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"Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 203 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle1" -1 200 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{CSTYLE "_cstyle1" -1 204 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle2" -1 201 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle3" -1 202 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 56 "Math 2250 Maple Project 7, Spring 2008. Tacoma Narrows." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "NAME _______________ ________ CLASSTIME ________ VERSION A-K or L-Z" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "Cir cle the version - see problem L7.1. There are three (3) problems in" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "this project. Please \+ answer the questions A, B, C , ... associated with" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 65 "each problem. The original worksheet \+ \"2250mapleL7-S2008.mws\" is a" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "template for the solution; you must fill in the code and all comments." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "Samp le code can be copied with the mouse. Use pencil freely to annotate" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 50 "the worksheet and to \+ clarify the code and figures." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "The problem headers f or the Spring 2008 revision of David Eyre's project" }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 25 "(original was year 2000)." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 40 "__________L7.1. NONLINEAR M CKENNA MODELS" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 51 "_____ _____L7.2. MCKENNA NON-HOOKES LAW CABLE MODEL." }{TEXT 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 40 "L7.1. PROBLEM (NONLINEAR MCKENNA MODELS)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "There are three (3) parts L7.1A to L7.1C to complete. Mostly, this is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "mouse copying. \+ Retyping the maple code by hand is not recommended." }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 49 "NO NLINEAR TORSIONAL MODEL WITH GEOMETRY INCLUDED." }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 72 "Consider the nonlinear, forced, damped oscillator equation for torsional" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 38 "motion, with bridge geometry included," }{TEXT 204 0 " " }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 58 " x'' + 0.05 x' + 2.4 sin(x)cos(x) = 0.06 cos (12 t/10) ," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 25 " x(0) = x0, x'(0) = v0" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 41 "and its corresponding linearized equation" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 46 " x'' + 0.05 x' + 2.4 x = 0.06 cos (12 t/10) ," }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 25 " x(0) = x0, x'(0) = v0." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "The spring-mass system parameters are m=1, c = 0.05, k = 2.4, w = \+ 1.2 ," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "F = 0.06. Map le code used to solve and plot the solutions appears below." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 44 "# # Use \"copy as maple text\" for maple 6+." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 63 "# x 0:=0: a:=200: b:=300: # For part A. Change it for part B!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 64 "# v0:=0: m:= 1: F := 0.06: w := 1.2: m:=1: c:= 0.05: k:= 2.4:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 39 "# with(DEtools): op ts:=stepsize=0.1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 71 "# deLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) + k* x(t) = F*cos(w*t):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 37 "# IClinear:=[[x(0)=x0,D(x)(0)=v0]]:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 62 "# DEplot(deLinea r,x(t),t=a..b,IClinear,opts,title='Linear');" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 53 "# deNonLinear:= m*diff(x (t),t,t) + c*diff(x(t),t) +" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 53 "# k*sin(x(t))*cos(x(t)) = F*c os(w*t):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 40 "# ICnonlinear:=[[x(0)=x0,D(x)(0)=v0]]:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 71 "# DEplot(deNonLinear,x(t ),t=a..b,ICnonlinear,opts,title='NonLinear');" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 62 " 7.1A. Let x0=0, v0=0. Plot the solutions of the l inear and" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " \+ nonlinear equations from t=200 to t=300. These plots represent" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 57 " the steady s tate solutions of the two equations." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 " 7.1B. Let x0=1 .2, v0=0. Plot the solutions of the linear and" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 71 " nonlinear equations from t=22 0 to t=320. These plots represent" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 " the steady state solutions of the two equation , with new" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 " \+ starting value x0=1.2. [You must modify line 1 of the maple" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 15 " code!]" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 " The two linear plots in A and B have to be identical \+ to the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " plo t of xss(t). The reason is the superposition formula (see" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " E&P) x(t)=xh(t)+xss(t) , even though the homogeneous solution" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " xh(t) is different for the two plots. Th is is because xh(t)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 38 " has limit zero at t=infinity." }{TEXT 204 0 "" }}{PARA 200 " " 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 " 7.1C. Deter mine the ratio of the apparent amplitudes (a number > 1)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 " for the nonlinear plot s in A and B. Do \"large sustained" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 58 " oscillations\" appear in the plot of the nonl inear" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 23 " stea dy-state? " }{TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 7 "#L7.1-A" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 7 "#L7.1-B" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 7 "#L7.1-C" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 204 54 "L7.2. PROBLEM ( MCKENNA'S NON-H OOKE'S LAW CABLE MODEL)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "There are three (3) parts L 7.2A to L7.2C to complete. Mostly, this is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "mouse copying. Retyping the maple code by ha nd is not recommended." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "The model of McKenna studies th e bridge with a nonlinear, forced, damped" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 62 "oscillator equation for torsional motion tha t accounts for the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 " non-Hooke's law cables coupled to the equations for vertical motion. T he" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "equations in thi s case couple the torsional motion with the vertical" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 26 "motion. The equations are:" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 62 " x'' + c x' - k G(x,y) = F sin wt, x(0) = x0, x'(0) = x1, " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 62 " y'' + c y' + (k /3) H(x,y) = g , y(0) = y0, y'(0) = y1," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "where x(t ) is the torsional motion and y(t) is the vertical motion. The" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "functions G(x,y) and \+ H(x,y) are the models of the force generated by the" }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 67 "cable when it is contracted and stret ched. Below is sample code for" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "writing the differential equations and for plotting the \+ solutions. It is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 29 "re ady to copy with the mouse." }{TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 15 "#with(DEtools):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 43 "#w := 1.3: F := 0.05: f(t) := F*sin(w *t):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 55 "#c := 0.01: k1 := 0.2: k2 := 0.4: g := 9.8: L := 6:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 29 "#STEP :=x->piecewise(x<0,0,1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 29 "#fp(t) := y(t)+(L*sin(x(t))):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 29 "#fm(t) := y(t)-(L*sin (x(t))):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 28 "#Sm(t) := STEP(fm(t))*fm(t):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 28 "#Sp(t) := STEP(fp(t))*fp(t):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 9 "#sys : = \{" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 72 "# diff(x(t),t,t) + c*diff(x(t),t) - k1*cos(x(t))*(Sm(t)-Sp(t)) =f(t)," }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 62 "# diff(y(t),t,t) + c*diff(y(t),t) + k2*(Sm(t)+Sp(t)) = g\}:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 52 "#ic : = [[x(0)=0, D(x)(0)=0, y(0)=27.25, D(y)(0)=0]]:" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 19 "#vars:=[x(t),y(t)]:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 20 "#opts :=stepsize=0.1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 47 "#DEplot(sys,vars,t=0..300,ic,opts,scene=[t,x]);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 204 67 "The amazing thing that happens \+ in this simulation is that the large" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "vertical oscillations take all the tension out of th e springs and they" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 36 " induce large torsional oscillations." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "L7.2A. TORSIONAL \+ OSCILLATION PLOT. Get the sample code above to produce" }{TEXT 204 0 " " }}{PARA 200 "" 0 "" {TEXT 204 56 " the plot of x(t) [that's wh at scene=[t,x] means]." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "L7.2B. ROADWAY TILT ANGLE. Esti mate the number of degrees the roadway" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " tilts based on the plot. Recall that x in t he plot is reported" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 " in radians. Comment on the agreement of this result with" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 " historical dat a and the video evidence in the film clip." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 " Ti p: Average the five largest amplitudes in the plot to find an" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " average maximum ampl itude for t=0 to t=300. Convert to degrees" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " using Pi radians = 180 degrees. The f ilm clip shows roadway" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 55 " maximum tilt of 30 to 45 degrees, approximately." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "L7.2C. VERTICAL OSCILLATION PLOT. Modify the DEplot code to sc ene=[t,y]" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 " an d plot the oscillation y(t) on t=0 to t=300. The plot is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " supposed to show 30-foot vertical oscillations along the roadway" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " that dampen to 7-foot vertical oscillatio ns after 300 seconds." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 62 " The agreement between th ese oscillation results and the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " historical data for Tacoma Narrows, especially th e visual data" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " \+ present in the film clip of the bridge disaster, should be clear" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " from the plots . This is your only answer check for the plot" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 16 " results. " }{TEXT 204 0 "" }} {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 33 "#L7.2-A Torsional plot t-versus-x" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 58 "#L7.2-b Roadway tilt a ngle estimate in degrees + comments." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 33 "#L7.2-C Vertical plot t-versus-y. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 202 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }