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t_dawolf
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Homework Statement
The center of mass (C) of the circular rolling disc is offset from its centroid (O) by an eccentric distance [tex]OC = \epsilon[/tex]. The radius of the wheel is R. The disc is rolling without slipping at a constant rotational speed [tex]\omega[/tex] by a variable torque T. Solve for this torque T by:
(a) Newton-Euler equations, taking moments about C
(b) N-E eqns, taking moments about P
(c) Lagrange's equations
(please note in the diagram Th = [tex]\theta[/tex] and ep = [tex]\epsilon[/tex])
Homework Equations
[tex] \phi = (\theta + \frac{\pi}{2}) [/tex]
[tex]r_{C} = -\epsilon cos(\phi) i + (R - \epsilon sin(\phi))j[/tex]
[tex]v_{C} = \epsilon \omega sin(\phi)i - \epsilon \omega cos(\phi))[/tex]
[tex]a_{C} = \epsilon \omega^{2} cos(\phi)i + \epsilon \omega^{2} sin(\phi)[/tex]
[tex][F_x = 0 ; F_y = mg ; T] = Diag([m m I_c])[\ddot{x} ; \ddot{y} ; \alpha = 0] + [ 1 0 ; 0 1 ; -\epsilon cos(\phi) -\epsilon sin(\phi)][R_x ; R_y][/tex]
[tex]T = \epsilon m (w^2(cos^2(\phi)i + sin^2(\phi)j) - g sin(\phi)j)[/tex]
The Attempt at a Solution
First I'm just trying to solve for (a), and I'm not sure that I've set up the Reactions coefficients matrix correctly for the Torque equation. I can imagine that what I've derived for the torque is correct, but I'm still wrapping my head around the concepts and processes and would really appreciate some guidance regarding my equation setup and approach.
Thanks!
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