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Nonlinear DynamicsAn International Journal of NonlinearDynamics and Chaos in EngineeringSystems ISSN 0924-090XVolume 73Combined 1-2 Nonlinear Dyn (2013) 73:853-867DOI 10.1007/s11071-013-0837-4
Application of piezoelectric actuationto regularize the chaotic response of anelectrostatically actuated micro-beam
Saber Azizi, Mohammad-Reza Ghazavi,Siamak Esmaeilzadeh Khadem, GhaderRezazadeh & Cetin Cetinkaya
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Nonlinear Dyn (2013) 73:853–867DOI 10.1007/s11071-013-0837-4
O R I G I NA L PA P E R
Application of piezoelectric actuation to regularizethe chaotic response of an electrostatically actuatedmicro-beam
Saber Azizi · Mohammad-Reza Ghazavi ·Siamak Esmaeilzadeh Khadem ·Ghader Rezazadeh · Cetin Cetinkaya
Received: 24 September 2012 / Accepted: 18 February 2013 / Published online: 14 March 2013© Springer Science+Business Media Dordrecht 2013
Abstract The impetus of this study is to investi-gate the nonlinear chaotic dynamics of a clamped–clamped micro-beam exposed to simultaneous elec-trostatic and piezoelectric actuation. The micro-beamis sandwiched with piezoelectric layers throughout itslength. The combined DC and AC electrostatic actu-ation is imposed on the micro-beam through two up-per and lower electrodes. The piezoelectric layers areactuated via a DC electric voltage applied in the di-rection of the height of the piezoelectric layers, whichproduces an axial force proportional to the appliedDC voltage. The governing differential equation ofthe motion is derived using Hamiltonian principle anddiscretized to a nonlinear Duffing type ODE usingGalerkin method. The governing ODE is numerically
S. Azizi · M.-R. Ghazavi (�) · S. Esmaeilzadeh KhademTarbiat Modares University, Tehran, Irane-mail: [email protected]
S. Azizie-mail: [email protected]
S. Esmaeilzadeh Khademe-mail: [email protected]
G. RezazadehUrmia University, Urmia, Irane-mail: [email protected]
C. CetinkayaMechanical and Aeronautical Engineering Department,Clarkson University, Potsdam, NY, USAe-mail: [email protected]
integrated to get the response of the system in terms ofthe governing parameters. The results show that the re-sponse of the system is greatly affected by the amountsof DC and AC electrostatic voltages applied to the up-per and lower electrodes. The results show that the re-sponse of the system can be highly nonlinear and insome regions chaotic. Evaluating the K–S entropy ofthe system, based on several initial conditions given tothe system, the chaotic response is distinguished fromthe periodic or quasiperiodic ones. The main objectiveis to passively control the chaotic response by applyingan appropriate DC voltage to the piezoelectric layers.
Keywords Duffing equation · Chaotic dynamics ·MEMS · Piezoelectric layers · Electrostatic actuation
1 Introduction
Analysis, modeling and experimental results relatedto the nonlinear behavior of MEM/NEM devices havenumerously been reported [1–10]. Wang et al. ob-served chaotic response in a bistable MEMS; theypresented theoretical analysis to demonstrate the exis-tence of a strange attractor and performed model ver-ification using experimental data [1]. Luo and Wangstudied the chaotic response of a mechanical modelfor MEMS with time-varying capacitors; they re-ported chaotic response in a certain frequency bandof the MEM device; the chaotic motion is also in-vestigated in the vicinity of a specified resonant sep-aratrix both analytically and numerically [2]. Liu et
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854 S. Azizi et al.
al. discovered period doubling and chaos in a simu-lated MEMS cantilever system with electrostatic sens-ing and actuation [11]. The period doubling route tochaos was also reported in refs. [1, 4, 12]. Sudipto etal. published a paper on the nonlinear dynamic prop-erties of electrostatically actuated microstructures un-der superharmonic excitations using numerical simu-lations; banded chaotic response was observed duringthe period doubling bifurcation. Rhoads et al. stud-ied the dynamic response of a class of electrostaticallydriven MEM oscillators [12]; cubic type of nonlin-earity due to the nonlinear spring and time-varyinglinear and nonlinear stiffness due to electrostatic ac-tuation were included in their formulation. DeMar-tini et al. [13] studied the chaotic behavior of a sin-gle degree of freedom MEM oscillator, governed bynonlinear Mathieu type oscillator. Using Melnikov’smethod they described the region of parameter spacewhere the response was chaotic. In the literature, Mel-nikov’s method is applied in several published papersto investigate the chaotic response [13–15]. Shabaniet al. investigated the development of superharmon-ics and chaotic response in an electrostatically ac-tuated torsional micro-mirror near pull-in condition[16]. They reported DC and AC symmetry breaking intheir model, which led to chaotic response by increas-ing the amplitude of the harmonic excitation. DC andAC symmetry breaking in MEM devices was previ-ously reported by De and Aluru [4]. Controlling thechaotic response of the MEM devices and regular-izing it is one of the outstanding targets of the re-cently published papers. Chavarette et al. [17] stud-ied the same mathematical model previously proposedby Luo and Wang [2] and controlled the chaotic re-sponse of a periodic orbit using optimal linear controltheory. Polo et al. [18] studied the nonlinear chaoticbehavior of a MEM device. Their model was a 2DOFmechanical system including a nonlinear spring withcubic nonlinearity, whose equations of motion endedup in a coupled Duffing type ODEs. They controlledthe chaotic response applying static output feedbackand geometric nonlinear control strategies. Haghighiand Markazi [15] proposed a MEM SDOF systemwith electrostatic actuation on both sides of the proofmass. Using Melnikov’s theorem they investigated thechaotic response of the system in terms of the gov-erning parameters. They proposed a robust adaptivefuzzy control algorithm to regularize the chaotic re-sponse of the system. The model studied in the present
study is a clamped–clamped micro-beam, sandwichedwith two piezoelectric layers through the length of themicro-beam. The composite micro-beam is subjectedto a pure DC and a combination of DC–AC voltagesthrough lower and upper electrodes, respectively. Thecombination of DC–AC actuation is mainly used inMEMS RF switches to solve the problem of high driv-ing voltage [8, 19]. The main objective of the presentstudy is to convert the irregular chaotic response of thesystem to a regular periodic one by applying an appro-priate voltage to the piezoelectric layers. Piezoelec-tric actuation was previously applied by the authorsto stabilize the pull-in [6] and flutter instabilities [20]of electrostatically actuated MEM devices. Piezoelec-trically sandwiched micro-beams were first proposedby Rezazadeh et al. [21] to control the static pull-ininstability of a MEM device, and later on similar mod-els were studied (see [6, 20, 22–24]). In this studythe equations of motion are derived and discretizedto a single degree of freedom ODE using Galerkinmethod; the governing ODE is a Duffing type differ-ential equation with a nonlinear electrostatic force inwhich cubic nonlinearity arises due to the mid-planestretching; this type of nonlinearity is reported in sev-eral papers [1, 3, 5, 12–15, 18, 25–27]; however, mostof them, excluding [26], start with single degree offreedom equation without mentioning the source ofthe cubic nonlinearity. The phase space in the presentstudy is three-dimensional, but the response of the sys-tem is explored by reducing the phase space to two-dimensional space by an appropriate Poincare section.Depending on the amounts of the applied electrostaticDC and AC voltages, qualitatively different type ofresponses, including chaotic response, are observed.The chaotic response is recognized by evaluating theK–S entropy [28, 29] and regularized by applying anappropriate voltage to the piezoelectric layers.
2 Modeling
As illustrated in Fig. 1, the studied model is anisotropic clamped–clamped micro-beam of length l,width a, thickness h, density ρ, and Young’s modu-lus E. The micro-beam is sandwiched with two piezo-electric layers throughout the length of the micro-beam. The piezoelectric layers are of thickness hp anddensity ρp . The Young’s modulus of the piezoelectriclayers is denoted by Ep and the equivalent piezoelec-tric coefficient is denoted by e31. Two electrodes are
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Fig. 1 Schematics of the clamped–clamped piezoelectricallysandwiched micro-beam and the electrodes
placed underneath and on top of the micro-beam. Ini-tial gaps between the micro-beam and the electrodesare both g0 and the applied electrostatic voltages bythe upper and lower electrodes are denoted by Vu
and Vl , respectively. The applied voltage through theupper electrode is supposed to be a combination of aDC voltage VDC and an AC voltage with amplitudeVAC and frequency Ω ; the voltage applied through thelower electrode is a pure DC voltage, the same as theDC component of the upper electrode. The coordinatesystem as illustrated in Fig. 1 is attached to the mid-plane of the very left end of the micro-beam, where x
and z are respectively the horizontal and vertical coor-dinates. The deflection of the micro-beam along the z
axis is denoted by w(x, t).When a clamped–clamped beam undergoes bend-
ing, the extended length of the beam (l′) becomeslarger than its initial length l, leading to the introduc-tion of an axial force as follows [30]:
Fa = Eah + 2Epahp
l
(l′ − l
)
≈ Eah + 2Epahp
2l
∫ l
0
(∂w
∂x
)2
dx (1)
here l′ is estimated based on the integration of the arclength ds as [31]:
l′ =∫ l
0ds ≈
∫ l
0
√
1 +(
∂w
∂x
)2
dx
= l + 1
2
∫ l
0
(∂w
∂x
)2
dx (2)
The governing equation of the transverse motioncan be obtained by the minimization of the Hamil-tonian using variational principle. The total potentialstrain energy of the micro-beam includes the bending
and axial strain energies (Ub,Ua) and the electricalenergy Ue as [6]:
U(t) = Ub + Ua + Ue
= EI
2
∫ x=l
x=0
(∂2w
∂x2
)2
dx
+ Ephahp(h2 + hp)
2
∫ x=l
x=0
(∂2w
∂x2
)2
dx
+ ae31VP
∫ x=l
x=0
(∂w
∂x
)2
dx
+ Eah + 2Epahp
8l
(∫ l
0
(∂w
∂x
)2
dxa
)2
× (l′ − l
) + ε0aV 2u
2
∫ l
0
dx
(g0 − w)
+ ε0aV 2l
2
∫ l
0
dx
(g0 + w)(3)
where I and VP denote respectively the moment ofinertia of the cross section about the horizontal axispassing through the center of the surface for the crosssection of the micro-beam, and the applied voltage tothe piezoelectric layers. In Eq. (3) the first two termsare the strain energies due to the bending of the micro-beam, the third term is the strain energy due to theaxial force of the piezoelectric layers, the fourth termis the strain energy due to the stretching of the mid-plane and the last two terms indicate the electrical po-tential energy stored between the micro-beam and thetwo substrates, underneath and above; ε0 is the dielec-tric constant of the gap medium.
The kinetic energy of the micro-beam is repre-sented as [6]:
T = ρah
2
∫ x=l
x=0
(∂w
∂t
)2
dx
+ ρpahp
∫ x=l
x=0
(∂w
∂t
)2
dx (4)
The Hamiltonian is represented in the followingform:
H = T − U (5)
Substituting Eqs. (3) and (4) into Eq. (5), the Hamilto-nian reduces to
H = 1
2ρah
∫ l
0
(∂w
∂t
)2
dx
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856 S. Azizi et al.
− 1
2EI
∫ l
0
(∂2w
∂x2
)2
dx
− Eah
8l
(∫ l
0
(∂w
∂x
)2
dx
)2
− ε0aV 2u (t)
2
∫ l
0
dx
(g0 − w)
− ε0aV 2l (t)
2
∫ l
0
dx
(g0 + w)(6)
Based on the fact that the variation of the integral ofthe Hamiltonian over the time period [0, t] vanishes,namely, δ
∫ t
0 (T − U)dt = 0, the governing equationof motion and the corresponding boundary conditionsare obtained as
(EI)eq∂4w(x, t)
∂x4+ (ρA)eq
∂2w(x, t)
∂t2
−(
FP + (EA)eq
2l
∫ l
0
(∂w(x, t)
∂x
)2
dx
)
× ∂2w(x, t)
∂x2
= ε0a(VDC + VAC sin(Ωt))2
2(g0 − w(x, t))2− ε0aV 2
DC
2(g0 + w(x, t))2
(7)
subject to the following boundary conditions:
w(0, t) = w(l, t) = 0,∂w(0, t)
∂x= ∂w(l, t)
∂x= 0
(8)
where:
(EI)eq = EI + EP hahP
(h
2+ hp
)
FP = 2ae31VP(9)
(ρA)eq = ρah + 2ρP hP a
(EA)eq = Eah + 2EP ahP
The integral term in Eq. (7) represents the mid-plane stretching of the micro-beam due to the im-movable edges. Nonlinearities in resonant micro-systems generally arise from three sources: (i) large(finite) structural deformations, (ii) displacement-dependent excitations (stiffness parametric excita-tion), and (iii) tip/sample interaction potentials (e.g.
electrostatic interactions, and the Lennard–Jones po-tential). According to Eq. (8), two types of nonlin-earities exist in this model. The nonlinearity of thedynamics of the structure adds interesting behavior tothe response of the system.
For convenience the following non-dimensional pa-rameters (with over-hats) are introduced:
w = w
g0, x = x
l, τ = t
t, Ω = Ωt (10)
where t is a timescale defined as follows:
t =√
(ρA)eql4
(EI)eq(11)
Substituting Eq. (10) into Eq. (7) and dropping thehats and assuming the amplitude of the AC voltage tobe much less than the DC voltage, the equation of themotion in the non-dimensional form is obtained:
∂4w(x, τ)
∂x4+ ∂2w(x, τ)
∂t2
−(
α1 + α2
∫ l
0
(∂w(x, τ )
∂x
)2
dx
)∂2w(x, τ)
∂x2
= α3V2DC
(1
(1 − w)2− 1
(1 + w)2
)
+ 2α3VDCVAC sin(Ωτ)
(1 − w)2 (12)
where
α1 = FP l2
(EI)eq, α2 = (EA)eqg
20
2(EI)eq(13)
α3 = ε0al4
2g30(EI)eq
3 Numerical solution
To approximate the homoclinic trajectory of Eq. (12)with the homoclinic orbit of the well-known Duff-ing equation, the first term on the right-hand side ofEq. (12) in a given time is expanded in Taylor seriesup to the fourth order [15]; the resultant is:
∂4w(x, τ)
∂x4+ ∂2w(x, τ)
∂t2
−(
α1 + α2
∫ l
0
(∂w(x, τ )
∂t
)2
dx
)∂2w(x, τ)
∂x2
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= α3V2DC
(4w(x, τ) + 8w3(x, τ ) + O
(w5(x, τ )
))
+ 2α3VDCVAC sin(Ωτ)
(1 − w)2 (14)
Galerkin method is used to discretize Eq. (14);therefore the approximate solution is supposed to bein the form
w(x, τ) =n∑
i=1
qi(τ )ϕi(x) (15)
where ϕi(x) and qi(τ ) are respectively the linear shapefunction of a clamped–clamped micro-beam and thecorresponding amplitude. Substituting Eq. (15) intoEq. (14) and based on the Galerkin method multiply-ing both sides by ϕj (x), and integrating the resultantover the length of the micro-beam reduces to
n∑
i=1
qi(τ )
∫ 1
0ϕIV
i (x)ϕr(x) dx
+n∑
i=1
qi (τ )
∫ 1
0ϕr(x)ϕi(x) dx
− α1
n∑
i=1
qi(τ )
∫ 1
0ϕ′′
i (x)ϕr(x) dx
− α2
n∑
i=1
n∑
j=1
n∑
p=1
qi(τ )qj (τ )qp(τ )
×∫ 1
0ϕ′′
i (x)ϕr(x) dx
∫ 1
0ϕ′
j (x)ϕ′p(x) dx
= α3V2DC
(
4n∑
i=1
qi(τ )
∫ 1
0ϕr(x)ϕi(x) dx
+ 8n∑
i=1
n∑
j=1
n∑
p=1
qi(τ )qj (τ )qp(τ )
×∫ 1
0ϕr(x)ϕi(x)ϕj (x)ϕp(x) dx
)
× 2α3VDCVAC sin(Ωτ)
×∫ 1
0
ϕr(x) dx
(1 − ∑ni=1 qi(τ )ϕi(x))2
(16)
Equation (16) can be written in the matrix form:
n∑
i=1
qi (τ )Mir +n∑
i=1
qi(τ )Kir
+n∑
i=1
n∑
j=1
n∑
p=1
qi(τ )qj (τ )qp(τ )Kijpr = Fir (17)
where:
Mir =∫ 1
0ϕr(x)ϕi(x) dx
Kir =∫ 1
0ϕIV
i (x)ϕr(x) dx
− α1
∫ 1
0ϕ′′
i (x)ϕr(x) dx
− 4α3V2DC
∫ 1
0ϕr(x)ϕi(x) dx
(18)
Kijpr = α2
∫ 1
0ϕ′′
i (x)ϕr(x) dx
∫ 1
0ϕ′
j (x)ϕ′p(x) dx
− 8α3V2DC
∫ 1
0ϕr(x)ϕi(x)ϕj (x)ϕp(x) dx
Fir = 2α3VDCVAC sin(Ωτ)
×∫ 1
0
ϕr(x) dx
(1 − ∑ni=1 qi(τ )ϕi(x))2
Equation (17) is in the form of Duffing equation withnonlinear position dependent force due to the electro-static actuation. Consider the eigen-mode in the re-sponse of the system and define the following phasespace variables:
⎧⎨
⎩
S1
S2
S3
⎫⎬
⎭=
⎧⎨
⎩
q(τ)
q(τ )
Ωτ
⎫⎬
⎭(19)
The non-autonomous equation (17) reduces to thefollowing so-called autonomous first-order differentialequations:
S1 = S2
S2 = 1
m
(F − klS1 − knS
31
)(20)
S3 = Ω
where kl = K11, kn = K1111,m = M11.
4 Kolmogorov–Sinai entropy
The concept of the entropy in dynamical systems wasintroduced by Kolmogorov (1958) and Sinai in (1959)[28]; they were able to prove that the K–S entropynamed after them is a topologically invariant. From the
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858 S. Azizi et al.
statistical mechanics point of view entropy is definedby the number of accessible states for the system underconsideration. The relationship between K–S entropy,the Lyapunov exponents, and the traditional thermo-dynamic entropy has variously been explored [32]. Inorder to determine the K–S entropy, an appropriatePoincare section in the three-dimensional phase spaceis selected. Basically the Poincare section based on theidea of Henri Poincare reduces the dimension of thephase space by reducing a continuous trajectory as asequence of discrete points observed at constant timeintervals. In the present problem we are assuming thisconstant time interval, a so-called sampling time, beequal to the period of the AC excitation voltage. Toobtain the K–S entropy, a single trajectory is allowedto run for a long time to map up the Poincare section;then the Poincare section is covered with cells. Westart a trajectory in one cell and label it c(0); at a latertime τ0 = 2π/Ωt the trajectory will be in cell c(1),at t = 2τ0 it will be in cell c(2), and this is to be con-tinued up to t = Nτ0 when the trajectory will occupycell c(N) on the Poincare section. The sequence of theoccupied cells c(0), c(1), . . . , c(N) is recorded. Thenwe start off with another trajectory from the same ini-tial cell c(0) but with another initial condition (in thevicinity of the previous one) and let the trajectory torun up to time t = Nτ0 and therefore record anothersequence of N cells. This process is repeated manytimes; thereby a large number of N sequences arerecorded. To determine the entropy of the system, therelative number of times a particular sequence of N
cell labels occurs is determined as
p(i) = ni
Σ(21)
where ni is the number indicating how many times theith sequence occurs, and Σ is the number of distinctsequences [32]. Finally the K–S entropy is defined asfollows:
Ks = limN→∞
1
N(SN − S0) (22)
where the entropy SN is defined to be
SN = −∑
i
p(i) lnp(i) (23)
In Eq. (22) the sum is taken over all sequences of N
cells that start with c(0) [28, 32]. If all the sequencesstarting from the same initial cell track each other as
time goes on and occupy the same cells, this meansthat the motion is regular and consequently SN = 0.Assuming the other extreme (purely random) wherenone of the sequences are similar, it can be shown thatSN = lnΥ , where Υ is the number of sequences. Con-sequently, SN grows as Υ increases.
5 Results and discussion
The geometrical and mechanical properties of the casestudy are represented in Table 1.
Assuming VAC = Vp = 0, it can be shown that thetypes of the equilibrium points of the system directlydepend on the applied electrostatic DC voltage (VDC)
as follows:
S∗1
= S∗2 = 0
(24)
S∗2 = 0, S∗
1= ±
√√√√−
∫ 10 ϕIV
i (x)ϕr(x) dx − 4α3V2DC
∫ 10 ϕr(x)ϕi(x) dx
α2∫ 1
0 ϕ′′i (x)ϕr(x) dx
∫ 10 ϕ′
j (x)ϕ′p(x) dx − 8α3V
2DC
∫ 10 ϕr(x)ϕi(x)ϕj (x)ϕp(x) dx
where S∗1
and S∗2 correspond to the equilibrium posi-
tions. Figure 2 depicts the values of the non-dimen-sional linear and nonlinear stiffness terms versus theapplied DC electrostatic voltage. In region I, wherethe linear and nonlinear stiffness terms are of the samesign, there exists only one center fixed point; however,
in region II, where the linear and nonlinear stiffnessterms are of opposite signs, two other fixed points in-cluding one saddle node and one additional center ap-pear.
Figure 3 illustrates the position and the types of theequilibrium points based on the DC electrostatic volt-
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Table 1 Geometrical and material properties of the micro-beam and piezoelectric layers
Geometrical and material properties Micro-beam Piezoelectric layers
Length (L) 600 µm 600 µm
Width (a) 30 µm 30 µm
Height (h) 3 µm 0.01 µm
Initial gap (w0) 2 µm –
Young’s modulus (E) 169.61 GPa 76.6 GPa
Density (ρ) 2331 kg/m3 7500 kg/m3
Piezoelectric constant (e31) – −9.29 [21]
Permittivity constant (ε0) 8.845 × 10−12 F/m
Mass (ng) 41.958 2.7
Fig. 2 Linear andnonlinear stiffness termsversus the appliedelectrostatic voltage, andthe type of equilibriumpoints, assumingVAC = Vp = 0
age. The center type fixed point (S∗1
= S∗2 = 0) in re-
gion I becomes a saddle node in region II through apitchfork bifurcation point. As the DC voltage in re-gion II increases, the two center points move awayfrom the saddle node; for VDC > 6.2 V, the centerpoints physically disappear since they move to theother sides of the substrates, though mathematicallythey still exist.
Mobki et al. [33] studied a similar system: in theirmodel, the mid-plane stretching term is neglected andaccordingly the nonlinear stiffness term does not ap-pear in the discretized equation of the motion; as aresult, the two center fixed points in region II disap-pear. According to their results in region I, there aretwo more saddle nodes and two singular points on ei-ther side of the center point, which do not appear here;this is due to the Taylor expansion applied to Eq. (12),
which excludes two unstable saddle nodes and singu-lar points from the group of fixed points. The essenceof the present study is to investigate the chaotic re-sponse of the micro-beam, which occurs in region II.Since the behavior of the system in region I and in thevicinity of the unstable nodes is out of the scope of thepresent study, Taylor expansion not only does not af-fect the qualitative response of the system [15] but alsoconsiderably reduces the numerical integration time.
Figure 4 illustrates the phase plane of the systemwith various initial conditions (S2 = 0 and S1 is sweptfrom −0.8 to 0.8), Vp = 0.0 V and VDC = 3.0 V. Ac-cording to Fig. 2 with VDC = 3.0 V, the system is inregion I and periodic or quasiperiodic response is ex-pected. As clearly pertains to each individual initialcondition, the response is quasiperiodic. In this study,
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860 S. Azizi et al.
unless mentioned otherwise, the excitation frequencyis assumed to be 7 × 104 rad/s = 11.14 kHz.
Figure 5 illustrates the phase response of the systemwith VAC = Vp = 0, VDC = 5.5 V and various initialconditions (S2 = 0 and S1 is swept from −0.8 to 0.8).
As the phase plane of the system illustrates, thecenter points are symmetrically located on either sideof the saddle node. The homoclinic orbit which liesin the intersection of the stable and unstable mani-folds joins the saddle equilibrium point to itself. Ho-
Fig. 3 Position and types of the fixed points versus applied DCvoltage (bifurcation diagram)
moclinic orbits are common in conservative systemsand rare otherwise [9, 10]. It is worth noting that thisorbit does not correspond to a periodic solution sincethe trajectory takes forever to reach the fixed point.In nonlinear systems in which the linearized systemhas a homoclinic orbit, mostly the chaotic responseis originated from the homoclinic orbit where the sta-ble and unstable manifolds intersect (Melnikov’s theo-rem) [28]. Figures 6, 7 and 8 depict the phase trajecto-ries and the Poincare sections of the response with the
Fig. 5 Phase plane of the response of the system in region II,with VAC = Vp = 0 and VDC = 5.5 V. Homoclinic orbit isdashed
Fig. 4 Phase plane of the response of the system in region I, Vp = 0, VDC = 3.0 V. (a) VAC = 0, (b) VAC = 5 mV
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Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated 861
Fig. 6 Response of the system in region II, with Vp = 0,VAC = 1 mV and VDC = 5.5 V. (a) Phase plane; (b) Poincare section
Fig. 7 Response of the system in region II, with Vp = 0,VAC = 5 mV and VDC = 5.5 V. (a) Phase plane; (b) Poincare section
same amount of applied DC electrostatic voltage as ofFig. 5, and various levels of AC electrostatic voltages.
Applying AC voltage to the system increases thedimension of the phase space from two to three wherethe incidence of chaotic response is possible. As men-tioned, the Poincare section reduces the dimension ofthe phase space by reducing a continuous trajectoryas a sequence of discrete points observed at constanttime intervals equal to the period of the AC excitationvoltage.
As parts (b) of Figs. 6–8 illustrate, based on theinitial condition applied to the system, the response
may either be periodic, quasiperiodic or chaotic. Thechaotic response is originated around the saddle nodewhere the homoclinic orbit is originated. Qualitativelythe chaotic response is distinguishable by the exis-tence of fractal in the Poincare section. The general-ized area of the fractal increases as the amplitude ofthe AC voltage increases. Two points in the vicinityof each other on the fractal at a given time will ar-bitrarily be far apart as time goes on. Figures 9, 10and 11 illustrate the spectral and temporal responsesalong with the Poincare section of the system with var-ious initial conditions and three different levels of AC
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Fig. 8 Response of the system in region II, with Vp = 0,VAC = 10 mV and VDC = 5.5 V. (a) Phase plane; (b) Poincare section
Fig. 9 Frequencyspectrum, temporalresponse (inset) along withthe Poincare section (inset)of the system response inregion II withVp = 0,VAC = 1 mV,VDC = 5.5 V.(a) S1 = 0.01, S2 = 0.00,(b) S1 = 0.22, S2 = 0.00 asinitial conditions
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Fig. 10 Frequencyspectrum, temporalresponse (inset) along withthe Poincare section (inset)of the system response inregion II withVp = 0,VAC = 5 mV,VDC = 5.5 V.(a) S1 = 0.01, S2 = 0.00,(b) S1 = 0.22, S2 = 0.00 asinitial conditions
voltage amplitude as those of Figs. 5–8 with Vp = 0,VDC = 5.5 V.
Figures 9–11 reveal that, for any arbitrarily chosenset of initial conditions (S1 and S2) in the near enoughvicinity of the saddle node bifurcation point on thePoincare section, the system exhibits chaotic response;as the amplitude of the applied AC voltage increases,the response becomes chaotic even with initial condi-tions farther from the saddle node bifurcation. WithS1 = 0.01, S2 = 0.00 as the initial conditions, the re-sponse is chaotic for all three amplitudes of AC volt-age (1, 5 and 10 mV); however, with the other set ofinitial conditions (S1 = 0.22, S2 = 0.00), which is far-ther than the previous set from the saddle node bifur-cation point, the response is regular for the lower am-plitudes of AC voltage (1, 5 mV) but still chaotic for10 mV. In order to quantitatively determine the chaoticnature of the response, corresponding to each individ-
ual level of AC voltage amplitude (1, 5 and 10 mV),a set of close enough initial conditions in the vicinityof saddle node bifurcation are chosen and, based onthem, the K–S entropy of the system is investigated.The K–S entropy of the system in the vicinity of sad-dle node for 1, 5 and 10 mV as the amplitude of the ACvoltage is 0.146, 0.172 and 0.177, respectively. Thepositive value of the K–S entropy reveals the chaoticnature of the response. As clearly, the larger the am-plitude of the AC voltage the larger is the K–S entropyof the system. The K–S entropy of the system corre-sponding to the sets of initial conditions in the vicinityof initial conditions (S1 = 0.22, S2 = 0.00) is respec-tively 0, 0 and 0.177, corresponding to the three lev-els of AC voltage amplitudes (1, 5 and 10 mV). Fig-ure 12 illustrates the response of the micro-beam withthe same number of parameters as in Fig. 10 but withVp = −10 mV. Figure 12 reveals that applying voltage
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Fig. 11 Frequencyspectrum, temporalresponse (inset) along withthe Poincare section (inset)of the system response inregion II withVp = 0,VAC = 10 mV,VDC = 5.5 V. (a) S1 = 0.01,S2 = 0.00,(b) S1 = 0.22, S2 = 0.00 asinitial conditions
to the piezoelectric layers (Vp = −10 mV) regular-izes the response initiated with S1 = 0.22, S2 = 0.00(Fig. 12a); however, the system still exhibits chaoticresponse with S1 = 0.01, S2 = 0.00 given as the ini-tial conditions. As Fig. 13 illustrates, increasing theamount of piezoelectric voltage results in the regular-ization of the response with S1 = 0.01, S2 = 0.00 asinitial conditions.
Figure 14 illustrates the K–S entropy versus theamplitude of AC voltage with three different levelsof piezoelectric voltage (0, −10, and −30 mV) andS1 = 0.01, S2 = 0.00 as the initial conditions.
6 Conclusion
The response of fully clamped piezoelectrically sand-wiched micro-beam exposed to two sides of electro-
static actuation was investigated. The electrostatic ac-tuation on the lower side electrode was a pure DC volt-age and on the upper side was a combination of sameDC and another AC voltage. The equations of the mo-tion were derived using Hamiltonian principle and dis-cretized to an equivalent ODE using Galerkin method.The single degree of freedom model was a Duffingtype ODE with nonlinear force due to the electrostaticactuation and stretching effects. The equation of themotion was numerically integrated over the time do-main and the corresponding temporal and spectral re-sponses were obtained. It was shown that the qualityof the response is highly dependent on the values ofthe DC and piezoelectric voltages applied to the sys-tem. Without application of piezoelectric voltage de-pendent on the value of the applied DC voltage, typesand quantities of the equilibrium points vary. In the
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Fig. 12 Frequencyspectrum, temporalresponse (inset) along withthe Poincare section (inset)of the system response inregion II withVp = −10 mV,VAC = 10 mV,VDC = 5.5 V.(a) S1 = 0.01, S2 = 0.00,(b) S1 = 0.22, S2 = 0.00 asinitial conditions
Fig. 13 Frequencyspectrum, temporalresponse (inset) along withthe Poincare section (inset)of the system response inregion II withVp = −30 mV,VAC = 10 mV,VDC = 5.5 V,S1 = 0.01, S2 = 0.00 asinitial conditions
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Fig. 14 K–S entropy versus the amplitude of the AC voltagewith three different levels of piezoelectric voltage (0, −10 and−30 mV) (the dashed lines are interpolated)
range 0.0 < VDC < 4.7, the system had only one center,exhibiting regular response regardless of the appliedinitial condition to the system; however, in the range4.7 < VDC < 7.2, the system had one saddle node andtwo centers on the either side of the saddle node. Ap-plying AC voltage to the system changed the natureof the response in the vicinity of saddle node bifur-cation point to chaotic motion, which was quantita-tively and qualitatively determined with K–S entropyand spectral response of the system. It was shown thatthe K–S entropy of the system in the chaotic region ispositive, indicating that the system is super-sensitiveto the applied initial condition in the chaotic region.It was concluded that the K–S entropy in the chaoticregion increases as the amplitude of the AC voltage in-creases. Applying an appropriate negative sign, piezo-electric voltage led to positive linear and nonlinearstiffness coefficients, meaning that the saddle node bi-furcation point, and accordingly the homoclinic orbit,disappears from the phase plane of the response; thisled to the regularization of the chaotic response of thesystem.
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