1EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 1EE C245 – ME C218Introduction to MEMS DesignFall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture 15: Beam CombosEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 2Announcements• Turn part (a) of your HW#4 in as an emailed gds file to the TA of your choiceLi-Wen Hung: [email protected] Lin: [email protected]• Midterm Exam will be Thursday, Nov. 12EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 3Lecture Outline• Reading: Senturia Chpts. 9, 10• Lecture Topics:ª Beam Bendingª Stress Gradients in Cantileversª Folded-Flexure Suspensionsª Stressed Folded-Flexuresª Energy Methods( Virtual Work( Energy FormulationsEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 4Beam Segment in Pure BendingSmall section of a beam bent in response to a tranverse loadR3EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 5Beam Segment in Pure Bending (cont.)EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 6Internal Bending MomentSmall section of a beam bent in response to a transverse loadR4EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 7Differential Beam Bending EquationNeutral axis of a bent cantilever beamEE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 8Cantilever Beam w/ a Concentrated LoadFxLxClamped end condition:At x=0:w=0dw/dx = 0Free end conditionh5EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 9Cantilever Beam w/ a Concentrated LoadFxLxClamped end condition:At x=0:w=0dw/dx = 0Free end conditionhEE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 10Maximum Stress in a Bent Cantilever6EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 11Stress Gradients in CantileversEE C245: Introduction to MEMS Design Lecture 4 C. Nguyen 10/16/2007 12Vertical Stress Gradients• Variation of residual stress in the direction of film growth• Can warp released structures in z-direction7EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 13Stress Gradients in CantileversAverage stressStress gradient• Below: surface micromachined cantilever deposited at a high temperature then cooled → assume compressive stressOnce released, beam length increases slightly to relieve average stressBut stress gradient remains → induces moment that bends beamAfter which, stress is relievedEE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 14Stress Gradients in Cantilevers (cont)8EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 15Measurement 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 = ΓL2/2[P. Krulevitch Ph.D.]EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 16Folded-Flexure Suspensions9EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 17Folded-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 ratioComb-Driven Folded Beam ActuatorFolding TrussxyzEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 18Beam End Conditions[From Reddy, Finite Element Method]10EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 19Common 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 7 C. Nguyen 10/10/07 20Series Combinations of Springs• For springs in series w/ one loadª Deflections addª Spring constants combine like “resistors in parallel”Compliances effectively add:1/k = 1/kc+ 1/kcY(L) = F/k = 2 y(Lc) = 2 (F/kc) = F(1/kc+ 1/kc)k = kc||kc→11EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 21Parallel 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 constantsY(L) = F/k = Fa/ka= Fb/kb= (F/2) (1/ka)k = 2 kaEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 22Folded-Flexure Suspension Variants[From Michael Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994]• Below: just a subset of the different versions• All can be analyzed in a similar fashion12EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 23Deflection of Folded FlexuresHalf of F absorbed in other half (symmetrical)This equivalent to two cantilevers of length L/2Composite cantilever free ends attach here 4 sets of these pairs, each of which gets ¼ of the total force FEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 24Constituent Cantilever Spring Constant• From our previous analysis:()yLEIyFLyyEILFyxczcczcc−=⎟⎟⎠⎞⎜⎜⎝⎛−= 36312)(2233)(czcccLEILxFk ==• From which the spring constant is:• Inserting Lc= L/23324)2/(3LEILEIkzzc==13EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 25Overall 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/4FpairRigid
View Full Document
Unlocking...