User:Egm3520.s12.Team 5.Shankwitz

Problem Statement

In the image below, there are two steel shafts, ABC and DEF, for which the maximum allowable shear stress is 8500 psi. They are connected by gears at A and D of given radii 4 in. and 2.5 in., respectively. There is a known applied torque at C, T_C of 5 kips•in. and an unknown torque, T_F, applied at F

shafts AC and DF

Determine the minimum diameter of bars BC and EF.

Given

$T_{C}=5\,kips\cdot in.$
$\tau _{max}=8500\,psi.$
$r_{A}=4\,in.$
$r_{D}=2.5\,in.$

Solution

Step One: Draw Free Body Diagrams

Bars ABC and DEF

Gears A and D

Step Two: Statics

From the free body diagrams of bars AC and DF, we can deduce the following:

$\displaystyle T_{A}=T_{C}$

(4.2-1)

$\displaystyle T_{F}=T_{D}$

(4.2-2)

From the definition of torque, $T=F\times r$ we get

$\displaystyle F_{A}={\frac {T_{A}}{r_{A}}}$

(4.2-3)

$\displaystyle F_{D}={\frac {T_{D}}{r_{D}}}$

(4.2-4)

So, setting (4.2-4) equal to (4.2-3) $T_{D}$ can be represented as

$\displaystyle T_{D}={\frac {r_{D}}{r_{A}}}T_{A}$

(4.2-5)