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Equations of Motions of a Wheel Axle Set

  1. Sep 15, 2014 #1
    Hello all,

    I am currently studying dynamics of a wheel-axle set for my research. My problem is I could not find the same equation for the rate of the change of the momentum in the book, book is a little bit old and I could not find any errata about the book or any other references that explains the derivation of equations. Thank you in advance for your help.

    I am trying to obtain the general wheel axle set equations of motion given in the 5th chapter of the book (all the equations and figures are taken from this book):

    http://books.google.cz/books?id=TVenrrNeB4kC&printsec=frontcover&hl=tr&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

    I am giving the axes systems in the book used:

    https://imagizer.imageshack.us/v2/965x464q90/661/dLDE2P.png [Broken]

    The first axes is used as fixed inertial reference frame. The second one is an intermediate frame rotated through an angle [itex]\psi[/itex] about the z axis of the third axes system (which is attached to the mass center of wheelset) Transformation equations between coordinate axes given in the book:

    [itex]
    \begin{Bmatrix}
    i^{'}\\j^{'} \\ k^{'}
    \end{Bmatrix}=\begin{bmatrix}
    1 &0 &0 \\
    0 &cos\phi &sin\phi \\
    0 &-sin\phi & cos\phi
    \end{bmatrix}\begin{Bmatrix}
    i^{''}\\j^{''} \\ k^{''}
    \end{Bmatrix}
    [/itex]

    [itex]
    \begin{Bmatrix}
    i^{''}\\j^{''} \\ k^{''}
    \end{Bmatrix}=\begin{bmatrix}
    cos\psi &sin\psi &0 \\
    -sin\psi &cos\psi &0 \\
    0 &0 & 1
    \end{bmatrix}\begin{Bmatrix}
    i^{'''}\\j^{'''} \\ k^{'''}
    \end{Bmatrix}
    [/itex]

    [itex]
    \begin{Bmatrix}
    i^{'}\\j^{'} \\ k^{'}
    \end{Bmatrix}=\begin{bmatrix}
    cos\psi &sin\psi &0 \\
    -cos\phi sin\psi &cos\phi cos\psi &0 \\
    sin\phi sin\psi &-sin\phi cos\psi & 1
    \end{bmatrix}\begin{Bmatrix}
    i^{'''}\\j^{'''} \\ k^{'''}
    \end{Bmatrix}
    [/itex]



    for small [itex]\psi[/itex] and [itex]\phi[/itex]


    [itex]
    \begin{Bmatrix}
    i^{'}\\j^{'} \\ k^{'}
    \end{Bmatrix}=\begin{bmatrix}
    1 &\psi &0 \\
    -\psi &1 &0 \\
    0 &-phi & 1
    \end{bmatrix}\begin{Bmatrix}
    i^{'''}\\j^{'''} \\ k^{'''}
    \end{Bmatrix}
    [/itex]




    https://imagizer.imageshack.us/v2/773x270q90/661/B4L8It.png [Broken]​

    The angular velocity [itex]\mathbf{\omega}[/itex] of the axle wheelset is given by:

    [itex]\mathbf{\omega}=\dot{\phi }i^{''}+\left ( \Omega +\dot{\beta } \right )j^{'}+\dot{\psi }k^{''}[/itex]​

    The angular velocity [itex]\mathbf{\omega}[/itex] expressed in body coordinate axis is given by:

    [itex]\mathbf{\omega}=\dot{\phi }i^{'}+\left ( \Omega +\dot{\beta }+\dot{\psi }sin\phi \right )j^{'}+\dot{\psi }cos\phi k^{'}[/itex]

    [itex]\mathbf{\omega}=\omega_{x}i^{'}+\omega_{y}j^{'}+\omega_{z}k^{'}[/itex]​

    where [itex]\omega_{x}=\dot{\phi }, \omega_{y}=\left ( \Omega +\dot{\beta }+\dot{\psi }sin\phi \right ), \omega_{z}=\dot{\psi }cos\phi[/itex] and the angular momentum of the wheel axle set in the body coordinate system

    [itex]\mathbf{H}=I_{wx}\omega_{x}i^{'}+I_{wy}\omega_{y}j^{'}+I_{wz}\omega_{z}k^{'}[/itex]

    please note that because of symmetry(principal mass moments) [itex]I_{wx}=I_{wz}[/itex].

    Angular velocity of coordinate axes

    ω_axis×H=(ψ ̇sinφI_wx ψ ̇cosφi^'-ψ ̇cosφI_wy (Ω+β ̇+ψ ̇sinφ) i^' )+(φ ̇I_wy (Ω+β ̇+ψ ̇sinφ) k^'-ψ ̇sinφI_wx φ ̇k^' )

    [itex]\mathbf{\omega_{axis}}=\dot{\phi }i^{'}+\dot{\psi }k^{''}=\dot{\phi }i^{'}+\dot{\psi }sin\phi j^{'}+\dot{\psi }cos\phi k^{'}[/itex]

    The rate of change of momentum is given as

    [itex]\mathbf{dH/dt}=I_{wx}\dot{\omega_{x}}i^{'}+I_{wy}\dot{\omega_{y}}j^{'}+I_{wz}\dot{\omega_{z}}k^{'}+\mathbf{\omega_{axis}}\times\mathbf{H} [/itex]

    This point is where I can not get the same equation in the book for rate of change of momentum. The rate of change of momentum given in fixed intertial frame is:

    [itex]\mathbf{dH/dt}=\left (I_{wx}\ddot \phi- I_{wy}\Omega \dot\psi \right )i^{'''}+I_{wy}\ddot \beta j^{'''}+\left (I_{wy}\Omega\dot \phi+ I_{wx}\ddot\psi \right ) k^{'''} [/itex]

    Probably I am missing a simple point but I could not find what it is.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Sep 16, 2014 #2
    I am sorry, please do not consider this part:

    ω_axis×H=(ψ ̇sinφI_wx ψ ̇cosφi^'-ψ ̇cosφI_wy (Ω+β ̇+ψ ̇sinφ) i^' )+(φ ̇I_wy (Ω+β ̇+ψ ̇sinφ) k^'-ψ ̇sinφI_wx φ ̇k^' )

    Probably I wrote (copy and paste from my notes) it by mistake.
     
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