$(2$ pts$)$ This problem considers the design space and objective space for the cantilever beam spring provided below. You are trying to target a deflection of $2$ mm at the beam tip and minimize stress, $,$ at the location shown in the figures below. Given the information provided in the figures, $...$

a$)$ Determine the width $(b)$ and thickness $(h)$ required $($i$.$e$.$ your design space coordinates$)$ for each of the two Pareto optimal solutions shown in the objective space.

b$)$ One design places all the emphasis on stress minimization, the other places all the emphasis on hitting the target deflection. Which one is which?

$=\frac{4P{L}^3}{Eb{h}^3}$

$=\frac{6PL}{b{h}^2}$

$\backslash$table$[[$Parameter$,$Value,Units $($SI$)],[E,207{10}^9,$Pa$],[P,10,$N$],[L,60{10}^{-3},$m$],[b,10{10}^{-3}$ to $16{10}^{-3},$m$],[h,1{10}^{-3}$ to $1\mathrm{.}5{10}^{-3},$m$]]$

Hints:

Watch your units. I recommend doing everything in SI units $($i$.$e$.,$ in this case use $m,Pa,N)$

Design A is deflecting less than the target value $($i$.$e$.$ actual tip deflection is $2-1\mathrm{.}227=0\mathrm{.}773mm)$

With some algebra, you can show

$h=\frac{2L^2}{3E}$