Q$4(20$ Points$)$ Design of a Lightweight Support Column for Solar Panels

A concept for a lightweight support column is being considered for mounting a solar panel. The column must support a mass m of $200$ kg $,$ which represents the combined weight of the solar panel and mounting hardware. The mass is applied at a height H from the base, and the column must resist elastic buckling. The column is to be constructed using stock tubing and must be as lightweight as possible.

a$.$ Complete the following for deriving an index that allows you to select the material and shape for the support column:

Function:

Constraints:

Objective:

Free Variables:

Fall $2024$

EXAM $1$

ME $4170/5170$

b$.$ Derive a merit index for selecting the optimal shape and material that minimizes weight while meeting the constraints.

c$.$ Using the following material properties, evaluate the merit index for each material and recommend a suitable choice for the support column.

$\backslash$table$[[$Material$,$Modulus $E(GPa),$Density $(k\frac{g}{m^3}),$Radius $($mm$),$Thickness$/$Radius$],[$Aluminum$,69,2700,30,0\mathrm{.}08],[$Steel$,210,7900,25,0\mathrm{.}05],[$Titanium$,110,4500,20,0\mathrm{.}07],[$Polycarbonate$,3,1200,40,0\mathrm{.}12],[$Treated Bamboo,$15,700,50,$Solid Section$]]$