# Magnetic Levitation: An Academic Prototype
## 1 Introduction  
Vibration caused by mass imbalance that occurs in rotating mechanical systems is a common problem and entails losses of different types, but they mainly involve friction and wear. These losses translate into poor operation, low efficiency, and energy and economic costs derived from the maintenance or replacement of system elements. The use of rotating mechanical systems is widespread, especially in the industrial world. We can cite many rotating mechanical systems: turbines, used to obtain electrical energy through hydraulic or thermal energy; compressors that increase the pressure of a fluid while reducing its volume; generators that convert a type of energy (chemical, mechanical, light, etc.) into electrical energy; aircraft engines and engines in general; or wind turbines, among many other applications [1]. When considering the friction of a rotating shaft in the context of traditional technology, it is important to examine how friction occurs in these types of systems and how magnetic levitation can mitigate or eliminate some of these friction effects. The simplest system that currently exists to limit the vibration of a shaft is through a larger diameter tube, inside which the motor shaft is located. Its movement is physically limited by the tube that surrounds it (see Figure 1). To further limit the movement of the shaft to be controlled, a lubricating element is placed between them. Due to the high presence of applications such as those mentioned, there are different systems that allow mitigating the effect of unwanted vibrations, both improving the mechanical performance of the mechanisms and extending their useful life. These systems usually incorporate actuators and control elements, because controlling vibration amplitudes within acceptable limits is crucial to ensuring the safe and reliable operation of mechanical systems. In modern machinery, particularly in rotating components, managing vibrations is vital to maintaining performance and preventing premature wear or failure. To achieve this, rotating elements are often equipped with transducers that monitor vibrations, actuators that can apply corrective forces, and sophisticated control systems that adjust theoperation in real time to minimize vibrations. Numerous studies have focused on mitigating the intrinsic challenges associated with rotating systems, particularly in addressing the issue of mass imbalance. While it is impossible to entirely eliminate mass imbalance, researchers have made significant progress in developing methods to reduce its impact. These efforts aim to minimize vibration and other related issues, enhancing the stability, efficiency, and longevity of rotating machinery. Magnetic shafts are increasingly used in many industrial applications in order to replace rolling elements, which is a great potential option for vibration reduction in rotating machines [2]. Finding a good control system for the mass imbalance that happens in these systems leads to significant energy and economic savings [3]. As is well known, levitation consists of keeping an object suspended in the air without any type of mechanical contact. Magnetic bearings are a technology that uses magnetic levitation to hold and rotate a metal object without physical contact [4]. It is produced due to the repulsion between or through two poles of magnets to attract or repel metallic objects. There are several ways in which the levitation of an object occurs, although not all of them are valid for the levitation of a shaft. In recent years, there has been enormous research efforts related to the vibration control of rotating devices [5; 6], which can be classified into two large blocks: (1) active vibration control [7] and (2) passive vibration control [8]. Precise control of magnetic fields is crucial for stable levitation and movement. Different technologies have been established to mitigate the effect of vibrations [9; 10] with the aim of obtaining a good system performance [11].