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ASE 365 - HW 6 - Homework 6due 5:00 p.m Friday March 12...

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Homework 6 —due 5:00 p.m. Friday, March 12 These problems are concerned with the motion of a mass m suspended by three springs of stiffness k in a moving frame: 4 k k m k ¯ y ( t ) ¯ x ( t ) y ( t ) 3 x ( t ) The suspended mass has displacements x ( t ) and y ( t ) relative to an inertially fixed reference frame, and the outer supporting frame has displacements ¯ x ( t ) and ¯ y ( t ) relative to the inertially fixed reference frame. One spring is horizontal, one is vertical, and the third is inclined so that it has a slope of 4/3, as shown. Ignore gravity for this entire homework assignment. 1. Derive equations of motion that govern x ( t ) and y ( t ) . 2. Find natural frequencies and natural modes of free vibration for the mass m . Show each mode visually and describe it in words. 3. Transform the equations of motion using the modes, so that they are decoupled. 4. Find the free response to these initial conditions: (a) x (0) = x 0 , y (0) = ˙ x (0) = ˙ y (0) = 0 . Plot this response in the x - y plane over the first two periods of the lower-frequency mode. (This will be a parametric plot, with
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