Q$3($text $2-1\mathrm{.}27$m$)$

Formulate the problem of minimum weight design of a power screw subjected to an axial load, F $,$ as shown using the pitch $\backslash (($p$)\backslash ),$ major diameter $\backslash (($d$)\backslash ),$ nut height $(\backslash ($ h $\backslash )),$ and screw length $\backslash (($s$)\backslash )$ as design variables. Consider the following constraints in the formulation:

$1.$ The screw should be self$-$locking $[1\mathrm{.}117].$

$2.$ The shear stress in the screw should not exceed the yield strength of the material in shear. Assume the shear strength in shear $($according to distortion energy theory$),$ to be $\backslash (0\mathrm{.}577\backslash$sigma$\_\{$y$\}\backslash ),$ where $\backslash (\backslash$sigma$\_\{$y$\}\backslash )$ is the yield strength of the material.

$3.$ The bearing stress in the threads should not exceed the yield strength of the material, $\backslash (\backslash$sigma$\_\{$y$\}\backslash ).$

$4.$ The critical buckling load of the screw should be more than the applied load, $\backslash ($ F $\backslash ).$

$($a$)$

$($b$)$

Figure $1\mathrm{.}28$ Power screw.

Hint: for power screw relations, see Shigley's Mechanical Engineering Design, $\backslash (10^\{\backslash$text $\{$th $\}\}\backslash )$ ed$.$ Sec $8-2$ Mechanics of power screw, pages $406-411.$ Show your approach.