1) You have been asked to select the best material for the redesign of a dog’s throw toy. It is basically a long pole of fixed length, L, and unknown cross section, A, that is used to pick up a tennis ball and throw it (without getting dog saliva on the user’s hands). In order to throw the ball a long distance, the pole must store as much elastic energy as possible. It must be able to pick up the ball, under an end load, without buckling. In addition, the manufacturer has a limit on the material costs, so that the pole must cost less than $_{o} [in US$]. Use this information, the equations below, your grey matter, and all your pet care skills to answer the questions that follow.

1) (10 points) What is the measure of performance, P, for this design?
2) (5 points) Is this design UNDERCONSTRAINED, FULLY DETERMINED, OR OVERCONSTRAINED, and why?
3) (25 points) Derive the first performance index.
4) (25 points) Derive the second performance index.
5) (25 points) Derive the coupling equation that links them.
6) (5 points) Can we get an increase in performance by using a complex cross sectional shape for this design? EXPLAIN.
7) (5 points) What animal makes the best pet? Why?1) (10 points) What is the measure of performance, P, for this design?
The measure of performance is givenin the problem statement as the feature that must be maximized or minimized in the design. In this case, it is stated that "...the pole must store as much elastic energy as possible..." Therefore, the MOP = P = U = stored elastic energy.
2) (5 points) Is this design UNDERCONSTRAINED, FULLY DETERMINED, OR OVERCONSTRAINED, and why?
This is an OVERCONSTRAINED problem. We have one free parameter (A) and two constraints (no buckling failure, cost of $_{o}). Since there are more constraints than free parameters, it is an overconstrained problem.
3) (25 points) Derive the first performance index.
I'll start with the buckling constraint:
4) (25 points) Derive the second performance index.
Now for the cost constraint:
5) (25 points) Derive the coupling equation that links them.
Setting the two equations for performance equal and finding the ratio of M's gives the following coupling equation:
6) (5 points) Can we get an increase in performance by using a complex cross sectional shape for this design? EXPLAIN.
Using a complex cross sectional shape will increase the shape factor, PHI, for elastic bending. In this case, the shape factor appears only in the equation for M_{1}, where it appears in the denominator. Thus, increasing the shape factor will actually DECREASE our performance! The answer is NO.
WARNING! ONLY READ THE REST OF THIS EXPLANATION IF YOU WANT TO KNOW WHAT IS GOING ON WITH THIS ANSWER. IT DOES NOT CHANGE THE CORRECT ANSWER GIVEN ABOVE, BUT ONLY TRIES TO EXPLAIN THE RESULT.
This seemingly bizarre result comes about because of the model I chose to use for the analysis. The elastic energy equation I chose to use is based on loading the throw toy up to the yield stress (SIGMAf appears in the equation). With this definition of elastic strength, the model can make a larger stored elastic energy by increasing the volume of the beam. Since the length, L, is fixed, the only way to do this with geometry is to increase A. Using a complex shape makes the cross section more efficient, thereby lowering A and lowering the stored elastic energy.
Could we have done a better analysis? Yep...can you see how to do it?
7) (5 points) What animal makes the best pet? Why?
The results for the class (both 480 and 580) broke down as follows:
Animal 
Number of Answers 
Percentage of Class 
Dog 
42 
54% 
Cat 
16 
21% 
Fish 
3 
4% 
Other 
17 
21% 
Animal entries in the "Other" category included llamas, miniature pigs, monkeys, turtles, dolphins, and "someone else's pet". Human entries included Graduate students and little sisters. Nonanimal entries included plants, rocks, and (naturally) DIAMOND. Fantasy entries included killer african bees, giant eagles, rhinos, and the liger. No one got the correct answer, which is "leopards".