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 Dynamics of Electromechanical Continua essentially the same features as that described here will result Often the twisting motions that characterize the dynamics of the wire are found in other electromechanical systems that involve an imposed magnetic field An example is the cyclotron wave of electron beam theory t 10 5 DISCUSSION In this chapter we have explored the consequences of continuum electromechanical coupling with simple elastic continua This has produced mathematical analyses and physical interpretations of evanescent waves absolute instabilities and waves and instabilities in convecting systems The unifying mathematical concept is the dispersion relation presented graphically in the w k plots Although our examples have been framed in terms of simple physical situations the phenomena we have discussed occur in the wide variety of practical situations indicated in Section 10 0 and throughout the chapter PROBLEMS 10 1 The current carrying wire described in Section 10 1 2 is attached to a pair of dashpots with 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 o 0 and other system parameters x x Fig 10P 1 Film Cartridge produced by the National Committee for Fluid Mechanics Films Current InducedInstabilityofa Mercury Jet may be obtained from Education Development Center Inc Newton Mass The instability seen in this film is convective as would be the case here if the string were in motion with U v f W H Louisell Coupled Modes and ParametricElectronics Wiley 1960 p 51 Problems 10 2 Consider the same physical situation as that described in Section 10 1 2 except with the current carrying wire constrained at x 0 so that a8 ax O t 0 and driven at x 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 d od CO C 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 and 10 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 a physical argument in support of your answer 10 4 In Section 10 1 2 a current carrying wire in a magnetic field was described by the equation of motion 82 m a28 f x Ib F x t a where Fis an externally applied force unit length We wish to consider the flow of power on the string Because F la 8t is the power input unit length to the string we can find a conservation of power equation by multiplying a by a8 at Show that aW aP Pin b where Pin F a8 at W energy stored unit length 4m 2 Ib P power flux a a8 8t f atx t 10 5 Waves on the string in Problem 10 4 have the form E 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 ordinary waves in contrast to evanescent waves The instantaneous power P carried by the string is given in Problem 10 4 Sinusoidal steady state conditions prevail a Compute the time average power carried by the waves under the assumption that k is real Your answer should show that the powers carried by the forward and Dynamics of Electromechanical Continua backward waves are independent that is kf P where 4 is the complex conjugate of b Show that if k jfl real we obtain by contrast KP A single evanescent wave cannot carry power c Physically how could it be argued that b must be the case rather than a for an evanescent wave 10 6 Use the results of Problem 10 4 to show that the group velocity vg dowdk is given by the ratio of the 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 other and from parallel rigid walls Fig 10P 7 The membranes and walls support currents such Ho Fig 10P 7 that 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 conductors is conserved a Assume that both membranes have tension S and mass unit area am Write two equations of motion for and 5 X1 b Assume that 5z Re i expj wt kx and 25 Re expj wt kx and find the dispersion equation c Make an w k plot to show complex values of k for real values of wto Show which branch of this plot goes with E1 and which with 1 What are the respective cutoff frequencies for these odd and even modes d The membranes are fixed at x 0 and given the displacements 5 1 t aXnd 2 and sketch for wt 0 expojwt Find 1E 2 1 t Re 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 is illustrated by the following simple example of a cutoff wave In dealing with electromagnetic waves we require that both the electric displacement current in Ampere s law and the Problems Ex z t ov z 0 Fig 10P 8 magnetic induction in Faraday s law see Section B 2 1 be accounted for We consider one dimensional plane waves in which E iE z t and H i H z t a Show that Maxwell s equations require that aE az pyH at 8aH az 8eoE 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 electron number density whereas e and m are the electronic charge and mass Use a linearized force equation to relate E and v where v is the average electron velocity in the x direction c …
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