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Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation
Applied Mathematical Modelling 35 (2011) 4796–4815
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Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation Saber Azizi a, Ghader Rezazadeh b,⇑, Mohammad-Reza Ghazavi a, Siamak Esmaeilzadeh Khadem a a b
Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran Mechanical Engineering Department, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 26 July 2010 Received in revised form 19 March 2011 Accepted 31 March 2011 Available online 19 April 2011 Keywords: Micro-beam Piezoelectric layer Stability Mathieu equation MEMS/NEMS
a b s t r a c t In the present article an investigation is presented into the stability of an electro-statically deflected clamped–clamped micro-beam sandwiched by two piezoelectric layers undergoing a parametric excitation applying an AC voltage to these layers. Applying an electrostatic actuation not only deflects the micro-beam but also decreases the bending stiffness of the structure, which can lead the structure to an unstable position by undergoing a saddle node bifurcation. Utilizing an appropriate AC actuation voltage to the piezoelectric layers produces a time varying axial force, which can play the role of a stabilizer exciting the system parameter. The governing equation of the motion is a nonlinear electro-mechanically coupled type PDE, which is derived using variational principle and discretized, applying Eigenfunction expansion method. The resultant is a Mathieu type equation in its damped form. Using Floquet theory for single degree of freedom system the stable and unstable regions of the problem are investigated. The effects of viscous damping and electrostatic actuation on the stable regions of the problem are also studied. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Micro-electromechanical systems (MEMS) are widely being used in today’s technology. So investigating the problems referring to MEMS, owns a great importance. One of the significant fields of study is the stability analysis of the parametrically excited systems. Parametrically excited micro-electromechanical devices are ever increasingly being used in radio, computer and laser engineering [1]. Parametric excitation occur in a wide range of mechanics, due to time dependent excitations, especially periodic ones; some examples are columns made of nonlinear elastic material, beams with a harmonically variable length, parametrically excited pendulums and so forth. Investigating stability analysis on parametrically excited MEM systems is of great importance. In 1995 Gasparini et al. [2] studied on the transition between the stability and instability of a cantilevered beam exposed to a partially follower load. Piezoelectric micro cantilevers were first utilized for sensing purposes [3] and then utilized as actuators [4]. In 1999 Cattan et al. [5] used an unusual experimental method to study on the piezoelectric properties of PZT films for micro cantilever. In 2000 Mitrovic et al. [6] investigated the response of six piezoelectric stack actuators under electrical, mechanical, and combined electromechanical loading. They focused on understanding the behavior of piezoelectric materials under combined electromechanical loading, and determining fundamental properties necessary to model the constitutive response and optimizing the actuator performance. In 2002 Li et al. [7] studied on piezoelectric AlGa bimorph micro actuators and verified their results using experimental set ups.
⇑ Corresponding author. Tel.: +98 914 145 1407. E-mail addresses: [email protected] (S. Azizi), [email protected] (G. Rezazadeh), [email protected] (M.-R. Ghazavi), s.khadem@ modares.ac.ir (S.E. Khadem). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.03.049
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In 2005 Zhang and Meng [8] studied on the behavior of a micro cantilever under combined parametric and forcing excitation. They used harmonic balance method to simulate the resonance amplitude frequency of the system under the combined parametric and forcing excitation. In 2006 Rezazadeh et al. [9] presented a novel method to control the pull-in voltage and to measure the residual stress in the fixed–fixed and cantilever MEM actuators. In 2006 Rhoads et al. [10] investigated generalized parametric resonance in electro-statically actuated micro-electromechanical oscillators. They used perturbation analysis to study the behavior of the proposed model. Their results include a wide array of interesting dynamical behavior, most of which can be attributed to the existence of nonlinear parametric excitation in their equation of motion. In 2007 Zhang and Meng [11] studied on the nonlinear response and dynamics of the electro-statically actuated MEMS resonant sensors under two parametric and external excitations. In 2007 Zhu et al. [12] investigated the parametric resonance of coupled micro-electromechanical oscillators under periodically varying nonlinear coupling forces. They used harmonic balance method combined with Newton iteration method to find the steady state periodic solutions. In 2007 DeMartini et al. [13] investigated linear and nonlinear tuning of parametrically excited MEMS oscillators. The governing equation of motion was a nonlinear Mathieu type with time varying linear and nonlinear stiffness term. They used perturbation technique to determine the behavior of the system. In 2008 Goyal and Kapania [14] studied on the dynamic stability of laminated beams subject to a combination of conservative and non-conservative tangential follower loads. They also investigated the divergence and flutter instabilities. In 2008 Djondjorov and Vassilev [15] investigated the dynamic stability of a cantilevered Timoshenko beam lying on an elastic foundation of Winkler type. Their model was subjected to a tangential follower force. In 2008 Gayesh and Balar [16] studied on the non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. They applied the method of multiple scales to the governing equations of motion, and considering the solvability condition obtained the linear and nonlinear frequencies and mode shapes of the system. They also investigated the stability and bifurcation points of the system through parametric studies. In 2008 Kim et al. [17] performed stability analysis on thin-walled composite columns. They compared their numerical solutions with those of analytical ones and finite element method using Hermitian beam element. In 2008 Ding and Chen [18] investigated the stability of axially accelerating viscoelastic beams subjected to parametric vibration, using multiple scales method. They also performed numerical investigations applying the finite difference method. In 2008 Zamanian et al. [19] studied on the deflection and natural frequency of a micro-beam under combined electrostatic and piezoelectric actuation. They solved the nonlinear governing differential equation using Galerkin method and showed that a new sensor-actuator system may be actuated by applying the voltage to the piezoelectric layer, and the actuation may be sensed by the value of the output electric current induced from the movement of the polarized micro-beam. In 2009 Rezazadeh et al. [20] investigated the electromechanical behavior of microbeams with piezoelectric and electrostatic actuation. They solved the nonlinear governing differential equation using step by step linearization method. They could control the trigger time of the micro switch by applying appropriate voltage to the piezoelectric layers. In 2009 Rezazadeh et al. [21] studied on the static and dynamic stabilities of a micro-beam with various boundary conditions actuated by a DC piezoelectric voltage. In 2009 Guo et al. [22] performed instability analysis on torsional MEMS/NEMS actuators under capillary force. They qualitatively analyzed the nonlinear equation, and bifurcation phenomenon. In 2009 Yoo et al. [23] proposed a dynamic modeling method of an axially oscillating beam subjected to periodic impulsive force. They investigated the effect of various parameters governing the problem including the impulse magnitude, the oscillating frequency, the oscillating speed amplitude and the model damping ratio on the stability of the beam. Chang et al. [24] studied on the vibration and stability analysis of an axially moving Rayleigh beam. They considered two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion. The authors used direct time numerical integration, based on Runge–Kutta algorithm, and Floquet theory for vibration and stability analysis respectively. They also investigated the divergent and flutter instabilities. In 2009 Mahmoodi et al. [25] studied on the nonlinear flexural response of piezoelectrically driven micro cantilever sensors they used multiple scales method to study the asymptotic behavior of the sensor’s response. In 2009 Mahmoodi and Jalili [26] performed an experimental nonlinear vibration analysis on piezoelectrically actuated micro cantilevers. In 2009 Susanto [27] analyzed vibration of piezoelectric laminated slightly curved beams using distributed transfer function method. In 2010 Yang et al. [28] in another study investigated the stability of axially accelerating Timoshenko beam exploiting averaging method. Studying the behavior of MEM structures covered with piezoelectric layers actuated with AC voltage will also lead in parametric vibration and investigating its stability is of great importance due to its ever increasing applications. In 2010 Ghazavi et al. [29] studied on the pure parametric excitation of a cantilever micro-beam actuated with piezoelectric actuation; they investigated the stability using Floquet theory for single degree of freedom systems. Although few studies have focused on [19,30], simultaneous piezoelectric and electrostatic actuations add interesting qualifications including shifting frequency operation range, to the MEM and NEM structures. In the present study the stability analysis is carried out on a clamped–clamped micro-beam sandwiched with two piezoelectric layers located on its upper and lower sides. The micro-beam is subjected to an electrostatic field. Applying an AC voltage to the piezoelectric layers, and considering small vibrations around the electro-statically deflected equilibrium position, the stability analysis is performed on the governing Mathieu type equation of the motion using the Floquet theory for single degree of freedom system. The stable and unstable regions of the problem are investigated, and the transition curves separating stable from unstable solutions of the system in the parameter plane and in terms of amplitude and frequency of internal parametric excitation are illustrated. The effect of the damping coefficient and electrostatic voltage on the stability of the structure is also investigated. And, the possibility of stabilizing pull-in instability using a parametric excitation is investigated.
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2. Modeling As illustrated in Fig. 1 the studied model is an isotropic micro-beam of length l, width a, thickness h, density q with Young’s modulus, E sandwiched with two piezoelectric layers having thickness hp, density qp throughout the micro-beam length. The Young modulus of the piezoelectric layers is denoted by Ep and the equivalent piezoelectric coefficient is supposed to be e31 . The initial gap between the micro beam and the substrate is w0 and the applied electrostatic voltage is denoted by Ves. The coordinate system as illustrated, is attached to the middle of the left end of the micro-beam where x and z refer to the horizontal and vertical coordinates, respectively. The governing equation of the transverse motion can be obtained by the minimization of the Lagrangian using variational principle. The total potential energy includes the bending and axial strain energies (Ub, Ua) and the electrical energy (Ue) as following [29]:
!2 Z l Z 2 1 @2w 1 l @w dx þ F dx þ U e ; ðEIÞeq P 2 @x2 2 0 @x 0 h þ hp ; F P ¼ 2ae31 V ac cosð2xtÞ; ðEIÞeq ¼ EI þ Ep hahp 2
UðtÞ ¼ U b þ U a þ U e ¼
ð1Þ
where I denotes the moment of inertia of the cross section about the horizontal axis passing through the center of the surface of the cross section of the micro-beam, w is the mid plane deflection, Ua indicates the strain energy due to the axial load (Fp),
Piezoelectric Layers h
x
Microbeam
z
Ves
V = Vac cos (2ωt ) p
h hp
Ves
V = Vac cos (2ωt )
l
a
Fig. 1. Schematic view of the studied model (A) Front view. (B) Side view.
keq meq w(l / 2, t )
w0
Ves
Fig. 2. Schematic view of the single degree of freedom system.
Table 1 Geometrical and material properties of the micro-beam and piezoelectric layers. Geometrical and material properties
Micro-beam
Piezoelectric layers
Length (L) Width (a) Height (h) Initial gap (w0) Young’s modulus (E) Density (q) Piezoelectric constant e31 Permittivity constant (e0)
600 lm 50 lm 3 lm 1 lm 169.61 GPa 2331 kg/m3 _ 8.845 1012(F/m)
600 lm 50 lm 0.01 lm – 76.6 GPa 7500 kg/m3 9.29 [21]
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which is the result of piezoelectric actuation. Vac and x refer to the amplitude of the alternative and the frequency of the applied voltage to the piezoelectric layers respectively. Ue indicates the electrical potential energy stored between the capacitor plates. It must be noted that U e is the electrical co-energy and denoted as [31]:
U e ¼
1 2
Z 0
e0 aV 2es
l
ðw0 wÞ
dx;
ð2Þ
where e0 is the permittivity constant of the dielectric material between the micro beam and the substrate, Ves indicates the applied voltage between the micro beam and the substrate, and w0 is the initial gap between the micro beam and the substrate. The kinetic energy is expressed as follows:
T¼
1 ðqAÞeq 2
Z l 2 @w dx; @t 0
ð3Þ
ðqAÞeq ¼ aðqh þ 2qp hp Þ; where q and qp refer to the densities of the micro-beam and the piezoelectric layers respectively. The Lagrangian may be consisted as:
1 0.8
Middle gap) (μ m)
0.6
Stable Unstable Stable,Physically imposible
Pull-in
•
0.4 0.2 0 -0.2 -0.4 0
2
4
6
Ves(V) Fig. 3. Center gap versus applied electrostatic voltage.
Fig. 4a. Phase plain corresponding to center gap and Ves = 0(V).
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Fig. 4b. Phase plain corresponding to center gap and Ves = 2(V).
Fig. 4c. Phase plain corresponding to center gap and Ves = 4(V).
1 LðtÞ ¼ TðtÞ U b U a U e ¼ ðqAÞeq 2
Z l 0
!2 2 Z 2 Z l @w 1 @2w 1 l @w dx ðEIÞeq dx F dx U e : P @t 2 @x2 2 0 @x 0
ð4Þ
Extermizing the Lagrangian and using Rayleigh’s dissipation function to exploit Lagrange equation for non-conservative systems, and considering dU e ¼ dU e [31] the dynamic equation of the motion considering viscous damping will be obtained as Eq. (5):
ðEIÞeq
@4w @2w @ 2 w e @w e0 aV 2es ¼ þ ðqAÞeq 2 F P 2 þ C ; 4 @x @x @t @t 2ðw0 wÞ2
ð5Þ
e stands for the viscous damping coefficient per unit length of the micro-beam. where C 3. Numerical solution The solution of the system can be considered as follows:
wðx; tÞ ¼ w ðxÞ þ wd ðx; tÞ; ⁄
where w (x) refers to the static deflection and
ð6Þ wd⁄(x,
t)corresponds to the dynamic response of the micro-beam due to the Vac.
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Fig. 4d. Phase plain corresponding to center gap and Ves = 6(V).
Fig. 4e. Phase plain corresponding to center gap and Ves = Vpull-in = 6.9(V).
Considering the static equilibrium position due to the electrostatic voltage as denoted in Eq.(7) and substituting Eq. (6) in (5) and expanding the right hand side of the Eq. (5) up to the first term of Tailor series, the equation of motion will be obtained as indicated in Eq.(8):
ðEIÞeq
@ 4 w e0 aV 2es ¼ ; @x4 2ðw0 w Þ2
ð7Þ
ðEIÞeq
@ 4 wd @ 2 wd e @wd @ 2 ðwd Þ e0 aV 2es þ ðqAÞeq þC wd ¼ 0: FP 2 4 2 @x @x @t @t 2ðw0 w Þ3
ð8Þ
There is no exact solution to the Eq. (8) [32] so an approximate solution is supposed to be in the form as follows:
wd ¼
n X
uj ðxÞqj ðtÞ;
ð9Þ
j¼1
where uj and qj(t) refers to the shape functions of a clamped–clamped beam and time dependant amplitudes respectively. Substituting Eq. (9) in Eq. (8) results in:
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e ¼ 0:0. Fig. 5a. V es ¼ 0:0ðVÞ; C
e ¼ 0:5. Fig. 5b. V es ¼ 0:0ðVÞ; C
e ¼ 1:0. Fig. 5c. V es ¼ 0:0ðVÞ; C
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e ¼ 0:0. Fig. 6a. V es ¼ 2:0ðVÞ; C
e ¼ 0:5. Fig. 6b. V es ¼ 2:0ðVÞ; C
e ¼ 1:0. Fig. 6c. V es ¼ 2:0ðVÞ; C
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ðEIÞeq
n X
uIVj ðxÞqj ðtÞ þ ðqAÞeq
n X
j¼1
n X
uj ðxÞq€j ðtÞ F P
j¼1
n X
u00j ðxÞqj ðtÞ þ Ce
j¼1
n X
_ e0 aV 2es ðw0 w Þ3 uj ðxÞqðtÞ
j¼1
uj ðxÞqj ðtÞ ¼ Rðx; tÞ;
ð10Þ
j¼1
where R(x, t) is the residual. Multiplying Eq. (10) in ui(x)and integrating the outcome in domain [0, l] and considering the orthogonality of the distinct shape functions one will have: n X
n X
qj ðtÞK ij þ
j¼1
€j ðtÞM ij þ q
j¼1
n X
_ qðtÞC ij ¼ 0;
ð11Þ
j¼1
where:
K ij ¼ ðEIÞeq
Z
M ij ¼ ðqAÞeq e C ij ¼ C
Z 0
l
0
ui ðxÞuIVj ðxÞdx F P
Z 0
Z 0
l
u00j ðxÞui ðxÞdx e0 aV 2es
Z 0
l
ðw0 w Þ3 ui ðxÞuj ðxÞdx;
l
ð12Þ
ui ðxÞuj ðxÞdx;
l
ui ðxÞuj ðxÞdx:
e ¼ 0:0. Fig. 7a. V es ¼ 4:0ðVÞ; C
e ¼ 0:5. Fig. 7b. V es ¼ 4:0ðVÞ; C
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e ¼ 1:0. Fig. 7c. V es ¼ 4:0ðVÞ; C
In the structures subjected to an electrostatic actuation, increasing voltage, applied to the electrostatic areas, decreases equivalent stiffness of the structure, therefore for a given applied voltage called as pull-in voltage in the MEMS literature, a stationary instability is occurred by undergoing system to a saddle node bifurcation. This kind of instability can be even observed in single degree of freedom systems. Therefore applying the first shape function of the micro-beam, this kind of instability of electro-statically deflected micro-beams qualitatively can be studied. For a clamped–clamped micro-beam the first Eigen-function is given as follows [33]:
u1 ðxÞ ¼ 1:0178 cos 4:73
x x x x cosh 4:73 þ sin 4:73 sinh 4:73 : l l l l
ð13Þ
Expanding Eq. (11) for j = 1, and considering xt = s one will have: 2
d q ~cðqðtÞÞ þ þ ðd þ 2e cosð2sÞÞq ¼ 0; ds2 ds
ð14Þ
where:
d ¼ 12:36
e ¼ 0:858
ðEIÞeq 4
x qAÞeq l 2ð
ae31 V ac
x2 ðqAÞeq l2
index ; 1:856x2 ðqAÞeq l
ð15Þ
;
e ¼ 0:0. Fig. 8a. V es ¼ 6:0ðVÞ; C
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e ¼ 0:5. Fig. 8b. V es ¼ 6:0ðVÞ; C
e ¼ 1:0. Fig. 8c. V es ¼ 6:0ðVÞ; C
e c¼
C ; ðqAÞeq x
where the parameter index, which is in direct relation with the electrical stiffness, stands for:
index ¼ e0 aV 2es
Z 0
l
ðw0 w ðxÞÞ3 ui ðxÞuj ðxÞdx:
ð16Þ
The static deflection w⁄(x) depends on the applied electrostatic voltage. For any applied electrostatic voltage, Eq. (7) is solved numerically using FE method with Hermitian shape functions [31]. Having obtained the nodal values of the static deflection, the functionality of the static deflection on the coordinate x is determined using the least square method. To determine the parameter ‘‘index’’ the integral in Eq. (16) is evaluated numerically.
4. Stability analysis In this part the stability analysis once considering electrostatic actuation and once considering piezoelectric actuation, on electro-statically deflected micro-beam is performed on the governing equation of the motion.
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e ¼ 0:0. Fig. 9a. Pull-in voltage, C
e ¼ 0:5. Fig. 9b. Pull-in voltage, C
e ¼ 1:0. Fig. 9c. Pull-in voltage, C
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4.1. Electrostatic actuation To investigate the behavior of the system the equivalent single degree of freedom system located at the middle length of the micro-beam is considered as illustrated schematically in Fig. 2. Considering electrostatic actuation without that of piezoelectric and using Eigen-function expansion method to derive the equivalent single degree of freedom system at the center of the micro-beam, the equation of the motion and the corresponding equivalent mass and the stiffness will be obtained as follows:
l l e0 AV 2es € ; t þ keq w ; t ¼ meq w ; 2 2 2ðw0 wðl=2; tÞÞ Z
meq ¼ ðqAÞeq
keq ¼ ðEIÞeq
l
0
Z 0
u1 ðxÞu1 ðxÞdx;
l
u1 ðxÞuIV1 ðxÞdx;
Fig. 10a. Phase portrait corresponding to point ‘B’.
Fig. 10b. Time history corresponding to point ‘B’.
ð17Þ
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where:
wðl=2; tÞ ¼ qðtÞu
l : 2
ð18Þ
In Eq. (17) parameter (A) is the area of the equivalent mass exposed to the electrostatic actuation and is determined in a manner that the static pull-in voltage of the continuous system to be equal to that of the discritized one; so one will have:
fðEIÞeq
Rl
0
u1 ðxÞuIV1 ðxÞdxg2ðwðl=2; tÞ w0 Þwðl=2; tÞ ¼ A: e0 V 2Pullin
ð19Þ
4.2. Electrostatic and piezoelectric actuation Eq. (14) is in the form of damped Mathieu equation, which was discussed in its damp-less form by Mathieu (1868) in connection with the problem of vibrations of elliptic membrane [34]. When the governing differential equation of motion
Fig. 11a. Phase portrait corresponding to point ‘C’.
Fig. 11b. Time history corresponding to point ‘C’.
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for the system is of Mathieu type, a few commonly used well known methods are perturbation and iteration techniques and Bolotin method, which is based on Floquet theory [32]. In this work to establish stability analysis, the Floquet theory for single degree of freedom system is applied. Eq. (14) is a second order homogenous differential equation, so there are two linear and non-vanishing independent solutions called q1(t), q2(t). It can be proved that if q1(t), q2(t)are a fundamental set of solutions of Eq. (14), q1(t + s)andq2(t + s) are also a fundamental set of solutions of Eq. (14) [29]; so one may have:
q1 ðt þ TÞ q2 ðt þ TÞ
¼
a11
a12
a21
a22
q1 ðtÞ q1 ðtÞ
:
a11 a12 is not unique and it depends on the fundamental solution being used. There exist a fundamental set of a21 a22 solution in which one has: The matrix
ð20Þ
e ¼ 0:0. Fig. 12a. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 0:0. Fig. 12b. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
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e ¼ 0:5. Fig. 12c. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 0:5. Fig. 12d. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
Q 1 ðs þ pÞ ¼ k1 Q 1 ðsÞ; Q 2 ðs þ pÞ ¼ k2 Q 2 ðsÞ
ð21Þ
in which Q1(s + p) and Q2(s + p) are denoted as normal or Floquet solutions [29]; the corresponding k1 and k2 are constants, which may be complex. It follows from Eq. (21) that
Q i ðs þ npÞ ¼ kni Q i ðsÞ;
ð22Þ
where n is an integer. The stable (bounded) and unstable solutions will be determined as soon as the values of the corresponding k1 ; k2 are investigated. Consequently jki j < 1 and jki j < 1 correspond to bounded (Stable) and unbounded (unstable) solutions respectively. As reasonable ki ¼ 1, and ki ¼ 1 are respectively relative to periodic solutions with periods p and 2p [29].
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e ¼ 1:0. Fig. 12e. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 1:0. Fig. 12f. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
5. Results and discussions In this article as a case study a silicon micro-beam sandwiched with PZ-4 layers is considered. The geometrical and material properties of the micro-beam and piezoelectric layers involved in the problem are given in Table 1.
5.1. Electrostatic actuation Considering Vac = 0 the problem is a clamped–clamped micro-beam sandwiched with piezoelectric layers subjected to electrostatic actuation which was studied in the references [9,20]. According to Eq. (7) growing the electrostatic voltage increases the static deflection w⁄(x), and consequently raises the absolute value of the parameter ‘index’. Fig. 3 illustrates w0 wjx¼ l denoted as middle gap, versus Ves. z The phase portraits corresponding to Eq. (14) considering e = 0 with various applied electrostatic voltages are given in Fig. 4. The superscript dot ( ) stands for the differentiation with respect to time (t).
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In Fig. 3 dashed lines correspond to physically impossible solutions. As Fig. 4a claims with Ves = 0(V) there exists a stable equilibrium point and with any initial condition the response is periodic. Applying electrostatic voltage leads in the appearance of two extra equilibrium points including a saddle node, and a stable but physically impossible centre in the phase space. The impossibility of the stable solutions refers to the existence of the substrate beneath, which limits the amplitude of the motion of the micro-beam. As clear increasing the electrostatic voltage lessens the vastity of the periodic orbits and enlarges to those of non-periodic ones; this is due to the fact that increasing electrostatic voltage leads in the softening of the micro-beam, which increases the absolute value of the parameter ‘index’ (Electrical stiffness). This behavior is continued up to a critical voltage denoted as pull-in, in which no periodic orbits are seen in the phase space except those of physically impossible. In the coming section it is tried to stabilize the pull-in instability using piezoelectric actuation. 5.2. Electrostatic and piezoelectric actuations Applying simultaneous electrostatic and piezoelectric actuation causes the micro-beam to have a constant and a periodically time varying stiffness components. In this section the dependency of the stability of the micro-beam, on the applied electrostatic voltage, amplitude of Vac, and damping coefficient is investigated. Considering Ves = 0(V), the problem leads to that type studied by Ghazavi et al. [29] where a good agreement is seen. The stable and unstable (dashed) regions in the plane of amplitude of Vacand the excitation frequency are given in Figs. 5–9. Figs. 5–10 claim that increasing the electrostatic voltage enlarges the unstable region which is in a good agreement with those given in Fig. 4. This is due to the fact that applying electrostatic voltage decreases the constant component of the stiffness, which leads in the increase of the amplitude of the motion of the micro-beam; This behavior is continued up to pull-in voltage in which almost everywhere in the plane of the amplitude of Vac, and the excitation frequency is unstable, except those of high excitation frequency and low amplitude of Vac; This is considerably important, because one may excite the micro-beam with a suitable amplitude and frequency of piezoelectric actuation, to stabilize the pull-in instability. Also increasing the amplitude of Vac have a destabilizing effect, this is reasonable due to the growth of the amplitude of the time varying component of the stiffness. As reasonable increasing the value of the damping coefficient lessens the amplitude of the motion of the micro-beam and has a stabilizing effect in the behavior of the structure. As obvious the transition curve in Fig. 9 asymptotically lies on the horizontal axis as the excitation frequency reduces; This is due to the unstable nature of the pull-in point; however applying ‘AC’ voltage even with a small amplitude but a high enough frequency results in the stability of the micro-beam. Fig. 10 refers to the phase portrait and the corresponding time history of point ‘ B’ illustrated in Fig. 9a, with coordinates x ¼ 1:5 105 ðrad Þ; V ac ¼ 3ðVÞ.As the results of the stability analysis claim this point is located in unstable region s and the corresponding time history is boundless. Fig. 11 illustrates the time history and the phase plain corresponding to
point ‘C’ with the same amplitude of ‘Vac’ with point ‘B’, but with a high enough excitation frequency, x ¼ 3 105 rad which s stabilizes the pull-in instability. Fig. 12 shows the phase portrait and the corresponding time histories of point ‘A’ with coor dinate x ¼ 2:5 105 rad ; V ac ¼ 5ðVÞ with various damping coefficients. As the results claim, damping coefficient limits the s amplitude of the motion and has a stabilizing effect on the response of the system. Fig. 13 illustrates the phase portrait and the corresponding time histories of point ‘A’ with Ves = 6(V) and various damping coefficients. Although the damping coefficient considerably limits the amplitude of the motion but it is not high enough to
Fig. 13a. Phase portrait corresponding to point ‘A’ with Ves = 6(V) and various damping coefficients.
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Fig. 13b. Time history corresponding to point ‘A’ with Ves = 6(V) and various damping coefficients.
make the micro-beam undergo a stable response. However, in this case, one may actuate the piezoelectric layers with a high enough frequency to stabilize the response. 6. Conclusion In this paper the stability analysis was performed on a piezoelectrically actuated clamped–clamped micro-beam deflected due to an electrostatic actuation. The governing differential equation of the motion was derived by the minimization of the Lagrangian and developed to non-conservative systems applying Rayleigh’s dissipation function. The nodal values of the static deflection of the micro-beam was determined by numerically solving the governing differential equation and the functionality of the deflection on the coordinate ‘x’ was determined using least square method. The governing PDE was discritized using Eigen-function expansion method and the behavior of the micro-beam with various applied electrostatic voltages was investigated. The results showed that in the presence of no electrostatic voltages with any initial condition the response is periodic. Applying electrostatic voltage resulted in the decrease of the area of the periodic solutions in the phase portrait and the greater electrostatic voltage led in the greater area of the non-periodic solutions; this behavior was continued up to pull-in voltage in which no periodic orbits was seen in the phase portrait. Applying simultaneous electrostatic and piezoelectric voltage led in the addition of a time varying component to the stiffness of the system which was due to the time varying actuation of the piezoelectric layers. The achieved equation was a Mathieu type differential equation in its damped form. Using Floquet theory for single degrees of freedom systems the stability analysis was performed and the stable regions were investigated in the plane of amplitude of Vac and the excitation frequency. The effect of the damping coefficient and the applied electrostatic voltage on the stability was determined. The results showed that increasing electrostatic voltage led in the increase of the vastity of the unstable region; in pull-in instability the constant component of the stiffness equaled zero and the response was unstable except in low enough amplitude of Vac and high enough excitation frequency. According to the results the more the damping coefficient increases the more does the vastity of the stable region increase, which is due to the reduction of the amplitude of the motion of the micro-beam. References [1] . I. Khatami, M. H. Pashai, and N. Tolou, Comparative vibration analysis of a parametrically nonlinear excited oscillator using HPM and numerical method, Mathematical Problems in Engineering, doi:10.1155/2008/956170. [2] A.M. Gasparini, A.V. Saetta, R.V. Vitaliani, On the stability and instability regions of non-conservative continuous system under partially follower forces, Comput. Meth. Appl. Mech. Eng. 124 (1–2) (1995) 63–78. [3] T. Itoh, T. Suga, Scanning force microscopes using a piezoelectric micro cantilever, J. Vacuum Sci. Technol. 12 (3) (1994) 1581–1585. [4] T. Itoh, C. Lee, T. Suga, Deflection detection and feedback actuation using a selfexcitedpiezoelectric Pb(Zr, Ti) O3 micro cantilever for dynamic scanning force microscopy, Appl. Phys. Lett. 69 (14) (1996) 2036–2038. [5] E. Cattan, T. Haccart, G. Velu, D. Remiens, C. Bergaud, L. Nicu, Piezoelectric properties of PZT films for microcantilever, J. Sens. Actuat. 74 (1999) 60–64. [6] M. Mitrovic, G.P. Carman, F.K. Straub, Response of piezoelectric stack actuators under combined electro-mechanical loading, J. Solids Struct. 38 (2001) 4357–4374. [7] L. Li, P. Kumar, S. Kanakraju, D.L. DeVoe, Piezoelectric ALGaAs bimorph microactuators, J. Micromech. Microeng. 16 (2002) 1062–1066. [8] W. Zhang, G. Meng, Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS, J. Sens. Actuat. A 119 (2005) 291–299. [9] Gh. Rezazadeh, A. Tahmasebi, M. Zubstov, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage, J. Microsyst. Technol. 12 (2006) 1163–1170.
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[10] J.F. Rhoads, S.W. Shaw, K.L. Turner, J. Moehlis, B.E. DeMartini, W. Zhang, Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators, J. Sound Vibrat. 296 (2006) 797–829. [11] W.M. Zhang, G. Meng, Nonlinear dynamic analysis of electrostatically actuated resonant MEMS sensors under parametric excitation, J. IEEE Sens. 7 (2007) 370–380. [12] J. Zhu, C.Q. Ru, A. Mioduchowski, High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators, The Eur. Phys. J. B 58 (2007) 411–421. [13] B.E. De Martini, J.F. Rhoads, K.L. Turner, S.W. Shaw, J. Moehlis, Linear and nonlinear tuning of parametrically excited MEMS oscillators, J. Microelectromech. Syst. 16 (2007) 310–318. [14] V. Goyal, R. Kapania, Dynamic stability of laminated beams subjected to non-conservative loading, J. Thin-walled Struct. 46 (2008) 1359–1369. [15] P.A. Djondjorov, V.M. Vassilev, On the dynamic stability of a cantilever under tangential follower force according to Timoshenko beam theory, J. Sound Vib. 311 (2008) 1431–1437. [16] M.H. Ghayesh, S. Balar, Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams, J. Solids Struct. 45 (2008) 6451– 6468. [17] N.I. Kim, D.K. Shin, M.Y. Kim, Flexural-torsional buckling loads for spatially coupled stability analysis of thin-walled composite columns, J. Adv. Eng. Softw. 39 (2008) 949–961. [18] H. Ding, L.Q. Chen, Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations, J. Mech. A/Solids 27 (2008) 1108–1120. [19] M. Zamanian, S.E. Khadem, S.N. Mahmoodi, The effect of a piezoelectric layer on the mechanical behavior of an electrostatic actuated micro-beam, J. Smart Mater. Struct. 17 (2008) 065024 (15pp). [20] G. Rezazadeh, A. Tahmasebi, Electromechanical behavior of micro-beams with piezoelectric and electrostatic actuation, Sens. Imag. 10 (2009) 15–30. [21] G. Rezazadeh, M. Fathalilou, R. Shabani, Static and dynamic stabilities of a micro-beam actuated by a piezoelectric voltage, J. Microsyst. Technol. 15 (2009) 1785–1791. [22] J.G. Guo, L.J. Zhou, Y.P. Zhao, Instability analysis of torsional MEMS/NEMS actuators under capillary force, J. Colloid Interface 331 (2009) 458–462. [23] H.H. Yoo, S.D. Kim, J. Chung, Dynamic modeling and stability analysis of an axially oscillating beam undergoing periodic impulsive force, J. Sound Vib. 320 (2009) 254–272. [24] J.R. Chang, W.J. Lin, Ch. J. Huang, S.T. Choi, Vibration and stability of an axially moving Rayleigh beam, J. Appl. Math. Model., doi:10.1016/ j.apm.2009.08.022 [25] S.N. Mahmoodi, M.F. Daqaq, N. Jalili, On the nonlinear-flexural response of piezoelectrically driven microcantilever sensors, J. Sens. Actuat. A 153 (2009) 171–179. [26] S.N. Mahmoodi, N. Jalili, Piezoelectrically actuated microcantilevers: An experimental nonlinear vibration analysis, J. Sens. Actuat. A 150 (2009) 131– 136. [27] K. Susanto, Vibration analysis of piezoelectric laminated slightly curved beams using distributed transfer function method, J. Solids Struct. 46 (2009) 1564–1573. [28] X.D. Yang, Y.Q. Tang, L.Q. Chen, C.W. Lim, Dynamic stability of axially accelerating Timoshenko beam: averaging method, Eur. J. Mech. A/Solids 29 (2010) 81–90. [29] M.R. Ghazavi, Gh. Rezazadeh, S. Azizi, Pure parametric excitation of a micro cantilever beam actuated by piezoelectric layers, Appl. Math. Model. 34 (2010) 4196–4207. [30] M. Zamanian, S.E. Khadem, Nonlinear vibration of an electrically actuated microresonator tuned by combined DC piezoelectric and electric actuations, Smart Mater. Struct. 19 (1) (2010) 015012, doi:10.1088/0964-1726/19/1/015012. [31] M.R. Ghazavi, Gh. Rezazadeh, S. Azizi, Finite element analysis of static and dynamic pull-in instability considering damping effects, J. Sensors Transducers 103 (2009) 132–143. [32] L. Mohanta, Dynamic stability of a sandwich beam subjected to parametric excitation, M.Sc. thesis, Department of Mechanical Engineering, National University of Technology, Deemed University, 2006. [33] S. Tse, Ivan E. Morse, Rolland, T. Hinkle, Mechanical vibrations theory and applications, second edition Francis. [34] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, John Willey & Sons, Inc., New York, 1979.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation Saber Azizi a, Ghader Rezazadeh b,⇑, Mohammad-Reza Ghazavi a, Siamak Esmaeilzadeh Khadem a a b
Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran Mechanical Engineering Department, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 26 July 2010 Received in revised form 19 March 2011 Accepted 31 March 2011 Available online 19 April 2011 Keywords: Micro-beam Piezoelectric layer Stability Mathieu equation MEMS/NEMS
a b s t r a c t In the present article an investigation is presented into the stability of an electro-statically deflected clamped–clamped micro-beam sandwiched by two piezoelectric layers undergoing a parametric excitation applying an AC voltage to these layers. Applying an electrostatic actuation not only deflects the micro-beam but also decreases the bending stiffness of the structure, which can lead the structure to an unstable position by undergoing a saddle node bifurcation. Utilizing an appropriate AC actuation voltage to the piezoelectric layers produces a time varying axial force, which can play the role of a stabilizer exciting the system parameter. The governing equation of the motion is a nonlinear electro-mechanically coupled type PDE, which is derived using variational principle and discretized, applying Eigenfunction expansion method. The resultant is a Mathieu type equation in its damped form. Using Floquet theory for single degree of freedom system the stable and unstable regions of the problem are investigated. The effects of viscous damping and electrostatic actuation on the stable regions of the problem are also studied. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Micro-electromechanical systems (MEMS) are widely being used in today’s technology. So investigating the problems referring to MEMS, owns a great importance. One of the significant fields of study is the stability analysis of the parametrically excited systems. Parametrically excited micro-electromechanical devices are ever increasingly being used in radio, computer and laser engineering [1]. Parametric excitation occur in a wide range of mechanics, due to time dependent excitations, especially periodic ones; some examples are columns made of nonlinear elastic material, beams with a harmonically variable length, parametrically excited pendulums and so forth. Investigating stability analysis on parametrically excited MEM systems is of great importance. In 1995 Gasparini et al. [2] studied on the transition between the stability and instability of a cantilevered beam exposed to a partially follower load. Piezoelectric micro cantilevers were first utilized for sensing purposes [3] and then utilized as actuators [4]. In 1999 Cattan et al. [5] used an unusual experimental method to study on the piezoelectric properties of PZT films for micro cantilever. In 2000 Mitrovic et al. [6] investigated the response of six piezoelectric stack actuators under electrical, mechanical, and combined electromechanical loading. They focused on understanding the behavior of piezoelectric materials under combined electromechanical loading, and determining fundamental properties necessary to model the constitutive response and optimizing the actuator performance. In 2002 Li et al. [7] studied on piezoelectric AlGa bimorph micro actuators and verified their results using experimental set ups.
⇑ Corresponding author. Tel.: +98 914 145 1407. E-mail addresses: [email protected] (S. Azizi), [email protected] (G. Rezazadeh), [email protected] (M.-R. Ghazavi), s.khadem@ modares.ac.ir (S.E. Khadem). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.03.049
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In 2005 Zhang and Meng [8] studied on the behavior of a micro cantilever under combined parametric and forcing excitation. They used harmonic balance method to simulate the resonance amplitude frequency of the system under the combined parametric and forcing excitation. In 2006 Rezazadeh et al. [9] presented a novel method to control the pull-in voltage and to measure the residual stress in the fixed–fixed and cantilever MEM actuators. In 2006 Rhoads et al. [10] investigated generalized parametric resonance in electro-statically actuated micro-electromechanical oscillators. They used perturbation analysis to study the behavior of the proposed model. Their results include a wide array of interesting dynamical behavior, most of which can be attributed to the existence of nonlinear parametric excitation in their equation of motion. In 2007 Zhang and Meng [11] studied on the nonlinear response and dynamics of the electro-statically actuated MEMS resonant sensors under two parametric and external excitations. In 2007 Zhu et al. [12] investigated the parametric resonance of coupled micro-electromechanical oscillators under periodically varying nonlinear coupling forces. They used harmonic balance method combined with Newton iteration method to find the steady state periodic solutions. In 2007 DeMartini et al. [13] investigated linear and nonlinear tuning of parametrically excited MEMS oscillators. The governing equation of motion was a nonlinear Mathieu type with time varying linear and nonlinear stiffness term. They used perturbation technique to determine the behavior of the system. In 2008 Goyal and Kapania [14] studied on the dynamic stability of laminated beams subject to a combination of conservative and non-conservative tangential follower loads. They also investigated the divergence and flutter instabilities. In 2008 Djondjorov and Vassilev [15] investigated the dynamic stability of a cantilevered Timoshenko beam lying on an elastic foundation of Winkler type. Their model was subjected to a tangential follower force. In 2008 Gayesh and Balar [16] studied on the non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. They applied the method of multiple scales to the governing equations of motion, and considering the solvability condition obtained the linear and nonlinear frequencies and mode shapes of the system. They also investigated the stability and bifurcation points of the system through parametric studies. In 2008 Kim et al. [17] performed stability analysis on thin-walled composite columns. They compared their numerical solutions with those of analytical ones and finite element method using Hermitian beam element. In 2008 Ding and Chen [18] investigated the stability of axially accelerating viscoelastic beams subjected to parametric vibration, using multiple scales method. They also performed numerical investigations applying the finite difference method. In 2008 Zamanian et al. [19] studied on the deflection and natural frequency of a micro-beam under combined electrostatic and piezoelectric actuation. They solved the nonlinear governing differential equation using Galerkin method and showed that a new sensor-actuator system may be actuated by applying the voltage to the piezoelectric layer, and the actuation may be sensed by the value of the output electric current induced from the movement of the polarized micro-beam. In 2009 Rezazadeh et al. [20] investigated the electromechanical behavior of microbeams with piezoelectric and electrostatic actuation. They solved the nonlinear governing differential equation using step by step linearization method. They could control the trigger time of the micro switch by applying appropriate voltage to the piezoelectric layers. In 2009 Rezazadeh et al. [21] studied on the static and dynamic stabilities of a micro-beam with various boundary conditions actuated by a DC piezoelectric voltage. In 2009 Guo et al. [22] performed instability analysis on torsional MEMS/NEMS actuators under capillary force. They qualitatively analyzed the nonlinear equation, and bifurcation phenomenon. In 2009 Yoo et al. [23] proposed a dynamic modeling method of an axially oscillating beam subjected to periodic impulsive force. They investigated the effect of various parameters governing the problem including the impulse magnitude, the oscillating frequency, the oscillating speed amplitude and the model damping ratio on the stability of the beam. Chang et al. [24] studied on the vibration and stability analysis of an axially moving Rayleigh beam. They considered two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion. The authors used direct time numerical integration, based on Runge–Kutta algorithm, and Floquet theory for vibration and stability analysis respectively. They also investigated the divergent and flutter instabilities. In 2009 Mahmoodi et al. [25] studied on the nonlinear flexural response of piezoelectrically driven micro cantilever sensors they used multiple scales method to study the asymptotic behavior of the sensor’s response. In 2009 Mahmoodi and Jalili [26] performed an experimental nonlinear vibration analysis on piezoelectrically actuated micro cantilevers. In 2009 Susanto [27] analyzed vibration of piezoelectric laminated slightly curved beams using distributed transfer function method. In 2010 Yang et al. [28] in another study investigated the stability of axially accelerating Timoshenko beam exploiting averaging method. Studying the behavior of MEM structures covered with piezoelectric layers actuated with AC voltage will also lead in parametric vibration and investigating its stability is of great importance due to its ever increasing applications. In 2010 Ghazavi et al. [29] studied on the pure parametric excitation of a cantilever micro-beam actuated with piezoelectric actuation; they investigated the stability using Floquet theory for single degree of freedom systems. Although few studies have focused on [19,30], simultaneous piezoelectric and electrostatic actuations add interesting qualifications including shifting frequency operation range, to the MEM and NEM structures. In the present study the stability analysis is carried out on a clamped–clamped micro-beam sandwiched with two piezoelectric layers located on its upper and lower sides. The micro-beam is subjected to an electrostatic field. Applying an AC voltage to the piezoelectric layers, and considering small vibrations around the electro-statically deflected equilibrium position, the stability analysis is performed on the governing Mathieu type equation of the motion using the Floquet theory for single degree of freedom system. The stable and unstable regions of the problem are investigated, and the transition curves separating stable from unstable solutions of the system in the parameter plane and in terms of amplitude and frequency of internal parametric excitation are illustrated. The effect of the damping coefficient and electrostatic voltage on the stability of the structure is also investigated. And, the possibility of stabilizing pull-in instability using a parametric excitation is investigated.
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2. Modeling As illustrated in Fig. 1 the studied model is an isotropic micro-beam of length l, width a, thickness h, density q with Young’s modulus, E sandwiched with two piezoelectric layers having thickness hp, density qp throughout the micro-beam length. The Young modulus of the piezoelectric layers is denoted by Ep and the equivalent piezoelectric coefficient is supposed to be e31 . The initial gap between the micro beam and the substrate is w0 and the applied electrostatic voltage is denoted by Ves. The coordinate system as illustrated, is attached to the middle of the left end of the micro-beam where x and z refer to the horizontal and vertical coordinates, respectively. The governing equation of the transverse motion can be obtained by the minimization of the Lagrangian using variational principle. The total potential energy includes the bending and axial strain energies (Ub, Ua) and the electrical energy (Ue) as following [29]:
!2 Z l Z 2 1 @2w 1 l @w dx þ F dx þ U e ; ðEIÞeq P 2 @x2 2 0 @x 0 h þ hp ; F P ¼ 2ae31 V ac cosð2xtÞ; ðEIÞeq ¼ EI þ Ep hahp 2
UðtÞ ¼ U b þ U a þ U e ¼
ð1Þ
where I denotes the moment of inertia of the cross section about the horizontal axis passing through the center of the surface of the cross section of the micro-beam, w is the mid plane deflection, Ua indicates the strain energy due to the axial load (Fp),
Piezoelectric Layers h
x
Microbeam
z
Ves
V = Vac cos (2ωt ) p
h hp
Ves
V = Vac cos (2ωt )
l
a
Fig. 1. Schematic view of the studied model (A) Front view. (B) Side view.
keq meq w(l / 2, t )
w0
Ves
Fig. 2. Schematic view of the single degree of freedom system.
Table 1 Geometrical and material properties of the micro-beam and piezoelectric layers. Geometrical and material properties
Micro-beam
Piezoelectric layers
Length (L) Width (a) Height (h) Initial gap (w0) Young’s modulus (E) Density (q) Piezoelectric constant e31 Permittivity constant (e0)
600 lm 50 lm 3 lm 1 lm 169.61 GPa 2331 kg/m3 _ 8.845 1012(F/m)
600 lm 50 lm 0.01 lm – 76.6 GPa 7500 kg/m3 9.29 [21]
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which is the result of piezoelectric actuation. Vac and x refer to the amplitude of the alternative and the frequency of the applied voltage to the piezoelectric layers respectively. Ue indicates the electrical potential energy stored between the capacitor plates. It must be noted that U e is the electrical co-energy and denoted as [31]:
U e ¼
1 2
Z 0
e0 aV 2es
l
ðw0 wÞ
dx;
ð2Þ
where e0 is the permittivity constant of the dielectric material between the micro beam and the substrate, Ves indicates the applied voltage between the micro beam and the substrate, and w0 is the initial gap between the micro beam and the substrate. The kinetic energy is expressed as follows:
T¼
1 ðqAÞeq 2
Z l 2 @w dx; @t 0
ð3Þ
ðqAÞeq ¼ aðqh þ 2qp hp Þ; where q and qp refer to the densities of the micro-beam and the piezoelectric layers respectively. The Lagrangian may be consisted as:
1 0.8
Middle gap) (μ m)
0.6
Stable Unstable Stable,Physically imposible
Pull-in
•
0.4 0.2 0 -0.2 -0.4 0
2
4
6
Ves(V) Fig. 3. Center gap versus applied electrostatic voltage.
Fig. 4a. Phase plain corresponding to center gap and Ves = 0(V).
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Fig. 4b. Phase plain corresponding to center gap and Ves = 2(V).
Fig. 4c. Phase plain corresponding to center gap and Ves = 4(V).
1 LðtÞ ¼ TðtÞ U b U a U e ¼ ðqAÞeq 2
Z l 0
!2 2 Z 2 Z l @w 1 @2w 1 l @w dx ðEIÞeq dx F dx U e : P @t 2 @x2 2 0 @x 0
ð4Þ
Extermizing the Lagrangian and using Rayleigh’s dissipation function to exploit Lagrange equation for non-conservative systems, and considering dU e ¼ dU e [31] the dynamic equation of the motion considering viscous damping will be obtained as Eq. (5):
ðEIÞeq
@4w @2w @ 2 w e @w e0 aV 2es ¼ þ ðqAÞeq 2 F P 2 þ C ; 4 @x @x @t @t 2ðw0 wÞ2
ð5Þ
e stands for the viscous damping coefficient per unit length of the micro-beam. where C 3. Numerical solution The solution of the system can be considered as follows:
wðx; tÞ ¼ w ðxÞ þ wd ðx; tÞ; ⁄
where w (x) refers to the static deflection and
ð6Þ wd⁄(x,
t)corresponds to the dynamic response of the micro-beam due to the Vac.
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Fig. 4d. Phase plain corresponding to center gap and Ves = 6(V).
Fig. 4e. Phase plain corresponding to center gap and Ves = Vpull-in = 6.9(V).
Considering the static equilibrium position due to the electrostatic voltage as denoted in Eq.(7) and substituting Eq. (6) in (5) and expanding the right hand side of the Eq. (5) up to the first term of Tailor series, the equation of motion will be obtained as indicated in Eq.(8):
ðEIÞeq
@ 4 w e0 aV 2es ¼ ; @x4 2ðw0 w Þ2
ð7Þ
ðEIÞeq
@ 4 wd @ 2 wd e @wd @ 2 ðwd Þ e0 aV 2es þ ðqAÞeq þC wd ¼ 0: FP 2 4 2 @x @x @t @t 2ðw0 w Þ3
ð8Þ
There is no exact solution to the Eq. (8) [32] so an approximate solution is supposed to be in the form as follows:
wd ¼
n X
uj ðxÞqj ðtÞ;
ð9Þ
j¼1
where uj and qj(t) refers to the shape functions of a clamped–clamped beam and time dependant amplitudes respectively. Substituting Eq. (9) in Eq. (8) results in:
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e ¼ 0:0. Fig. 5a. V es ¼ 0:0ðVÞ; C
e ¼ 0:5. Fig. 5b. V es ¼ 0:0ðVÞ; C
e ¼ 1:0. Fig. 5c. V es ¼ 0:0ðVÞ; C
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e ¼ 0:0. Fig. 6a. V es ¼ 2:0ðVÞ; C
e ¼ 0:5. Fig. 6b. V es ¼ 2:0ðVÞ; C
e ¼ 1:0. Fig. 6c. V es ¼ 2:0ðVÞ; C
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ðEIÞeq
n X
uIVj ðxÞqj ðtÞ þ ðqAÞeq
n X
j¼1
n X
uj ðxÞq€j ðtÞ F P
j¼1
n X
u00j ðxÞqj ðtÞ þ Ce
j¼1
n X
_ e0 aV 2es ðw0 w Þ3 uj ðxÞqðtÞ
j¼1
uj ðxÞqj ðtÞ ¼ Rðx; tÞ;
ð10Þ
j¼1
where R(x, t) is the residual. Multiplying Eq. (10) in ui(x)and integrating the outcome in domain [0, l] and considering the orthogonality of the distinct shape functions one will have: n X
n X
qj ðtÞK ij þ
j¼1
€j ðtÞM ij þ q
j¼1
n X
_ qðtÞC ij ¼ 0;
ð11Þ
j¼1
where:
K ij ¼ ðEIÞeq
Z
M ij ¼ ðqAÞeq e C ij ¼ C
Z 0
l
0
ui ðxÞuIVj ðxÞdx F P
Z 0
Z 0
l
u00j ðxÞui ðxÞdx e0 aV 2es
Z 0
l
ðw0 w Þ3 ui ðxÞuj ðxÞdx;
l
ð12Þ
ui ðxÞuj ðxÞdx;
l
ui ðxÞuj ðxÞdx:
e ¼ 0:0. Fig. 7a. V es ¼ 4:0ðVÞ; C
e ¼ 0:5. Fig. 7b. V es ¼ 4:0ðVÞ; C
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e ¼ 1:0. Fig. 7c. V es ¼ 4:0ðVÞ; C
In the structures subjected to an electrostatic actuation, increasing voltage, applied to the electrostatic areas, decreases equivalent stiffness of the structure, therefore for a given applied voltage called as pull-in voltage in the MEMS literature, a stationary instability is occurred by undergoing system to a saddle node bifurcation. This kind of instability can be even observed in single degree of freedom systems. Therefore applying the first shape function of the micro-beam, this kind of instability of electro-statically deflected micro-beams qualitatively can be studied. For a clamped–clamped micro-beam the first Eigen-function is given as follows [33]:
u1 ðxÞ ¼ 1:0178 cos 4:73
x x x x cosh 4:73 þ sin 4:73 sinh 4:73 : l l l l
ð13Þ
Expanding Eq. (11) for j = 1, and considering xt = s one will have: 2
d q ~cðqðtÞÞ þ þ ðd þ 2e cosð2sÞÞq ¼ 0; ds2 ds
ð14Þ
where:
d ¼ 12:36
e ¼ 0:858
ðEIÞeq 4
x qAÞeq l 2ð
ae31 V ac
x2 ðqAÞeq l2
index ; 1:856x2 ðqAÞeq l
ð15Þ
;
e ¼ 0:0. Fig. 8a. V es ¼ 6:0ðVÞ; C
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e ¼ 0:5. Fig. 8b. V es ¼ 6:0ðVÞ; C
e ¼ 1:0. Fig. 8c. V es ¼ 6:0ðVÞ; C
e c¼
C ; ðqAÞeq x
where the parameter index, which is in direct relation with the electrical stiffness, stands for:
index ¼ e0 aV 2es
Z 0
l
ðw0 w ðxÞÞ3 ui ðxÞuj ðxÞdx:
ð16Þ
The static deflection w⁄(x) depends on the applied electrostatic voltage. For any applied electrostatic voltage, Eq. (7) is solved numerically using FE method with Hermitian shape functions [31]. Having obtained the nodal values of the static deflection, the functionality of the static deflection on the coordinate x is determined using the least square method. To determine the parameter ‘‘index’’ the integral in Eq. (16) is evaluated numerically.
4. Stability analysis In this part the stability analysis once considering electrostatic actuation and once considering piezoelectric actuation, on electro-statically deflected micro-beam is performed on the governing equation of the motion.
S. Azizi et al. / Applied Mathematical Modelling 35 (2011) 4796–4815
e ¼ 0:0. Fig. 9a. Pull-in voltage, C
e ¼ 0:5. Fig. 9b. Pull-in voltage, C
e ¼ 1:0. Fig. 9c. Pull-in voltage, C
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4.1. Electrostatic actuation To investigate the behavior of the system the equivalent single degree of freedom system located at the middle length of the micro-beam is considered as illustrated schematically in Fig. 2. Considering electrostatic actuation without that of piezoelectric and using Eigen-function expansion method to derive the equivalent single degree of freedom system at the center of the micro-beam, the equation of the motion and the corresponding equivalent mass and the stiffness will be obtained as follows:
l l e0 AV 2es € ; t þ keq w ; t ¼ meq w ; 2 2 2ðw0 wðl=2; tÞÞ Z
meq ¼ ðqAÞeq
keq ¼ ðEIÞeq
l
0
Z 0
u1 ðxÞu1 ðxÞdx;
l
u1 ðxÞuIV1 ðxÞdx;
Fig. 10a. Phase portrait corresponding to point ‘B’.
Fig. 10b. Time history corresponding to point ‘B’.
ð17Þ
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where:
wðl=2; tÞ ¼ qðtÞu
l : 2
ð18Þ
In Eq. (17) parameter (A) is the area of the equivalent mass exposed to the electrostatic actuation and is determined in a manner that the static pull-in voltage of the continuous system to be equal to that of the discritized one; so one will have:
fðEIÞeq
Rl
0
u1 ðxÞuIV1 ðxÞdxg2ðwðl=2; tÞ w0 Þwðl=2; tÞ ¼ A: e0 V 2Pullin
ð19Þ
4.2. Electrostatic and piezoelectric actuation Eq. (14) is in the form of damped Mathieu equation, which was discussed in its damp-less form by Mathieu (1868) in connection with the problem of vibrations of elliptic membrane [34]. When the governing differential equation of motion
Fig. 11a. Phase portrait corresponding to point ‘C’.
Fig. 11b. Time history corresponding to point ‘C’.
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for the system is of Mathieu type, a few commonly used well known methods are perturbation and iteration techniques and Bolotin method, which is based on Floquet theory [32]. In this work to establish stability analysis, the Floquet theory for single degree of freedom system is applied. Eq. (14) is a second order homogenous differential equation, so there are two linear and non-vanishing independent solutions called q1(t), q2(t). It can be proved that if q1(t), q2(t)are a fundamental set of solutions of Eq. (14), q1(t + s)andq2(t + s) are also a fundamental set of solutions of Eq. (14) [29]; so one may have:
q1 ðt þ TÞ q2 ðt þ TÞ
¼
a11
a12
a21
a22
q1 ðtÞ q1 ðtÞ
:
a11 a12 is not unique and it depends on the fundamental solution being used. There exist a fundamental set of a21 a22 solution in which one has: The matrix
ð20Þ
e ¼ 0:0. Fig. 12a. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 0:0. Fig. 12b. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
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e ¼ 0:5. Fig. 12c. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 0:5. Fig. 12d. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
Q 1 ðs þ pÞ ¼ k1 Q 1 ðsÞ; Q 2 ðs þ pÞ ¼ k2 Q 2 ðsÞ
ð21Þ
in which Q1(s + p) and Q2(s + p) are denoted as normal or Floquet solutions [29]; the corresponding k1 and k2 are constants, which may be complex. It follows from Eq. (21) that
Q i ðs þ npÞ ¼ kni Q i ðsÞ;
ð22Þ
where n is an integer. The stable (bounded) and unstable solutions will be determined as soon as the values of the corresponding k1 ; k2 are investigated. Consequently jki j < 1 and jki j < 1 correspond to bounded (Stable) and unbounded (unstable) solutions respectively. As reasonable ki ¼ 1, and ki ¼ 1 are respectively relative to periodic solutions with periods p and 2p [29].
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e ¼ 1:0. Fig. 12e. Phase portrait corresponding to point ‘A’ V es ¼ 2ðVÞ; C
e ¼ 1:0. Fig. 12f. Time history corresponding to point ‘A’ V es ¼ 2ðVÞ; C
5. Results and discussions In this article as a case study a silicon micro-beam sandwiched with PZ-4 layers is considered. The geometrical and material properties of the micro-beam and piezoelectric layers involved in the problem are given in Table 1.
5.1. Electrostatic actuation Considering Vac = 0 the problem is a clamped–clamped micro-beam sandwiched with piezoelectric layers subjected to electrostatic actuation which was studied in the references [9,20]. According to Eq. (7) growing the electrostatic voltage increases the static deflection w⁄(x), and consequently raises the absolute value of the parameter ‘index’. Fig. 3 illustrates w0 wjx¼ l denoted as middle gap, versus Ves. z The phase portraits corresponding to Eq. (14) considering e = 0 with various applied electrostatic voltages are given in Fig. 4. The superscript dot ( ) stands for the differentiation with respect to time (t).
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In Fig. 3 dashed lines correspond to physically impossible solutions. As Fig. 4a claims with Ves = 0(V) there exists a stable equilibrium point and with any initial condition the response is periodic. Applying electrostatic voltage leads in the appearance of two extra equilibrium points including a saddle node, and a stable but physically impossible centre in the phase space. The impossibility of the stable solutions refers to the existence of the substrate beneath, which limits the amplitude of the motion of the micro-beam. As clear increasing the electrostatic voltage lessens the vastity of the periodic orbits and enlarges to those of non-periodic ones; this is due to the fact that increasing electrostatic voltage leads in the softening of the micro-beam, which increases the absolute value of the parameter ‘index’ (Electrical stiffness). This behavior is continued up to a critical voltage denoted as pull-in, in which no periodic orbits are seen in the phase space except those of physically impossible. In the coming section it is tried to stabilize the pull-in instability using piezoelectric actuation. 5.2. Electrostatic and piezoelectric actuations Applying simultaneous electrostatic and piezoelectric actuation causes the micro-beam to have a constant and a periodically time varying stiffness components. In this section the dependency of the stability of the micro-beam, on the applied electrostatic voltage, amplitude of Vac, and damping coefficient is investigated. Considering Ves = 0(V), the problem leads to that type studied by Ghazavi et al. [29] where a good agreement is seen. The stable and unstable (dashed) regions in the plane of amplitude of Vacand the excitation frequency are given in Figs. 5–9. Figs. 5–10 claim that increasing the electrostatic voltage enlarges the unstable region which is in a good agreement with those given in Fig. 4. This is due to the fact that applying electrostatic voltage decreases the constant component of the stiffness, which leads in the increase of the amplitude of the motion of the micro-beam; This behavior is continued up to pull-in voltage in which almost everywhere in the plane of the amplitude of Vac, and the excitation frequency is unstable, except those of high excitation frequency and low amplitude of Vac; This is considerably important, because one may excite the micro-beam with a suitable amplitude and frequency of piezoelectric actuation, to stabilize the pull-in instability. Also increasing the amplitude of Vac have a destabilizing effect, this is reasonable due to the growth of the amplitude of the time varying component of the stiffness. As reasonable increasing the value of the damping coefficient lessens the amplitude of the motion of the micro-beam and has a stabilizing effect in the behavior of the structure. As obvious the transition curve in Fig. 9 asymptotically lies on the horizontal axis as the excitation frequency reduces; This is due to the unstable nature of the pull-in point; however applying ‘AC’ voltage even with a small amplitude but a high enough frequency results in the stability of the micro-beam. Fig. 10 refers to the phase portrait and the corresponding time history of point ‘ B’ illustrated in Fig. 9a, with coordinates x ¼ 1:5 105 ðrad Þ; V ac ¼ 3ðVÞ.As the results of the stability analysis claim this point is located in unstable region s and the corresponding time history is boundless. Fig. 11 illustrates the time history and the phase plain corresponding to
point ‘C’ with the same amplitude of ‘Vac’ with point ‘B’, but with a high enough excitation frequency, x ¼ 3 105 rad which s stabilizes the pull-in instability. Fig. 12 shows the phase portrait and the corresponding time histories of point ‘A’ with coor dinate x ¼ 2:5 105 rad ; V ac ¼ 5ðVÞ with various damping coefficients. As the results claim, damping coefficient limits the s amplitude of the motion and has a stabilizing effect on the response of the system. Fig. 13 illustrates the phase portrait and the corresponding time histories of point ‘A’ with Ves = 6(V) and various damping coefficients. Although the damping coefficient considerably limits the amplitude of the motion but it is not high enough to
Fig. 13a. Phase portrait corresponding to point ‘A’ with Ves = 6(V) and various damping coefficients.
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Fig. 13b. Time history corresponding to point ‘A’ with Ves = 6(V) and various damping coefficients.
make the micro-beam undergo a stable response. However, in this case, one may actuate the piezoelectric layers with a high enough frequency to stabilize the response. 6. Conclusion In this paper the stability analysis was performed on a piezoelectrically actuated clamped–clamped micro-beam deflected due to an electrostatic actuation. The governing differential equation of the motion was derived by the minimization of the Lagrangian and developed to non-conservative systems applying Rayleigh’s dissipation function. The nodal values of the static deflection of the micro-beam was determined by numerically solving the governing differential equation and the functionality of the deflection on the coordinate ‘x’ was determined using least square method. The governing PDE was discritized using Eigen-function expansion method and the behavior of the micro-beam with various applied electrostatic voltages was investigated. The results showed that in the presence of no electrostatic voltages with any initial condition the response is periodic. Applying electrostatic voltage resulted in the decrease of the area of the periodic solutions in the phase portrait and the greater electrostatic voltage led in the greater area of the non-periodic solutions; this behavior was continued up to pull-in voltage in which no periodic orbits was seen in the phase portrait. Applying simultaneous electrostatic and piezoelectric voltage led in the addition of a time varying component to the stiffness of the system which was due to the time varying actuation of the piezoelectric layers. The achieved equation was a Mathieu type differential equation in its damped form. Using Floquet theory for single degrees of freedom systems the stability analysis was performed and the stable regions were investigated in the plane of amplitude of Vac and the excitation frequency. The effect of the damping coefficient and the applied electrostatic voltage on the stability was determined. The results showed that increasing electrostatic voltage led in the increase of the vastity of the unstable region; in pull-in instability the constant component of the stiffness equaled zero and the response was unstable except in low enough amplitude of Vac and high enough excitation frequency. According to the results the more the damping coefficient increases the more does the vastity of the stable region increase, which is due to the reduction of the amplitude of the motion of the micro-beam. References [1] . I. Khatami, M. H. Pashai, and N. Tolou, Comparative vibration analysis of a parametrically nonlinear excited oscillator using HPM and numerical method, Mathematical Problems in Engineering, doi:10.1155/2008/956170. [2] A.M. Gasparini, A.V. Saetta, R.V. Vitaliani, On the stability and instability regions of non-conservative continuous system under partially follower forces, Comput. Meth. Appl. Mech. Eng. 124 (1–2) (1995) 63–78. [3] T. Itoh, T. Suga, Scanning force microscopes using a piezoelectric micro cantilever, J. Vacuum Sci. Technol. 12 (3) (1994) 1581–1585. [4] T. Itoh, C. Lee, T. Suga, Deflection detection and feedback actuation using a selfexcitedpiezoelectric Pb(Zr, Ti) O3 micro cantilever for dynamic scanning force microscopy, Appl. Phys. Lett. 69 (14) (1996) 2036–2038. [5] E. Cattan, T. Haccart, G. Velu, D. Remiens, C. Bergaud, L. Nicu, Piezoelectric properties of PZT films for microcantilever, J. Sens. Actuat. 74 (1999) 60–64. [6] M. Mitrovic, G.P. Carman, F.K. Straub, Response of piezoelectric stack actuators under combined electro-mechanical loading, J. Solids Struct. 38 (2001) 4357–4374. [7] L. Li, P. Kumar, S. Kanakraju, D.L. DeVoe, Piezoelectric ALGaAs bimorph microactuators, J. Micromech. Microeng. 16 (2002) 1062–1066. [8] W. Zhang, G. Meng, Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS, J. Sens. Actuat. A 119 (2005) 291–299. [9] Gh. Rezazadeh, A. Tahmasebi, M. Zubstov, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage, J. Microsyst. Technol. 12 (2006) 1163–1170.
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