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/termsDynamics of Electromechanical Continuaessentially the same features as that described here will result.* Often thetwisting motions that characterize the dynamics of the wire are found inother electromechanical systems that involve an imposed magnetic field.An example is the cyclotron wave of electron beam theory.t10.5 DISCUSSIONIn this chapter we have explored the consequences of continuum electro-mechanical coupling with simple elastic continua. This has produced mathe-matical analyses and physical interpretations of evanescent waves, absoluteinstabilities, and waves and instabilities in convecting systems. The unifyingmathematical concept is the dispersion relation presented graphically in thew-k plots.Although our examples have been framed in terms of simple physicalsituations, the phenomena we have discussed occur in the wide variety ofpractical situations indicated in Section 10.0 and throughout the chapter.PROBLEMS10.1. The current-carrying wire described in Section 10.1.2 is attached to a pair of dashpotswith damping coefficients B and driven at x = -1, as shown in Fig. 10P.1.(a) What is the boundary condition at x = 0?(b) Compute the power absorbed in the dashpots for o < co, given the amplitude 0oand other system parameters.xFig. 10P.1* Film Cartridge, produced by the National Committee for Fluid Mechanics Films,Current-InducedInstabilityofa Mercury Jet, may be obtained from Education DevelopmentCenter, Inc., Newton, Mass. The instability seen in this film is convective, as would be thecase here if the string were in motion with U > v,.f W. H. Louisell, Coupled Modes andParametricElectronics, Wiley, 1960, p. 51.x__Problems10.2. Consider the same physical situation as that described in Section 10.1.2, except withthe current-carrying wire constrained at x = 0, so that (a8/ax)(O, t) = 0, and driven atx = -1 such that (--1, t) = E cos wat.(a) Find analytical expressions for C(x, t) with wc > w~ and c, < we.(b) Sketch the results of (a) at an instant in time for cases in which wa = 0, w, < w,dod > COC.(c) How could the boundary condition at z = 0 be realized physically?10.3. The ends of the spring shown in Fig. 10.1.2 and discussed in Sections 10.1.2 and10.1.3 are constrained such that-(0, t) = 0,(-1, t) = 0.(a) What are the eigenfrequencies of the spring with the current as shown in Fig.10.1.2?(b) What are these frequencies with I as shown in Fig. 10.1.9?(c) What current I is required to make the equilibrium with & = 0 unstable? Give aphysical argument in support of your answer.10.4. In Section 10.1.2 a current-carrying wire in a magnetic field was described by theequation of motion82& a28m = f _x -_ Ibý + F(x, t), (a)where Fis an externally applied force/unit length. We wish to consider the flow of power onthe string. Because F la/8t is the power input/unit length to the string, we can find aconservation of power equation by multiplying (a) by a8/at. Show thataW aPPin =-- + -, (b)where Pin = F a8/at,W = energy stored/unit length4m + + Ib(2,P = power fluxaý a8= -f 8t-atx t10.5. Waves on the string in Problem 10.4 have the formE(x, t) = Re[t+ei(Wt-k) + .ei(wt+kz)J.This problem makes a fundamental point of the way in which power is carried by ordinarywaves in contrast to evanescent waves. The instantaneous power P carried by the string isgiven in Problem 10.4. Sinusoidal steady-state conditions prevail.(a) Compute the time average power carried by the waves under the assumption thatk is real. Your answer should show that the powers carried by the forward and·Dynamics of Electromechanical Continuabackward waves are independent; that is,(P> = kf -where 4* is the complex conjugate of ý.(b) Show that if k =jfl,# real we obtain by contrastKP) _=-_-A single evanescent wave cannot carry power.(c) Physically, how could it be argued that (b) must be the case rather than (a) for anevanescent wave?10.6. Use the results of Problem 10.4 to show that the group velocity vg = dowdk is givenby the ratio ofthe time average power to the time average energy/unit length: v, = (P)/( W).Attention should be confined to the particular case of Problem 10.4 with F = 0.10.7. A pair of perfectly conducting membranes has equilibrium spacing d from each otherand from parallel rigid walls (Fig. 10P.7). The membranes and walls support currents suchHoFig. 10P.7that with & = 0 and &z= 0 the static uniform magnetic field intensities H0 are as shown.As the membranes deform, the flux through each of the three regions between conductorsis conserved.(a) Assume that both membranes have tension S and mass/unit area am. Write twoequations of motion for X1and 5,.(b) Assume that 5z = Re i, expj(wt -kx) and 25,= Re , expj(wt -kx) and findthe dispersion equation.(c) Make an w-k plot to show complex values of k for real values of wto.Show whichbranch of this plot goes with E1 =--and which with 1-= -- .What are therespective cutoff frequencies for these odd and even modes?(d) The membranes are fixed at x = 0 and given the displacements 5,(-1, t) =--2(-1, t) = Re expojwt. Find 1EandX 2 and sketch for wt = 0.10.8. An electromagnetic wave can be transmitted through or reflected by a plasma,depending on the frequency of the wave relative to the plasma frequency ow,. This phenom-enon, which is fundamental to the propagation of radio signals in the ionosphere, isillustrated by the following simple example of a cutoff wave. In dealing with electromagneticwaves, we require that both the electric displacement current in Ampere's law and theProblemsEx (z, t)ov (z, 0)Fig. 10P.8magnetic induction in Faraday's law (see Section B.2.1) be accounted for. We considerone-dimensional plane waves in which E = iE#(z, t) and H = i,H,(z, t).(a) Show that Maxwell's equations require thataE _ -pyH, 8aH, 8eoE,az at az at(b) The space is filled with plasma composed of equal numbers of ions and electrons.Assume that the more massive ions remain fixed and that n, is the electronnumber density, whereas e and m are the electronic charge and mass. Use
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