Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational methods and structural optimization Aug.-Dec. 2005
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
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Homework #4
Assigned: Aug. 25th, 2005
Due: Sep. 1st, 2005
Points: 20

  1. 10 points
    In the lectures, the proofs for the extension of the Euler-Lagrange necesary conditions were discussed for the case of global (integral form) constraints as well as local (pointwise) algebraic form of the constraints. Along the same lines, write and prove the necessary conditions when the constraint is a differential equation. For a proof, take the case of two functions (y1(x) and y2(x)) and one constraint of the form g(x,y1,y2,y1', y2').
  2. 5 points
    The equation of an single-sheet hyperboloid is given by
    (x2/a2) + (y2/b2) - (z2/c2) = 1.
    Find the equations using which the geodesic on this surface for any two points (x1, y1, z1) and (x2, y2, z2) could be solved.
  3. 5 points
    A horizontal beam of length L is pinned at (0,0) and is guided at the other end along a straight line y = mx + c. There is a torsional spring of spring constant k at the pin. Write down the governing equations and the boundary conditions if there is a vertical load F at the mid-point, i.e., at (L/2,0), in the downward direction.