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Determining the angular and linear accelerations of a Rigid Body

  1. Aug 9, 2009 #1
    Hello, all!

    I am currently working with the governing equation of a rotating, translating rigid body. That is:

    [tex] \vec{a_i} = \vec{a_c} + \vec{\alpha} \times \vec{r} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r}} \right)[/tex]

    Where [tex]a_i[/tex] is the linear acceleration of some point on the body [tex]i[/tex], [tex]a_c[/tex] is the linear acceleration of the centroid of the body, [tex]\alpha[/tex] is the angular acceleration of the body, [tex]\omega[/tex] is the angular velocity of the body, and [tex]r[/tex] is the vector from the centroid to the point [tex]i[/tex].

    Given some [tex]a_1[/tex] and [tex]a_2[/tex], how can I rework a system of two of those equations such that I can back out [tex]\alpha[/tex] (related to omega by the differential) and [tex]a_c[/tex]? That is:
    [tex] \vec{a_1} = \vec{a_c} + \vec{\alpha} \times \vec{r_1} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_1}} \right)[/tex]
    [tex]
    \vec{a_2} = \vec{a_c} + \vec{\alpha} \times \vec{r_2} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_2}} \right)
    [/tex]

    To give another way of viewing the problem: If I mount two accelerometers on a body and then rotate and translate the body in space, how can I determine what those body rotations and translations are based off the readings of my two accelerometers? I should only need two accelerometers (read: two equations) because there are only two unknowns, [tex]a_c[/tex] and [tex]\alpha[/tex].

    Thank you for the help!
     
  2. jcsd
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