Problem Statement: A bead slides down on a frictionless wire between point A (x=0) and point B (x=1) in a constant gravity field. Find the trajectory that enables the bead to travel from point A to point B in the shortest time for the following three cases:

1. Case I. y(1)=free (no constraint)
2. Case II. y(1)=0.5
3. Case III. Total length traveled = 2
4. Case IV. y(0.5)=0.1 (i.e., pin down the wire at a mid-point)
5. Case V. y'(1)=0.1 (i.e., the slope at the end)
6. Case VI. Speed at x=1 is 2

Solution:

Common sense tells us that additional constratints always make matters worse. In this problem, the first case with no constraint results in the shortest time to travel from x=0 to x=1. Additional constraints can be satisfied, but at the expense of raising the objective function.

• Optimal Control: Bead Sliding Down a Frictionless Wire
  • Snapshot of Mathcad Screen (slide.gif)
  • Mathcad File (slide.mcd)

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Computer Methods in Chemical Engineering -- Dynamic Optimization: Sliding Bead
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Nam Sun Wang Department of Chemical & Biomolecular Engineering University of Maryland College Park, MD 20742-2111 301-405-1910 (voice) 301-314-9126 (FAX) e-mail: nsw@umd.edu ©1996-2006 by Nam Sun Wang