{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 Output" 2 20 "" 0 1 0 0 255 1 0 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 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "This Maple Worksheet produces the solution to the rocket launch linear program." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "It proceeds by first constructing the constraints using t he \"sequence\" command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(simplex):" }} {PARA 7 "" 1 "" {TEXT -1 87 "Warning, the protected names maximize and minimize have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Define the paramete rs N (number of intervals), T (total time), g (gravity), a0=maximum ac celeration, h=desired height" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=25;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "T:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g:=32;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a0:=48; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G\"#[" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "h:=300;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" hG\"$+$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tau:=T/N;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG#\"\"#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We have 4N constr aints total, constructed via eqns (3.26, 3.27, 3.28) in the text:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "CS1:=seq(x[N+i]<=a0,i=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "CS2:=seq(-sum((1/2+i-j)*x[j],j=1..i) <=-(i^2/2)*g,i=1 ..N-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "CS3:=sum( (1/2+N -j)*x[j],j=1..N )=(N^2/2)*g+ h/(tau^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "CS4:=seq(x[i]<=x[i+N],i=1..N):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "CS5:=seq(-x[i]<=x[i+N],i=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "CS:=[CS1,CS2,CS3,CS4,CS5]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Obj:=sum(x[j],j=N+1..2*N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Z:=minimize(Obj,CS);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"ZGF2/&F(6#\"#?F2/&F(6#\"#\\F2/&F(6#\"#]F2/&F(6#\"#=F2/&F(6#\"#C F2/&F(6#\"#DF2/&F(6#\"#@F2/&F(6#\"#AF2/&F(6#\"# " 0 "" {MPLTEXT 1 0 10 "assign(Z); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=seq(x[i],i=1..N);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG6;\"#[F&F&F&F&F&F&F&F&F&F&F&F&# \"$i'\"#B\"\"!F*F*F*F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "V:=seq(tau*(sum(A[j],j=1..i)-i*g) ,i=1..N):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Y:=seq(tau^2*( sum( (((1/2)+ i-j)*A[j] ),j=1..i)-i^2*g/2),i=1..N):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the acceleration, vel ocity and position functions, we'll use Heaviside functions rather tha n" }}{PARA 0 "" 0 "" {TEXT -1 70 "piecewise defined functions... Just a matter of taste, not substance." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "Vall:=(a0-g)*t*(Heaviside(t)-Heaviside(t-tau))+sum( ( (A[j]-g)* (t-(j-1)*tau) + V[j-1])*(Heaviside(t-(j-1)*tau)-Heaviside(t-j*tau)),j= 2..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "Yall:=(1/2)*(a0- g)*t^2*(Heaviside(t)-Heaviside(t-tau))+sum( ((1/2)*(A[j]-g)*(t-(j-1)*t au)^2+V[j-1]*(t-(j-1)*tau)+Y[j-1])* (Heaviside(t-(j-1)*tau)-Heaviside( t-j*tau)),j=2..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Aall: =sum(A[j]*(Heaviside(t-(j-1)*tau)-Heaviside(t-j*tau)),j=1..N):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(Yall,t=0..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(Vall,t=0..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(Aall,t=0..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "32" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }