A Pendulum Whose Amplitude Increases with Wind!? ~ Why Did the Tacoma Narrows Bridge Collapse? ~

This presentation explores the fascinating physics behind the 1940 Tacoma Narrows Bridge collapse through the lens of self-excited vibration. The slides explain how a seemingly small change in a differential equation - from positive to negative damping - can dramatically transform vibration into increasingly amplified motion.

Key topics covered include
- Basic pendulum motion and harmonic vibration
- Damped oscillation and its mathematical modelling
- Bernoulli's Law and the concepts of lift and negative lift
- Self-excited oscillation mechanism

The presentation uses simple models to demonstrate how aerodynamic forces interacted with the bridge structure, while noting that the actual collapse involved more complex phenomena, including aeroelastic flutter and torsional vibration. An approachable introduction to how small changes in physical systems can produce dramatically different results.

References

K. Yusuf Billah; Robert H. Scanlan, Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, Am. J. Phys. 59, 118–124 (1991),
https://doi.org/10.1119/1.16590

- https://courses.washington.edu/cee517/scanlanTN.pdf

Transcript

1. A PENDULUM WHOSE AMPLITUDE INCREASES WITH WIND!? WHY DID THE TACOMA NARROWS BRIDGE COLLAPSE?
2. TODAY'S TOPICS Why did the Tacoma Narrows Bridge collapse? The bridge collapsed on November 7, 1940 Equations of motion for harmonic and damped oscillations A simple pendulum model Bernoulli's theorem and lift, negative lift Vibration equation with negative damping due to lift Mechanism of self-excited vibration A small difference in the equation of motion can drastically change the result!
3. THE COLLAPSE OF THE TACOMA NARROWS BRIDGE Collapsed on November 7, 1940 The bridge undulated significantly due to wind! Let's watch the actual video!
4. THE CURRENT TACOMA NARROWS BRIDGE Rebuilt in 1950 A second bridge was constructed in 2007
5. IMPORTANT NOTES The actual cause of the bridge collapse is more complex Wind tunnel tests and numerical simulations are used in bridge design But today we'll consider the simplest model We'll focus on the mechanism of self-excited vibration!
6. EQUATIONS OF MOTION FOR VIBRATION
7. PENDULUM MOTION Lift the pendulum and release it A restoring force acts to move it toward the center due to gravity Swings to the opposite end This repetition causes the pendulum to oscillate
8. EQUATION OF MOTION FOR HARMONIC OSCILLATION This motion is called harmonic oscillation m ​ + dt2 d x 2 ​ = restoring force ​ kx 0 x(t) = A cos ​ t + φ ( ​ m k )
9. EQUATION OF MOTION FOR DAMPED OSCILLATION A real pendulum doesn't continue oscillating forever The oscillation gradually decreases due to damping forces like air resistance and friction m ​ + dt2 d x 2 ​ + damping force ​ β ​ dt dx kx = 0
10. DAMPED OSCILLATION The oscillation gradually decreases! x(t) = A exp(−βt) cos ​ t + φ ( ​ m k )
11. LIFT AND NEGATIVE LIFT
12. FLUID MOTION Air and water are examples of "flowing substances" called fluids Forces exerted on objects by wind or water flow can be understood through fluid motion
13. BERNOULLI'S THEOREM A theorem that describes the relationship between fluid velocity and pressure When fluid velocity increases, pressure decreases When fluid velocity decreases, pressure increases Since air is light, we'll ignore potential energy today p + ​ ρv + 2 1 2 ​ = ρgh const
14. LIFT Fluid velocity increases on the upper side of the plate, pressure decreases Fluid velocity decreases on the lower side of the plate, pressure increases The pressure difference creates an upward force on the plate (lift) 左側が上がっている状態 速い流れ（低圧） 遅い流れ（⾼圧） ⾵ 上向きの⼒
15. NEGATIVE LIFT Fluid velocity decreases on the upper side of the plate, pressure increases Fluid velocity increases on the lower side of the plate, pressure decreases The pressure difference creates a downward force on the plate (negative lift) 右側が上がっている状態 遅い流れ（⾼圧） 速い流れ（低圧） ⾵ 下向きの⼒
16. APPLICATIONS OF LIFT AND NEGATIVE LIFT Lift creates an upward force that allows an airplane to fly Negative lift creates a downward force that keeps an F1 car pressed against the track
17. SELF-EXCITED VIBRATION
18. SELF-EXCITED VIBRATION The pendulum is subjected to lift due to wind The lift creates a vibration that amplifies the motion! (self-excited vibration) ⾵ 下向きの⼒ 上向きの⼒
19. FAMILIAR EXAMPLE If you swing the swing with the vibration, the vibration becomes larger! The same thing happens with lift!
20. EQUATION OF MOTION FOR SELF- EXCITED VIBRATION Vibration with negative damping Just a change in sign... What happens? m ​ ​ + dt2 d x 2 opposite sign! ​ −β ​ dt dx kx = 0
21. GENERAL SOLUTION Vibration becomes larger! x(t) = A exp(βt) cos ​ t + φ ( ​ m k )
22. WHY DID THE TACOMA NARROWS BRIDGE COLLAPSE? Self-excited vibration mechanism worked! The bridge undulated significantly due to wind!
23. CONCLUSION Equations of motion for harmonic oscillation can represent the motion of a pendulum! Equations of motion for damped oscillation can represent the motion of a pendulum that gradually decreases in amplitude! Forces due to fluid flow and plate tilt create upward lift or downward negative lift! Equations of motion for vibration with negative damping can represent the mechanism of self- excited vibration!
24. ONE SIGN CHANGES THE WORLD! m ​ + dt2 d x 2 β ​ + dt dx kx = 0 m ​ − dt2 d x 2 β ​ + dt dx kx = 0
25. ADDITIONAL NOTES The actual fluid dynamics of bridge structures is more complex It is believed that the Tacoma Narrows Bridge collapsed due to aeroelastic flutter Torsional vibration is also a significant cause Self-excited vibration alone is not the cause! But today we focused on the simplest self- excited vibration!
26. LT SPEAKER RECRUITMENT We are recruiting LT speakers at the Physics Meeting! Any genre is OK! If there is no application, the host will open a Jaiyan Ressaitaru under the guise of LT... If you are interested, please join the Physics Meeting Discord server!
27. ANNOUNCEMENT Next meeting is scheduled for May 17 We want everyone to watch the physics video together We welcome any proposals to watch the video together!