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{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Math 2280 Maple P roject 9, S2015. Tacoma Narrows." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " # " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# NAME ______________________ _ CLASSTIME ________" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# There are two (2) problems in thi s project. Please answer the questions" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# A, B, C , ... associated with each problem. The original wor ksheet" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# \"2280mapleL9-S2015.mws \" is a template for the solution; you must fill in" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# the code and all comments. Sample code can be co pied with the mouse. Use" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# penci l freely to annotate the worksheet and to clarify the code and" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# figures." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# The problem headers for the S2015 revision of David Eyre's project" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# (original was \+ year 2000)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# __________L9.1. N ONLINEAR MCKENNA MODELS" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# _____ _____L9.2. MCKENNA NON-HOOKES LAW CABLE MODEL." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# L9-1. PROBL EM (NONLINEAR MCKENNA MODELS)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# There are three (3) parts L9-1 A, L9-1B, L9-1C to complete. Mostly, this" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# is mouse copying. Retyping the maple code by hand i s not recommended." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# NONLINEAR TORSIONAL MODEL WITH GEOMETRY I NCLUDED." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# Consider the nonlinea r, forced, damped oscillator equation for torsional" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# motion, with bridge geometry included," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# x'' + 0.05 x' + 2.4 sin(x)cos(x) = 0.06 cos (12 t/10) ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# x(0) = x0, x'(0) = v0" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# an d its corresponding linearized equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# x'' + 0.05 x' + 2 .4 x = 0.06 cos (12 t/10) ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# \+ x(0) = x0, x'(0) = v0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# The spring-mass system parameters are m=1, c = 0.05, k = 2.4, w = 1.2 ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# F = 0.06. Maple code used to solve and plot the solutions ap pears below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " # Mouse copy into a maple worksheet." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " x0:=0: a:=200: b:=300: # For part A. \+ Change it for part B!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " v0:=0: \+ m:=1: F := 0.06: w := 1.2: m:=1: c:= 0.05: k:= 2.4:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " with(DEtools): opts:=stepsize=0.1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " deLinear:= m*diff(x(t),t,t) + c*diff(x (t),t) + k*x(t) = F*cos(w*t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " \+ IClinear:=[[x(0)=x0,D(x)(0)=v0]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " DEplot(deLinear,x(t),t=a..b,IClinear,opts,title='Linear');" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " deNonLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) +" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " \+ k*sin(x(t))*cos(x(t)) = F*cos(w*t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " ICnonlinear:=[[x(0)=x0,D(x)(0)=v0]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " DEplot(deNonLinear,x(t),t=a..b,ICnonlinear,opts,ti tle='NonLinear');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# \+ 9-1A. Let x0=0, v0=0. Plot the solutions of the linear and" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# nonlinear equations fro m t=200 to t=300. These plots represent" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# the steady state solutions of the two equations." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "# 9-1B. Let x0=1.2, v0=0. Plot the solutions of the linear and" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# nonlinear equat ions from t=220 to t=320. These plots represent" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# the steady state solutions of the two equ ation, with new" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# star ting value x0=1.2. [You must modify line 1 of the maple" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# code!]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# T he two linear plots in A and B have to be identical to the" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# plot of xss(t). The reason is t he superposition formula (see" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# \+ E&P) x(t)=xh(t)+xss(t), even though the homogeneous solution " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# xh(t) is different \+ for the two plots. This is because xh(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# has limit zero at t=infinity." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# \+ 9-1C. Determine the ratio of the apparent amplitudes (a number > 1) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "# for the nonlinear p lots in A and B. Do \"large sustained" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# oscillations\" appear in the plot of the nonlinear" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# steady-state? " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# L9-2. PROBLEM ( MCKENNA'S NON-HOOKE'S LAW CABLE MODEL)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# There are three (3) parts L9-2A, L9-2B, L9.2C to complete. Mos tly, this" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# is mouse copying. Re typing the maple code by hand is not recommended." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# The model \+ of McKenna studies the bridge with a nonlinear, forced, damped" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# oscillator equation for torsional motion that accounts for the" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# \+ non-Hooke's law cables coupled to the equations for vertical motion. T he" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# equations in this case coup le the torsional motion with the vertical" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# motion. The equations are:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# x'' + c x' - k G(x,y) = F sin wt, x(0) = x0, x'(0) = x1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# y'' + c y' + (k/3) H(x,y) = g , y(0) = \+ y0, y'(0) = y1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# where x(t) is the torsional motion and y(t) is the vertical motion. The" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# f unctions G(x,y) and H(x,y) are the models of the force generated by th e" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# cable when it is contracted \+ and stretched. Below is sample code for" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# writing the differential equations and for plotting the solu tions. It is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# ready for mouse c opy." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " w := 1.3: F := 0.05: f(t) := F*sin(w*t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "c := 0.01: k1 := 0.2: k2 := 0.4: g := 9.8: L := 6 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "STEP:=x->piecewise(x<0,0,1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fp(t) := y(t)+(L*sin(x(t))):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fm(t) := y(t)-(L*sin(x(t))):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Sm(t) := STEP(fm(t))*fm(t):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Sp(t) := STEP(fp(t))*fp(t):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sys := \{" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " diff(x(t),t,t) + c*diff(x(t),t) - k1*cos(x(t))*( Sm(t)-Sp(t))=f(t)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " diff(y(t ),t,t) + c*diff(y(t),t) + k2*(Sm(t)+Sp(t)) = g\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ic := [[x(0)=0, D(x)(0)=0, y(0)=27.25, D(y)(0)=0]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vars:=[x(t),y(t)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "opts:=stepsize=0.1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "DEplot(sys,vars,t=0..300,ic,opts,scene=[t,x]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# The amazing thing that happens in this simulation is that the \+ large" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# vertical oscillations ta ke all the tension out of the springs and they" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# induce large torsional oscillations." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# L9- 2A. TORSIONAL OSCILLATION PLOT. Get the sample code above to produce" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# the plot of x(t) [that' s what scene=[t,x] means]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# L9-2B. ROADWAY TILT ANGLE. Estim ate the number of degrees the roadway" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# tilts based on the plot. Recall that x in the plot is \+ reported" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "# in radians. C omment on the agreement of this result with" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "# historical data and the video evidence in t he film clip." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# Tip: Average the five largest amplitud es in the plot to find an" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# \+ average maximum amplitude for t=0 to t=300. Convert to degrees" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# using Pi radians = 180 de grees. The film clip shows roadway" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# maximum tilt of 30 to 45 degrees, approximately." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# L9-2C. VERTICAL OSCILLATION PLOT. Modify the DEplot code to s cene=[t,y]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# and plot th e oscillation y(t) on t=0 to t=300. The plot is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# supposed to show 30-foot vertical oscillati ons along the roadway" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# \+ that dampen to 7-foot vertical oscillations after 300 seconds." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# The agreement between these oscillation results and th e" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# historical data for \+ Tacoma Narrows, especially the visual data" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# present in the film clip of the bridge disa ster, should be clear" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# \+ from the plots. This is your only answer check for the plot" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# results. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }