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Unformatted text preview: kx c x kR kR f f ME 563 Fall 2011 SOLUTION Homework Prob. 3.1 For the two-DOF system shown below, the springs are unstretched when x = = . a) Use the Newton-Euler equations to develop the EOMs for this system using the generalized coordinates of x and . b) Use Lagranges equations to develop the EOMs for this system using the generalized coordinates of x and . c) Compare the mass, damping and stiffness matrices found in a) and b) above. Are they the same? Which formulation, if either, produces symmetric matrices? Newton-Euler Kinematics: a Gx = x + R Wheel: (1) F x = f kR = ma Gx f = kR m x + R ( ) (2) M G = fR = I G = 1 2 mR 2 f = 1 2 mR L-shaped block: (3) F x = f + kR kx c x = m x f = m x + c x + kx kR Combining (2) and (3): m x 1 2 mR + c x + kx kR = Combining (1) and (2): m x + 3 2 mR + kR = In matrix form: k x smooth no slip R k G A c m mR 2 m 3 mR 2 x + c x + k kR kR x = The mass and stiffness matrices are not symmetric, whereas the damping matrix is symmetric. NOTE : Depending on how you eliminated the friction constraint force f, you can get a set of EOMs with different symmetry properties than those found above. can get a set of EOMs with different symmetry properties than those found above....
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This document was uploaded on 12/23/2011.
- Fall '09