{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 Plo t" -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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 405 "MATH \+ 2280 - 1 THIRD MAPLE ASSIGNMENT\nTreibe rgs Due Wednesday, October 26, 2005 .\n\nIn this lab we study the forced vibrations and resonance in the d amped oscillator based on computing project 3.6 from the text. The oth er sections of 2280 and 2250 are doing a similar project. You may peru se the related assigment from Prof. Gustafson's class" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {URLLINK 17 "http://www.math.utah.e du/~gustafso/tnarrows/index.html" 4 "http://www.math.utah.edu/~gustafs o/tnarrows/index.html" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 888 "\nThere you will find links to a movie of the s pectacular catastophic collapse of the Tacoma Narrows Bridge, as well \+ as mathematical models of vibrations on that bridge that you can run o n MAPLE.\n\nWe begin by studying how the forcing frequency affects the solution to the undamped equation and see the phenomenon of resonance : as the forcing frequency w approaches the natural frequency w0, \+ the amplitude of the steady wave increases. When w=w0 the system is \+ in resonance and the solution blows up. When w is close but not equa l to w0, the solution will exhibit beats.\n\nThen we consider the dam ped equation. This time the amplitude of the steady solution will be g reatest when the forcing frequency is the practical resonant frequency of the system, which will be less than the forcing frequency. First \+ we illustrate with particular values of the physical constants m, k a nd c." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" } }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "\n(1.) Consider the case \+ without damping first. Let m=k=1 and c=0." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "ode1:=diff(y(t),t,t)+y(t)=0;\ndsolve(ode1,y(t));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/,&-%%diffG6$-%\"yG6#%\"tG-% \"$G6$F-\"\"#\"\"\"F*F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"tG,&*&%$_C1G\"\"\"-%$sinGF&F+F+*&%$_C2GF+-%$cosGF&F+F+" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 146 "\n\n\nThe system has a natur al frequency w0=1. Now consider the inhomogeneous equation with a for cing term of fixed amplitude F0 and frequency w." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "w:='w'; F0:='F0'; f:=t->F0*cos(w*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F0GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$ %)operatorG%&arrowGF(*&%#F0G\"\"\"-%$cosG6#*&%\"wGF.9$F.F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " ode2:=diff(y(t),t,t)+y(t)=f (t);\n sol2:=simplify(dsolve(\{ode2,y(0)=0,D(y)(0)=0\},y(t)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode2G/,&-%%diffG6$-%\"yG6#%\"tG-%\" $G6$F-\"\"#\"\"\"F*F2*&%#F0GF2-%$cosG6#*&%\"wGF2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/-%\"yG6#%\"tG,$*(%#F0G\"\"\",&-%$cosGF(! \"\"-F06#*&%\"wGF-F)F-F-F-,&F-F1*$)F5\"\"#F-F-F1F1" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 188 "Using cos(A+B)=cos A cos B - sin A sin B a nd cos(A-B)=cos A cos B + sin A sin B where A+B=wt and A-B = t we \+ find the trig substitution reduces the difference to a product of sinu soids" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "subs(-cos(t)+cos(w *t)=2*sin((1+w)*t/2)*sin((w-1)*t/2),sol2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*,\"\"#\"\"\"%#F0GF+-%$sinG6#,$*(F*!\" \",&F+F+%\"wGF+F+F'F+F+F+-F.6#,$*(F*F2,&F+F2F4F+F+F'F+F+F+,&F+F2*$)F4F *F+F+F2F2" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 161 "Thus the slowl y varying function env(t) may by regarded as the enveloping amplitud e of the quickly varying function sin((1+w)*t/2). The solution shows beats." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "env:=t->2*F0*si n((w-1)*t/2)/(w^2-1);\nwith(plots):\npsol1:=subs(\{w=1.1,F0=1\},sol2); \npeng:=subs(\{w=1.1,F0=1\},env(t));\nplot([rhs(psol1),peng,-peng],t=0 ..200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$envGf*6#%\"tG6\"6$%)oper atorG%&arrowGF(,$**\"\"#\"\"\"%#F0GF/-%$sinG6#,$*&#F/F.F/*&,&F/!\"\"% \"wGF/F/9$F/F/F/F/,&F/F9*$)F:F.F/F/F9F/F(F(F(" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psol1G/-%\"yG6#%\"tG,&*&$\"+iZ!>w%! \"*\"\"\"-%$cosGF(F/F/*&$\"+iZ!>w%F.F/-F16#,$*&$\"#6!\"\"F/F)F/F/F/F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pengG,$*&$\"+C&4Q_*!\"*\"\"\"-% $sinG6#,$*&$\"+++++]!#6F*%\"tGF*F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7ccn7$$\"\"!F)F(7$$\"3RLLL3x&)* 3\"!#<$\"3j(*>]B3nAZ!#=7$$\"3ymmm;arz@F-$\"3R(*42>dH-yF07$$\"3(****** \\7t&pKF-$!3(o--L#o'RX%F07$$\"3eLLLL3VfVF-$!3I!zY$**[yS?F-7$$\"3^]i!Rv 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What happens if w=w0? 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The damped c ase. " }{TEXT -1 0 "" }{MPLTEXT 0 21 86 "This time, lets stiffen and d amp the dashpot. Find the characteristic equation's roots" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart:with(DEtools):with(plots):" }}{PARA 7 "" 1 " " {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "m:=5;c:=2;k:=2;solve(m*r^2+ c*r+k=0,r);\n\n " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"mG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^$#!\"\"\"\"&#\"\"$F&^$F$#!\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 41 "so the pseudofrequency is w1. Solve ODE." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "w1^2= k/m -c^2/(4*m^2);evalf (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%#w1G\"\"#\"\"\"#\"\"*\"#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%#w1G\"\"#\"\"\"$\"+++++O!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ode3:=m*diff(y(t),t,t)+c* diff(y(t),t)+k*y(t)=0;\ndsolve(ode3,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode3G/,(*&\"\"&\"\"\"-%%diffG6$-%\"yG6#%\"tG-%\"$G6$ F0\"\"#F)F)*&F4F)-F+6$F-F0F)F)*&F4F)F-F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*(%$_C1G\"\"\"-%$expG6#,$*&\"\"&!\"\"F' F+F2F+-%$sinG6#,$*(\"\"$F+F1F2F'F+F+F+F+*(%$_C2GF+F,F+-%$cosGF5F+F+" } }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 144 "The system has a natural f requency w1=0.6. Now consider the inhomogeneous equation with a forcin g term of fixed amplitude F0 and frequency w." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f:=t->F0*cos(w*t);\node4:=m*diff(y(t),t,t)+c *diff(y(t),t)+k*y(t)=f(t);\nsol4:=dsolve(\{ode4,y(0)=0,D(y)(0)=1\},y(t ));\n\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)ope ratorG%&arrowGF(*&%#F0G\"\"\"-%$cosG6#*&%\"wGF.9$F.F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode4G/,(*&\"\"&\"\"\"-%%diffG6$-%\"yG6#% \"tG-%\"$G6$F0\"\"#F)F)*&F4F)-F+6$F-F0F)F)*&F4F)F-F)F)*&%#F0GF)-%$cosG 6#*&%\"wGF)F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol4G/-%\"yG6# %\"tG,(*&#\"\"\"\"\"$F-**-%$expG6#,$*&\"\"&!\"\"F)F-F6F--%$sinG6#,$*(F .F-F5F6F)F-F-F-,,\"#?F6*&\"#!)F-)%\"wG\"\"#F-F-*&\"$D\"F-)FA\"\"%F-F6* (F5F-%#F0GF-F@F-F-*&FBF-FHF-F-F-,(FFF-*&\"#;F-F@F-F6*&\"#DF-FEF-F-F6F- F6*,F0F--%$cosGF9F-FHF-,&FBF6*&F5F-F@F-F-F-FJF6F-*(FHF-,(*&FBF--FQ6#*& FAF-F)F-F-F6*(FBF--F8FXF-FAF-F6*(F5F-F@F-FWF-F-F-FJF6F6" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 92 "The exponential terms die away and t he remaining particular solution has an amplitude amp. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "A:=F0*(2-5*w^2)/(4-16*w^2+25*w^4); \nB:=F0*2*w/(4-16*w^2+25*w^4);\nC:=simplify(sqrt(A^2+B^2),sqrt,symboli c);\namp:= subs(F0=1,C);\nplot([rhs(subs(\{F0=1,w=1\},sol4)), subs(\{F 0=1,w=1\},amp), -subs(\{F0=1,w=1\},amp)], t=0..35,title=\"Forced Oscil lations in the Damped Oscillator\");\nplot(amp,w=0..6,title=\"Amplitud e vs Frequency\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*(%#F0G\" \"\",&\"\"#F'*&\"\"&F')%\"wGF)F'!\"\"F',(\"\"%F'*&\"#;F'F,F'F.*&\"#DF' )F-F0F'F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$**\"\"#\"\"\"%# F0GF(%\"wGF(,(\"\"%F(*&\"#;F()F*F'F(!\"\"*&\"#DF()F*F,F(F(F0F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG*&%#F0G\"\"\",(\"\"%F'*&\"#;F')% \"wG\"\"#F'!\"\"*&\"#DF')F-F)F'F'#F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ampG*&\"\"\"F&*$,(\"\"%F&*&\"#;F&)%\"wG\"\"#F&!\"\"*&\"#DF&)F -F)F&F&#F&F.F/" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6(-%'CURVESG6$7cu7$$\"\"!F)F(7$$\"3clm\"z%\\DO&*!#>$\"3SK6siJpS%*F-7$$ 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This practical resonance point .5656 854248 is somewhat less than the pseudo" }{TEXT -1 0 "" }{MPLTEXT 0 21 15 "frequency w1=.6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "d iff(amp,w);solve(%=0,w);\nevalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\",(\"\"%\"\"\"*&\"#;F))%\"wGF%F)F&*&\"#DF))F-F(F)F)#! \"$F%,&*&\"#KF)F-F)F&*&\"$+\"F))F-\"\"$F)F)F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!,$*(\"\"#\"\"\"\"\"&!\"\"F&#F'F&F',$*(F&F'F(F)F&F* F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F$$\"+[U&ol&!#5$!+[U&ol&F' " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 1694 "PROBLEMS\n\nWrite up s olutions in a self contained manner. Write enough to explain your anal ysis. Have MAPLE produce the solutions and graphs. Label these by hand if necessary. \n\n1. Choose some other values of F0, k, m and w \+ in the undamped equation (as in ode1). Show that when w is not eq ual to but is close the natural frequency, then some solutions exhibit beats. Show that as the driving frequency approaches the resonant fre quency, the amplitude of the beats become large. Show that if the driv ing frequency equals the resonant frequency then the solution blows up . Use both symbolic computation and plots of your functions to demonst rate these phenomena.\n\n2. Choose some other values of F0, c, k, \+ m, w and initial values (not those in either handout!) for the damp ed equation (as in ode3). Find the amplitude of the forced steady sol ution as a function of the driving frequency. Find the practical reson ant frequency -- the w for which the amplitide is max. (Be careful to choose m, c, and k so that the practical resonant frequency is posit ive.) Show an example where the initial data are large compared to the driving force and the natural frequency is much less than the driving frequency such that the natural oscillations dominate for a while unt il the slaved forced oscillations dominate. Use both symbolic computa tion and plots of your functions to demonstrate these phenomena.\n\n3. Choose a driving force F(t) = g(t)cos(wt) whose amplitude g(t) i s not constant, not sine or cosine (because then F(t) would be a sin usiod) and not from the text. (e.g. see p. 221-222 3rd ed.) Solve and plot an example of damped vibrations that are forced by your F(t).\n \n" }}}}{MARK "0 2 0" 23 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }