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Simulating earth rotation and (excess) Lenght of Day calculation

  1. Nov 2, 2012 #1
    Hello everybody :smile:,

    I'm new here and hope you can help me with this problem. I have to simulate the earth rotation with eulers equations of motion (without external torques at first).

    I have given:

    Solution of eulers equation without external torques:

    [itex]\omega = (x, y, z)' \left[\frac{rad}{s}\right][/itex] (angle velocity vector of earth)

    with:

    [itex]x = r_{earth} \cdot cos( C \cdot (t-t_0) ) [/itex]
    [itex]y = r_{earth} \cdot sin( C \cdot (t-t_0) )[/itex]
    [itex]z = D~[/itex]

    Meaning:
    • C is a constant dependent on z (is also constant)
    • D is constant in case of no external torques
    • ω is a time dependent vector. I have many ω (e.g. every minute for a whole year). So every minute I have a new ω.
    • ΩN is the nominal earth rotation rate (which I am not sure how to calculate, I have taken 2*PI/(24*60*60)).
    • T is the period of the day in s (24*60*60).

    With every new ω (I think I have to take this ω but I am not sure!?) and formular:

    [itex]\Delta LOD = \frac{(\Omega^N - \omega) \cdot T}{\Omega^N} [/itex]

    I have to calculate the LOD (so every minute a new one).

    My problem is, that ω is a vector and not a scalar to calculate the LOD. In this special case (without torques), LOD should be constant I think.

    Questions
    • I think, without external torques the LOD is constant, right?
    • Can I calculate ΩN or should I use 2∏/(60*60*60) or should I take this constant from the internet?
    • How can I calculate the ω in LOD formular? I tried to calculate the distance (sqrt(x²+y²+z²)), but I think this is the wrong way to solve this.

    Ideas
    • I can look, when ω has turned around 2∏? That would be one day. Then calculate how long it takes for one sec and use this result in the LOD calculation for ω. But how can I realize that with the stuff I've given? Or is this idea stupid?

    Please help me :confused:.

    Best regards!
     
    Last edited: Nov 2, 2012
  2. jcsd
  3. Nov 3, 2012 #2
    Ok, found the mistake after controlling several times all equations. There was an error in my calculation.
     
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