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# Introduction to MEMS Design MEC ENG C218

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This 15 page Class Notes was uploaded by Dayna Kessler on Thursday October 22, 2015. The Class Notes belongs to MEC ENG C218 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/226684/mec-eng-c218-university-of-california-berkeley in Mechanical Engineering at University of California - Berkeley.

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Date Created: 10/22/15

EE 6245 ME 6218 Introduction to MEMS Design Fall 2003 Roger Howe and Thara Srinivasan Lecture 7 Microstructural Elements Mostly for EE s but there may be a few new insights EE 0245 ME C218 Fall 2003 Lecture 7 for the ME s Today s Lecture The cantilever beam under small deflections Combining cantilevers in series and parallel folded suspensions More accurate models large deflections shear Design implications of residual stress and stress gradients Reading Senturia S D Microsystem Design Kluwer Academic Publishers 2001 Chapter 9 pp 201219 222231 J D Grade H Jerman and T W Kenny Design of large deflection electrostatic actuators Journal of Microelectromechanica Systems 12 335343 2003 EE C245 ME C218 Fall 2003 Lecture 7 Macro and Milli Suspensions 2000 Ford Focus Hard Disk Suspensions mostly 3 D steel parts and stamped 20 um stainlesssteel assembly line production With laminated 10 pm polyimide 100000 5 per year 15 um copper interconnect EE c245 ME c213 Fall 2003 Lecture 7 1 000 000 5 per week Springs in MEMS Coils 3D is tough for planar processing Flexures straightforward to make using surface or bulk micromachining but details of fabrication process constrain dimensions and anchorsjoints Simplest flexure a clampedfree cantilever beam aka a diving board EE C245 ME C218 Fall 2003 Lecture 7 A Cantilever Beam Clamped X F Goal nd relation between tip de ection yx LC and applied load F A ssumptions Tip de ection is small compared with beam length Plane sections normal to beam s axis remain plane and normal during bending pure bending 3 Shear stresses are negligible EE 3245 ME C218 Fall 2003 Lecture 7 Checking the Assumptions 5 JA Schweitz Uppsala University EE 3245 ME C218 Fall 2003 Lecture 7 A Beam Segment in Pure Bending y top is in tension is sx0 compression l72 72 EE C245 ME C218 Fall 2003 Lecture 7 Bending Moment M2 Concept of moment basic physics force Xdistance Integrate stress through thickness of beam Mz Why a minus sign See Senturia pp 20amp210 EE C245 ME C218 Fall 2003 Lecture 7 Bending Strain and Beam Curvature Mz Radius of curvature 9 geometric connection to strain EE C245 ME C218 Fall 2003 Lecture 7 Curvature and Strain cont 1 R result from basic calculus ab2 Combining the curvature and moment results h2 3 5x max and M EW Sxmax R h23 2 M 2 EE C245 ME C218 Fall 2003 Lecture 7 Flexural Rigidity Moment of Inertia I The term Wh312 is defined as the flexural rigidity Z Senturia uses moment of inertia Large flexural rigidity 9 low curvature 9 small deflections 9 stiff 2 7M2 EIZ Design implications 1 rigidity increases as the cube of the beam s thickness in the direction of bending the aspectratio h Wdetermines the ratio of bending rigidity in they and thez directions D N EE C245 ME C218 Fall 2003 Lecture 7 Revisit Cantilever Deflection due to Residual Stress Gradients Model the strain by a linear profile 86501 565 Ty 77777 W 7W 77 s 7 a Ea 400 7 E E E E 7 L g a u 1 o 5 u a a 5 o L a 350 7 g E g E g E g E r 39u m o 1 lt i w 5 m r m E 300 r C I 39 lt a p 7x 250 3 quot 20 LPCVD polySi Peter Krulevitch lSO l measured stress profiles PhD thesis ME l g l UC Berkeley 1994 E t 4 xth l 1 Inferred from 03 04 05 06 0 I 0 l 09 1 wafer curvature W Z i after incremental thinning of polySi nepm Below Surface uml EE C245 ME C218 Fall 2003 Lecture 7 Builtin Bending Moment Integrate differential moment through film thickness sign M IGXWOWy rhZ hZ 3 M EW j5 ryydy 0EWh 1 412 12 Apply moment to the cantilever 9 constant curvature dzy E Z dx2 EE C245 ME C218 Fall 2003 Lecture 7 TipDeflection Small Deflections Integrate to find the tip deflection yx L yx A o The strain gradient F can be found from the tip deflection A EE C245 ME C218 Fall 2003 Lecture 7 Boundary Conditions A stepup anchor will result in the average strain causing an offset angle at y 0 r l u39 EE C245 ME C218 Fall 2003 Lecture 7 Approach to suppressing initial offset angle The Cantilever with a Concentrated Load Clamped y 0 dydx 0 at X 0 L x c x V F Find the tip deflection yx LC and applied load F get effective spring constant kc dzy abc2 Mzx EIZ The moment varies linearly with x Mzltxgt EE C245 ME C218 Fall 2003 Lecture 7 Tip Deflection Integrate ODE twice and apply boundary conditions zero displacement zero slope at anchor F 3L xx2 6E4 yx Tip deflection yLc yL F JLS 3EIZ Spring constant kc Nm pN pm EE C245 ME C218 Fall 2003 Lecture 7 Summary of Common Loading and Boundary Conditions F Compendium of useful results httpwwwroarksformulascom caniilever beam concentrated raga u gurdsdrsnd beam Concentrated ma Ref G K Fedder PhD thesrs Dept of EECS UC Berkeley 1994 EE C245 ME C218 Fall 2003 Lecture 7 Series Combinations of Cantilevers Springs in series 9 same load deflections add yL Fk EE C245 ME C218 Fall 2003 Lecture 7 19 Parallel Combinations of Springs Same displacement 9 load is shared and the spring constant is the sum ofthe individual spring constants yL yL Fk EE C245 ME C218 Fall 2003 Lecture 7 20 FoldedFlexure Suspension Variants mixnan m lug4quot l lwulum l mmw illlmmmium sumum n camIIan mu vhllnmxlmn Michael Judy PhD Thesis EECS Dept UC Berkeley 1994 EE 3245 ME C218 Fall 2003 Lecture 7 Overall Spring Constant Four pairs of clampedguided beams each of which bend in series assume that trusses are inflexible Force is shared by each pair 9 Fm Displacement oftwo legs add springs in series 9 rigid truss leg y Fugkm 1keg Fpar y EE 3245 ME C218 Fall 2003 Lecture 7 Selected Goals for Suspension Design Compliance ratios are often required to be large eg the comb drive s maximum force is determined by lateral instability which is in turn directly related to the lateral spring constant Undesirable resonant modes ofthe structure are often required to be at significantly higher frequencies which translates to stiffer spring constants Robustness against residual stress and stress gradients eg folded flexures release most of the residual stress and cancel deflections due to gradients EE 0245 ME 0218 Fall 2003 Lecture 7 Folded Flexure Suspension with Residual Stress Gradient legs warp together EE 0245 ME 0218 Fall 2003 Lecture 7 Michael Judy PhDThesis EECS Dept UC Berkeley 1994 24 ADXL50 Suspension Cl Straight tethers are pulled at under tension quotbackbonequot domes up about 05 pm sense fingers are curled downward off of backbone A lllllll ill I ll do can we as goo no mo so transaction YWIM EE C245 ME C218 Fall 2003 Lecture 7 ADXL05 Suspension EE C245 ME C218 Fall 2003 Lecture 7 EE C245 ME C218 Fall 2003 Lecture 7 Motorola zAxis Accelerometer EE C245 ME C218 Fall 2003 Lecture 7 Normalized displacement yL DD NarmElIZed force Limits to Linearity b linearity for mu ch act theory ear theory 00 05 10 15 20 25 30 F Michael Judy PhD Thesis Ehw3 EECS Dept UC Berkeley 1994 4 EE 0245 ME 0218 Fall 2003 Lecture 7 Cantilever beams stiffen as the deflection exceeds about 10 of the length of the eam Note clampedclamped beams deviate from non smaller de ection s n ext lecture When is Shear Significant Fl Fixedguided beam under point load two cantilevers in series 1 12E I GWh kx1 kxb kx kxb L37 sz km Solvmg for kx K 1 glamy LJZ Roi D Hora cy Ph D thesis Dept oiME UC Berkeley 1998 EE 0245 ME 0218 Fall 2003 Lecture 7 E Wh 715L m Lgt10Wgt2 5 or less error

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