Stability and Bifurcation Analysis of an Asymmetricall
Hadi Madinei Mechanical Engineering Department,
Urmia University,
Urmia51818-57561,Iran e-mail:h.madinei@https://www.360docs.net/doc/af11328504.html, Ghader Rezazadeh1 Mechanical Engineering Department,
Urmia University,
Urmia51818-57561,Iran e-mail:g.rezazadeh@urmia.ac.ir
Saber Azizi Mechanical Engineering Department, Urmia University of Technology,
Urmia5716617165,Iran e-mail:s.azizi@mee.uut.ac.ir Stability and Bifurcation Analysis of an Asymmetrically Electrostatically Actuated Microbeam
This paper deals with the study of bifurcational behavior of a capacitive microbeam actu-ated by asymmetrically located electrodes in the upper and lower sides of the microbeam.
A distributed and a modi?ed two degree of freedom(DOF)mass–spring model have been implemented for the analysis of the microbeam behavior.Fixed or equilibrium points of the microbeam have been obtained and have been shown that with variation of the applied voltage as a control parameter the number of equilibrium points is changed.The stability of the?xed points has been investigated by Jacobian matrix of system in the two DOF mass–spring model.Pull-in or critical values of the applied voltage leading to qual-itative changes in the microbeam behavior have been obtained and has been shown that the proposed model has a tendency to a static instability by undergoing a pitchfork bifur-cation whereas classic capacitive microbeams cease to have stability by undergoing to a saddle node bifurcation.[DOI:10.1115/1.4028537]
Keywords:bifurcation,microbeam,electrostatic actuation,stability,MEMS
1Introduction
Micro-electromechanical-systems(MEMS)technology has been quickly growing since its beginning in1980s as sensors and actuators.The ability of MEMS to miniaturize,reduce the cost and energy consumption causes it to be interesting object for researches[1–4].Electrostatic actuation of conductive?exible microbeams due to their simplicity,as they require few mechani-cal components and small voltage levels for actuation are mostly used in many?elds[5].Electrostatically actuated microbeams are in?uenced to instability,which is known as pull-in phenomenon in MEMS literature.Pull-in instability greatly limits the stable range of operation of microbeams.Then,one of the important issues in the design of electrostatically actuated microbeam is to tune the electric load away from the pull-in instability[6],which leads to failure of the device.
The earliest mathematical analysis of pull-in instability may be found in the pioneering work of Nathanson et al.[7]who con-structed and analyzed a mass–spring model of electrostatic actua-tion and offered the?rst theoretical explanation of pull-in instability.At roughly the same time,Taylor[8]studied the elec-trostatic de?ection of two oppositely charged soap?lms,and he predicted that when the applied voltage was increased beyond a certain critical voltage,the two soap?lms would make contact. Since Nathanson and Taylor’s seminal work,numerous investiga-tors have analyzed and developed mathematical models of elec-trostatic actuation in attempts to understand further and control pull-in instability.
An overview of the physical phenomena of the mathematical models associated with the rapidly developing?eld of MEMS technology is given in Osterberg[9].Other important work on the stability of this problem was done by Mukherjee[10]that analyzed the dynamic motion of MEMS.Nemirovsky and Bochobza-Degani[11]found the model for the pull-in parameters of electrostatic actuators.Krylov and Maimon[12]studied on the pull-in dynamics of an elastic beam actuated by distributed elec-trostatic force.Chowdhury et al.[13]found a closed-form model for the pull-in voltage of electrostatically actuated cantilever beams.Batra et al.[14]studied on the vibrations of narrow microbeams predeformed by an electric?eld.Sadeghian and Rezazadeh[15]studied a comprehensive comparison between the generalized differential quadrature and Galerkin method for analysis of pull-in behavior of micro-electromechanical coupled systems.Talebian et al.[16]studied temperature effects on me-chanical behavior of an electrostatically actuated microplate and showed that temperature reduction from room temperature increases the pull-in voltage.Saeedivahdat et al.[17]studied the effects of thermal stresses on stability and frequency response of a fully clamped circular microplate and showed that the decrement of the microplate temperature,which leads to a tensile thermal stress in the microplate,increases the stable region of the capaci-tive microplate and increment of the microplate temperature vice versa.
In MEMS structure,due to distributed mechanical characters of microbeam,these systems generally have an in?nite number of modes of vibration.By the analytical method of modal analysis [18],it can be made clear that the modes,which are required to describe the vibration,are closely related to spatial dependence of the excitation load.Most excitation methods utilize on-chip elec-trodes.By shaping these electrodes,the excitation load can be controlled,giving us control over which modes are excited and how strong they are.The modes that will be detected depend also strongly on the shape of the detection elements.When no special attention is paid to the design of excitation and detection ele-ments,multiple modes will be excited and detected.However, vibration systems are usually operated at one eigenfrequency (usually the?rst).The appearance of other modes can lead to degrading system performance.By using selective mode excita-tion and selective mode detection,the condition for oscillation is met for an unambiguous frequency over a wide range[19,20]. Recently,several authors studied the selective mode excitation. Dominguez-Pumar et al.[21]studied the activation of different resonant modes of a MEMS resonator,by simply changing some parameters in the feedback?lter of Pulsed Digital Oscillators. Mohammadi et al.[22]studied a PnC slab micromechanical
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the
J OURNAL OF C OMPUTATIONAL AND N ONLINEAR D YNAMICS.Manuscript received January
29,2013;?nal manuscript received September9,2014;published online January12,
2015.Assoc.Editor:Carmen M.Lilley.
Journal of Computational and Nonlinear Dynamics MARCH2015,Vol.10/021002-1
Copyright V C2015by ASME
resonator with direct piezoelectric excitation to selectively excite and ef?ciently con?ne lateral ?exural and extensional vibrations at very high frequencies ($130MHz).Gil et al.[23]studied the selective excitation of different modes of two coupled microcanti-levers with piezoelectric layer.Although in other type of actuating system researchers have studied selective mode excitation but in many studies about electrostatically actuated microbeams,authors have excited the ?rst mode of vibration more than other modes and due to this excitation;microbeam is oscillated on frequency range,from ?rst natural frequency to frequency that related to 60–70%pull-in voltages of the ?rst mode of vibration.These sys-tems have a tendency to static instability by undergoing a saddle node bifurcation in related to the ?rst mode of the system.
So,in this paper,we show that by introducing a new type of electrostatic actuation mechanism by shaping the electrodes and locating them in the two side of the microbeam asymmetrically and exciting the second mode of vibration more than other modes,frequency range of vibration extended from second natural fre-quency to frequency that related to 60–70%pull-in voltages of second mode of vibration and by increasing the electrical load,the system has a tendency to static instability by undergoing a pitch-fork bifurcations.To demonstrate and analyze the possibility of this fact,a clamped microbeam,which is excited by two electro-des,is considered.To show characteristics of this system,distrib-uted and lumped two DOF models are considered.To solve the equation of motion in distributed model,a Galerkin based weighted residual method is used and by choosing suitable discre-tized model for this system,the results of two models are reason-ably seen to satisfy each other.
2Model Description and Mathematical Modeling
As illustrated in Fig.1,the studied model is a clamped–clamped
microbeam with length L ,width b ,thickness h ,density q with Young’s modulus E ,which is located between four electrodes with lengths L =2.In this model,the desirable mode is excited by choosing appropriate electrodes as driven ones.In the ?rst case,the same DC voltage is applied to electrode 1and electrode 2(or electrode 3and electrode 4).In this condition,classical electro-statically actuated microbeam is achieved and the ?rst mode of vibration is excited more than other modes.In the second case,the DC voltage is applied to all of the electrodes simultaneously and another type of electrostatically actuated microbeam is attained which was recently studied by several authors [24,25].In the third case that is propose of this paper,electrode 2and elec-trode 3(or electrode 1and electrode 4)are actuated with a same DC voltage appliance and new type of electrostatic actuation
mechanism is devised.In this case,according to the electrodes arrangement,second mode of vibration is excited more than other modes.So,three types of electrostatically actuated microbeam are illustrated using presented model in Fig.1.
2.1Distributed Model.The governing differential equation of motion for transverse motions of the microbeam is as follows [24]:
EI @4b w @b x tc
@b w @t tq A @2b w
@t
2?f eV ;b w T(1)
where ^w
is the mid plane de?ection,A and I are the area and moment of inertia of the cross section,c is a equivalent viscous damping coef?cient because of external squeeze ?lm damping
and internal thermo-elastic damping,b x and b t are the spatial coordi-nate along the beam and time,respectively;and f eV ;b w
Tis the electrostatic force which can be de?ned for the proposed microbeam as follows:
f V ;b w eT?e 0bV 2DC 2H b c 1àb x
? g 0àb w eTàH b x àb c 2? g 0tb w eT !;0
? is the Heaveside function,V DC is the applied voltage between the microbeam and the upper and lower electrodes,g 0is the initial gap between the undeformed microbeam and the elec-trodes (see Fig.1),^c
1and ^c 2show which electrodes are actuated by DC voltage and e 0?8:854?10à12eFm à1Tis the permittivity of free space.The accompanying boundary conditions are
b w 0;b t àá?b w L ;b t àá?0;
@b w eb x ;b t T@b x b x ?0?@b w eb x ;b t T@b x
b x ?L ?0(3)For convenience,we introduce the following nondimensional
variables:
w ?b w g 0;x ?b x L ;c 1?b c 1L ;c 2?b c 2L ;t ?b
t ?????????????
EI
q bhL s (4)Substituting Eqs.(4)and (2)into Eq.(1),yields
@4w @x 4tc @w @t t@2w @t 2?a V 2
DC
H c 1àx ? 1àw eT2àH x àc 2? 1tw eT2
!
;0 @w ex ;t T@x x ?0?@w ex ;t T@x x ?1?0(5)The parameters that appeared in Eq.(5)are c ? ^c L 4EIT ;a ? 6e 0L 4Eh 3g 30 (6) To obtain the static solution of Eq.(5),the time varying terms are considered to be zero.So the following equation correspond-ing to the static de?ection is achieved: @4w st @x 4?a V 2 DC H ?c 1àx 1àw st eT2àH ?x àc 2 1tw st eT2 ! ;0 Galerkin method or ?nite difference method creates a set of non-linear algebraic equations.In this paper,a two step method is used Fig.1Schematic of a clamped microbeam located between four electrodes 021002-2/Vol.10,MARCH 2015 Transactions of the ASME to solve the governing nonlinear algebraic equations.In the?rst step,step by step linearization method(SSLM)is applied[26]and in the second one,Galerkin method is used for solving the obtained linear equation.To use SSLM,it is supposed that w k st is the displacement of the beam due to the applied voltage,V k DC.So, by increasing the applied voltage to a new value,the displacement can be written as w kt1 st ?w k sttd w st?w k sttwexT(8) when V kt1 DC ?V k DCtd V DC(9) Therefore,Eq.(7)in keTth and kt1 eTth steps can be rewritten as follows: @4w k st @x ?aeV k DCT2 H?c1àx 1àw k st eT à H?xàc2 1tw k st eT ! (10a) @4w kt1 st @x4?aeV kt1 DC T2 H?c1àx 1àw kt1 st àá2à H?xàc2 1tw kt1 st àá2 ! (10b) Substituting Eq.(8)into Eq.(10b),it may be concluded @4w k st @x4t @4w @x4 ?aeV kt1 DC T2 H?c1àx 1àw k stàwexT eT2 à H?xàc2 1tw k sttwexT eT2 ! (11) By considering small value of d V DC,it is expected that the wexTwould be small enough,hence using of Calculus of Variation theory and Taylor’s series expansion about w k st,the linear coupled electrostatic forces can be written as @4w k st @x4t @4w @x4 ?aeV kt1 DC T2 H?c1àx 1àw k st eT2 à H?xàc2 1tw k st eT2 ! t2aeV kt1 DC T2 H?c1àx 1àw k st eT3 t H?xàc2 1tw k st eT3 ! w(12) Substituting Eq.(10a)into Eq.(12),the following equation to cal-culate wexTcan be obtained.The linearized equation to calculate wexTcan be expressed as L weT?d4w dx4 à2aeV kt1 DC T2 H?c1àx 1àw k st eT3 t H?xàc2 1tw k st eT3 ! w àaeV kt1 DC T2àeV k DCT2 H?c 1 àx 1àw k st eT à H?xàc2 1tw k st eT ! ?0 (13) The obtained linear differential equation is solved by Galerkin method.wexTbased on function spaces can be expressed as w xeT? X N m?1 s m u mexT(14) The approximate solution is constructed by expressing the wexTas a linear combination of a complete set of linearly independ-ent shape functions u mexT,which satisfy the boundary condi-tions.Substituting Eq.(14)into Eq.(13),and multiplying by u nexTas a weight function in Galerkin method and integrating the outcome from x?0to1,a set of linear algebraic equation is generated as X N m?1 K nm s m?F n;n?1;…;N(15a) K nm?K mech nm àK elec nm (15b) where K mech nm ? e1 u IV n u m dx K elec nm ?2aeV kt1 DC T2 e1 H?c1àx 1àw k st eT3 t H?xàc2 1tw k st eT3 ! u n u m dx F n?aeV kt1 DC T2àeV k DCT2 e1 H?c1àx 1àw k st eT2 t H?xàc2 1tw k st eT2 ! u n dx (16) In each step,by substituting calculated wexTin to Eq.(8),w stexTis obtained for a given V DC.According to Eq.(15b)by increasing the applied DC voltage,the equivalent stiffness of the structure decreases and when the equivalent stiffness of the structure equals zero,the obtained solution is unstable. 2.2Lumped Model.In this section,it is purposed to convert the continuous model of new type of electrostatic actuation mech-anism to a lumped model.For this end,2DOF lumped model of the structure is shown in Fig.2. By choosing suitable stiffness and multiplying corrective coef-?cient on forcing terms,characteristics of the structure in the lumped model is closed to the distributed model.For this aim, symmetric lumped model is investigated.Then,the governing equation of motion is written as m @2b x1 @b t2 tk1tk2 eTb x1àk2b x2?beL=2Te0bV 2 DC 2eg0àb x1T2 (17) m @2b x2 @b t2 àk2b x1tek1tk2Tb x2?àbeL=2Te0bV 2 DC 2eg0tb x2T2 (18) where b is corrective coef?cient that is multiplied to forcing term. The electrostatic force is continuous one and rate of this force is related to de?ection of the microbeam along its length.So,it is necessary to have a corrective coef?cient when it is replaced by equivalent centralized force[25].For convenience,the nondimen-sional variables are introduced as x1? b x1 g0 ;x2? b x2 g0 ;t?b t ??????????????? k1tk2 m r (19) Substituting Eq.(19)into Eqs.(17)and(18) yields Fig.2A equivalent lumped2-DOF model for new type of elec-trostatic actuation mechanism Journal of Computational and Nonlinear Dynamics MARCH2015,Vol.10/021002-3 @2x1 @t2tx1à k2 k1tk2 x2?b c V2 DC e1àx1T2 (20) @2x2 @t2à k2 k1tk2 x1tx2?àb c V2 DC e1tx2T2 (21) The parameter that appeared in Eqs.(20)and(21)is c? L e0b 4g0ek1tk2T (22) To obtain static solution in lumped model,the time varying terms in Eqs.(20)and(21)are considered zero.So the following equations to analysis the static de?ection are achieved: x1à k2 12 x2?b c V2 DC e1àx1T2 (23) à k2 12 x1tx2?àb c V2 DC e1tx2T2 (24) Introducing Z1?x1;Z2?x2;Z3?_x1;Z4?_x2,the governing equations of dynamic motion can be rewritten as _Z?F Z;V DC eT(25) where F Z;V DC eT? 00Z30 000Z4 bc V2 DC e1àZ1T2 àZ1 k2 12 Z200 k2 k1tk2 Z1à bc V2 DC e1tZ2T àZ200 2 66 66 66 66 4 3 77 77 77 77 5 (26) The?xed points are de?ned by the vanishing of the vector?eld; that is F Z;V DC eT?0.Locations in the state space where this con-dition is satis?ed are called singular or?xed points.Fixed points are also called stationary solutions,critical points,constant solu-tions,and sometimes steady-state solutions.Physically,a?xed point corresponds to an equilibrium position of a system[27]. Let the solution of F Z;V DC eT?0for V DC?V?be Z?.To deter-mine the stability of this equilibrium solution,a small disturbance denoted as Y is superimposed on Z teTit as follows: Z teT?Z?tY teT(27) Substituting Eq.(27)into Eq.(25)yields _Y?F Z?tY teT;V? eT(28) Expanding Eq.(28)in a Taylor series about Z?and retaining only linear terms in the disturbance leads to _Y?F Z?;V? eTtD Z F Z;V DC eTjeZ?;V?TYtOeY2T _Y%A Z?;V? eTY (29) The matrix of?rst partial derivatives is called the Jacobian ma-trix and represented as follows: A?@F @Z ? 0010 0001 2bceV?DCT2 e1àZ?1T à1 k2 k1tk2 00 k2 12 à 2bceV?DCT2 e1tZ?2T3 à100 2 66 66 66 64 3 77 77 77 75 (30) The eigenvalues of the constant matrix A provide information about the local stability of the?xed point Z?[27]. 3Numerical Results and Discussion To show the numerical results of the analysis presented in Section2,a clamped–clamped microbeam is considered with the characteristics introduced in Table1. Shape functions,which satisfy the boundary conditions of the clam- ped–clamped microbeam,are considered in the form below[28] u nexT?sin k n xàsinh k n x eTtB cos k n xàcosh k n x eT where B? sinh k nàsin k n ? n n ;k n?4:73;7:853;10:9956; (31) According to solution of Eq.(7)by increasing the voltage applied to the electrostatic areas,the electrical stiffness of the structure is increased and leads to the decrease of the equivalent stiffness of the structure.Therefore,for a given applied voltage called as pull-in voltage in the MEMS literature,a static instabil- ity is occurred.Figs.3and4show the de?ection of the microbeam at x?0:25and x?0:75. As shown in Figs.3and4at V DC?34:9eVT,the system has a tendency to static instability by undergoing a pitchfork bifurcation. This kind of bifurcation can be observed in dynamical systems with an inversion or re?ection symmetry.On the other hand,because of the jump from little amplitude to large one,we can classify this type of bifurcation as subcritical pitchfork bifurcation.In the classic elec- trostatically actuated microbeam,a static instability is occurred by undergoing system to a saddle node bifurcation(Fig.5). The obtained results are veri?ed in two cases.In the?rst case, considering c1;c2>1in Eq.(7)the problem is converted to a Table1Geometrical and material properties of the microbeam Design variable Value Length LeT350l m Width beT50l m Thickness heT3l m Initial gapeg0T1l m Young’s moduluseET169:6GPa Poisson’s ratioetT0:06 DensityeqT2331kg=m 3 Fig.3De?ection of the microbeam(at x?0:25)versus the applied voltage 021002-4/Vol.10,MARCH2015Transactions of the ASME classically actuated microbeam that was studied by Rezazadeh et al.[26].As shown,there is a good agreement between the results of the presented and the published one.In the second case,by considering c 1;c 2?0:5,dual type actuated microbeam is achieved and in this case,a ?nite element analysis was used to validate the obtained results.ANSYS software has been used for simulation.As shown in Fig.6,microbeam and air gap have been considered as two composite layers which glued together.The properties of each element are illustrated in Table 2.The simulation result for de?ection of the microbeam is shown in Fig 7.According to ANSYS results,the de?ection of the microbeam is increased by increasing the applied DC voltage until V DC ?36eV Tand for voltages greater than V DC ?36eV TANSYS results diverge,therefore V DC ?36eV Tis considered as pull-in voltage in ANSYS model,which are closely agreed with the results of our distributed model (see Fig.8).The pull-in voltage that is obtained from ANSYS model is greater than that obtained from our distributed model,because in ANSYS modeling,stretching force is considered,which is neglected in our model and it caused to increase the stiffness of the microbeam. As illustrated in Table 3for a classically actuated microbeam in which the ?rst mode of de?ection is excited,the microbeam has a tendency to a saddle node bifurcation but by exciting second mode of de?ection in the proposed microbeam,the microbeam loses stability by undergoing a subcritical pitchfork bifurcation.The value of the critical voltage leading the structure to an unsta-ble condition—pull-in voltage—for the proposed microbeam is 34:9eV T,which is 74%greater than that of the classically actuated microbeam. On the other hand,in Sec.2.2,the results of the lumped model for static solution lead to Eqs.(23)and (24).By solving these equations for different applied voltages,displacement of proof masses is speci?ed. As shown in Fig.9,by choosing suitable stiffness and correc-tive coef?cient (k 1?124:7N =m 2;k 2?381N =m 2;b ?0:55),the behavior of system in lumped model has good agreement with results of distributed model.In Fig.10,displacement of x 1in lumped model is compared with de?ection of microbeam at x ?0:25. Now by considering the Jacobian matrix of the system in the lumped model,local stability of ?xed points is investigated.According to eigenvalues of Jacobian matrix,the stability of the ?xed points can be speci?ed.When the eigenvalues of Jacobian matrix have nonzero real parts,the corresponding ?xed point is called a hyperbolic ?xed point,regardless of the values of the imaginary parts;otherwise it is called a nonhyperbolic ?xed point.As shown in Figs.11and 12,the eigenvalues of Jacobian matrix in branch 1by varying the control parameter (applied voltage)from 0to 34.9are purely imaginary and nonzero.The correspond-ing ?xed point in this state is called a stable nonhyperbolic ?xed point (stable center)but by increasing voltage the eigenvalues of Jacobian have positive real parts and we have unstable nonhyper-bolic ?xed point [27 ]. Fig.4De?ection of the microbeam (at x ?0:75)versus the applied voltage Fig.5De?ection of the classically actuated microbeam (at x ?0:5)versus the applied voltage Fig.6Simulation of the microbeam and air gap in ANSYS Table 2The properties of each element Microbeam Air gap Element type Structural Solid 95 Solid 226Degree of fredom U x ,U y ,U z U x ,U y ,U z ,voltage Number of elements 1144 15,325 Journal of Computational and Nonlinear Dynamics MARCH 2015,Vol.10/021002-5 By probing the eigenvalues of Jacobian matrix in branches 2and 3,as illustrated in Figs.13–16in both of the branches we have unstable nonhyperbolic ?xed point. 4Conclusion In the present study,stability and bifurcational behavior of a microbeam driven by asymmetrically located upper and lower electrodes were studied.Equation of static de?ection using a dis-tributed and a lumped 2-DOF was studied and positions of ?xed points versus voltages applied to the upper and lower electrodes were determined.The stability of the solutions was determined by eigenvalue analysis of local Jacobian matrix.It was shown that in the classically actuated microbeam the ?rst mode and in the proposed one the second mode of the de?ection were exited.In addition,it was shown that increasing the applied voltage leads the structure to an unstable condition by undergoing system to a subcritical pitch-fork bifurcation where in the classically actuated microbeam transition from a stable to an unstable condition is occurred through a saddle-node bifurcation.The obtained critical or pull-in voltage in the proposed case is 74%greater than that of Fig.7(a )and (b )De?ection of the microbeam simulated by ANSYS Fig.8(a )and (b )Comparing results of the distributed model and ANSYS model Table 3Comparing two structures Type of actuation Pull-in voltage and bifurcation type (a)Dual type (c 1,c 2?0.5) V pull àin ?34:9eV T Subcritical pitchfork bifurcation (b)Classic type (c 1,c 2>0.5) V pull àin ?20:1eV T Saddle node bifurcation Fig.9Displacement of x 1versus the applied voltage 021002-6/Vol.10,MARCH 2015 Transactions of the ASME Fig.10Comparing results of the lumped model and distrib-uted model Fig.11Imaginary parts of eigenvalues versus the applied volt-age (for branch 1)Fig.12Real parts of eigenvalues versus the applied voltage (for branch 1)Fig.13Imaginary parts of eigenvalues versus the applied volt-age (for branch 2) Fig.14Real parts of eigenvalue versus the applied voltage (for branch 2) Fig.15Imaginary parts of eigenvalue versus the applied volt-age (for branch 3) Journal of Computational and Nonlinear Dynamics MARCH 2015,Vol.10/021002-7 classic microbeam.The obtained results can be useful for MEMS community especially when a selective mode of excitation in microbeam resonators is expected. 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Fig.16Real parts of eigen-value versus the applied voltage (for branch 3) 021002-8/Vol.10,MARCH 2015Transactions of the ASME