*3.2. Cycloidal Blades*

These elements constitute the key to the success of this pendulum clock. The cycloid is traced by faithfully following the method explained by Christiaan Huygens in his book *Horologium Oscillatorium*, where it is defined as the cyclic curve that is generated by a point when rolling (without sliding) a circle along a line (Figure 3).

**Figure 3.** Tracing the cycloid with CATIA V5 according to *Horologium Oscillatorium* and a 3D CAD model.

This curve can be obtained in the DMU Kinematics Simulator module of CATIA V5, in which the software is instructed to draw the trajectory of a previously defined point of the circle, whereby this cycloidal curve describes a succession of points joined by a spline (Figure 4).

**Figure 4.** Generation of the cycloid by means of a spline.

Once the cycloidal curve is obtained, and by extrusion of a closed sketch that is created from the cycloid, the 3D models of the two cycloid blades can be obtained (Figure 5).

**Figure 5.** Cycloidal blade.

#### *3.3. Assembly of Subsets and Final Set*

This operation was carried out in the CATIA V5 Assembly Design module through the application of a series of geometric and movement restrictions (degrees of freedom) to the different parts, which place them in their final positions. Thus, once each of the subsets was assembled, the final set could be assembled.

Among all the subsets, the pendulum is perhaps the most complicated. Since the pendulum is suspended by two strings between the cycloidal blades (Figure 6), and since CATIA V5 cannot model flexible elements such as the string, this difficulty was resolved by assimilating it into a chain formed by seven links, which represents an acceptable approximation of the real behaviour of the string. Figure 7 shows the pendulum geometry tree, where the various parts of this subset can be observed.

**Figure 6.** Pendulum between cycloidal blades according to *Horologium Oscillatorium*.

CATIA V5 considers each sub-assembly in a rigid way; if it becomes flexible and becomes an independent mechanism, then the possibility of relating it to another mechanism is lost, which is inconvenient for kinematic simulation.

Likewise, the model of the chain (simulated string) was given a real look (Figure 8), in order to attain a satisfactory result.

Finally, before obtaining the complete assembly of the isochronous pendulum clock (Figure 9), various sub-assemblies were made, such as that of the spheres, the second disk, and the pendulum.

**Figure 7.** Pendulum geometry tree.

**Figure 8.** Rendering of the chain with a real string appearance.

**Figure 9.** Rendering of the complete assembly of the isochronous pendulum clock.

## **4. Simulation of Pendulum Kinematics**

For the kinematic simulation of the pendulum (the movement of which posed the most problematic phase), the DMU Kinematics Simulator module of CATIA V5 was applied. This module requires the existence of a fixed part for the simulation of a mechanism, and it was established at the beginning as a restriction, and subsequently the unions between the different parts were defined. Without doubt, the simulation of the movement of the pendulum presented the greatest difficulty.

This simulation raised two main drawbacks. In the first place, CATIA V5 fails to allow flexible elements, such as the string, to be modelled, which has been solved by approaching the string as a chain of seven links. Secondly, the possibility that the chain adapts to the cycloidal blades in its periodic movement should also be borne in mind. CATIA has the capability for contact detection, but if this option is activated, then at the moment it detects a contact, the simulation would halt.

The movement of the pendulum was resolved in two ways: By means of commands, that is, by conveniently moving each degree of freedom; and by means of functions, that is, by imposing a function on each degree of freedom of the driving mechanism. Therefore, since the string was modelled as a seven-link chain, at each point of connection between the links there is a degree of freedom of rotation, which are the drivers of the pendulum movement.

By means of command simulation (Figure 10), one of the properties of the cycloid discovered by Huygens is verified: "The evolute of a cycloid is the displaced cycloid itself", since it can be observed that the cycloid of the wooden base and the cycloid of the cycloidal blades are the same but displaced.

**Figure 10.** Command simulation.

To this end, the simulation of a single movement period is edited with commands. One by one the degrees of freedom are moving, in the most similar way to reality, in the 'Kinematic Simulation' window. At the same time, in the 'Edit Simulation' window, frames are successively inserted for each movement of a degree of freedom made.

Subsequently, the commands were gradually moved, so that when they reach the cycloidal blades, the links adapted to them as much as possible, thereby ensuring, as far as possible, the tangency of the links to the corresponding cycloidal blade, in such a way that the behaviour closely resembled that of a string. The simulation was then compiled, and an animated film generated, thereby obviating the need for the computer to create it every time a visualization is desired, which would greatly slow down the movement.

Subsequently, the generated animation was cyclically reproduced in order to attain the periodic movement of the pendulum, since only one period of the movement was inserted, and the film is captured in a video.

Finally, in this video it can be seen that the property enunciated by Huygens is fulfilled and that the pendulum model is a very good approximation for the result obtained, although a small error is introduced due to the existence of the clamp that must be used for the chain to exit between the cycloid blades, which means that a small part of the cycloid is neglected (Figure 11).

**Figure 11.** Detail of the chain clamp between the cycloidal blades.

By means of simulation through functions, the animation of the clock as a whole is made possible; however, due to the high number of parts that each move at a different speed, and with respect to the

other parts, a discrete synchronization by means of commands of the complete mechanism becomes practically impossible.

When a mechanism is created with CATIA V5, the reference between elements is set as the position they have at the time of creating the joint, and hence, before creating the joints in the Assembly Design module, the links are placed tangentially to one of the cycloidal blades (Figure 12), and it is at that moment when the joints between the different links are created.

**Figure 12.** Position of the links tangential to the cycloid blade.

The function (Figure 13) that describes the movement of a simple pendulum (simple periodic movement or simple harmonic movement) is [41]

$$\mathbf{y} \text{ (t)} = \text{A } \sin \left( \omega \mathbf{t} + \Phi \right)$$

where A is the amplitude of the periodic movement and Φ is the offset.

**Figure 13.** Function describing the movement of the simple pendulum.

For the pendulum to start moving at its point of maximum amplitude (Φ = ± 90◦), and since the pendulum must mark seconds, ω must be 180◦/s.

In this case, the chain of seven links has been modelled as a composite pendulum, in which each link describes a simple periodic movement with respect to the previous link. Therefore, the angular functions that simulate the movement of these links are

For the first link: θ<sup>1</sup> (t) = θ<sup>1</sup> sin (180t − 90) + θ<sup>1</sup>

For the remaining links: θ<sup>j</sup> (t) = (θ<sup>j</sup> − θi) sin (180t − 90) + (θ<sup>j</sup> − θi)

where j is a link and i is the link immediately before. In order to fully define the links, it suffices to measure the angles with respect to the vertical in the initial position of tangency to the cycloidal blade (Figures 14 and 15).

**Figure 14.** Modelling of the seven-link chain as a composite pendulum.

**Figure 15.** Values of the angles that form the links with respect to the vertical in their initial position of tangency to the cycloidal blade.

Therefore, the laws that describe the movement of the links are:

$$\begin{aligned} \theta\_1 \text{ (t)} &= (180 - 167.241) \sin \left( 180t - 90 \right) + \theta\_1 = 12.759 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_2 \text{ (t)} &= (167.241 - 164.460) \sin \left( 180t - 90 \right) + \theta\_1 = 2.781 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_3 \text{ (t)} &= (164.460 - 160.409) \sin \left( 180t - 90 \right) + \theta\_1 = 4.051 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_4 \text{ (t)} &= (160.409 - 158.457) \sin \left( 180t - 90 \right) + \theta\_1 = 1.952 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_5 \text{ (t)} &= (158.457 - 155.726) \sin \left( 180t - 90 \right) + \theta\_1 = 2.731 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_6 \text{ (t)} &= (155.726 - 155.632) \sin \left( 180t - 90 \right) + \theta\_1 = 0.094 \sin \left( 180t - 90 \right) + \theta\_1 \\ \theta\_7 \text{ (t)} &= (155.632 - 152.049) \sin \left( 180t - 90 \right) + \theta\_1 = 3.583 \sin \left( 180t - 90 \right) + \theta\_1 \end{aligned}$$

Finally, the process that follows is analogous to the previous process: In the simulation with commands a simulation is edited, and a film is then created or a video is generated. As can be observed in Figure 16, the string is curved before reaching the cycloidal blade, and precisely at the moment of arrival it adopts the shape of the blade, which represents a good approximation.

**Figure 16.** Sequence of chain curvature (simulated string) before reaching the cycloidal blade.

In the screenshots of the animation of Figure 16, the good approximation of this model can be appreciated: The chain (simulated string) is curved before reaching the cycloidal blade, and, exactly at the moment of arrival, its form is adopted. Although the chain does not move exactly like a string, the pendulum shaft carries the necessary movement to regulate the march of the clock. The reason why a greater number of links was not been chosen, which in principle would appear to guarantee a greater reliability of the model, 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. Thus, seven links provides a working scenario of relative comfort and, as seen in the videos and animations obtained, the pendulum describes the cycloid very closely.

Furthermore, the kinematic relationships between the various gears are established by their transmission ratios and it is, therefore, only necessary to establish motion simulation functions for the remaining degrees of freedom. Once these degrees of freedom have been set, the clock can then be simulated and various animations of its operation can be generated.
