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

37# Constraint # Constraint1. h 1 ≥ h min 6. L 2 ≥ 10h 22. h 2 ≥ h min 7. X ≥ 2L 2 + 2h 1 + b m3. b m ≥ h min 8. Y ≥ 2L 1 + h m4. h m ≥ 2h 2 + h min 9. k y ≥ αk x5. L 1 ≥ 10h 1 10. β max ≥ βTable 5.2: Design constraints for crab-leg resonator.enforce that the stress, β, at the joint (where the two beams join to form a leg) doesnot exceed a maximum. The nonlinear expression for β is given byβ =12Eh 3 1h 3 2L 1 Dh 2 2L 2 1(4h 3 1L 2 + h 3 2L 1 )where D is the maximum allowable deflection of the structure in the x-direction.Once again, through simple algebraic manipulation the constraints can be written aspolynomials in the form g i (x) ≤ 0. Table (5.3) contains the values of constants usedfor this example. All of the design parameters were chosen to be realistic.Design Parameter - DescriptionValuet - thickness of proof mass and beams 2.0 µmρ - density of silicon (2330 kg/m 3 ) 2.3 × 10 −12 gm/µm 3E - Elastic Modulus of silicon (160 GPa) 1.6 × 10 8 gm/µms 2h min - minimum dimension 2.0 µmD - maximum deflection in the x-direction 2.0 µmβ max - maximum allowable stress (1.6 GPa) 1.6 × 10 6 gm/µms 2X - maximum size of structure in the x-direction 600µmY - maximum size of structure in the y-direction 600µmα - minimum stiffness ratio k y /k x 16Table 5.3: Design parameters for crab-leg resonator.5.2.1 Determining the set of feasible resonant frequenciesCurrently we have a rational polynomial that describes the resonant frequency ofthe crab-leg and a set of constraints. There is no notion of uncertainty yet, but the