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hwk03_soln

# hwk03_soln - ME 563 Fall 2011 Homework Prob 3.1 SOLUTION...

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kx c x kR θ kR θ f f ME 563 – Fall 2011 SOLUTION Homework Prob. 3.1 For the two-DOF system shown below, the springs are unstretched when x = θ = 0 . a) Use the Newton-Euler equations to develop the EOMs for this system using the generalized coordinates of x and θ . b) Use Lagrange’s equations to develop the EOMs for this system using the generalized coordinates of x and θ . 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  θ Wheel: (1) F x = f kR θ = ma Gx f = kR θ m  x + R  θ ( ) (2) M G = fR = I G  θ = 1 2 mR 2  θ f = 1 2 mR  θ L-shaped block: (3) F x = f + kR θ kx c x = m  x f = m  x + c x + kx kR θ Combining (2) and (3): m  x 1 2 mR  θ + c x + kx kR θ = 0 Combining (1) and (2): m  x + 3 2 mR  θ + kR θ = 0 In matrix form: k x smooth no slip R θ k G A c

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m mR 2 m 3 mR 2  x  θ + c 0 0 0 x θ + k kR 0 kR x θ = 0 0 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.
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hwk03_soln - ME 563 Fall 2011 Homework Prob 3.1 SOLUTION...

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