EE C245 ME C218 Introduction to MEMS Design Fall 2007
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1 EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA Lecture 15: Beam Combos EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 1 Announcements Turn part (a) of your HW#4 in as an ed gds file to the TA of your choice Li-Wen Hung: lwhung@eecs.berkeley.edu Yang Lin: linyang@eecs.berkeley.edu Midterm Exam will be Thursday, Nov. 1 EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 2 1
2 Lecture Outline Reading: Senturia Chpts. 9, 10 Lecture Topics: Beam Bending Stress Gradients in Cantilevers Folded-Flexure Suspensions Stressed Folded-Flexures Energy Methods Virtual Work Energy Formulations EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 3 Beam Segment in Pure Bending Small section of a beam bent in response to a tranverse load R EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 4 2
3 Beam Segment in Pure Bending (cont.) EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 5 Internal Bending Moment Small section of a beam bent in response to a transverse load R EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 6 3
4 Differential Beam Bending Equation Neutral axis of a bent cantilever beam EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 7 Cantilever Beam w/ a Concentrated Load Free end condition F Clamped end condition: At x=0: w=0 dw/dx = 0 x L h x EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 8 4
5 Cantilever Beam w/ a Concentrated Load Free end condition F Clamped end condition: At x=0: w=0 dw/dx = 0 x L h x EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 9 Maximum Stress in a Bent Cantilever EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/
6 Stress Gradients in Cantilevers EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 11 Vertical Stress Gradients Variation of residual stress in the direction of film growth Can warp released structures in z-direction EE C245: Introduction to MEMS Design Lecture 4 C. Nguyen 10/16/
7 Stress Gradients in Cantilevers Below: surface micromachined cantilever deposited at a high temperature then cooled assume compressive stress Average stress Stress gradient Once released, beam length increases slightly to relieve average stress After which, stress is relieved But stress gradient remains induces moment that bends beam EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 13 Stress Gradients in Cantilevers (cont) EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/
8 Measurement of Stress Gradient Use cantilever beams Strain gradient (Γ = slope of stress-thickness curve) causes beams to deflect up or down Assuming linear strain gradient Γ, z = ΓL 2 /2 [P. Krulevitch Ph.D.] EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 15 Folded-Flexure Suspensions EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/
9 Folded-Beam Suspension Use of folded-beam suspension brings many benefits Stress relief: folding truss is free to move in y- direction, so beams can expand and contract more readily to relieve stress High y-axis to x-axis stiffness ratio Folding Truss y z x Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 17 Beam End Conditions [From Reddy, Finite Element Method] EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/
10 Common 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, 1994 EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 19 Series Combinations of Springs For springs in series w/ one load Deflections add Spring constants combine like resistors in parallel Y(L) = F/k = 2 y(l c ) = 2 (F/k c ) = F(1/k c + 1/k c ) Compliances effectively add: 1/k = 1/k c + 1/k c k = k c k c EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/
11 Parallel Combinations of Springs For springs in parallel w/ one load Load is shared between the two springs Spring constant is the sum of the individual spring constants Y(L) = F/k = F a /k a = F b /k b = (F/2) (1/k a ) k = 2 k a EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 21 Folded-Flexure Suspension Variants Below: just a subset of the different versions All can be analyzed in a similar fashion [From Michael Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/
12 Deflection of Folded Flexures This equivalent to two cantilevers of length L/2 Composite cantilever free ends attach here Half of F absorbed in other half (symmetrical) 4 sets of these pairs, each of which gets ¼ of the total force F EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 23 Constituent Cantilever Spring Constant From our previous analysis: 2 Fc Lc y Fc y x y = 2 ( ) y = EI 1 z L c 6EI z F k c = = x ( L y) From which the spring constant is: Inserting L c = L/2 c 3EI c z 3 ( Lc ) Lc k 3EI z c = = 3 ( L / 2) 24EI L 3 z EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/
13 Overall Spring Constant Rigid Truss Four pairs of clamped-guided beams In each pair, beams bend in series (Assume trusses are inflexible) Force is shared by each pair F pair = F/4 Leg L F pair EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/
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