1EE C245 – ME C218 Fall 2003 Lecture 7EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 7MicrostructuralElements** Mostly for EE’s, but theremay be a few new insightsfor the ME’s2EE C245 – ME C218 Fall 2003 Lecture 7Today’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. 201-219, 222-231.J. D. Grade, H. Jerman, and T. W. Kenny, “Design of large deflection electrostatic actuators,” Journal of Microelectromechanical Systems, 12, 335-343 (2003).23EE C245 – ME C218 Fall 2003 Lecture 7Macro and Milli Suspensions2000 Ford Focus(mostly 3-D steel parts andassembly-line production)… 100,000’s per year Hard Disk Suspensions(stamped 20 µm stainless steelwith laminated 10 µm polyimide+ 15 µm copper interconnect)… 1,000,000’s per week4EE C245 – ME C218 Fall 2003 Lecture 7Springs in MEMS• Coils: 3-D is tough for planar processing!• Flexures: straightforward to make using surface or bulk micromachining, but details of fabrication process constrain dimensions and anchors/joints• Simplest flexure: a “clamped-free” cantilever beam … a.k.a. a diving board35EE C245 – ME C218 Fall 2003 Lecture 7A Cantilever BeamClamped:xx = LcGoal: find relation between tip deflection y(x = Lc) and applied load FAssumptions:1. Tip deflection is small compared with beam length2. Plane sections (normal to beam’s axis) remain plane and normalduring bending … “pure bending”3. Shear stresses are negligibleF6EE C245 – ME C218 Fall 2003 Lecture 7Checking the AssumptionsJ.-A. Schweitz, Uppsala University47EE C245 – ME C218 Fall 2003 Lecture 7A Beam Segment in Pure Bendingbottom is in compressiontop is in tensionneutral axis (εx= 0)εxyyh/2-h/28EE C245 – ME C218 Fall 2003 Lecture 7Bending Moment Mz• Concept of moment (basic physics): force X distance• Integrate stress through thickness of beam=−zMWhy a minus sign? See Senturia, pp. 208-21059EE C245 – ME C218 Fall 2003 Lecture 7Bending Strain and Beam Curvature=−zMRadius of curvature à geometric connection to strainRR + h/2dθ=max,xε10EE C245 – ME C218 Fall 2003 Lecture 7Curvature and Strain (cont.)221dxydR =−=−zMRhx2/max,=ε… result from basic calculus Combining the curvature and moment results:and( )max,32322/xzhhEWM ε=−611EE C245 – ME C218 Fall 2003 Lecture 7Flexural Rigidity (Moment of Inertia) Iz• The term Wh3/12 is defined as the flexural rigidity, Iz(Senturia uses “moment of inertia”)• Large flexural rigidity à low curvature à small deflections à stiff• Design implications: 1. rigidity increases as the cube of the beam’s thickness(in the direction of bending)2. the aspect ratio h / W determines the ratio of bending rigidity in the y and the z directions22dxydEIMzz=−12EE C245 – ME C218 Fall 2003 Lecture 7Revisit Cantilever Deflectiondue to Residual Stress Gradients• Model the strain by a linear profile yyresresΓ+= εε )(zPeter Krulevitch,Ph.D. thesis, ME, UC Berkeley, 1994* inferred from wafer curvature after incremental thinning of poly-SiLPCVD poly-Si:measured* stress profiles713EE C245 – ME C218 Fall 2003 Lecture 7Built-in Bending Moment• Integrate differential moment through film thickness (sign?)( )==∫−yWdyMhhrr2/2/σ( )Γ+=⋅Γ+=∫−12032/2/EWhdyyyEWMhhrrε• Apply moment to the cantilever à constant curvature=22dxydEIz14EE C245 – ME C218 Fall 2003 Lecture 7Tip-Deflection (Small Deflections)• The strain gradient Γ can be found from the tip deflection ∆:=∆==)( Lxy• Integrate to find the tip deflection y(x =L)=Γ815EE C245 – ME C218 Fall 2003 Lecture 7Boundary Conditions• A “step-up” anchor will result in the average strain causing an offset angle at y = 0Approach tosuppressinginitial offsetangle 16EE C245 – ME C218 Fall 2003 Lecture 7The Cantilever with a Concentrated LoadClamped: y = 0, dy/dx = 0 at x= 0xx = LcFind the tip deflection y(x = Lc) and applied load F … get effective springconstant kcF=− )(xMzThe moment varies linearly with x22)(dxydEIxMzz=−917EE C245 – ME C218 Fall 2003 Lecture 7Tip Deflection• Integrate ODE twice and apply boundary conditions (zero displacement, zero slope) at anchor2)3(6)( xxLEIFxycz−=• Tip deflection: y(Lc)33)(czcLEIFLy=• Spring constant: kc(N/m) … = (µN/ µm)=ck18EE C245 – ME C218 Fall 2003 Lecture 7Summary of Common Loadingand Boundary ConditionsCompendium of useful results:http://www.roarksformulas.com1019EE C245 – ME C218 Fall 2003 Lecture 7Series Combinations of Cantilevers• Springs in series à same load; deflections addy(L) = F/k =y(L)#1#2F LcLcL = 2Lc20EE C245 – ME C218 Fall 2003 Lecture 7Parallel Combinations of Springs• Same displacement à load is shared and the spring constant is the sum of the individual spring constantsy(L)abF y(L) = F/kF/2F/21121EE C245 – ME C218 Fall 2003 Lecture 7Folded-Flexure Suspension VariantsMichael Judy, Ph.D. Thesis, EECS Dept., UC Berkeley, 199422EE C245 – ME C218 Fall 2003 Lecture 7Overall Spring Constant• Four pairs of clamped-guided beams, each of which bend in series (assume that trusses are inflexible)Force is shared by each pair à Fpair=Displacement of two legs add (springs in series) àFpairlegrigid trussy = Fpair/kpair=1/kleg= y =1223EE C245 – ME C218 Fall 2003 Lecture 7Selected Goals for Suspension Design• Compliance ratios are often required to be large(e.g., 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 of the structure are often required to be at significantly higher frequencies, which translates to stiffer spring constants• Robustness against residual stress and stress gradients (e.g., folded flexures release most of the residual stress and cancel deflections due to gradients)24EE C245 – ME C218 Fall 2003 Lecture 7Folded Flexure Suspension withResidual Stress Gradientlegs warp togethercomb teeth areengagedMichael Judy, Ph.D. ThesisEECS Dept., UC Berkeley, 19941325EE C245 – ME C218 Fall 2003 Lecture 7ADXL-50 Suspension26EE C245
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