Equations of Motions of a Wheel Axle Set

[altanonat]

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

The first axes is used as fixed inertial reference frame. The second one is an intermediate frame rotated through an angle \psi 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:

<br /> \begin{Bmatrix}<br /> i^{&#039;}\\j^{&#039;} \\ k^{&#039;} <br /> \end{Bmatrix}=\begin{bmatrix}<br /> 1 &amp;0 &amp;0 \\ <br /> 0 &amp;cos\phi &amp;sin\phi \\ <br /> 0 &amp;-sin\phi &amp; cos\phi<br /> \end{bmatrix}\begin{Bmatrix}<br /> i^{&#039;&#039;}\\j^{&#039;&#039;} \\ k^{&#039;&#039;} <br /> \end{Bmatrix}<br />

<br /> \begin{Bmatrix}<br /> i^{&#039;&#039;}\\j^{&#039;&#039;} \\ k^{&#039;&#039;} <br /> \end{Bmatrix}=\begin{bmatrix}<br /> cos\psi &amp;sin\psi &amp;0 \\ <br /> -sin\psi &amp;cos\psi &amp;0 \\ <br /> 0 &amp;0 &amp; 1<br /> \end{bmatrix}\begin{Bmatrix}<br /> i^{&#039;&#039;&#039;}\\j^{&#039;&#039;&#039;} \\ k^{&#039;&#039;&#039;} <br /> \end{Bmatrix}<br />

<br /> \begin{Bmatrix}<br /> i^{&#039;}\\j^{&#039;} \\ k^{&#039;} <br /> \end{Bmatrix}=\begin{bmatrix}<br /> cos\psi &amp;sin\psi &amp;0 \\ <br /> -cos\phi sin\psi &amp;cos\phi cos\psi &amp;0 \\ <br /> sin\phi sin\psi &amp;-sin\phi cos\psi &amp; 1<br /> \end{bmatrix}\begin{Bmatrix}<br /> i^{&#039;&#039;&#039;}\\j^{&#039;&#039;&#039;} \\ k^{&#039;&#039;&#039;} <br /> \end{Bmatrix}<br />


​

for small \psi and \phi

<br /> \begin{Bmatrix}<br /> i^{&#039;}\\j^{&#039;} \\ k^{&#039;} <br /> \end{Bmatrix}=\begin{bmatrix}<br /> 1 &amp;\psi &amp;0 \\ <br /> -\psi &amp;1 &amp;0 \\ <br /> 0 &amp;-phi &amp; 1<br /> \end{bmatrix}\begin{Bmatrix}<br /> i^{&#039;&#039;&#039;}\\j^{&#039;&#039;&#039;} \\ k^{&#039;&#039;&#039;} <br /> \end{Bmatrix}<br />




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

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

\mathbf{\omega}=\dot{\phi }i^{&#039;&#039;}+\left ( \Omega +\dot{\beta } \right )j^{&#039;}+\dot{\psi }k^{&#039;&#039;}​

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

\mathbf{\omega}=\dot{\phi }i^{&#039;}+\left ( \Omega +\dot{\beta }+\dot{\psi }sin\phi \right )j^{&#039;}+\dot{\psi }cos\phi k^{&#039;}

\mathbf{\omega}=\omega_{x}i^{&#039;}+\omega_{y}j^{&#039;}+\omega_{z}k^{&#039;}​

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

\mathbf{H}=I_{wx}\omega_{x}i^{&#039;}+I_{wy}\omega_{y}j^{&#039;}+I_{wz}\omega_{z}k^{&#039;}
​

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

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^' )

\mathbf{\omega_{axis}}=\dot{\phi }i^{&#039;}+\dot{\psi }k^{&#039;&#039;}=\dot{\phi }i^{&#039;}+\dot{\psi }sin\phi j^{&#039;}+\dot{\psi }cos\phi k^{&#039;}
​

The rate of change of momentum is given as

\mathbf{dH/dt}=I_{wx}\dot{\omega_{x}}i^{&#039;}+I_{wy}\dot{\omega_{y}}j^{&#039;}+I_{wz}\dot{\omega_{z}}k^{&#039;}+\mathbf{\omega_{axis}}\times\mathbf{H}
​

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:

\mathbf{dH/dt}=\left (I_{wx}\ddot \phi- I_{wy}\Omega \dot\psi \right )i^{&#039;&#039;&#039;}+I_{wy}\ddot \beta j^{&#039;&#039;&#039;}+\left (I_{wy}\Omega\dot \phi+ I_{wx}\ddot\psi \right ) k^{&#039;&#039;&#039;}
​

Probably I am missing a simple point but I could not find what it is.

 

[altanonat]

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.