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/termsAppendix GSUMMARY OF PARTS I AND IIAND USEFUL THEOREMSIDENTITIESAxB.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 V V- V2= ,V.V xA = 0,V xV= 0,V x (Vx A)= V(V -A) - V2A,(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 + V Vx A,V x (A x B) = A(V B) -B(V. A) + (B V)A -(A. V)B.-I·-----·l~·l~---·U~ZI---_ ___THEOREMSý d = -ka.dDivergence theoremIs A-n = nd .A dVStokes's theorem A dl =f(V x A).ndabC~ciiGScndaTable 1.2 Summary of Quasi-Static Electromagnetic EquationsDifferential EquationsIntegral EquationsMagnetic field systemElectric field systemVX H = JV.B = 0V.J =0aBV x E =TtVx E=OV D = pv J,--aaDV x H=J,f+(1.1.1)(1.1.11)(1.1.12)(1.1.14)(1.1.15)H -dl = fS J n daBB.nda = 0J -n da = 0E' dl =--B.ndawhere E' = E + v X BE .dl = 0SD -nda = fvp, dVfJ3' n da = -p dVdH'.dl = J, -n da + D .n dawhere J' = Jf -pfvH'=H-v x D(1.1.20)(1.1.21)(1.1.22)(1.1.23)(1.1.24)(1.1.25)(1.1.26)(1.1.27)__Table 2.1 Summary of Terminal Variables and Terminal RelationsMagnetic field system Electric field systemDefinition of Terminal VariablesChareeA, = BB.ndaCurrenti-f= Jy n'daýk"Iqk = f pidVVoltagevk E •dlTerminal ConditionsdtAk = Ak(i1 ""i-; geometry)ik = ik '"... AN; geometry)ik= dqkdtqk = qk( '...'VN; geometry)vk= CV(ql '.qN; geometry)Definitionof Terminal VariablesTable 3.1 Energy Relations for an Electromechanical Coupling Network with N Electricaland M Mechanical Terminal Pairs*Magnetic Field Systems Electric Field SystemsConservation of EnergyN MdWm = I ij dA, -f dxj5j1 j1N MdW -= di + e dfej=1 -1f5• = -- •-Wax .a (i. i; x1.... X )f e= ax,Nw,+ w.'=J=1N Jf(a) dWe = v dq -fij dxN I(c) dWe >= q1 du1+ :ý fe dxjj=l j=1Forces of Electric Origin, j= 1 ..., M(eW,(ql, qN; Xl... .x ,)(e) fe = -wei qx .Sax.Sa W(v,, ... ..V; xl ....,X31)(g) =iRelation of Energy to Coenergy(i) W + We = jqjj=1Energy and Coenergy from Electrical Terminal RelationsNlN (OWm i(A, ... j-,', 0 ... , 0; x ..... XM ) di' (k) We .( .q 1,•, 0 ...,...1 0The mechanical variables and can be regarded as theth force and displacement or theth torque T and angular displacement 0(n)T7hemechanicalva 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 withMoving MediaDifferential Equations Transformations Boundary ConditionsV x H = J, (1.1.1) H' = H (6.1.35) n x (Ha -Hb) = Kf (6.2.14)V. B = 0 (1.1.2) B' = B (6.1.37) n. (Ba -Bb) = 0 (6.2.7)field V. J! = 0 (1.1.3) J,= J, (6.1.36) n .(Jfa -Jb) + Vy• K, = 0 (6.2.9)systems aBV x E = -(1.1.5) E' = E + vr x B (6.1.38) n X (Ea -Eb) = vn(Ba -Bb) (6.2.22)B = Io(H + M) (1.1.4) M' = M (6.1.39)V X E = 0 (1.1.11) E' =E (6.1.54) n (E -Eb) =-0 (6.2.31)V.D = pf (1.1.12) D' = D (6.1.55) n (Da -- Db) = a (6.2.33); = Pf (6.1.56)Electric , ao81. bElectric J= -(1.1.14) J,= J= -pvr (6.1.58) n * (Ja _-Jb)+ V~. K, = -V(pla ) -(6.2.36)field at atsystems DV x H = Jf + (1.1.15) H' = H - v' X D (6.1.57) n X (Ha -Hb) = K + vn X [n x (Da -Db)] (6.2.38)D = CoE + P (1.1.13) P' = P (6.1.59)Appendix GFrom Chapter 8; The Stress Tensor and Related Tensor ConceptsIn what follows we assume a right-hand cartesian coordinate systemxL, x2, x. The component of a vector in the direction of an axis carries thesubscript of that axis. When we write F, we mean the mth component of thevector F, where m can be 1, 2, or 3. When the index is repeated in a singleterm, it implies summation over the three values of the indexaH, 8aH 8H, aH,andand a a a aH, = H1 + H1 2 + Hs H V.8X, 8ax 8x axThis illustrates the summation convention. On the other hand, 8H,/ax,represents any one of the nine possible derivatives of components of H withrespect to coordinates. We define the Kronecker delta 68,,, which has the values1, when m = n,6,, = (8.1.7)0, when m 0 n.The component Tn,, of the stress tensor can be physically interpreted as themth component of the traction (force per unit area) applied to a surface witha normal vector in the n-direction.ix1x3X2Fig. 8.2.2 Rectangular volume with center at (z@, x, Xs) showing the surfaces and direc-tions of the stresses T,,.I-*··~-·····)-·LIUI··~11111Summary of Parts I and IIThe xl-component of the total force applied to the material within thevolume of Fig. 8.2.2 is= T + , x3 Ax2Ax3 -rT1x -l I , x Ax2 Ax.+12 12 + 3 T12\1. 2 2 1+T (xiX2,z, x + 2 AxAX -T xx1,-4,-2Ax3) Ax3\(8.2.3)Here we have evaluated the components of the stress tensor at the centersof the surfaces on which they act; for example, the stress component T11acting on the top surface is evaluated at a point having the same x2-and x3-coordinates as the center of the volume but an x1 coordinate Ax1/2 above thecenter.The dimensions of the volume have already been specified as quite small.In fact, we are interested in the limit as the dimensions go to zero. Con-sequently, each component of the stress tensor is expanded in a Taylor seriesabout the value at the volume center with only linear terms in each seriesretained to write (8.2.3) as( Ax1 ITn1 Ax aT1=T + T T11 + -1 a ,,AAx32 8x1 2 ax1A,,x2T12 T1 2+ Ax i-T Ax1AxA+ L_ Ax,Ax,+_x (T3_.aT..__T13 +A3 aTh) ATx1Ax2 ax, 2 ax3orf = + a +T-xAxAx3.(8.2.4)All terms in this expression are to be evaluated at the center of the volume(x1, x,, xa). We have thus verified our physical intuition that space-varyingstress tensor components are necessary to obtain a net force.From (8.2.4) we can obtain the x,-component of the force density F at thepoint (x1, X2, x3) by writingF1 = lim T11 + + aT13 (8.2.5)Ax1Ayx,,Ax-OAxAxAxz, ax, ax2 ax,The limiting process makes the expansion of (8.2.4) exact. The summationconvention is used to write (8.2.5) asAppendix GF1 = T- (8.2.6)ax"A similar process for the other two components of the force and force densityyields the
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