Hypocycloidal Movement

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Greetings to the entire academic and scientific community of hive.

Introduction


How much mobility we can observe around us so that this characteristic makes this phenomenon indispensable for all humanity, the above expressed says a lot, since with us there are countless natural phenomena that allow us not only to learn from them, but also offers us power to extend our stay in this majestic universe, at this point we can say and affirm that when we refer to the phenomenon of movement we are talking primarily about life, but also comfort and therefore welfare for all of us.

Therefore, through this article we will continue in the search for another form of mobilities present in some of our daily activities, it is important to remember and highlight where we come from, that is, the fundamental bases with which we have been able to structure any type of knowledge related to such an essential phenomenon of the movement, therefore, we must mention base mobilities such as circular, parabolic, elliptical and hyperbolic.

From the previous base movements we have been able to find other movable phenomena such as; oscillatory, vibratory, undulatory, chaotic, muffled forced harmonic, cycloidal, epicycloidal and even combinations of these, as I represent it the case of the rectilinear-curvilinear movement, this last one constitutes complex phenomena of mobilities and that very often we can find it so much in our natural environment as in any of our activities.

It is also worth highlighting other forms of intrinsic movement such as periodic movement and this in turn is an essential aspect or characteristic of the recognized simple harmonic movement, and such harmonic mobility as we could verify represents the most prominent and important model among the oscillatory phenomena, and without forgetting the fact which tells us that any particle, body or object when describing some movement will transit through the geometric place of some kind of shape or figure designed by our geometry.

Before talking about our mobility to be studied or analyzed in this article it is important to be able to remember the movement of a certain circumference on a straight line with a constant speed and without slipping, where, as we could observe we obtained the cycloid curve which when transiting it any particle, body or object developed the phenomenon of cycloidal mobility, highlighting the relationship between the trajectory curve and the name of the respective movement.

But we then had the fabulous circumference rolled over another circumference (external mobility), where, due to this characteristic, we used the prefix Epi and thus gave rise to another form of movement, in this case the epicycloidal, and the same linked to extraordinary curves which are called epicycloidal trajectories in this case because of their relationship with the aforementioned mobility.

In this article we are going to roll that circumference into another one, that is, inside it, in this way we can express that these curves trajectories will be different from the epicycloidal ones due to the movement of the generating circumference inside the other circumference called directrix, and in this way we will obtain the referred curves trajectories called hypocycloidal and consequently the hypocycloidal movement.

It is important to emphasize that any point belonging to the generator or mobile circumference will be able to describe the figure of a certain hypo-cycloid curve, where, if the radius of the directive or fixed circumference is 2, 3, 4 or n times greater than that of the generator or mobile circumference, then, we will obtain hypo-cycloid curves of 2, 3, 4 or n points corresponding to the radii mentioned above.

Therefore, from a geometric vision we can say that a hypo-cycloid curve constitutes that geometric place of the different positions of any point belonging to the circumference of a certain circle, and which turns or rolls in an inner way, without slipping or skidding in the directive or fixed circumference, this last one we express is of greater size than the generator.

These hypo-cycloid curves are also known as cyclic curves, rolling curves or mechanical curves, this is due to their close link with the manufacture of parts for movement transmission mechanisms, therefore, these curves are important when developing or designing gear teeth and thus relate to their raw curves, ie epicycloidal.

Hypocycloidal Movement


With the present mobility we continue to locate any type of movement present around us, where, as we know we develop our existence, and we can say then, that the hypo-cycloidal movement contributes its grain of sand along with the other mobilities for the purpose described above, in this way we have been able to raise as we know our intellectual and social level.

At the beginning of this publication they were able to observe some examples of this type of movement, where, a circle rolls or turns inside another circle in a tangential way giving as a result each one of the curves already described as hypocycloid, it is important to emphasize that thanks to the phenomenon of the movement and the circle we have managed to decipher important curves as it is it the cycloid, epicicloid and in this occasion the hypocycloid.

As in all mobility, any particle, body or object when carrying out this phenomenon will draw a certain figure which represents the trajectory of the movement in particular, therefore, it is important to know the origin of the development of both a hypo-cycloid curve and the movement that thanks to this curve we call hypo-cycloid as we can see in the following figure 1.

Figure 1. Guiding and generating circumference at the beginning of a hypocycloidal trajectory


In the previous figure 1, we can visualize how the formation of a type of curve and of hypo-cycloidal movement begins, in which it is important to take into account the proportion between the radii of the related circles, that is, mobile (generator) and fixed (directrix), since if we had a fixed or directional radius of 2, 3, 4 or even n times bigger than the radius of the mobile or generator circumference, then, we would have hypocyloids with 2, 3, 4 points and so on respectively, next we will show an example of a hypocycloid called astroid as shown in the following figure 2.

Figure 2. Generation of a hypocycloid, astroid


In the previous figure 2, they were able to see the obtaining of a type of hypo-cycloid curve called astroid, and depending on the relation of the radius of the circles involved (generator-directive) we will obtain many other hypo-cycloids, forming in this way a wide family of these curves and that at the same time represent trajectories of the hypo-cycloidal movement.

The prefix Hiccup is implemented to indicate that some particle, body or object is below something as it happens in the case of the generator circumference at the moment of originating the hypocycloid curves and thus developing the referred hypocycloidal movement, We must highlight from these wonderful curves their cyclical character since due to this their trajectory repeats itself cyclically, and in this way as the epicycloids are fundamental in the design of parts for gears, next we will show an example of the implementation of these curves in gear teeth as we can see in the following figure 3.

Figure 3. Implementation of hypocyloids in gear tooth design


For this reason these curves are also called mechanical curves, and their essential relation as far as the transmission of innumerable types of movements is concerned, where, we find ourselves mainly in the circular movement carried out by the mobility of one or several wheels within another as we can visualize in the previous figure 3.

The hypo-cycloidal movement has allowed to deepen in the development of any type of element or mechanism with which we can generate other mobilities and with it to help to reduce enormously our daily tasks, next we will show another example of the implementation as much of these mechanical curves as of the movement developed through these curves trajectories as we can observe in the following figure 4.

Figure 4. Design of elements or components with compound, epicycloidal-hypocycloidal movement


When our eyes can visualize a certain movement of a wheel inside another we will know that the developed phenomenon is the hypocycloidal one, and without concerning the activity carried out in that moment, since in certain occasions we can find the audacity of acrobats rolling with their motorcycles inside a sphere and that if we observe it from a frontal rectangular plane this motorcycle turning inside the circle we will observe that the wheels of such mechanical system (motorcycle) originated hypocycloids as we can observe in the following figure 5.

Figure 5. Generation of hypocycloids through the wheels of a motorcycle


Conclusion

We manage in this way to add another form of mobility developed in our environment, either naturally or artificially, thus proving that the phenomenon of movement will always be indispensable in our development and consequently in our lives, and that is why it is important to be able to obtain any kind of knowledge related to the referred phenomenon of movement.

We can emphasize that the hipocicloid curves like any other curve we will be able to configure them by means of a certain movement like the carried out during the movement of two circumferences a denominated directrix that is, where, it rolls or turns the other denominated generator as we could observe it in each one of the previous examples.

It is important to take into account the radii of the related circumferences since the types of hypocycloids to be obtained will depend on this, therefore, this proportion will determine the number of arcs or points of a given hypocycloid and in this way the hypocycloidal movement will be more or less complex, remembering that the movement described above is used in the design of important movement transmission systems and the generated trajectory curve, the hypocycloid, has served as a basis for the design or manufacture of parts or gear teeth for mechanical systems.

Therefore, we can say that these curves and the movement generated by them are of great interest in the world of physics, especially in the area of mechanics because of their close link to the gears through the design of their tooth profiles, the latter being vital in the transmission systems of any type of movement.

Until another opportunity my appreciated friends and readers of Hive.blog, specially to the members of the big communities friends of #stemsocial @minnowbooster and #curie, reason why I recommend widely to be part of these exemplary projects, because they emphasize the valuable task of the academy and this way of all the scientific field.

Note: All images were made using the Power Point application and the animated gif with the PhotoScape application.

Bibliographic References

[1]Charles H. Lehmann. Geometría analítica


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