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MIT 6 003 - Study Notes

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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/termsChapter 9SIMPLE ELASTIC CONTINUA9.0 INTRODUCTIONThe study of the effects of motion on electric and magnetic fields (Chapter7) and ofelectromagnetic force densities (Chapter 8)provides the backgroundnecessary for an introduction to the electromechanics of continuous media.To someone familiar with the dynamics of continuous media this is a pre-tentious statement, for it implies that the description of distributed mechan-ical systems requires only a minor addition to the largely electromagneticconsiderations so far introduced. In general, this is far from the case; forexample, does the mechanical medium consist of a solid or a fluid? In eithercase the equations of motion vary considerably with the particular fluid orsolid under study. These equations generally involve three-dimensionaldeformations, hence are likely to be at least as complicated as the electro-magnetic field equations if not more so.Fortunately, many of the most significant and practical interactions withcontinuous media can be modeled in terms of one or two-dimensionalstructures that not only retain the salient features of the three-dimensionaldynamics but represent idealizations that we should like to approach inpractice. In this and the next chapter attention is confined to situations inwhich the mechanical side of the electromechanical problem takes the formof one of two simple models: the thin rod subject to longitudinal motions andwires and membranes undergoing transverse motions. The derivation of theone-and two-dimensional equations of motion for these simple cases servesto illustrate the essential steps required to write the more general expressionsfor elastic media and fluids, as undertaken in Chapters 11 and 12. At thesame time the continuum electromechanical dynamics studied in this and thenext chapter give a preview of types of dynamics found in acoustics, fluiddynamics, electron beam-plasma dynamics, magnetohydrodynamics, electro-hydrodynamics, and microwave magnetics.In this chapter the discussion is limited to electromechanical interactionsSimple Elastic Continuawith continuous media that occur through boundary conditions representableby terminal pairs. In Chapter 10 we consider physical situations in which theelectromechanical coupling is itself distributed and in which our lumpedparameter concept of a terminal pair can no longer account for the coupling.9.1 LONGITUDINAL MOTION OF A THIN RODLongitudinal motion of a thin elastic rod provides a logical first topic indiscussing the dynamics of elastic continua. This is true because we emphasizethe wavelike nature of the dynamics; and in a thin rod longitudinal waveshave a particularly simple form. As we shall see, waves in a thin rod canpropagate without changing their shapes; hence they can be understood bymeans of comparatively simple mathematical techniques. This distortion-freebehavior of the thin rod is used in applications such as acoustic delay linesand electromechanical filters in which the properties of the electromechanicalsystem are especially attractive. We discuss some applications later in thissection.To describe longitudinal motion in an elastic rod we must make a mathe-matical model. This process consists essentially of two steps: (a) a mathe-matical description of force equilibrium for a small element of the rod and(b) a description of the elastic property of the rod.We consider the long thin rod shown in Fig. 9.1.1a. The rod has a uniformcross section of area A perpendicular to the longitudinal (xl)-direction.We apply forces in the xj-direction and observe motion in the x1-direction.By "thin" we mean that the dimensions of the rod perpendicular to x1 aresmall enough that effects of any transverse motion are negligible. Thef=-I1 Txl x, + --pX1 2 X1 x 2+(b)Fig. 9.1.1 Thin elastic rod with axis in the xx-direction and uniform cross section of area A:(a) the rod; (b) force and tractions applied to an element of length Ax1centered at xz.Longitudinal Motion of a Thin Rodcriterion for making this assumption is obtained from the treatment of three-dimensional elasticity in Chapter 11.To describe force equilibrium at each point along the rod we write Newton'ssecond law for a small element of length Azx centered at xx, as illustrated inFig. 9.1.lb. There are two kinds of forces applied to this element of material:body forces, such as those due to gravity and electromagnetic fields, that actthroughout the volume of the element and surface forces applied to thetransverse surfaces of the element by the adjacent material.When we specify a volume force density of magnitude F, in the xx-directionand require that over the length Ax, the force density shall not vary appreci-ably, we can write the total body force f asfx = F1A Azx. (9.1.1)This force is indicated in Fig. 9.1.lbThe forces applied at the surfaces of the element by adjacent matter aredescribed in the following way. Consider first the situation in Fig. 9.1.2ain which a rod is at rest and subjected to equal and opposite forces of magni-tudef. When an imaginary transverse cut is made in the rod, as illustratedin Fig. 9.1.2b, each segment must still be in equilibrium. If there are noexternally applied body forces, the force f is applied to the two pieces ofmaterial at the cut, as shown. The vector force per unit area (or traction r,as discussed in Section 8.2.1) applied to the left-hand segment by the right-hand segment is7 = il -.(9.1.2)A-'ifAiffx1x(b)Fig. 9.1.2 An elastic rod in static equilibrium: (a) the rod with applied forces; (b) equi-librium conditions at an imaginary cut.Simple Elastic ContinuaThe traction applied to the right-hand segment by the left-hand segment isT = -i-.(9.1.3)AWe define the stress T1, (as in Section 8.2.1) transmitted by the rod asTi A= .(9.1.4)Then we obtain the xl-component of the mechanical traction rl asT1 = T11n1, (9.1.5)where n, is the magnitude of the x1-component of the outwarddirected unitnormal vector for the segment of rod to which the traction is applied.


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MIT 6 003 - Study Notes

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