{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 17 "Euler's equations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 290 "We demon strate the free rotation of a rigid body (e.g., our textbook) by solvi ng Euler's equations for the angular velocity vector that defines the \+ motion relative to an inertial frame, but expressing it using componen ts in the body frame (such that the moments of inertia remain constant )." }}{PARA 0 "" 0 "" {TEXT -1 149 "The numerical solution demonstrate s the instability of the motion around the principal axis associated w ith the intermediate moment of inertia value." }}{PARA 0 "" 0 "" {TEXT -1 149 "Euler's equations are given in the text at the bottom of page 329, i.e., in Knudsen-Hjorth: Elements of Newtonian Mechanics, S pringer 2001 (3rd ed.)." }}{PARA 0 "" 0 "" {TEXT -1 107 "We use [w1,w2 ,w3] for the angular velocity vector, and I1,I2,I3 for the three princ ipal moments of inertia." }}{PARA 0 "" 0 "" {TEXT -1 92 "N1,N2,N3 are \+ the body-frame components of the torque which we set to zero for free \+ rotation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "E1:=I1*diff(w1 (t),t)-(I2-I3)*w2(t)*w3(t)=N1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E 1G/,&-%%diffG6$-%#w1G6#%\"tGF-$\"+n\"z1E$!#7*($\"+.v;#=$F0\"\"\"-%#w2G F,F4-%#w3GF,F4F4%#N1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "E2 :=I2*diff(w2(t),t)-(I3-I1)*w3(t)*w1(t)=N2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G/,&-%%diffG6$-%#w2G6#%\"tGF-$\"+n;lSv!#7*($\"+.+9 iuF0\"\"\"-%#w3GF,F4-%#w1GF,F4!\"\"%#N2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "E3:=I3*diff(w3(t),t)-(I1-I2)*w1(t)*w2(t)=N3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G/,&-%%diffG6$-%#w3G6#%\"tGF-$\"+ <>Gs5!#6*($\"++D(*zU!#7\"\"\"-%#w1GF,F5-%#w2GF,F5F5%#N3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Free rotation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "E1:=subs(N1=0,E1): E2:=subs(N2=0,E2): E3:=subs(N3= 0,E3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "For our textbook we hav e in SI units:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "M:=1.63;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"$j\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=235*10^(-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"#Z\"$+#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=154*10^(-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"#x \"$+&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "c:=17*10^(-3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG#\"#<\"%+5" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "We can use the result for a rectangular plate spi nning about the CM to find the three principal moments of inertia:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I1:=M/12*(b^2+c^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G$\"+n\"z1E$!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I2:=M/12*(a^2+c^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G$\"+n;lSv!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I3:=M/12*(a^2+b^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#I3G$\"+<>Gs5!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "The equa tions are ready for numerical solution, and we can specify the initial conditions. We start the motion about the 1-axis (smallest moment of \+ inertia, should be stable), and admix small contributions about the ot her axes (it is never possible to hit the symmetry axis perfectly when throwing the book)." }}{PARA 0 "" 0 "" {TEXT -1 120 "We give the main initial spin axis a value of about one revolution per second, and the others about one percent of that." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "E1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-% #w1G6#%\"tGF+$\"+n\"z1E$!#7*($\"+.v;#=$F.\"\"\"-%#w2GF*F2-%#w3GF*F2F2 \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "IC:=w1(0)=6.28,w2( 0)=5/100,w3(0)=5/100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6%/-%#w 1G6#\"\"!$\"$G'!\"#/-%#w2GF)#\"\"\"\"#?/-%#w3GF)F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "sol:=dsolve(\{E1,E2,E3,IC\},\{w1(t),w2(t),w 3(t)\},numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "W1:=subs(sol,w1(t)): W2:=subs(sol,w2(t)): W3:=subs(so l,w3(t)):" }}}{EXCHG {PARA 0 "> " 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