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Introduction

We continue to observe our environment with the aim of being able to look for any kind of mobility, in this way we have already found vital movements that are part of our lives such as; circular, parabolic, elliptical, hyperbolic, oscillatory, vibratory, wave, chaotic and cycloidal movement, but we can also find the combination of these phenomena and in this way raise the degree of complexity of this phenomenon.

It is important to always take into account each of the trajectories traced during the development of these mobilities, since with these mobilities we have described essential geometric places of forms or figures such as; the circumference, parabola, ellipse, hyperbola, cycloid, as well as portions or combinations of the previous geometric figures.

We can affirm that we have demonstrated the essential utility of each one of the acquired knowledge of the phenomenon of the movement, these learnings the man with the aid of the scientific field has implemented them effectively in important physical or mechanical systems that we used to be able to develop great part of our more daily activities.

In the present article we will analyze the damped forced harmonic movement, a very complex mobility since to carry it out we must inject some kind of force or energy to a certain mechanical system, and in this way we force the realization of a certain movement and also we will help it to its damping, extraordinary combinations designed by the man with the purpose of being able to develop activities related to his daily life.

When observing our environment we can say that there are very complex mechanical systems, and where, the damping is of vital importance, for example, any of us has been mounted on a vehicle (motorcycle or car) and we can feel that the suspension of the same is fundamental to that system, since the suspension is directly related to the trajectory or travel of the car.

It is important to be able to state that the damping system is generally made up of shock absorbers, bearings with suspension stops and of course the stabilizer bars. This system will absorb every oscillating-vibrating movement whether it is slight, regular or sudden, this when travelling on any road but especially when it is in poor condition.

When we refer to damping, we are referring to the capacity to gradually stop a certain movement or to lower the impact in terms of its speed at a specific moment. A practical example of natural damping can be seen in the oscillatory movement when when we swing and gradually become paralyzed when we stop supplying force or energy to this movement.

Therefore, in the previous example, we noticed the existence of a friction which will lead us to the state of equilibrium, one of these factors we can express that it is the air which works together with the gravitational force of the terrestrial nucleus, but to avoid the previous factors of natural origin, the man has designed mechanical systems where it is possible to control its mobilities, with the combination of damping force.

Then, we could express that in an open mechanical system we could consider a double damping, one of natural character (the factors before mentioned) and the other one of artificial character carried out by certain instruments equipped with some liquid, springs, among others, with the purpose of diminishing the speed of movement or of mitigating some disturbance at the moment of developing any type of mobility.

Forced harmonic movement with damping

From our universe we have been able to verify that it is full of any type of mobility, either of oscillatory character or of any other type of movement previously indicated, in this opportunity we will focus on complex mobilities such as the case of a muffled forced harmonic movement, and we can say that this combination represents an important factor of stability for any mechanical system designed by man.

We know from the previous articles that all real or natural free harmonic mobility will be cushioned by the frictions present in our environment (such as the air and gravitational action), these natural conditions make this movement stop little by little until it stops completely, therefore, by not contributing some type of energy to this natural mobility it will lose its kinetic energy as time goes by, and in this way the mobility that we call naturally cushioned harmonic movement is originated.

It is important to be able to relate the trajectories of harmonic mobilities starting from our reference, the simple harmonic movement, all types of harmonic movement either forced to no, as it loses energy it will behave according to the curves that we will observe in the following figure 1.

Figure 1. Harmonic and damped path curves

In the previous figure 1, we observe that in the second and third curve the energy related to a certain system in harmonic movement is dissipated, so it is important to take into account the types of friction present during the harmonic movement analyzed, taking into account that if we force a certain mechanical system to perform any mobility the cave described would be the first one observed.

If we notice that the energy supplied begins to dissipate, in this case we would go to the second and then the third curve respectively until reaching the point of stable equilibrium of the piece or set of objects involved in the movement or movements, this highlights the enormous usefulness we have achieved by forcing and cushioning any system to make movements.

We could clearly express that any mechanical system will be subject to the action of a force, and this allows a forced movement to be generated, as the examples that we will see later in the development of this article, therefore, we will emphasize that damping supports are always necessary (in mechanical systems) in order to be able to balance the external forces provided and avoid a possible collapse of the system.

It is important to take into account that when talking about any type of harmonic movement, we will always find the oscillatory mobilities and their elemental characteristics, but above all the most important of them, that is, the simple harmonic movement which we have been able to describe mathematically, thus being able to interpret innumerable movements present in our complex universe.

Then, it is necessary to know two important formulations related to this type of harmonic mobility, the first one referring to a certain system without damping, that is to say, that mobility without the presence of some type of friction and which carries by expression or mathematical formulation the following differential equation.

In the previous formulation, we have that the variable (X) represents the movement, (ωo) to the speed or angular frequency, and of being expected (t) to the time expressed in seconds, the solution of this equation which indicates the differentiation of the indicated magnitude (X) with respect to the time is the following one:

From the previous equation we can express, that (X) constitutes the length or elongation of the particle, body or object in relation to its position of stable equilibrium, the variable (A) represents the amplitude, that is, the maximum displacement, the set (ω.t+δ) represents the phase of this mobility, where, δ constitutes the phase angle.

It is important to start relating to practical examples of damped forced harmonic mobilities which we can observe in our environment, but first it is important to visualize some essential elements implemented in mechanical systems with movements described above, in the following figure 2.

Figure 2. Some types of shock absorbers

Now we can visualize an example of the application of these damping components as we can see in the following figure 3.

Figure 3. Mechanical system with damping

Many are the activities of our daily life that implement these elements of cushioning as we can continue to observe in the following figure 4.

Figure 4. Mechanical systems with double damping

These cushioning applications have been taken to other activities such as the one we will see below in figure 5.

Figure 5. Further application of the damping elements

In this way we can find many examples, where, muffled forced harmonic mobilities are developed, representing this complex movement one of the most used tools in the design of essential mechanical systems that the man every day develops in favor of all the humanity, demonstrating with it that the phenomenon of the movement forms intrinsic part of our lives.

Conclusion

Any of us when we refer or talk about cushioning immediately think of softness, elasticity, that is, balance in the functioning of a certain natural system, physical or mechanical, with this essential aspect we highlight the importance of this type of complex mobility in many activities that frame our daily life.

As in many cases, this type of movement goes unnoticed because when we start to detail our environment we would realize that many are the muffled forced harmonic movements that are part of our lives, especially in those mechanical systems with artificial mobility designed by man, implementing the essential tool of science and its invaluable creativity.

It is important to emphasize that many movements can be observed when we analyze a certain mechanical system which develops a whole cycle of muffled forced harmonic mobility. For this reason, we consider this mobility to be ultra-complex since we could study several types of movements at the same time, confirming that we have managed to configure systems that develop complex movements from the simple.

In general, we can express that every mechanical system will need the action of a certain force which will force it, that is to say, it will force some kind of movement which will allow it to do a specific job, therefore, we can say that both natural and mechanical (artificial) damping play an essential role in the proper functioning of many systems as much natural as those designed by man.

We continue to verify that our universe is synonymous with movement, and thus, synonymous with life, where, the living species of that space-time described above do not escape such influence, in this way we describe and know in a general way to another type of mobility that is part of our most daily activity.

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

Bibliographic References

[1]Charles H. Lehmann. Geometría analítica.

[2]Damped harmonic motion.

[3]Damped Simple Harmonic Motion.

[4]Amortiguadores hidráulicos.