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 Chapter 9 SIMPLE ELASTIC CONTINUA 9 0 INTRODUCTION The study of the effects of motion on electric and magnetic fields Chapter 7 and ofelectromagnetic force densities Chapter 8 provides the background necessary for an introduction to the electromechanics of continuous media To someone familiar with the dynamics of continuous media this is a pretentious statement for it implies that the description of distributed mechanical systems requires only a minor addition to the largely electromagnetic considerations so far introduced In general this is far from the case for example does the mechanical medium consist of a solid or a fluid In either case the equations of motion vary considerably with the particular fluid or solid under study These equations generally involve three dimensional deformations hence are likely to be at least as complicated as the electromagnetic field equations if not more so Fortunately many of the most significant and practical interactions with continuous media can be modeled in terms of one or two dimensional structures that not only retain the salient features of the three dimensional dynamics but represent idealizations that we should like to approach in practice In this and the next chapter attention is confined to situations in which the mechanical side of the electromechanical problem takes the form of one of two simple models the thin rod subject to longitudinal motions and wires and membranes undergoing transverse motions The derivation of the one and two dimensional equations of motion for these simple cases serves to illustrate the essential steps required to write the more general expressions for elastic media and fluids as undertaken in Chapters 11 and 12 At the same time the continuum electromechanical dynamics studied in this and the next chapter give a preview of types of dynamics found in acoustics fluid dynamics electron beam plasma dynamics magnetohydrodynamics electrohydrodynamics and microwave magnetics In this chapter the discussion is limited to electromechanical interactions Simple Elastic Continua with continuous media that occur through boundary conditions representable by terminal pairs In Chapter 10 we consider physical situations in which the electromechanical coupling is itself distributed and in which our lumped parameter concept of a terminal pair can no longer account for the coupling 9 1 LONGITUDINAL MOTION OF A THIN ROD Longitudinal motion of a thin elastic rod provides a logical first topic in discussing the dynamics of elastic continua This is true because we emphasize the wavelike nature of the dynamics and in a thin rod longitudinal waves have a particularly simple form As we shall see waves in a thin rod can propagate without changing their shapes hence they can be understood by means of comparatively simple mathematical techniques This distortion free behavior of the thin rod is used in applications such as acoustic delay lines and electromechanical filters in which the properties of the electromechanical system are especially attractive We discuss some applications later in this section To describe longitudinal motion in an elastic rod we must make a mathematical model This process consists essentially of two steps a a mathematical 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 uniform cross 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 are small enough that effects of any transverse motion are negligible The f I1 T xl X1 2 x p 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 Ax1 centered at xz Longitudinal Motion of a Thin Rod criterion for making this assumption is obtained from the treatment of threedimensional elasticity in Chapter 11 To describe force equilibrium at each point along the rod we write Newton s second law for a small element of length Azx centered at xx as illustrated in Fig 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 act throughout the volume of the element and surface forces applied to the transverse surfaces of the element by the adjacent material When we specify a volume force density of magnitude F in the xx direction and require that over the length Ax the force density shall not vary appreciably we can write the total body force f as fx F1 A Azx 9 1 1 This force is indicated in Fig 9 1 lb The forces applied at the surfaces of the element by adjacent matter are described in the following way Consider first the situation in Fig 9 1 2a in which a rod is at rest and subjected to equal and opposite forces of magnitudef When an imaginary transverse cut is made in the rod as illustrated in Fig 9 1 2b each segment must still be in equilibrium If there are no externally applied body forces the force f is applied to the two pieces of material 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 righthand segment is 7 il A 9 1 2 if Aiff x1x b Fig 9 1 2 An elastic rod in static equilibrium a the rod with applied forces b equilibrium conditions at an imaginary cut Simple Elastic Continua The traction applied to the right hand segment by the left hand segment is T i A 9 1 3 We define the stress T1 as in Section 8 2 1 transmitted by the rod as 9 1 4 Ti A Then we obtain the xl component of the mechanical traction rl as T1 9 1 5 T11n1 where n is the magnitude of the x1 component of the outwarddirected unit normal vector for the segment of rod to which the traction is applied For this one dimensional case illustrated in Fig 9 1 2 n 1 for the left hand segment and n 1 for the
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