**2. The Isochronous Pendulum Clock**

As previously stated, in 1673, Huygens published his masterpiece *Horologium Oscillatorium*, which was divided into five parts, wherein he describes the isochronous pendulum clock and studies the properties of the cycloid and geometric curves in general. He also indicated a certain length of the pendulum to produce a certain number of movements, and observed that the pendulum that marked seconds should measure approximately 99.42 cm.

This research has been carried out based on the first three parts of the book. The first describes the clock (Figure 1) as consisting of three main subsets: The gear train, the mechanism of transformation of the circular movement into an oscillatory movement, and the elements associated with the movement of the pendulum.

**Figure 1.** Isochronous pendulum clock of Christiaan Huygens according to *Horologium Oscillatorium*.

The gear train is configured on three axes interconnected by wheel-pinion assemblies that are responsible for managing the clock hands. The mechanism of the transformation of the circular movement into an oscillatory movement is composed of a crown wheel, K, that rotates around the vertical axle and transforms its rotation into oscillation of the rod, S, thanks to a horizontal axle with vanes, LM, held by the so-called 'Gnomons' (parts where the axles are housed). Finally, the elements associated with the pendulum are the pendulum itself, its support, and the cycloidal blades that limit its movement on either side, transmitted thanks to the rod, S.

In the second and third parts of the work, the theoretical foundations are enunciated and demonstrated to achieve the isochrony of the pendulum, by forcing the pendulum to move tangent to two cycloidal blades, which describe its extremes, the cycloid, and, therefore, a tautochronous trajectory (isochronous movement). In this way, the pendulum achieves a constant period (one second) independent of the amplitude of the movement.

The understanding of the operation of the mechanism was not easy, although the main challenges faced in this research include the creation of the designs of the gears of different tooth profiles, the design of the two cycloidal blades (two coupled escapement-driven blades), the simulation of the pendulum movement, and especially, the behaviour of the strings that support the bob of the pendulum.

#### **3. Sizing, Design, and Assembly of the Elements of the Isochronous Pendulum Clock**

The clock design has been carried out independently for each of the three component parts: The gear train; the pendulum with the cycloidal blades; and the clamp (Gnomon P), the dimensions of which are explained in detail in the *Horologium Oscillatorium*. For this modelling phase, the Sketcher and Part Design modules of CATIA V5 were applied. These 3D CAD modelling techniques provide a fundamental tool in the process of designing of technical historical heritage, for example, in aerospace heritage [33], as a previous stage of CAE (computer-aided engineering) analysis [34,35] and, in general, as an application for any example of cultural heritage [36].

The only information available regarding the operation of the pendulum clock appears as a brief description of the component parts: The number of teeth of each wheel, the speed that each axle must have, and the dimensions that the pendulum must have in order to mark seconds. However, there is no information available regarding the shape and dimension of each piece, and hence the design proposed in this research has been made only with the graphic information available in Figure 1.

It is well-known that pendulum clocks are characterized by using an oscillating weight to measure time accurately, and that they enjoy the advantage that the pendulum behaves like a harmonic oscillator, that is, its oscillation cycles are produced in equal time intervals (periods), and only depend on its length (isochrony). For this reason, these pendulum clocks must remain in a fixed position, since any displacement would affect the movement of the pendulum and the accuracy of its operation. Therefore, the sizing of a pendulum clock is completely parametric, and depends only on the length of the pendulum, which uniquely implies a gear ratio.

The pendulum clock can therefore be considered as being divided into two distinct parts: on the one hand, there is gear train and, on the other hand, the pendulum which is made up of the cycloidal blades and the clamp (Gnomon P), the sizing of which is explained in detail in the book.
