DYNAMIC ANALYSIS OF A DIGITAL MICROMIRROR DEVICE

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1 Proceedings of IMECE ASME International Mechanical Engineering Congress and Exposition November -,, Chicago, Illinois, USA Proceedings of IMECE ASME International Mechanical Engineering Congress and Exposition November -,, Chicago, Illinois, USA IMECE- IMECE- DYNAMIC ANALYSIS OF A DIGITAL MICROMIRROR DEVICE G. Chaabane Applied Mechanics & Systems Research Laboratory Tunisia Potechnic School, BP La Marsa, Tunisia E. M. Abdel-Rahman Dept. of Systems Design Engineering University of Waterloo Waterloo, ON NL G, Canada A. H. Nayfeh Dept. of Engineering Science & Mechanics, MC Virginia Tech Blacksburg, Virginia, USA S. Choura, S. El-Borgi, H. Jammoussi Applied Mechanics & Systems Research Laboratory Tunisia Potechnic School, BP La Marsa, Tunisia Abstract We developed a distributed-parameter model partial differential equations and associated boundary conditions that describe the coupled torsion and bending motions of the Digital Micromirror Device DMD using the extended Hamilton principle. The work done by the electrostatic field is expressed in the form of a potential energy. It is found that coupling between the torsion and bending motions appears in the boundary conditions. The nonlinearity is main due to the application of the electrostatic forces and moments. Nonlinear terms appear on in the boundary conditions. The developed model provides a basis for a thorough study of the static and dynamic behaviors of the electromechanical device. The static response of the DMD for different DC loads shows the occurrence of pull-in snap-down instability at critical voltage values corresponding to the collapse of the yoke to mechanical stops. Estimates of the voltage, angle, and deflection at pull-in are given. The dynamic behavior of the DMD is anazed by plotting the natural frequencies versus the applied DC voltage. We conducted a study of the sensitivity of the static and dynamic behaviors of the micromirror to variations in the geometric parameters of the DMD. It is found that the thickness and width of the hinges are the key parameters in- fluencing the occurrence of static pull-in and the values of the voltage, angle, and deflection at pull-in. Introduction Several methods have been proposed over the last decade for the modeling of micromirrors and torsional microactuators. Evoy et al. [] modeled a paddle oscillator by a spring-mass system in which the resistance to translation is represented by a nonlinear third-order spring and the resistance to rotation is represented by a linear torsional spring. They obtained two decoupled equations of motion: an equation of a damped translational oscillator and an equation of a damped torsional oscillator. Nemirovsky and co-workers [, ] and Zhang et al. [] modeled the static rotation tilt angle of a micromirror by a rigid plate suspended from linear torsional springs; they neglected the translational bending motion. Lee [] and Bhaskar et al. [] used this approach to model the motion of a torsional micromirror as a damped angular oscillator. Tay et al. [] used laser interferometery to measure the tilt angle of a micromirror and determine its relationship to the applied voltage. Lee [] used measurements of the scanning pattern of a micromirror to construct frequencyresponse curves of its tilt angle. They are bent to the left, indi- Address all correspondence to this author <eihab@engmail.uwaterloo.ca> Copyright c by ASME Downloaded From: on // Terms of Use:

2 cating softening-type nonlinearity, and hence exhibit a hysteretic response. Yao et al. [] idealized a biaxial torsional micromirror as a rigid mass suspended from four linear torsional springs undergoing two-degree-of-freedom rotations two tilt angles; coupling between these -dof appear in the electrostatic field. Zhou et al. [] extended the work of Yao et al. by considering -dof motions. Chiou and Lin [] and Huang et al. [] modeled the static transverse deflections and rotations of a torsional micromirror as a rigid plate suspended from linear translational and torsional springs representing the bending and torsional stiffness of the support beams. Zhao et al. [] used this approach to model a torsional micromirror by two translational and torsional damped oscillators nonlinear coupled through the electrostatic field. Xiao et al. [] used a similar model augmented by a cubic stiffness term to more accurate represent the resistance of the suspension beams to transverse translational motions. Jung, Lee, and Choi [] anazed the torsional motion of a simplified version rigid mass connected to linear torsional springs of a DMD using the commercial code ANSYS and calculated the static rotations and the static pull-in voltage. Xue et al. [] anazed the static and dynamic responses of the DMD tilt angle using a rigid mass connected to nonlinear torsional springs taking into account plasticity in the hinge response. In the present work, we study the static and dynamic behaviors of a DMD using a rigorous distributed-parameter system, taking into account coupling between its bending and torsional motions. This anasis is used to investigate the feasibility of developing reduced-order models for predicting the static and dynamic behaviors of DMDs and designing control strategies to optimize their responses. Modeling of the DMD The DMD, Figure, is composed of two flexible hinges, a rigid yoke, a rigid post, a rigid mirror, four landing spring tips, and four electrodes actuators. Throughout the anasis, the subscripts m and y refer to mirror and yoke parameters and states. The hinges are two microbeams, each of length l, width w, and thickness h. These beams are fixed at one side and connected on to the yoke at the other end: a rigid H-shaped plate of length L y, thickness h y, and width w. On top of the yoke sits a rigid bar of length L p and cross-section area A, which is the post. Above this bar sits a rigid square plate the mirror of length L m and thickness h m. Beneath the micromirror are two rectangular electrodes each of length e m and width b b, where b and b are the distances between the x-axis and the inner and outer edges of the electrodes, respective. Two more rectangular electrodes are located under the yoke, each is of length e y and width a a, where a and a are the distances between the x- axis and the inner and outer edges of the electrodes, respective. The yoke electrodes are at the substrate level, whereas the mirror electrodes and the yoke are a distance H above the substrate. The mirror is set at a distance L p above the yoke and mirror electrodes. The numerical values of the model parameters are listed in Table. Table. Material and geometric properties of the microbeam. ρ E ν ε o g. g/cm GPa. π. m/s h m L p h y H h.µm µm.µm.µm.µm L m L y l y l l o µm.µm µm µm.µm A a a b b µm.µm.µm.µm.µm e m e y w w w µm.µm. µm. µm. µm h s l s w s. µm. µm. µm Figure. A schematic of the DMD model. The micromirror is actuated by supping a voltage V to the electrodes on the side of the desired tilt. This results in an electrostatic potential between the electrodes and the upper structure yoke - mirror and generates two electrostatic pressures P m and P y on the lower surfaces of the mirror and yoke, respective. The pressure produces an electrostatic force F m + F y, and hence an electrostatic moment M m + M y around the hinges. Hence, the Copyright c by ASME Downloaded From: on // Terms of Use:

3 rigid structure yoke - post - mirror rotates an angle θ o and deflects downward a distance u o simultaneous. The spring tips are modeled by four linear springs of undeformed length l o and equivalent stiffness k determined from the assumption that the spring tips deform in their first mode of vibration. In addition, it is assumed that the sliding friction of the spring tips against the substrate is negligible. Because there are four spring tips, the DMD structure may have one of the following contact configurations: where T hinges = I o Z l + ρwh Z θ dx+ l Z l+ u dx+ θ dx Z l+ u dx T yoke = ρh y [L y w L y l y w ] u o + ρh y. Mod: no contact between the springs and the landing area;. Mod: contact of two springs with one side of the landing area one side contact;. Mod: contact of all four springs with the landing area both sides contact. Therefore, the motion can be described by one of three sets of equations associated with the above configurations. In practice, because the flexural deformation is smaller than the angular deformation, it is reasonable to assume that the third configuration is unlike to occur. Thus, the DMD dynamics is described by one of two sets of equations associated with the first and second configurations. For this, we seek the development of two sets of dynamics models: one without contact and another with contact of one of the two pairs of springs with the landing area. Variation of the integral of the total energy in the DMD between two arbitrary time instants t and t can be written as Z t t [δt δπ +U]dt = where T is the total kinetic energy, π is the potential energy due to gravitation and elastic deformations, and U is the potential energy of the electrostatic field. The energy terms are given by: T =T hinges + T yoke + T post + T mirror { Mod π =π hinges + π yoke + π post + π mirror + π springs Mod U = ε oe y H V uo a tanθ o ln θ o H u o a tanθ o ε oe m V Lp u o b tanθ o ln θ o L p u o b tanθ o [ l y L y w h y + w + Ly w h y + w ] θ o T post = ρal p u o + ρal p θ o + ρal p u o θ o sinθ o T mirror =ρh m L m A [ u o + ] + L p u o θ o sinθ o Z π hinges = GJ p + EI b l Z l h m + L p + L m θ o Z l+ θ x dx+ θ x dx Z l+ u xx dx+ u xx dx π yoke = ρh y g[l y w L y l y w ]u o π post =ρal p g L p cosθ o u o π mirror =ρ L m A h m gl p cosθ o u o π spings = Ew sh s ls H + u o + w tanθ o + l o cosθ o and I b = wh, I o = ρwh w + h, J p = hw π w tanh πh w Here, E, ρ, and ν are the aluminum modulus of elasticity, density, and Poisson s ratio; and g is the gravity acceleration; l s,w s and h s are, respective, the length, width, and thickness of the spring tips. Apping the extended Hamilton principle to equation and integrating by parts yields two coupled partial-differential equations of motion and six boundary conditions BCs describing the torsion and bending in each of the two spatial domains: - Left Hinge: l l y x l y - Right Hinge: l y x l y + l We note that the two sets of equations are equivalent because the model structure and applied loads are symmetric with respect to the yz plane. As a result, the equations of motion associated with the left domain are adopted for anasis and simulation purposes. These equations and associated boundary conditions BCs in Copyright c by ASME Downloaded From: on // Terms of Use:

4 nondimensional form are expressed as follows: θ + θ x = and κü + u = x u x =, u = and θ = at x = l y l l { EI b H u x = at x = l y l Λu,θ + αθ cr τ θ cosθ cr θ + αθ cr θ τ sinθ cr θ +M H ü τ g H EI b u l x + k[ Hu H +l o cosθ cr θ + w tanθ cr θ ]} x= = { l Γu,θ +α H ü GJ p θ cr τ g sinθ cr θ + J τ θ cr θ + GJ p θ θ cr l x khu Hl o sinθ cr θ where +k [ w Hu H + w tanθ cr θ sec θ cr θ +w l o cosθ cr θ l o sinθ cr θ ]} x= = θ cr = H, θ = θ, u = u w θ cr H, x = x l, t = t τ, I o l τ =, κ = ρhwl GJ p EI b τ, Λu,θ = U u, Γu,θ = U θ, M = ρh y [L y w L y l y w ]+ρal p + ρh m L m A, [ L y w h y + w + Ly w h y + w ] J = ρh y + ρal p + ρh ml m [ and α = ρl p ALp + h m L m A ] h m + L p + L m, The spring tips stiffness k is set equal to zero in the first configuration. Static Anasis of the DMD The equilibrium positions of the DMD under a DC voltage are obtained by setting the time derivatives equal to zero in equations,, and. Integrating the ordinary-differential equations resulting from equation over x, we rewrite the spatial distributions of the torsion and bending along the hinge Deflection μm Torsion degrees length as Figure. The static deflection u os vs voltage. Figure. The static tilt angle θ os vs voltage. θ s x =λ x + λ u s x =λ x + λ x + λ x + λ The coefficients λ i are determined by substituting equations and into equations to and numerical solving the resulting nonlinear -dimensional algebraic system. For each DC voltage, two sets of λ i are obtained except at V = and pullin V = V p, where on one set of solutions is found. No real solutions are found for voltages larger than the pull-in voltage V > V p. The static response of the DMD, corresponding to the torsion θ os and deflection u os at the tip of the hinge x = l y, are given in Figures and as functions of the applied DC voltage. For voltages below V p, the bending and torsion increase with the applied DC voltage, thereby tracing the lower branch of equi- Copyright c by ASME Downloaded From: on // Terms of Use:

5 libriums in Figures and. When the applied voltage reaches V p, the tilt angle and deflection attain their critical values θ p =. and u p =.µm. When the applied voltage exceeds V p =. Volts, the yoke collapses abrupt to the mechanical stop because the hinges can no longer resist the electrostatic forces and moments. The equilibrium positions corresponding to the upper branches in Figures and are unstable and can not be observed experimental. Deflection μm h=.μm h=.μm h=.μm. w=.μm. h=.μm. w=.μm. Deflection μm... w=.μm w=.μm w=.μm h=.μm Figure. Sensitivity of the deflection u os to variations in h... h=.μm h=.μm Figure. Sensitivity of the deflection u os to variations in w. Torsion degrees h=.μm h=.μm h=.μm Torsion degrees w=.μm w=.μm w=.μm w=.μm w=.μm Figure. Sensitivity of the tilt angle θ os to variations in w. Figures - show the effect of variations of the width w and thickness h of the hinges on the deflection and tilt angle. As expected, decreasing the stiffness of the hinges by decreasing w and/or h increases the tilt angle and the deflection and decreases the pull-in voltage. Moreover, increasing the hinge width w increases the pull-in deflection and decreases the pull-in angle. On the other hand, increasing the hinge thickness h has the opposite effect. This is consistent with the results of lumped-mass models Figure. Sensitivity of the tilt angle θ os to variations in h. where it can be seen that increasing the hinge width w increases the torsional stiffness in proportion to the bending stiffness and vice versa for the hinge thickness h. The effects of variations in the position of the inner edge a from. to.µm in steps of.µm and outer edge a from. to.µm in steps of.µm of the yoke electrodes on the deflection and tilt angle are shown in Figures -. Increasing a or decreasing a expands the area of the electrodes and therefore decreases the pull-in voltage. However, increasing a has proportional more impact than decreasing a because it also moves the centroids of the electrodes further away from the axis of rotation. Further, the impact of variations in the areas and locations of the yoke electrodes on the DMD response is much less than that due to variations in the hinge dimensions. Copyright c by ASME Downloaded From: on // Terms of Use:

6 Deflection μm.. increasing a Deflection μm.. increasing a.... Figure. Sensitivity of the deflection u o to variations in a. Figure. Sensitivity of the deflection u o to variations in a. Torsion degrees increasing a Torsion degrees increasing a Figure. Sensitivity of the tilt angle θ o to variations in a. Figure. Sensitivity of the tilt angle θ o to variations in a. into these linear equations of motion yields Linear Vibrations Anasis of the DMD To determine the natural frequencies and associated mode shapes around the stable equilibrium configuration θ s,u s,we replace θ and u with θ s + θ d and u s + u d in equations to, expand the result in Taylor series in θ d and u d, drop all nonlinear terms, and obtain a set of linear equations. Substituting the harmonic solution d φ dx + ω φ = d ξ dx κω ξ = where φx and ξx are respective the torsion and bending parts of the mode shape and ω is the corresponding natural frequency. The general solutions of equations and can be expressed as θ d x,t =φx cosωt u d x,t =ξx cosωt φx =C sinωx +C cosωx ξx =C cosβx +C sinβx +C coshβx +C sinhβx Copyright c by ASME Downloaded From: on // Terms of Use:

7 where β = κω. Substituting equations and into the BCs and setting the determinant of the coefficient matrix of the resulting linear algebraic system in the C i equal to zero yields the characteristic equation of the system. The lowest two solutions of the characteristic equation the natural frequencies of the micromirror are shown in Figures and as functions of the applied voltage. The first natural frequency goes through zero at pull-in and the second natural frequency decreases significant as the voltage increases, whereas all of the higher natural frequencies are insensitive to variations in the electrostatic field. Throughout the voltage variation, the third natural frequency rad/s is two orders of magnitude higher than the first two natural frequencies. First torsional and bending modes first torsional mode shape first bending mode shpe..... x *. x Figure. The first mode shape at zero voltage.. ω... Figure. The first natural frequency ω vs voltage. Second torsional and bending modes second bending mode shape second torsional mode shape..... x * Figure. The second mode shape at zero voltage. ω. x... Figure. The second natural frequency ω vs voltage. The associated first two mode shapes are shown in the absence, Figure and, and in the presence, Figures and, of the electrostatic field. In the absence of an electrostatic field, the first mode corresponds to a pure torsional mode, whereas the second mode is a pure bending mode. Increasing the voltage develops an electrostatic field which couples the original uncoupled EVPs. As a result, both modes develop complimentary components: bending for the first mode and torsional for the second mode. Exciting either of these two modes produces bending and torsion motions. The shapes of these motions are similar in both modes but their proportions are different. The dominance of torsion in the first mode and bending in the second mode diminishes as the voltage increases, but it does not disappear up to pull-in. All of the higher modes remain uncoupled: bending-on and torsion-on modes. Figures - show the effect of variations in the hinge width w and thickness h on the first and second natural frequen- Copyright c by ASME Downloaded From: on // Terms of Use:

8 First torsional and bending modes first torsional mode shape first bending mode shape ω x.. w=.μm. w=.μm w=.μm w=.μm w=.μm x * Figure. The first mode shape at V = Volts. Figure. Sensitivity of ω to variations in w.. Second torsional and bending modes second bending mode shape second torsional mode shape x * Figure. The second mode shape at V = Volts. cies. Increasing w and h increases the stiffness of the DMD, thereby increasing the pull-in voltage and the value of the second natural frequency ω at pull-in. The effect of variations in the area and location of the yoke electrodes, parameters a and a, on the linear vibrations of the DMD is similar to those observed in the above static anasis. Conclusions In this paper, a continuous model distributed-parameter system describing the coupled torsion and bending motions of the Digital Micromirror Device DMD was developed using the extended Hamilton principle. It was shown that torsion and bending motions are nonlinear coupled through the boundary conditions. Closed-form expressions for the static torsion and deflection of the DMD under DC voltages were developed. They can be used to predict the voltage, tilt angle, and deflection at pull-in. The DMD response exhibits the static pull-in instability. The linear vibrations problem of the DMD around the static ω x w=.μm w=.μm w=.μm w=.μm w=.μm Figure. Sensitivity of ω to variations in w. equilibrium positions was developed and the natural frequencies and corresponding mode shapes were calculated. The electrostatic field couples the linear torsional and bending vibrations. The strength of this coupling increases as the strength of the filed applied voltage increases. As a result, an excitation of either of the first two modes leads to combined torsional and bending motions. The DMD response is more sensitive to variations in the thickness and width of the hinges than variations in the locations and sizes of the yoke electrodes. The third natural frequency is two orders of magnitude larger than the first and second natural frequencies, and their mode shapes are not involved in the electrostatic coupling phenomenon. As a result, it is reasonable to use model reduction methods to create a two-degree-of-freedom compact model, which describes the static and dynamics behaviors of DMDs. A model along these lines is current under development. Copyright c by ASME Downloaded From: on // Terms of Use:

9 ω ω. x... h=.μm h=.μm h=.μm h=.μm h=.μm x Figure. Sensitivity of ω to variations in h. h=.μm h=.μm h=.μm h=.μm h=.μm Figure. Sensitivity of ω to variations in h. REFERENCES [] Evoy, S., Carr, D. W., Sekaric, L., Olkhovets, A., Parpia, J. M., Craighead, H. G.,, Nanofabrication and electrostatic operation of single-crystal silicon paddle oscillators, Journal of Applied Physics,, pp.. [] Degani, O., Socher, E., Lipson, A., Leitner, T., Setter, D. J., Kaldor, S., Nemirovsky, Y.,, Pull-in study of an electrostatic torsion microactuator, Journal of Microelectromechanical Systems,, pp.. [] Nemirovsky, Y. and Bochobza-Degani, O.,, A methodology and model for the pull-in parameters of electrostatic actuators, Journal of Microelectromechanical Systems,, pp.. [] Zhang, X. M., Chau, F. S., Quan, C., Lam, Y. L., Liu, A. Q.,, A study of the static characteristics of a torsional micromirror, Sensors and Actuators A,, pp.. [] Lee, C.,, Design and fabrication of epitaxial silicon micromirror devices, Sensors and Actuators A,, pp.. [] Bhaskar, A. K., Packirisamy, M., Bhat, R. B.,, Modeling switching response of torsional micromirrors for optical microsystems, Mechanism and Machine Theory,, pp.. [] Tay, C. J., Quan, C., Wang, S. H., Shang, H.M.,, Determination of a micromirror angular rotation using laser interfeometric method, Optics Communications,, pp.. [] Yao, Y., Zhang, X., Wang, G., Huang, L.,, Efficient modeling of a biaxial micromirror with decoupled mechanism, Sensors and Actuators A,, pp.. [] Zhou, G., Tay, F. E. H., Chau, F. S.,, Macromodelling of a double-gimballed electrostatic torsional micromirror, Journal of Micromechanics and Microengineering,, pp.. [] Chiou, J. C. an Lin, Y. C.,, A multiple electrostatic electrodes torsion micromirror device with linear stepping angle effect, Journal of Microelectromechanical Systems,, pp.. [] Huang, J. M., Liu, A. Q., Deng, Z. L., Zhang, Q. X., Ahn, J., Asundi, A.,, An approach to the coupling effect between torsion and bending for electrostatic torsional micromirrors, Sensors and Actuators A,, pp.. [] Zhao, J. P., Chen, H.L., Huang, J. M., Liu, A. Q.,, A study of dynamic characteristics and simulation of MEMS torsional micromirrors, Sensors and Actuators A,, pp.. [] Xiao, Z., Peng, W., Farmer, K. R.,, Anatical behavior of rectangular electrostatic torsion actuators with nonlinear spring bending, Journal of Microelectromechanical Systems,, pp.. [] Jung, K., Lee, J., Choi, B.,, Numerical anasis of the micromirror for projection TV using FEM, Microsystem Technologies,, pp.. [] Xue, Z., Saif, M.T.A., Huang, Y.,, The strain gradient effect in microelectromechanical systems MEMS, Journal of Microelectromechanical Systems,, pp.. Copyright c by ASME Downloaded From: on // Terms of Use:

CHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS

61 CHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS 4.1 INTRODUCTION The analysis of cantilever beams of small dimensions taking into the effect of fringing fields is studied and