Dynamics_Part78

Dynamics_Part78 - Problem Set 10: Problem 1. Problem: The...

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Unformatted text preview: Problem Set 10: Problem 1. Problem: The motion of a small cart of mass m is governed by two springs, a dashpot and an oscillating attachment B, which moves horizontally with xB = b cos t. Ignore effects of rolling friction on the cart's motion. (a) Derive the differential equation governing the cart's motion. HINT: Check that your equation makes sense for the limiting case k1 = k2 and Point B not moving. (b) Determine the natural frequency of the system. (c) Determine the critical damping coefficient for the system. Solution: (a) The forces acting on the cart are as shown in the following figure. Clearly, for motion in the positive x direction, both spring forces (relative to a displacement x) and the dashpot force are directed in the negative x direction. Attachment B's motion adds a force in the positive x direction for positive xB . ................................................................................................. ................................................................................................. ................................................................................................. .......................................... . ... ................................................................................................. ... ................................................................................................. ........................................................................................................... ........................................................................................................... 2 B . ................................................................................................. ................................................................................................. ... ................................................................................................ .. ................................................................................................. ................................................................................................. ................... . ... . .................................................................................................................... . ................................................................................................. . ................................................................................................................ ................................................................................................. .. . . ................................................................................................. .... ................................................................................................. ..... ......... ................................................................................................. ........................................................................................................... ................................................................................................. ........................................................................................................... . ... . ...... . ....... ................................................................................................. ................................................................................................. ................................................................................................. 2 ................................................................................................. ................................................................................................. 1 . . .............. .. . . . . ........... . . cx m x k (x - xo ) k (x - xo ) k x The basic equation of motion for this system is m = -cx - k1 x - k2 (x - xo ) + k2 (xB - xo ) x where xo is the equilibrium displacement of Spring 2. This equation can be rearranged to read m + cx + (k1 + k2 )x = k2 xB x Finally, using the fact that the displacement of attachment B is xB = b cos t, the differential equation governing the cart's motion is m + cx + (k1 + k2 )x = k2 b cos t x In equilibrium with xB = 0 and k1 = k2 = k, the equation of motion would simplify to m + cx + 2kx = 0 x This is physically correct since we expect to have the springs in series for this limiting case. (b) By inspection, the natural frequency of this system is given by 2 n = k1 + k2 m = n = k1 + k2 m ...
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.

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