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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 399 "MATH \+
2280 - 1                                THIRD MAPLE ASSIGNMENT\nTreibe
rgs                                    Due Wednesday, October 15, 2003
.\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 a previous class\n" }{URLLINK 17 "http:/
/www.math.utah.edu/~treiberg/M2250LA2.html" 4 "http://www.math.utah.ed
u/~treiberg/M2250LA2.html" "" }{MPLTEXT 0 21 1 "\n" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 887 "There you will find links to a movie of the spectac
ular catastophic collapse of the Tacoma Narrows Bridge, as well as mat
hematical models of vibrations on that bridge that you can run on MAPL
E.\n\nWe begin by studying how the forcing frequency affects the solut
ion to the undamped equation and see the phenomenon of resonance: as t
he forcing frequency  w  approaches the natural frequency  w0,  the am
plitude of the steady wave increases. When  w=w0  the system is in res
onance and the solution blows up.  When  w is close but not equal to w
0,  the solution will exhibit beats.\n\nThen we consider the damped eq
uation. This time the amplitude of the steady solution will be greates
t when the forcing frequency is the practical resonant frequency of th
e system, which will be less than the forcing frequency.  First we ill
ustrate with particular values of the physical constants  m, k and  c.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "\n with(DEtools):" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "\n(1.)   Consider the case wi
thout damping first. Let m=k=1 and c=0." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "\n ode1:=diff(y(t),t,t)+y(t)=0;\n" }{MPLTEXT 1 0 20 "
\n dsolve(ode1,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/,&-%
%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"F*F2\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&%$_C1G\"\"\"-%$sinGF&F+F+*&%$_C2GF+
-%$cosGF&F+F+" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 146 "\n\n\nThe \+
system has a natural frequency  w0=1. Now consider the inhomogeneous e
quation with a forcing term of fixed amplitude  F0  and frequency  w.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "w:='w'; F0:='F0'; f:=t-
>F0*cos(w*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGF$" }}{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*$)F5\"\"#F-F-F1F1" }}
}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 188 "Using cos(A+B)=cos A cos B \+
- sin A sin B  and cos(A-B)=cos A cos B + sin A sin B  where  A+B=wt  \+
and  A-B = t we find the trig substitution reduces the difference to a
 product of sinusoids" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "\n
 subs(-cos(t)+cos(w*t)=2*sin((1+w)*t/2)*sin((w-1)*t/2),sol2);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*,\"\"#\"\"\"%#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 slowly varying function  env(t)  may by regarded as the \+
enveloping amplitude 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*sin((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(p
sol1),peng,-peng],t=0..200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$env
Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$**\"\"#\"\"\"%#F0GF/-%$sinG6#,$*
&#F/F.F/*&,&F/!\"\"%\"wGF/F/9$F/F/F/F/,&F/F9*$)F:F.F/F/F9F/F(F(F(" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
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{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 210 "Note that the amplitude of t
he beats is 2 F0 / |w^2-1|  which blows up as w-> w_0=1 the resonant f
requency. What happens if  w=w0? The forcing term adds energy to each \+
cycle so the solution grows without bound." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 175 "ode3:=diff(y(t),t,t)+y(t)=F0 *cos(t);\nsol3:=simpl
ify(dsolve(\{ode3,y(0)=0,D(y)(0)=0\},y(t)));\nplot([rhs(subs(F0=1,sol3
)),t/2,-t/2],t=0..200);\nrestart:with(DEtools):with(plots):\n" }}
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$F`dr$!3$3++]P+V)oFS7$Fjir$!3@mm\"zpe*zqFS7$F\\bs$!3;,++D\\'QH(FS7$Fdi
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DJXaE\")FS7$F\\ju$!3%zmmm'*RRL)FS7$F^bv$!3(ommTvJga)FS7$F`jv$!3'[L$e9t
Oc()FS7$Fhaw$!32,++]Qk\\*)FS7$Fdjw$!3uLL$3dg6<*FS7$Ffbx$!3]nmmmxGp$*FS
7$Fb[y$!3J,+D\"oK0e*FS7$F^dy$!3Y,+v=5s#y*FS7$F`\\z$!$+\"F)-Fe\\z6&Fg\\
zFh\\zFh\\zF(-%+AXESLABELSG6$Q\"t6\"Q!Fg_[l-%%VIEWG6$;F(F`\\z%(DEFAULT
G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "
Curve 2" "Curve 3" }}}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c
hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
0 21 118 "(2.) The damped case. This time, lets stiffen and damp the d
ashpot m=1,c=2, k=5. The characteristic equation has roots" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve(r^2+2*r+5=0,r);\n\n          \+
           " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^$!\"\"\"\"#^$F$!\"#" }
}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 47 "so the pseudofrequency is  \+
w1^2=k/m-c^2/4m^2=4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode
3:=diff(y(t),t,t)+2*diff(y(t),t)+5*y(t)=0;\ndsolve(ode3,y(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode3G/,(-%%diffG6$-%\"yG6#%\"tG-%\"
$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F2\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*(%$_C1G\"\"\"-%$expG6#,$F'!\"\"
F+-%$sinG6#,$*&\"\"#F+F'F+F+F+F+*(%$_C2GF+F,F+-%$cosGF3F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 142 "The system has a natural fre
quency w1=1. Now consider the inhomogeneous equation with a forcing te
rm of fixed amplitude  F0  and frequency  w." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 117 "f:=t->F0*cos(w*t);\node4:=diff(y(t),t,t)+2*diff
(y(t),t)+5*y(t)=f(t);\nsol4:=dsolve(\{ode4,y(0)=0,D(y)(0)=1\},y(t));\n
\n\n\n  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)oper
atorG%&arrowGF(*&%#F0G\"\"\"-%$cosG6#*&%\"wGF.9$F.F.F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%ode4G/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-
\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F2*&%#F0GF2-%$cosG6#*&%\"wG
F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol4G/-%\"yG6#%\"tG,(*&
#\"\"\"\"\"#F-**-%$expG6#,$F)!\"\"F--%$sinG6#,$*&F.F-F)F-F-F-,,*&%#F0G
F-)%\"wGF.F-F-*&\"\"'F-F=F-F-*$)F>\"\"%F-F4\"#DF4*&\"\"&F-F<F-F-F-,(FD
F-*&F@F-F=F-F4FAF-F4F-F4*,F0F--%$cosGF7F-F<F-,&FFF4*$F=F-F-F-FGF4F-*(F
<F-,(*&FFF--FK6#*&F>F-F)F-F-F4*(F.F--F6FRF-F>F-F4*&F=F-FQF-F-F-FGF4F4
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 85 "The exponential terms di
e away and the remaining particular solution has an amplitude" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "A:=F0*(5-w^2)/(25-6*w^2+w^4
);\nB:=F0*2*w/(25-6*w^2+w^4);\nC:=simplify(sqrt(A^2+B^2),sqrt,symbolic
);\namp:= subs(F0=1,C);\nplot([rhs(subs(\{F0=1,w=sqrt(3)\},sol4)), sub
s(\{F0=1,w=sqrt(3)\},amp), -subs(\{F0=1,w=sqrt(3)\},amp)], t=0..20);\n
plot(amp,w=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*(%#F0G\"
\"\",&\"\"&F'*$)%\"wG\"\"#F'!\"\"F',(\"#DF'*&\"\"'F'F+F'F.*$)F,\"\"%F'
F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$**\"\"#\"\"\"%#F0GF(%
\"wGF(,(\"#DF(*&\"\"'F()F*F'F(!\"\"*$)F*\"\"%F(F(F0F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"CG*&%#F0G\"\"\",(\"#DF'*&\"\"'F')%\"wG\"\"#F'!
\"\"*$)F-\"\"%F'F'#F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ampG*&\"
\"\"F&*$,(\"#DF&*&\"\"'F&)%\"wG\"\"#F&!\"\"*$)F-\"\"%F&F&#F&F.F/" }}
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" 0 "" {MPLTEXT 0 21 172 "The maximum amplitude of the resonance occur
s at the maximum of the function amp(w). This practical resonance poin
t  1.732 is somewhat less than the natural frequency w1=2." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(diff(amp,w)=0,w);\nevalf(%);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!*$\"\"$#\"\"\"\"\"#,$F$!\"
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {MPLTEXT 0 21 1468 "\n\nPROBLEMS\n\n1.  Choose some other values \+
of  F0,  k, m  and  w   in the undamped equation (as in  ode1). Show t
hat when  w  is not equal to but is close the natural frequency, then \+
some solutions exhibit beats. Show that as the driving frequency appro
aches the resonant frequency, the amplitude of the beats become large.
 Show that if the driving frequency equals the resonant frequerncy the
n the solution blows up. Use both symbolic computation and plots of yo
ur functions to demonstrate these phenomena.\n\n2.   Choose some other
 values of  F0,  c, k, m,  w   and initial values (not those in either
 handout!) for the damped equation (as in  ode3). Find the amplitude o
f the forced steady solution as a function of the driving frequency. F
ind the practical resonant frequency -- the w  for which the amplitide
 is max. (Be careful to choose  m, c, and k so that the practical reso
nant frequency is positive.) Show an example where the initial data ar
e large compared to the driving force and the natural frequency is muc
h less than the driving frequency such that the natural oscillations d
ominate for a while until the slaved forced oscillations dominate.  Us
e both symbolic computation and plots of your functions to demonstrate
 these phenomena.\n\n3. Choose a driving force  F(t)  = g(t)cos(wt)  w
hose amplitude  g(t)  is not constant and not from the text. (e.g. see
 p. 221-222 3rd ed.)  Solve and plot an example of damped vibrations t
hat are forced by your  F(t).\n\n" }}}}{MARK "0 0 1" 84 }{VIEWOPTS 1 
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