{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 278 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 279 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 260 280 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey \+ Emphasis" -1 281 "Times" 1 12 96 52 84 1 0 1 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 "Heading 1" -1 3 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "H eading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 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 "Time s" 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 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" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "Resonance of an under-damped osci llatory system " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaim o, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 281 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives \+ its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\ \\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Introduction " }}{PARA 0 "" 0 "" {TEXT -1 150 "In practice friction is always present in any physical oscilla tory system so the simple treatment of resonance, involving no damping , is unrealistic. " }}{PARA 0 "" 0 "" {TEXT -1 171 "We now investigate the steady-state part of the solution of the differential equation fo r an under-damped oscillatory system which is subject to a periodic dr iving force. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Acknowledgement: The basic ideas in this worksheet are t aken from a Maple worksheet by Robert Lopez. " }}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }{URLLINK 17 "http://www.mapleapps.com/powertools/e ngineeringmath/html/Section%2005-11.html" 4 "http://www.mapleapps.com/ powertools/engineeringmath/html/Section%2005-11.html" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Genuine resonance and resonant angular frequency .. example " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$ %\"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+10*x = 10* cos*gamma*t;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"#5F'%\"xGF'F'**F+F'%$ cosGF'%&gammaGF'%\"tGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "models a damped oscillator subject to a driving force with angular frequency " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We inves tigate how the amplitude of the steady-state part of the solution depe nds on the angular frequency " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "To solve this differential equation consider the homogene ous differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+10*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"#5F'%\"xGF'F'\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+10 = 0;" "6#/,(*$% \"mG\"\"#\"\"\"*&F'F(F&F(F(\"#5F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Completing the square gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+1 = -10+1;" "6#/,(*$%\"mG\"\"# \"\"\"*&F'F(F&F(F(F(F(,&\"#5!\"\"F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(m+1)^2 = -9;" "6#/*$,&%\"mG\"\"\"F'F'\"\"#,$\"\"*!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m+1 = ``;" "6#/,&%\"mG\"\"\"F&F& %!G" }{TEXT 262 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3;" "6#\"\"$" } {TEXT -1 1 " " }{TEXT 264 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "m = -1;" "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 1 " " } {TEXT 263 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3;" "6#\"\"$" }{TEXT -1 1 " " }{TEXT 265 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The complementary solution is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(t) = C[1]*exp(-t)* cos(3*t)+C[2]*exp(-t)*sin(3*t);" "6#/-%\"cG6#%\"tG,&*(&%\"CG6#\"\"\"F- -%$expG6#,$F'!\"\"F--%$cosG6#*&\"\"$F-F'F-F-F-*(&F+6#\"\"#F--F/6#,$F'F 2F--%$sinG6#*&F7F-F'F-F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The appropriate form for a trial particular solution is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-% \"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*gamma*t+B*sin*gamma*t;" "6#/%! G,&**%\"AG\"\"\"%$cosGF(%&gammaGF(%\"tGF(F(**%\"BGF(%$sinGF(F*F(F+F(F( " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=``" "6#/*&%#dxG\"\"\"%#dtG !\"\"%!G" }{XPPEDIT 18 0 "`p '`(t) = -gamma*A*sin*gamma*t+gamma*B*cos* gamma*t;" "6#/-%$p~'G6#%\"tG,&*,%&gammaG\"\"\"%\"AGF+%$sinGF+F*F+F'F+! \"\"*,F*F+%\"BGF+%$cosGF+F*F+F'F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)=``" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"%! G" }{XPPEDIT 18 0 "`p ''`(t) = -gamma^2*A*cos*gamma*t-gamma^2*B*sin*ga mma*t;" "6#/-%%p~''G6#%\"tG,&*,%&gammaG\"\"#%\"AG\"\"\"%$cosGF-F*F-F'F -!\"\"*,F*F+%\"BGF-%$sinGF-F*F-F'F-F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6 #*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 5 " and " }{TEXT 266 1 "x" }{TEXT -1 37 " in the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-gamma^2*A*cos*gamm a*t-gamma^2*B*sin*gamma*t+2*(-gamma*A*sin*gamma*t+gamma*B*cos*gamma*t) +10*(A*cos*gamma*t+B*sin*gamma*t) = 10*cos*gamma*t;" "6#/,**,%&gammaG \"\"#%\"AG\"\"\"%$cosGF)F&F)%\"tGF)!\"\"*,F&F'%\"BGF)%$sinGF)F&F)F+F)F ,*&F'F),&*,F&F)F(F)F/F)F&F)F+F)F,*,F&F)F.F)F*F)F&F)F+F)F)F)F)*&\"#5F), &**F(F)F*F)F&F)F+F)F)**F.F)F/F)F&F)F+F)F)F)F)**F5F)F*F)F&F)F+F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "By comparing coefficients of the cosine terms we obtain t he equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-gam ma^2*A+2*gamma*B+10*A = 10;" "6#/,(*&%&gammaG\"\"#%\"AG\"\"\"!\"\"*(F' F)F&F)%\"BGF)F)*&\"#5F)F(F)F)F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "( 10-gamma^2)*A+2*gamma*B = 10;" "6#/,&*&,&\"#5\"\"\"*$%&gammaG\"\"#!\" \"F(%\"AGF(F(*(F+F(F*F(%\"BGF(F(F'" }{TEXT -1 14 " ------- (i). " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 68 "By compa ring coefficients of the sine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-gamma^2*B-2*gamma*A+10*B = 0;" "6#/,(*&%&gammaG\"\"#%\"BG\"\"\"!\"\"*(F'F)F&F)%\"AGF)F**&\"#5F)F (F)F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(10-gamma^2)*B-2*gam ma*A = 0;" "6#/,&*&,&\"#5\"\"\"*$%&gammaG\"\"#!\"\"F(%\"BGF(F(*(F+F(F* F(%\"AGF(F,\"\"!" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Multiplying equation (i) \+ by " }{XPPEDIT 18 0 "2*gamma;" "6#*&\"\"#\"\"\"%&gammaGF%" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*gamm a*(10-gamma^2)*A+4*gamma^2*B = 20*gamma;" "6#/,&**\"\"#\"\"\"%&gammaGF ',&\"#5F'*$F(F&!\"\"F'%\"AGF'F'*(\"\"%F'*$F(F&F'%\"BGF'F'*&\"#?F'F(F' " }{TEXT -1 17 " ------- (iii), " }}{PARA 0 "" 0 "" {TEXT -1 33 "and \+ multiplying equation (ii) by " }{XPPEDIT 18 0 "10-gamma^2;" "6#,&\"#5 \"\"\"*$%&gammaG\"\"#!\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "(10-gamma^2)^2*B-2*gamma*(10-gamma^2) *A = 0;" "6#/,&*&,&\"#5\"\"\"*$%&gammaG\"\"#!\"\"F+%\"BGF(F(**F+F(F*F( ,&F'F(*$F*F+F,F(%\"AGF(F,\"\"!" }{TEXT -1 16 " ------- (iv). " }} {PARA 0 "" 0 "" {TEXT -1 40 "Adding equations (iii) and (iv) gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(gamma^4-16*gamma ^2+100)*B = 20*gamma;" "6#/*&,(*$%&gammaG\"\"%\"\"\"*&\"#;F)*$F'\"\"#F )!\"\"\"$+\"F)F)%\"BGF)*&\"#?F)F'F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B = 20*gamma/(gamma^4-16*gamma^2+100);" "6#/%\"BG*(\"#?\"\"\"%&g ammaGF',(*$F(\"\"%F'*&\"#;F'*$F(\"\"#F'!\"\"\"$+\"F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = (100-10*gamma^2)/(gamma^4-16*gamma^ 2+100);" "6#/%\"AG*&,&\"$+\"\"\"\"*&\"#5F(*$%&gammaG\"\"#F(!\"\"F(,(*$ F,\"\"%F(*&\"#;F(*$F,F-F(F.F'F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "This gives the particular solution: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "p(t) = ``((100-10*gamma^2)/(gamma^4-1 6*gamma^2+100))*cos*gamma*t+``(20*gamma/(gamma^4-16*gamma^2+100))*sin* gamma*t;" "6#/-%\"pG6#%\"tG,&**-%!G6#*&,&\"$+\"\"\"\"*&\"#5F0*$%&gamma G\"\"#F0!\"\"F0,(*$F4\"\"%F0*&\"#;F0*$F4F5F0F6F/F0F6F0%$cosGF0F4F0F'F0 F0**-F+6#*(\"#?F0F4F0,(*$F4F9F0*&F;F0*$F4F5F0F6F/F0F6F0%$sinGF0F4F0F'F 0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The general solut ion is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[ 1]*exp(-t)*cos(3*t)+C[2]*exp(-t)*sin(3*t)+``((100-10*gamma^2)/(gamma^4 -16*gamma^2+100))*cos*gamma*t+``(20*gamma/(gamma^4-16*gamma^2+100))*si n*gamma*t;" "6#/-%\"xG6#%\"tG,**(&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-- %$cosG6#*&\"\"$F-F'F-F-F-*(&F+6#\"\"#F--F/6#,$F'F2F--%$sinG6#*&F7F-F'F -F-F-**-%!G6#*&,&\"$+\"F-*&\"#5F-*$%&gammaGF;F-F2F-,(*$FM\"\"%F-*&\"#; F-*$FMF;F-F2FIF-F2F-F4F-FMF-F'F-F-**-FE6#*(\"#?F-FMF-,(*$FMFPF-*&FRF-* $FMF;F-F2FIF-F2F-F@F-FMF-F'F-F-" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 56 "________________________________________ ________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "No matter what initial conditions we take , the " }{TEXT 259 17 "steady-state part" }{TEXT -1 21 " of the soluti on is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = ``( (100-10*gamma^2)/(gamma^4-16*gamma^2+100))*cos*gamma*t+``(20*gamma/(ga mma^4-16*gamma^2+100))*sin*gamma*t;" "6#/-%\"xG6#%\"tG,&**-%!G6#*&,&\" $+\"\"\"\"*&\"#5F0*$%&gammaG\"\"#F0!\"\"F0,(*$F4\"\"%F0*&\"#;F0*$F4F5F 0F6F/F0F6F0%$cosGF0F4F0F'F0F0**-F+6#*(\"#?F0F4F0,(*$F4F9F0*&F;F0*$F4F5 F0F6F/F0F6F0%$sinGF0F4F0F'F0F0" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 34 "__________________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "This is a sine wave with amplitude " }{XPPEDIT 18 0 "C=sq rt(A^2+B^2)" "6#/%\"CG-%%sqrtG6#,&*$%\"AG\"\"#\"\"\"*$%\"BGF+F," } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "A = (100-10*gamma^2)/(gamma^4-16 *gamma^2+100);" "6#/%\"AG*&,&\"$+\"\"\"\"*&\"#5F(*$%&gammaG\"\"#F(!\" \"F(,(*$F,\"\"%F(*&\"#;F(*$F,F-F(F.F'F(F." }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B = 20*gamma/(gamma^4-16*gamma^2+100);" "6#/%\"BG*(\"#? \"\"\"%&gammaGF',(*$F(\"\"%F'*&\"#;F'*$F(\"\"#F'!\"\"\"$+\"F'F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforward t o check that the " }{TEXT 259 9 "amplitude" }{TEXT -1 5 " is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C=10/(gamma^4-16*gamm a^2+100)" "6#/%\"CG*&\"#5\"\"\",(*$%&gammaG\"\"%F'*&\"#;F'*$F*\"\"#F'! \"\"\"$+\"F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "We can perform this calculation with Mapl e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "de := diff(x(t),t$2)+2*diff(x(t),t)+10*x(t)=10*cos(g amma*t);\nunassign('A','B','p'):\np := t -> A*cos(gamma*t)+B*sin(gamma *t):\n'p(t)'=p(t);\nsubs(x(t)=p(t),de);\nsimplify(%);\nsolve(identity( %,t),\{A,B\});\nassign(%):\n'p(t)'=p(t);\n`amplitude`=simplify(sqrt(A^ 2+B^2),symbolic);\nG := unapply(rhs(%),gamma):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*& F1F2-F(6$F*F-F2F2*&\"#5F2F*F2F2,$*&F7F2-%$cosG6#*&%&gammaGF2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"tG,&*&%\"AG\"\"\"-%$cosG 6#*&%&gammaGF+F'F+F+F+*&%\"BGF+-%$sinGF.F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*&%\"AG\"\"\"-%$cosG6#*&%&gammaGF+%\"tGF +F+F+*&%\"BGF+-%$sinGF.F+F+-%\"$G6$F1\"\"#F+*&F9F+-F&6$F(F1F+F+*(\"#5F +F*F+F,F+F+*(F>F+F3F+F4F+F+,$*&F>F+F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*(%\"AG\"\"\"-%$cosG6#*&%&gammaGF'%\"tGF'F')F,\"\"#F '!\"\"*(%\"BGF'-%$sinGF*F'F.F'F0**F/F'F&F'F3F'F,F'F0**F/F'F2F'F(F'F,F' F'*(\"#5F'F&F'F(F'F'*(F8F'F2F'F3F'F',$*&F8F'F(F'F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/%\"AG,$*(\"#5\"\"\",(*$)%&gammaG\"\"%F)F)*&\"#;F)) F-\"\"#F)!\"\"\"$+\"F)F3,&*$F1F)F)F(F3F)F3/%\"BG,$*(\"#?F)F-F)F*F3F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"tG,&**\"#5\"\"\",(*$)%&g ammaG\"\"%F+F+*&\"#;F+)F/\"\"#F+!\"\"\"$+\"F+F5,&*$F3F+F+F*F5F+-%$cosG 6#*&F/F+F'F+F+F5**\"#?F+F/F+F,F5-%$sinGF;F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*amplitudeG,$*&\"#5\"\"\",(*$)%&gammaG\"\"%F(F(*&\"#; F()F,\"\"#F(!\"\"\"$+\"F(#F2F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The maximum amplitude occurs where " }{XPPEDIT 18 0 "Diff ([10/sqrt(gamma^4-16*gamma^2+100)],gamma)=0" "6#/-%%DiffG6$7#*&\"#5\" \"\"-%%sqrtG6#,(*$%&gammaG\"\"%F**&\"#;F**$F0\"\"#F*!\"\"\"$+\"F*F6F0 \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff( [10/sqrt(gamma^4-16*g amma^2+100)],gamma)=-5/(gamma^4-16*gamma^2+100)^(3/2)*(4*gamma^3-32*ga mma)" "6#/-%%DiffG6$7#*&\"#5\"\"\"-%%sqrtG6#,(*$%&gammaG\"\"%F**&\"#;F **$F0\"\"#F*!\"\"\"$+\"F*F6F0,$*(\"\"&F*),(*$F0F1F**&F3F**$F0F5F*F6F7F **&\"\"$F*F5F6F6,&*&F1F**$F0FAF*F**&\"#KF*F0F*F6F*F6" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The p ositive value of " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 37 " \+ for which the derivative is zero is " }{XPPEDIT 18 0 "gamma=2*sqrt(2) " "6#/%&gammaG*&\"\"#\"\"\"-%%sqrtG6#F&F'" }{TEXT -1 1 " " }{TEXT 267 1 "~" }{TEXT -1 8 " 2.828. " }}{PARA 0 "" 0 "" {TEXT -1 14 "This value of " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 47 " gives the max imum amplitude and is called the " }{TEXT 259 26 "resonant angular fre quency" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 134 "The resonant \+ angular frequency is a bit less than the both the quasi angular freque ncy of the underlying oscillatory system, which is " }{XPPEDIT 18 0 "3 ;" "6#\"\"$" }{TEXT -1 90 ", and the natural angular frequency of the \+ corresponding system with no damping, which is " }{XPPEDIT 18 0 "sqrt( 10)" "6#-%%sqrtG6#\"#5" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 8 " 3.162. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The following graph shows how the amplitude changes as the angula r frequency of the driving force changes. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 256 26 "resonant angular frequency" }{TEXT -1 37 " \+ is indicated by the vertical dotted " }{TEXT 256 4 "blue" }{TEXT -1 17 " line, while the " }{TEXT 261 23 "quasi angular frequency" }{TEXT -1 74 " of the underlying oscillatory system is indicated by the verti cal dotted " }{TEXT 261 5 "brown" }{TEXT -1 14 " line and the " } {TEXT 257 25 "natural angular frequency" }{TEXT -1 86 " of the corresp onding system with damping removed is indicated by the vertical dotted " }{TEXT 257 5 "green" }{TEXT -1 6 " line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "G := gamma -> 10/ sqrt(gamma^4-16*gamma^2+100):\np1 := plot(G(gamma),`gamma`=0..5,0..1.8 ,labels=[``,amplitude]):\na := 3.: b := evalf(2*sqrt(2)): c := evalf(s qrt(10)):\np2 := plot([[[a,0],[a,1.8]],[[b,0],[b,1.8]],[[c,0],[c,1.8]] ],\n color=[brown,blue,COLOR(RGB,0,.7,0)],linestyle=2):\nt1 : = plots[textplot]([3.5,-.16,g],font=[SYMBOL,10]):\nplots[display]([p1, p2,t1],view=[0..5,-.16..1.8]);" }}{PARA 13 "" 1 "" {GLPLOT2D 545 411 411 {PLOTDATA 2 "6)-%'CURVESG6$7Y7$$\"\"!F)$\"\"\"F)7$$\"3GLLL3x&)*3\" !#=$\"3%en\")4)3&4+\"!#<7$$\"3umm\"H2P\"Q?F/$\"3uRn[X6L.5F27$$\"3MLL$e RwX5$F/$\"3MxVk=Nv25F27$$\"33ML$3x%3yTF/$\"3v&fJ4p0T,\"F27$$\"3emm\"z% 4\\Y_F/$\"3'RxqmeqB-\"F27$$\"3`LLeR-/PiF/$\"3FJX=w2#=.\"F27$$\"3]***\\ il'pisF/$\"3UD#[m*p[V5F27$$\"3>MLe*)>VB$)F/$\"3Ki..2?ld5F27$$\"3Y++DJb w!Q*F/$\"3-vnq#R**R2\"F27$$\"3%ommTIOo/\"F2$\"3q4[1(=eK4\"F27$$\"3YLL3 _>jU6F2$\"3\")ofQEHQ76F27$$\"37++]i^Z]7F2$\"3[(=I%ebYO6F27$$\"33++](=h (e8F2$\"3/XGs)z2N;\"F27$$\"3/++]P[6j9F2$\"33Nx:o)4C>\"F27$$\"3UL$e*[z( yb\"F2$\"3ZfG/W8;@7F27$$\"3wmm;a/cq;F2$\"3E8lckL_e7F27$$\"3%ommmJF2$\"3]$R\"4UmGw8F27$$\"3K+]i!f#=$3#F2$\"3\\.*o#>C!GU\"F27$$\"3?+](=x pe=#F2$\"3_(e&oN%\\$o9F27$$\"37nm\"H28IH#F2$\"32r.lpR'e^\"F27$$\"3um;z pSS\"R#F2$\"3%4\\$z4.(yb\"F27$$\"3GLL3_?`(\\#F2$\"35=3*ea8\"*f\"F27$$ \"3fL$e*)>pxg#F2$\"3)y.WxADVj\"F27$$\"3%omm\"z+vbEF2$\"3y5ANPvGY;F27$$ \"33+]Pf4t.FF2$\"3%fQg0+fdl\"F27$$\"3om\"zWi^bv#F2$\"3aY50$fYGm\"F27$$ \"3uLLe*Gst!GF2$\"3%Q=Xs#3Mm;F27$$\"3J+vV)>ST$GF2$\"3gB>9TCkm;F27$$\"3 )om\"H2\"34'GF2$\"3cT%fjtwem\"F27$$\"3YLe9;gn()GF2$\"3]N];>z,k;F27$$\" 30+++DRW9HF2$\"3=3?z>m/h;F27$$\"3K+]7y#=o'HF2$\"3o)[>s)Q(>l\"F27$$\"3: ++DJE>>IF2$\"3^eM.WreQ;F27$$\"3F+]i!RU07$F2$\"3>*yMKCt3g\"F27$$\"3+++v =S2LKF2$\"3`Mx`1%QFa\"F27$$\"3Jmmm\"p)=MLF2$\"3odIfyf,z9F27$$\"3B++](= ]@W$F2$\"3mXA5hM*GS\"F27$$\"35L$e*[$z*RNF2$\"3X*fnbBz*H8F27$$\"3e++]iC $pk$F2$\"3UBMF(**4\"\\7F27$$\"3[m;H2qcZPF2$\"3a]w3aFBs)4\"F27$$\"3Ymm;/OgbRF2$\"3[G8D,I$)G5F27$$\"3w**\\i lAFjSF2$\"3B*ev%3sq.'*F/7$$\"3yLLL$)*pp;%F2$\"3O@3l[u&>**)F/7$$\"3)RL$ 3xe,tUF2$\"3n&\\xc5bVT)F/7$$\"3Cn;HdO=yVF2$\"3!HkKrw#f()yF/7$$\"3a+++D >#[Z%F2$\"30l`-;ajTuF/7$$\"3SnmT&G!e&e%F2$\"3Gxe&HT@@(pF/7$$\"3#RLLL)Q k%o%F2$\"3WE0_CN#oe'F/7$$\"37+]iSjE!z%F2$\"3M#>)3Kj()3iF/7$$\"3a+]P40O \"*[F2$\"3%)=X^.u1weF/7$$\"\"&F)$\"3[\"H_A'>+ZbF/-%'COLOURG6&%$RGBG$\" #5!\"\"F(F(-F$6%7$7$$\"\"$F)F(7$Fa]l$\"3/+++++++=F2-Fg\\l6&Fi\\l$\")#) eqk!\")$\"))eqk\"Fj]lF[^l-%*LINESTYLEG6#\"\"#-F$6%7$7$$\"32+++CrUGGF2F (7$Fe^lFd]l-Fg\\l6&Fi\\lF(F($\"*++++\"Fj]lF]^l-F$6%7$7$$\"3++++gwFiJF2 F(7$F`_lFd]l-%&COLORG6&Fi\\lF($\"\"(F\\]lF(F]^l-%%TEXTG6%7$$\"#NF\\]l$ !#;!\"#Q\"g6\"-%%FONTG6$%'SYMBOLGF[]l-%+AXESLABELSG6%%!G%*amplitudeG-F d`l6#%(DEFAULTG-%%VIEWG6$;F(Fb\\l;F^`l$\"#=F\\]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The foll owing animation illustrates the dependence of the steady state solutio n on the angular frequency " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" } {TEXT -1 23 " of the driving force. " }}{PARA 0 "" 0 "" {TEXT -1 254 " To play the animation, click on the graphic and use the controls in th e context bar, use the animation menu, or right click on the graphic t o obtain a context-sensitive menu. You can advance one frame at a time by using the ->| button in the context bar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "phi := unapply(p( t),gamma,t): h := .1:\n'phi(gamma,t)'=phi(gamma,t);\nplots[display]([s eq(plots[display]([plot(phi(h*i,t),t=0..4*Pi,color=COLOR(HUE,h*i/7),\n labels=[`t`,`x(t)`]),plots[textplot]([6.28,2,g=h*i],font=[SYMBOL,11] )]),i=0..60)],\n insequence=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6$%&gammaG%\"tG,&**\"#5\"\"\",(*$)F'\"\"%F,F,*&\"#;F,)F' \"\"#F,!\"\"\"$+\"F,F5,&*$F3F,F,F+F5F,-%$cosG6#*&F'F,F(F,F,F5**\"#?F,F 'F,F-F5-%$sinGF;F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 649 276 276 {PLOTDATA 2 "6#-%(ANIMATEG6in7&-%'CURVESG6$7S7$$\"\"!F-$\"\"\"F-7$$\"3 o#e#[q66RF!#=F.7$$\"3>O;(R:(RA^F3F.7$$\"383\\]-:l-yF3F.7$$\"3o)on;Bn+0 \"!#m!>4#F=F.7$$\"3EA;_@NkdBF=F.7$$\"3flt())e')4j#F=F.7$$\"3uO!) 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "0F?F-*$F3F?F4F-F'F-F-F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The appropriate form for a trial particular solution is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\" xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*gamma*t+B*sin*gamma*t;" "6 #/%!G,&**%\"AG\"\"\"%$cosGF(%&gammaGF(%\"tGF(F(**%\"BGF(%$sinGF(F*F(F+ F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=``" "6#/*&%#dxG\"\"\" %#dtG!\"\"%!G" }{XPPEDIT 18 0 "`p '`(t) = -gamma*A*sin*gamma*t+gamma*B *cos*gamma*t;" "6#/-%$p~'G6#%\"tG,&*,%&gammaG\"\"\"%\"AGF+%$sinGF+F*F+ F'F+!\"\"*,F*F+%\"BGF+%$cosGF+F*F+F'F+F+" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)=``" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(! \"\"%!G" }{XPPEDIT 18 0 "`p ''`(t) = -gamma^2*A*cos*gamma*t-gamma^2*B* sin*gamma*t;" "6#/-%%p~''G6#%\"tG,&*,%&gammaG\"\"#%\"AG\"\"\"%$cosGF-F *F-F'F-!\"\"*,F*F+%\"BGF-%$sinGF-F*F-F'F-F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2 )" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 5 " and \+ " }{TEXT 276 1 "x" }{TEXT -1 37 " in the differential equation gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-gamma^2*A*cos*ga mma*t-gamma^2*B*sin*gamma*t+2*lambda*(-gamma*A*sin*gamma*t+gamma*B*cos *gamma*t)+omega^2*(A*cos*gamma*t+B*sin*gamma*t) = F[0]*cos*gamma*t;" " 6#/,**,%&gammaG\"\"#%\"AG\"\"\"%$cosGF)F&F)%\"tGF)!\"\"*,F&F'%\"BGF)%$ sinGF)F&F)F+F)F,*(F'F)%'lambdaGF),&*,F&F)F(F)F/F)F&F)F+F)F,*,F&F)F.F)F *F)F&F)F+F)F)F)F)*&%&omegaGF',&**F(F)F*F)F&F)F+F)F)**F.F)F/F)F&F)F+F)F )F)F)**&%\"FG6#\"\"!F)F*F)F&F)F+F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "By comparing coefficients of the cosine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-gamma^2*A+2*lambda*gamma*B+omega^2*A = F[0];" "6#/,(*&%&gammaG\"\"#%\"AG\"\"\"!\"\"**F'F)%'lambdaGF)F&F)%\"B GF)F)*&%&omegaGF'F(F)F)&%\"FG6#\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(omega^2-gamma^2)*A+2*lambda*gamma*B = F[0];" "6#/,&*&,&*$%&omegaG\" \"#\"\"\"*$%&gammaGF)!\"\"F*%\"AGF*F***F)F*%'lambdaGF*F,F*%\"BGF*F*&% \"FG6#\"\"!" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 68 "By comparing coefficients of the \+ sine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-gamma^2*B-2*lambda*gamma*A+omega^2*B = 0;" "6#/,(*& %&gammaG\"\"#%\"BG\"\"\"!\"\"**F'F)%'lambdaGF)F&F)%\"AGF)F**&%&omegaGF 'F(F)F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(omega^2-gamma^2)*B- 2*lambda*gamma*A = 0;" "6#/,&*&,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF)!\" \"F*%\"BGF*F***F)F*%'lambdaGF*F,F*%\"AGF*F-\"\"!" }{TEXT -1 15 " ----- -- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Multiplying equation (i) by " }{XPPEDIT 18 0 "2*lambda*gamma;" "6# *(\"\"#\"\"\"%'lambdaGF%%&gammaGF%" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*lambda*gamma*(omega^2-gam ma^2)*A+4*lambda^2*gamma^2*B = 2*lambda*gamma*F[0];" "6#/,&*,\"\"#\"\" \"%'lambdaGF'%&gammaGF',&*$%&omegaGF&F'*$F)F&!\"\"F'%\"AGF'F'**\"\"%F' *$F(F&F'F)F&%\"BGF'F'**F&F'F(F'F)F'&%\"FG6#\"\"!F'" }{TEXT -1 17 " -- ----- (iii), " }}{PARA 0 "" 0 "" {TEXT -1 33 "and multiplying equation (ii) by " }{XPPEDIT 18 0 "omega^2-gamma^2;" "6#,&*$%&omegaG\"\"#\"\" \"*$%&gammaGF&!\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(omega^2-gamma^2)^2*B-2*lambda*gamma*(omega^2 -gamma^2)*A = 0;" "6#/,&*&,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF)!\"\"F)% \"BGF*F**,F)F*%'lambdaGF*F,F*,&*$F(F)F**$F,F)F-F*%\"AGF*F-\"\"!" } {TEXT -1 17 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 39 "Adding \+ equations (iii) and (iv) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "((omega^2-gamma^2)^2+4*lambda^2*gamma^2)*B = 2*lambda*g amma*F[0]" "6#/*&,&*$,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF*!\"\"F*F+*(\" \"%F+*$%'lambdaGF*F+F-F*F+F+%\"BGF+**F*F+F2F+F-F+&%\"FG6#\"\"!F+" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B = 2*lambda*gamma*F[0]/((omega^ 2-gamma^2)^2+4*lambda^2*gamma^2);" "6#/%\"BG*,\"\"#\"\"\"%'lambdaGF'%& gammaGF'&%\"FG6#\"\"!F',&*$,&*$%&omegaGF&F'*$F)F&!\"\"F&F'*(\"\"%F'*$F (F&F'F)F&F'F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similar ly: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = (omega^2- gamma^2)*F[0]/((omega^2-gamma^2)^2+4*lambda^2*gamma^2);" "6#/%\"AG*(,& *$%&omegaG\"\"#\"\"\"*$%&gammaGF)!\"\"F*&%\"FG6#\"\"!F*,&*$,&*$F(F)F** $F,F)F-F)F**(\"\"%F**$%'lambdaGF)F*F,F)F*F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "This gives the particular solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(t) = ``((omega^2-gamma^2) *F[0]/((omega^2-gamma^2)^2+4*lambda^2*gamma^2))*cos*gamma*t+``(2*lambd a*gamma*F[0]/((omega^2-gamma^2)^2+4*lambda^2*gamma^2))*sin*gamma*t;" " 6#/-%\"pG6#%\"tG,&**-%!G6#*(,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF1!\"\"F2 &%\"FG6#\"\"!F2,&*$,&*$F0F1F2*$F4F1F5F1F2*(\"\"%F2*$%'lambdaGF1F2F4F1F 2F5F2%$cosGF2F4F2F'F2F2**-F+6#*,F1F2FBF2F4F2&F76#F9F2,&*$,&*$F0F1F2*$F 4F1F5F1F2*(F@F2*$FBF1F2F4F1F2F5F2%$sinGF2F4F2F'F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The general solution is: " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*ex p(-lambda*t)*cos(sqrt(omega^2-lambda^2)*t)+C[2]*exp(-lambda*t)*sin(sqr t(omega^2-lambda^2)*t)+``;" "6#/-%\"xG6#%\"tG,(*(&%\"CG6#\"\"\"F--%$ex pG6#,$*&%'lambdaGF-F'F-!\"\"F--%$cosG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#F- *$F3F?F4F-F'F-F-F-*(&F+6#F?F--F/6#,$*&F3F-F'F-F4F--%$sinG6#*&-F:6#,&*$ F>F?F-*$F3F?F4F-F'F-F-F-%!GF-" }{TEXT -1 2 " \n" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``((omega^2-gamma^2)*F[0]/((omega^2-gam ma^2)^2+4*lambda^2*gamma^2))*cos*gamma*t+``(2*lambda*gamma*F[0]/((omeg a^2-gamma^2)^2+4*lambda^2*gamma^2))*sin*gamma*t;" "6#,&**-%!G6#*(,&*$% &omegaG\"\"#\"\"\"*$%&gammaGF,!\"\"F-&%\"FG6#\"\"!F-,&*$,&*$F+F,F-*$F/ F,F0F,F-*(\"\"%F-*$%'lambdaGF,F-F/F,F-F0F-%$cosGF-F/F-%\"tGF-F-**-F&6# *,F,F-F=F-F/F-&F26#F4F-,&*$,&*$F+F,F-*$F/F,F0F,F-*(F;F-*$F=F,F-F/F,F-F 0F-%$sinGF-F/F-F?F-F-" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 37 "_____________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "N o matter what initial conditions we take, the " }{TEXT 259 17 "steady- state part" }{TEXT -1 21 " of the solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = ``((omega^2-gamma^2)*F[0]/((omeg a^2-gamma^2)^2+4*lambda^2*gamma^2))*cos*gamma*t+``(2*lambda*gamma*F[0] /((omega^2-gamma)^2+4*lambda^2*gamma^2))*sin*gamma*t;" "6#/-%\"xG6#%\" tG,&**-%!G6#*(,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF1!\"\"F2&%\"FG6#\"\"!F 2,&*$,&*$F0F1F2*$F4F1F5F1F2*(\"\"%F2*$%'lambdaGF1F2F4F1F2F5F2%$cosGF2F 4F2F'F2F2**-F+6#*,F1F2FBF2F4F2&F76#F9F2,&*$,&*$F0F1F2F4F5F1F2*(F@F2*$F BF1F2F4F1F2F5F2%$sinGF2F4F2F'F2F2" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 278 39 "______________________________________ _" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "This is a sine wave with amplitude " }{XPPEDIT 18 0 "C=sq rt(A^2+B^2)" "6#/%\"CG-%%sqrtG6#,&*$%\"AG\"\"#\"\"\"*$%\"BGF+F," } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "A = (omega^2-gamma^2)*F[0]/((ome ga^2-gamma^2)^2+4*lambda^2*gamma^2);" "6#/%\"AG*(,&*$%&omegaG\"\"#\"\" \"*$%&gammaGF)!\"\"F*&%\"FG6#\"\"!F*,&*$,&*$F(F)F**$F,F)F-F)F**(\"\"%F **$%'lambdaGF)F*F,F)F*F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B = 2*la mbda*gamma*F[0]/((omega^2-gamma^2)^2+4*lambda^2*gamma^2);" "6#/%\"BG*, \"\"#\"\"\"%'lambdaGF'%&gammaGF'&%\"FG6#\"\"!F',&*$,&*$%&omegaGF&F'*$F )F&!\"\"F&F'*(\"\"%F'*$F(F&F'F)F&F'F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforward to check that the " }{TEXT 259 9 "amplitude" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F[0]/sqrt((omega^2-gamma^2)^2+4*lambda^2*gamma^2); " "6#*&&%\"FG6#\"\"!\"\"\"-%%sqrtG6#,&*$,&*$%&omegaG\"\"#F(*$%&gammaGF 1!\"\"F1F(*(\"\"%F(*$%'lambdaGF1F(F3F1F(F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "We can perform this calculation with Maple. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "de := diff(x(t),t$2)+2*lambda*diff(x(t),t)+omega^2*x(t)=F[0]*co s(gamma*t);\nunassign('A','B','p'):\np := t -> A*cos(gamma*t)+B*sin(ga mma*t):\n'p(t)'=p(t);\nsubs(x(t)=pt,de);\nsimplify(%);\nsolve(identity (%,t),\{A,B\});\nassign(%):\n'p(t)'=p(t);\n`amplitude`=simplify(sqrt(A ^2+B^2),symbolic);\nG := unapply(rhs(%),gamma):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*( F1F2%'lambdaGF2-F(6$F*F-F2F2*&)%&omegaGF1F2F*F2F2*&&%\"FG6#\"\"!F2-%$c osG6#*&%&gammaGF2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#% \"tG,&*&%\"AG\"\"\"-%$cosG6#*&%&gammaGF+F'F+F+F+*&%\"BGF+-%$sinGF.F+F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$,&*&%\"AG\"\"\"-%$cos G6#*&%&gammaGF+%\"tGF+F+F+*&%\"BGF+-%$sinGF.F+F+-%\"$G6$F1\"\"#F+*(F9F +%'lambdaGF+-F&6$F(F1F+F+*&)%&omegaGF9F+F(F+F+*&&%\"FG6#\"\"!F+F,F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*()%&gammaG\"\"#\"\"\"%\"AGF)-%$co sG6#*&F'F)%\"tGF)F)!\"\"*(F&F)%\"BGF)-%$sinGF-F)F0*,F(F)%'lambdaGF)F'F )F*F)F3F)F0*,F(F)F6F)F'F)F2F)F+F)F)*()%&omegaGF(F)F*F)F+F)F)*(F9F)F2F) F3F)F)*&&%\"FG6#\"\"!F)F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"B G,$*,\"\"#\"\"\"&%\"FG6#\"\"!F)%'lambdaGF)%&gammaGF),**$)F/\"\"%F)F)*( F(F))F/F(F))%&omegaGF(F)!\"\"*(F3F))F.F(F)F5F)F)*$)F7F3F)F)F8F)/%\"AG, $*(F*F)F0F8,&*$F5F)F)*$F6F)F8F)F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"pG6#%\"tG,&**&%\"FG6#\"\"!\"\"\",**$)%&gammaG\"\"%F.F.*(\"\"#F.)F2 F5F.)%&omegaGF5F.!\"\"*(F3F.)%'lambdaGF5F.F6F.F.*$)F8F3F.F.F9,&*$F6F.F .*$F7F.F9F.-%$cosG6#*&F2F.F'F.F.F9*.F5F.F*F.F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The maximum amplitude occurs where " }{XPPEDIT 18 0 "d/(d*gamma)" "6#*&%\"dG\"\"\"*&F$F%%&gammaGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[F[0]/sqrt((omega^2-gamma^2)^2+4*lambda^2*gamma^2)] = 0;" "6#/7#*&&% \"FG6#\"\"!\"\"\"-%%sqrtG6#,&*$,&*$%&omegaG\"\"#F**$%&gammaGF3!\"\"F3F **(\"\"%F**$%'lambdaGF3F*F5F3F*F6F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/(d*gamma)" "6#*&%\"dG\"\"\"*&F$F%%&gammaGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[F[0]/sqrt((omega^2-gamma^2)^2+4*lambda^2*gamma^2)] = - 1/2;" "6#/7#*&&%\"FG6#\"\"!\"\"\"-%%sqrtG6#,&*$,&*$%&omegaG\"\"#F**$%& gammaGF3!\"\"F3F**(\"\"%F**$%'lambdaGF3F*F5F3F*F6,$*&F*F*F3F6F6" } {XPPEDIT 18 0 "``((2*(omega^2-gamma^2)*(-2*gamma)+8*lambda^2*gamma)*F[ 0]/(((omega^2-gamma^2)^2+4*lambda^2*gamma^2)^(3/2)));" "6#-%!G6#*(,&*( \"\"#\"\"\",&*$%&omegaGF)F**$%&gammaGF)!\"\"F*,$*&F)F*F/F*F0F*F**(\"\" )F**$%'lambdaGF)F*F/F*F*F*&%\"FG6#\"\"!F*),&*$,&*$F-F)F**$F/F)F0F)F**( \"\"%F**$F6F)F*F/F)F**&\"\"$F*F)F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*gamma*(omega^2-2*lambda^2-gamma^2)*F[0]/(((omega^2-gamma^2)^2+4*l ambda^2*gamma^2)^(3/2));" "6#/%!G*,\"\"#\"\"\"%&gammaGF',(*$%&omegaGF& F'*&F&F'*$%'lambdaGF&F'!\"\"*$F(F&F/F'&%\"FG6#\"\"!F'),&*$,&*$F+F&F'*$ F(F&F/F&F'*(\"\"%F'*$F.F&F'F(F&F'*&\"\"$F'F&F/F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "This deri vative is zero when " }{XPPEDIT 18 0 "gamma=0" "6#/%&gammaG\"\"!" } {TEXT -1 10 " and when " }{XPPEDIT 18 0 "gamma^2=omega^2-2*lambda^2)" "6#/*$%&gammaG\"\"#,&*$%&omegaGF&\"\"\"*&F&F**$%'lambdaGF&F*!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Since the " }{TEXT 259 26 "resonant angular frequency" }{TEXT -1 26 " must be positive, it is : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "gamma[resonant] = sqrt(omega^2-2*lambda^2);" "6#/&%&gammaG6#%)resonantG-%%sqrtG6#,&*$ %&omegaG\"\"#\"\"\"*&F.F/*$%'lambdaGF.F/!\"\"" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 59 ": This resonant angular frequency is defined pr ovided that " }{XPPEDIT 18 0 "lambda " 0 "" {MPLTEXT 1 0 180 "F[0]/sqrt(gamma^4-2*gamma^2*omega^ 2+4*lambda^2*gamma^2+omega^4):\nDiff(%,gamma)=simplify(diff(%,gamma)); \nop(select(_U->evalb(sign(_U)=1 and _U<>0),[solve(subs(gamma=g,rhs(%) ),g)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&&%\"FG6#\"\"! \"\"\",**$)%&omegaG\"\"%F,F,*(\"\"#F,)F0F3F,)%&gammaGF3F,!\"\"*$)F6F1F ,F,*(F1F,)%'lambdaGF3F,F5F,F,#F7F3F6,$*,F3F,F6F,,(*$F4F,F7*$F5F,F,*&F3 F,F;F,F,F,F(F,F-#!\"$F3F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)%& omegaG\"\"#\"\"\"F)*&F(F))%'lambdaGF(F)!\"\"#F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "The dependence of the resonant angular \+ frequency on damping " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 41 "Consider again the differential equatio n:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2 *lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(%' lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = F[0]*co s*gamma*t;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF'F'**& %\"FG6#\"\"!F'%$cosGF'%&gammaGF'%\"tGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "which models a damped oscillator subject to a drivin g force with angular frequency " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 148 "In order to obtain an \+ example to illustrate how the resonant frequency and the amplitude of \+ the steady-state oscillations depend on the damping let " }{XPPEDIT 18 0 "omega^2=10" "6#/*$%&omegaG\"\"#\"#5" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "F[0]=10" "6#/&%\"FG6#\"\"!\"#5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 47 "In each case the resonant angular frequen cy is " }{XPPEDIT 18 0 "sqrt(omega^2-2*lambda^2) = sqrt(10-2*lambda^2) ;" "6#/-%%sqrtG6#,&*$%&omegaG\"\"#\"\"\"*&F*F+*$%'lambdaGF*F+!\"\"-F%6 #,&\"#5F+*&F*F+*$F.F*F+F/" }{TEXT -1 116 ", while the amplitude of the steady-state oscillations when the angular frequency of the driving f orce is gamma is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F[0]/sqrt((omega^2-gamma^2)^2+4*lambda^2*gamma^2)=10/sqrt((10-gamma^2 )^2+4*lambda^2*gamma^2)" "6#/*&&%\"FG6#\"\"!\"\"\"-%%sqrtG6#,&*$,&*$%& omegaG\"\"#F)*$%&gammaGF2!\"\"F2F)*(\"\"%F)*$%'lambdaGF2F)F4F2F)F5*&\" #5F)-F+6#,&*$,&F;F)*$F4F2F5F2F)*(F7F)*$F9F2F)F4F2F)F5" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 95 "The associated amplitude of the stead y-state oscillations at the resonant angular frequency is " }{XPPEDIT 18 0 "F[0]/(2*lambda*sqrt(omega^2-lambda^2)) = 5/(lambda*sqrt(10-lambd a^2));" "6#/*&&%\"FG6#\"\"!\"\"\"*(\"\"#F)%'lambdaGF)-%%sqrtG6#,&*$%&o megaGF+F)*$F,F+!\"\"F)F4*&\"\"&F)*&F,F)-F.6#,&\"#5F)*$F,F+F4F)F4" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The following picture shows curves " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(gamma)=10/sqrt((10-gamma^2)^2+4*lambd a^2*gamma^2)" "6#/-%\"fG6#%&gammaG*&\"#5\"\"\"-%%sqrtG6#,&*$,&F)F**$F' \"\"#!\"\"F2F**(\"\"%F**$%'lambdaGF2F*F'F2F*F3" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 22 "for various values of " }{XPPEDIT 18 0 "l ambda" "6#%'lambdaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "T he dotted " }{TEXT 271 5 "black" }{TEXT -1 31 " curve joins the maximu m points" }{XPPEDIT 18 0 " ``(sqrt(10-2*lambda^2),5/(lambda*sqrt(10-l ambda^2)))" "6#-%!G6$-%%sqrtG6#,&\"#5\"\"\"*&\"\"#F+*$%'lambdaGF-F+!\" \"*&\"\"&F+*&F/F+-F'6#,&F*F+*$F/F-F0F+F0" }{TEXT -1 34 " on the variou s curves, while the " }{TEXT 257 25 "natural angular frequency" } {TEXT -1 1 " " }{XPPEDIT 18 0 "omega = sqrt(10)" "6#/%&omegaG-%%sqrtG6 #\"#5" }{TEXT -1 86 " of the corresponding system with damping removed is indicated by the vertical dashed " }{TEXT 257 5 "green" }{TEXT -1 6 " line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 652 "G := (gamma,lambda) -> 10/sqrt((10-gamma^2)^2+4*lamb da^2*gamma^2):\np1 := plot([seq(G(gamma,lambda),lambda=[2,1.5,1,.75,.5 ,.33])],`gamma`=0..5,\n color=[blue,aquamarine,red,magenta,c oral,COLOR(RGB,.4,0,.9)]):\np2 := plot([sqrt(10-2*lambda^2),5/(lambda* sqrt(10-lambda^2)),lambda=.25..10],\n linestyle=2,color=black) :\nc := evalf(sqrt(10)):\np3 := plot([[c,0],[c,5.5]],color=COLOR(RGB,0 ,.6,0),linestyle=3):\nt1 := plots[textplot]([[3.5,-.16,g],[2.8,.7,l=2] ,[3,1.3,l=1.5],[3.13,1.8,l=1],\n [3.23,2.32,l=.75],[3.18,3.38 ,l=.5],[3.55,4.7,l=.33]],font=[SYMBOL,10]):\nplots[display]([p1,p2,p3, t1],view=[0..5,-.16..5.5],labels=[``,amplitude]);" }}{PARA 13 "" 1 "" {GLPLOT2D 592 579 579 {PLOTDATA 2 "63-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F) 7$$\"3GLLL3x&)*3\"!#=$\"3XrWsOpB+5!#<7$$\"3umm\"H2P\"Q?F/$\"3tR&>!)=B3 +\"F27$$\"3MLL$eRwX5$F/$\"3sg2$[b')=+\"F27$$\"33ML$3x%3yTF/$\"3g\"p&zJ dN.5F27$$\"3emm\"z%4\\Y_F/$\"3#4'4dgg;05F27$$\"3`LLeR-/PiF/$\"39gAVu$) 425F27$$\"3]***\\il'pisF/$\"3XIpx.gG45F27$$\"3>MLe*)>VB$)F/$\"3wt\\Qwn l65F27$$\"3Y++DJbw!Q*F/$\"3;x=gpr,95F27$$\"3%ommTIOo/\"F2$\"32gZW%*GI; 5F27$$\"3YLL3_>jU6F2$\"3WrSH)>n!=5F27$$\"37++]i^Z]7F2$\"3-7$)*HG5'>5F2 7$$\"33++](=h(e8F2$\"3\"y+Z41&\\?5F27$$\"3/++]P[6j9F2$\"3N@&f?]:0-\"F2 7$$\"3UL$e*[z(yb\"F2$\"3C%>ig'Hl>5F27$$\"3wmm;a/cq;F2$\"3QjE`RGJ<5F27$ $\"3%ommmJ6.95F27$$\"3/+]iSj0x=F2$\"3$3(G_RW]35F27$$ \"3gmmm\"pW`(>F2$\"3sqjROx\">+\"F27$$\"3K+]i!f#=$3#F2$\"3NC!yIz1r#**F/ 7$$\"3?+](=xpe=#F2$\"3Pq+]Y^:>)*F/7$$\"37nm\"H28IH#F2$\"3_g!>o$4&[o*F/ 7$$\"3um;zpSS\"R#F2$\"3u6M+X1-U&*F/7$$\"3GLL3_?`(\\#F2$\"3/Q$o1W\"fn$* F/7$$\"3fL$e*)>pxg#F2$\"3'zn>\"4YFl\"*F/7$$\"33+]Pf4t.FF2$\"398m,z26t* )F/7$$\"3uLLe*Gst!GF2$\"3wSnFr,u]()F/7$$\"30+++DRW9HF2$\"3szdih3F2&)F/ 7$$\"3:++DJE>>IF2$\"3YUb>G8Ce#)F/7$$\"3F+]i!RU07$F2$\"3%*)*yxP1m4!)F/7 $$\"3+++v=S2LKF2$\"3>4$\\O>Wys(F/7$$\"3Jmmm\"p)=MLF2$\"3/3*fP&R#>Z(F/7 $$\"3B++](=]@W$F2$\"3#G1-yVE$)>(F/7$$\"35L$e*[$z*RNF2$\"3Y3k(yTm>&pF/7 $$\"3e++]iC$pk$F2$\"3mI^fA*>ho'F/7$$\"3[m;H2qcZPF2$\"3*eq6f&*32W'F/7$$ \"3O+]7.\"fF&QF2$\"3[yJW^aH!>'F/7$$\"3Ymm;/OgbRF2$\"3AZaqH@V_fF/7$$\"3 w**\\ilAFjSF2$\"3g:Kp#4K:r&F/7$$\"3yLLL$)*pp;%F2$\"3)GMut&3(y[&F/7$$\" 3)RL$3xe,tUF2$\"3i0npj^%zE&F/7$$\"3Cn;HdO=yVF2$\"3]:`*R!))ye]F/7$$\"3a 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%&omegaG\"\"#F(*$%&gammaGF1!\"\"F1F(*(\"\"%F(*$F3F1F(%'lambdaGF1F(F4" }{TEXT -1 92 " of the steady-state oscillations tends to 0, and, in an y case, the amplitude is small when " }{XPPEDIT 18 0 "gamma" "6#%&gamm aG" }{TEXT -1 22 " is large compared to " }{XPPEDIT 18 0 "omega" "6#%& omegaG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 348 "Physically, this means that for a sufficiently hi gh driving frequency, the system cannot react quickly enough to the ra pidly changing driving force imposed on it and the system becomes \"pa ralysed\". If you have a motor which is shaking itself to pieces becau se of resonance, either inrease the amount of damping, or increase the speed of the motor. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 83 "Given an oscillatory system whose motion is govern ed by the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"* &F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG \"\"#%\"xGF'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 14 "the motion is " }{TEXT 259 12 "under- damped" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "0YZ7F%GG!#FAFEF6-F$6&7$7$$\"\"%F)F(7$FUF>FAF EF6-F$6%7$F'7$F($!3++++++++]F@-F.6&F0F)F)F)-%*LINESTYLEGF8-F$6%7$FT7$F UFfnFhnFjn-F$6'7$7$$\"\"\"F)$\"#N!\"#7$F(Ffo7%7$$\")+++!*!\"*$\"+++++I !#5Fio7$F\\p$\"+++++SFap-FF6#%,PATCHNOGRIDG-F.6&F0$\")#)eqkF3$\"))eqk \"F3F\\q-F[o6#\"\"&-F$6'7$7$$\"$v\"FhoFfo7$$\"$#GFhoFfo7%7$$\"+++qBFF^ pFcpFgq7$F\\rF_pFepFhpF^q-F$6'7$7$$\"#:!\"\"$!\"$Fer7$F(Ffr7%7$F\\p$!+ ++++NFapFhr7$F\\p$!+++++DFapFepFhpF^q-F$6'7$7$$\"#DFerFfr7$$\"#SFerFfr 7%7$$\"++++5RF^pF^sFfs7$F[tF[sFepFhpF^q-F$6'7$7$$F`qF)Ffr7$FUFfr7%7$$ \"++++!4%F^pF[sFct7$FftF^sFepFhpF^q-F$6'7$7$$\"\"'F)Ffr7$F+Ffr7%7$$\"+ +++5pF^pF^sF_u7$FbuF[sFepFhpF^q-%%TEXTG6%7$FU$F,FerQ\"w6\"-%%FONTG6$%' SYMBOLG\"#7-Ffu6%7$FhqFdoFjuF\\v-Ffu6%7$F(FiuQ\"0F[vF\\v-Ffu6%7$$\"#sF erF(Q\"lF[vF\\v-Ffu6%7$$F9F)FfrQ-under-dampedF[v-F]v6$%*HELVETICAG\"#5 -Ffu6%7$$\"#bFerFfrQ,over-dampedF[vFcw-Ffu6%7$FU$!\"'FerQ+criticallyF[ vFcw-Ffu6%7$FU$F^pFerQ'dampedF[vFcw-Ffu6%7$$\"$T\"Fho$\"\"$FerQ)possib leF[vFcw-Ffu6%7$F[y$F`qFerQ*resonanceF[vFcw-Ffu6%7$Fhq$\"#lFhoQ\"2F[v- F]v6$Few\"\"*-Ffu6%7$Fhq$F]zFerQ\"_F[v-F]v6$Few\"#8-Ffu6%7$Fhq$\"#'*Fh oFbzFcz-Ffu6%7$$\"$u#FhoFiuQ\"/F[vFcz-Ffu6%7$$\"#FFer$\"#jFhoQ\"\\F[v- F]v6$FewF^u-%+AXESLABELSG6%Q!F[vF^\\l-F]v6#%(DEFAULTG-%*AXESSTYLEG6#%% NONEG-%%VIEWG6$Fa\\lFa\\l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 56.000000 381.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curv e 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "C urve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25 " "Curve 26" "Curve 27" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 26 "The differential equation:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2;" "6#,&*(%\"dG\"\"#% \"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+10*x = 10*cos(2*sqrt(2)*t);" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'* &\"#5F'%\"xGF'F'*&F+F'-%$cosG6#*(\"\"#F'-%%sqrtG6#F2F'%\"tGF'F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 118 "models a damped oscilla tor subject to a driving force with angular frequency equal to the res onant angular frequency. " }}{PARA 0 "" 0 "" {TEXT -1 86 "(a) Find the solution of this differential equation subject to the initial conditi ons " }{XPPEDIT 18 0 "x(0)=0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "Eval(dx/dt,t=0)=0 " "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\" \"/%\"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 101 "(b) \+ Using the same initial conditions as in part (a), find the solution of the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+10*x = 10*c os(4*sqrt(2)*t);" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"#5F'%\"xGF'F'*&F+ F'-%$cosG6#*(\"\"%F'-%%sqrtG6#\"\"#F'%\"tGF'F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 144 " which models the same system as befo re except that the angular frequency of the driving force is now doubl e the resonant angular frequency. " }}{PARA 0 "" 0 "" {TEXT -1 106 "(c ) Plot the graphs of the two solutions found in parts (a) and (b) in t he same diagram. Use the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\" tG" }{XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "_________________ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "de1 := diff(x(t),t$2)+2*diff(x(t), t)+10*x(t)=10*cos(2*sqrt(2)*t);\nde2 := diff(x(t),t$2)+2*diff(x(t),t)+ 10*x(t)=10*cos(4*sqrt(2)*t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de1,ic \},x(t));\ng1 := unapply(rhs(%),t):\ndsolve(\{de2,ic\},x(t));\ng2 := u napply(rhs(%),t):\nplot([g1(t),g2(t)],t=0..10,color=[red,brown]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/,(-%%diffG6$-%\"xG6#%\"tG-%\"$ G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"#5F2F*F2F2,$*&F7F2-%$cosG6#,$*( F1F2F1#F2F1F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(-%% diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"#5F2F* F2F2,$*&F7F2-%$cosG6#,$*(\"\"%F2F1#F2F1F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,**&#\"\"&\"\"$\"\"\"*&-%$expG6#,$ F'!\"\"F--%$sinG6#,$*&F,F-F'F-F-F-F-F3*&#F+\"\"*F-*&F/F--%$cosGF6F-F-F 3*&#F+F;F--F>6#,$*(\"\"#F-FE#F-FEF'F-F-F-F-*&#\"#5F;F-*&-F5FBF-FEFFF-F -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,**&#\"#N\"$`\"\"\" \"*&-%$expG6#,$F'!\"\"F--%$sinG6#,$*&\"\"$F-F'F-F-F-F-F3*&#\"#bF,F-*&F /F--%$cosGF6F-F-F-*&#\"#?F,F-*&-F56#,$*(\"\"%F-\"\"##F-FIF'F-F-F-FIFJF -F-*&#F>F-7$$\"3[LL$e9ui2%F-$\"3'\\co42JJ$\\F-7$$\"3\"pm\"H2Q\\4YF-$\"30VQ 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qkF]hn$\"))eqk\"F]hnF``t-%+AXESLABELSG6$Q\"t6\"Q!Ff`t-%%VIEWG6$;F(Fcgn %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 47 "_________ ______________________________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 26 "The differential equati on:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+ 4;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\"%F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+16*x = cos*gamma*t;" "6#/,&*&%#dx G\"\"\"%#dtG!\"\"F'*&\"#;F'%\"xGF'F'*(%$cosGF'%&gammaGF'%\"tGF'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "models a damped oscillat or subject to a driving force with angular frequency " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "( a) Find the steady-state solution of this differential equation. " }} {PARA 0 "" 0 "" {TEXT -1 86 "(b) Find an expression for the amplitude \+ of the steady-state oscillations in terms of " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 59 " and plot a graph which shows how this ampl itude varies as " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 9 " va ries. " }}{PARA 0 "" 0 "" {TEXT -1 107 "(c) Find the resonant angular \+ frequency of the system and the corresponding amplitude of the oscilla tions. " }}{PARA 0 "" 0 "" {TEXT -1 47 "______________________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________ ________" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for picture " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for summary picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1305 "p1 := plot([[0,0],[7,0]],t hickness=2,color=navy):\np2 := plot([[[0,0],[0,1/2]],[[2*sqrt(2),0],[2 *sqrt(2),1/2]],[[4,0],[4,1/2]]],style=line, color=blue,thickness =2):\np3 := plot(\{[[0,0],[0,-1/2]],[[4,0],[4,-1/2]]\},color=black,lin estyle=2):\np4 := plottools[arrow]([1,.35],[0,.35],0,.1,.09,arrow,colo r=brown,linestyle=5):\np5 := plottools[arrow]([1.75,.35],[2.82,.35],0, .1,.09,arrow,color=brown,linestyle=5):\np6 := plottools[arrow]([1.5,-. 3],[0,-.3],0,.1,.06,arrow,color=brown,linestyle=5):\np7 := plottools[a rrow]([2.5,-.3],[4,-.3],0,.1,.06,arrow,color=brown,linestyle=5):\np8 : = plottools[arrow]([5,-.3],[4,-.3],0,.1,.09,arrow,color=brown,linestyl e=5):\np9 := plottools[arrow]([6,-.3],[7,-.3],0,.1,.09,arrow,color=bro wn,linestyle=5):\nt1 := plots[textplot](\{[0,.7,`0`],[2.82,1,`w`],[4,. 7,`w`]\},font=[SYMBOL,12]):\nt2 := plots[textplot]([7.2,0,`l`],font=[S YMBOL,12]):\nt3 := plots[textplot]([[2,-.3,`under-damped`],[5.5,-.3,`o ver-damped`],\n [4,-.6,`critically`],[4,-.9,`damped`],[1.41,.3,`possi ble`],\n [1.41,.5,`resonance`]],font=[HELVETICA,10]):\nt4 := \+ plots[textplot]([2.82,.65,`2`],font=[HELVETICA,9]):\nt5 := plots[textp lot]([[2.82,.9,`_`],[2.82,.96,`_`],[2.74,.7,`/`]],font=[HELVETICA,13]) :\nt6 := plots[textplot]([2.7,.63,`\\\\`],font=[HELVETICA,6]):\nplots[ display]([p||(1..9),t||(1..6)],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }