{VERSION 5 0 "IBM INTEL NT" "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 "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 "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 25 "Tacoma Narrows Bridge La b" }}{PARA 258 "" 0 "" {TEXT 256 46 "MTH 286 Introduction to Different ial Equations" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "This lab is bas ed on section 4.5 of the text. Additional information can be found a t www.me.utexas.edu/~uer/papers/paper_jk.html and www.tpctourism.org/b ridge.htm." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 690 "The original 5,939 foot long Tacoma Narrows Bridge, popularly known as \"Galloping Gerti e\", opened to traffic on July 1, 1940 after two years of construction , linking Tacoma and Gig Harbor. It collapsed just four months later \+ during a 42 mile per hour wind storm on Nov. 7, 1940. The bridge earn ed the nickname \"Galloping Gertie\" from its rolling, undulating beha vior. Motorists crossing the 2,800 foot center span sometimes felt as though they were traveling on a giant roller coaster, watching the ca rs ahead disappear completely for a few moments as if they had been dr opped in a trough by a giant wave. The cover of the book has a pictur e of the bridge shortly before its collapse. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "The following is an eye-witness account provided by Leonard Coatsworth, a Tacoma newspaper editor (from www.tpctourism.or g/bridge.htm.):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 256 "\"Just as I d rove past the towers, the bridge began to sway violently from side to \+ side. Before I realized it, the tilt became so violent that I lost co ntrol of the car... I jammed on the brakes and got out, only to be thr own onto my face against the curb." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "\"Around me I could hear the concrete cracking. I started to \+ get my dog Tubby, but was thrown again before I could reach the car. \+ The car itself began to slide from side to side of the roadway." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 401 "\"On hands and knees most of the \+ time, I crawled 500 yards or more to the towers... My breath was comin g in gasps; my knees were raw and bleeding, my hands bruised and swoll en from gripping the concrete curb... Toward the last, I risked rising to my feet and running a few yards at a time... Safely back at the to ll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "The brid ge was constructed using the most advanced engineering knowledge of th e time. It was considered so safe when constructed that the insurance agent felt confident in embezzling the premiums so that the ill-fated bridge was uninsured." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "As a result of the Tacoma Na rrows bridge disaster, wind tunnel testing is now standard engineering practice in suspension bridge design." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1416 "Galloping Gertie gave evidence of design problems even \+ while under construction. Workers noticed that the bridge oscillated \+ as much as 3 feet with even the slightest breeze. The carnival ride l ike experience for motorists crossing the span was a serious concern f or the Washington Toll Bridge Authority and they asked the University \+ of Washington Engineering Department to study the problem and recommen d solutions. Because of this, Prof. Farquharson closely monitored the bridge, measuring wind velocities and noting the shape of wave motion s in the structure. During the morning of November 7, 1940 Prof. Farq uharson was filming the bridge in 38 mph winds. By 10:00AM the winds \+ had increased to 42mph. The deck was rising and falling at a rate of \+ 38 times per minute with an amplitude of 3 feet. Suddenly, the freque ncy changed, slowing to 12 cycles per minute. The rhythmic vertical m otion shifted to a twisting torsional motion and the amplitude increas ed from 3 feet to 28 feet. The span first tilted 45 degrees to the ri ght, then 45 degrees to the left. David Steinman, a preeminent suspen sion bridge designer wrote in viewing Farquharson's film \"it was diff icult to realize that the girders were made not of rubber, but of stru ctural steel having a modulus of elasticity of 29,000,000 pounds per s quare inch.\" (quoted from Historic American Engineering Record: Tacom a Narrows Bridge HAER No. WA-99)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "At 11:00AM a 600 foot section of roadway fell away; the rhythmic \+ motion continued, and in less than 10 minutes the entire center span f ell into Puget Sound." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 418 "Debate \+ continues today as to what caused the bridge to fail. It is generally agreed that modelling the vertical motion of the bridge as a mass-spr ing system is too simple. In this lab, we analyse the behavior that c aused the failure of the bridge using the model from the text based on the work of Lazer and McKenna. The modification is to incorporate a \+ nonlinear term to account for slack in the suspension cables." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Our simple model is " }{XPPEDIT 18 0 "d^2y/dt^2+alpha*dy/dt+beta*y+c(y)=-g" "6#/,**(%\"dG\"\"#%\"yG\" \"\"*$%#dtGF'!\"\"F)*(%&alphaGF)%#dyGF)F+F,F)*&%%betaGF)F(F)F)-%\"cG6# F(F),$%\"gGF," }{TEXT -1 7 " where " }{TEXT 260 1 "y" }{TEXT -1 45 " i s the vertical displacement of the bridge, " }{XPPEDIT 18 0 "alpha" "6 #%&alphaG" }{TEXT -1 32 " is the coefficient of damping, " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 63 " is a parameter corresponding to the stiffness of the roadbed, " }{TEXT 259 1 "g" }{TEXT -1 49 " is th e gravitational constant, and the function " }{XPPEDIT 18 0 "c(y)" "6# -%\"cG6#%\"yG" }{TEXT -1 54 " accounts for the pull of the cables. Th e function c(" }{TEXT 262 1 "y" }{TEXT -1 14 ") is given by " } {XPPEDIT 18 0 "c(y)=gamma*y" "6#/-%\"cG6#%\"yG*&%&gammaG\"\"\"F'F*" } {TEXT -1 4 " if " }{XPPEDIT 18 0 "y<0" "6#2%\"yG\"\"!" }{TEXT -1 6 ", \+ and " }{XPPEDIT 18 0 "c(y)=0" "6#/-%\"cG6#%\"yG\"\"!" }{TEXT -1 219 ", otherwise. The idea is that if the bridge is sagging, the cables wil l act like a spring to pull the bridge back into place, but if the bri dge has risen, the cables will be slack and will have no effect on the bridge." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Following the text, \+ we transform our differential equation into the following system, wher e the function c(" }{TEXT 261 1 "y" }{TEXT -1 34 ") has been combined \+ with the term " }{XPPEDIT 18 0 "beta*y" "6#*&%%betaG\"\"\"%\"yGF%" } {TEXT -1 23 " to yield the function " }{XPPEDIT 18 0 "h(y)" "6#-%\"hG6 #%\"yG" }{TEXT -1 10 " which is " }{XPPEDIT 18 0 "a*y" "6#*&%\"aG\"\" \"%\"yGF%" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "y<0" "6#2%\"yG\"\"!" } {TEXT -1 6 ", and " }{XPPEDIT 18 0 "b*y" "6#*&%\"bG\"\"\"%\"yGF%" } {TEXT -1 11 " otherwise." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sys:=[diff(y(t),t)=v(t),diff(v(t),t)=-piecewise(y(t) < 0,a*y(t),b*y(t ))-alpha*v(t)-g];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG7$/-%%diff G6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,(-%*PIECEWISEG6$7$*&%\"aG\"\"\"F* F:2F*\"\"!7$*&%\"bGF:F*F:%*otherwiseG!\"\"*&%&alphaGF:F.F:FA%\"gGFA" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Using the constants used by Laze r and McKenna (which don't model any particular bridge), we have" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sysa:=subs(a=17,b=13,alpha=0 .01,g=10,sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sysaG7$/-%%diffG6 $-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,(-%*PIECEWISEG6$7$,$F*\"#<2F*\"\"!7 $,$F*\"#8%*otherwiseG!\"\"F.$F@!\"#!#5\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 45 "Below is the direction field for this system:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(DEtools):\nwith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "dfieldplot(sysa,[y(t),v (t)],t=0..12,y=-1..1,v=-1..1,\\\ntitle=`Direction Field for Bridge`): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phaseportrait(sysa,[y( t),v(t)],t=0..12,[[y(0)=-8/17,v(0)=0],[y(0)=-6/17,v(0)=0],[y(0)=-4/17, v(0)=0],[y(0)=-2/17,v(0)=0],[y(0)=-10/17,v(0)=0]],y=-1..1,v=-1..1,step size=.05,\\\ntitle=`Direction Field for Bridge`);" }}{PARA 13 "" 1 "" {GLPLOT2D 304 304 304 {PLOTDATA 2 "6(-%'CURVESG6]dl7%7$$!+oN[ ))F*$!+%y\"p^5F-7$$!+W[QY!*F*$!+b@3$[*F*7$$!+O*QF8*F*$!+`#o;t*F*7%7$$! +XmYZxF*$!+!=H00\"F-7$$!+Rq+U!)F*$!+/#3Z\\*F*7$$!+A,c/\")F*$!+>NK](*F* 7%7$$!+]NHolF*$!+a$[\\/\"F-7$$!+wp\"f6(F*$!+ck^]&*F*7$$!+7Rs4rF*$!+o9g 8)*F*7%7$$!+s6#)o_F*$!+M#)*H#**F*7$$!+'>E,J'F*$!+x,q25F-7$$!+.mUlgF*$! +;AQ<5F-7%7$$!+\\\"Qa\\%F*$!+*f6B`*F*7$$!+hgCy\\F*$!+S)on/\"F-7$$!+\") HzcZF*$!+PCbK5F-7%7$$!+qB\"ya$F*$!+4cm\"\\*F*7$$!+#o31#QF*$!+RM$30\"F- 7$$!+]HYMOF*$!+o=BK5F-7%7$$!+:$)pPDF*$!+'yC@[*F*7$$!+z&fas#F*$!+@vy^5F -7$$!+p#R`b#F*$!+#))4<.\"F-7%7$$!+t>d2:F*$!+%RY&y%*F*7$$!+jFK];F*$!+h` 9_5F-7$$!+4H0*[\"F*$!+U,NJ5F-7%7$$!+%)Q&zo%!#6$!+>o$oZ*F*7$$!+'*=OQeFf t$!+=jJ_5F-7$$!+-LQ\"G%Fft$!+)p+6.\"F-7%7$$\"+813adFft$!+h(yfZ*F*7$$\" +(=NAx%Fft$!+C@S_5F-7$$\"+bS'[H'Fft$!+Y'Q4.\"F-7%7$$\"+\\=]A;F*$!+\\%* 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mF\\f]m7$$!1#z$=THN#)eFa\\dl$\"1Q(\\#y%[@&oF\\f]m7$$!1F'\\6%HN#)eFa\\d l$\"1XaS->XlnF\\f]m7$$!1:m6THN#)eFa\\dl$\"1W)G^>iBR'F\\f]m7$$!1bh3THN# )eFa\\dl$\"1q9;=x\")[dF\\f]m7$$!1M&f5%HN#)eFa\\dl$\"1%G&)=`*>i[F\\f]m- %&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-Fd[dl6&%$RGBG$\"\"*!\"\"Feabm$\" \"#Fgabm-%%VIEWG6$;$!+++++6F-$\"+++++6F-F]bbm-%+AXESLABELSG6$%\"yG%\"v G-%&TITLEG6#%;Direction~Field~for~BridgeG" 1 2 0 1 0 2 6 0 4 2 1 0.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "For exercises 1-3, (a) discuss what parameters are changed, and (b) discuss how you expect a change in the paramete r values to affect solutions. (For example, does the modification mak e the system look more or less like a linear system?)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 1 "1" }{TEXT -1 275 ". The stiffness of the \+ roadbed is increased, for example, by reinforcing the concrete or addi ng extra material that makes it harder for the roadbed to bend. (This approach was planned for the Tacoma Narrows Bridge, but the bridge co llapsed before the changes could be made.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 1 "2" }{TEXT -1 42 ". The coefficient of damping is increase d." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 1 "3" }{TEXT -1 187 ". The str ength of the cables is increased. (More cables were added to the Tacom a Narrows Bridge, but they snapped due to the oscillations of the brid ge shortly after they were installed.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 1 "4" }{TEXT -1 295 ". Refer to exercise 4 in the text for a diagram of the Lazer-McKenna light flexible long span suspension brid ge. Why does this design avoid the problems of the standard suspensio n bridge design? Discuss this in a paragraph and give model equations similar to those in the text for this design." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 1 "5" }{TEXT -1 268 ". The mathematical model that we a re employing is an approximation. Read section 4.5 of the text and th e description above of the Tacoma Narrows bridge failure and discuss s ome of the shortcomings of the model, i.e. what behavior does the mode l not take into account?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 738 "Next we consider the effects of the action of the wind on the bridge. One theory regarding the collapse of the Tacoma Narrows Bridge is that th e bridge experienced a phenomenon known as vortex shedding. Vortex sh edding is a condition where objects like airplane wings and bridge de cks displace air flowing around them creating wind currents that form \+ eddies or vortices swirling around the object. These vortices then po und rhythmically against the object even though the winds themselves m ay be nearly constant. If this vortex shedding is resonant with the n atural frequency of the bridge, then violent oscillations can result. \+ We model the vortex shedding by adding a periodic forcing term to the original differential equation: " }{XPPEDIT 18 0 "d^2y/dt^2+alpha*dy /dt+beta*y+c(y)=-g+lambda*sin(mu*t)" "6#/,**(%\"dG\"\"#%\"yG\"\"\"*$%# dtGF'!\"\"F)*(%&alphaGF)%#dyGF)F+F,F)*&%%betaGF)F(F)F)-%\"cG6#F(F),&% \"gGF,*&%'lambdaGF)-%$sinG6#*&%#muGF)%\"tGF)F)F)" }{TEXT -1 28 ". Our system then becomes: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 " sys2:=\{diff(y(t),t)=v(t),diff(v(t),t)=-piecewise(y(t)<0,a*y(t),b*y(t) )-alpha*v(t)-g+lambda*sin(mu*t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%sys2G<$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,*-%*PIECEWISEG6 $7$*&%\"aG\"\"\"F*F:2F*\"\"!7$*&%\"bGF:F*F:%*otherwiseG!\"\"*&%&alphaG F:F.F:FA%\"gGFA*&%'lambdaGF:-%$sinG6#*&%#muGF:F-F:F:F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Substituting the constants used in the te xt, we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sysb:=subs(a= 17,b=13,alpha=0.01,g=10,mu=4,lambda=0.03,sys2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sysbG<$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-, *-%*PIECEWISEG6$7$,$F*\"#<2F*\"\"!7$,$F*\"#8%*otherwiseG!\"\"F.$F@!\"# !#5\"\"\"-%$sinG6#,$F-\"\"%$\"\"$FB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "for the case when " }{XPPEDIT 18 0 "lambda<0.05" "6#2%'lambdaG- %&FloatG6$\"\"&!\"#" }{TEXT -1 206 ". We can get a numerical solution for this system similar to figure 4.31 in the text. With a larger fo rcing function, we still experience oscillations that are fairly small in magnitude, but with a large " }{XPPEDIT 18 0 "v(0)" "6#-%\"vG6#\" \"!" }{TEXT -1 271 ", corresponding to a sudden gust of wind, the line ar model is no longer valid (because the cable goes slack) and the bri dge experiences larger oscillations which do not die down. All three \+ situations are included in the graph below of vertical displacement ag ainst time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sp:=dsolve( \{op(sysb),y(0)=-8/17,v(0)=0\},[y(t),v(t)],type=numeric):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "small:=odeplot(sp,[t,y(t)],0..5,lab els=[t,y] ):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "sysc:=su bs(a=17,b=13,alpha=0.01,g=10,mu=4,lambda=.3,sys2):\nfp:=dsolve(\{op(sy sc),y(0)=-8/17,v(0)=0\},[y(t),v(t)],type=numeric):\nlargeforce:=odeplo t(fp,[t,y(t)],0..5,labels=[t,y] ):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "fp2:=dsolve(\{op(sysc),y(0)=-8/17,v(0)=4\},[y(t),v(t )],type=numeric):\nlargeosc:=odeplot(fp2,[t,y(t)],0..5,labels=[t,y] ): \ndisplay(small,largeforce,largeosc);" }}{PARA 13 "" 1 "" {GLPLOT2D 304 304 304 {PLOTDATA 2 "6&-%'CURVESG6$7T7$\"\"!$!+`B)eq%!#57$$\"+k\"3 /-\"F+$!1\\pcl^B3[!#;7$$\"+Gj\"3/#F+$!1Q'Gw&\\_'4&F17$$\"+#\\C71$F+$!1 'omby*H>bF17$$\"+cEj\"3%F+$!1%Q#>Q#z?+'F17$$\"+?3/-^F+$!1A%G)ztYgkF17$ $\"+%)*[C7'F+$!1$eZEy.\\\"oF17$$\"+[r&G9(F+$!1'[1>)[Z/qF17$$\"+7`Ej\") F+$!1L,p5wQ(*pF17$$\"+wMn$=*F+$!1TPDPOI'z'F17$$F.!\"*$!1)Rx7')QwV'F17$ $\"+!)*[C7\"Ffn$!1/;sJ&*)\\)fF17$$\"+'z*[C7Ffn$!1#e60Qoy^&F17$$\"+71`E 8Ffn$!17z=\\Pu<^F17$$\"+G9dG9Ffn$!1onJ7>$Q&[F17$$\"+WAhI:Ffn$!1L62n,7r ZF17$$\"+gIlK;Ffn$!17AIgPp#)[F17$$\"+wQpMoE1ww;&F17$$\"+#pMn$ =Ffn$!1#[p^N/^d&F17$$\"+3bxQ>Ffn$!1)y'H\\j-LgF17$$\"+Cj\"3/#Ffn$!1Lq)) Hl>hkF17$$\"+Sr&G9#Ffn$!15;\"ekL^y'F17$$\"+cz*[C#Ffn$!1E)=Vdx!\\pF17$$ \"+s(QpM#Ffn$!1#QUrQ\"fDpF17$$\"+)ez*[CFfn$!1\\c(3\\a,s'F17$$\"+//-^DF fn$!1vr-!ox*pjF17$$\"+?71`EFfn$!1FGB-[GPfF17$$\"+O?5bFFfn$!1B)4]dq#)\\ &F17$$\"+_G9dGFfn$!1+](>v\"pH^F17$$\"+oO=fHFfn$!1@_zL*\\a*[F17$$\"+%[C 71$Ffn$!1L!>\"e3^N[F17$$\"++`EjJFfn$!1\\bor>3f\\F17$$\"+;hIlKFfn$!1Z;j oM@V_F17$$\"+KpMnLFfn$!1N1xTC.PcF17$$\"+[xQpMFfn$!1uU73L$32'F17$$\"+k& G9d$Ffn$!1aW]!4.%okF17$$\"+!QpMn$Ffn$!1ijU!>//w'F17$$\"+'>5bx$Ffn$!1>c (4*eY'*oF17$$\"+75bxQFfn$!1wXn*4wQ&oF17$$\"+G=fzRFfn$!1G2/LJPTmF17$$\" +WEj\"3%Ffn$!1dn#*)R/uH'F17$$\"+gMn$=%Ffn$!1M(\\$\\<>$)eF17$$\"+wUr&G% Ffn$!1Kp3U3&=Z&F17$$\"+#4bxQ%Ffn$!1-%o%=d\\N^F17$$\"+3fz*[%Ffn$!1#4(z! >,E$\\F17$$\"+Cn$=f%Ffn$!16+J&)o$y*[F17$$\"+Sv(Qp%Ffn$!1BY2&QSh.&F17$$ \"+c$=fz%Ffn$!1nm:%4V?K&F17$$\"+s\"fz*[Ffn$!10sY<`L/dF17$$\"+))****** \\Ffn$!1!pi$\\*>_6'F1-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7TF'7$F-$!1v YYKeN1[F17$F3$!1&p!y<%eA3&F17$F8$!1+3I*Q__Z&F17$F=$!1#)e#R9z-\"fF17$FB $!1\\`1jtr4jF17$FG$!1$*3f.K?2mF17$FL$!1>$)QJnlenF17$FQ$!1M*zuQ#o[nF17$ FV$!1XU)foR9f'F17$Fen$!1)p(zm58EjF17$Fjn$!1<7bE)H#3gF17$F_o$!1![ff'eH) p&F17$Fdo$!12Xf]ud]aF17$Fio$!1`Qw#\\BNI&F17$F^p$!1'fMT]kQF&F17$Fcp$!1n xO^V;b`F17$Fhp$!1`ib%4`3_&F17$F]q$!1EN1#e(*3t&F17$Fbq$!1%32&[;hSfF17$F gq$!1]h\\R#p(4hF17$F\\r$!1fEW-\\@5iF17$Far$!1Kp@#\\Z0B'F17$Ffr$!1[mX#R )4xhF17$F[s$!1&H:2c+72'F17$F`s$!1)\\o'>E\\VfF17$Fes$!1\\-(oEwl#eF17$Fj s$!1D0n\")fjZdF17$F_t$!1\"QAe2'eAdF17$Fdt$!1KcK@y%Hv&F17$Fit$!1GRcBu0E eF17$F^u$!1h*=C&)*[=fF17$Fcu$!1ofa^4!>+'F17$Fhu$!1Bg_(zz*\\gF17$F]v$!1 Tj\"yho\\/'F17$Fbv$!1N$>99'H#)fF17$Fgv$!1k3t(\\kA(eF17$F\\w$!1nX:)=]$Q dF17$Faw$!1PRZX.S7cF17$Ffw$!1aya@&zw_&F17$F[x$!1s6BH,@6bF17$F`x$!1t$=K +[qd&F17$Fex$!1$zj%yi*>s&F17$Fjx$!12D9bg![#fF17$F_y$!1\"Qn_UD#\\hF17$F dy$!1^#od3s/N'F17$Fiy$!15(Q*f](R['F17$F^z$!1'*[Jwrq9lF17$Fcz$!1r:k*)RA DkF17$Fhz$!1P%3$Q**o?iF1F\\[l-F$6$7TF'7$F-$!1l'=\\&[(4Y)!#<7$F3$\"1rQ! )*f(*G:#F17$F8$\"1\"oRVSdV&QF17$F=$\"1P$zc\"e4QSF17$FB$\"1Zr;g-$Go#F17 $FG$!1T&=B\\K;=$!#=7$FL$!1zbG+$\\Ts$F17$FQ$!1mP)zYL/y(F17$FV$!1-(p:AW0 :\"!#:7$Fen$!1&[DoBzhU\"Fifl7$Fjn$!1>QGS%3zb\"Fifl7$F_o$!1'[.(HPUB:Fif l7$Fdo$!1I$4t))G*G8Fifl7$Fio$!1Kt/%HD!35Fifl7$F^p$!1tTFQ<.fhF17$Fcp$!1 30hw:V)>#F17$Fhp$\"1+@;:!zW7\"F17$F]q$\"1^6wqm())G$F17$Fbq$\"1lWCpOq3S F17$Fgq$\"1u$*>+:#G>$F17$F\\r$\"1.VL`\\U1&*F_el7$Far$!1jbMiw\"\\T#F17$ Ffr$!1BVf&p#ynjF17$F[s$!1XaJ*>(3B5Fifl7$F`s$!17$Fhu$ \"1G@&G]hAc#F17$F]v$\"1/,B^%=?@t$[,&F17$Ffw$!1HA^DovD*)F17$F[ x$!1(\\.LI[3B\"Fifl7$F`x$!1ZrD[:^e9Fifl7$Fex$!1,_`os#o`\"Fifl7$Fjx$!1! yblQ0FX\"Fifl7$F_y$!1r@#QI#*3A\"Fifl7$Fdy$!1jZrQh&Q\"))F17$Fiy$!1Xa;wS GD\\F17$F^z$!1&fNZ#>w47F17$Fcz$\"1M&4C&>I-