Journal of Sound and Vibration 331 (2012) 3617–3627
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
Harmonic balance analysis of the bistable piezoelectric
inertial generator
Samuel C. Stanton, Benjamin A.M. Owens n, Brian P. Mann
Department of Mechanical Engineering, Duke University, Durham, NC 27708, USA
a r t i c l e i n f o
abstract
Article history:
Received 5 February 2011
Received in revised form
12 March 2012
Accepted 13 March 2012
Handling Editor: M.P. Cartmell
Available online 13 April 2012
This paper applies the method of Harmonic Balance to analytically predict the
existence, stability, and influence of parameter variations on the intrawell and interwell
oscillations of bistable piezoelectric inertial generator. Existing work on the bistable
piezoelectric harvester in the presence of varying harmonic environmental loading has
been relegated to simulation and experimental analyses. Furthermore, linear piezoelectric behavior and linear damping has always been presumed. This paper improves
upon an existing model for the bistable piezoelectric harvester by incorporating
nonlinear dissipation and cubic softening influences in the electroelastic laminates
before applying analytical methods. A framework for theoretically predicting empirical
observations, such as optimal impedance loads for steady-state motions, is provided as
well as other dynamic considerations such as potential well escape phenomena.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
The bistable piezoelectric inertial generator has recently emerged as a popular mechanism by which one of the
foremost challenges in vibratory energy harvesting, consistent performance in complex spectral environments, may be
solved. However, other strategies such as regenerative optimal control [1], resonant frequency tuning [2], designs for
nonlinear mechanical compliance [3–5], among many others, are gaining momentum as well. The recent review articles by
Zhu et al. [6] and Tang et al. [7] provide concise yet detailed analysis of much of the literature concerning broadband
devices for inertial power generation.
Specifically concerning the bistable design, the last few years have witnessed a dramatic increase in the design,
validation, and analysis of such devices. Shahruz [8], Ramlan et al. [9], and McInnes et al. [10] were among some of the first
to propose exploiting a snap action instability for enhanced vibratory energy harvesting. While all these works provided
novel ideas such as energetic oscillations in the presence of noise [8], robustness to mistuning and low frequency
advantages [9], and stochastic resonance [10], each investigation chose to neglect the coupled electrical network modeling
to emphasize the potential mechanical advantages. This is problematic in view of the fact that the energy transfer and
pooling through harvesting systems is not uni-directional and may impinge otherwise promising results. Also, while Ref.
[10] draws attention to the promising dynamical interaction of stochastic resonance, achieving this phenomenon requires
a dynamic bistable potential well. The authors suggest this could simply be accomplished by applying a harmonic axial
load on a buckled beam, but without consideration of the actuation energy for doing so, the feasibility of realizing this
phenomenon in practice becomes an open question.
n
Corresponding author. Tel.: þ1 512 797 7456.
E-mail address: benjamin.owens@duke.edu (B.A.M. Owens).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jsv.2012.03.012
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S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
Cottone et al. [11] and Gammaitoni et al. [12] provided experimentally validated models with electromechanical
coupling considerations for bistable piezoelectric inertial generators. Cottone et al. demonstrated through numerics and
experimentation the clear advantage of oscillations about a double well potential in the presence of exponentially
correlated noise while Gammaitoni et al. extended the analysis to demonstrate that optimal nonlinear coefficients in the
governing equations could yield marked improvement over the standard linear approach for both bistable and monostable
nonlinear systems. About the same time, Erturk et al. [13] and Stanton et al. [14] theoretically and experimentally studied
harmonically forced bistable piezoelectric generators with an emphasis on broadband response. The device due to Erturk
et al. [13] utilized a ferromagnetic structure buckled by adjacent attracting magnets inspired by Moon and Holmes [15]
while Stanton et al. employed a permanent magnet proof mass oriented in opposition to a fixed magnet to induce the
supercritical bifurcation. Both studies indicate that a harmonic cross-well attractor exists over a broad frequency range
and can achieve similar marked improvement over the linear oscillator as discussed in Refs. [11,12]. The relative
performance of mono and bi-stable potentials was further explored by Masana and Daqaq [16]. Recently, Erturk and Inman
[17] provided a phenomenological discussion on the high-energy orbits of the bistable piezoelectric harvesters to include
ideas for alternative transduction mechanisms. Ferrari et al. subjected the proof mass magnet bistable design to white
noise stochastic excitation both numerically and experimentally and found an 88 percent improvement in the root-meansquare voltage response [18], while Daqaq [19] noted a power enhancement over linear systems for specific potential
configurations in the presence of exponentially correlated Gaussian noise, but none for white noise. Recently, Ando et al.
fabricated and validated all of the preceding advantages in a true MEMS-scale device, thus extending the advantages
predicted and observed at the macroscopic scale towards the microscale regime [20]. Karami and Inman [21] used
approximation methods to explore changes in damping and excitation frequency in nonlinear energy harvesting systems.
In all these studies, linear response within the piezoceramics and linear damping is assumed. Mann and Owens [22]
recently modeled and experimentally studied a bistable electromagnetic harvester and found similar attributes for
harvesting energy from harmonic excitations with varying amplitude and frequency.
The intent of this paper is to shift away from purely numerical analyses that have dominated the bistable piezoelectric
inertial generator literature by (1) applying analytical methods to ascertain performance and (2) enhance the intuition of
the modeling framework by updating the widely applicable model within Ref. [14] to include recent observations
concerning nonlinear elasticity and damping within piezoelectric harvester materials [23–25]. Through analytical studies,
new insight can be obtained, serving as a basis for preliminary design, analysis, and optimization. We begin in Section 2 by
describing the harvester under investigation, add nonlinear piezoelectric effects, and generate a polynomial approximation
for the complex magnetoelastic potential derived in Ref. [14]. Section 3 applies the method of harmonic balance to study
both the existence and stability of low-energy and high-energy orbits as well as the influence of variations in system
parameters. Section 4 discusses the theoretical implications in the context of the energy harvesting problem. Conclusions
follow in Section 5.
2. Nonlinear harvester model
The piezoelectric harvester investigated has a mechanical nonlinearity induced by repelling magnetic dipoles as shown
in Fig. 1. The same design was a subject of thorough numerical and phenomenological discussion within Ref. [14] and
serves as a basis for the analytical study that follows. Upon reaching a critical magnet separation distance between the
dipole affixed to the cantilever as an end mass and the dipole fixed to the inertial reference frame, the harvester will
oscillate between two potential wells due to a pitchfork bifurcation. While a detailed mathematical model was derived
from energy principles in Ref. [14], the nonlinear system was not suitable for approximate analytical studies due to the
complexity of the magnet model. Fortunately, however, the bistable potential well can be sufficiently approximated by a
fourth-order potential function. Before making this approximation, however, we supplement the linear piezoelectricity
z
x
S
x = Lp
RL
z(t)
N
w (L, t)
x=L
Fig. 1. Illustration of the bistable harvester concept.
s
N
S
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
3619
model previously applied with nonlinear elasticity. The reason for doing so is in light of several recent studies
demonstrating the importance of modeling inherent nonlinearities within the piezoelectric laminates [23,24]. These
nonlinear material effects have been shown to be primarily due to higher order elastic effects, with strong nonlinear
damping also present. Furthermore, the addition of a proof mass was shown to incite nonlinear interactions at much lower
excitation thresholds than those for a fixed-free cantilever [25]. Accordingly, the Lagrangian for the first vibration mode of
the harvester in Ref. [14] is improved by including elastic nonlinearities as
2
2
L ¼ 12 mx_ þ f z x_ z_ 12 kx 2 14 gx 4 UðxÞ þ y x_ l_ þ 12C l_ ,
(1)
where xðtÞ is the modal displacement coordinate, l_ ðtÞ is the circuit flux linkage, zðtÞ is an external base motion, m is the
modal mass, fz is the modal base excitation coefficient, k is the modal linear stiffness, g is the nonlinear modal contribution
from the nonlinear elasticity of the piezoceramics, y is the modal coupling coefficient, and C is the series capacitance of the
piezoceramics. The function UðxÞ is the nonlinear potential field due to the magnet interactions. Note that the axial load
imparted by the repelling magnetic load through the cantilever is presumed insignificant in our analysis, although the
bifurcation analysis in Ref. [14] indicated that this would induce a hardening potential field. This phenomenon has recently
been discussed as a tuning strategy for linear piezoelectric harvesters by Mansour et al. [26], who also verified that a
stiffening effect will occur prior to the supercritical pitchfork bifurcation.
The nonconservative generalized forces in the system are due to dissipative mechanisms, namely those due to material
losses in the beam and the electrical impedance load. These are introduced through a dissipation function
D¼
1
2
11 _2
ðda þ db x 2 Þx_ þ
l ,
2
2R
(2)
where R is the electrical impedance, da represents linear damping and amplitude dependent damping is in proportion to db
following [23].
The total potential energy of the harvester is composed of the bending potential of the cantilever and the magnetic
interactions. Fortunately, upon surpassing the separation distance s, shown in Fig. 1, the system becomes bistable and the
total potential energy may be re-written in the form
U tot ¼ 12 ax 2 þ 14bx 4 þ U b ,
(3)
pffiffiffiffiffiffiffiffiffi
where a, b 4 0, the two potential well minima are at x ¼ 7 a=b, and Ub is the potential barrier height. Fig. 2 illustrates the
fourth-order approximation against the exact expression (found in Ref. [14]) for a hypothetical harvester. The approximate
potential field of Eq. (3) can be seen to be accurate in the vicinity of the equilibrium points and near the potential barrier.
Considering the analysis that follows and that bistable harvesters are designed to operate near these regions, this
approximation is sufficient and enables analytical studies.
Applying Lagrange’s equations yields the equations of motion for the harvester as
mx€ þ ðda þ db x 2 Þx_ ax þ bx 3 y v ¼ f z z€ ,
C v_ þ
(4a)
1
v þ y x_ ¼ 0,
R
(4b)
l_ ffiffiffiffiffiffiffiffiffi
to ffi its voltage
These equations can be nonwhere we transform flux linkage p
pffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffi equivalent
pffiffiffiffiffiffiffiffiffi v for convenience.
dimensionalized by substituting t ¼ t a=m, x ¼ x b=a, z ¼ z b=a, and v ¼ v bC =a where (t,x,z,v) are the dimensionless
Magnetoelastic Potential (mJ)
2.84
2.82
2.8
2.78
2.76
2.74
2.72
Ub
2.7
−15
−10
−5
0
5
Tip Displacement (mm)
10
15
Fig. 2. Comparison of the fully nonlinear magnetoelastic potential field (solid line) with the quartic approximation (dotted line). Full potential field
equation found in Ref. [14].
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S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
time, modal displacement, base displacement, and voltage. The dimensionless equations are therefore
_
x€ þ ðma þ mb x2 Þxx
þx3 yv ¼ f ðtÞ,
(5a)
v_ þ mc v þ yx_ ¼ 0,
(5b)
where the overdot now denotes a derivative with respect to dimensionless time. In terms of the original physical
parameters,
rffiffiffiffiffi
rffiffiffiffiffi
da
d
a
1
m
y
, mc ¼
ma ¼ pffiffiffiffiffiffiffi
, y ¼ pffiffiffiffiffiffi ,
(6)
, mb ¼ b
RC a
b m
ma
aC
and f ðtÞ ¼ f z z€ =m.
The rest of this paper investigates the response of the bistable harvester as described by Eqs. (5a) and (5b) to harmonic
excitation.
3. Harmonic balance analysis
This section applies the method of harmonic balance to study both the intrawell and interwell dynamics of the bistable
generator to external harmonic loading. The following analysis is of value in that analytical methods are provided for
characterizing the response of the harvester to a wide range of frequencies and for various parameter variations.
Considering the accelerating interest in this type of design, analytical expressions for the generator response provide for
fast and convenient methods for optimization and preliminary suitability analyses. Furthermore, the response of a strongly
nonlinear harvester to a harmonic load is non-trivial and deserving of a more thorough analysis that does not yet exist in
the literature. Upon introducing a harmonic load of z€ ¼ A cos Ot, the dimensionless equations are modified as
_
x€ þ ðma þ mb x2 Þxx
þ x3 yv ¼ f cos ot,
(7a)
v_ þ mc v þ yx_ ¼ 0,
(7b)
pffiffiffi
pffiffiffiffiffiffiffiffiffiffi
3=2
where o ¼ O m=a is a dimensionless excitation frequency and f ¼ f z A b=a .
In the method of harmonic balance, the harvester response is presumed to be accurately modeled by a truncated
Fourier series, where the number of terms dictates the accuracy of the intended solution. In previous analyses, [13,14,27]
there was strong interest in the large orbit, well-mixing harmonic solutions. This type of motion maintains a dominant
fundamental frequency at the frequency of excitation. Hence, to analytically characterize such motion we assume that the
response of the harvester can be modeled as
x ¼ cðtÞ þ a1 ðtÞsin ot þb1 ðtÞcos ot,
(8a)
v ¼ a2 ðtÞsin ot þb2 ðtÞcos ot
(8b)
x_ ¼ c_ þðb_ 1 þ a1 oÞcos ot þða_ 1 b1 oÞsin ot,
(9a)
v_ ¼ ðb_ 2 þ a2 oÞcos ot þ ða_ 2 b2 oÞsin ot
(9b)
x€ ¼ ð2a_ 1 ob1 Þo cos otð2b_ 1 þ oa1 Þo sin ot:
(9c)
with slowly varying coefficients such that
and
Substituting the above expressions into the dimensionless system, neglecting higher harmonics and balancing constant
terms and those multiplied by sin ot and cos ot, we obtain from the mechanical equation the following:
cðc2 1 þ 32 r 2 Þ ¼ ½ma þ mb ð12r 2 þc2 Þc_ þ mb ca1 a_ 1 þ mb cb1 b_ 1 ,
(10a)
2
_
Q a a1 þ Q b b1 þ ya2 ¼ ½ma þ mb ð34 a21 þ 14 b1 þc2 Þa_ 1 þð12mb a1 b1 2oÞb_ 1 þ 2mb a1 cc,
(10b)
_
Q a b1 Q b a1 þ yb2 þ f ¼ ½ma þ mb ðc2 þ 14 a21 þ 34 b1 Þb_ 1 þ ð12mb a1 b1 þ2oÞa_ 1 þ 2mb b1 cc,
(10c)
Q a ¼ 1o2 þ 3c2 þ 34r 2
(11a)
Q b ¼ o½ma þ mb ðc2 þ 14r 2 Þ
(11b)
2
where
and
and r 2 ¼
2
a21 þ b1 .
Applying the same procedure to the electrical network equation yields
ya_ 1 þ a_ 2 ¼ yob1 þ ob2 mc a2 ,
(12a)
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
yb_ 1 þ b_ 2 ¼ yoa1 oa2 mc b2 :
3621
(12b)
In steady state, all time derivatives vanish so that we can re-write the mechanical amplitude equations as
0 ¼ cðc2 1þ 32r 2 Þ,
(13a)
0 ¼ Q a a1 Q b b1 ya2 ,
(13b)
f ¼ Q a b1 þQ b a1 yb2 ,
(13c)
ob2 þ mc a2 ¼ yob1 ,
(14a)
oa2 mc b2 ¼ yoa1 :
(14b)
and the electrical amplitude equations as
Because Eqs. (14a) and (14b) are a linear system, the electrical coefficients a2 and b2 can be solved for in terms of the
mechanical coefficients as
oy
ðm b oa1 Þ,
o2 þ m2c c 1
a2 ¼
b2 ¼ (15a)
oy
ðob1 þ mc a1 Þ:
o2 þ m2c
(15b)
Substituting the steady-state solutions for a2 and b2 into the steady-state equations for a1 and a2 and squaring and adding
the expressions gives the sixth order in r nonlinear algebraic equation
2
f ¼ r 2 ðL2a þ L2b Þ,
where
"
1
4
#
La ¼ o ma þ mb c2 þ r 2 þ
Lb ¼ 1o
2
1
y2
o2 þ m2c
(16)
mc y2
,
o2 þ m2c
(17a)
3 2
r
4
(17b)
!
þ 3c2 þ
and the frequency response can be determined by numerically finding the positive real roots. Similarly, by squaring and
adding Eqs. (15a) and (15b), the dimensionless response voltage can be written in terms of the mechanical amplitude as
!
s¼
oy
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r:
o2 þ m2c
(18)
The power through the impedance can then be written as P ¼ mc s2 and the subsequent average power can be found over a
period T ¼ o=2p as
!
2
1 mc o2 y
(19)
Pa ¼
r2 ,
2 o2 þ m2c
where r is an implicit function of the forcing amplitude, damping, electromechanical coupling, and electrical dissipation as
derived from the roots of Eq. (16).
3.1. Stability analysis
A stability analysis is required because only a few physically meaningful solutions of the six total roots for r in Eq. (16)
can be realized in practice. To therefore ascertain the stability of solutions, it is first necessary to rewrite Eqs. (10a)–(10c)
as well as Eqs. (12a) and (12b) in the matrix form
Gx_ ¼ FðxÞ,
(20)
T
with the vector x ¼ ½c,a1 ,b1 ,a2 ,b2 . By matrix inversion, Eq. (20) becomes
x_ ¼ PðxÞ,
1
(21)
where PðxÞ ¼ G FðxÞ. Stability can be determined by taking the Jacobian of PðxÞ and calculating its value at the steadystate values for x, which we denote as xss ,
@P
J ¼ :
(22)
@x
x ¼ xss
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S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
The values for xss are known through solutions of Eqs. (13a)–(13c) and Eqs. (15a) and (15b). However, from Eq. (13a) there
exist two types of solutions for c. In particular, oscillations are centered about the origin for c¼0 and their trajectories lie
on manifolds above the potential energy level of the separatrix. For the solution c2 ¼ 1 32 r 2 , the steady-state solutions are
centered within one of the two symmetric potential wells. Accordingly, the stability assessment requires analysis of the
cross-well and intrawell solutions separately.
For motions in high-energy orbits, or interwell oscillations, the procedure for calculating stability is as follows: for a
given forcing amplitude f and frequency o, Eq. (16) with c¼0 yields six possible solutions for r. The values for a1 and b1 can
then be found through the relations
!
a1 ¼
La
f
L2a þ L2b
b1 ¼
Lb
f,
L2a þ L2b
(23a)
and
!
(23b)
which then allows for a2 and b2 to be determined from Eqs. (15a) and (15b). Thus, the Jacobian matrix can be populated
and finding the signs of its eigenvalues allows for stability to be determined.
For oscillations confined to a single well, the preceding discussion applies with the added complication due to the nontrivial solution
c2 ¼ 132r 2 :
(24)
Inserting this expression into Eq. (16) and solving for new solutions for r provides a basis for which all remaining steadystate values for c, a1, b1, a2 and b2 can be determined as previously described. Stable solutions likewise correspond to all
eigenvalues of the Jacobian having negative real parts.
4. Theoretical implementation and energy harvesting implications
The previous section described in detail the theoretical procedure for determining the amplitude and stability of
harmonic solutions for the bistable harvester. That is, while there are many complicated motions ranging from period
doubling cascades en route to chaos, intrawell chaos, interwell chaos, period-n motions to other quasi-periodic motions,
many of which may also coexist [14,15,28–31], the large-orbit nearly harmonic solutions are of considerable interest for
their ease of conversion for power storage, impedance matching, or SSHI circuit implementation [32].
In this section, we apply the harmonic balance solutions derived in the previous section to study the existence and
stability of steady-state harmonic motions, as well as the influence of variations in several model parameters. While the
presence of superharmonics in systems with well-escape and bistable characteristics causes a divergence between
approximate analytical solutions and experimental or numerical studies [28,33–37], they provide initial intuition about
the phenomenological behavior of the system. For this study, the parameter values do not directly correlate to a particular
experimental device; however, the magnet configuration and modeling is inspired by Stanton et al. [14], while the partially
laminated cantilever design is inspired by Erturk et al. [13] and Erturk and Inman [17]. In particular, we simulate a brass
cantilever (length 101.6 mm, width 6.4 mm, thickness 0.25 mm) with bimorph PZT-5H laminates (each with length
25.4 mm, width 6.4 mm, and thickness 0.27 mm) and a 3 g permanent magnet proof mass with a residual flux density of
1.48 T. The PZT-5H laminates are modeled with the linear and nonlinear stiffness coefficients from Ref. [25] as well as the
widely available linear coupling and capacitance characteristics. Thus, the numerical values are representative of a device
that could easily be realized.
Oscillations within one of the two symmetric potential wells are known to be of the softening type [28]. For the two
most common piezoceramics utilized in harvester design, PZT-5A and PZT-5H, material elasticity is strongly softening as
well, so much so that normal first mode hardening oscillations for large amplitude beam motions rarely offset the
piezoelectric softening effect [25]. Therefore, it stands to reason that the intrawell motion of a bistable piezoelectric
harvester will exhibit marked softening nonlinear resonance curves and exhibit linear response for nearly infinitesimal
excitation ranges. Of equal importance to the higher order piezoelectric elasticity influence is nonlinear material damping.
The influence of these combined effects is shown in Fig. 3a. As the nonlinear damping parameter mb is increased, a
predicted saddle-node bifurcation is suppressed and is in agreement with experimental studies in Refs. [23–25]. In Fig. 3b,
the trend in weakening jump phenomena can also be seen for a forcing frequency of o ¼ 1:5. For a dimensionless electrical
impedance of mc ¼ 0:01 (near open circuit conditions) and no nonlinear damping, the intrawell solution goes unstable and
escapes over the potential barrier when f¼0.617. However, for mb ¼ 0:08, which corresponds to the nonlinear resonance
curve in Fig. 3a with no multi-valued solutions, the escape amplitude decreases to f¼ 0.517. However, from Fig. 3a, note
that the low-energy intrawell solutions are stable to larger amplitudes as nonlinear damping increases. Likewise, the highenergy orbits are affected by an increasing level of nonlinear damping. Fig. 4 illustrates the increased forcing threshold for
increasing values of nonlinear damping. Therefore, the presence of nonlinear damping must be taken into consideration in
designing a harvester to exhibit cross-well response in response to harmonic excitation.
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
3623
0.5
0.4
r
0.3
0.6
0.2
0.1
0.5
0
0.05
0.4
0.1
0.15
0.2
0.25
f
r
increasing b
0.3
0.2
0.1
0
1.1
1.2
1.3
1.4
1.5
1.6
ω
1.7
1.8
Fig. 3. Influence of nonlinear damping on suppressing (a) the intrawell frequency response and (b) the intrawell force response curve. Stable solutions
represented by thick lines, unstable by thin. Parameter values are ma ¼ 0:08, mb ¼ ½0:01,0:03,0:08, mc ¼ 0:01, y ¼ 0:9 for both and f ¼0.07 for (a) and o ¼ 1:5
for (b).
1.6
increasing μb
1.4
1.2
r
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
f
0.6
0.7
0.8
0.9
1
Fig. 4. Influence of nonlinear damping on the interwell force response curve. Stable solutions represented by thick lines, unstable by thin. Parameter
values are ma ¼ 0:08, mb ¼ ½0:0,0:05,0:1,0:2, mc ¼ 0:01, y ¼ 0:9, o ¼ 1:1.
The analytic solution also enables convenient analysis of the influence of dissipation through electronics. Still focusing on
the intrawell motion, Fig. 5 illustrates the influence of varying mc for a constant excitation and mechanical damping.
Nonintuitive trends in the frequency response are shown in Fig. 5a for four values of mc . Near open circuit conditions
(mc ¼ 0:01) barely influence the response, while ranges from 0:1o mc o 1:0 are shown to dramatically suppress the resonance
curves. For mc beyond this critical threshold, the electrical network approaches short-circuit conditions and the harvester begins
to mimic the behavior of an uncoupled system. The plot for mc ¼ 10 in Fig. 5a illustrates this trend as the response begins to
rise. Variations in mc also have interesting influence on the respective average power Pa for each frequency. Despite yielding the
most suppressed nonlinear resonance curve of all mc shown, the average power for mc ¼ 1:0 is significantly larger than the
average power for the largest mechanical response when mc ¼ 0:01. It can be inferred that for a certain set of excitation and
mechanical parameters, there exists an optimal impedance load to extract maximum power, which will be discussed shortly.
Erturk and Inman [17] experimentally demonstrated that there exists an optimal impedance load for the steady-state
large orbit dynamics of the bistable generator (see Ref. [17, Fig. 13]). This operating condition can also be theoretically
predicted by the method of Harmonic Balance. Figs. 6 and 7 focus on the high-energy orbit solutions for forcing amplitudes
that drive the system above the potential well barrier for different ranges of mc . Sample amplitude responses are shown in
Figs. 6b and 7b for f¼0.4 and varying frequency, which corresponds to a realistic vibration amplitude. The distinct
advantage in the bistable design lies in the existence of a large-orbit attractor over a very wide frequency range, which in
this case is shown over o ¼ 0:2 to o ¼ 2. For a range of frequencies, however, the stable intrawell harmonic solution does
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S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
x 10
c = 0.01
−3
c = 0.01
7
0.6
c = 10
c = 1
6
0.5
c = 10
5
r
Pa
0.4
c = 0.1
0.3
c = 1
4
3
0.2
c = 0.1
2
0.1
0
1.1
1
1.2
1.3
1.4
1.5
1.6
1.7
0
1.1
1.8
1.2
1.3
1.4
ω
1.5
1.6
1.7
1.8
ω
Fig. 5. Variations in impedance on (a) the intrawell response amplitude and (b) the average power. Parameter values are ma ¼ 0:08, mb ¼ 0:001, y ¼ 0:9,f ¼ 0:07.
0.4
increasing c
(b)
0.35
0.3
Pa
0.25
0.2
0.15
0.1
(a)
0.05
0
0.2
0.4
0.6
0.8
1
ω
1.2
1.4
1.6
1.8
2
Fig. 6. Part (a) depicts the effect of increasing electrical impedance from mc ¼ 0:01 to 0.3 on average power generated by the large orbit motion. Part (b)
shows a frequency response for large forcing and mc ¼ 0:05 such that a large orbit solution manifests. Other parameter values are
ma ¼ 0:05, mb ¼ 0:001, y ¼ 0:9,f ¼ 0:4.
not coexist with the large orbit solution and may almost assure the desired vigorous response in this parameter range.
When there are coexisting solutions, the major design challenge is to realize large-orbit solutions amidst sensitivity to
initial conditions, basins of attraction which may be extremely restricted in some instances, and easily accessible in others,
and even interwell chaotic responses. Figs. 6a and 7a show the average power for the same large-orbit stable response in
Figs. 6b and 7b, but for varying values of mc . Fig. 6a shows values for low mc and indicates a peak power within the given
range. Fig. 6a similarly indicates a peak power, however within a different range of higher mc values. This highlights the
need to tune a system for optimal power generation. Fortunately, the harmonic balance method provides a method for
finding such optimal solutions and is discussed next.
For a given f and o, calculating r from Eq. (16) and subsequently the average power from Eq. (19) for varying mc reveals
an optimal impedance load for a given steady-state oscillation. Since r can be solved for both in the interwell and intrawell
cases, Fig. 8 illustrates typical curves for o ¼ 1:1 and values of f¼0.02 and 0.5 for the intra and interwell solutions,
respectively. For each parameter set, there exist stable intrawell and interwell steady-state responses whose respective
average power is shown in Fig. 8. Although Fig. 8 indicates similar optimal values for mc for the intrawell and interwell
harmonic motions, this is not always the case due to the fact that c ¼0 for interwell solution and c2 ¼ 1 32 r for the
intrawell solution in determining r for Eq. (19). For different values of mc ¼ 0:4, the fully nonlinear equations were
simulated and the average power was calculated over one period upon reaching a steady-state harmonic solution.
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
3625
2
1.8
1.6
1.4
0.4
r
1.2
1
0.35
0.8
0.6
0.4
0.3
0.2
0
0.2
0.4
0.6
0.8
1
Pa
0.25
ω
1.2
1.4
1.6
1.8
2
0.2
0.15
increasing c
0.1
0.05
0
0.2
0.4
0.6
0.8
1
1.2
ω
1.4
1.6
1.8
2
Fig. 7. Part (a) depicts the effect of increasing electrical impedance from mc ¼ 0:4 to 22 on average power generated by the large orbit motion. Part (b)
shows a frequency response for large forcing and mc ¼ 10 such that a large orbit solution manifests. Other parameter values are
ma ¼ 0:05, mb ¼ 0:001, y ¼ 0:9,f ¼ 0:4.
x 10−5
7
6
Pa
5
4
3
2
1
0
0.4
0.35
0.3
Pa
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
c
Fig. 8. Average power curves for o ¼ 1:1 for (a) the cross-well solution and (b) the intrawell solution for two different amplitudes f ¼ 0.02 and f¼ 0.5,
respectively. The circles are values determined from simulation. Other parameter values are ma ¼ 0:03, mb ¼ 0:001, y ¼ 0:9.
3626
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
1.8
0.02
1.6
0.018
0.016
1.4
0.014
Pa
1.2
1
0.012
r
0.01
0.8
0.008
0.6
0.006
0.4
0.004
0.2
0.002
0
0
0.2 0.4 0.6 0.8
1
f
1.2 1.4 1.6 1.8
2
0.2 0.4 0.6 0.8
1
f
1.2 1.4 1.6 1.8
2
Fig. 9. Variations in electromechanical coupling on (a) the interwell response amplitude and (b) the average power. Numerical simulations indicated by
circles with connecting dashed line. Parameter values are ma ¼ 0:08, mb ¼ 0:001, mc ¼ 0:01, o ¼ 1:1.
The simulation results correlate well with the analytical predictions for the small intrawell oscillations, however deviate
for the cross-well solutions. This deviation was expected due to the fact that interwell solutions would also contribute
strong responses at 3o and various integer and non-integer multiples of o, while the intrawell dynamics would contribute
signal energy at 2o, in addition to energy at 3o and other nonlinear resonances.
The level of coupling between the two systems can also affect the response of the system. Fig. 9 illustrates how
variations in electromechanical coupling affects the forcing threshold amplitude for interwell vibrations, as well as the
average power level. Additionally, Fig. 9 includes several numerically simulated values for confirmation of the analytical
results. Variations in numerically simulated and analytical values are the result of the simplifications inherent in harmonic
balance. At low levels of coupling, the forcing threshold is effectively constant, as shown in Fig. 9a, approaching the
interwell response of an uncoupled system as y approaches 0. Small increases in the lower coupling range only serve to
decrease the system response amplitude. Beyond an optimal level, increases in coupling push the forcing threshold up,
destabilizing the interwell solutions at lower forcing levels. The changing coupling has a different effect on average power,
as in Fig. 9b. For low coupling levels, the system produces lower peak power, although with the benefit of a smaller forcing
threshold. For different forcing amplitudes, Fig. 9b illustrates that there exists an optimal coupling coefficient for peak
power. This further highlights the necessity of tuning the system, electrical and mechanical, to the excitation environment.
5. Summary and conclusions
To date, a majority of the work concerning the bistable harvester has been relegated to numerical simulation and
experimental testing. Thus, this paper supplies an analytical framework for predicting the response of the bistable
generator to enable swift evaluation, suitability, and optimization for applications. This paper began by updating the
model in Stanton et al. [14] to include nonlinear piezoelectric effects and reduced the complexity by approximating the
magnetoelastic potential by a quartic function.
This paper analyzed the ability of the harmonic balance method to model key trends in both the mechanical and
electrical domain for the bistable harvester experiencing motions both within and across potential wells. Intrawell motion
is shown to be highly nonlinear and very sensitive to variations in nonlinear damping, impedance loads, and
electromechanical coupling. More importantly, the high-energy orbit is shown to theoretically exist of a very wide
frequency range, especially at lower frequencies, where most ambient mechanical energy is available. When in harmonic
response, there also exists an optimal impedance load and coupling for extracting maximum power for both interwell and
intrawell oscillations. While the method of harmonic balance does ignore the more complicated behavior of the system, it
does serve as an effective first approximation for harvester design and optimization and can also be very accurate in
comparison to simulation results in Figs. 8 and 9.
Acknowledgments
S.C. Stanton would like to particularly acknowledge support received from the Duke University Center for Theoretical
and Mathematical Sciences and all authors gratefully acknowledge financial support from the Army Research Office and
Dr. Ronald Joslin through an ONR Young Investigator Award.
S.C. Stanton et al. / Journal of Sound and Vibration 331 (2012) 3617–3627
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