1. Tacoma Narrows Bridge From TPT

2. Vertical Oscillations The only observed oscillation up until an hour before the collapse was vertical. Example of simple forced harmonic motion (sinusoidal motion of the deck due to an external sinusoidal force.)

3. Forced Harmonic Motion 1.The bridge deck oscillates at ω ex, the angular frequency of the external sinusoidal force, not ω 0, the natural angular frequency of the bridge deck.

4. Forced Harmonic Motion 2.The amplitude of the oscillation has a maximum when ω ex = ω 0. In other words, a resonance behavior as a function of wind velocity ω ex is proportional to the wind velocity.

5. Forced Harmonic Motion The wind tunnel displayed both characteristics of forced harmonic motion. 1.The frequency of the vertical oscillations varies linearly with the wind velocity.

6. Oscillating External Force The OEF involved in the vertical oscillations is due to vortex shedding by the constant velocity wind separated by the deck of the bridge.

7. Oscillating External Force Insert figure 2 These vortex arrays are two rows of vortices with opposite dirctions of circulation and with a frequency that is approximately proportional to the wind veocity.

8. Oscillating External Force Insert figure 2 At the time the vertex breaks loose from the top (bottom) of the bridge deck, a downward (upward) force is exerted on the bridge deck.

9. Vertex Shedding Examples 1.Automobile radio antenna will oscillate ┴ direction of motion of a car at certain velocities. 2.Stop lights strung across the street will oscillate ┴ to the direction of the wind 3.Each flutter of a dropped piece of paper is one vortex being shed

10. Torsional Oscillations Last 45 minutes – new twisting, torsional motion with very large amplitude –is the cause of the bridge collapse

11. Torsional Oscillations is an aerodynamically induced self-excitation (or aerodynamic flutter). This would give a steady-state solution.

12. Torsional Oscillations The two characteristics of this steady state solution are motion at the natural frequency ω 0 (not ω ex, since there is no external sinusoidal torque), and no resonance behavior in the amplitude as a function of the wind velocity.

13. Torsional Oscillations The wind tunnel model has both these characteristics. First, the frequency of the torsional motion does not change with wind velocity. No resonance behavior is observed.

15. Vertical Oscillation Vertical oscillation – example of forced harmonic oscillations, where the vortex train created by the wind separated by the bridge deck is the sinusoidal external force.

16. Vertical Oscillation Major Characteristics 1.Frequency of the vertical oscillation of the bridge deck is at the vortex shedding frequency. 2.The amplitude of the oscillation has a maximum when the vortex shedding frequency equals the natural vertical frequency of the bridge deck.

17. Torsional Oscillations It is an aerodynamically induced self-excitation phenomenon, where the amplification mechanism is still being debated by the engineers.

18. Torsional Oscillations Major characteristics of self- excitation oscillations are that the oscillating frequency is the natural torsional frequency of the bridge deck and there is no resonance phenomenon in the amplitude as a function of wind velocity.