Comb Resonator Design (4)

Size: px

Start display at page:

Download "Comb Resonator Design (4)"

Collin Lloyd
5 years ago
Views:

1 Lecture 8: Comb Resonator Design (4) - Intro. to Fluidic Dynamics (Damping) - School of Electrical Engineering and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory dicho@snu.ac.kr URL:

2 Two Types of Fluids Liquid : molecules are free to change position but are bound by cohesive forces. (fixed volume) Gas : molecules are unrestricted. (no shape or volume) Liquid Gas usage and possession is in violation of copyright laws 2

3 System Methods of Analysis It is defined as a fixed, identifiable quantities of mass. No mass crosses the system boundary. Heat and work may cross the boundary of the system. System boundary Piston PV = nrt usage and possession is in violation of copyright laws 3

4 Control volume Methods of Analysis (cont d) An arbitrary volume in space through which fluid flows. Control surface The geometric boundary of the control volume. usage and possession is in violation of copyright laws 4

5 Mean free path (λ) Mean Free Path A distance that the molecule travels before it reacts with another molecules. λ λ = vt v : average velocity of the molecule t : mean free time mean free path : λ [ m] = ( in air, at room temperature ) P [ Torr] ex) P = 1 atm λ = 60 nm usage and possession is in violation of copyright laws 5

6 Absolute viscosity (µ) Viscosity The shearing stress (τ) and rate of shearing strain (du/dy) can be related with a relationship of the form. τ u f y τ = du μ dy μ : viscosity (absolute) 2 [ μ ] = ( N / m ) ( s ) f =force, u=velocity, A= area of the plate, d =distance between the plates, = shear stress τ usage and possession is in violation of copyright laws 6

7 Kinetic viscosity ( v ) μ ρ 2 v= : [ v] = m / s Viscosity (cont d) μ :absolute viscosity, it ρ : density For gases, kinetic viscosity increases with temperature, whereas for liquids, kinetic viscosity decreases with increasing temperature. usage and possession is in violation of copyright laws 7

8 Viscous flow Fluid Motions The shear stress gives rise to change the velocity of the fluid at the interface between the solid and fluid. Inviscid flow The effect of shear stress is so small as to be negligible. usage and possession is in violation of copyright laws 8

9 Dissipated energy (D) Dissipation in Fluid Flow Work is required to move the plate, but this work is not stored as potential or kinetic energy in the plate or fluid. The differential volume is deformed by the shear force. The work done on the differential volume. D = τ du dxdz x usage and possession is in violation of copyright laws 9

10 Surface force Stress Field It acts on the boundaries of a medium through direct contact. (pressure, stress etc) Body force Forces developed without physical contact, and distributed over the volume of fluid. (gravitational force etc) usage and possession is in violation of copyright laws 10

11 Momentum Equation Forces acting on a fluid particle <Surface forces in the x direction on a fluid element> usage and possession is in violation of copyright laws 11

12 Momentum Equation (cont d) Surface force (Fs) for x-axis σ xx dx σ xx dx dfsx = ( σxx + ) dydz ( σxx + ) dydz x 2 x 2 τ yx dy τ yx dy + ( τ yx + ) dxdz ( τ yx + ) dxdz y 2 y 2 τzx dz τzx dz + ( τzx + ) dxdy ( τzx + ) dxdy z 2 z 2 σ τ xx yx τ yx = ( + + ) dxdydz (1) x y y usage and possession is in violation of copyright laws 12

13 Momentum Equation (cont d) Body force (FB) for x-axis : gravity df Bx = ρg dxdydz (2) x Then, the total force in x direction ((1) + (2)) df = df + df x sx Bx σ τ xx yx τzx = ( ρgx ) dxdydz (3) x y z σ yy τxy τzy similarily, df y = ( ρgy ) dxdydz y x z σ zz τ xz yz dfz = ( ρgz ) dxdydz z x y usage and possession is in violation of copyright laws 13

14 Differential Momentum Equation From the Newton s 2 nd law of motion dv df = dm dt σ τ xx yx τ zx ( ρ gx ) dxdydz ddd x y z u u u u = ρddd dxdydz ( + u + v + w ) t x y z σ τ xx yx τ zx u u u u ρgx = ρ ( + u + v + w ) x y z t x y z Similarly, we can have y and z differential equation of motion. usage and possession is in violation of copyright laws 14

15 Newtonian fluid Navier-Stokes Equation Fluids for which the shearing stress is linearly related to the rate of the shearing strain. 2 ( v u τ xy = τ yz = μ + ), σ xx = ρ μ v + 2μ u x y 3 x v w 2 v τ yz = τ zy = μ( + ), σ yy = ρ μ v + 2μ z y 3 y 2 ( w u τ zx = τ xz = μ + ), σ zz = ρ μ v + 2 μ w x z 3 z Navier - stokes Equation x - momentum Equation σ u p du 2 ρ = ρgx + μ(2 v) σ t x x dx 3 du v dw u + μ( ) + μ( ) y dy x z dx z usage and possession is in violation of copyright laws 15

16 Navier-Stokes Equation (cont d) Navier-Stokes Equation Equation of motion governing fluid behavior u 2 ρ + ( u) u = ρ g ( p) + μ u t ρ: mass density μ: absolute vis cosity p : pressure u : velocity g : gravity [Ref.] Bruce R. Munson Fundamentals of Fluid Mechanics, 2 nd ed. usage and possession is in violation of copyright laws 16

17 Navier-Stokes Equation (cont d) Apply the Navier-Stokes equation to some special cases Couette flow Stokes flow usage and possession is in violation of copyright laws 17

18 Couette flow Couette Flow We obtain a linear velocity profile for the fluid film underneath the plate, as shown in Figure. [Ref.] Y.-H. Cho, A. P. Pisano, and R. T. Howe, "Viscous damping model for laterally oscillating microstructures, IEEE JMEMS, Vol. 3, No. 2, pp , X. Zhang and W. C. Tang, Viscous air damping in laterally driven microresonators, MEMS 1994, pp , 1994 usage and possession is in violation of copyright laws 18

19 Couette Flow (cont d) ρ u + iu u = g p u t ρ + μ Navier- Stokes Equation : ( ) ( ) 2 Neglecting the time dependent term, du/dt=0. And we assume no pressure gradient, dp/dx=0. Then we have D ux ( y ) u = 0 = 0 2 y No-slip boundary conditions (zero velocity at the surface of the stationary plate), than we have u x = y d u 0 Dissipated energy : π w μ d D u 2 = ( ) 0 usage and possession is in violation of copyright laws 19

20 Stokes flow Stokes Flow The steady state velocity profile, u(y,t), in the fluid is governed by time term of the Navier-Stokes Equation. [Ref.] Y.-H. Cho, A. P. Pisano, and R. T. Howe, "Viscous damping model for laterally oscillating microstructures, IEEE JMEMS, Vol. 3, No. 2, pp , X. Zhang and W. C. Tang, Viscous air damping in laterally driven microresonators, MEMS 1994, pp , 1994 usage and possession is in violation of copyright laws 20

21 Stokes Flow (cont d) ρ u + iu u = g p u t ρ + μ Navier-Stokes Equation : ( ) ( ) 2 And we assume no pressure gradient. Then we have 2 2 u μ u u u =, = v v : kinetic vis cosity 2 2 t ρ y t y No-slip boundary conditions (i.e., zero velocity at the surface of the stationary plate, Than we have u = u e where = w v x j( β y wt) 0 β /2 Dissipated energy : D = π 2 sinh 2 d sin 2 d u0 ( β + μβ β ) w cosh 2β d cos 2βd usage and possession is in violation of copyright laws 21

22 Squeeze-film damping Dissipated force Squeeze Film Vertical Motion of upper plate relative to fixed bottom plate with viscous fluid between plates Viscous drag during flow creates dissipative force on plate opposing motion F Moving plate d dh dt Fixed plate usage and possession is in violation of copyright laws 22

23 Assumption Squeeze Film (cont d) No pressure gradient transverse to the plate Gap, h, is much smaller than the lateral dimensions of plate. F d Moving plate Fixed plate dh dt 3 L /2 μwl dh F = pwdx = = bh L /2 3 d dt μwl damping coefficient : b = 3 d 3 usage and possession is in violation of copyright laws 23

24 Quality Factor Analysis Laterally driven microresonator Stokes flow model Qd, Qc Squeeze film model Qs usage and possession is in violation of copyright laws 24

25 Quality Factor Analysis (cont d) Damping Coefficient & Quality factor System equation F = Mx + bx + Kx ext b K 2 x'' + x' + x= x'' + 2ζωn + ωn M M 1 ζ = K ω n = 2Q M Damping ratio :, Natural frequency : Damping coefficient : 2 n b = M ζω = M K Q Quality factor : Q = MK b usage and possession is in violation of copyright laws 25

26 Quality Factor Analysis (cont d) Parameter (1 atm, at room temperature) Natural frequency : f = khz Absolute viscosity : µ= 1.8 x 10-5 kg/m sec Kinetic viscosity : v = 1.5 x 10-5 m 2 /sec Density of silicon : 2330 kg/m 3 Distance of inter-plate (sacrificial gap) d=20 μm Distance of inter-plate (between structure and cap) h=150 μm Distance of inter-combs : dc=2 μm Mass: 42 μg Spring stiffness: N/m usage and possession is in violation of copyright laws 26

27 Stokes Flow Model (Qd) d h 2 cap Y direction Plate u x direction y direction Substrate put u = u e = v 1 π du D = τ 0 0ud( wt), τ 0 μ ( frictional shear) w = dy j( β y- wt) w 0 where β, π 2 sinh 2βy+ sin 2βy D = u0 μβ ( ) w cosh 2 β y cos 2 β y ( a) 0< y < d 1 π 2 sinh 2βd + sin 2βd D = u0 μβ ( ) w cosh 2 β d cos 2 β d Qd 1 2πW 1 cosh 2βd - cos 2βd = = MK ( ) (1) AD μ Aβ sinh 2βd + sin 2βd W : strain energy, A: plate area, D : dissipate energy ( b) d Qd 2 2 < y < d h = d - d πW 1 cosh2βh cos2βh = = MK ( ) (2) AD μaβ sinh 2βh + sin 2βh [Ref.] T. Y. Song, S. J. Park, Y. H. Park, D. H. Kwak, H. H. Ko, and D. I. Cho, Quality Factor in Micro- gyroscopes, APCOT MNT 2004, pp , 2004 usage and possession is in violation of copyright laws 27

28 Stokes Flow Model (Qc) Air damping between the comb fingers Qc v d Plate v Qd Substrate dc Qc Air damping underneath the plate put u = u0e where β =, v 1 2π du D = 0 ud ( wt ), 0 ( frictional shear ) w τ τ = μ 0 dy j( β y- wt) w π 2 sinh 2βy+ sin 2βy D= u0 μβ ( ) w cosh 2βy cos 2βy () c 0< y< dc 2 sinh 2 dc sin 2 dc D π u0 ( β + = μβ β ) w cosh 2βdc cos 2βdc 2πW 1 cosh2β dc cos 2βdc Qc = = MK ( ) (3) AD μaβ sinh 2βdc + sin 2βdc W: strain energy, A: plate area, D: dissipate energy Qc between comb fingers usage and possession is in violation of copyright laws 28

29 Squeeze Film Model (Qs) F 3 L /2 μ wl dh F = pwdx= L /2 3 d dt 3 μwl b = 3 d 3 d ( d ) Qs = M K (4) 3 μwl h dh dt x L/2 L/2 usage and possession is in violation of copyright laws 29

Quality Factor Analysis Results Quality factor versus viscosity Quality factor Quality factor Damping coefficient Quality facto or Q d2 Q c Q d1 Q total (a) 93 8.15e -4 (b) 561 1.35e -4 (c) 497 1.

30 Quality Factor Analysis Results Quality factor versus viscosity Quality factor Quality factor Damping coefficient Quality facto or Q d2 Q c Q d1 Q total (a) e -4 (b) e -4 (c) e -4 Viscosity it [kg/m*sec] total Quality factor : Q total MK = + + +, b= Q Qd Qd Qc Qs Q total 1 2 (d) 5.12e e -12 Total e -3 [Ref.] T. Y. Song, S. J. Park, Y. H. Park, D. H. Kwak, H. H. Ko, and D. I. Cho, Quality Factor in Micro- gyroscopes, APCOT MNT 2004, pp , usage and possession is in violation of copyright laws 30

31 Quality Factor Analysis Result (cont d) Quality factor g/m*sec] Viscosity [kg μ effi μ = ( ) 0 K n Pressure[mTorr] The viscosity versus the pressure μ μeffi = ( ) LP λ, λ : mean free path, L : dis tan λ = Kn = P L Kn : Knudsen number ce between of the parallel plate [Ref.] M. Bao, H. Yang, H. Yin, and Y. Snu Energy transfer model for squeeze-film air damping in low vacuum, IOP JMM, Vol. 12, pp , 2002 usage and possession is in violation of copyright laws 31

32 Quality Factor Analysis Result (cont d) The Quality factor versus the pressure Sacrificial gap Quality factor Damping Coefficient (b) P = 200mTorr 2 um 4.1 x x 10-6 P = 200mTorr 20 um 1.2 x x 10-7 usage and possession is in violation of copyright laws 32

33 Quality Factor Analysis Result (cont d) Quality factor y factor Quality P=100mTorr P=200mTorr Sacrificial gap [m] The Quality factor vs. the sacrificial gap Sacrificial gap Quality factor Damping Coefficient (b) P = 100mTorr 20 um 3.8 x x 10-7 P = 200mTorr 20 um 1.2 x x 10-7 usage and possession is in violation of copyright laws 33

34 Reference Bruce R. Munson Fundamentals of Fluid Mechanics, 2nd ed. Y.-H. Cho, A. P. Pisano, and R. T. Howe, "Viscous damping model for laterally oscillating microstructures, IEEE JMEMS, Vol. 3, No. 2, pp , X. Zhang and W. C. Tang, Viscous air damping in laterally driven microresonators, MEMS 1994, pp , 1994 T. Y. Song, S. J. Park, Y. H. Park, D. H. Kwak, H. H. Ko, and D. I. Cho, Quality Factor in Micro-gyroscopes, APCOT MNT 2004, pp , 2004 M. Bao, H. Yang, H. Yin, and Y. Snu Energy transfer model for squeeze-film air damping in low vacuum, IOP JMM, Vol. 12, pp , 2002 usage and possession is in violation of copyright laws 34

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says