# Math 2250 Maple Project 7, F2010. Tacoma Narrows.
#
# NAME _______________________ CLASSTIME ________ VERSION A-K or L-Z
#
# Circle the version - see problem L7.1. There are three (3) problems in
# this project. Please answer the questions A, B, C , ... associated
# with
# each problem. The original worksheet "2250mapleL7-F2010.mws" is a
# template for the solution; you must fill in the code and all comments.
# Sample code can be copied with the mouse. Use pencil freely to
# annotate
# the worksheet and to clarify the code and figures.
#
# The problem headers for the F2010 revision of David Eyre's project
# (original was year 2000).
# __________L7.1. NONLINEAR MCKENNA MODELS
# __________L7.2. MCKENNA NON-HOOKES LAW CABLE MODEL.
>
# L7.1. PROBLEM (NONLINEAR MCKENNA MODELS)
#
# There are three (3) parts L7.1A to L7.1C to complete. Mostly, this is
# mouse copying. Retyping the maple code by hand is not recommended.
#
# NONLINEAR TORSIONAL MODEL WITH GEOMETRY INCLUDED.
# Consider the nonlinear, forced, damped oscillator equation for
# torsional
# motion, with bridge geometry included,
#
# x'' + 0.05 x' + 2.4 sin(x)cos(x) = 0.06 cos (12 t/10) ,
# x(0) = x0, x'(0) = v0
#
# and its corresponding linearized equation
#
# x'' + 0.05 x' + 2.4 x = 0.06 cos (12 t/10) ,
# x(0) = x0, x'(0) = v0.
#
# The spring-mass system parameters are m=1, c = 0.05, k = 2.4, w = 1.2
# ,
# F = 0.06. Maple code used to solve and plot the solutions appears
# below.
#
> # # Use "copy as maple text" for maple 6+.
> # x0:=0: a:=200: b:=300: # For part A. Change it for part B!
> # v0:=0: m:=1: F := 0.06: w := 1.2: m:=1: c:= 0.05: k:= 2.4:
> # with(DEtools): opts:=stepsize=0.1:
> # deLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) + k*x(t) =
> F*cos(w*t):
> # IClinear:=[[x(0)=x0,D(x)(0)=v0]]:
> # DEplot(deLinear,x(t),t=a..b,IClinear,opts,title='Linear');
> # deNonLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) +
> # k*sin(x(t))*cos(x(t)) = F*cos(w*t):
> # ICnonlinear:=[[x(0)=x0,D(x)(0)=v0]]:
> #
> DEplot(deNonLinear,x(t),t=a..b,ICnonlinear,opts,title='NonLinear');
>
# 7.1A. Let x0=0, v0=0. Plot the solutions of the linear and
# nonlinear equations from t=200 to t=300. These plots
# represent
# the steady state solutions of the two equations.
#
# 7.1B. Let x0=1.2, v0=0. Plot the solutions of the linear and
# nonlinear equations from t=220 to t=320. These plots
# represent
# the steady state solutions of the two equation, with new
# starting value x0=1.2. [You must modify line 1 of the maple
# code!]
#
# The two linear plots in A and B have to be identical to the
# plot of xss(t). The reason is the superposition formula (see
# E&P) x(t)=xh(t)+xss(t), even though the homogeneous solution
# xh(t) is different for the two plots. This is because xh(t)
# has limit zero at t=infinity.
#
# 7.1C. Determine the ratio of the apparent amplitudes (a number > 1)
# for the nonlinear plots in A and B. Do "large sustained
# oscillations" appear in the plot of the nonlinear
# steady-state?
>
> #L7.1-A
> #L7.1-B
> #L7.1-C
>
# L7.2. PROBLEM ( MCKENNA'S NON-HOOKE'S LAW CABLE MODEL)
#
# There are three (3) parts L7.2A to L7.2C to complete. Mostly, this is
# mouse copying. Retyping the maple code by hand is not recommended.
#
# The model of McKenna studies the bridge with a nonlinear, forced,
# damped
# oscillator equation for torsional motion that accounts for the
# non-Hooke's law cables coupled to the equations for vertical motion.
# The
# equations in this case couple the torsional motion with the vertical
# motion. The equations are:
#
# x'' + c x' - k G(x,y) = F sin wt, x(0) = x0, x'(0) = x1,
# y'' + c y' + (k/3) H(x,y) = g , y(0) = y0, y'(0) = y1,
#
# where x(t) is the torsional motion and y(t) is the vertical motion.
# The
# functions G(x,y) and H(x,y) are the models of the force generated by
# the
# cable when it is contracted and stretched. Below is sample code for
# writing the differential equations and for plotting the solutions. It
# is
# ready to copy with the mouse.
> #
> #with(DEtools):
> #w := 1.3: F := 0.05: f(t) := F*sin(w*t):
> #c := 0.01: k1 := 0.2: k2 := 0.4: g := 9.8: L := 6:
> #STEP:=x->piecewise(x<0,0,1):
> #fp(t) := y(t)+(L*sin(x(t))):
> #fm(t) := y(t)-(L*sin(x(t))):
> #Sm(t) := STEP(fm(t))*fm(t):
> #Sp(t) := STEP(fp(t))*fp(t):
> #sys := {
> # diff(x(t),t,t) + c*diff(x(t),t) -
> k1*cos(x(t))*(Sm(t)-Sp(t))=f(t),
> # diff(y(t),t,t) + c*diff(y(t),t) + k2*(Sm(t)+Sp(t)) = g}:
> #ic := [[x(0)=0, D(x)(0)=0, y(0)=27.25, D(y)(0)=0]]:
> #vars:=[x(t),y(t)]:
> #opts:=stepsize=0.1:
> #DEplot(sys,vars,t=0..300,ic,opts,scene=[t,x]);
>
# The amazing thing that happens in this simulation is that the large
# vertical oscillations take all the tension out of the springs and they
# induce large torsional oscillations.
#
# L7.2A. TORSIONAL OSCILLATION PLOT. Get the sample code above to
# produce
# the plot of x(t) [that's what scene=[t,x] means].
#
# L7.2B. ROADWAY TILT ANGLE. Estimate the number of degrees the roadway
# tilts based on the plot. Recall that x in the plot is reported
# in radians. Comment on the agreement of this result with
# historical data and the video evidence in the film clip.
#
# Tip: Average the five largest amplitudes in the plot to find an
# average maximum amplitude for t=0 to t=300. Convert to degrees
# using Pi radians = 180 degrees. The film clip shows roadway
# maximum tilt of 30 to 45 degrees, approximately.
#
# L7.2C. VERTICAL OSCILLATION PLOT. Modify the DEplot code to
# scene=[t,y]
# and plot the oscillation y(t) on t=0 to t=300. The plot is
# supposed to show 30-foot vertical oscillations along the
# roadway
# that dampen to 7-foot vertical oscillations after 300 seconds.
#
# The agreement between these oscillation results and the
# historical data for Tacoma Narrows, especially the visual data
# present in the film clip of the bridge disaster, should be
# clear
# from the plots. This is your only answer check for the plot
# results.
>
> #L7.2-A Torsional plot t-versus-x
> #L7.2-b Roadway tilt angle estimate in degrees + comments.
> #L7.2-C Vertical plot t-versus-y.
>