MIT OpenCourseWare http ocw mit edu Electromechanical Dynamics For any use or distribution of this textbook please cite as follows Woodson Herbert H and James R Melcher Electromechanical Dynamics 3 vols Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms Chapter 10 DYNAMICS OF ELECTROMECHANICAL CONTINUA 10 0 INTRODUCTION In Chapter 9 we treated simple examples of mechanical continua to establish the basic techniques of making mathematical models and to illustrate the kinds of dynamic behavior and the mathematical methods needed in analyses In that chapter simple elastic continua at rest were excited at boundaries so that the resulting continuum dynamics were determined by mechanical characteristics alone In this chapter we still restrict our attention to simple elastic continua but we generalize on the treatment of Chapter 9 to include the effects of distributed forces of electric origin and material motion By the use of simple models we illustrate the basic phenomena that occur in a wide variety of physical systems and the analytical techniques used in their mathematical description In spite of the diversity of physical situations in which continuum electromechanical interactions are important a unity results from mathematical techniques that are common to all of the situations It is our purpose here to illuminate in the simplest context possible these mathematical techniques and the physical phenomena they describe As stated earlier the techniques presented are fundamental to a wide variety of physical situations It is therefore helpful for the purpose of appreciating our objectives to review some of the technical areas concerned with continuum electromechanics Magnetohydrodynamics MHD is concerned with the interactions of free currents and magnetic fields in fluids liquids and gases which have high enough electrical conductivity that a quasi static magnetic field model is appropriate for describing the electromagnetic part of the system To reflect more accurately the nature of the mechanical medium this area is sometimes referred to as magnetogasdynamics MGD or as magnetofluiddynamics MFD Areas of application include pumping and levitation of liquids usually Dynamics of Electromechanical Continua metals orientation and confinement of extremely hot ionized gases or plasmas as for example in thermonuclear fusion experiments electric power generation from ionized gases produced by combustion of fossil fuels or from heat produced in a fission reactor t and space propulsion achieved by electromagnetic acceleration of ionized gases Scientific interest in this area includes such geophysical and astrophysical topics as the origin of the earth s magnetic field in its liquid metal core and the dynamics of stellar structures composed of highly ionized gases A similar area isferrohydrodynamics which is concerned with magnetization interactions of magnetic fields with a ferromagnetic fluid Electrohydrodynamics EHD is concerned with interaction between electric fields and free or bound polarization charges in fluids The fluids may be extremely good insulators slightly conducting or even highly conducting The distinguishing feature is that the electromagnetic part of the system is described by a quasi static electric field model Applications of EHD include pumping and levitation of liquids and gases extraction of contaminants from gases such as smoke mixing of liquids orientation of liquids in near zerogravity environments augmentation of heat transfer and property measurements in fluid systems EHD interactions also occur in meteorology in which charge distribution in the atmosphere as in a thunderstorm is important and in surface physics in which the distribution of charges at an interface is significant as in frictional electrification tt The engineering and scientific applications ofelectron andion beams involve continuum electromechanical interactions Electron beams confined by magnetic fields and interacting through electric fields with distributed electric circuits are commonly used to generate power at microwave frequencies 1T In such applications the beam is represented by quasi static equations but the distributed electric circuits support electromagnetic waves and are not amenable to quasi static analysis Electron beams are also used for heating welding forming and purifying metals Charged particle beams electrons and ions are used for medical treatment for measuring collision cross sections and for heating plasmas D J Rose and M Clark Jr Plasmas and Controlled Fusion M I T Press and Wiley New York 1961 t G W Sutton and A Sherman Engineering Magnetohydrodynamics McGraw Hill New York 1965 1 Ibid p 447 R E Rosensweig Magnetic Fluids Intern Sci Technol 55 48 56 90 July 1966 H J White IndustrialElectrostatic Precipitation Addison Wesley Reading Mass 1963 tt L B Loeb Static Electrification Springer Berlin 1958 C C Johnson Fieldand Wave Electrodynamics McGraw Hill New York 1965 p 275 T H Stix The Theory ofPlasma Waves McGraw Hill New York 1962 p 107 Introduction Plasma dynamics is concerned with the behavior of gases composed at least in part of charged particles Thus continuum electromechanical interactions will affect the behavior of a plasma Probably the most common example of a plasma is the ionized gas in a fluorescent lamp Other examples are gas filled rectifiers flames such as rocket exhausts and the sun The physical characteristics of ionized gases can assume many forms In certain cases such as proposed fusion devices and MHD generators the plasma behaves as a highly conducting fluid and its dynamic behavior is described by a magnetohydrodynamic approximation In other cases the plasma is only slightly ionized and the electrohydrodynamic equations are appropriate In still other cases the plasma may be so tenuous that it is best described as a collection of noninteracting particles in imposed magnetic and electric fields In all of these regimes the plasma exhibits the basic phenomena of wave propagation and instability subjects that are treated in this chapter Electrons and holes in semiconductors usefully modeled as plasmas give rise to the name solid stateplasmas t These charges behave collectively like gaseous plasmas and are thus amenable to analysis with the same types of
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