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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 21 "Tacoma Narrows Bridge" }}
{PARA 200 "" 0 "" {TEXT -1 125 "We model the motion of the center of a
 suspension bridge as if it were a spring-mass system with the followi
ng modifications:" }}{PARA 200 "" 0 "" {TEXT -1 124 "(1) The \"constan
t\" of proportionality for the spring force is smaller when the cable \+
is slack versus when the cable is taut." }}{PARA 200 "" 0 "" {TEXT -1 
59 "(2) Gravity provides an additional constant force downward." }}
{PARA 200 "" 0 "" {TEXT -1 63 "(3) The wind generally provides an addi
tional sinusoidal force." }}{PARA 200 "" 0 "" {TEXT -1 53 "(4) Gusts o
f wind provide a nonzero initial velocity." }}{PARA 200 "" 0 "" {TEXT 
-1 0 "" }}{PARA 200 "" 0 "" {TEXT -1 118 "Let y(t) be the upward dista
nce of the bridge center from the point at which the cables transition
 from taut to slack." }}{PARA 200 "" 0 "" {TEXT -1 0 "" }}{PARA 200 "
" 0 "" {TEXT -1 80 "There were problems in computing solutions over a \+
sufficiently long time domain." }}{PARA 200 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 123 "restart:\nspring := `if`
(y(t)<0, k1, k2);\neqn := m*diff(y(t),t$2) + b*diff(y(t),t) + spring*y
(t) = -g + lambda*sin(omega*t);" }}}{EXCHG {PARA 200 "" 0 "" {TEXT -1 
74 "Note that the parameter values within the `if` operator must be fi
lled in." }}{PARA 201 "> " 0 "" {MPLTEXT 1 -1 212 "m:=1: b:=0.01: k1:=
17: k2:=13: g:=10: omega:=4: lambda:=0.02:\nspring := `if`(y(t)<0, 17,
 13);\neqn := m*diff(y(t),t$2) + b*diff(y(t),t) + spring*y(t) = -g + l
ambda*sin(omega*t);\nics := \{y(0) = -g/k2, D(y)(0) = 0\};" }}}{EXCHG 
{PARA 200 "" 0 "" {TEXT -1 19 "This does not work." }}{PARA 201 "> " 
0 "" {MPLTEXT 1 -1 24 "dsolve(\{eqn\} union ics);" }}}{EXCHG {PARA 
200 "" 0 "" {TEXT -1 34 "Instead obtain a numeric solution." }}{PARA 
201 "> " 0 "" {MPLTEXT 1 -1 41 "soln := dsolve(\{eqn\} union ics, nume
ric);" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 17 "soln(0); soln(3
);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 200 "" 
0 "" {TEXT -1 27 "Library needed for odeplot." }}{PARA 201 "> " 0 "" 
{MPLTEXT 1 -1 12 "with(plots):" }}}{EXCHG {PARA 200 "" 0 "" {TEXT -1 
76 "Plots of y versus t, y and v versus t, phase plot, and phase plot \+
animation." }}{PARA 201 "> " 0 "" {MPLTEXT 1 -1 200 "soln := dsolve(\{
eqn\} union ics, numeric, range=0..6);\nodeplot(soln, refine=4);\nodep
lot(soln, [[t,y(t)],[t,D(y)(t)]]);\nodeplot(soln, [y(t),D(y)(t)], refi
ne=4);\nodeplot(soln, [y(t),D(y)(t)], frames=20);" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 -1 315 "m:=1: b:=0.01: k1:=17: k2:=13: g:=10:
 omega:=4: lambda:=.1:\nspring := `if`(y(t)<0, 17, 13):\neqn := m*diff
(y(t),t$2) + b*diff(y(t),t) + spring*y(t) = -g + lambda*sin(omega*t):
\nics := \{y(0) = -g/k2, D(y)(0) = 3\}:\nsoln := dsolve(\{eqn\} union \+
ics, numeric, range=0..100):\nodeplot(soln, [[t,y(t)],[t,D(y)(t)]], re
fine=4);" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "13" 0 
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