**5. Conclusions**

In this article, the three-dimensional modelling of the isochronous pendulum clock of Christiaan Huygens is presented, together with the kinematic simulation of the pendulum using the CATIA V5 software.

The 3D model of the isochronous pendulum clock has been obtained with exactly the same proportions as those described in the work *Horologium Oscilatorium*, thanks to the sizing criteria adopted. However, the only information available regarding the operation of the pendulum clock appears as a brief description of the parts that compose it, the number of teeth of each wheel, the speed at which each axle should move, and the dimensions that the pendulum should have to mark the seconds. There is no information regarding the shape or dimensions of each piece, and hence the design proposed in this research has been created with only the graphic information available in Figure 1. Specifically, given the absence of detailed information regarding the dimensions of the elements, it has been assumed that the gear train as a whole measured vertically one third or, at most, half the length of the pendulum, which has enabled the primitive diameter of the largest wheel (crown wheel), C, of the gear train to be estimated as being 240 mm. From this dimension, it has been possible to define the dimensions of the rest of the wheels and the positioning of the axles, since the dimensioning of a pendulum clock is totally parametric, and depends solely on the length of the pendulum, which univocally implies a relationship of gears.

On the other hand, the main challenges faced in this investigation include the design of the gears of different tooth profiles, the design of the two cycloidal blades, the simulation of the pendulum movement, and, especially, the behaviour of the strings that support the bob of the pendulum. Moreover, one of the main limitations is that presented by CATIA V5 for the modelling of flexible elements such as strings, an inconvenience successfully resolved by approaching the string as a simple chain of seven links that behave as a composite pendulum. The rationale for choosing the number of links lies in dimensional and geometric causes. From the dimensional point of view, the pendulum section that was represented (which was small in size), could not be easily represented with more than seven links, while from the geometric point of view, seven links sufficiently rectified the cycloid, a curve to which the pendulum had to adapt geometrically.

The kinematic simulation of the pendulum was also performed, whereby the exactitude of the clock in the measurement of time was verified and the pendulum was described with close approximation to the cycloid. Furthermore, thanks to CAD and kinematic simulation techniques, it has been possible to understand and interpret the operation of Huygens's isochronous pendulum clock, which in turn has enabled virtual recreations to be generated that can serve as educational tools in the museum of Science History, and has facilitated the dissemination of technical historical heritage. Moreover, future uses may involve the development of applications of virtual reality and augmented reality, the incorporation of the WebGL model into a website, and the ability to print in 3D using additive manufacturing techniques.

Finally, the results of this research regarding the CAD 3D model can be employed by other researchers, and shared online via maker communities, such as GrabCAD [42] and Thingiverse [43].

**Author Contributions:** Investigation, J.I.R.-S. and F.J.G.-C.; Methodology, F.J.G.-C. and G.D.R.-C.; Validation, F.J.G.-C.; Writing—original draft, J.I.R.-S. and F.J.G.-C.; Writing—review & editing, J.I.R.-S., F.J.G.-C. and G.D.R.-C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We sincerely appreciate the work of the reviewers of this manuscript.

**Conflicts of Interest:** The authors declare there to be no conflicts of interest.
