Project Three

Tan Tran, Sarah Polzer, Joshua Cho

The Tacoma Narrows Bridge

A mathematical model was created by McKenna and Tuama to capture the Tacoma Narrows Bridge incident. The goal of the model was to explain how torsional oscillations can be magnified by force that is strictly vertical.

Assume that a roadway of width 2l is hanging between two cables. The roadway sits at an equilibrium height at rest, and y represents the current distance that the roadway hangs below this equilibrium. Theta is the angle that the roadway makes with the horizontal and the two cables stretch from equilibrium at y + lsinθ and y - lsinθ. It is also assumed that there is a damping coefficient that is proportional to the velocity.

Hooke’s law proposes a linear response, or that the restoring force of the cables will be proportional to the deviation. Hooke’s constant K and Newton’s law (F=ma) combined made the two equations of motion for y and theta below.

However, Hooke’s law was designed for springs with restoring forces that are equal when the spring is compressed or stretched. McKenna and Tuama believed that the cables pulled back with more force when stretched than when compressed. They replaced the linear restoring force in Hooke’s law with a nonlinear force. Their equations of motion are pictured below.

Question 1)

Run tacoma.m with wind speed W = 80 km/hr and initial conditions y = y′ = θ′ = 0, θ = 0.001. The bridge is stable in the torsional dimension if small disturbances in θ die out; unstable if they grow far beyond original size. Which occurs for this value of W?

https://youtu.be/KMYpJ9MTFqI

For a windspeed of 80 km/hr, theta initially increased 300-fold and then came back to about 200 magnification. In the image, at around time = 0.2, theta shot up to 0.3 from zero. Theta increased sharply and then oscillated and eventually settled down at 0.2; the bridge was very unstable and then stabilized at a theta of 0.2. Theta still grew far beyond its original size, so the bridge was unstable in the torsional dimension. We used Runge-Kutta in this part as well as the Trapezoid Method because we are committed to scientific accuracy.

Question 2)

Replace the Trapezoid Method by fourth-order Runge–Kutta to improve accuracy. Also, add new figure windows to plot y(t) and θ(t).

https://youtu.be/AweAXfxRAT4

y(t) stabilized at about 2 meters, with a delta of 4 meters.

Again, our max magnification using the Runge-Kutta method was 301.4491. We next look at the inferior Trapezoid method which returned us a magnification of 305.6322.

As we can see, the graphs look very similar at visual inspection, but the raw numbers tell us otherwise.

Question 3)

The system is torsionally stable for W = 50 km/hr. Find the magnification factor for a small initial angle. That is, set θ(0) = 10−3 and find the ratio of the maximum angle θ(t), 0 ≤ t < ∞, to θ(0). Is the magnification factor approximately consistent for initial angles θ(0) = 10−4, 10−5,...?

Output for Theta vs. Magnification:

Initial Theta	Magnification
0.001	20.6062
0.0001	20.4397
10^-5	20.438
10^-6	20.438
10^-7	20.438
10^-8	20.438
10^-9	20.438
10^-10	20.438

The magnification factor for an initially small angle (0.001) was 20.6062. The magnification factor was in fact consistent for θ(0) = 10−4, 10−5, not deviating far from 20.44.

Question 4)

Find the minimum wind speed W for which a small disturbance θ(0) = 10−3 has a magnification factor of 100 or more. Can a consistent magnification factor be defined for this W?

The minimum wind speed for which a small disturbance had a magnification factor of 100 or more was 59.5193 km/hr. A consistent magnification factor cannot absolutely be defined for this W because when theta is changed, there can be drastic changes in the magnification factor. For example, for a theta of 0.001, the magnification factor at W=59.5193 km/hr was 99.9994; whereas, for a theta of 0.0001, the magnification factor was 74.7284. However, at thetas smaller than 0.0001, the magnification factor stayed stable at around 74.6. A consistent magnification factor can be defined for thetas smaller than 0.0001.

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Question 5)

Design and implement a method for computing the minimum wind speed in Step 4, to within 0.5 × 10−3 km/hr. You may want to use an equation solver from Chapter 1

Running the bisection method, we found that the minimum windspeed was 59.519 km/hr.

Question 6)

Try some larger values of W. Do all extremely small initial angles eventually grow to catastrophic size?

No, we found that at 100km/h winds, the max theta became safe at around 10^-6 initial theta. However, when we bump the windspeed to a somewhat ridiculous 300km/h, not even an initial theta of 10^-15 would be able to save this bridge.

Question 7)

What is the effect of increasing the damping coefficient? Double the current value and find the change in the critical wind speed W. Can you suggest possible changes in design that might have made the bridge less susceptible to torsion?

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As we can see, doubling the damping coefficient allows the bridge to remain relatively stable in winds exceeding 90km/h. Some possible changes may be to have a heavier road or thicker cables that stretch less while also having more mass.