Robust Optimization: Design in MEMS - University of California ...
Robust Optimization: Design in MEMS - University of California ...
Robust Optimization: Design in MEMS - University of California ...
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355.2 Crab-Leg ResonatorThe second example that we will discuss is a six variable crab-leg resonator. Thisproblem was orig<strong>in</strong>ally taken from [13] and has been modified and extended. Thegoal is identical to that for the problem presented <strong>in</strong> section 5.1. We would like torobustly design the resonant frequency <strong>in</strong> the presence <strong>of</strong> geometric process variations.The design for the two DOF resonator was rather <strong>in</strong>tuitive, however this problem issufficiently complex that numerical optimization is necessary to f<strong>in</strong>d a robust design.There is significant freedom <strong>in</strong> how the design variables can be chosen and we willshow that the optimized solution is superior to the average design.The crab-leg resonator is shown <strong>in</strong> figure (5.6). The structure exhibits four-foldsymmetry and thus there are six design variables <strong>of</strong> <strong>in</strong>terest. The design variables h mand b m are the height and width <strong>of</strong> the pro<strong>of</strong> mass, respectively. The legs that supportthe pro<strong>of</strong> mass are anchored to the substrate. The design variables for the four legsare h 1 , h 2 , L 1 and L 2 . We are assum<strong>in</strong>g that this resonator will be fabricated us<strong>in</strong>ga surface micro-mach<strong>in</strong><strong>in</strong>g technology and therefore the entire structure will have afixed thickness t. The variables X and Y represent the maximum dimensions <strong>of</strong> theoverall structure. F<strong>in</strong>ally, we will def<strong>in</strong>e k x and k y to be the stiffness <strong>of</strong> the structure<strong>in</strong> the x and y-direction, respectively.We will use the same approximation for natural frequency as <strong>in</strong> the preced<strong>in</strong>gsection, assum<strong>in</strong>g that w n = √ k x /M. Our goal is to design the resonant frequency <strong>in</strong>the x-direction, and thus k x is the stiffness <strong>of</strong> <strong>in</strong>terest. We will also neglect the mass<strong>of</strong> the four legs and therefore the expression for M isM = ρh m b m twhere ρ is the density. The expressions for k x and k y are slightly more complex butare given byk x = 16Etk y = 16Et( ) 3 ( )h 1 h 31 L 2 +h 3 2 L 1L 1 4h 3 1 L 2+h 3 2 L 1( ) 3 ( )h 2 h 31 L 2 +h 3 2 L 1L 2 h 3 1 L 2+4h 3 2 L 1where E is the Young’s Modulus <strong>of</strong> the material. The expressions for k x and k y werederived based on the several assumptions. The derivations can be found <strong>in</strong> [13]. It