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Euler's Equations, a freely rotating lamina

  1. Mar 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a lamina rotating freely (no torques) about a point O of the lamina. Use Euler's equations to show that the component of [tex]\omega[/tex] in the plane of the lamina has constant magnitude.

    [Hint: Use the reults of Problems 10.23 and 10.30. According to Problem 10.30, if you choose the direction e3 normal to the plane of the lamina, e3 points along a principal axis. Then what you have to prove is that the time derivative of [tex]\omega[/tex][tex]_{1}[/tex][tex]^{2}[/tex] + [tex]\omega[/tex][tex]_{2}[/tex][tex]^{2}[/tex] is zero.]


    2. Relevant equations
    The result of 10.23 is that Izz = Ixx + Iyy
    The result of 10.30 is that for a lamina rotating about a point O in the body, the axis through O and perpendicular to the plane is a principal axis.

    Euler's equations, as given in my book, are
    y1W1 - (y2 - y1)w2w3 = N1
    y2W2 - (y3 - y1)w3w1 = N2
    y3W3 - (y1 - y2)w1w2 = N3
    where y is lambda, the eigenvalue, w is omega, W is omega dot, and N is the torque.

    3. The attempt at a solution
    If the lamina is rotating freely, the N1=N2=N3=0.
    If I choose e3 to be normal to the plane of the lamina, then e3 points along a principal axis, and that means that

    w3 = constant,
    so
    W3 = 0.

    Those last two statements I'm not sure about.

    However, I still know that I would like to prove that the time derivative of w1^2 + w2^2 = 0. That is,

    2w1W1 + 2w2W2 = 0

    And solving Euler's equations for W1 and W2, with N1,2,3 = 0, you have

    W1 = ((y2-y3)/y1)w2w3
    W2 = ((y3-y1)/y2)w3w1

    Plugging these in to 2w1W1 + 2w2W2 = 0 and doing algebra, I found that, if 2w1W1 + 2w2W2 = 0 is true, then

    y2^2 - y2y3 + y1y3 - y1^2 = 0.

    Which doesn't seem to necessarily be true.

    Help? Thank you!
     
  2. jcsd
  3. Apr 27, 2012 #2
    I don't believe that you can assume that w3 is constant. The problem does not state that the lamina is rotating about a principal axis. Instead, use the result you quoted ([itex]\lambda_{3}[/itex]=[itex]\lambda_{1}[/itex]+[itex]\lambda_{2}[/itex]) and use Euler's equations to show that

    [itex]\omega_{1}[/itex][itex]\omega_{1}^{.}[/itex]+[itex]\omega_{2}[/itex][itex]\omega_{2}^{.}[/itex]=0

    Also, if the lamina WERE rotating about its principal axis that is perpendicular to the surface then the angular velocity would always point along this direction in the absence of torques and[itex]\omega_{1}[/itex] and [itex]\omega_{2}[/itex] would be zero.
     
    Last edited: Apr 27, 2012
  4. Apr 27, 2012 #3
    Oops. Those dots were supposed to be above omega one and omega two. I'm sure you get the idea though.
     
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