Non linear behavior of electrostatically actuate micro-structures Dr. Ir. Stéphane Paquay, Open Engineering SA Dr. Ir. Véronique Rochus, ULg (LTAS-VIS) Dr. Ir. Stefanie Gutschmit, ULg (LTAS-VIS)
Outline Introuction Basic principles Finite element formulation Nonlinear algorithms Valiation & examples Oofelie::MEMS, riven by SAMCEF Fiel Perspectives
Introuction Electrostatics is often use in micro-systems RF Switches Micro-resonators (gyrometers,...) Micro-lens for biomeical application Aaptative optics... 1D Reference problem With courtesy of University of British of Columbia an British Columbia Cancer Research Centre
Basic Principles: Static analysis Static equilibrium equation: Static Pull-in Voltage: V Stable point Unstable point The voltage amplitue for which only one equilibrium position exists u
Electromechanical Problem: Dynamic Analysis Dynamic Equation: V = 0, the phase iagram is an ellipsoi V = V*, an unstable zone appears V, the stability zone is reuce an isappears when the static pull-in voltage is reache 2 new parameters
Basic principles: Dynamic Pull-In Voltage Dynamic Pull-in Voltage: When the voltage is applie suenly, the system becomes unstable Step of voltage
Basic principles: Dynamic Pull-in time Dynamic Pull-in Time: The time neee for the plates to stick together when the static pull-in voltage is applie Inepenent of the gap an the permittivity
Finite Element: Strongly Couple Formulation Analytical expression of the tangent stiffness matrix Avantages Faster convergence to the non-linear solutions Accurate evaluation of the pull-in voltage Moal analysis Time integration Full explanation for 1D formulation 2D an 3D extension is "obvious" using "Virtual Work" principle
1D Formulation Gibb s energy ensity: with Mechanical force: Electrical charge:
1D Formulation Electrostatic force: δ δ e e e e W W u W f = = * 0 lim with V x x V x V W e ² 2 1 2 1 0 0 ε 0 ε = = x x V x V W e 2 1 0 0 * ε δ = + V V V V V x x ² 2 1 2 1 2 1 2 0 0 0 0 0 0 δ ε ξ ξ ξ ε δ ξ δ ξ δ ξ δ ε + = + = + + + = ² ² 2 1 0 V f e ε =
1D Formulation Equilibrium Equations ku + qe Linearisation aroun a position k( u ~ + u) + fe( u, V ) = fext + f qe( u, V ) = qext qext Electric force Electric charge Tangent stiffness Matrix ext f e = = q f ext ext ~ ( u~, V ) ε 0V ε 0V ~ 2 ~ ~ 0V 0 V f u V fe( u ~ ε ε (, ) =, V ) u V e ~ + 3 ~ 2 ~ ~ V qe u V qe( u ~ ε (, ), V ) 0 ε 0 = ~ u + ~ V 2 f q e e = = 2 2
General FE Formulation (2D an 3D) Gibbs energy ensity On a volume Mechanical omain Electric omain Unknowns
General FE Formulation (2D & 3D) Virtual work Equilibrium equations Pure Mech. Electrostatic force =0 Pure Electric
Nonlinear solvers: Staggere vs Monolithical Staggere Metho Monolithical Metho
Nonlinear solvers for Pull-In computation Newton Raphson Riks-Crisfiel V pi V pi
Nonlinear solver: Moal analysis In FEM formulation: Mu&& + Ku = f e ( φ( u)) 1 st metho: Projection on the first mech. eigenmoe Natural frequency 2 n metho: Linearisation aroun an equilibrium position 2 ( K ( q ) ω M) q = tan 0 0
Integration of flui amping (in film) Assumptions: laminar an fully evelope flow (low Reynols number, viscous ominate flow) pressure oes not vary in z-irection the flui oes not slip at the walls very low Knusen number ( Kn = λ /h0, where λ mean free path of molecules, inversely proportional to P). Nonlinear Reynols equation: Last assumption for linear amping: small amplitue motion in comparison with h0
Application 1: Thin Beam, large gap The stiffness of the beam becomes non-linear with respect to isplacement (cable effect) Large isplacements FE formulation taking into account geometric NL was use
Application 1: Thin Beam, large gap Static Equilibrium position 96.4 Volt 18.4 Volt
Application 1: Thin Beam, large gap Natural frequency
Application 1: Thin beam, large gap Dynamic Behaviour Static Pull-in Voltage Dynamic Pull-in Voltage 96.4 V 83.6 V Difference 13%
Application 1: Smaller electroe Moal solution : Projection vs. Monolithical Metho Nearly same results for uniform electroes Difference when the lower electroe is reuce
Application 1: Smaller electroe First Natural Frequency Projection metho F(Hz) Couple metho V
Application 2: Micro-resonators Stuie Micro-evices Electrostatically actuate micro-brige (left) Electrostatically actuate cantilever micro-beam (right)
Application 2: Micro-resonators
Application 2: Micro-resonators Parameters to consier: Pre-stress (ue to manufacturing) Shape of anchor Moel upating on E Micro-Brige Cantilever Micro-Beams V V Poly1 Poly2 Displacement [m] Displacement [m]
Application 2: Micro-resonators
Application 2: Dynamic measurements in air cantilever: L=175 µm, W=30 µm, poly 1
Application 2: Dynamic measurements in vacuum cantilever: L=175 µm, W=30 µm, poly 1 29
Application 2: Dynamic measurements in vacuum
Integration in Oofelie::MEMS, riven by SF Electrostatic effect ae With BEM (no tangent stiffness) With FEM (with tangent stiffness) With FEM/BEM (with tangent stiffness) Structural isplacements an electrostatic noal forces Electric potential istribution
Oofelie::MEMS, BEM using FMM Electrostatically actuate micro-lens for biomeical application (With courtesy of University of British of Columbia an British Columbia Cancer Research Centre, CANADA)
Perspectives Enhancement of amping moeling Thermo-elastic amping (alreay available) Flui Molecular regime implementation Support loss... Better efinition of manufacturing pre-stress Introuction of a NL harmonic solver Extraction of tangent stiffness for BEM