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 Lumped Parameter Electromechanics We carry out the indicated differentiations and include these two forces as sources in writing Newton s second law for the two mechanical nodes O 2w I 1 n2A 2 2 1X eow l 2 X2 2 EOw I x 2 2 X 1 d2x M w 12 q W x o w 2 i 2 o O 22 21 1 dx B d2q X L dxz dt2 sKlz m n dt Equations h i m and n are the four equations of motion for the system in Fig 3 2 2 Several important aspects of these equations should be examined First we note that all four equations are coupled that is each equation contains all four dependent variables We also note that there is no external coupling between electrical terminal pairs and between mechanical terminal pairs thus all the coupling occurs through the electric fields We note further that the coupling between the two mechanical terminal pairs see m and n results in terms that are functions of mechanical positions and voltages Thus these coupling terms appear essentially as nonlinear elements whose properties depend on the electrical variables voltages 3 3 DISCUSSION In this chapter we have learned some of the general properties of conservative electromechanical coupling networks In the process we have indicated techniques for finding mechanical forces of electric origin once electrical terminal relations are known We have also introduced techniques for studying the energy conversion properties of coupling fields and illustrated the method of writing complete equations of motion for electromechanical systems In Chapter 5 we complete our study of lumped parameter electromechanical systems by introducing techniques for solving the equations of motion and by emphasizing some of the more important phenomena that occur in these systems PROBLEMS 3 1 A simple plunger type solenoid for the operation of relays valves etc is represented in Fig 3P l Assume that it is a conservative system and that its electrical equation of state is 1 x a a Find the force that must be appliedto the plunger to hold it in equilibrium at a displacement x and with a current i Problems Plunger ypermeable iron Fig 3P 1 b Make a labeled sketch of the force of part a as a function of x with constant i c Make a labeled sketch of the force of part a as a function of x with constant A 3 2 An electrically linear electric field system with two electrical terminal pairs is illustrated in Fig 3P 2 The system has the electrical equations of state v 1 S91q1 S1 92a and v2 S21 q Saq 2 See Example 3 1 1 for a physical case of this type a Calculate the energy input to the system over each of the three paths A B and C in the q q2 plane illustrated in Fig 3P 2b b What is the relation between coefficients S1 2 and S 21 to make these three values of energy the same c Derive the result of b by assuming that the system is conservative and applying reciprocity q2 q1 0vI Electric field 0 system V2 a Fig 3P 2 3 3 A slab of dielectric slides between plane parallel electrodes as shown The dielectric obeys the constitutive law D oc E E E EOE where Eois the permittivity of free space and a is a constant Find the force of electrical origin on the slab Your answer should take the formf e f v Lumped Parameter Electromechanics Depth d into paper Fig 3P 3 3 4 A magnetic circuit including a movable plunger is shown in Fig 3P 4 The circuit is excited by an N turn coil and consists of a perfectly permeable yoke and plunger with a variable air gap x t and a fixed nonmagnetic gap d The system with the cross section shown has a width w into the paper The following parts lead to a mathematical formulation of the equations of motion for the mass M given the excitation I t a Find the terminal relation for the flux 2 i z linked by the electrical terminal pair Ignore fringing in the nonmagnetic gaps Note that the coil links the flux through the magnetic material N times b Find the energy WQ 2 x stored in the electromechanical coupling This should be done by making use of part a c Use the energy function Wm x to compute the force of electrical origin f acting on the plunger d Write an electrical circuit equation of motion involving A and x as the only dependent variables and I t as a driving function e Write the mechanical equation of motion for the mass This differential equation should have Aand x as the only dependent variables hence taken with the result of d should constitute a mathematical formulation appropriate for analyzing the system dynamics Width w into paper I lct RX Mass Fig 3P 4 M Problems r Fig 3P 5 3 5 A magnetic circuit with a movable element is shown in Fig 3P 5 With this element centered the air gaps have the same length a Displacements from this centered position are denoted by x a Find the electrical terminal relations Al il i2 x and A2 il i2 x in terms of the parameters defined in the figure x stored in the electromechanical coupling b Compute the coenergy WQ i 1 i2 3 6 An electrically nonlinear magnetic field coupling network illustrated in Fig 3P 6 has the equations of state I f i 1 x a o 0 Fr 2 P J4 1 x a 2 a j where I0 Ar and a are positive constants a Prove that this system is conservative b Evaluate the stored energy at the point Ax x in variable space fe i A Magnetic field coupling system Fig 3P 6 x Lumped Parameter Electromechanics i 1 0i2 0 Magnetic field coupling e o x2 X 2 0 0 Fig 3P 7 3 7 The electrical terminal variables of the electromechanical coupling network shown in Fig 3P 7 are known to be A2 axi 3 bxlxpi2 and 22 bxlx4i 1 cx2i2 where a b and c are constants What is the coenergy Wm ii i2 x 1 x2 stored in the coupling network 3 8 A schematic diagram of a rotating machine with a superconducting rotor moment of inertia J is shown in Fig 3P 8 Tests have shown that 2 i1 L1 izLm cog 0 and A iLm cos 0 i2L2 where O t is the angular deflection of the shaft to which coil 2 is attached The machine is placed in operation as follows a With the 2 terminals open circuit and the shaft at 0 …
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