# The Design and Simulation of an Astronomical Clock
## 7 The Clock Gear Trains  
In accordance with the presented astronomical data and their approximated ratios, the design of the astronomical clock mechanism is realized and a fully operational 3D model of this mechanical device is created by using the SolidWorks 2016 application (Dassault Systemes, Dassault Group, Velizy-Villacoublay, France). As is shown in Figure 8, the mechanism of the clock consists of five gear trains. Each gear train had its specific speed ratio determined in the previous section by the method of continued fractions as the best rational and practically applicable approximation of the real ratios. The prime factors of the numerator and denominator of these approximated ratios are also calculated and the proper gear modulus and the number of teeth are determined. These ratios, the number of gear teeth and gear modulus are given in Table 6. The problem of selecting the best but practically applicable convergents or semi-convergents of the continued fraction approximation for the construction of the clock gears must be emphasized again. Obviously, higher terms of this rational approximation produce smaller errors. However, the higher terms have larger numerators and denominators and, very often, their prime factors are also huge. Since these prime factors have to be realized directly and precisely by the number of gear teeth, extremely high terms of continued fraction approximations are not practical and useful. On the other hand, lower terms of rational approximation are usable but produce larger errors. Thus, the problem of selecting the best but practically applicable convergents or semi-convergents of the continued fraction approximation must be solved as a compromise between two opposite demands. The mechanism of the astronomical clock has four coaxial shafts which hold the lunar nodes, the Sun, the zodiac and the Moon pointers. Additionally, the clock mechanism is equipped with one separate shaft which controls the rotation of the sphere, which is painted \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Approximation** & **Type** & **Approximation** \\ \hline C & \(0/1\) & C & \(346/347=\left(2\times 173\right)/347\) \\ C & \(1/1\) & C & \(347/348=347/\left(2\times 3\times 29\right)\) \\ SC & \(173/174=173/\left(2\times 3\times 29\right)\) & C & \(693/695=\left(3^{2}\times 7\times 11\right)/\left(5\times 139\right)\) \\ SC & \(174/175\) & C & \(1040/1043=\left(2^{4}\times 5\times 133\right)/\left(7\times 149\right)\) \\ SC & \(\ldots\)\(\ldots\)\(\ldots\) & C & \(1733/1738=1733/\left(2\times 11\times 79\right)\) \\ SC & \(345/346=\left(3\times 5\times 23\right)/\left(2\times 173\right)\) & C & \(2773/2781=\left(47\times 59\right)/\left(3^{3}\times 103\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The best rational approximations of the ratio \(R_{SN}\). half white and half dark blue and displays the lunar phase. As was already emphasized and shown in the operational scheme of the clock mechanism in Figure 8, the shafts are mutually connected by five previously mentioned gear trains. The clock mechanism has two five-pointed frames which support the astronomical face, with all shafts and gear trains as one functional block. The gear trains which control the motion of the Sun pointer and the lunar phase's spherical display are shown in Figure 9 and the related gear numbers are given in Table 6. The gear modules for both trains are 2 mm. The Sun gear train consists of three gears, the first of which is connected to the hour shaft of the ordinary clock. This shaft holds the Sun pointer equipped with a small gold-plated symbol of the Sun. The hour shaft of this clock drives all gear trains of the astronomical clock. The lunar phase train has four pairs of gears and takes the power from the Sun shaft. The last shaft of the lunar phase's gear train is connected to the small spherical pointer which displays the current Moon phase.