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Rigid body orientation using Euler angles confusion

  1. Mar 15, 2017 #1

    Tar

    User Avatar

    Hello,

    1. The problem statement, all variables and given/known data
    I'm given the following exercise:

    "A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation?
    α is defined as the angle between the rod and its projection on the XY plane.
    β is defined as the angle between the X-axis and the rod's projection on the XY plane.
    (Note: We are using ZXZ Euler angles convention, having the following order of angles: ϕ,θ,ψ.)"
    1iPlX.png

    The answer is: α=θ+0.5π, β=ϕ
    I don't understand why.

    3. The attempt at a solution
    As far as I know, usually the body points in the e3 direction in the body frame. Initially the body frame points towards the original Z direction, after rotation of ϕ around the Z axis, and then rotation of θ around the x′ axis, we will receive the following:
    0yPi6.png
    Note that the new z′ axis doesn't point to where it should be (according to the answer) and ϕ doesn't represent the angle between z' axis' projection on the XY plane and X-axis.

    What am I missing?

    Thank you.
     
  2. jcsd
  3. Mar 15, 2017 #2

    kuruman

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    Homework Helper
    Gold Member

    This is not clear from the statement of the problem, but let's assume that it is true. If e3 is the Z direction, a rotation about that axis by any angle will not change the orientation of the rod, but will change the orientation of the x and y axes to x' and y'. Now to get the rod below the x'y' plane, you will need angle θ greater than π/2. The solution says that's not the case. So either your assumption about the original orientation of the rod is incorrect or the solution is incorrect.

    On edit: Welcome to PF.
    :welcome:
     
  4. Mar 15, 2017 #3

    Tar

    User Avatar

    You're right. the lecturer have verified again (for the 3rd time) and there's a mistake that he hadn't noticed on his first verification.

    I think this thread should be deleted as it doesn't help anyone,

    Thanks for your help.
     
    Last edited: Mar 15, 2017
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