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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 23 59 "Math 2250 Maple Project 3a
, NOVEMBER 2004. Tacoma Narrows." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}
{PARA 0 "" 0 "" {TEXT 23 68 "NAME _______________________ CLASSTIME _
_______ VERSION A-K or L-Z" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA
0 "" 0 "" {TEXT 23 69 "Circle the version - see problem 3.1. There are
three (3) problems in" }}{PARA 0 "" 0 "" {TEXT 23 71 "this project. P
lease answer the questions A, B, C , ... associated with" }}{PARA 0 "
" 0 "" {TEXT 23 66 "each problem. The original worksheet \"2250mapleL3
a-F2004.mws\" is a" }}{PARA 0 "" 0 "" {TEXT 23 70 "template for the so
lution; you must fill in the code and all comments." }}{PARA 0 "" 0 "
" {TEXT 23 71 "Sample code can be copied with the mouse. Use pencil fr
eely to annotate" }}{PARA 0 "" 0 "" {TEXT 23 50 "the worksheet and to \+
clarify the code and figures." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}
{PARA 0 "" 0 "" {TEXT 23 70 "The problem headers for the FALL 2004 rev
ision of David Eyre's project" }}{PARA 0 "" 0 "" {TEXT 23 25 "(origina
l was year 2000)." }}{PARA 0 "" 0 "" {TEXT 23 45 " __________3.1. UNDA
MPED FORCED OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT 23 36 " __________3
.2. PRACTICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT 23 51 " __________3.
3. MCKENNA NON-HOOKES LAW CABLE MODEL." }}{PARA 0 "" 0 "" {TEXT 23 0 "
" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 44 "3.1. \+
PROBLEM (UNDAMPED FORCED OSCILLATIONS )" }}{PARA 0 "" 0 "" {TEXT 23 0
"" }}{PARA 0 "" 0 "" {TEXT 23 65 "FORCED LINEAR OSCILLATIONS. Conside
r the undamped forced problem" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}
{PARA 0 "" 0 "" {TEXT 23 30 " mx'' + k x = 7 cos(wt)," }}{PARA
0 "" 0 "" {TEXT 23 23 " x(0)=0, x'(0)=0," }}{PARA 0 "" 0 ""
{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "where m, k and w are non-
negative constants. Values m, k are defined by" }}{PARA 0 "" 0 ""
{TEXT 23 35 "the first letter of your last name:" }}{PARA 0 "" 0 ""
{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 58 " Version A-K: m=1, k=2.5 \+
Version L-Z: m=2, k=3.5" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}
{PARA 0 "" 0 "" {TEXT 23 72 " A. Choose the forcing angular freque
ncy w to be 5 times larger than" }}{PARA 0 "" 0 "" {TEXT 23 72 " \+
the natural angular frequency w0, w0^2=k/m. Solve for x(t) using" }}
{PARA 0 "" 0 "" {TEXT 23 72 " dsolve(). Plot the solution x(t) \+
on a suitable interval in order" }}{PARA 0 "" 0 "" {TEXT 23 68 " \+
to show the global behavior of the solution x(t). See Figure" }}
{PARA 0 "" 0 "" {TEXT 23 56 " 5.6.2, page 350. Reference: Examp
le 1, page 350." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 ""
{TEXT 23 70 " B. The solution x(t) in part A is the sum of two func
tions, one of" }}{PARA 0 "" 0 "" {TEXT 23 70 " \200 period 2Pi/w \+
and the other of period 2Pi/w0. Display the exact" }}{PARA 0 "" 0 ""
{TEXT 23 70 " period, as calculated from the solution formula fo
r x(t) -- see" }}{PARA 0 "" 0 "" {TEXT 23 69 " page 350, Example
1, for details about least common multiples." }}{PARA 0 "" 0 ""
{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 62 " C. Suggest a value fo
r the forcing frequency w so that the" }}{PARA 0 "" 0 "" {TEXT 23 67 "
oscillations exhibit resonance. Show resonant behavior on a" }
}{PARA 0 "" 0 "" {TEXT 23 51 " graph. Check against Figure 5.6.4
, page 352." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT
23 43 "# Use semicolons to see what you have done." }}{PARA 0 "" 0 ""
{TEXT 23 72 "m:=???; k:=???; # Define mass and
Hooke's const." }}{PARA 0 "" 0 "" {TEXT 23 56 "w:=5*sqrt(k/m): \+
# w = 5 times w0" }}{PARA 0 "" 0 "" {TEXT 23 74 "de:
=m*diff(x(t),t,t)+k*x(t)=7*cos(w*t); # Define the differential equatio
n" }}{PARA 0 "" 0 "" {TEXT 23 71 "ic:=x(0)=0,D(x)(0)= 0: \+
# Define the initial conditions" }}{PARA 0 "" 0 "" {TEXT 23 69 "p:
=dsolve(\{de,ic\},x(t),method=laplace): # Symbolically solve for x(t)
" }}{PARA 0 "" 0 "" {TEXT 23 70 "X:=unapply(rhs(p),t): \+
# Make X(t)= the dsolve answer" }}{PARA 0 "" 0 "" {TEXT 23 64 "a:=0
: b:=???: # Define the plot domain" }}{PARA
0 "" 0 "" {TEXT 23 59 "plot(X(t),t=a..b); # Plot \+
the solution" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 6 "#3.1-A" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 6 "#3.1-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#3
.1-C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"" 0 "" {TEXT 23 34 "2.2. PROBLEM (PRACTICAL RESONANCE)" }}{PARA 0 ""
0 "" {TEXT 23 37 " Consider the damped forced problem" }}{PARA 0 ""
0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 37 " mx'' + c x' + \+
k x = 7 cos(w t)," }}{PARA 0 "" 0 "" {TEXT 23 23 " x(0)=0, x'(0)
=0." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 56 "De
pending on the first letter of your last name, assume:" }}{PARA 0 ""
0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 55 " Version A-K: m=1, k
=25 Version L-Z: m=2, k=36" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }
}{PARA 0 "" 0 "" {TEXT 23 70 " A. Consider the damping constants c=
2, c=1 and c=1/2. Compute the" }}{PARA 0 "" 0 "" {TEXT 23 63 " \+
amplitude function C(w) [(21), page 355] for these three" }}{PARA 0 "
" 0 "" {TEXT 23 70 " equations, then plot for w=0 to w=20 the th
ree amplitude graphs" }}{PARA 0 "" 0 "" {TEXT 23 72 " on a singl
e set of axes. Compare against Figure 5.6.9 page 357 of" }}{PARA 0 ""
0 "" {TEXT 23 50 " E&P (it has one curve, yours has 3 curves)."
}}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 " B. \+
For each case c=2, c=1, c=1/2, print the values w*, C* where" }}{PARA
0 "" 0 "" {TEXT 23 70 " C*=C(w*)=max \{C(w) : 0 <= w <= 20\}. Th
e three data pairs should" }}{PARA 0 "" 0 "" {TEXT 23 70 " show \+
that C* becomes larger as c tends to zero. SAVE YOUR MAPLE" }}{PARA 0
"" 0 "" {TEXT 23 22 " FILE FREQUENTLY" }}{PARA 0 "" 0 "" {TEXT
23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 " Maple Hint: Use Maple's mou
se interface on the graphic of Part C." }}{PARA 0 "" 0 "" {TEXT 23 69
" Specifically, click on a possible maximum (horizontal tangent) in
" }}{PARA 0 "" 0 "" {TEXT 23 66 " the graph to display the values w
*, C* on the screen. Copy the" }}{PARA 0 "" 0 "" {TEXT 23 36 " valu
es into a worksheet comment." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA
0 "" 0 "" {TEXT 23 9 "# EXAMPLE" }}{PARA 0 "" 0 "" {TEXT 23 42 "F:=???
: m:=???: k:=???: unassign('c','w'):" }}{PARA 0 "" 0 "" {TEXT 23 46 "C
:=unapply(F/sqrt((k-m*w*w)^2+(c*w)^2),(w,c));" }}{PARA 0 "" 0 ""
{TEXT 23 51 "plot(\{C(w,2),C(w,1),C(w,1/2)\},w=0..15,color=black);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 51 "#3.2-A Plot C(w), three graphics on one set of axes"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#3.2-B Table of six data \+
values for w*, C*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 23 39 "3.3. PROBLEM (NONLINEAR MCKENNA MO
DELS)" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "
There are six (6) parts 3.3A to 3.3F to complete. Mostly, this is mous
e" }}{PARA 0 "" 0 "" {TEXT 23 60 "copying. Retyping the maple code by \+
hand is not recommended." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "
" 0 "" {TEXT 23 49 "NONLINEAR TORSIONAL MODEL WITH GEOMETRY INCLUDED.
" }}{PARA 0 "" 0 "" {TEXT 23 72 "Consider the nonlinear, forced, dampe
d oscillator equation for torsional" }}{PARA 0 "" 0 "" {TEXT 23 38 "mo
tion, with bridge geometry included," }}{PARA 0 "" 0 "" {TEXT 23 0 ""
}}{PARA 0 "" 0 "" {TEXT 23 59 " x'' + 0.05 x' + 2.4 sin(x)cos(x) = \+
0.06 cos (12 t/10) ," }}{PARA 0 "" 0 "" {TEXT 23 26 " x(0) = x0, x
'(0) = v0" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23
41 "and its corresponding linearized equation" }}{PARA 0 "" 0 ""
{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 " x'' + 0.05 x' + 2.4 x \+
= 0.06 cos (12 t/10) ," }}{PARA 0 "" 0 "" {TEXT 23 26 " x(0) = x0, \+
x'(0) = v0." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT
23 71 "The spring-mass system parameters are m=1, c = 0.05, k = 2.4, w
= 1.2 ," }}{PARA 0 "" 0 "" {TEXT 23 72 "F = 0.06. Maple code used to \+
solve and plot the solutions appears below." }}{PARA 0 "" 0 "" {TEXT
23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 43 " # Use \"copy as maple text\"
for maple 6+." }}{PARA 0 "" 0 "" {TEXT 23 62 " x0:=0: a:=200: b:=30
0: # For part A. Change it for part B!" }}{PARA 0 "" 0 "" {TEXT 23
63 " v0:=0: m:=1: F := 0.06: w := 1.2: m:=1: c:= 0.05: k:= 2.4:" }
}{PARA 0 "" 0 "" {TEXT 23 38 " with(DEtools): opts:=stepsize=0.1:"
}}{PARA 0 "" 0 "" {TEXT 23 70 " deLinear:= m*diff(x(t),t,t) + c*diff
(x(t),t) + k*x(t) = F*cos(w*t):" }}{PARA 0 "" 0 "" {TEXT 23 36 " ICl
inear:=[[x(0)=x0,D(x)(0)=v0]]:" }}{PARA 0 "" 0 "" {TEXT 23 61 " DEpl
ot(deLinear,x(t),t=a..b,IClinear,opts,title='Linear');" }}{PARA 0 ""
0 "" {TEXT 23 52 " deNonLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) +
" }}{PARA 0 "" 0 "" {TEXT 23 52 " k*sin(x(t))*cos(x(t)
) = F*cos(w*t):" }}{PARA 0 "" 0 "" {TEXT 23 39 " ICnonlinear:=[[x(0)
=x0,D(x)(0)=v0]]:" }}{PARA 0 "" 0 "" {TEXT 23 70 " DEplot(deNonLinea
r,x(t),t=a..b,ICnonlinear,opts,title='NonLinear');" }}{PARA 0 "" 0 ""
{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT
23 71 " A. Let x0=0, v0=0. Plot the solutions of the linear and n
onlinear" }}{PARA 0 "" 0 "" {TEXT 23 71 " equations from t=200 \+
to t=300. These plots represent the steady" }}{PARA 0 "" 0 "" {TEXT
23 45 " state solutions of the two equations." }}{PARA 0 "" 0 "
" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " B. Let x0=1.2, v0=
0. Plot the solutions of the linear and nonlinear" }}{PARA 0 "" 0 ""
{TEXT 23 71 " equations from t=220 to t=320. These plots repres
ent the steady" }}{PARA 0 "" 0 "" {TEXT 23 68 " state solutions
of the two equation, with new starting value" }}{PARA 0 "" 0 ""
{TEXT 23 59 " x0=1.2. [You must modify line 1 of the maple code
!]" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " \+
The two linear plots in A and B have to be identical to the plot
" }}{PARA 0 "" 0 "" {TEXT 23 68 " of xss(t). The reason is the \+
superposition formula (see E&P)" }}{PARA 0 "" 0 "" {TEXT 23 72 " \+
x(t)=xh(t)+xss(t), even though the homogeneous solution xh(t) is" }}
{PARA 0 "" 0 "" {TEXT 23 69 " different for the two plots. Thi
s is because xh(t) has limit" }}{PARA 0 "" 0 "" {TEXT 23 27 " z
ero at t=infinity." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 ""
{TEXT 23 72 " C. Determine the ratio of the apparent amplitudes (a \+
number > 1) for" }}{PARA 0 "" 0 "" {TEXT 23 72 " the nonlinear p
lots in A and B. Do \"large sustained oscillations\"" }}{PARA 0 "" 0 "
" {TEXT 23 56 " appear in the plot of the nonlinear steady-state
?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 6 "#3.3-A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#3
.3-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#3.3-C" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
23 68 " MCKENNA'S NON-HOOKE'S LAW CABLE MODEL FOR THE TACOMA NARROWS B
RIDGE" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "
The model of McKenna studies the bridge with a nonlinear, forced, damp
ed" }}{PARA 0 "" 0 "" {TEXT 23 62 "oscillator equation for torsional m
otion that accounts for the" }}{PARA 0 "" 0 "" {TEXT 23 68 "non-Hooke'
s law cables coupled to the equations for vertical motion." }}{PARA 0
"" 0 "" {TEXT 23 72 "The equations in this case couple the torsional m
otion with the vertical" }}{PARA 0 "" 0 "" {TEXT 23 26 "motion. The eq
uations are:" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT
23 63 " x'' + c x' - k G(x,y) = F sin wt, x(0) = x0, x'(0) = x1,
" }}{PARA 0 "" 0 "" {TEXT 23 63 " y'' + c y' + (k/3) H(x,y) = g , \+
y(0) = y0, y'(0) = y1," }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0
"" 0 "" {TEXT 23 71 "where x(t) is the torsional motion and y(t) is th
e vertical motion. The" }}{PARA 0 "" 0 "" {TEXT 23 72 "functions G(x,y
) and H(x,y) are the models of the force generated by the" }}{PARA 0 "
" 0 "" {TEXT 23 67 "cable when it is contracted and stretched. Below i
s sample code for" }}{PARA 0 "" 0 "" {TEXT 23 72 "writing the differen
tial equations and for plotting the solutions. It is" }}{PARA 0 "" 0 "
" {TEXT 23 29 "ready to copy with the mouse." }}{PARA 0 "" 0 "" {TEXT
23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 14 "with(DEtools):" }}{PARA 0 "" 0
"" {TEXT 23 42 "w := 1.3: F := 0.05: f(t) := F*sin(w*t):" }}{PARA 0
"" 0 "" {TEXT 23 54 "c := 0.01: k1 := 0.2: k2 := 0.4: g := 9.8: L \+
:= 6:" }}{PARA 0 "" 0 "" {TEXT 23 28 "STEP:=x->piecewise(x<0,0,1):" }}
{PARA 0 "" 0 "" {TEXT 23 28 "fp(t) := y(t)+(L*sin(x(t))):" }}{PARA 0 "
" 0 "" {TEXT 23 28 "fm(t) := y(t)-(L*sin(x(t))):" }}{PARA 0 "" 0 ""
{TEXT 23 27 "Sm(t) := STEP(fm(t))*fm(t):" }}{PARA 0 "" 0 "" {TEXT 23
27 "Sp(t) := STEP(fp(t))*fp(t):" }}{PARA 0 "" 0 "" {TEXT 23 8 "sys := \+
\{" }}{PARA 0 "" 0 "" {TEXT 23 71 " diff(x(t),t,t) + c*diff(x(t),t
) - k1*cos(x(t))*(Sm(t)-Sp(t))=f(t)," }}{PARA 0 "" 0 "" {TEXT 23 61 " \+
diff(y(t),t,t) + c*diff(y(t),t) + k2*(Sm(t)+Sp(t)) = g\}:" }}
{PARA 0 "" 0 "" {TEXT 23 51 "ic := [[x(0)=0, D(x)(0)=0, y(0)=27.25, D(
y)(0)=0]]:" }}{PARA 0 "" 0 "" {TEXT 23 18 "vars:=[x(t),y(t)]:" }}
{PARA 0 "" 0 "" {TEXT 23 19 "opts:=stepsize=0.1:" }}{PARA 0 "" 0 ""
{TEXT 23 46 "DEplot(sys,vars,t=0..300,ic,opts,scene=[t,x]);" }}{PARA
0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "The amazing thi
ng that happens in this simulation is that the large" }}{PARA 0 "" 0 "
" {TEXT 23 70 "vertical oscillations take all the tension out of the s
prings and they" }}{PARA 0 "" 0 "" {TEXT 23 36 "induce large torsional
oscillations." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 ""
{TEXT 23 72 " D. TORSIONAL OSCILLATION PLOT. Get the sample code above
to produce the" }}{PARA 0 "" 0 "" {TEXT 23 49 " plot of x(t) [that
's what scene=[t,x] means]." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA
0 "" 0 "" {TEXT 23 71 " E. Estimate the number of degrees the roadway \+
tilts based on the plot." }}{PARA 0 "" 0 "" {TEXT 23 68 " Recall th
at x in the plot is reported in radians. Comment on the" }}{PARA 0 ""
0 "" {TEXT 23 72 " agreement of this result with historical data an
d the video evidence" }}{PARA 0 "" 0 "" {TEXT 23 21 " in the film c
lip." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 68 " \+
Tip: Average the five largest amplitudes in the plot to find an" }
}{PARA 0 "" 0 "" {TEXT 23 67 " average maximum amplitude for t=0 t
o t=300. Convert to degrees" }}{PARA 0 "" 0 "" {TEXT 23 72 " using
Pi radians = 180 degrees. The film clip shows roadway maximum" }}
{PARA 0 "" 0 "" {TEXT 23 45 " tilt of 30 to 45 degrees, approximat
ely." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " \+
F. VERTICAL OSCILLATION PLOT. Modify the DEplot code to scene=[t,y] an
d" }}{PARA 0 "" 0 "" {TEXT 23 70 " plot the oscillation y(t) on t=0
to t=300. The plot is supposed to" }}{PARA 0 "" 0 "" {TEXT 23 71 " \+
show 30-foot vertical oscillations along the roadway that dampen to"
}}{PARA 0 "" 0 "" {TEXT 23 51 " 7-foot vertical oscillations after \+
300 seconds." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT
23 70 " The agreement between these oscillation results and the his
torical" }}{PARA 0 "" 0 "" {TEXT 23 70 " data for Tacoma Narrows, e
specially the visual data present in the" }}{PARA 0 "" 0 "" {TEXT 23
69 " film clip of the bridge disaster, should be clear from the plo
ts." }}{PARA 0 "" 0 "" {TEXT 23 56 " This is your only answer check
for the plot results." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#3.3-D Torsional plot t-vers
us-x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "#3.3-E Roadway osc
illation estimate in degrees + comments." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 32 "#3.3-F Vertical plot t-versus-y." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "17 0 0" 2 }{VIEWOPTS 1 1 0 1 1
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }