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\begin{center}
    {\Large \bf CONTACT OF RACK AND PINION TOOTH FLANKS } \\ [4mm]
    {\em Martin Kratochvil} \\ [2mm]
    University of West Bohemia
\end{center}

\begin{abstract}
{\bf \noindent The paper is devoted to creation of tooth geometry of rack and pinion mechanism using
MATLAB. The aim is to create geometry of pinion tooth based on known geometry of rack tooth.
Geometry of rack tooth flank surface is described by position vector and matrix transformation.
Mutual movement of rack and pinion is determined by matrix transformations. Then, normal vector of
rack tooth flank surface and tangent vector of movement are created. Calculation of contact position is
done using numeric equation solver. The solution is contact angle, which determines position of
contact. Geometry of pinion tooth can be calculated using position vector of rack tooth and contact
angle. }
\end{abstract}
% ----- ZAČÁTEK HLAVNÍ ČÁSTI ------
\vspace{-3mm}
\section {Rack tooth flank surface } % ----- KAPITOLA ------
\vspace{-3mm}

The geometry of~the~rack tooth flank surface is~defined in~the standard CSN~01~4607 \cite{CSN}. % DODELAT
For the purposes of~calculation of the pinion gear tooth flank surface it is~necessary to~create the position vector of~the rack tooth flank. The diameter of the pinion and~the tooth size are determined by~the number of the pinion teeth $z$ and modulus $m$. 
\begin{figure}[!hb]
   \centering
   \epsfysize=68mm
   \epsfxsize=102mm
   \epsffile{hreben_profil.eps}
   \caption{Rack tooth profile }
\label{profile}
\end{figure}
\\
Equation of {\it tooth profile} in~XY plane
\begin{equation} \label{profile}
\tan{\alpha} = \frac{k}{y}
\end{equation}
Equation of {\it slope} in~XZ plane
\begin{equation} \label{slope}
\tan{\beta} = \frac{x}{z}
\end{equation}
% Substituce
For reason of~clarity of~matrices, substitution is used:
\begin{equation} \label{A}
A = \frac{1}{\tan{\alpha}}
\end{equation}
\begin{equation}
B = \tan{\beta} \label{B}
\end{equation}
\begin{equation}
R = \frac{mz}{2} \label{R}
\end{equation}
% Polohovy vektor
The position vector is created using equations (\ref{profile}), (\ref{slope}) and substitution by (\ref{A}), (\ref{B}), (\ref{R})
\begin{equation} \label{profil}
\mathbf{s}_{H} = 
\left[ \begin{array}{cccc}
k + zB \\
kA + R\\
z
\end{array} \right]
\end{equation}
where $k$ is curve parameter and $z$ is position at Z axis. The first value of~the curve parameter $k$ should be:
\begin{equation}
k_0 = m \tan{\alpha}
\end{equation}
% Tečny
The next step is to~determine the normal vector of~the rack tooth flank surface. It~is based on two tangent vectors, the tangent vector of the profile curve (\ref{tp}) and the tangent vector of slope (\ref{ts}).
\begin{equation} \label{tp}
\mathbf{t}_1 = \frac{d\mathbf{s}_{H}}{d\rho} = 
\left[ \begin{array}{cccc}
1 \\
A \\
0
\end{array} \right]
\end{equation}
\begin{equation} \label{ts}
\mathbf{t}_2 =  
\left[ \begin{array}{cccc}
\sin{\beta} \\
0 \\
\cos{\beta}
\end{array} \right]
\end{equation}
% Normála
The normal vector of~the rack tooth surface is~cross product of the two tangent vectors:
\begin{equation} \label{normala}
\mathbf{n}_H = \mathbf{t}_1 \times \mathbf{t}_2 = 
\left[ \begin{array}{cccc}
A \\
-1 \\
0
\end{array} \right]
\times
\left[ \begin{array}{cccc}
\sin{\beta} \\
0 \\
\cos{\beta}
\end{array} \right]
=
\left[ \begin{array}{cccc}
A\cos{\beta} \\
-\cos{\beta} \\
-A\sin{\beta}
\end{array} \right]
\end{equation}
%
\vspace{-3mm}
\section {Transformation matrices } % ----- KAPITOLA ------
\vspace{-3mm}
The transformation matrices are used to translate or rotate a vector. For the purpose of calculation of~the mating profile of the pinion gear, the transformation matrices describes the motion of~the rack.
% Do středu
\begin{figure}[!hb]
   \centering
   \epsfysize=59mm
   \epsfxsize=55mm
   \epsffile{hreben_pohyb.eps}
   \caption{Vector transformations }
\label{transformace}
\end{figure}
\\
The motion of~the rack is~described by three matrix transformations.
The first matrix describes the translation to~origin and~the relative displacement during rolling:
\begin{equation}
\mathbf{T}_{pos} =
\left[ \begin{array}{cccc}
1 & 0 &0 & -R\rho \\
0 & 1 &0 & -R \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\end{equation}
% Natočení
The second matrix describes rotation clockwise by~angle $\rho$:
\begin{equation} 
\label{rot}
\mathbf{T}_{rot} =
\left[ \begin{array}{cccc}
\cos{\rho}& \sin{\rho} &0 & 0 \\
-\sin{\rho} & \cos{\rho} &0 & 0 \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\end{equation}
% Kružnie - valivý bod
The last transformation is translation to involute rolling point:
\begin{equation}
\mathbf{T}_{pol} =
\left[ \begin{array}{cccc}
1 & 0 &0 & R\sin{\rho} \\
0 & 1 &0 & R\cos{\rho} \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\end{equation}
The transformation that describes the motion of~the rack is the dot~product of these three matrices:
\begin{equation} \label{celk}
\mathbf{T}_{celk} = \mathbf{T}_{pos} \cdot \mathbf{T}_{rot} \cdot \mathbf{T}_{pol} =
\left[ \begin{array}{cccc}
\cos{\rho}& \sin{\rho} &0 & -R\rho\cos{\rho} \\
-\sin{\rho} & \cos{\rho} &0 & R\rho\sin{\rho} \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\end{equation}
%
\vspace{-3mm}
\section {Vector transformations } % ----- KAPITOLA ------
\vspace{-3mm}
% Transformace polohového vektoru
The rack rolling on~the pinion is described by~the dot product of matrix (\ref{celk}) and vector (\ref{profil}):
\begin{equation}
\mathbf{s}_{Hb} = \mathbf{T}_{celk} \cdot \mathbf{s}_{H} = 
\left[ \begin{array}{cccc}
\cos{\rho}& \sin{\rho} &0 & -R\rho\cos{\rho} \\
-\sin{\rho} & \cos{\rho} &0 & R\rho\sin{\rho} \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\cdot
\left[ \begin{array}{cccc}
k +zB\\
kA + R\\
z \\
1
\end{array} \right]
\end{equation}
%
\begin{equation} \label{transprofil}
\mathbf{s}_{Hb}
=
\left[ \begin{array}{cccc}
(k+zB)\cos{\rho} + (kA+R)\sin{\rho} - R\rho\cos{\rho} \\
-(k+zB)\sin{\rho} + (kA+R)\cos{\rho}+ R\rho\sin{\rho}\\
z \\
1
\end{array} \right]
\end{equation}
% Transformace normály
The direction of~the normal vector (\ref{normala}) of~the rack tooth flank varies with the rolling of~the rack. Because it~is a directional vector, to preserve direction, only the matrix~of rotation (\ref{rot}) can be~used:
\begin{equation}
\mathbf{n}_{Hb} = \mathbf{T}_{rot} \cdot \mathbf{n}_{H} = 
\left[ \begin{array}{cccc}
\cos{\rho}& \sin{\rho} &0 & 0 \\
-\sin{\rho} & \cos{\rho} &0 & 0 \\
0 & 0 &1 & 0 \\
0 & 0 &0 & 1 \\
\end{array} \right]
\cdot
\left[ \begin{array}{cccc}
A\cos{\beta} \\
-\cos{\beta} \\
-A\sin{\beta} \\
1
\end{array} \right]
\end{equation}
\begin{equation}
\mathbf{n}_{Hb}
=
\left[ \begin{array}{cccc}
(A\cos{\rho} - \sin{\rho})\cos{\beta} \\
(-A\sin{\rho} - \cos{\rho})\cos{\beta} \\
-A\sin{\beta} \\
1
\end{array} \right]
\end{equation}
\vspace{-3mm}
\section{Tangent vector of trajectory} % ----- KAPITOLA ------
\vspace{-3mm}
The transformed position vector of~the rack tooth flank surface $\mathbf{s}_{Hb} $ (\ref{transprofil}) describes the trajectory of every point of~the rack tooth flank surface. To calculate the contact angle, it is necessary to calculate the tangent vector of trajectory:
\begin{equation}
\mathbf{t}_{sHb} = \frac{d\mathbf{s}_{Hb}}{d\rho} = 
\left[ \begin{array}{cccc}
-(k+zB)\sin{\rho} + (kA+R)\cos{\rho} + R(\rho\sin{\rho} - \cos{\rho}) \\
-(k+zB)\cos{\rho} - (kA+R)\sin{\rho} + R(\rho\cos{\rho} + \sin{\rho}) \\
0 \\
1
\end{array} \right]
\end{equation}

\section{Calculation of contact angle} % ----- KAPITOLA ------
The point~of~the rack tooth flank surface is~in~contact, when its~normal vector $\mathbf{n}_{Hb}$ and~tangent vector $\mathbf{t}_{sHb}$ of~trajectory are perpendicular. The condition of~vector perpendicularity is~the zero dot~product of~ these vectors. The equation \label{perp} is~a~nonlinear goniometric equation, it is necessary to~use the numerical {\it fsolve} solver.  
%
\begin{equation} \label{perp}
\mathbf{t}_{sHb} \cdot \mathbf{n}_{Hb} = 0 \Rightarrow \rho_k
\end{equation}
%
\vspace{-3mm}
\section{Geometry of pinion tooth flank}% ----- KAPITOLA ------
\vspace{-3mm}
The surface of~the pinion tooth flank can be calculated using the equation of~rack tooth flank surface~(\ref{transprofil}) and~contact angle $\rho_k$:
\begin{equation}
\mathbf{s}_{Kb} = \mathbf{s}_{Hb}(k, z, \rho_k)
\end{equation}
The figure \ref{flank} below is~an~example of~a~solution created using the presented approach. The number of~teeth $z$ is 24, modulus $m$ 3 mm, angle $\beta$ $15^\circ$.
\begin{figure}[!hb]
   \centering
   \epsfysize=60.8mm
   \epsfxsize=98.3mm
   \epsffile{flank.eps}
   \caption{Calculated pinion tooth flank surface, red arcs - pitch circle, green arcs - base circle }
\label{flank}
\end{figure}
\begin{figure}[!hb]
   \centering
   \epsfysize=59.2mm
   \epsfxsize=97.3mm
   \epsffile{kontakt.eps}
   \caption{Contact of rack (blue) and pinion (black) profile, red arc - pitch circle, green arc - base circle }
\label{flank}
\end{figure}

\vspace{-3mm}
\section{Conclusion}% ----- KAPITOLA ------
\vspace{-3mm}
Creation of geometry based on~this contact theorem is one way to~create the geometry of the mating gear. It is relatively simple, but requires a numerical solver.


% ----- KONEC HLAVNÍ ČÁSTI ------
\vspace{-3mm}
\section*{Acknowledgments}
\vspace{-3mm}

The research work reported here was made possible by project SGS-2016-012.

\bibliography{DSP}

\vspace{-6mm}

\noindent \rule[-6mm]{160mm}{0.2mm}

\noindent
  Martin Kratochvil \\
  Department of Machine Design, University of West Bohemia, Univerzitni 2732/8, 301 00 Pilsen, Czech Republic,
  kratochv@kks.zcu.cz


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