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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Math 2250 Maple Project 2
\nSpring 2001 revision of David Eyre's project (2000 version).\n" }}
{PARA 0 "" 0 "" {TEXT -1 99 "There are six (6) problems in this projec
t. You are expected to answer the questions A, B, C , ... " }}{PARA 0 
"" 0 "" {TEXT -1 100 "associated with each problem.  The original work
sheet \"project2.mws\" is a template for the solution;" }}{PARA 0 "" 
0 "" {TEXT -1 104 " you must fill in the code and all comments. Sample
 code can be copied with the mouse. Please use pencil" }}{PARA 0 "" 0 
"" {TEXT -1 85 " freely to annotate the worksheet and to clarify any c
ode or figure presented herein." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "The problem headers:" }}{PARA 0 "" 0 "" 
{TEXT -1 432 " 2.1. OVERDAMPED FREE OSCILLATIONS.                    T
he first four problems study the\n 2.2. UNDERDAMPED FREE OSCILLATIONS.
                  simplest linear model for x(t).\n 2.3. UNDAMPED FORC
ED OSCILLATIONS (c=0).\n 2.4. DAMPED FORCED OSCILLATIONS (c>0).\n 2.5.
 LARGE SUSTAINED OSCILLATIONS.                      This is the second
, nonlinear model.\n 2.6. MCKENNA NON-HOOKES LAW CABLE MODEL.    This \+
is the third, nonlinear model.\n" }}{PARA 0 "" 0 "" {TEXT -1 77 " FREE
 OSCILLATIONS. Consider the general problem of free linear oscillation
s\n" }}{PARA 0 "" 0 "" {TEXT -1 63 "           m x'' + c x' + k x=0,\n
           x(0)=x0,  x'(0)=v0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 65 " where  m,  c  and  k  are non-negative c
onstants.  The symbols  " }}{PARA 0 "" 0 "" {TEXT -1 76 " x0  and  v0 \+
 are the initial position and initial velocity,  respectively.\n" }}
{PARA 0 "" 0 "" {TEXT -1 44 "2.1.  PROBLEM (OVERDAMPED FREE OSCILLATIO
NS)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 " \+
   A. Let m=k=1. Suggest a value for parameter c  > 0 so that the free
 oscillations are overdamped." }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}
{PARA 0 "" 0 "" {TEXT -1 100 "    B. Use x(0)=1 and x'(0)= -2 for the \+
initial conditions and Maple's \"dsolve\" to find the explicit" }}
{PARA 0 "" 0 "" {TEXT -1 95 "        real solution x(t). Plot the solu
tion x(t) for t=0 to t=5 using Maple's \"plot\" command." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "    C. Suggest ne
w values for x(0) and x'(0) such that the solution is non-negative, in
creases near t=0 but" }}{PARA 0 "" 0 "" {TEXT -1 114 "        eventual
ly decreases with limit zero at t=infinity. Find the explicit solution
 and plot it for t=0 to t=5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 107 "    D. Suggest new values for x(0) and x'(0) s
uch that the solution changes sign exactly once on t>0, then " }}
{PARA 0 "" 0 "" {TEXT -1 96 "         decreases to zero at t=infinity.
 Find the explicit solution and plot it for t=0 to t=5." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "EXAMPLE(Wrong param
eters! Change it!)\nde:=diff(x(t),t,t)-diff(x(t),t)+x(t)=0:\nic:=x(0)=
1,D(x)(0)= -2:\ndsolve(\{de,ic\},x(t),method=laplace);\nX:=unapply(rhs
(%),t):\nplot(X(t),t=0..5);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 160 "#2.1-A and 2.1-B\nde:=diff(x(t),t,t)+2*diff(x(t),t)+x(t)=0:\nic
:=x(0)=1,D(x)(0)= -2:\ndsolve(\{de,ic\},x(t),method=laplace);\nX:=unap
ply(rhs(%),t):\nplot(X(t),t=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"xG6#%\"tG,&*&F'\"\"\"-%$expG6#,$F'!\"\"F*F/F+F*" }}{PARA 13 "" 1 
"" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7X7$$\"\"!F)$\"\"
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io#[rb1%>#*F/7$$\"3/++]P[6j9Fap$!3/-;M'yv@2\"F27$$\"3UL$e*[z(yb\"Fap$!
3(*)o*yJJzu6F27$$\"3vmm;a/cq;Fap$!3]qeS4Jgh7F27$$\"3$ommmJ<gw\"Fap$!3s
Zs_yq)*48F27$$\"3.+]iSj0x=Fap$!31&>XX]YAM\"F27$$\"3gmmm\"pW`(>Fap$!3i#
QhtmMHN\"F27$$\"3K+]i!f#=$3#Fap$!3R'[#Q-C#*[8F27$$\"3>+](=xpe=#Fap$!3I
zJ%=\")zEL\"F27$$\"37nm\"H28IH#Fap$!3y7-qW#\\aI\"F27$$\"3um;zpSS\"R#Fa
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$!3QWZi&R\"*44\"F27$$\"3/+++DRW9HFap$!3oxh#>)HGQ5F27$$\"39++DJE>>IFap$
!3$y#=a**\\'=')*F/7$$\"3F+]i!RU07$Fap$!3RuKRMtje$*F/7$$\"3+++v=S2LKFap
$!3>GAi#opj!))F/7$$\"3Jmmm\"p)=MLFap$!3n&=:mnj)>$)F/7$$\"3B++](=]@W$Fa
p$!3e^:.#ReQ\"yF/7$$\"35L$e*[$z*RNFap$!3i,y9\\tZptF/7$$\"3e++]iC$pk$Fa
p$!3]\"o*H;B\"3!pF/7$$\"3Zm;H2qcZPFap$!36)zwwc)RxkF/7$$\"3O+]7.\"fF&QF
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$!3]&>lHI#em_F/7$$\"3yLLL$)*pp;%Fap$!3a,Y;J[`3\\F/7$$\"3(RL$3xe,tUFap$
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3n(QXEhXJS$F/7$$\"37+]iSjE!z%Fap$!3;X.%)4P!)\\JF/7$$\"3a+]P40O\"*[Fap$
!3ns7,Jt(G#HF/7$$\"\"&F)$!3!o=M'*zy^p#F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(
F(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F(F^\\l%(DEFAULTG" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "#2.1-C\nic:=x(0)=1,D(x)(0)= 2:\nds
olve(\{de,ic\},x(t),method=laplace);\nX:=unapply(rhs(%),t):\nplot(X(t)
,t=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&F'\"\"
\"-%$expG6#,$F'!\"\"F*\"\"$F+F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6%-%'CURVESG6$7Y7$$\"\"!F)$\"\"\"F)7$$\"3VmmmT&)G\\a!
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f1O\"F27$$\"32++](=h(e8F2$\"34&GT4\"z\\/8F27$$\"3/++]P[6j9F2$\"3GN\"GN
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\\u!p4G!*F67$$\"3>+](=xpe=#F2$\"3C.6B%GPK\\)F67$$\"37nm\"H28IH#F2$\"3B
i/jZ'>[&zF67$$\"3um;zpSS\"R#F2$\"3X+j#pp&\\zuF67$$\"3GLL3_?`(\\#F2$\"3
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?Nd&4U+hF67$$\"3uLLe*Gst!GF2$\"32jaUb-^(o&F67$$\"3/+++DRW9HF2$\"3^Z1k#
=;UG&F67$$\"39++DJE>>IF2$\"3YPp(Q*[=7\\F67$$\"3F+]i!RU07$F2$\"3-a9]z,#
Hd%F67$$\"3+++v=S2LKF2$\"3)Rw#*\\Ia$>UF67$$\"3Jmmm\"p)=MLF2$\"3UJ]$oz)
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a-SrLF67$$\"3e++]iC$pk$F2$\"3e'p_H*G38JF67$$\"3Zm;H2qcZPF2$\"3sEU@J4A'
)GF67$$\"3O+]7.\"fF&QF2$\"385nUN^+lEF67$$\"3Xmm;/OgbRF2$\"3o.\\+N@ijCF
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#4#F67$$\"3(RL$3xe,tUF2$\"3Cc51F4LE>F67$$\"3Cn;HdO=yVF2$\"3XX\\yz<it<F
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F67$$\"3\"RLLL)Qk%o%F2$\"3-s.$*3YQ!R\"F67$$\"37+]iSjE!z%F2$\"3=wu@i3Nx
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LOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F(Fc
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0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "#2.1-D\nic:
=x(0)= -1,D(x)(0)= 2:\ndsolve(\{de,ic\},x(t),method=laplace);\nX:=unap
ply(rhs(%),t):\nplot(X(t),t=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"xG6#%\"tG,&*&F'\"\"\"-%$expG6#,$F'!\"\"F*F*F+F/" }}{PARA 13 "" 1 
"" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7X7$$\"\"!F)$!\"\"
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LQF27$$\"3h+]PfyG7ZF2$!3(3[9Zkt2I$F27$$\"3emm\"z%4\\Y_F2$!3?+q?&o]H\"G
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$\"3%ommTIOo/\"!#<$\"3+Y^)y_sTk\"F/7$$\"3YLL3_>jU6Fap$\"3-4tQ/4l\\XF/7
$$\"37++]i^Z]7Fap$\"3<?l(z[CG<(F/7$$\"32++](=h(e8Fap$\"3io#[rb1%>#*F/7
$$\"3/++]P[6j9Fap$\"3/-;M'yv@2\"F27$$\"3UL$e*[z(yb\"Fap$\"3(*)o*yJJzu6
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$$\"3`+++D>#[Z%Fap$\"382NdAUgeRF/7$$\"3RnmT&G!e&e%Fap$\"3x]Z9qR^cOF/7$
$\"3\"RLLL)Qk%o%Fap$\"3n(QXEhXJS$F/7$$\"37+]iSjE!z%Fap$\"3;X.%)4P!)\\J
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urve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "2.2. PROBLEM (UNDERDAM
PED FREE OSCILLATIONS)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 78 "    A.  Let m=1, k=5. Find a  parameter value c > 0 \+
 so that the solution x(t)" }}{PARA 0 "" 0 "" {TEXT -1 77 "         ch
anges sign infinitely many times and decays to zero at t=infinity." }}
{PARA 0 "" 0 "" {TEXT -1 78 "         Plot the solution for initial va
lues x(0)=0,  x'(0)= 1 on t=0 to t=5." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 64 "    B. Estimate from the graph the ps
eudoperiod of the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 170 "#2.2-A Choose m=1,c=2,k=5.\nde:=diff(x(t
),t,t)+2*diff(x(t),t)+5*x(t)=0:\nic:=x(0)=0,D(x)(0)=1:\ndsolve(\{de,ic
\},x(t),method=laplace);\nX:=unapply(rhs(%),t):\nplot(X(t),t=0..5);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&-%$expG6#,$F'!\"\"
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urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#2.2-B The pseud
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 302 "FORCED LINEAR OSCILLATIONS.\n    \+
  Consider the forced problem\n\n      m x'' + c x' + k x = Fcos(wt),
\n      x(0)=x0,  x'(0)=v0,\n\n     where m, c, k, w are non-negative \+
constants and F is a nonzero constant.\n     The symbols  x0  and  v0 \+
 are the initial position and initial velocity,\n     respectively.\n
" }}{PARA 0 "" 0 "" {TEXT -1 49 "2.3. PROBLEM (UNDAMPED FORCED OSCILLA
TIONS (c=0))" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}{PARA 0 "" 0 "" 
{TEXT -1 92 "    A.  Choose   m=1,  c=0,  k=4,  w=6,  so that the forc
ing  frequency w is 3 times larger " }}{PARA 0 "" 0 "" {TEXT -1 98 "  \+
        than the natural frequency w0=2.  Let  F=1,  x0=0,  v0=0. Solv
e for x(t) using dsolve()." }}{PARA 0 "" 0 "" {TEXT -1 95 "          P
lot the solution x(t) on a suitable interval in order to show the glob
al behavior of" }}{PARA 0 "" 0 "" {TEXT -1 28 "          the solution \+
x(t)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 
"    B. The solution x(t) is the sum of two functions, one of period  \+
2Pi/w  and the other of period  2Pi/w0." }}{PARA 0 "" 0 "" {TEXT -1 
82 "         Find the period of x(t) by examining the graph and the eq
uation for x(t)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 113 "    C. Let c=0,  m=1,  k=4,  F=10. Suggest a value for t
he forcing frequency  w  so that the oscillations exhibit" }}{PARA 0 "
" 0 "" {TEXT -1 92 "         resonance.  Show resonant behavior on a g
raph using initial values x(0)=0, x'(0)=0." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "#2.3-A \nde:=d
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "2.4. PROBLEM (DAMPED FORCED O
SCILLATIONS (c>0))" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }}{PARA 0 
"" 0 "" {TEXT -1 75 "    A.  Choose parameter values m=1, c=2,  k=25, \+
 F=10,  w=1,  x0=0,  v0=0." }}{PARA 0 "" 0 "" {TEXT -1 62 "          S
olve for x(t) and plot the solution on t=0 to t=10." }}{PARA 0 "" 0 "
" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 80 "    B.  Extract fr
om the solution x(t) the steady-state solution xss(t). Plot it" }}
{PARA 0 "" 0 "" {TEXT -1 24 "         on t=0 to t=10." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "    C. Choose F = 10, \+
m=1, k = 25, and consider c=2, c=1, c=1/2.  Compute the amplitude func
tion" }}{PARA 0 "" 0 "" {TEXT -1 106 "        C(w) of xss(t) [page 346
] for these three equations, then plot for w=0 to w=20 the three ampli
tude" }}{PARA 0 "" 0 "" {TEXT -1 39 "        graphs on a single set of
 axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 
"    D. Print the natural frequency w and the maximum amplitude C(w) f
or each case c=2, c=1, c=1/2." }}{PARA 0 "" 0 "" {TEXT -1 89 "        \+
The three data pairs should show that max C(w) becomes larger as c ten
ds to zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "EXAMPLE(Beware! Wrong values!)\nF:=5: m:=1: k:=25: c:='c': w:='
w':" }}{PARA 0 "" 0 "" {TEXT -1 88 "C:=(w,c)->F/sqrt((k-m*w*w)^2+(c*w)
^2):\nplot(\{C(w,4),C(w,3),C(w,2)\},w=0..15,color=black);" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "#2.4-A\nde:=diff(x(t),t,t)+2*diff(
x(t),t)+25*x(t)=10*cos(t):\nic:=x(0)=0,D(x)(0)=0:\ndsolve(\{de,ic\},x(
t),method=laplace);\nX:=unapply(rhs(%),t):\nplot(X(t),t=0..10);" }}
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{TEXT -1 796 "NONLINEAR MODEL WITH GEOMETRY INCLUDED.\nConsider the no
nlinear, forced, damped oscillator equation for torsional motion, with
 bridge geometry\nincluded,\n\n    mx'' + c x' + k sin(x)cos(x) = F co
s wt ,\n    x(0) = x0,  x'(0) = v0\n\nand its corresponding linearized
 equation\n\n   mx'' + c x' + k x = F cos wt ,  \n   x(0) = x0,  x'(0)
 = v0.\n\n  Maple code used to solve and plot the solutions appears be
low.\n\n   m:=1:  F := 10:  w := 5: k := 25: c:= 7:\n   deNonLinear:= \+
m*diff(x(t),t,t) + c*diff(x(t),t) + k*sin(x(t))*cos(x(t)) = F*cos(w*t)
:\n   deLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) + k*x(t) = F*cos(w*
t):\n   with(DEtools): opts:=stepsize=0.1:\n   DEplot(deNonLinear,x(t)
,t=0..2,[[x(0)=0,D(x)(0)=0]],opts,title='NonLinear');\n   DEplot(deLin
ear,x(t),t=0..2,[[x(0)=0,D(x)(0)=0]],opts,title='Linear');\n" }}{PARA 
0 "" 0 "" {TEXT -1 43 "2.5. PROBLEM (LARGE SUSTAINED OSCILLATIONS)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "    A. Se
t the parameters in the linear and nonlinear equations equal to the fo
llowing:" }}{PARA 0 "" 0 "" {TEXT -1 88 "         m=1, c = 0.05,  k = \+
2.4,  w = 1.2 ,  F = 0.06,  x0=0,  v0=0. Solve and plot the" }}{PARA 
0 "" 0 "" {TEXT -1 76 "        solutions of the linear and nonlinear e
quations from t=150 to t=250." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 256 "    B. Set the parameters in the linear \+
and nonlinear equations equal to the following:\n         m=1, c = 0.0
5,  k = 2.4,  w = 1.2 ,  F = 0.06,  x0=1.2,  v0=0. Solve and plot the
\n        solutions of the linear and nonlinear equations from t=200 t
o t=300.\n" }}{PARA 0 "" 0 "" {TEXT -1 90 "    C. The plots made in A \+
and B represent the steady-state solutions of the two equations" }}
{PARA 0 "" 0 "" {TEXT -1 95 "         under two different sets of init
ial conditions. Argue in a sentence why the two linear" }}{PARA 0 "" 
0 "" {TEXT -1 92 "         plots have to be identical, based upon the \+
superposition formula x(t)=xh(t)+xss(t)," }}{PARA 0 "" 0 "" {TEXT -1 
92 "         even though the homogeneous solution xh(t) is different f
or the two plots. Finally," }}{PARA 0 "" 0 "" {TEXT -1 94 "         de
termine the ratio of the apparent amplitudes (a number > 1) for the no
nlinear plots" }}{PARA 0 "" 0 "" {TEXT -1 92 "         and explain why
 \"large sustained oscillations\" is an appropriate description of the
" }}{PARA 0 "" 0 "" {TEXT -1 41 "         nonlinear steady-state behav
ior." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 413 "#2.5-A\nm:=1:  F := 0.06:  w := 1.2: k := 2.4: c:=
 0.05:\nopts:=stepsize=0.1:\ndeNonLinear:= m*diff(x(t),t,t) + c*diff(x
(t),t) + k*sin(x(t))*cos(x(t)) = F*cos(w*t):\n   deLinear:= m*diff(x(t
),t,t) + c*diff(x(t),t) + k*x(t) = F*cos(w*t):\n   with(DEtools):\n   \+
DEplot(deNonLinear,x(t),t=150..250,[[x(0)=0,D(x)(0)=0]],opts,title='No
nLinear');\n   DEplot(deLinear,x(t),t=150..250,[[x(0)=0,D(x)(0)=0]],op
ts,title='Linear');\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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