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MIT2_017JF09_p15 - 15 BOUNCING ROBOT 15 37 Bouncing Robot...

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15 BOUNCING ROBOT 37 15 Bouncing Robot You are asked to explore the simple dynamics of a bouncing robot device, using simulation. There is a single mass with a very light helical spring attached on the bottom of it; the mass is 40 kg , and the spring constant is 10400 N/m . The spring additionally has a little bit of damping, 35 Ns/m . The spring is NOT attached to the ground. The initial condition has zero vertical velocity and a 15 cm compression of the spring. 1. Formulate the system dynamics with an annotated drawing and a statement of the governing equations. Solution: Let us say that vertical position y = 0 corresponds with the spring in its natural, uncompressed state, and it is just touching the ground. Then when y < 0 , there is a compression force pushing the mass upward at ky , and the associated damping by ˙ . When y > 0 , there is no spring force at all. In all cases, there is a steady gravity force acting. We have then: my ¨ + by ˙ + ky = mg when y < 0 , or my ¨ = mg when y > 0 .
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