1EE C245 – ME C218Introduction to MEMS DesignF ll 200Fall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 1University of California at BerkeleyBerkeley, CA 94720Lecture 15: Beam CombosLecture Outline• Reading: Senturia, Chpt. 9• Lecture Topics:ªBending of beamsªBending of beamsª Cantilever beam under small deflectionsª Combining cantilevers in series and parallelª Folded suspensionsª Design implications of residual stress and stress gradientsEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 22Stress Gradients in CantileversEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 3Vertical Stress Gradients• Variation of residual stress in the direction of film growth• Can warp released structures in z-directionEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 43Stress Gradients in Cantilevers• Below: surface micromachined cantilever deposited at a high temperature then cooled → assume compressive stressAverage stressAfter EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 5Stress gradientOnce released, beam length increases slightly to relieve average stressBut stress gradient remains → induces moment that bends beamwhich, stress is relievedStress Gradients in Cantilevers (cont)EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 64Radius of Curvature f/ Stress GradientEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 7Measurement of Stress Gradient• Use cantilever beamsª Strain gradient (Γ = slope of stress-thickness curve) causes beams to deflect up or downª Assuming linear strain gradient Γ, z = ΓL2/2gg,[P. Krulevitch Ph.D.]EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 85Tip Bending DistanceEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 9Folded-Flexure SuspensionsEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 106Folded-Beam Suspension• Use of folded-beam suspension brings many benefitsª Stress relief: folding truss is free to move in y-direction, so beams can expand and contract more readily to relieve stressª High y-axis to x-axis stiffness ratioFolding TrussxyzEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 11Comb-Driven Folded Beam ActuatorBeam End ConditionsEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 12[From Reddy, Finite Element Method]7Common Loading & Boundary Conditions• Displacement equations derived for various beams with concentrated load F or distributed load f• Gary Fedder Ph.D. Thesis, EECS, UC Berkeley, 1994EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 13Series Combinations of Springs• For springs in series w/ one loadª Deflections addª Spring constants combine like “resistors in parallel”xyzEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 148Parallel Combinations of Springs• For springs in parallel w/ one loadª Load is shared between the two springsª Spring constant is the sum of the individual spring constantsxyzEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 15Folded-Flexure Suspension Variants• Below: just a subset of the different versions• All can be analyzed in a similar fashionEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 16[From Michael Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994]9Deflection of Folded FlexuresThis equivalent to two cantilevers of length Lc/2Composite cantilever Composite cantilever free ends attach here EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 17Half of F absorbed in other half (symmetrical)4 sets of these pairs, each of which gets ¼ of the total force FConstituent Cantilever Spring Constant• From our previous analysis:()yLEIyFLyyEILFyxczcczcc−=⎟⎟⎠⎞⎜⎜⎝⎛−= 36312)(22zcz⎠⎝33)(czcccLEILxFk ==• From which the spring constant is:•Inserting L= L/2EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 18Inserting Lc= L/23324)2/(3LEILEIkzzc==10Overall Spring Constant• Four pairs of clamped-guided beamsª In each pair, beams bend in seriesª (Assume trusses are inflexible)•Force is shared by each pair →Fpair= F/4Rigid TrussForce is shared by each pair →Fpair F/4LegLEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 19FpairFolded-Beam Stiffness Ratios• In the x-direction:•In the zdirection:Folded-beam suspension324LEIkzx=In the z-direction:ª Same flexure and boundary conditions• In the y-direction:Shuttle324LEIkxz=EWh8EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 20• Thus: AnchorFolding trussLEWhky8=24⎟⎠⎞⎜⎝⎛=WLkkxyMuch stiffer in y-direction![See Senturia, §9.2]11Folded-Beam Suspensions Permeate MEMSGyroscope [Draper Labs.]Accelerometer [ADXL-05, Analog Devices]EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 21Micromechanical Filter [K. Wang, Univ. of Michigan]Folded-Beam Suspensions Permeate MEMS• Below: Micro-Oven Controlled Folded-Beam ResonatorEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 2212Stressed Folded-FlexuresEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 23Clamped-Guided Beam Under Axial Load• Important case for MEMS suspensions, since the thin films comprising them are often under residual stress• Consider small deflection case: y(x) « LxzGoverning differential equation: (Euler Beam Equation)LWxyEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 24Governing differential equation: (Euler Beam Equation)Unit impulse @ x=LAxial Load13The Euler Beam Equation• Axial stresses produce no net horizontal force; but as soon as the beam is bent, there is a net downward forceª For equilibrium, must postulate some kind of upward load on the beam to counteract the axial stressderived forceRAxial StressEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 25on the beam to counteract the axial stress-derived forceª For ease of analysis, assume the beam is bent to angle πThe Euler Beam EquationNote: Use of the full bend angle of π to establish conditions for load balance; but this returns us to case of small displacements and small anglesEE
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