Coupled optimization of tuned-mass energy harvestersaccounting for host structure dynamics

Paolo Bisegnaa, Giovanni Caruso∗,b, Giuseppe Vairoa

aDepartment of Civil Engineering and Computer Science, University of Rome “TorVergata”, 00133 Rome, Italy

bConstruction Technologies Institute - Italian National Research Council (ITC-CNR),00016 Monterotondo stazione, Italy

Abstract

A tuned-mass electromagnetic energy harvester mounted on a vibrating struc-

ture is here studied, accounting for the dynamic coupling between harvester and

host structure. The latter is modeled as a modal mass-spring-dashpot system,

whereas the harvester device is composed by an electromagnetic transducer

and a tunable secondary mass-spring-dashpot system. A thorough analytical

coupled optimization of the harvested power with respect to the harvester com-

ponents is here presented. Closed-form design formulas are supplied for the

optimal values of the electromagnetic damping coefficient and tuning frequency,

as functions of the excitation frequency, mass ratio, and mechanical damping

coefficients. The optimized parameters yield a wide effective harvesting band-

width when proper values of the mass ratio and device mechanical damping are

chosen. It is shown that neglecting in the design process the dynamic coupling

between harvesting device and vibrating host structure, i.e. treating the latter

as a mere vibration source as generally assumed in the literature, leads to a

significant degradation of the harvesting performance.

Key words: vibration energy harvesting, harvester/host-structure coupling,

tuned-mass harvester, optimal design parameters, effective bandwidth

∗Corresponding author.E-mail addresses: bisegna@uniroma2.it (P. Bisegna); g.caruso@itc.cnr.it (G. Caruso);vairo@ing.uniroma2.it (G. Vairo).

Preprint submitted to Journal of Intelligent Material Systems and Structures (SAGE) July 4, 2013

1. Introduction

Recent advances in the field of micro-electro-mechanical systems (MEMS)

have contributed to the conception of new wireless and low-power remote sen-

sors and actuators. A challenging task in this context is represented by the

development of self-powered MEMS, not requiring batteries replacement, which

can be employed in hostile and/or inaccessible environments. Several strate-

gies and devices have been recently proposed, aiming to harvest energy from

available environmental sources and to convert it into electrical energy (Cook-

Chennault et al., 2008). In particular, energy harvesting from vibrations is

especially attractive, since vibration sources widely occur in a large number of

technical scenarios and physical contexts. Several physical principles can be

exploited to harvest energy from environmental vibrations, the most common

being electrostatic generation (e.g., Kiziroglou et al. (2009); Lee et al. (2009);

Kiziroglou et al. (2010); Le et al. (2012)), piezoelectricity (e.g., Erturk and In-

man (2008); Anton and Sodano (2007); Liao and Sodano (2008); Renno et al.

(2009); Ali et al. (2010, 2011)), and electromagnetic induction (e.g., El-hami

et al. (2001); Stephen (2006); Tang and Zuo (2011)). This paper focuses the

attention on the latter. Most of the relevant contributions existing in the lit-

erature (e.g., Williams and Yates (1996); El-hami et al. (2001); Roundy et al.

(2003); Stephen (2006); Halvorsen (2008); Cassidy et al. (2011)) deal with a

harvester scheme comprising a seismic magnetic mass connected to a housing

by a spring, and moving with respect to a coil which is attached to the hous-

ing. The coil is connected in series with the harvesting circuit, which is usually

modeled by an electric resistance (e.g., El-hami et al. (2001); Roundy et al.

(2003); Stephen (2006)), though more complex electrical circuits have been an-

alyzed (e.g. Zhu et al. (2012) for electromagnetic energy harvesting and Kong

et al. (2010) for piezoelectric energy harvesting). That electro-mechanical ar-

rangement has been conveniently modeled by the classical mass-spring-dashpot

linear system, in which the damper element accounts for both the mechanical

friction and the electromagnetic force. The latter arises from the interaction

2

between the magnetic field produced by the moving magnetic mass and that

generated by the electric current flowing in the coil, and can be generally mod-

eled as a linear viscous contribution (Stephen, 2006). Such a simple mechanical

model can be used to analyze different harvesting scenarios, such as a harvesting

device fixed to a vibrating/moving structure and excited via a base motion, or

a harvesting device directly excited by an external load (e.g., unsteady pressure

loads induced by wind or water flows). In the case of an excitation source that

is harmonic and unaffected by the harvesting process, the maximum power level

which can be extracted by the device is inversely proportional to its mechanical

damping. It is attained when the system is excited at resonance, by choosing

the electromagnetic damping coefficient equal to the mechanical one (Williams

and Yates, 1996; Roundy et al., 2003; Stephen, 2006). The harvestable power

sharply decreases down to vanishing levels with increase of the excitation mis-

tuning. Accordingly, the device allows the harvesting of significant energy levels

only in a narrow frequency band centered around the resonance condition. The

effective bandwidth may be widened by increasing the mechanical damping, but

this amounts at reducing the value of the maximum harvestable power (Stephen,

2006). Possible strategies for improving the harvesting performance have been

recently explored in the literature (Tang et al., 2010). For example, the addi-

tion of a magnetic spring to the standard device (Mann and Sims, 2009; Karami

and Inman, 2011; Erturk and Inman, 2011) introduces nonlinear effects which

can be conveniently exploited in order to increase the harvesting bandwidth. A

similar concept has been recently applied in Friswell et al. (2012), relevant to

the case of a piezoactuated cantilever-beam harvester equipped with a tip mass.

Another strategy is based on the use of multi-degrees-of-freedom harvesting ar-

rangements. That approach has been considered, e.g., by Harne (2013), wherein

such a technique was proved to be effective for achieving a broadband harvesting

efficiency. A similar result has been obtained by Zhou et al. (2011), wherein a

piezoelectric harvester was connected to the vibration source by a multi-mode

oscillating system. Finally, dynamic magnifiers have been proposed, mainly in

the context of piezoelectric harvesting devices, with the aim to improve the

3

amount of the harvested power and the effective bandwidth (Cornwell et al.,

2005; Aldraihem and Baz, 2011; Aladwani et al., 2012). The method consists of

connecting the harvesting oscillating system to the vibration source by means of

a supplementary undamped resonant structure. The latter, suitably tuned on

the vibration source frequency, allows to amplify the strain experienced by the

piezoelectric transducer. A similar concept has been recently applied by Tang

and Zuo (2011, 2012) to the case of an electromagnetic harvesting system based

on a dual mass scheme, which mimics the classical TMD (Tuned Mass Damper)

configuration (e.g., Bisegna and Caruso (2012)) used to damp structural vi-

brations. When the vibration source is a structure, as it is usual in practical

applications, power is harvested at the expense of its vibration energy. Hence,

two desirable targets can be simultaneously pursued: energy harvesting and vi-

bration damping (Lesieutre et al., 2004; Chtiba et al., 2010; Harne, 2013; Tang

and Zuo, 2012; Ali and Adhikari, 2013). Although optimization procedures with

respect to both the requirements could be conceived, optimal strategies are gen-

erally defined addressing either one of them. In order to effectively design the

harvesting/damping device, it is important to take into account the dynamic

coupling between the two systems. This issue, only recently taken into account

in the context of the energy harvester design and optimization (Tang and Zuo,

2011; Harne, 2012; Ali and Adhikari, 2013), deserves further investigations.

In this paper, an oscillating magnetic harvesting device mounted on a given

vibrating structure is analyzed (Section 2), with the aim of maximizing the

harvested power, rigorously accounting for the coupling with the host struc-

ture. The latter is modeled as a modal mass-spring-dashpot system (Fig. 1a),

excited by a harmonic source whose frequency is not at the designer’s disposal.

The harvester is composed by a mass and connecting elements, including the

electromagnetic transducer. The mechanical scheme here considered is simi-

lar to the one studied in Tang and Zuo (2011), but the more realistic case of

presence of damping on the primary mass is herein accounted for. This latter

assumption, as shown in the analysis, removes the singularity in the expres-

sion of the harvested power given in Tang and Zuo (2011), where an infinite

4

level of harvestable power was computed in correspondence of a particular ex-

citation frequency. Moreover, here a different perspective is adopted in the

optimization process. In fact, the harvesting device is intended as a tunable

oscillating secondary system, providing the whole coupled system with an ad-

ditional resonance frequency. The latter has to be properly tuned, depending

on the vibration source frequency, in order to get a wide effective bandwidth of

the harvesting device. Accordingly, a thorough analytical optimization of the

coupled system with respect to the tuning frequency, other than the electrical

damping parameter, is herein presented (Sections 3 and 4). Original closed-form

design formulas for these optimization parameters are obtained, as functions of

the excitation frequency, mass ratio, and mechanical damping coefficients. The

performance of the optimized tuned mass harvester, denoted in the foregoing

with the acronym TMH, is compared with the performance obtained by design-

ing the device under the assumption that it does not affect the dynamics of

the primary vibrating system, treated as a mere vibration source. The latter

case is referred to as TMH-U (Tuned Mass Harvester-Uncoupled), and is briefly

recalled in Section 3.4. A significant performance degradation is demonstrated

when the harvester is optimized according to the TMH-U approach instead of

employing the present coupled optimization. For the sake of comparison, also

the harvesting scheme involving an electromagnetic transducer directly inserted

between the main mass and the ground, without employing a secondary oscillat-

ing system, is considered. In that case, denoted with the acronym DH (Direct

Harvester) and briefly summarized in Appendix, the harvested power is opti-

mized with respect to the electromagnetic transducer parameter. However, such

a solution is seldom feasible in practical applications involving large vibrating

structures. The comparison results show that the TMH can provide the same

power peak of the DH at resonant excitation, but exhibiting a larger effective

bandwidth, which can be significantly widened by adopting small enough val-

ues of the mechanical damping coefficient on the secondary mass and/or large

enough mass ratio. In particular, if the mechanical damping coefficient of the

secondary system approaches zero, a constant power can be extracted for any

5

k1 c1

k2c2

ce

m1

m2

x1

x2

b) c)m1

k1 c1

x1

a)m1

k1c1

ce

x1

Figure 1: a) Modal description of the main vibrating structure. Harvesters configurations:b) Tuned Mass Harvester optimized accounting (TMH) or not (TMH-U) for dynamic cou-pling with the primary system; c) primary system directly equipped with the electromagnetictransducer (DH: Direct Harvester). Gray-shaded regions identify the harvester equipment.

excitation frequency, similarly to the result obtained by Renno et al. (2009),

where a piezoelectric harvesting system connected to a tunable resonant elec-

tric circuit was studied and optimized. The ability of significantly widening the

effective bandwidth of the harvesting device is thwarted if the harvesting device

is designed following the uncoupled optimization approach TMH-U.

2. Modeling assumptions

The harvester device sketched in Fig. 1b (namely, TMH) is herein analyzed.

For the sake of comparison, the DH scheme is also reported in Fig. 1c. With

reference to the notation introduced in Fig. 1, the given vibrating structure

(namely, the main system), is modeled by a modal mass m1, a linear elastic

spring k1, and a linear viscous dashpot c1. The main system is assumed to be

harmonically excited by either a force f(t) = foRe [exp(iωt)] acting on m1 or a

displacement δ(t) = δoRe [exp(iωt)] applied at the basement, where fo and δo

are, respectively, the force and displacement amplitudes, ω is the circular exci-

tation frequency, i is the imaginary unit and t is the time. The harvester is a

secondary system comprising a mass m2, a linear elastic spring k2, and a linear

viscous dashpot c2. The dashpots c1 and c2 take into account the mechanical

friction arising, e.g., within the elastic elements k1 and k2, or any other possible

6

mechanical viscous damping. Power is extracted from the coupled system by

interconnecting the two masses with a linear electromagnetic transducer, char-

acterized by an electromagnetic damping coefficient ce. It is remarked that ce

is an equivalent damping coefficient that allows to describe the damping effect

due to the electromagnetic forces arising within the transducer, whenever the

inductive voltage drop arising in the transducer coils can be neglected with re-

spect to the resistive one arising in the harvesting circuit. The coefficient ce is

given by (Stephen, 2006):

ce =K2

Rint + Rharv, (1)

where K is the transducer electromechanical coupling coefficient, Rint is the

electrical resistance of the transducer coil, and Rharv is the resistance of the

harvesting circuit. Recently, Mann and Sims (2010) and Cui et al. (2013) in-

cluded the inductive voltage drop relevant to the transducer coils in the analysis

of electromagnetic energy harvesters, highlighting situations in which its pres-

ence cannot be neglected in the analysis. The mean power pe absorbed by the

electromagnetic transducer in the steady-state regime of the system is given by

pe =ω

2π

∫ 2π/ω

0

ce(x1 − x2)2 dt , (2)

where x1 and x2 denote the displacement of the masses m1 and m2, respectively,

and a dot denotes time differentiation. As a matter of fact, pe is the sum of

the powers dissipated by the transducer coil resistance and by the harvesting

circuit resistance, but only the latter contribution represents the power really

harvested (Stephen, 2006). Nevertheless, assuming that Rint is much smaller

than Rharv, the mean power pe is a reasonable estimate of the harvested power

(Tang and Zuo, 2011). The harvested power pe will be optimized with respect to

the harvester parameters k2 and ce, keeping fixed the primary mass parameters

m1, c1, k1, and the harvester parameters m2, c2. In order to simplify the

analysis, the following dimensionless quantities are introduced by rescaling the

involved dimensional parameters by scales independent from the optimization

7

parameters k2 and ce:

µ =m2

m1, φ =

ω2

ω1, α =

ω

ω1,

ζ1 =c1

2m1ω1, ζ2 =

c2

2m1ω1, ζe =

ce

2m1ω1, (3)

with ω1 =√

k1/m1 and ω2 =√

k2/m2. Moreover, µ is the mass ratio, φ is the

tuning parameter, α is the dimensionless frequency of the harmonic excitation,

ζ1, ζ2 are the viscous damping coefficients, and ζe is the electromagnetic damping

coefficient. Accordingly, the optimization procedure described in the foregoing

will be performed with respect to φ and ζe, assuming that µ, α, ζ1 and ζ2 are

assigned. Typical values of the latter parameters are in the order of: µ ∼ 10−3–

10−2, α ∼ 10−1–101, ζ1 ∼ 10−3–10−2, and ζ2 ∼ 10−5–10−4, respectively. The

seemingly small value of ζ2 is due to the chosen scaling, involving the primary

mass m1.

3. Harmonic force excitation

The differential equations describing the dynamics of the TMH, excited by

the harmonic force f(t) acting on the main mass m1, are:

m1x1 + c1x1 + k1x1 + (c2 + ce)(x1 − x2) + k2(x1 − x2) = f(t) ,

m2x2 + (c2 + ce)(x2 − x1) + k2(x2 − x1) = 0 . (4)

It is worth noting that in writing (4) it has been implicitly assumed that the

modal shape of the primary system has unitary amplitude where the secondary

oscillating system is attached. Similarly, in order to make consistent the com-

parison between DH and TMH throughout this paper, the magnetic transducer

in the DH scheme (see Appendix) is also assumed to be connected to a point

of the main structure exhibiting unitary modal amplitude. By substituting

x(1,2)(t) = X(1,2) exp(iωt) into (4), and using the dimensionless parameters in-

troduced in (3), the following equations are obtained, describing the system

harmonic response:

−α2X1 + 2iαζ1X1 + X1 + 2iα(ζ2 + ζe)(X1 −X2) + µφ2(X1 −X2) = 1 ,

−µα2X2 + 2iα(ζ2 + ζe)(X2 −X1) + µφ2(X2 −X1) = 0 , (5)

8

where X(1,2) = X(1,2)/(fo/k1). The dimensionless counterpart of the mean

harvested power pe given in (2) is:

Pe =m1ω1pe

f2o

= ζeα2|X2 −X1|2 . (6)

After simple algebra, the solution of (5) yields the dimensionless amplitude of

the displacement |∆X| = |X2 − X1| of the harvester mass m2 relative to the

main mass m1, in the steady-state regime of the system:

|∆X| = µα2

√A (ζ2 + ζe)

2 + B (ζ2 + ζe) + C, (7)

where

A = 4α2{[(1 + µ)α2 − 1]2 + 4α2ζ2

1

},

B = 8µ2α6ζ1 ,

C = µ2{[(1 + µ)α2 − 1]φ2 − α2(α2 − 1)

}2+ 4µ2α2ζ2

1 (α2 − φ2)2 . (8)

Hence, from (6) it turns out that:

Pe =µ2α6ζe

A (ζ2 + ζe)2 + B (ζ2 + ζe) + C

. (9)

3.1. Optimization with respect to φ and ζe

In what follows, the dimensionless mean harvested power Pe is optimized

with respect to the tuning parameter φ and the electromagnetic damping co-

efficient ζe, under the constraint φ ≥ 0, ζe ≥ 0. The dimensionless excitation

frequency α, the mass ratio µ, and the mechanical damping coefficients ζ1 and

ζ2 are regarded as given positive parameters. By noting that Pe = 0 for ζe = 0,

and Pe → 0 for ζe → +∞ or φ → +∞, it follows that the global maximum of

Pe(φ, ζe) is attained either at interior points satisfying the equations

∂Pe

∂φ= 0,

∂Pe

∂ζe= 0 , (10)

or at boundary points satisfying the equations

φ = 0,∂Pe

∂ζe

∣∣∣∣φ=0

= 0 . (11)

9

3.1.1. Interior optimality

Equations (10) yield:

φi = α

√1− µα2[(1 + µ)α2 − 1]

[(1 + µ)α2 − 1]2 + 4ζ21α2

, (12)

and

ζ ie = ζ2 +

µ2α4ζ1

[(1 + µ)α2 − 1]2 + 4α2ζ21

. (13)

The relevant value of the harvested power is given by:

P ie =

116ζ1

1

1 +4ζ1ζ2

µ2α2+

ζ2

ζ1

[(1 + µ)α2 − 1]2

µ2α4

, (14)

and the amplitude of the relative displacement turns out to be:

|∆X i| = 4P ie

α

√4ζ2

1

µ2α2+

[(1 + µ)α2 − 1]2

µ2α4. (15)

It is interesting to compute the above quantities at α = 1 (i.e., α2 − 1 = 0)

and at α = 1/√

1 + µ (i.e., (1 + µ)α2 − 1 = 0), which respectively identify

the resonance frequencies of the main mass m1 if the secondary mass m2 were

removed, or bonded to it. In particular, at α = 1 it turns out that:

φi =2ζ1√

µ2 + 4ζ21

, ζ ie = ζ2 +

µ2ζ1

µ2 + 4ζ21

,

P ie =

116ζ1

1

1 +4ζ1ζ2

µ2+

ζ2

ζ1

, |∆X i| = 4P ie

√1 + 4ζ2

1/µ2 , (16)

whereas at α = 1/√

1 + µ it turns out that:

φi =1√

1 + µ, ζ i

e = ζ2 +µ2

4(1 + µ)ζ1,

P ie =

116ζ1

1

1 +4ζ1ζ2(1 + µ)

µ2

, |∆X i| = 8(1 + µ)µ

ζ1Pie . (17)

The analysis of previous relationships indicate that, owing to the coefficient

ζ1 multiplying P ie in the latter expression of |∆X i|, a significant power can be

harvested at α = 1/√

1 + µ even with limited relative motion between primary

and secondary mass.

10

3.1.2. Boundary optimality at φ = 0

Enforcing equations (11), it turns out that:

ζbe =

√ζ22 +

µ2α2[(α2 − 1)2 + 4α2ζ21 + 8ζ1ζ2α2]

4{[(1 + µ)α2 − 1]2 + 4α2ζ21}

. (18)

The relevant expression of the harvested power P be is obtained by substituting ζe

from (18) into (9). It turns out to be somewhat lengthy and is not reported here

for the sake of brevity. However, the leading-order term of its Taylor expansion

for ζ1 = O(ε) and ζ2 = O(ε), assuming that α2 − 1 6= 0 and (1 + µ)α2 − 1 6= 0,

is:

P be =

µα3

4|α2 − 1||(1 + µ)α2 − 1| + o(1) , (19)

and the leading-order term of the Taylor expansion of the amplitude of the

relative displacement is:

|∆Xb| =√

22|α2 − 1| + o(1) . (20)

The previous expansions do not hold at α = 1 and α = 1/√

1 + µ. Indeed, the

following expansions respectively prevail at those points:

ζbe = ζ1 + ζ2 + o(ε) , P b

e =1

16(ζ1 + ζ2)+ o(1) , |∆Xb| = 4P b

e + o(1) , (21)

and

ζbe =

µ2

4(1 + µ)ζ1+ o(1) , P b

e =1

16ζ1+ o(1) , |∆Xb| = 8(1 + µ)

µζ1P

be + o(ε) .

(22)

These expressions respectively coincide with the leading-order terms resulting

from equations (16) and (17), relevant to interior optimality.

3.1.3. Global Optimality

In order to compute the maximum power P opte harvestable at a given ex-

citation frequency α, the optimality conditions derived above are compared

with each other. Accordingly, the relevant optimal tuning parameter φopt and

electromagnetic damping coefficient ζopte are obtained, together with the cor-

responding amplitude of the relative displacement |∆X|, as functions of α. It

11

Figure 2: Optimality regions. Interior optimality (I): yellow. Boundary optimality (B): green.Separation curve (C): blue. ζ1 = 1 · 10−2 (µcr = 0.0404) and ζ2 = 5 · 10−5.

turns out that the interior optimality prevails whenever φi, reported in equation

(12), is well defined. That occurs at points (µ, α) which satisfy the inequality:

[(1 + µ)α2 − 1](α2 − 1) + 4α2ζ21 ≥ 0 . (23)

The boundary optimality prevails elsewhere. When

µ < µcr := 4ζ1(1 + ζ1) , (24)

inequality (23) turns out to be satisfied for any value of α, and hence interior

optimality prevails irrespective of α. For assigned values of the damping coef-

ficients ζ1 and ζ2, Fig. 2 shows the interior (I) and boundary (B) optimality

regions, separated by the curve C, given by (23) with equality sign. In partic-

ular, on that curve the condition φi = 0 prevails, and its minimum identifies

µcr in (24). For small values of ζ1, the left and right branches of the boundary

curve C respectively approaches the curves α = 1/√

1 + µ and α = 1, as shown

by equation (23). It is worth observing that the case µ > µcr can be retained

as quite unusual in applications involving large structures.

3.2. Further optimization with respect to α

Plots of φopt, ζopte , P opt

e , and relevant |∆X| are reported in Figs. 3, 4, 5, and

6, respectively, as functions of the dimensionless frequency α, for several values

12

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

0.75

1

1.25

1.5

α

φopt

µ=0.005µ=0.01µ=0.015µ=0.02

Figure 3: TMH: optimized tuning parameter φopt vs. dimensionless excitation frequency α,for several values of mass ratio µ. ζ1 = 1 · 10−2 (µcr = 0.0404) and ζ2 = 5 · 10−5.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

10−4

10−3

10−2

α

ζ eopt

µ=0.005µ=0.01µ=0.015µ=0.02

Figure 4: TMH: optimized electromagnetic damping coefficient ζopte vs. dimensionless ex-

citation frequency α, for several values of mass ratio µ. ζ1 = 1 · 10−2 (µcr = 0.0404) andζ2 = 5 · 10−5.

13

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7

α

Peop

t

TMH, µ=0.005TMH, µ=0.01TMH, µ=0.015TMH, µ=0.02DH

Figure 5: TMH: optimized harvested power P opte vs. dimensionless excitation frequency α,

for several values of mass ratio µ. The optimized harvested power relevant to the DH is alsoreported. ζ1 = 1 · 10−2 (µcr = 0.0404) and ζ2 = 5 · 10−5.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

50

100

150

200

250

α

|∆ X

|

µ=0.005µ=0.01µ=0.015µ=0.02

Figure 6: Relative displacement |∆X| relevant to the optimized TMH system vs. dimension-less excitation frequency α, for several values of mass ratio µ. ζ1 = 1 · 10−2 (µcr = 0.0404)and ζ2 = 5 · 10−5.

14

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

30

α

Peop

t

ζ1=2.5⋅10−3

ζ1=5⋅10−3

ζ1=7.5⋅10−3

ζ1=1⋅10−2

1/(16ζ1)

Figure 7: TMH: optimized power P opte versus dimensionless frequency α, for several values of

ζ1. ζ2 = 5 · 10−5, µ = 0.01.

0.8 0.9 1 1.1 1.23.5

4

4.5

5

5.5

6

6.5

α

Peop

t

ζ2=1⋅10−4

ζ2=5⋅10−4

ζ2=1⋅10−3

1/(16ζ1)

Figure 8: TMH: detail of the optimized power P opte versus dimensionless frequency α around

α = 1, for several values of ζ2. The curve segments delimited by diamonds are relevant toboundary optimality. ζ1 = 1 · 10−2 (µcr = 0.0404) and µ = 0.1.

15

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7

α

Peop

t

αmax

ζ2=1⋅10−5

ζ2=5⋅10−5

ζ2=1⋅10−4

ζ2=5⋅10−4

1/(16ζ1)

Figure 9: TMH: optimized power P opte versus dimensionless frequency α, for several values

of ζ2. The optimized harvested power relevant to the DH is also reported. ζ1 = 1 · 10−2

(µcr = 0.0404) and µ = 0.01.

of the mass ratio µ, all below the critical value µcr = 0.0404 relevant to the

choice ζ1 = 1 · 10−2. Therefore, all the curves refer only to interior optimality

case. According to (14), the curves in Fig. 5 exhibit a peak, attained at

αmax =1√

1 + µ− 2ζ21

. (25)

provided that ζ21 < (1 + µ)/2. It is easy to show that for α = αmax and for any

µ the interior optimality condition holds. Hence, from (12)–(15), it turns out

that:

φmax =

√1 + µ/2− ζ2

1√(1 + µ− ζ2

1 )(1 + µ− 2ζ21 )

, ζmaxe = ζ2 +

µ2

4(1 + µ− ζ21 )ζ1

,

Pmaxe =

116ζ1

1

1 +4ζ1ζ2(1 + µ− ζ2

1 )µ2

, |∆X| = 8√

(1 + µ/2− ζ21 )

µφmaxζ1P

maxe .

(26)

In particular, equation (26)3 shows that the power peak increases with the

inverse of the mechanical damping coefficient ζ1, as also depicted in Fig. 7.

When µ > µcr and ζ2 is sufficiently small, another peak appears at α ≈ 1 lying

in the boundary-optimality region, as showed in Fig. 8. Asymptotic expansions

16

of α and the relevant power peak level, for small damping coefficients ζ1 and ζ2,

are given by:

α = 1− 2(ζ1 + 2ζ2)2

µ+ o(ε2) , (27)

Popt

e =1

16(ζ1 + ζ2)

[1 +

4ζ2(ζ1 + 2ζ2)2

µ2(ζ1 + ζ2)

]+ o(ε) . (28)

The power peak Popt

e at α = α, behaving as 1/[16(ζ1 + ζ2)], is lower than

Pmaxe and approaches it when ζ2 → 0, as depicted in Fig. 8. Therefore, Pmax

e

given in (26) is the maximum power which can be extracted over the full range

of α values. It is easy to show that the maximum power Pmaxe , extracted at

α = αmax, is lower than the maximum power (50) which can be extracted by the

DH at α = 1, and approaches the latter as ζ1ζ2/µ2 approaches zero, as depicted

in Fig. 9.

Finally, letting α →∞ into (14), it can be verified that P opte monotonically

approaches from above the value:

P∞e =1

16ζ1

1

1 +ζ2

ζ1

(1 + µ)2

µ2

. (29)

3.3. Effective bandwidth of harvesting

It is of great interest to estimate the effective bandwidth Lb of the TMH,

here defined as the frequency range where the optimized extracted power P opte

is greater than half of its maximum Pmaxe . In formula, Lb = αH−αL, where αL

and αH are, respectively, the smallest and largest real positive roots of

P opte (α) =

Pmaxe

2. (30)

The analytical expressions of αL,H is somewhat lengthy. For the sake of simplic-

ity, the asymptotic expansions of these quantities for ζ1 = O(ε) and ζ2 = O(ε)

are derived. In detail, αL corresponds to interior optimality and its asymptotic

expansion reads as:

αL =1√

1 + µ + µ√

ζ1ζ2

+ o(ε) , (31)

17

0 1 2 3 40

1

2

3

4

5

6

7

α

Peop

t

αL

αH

ζ2=0.5 ζ

2cr

ζ2=2 ζ

2cr

Figure 10: Evaluation of the bandwidth Lb, for several values of ζ2. The horizontal dashed lineindicates the level 1/(32ζ1), approximating the half of Pmax

e provided that ζ1ζ2/µ2 ∼ 10−4

(equation (26)). ζ1 = 1 · 10−2 and µ = 0.05 (ζ2cr = 2.3 · 10−5).

whereas, by setting

ζ2cr = ζ1µ2/(1 + µ)2 , (32)

and referring to Fig. 10, the following cases can be distinguished for the asymp-

totic expansion of αH:

(i) ζ2 ≤ ζ2cr. In that range αH becomes unbounded, since the horizontal

asymptote P∞e given in (29) of the optimized power P opte is greater than

Pmaxe /2, provided that ζ1 <

√1 + µ.

(ii) ζ2cr < ζ2 ≤ ζ1. In that range αH corresponds to interior optimality and

the following expression is obtained:

αH =1√

1 + µ− µ√

ζ1ζ2

+ o(ε) ; (33)

(iii) ζ2 > ζ1. In that range, of limited interest for practical applications, αH

corresponds to boundary optimality and its asymptotic expansion reads

as:

αH =1√

1 + µ+ 2

√2

ζ1

1 + µ+ o(ε) . (34)

18

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7

α

Peop

t

µ=0.005µ=0.0075µ=0.01TMHTMH−U

Figure 11: Optimized harvested power P opte vs. dimensionless excitation frequency α, for

several values of mass ratio µ. Comparison between the TMH and TMH-U performances.ζ1 = 1 · 10−2 and ζ2 = 5 · 10−5.

3.4. Uncoupled optimization of the TMH

In this section, referring to Fig. 1b, the assumption that the harvester does

not affect the dynamics of the primary vibrating system is enforced (Stephen,

2006). Neglecting the coupling with the TMH (i.e., with the secondary sys-

tem), the first equation in (4) reduces to the differential equation describing the

dynamics of the main system excited by the harmonic force f(t):

m1x1 + c1x1 + k1x1 = f(t) , (35)

which, under stationary hypothesis and in dimensionless form, reads as:

−α2X1 + 2iαζ1X1 + X1 = 1 . (36)

This yields:

X1 =1

1− α2 + 2iαζ1. (37)

The latter quantity is now regarded as an imposed motion of the basement of the

harvester device. It is substituted in the second of (5), governing the dynamics

of the TMH, which in turn yields X2. Those values are then substituted into

(6), yielding the harvested power P unce :

P unce =

µ2α6ζe

[(α2 − 1)2 + 4ζ21α2][µ2(α2 − φ2)2 + 4α2(ζ2 + ζe)2]

, (38)

19

Its optimization is straightforward, yielding:

φ = α , ζe = ζ2 , P unce =

116ζ1

14ζ1ζ2

µ2α2+

ζ2

ζ1

(α2 − 1)2

µ2α4

. (39)

It is pointed out that the value of P unce derived above is correct under the as-

sumption that the main mass dynamics is unaffected by the harvester device.

As a matter of fact, the power really harvestable from the coupled system opti-

mized under uncoupled assumption is obtained by substituting the values of φ

and ζe from (39) into (9). It turns out to be:

P unce =

116ζ1

1

1 +4ζ1ζ2

µ2α2+

ζ2

ζ1

[(1 + µ)α2 − 1]2

µ2α4+

µ2α2

16ζ1ζ2

. (40)

It is interesting to compare (40) with (14), obtained by using the values of φ and

ζe respectively given by (12) and (13), resulting from the optimization of the

coupled system. It appears that the coupled optimization is mandatory when-

ever the extra term (µ2α2)/(16ζ1ζ2) in the denominator of (40) is not negligible

with respect the remaining ones. This feature is highlighted in Fig. 11, by com-

paring the optimized harvested power P opte with the power P unc

e harvestable

under uncoupled assumption.

4. Harmonic base excitation

The differential equations describing the evolution of the TMH in the time

domain, when the main structure is excited by a harmonic displacement acting

on its basement, are:

m1x1 + c1x1 + k1x1 + (c2 + ce)(x1 − x2) + k2(x1 − x2) = −m1δ(t) ,

m2x2 + (c2 + ce)(x2 − x1) + k2(x2 − x1) = −m2δ(t) .

The harmonic response is governed by the following dimensionless equations:

−α2X1 + 2iαζ1X1 + X1 + 2iα(ζ2 + ζe)(X1 −X2) + µφ2(X1 −X2) = α2 ,

−µα2X2 + 2iα(ζ2 + ζe)(X2 −X1) + µφ2(X2 −X1) = µα2 ,

20

where X(1,2) = X(1,2)/δo. The extracted mean power pe harvested in the steady-

state regime, given by (2), is normalized as by Stephen (2006) yielding the

following dimensionless expression:

Pe =pe

δ2oω3m1

=ζe

α|X2 −X1|2 (41)

Accordingly, the dimensionless mean harvested power is given by:

Pe =µ2α3(1 + 4ζ2

1α2)ζe

A (ζ2 + ζe)2 + B (ζ2 + ζe) + C

(42)

where the quantities A, B and C are the same as in (8). The harvested power

(42) coincides with the expression in (6), relevant to the force excitation case,

up to the factor (1 + 4ζ21α2)/α3 not depending on the optimization parameters

φ and ζe. As a consequence, their optimal expressions coincide with the ones

derived in the case of force excitation. Accordingly, the optimized power for

base excitation exhibits features similar to the force excitation case, previously

studied. For the sake of conciseness, further details are omitted.

5. Discussion

The optimization results derived in the previous sections highlight the fea-

tures of the TMH harvester, by also allowing the comparison with other har-

vesting approaches, as the TMH-U (Section 3.4) and DH (Appendix). The

maximum harvestable power relevant to the TMH system occurs when the sys-

tem is excited at the frequency αmax given in (25). Assuming small values of the

mechanical damping ζ1, that frequency is very close to 1/√

1 + µ, which would

be the resonant frequency of the system if m2 were bonded to m1. At that

excitation frequency αmax, the power peak Pmaxe given in (26) is attained. It is

lower than the value 1/(16ζ1), which is the maximum power harvestable by the

DH approach (equation (50)), and approaches the latter when ζ1ζ2/µ2 << 1,

i.e., when the mass ratio µ is large enough, and/or the harvester mechanical

damping coefficient ζ2 is small enough, for a given ζ1 (Fig. 5). In the limiting

case of vanishing mechanical damping on the primary mass, i.e., ζ1 = 0, from

21

(26), or equivalently from (17), it can be also observed that the maximum har-

vestable power at excitation frequency αmax|ζ1=0 = 1/√

1 + µ goes to infinity

and correspondingly ζmaxe → ∞. A similar conclusion was observed by Tang

and Zuo (2011), studying a dual mass harvester, whose dynamical behavior is

described by equations (5) setting ζ1 = 0. As a matter of fact, in the more

realistic case of a finite (even very small) value of ζ1, the above singularity is re-

moved, and the harvested power provided by the TMH device is upper bounded

by 1/(16ζ1) and approaches zero for ζe →∞ for any value of α (equation (9)).

Fig. 4 reports the behavior of the optimal value of ζe. It exhibits a peak value

at αmax (equation (26)), sharply increasing when µ increases. As a consequence,

the use of too large values of µ could be in contrast with practical viability of

the device, implying the use of large-size electromagnetic transducers. The di-

mensionless relative displacement amplitude |∆X| between the masses m1 and

m2 is reported in Fig. 6. It shows that an antiresonance appears close to the

optimal excitation frequency αmax. That figure also shows that |∆X| exhibits

peaks ranging from 150 to 200. These values do not jeopardise the feasibility of

the harvesting device, since they correspond to a dimensional relative displace-

ment of 150÷ 200 times the static deflection fo/k1 of the main structure, being

usually very small in practical applications. Furthermore, when α is far from the

resonance frequency region (1/√

1 + µ, 1) the required optimal values of ζe and

|∆X| sharply decrease from their peak values, as Figs. 4 and 6 reveal, favoring

the feasibility of the optimized TMH in that frequency range. The main advan-

tage of using a tunable oscillating harvester relies on the feasibility of extracting

an appreciable power level over a wide excitation frequency band. As Figs. 5

and 9 highlight, the effective bandwidth of the TMH is significantly widened by

increasing µ and decreasing ζ2, respectively, outperforming over the DH, whose

effectiveness is limited to excitation frequencies very close to the resonance fre-

quency α = 1 of the main system. In particular, if the condition ζ2 < ζ2cr (equa-

tion (32)) is satisfied, the effective bandwidth of the TMH becomes unbounded,

and a significant power level can be extracted for any α > αL. Moreover, as

ζ2 approaches zero, the optimized power P opte relevant to the interior optimal-

22

10−3

10−2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

µ

Lb·P

max

e

ζ2=1⋅10−5

ζ2=5⋅10−5

ζ2=1⋅10−4

TMHTMH−UDH

Figure 12: Performance parameter LbPmaxe vs. mass ratio µ for several values of ζ2. Com-

parison among TMH, TMH-U and DH performances. ζ1 = 1 · 10−2.

ity region becomes independent of α and equal to its maximum value 1/(16ζ1)

(equation (14)). A similar behavior was observed by Renno et al. (2009), where

a resistive-inductive electric circuit connected to a piezoelectric device was cho-

sen as a secondary oscillating system to be tuned on the excitation frequency.

The wide effective bandwidth of the TMH is achieved as a result of the ad-

ditional resonance frequency provided by the secondary harvesting oscillating

system, to be tuned depending on the vibration source frequency. This point can

be further highlighted by considering the limiting case of vanishing mechanical

damping coefficients. In that case, equation (12), prevailing outside the reso-

nance frequency region (1/√

1 + µ, 1), amounts to making one of the undamped

natural frequencies of the coupled TMH system coincide with the dimensionless

excitation frequency α. The TMH performances are compared with the TMH-

U’s in Fig. 11, in terms of harvested power. The TMH-U performances have

been optimized under the simplifying hypothesis, usually assumed in previous

studies dealing with single-degree-of-freedom harvester excited at the basement,

that the dynamics of the main mass m1 is unaffected by the oscillating mass m2

(i.e., by setting φ and ζe respectively equal to α and ζ2, equation (39)). It turns

out that, even for quite small values of µ, the dynamic interaction between the

23

two masses cannot be considered as negligible. In those cases, the optimization

based on the uncoupled assumption yields significantly worse performance than

the fully coupled optimization. This effect is due to the term (µ2α2)/(16ζ1ζ2),

appearing in the denominator of (40) (TMH-U) and lacking in the denominator

of (14) (TMH). That term could be made small around the resonance frequency

region, by choosing very small values of µ and/or large mechanical damping

coefficients; but it becomes anyway important as α increases, sharply reducing

the effective bandwidth of the TMH-U system compared to the TMH. Finally,

the TMH, TMH-U and DH are compared in terms of the overall performance

parameter LbPmaxe , that gives a straight indication on both the effective band-

width and harvestable power level. In particular, Fig. 12 highlights the influence

of the mass ratio µ and of the secondary mechanical damping ζ2 on LbPmaxe .

It turns out that the TMH outperforms the DH if µ is large enough and/or ζ2

is small enough, in accordance with results shown in Figs. 5 and 9. Moreover,

the TMH scheme exhibits values of the performance parameter undoubtedly

better than the TMH-U’s, revealing that a dramatic performance enhancement

can be achieved if the optimization process is performed taking into account

the dynamic coupling between the main structure and the harvesting oscillating

device. This effect is significant even for values of the mass ratio in the order of

some thousandths.

6. Concluding remarks

The optimization of an oscillating magnetic harvesting device mounted on a

given vibrating structure has been performed, accounting for the dynamic cou-

pling between the harvester and the primary structure. The structure, modeled

as a modal mass-spring-dashpot system representing its main vibration mode,

was harmonically excited by either a force or a vibration applied at the base-

ment. The harvester, composed by a secondary mass-spring-dashpot system

equipped with an electromagnetic transducer, was considered as a tunable os-

cillating system to be optimized depending on the excitation frequency. The

main original contribution of this paper is the analysis of the effects related

24

to the dynamic coupling between the harvester and the main structure in the

optimization of the harvester device. It was proved that significantly enhanced

harvesting performances can be obtained with respect to those obtained under

uncoupled assumption, i.e. considering the main structure as a mere vibration

source unaffected by the harvesting process. The analytical optimization pro-

cedure yielded original closed-form design formulas for the optimal electromag-

netic damping coefficient and tuning frequency, as functions of the excitation

frequency, mass ratio, and mechanical damping coefficients. It was shown that

the maximum power can be extracted when the excitation frequency is close to

the resonant frequency of the main structure with the harvester mass bonded

on it. The optimized device can exhibit a wide effective bandwidth if the addi-

tional resonance frequency provided by the harvesting system is properly tuned

on the vibration source frequency, thus enhancing the extracted power over a

wide frequency range. Moreover, it turned out that the effective bandwidth of

the harvesting system can be significantly widened by reducing the mechanical

damping on the secondary mass, becoming theoretically unbounded when the

secondary mechanical damping is below a critical value depending on the mass

ratio. This possibility is thwarted if the harvester/main-structure dynamic cou-

pling is neglected in the optimization process. The proposed analytical results

open to the possibility of developing a semi-active control system able to opti-

mize in real time the TMH performance as a function of the excitation harmonic

content.

Acknowledgements

The Authors express their sincere gratitude to Professor Franco Maceri for

valuable comments on this work. This work was developed within the framework

of Lagrange Laboratory, a European research group comprising CNRS, CNR,

the Universities of Rome “Tor Vergata”, Calabria, Cassino, Pavia, and Salerno,

Ecole Polytechnique, University of Montpellier II, ENPC, LCPC, and ENTPE.

25

Funding

This work was supported by MIUR [PRIN, grant number F11J12000210001],

and by Italian Civil Protection Department [RELUIS-DPC 2010-13].

Appendix: Direct Harvester (DH)

In this section the scheme involving an electromagnetic transducer directly

inserted between the main mass and the ground without employing a secondary

oscillating system is considered (Fig. 1c). The differential equation describing

the evolution of the primary mass m1, excited by a harmonic force and acted

upon by the transducer, is:

m1x1 + (c1 + ce)x1 + k1x1 = f(t) , (43)

which, under stationary hypothesis and in dimensionless form, reads as:

−α2X1 + 2iα(ζ1 + ζe)X1 + X1 = 1 . (44)

The harvested power can be written as

pe =ω

2π

∫ 2π/ω

0

cex21 dt =

ce

2ω2|X1|2 . (45)

The same quantity, written in dimensionless form, reads as:

Pe =m1ω1pe

f2o

= ζeα2|X1|2 . (46)

Substituting the solution of (44) into (46) yields

Pe =α2ζe

(α2 − 1)2 + 4α2(ζe + ζ1)2. (47)

The optimal ζe maximizing (47) is:

ζopte =

√ζ21 +

(α2 − 1

2α

)2

. (48)

The relevant expression of the optimized extracted power reads as:

P opte =

18(ζ1 + ζopt

e ). (49)

26

It is straightforward to verify that P opte exhibits a maximum for α = 1, when

ζe = ζ1, equal to (e.g., Williams and Yates (1996); El-hami et al. (2001); Stephen

(2006); Halvorsen (2008); Tang and Zuo (2011); Cassidy et al. (2011)):

Pmaxe =

116ζ1

. (50)

The excitation frequencies αL and αH where P opte is reduced at Pmax

e /2 turn

out to be:

αL,H =

√1 + 16ζ2

1 ∓ 4ζ1

√2 + 16ζ2

1 = 1∓ 2ζ1

√2 + o(ζ1) , (51)

so that the effective bandwidth Lb of the DH is:

Lb = αH − αL = 4ζ1

√2 + o(ζ1) ' 1

2√

21

Pmaxe

. (52)

Thus, a tradeoff between maximum harvestable power Pmaxe and effective band-

width Lb is unavoidable, resulting in a constant performance parameter LbPmaxe .

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