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Modeling and Analysis of Mechanical Systems All students, please submit a pdf file with your solutions (can be scanned) to c anvas

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Modeling and Analysis of Mechanical Systems

All students, please submit a pdf file with your solutions (can be scanned) to c anvas.

You must show all of your work or you will not receive credit.

Problem 1. For the truss shown below, write down the equations A T y = f in three unknowns y1, y2, y3 to balance the four external forces, f 1 H , f 2 H , f 1 V , f 2 V . Under what conditions on these forces will the equations have a solution (allowing the truss to avoid collapse)?

Problem 2. For the square truss shown below, (a) Write down the 7 by 6 matrix in e = Ax. Define x as x = (x 1 H , x 1 V , x 2 H , x 2 V , x 3 H , x 3 V ). (b) Which type of truss is this? (statically determinate, statically indeterminate, rigid motion, mechanism) (c) What is the rank of A and what are the solutions to Ax = 0? 1 (d) What are the solutions to A T y = 0?

Problem 3. For the truss shown below, which is the truss from Lecture notes 14, example 3, let the forces be f1 = f2 = f4 = f6 = 0, f3 = 1, and f5 = −1. These forces satisfy the conditions for no rigid motion. What are y1, y2, and y3?

Problem 4. For a hanging bar with constant f but weakening elasticity c(x) = 1−x, find the displacement u(x).

Problem 5. Suppose a bar is free at both ends: w(0) = w(1) = 0. This allows rigid motion. What is the condition on the external force f in order that − dw dx = f(x), w(0) = w(1) = 0 has a solution? (Hint: integrate both sides of the equation from 0 to 1).

Problem 6. For a hanging bar with constant c but with decreasing f = 1−x, find u(x). 2