Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational methods and structural optimization Jan.-May, 2010
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
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Homework #1
Assigned: Jan. 5th, 2010
Due: Jan. 12th, 2010
Points: 30

  1. 15 points
    Optimization is about making compromises--good compromises. Let us consider a structural engineering problem to get a feel for it. We have a micromachined out-of-plane accelerometer with a proof-mass and four beams suspending it over a substrate at a height of 2 microns. Each of the four beams connect to the mid-points of the four sides of the square-shaped proof-mass as if it is fixed-guided beam. The proof-mass has a thickness of 4.75 microns and the thickness of the beams is 2.5 microns.
    Let us denote its side of the square of the proof-mass with 's'. The length of each beam is denoted by 'L' and their width with 'w'.
    We need to determine these three (s, L, and w) parameters such that the entire accelerometer fits within an area of 1 mm by 1 mm and satisfies the following requirments:
    (i) it should have a sensitivity of 1E-5 or larger, where the sensitivity is defined as (2)*(amp)*(x)/( (2E-6) - x) with x = (m)*(9.81)*(1E-2)/(k) with m = (s2)*(4.75E-6)*(2300). k is the stiffness of the proof-mass and is given by (4)*(150E9)*(w)*(2.5E-6)3/(L3). Take amp = 1.0 first. (ii) it should have a resonance frequency, sqrt(k/m)/(2)/(pi), should be at least 30 kHz. (iii) L should be at least 10 times more than w and ten times more 2.5E-6 m. Find a solution and try to see if you need to make any compromise between the first two requirements. Try to find a solution to this problem. If you do not find a solution, try to increase 'amp' until you fidn a solution but keep it as low as possible.
  2. 15 points
    If the resonance frequency requirement is relaxed to 5 kHz, what is the highest sensitivity you can achieve? Pose it as an optimization problem in one variable (any one of s, L, and w), two variables (any two of s, L, and w), and all three variables (s, L, and w), and solve all three of them. Use Matlab to do this. There is a routine called 'fmincon' that you can use to do this. In one and two variables cases, show the 'design landscape' graphically and point to the minimim you have found.