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 Appendix G SUMMARY OF PARTS I AND II AND USEFUL THEOREMS IDENTITIES AxB C A Bx C A x B x C B A C C A B V V VO vv V A B V A V B Vx A B V x A V x B V Y Vy Y V V vA A VV VV A V AxB B VxA A VxB V VV V2 V V xA 0 V xV 0 V x Vx A V V A V2 A V x A x A A V A IV A A V A B A V B B V A A x V x B B x V x A V x A Vo x A VVx A V x A x B A V B B V A B V A A V B I l l U ZI THEOREMS bC d ka d cii G Divergence theorem Is A nnd A dV Sc Stokes s theorem A dl f V x A nda nda Table 1 2 Summary of Quasi Static Electromagnetic Equations Differential Equations Magnetic field system VX H J Integral Equations 1 1 1 H dl fS J V B 0 B nda 0 B 1 1 21 V J 0 J n da 0 1 1 22 V x E aB E dl Tt where E Electric field system 1 1 20 n da Vx E O 1 1 11 V D p 1 1 12 v J a V x H J f 1 1 14 aD 1 1 15 1 1 23 B nda E vXB 1 1 24 E dl 0 1 1 25 SD nda fvp dV J f 3 n da H dl d p d V J n da where J Jf pfv H H v x D 1 1 26 D n da 1 1 27 Table 2 1 Summary of Terminal Variables and Terminal Relations Magnetic field system Electric field system of Terminal Variables Definition Definition of Terminal Variables Charee A B nda B qk Current i f pidV Voltage f Jy n da vk E dl k I Terminal Conditions ik dqk dt dt Ak Ak i1 ik ik i geometry AN geometry qk qk VN geometry v k CV ql qN geometry Energy Relations for an Electromechanical Coupling Network with N Electrical and M Mechanical Terminal Pairs Table 3 1 Magnetic Field Systems Electric Field Systems Conservation of Energy N M dWm I ij dA 5j1 dxj a v dq fij dx N c e dfe di j 1 Jf dWe j1 M N dW N f I dWe q1 du 1 j l 1 fe dxj j 1 Forces of Electric Origin j 1 M f5 Wax f a e i ax e i x 1 X g fe i Sa wei eW ql W v qN qx Xl x aSx V xl X3 1 Relation of Energy to Coenergy N w w i W We J 1 j 1 jqj Energy and Coenergy from Electrical Terminal Relations Wm i A 1 al 0 variables The mechanic T7hemechanicalva NlN j and 0 0 x XM di k O We q 1 can be regarded as theth force and displacement or theth torque 0 T and angular displacement 0 n riables fi and xi can be regarded as thejth force and displaement or trejth torque Tj and angular displacement Oj Table 6 1 Differential Equations Transformations and Boundary Conditions for Quasi static Electromagnetic Systems with Moving Media Differential Equations V x H J field systems 1 1 1 Transformations H H 6 2 14 1 1 2 B B 6 1 37 n Ba Bb 0 V J 0 1 1 3 J 6 1 36 n Jfa Jb Vy 1 1 5 E E v r x B 6 1 38 n X E a Eb vn Ba Bb B Io H M 1 1 4 M M 6 1 39 V X E 0 1 1 11 E E 6 1 54 aB V x E J field 8 1 D CoE P D n 6 2 7 K 0 6 2 9 6 2 22 E Eb 0 a b 6 2 31 D D 1 1 14 J pvr J 6 1 58 1 1 15 H H v X D 6 1 57 n X Ha Hb K vn X n x D a Db 6 2 38 1 1 13 P P 6 1 59 at V x H Jf J 1 1 12 systems n x H a Hb Kf V B 0 V D pf Electric 6 1 35 Boundary Conditions Pf 6 1 55 n D D a 6 2 33 6 1 56 b n Ja Jb V K a V pl ao at 6 2 36 Appendix G From Chapter 8 The Stress Tensor and Related Tensor Concepts In what follows we assume a right hand cartesian coordinate system xL x 2 x The component of a vector in the direction of an axis carries the subscript of that axis When we write F we mean the mth component of the vector F where m can be 1 2 or 3 When the index is repeated in a single term it implies summation over the three values of the index aH 8aH 8H aH and and a H 8X H1 a 8ax a H1 2 8x Hs a H V ax This illustrates the summation convention On the other hand 8H ax represents any one of the nine possible derivatives of components of H with respect to coordinates We define the Kronecker delta 68 which has the values 1 when m n 8 1 7 6 0 when m 0 n The component Tn of the stress tensor can be physically interpretedas the mth component of the traction force per unit area applied to a surface with a normal vector in the n direction ix1 x3 X2 Fig 8 2 2 Rectangular volume with center at z x Xs showing the surfaces and directions of the stresses T I LIUI 11111 Summary of Parts I and II The xl component of the total force applied to the material within the volume of Fig 8 2 2 is T 12 T Ax 2 Ax3 x3 1 2 3 2 xiX2 z x AxAX I rT1lx T12 1 2 xx1 4 T Ax 2 Ax x 1 2 Ax3 23 Ax 8 2 3 Here we have evaluated the components of the stress tensor at the centers of the surfaces on which they act for example the stress component T11 acting on the top surface is evaluated at a point having the same x2 and x3coordinates as the center of the volume …
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