Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May., 2017
Instructor: G. K. Ananthasuresh , Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 10:00 AM; Venue: ME MMCR

Homework #1
Assigned: Jan. 5th, 2017
Due: Jan. 12th, 2017
Points: 50
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 20 points
   
This is an exercise for you to learn how to plot surfaces in Matlab and learn about the Hessian concept. Choose any two-variable function that has at least one minimum, one maximum, and one saddle point. Then, color the surface based on the nature of the Hessian. That is, put a green point if the Hessian is positive definite, a blue point if the Hessian is negative definite, and a black point if the Hessian is neither of the first two. Do the same using contour plot also.
2. 30 points
   The mechanical advantage of a compliant mechanism varies with the stiffness of a workpiece it interacts with. A typical relationship is shown in the figures below, graphically and mathematically.
   
   Here, MA_s is the bounding mechanical advantage when the stiffness of the workpiece (i.e., k_ext goes to infinitiy. We want to maximize this. On the other hand, s_k is the sensitivity coefficient. We have to minimize this so that the compliant mechanism's mechanical advantage is not dependent a lot on the workpiece stiffness. The optimization variables you have are three: input-side stiffness (k_ci), output-side stiffness (k_co), and intrinsic geometric amplication factor (n).

   Pose and solve two optimization problems by assuming reasonable values of s^* and MA^*.
   Problem 1: Maximize MA_s subject to (s_k - s^*) is less than or equal to zero.
   Problem 2: Minimize s_k subject to (MA^* - MA_s) is less than or equal to zero.
   Extra points of 30 if you plot the Pareto front for this problem.