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 3 LUMPED PARAMETER ELECTROMECHANICS 3 0 INTRODUCTION Having reviewed the derivations of lumped electric circuit elements and rigid body mechanical elements and generalized these concepts to allow inclusion of electromechanical coupling we are now prepared to study some of the consequences of this coupling In the analysis of lumped parameter electromechanical systems experience has shown that sufficient accuracy is obtained in most cases by making a lossless model of the coupling system Thus energy methods are used to provide simple and expeditious techniques for studying the coupling process After introducing the method of calculating the energy stored in an electromechanical coupling field we present energy methods for obtaining forces of electric origin We shall then study the energy conversion process in coupling systems and finally discuss the formalism of writing equations of motion for complete electromechanical systems The techniques for analyzing the dynamic behavior of lumped parameter electromechanical systems are introduced and illustrated in Chapter 5 3 1 ELECTROMECHANICAL COUPLING There 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 free charge a current 4 The force resulting from a magnetic field acting on magnetizable material Electromechanical Coupling K b Fig 3 1 1 a A magnetic field electromechanical system b its representation in terms of terminal pairs Note that the coupling network does not include mechanical energy storages M or electrically dissipative elements R Because of the restriction of our treatment to quasi static systems the fields 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 electric fields and the forces due to magnetic fields To illustrate how the coupling can be taken into account suppose the problem to be considered is the magnetic field system shown in Fig 3 1 1 The electromechanical coupling occurs between one electrical terminal pair with the variables i and A and one mechanical terminal pair composed of the node x acted on by the electrical forcefe It has been demonstrated in Sections 2 1 1 and 2 1 2 that the electrical terminal variables are related by an electrical terminal relation expressible in the form A 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 the value of A is known We now make a crucial assumption motivated by the form of the electrical equation given the current i and position x the force of electric origin has a certain single value 3 1 2 fe f i x pll 11 Lumped Parameter Electromechanics that is the forcefe exerted by the system in the box on the mechanical node is a function of the state i x This is reasonable if the box includes only those elements that store energy in the magnetic field Hence all purely electrical elements inductors that do not involve x capacitors and resistors and purely mechanical elements all masses springs and dampers are connected to the terminals externally Note that fe is defined as the force of electrical origin applied to the mechanical node in a direction that tends to increase the relative displacement x Because 3 1 1 can be solved for i to yield i i 0 x the forcef 6 can also be written as 3 1 3 3 1 4 It is well to remember that the functions of 3 1 2 and 3 1 4 are different because the variables are different however for a particular set of i A x the forcefe will have the same numerical value regardless of the equation used In a similar way the mechanical force of electric origin for an electric field system see Fig 3 1 2 can be written as fe fe q x 3 1 5 or 3 1 6 f fe v X fe f 2 x q 1 1 a K I Fig 3 1 2 b a An electric field electromechanical system b its representation in terms of terminal pairs Note that the coupling network does not include mechanical energy storage elements M or electrically dissipative elements G Electromechanical Coupling When the mechanical motion is rotational the same ideas apply We replace 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 and one mechanical terminal pair the discussion can be generalized to any arbitrary number of terminal pairs For instance if an electric field system has N electrical terminal pairs and M mechanical terminal pairs for which the terminal relations are specified by 2 1 36 then 3 1 6 is generalized to i21 v fie fie V1 VN X1 X 2 i 1 2 M XM 3 1 7 where the subscript i denotes the mechanical terminal pair at which f8 is applied to the external system by the coupling field The other forms off can be generalized in the same way The next question to be considered is how to determine the forcef for a particular system One method is to solve the field problem find force densities and then perform a volume integration to find the total force This process described in Chapter 8 supports our assumption that f has the form of 3 1 2 and 3 1 5 It is often impractical however to solve the field 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 fit an analytical curve to obtain a function in closed form This method also has obvious disadvantages It is shown in the next section that when the electrical terminal relations are known and the coupling system can be represented as lossless the forcef can be found analytically Because electrical lumped parameters are usually easier to calculate and or measure than mechanical forces this often provides the most convenient way of determining the mechanical forces of electric origin fe 3 1 1
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