Bifurcation-Based Stability Analysis of Electrostatically Actuated Micromirror as a Two Degrees of Freedom System

Transcription

1 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 Bifurcation-Based Stability Analysis of Electrostatically Actuated Micromirror as a Two Degrees of Freedom System Kuntao Ye, *, Yan Luo and Yingtao Jiang, * Abstract: Torsional micromirror devices have been widely used in micro dislays, RF switches, otical communications, and otical coherence tomograhy systems. In order to study the stability of trostatically driven torsional micromirror system with double bottom lates and two voltage sources, a dimensionless, two degrees of freedom (-DoF) dynamic model was constructed. Governed by the dimensionless hase sace model equation, the ull-in and bifurcation henomena were analyzed using the Hamiltonian method and numerical simulation. In articular, the influence of the daming coefficient and the torsion-bending couling effect on the hase trajectory was investigated. Furthermore, the conditions that can lead to ull-in were numerically determined for saddle-node, itchfork and Hof bifurcations in the framework of -DoF system. Result showed that the dynamic ull-in voltage as redicted by the roosed -DoF system model is considerably lower than that by the one degree of freedom (-DoF) system model. It was also confirmed that the ull-in voltage varies with the daming coefficient and/or the ratio of the two voltages alied to the bottom lates of the micromirror. The modelling method and stability analysis resented in this aer shall rovide valuable insight to the design and control of trostatically actuated micromirror systems. Keywords: MEMS, micromirror, bifurcation, ull-in, stability, hase trajectories, DoF. Introduction Electrostatically actuated torsional micromirror is widely utilized in various engineering or research fields due to its advantages of low energy consumtion, simle structure and high scanning frequency [Koay and Rahim (6)]. Like any other devices driven by trostatic force, a torsional micromirror system s to late can be attracted to and hysically get in touch with its bottom late. This so-called ull-in henomenon, whenever it occurs, is an indicator that the torsional micromirror system is at an unstable state. The ull-in henomenon in trostatically driven systems is well researched for the develoment of micro-tro-mechanical systems (MEMS) technology. An early study of ull-in concerned only the static equilibrium [Zhang, Chau, Quan et al. ()]. Very Jiangi University of Science and Technology, School of Science, Ganzhou, Jiangi, China 4 University of Nevada Las Vegas, Deartment of Electrical and Comuter Engineering, Las Vegas, NV, USA 8954 * Corresonding author: Kuntao Ye. kuntaoye@6.com; Yingtao Jiang. yingtao.jiang@unlv.edu CMES. doi:.97/cmes

2 6 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 quickly, it was realized that the ull-in of any trostatically actuated MEMS device is related to bifurcation seen in a nonlinear system, eemlified in a study to characterize the ull-in instability of a MEMS device using the bifurcation diagram obtained from a simle model [Pelesko ()]. In general, study of the ull-in of a torsional micromirror has been carried out by either solving the static equilibrium equation and dynamic differential equation directly [Zhang, Chau, Quan et al. (); Zhao, Chen, Huang et al. (5)], or alying hase sace analysis. Through the latter aroach, the bifurcation henomenon of micromirror system can be identified right in the state/hase sace by eamining the state change of the system, and those state changes are found to be directly tied to the stability or instability of the system. Following the hase sace analysis, Li et al. [Li, Xi and Hua ()] were able to mathematically reveal that the DMD system ehibits both the saddle node and the codimension bifurcations. Taghizadeh et al. [Taghizadeh and Mobki (4)] analyzed an trostatically driven double bottom late torsional micromirror system in hase sace using one-degree-of-freedom(-dof) dynamic model, and the system shows the saddle node and itchfork bifurcations. Li et al. [Li, Duan, Ma et al. (4)] roved that a sequence of Hof bifurcation occurs at the equilibria in hase sace analysis. Pratiher et al. [Pratiher (4); Li and Zhang (6)] studied the static and dynamic characteristics of trostatically driven MEMS microbeam also in the hase sace. Although these studies have imlicitly or elicitly recognized that subtle differences eist between the static ull-in and dynamic ull-in behaviors [Zhang, Yan, Peng et al. (4)], no study yet is conducted to eamine these differences through the hase sace analysis, and it remains unelored regarding how bifurcations of a MEMS micromirror system are linked to the dynamic ull-in. Another drawback of all the bifurcation studies on MEMS micromirror system is attributed to the fact that they were all based on the -DoF model, which ignored the torsion-couling effect. When this imortant effect is ushered in for more accurate analysis, a more comle two-degree-of-freedom (-DoF) dynamics model needs to be develoed and used. To address the aforementioned roblems in analyzing the ull-in henomena of the trostatically actuated MEMS torsional micromirror system with two voltage sources, the Newton-Euler method is alied to build a nonlinear -DoF dimensionless dynamic model, as detailed in Section. With the roosed -DoF model, both the static and dynamic characteristics of the micromirror can thus be studied through eamining the static and the dynamic bifurcations in hase sace, as reorted in Sections and 4. These results are comared with those obtained from the Hamiltonian method for undamed systems. -DoF Model for double bottom late micromirror system As shown in Fig. (a), the micromirror system is comosed of two torsional cantilever beams, a micromirror surface, a to late, two bottom lates, and two fied anchors that sit on to of the silicon substrate. The width, length and thickness of the to late are a, L, and t m, resectively. The thickness of the mirror is t Al. The width, length and thickness of the two cantilever beams are w, l, and t b, resectively. The size and osition of the bottom

3 Bifurcation-Based Stability Analysis of Electrostatically Actuated 6 late are determined by a and a. The initial (rest) sacing between the to late and the bottom lates is h. As far as a micromirror system with a single bottom late is concerned, a torsion micromirror model based on small angle aroimation [Wetsel and Strozewski (99)] could be used. If the bending-torsion couling effect, resulting from a driving voltage alied to the lates, needs to considered, a vertical dislacement has to be added into the model, along with the angular dislacement, as the case in Huang s work [Huang and Liu (4)]. In the same token, for the double-bottom-late torsional micromirror system driven by two voltages, V and V, the bending-torsion couling effect cannot be ignored. Consequently, rotation of the micromirror will be reflected in the angle of rotation θ and the vertical dislacement y, as illustrated in Fig. (b). (a) Figure : Schematic views of the torsional micromirror: (a) D view, and (b) crosssectional view Since the lengths and widths of the to and bottom lates are much greater than the distance between the to and the bottom lates, the edge effect can be safely neglected [Huang and Liu (4); Zhang, Meng and Chen (7)]. The trostatic forces between the to late with unit length and the two resective bottom lates, hereinafter referred as the left bottom and the right bottom lates, are given below df df L R r V L h y sin d r V L h y sin d where ε r is the relative ditric constant ( for air), and ε is the ditric constant of vacuum. If we integrate Eqs. () and () (see Eqs. () and (4)) and use small angle aroimation (i.e. θ and sinθ are assumed to be equal when θ is small), the trostatic force between a bottom late and the to late and the corresonding trostatic torque can be eressed as Eq. (5) through Eq. (8). F M F df df V L L r a a h y h y (b) () () () (4) (5)

4 64 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 M r V L h y h y h y a / ln h y a / h y a / h y a / L (6) V L R r a a h y h y F M V L r h y h y h y a / ln h y a / h y a / h y a / R (8) The total trostatic force and the total trostatic torque of the to late, at the angle of rotation θ and vertical dislacement of y, are given in Eqs. (9) and (). F L R F F (9) M L R M M () In addition to trostatic force, the micro-mirror system also is eerted on daming force F dam, elastic recovery force F elas and corresonding torques, as given in Eqs. () through Eq. (4) : F dam M dam dy Cy dt d C dt () (4) Felas K y M elas K where C y and K y reresent the daming factor and the stiffness of the vertical direction, resectively, and C θ and K θ reresent the daming factor and stiffness of the torsional direction, resectively [Zhao, Chen, Huang et al. (5)]. The micromirror system driven by an inut voltage has a -DoF of couling outut: the torsional angle of θ and vertical dislacement y. Following the Newton-Euler dynamics, the dynamics model of the micromirror system is constructed as Eq. (5): d d I C K M dt dt d y dy m C y K y y F dt dt Eq. (5) through Eq. () are substituted into Eq. (5), and then the variables are normalized by including the relationshis defined in Eq. (6). The dimensionless twodegrees-of-freedom dynamic equation of the micromirror system is thereby obtained as given in Eq. (7). a a, a a, y / h, a / h K K y V,,, t, V =V (6) I m V (7) () () (5) d d BV ( ) ( ) d d d d BV R br [ ] d d C C K I La La R,, br R [ ln( ) ln( )] y R I m y, B, B, r m K 6Kh 4mh (7)

5 Bifurcation-Based Stability Analysis of Electrostatically Actuated 65 where R, R are the equivalent daming coefficients, ω is the natural angular frequency, and ω is the natural frequency of the vertical bending movement. ω and ω can be calculated from the moment of inertia I, mass m, bending stiffness K θ, and torsional rigidity K y [Zhao, Chen, Huang et al. (5)]. Note that reresents the ratio of the two driving voltages alied to the two bottom lates [Taghizadeh and Mobki (4)]. As the system state variables are set to be =δ, =dδ/dτ, =φ, 4=dφ/dτ, the second order differential equations given in Eq. (7) can be transformed to the hase sace equations which are the first order differential equations, as listed in Eq. (8) br ln ) ( ) ( ln B V R B V R (8) As a dimensionless -DoF dynamics model, Eq. (8) can be used to comute the static ull-ins and bifurcations, dynamic ull-ins and bifurcations of the torsional micromirror in the hase sace. Table : Parameters of torsional micromirror (unit μm) Parameters Symbols Value Width of the to late a Length of the to late L Thickness of the to late tm.5 Thickness of the mirror surface tal.5 Width of the torsional Beam w Length of the torsional Beam l 65 Thicness of the torsional Beam tb.5 Ga between the bottom lates a 6 Total Width of all the bottom lates a 84 Distance between the to and bottom late h.75 The coefficients in the analysis rocess are calculated from the arameters tabulated in Tab., and values of the material roerties are taken from the ones used in [Zhao, Chen, Huang et al. (5)]. Note that if and in Eq. (8) take the value of zero, the system actually degenerates into a -DoF outut system that does not consider the bendingtorsion couling effect. Static ull-in and bifurcation As the driving voltage gradually reaches a critical value, beyond which the state of mechanical equilibrium is no longer maintained, the ull-in of the torsion micromirror will haen, along with a bifurcation henomenon as redicted by the static analysis. This so-called static bifurcation and the static characteristics are studied through Hamiltonian method and numerical simulation in the hase sace.

6 66 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8. Hamiltonian analysis When the -DoF system is in a static equilibrium state, not only does the elastic recovery moment balances the trostatic moment, but the elastic recovery force and the trostatic force in the vertical direction also need to balance each other. The static characteristics of the system thus can be analyzed using the Hamiltonian otential function that accounts for both balance of moments and balance of forces. When daming R =R =, the Hamiltonian [Azizi, Ghazavi, Khadem et al. (); Shang, Song and Wen (6)] of the system described in Eq. (8) is given in Eq. (9). 4 4 ln ln ) ( ),,, ( V B B H br (9) The otential function is: ln ln ) ( ), ( V B B U br () The equilibrium state of the system is comuted by having [Elata and Bamberger (6)]: )/, ( )/, ( U f and U f () When the system is in equilibrium, it needs to satisfy the condition given in Eq. (): br BV (a) ln ) ( ) ( ln B V (b) If the voltage ratio is known, the relationshi curves between the and can be obtained by solving Eqs. (a) and (b) for different driving voltages V, and the intersection oint of these two curves is actually the equilibrium oint of the system. When the driving voltage is increased to the critical voltage, the two relationshi curves only have one intersection oint and thus, the system has only one equilibrium state solution. When the driving voltage is greater than the critical voltage, these two relationshi curves do not intersect, and the static equilibrium condition governed by Eq. () cannot be satisfied. In a simle word, the system now enters the ull-in, and this critical voltage is thus referred as the ull-in voltage. The vs. relationshi curves obtained from solving Eqs. (a) and (b) are shown in Figs. (a) and (b), resectively, for the cases of = and =, and critical voltage V= v. When =, V=.5 v, there is a unique intersection oint between curve and curve, and the static ull-in occurs, as shown in Fig. (a). When =, V=.85 v, there is a unique intersection oint between curve and curve, and the static ull-in occurs, as shown in Fig. (b). Similarly, when the voltage ratio falls in between, i.e. <<, the static ull-in voltage lies in somewhere between the value seen in = case and that in = case.

7 Bifurcation-Based Stability Analysis of Electrostatically Actuated 67 (a) = (b) = Figure : The static ull-in of the -DoF micromirror system by the Hamiltonian method. Numerical solution in hase sace In Eq. (8), there are four state variables in the -DoF system, namely the two outut degrees of freedom and their derivatives, which together constitute a four-dimensional (4D) hase sace. In the dynamic hase sace built for the system at different driving voltages, the resective rojections of the hase diagram onto the (, ) lane and ( 4, ) lane can be used to study both the dynamic behavior and the static characteristics of the system, rather than using the 4D trajectory that is imossible to visualize and hard to analyze. In a nutshell, the hase lane made of the vertical dislacement (state variable ) and s derivative (state variable ) and the hase lane made of the torsion angle (state variable ) and s derivative (state variable 4) can be interreted as two windows to hel observe the 4D hase trajectory. After making the daming coefficient in Eq. (8) equal to zero and setting the state variables [,,, 4] with different initial values, for different and V values, the hase trajectories of the system s state variables can be obtained using the fourth order Runge-Kutta method, and the rojected trajectories for each given [,,, 4] are shown in Fig.. Note that the changes of the osition of the fied oint and the rojected trajectories shown in Fig. can be elored to determine if the system is globally stable. It can be seen from Fig. (a), when the driving voltage is zero, there is eactly one stable center oint in the system, as ehibited on the hase lane of 4=dφ/dτ vs. φ.

8 68 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 (a) =any number between and, V= (b) =, V=5 v (c) =, V=.5 v (d) =, V=5 v (e) =, V=.85 v Figure : The rojected hase trajectories of the -DoF system without daming In the case of =, V and the driving voltage smaller than a critical value of.5 v,

9 Bifurcation-Based Stability Analysis of Electrostatically Actuated 69 different from the line-shaed trajectories as redicted by the -DoF model, the actual trajectories, according to the roosed -DoF model, form a ring shae, as shown in Fig. (b). The center oint in this case is shifted to the right as the driving voltage increases, and there eists a saddle oint in the system. When the critical voltage is reached as the case in Fig. (c), it is revealed from the hase diagram that the overall instability arises. That is, the saddle oint overlas with the center oint and it actually gets disaeared, meaning that the saddle node bifurcation does occur and the critical voltage is indeed the static ull-in voltage. In the case of =, shown in Figs. (a), (d), and (e), the center remains at the origin, and the system has two saddle oints that are symmetric with resect to that center oint. Both oints are dragged closer to the center oint as the driving voltage increases. By the time when the driving voltage reaches the static ull-in voltage, the three equilibria finally merge into one, and thus, the itchfork bifurcation occurs and the overall instability henomenon can be seen from the rojections of their hase diagrams. It can also be seen from Fig. that the rojected unstable trajectories of the vertical dislacement and the angle of rotation aear at the same time. The static characteristics of the -DoF system can thus be studied by insecting the number and ositions of the main equilibrium oints on the hase diagram of the torsional direction. (a) = (b) = Figure 4: The static characteristic curve of -DoF system The summarized static characteristics are shown in Fig. 4. It becomes quite clear that the static ull-in voltage is equal to the static bifurcation voltage, which is.5 v for =, and.85 v for =. These results are consistent with those obtained from the Hamiltonian method. When the voltage ratio falls into the range of <<, the bifurcation voltage is in the range of.5 v to.85 v. 4 Dynamic ull-in and bifurcation Dynamic behavior of a torsional micromirror reflects how the system resonds to the driving voltage with a secial waveform, like a ste function. In this section, the ull-in and bifurcation henomena of the micromirror system driven by the ste voltage are analyzed based on -DoF model governed by Eq. (8) for two distinct scenarios: one with daming and one without.

10 7 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 4. System without daming The dynamic ull-in condition of the undamed -DoF micromirror system can always be obtained through the Hamiltonian method given in Eq. (9). The energy constraints of the system denoted as H can be set as given in Eq. () [Elata and Bamberger (6)]. H B B ) V ( () Since this is an undamed system, energy shall be conserved. That is, B B V H H 4 ln ln br (4) Let the state variables and 4 in Eq. (4) assume the value of zero. That is, both the kinetic energy of the vertical direction and the kinetic energy of the angular direction are equal to zero, which brings u the surface equation of V (, ) as eressed in Eq. (5). B B V ln ln br (5) The contour on the surface of V (, ) is rojected onto the lane (, ), as shown in Fig. 5. The cases of = and = are lotted in Figs. 5(a) and 5(b), resectively. Fig. 5(a) shows that the ideal dynamic bifurcation voltage, for the case of =, is. v. When = and V (, ) is less than the ideal dynamic bifurcation voltage, the trajectory on the dislacement lane (, ) defined by Eq. (5) is searated into two closed regions, which corresond to the stable and unstable equilibrium situations, resectively. The two regions are drawn closer to each other as the driving voltage increases, and they overla at the unstable equilibrium when the ideal dynamic bifurcation voltage is reached. When the voltage is further increased to surass the ideal dynamic bifurcation voltage, the two regions actually merge into one single region, and there eists an oen ath that connects the stable and unstable regions. The contour of the case of is similar to that of =, as shown in Fig. 5(b), where the ideal dynamic bifurcation voltage is.88 v for the case of =. The actual movement of the micromirror is different from that of the ideal situation, as shown in Fig. 5, because there eists a nonlinear couling between state variables and. Such nonlinear couling causes an unconstrained hase difference between the torsion movement and the vertical movement. As so, the kinetic energy of the vertical direction and the kinetic energy of the angular direction cannot be zero at the same time, which is different from the undamed situation discussed reviously through Eq. (5). Even if the micromirror acquires the same amount of inut energy, as shown in Fig. 5, its area enclosed by the actual moving trajectory is smaller than the area enclosed by the ideal trajectory. That is, the actual dynamic ull-in voltage should be greater than the ideal dynamic bifurcation voltage, as indicated in Fig. 5.

11 Bifurcation-Based Stability Analysis of Electrostatically Actuated 7 (a) = (b) = Figure 5: Ideal dynamic bifurcation characteristics of the undamed -DoF system In order to find the actual dynamic ull-in voltage of the undamed -DoF micromirror system, one can increment the driving voltage at a ste size of. v, beginning from v and setting the initial state to be [,,, ], after which the trajectory rojections for voltage are obtained. Some rojected hase trajectories are shown in Fig. 6. As the driving voltage increases, radius of the circle formed by the rojected hase trajectory increases as well, and an unstable trajectory aears when the voltage eceeds the ull-in voltage V (, )= v, as shown in Fig. 6. Figure 6: Projected dynamic hase trajectories of zero daming -DoF system The dynamic ull-in voltages obtained from the numerical simulation of the -DoF system with no daming at = and = are 9.9 v and.99 v, resectively, which are larger than their resective ideal dynamic bifurcation voltages of. v and.88 v. In Fig. 7, the dynamic characteristics of the system, for the case of = and voltage swing of v to v, are lotted together in the same figure. One can see that when the driving voltage is larger than v, the difference between the -DOF model and the -DOF model becomes quite noticeable. These results as lotted in Fig. 7 are in good agreement with those obtained from solving the deferential equations in time domain [Zhao, Chen, Huang et al. (5)]. Similar observation can be also be made for the cases when is not equal to zero. These differences clearly indicate that torsion-bending couling effect should not be omitted from the stability analysis of torsional micromirror system.

12 7 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 Figure 7: The comarison of dynamic characteristic curves between -DoF and -DoF system 4. System with daming In addition to the driving voltage, the daming coefficients of the system can also imact the system ull-in. Figure 8: The dynamic Hof bifurcation of the -DoF system with the daming factor R assuming V= and initial state of [.,.,.,.7]. As shown in Fig. 8, with both the driving voltage and the initial state are fied, say V= and the initial state of [.,.,.,.7], when the daming coefficient swees over a wide range, say R =R =R=.4, down to R =R =R=, there is a drastic shae change of the rojected trajectories, going from a siral shae with a focus oint, for the case of R=.4, to a limit cycle, for the case of R=. In the latter case, a Hof bifurcation actually occurs. The analysis method of the actual dynamic ull-in voltage of damed -DoF system is similar to that of the undamed system. As the driving voltage swees a ste size of.v along with the initial state of [,,, ], the trajectory of -DoF is calculated and rojected onto both the torsional direction and the vertical direction. A few hase trajectory rojection curves for a few daming coefficients, R=.,, are shown in Fig. 9.

13 Bifurcation-Based Stability Analysis of Electrostatically Actuated 7 (a) =, R=. (b) =, R= Figure 9: The rojection of hase trajectories of -DoF system with different daming When the equivalent daming coefficient R and the voltage is smaller than the critical voltage, the trajectory shows the corresonding focus oint. But when the critical voltage is reached, there eists an unstable trajectory, which indicates the occurrence of ull-in. Alying different driving voltages to the -DoF system with R=,.,, the ull-in angle and the ull-in voltage for -DoF system with different daming coefficients are lotted in Fig.. One can see that dynamic ull-in voltage increases along with the increase of the daming coefficient. In another word, if the driving voltage remains fied, the maimum deflection angle is smaller for higher daming coefficient, indicating that daming tends to make the system more stable. Fig. lots the dynamic ull-in voltages across a range of daming coefficients, as obtained from the -DoF micromirror system and the -DoF. One can see that the dynamic ull-in voltage redicted by the -DoF model is considerably lower than that is redicted by the -DoF model for the same daming coefficient and =.

14 74 Coyright 8 Tech Science Press CMES, vol.4, no.,.6-76, 8 Figure : Pull-in angle vs. ull-in voltage for -DoF system at daming R=,., Figure : The dynamic ull-in voltages vs. different the daming factors for -DoF and -DoF systems 5 Conclusion In this aer, two degrees of freedom dynamic model was constructed for an trostatically driven torsional micromirror system with double bottom lates and two voltage sources. Based on the dimensionless hase sace model equation thus constructed, the ull-in and bifurcation henomena were analyzed using the Hamiltonian method and numerical method. The following conclusions are drawn:. Due to the torsion-bending couling effect, when the driving voltage is smaller than a critical voltage but larger than zero, the -DoF system shows ring-shaed rojected trajectories whereas the line-shaed trajectory is redicted by the -DoF system. In addition, saddle node bifurcation, itchfork bifurcation and Hof bifurcation can all be redicted by the -DoF model for MEMS torsional micromirror. The ull-in conditions can be readily identified by eamining these bifurcation oints.. Static bifurcation and static ull-in voltage are essentially the same, and they have higher value than the dynamic ull-in voltage. The static ull-in voltage is nearly the same as the dynamic voltage when the voltage ratio is close to. The dynamic ullin voltage, on the other hand, falls between the ideal dynamic bifurcation voltage and the static bifurcation voltage. The driving voltage alied to the bottom gate

15 Bifurcation-Based Stability Analysis of Electrostatically Actuated 75 must be higher than the ideal dynamic bifurcation voltage for a dynamic ull-in to occur. In addition, dynamic ull-in voltage varies with the voltage ratio.. Even with the same initial state, the ull-in voltage of the damed system is always lower than that of the undamed system; the ull-in voltage increases with daming. When the same driving voltage is alied, maimum deflection angle dros as daming increases. At the daming coefficient R=, the system shows a dynamic Hof bifurcation. 4. Pull-in voltage redicted by the -DoF system is considerably lower than that by the -DoF, clearly indicating that for accurate stability analysis of a torsional micromirror, the torsion- bending couling effect should not be ignored. Acknowledgement: This work is suorted by the National Natural Science Foundation of China (Project 6684), the Fund of National High-level Overseas Talent Returning ([]48), and a scholarshi from China Scholarshi Council (No. 7). References Azizi, S.; Ghazavi, M.; Khadem, S. E.; Rezazadeh, G.; Cetinkaya, C. (): Alication of iezotric actuation to regularize the chaotic resonse of an trostatically actuated micro-beam. Nonlinear Dynamics, vol. 7, Elata, D.; Bamberger, H. (6): On the dynamic ull-in of trostatic actuators with multile degrees of freedom and multile voltage sources. Journal of Microtromechanical Systems, vol. 5, no.,. -4. Huang, J.; Liu, A. (4): An aroach to the couling effect between torsion and bending for trostatic torsional micromirrors. Sensors & Actuators A Physical, vol. 5, no., Koay, L. K.; Rahim, N. A. A. (6): Reviews: Torsional sring mechanism resonant scanner s technology. Journal of Mechanical Science and Technology, vol., no. 4, Li, L.; Zhang, Q. C. (6): Nonlinear dynamic analysis of trically actuated viscoelastic bistable microbeam system. Nonlinear Dynamics, vol. 87, no., Li, Q.; Xi, J.; Hua, C. (): Bifurcations of a micro-tromechanical nonlinear couling system. Communications in Nonlinear Science & Numerical Simulation, vol. 6, no., Li, Y.; Duan, W.; Ma, S.; Li, P. (4): Bifurcation of a microtro mechanical nonlinear couling system with delay feedback. Journal of Alied Mathematics, vol. 4,. -6. Pelesko, J. A. (): Mathematical modeling of trostatic MEMS with tailored ditric roerties. SIAM Journal on Alied Mathematics, vol. 6, no., Pratiher, B. (4): Stability and bifurcation analysis of an trostatically controlled highly deformable microcantilever based resonator. Nonlinear Dynamics, vol. 78, no.,

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