Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational methods and structural optimization Jan.-May, 2008
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
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Homework #4
Assigned: Feb. 4th, 2009
Due: Feb. 10, 2009
Points: 20

  1. 10 points
    Consider a curve in 3D given by the following parameterization.
    x = R cos(theta)
    y = R sin(theta)
    z = (R/4) cos(theta)
    Find the surface that is enclosed by the above curve such that the area of the surface is minimized.
  2. 10 points
    Consider the buckling of columns. The differential equation that determines the Euler buckling load is given by: (EIw'')''+Pcr w'' = 0. What functional, when minimized with respect to w, would give this differential equation. Derive also the boundary conditions. Extra points if you also interpret the terms in the expression of the functional.
  3. Extra 10 points
    Solve the Galileo's brachistochrone problem of Homework #1. Now that we have discussed calculus of variations, see if you can find the time-minimizing circular arc passing through two given points analytically. In other words, obtain an equation in terms of the radius of the circular arc and other given information in the problem. Remember that you solved it numerically at that time without even using the gradient.