Non-dimensional design approach for electrodynamic bearings

Electrodynamic bearings (EDBs) are passive magnetic bearings that exploit the interaction between eddy currents developed in a rotating conductor and a static magnetic field to generate forces. Similar to other types of magnetic suspensions, EDBs provide contactless support, thus avoiding problems with lubrication, friction and wear. The most interesting aspect of EDBs is that levitation can be obtained by passive means, hence, no electronic equipment, such as power electronics or sensors, are necessary. Despite their promising characteristics, rotors running on EDBs are still lacking a design procedure; furthermore, at present the static behavior of a bearing can only be defined by means of finite element analyses. The aim of the present paper is to present a methodology that allows performing a first approximation design without resorting to detailed FE analyses. The methodology is based on the use of non-dimensional parameters, similar to the analysis of fluid bearings (Sommerfeld number). The non-dimensional quantities are derived using dimensional analysis, and contain the main geometrical and physical parameters determining the EDBs' performance. The relation between the non-dimensional quantities characterizing the static performance of the EDB is derived using FE simulations and is presented in the form of graph

Test and Theory of Electrodynamic Bearings Coupled to Active Magnetic Dampers

Electrodynamic bearings (EDBs) are passive magnetic bearings that exploit the interaction between eddy currents developed in a rotating conductor and a static magnetic field to generate forces. Similar to other types of magnetic suspensions, EDBs provide contactless support, thus avoiding problems with lubrication, friction and wear. Electrodynamic bearings have also drawbacks such as the difficulty in insuring a stable levitation in a wide speed range. The paper presents a solution where the EDBs are coupled with active magnetic dampers (AMDs) to guarantee a stable levitation

Non-dimensional design approach for electrodynamic bearings

Electrodynamic bearings (EDBs) are passive magnetic bearings that exploit the interaction between eddy currents developed in a rotating conductor and a static magnetic field to generate forces. Similar to other types of magnetic suspensions, EDBs provide contactless support, thus avoiding problems with lubrication, friction and wear. The most interesting aspect of EDBs is that levitation can be obtained by passive means, hence, no electronic equipment, such as power electronics or sensors, are necessary. Despite their promising characteristics, rotors running on EDBs are still lacking a design procedure; furthermore, at present the static behavior of a bearing can only be defined by means of finite element analyses. The aim of the present paper is to present a methodology that allows performing a first approximation design without resorting to detailed FE analyses. The methodology is based on the use of non-dimensional parameters, similar to the analysis of fluid bearings (Sommerfeld number). The non-dimensional quantities are derived using dimensional analysis, and contain the main geometrical and physical parameters determining the EDBs' performance. The relation between the non-dimensional quantities characterizing the static performance of the EDB is derived using FE simulations and is presented in the form of graphs

Stability of a 4 degree of freedom rotor on electrodynamic passive magnetic bearings

Electrodynamic bearings exploit repulsive forces due to eddy currents to produce positive stiffness by passive means without violating the Earnshaw stability criterion. This remarkable characteristic makes this type of bearing a suitable alternative to active magnetic bearings in fields such as kinetic energy storage flywheels, turbo pumps, high speed compressors, among others. However, the suspension can become unstable due to rotating damping. To obtain deeper understanding of this instability phenomenon this paper presents the analysis of stability of a four degree of freedom (4dof) rotor supported by electrodynamic bearings. The 4dof rotor model is coupled to the dynamic model of the eddy current forces generated by the electrodynamic bearing and the stability of the complete system is analyzed. This model is used to study the stability of both cylindrical and conical whirling motion of the rotor. In addition to the well known cylindrical whirl instability the possible occurrence of conical instability is demonstrated. Finally the effects of two stabilization strategies are analyze

A new predictor-corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low-frequency integral formulation of the Maxwell equations coupled with Newton-Euler rigid-body dynamic equations. Two different integration schemes based on the predictor-corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single-step time marching algorithm, while the mechanical dynamics is studied by a predictor-corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor-corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two-degree-of-freedom levitating device based on permanent magnets is finally reported

Electrodynamic Bearings Modeling and Design

The number of high speed applications is sharply increasing in the last years so that the problem of the choice of the most suitable bearing must be handled. Considering the rotating machines, during the past ten years, high speed turbomolecular pumps have been mass produced showing that the high speed technology is mature. Since friction, wear, maintenance and lifetime represent the main issues of high speed applications, magnetic bearings, which are a contact free devices, could be considered as a possible solution. A magnetic bearing is a contact free bearing wherein the load is carried by magnetic forces. The magnetic field is generated in such a way that it provides necessary stiffness and damping to make the rotor hover safely during operation. Magnetic bearings are characterized by no wear, no maintenance and furthermore they don't require lubrication. The present work deals with a particular category of magnetic bearing: the electrodynamic bearings. The most interesting aspect of electrodynamic bearings is that levitation can be obtained by passive means, thus no electronic equipment, such as power electronics or sensors, are necessary. Therefore, electrodynamic suspensions are an advantageous alternative to active magnetic suspensions: they are less complex, less subject to failure, and possibly far lower in cost. On the other hand, the main drawback of electrodynamic bearings is that a stable levitation is provided only when the speed is above a threshold value, which opens stability issue at low speeds. Presently, the design of electrodynamic bearings is based on the force-to-angular speed characteristic obtained for a fixed eccentricity. Although this characteristic describes the behaviour of the bearing in quasi-static conditions, it is not suitable for dynamic conditions where the rotor is animated by a nonsynchronous whirl, or even a non periodic motion about the stator axis. A model that could take explicitly into account both the quasi-static and the dynamic conditions is still lacking. Therefore the first objective of the present work is to develop an analytical model of radial and axial electrodynamic bearings able to describe both quasi-static and dynamic behaviours. In this way not only the static characteristics such as achievable stiffness and force, but also the dynamic performance of a system involving electrodynamic bearings can be evaluated and used as design criteria. The developed models of radial and axial electrodynamic bearings have been experimentally validated. Two test rigs, the first based on a radial electrodynamic bearings, the second characterized by the presence of an axial electrodynamic bearing, have been designed and built to this end. Since the electrodynamic bearings are intrinsically unstable devices, the stability issue is examined in the dissertation. So far, the most common solution to achieve the stability is to add non-rotating damping between the rotor and the stator. It is natural to obtain the stability by adding non-rotating damping between the rotor and the stator. Nevertheless, this strategy is not as easy to apply as it seems, given that the stabilizing device must be contactless, consistently with the electrodynamic bearings issue. Although effective, this solution could imply the installation of a dedicated magnet on the rotor with a consequent increase of the rotor weight and complexity and the rising of some concerns about the mechanical resistance, and a conductor where eddy currents can arise on the statoric part of the system. In the dissertation a new stabilizing technique for electrodynamic bearings is proposed. Instead of introducing damping between the rotor and the non-rotating part of the bearing, the author propose to introduce an elastic and dissipative element between the statoric part of the bearing and the case of the machine. The innovative aspect of the proposed solution is that the stabilizing dissipation is introduced into the statoric part of the electrodynamic bearing, which does not rotate. The main advantages of this configuration are: reduced complexity from the constructive point of view since no rotating parts are involved; an easier tuning of the stabilizing system main parameters; an extremely effective contribution to the stability; conventional and low cost devices are suitable for the implementation of the proposed stabilizing system. The wide variety of practical solutions that can be adopted for the proposed stabilization techniques, such as rubber bushings or squeeze film dampers for example, is an indicative parameter of the true possibility of applying it in industrial applications. The impact of the proposed solution on the rotor stability is investigated, and it is demonstrated that a damping device between the rotor and the statoric part is no longer necessary. Although electrodynamic bearings are quite complex devices, they are still lacking a clear design procedure, therefore a procedure addressed to optimize the design of electrodynamic bearings for industrial applications could be useful. Since the developed models, able to describe the quasi-static and dynamic behaviours of electrodynamic bearings, have been experimentally validated, they can be assumed as useful tools oriented to the design of these devices. Therefore, at the end of the dissertation, a design procedure based on those models is presented. The developed models and the proposed stabilizing technique make the electrodynamic bearings ready for the industrial application

Climate change regulation: comparing the EU and the US legal approaches

Climate change regulation. Climate change regulators: comparing the EU and the US systems. Models of Climate change regulation. Legal challenges to Climate change regulation: the EUETS and clean power plan litigations. Global regulation of climate change.Climate change regulation. Climate change regulators: comparing the EU and the US systems. Models of Climate change regulation. Legal challenges to Climate change regulation: the EUETS and clean power plan litigations. Global regulation of climate change.LUISS PhD Thesi

I poteri di regolazione dell\u2019Aeeg, l\u2019autonomia contrattuale e i poteri impliciti

In un mercato aperto e concorrenziale l'attivit\ue0 di "micro" regolazione dell'Aeeg pu\uf2 incidere sul rapporto contrattuale se volta a ristabilire un contraddittorio paritario tra le parti del contratto di somministrazione (utente e fornitore) e quindi qualora sussista uno squilibrio contrattuale in danno del contraente debole (utente). Diversamente l'etero integrazione contrattuale ad opera di una direttiva dell'Aeeg sar\ue0 legittima soltanto se espressamente consentita da una norma di legge. In mancanza della norma attributiva del potere, \ue8 da escludere il ricorso alla teoria dei poteri impliciti in virt\uf9 di un concezione rigorosa del principio di legalit\ue0 di cui il Tar Lombardia \ue8 ancora una volta portatore