# Equations of Motions of a Wheel Axle Set

1. Sep 15, 2014

### 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):

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 $\psi$ 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:

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

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

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

for small $\psi$ and $\phi$

$\begin{Bmatrix} i^{'}\\j^{'} \\ k^{'} \end{Bmatrix}=\begin{bmatrix} 1 &\psi &0 \\ -\psi &1 &0 \\ 0 &-phi & 1 \end{bmatrix}\begin{Bmatrix} i^{'''}\\j^{'''} \\ k^{'''} \end{Bmatrix}$

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

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

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

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

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

$\mathbf{\omega}=\omega_{x}i^{'}+\omega_{y}j^{'}+\omega_{z}k^{'}$​

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

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

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^' )

$\mathbf{\omega_{axis}}=\dot{\phi }i^{'}+\dot{\psi }k^{''}=\dot{\phi }i^{'}+\dot{\psi }sin\phi j^{'}+\dot{\psi }cos\phi k^{'}$

The rate of change of momentum is given as

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

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:

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

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

Last edited by a moderator: May 6, 2017
2. Sep 16, 2014

### 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.