DEPARTAMENTO DE ENGENHARIA MECÂNICA ENERGY ANALYSIS OF ELECTROMECHANICAL SYSTEMS

  UNIVERSIDAD DE OVIEDO DEPARTAMENTO DE INGENIERÍA ELÉCTRICA
ESCUELA POLITÉCNICA SUPERIOR DEPARTAMENTO DE ESTADÍSTICA E INVESTIGACIÓN OPERATIVA
ESCUELA POLITÉCNICA SUPERIOR DE JAÉN DEPARTAMENTO DE ESTADÍSTICA E

FACULTAD DE CIENCIAS DEPARTAMENTO DE BIOLOGÍA ANIMAL B VEGETAL
FACULTAD DE CIENCIAS EXPERIMENTALES DEPARTAMENTO DE QUÍMICA FÍSICA Y
FACULTAD DE CIENCIAS SOCIALES Y JURÍDICAS DEPARTAMENTO DE ESTADÍSTICA

Foi feito um estudo das funções de forma para um elemento de viga e obtidas as matrizes

Departamento de Engenharia Mecânica

ENERGY ANALYSIS OF ELECTROMECHANICAL SYSTEMS


Student: Natasha Hirschfeldt

Advisor: Roberta Lima





Abstract

Electromechanical systems are composed by two subsystems with different origins: one mechanical and another electromagnetic. Therefore, the energies in an electromechanical system have also different origins: some are mechanical, as kinetic and potential, and others are electromagnetic, as magnetic and electrical. To properly describe the dynamics of an electromechanical system, it is not sufficient to describe each subsystem separately. It is necessary to consider coupling terms between them. These terms characterize the mutual influence between the two subsystems and their interplay of energies. The coupling terms can have a magnetic origin (as shown in [4,5,6]) or an electric origin. This research analyzes with an energetic approach electromechanical systems. It is shown how the equations that govern their dynamics can be obtained by the Lagrangian method [2,8] and some energetic analyses are performed. To exemplify, several electromechanical systems were analyzed, as for example, an electromagnetic loudspeaker.

Objective

The purpose of the research was to provide a step by step of how to describe the dynamics of an electromechanical system using the Lagrangian method without making common mistakes made in some books and published papers, such as using only mechanical variables to describe this kind of system or not considering a coupling term when describing the system. This topic is discussed in [3], [4] and [7]. To accomplish this goal, several electromechanical systems were analyzed and one paper was submitted to a congress with some of the obtained results. In the paper, an electromagnetic loudspeaker was analyzed and the interactions between the two subsystems that compose the system, i.e., the interplay of energies (kinetic, potential, magnetic and electrical), were discussed.

Methodology

The first step to make an energetic analysis of electromechanical systems was to recall some important concepts of purely electromagnetic systems [1]. This way, it was possible to describe the dynamics of a pure electromagnetic systems using the Lagrangian method and it was possible to describe the coupling energies between the mechanical and electromagnetic subsystems. The electromagnetic loudspeaker analyzed is composed by a mechanical subsystem (a mass m, a spring of constant k and a damper of constant b, where the last two simulate a membrane that dislocates the air), an electromagnetic subsystem (a voltage source υ in series with an RL circuit, i.e., an inductor of inductance l and a resistor of resistance r) and an element called moving-coil transducer [6] (with transducer constant ϱ) that couples the subsystems. Two variables were used to describe the system configuration. One of them is mechanical, it is called x the displacement of the mass m from the mechanical subsystem's equilibrium point, and the other is electromagnetic, the charge q passing through the circuit.

To analyze the interplay between the different types of energy in the system, a routine was developed using the software Matlab to integrate the initial value problem that gives the system dynamics during a defined time interval. The Runge-Kutta of 4th and 5th order method was used as time-integration scheme.

DEPARTAMENTO DE ENGENHARIA MECÂNICA ENERGY ANALYSIS OF ELECTROMECHANICAL SYSTEMS DEPARTAMENTO DE ENGENHARIA MECÂNICA ENERGY ANALYSIS OF ELECTROMECHANICAL SYSTEMS

Conclusions

In the research, it was possible to characterize the mutual influence between the two subsystems that compose an electromechanical system and the interplay of the energies between them. It was verified that these coupling terms can have a magnetic origin or an electric origin. In the analysis of the electromagnetic loudspeaker, it was possible to conclude that the moving-coil transducer is an element that does not store energy in system and, therefore, does not contribute to the energy balance. Which means that the transducer’s energy is only transmitted from one subsystem to another. That is, it was also explored in the research the fact that a coupling element does not necessarily have to store energy, it can only transform it, making the energy transition in this case a little bit different from a pure mechanical system.

The research has two products: a didactic text explaining how to describe the dynamics of electromechanical systems using the Lagrangian method (with several examples of different electromechanical systems) and a paper submitted to the congress XL National Congress of Applied and Computation Mathematics entitled “Energy analysis of an electromagnetic loudspeaker”.

References

1 - DAQAQ, M. F. Dynamics of Particles and Rigid Bodies: a self-learning approach. 1 ed. Hoboken, NJ : John Wiley & Sons, 2019.

2 - JELTSEMA, D., Scherpen, J. M. A. Multidomain modeling of nonlinear networks and systems. IEEE Control Systems, v. 29, n. 4, p. 28-59, 2009.

3 - LIMA, R., Sampaio, R., Hagedorn, P., Deü, J. Comments on the paper “On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors” published in this journal. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2019.

4 - MANHÃES, W., Sampaio, R., Lima, R., Hagedorn, P., Deü, J. Lagrangians for electromechanical systems. Mecánica Computacional, v. XXXVI, p. 1911-1934, 2018.

5 - MANHÃES, W., Sampaio, R., Lima, R., Hagedorn, P. Two coupling mechanisms compared by their Lagrangians. DINAME 2019, Proceedings of the XVIII International Symposium on Dynamic Problems of Mechanics, 2019.

6 - PREUMONT, A. Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems. v. 136, G.M.L. GLADWELL, University of Waterloo, Canada, 2006.

7 - SAMPAIO, R., Lima, R., Hagedorn, P. One alone makes no coupling. Mecánica Computacional, v. XXXVI, p. 931-944, 2018.

8 - WELLS, D. A. Schaum’s outline of theory and problems of Lagrangian Dynamics with a treatment of Euler’s Equations of Motion, Hamilton’s Equations and Hamilton’s Principle. New York: McGraw-Hill, 1967.


FACULTAD DE HUMANIDADES Y CIENCIAS DE LA EDUCACIÓN DEPARTAMENTO
FACULTADESCUELA DE CIENCIAS EXPERIMENTALES DEPARTAMENTO DE ESTADÍSTICA E INVESTIGACIÓN
FACULTADESCUELA DE HUMANIDADES Y CIENCIAS DE LA EDUCACIÓN DEPARTAMENTO


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