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Mechanics - moment, Euler's second law

  1. Feb 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Hello! :)
    I got stuck on a problem I got for my physics class, where I am supposed to derive the differential equations for a mechanical system (x'' =, θ'' =, and φ'' =) , and then simulate how the system behaves in matlab.

    Here is a picture: http://forumbilder.se/show.aspx?iid=c72201260720P7323

    As you can see, there is a ring, which can roll on the ground. The ring has mass m1, radius R and center of mass G1.
    Inside the ring is a half-circle, that can move frictionlessly inside the ring. It has a mass m2, radius R and center of mass G2.
    At the center of mass G2 a pole is articulated, so it can move freely relative to the half-sphere. It has a uniform mass m3 and length l.

    The hint for the problem is to be careful in selecting the moment points for Euler II, so that no unknown forces will have to be derived using Euler I (F=ma).


    2. Relevant equations

    Euler II:
    M = I*α + rxma

    3. The attempt at a solution

    I have finished the kinematical analysis, and it should be correct.

    First I try to calculate the moment for the pole, around the point G2, using Euler II:

    G2: -m3 * g * l/2 * sin(φ) = I_G3 * φ'' + (-l/2 * sin(φ) x^ + l/2 * cos(φ) y^) x (-m3*g y^)
    <=> φ'' = -(m3 * g * l * sin(φ)) / I_G3

    Unsure from here:
    The problem I have are for the systems half-circle + pipe, and for the whole system. I am also not quite sure how the pipe will affect the half-sphere, so I named the force ξ, to deal with it later. Here is my attempt though, for the half-circle + pipe:

    G1: -(m2*g + ξ)*d*sin(θ) = I_G2 * θ'' + (d*sin(θ) x^ - d*cos(θ) y^) x m*(a_G2x x^ + a_G2y y^)

    Here a_G2x and a_G2y are from the kinematical analysis, the acceleration for G2 in x and y respectively.

    Lastly, I try the same for the whole system, around the moment point M, which I put where the ring is in contact with the ground:

    M: -(m2*g + ξ)*d*sin(θ) = something + (d*sin(θ) x^ + (R-d)*cos(θ) y^) x m*(a_G2x x^ + a_G2y y^)

    So, this is how far I get... I am very grateful for any help that you can give me! :=)
    You don't need to give me the answer, as long as I am sure I have the correct moment equations, I should be able to derive the differential equations.

    Thank you! :)
     
  2. jcsd
  3. Feb 24, 2012 #2
    No one? :(
     
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