Robust Optimization: Design in MEMS - University of California ...

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27Similar to most non-convex optimization packages, when our algorithm is startedfrom different initial conditions, it is likely to end up at different solutions. In addition,our algorithm is not deterministic because of the randomness used in choosing asearch direction. This results in different solutions being found from the same initialcondition, whereas deterministic algorithms will always follow the same path to asolution.
28Chapter 5Robust Design Examples5.1 Two Degree of Freedom MEMS ResonatorThe first example we will consider is a simple resonator with two degrees of freedom.Our objective is to robustly design the resonant frequency of this structure.This is not a difficult problem to solve in closed form, but is a good example that wewill use to illustrate our robust design technique presented in chapter 2. Additionalmotivation for studying this problem is that with only two design variables it is easyto visualize.The two degree of freedom (DOF) resonator, illustrated in figure (5.1), consistsof a proof mass attached by two parallel beams to a fixed anchor. To simplify theproblem we assume that the proof mass has a fixed area A, and that the thickness ofthe entire device is t. The two design variables are the beam width, h, and length,L, as shown in figure (5.1). We will define the design vector to be x = [h L] T .The natural frequency, w n , of the simple resonator can be approximated as √ k/M,where k is the combined stiffness of the two parallel beams and M is the mass of theproof mass. The expressions for k and M arek = 2Eth3L 3M = ρAtwhere E is Elastic Modulus and ρ is the density of the material used to fabricate theresonator. We will assume additive uncertainty in the design variables, and therefore
Page 1 and 2: Robust Optimization:Design in MEMSb
Page 4 and 5: 36 Conclusion 46Bibliography 48A Al
Page 6 and 7: 5List of Tables3.1 Probabilities wi
Page 8 and 9: 7Chapter 1IntroductionMicro-electro
Page 11 and 12: 10performance, f(x, 0), is equal to
Page 13 and 14: 12function f(x), there are several
Page 15 and 16: 14Chapter 3Optimization AlgorithmIn
Page 17 and 18: 16paribus. This can be very effecti
Page 19 and 20: 18to the active linear constraints
Page 21 and 22: 20can all the constraints be satisf
Page 23 and 24: 22Chapter 4Optimization ExamplesWe
Page 25 and 26: 2475007300220200t 1t2t 37100180Cost
Page 27: 26design variables, other than the
Page 31 and 32: 30where ¯f is the w 2 design , Σ
Page 33 and 34: 32design along the line c 2 (x) = 0
Page 35 and 36: 34550500450(5)global solutionPath 1
Page 37 and 38: 36Figure 5.6: Diagram of Six Variab
Page 39 and 40: 38problem is well-defined. Before a
Page 41 and 42: 405.2.3 Robust DesignWe will use th
Page 43 and 44: 42evident by noting that the c 2 (x
Page 45 and 46: 44Robust Uncorrelated DesignAverage
Page 47 and 48: 46Chapter 6ConclusionIn this report
Page 49 and 50: 48Bibliography[1] What is MEMS Tech
Page 51: 50Appendix AAlkylation Constantsi c

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