðt1f Þl1 þ 4xXr5X1 a21 cosðt2f Þl2 glF

ð70Þ

f8xXr6X1 a22 a3 cosðt1c Þl1 þ 4xXr7X1 a22 cosðt2c Þl2 glc þ X29 a3 ; where,

tf ¼ xrT 1  b1 ; tc ¼ qrT 1  b2 ; t1f ¼ 2b3  2b1 þ 2xdT 1 ; t2f ¼ b3  2b1 þ 2xdT 1 ; t1c ¼ 2b3  2b2 þ 2qdT 1 ; t2c ¼ b3  2b2 þ 2qdT 1 ;

ð71Þ

Alternatively, one can express A and C, D in Cartesian form as

AðT 1 Þ ¼ 12 ðp1  iq1 ÞeixrT 1 ; CðT 1 Þ ¼ 12 ðp2  iq2 ÞeiqrT 1 ; DðT 1 Þ ¼ 12 ðp3  iq3 ÞfðeixðrdÞT 1 l1 þ e2ixðrdÞT 1 l2 ÞlF þ ðeiqðrdÞT 1 l1 þ e2iqðrdÞT 1 l2 Þlc g; By separating the real and imaginary parts in Eqs. (61)–(63), one obtain

ð72Þ

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

p01 ¼ xrq1 þ CC21 p1 þ 4xC3C1 ðq1 p23 þ q1 q23 Þ þ 4xC4C1 ðq1 p22 þ q1 q22 Þ þ 83xCC51 ðq1 p21 þ q31 Þ þf8xC6C1 ð2p1 p3 q3  q1 p23 þ q1 q23 Þl1 þ 4xC7C1 ðp1 q3  q1 p3 Þl2 glF ; q01 ¼ xrp1 þ CC21 q1  4xC3C1 ðp1 p23 þ p1 q23 Þ  4xC4C1 ðp1 p22 þ p1 q22 Þ  83xCC51 ðp1 q21 þ p31 Þ f8xC6C1 ð2q1 p3 q3  p1 q23 þ p1 p23 Þl1 þ 4xC7C1 ðp1 p3 þ q1 q3 Þl2 glF ; p02 ¼ qrq2 þ KK21 p2 þ 4qKK3 1 ðq2 p23 þ q2 q23 Þ þ 4qKK4 1 ðq2 p21 þ q2 q21 Þ þ 83qKK51 ðq2 p22 þ q32 Þ þf8qKK6 1 ð2p2 p3 q3  q2 p23 þ q2 q23 Þl1 þ 4qKK7 1 ðp2 q3  q2 p3 Þl2 glc ; q02 ¼ qrp2 þ KK21 q2  4qKK3 1 ðp2 p23 þ p2 q23 Þ  4qKK4 1 ðp2 p21 þ p2 q21 Þ  83qKK51 ðp2 q22 þ p32 Þ f8qKK6 1 ð2q2 p3 q3  p2 q23 þ p2 p23 Þl1 þ 4qKK7 1 ðp2 p3 þ q2 q3 Þl2 þ qKK81 glc ;

9

ð73Þ

ð74Þ

ð75Þ

ð76Þ

p03 ¼ ðxðr  dÞq3 l1  2xðr  dÞq3 l2 ÞlF þ ðqðr  dÞq3 l1  2qðr  dÞq3 l2 Þlc X2 4xr X1

ðp21 q3 þ q21 q3 Þ þ 4xXr3X1 ðp22 q3 þ q22 q3 Þ þ f8xXr4X1 ðq21 q3 þ 2p1 q1 p3  p21 q3 Þl1

þ 2xXr5X1 p1 q1 l2 glF þ f8xXr6X1 ð2p2 q2 p3  q3 p22 þ q22 q3 Þl1 þ 2xXr7X1 ðp2 q2 Þl2 glc ;

ð77Þ

þð2x1r X 1 ÞðX 8 p3  X 9 q3 Þ; q03 ¼ ðxðr  dÞp3 l1 þ 2xðr  dÞp3 l2 ÞlF þ ðqðr  dÞp3 l1 þ 2qðr  dÞp3 l2 Þlc X2 X3 X4 ðq2 p þ p21 p3 Þ  ðp2 p þ q22 p3 Þ  f ðp2 p  q21 p3 þ 2p1 q1 q3 Þl1 4xr X1 1 3 4xr X1 2 3 8xr X1 1 3 X5 X6 X7 1 þ ðp2  q21 Þl2 glF  f ðp p2 þ 2p2 q2 q3  q22 p3 Þl1 þ ðp2  q22 Þl2 glc ; þð ÞðX 8 q3 þ X 9 p3 Þ; 2xr X 1 4xr X1 1 8xr X1 3 2 4xr X1 2 ð78Þ 

The fixed points of the system are solved by setting p01 ¼ q01 ¼ p02 ¼ q02 ¼ p03 ¼ q03 ¼ 0 or a01 ¼ a02 ¼ a03 ¼ t0f ¼ t0c ¼ t01f ¼ t01c ¼ t02f ¼ t02c ¼ 0 into Eqs. (65)–(70). A pseudo arc length scheme is used to trace branches of equilibrium solutions which may indicate losing stability instances due to saddle-point or Hopf bifurcations. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The amplitudes a1, a2 and a3 can be obtained by substitution into relation ai ¼ p2i þ q2i . The stability of fixed points is ascertained by investigating eigenvalues of the Jacobian matrix of the right-hand sides of Eqs. (73)–(78).

5. Numerical results In this section, the steady-state response of a composite beam equipped with a sliding mass as an energy harvester is investigated. Parameters of the energy harvester listed in Table 2 has been designed in order to fulfill the internal resonance conditions; For this example, natural frequencies of flapwise-torsional and chordwise modes are calculated as x ¼ 3:40293157; q ¼ 6:6132826, respectively (See Table 3 in the Appendix).

Table 2 Parameters of the energy harvester. Parameters

Values

ms ks for 1-1 internal resonance ks for 2-1 internal resonance Rint Rload

0.05 0.5789 2.3160 188X

a n Rc LH

1  106 X 7:752 Vs=m 200 0.010 m 0.001 H

10

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Fig. 3. Flapwise 2-1 internal resonance with the transducer.

Fig. 4. Frequency response curves (2-1 internal resonance) for different values of excitation amplitude and d ¼ 0:01. amplitude, a3 amplitude, _ _ _ Unstable amplitudes.

a1 amplitude,

a2

5.1. Flapwise excitation Fig. 3 illustrates the two-to-one internal resonance case. Fig. 4 shows the frequency response curve for two-to-one internal resonance tuned at d ¼ 0:01 under flapwise excitation. Various external excitation amplitudes are considered, illustrated in parts (a)-(d). Fig. 4(a) shows the steady state response for a force amplitude f v ¼ 0:001. The input power is transferred to the moving mass while the detuning parameter is varied from 0.1 to 0.1. The steady state response for f v ¼ 0:005 is sketched in Fig. 4(b) where a jump appears at r ¼ 0:0462. Fig. 4(c) demonstrates two jumps occurring at r ¼ 0:012 and r ¼ 0:115. Two Hopf bifurcations are also figuring at r ¼ 0:00232 and r ¼ 0:00445 and saddle nodes occur at r ¼ 0:00449 and r ¼ 0:0129. Fig. 4(d) presents the steady amplitude versus the detuning parameter for f v ¼ 0:04: Hopf bifurcations appeared at r ¼ 0:0208 and r ¼ 0:0185 while saddle nodes occur at r ¼ 0:0167 and r ¼ 0:0217. A backbone curve is extending to r ¼ 2:87. As seen, the a1 resonance curve bends further to the right with the increase of the force amplitude f v but the a3 curve still presents a narrow-band frequency response, reflecting that the main goal is only partially achieved in this case. Fig. 5 shows the frequency response curve for 2-1 internal resonance occurring at d ¼ 0:05 under flapwise excitation: Fig. 5(a) shows the frequency response for f v ¼ 0:005. The amplitudes are stable in the whole domain. Fig. 5(b) presents the response for f v ¼ 0:04. Hopf bifurcations occur at r ¼ 0:0317 to r ¼ 0:0366. A first jump appears at r ¼ 0:0418. Another Hopf bifurcation occurs at r ¼ 0:0466. The second jump occurs at r ¼ 2:87. Fig. 6 shows the frequency responses at 1-1 internal resonance for d ¼ 0:01 and various force amplitudes under flapwise excitation: In Fig. 6(a), the steady state response is stable in the whole domain. As shown in Fig. 4(b), by slightly increasing the force amplitude, a consistent jump occurs between r ¼ 0:01 and r ¼ 0:0073. By further increasing the excitation force ampli-

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Fig. 5. Frequency response for different values of excitation amplitude and d ¼ 0:05. Unstable amplitudes.

a1 amplitude,

a2 amplitude,

11

a3 amplitude, _ _ _

tude from 0.003 to 0.01, the double jump extends from r ¼ 0:196 to r ¼ 0:179, Fig. 4(c). In Fig. 6(d) for f v ¼ 0:04, a saddle node bifurcation occurs at r ¼ 2:869. Fig. 7 shows the frequency response for 1-1 internal resonance and d ¼ 0:05 under flapwise excitation. Fig. 7(a) and (b) are plotted for f v ¼ 0:005 and f v ¼ 0:04, respectively. Fig. 8 shows the frequency response for 1-1 internal resonance and d ¼ 0:01 under flapwise excitation: In Fig. 8(a), the response is stable in the whole frequency domain. Fig. 8(b) illustrates a single jump appearing at r ¼ 0:115 but the other degree of freedom is not stimulated in either cases (a3 =0). Fig. 8(c) which is plotted for f v ¼ 0:02 indicates a double jump extending from r ¼ 0:82 to r ¼ 0:71 for a1 while the a3 branch presents a single jump inclining to the left. Moreover, the equilibrium points lose stability through a Hopf bifurcation at r ¼ 0:0339 and r ¼ 0:0345. Results reveal the influence of the external excitation amplitude on the amplitude response curves. It should be noticed that both amplitude and resonance range increase substantially with the excitation amplitude. In the absence of experimental data, the authors tried to verify the trend via a numerical investigation; In order to verify the frequency responses of the system under flapwise and chordwise excitations, the amplitudes obtained by the arc-length continuation method got verified through numerical integration of the set of differential Eqs. (65)–(70) via the Runge-Kutta method. The numerical amplitude–frequency response curves (actually only its stable portions) are compared with the analytical amplitude–frequency response curves, shown in Figs. (8, 9, 11) in which the solid lines represent the stable solutions and dash lines are for the unstable branches. Single dots denote results obtained via numerical integration. Different colors have been employed for distinguishing between different curves (a1 , a2 , a3 ).

Fig. 6. Frequency response for different values of excitation amplitude and d ¼ 0:01. Unstable amplitudes.

a1 amplitude,

a2 amplitude,

a3 amplitude, _ _ _

12

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Fig. 7. Frequency response for different values of excitation amplitude and d ¼ 0:05. Unstable amplitudes.

Fig. 8. Frequency response for different values of excitation amplitude and d ¼ 0:01. Unstable, s s s Num. Int.

a1 amplitude,

a1 amplitude,

a2 amplitude,

a2 amplitude,

a3 amplitude, _ _ _

a3 amplitude, _ _ _

5.2. Chordwise excitation In this section, the arclength continuation method is utilized to solve the modulation equations in Cartesian form which are given in Eqs. (65)–(70) for the chordwise excitation case. Numerical results are obtained for one-to-one and two-to-one internal resonances with various excitation amplitude values: Fig. 9 depicts the 1-1 internal resonance case. Fig. 10 shows the frequency response for 1-1 internal resonance with the internal detuning parameter equal to 0.01. As shown in parts (a) and (b), both the chordwise flexural mode (a2 ) and the moving mass oscillation (a3 ) become excited. Fig. 10(b) shows the frequency response for f w ¼ 0:005 and f w ¼ 0:008. Fig. 11 shows the frequency response for 1-1 internal resonance and detuning parameter equal to 0.01. Fig. 8(a) is plotted for f w ¼ 0:005 and f w ¼ 0:008 and Fig. 8(b) is plotted for f w ¼ 0:01 and f w ¼ 0:03. In the backward sweep shown in Fig. 10, a region of relatively large-amplitude oscillations is observed that spreads outward considerably with the force amplitude. Such phenomena are proved to be very desirable for energy harvesting purposes.

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

13

Fig. 9. Chordwise 1-1 internal resonance with the transducer.

Fig. 10. Frequency response for different values of excitation amplitude and d ¼ 0:01 under chordwise excitation (1-1 internal resonance) amplitude, a2 amplitude, a3 amplitude, _ _ _ Unstable, s s s Num. Int.

a1

5.3. Computation of harvested power As demonstrated, each design has a different resonance range and tuning is necessary for catching the internal resonance condition with the highest peak and widest bandwidth. Indeed, the average delivered power has to be considered to evaluate the merits of each design for harvestering from a broadband source. In this section, the power is evaluated for different inter    nal resonance designs by calculating the energy expended in the resistive load through equation jpj ¼ RI20  for 2-1 and 1-1 internal resonances in flapwise and chordwise excitations. Fig. 12 shows the harvested power versus frequency for 1-1 internal resonance in flapwise excitation mode. The values of amplitude force and internal detuning parameter are selected as f v ¼ 0:01 and d ¼ 0:01, corresponding to Fig. 6c. Fig. 13 shows the power extracted from the 2-1 internal resonance case with values f v ¼ 0:04 and d ¼ 0:01 in the flapwise direction. Fig. 14 shows the 1-1 internal resonance case in chordwise excitation mode. The amplitude force and internal detuning parameters are chosen as f w ¼ 0:03; d ¼ 0:01. In fact, the purpose of the energy harvester in question is mainly defined as extracting energy from a broadband source in an optimal manner. The problem should therefore reside in maximizing the area under the power spectrum within the available frequency interval of operation, nevertheless the maximum pick may suffice in case of narrowband sources. Based on the results of Figs. 12–14, it can be obtained that the electrical power generated by the internal resonance harvester depends not only on the harvester itself but also on the external electrical load, and there is an optimal electrical load for which the electrical power harvested by the harvester reaches maximum value under the given exciting ambient vibrations.

14

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

Fig. 11. Frequency response for different values of excitation amplitude at d ¼ 0:01 under chordwise excitation (1-1 internal resonance). amplitude, a2 amplitude, a3 amplitude, _ _ _ Unstable, s s s Num. Int.

Fig. 12. Transmitted power for 1-1 internal resonance (f v ¼ 0:01, d ¼ 0:01) in the flapwise excitation case.

Fig. 13. Power extracted for 2-1 internal resonance (f v ¼ 0:04, d ¼ 0:01) in the flapwise excitation case.

Fig. 14. Transmitted power for 1-1 internal resonance (f v ¼ 0:03, d ¼ 0:01) in the chord excitation case.

a1

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

15

    Let’s calculate the power expended in the resistive load through evaluating equation jpj ¼ RI20  for different resistance   2 values at internal resonance conditions. Since I0  ¼ ðRDc BÞl I0 , the power is rewritten as 33

jpj ¼ Rload ð

2

D33 2

ðRc BÞl

Þ I2 0

ð79Þ

By substituting the expression for intensity obtained in Eq. (50), one gets

  jke xr j I  ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi jDðT 1 Þj; jDðT 1 Þj ¼ ð1=2Þjðp3  iq3 Þj ¼ ð1=2Þja3 j; 0 2 ð1 þ x2r kq Þ

ð80Þ

where the maximum jpj is obtained at resonance xr where the quantity ja3 j reaches its maximum value for the 1-1 or 2-1 internal resonance cases. For given values of LH ; Rint ; B, the dependence of the maximum power on the load resistance can be characterized by substituting into the power expression as follows:

 

jpj ¼ Rload

2 x2r a23  ð2npRc BlÞ D33 LH   4 ; kq ¼ 2 2 Rint þ Rload 4ð1 þ x2r kq Þ ðRnt þ Rload Þ ml

sffiffiffiffiffiffiffiffi D33 ml

ð81Þ

4

which upon substitution results in

 

jpj ¼ Rload

x2r a23  2

4ððRint þ Rload Þ þ x

2

2 L2 D33 Þ r H ml4



ð2npRc BlÞ D33  4; 1 ml

ð82Þ

From Eq. (82), it can be concluded that the electrical power generated by the presented harvester depends on the external vibration characteristics (frequency and amplitude), the mechanical properties of the system (such as natural frequencies), and the electrical properties of the harvesting system. Note that the value of amplitude a3 is also implicitly dependent on the electrical load Rload; Therefore the value of the load has a great impact on the electrical power output of the harvester. As mentioned, it is of great importance to find out the optimal electrical load for a given exciting vibration, that permits to harvest the maximum value of electrical power. Under uniform input conditions, Fig. (15) shows the output power of the proposed electromagnetic harvester for different electrical loads, i.e. 10 kX, 20 kX, 30 kX, 50 kX, 100 kX and 200 kX plotted with respect to frequency variations. It can be seen that the power output of the harvester decreases with the external resistance, without affecting the effective frequency interval. This figure also indicates that further decreasing the external electrical load may lead to instability and hence the maximum output power of the proposed harvester under vibrational excitation is limited by this phenomenon. Results show that the proposed harvester can be designed according to the available broadband excitation source for powering low-power wireless sensors. It should be emphasized that the required output power from the harvester has to be calculated according to the average energy consumption of the wireless device, which may differ whether it is in sleep, transmitting or receiving mode, and also depends on the transmission distance and baud rate [1]. Fig. (15) indicates that, under chordwise internal resonance, the electrical power generated by the proposed harvester is almost linear with respect to the external excitation frequency. Furthermore, the maximum harvested power decreases with the external load resistance but the effective frequency range remains the same. It is worthy to maintain that the maximum harvesting efficiency under broadband source resides in maximizing the area under the power spectrum within the range of available frequencies. The curves in Figs. (10) and (11) present two separated peaks bending to the left and the right. The double jumping leads to broadband vibration-based energy harvesting which also means that tuning the harvester close to its maximum efficiency state is facilitated as the resonance is no more narrow-banded. Meanwhile, the optimal external resistive load for reaching maximum electrical power generation is limited by the instability verge.

Fig. 15. Power versus frequency for different values of electrical load under chordwise internal resonance.

Stable branches,

Unstable branches.

16

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

It is also worth to mention that the method of multiple-scales is known to conveniently apply at small oscillation amplitudes when the nonlinearity is weak. In many figures, however, relatively strong hardening or softening behaviors are shown that were predicted using that method. These results may not be reliable if they are contradicting the main assumption the method is based upon. Hence it is better to support some of the semi-analytical results by numerical integration. 6. Conclusion In order to broaden the frequency bandwidth of energy harvesting devices, an attempt to design a nonlinear prototype able to cover a broad low-frequency range is made, enhancing power extraction from ambient sources compared to traditional linear harvesters. More particularly, the internal resonance is exploited for realizing an energy harvester with desired characteristics. The beam is excited in flapwise and chordwise directions and the sliding mass which dually acts as an energy harvester/energy absorber, is tuned to one-to-one and two-to-one internal resonant conditions. The method of multiple scales is used to obtain the modulation equations describing the interaction between modes. The resulting modulation equations are derived in the first primary resonance for two assumed excitations. The following conclusions are obtained from the amplitudefrequency diagrams: 1) A double-jump behavior appears in frequency responses of flapwise and chordwise excitations, particularly due to inclusion of coil inductance in the analysis, permitting a broadband energy harvesting. 2) Both the amplitude of steady-state responses and resonance range increase in a nonlinear manner with the excitation amplitude in both flapwise and chordwise excitations, which is a desired property for the actual purpose. 3) The design has a two-fold particularity; Firstly, despite that the flexural mode and the transducer are uncoupled to the first-order, an inter-mode energy transfer is permitted via setting proper conditions for internal resonance. Secondly, tuning the harvester close to its maximum efficiency state is facilitated as the resonance is no more narrow-banded due to the apparition of the double jump phenomenon. 4) The optimal external resistive load for reaching maximum electrical power generation may become limited by the instability verge of the system. 5) Deliverable average power can be enough for a vast range of applications. 6) The compactness of proposed harvester permits to incorporate it within the body of main vibrational structures, acting as some hidden vibration absorbers. CRediT authorship contribution statement H. Karimpour: Conceptualization, Methodology, Supervision, Software, Writing - original draft, Writing - review & editing. M. Eftekhari: Software, Visualization, Validation, Investigation, Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix The functions F NV ; F NW ; F Nu ; F Nr in Eqs. (10)–(13) are defined as

F NV ¼ Q v ðtÞ  ðv 0 ðs  1ÞÞ Q u ðtÞ þ Hv ðs; tÞ þ Hv j ðs; tÞ 0

ðA:1Þ

0

F NW ¼ Q w ðtÞ  ðw0 ðs  1ÞÞ Q u ðtÞ þ Hw ðs; tÞ þ Hwj ðs; tÞ

ðA:2Þ

Table 3 Coefficients of Eqs. (61)–(63) and natural frequencies. Present study

Ref. [24]

x = 3.34465975

q = 6.68905

x = 3.34465987

q = 6.68906

C1 C2 C3 C4 C5 C6

K1 K2 K3 K4 K5

C1 C2 C3 C4 C5 C6

K1 K2 K3 K4 K5

¼ 1:000043 ¼ 0:035231 ¼ 41:37863 ¼ 20:78856 ¼ 39:00016 ¼ 0:39148 f v X2v

¼ 1:00008 ¼ 0:064994 ¼ 20:68937 ¼ 20:789006 ¼ 9:27943

¼ 1:00005 ¼ 0:035232 ¼ 41:37849 ¼ 20:78901 ¼ 39:00110 ¼ 0:39149f v X2v

¼ 1:00008 ¼ 0:064994 ¼ 20:68924 ¼ 20:78901 ¼ 9:28130

17

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

F Nu ¼ Q u ðtÞ þ Hu ðs; tÞ; F Nr ¼ kt I þ Hr ðtÞ;

ðA:3Þ

The functions Hv ðs; tÞ; Hw ðs; tÞ; Hu ðs; tÞ are defined in previous work [24] and the functions Hv j ðs; tÞ; Hwj ðs; tÞ; Hr ðtÞ are

n    o € þ ms€r 1  v202 þ w202  2ms r_ v_ 0 v 0 þ ks r 1  v202 þ w202 Hv j ðs; t Þ ¼ v 00 ðs; tÞ ms u s¼rðt Þ

 0    0  ms v_ r_ þ ms€r v 0 þ ms r_ r_ v 00 þ v_ þ ks r v 0 þ v 0 w02 dðs  r ðtÞÞ

ðA:4Þ

n    o € þ ms€r 1  v202 þ w202  2ms r_ v_ 0 v 0 þ ks r 1  v202 þ w202 Hwj ðs; t Þ ¼ w00 ðs; tÞ ms u s¼r ðt Þ n  o 02 0 0 00 0 v 0 _  ks r w þ 2 w dðs  rðt ÞÞ _  ms€r w  ms r_ ðr_ w þ wÞ  ms r_ w

ðA:5Þ

 €  2ms u_ w _ 0 w0  ms v€ v 0  ms ww € 0  ks rw02 s¼rðtÞ Hr ðtÞ ¼ ms u

ðA:6Þ

The functions and coefficients in Eqs. (26)–(30) are defined as

Z f V ¼ f ðE; V; W; ::Þ þ 1=2ð

1

0

Z f W ¼ gðE; V; W; ::Þ þ 1=2ð

1

0

Z f u ¼ hðE; V; W; ::Þ þ 1=2ð

0

uv ðsÞdsÞðf v X2v ÞðeiXv T 0 þ eiXv T 0 Þ;

1

uw ðsÞdsÞðf w X2w ÞðeiXw T 0 þ eiXw T 0 Þ;

ðA:8Þ

uu ðsÞdsÞðf u X2u ÞðeiXu T 0 þ eiXu T 0 Þ;

ðA:9Þ

f r ¼ kðE; V; W; ::Þ þ kt I;

cv 2 cv 3 cv 4

ðA:10Þ

R1 /2v ðsÞds þ J f 0 /02 v ðsÞds þ ms /v ðr e Þ; R1 ¼ cv 0 /2v ðsÞds; R1 ¼ b33 0 /002 v ðsÞds; R1 0 ¼ b13 0 // ðsÞ/00v ðsÞds:

cv 1 ¼

R1 0

R1

R1

0

0

/2w ðsÞds þ J g R1 cw2 ¼ cw 0 /2w ðsÞds; R1 cw3 ¼ b22 0 /002 w ðsÞds:

cw1 ¼

ðA:7Þ

c/1 ¼ J n

R1 0

ðA:11Þ

/02 w ðsÞds þ ms /w ðr e Þ; ðA:12Þ

/2/ ðsÞds;

R1

/2/ ðsÞds; 0 R 1 02 c/3 ¼ b11 0 // ðsÞds; R1 c/4 ¼ b13 0 /02 v ðsÞds: c/2 ¼ c/

ðA:13Þ

Functions fV 1 ; fW 1 ; f u1 ; fr 1 in Eqs. (41)–(44) are

Z

1

f V1 ¼ f 1 ðV 0 ; W 0 ; E0 ; r 0 ; . . .Þ þ 1=2 0

Z f W1 ¼ g 1 ðV 0 ; W 0 ; E0 ; r0 ; . . .Þ þ 1=2 0

Z f u1 ¼ h1 ðV 0 ; W 0 ; E0 ; . . .Þ þ 1=2

uw ðsÞds 

uu ðsÞds

f r1 ¼ 2ms D0 D1 r0 þ k1 ðV 0 ; W 0 ; r0 Þ þ kt I0 ; Coefficients of modulation equations:

  f w X2w eiXw T 0 þ eiXw T 0 ;

1

1

0

  f v X2v eiXv T 0 þ eiXv T 0 ;

uv ðsÞds

f u X2u



eiXu T 0 þ eiXu T 0 ;

ðA:14Þ

ðA:15Þ

ðA:16Þ ðA:17Þ

18

H. Karimpour, M. Eftekhari / Mechanical Systems and Signal Processing 139 (2020) 106614

C1 C2 C3 C4 C5 C6 C7

¼ cv 1  ac/1 ; ¼ ðcv 2 þ2ac/2 Þ ;

¼ x2 cv r2 V€  x2r cv r€rV þ cv r2 V þ x2r cv r_ 2 V ;

¼ cv W 2 V þ acv EW 2  x2 cv VW € 2 þ c/VW 2 þ ac/EW 2 ;

¼ a2 cv E2 V þ cv V 3 þ a3 cv E3 þ acv EV 2  ð4=3Þx2 cv LV þ a2 c/E2 V þ c/V 3 þ ac/EV 2 ;

¼ x2 cv r2 V€ þ xxr cv rr_ V_  x2r cv r€rV þ cv r2 V  x2r cv r_ 2 V ; ¼ x2 cv n1  x2r cv n6  ks cv n7 þ xxr cv n8 ; R1 C8 ¼ ð1=2Þf v X2v /v ðsÞds;

ðA:18Þ

0

K1 ¼ cw1 ; K2 ¼ ðcw2 2 Þ ; K3 ¼ q2 cwr2 W€ þ cwr2 W  x2r cwr€rW þ x2r cwr_ 2 W ; K4 ¼ cwWV 2 þ x2 cwW V_ 2 þ a2 cwE2 W þ acwEVW ; K5 ¼ cwW 3  ð4=3Þq2 cwLW ; K6 ¼ q2 cwr2 W€ þ cwr2 W  x2r cwr€rW  x2r cwr_ 2 W ; K7 ¼ q2 cwn1  x2r cwn6  ks cwn7 ; 2 R1 K8 ¼ f w2Xw 0 /w ðsÞds;

ðA:19Þ

X1 ¼ ms ; X2 ¼ x2 crrV_ 2  x2 cr rV V€ ; X3 ¼ q2 cr rW_ 2 þ cr rW 2  q2 cr rW W€ ; X4 ¼ x2 crrV_ 2  x2 cr rV V€ ; X5 ¼ x2 cr n1  x2 crn2 ;

ðA:20Þ

X6 ¼ q cr rW_ 2 þ cr rW 2  q cr rW W€ ; X7 ¼ q cr n3  q crn4 ; 2

2

2

2

References [1] X. Tang, X. Wang, R. Cattley, F. Gu, A.D. Ball, Energy harvesting technologies for achieving self-powered wireless sensor networks in machine condition monitoring: a review, Sensors 18 (2018) 4113. [2] C.N. Loong, C. Chang, E.G. Dimitrakopoulos, Circuit nonlinearity effect on the performance of an electromagnetic energy harvester-structure system, Eng. Struct. 173 (2018) 449–459. [3] T. Lin, J. Wang, L. Zuob, Efficient electromagnetic energy harvester for railroad transportation, Mechatronics 53 (2018) 277–286. [4] H.L. Dai, Y.W. Yang, A. Abdelkefic, L. Wang, Nonlinear analysis and characteristics of inductive galloping energy harvesters, Commun. Nonlinear Sci. Numer. Simul. 59 (2018) 580–591. [5] S. Zhou, L. Zuo, Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting, Commun. Nonlinear Sci. Numer. Simul. 61 (2018) 271–284. [6] M.F. Daqaq, R. Masana, A. Erturk, D.D. Quinn, On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion, ASME Appl. Mech. Rev. 66 (4) (2014) 040801. [7] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy harvesting via bistable systems, Smart Mater. Struct. 22 (2) (2013) 023001. [8] C. Wei, X. Jing, A comprehensive review on vibration energy harvesting: Modelling and realization, Renew. Sustain. Energy Rev. 74 (2017) 1–18. [9] N. Tran, M. Ghayesh, M. Arjomandi, Ambient vibration energy harvesters: A review on nonlinear techniques for performance enhancement, Int. J. Eng. Sci. 127 (2018) 162–185. [10] Z. Lu, Z. Wang, Y. Zhou, X. Lu, Nonlinear dissipative devices in structural vibration control: A review, J. Sound Vib. 423 (2018) 18–49. [11] S. Dhote, Z. Yang, K. Behdinan, J. Zu, Enhanced broadband multi-mode compliant orthoplanar spring piezoelectric vibration energy harvester using magnetic force, Int. J. Mech. Sci. 135 (2018) 63–71. [12] Q. Lu, L. Liu, F. Scarpa, J. Leng, Y. Li, A novel composite multi-layer piezoelectric energy harvester, Compos. Struct. 201 (2018) 121–130. [13] A.H. Nayfeh, Nonlinear Interactions: Analytical, Computational, and Experimental Methods, Wiley, Hoboken, NJ, 2000. [14] L.Q. Chen, Y.L. Zhang, G.C. Zhang, H. Ding, Evolution of the double-jumping in pipes conveying fluid flowing in the supercritical speed, Int. J. Nonlinear Mech. 58 (1) (2014) 11–21. [15] L.Q. Chen, W. Jiang, Internal resonance energy harvesting, J. Appl. Mech. 82 (3) (2015) 031004. [16] W. Yang, S. Towfighian, A hybrid nonlinear vibration energy harvester, Mech. Syst. Sig. Process. 90 (2017) 317–333. [17] F.M. Foong, C.K. Thein, D. Yurchenko, On mechanical damping of cantilever beam-based electromagnetic resonators, Mech. Syst. Sig. Process. 119 (2019) 120–137. [18] K. Fan, Q. Tan, H. Liu, Y. Zhang, M. Cai, Improved energy harvesting from low-frequency small vibrations through a monostable piezoelectric energy harvester, Mech. Syst. Sig. Process. 117 (2019) 594–608. [19] F.M. Foong, C.K. Thein, B.L. Ooi, D. Yurchenko, Increased power output of an electromagnetic vibration energy harvester through anti-phase resonance, Mech. Syst. Sig. Process. 116 (2019) 129–145. [20] G. Hu, J. Wang, Z. Su, G. Li, H. Peng, K.C.S. Kwok, Performance evaluation of twin piezoelectric wind energy harvesters under mutual interference, Appl. Phys. Lett. 115 (7) (2019) 073901. [21] K. Yang, J. Wang, D. Yurchenko, A double-beam piezo-magneto-elastic wind energy harvester for improving the galloping-based energy harvesting, Appl. Phys. Lett. 115 (19) (2019) 193901. [22] J. Wang, G. Hu, Z. Su, G. Li, W. Zhao, L. Tang, L. Zhao, A cross-coupled dual-beam for multi-directional energy harvesting from vortex induced vibrations, Smart Mater. Struct. (2019), https://doi.org/10.1088/1361-665X/ab5249. [23] J. Wang, S. Zhou, Z. Zhang, D. Yurchenko, High-performance piezoelectric wind energy harvester with Y-shaped attachments, Energy Convers. Manage. 181 (2) (2019) 645–652. [24] H. Karimpour, M. Eftekhari, Exploiting internal resonance for vibration suppression and energy harvesting from structures using an inner mounted oscillator, Nonlinear Dyn. 77 (3) (2014) 699–727. [25] A. Preumont, Mechatronics. Dynamics of Electromechanical and Piezoelectric Systems, Springer, 2006, ULB Active Structures Laboratory.