{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 23 56 "Math 2250 Maple Project 5, Spring 2006. 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 70 "Circle the version - see problem L5.1. There ar e three (3) problems in" }}{PARA 0 "" 0 "" {TEXT 23 71 "this project. \+ Please answer the questions A, B, C , ... associated with" }}{PARA 0 " " 0 "" {TEXT 23 65 "each problem. The original worksheet \"2250mapleL5 -S2006.mws\" is a" }}{PARA 0 "" 0 "" {TEXT 23 70 "template for the sol ution; 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 free ly to annotate" }}{PARA 0 "" 0 "" {TEXT 23 50 "the worksheet and to cl arify the code and figures." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 "The problem headers for the FALL 2004 revision \+ of David Eyre's project" }}{PARA 0 "" 0 "" {TEXT 23 25 "(original was \+ year 2000)." }}{PARA 0 "" 0 "" {TEXT 23 46 " __________L5.1. UNDAMPED \+ FORCED OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT 23 37 " __________L5.2. \+ PRACTICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT 23 52 " __________L5.3. \+ MCKENNA NON-HOOKES LAW CABLE MODEL." }}{PARA 0 "" 0 "" {TEXT 23 0 "" } }{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "L5.1. PR OBLEM (UNDAMPED FORCED OSCILLATIONS )" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "" {TEXT 23 65 "FORCED LINEAR OSCILLATIONS. Consider 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 co nstants. Values m, k are defined by" }}{PARA 0 "" 0 "" {TEXT 23 35 "th e 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 Versio n L-Z: m=2, k=3.5" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " A. Choose the forcing angular frequency w to be 5 ti mes larger than" }}{PARA 0 "" 0 "" {TEXT 23 72 " the natural an gular 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 in terval in order" }}{PARA 0 "" 0 "" {TEXT 23 68 " to show the gl obal behavior of the solution x(t). See Figure" }}{PARA 0 "" 0 "" {TEXT 23 56 " 5.6.2, page 350. Reference: Example 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 functions, one of" }} {PARA 0 "" 0 "" {TEXT 23 69 " period 2Pi/w and the other of peri od 2Pi/w0. Display the exact" }}{PARA 0 "" 0 "" {TEXT 23 70 " pe riod, as calculated from the solution formula for x(t) -- see" }} {PARA 0 "" 0 "" {TEXT 23 69 " page 350, Example 1, for details a bout least common multiples." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 62 " C. Suggest a value for the forcing frequenc y w so that the" }}{PARA 0 "" 0 "" {TEXT 23 67 " oscillations ex hibit 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 sem icolons 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 equation" }}{PARA 0 " " 0 "" {TEXT 23 71 "ic:=x(0)=0,D(x)(0)= 0: # Define t he initial conditions" }}{PARA 0 "" 0 "" {TEXT 23 69 "p:=dsolve(\{de,i c\},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 soluti on" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L5.1-A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "# L5.1-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L5.1-C" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 23 35 "L5.2. 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 "Dependi ng 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 " a mplitude 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 thre e amplitude graphs" }}{PARA 0 "" 0 "" {TEXT 23 72 " on a single \+ 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. Fo r 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\}. The \+ three data pairs should" }}{PARA 0 "" 0 "" {TEXT 23 70 " show th at 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 52 "#L5.2-A Plot C(w), three graphics on one set of axes " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#L5.2-B Table of six da ta values for w*, C*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 23 40 "L5.3. PROBLEM (NONLINEAR MCKENNA \+ MODELS)" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 73 "There are six (6) parts L5.3A to L5.3F to complete. Mostly, this i s mouse" }}{PARA 0 "" 0 "" {TEXT 23 60 "copying. Retyping the maple co de by hand is not recommended." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 49 "NONLINEAR TORSIONAL MODEL WITH GEOMETRY I NCLUDED." }}{PARA 0 "" 0 "" {TEXT 23 72 "Consider the nonlinear, force d, damped oscillator equation for torsional" }}{PARA 0 "" 0 "" {TEXT 23 38 "motion, with bridge geometry included," }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 59 " x'' + 0.05 x' + 2.4 s in(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 mapl e text\" for maple 6+." }}{PARA 0 "" 0 "" {TEXT 23 62 " x0:=0: a:=20 0: b:=300: # 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:=stepsi ze=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 " IClinear:=[[x(0)=x0,D(x)(0)=v0]]:" }}{PARA 0 "" 0 "" {TEXT 23 61 " DEplot(deLinear,x(t),t=a..b,IClinear,opts,title='Linear');" }} {PARA 0 "" 0 "" {TEXT 23 52 " deNonLinear:= m*diff(x(t),t,t) + c*dif f(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 " ICnonlin ear:=[[x(0)=x0,D(x)(0)=v0]]:" }}{PARA 0 "" 0 "" {TEXT 23 70 " DEplot (deNonLinear,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 nonlinear" }}{PARA 0 "" 0 "" {TEXT 23 71 " equat ions 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. L et 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. Th ese plots represent 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 o f 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 identi cal 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 so lution xh(t) is" }}{PARA 0 "" 0 "" {TEXT 23 69 " different for \+ the two plots. This is because xh(t) has limit" }}{PARA 0 "" 0 "" {TEXT 23 27 " zero 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 plots in A and B. Do \"large sustained oscillati ons\"" }}{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 7 "#L5.3-A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L5.3-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L5.3-C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 23 68 " MCKENNA'S NON-HOOKE'S LAW CABLE M ODEL FOR THE TACOMA NARROWS BRIDGE" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 72 "The model of McKenna studies the bridge w ith a nonlinear, forced, damped" }}{PARA 0 "" 0 "" {TEXT 23 62 "oscill ator equation for torsional motion that accounts for the" }}{PARA 0 " " 0 "" {TEXT 23 68 "non-Hooke's law cables coupled to the equations fo r vertical motion." }}{PARA 0 "" 0 "" {TEXT 23 72 "The equations in th is case couple the torsional motion with the vertical" }}{PARA 0 "" 0 "" {TEXT 23 26 "motion. The equations are:" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 63 " x'' + c x' - k G(x,y) = F s in 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 the vertical motion. The" }}{PARA 0 "" 0 "" {TEXT 23 72 "functions G(x,y) and H(x,y) are the models of the fo rce generated by the" }}{PARA 0 "" 0 "" {TEXT 23 67 "cable when it is \+ contracted and stretched. Below is sample code for" }}{PARA 0 "" 0 "" {TEXT 23 72 "writing the differential equations and for plotting the s olutions. 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 := 0.0 5: 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 thing that happens in this simulation is that the large" }}{PARA 0 "" 0 "" {TEXT 23 70 "vertical oscillations take \+ all the tension out of the springs 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 numbe r of degrees the roadway tilts based on the plot." }}{PARA 0 "" 0 "" {TEXT 23 68 " Recall that x in the plot is reported in radians. Com ment on the" }}{PARA 0 "" 0 "" {TEXT 23 72 " agreement of this resu lt with historical data and the video evidence" }}{PARA 0 "" 0 "" {TEXT 23 21 " in the film clip." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 68 " Tip: Average the five largest amplit udes in the plot to find an" }}{PARA 0 "" 0 "" {TEXT 23 67 " avera ge maximum amplitude for t=0 to 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, approximately." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 72 " F. VERTICAL OSCILLATION PLOT. Modify the DEplot code to scene=[t,y] and" }}{PARA 0 "" 0 "" {TEXT 23 70 " pl ot 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 al ong the roadway that dampen to" }}{PARA 0 "" 0 "" {TEXT 23 51 " 7-f oot 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 historical" }}{PARA 0 "" 0 "" {TEXT 23 70 " data for Tacoma Narrows, especially the visual data present in the" }}{PARA 0 "" 0 "" {TEXT 23 69 " film clip of the bridge disas ter, should be clear from the plots." }}{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 33 " #L5.3-D Torsional plot t-versus-x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "#L5.3-E Roadway oscillation estimate in degrees + co mments." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "#L5.3-F Vertical plot t-versus-y." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 37 0" 5 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }