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

355.2 Crab-Leg ResonatorThe second example that we will discuss is a six variable crab-leg resonator. Thisproblem was originally taken from [13] and has been modified and extended. Thegoal is identical to that for the problem presented in section 5.1. We would like torobustly design the resonant frequency in the presence of geometric process variations.The design for the two DOF resonator was rather intuitive, however this problem issufficiently complex that numerical optimization is necessary to find a robust design.There is significant freedom in 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 in figure (5.6). The structure exhibits four-foldsymmetry and thus there are six design variables of interest. The design variables h mand b m are the height and width of the proof mass, respectively. The legs that supportthe proof mass are anchored to the substrate. The design variables for the four legsare h 1 , h 2 , L 1 and L 2 . We are assuming that this resonator will be fabricated usinga surface micro-machining technology and therefore the entire structure will have afixed thickness t. The variables X and Y represent the maximum dimensions of theoverall structure. Finally, we will define k x and k y to be the stiffness of the structurein the x and y-direction, respectively.We will use the same approximation for natural frequency as in the precedingsection, assuming that w n = √ k x /M. Our goal is to design the resonant frequency inthe x-direction, and thus k x is the stiffness of interest. We will also neglect the massof 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 of the material. The expressions for k x and k y werederived based on the several assumptions. The derivations can be found in [13]. It