Electromechanical Dynamics (Part 1).0069

Electromechanical Dynamics (Part 1).0069 - d 20 ix[J d 2-2...

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Lumped Electromechanical Elements Using (2.2.20) in (2.2.21) and simplifying, we find that dT+dTA = p dV i,(x2 + y2) i[, z 2 d - dt dt' dt) F d 2 O d0 2 1 i t2 zzdz ( L, (2.2.22) where dT the torque from external sources, where dT = the torque from external sources, dT, = the torque from internal sources. To find the total acceleration torque on the body we must integrate (2.2.22) throughout the volume V of the body. For this purpose we find it convenient to define the moment of inertia about the z-axis as J.= f(z2 + y 2 )p dV (2.2.23) and the products of inertia* J., = xzp dV, (2.2.24) J,ý = yzp dV. (2.2.25) When we use these parameters and integrate (2.2.22) throughout the volume V of the body, the internal torque integrates to zero [see (2.2.14) and (2.2.15) and the associated footnote for arguments similar to those required here], and we obtain for the total acceleration torque T applied by external sources
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Unformatted text preview: d 20 ix[J d 2-2 dO 2 . d 2 0 9 dO\ 2 l T dt2, -is d) + J, . (2.2.26) With the restriction to rotation about a fixed axis, only the first term in this exoression affects the dynamics of the body. Thus we write the z component of (2.2.26) as d2O T,= Jd (2.2.27) dt 2 and represent this element in a mechanical circuit as in Fig. 2.2.14. Note that this Inertial reference circuit has exactly the same form as that Fig. 2.2.14 Circuit representation adopted earlier for representing mass in a of a moment of inertia, translational system (see Fig. 2.2.11). * See, for example, I. H. Shames, Engineering Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1960, pp. 187-188. A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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