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Berkeley ELENG C245 - Microstructural Elements

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EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/091Copyright © 2009 Regents of the University of CaliforniaEE C245 – ME C218Introduction to MEMS DesignF ll 2009Fall 2009Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 1University of California at BerkeleyBerkeley, CA 94720Lecture Module 8: Microstructural ElementsOutline• 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 LecM 8 C. Nguyen 9/28/07 2EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/092Copyright © 2009 Regents of the University of CaliforniaBending of BeamsEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 3Beams: The Springs of Most MEMS• Springs and suspensions very common in MEMSª Coils are popular in the macro-world; but not easy to make in the micro-worldª Beams: simpler to fabricate and analyze; become py“stronger” on the micro-scale → use beams for MEMSEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 4Comb-Driven Folded Beam ActuatorEE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/093Copyright © 2009 Regents of the University of CaliforniaBending a Cantilever BeamFClamped end condition:At x 0:Free end condition• Objective: Find relation between tip deflection y(x=Lc) and applied load Fx′LxAt x=0:y=0dy/dx = 0EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 5• Assumptions:1. Tip deflection is small compared with beam length2. Plane sections (normal to beam’s axis) remain plane and normal during bending, i.e., “pure bending”3. Shear stresses are negligibleReaction Forces and MomentsEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 6EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/094Copyright © 2009 Regents of the University of CaliforniaSign Conventions for Moments & Shear Forces(+) moment leads to deformation with a (+) radius of curvature (i.e., upwards)z(i.e., upwards)(-) moment leads to deformation with a (-) radius of curvature (i.e., downwards)R = (+)R = (-)(+) shear forces EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 7produce clockwise rotation (-) shear forces produce counter-clockwise rotation Beam Segment in Pure BendingSmall section of a beam bent in a beam bent in response to a tranverse loadREE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 8EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/095Copyright © 2009 Regents of the University of CaliforniaBeam Segment in Pure Bending (cont.)EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 9Internal Bending MomentSmall section of a beam bent in response to a transverse loadREE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 10EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/096Copyright © 2009 Regents of the University of CaliforniaDifferential Beam Bending EquationNeutral axis of a bent cantilever beamEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 11Example: Cantilever Beam w/ a Concentrated LoadEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 12EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/097Copyright © 2009 Regents of the University of CaliforniaCantilever Beam w/ a Concentrated LoadFClamped end condition:At x 0:Free end conditionhxLxAt x=0:w=0dw/dx = 0EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 13Cantilever Beam w/ a Concentrated LoadFClamped end condition:At x 0:Free end conditionhxLxAt x=0:w=0dw/dx = 0EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 14EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/098Copyright © 2009 Regents of the University of CaliforniaMaximum Stress in a Bent CantileverEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 15Stress Gradients in CantileversEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 16EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/099Copyright © 2009 Regents of the University of CaliforniaVertical Stress Gradients• Variation of residual stress in the direction of film growth• Can warp released structures in z-directionEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 17Stress Gradients in Cantilevers• Below: surface micromachined cantilever deposited at a high temperature then cooled → assume compressive stressAverage stressAfter which, EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 18Stress gradientOnce released, beam length increases slightly to relieve average stressBut stress gradient remains → induces moment that bends beamstress is relievedEE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/0910Copyright © 2009 Regents of the University of CaliforniaStress Gradients in Cantilevers (cont)EE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 19Measurement of Stress Gradient• Use cantilever beamsª Strain gradient (Γ = slope of strain-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 LecM 8 C. Nguyen 9/28/07 20EE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/0911Copyright © 2009 Regents of the University of CaliforniaFolded-Flexure SuspensionsEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 21Folded-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 LecM 8 C. Nguyen 9/28/07 22Comb-Driven Folded Beam ActuatorEE 245: Introduction to MEMSModule 8: Microstructural ElementsCTN 10/5/0912Copyright © 2009 Regents of the University of CaliforniaBeam End ConditionsEE C245: Introduction to MEMS Design LecM 8 C. Nguyen 9/28/07 23[From Reddy, Finite Element Method]Common Loading & Boundary Conditions• Displacement equations derived for various beams with concentrated load F or distributed load f•


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Berkeley ELENG C245 - Microstructural Elements

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