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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,
    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


    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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