Analysis
When designing the length of the support arms, the first step was to measure the frame (Fig A-1) to determine my envelope size for the device when the arms are folded up inside the frame. The triangle shaped frame of 22inch on the top by 25 ½ diagonal and 17inch vertical. The lengths of the arms were designed to fit inside the frame at 12 inches for the top arms and 7 inches for the bottom arm as shown in Fig A-2.
Using the arm lengths of the bike free body diagrams were created of the bike rack as shown in Fig A-3. The straps are there to hold the bike in place and prevent any moments. The top strap would help decrease the force on B caused by the moment around A. The straps were eliminated from the analysis to insure that if the straps had slack or were loose all the full weight would be on the support arms so they were over design to be able to hold the weight of the bike up even if the car was tilted so all the weight was on one are it would still support the weight. Using Fig A-4 the sum of forces in the Y coordinate the equation +Fy=0=FAy-50 lb simplified to FAy =50 lb. The sum of the moment around A was then calculated to +MA=50*cos(18)(13.416)-Fbx*sin(35)*11.93 which simplified to Fbx=93.23.
Using the Fay =50 lb the force perpendicular to the arm as Faup was calculated using the formula 50*cos(10)=49.24 lb as shown on Fig A-4. Next found the max moment of the top arm in Fig A-5 with the formula Mmax=Faup*L so Mmax=49.24*12 = 590.9 lb*in. Next we found the sectional modulus using the formula Sx=Mmax/(σmax), Fig A-6 shows that Mmax=590.9 lb*in and the max stress of 45000 psi as defined by the given 6061 t6 material properties, solving this formula we found the Sx=0.013in. Using the calculated sectional modulus a table was made as seen in Fig A-7 (Below) showing the different hollow square tube size and their sectional modulus and then their calculated factor of safety using the formula FOS=(Sx)/(Required Sx). Using the table from Fig A-7 I selected 1x1x0.125 since it has the smallest sectional modulus available while also being above the required FOS of 6 at 8.76.
