Homework #3
Assigned: Jan. 24th, 2010
Due: Feb. 2nd, 2010
Points: 30

1. 10 points
   Consider a tensegrity structure consisting of three compression members and nine tension members. See an excerpt from a children's magazine, the Young Voices of IISc . A tensegrity structure keeps its shape because it has the lowest strain energy in that shape. Pose it as a minimization problem and solve it to find the shape of the tensegrity structure that has the following as connected in the figures shown in the article of the Young Voices.
   
   • Lengths of the compression members = 50 mm
   • Stiffness of the tension members = 10 N/mm
   • The free length of each of the tension members is 30 mm.
   Since the strain energy stored by the tension members is much more than that in the compression members, consider only the strain energy stored in the tension members. You may change the values if you do not see a sufficiently tall (but not too tall) shape for the equilibrium tensegrity shape.
   You will need to use inequality constraints here to ensure that the tension members do not become slack. Can you interpret the physical meaning of the Lagrange multipliers corresponding to these inequality constraints in this problem?
2. 10 points
   We want to understand the difference between finite variable calculus and calculus of variations. First, we need to understand the need for calculus of variations. So, let us begin, as they say, at the very beginning which is a good place to start. Let us go back to a problem given in Class 5 of the National Council for Educational Research and Training (NCERT) of India. See the problem as described on page 157 (and page 12 in the pdf file) . Do this easy problem of the rectangle.

   Now, assume that the king asked Cheggu, the carpenter, to take a triangular, quadrilateral, pentagonal, etc., shaped land of given perimeter. In each case, find the optimal value of the area and plot that against the number of sides. Write a general Matlab program to solve it. Go up to as many sides as possible. You know the circle is the solution just as Cheggu's clever wife knew. Plot that area and see how close you get with your polygonal solutions. By the way, this is known as Queen Dido's problem. And it is the classic isoperimetric problem--same perimeter problem. We will see more of it as we go along.

3. 10 points
   The best way to demonstrate that you understood something well is to create a problem on your own. So, to show that you have understood what calculus of variations means, formulate a new problem of calculus of variations on your own. It can be about anything. Just make sure that you did it on your own. Just for this problem, you cannot consult anyone. Note that it is not difficult to find out if your problem is new or not. It may not be the greatest problem; just try to pose a new problem-- your problem.