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/termsLumped-Parameter ElectromechanicsWe carry out the indicated differentiations and include these two forces as sources inwriting Newton's second law for the two mechanical nodes.O2w[I1 + ('n2A 2•)] eow(l - X2) 2 EOw(I~ - x2)1X2 1-2 -X 2d2x, dx,= M + B,-+ Klz,s (m)q%w o._w.io(~ 1\ d2q, dxz-12 %-W 2£ +2 X2 +L (n)x O 22 1 dt2 dtEquations (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 thatall 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 betweenmechanical terminal pairs; thus all the coupling occurs through the electric fields. We notefurther 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 thesecoupling terms appear essentially as nonlinear elements whose properties depend on theelectrical variables (voltages).3.3 DISCUSSIONIn this chapter we have learned some of the general properties of conserva-tive electromechanical coupling networks. In the process we have indicatedtechniques for finding mechanical forces of electric origin once electricalterminal relations are known. We have also introduced techniques forstudying the energy conversion properties of coupling fields and illustratedthe method of writing complete equations of motion for electromechanicalsystems. In Chapter 5 we complete our study of lumped-parameter electro-mechanical systems by introducing techniques for solving the equations ofmotion and by emphasizing some of the more important phenomena thatoccur in these systems.PROBLEMS3.1. A simple plunger-type solenoid for the operation of relays, valves, etc., is representedin Fig. 3P.l. Assume that it is a conservative system and that its electrical equation ofstate is1 + x/a(a) Find the force that must be appliedto the plunger to hold it in equilibrium at adisplacement x and with a current i.Problems,-PlungerypermeableironFig. 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 illustratedin Fig. 3P.2. The system has the electrical equations of state v1 = S91q1 + S192a andv2 = S21q + Saq2.(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 Cin the q,-q2 plane illustrated in Fig. 3P.2b.(b) What is the relation between coefficients S12 and S21 to make these three values ofenergy the same?(c) Derive the result of (b) by assuming that the system is conservative and applyingreciprocity.q1 q2+0-Electric +vI field V2-0-system(a)Fig. 3P.23.3. A slab of dielectric slides between plane parallel electrodes as shown. The dielectricobeys the constitutive law D = oc(E .E)E + EOE, where Eois the permittivity of free spaceand a is a constant. Find the force of electrical origin on the slab. Your answer should takethe formf e =f (v, ).Lumped-Parameter ElectromechanicsDepth d into paperFig. 3P.33.4. A magnetic circuit, including a movable plunger, is shown in Fig. 3P.4. The circuit isexcited by an N-turn coil and consists of a perfectly permeable yoke and plunger with avariable air gap x(t) and a fixed nonmagnetic gap d. The system, with the cross sectionshown, has a width w into the paper. The following parts lead to a mathematical formulationof 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 throughthe magnetic material N times.(b) Find the energy WQ(2, x) stored in the electromechanical coupling. This shouldbe done by making use of part (a).(c) Use the energy function Wm(., x) to compute the force of electrical origin facting on the plunger.(d) Write an electrical (circuit) equation of motion involving A and x as the onlydependent variables and I(t) as a driving function.(e) Write the mechanical equation of motion for the mass. This differential equationshould have Aand x as the only dependent variables, hence taken with the resultof (d) should constitute a mathematical formulation appropriate for analyzingthe system dynamics.Width w into paper-Mass M&Fig. 3P.4I~>lct~ ~RXProblemsFig. 3P.53.5. A magnetic circuit with a movable element is shown in Fig. 3P.5. With this elementcentered, the air gaps have the same length (a). Displacements from this centered positionare denoted by x.(a) Find the electrical terminal relations Al(il, i2,x) and A2(il, i2,x)in terms of theparameters defined in the figure.(b) Compute the coenergy WQ(i1, i2,x) stored in the electromechanical coupling.3.6. An electrically nonlinear magnetic field coupling network illustrated in Fig. 3P.6 hasthe equations of state_i=I _ _+_,_ f '0Fr)2/P +=J4o1 + x-a a (1 + x/a)2 jwhere 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.i fe+ Magnetic field +A coupling x_ systemFig. 3P.6+;r-,Lumped-Parameter Electromechanicsi1+0-i2---X20-Magneticfieldcoupling+-0e----o+x2-0Fig. 3P.73.7. The electrical terminal variables of the electromechanical coupling network shown inFig. 3P.7 are known to be A2= axi,3 + bxlxpi2 and 22 = bxlx4i1 + cx2i2 , where a, b, andc are constants. What is the coenergy Wm(ii, i2, x1, x2) stored in the coupling network?3.8. A schematic diagram of a rotating machine with a superconducting rotor (moment ofinertia J) is shown in Fig. 3P.8. Tests have shown that )2 = i1L1 + izLm cog 0 and A• =iLm cos 0 + i2L2, where O(t) is the angular deflection of the shaft to which coil (2) isattached. The
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