*3.1. Gear Train*

The gear train design is undoubtedly the first step in the redesign phase of the clock: It is decisive in the size of the resulting clock, since the distance between the axles is directly related to their primitive diameter. The gear train can be differentiated from the rest of the gears because the speed ratio of each pair of gears that are geared is directly related to the ratio of the diameters of the wheels, and this, in turn, is related to the ratio of the number of teeth. Therefore, the different speeds of rotation of each axle will be the speeds of the hands of the clock, which depends on the relation of the number of teeth and not of the size.

To obtain the speed ratios between gears, it suffices to use wheels whose number of teeth are in an inverse relationship, and the size of these gears will be determined by their own modulus (the gearing wheels have the same modulus).

Given the absence of detailed information regarding the dimensions of the elements, it has been assumed that the gear train as a whole would measure vertically one third or, at most, half the length of the pendulum. This assumption has enabled us to estimate that the primitive diameter, d, of the largest wheel (C—crown wheel), which has 80 teeth, is 240 mm, for which the modulus, m (the quotient between the primitive diameter and the number of teeth), is 3. It is known that the modulus is a standardized parameter, since it is decisive in the construction and calculation of gears, in such a way that all the data of the gearwheels is expressed as a function of said parameter. Thus, the complete pendulum clock design is based on this unit of measure.

Once all the gearing wheels were defined, the distance between the axles could be determined as the semi-sum of the primitive diameters of said wheels. Additionally, the decision was taken that the gear pressure angle should be the same, with the objective of transmitting movements and not forces and, on the other hand, considering that the distance between the AA and BB plates that form the structure of the clock (Figure 1) is 150 mm and the dimensions are 200 mm wide by 552 mm high.

Since the transmission between the gears must be carried out continuously to allow uniform movements, any of several curves could be chosen for the design of the tooth profile: The cycloid, the epicycloid, the hypocycloid, and the involute of the circle [37]. Finally, the profile of the involute of the circle was chosen because it can be traced with arcs of circumference, thereby obviating the freehand trace, and with it the error introduced [38].

To this end, our method of choice was that of Grant's odontograph [39], which results in a curve that is a very good approximation to the theoretical profile. The design parameters of wheels and sprockets were chosen while taking into account both the requirements of the chosen tracing method and the basic conditions of the wheels and sprockets that engage [39,40].

Likewise, the 3D modelling of all teeth followed a common pattern of operation: First, a sketch was made, in which the tooth profile was drawn; secondly, an extrusion (pad) of that sketch was performed, to subsequently produce a circular matrix (circular pattern) of the extrusion; finally, a hole was made to accommodate the corresponding axle.

Once all the gears were defined, the positions of each of the five axles of the mechanism were determined by means of the semi-sum of the primitive diameters of each gear. This design was completely satisfactory, since it was verified that there were interferences between the gears by sliding the teeth between them, and allowing the precise operation of the clock.

Due to the complexity of the final model (consisting of 49 different components), and to the fact that presenting the assembly plan would be extremely extensive, as would the detailed drawings of each dimensioned piece and the exploded view of the assembly to show its order of assembly in the search for the reproducibility of the work, we opted to include Figure 2, which shows only the front view of the 3D CAD model and indicates the nine wheels (I, H, G, F, E, C, B, Y, and P) and the pinion, S, as well as other main parts of the assembly. Likewise, Table 1 shows the main dimensional values determined in this study for all nine wheels and the pinion.

**Figure 2.** Front view of the 3D CAD model with an indication of all wheels and the pinion, as well as the main parts.


**Table 1.** Main dimensional values determined for all wheels and pinion.
