{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 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 }