Project: Christina Gruenhagen - Math 301: Ordinary Differential Equations

Path: Labs/Narrows/Modeling The Tacoma Narrows Bridge.ipynb

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This lab is based on a project in our text (http://faculty.sfasu.edu/judsontw/ode/html-snapshot/secondorder05.html) and a lesson from $\textit{Differential Equations, 4th Ed.}$ by Blanchard, Devaney, and Hall.

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The Tacoma Narrows Bridge: A mere 8 minutes from our campus. You may have even driven or biked over it!

This bridge is world-famous example widely discussed in differential equations and civil engineering courses. Why? It first opened July 1, 1940. From the first day, the road bed was observed to oscillate significantly up and down. It earned the nickname "Galloping Gertie" for it's bouncing behavior. Only a few months later, on the morning of November 7, the bridge began to undulate up and down as much as three feet. It eventually began to experience twisting oscillations in addition. At 11:10 a.m., the bridge collapsed into the Sound. Famous videos document the unsettling motion (https://www.youtube.com/watch?v=j-zczJXSxnw&feature=youtu.be).

Things get even more interesting, though! The Tacoma Narrows Bridge was actually not insured due to insurance fraud committed by local travel agent, who pocketed the premium and failed to report the $800,000 policy to his employers. The bridge officials had planned to cancel the policy the following week, in which case the man would have never been caught. The governor of the state of Washington announced later "We are going to build the exact same bridge, exactly as before," prompting Theodore Von Karman, director of the Guggenheim Aeronautical Laboratory at the California Institute of Technology and a member of the board of inquiry into the collapse to send a telegram stating, "If you build the exact same bridge exactly as before, it will fall into the exact same river exactly as before."

The collapse of the bridge has been widely attributed to a resonance phenomenon. However, we will see in this lab that the truth is somewhat more complicated.

Initial Model

The Narrows bridge is a suspension bridge, like the Golden Gate Bridge in San Francisco. A series of support pillars hold large cables. Additional cables hang vertically from the primary cables and hold the road bed. Simplifying things, we can think of the road bed as a mass being held by a cable in tension. This cable is similar to a spring, in that it opposes stretching. This tensile force balances out the gravitational force pulling the mass of the roadbed down.

Finally, all real-world oscillators have at least some degree of damping (alas, no perpetual motion for us). Therefore, the cables and roadbed experience a force opposing their own motion. All together, we get a harmonic oscillator system by balancing the forces:

m\frac{d^2y}{dt^2} + \alpha \frac{dy}{dt} + \beta y = -g

In the cell below, define every variable and parameter in the above equation in context. Include units when possible.

$m =$ mass of the roadbed (kilograms or some greater unit prefix), $\alpha =$ damping coefficient of the suspended roadbed system (newton seconds per meter), $\beta =$ spring constant of the cables supporting the roadbed (newtons per meter), $g =$ gravity (meters per second squared or newtons per kilogram)

Rewrite this differential equation as a first-order system of differential equations.

$\frac{dy}{dt}$ = $v$

$\frac{dv}{dt}$ = (- $\alpha$ v - $\beta$ y - g)( $\frac{1}{m}$ )

Take the mass of the bridge to be one, the damping coefficient to be small (0.01), the spring constant to be 17, and the force of gravity to be 10.

Plot a vector field of this system.

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y, v, t = var('y v t') # define variables
a = 0.01               # set parameters
b = 17
g = 10
# m = 1 and can therefore be ignored
F = [v, -a*v - b*y - g] # system of equations
n = sqrt(F[0]^2 + F[1]^2) # normalize the vectors
p = plot_vector_field((F[0]/n, F[1]/n), (y, -4, 4), (v, -4, 4))  # plot the vector field
p

In the cells below, use the tools we have practiced to summarize the dynamics of this system.

With any given initial conditions, the bridge will return to a stable and unmoving position. All solutions will approach the specific solution $v=0, y= \frac{10}{17}$ as $t \rightarrow \infty$ .

Suppose the wind provides a periodic forcing with amplitude $\lambda$ and frequence $\omega$ . Even a constant wind can provide a periodic forcing to a flat roadbed because the wind produces vortices and eddies above and below the surface which can dissipate and jolt the surface up and down in rhythm. The winds in the Narrows are famously strong.

Describe the motion of the bridge under such a forcing function. Be sure to comment upon what types of forcing would lead to pronounced oscillations and why. Finally, comment upon how realistic it is for the wind to produce a sustained forcing of the necessary frequency to produce large oscillations in which the bridge frequently rises above the horizontal.

m\frac{d^2y}{dt^2} + \alpha \frac{dy}{dt} + \beta y = -g + F(t)

Suppose the wind produces a forcing function with amplitude 0.03 and frequency 4. Plot the solution up to time t = 1000 for an initial condition that is at equilibrium and with an initial velocity of 200 (caused by a large sudden gust of wind). Comment on the behavior.

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# Set up the variables
y, v, t = var('y, v, t')
a = 0.01
b = 17
g = 10

# System with forcing function 0.3*sin(4t)
F = [v, -g + 0.03*sin(4*t) - a*v - b*y]

# Solve the system
P = desolve_system_rk4(F, [y, v], ics = [0, 0, 200], ivar = t, end_points = 1000, step = 0.1)

# Plot y(t)
Q = [[i, j] for i,j,k in P]
p = line(Q, color= 'blue', axes_labels=['$t$','$y(t)$'], legend_label = '$y(t)$', legend_color = 'blue', fontsize=12)
p

This plot indicates that, given the initial velocity and forcing function $F(t)=0.03\sin 4t$ , the sudden gust of wind causes oscillations that dissipate gradually as t approaches 1000.

$\LaTeX$

An Improved Model

In fact, the cables holding the bridge do not act as a simple spring. They are more like rubber bands in that they exert force when in tension (when the mass of the bridge lies below the horizontal), but not when they are slack (when the mass of the bridge is above the horizontal). Image below from Blanchard, Devaney, and Hall.

The system of differential equations now becomes:

\begin{align*} \frac{dy}{dt} &= v\\ \frac{dv}{dt} &= h(y) - 0.01v - 10 + F(t) \end{align*}

where F(t) is the forcing function.

What is the function h(y) that captures the rubber-band effect? Assume the spring constant is 17 when the cables are in tension, but only 13 when they are slack.

Hint: You will need to define a piecewise function with a discontinuity at y = 0!

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#assuming the spring constant is 17 when cables are in tension
#assuming the spring constant is 13 when cables are slack
#discontinuity at y = 0

$h(y)$ =

\begin{cases} 17y & \text{if $y<0$, cables are in tension} \\ 13y & \text{if $y>0$, cables are slack} \\ \end{cases}

Plot the vector field of this modified, unforced system. How does it differ from the original vector field?

Hint: You will need to code up your piecewise function. This can be accomplished with a step function unit_step(x) whose value is 0 when $x < 0$ and 1 when $x > 0$ . For example, the absolute value function can be written as:

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x = var('x') #declare variable

myAbs = -x + 2*x*unit_step(x)

plot(myAbs, (x,-2,2)) #plot the vector field

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y = var('y') #declare variable

cable_spring_force = 17*y  - 4*y*unit_step(y) # When y is greater than 0 add -4y to 17y to get the 13y we were looking for

plot(myAbs2, (y, -10, 10)) #plot the vector field

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y, v, t = var('y v t') # define variables

# construct our piecewise function:
cable_spring_force = 17*y - 4*y*unit_step(y)

F = [v, -cable_spring_force - 0.01*v -10] # system of equations, including the piecewise cable force
n = sqrt(F[0]^2 + F[1]^2) # normalize the vectors
p = plot_vector_field((F[0]/n, F[1]/n), (y, -4, 4), (v, -4, 4))  # plot the vector field
p

This differs from our original vector field because our original vector field treated the cables as if they were springs with a constant spring constant. The new vector field treats the cables like rubber bands which will have a spring constant of 17 when stretched beneath horizontal, and 13 when above horizontal. This model will be a better approximation of the actual behavior of the bridge system.

Plot the solution to this new system using the same parameters and intial conditions as before. How is the long-term behavior different and what consequences does this have for the bridge? (try setting your y-limits to go from -5 to 5 so you can see what happens at the end more easily).

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#Rowen
#

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#initial velocity is 200
y, v, t = var('y, v, t')
myAbs2 = 17*y - 4*y*unit_step(y)
F = [v, -myAbs2 - 0.01*v -10 +.03*sin(4*t)] #Assuming forcing function 0.3*sin(4t)
P = desolve_system_rk4(F, [y, v], ics = [0, 0, 200], ivar = t, end_points = 1000, step = 0.1)
Q = [[i, j] for i,j,k in P]
p = line(Q, color= 'blue', axes_labels=['$t$','$y(t)$'], legend_label = '$y(t)$', ymin=-5, ymax=5, legend_color = 'blue', fontsize=12)
p

What are some possible bridge design solutions to avoid these problematic dynamics?

A potential structural solution to these problematic dynamics is to have openings in the roadway so that wind may pass through (therefore accounting for the forcing function), which was the solution utilized when the Tacoma Narrows Bridge was originally rebuilt. Another solution could be structures to increase the damping coefficient similar in function to a dashpot.

Bibliography

(1) Judson, Thomas W. The Ordinary Differential Equations Project. http://faculty.sfasu.edu/judsontw/ode/html-snapshot/secondorder05.html. Accessed 7 July 2020.

(2) Blanchard, P., Devaney, R., and Hall, G. Differential Equations. 4th Ed. Brooks / Cole: Boston.

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