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/termsChapter 3LUMPED-PARAMETERELECTROMECHANICS3.0 INTRODUCTIONHaving reviewed the derivations of lumped electric circuit elements andrigid-body mechanical elements and generalized these concepts to allowinclusion of electromechanical coupling, we are now prepared to study someof the consequences of this coupling.In the analysis of lumped-parameter electromechanical systems experiencehas shown that sufficient accuracy is obtained in most cases by making alossless model of the coupling system. Thus energy methods are used toprovide simple and expeditious techniques for studying the coupling process.After introducing the method of calculating the energy stored in anelectromechanical coupling field, we present energy methods for obtainingforces of electric origin. We shall then study the energy conversion process incoupling systems and finally discuss the formalism of writing equations ofmotion for complete electromechanical systems. The techniques for analyzingthe dynamic behavior of lumped-parameter electromechanical systems areintroduced and illustrated in Chapter 5.3.1 ELECTROMECHANICAL COUPLINGThere are four technically important forces of electric origin.1. The force resulting from an electric field acting on free charge.2. The force resulting from an electric field acting on polarizable material.3. The force resulting from a magnetic field acting on a moving freecharge (a current).4. The force resulting from a magnetic field acting on magnetizablematerial.Electromechanical CouplingK,(b)Fig. 3.1.1 (a) A magnetic field electromechanical system; (b) its representation in termsof terminal pairs. Note that the coupling network does not include mechanical energystorages (M) or electrically dissipative elements (R).Because of the restriction of our treatment to quasi-static systems, thefields that give rise to forces in a particular element are electric or magnetic,but not both. Thus we can consider separately the forces due to electricfields and the forces due to magnetic fields.To illustrate how the coupling can be taken into account suppose theproblem to be considered is the magnetic field system shown in Fig. 3.1.1.The electromechanical coupling occurs between one electrical terminalpair with the variables i and A and one mechanical terminal pair composedof the node x acted on by the electrical forcefe. It has been demonstrated inSections 2.1.1 and 2.1.2 that the electrical terminal variables are related by anelectrical terminal relation expressible in the formA = 2(i, x). (3.1.1)This relation tells us the value of A, given the values of i and x. We can say,given the state (i, x) of the magnetic field system enclosed in the box, that thevalue of A is known.We now make a crucial assumption, motivated by the form of the electricalequation: given the current i and position x, the force of electric origin has acertain single value---pll~-11__·fe =f"(i, x);(3.1.2)Lumped-Parameter Electromechanicsthat is, the forcefe exerted by the system in the box on the mechanical nodeis a function of the state (i, x). This is reasonable if the box includes onlythose elements that store energy in the magnetic field. Hence all purelyelectrical elements (inductors that do not involve x, capacitors, and resistors)and purely mechanical elements (all masses, springs, and dampers) areconnected to the terminals externally.Note thatfe is defined as the force of electrical origin applied to themechanical node in a direction that tends to increase the relative displacementx. Because (3.1.1) can be solved for i to yieldi= i(0, x), (3.1.3)the forcef6 can also be written asfe =f'(2, x). (3.1.4)It is well to remember that the functions of (3.1.2) and (3.1.4) are differentbecause the variables are different; however, for a particular set of i, A, xthe forcefe will have the same numerical value regardless ofthe equation used.In a similar way the mechanical force of electric origin for an electric fieldsystem (see Fig. 3.1.2) can be written asfe fe(q, x) (3.1.5)orf" =fe(v, X). (3.1.6)q -------------- 1KI(b)Fig. 3.1.2 (a) An electric field electromechanical system; (b) its representation in termsof terminal pairs. Note that the coupling network does not include mechanical energystorage elements (M) or electrically dissipative elements (G).1(a)Electromechanical CouplingWhen the mechanical motion is rotational, the same ideas apply. Wereplace force f by torque T' and displacement x by angular displacement 0.Although the systems of Figs. 3.1.1 and 3.1.2 have only one electrical andone mechanical terminal pair, the discussion can be generalized to anyarbitrary number of terminal pairs. For instance, if an electric field systemhas N electrical terminal pairs and M mechanical terminal pairs for which theterminal relations are specified by (2.1.36), then (3.1.6) is generalized tofie =fie(V1, i21(v *, VN; X1, X2 .... XM),(3.1.7)i= 1,2, .. ,M,where the subscript i denotes the mechanical terminal pair at which f8 isapplied to the external system by the coupling field. The other forms off canbe generalized in the same way.The next question to be considered is how to determine the forcef" for aparticular system. One method is to solve the field problem, find forcedensities, and then perform a volume integration to find the total force. Thisprocess, described in Chapter 8, supports our assumption that f has theform of (3.1.2) and (3.1.5). It is often impractical, however, to solve thefield problem. A second method of determining f is experimental; that is,if the device exists, we can measuref" as a function of the variables (i and x,2 and x, v and x, or q and x) on which it depends, plot the results, and fitan analytical curve to obtain a function in closed form. This method alsohas obvious disadvantages.It is shown in the next section that when the electrical terminal relationsare known and the coupling system can be represented as lossless the forcef"can be found analytically. Because electrical lumped parameters are
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