You don’t need to understand the mathematics to be able to appreciate physics. For physics is the study of nature. Albeit the physics gets lost in the amusement, its existence cannot be dismissed.
The constructive interference of waves leads to the formation of that huge amplitude in the middle of the pool. And only when you actuate the pool by the driver at particular frequencies do you get the maximum amplitude for vibration- Resonance!
On November 7 the Tacoma Narrows was seen dancing happily to the song of the winds. But unfortunately, the bridge collapsed only after an hour of swaying.
The reason for collapse of the Tacoma Narrows bridge is often attributed by many physics textbooks as Resonance. But it is incorrect. Atleast, the way in which it is explained in textbooks.
This real reason for it’s collapse is what engineer’s call as Flutter.
The collapse Mechanism.
I’ll start explaining this by highlighting some key points in the bridges design.
The bridge itself had a span of 4944 ft (1506.9 m) and connected the city of Tacoma to the Kitsap Peninsula. The bridge consisted of two pillars which suspended the central span which itself was 2800 (852.4 m) feet long and 39 ft (11.9 m) wide. During the construction of the bridge itself, it was reported that some transverse (vertical) oscillations occurred across all three segments of the bridge with the two pillars and the connection to the shore acting as nodes (areas of no oscillation).
To counteract this, the left and right sections of the bridge were reinforced by diagonal ties and hydraulic buffers that damped the oscillations but the center of the bridge was still free to vertically oscillate.
During the short commercial lifetime of the bridge (between it’s completion on 1st July, 1940, until 7th November 1940), it earned the affectionate name of “Gallopin’ Gertie” as it frequently oscillated with a range of 0 – 8 nodes between the two pillars. The maximum amplitude before the gale that caused the collapse of the bridge was recorded to be 5 ft (1.52 m) from crest to trough at a frequency of 0.13 Hz and well within the range of maximum stress the bridge was designed to withstand.
Several days before the 7th November, it is believed that that K-bracing under the deck and diagonal ties at the support pillars had been weakened during a storm; with one witness reporting to have seen the bridge behaving differently (this is later interpreted to mean that the bridge had been displaying larger than normal transverse oscillations).
On the morning of 7th November, the Tacoma Narrow bridge was buffeted by wind velocity of 42 miles per hour. The high winds, suspected structural damage and the loosening of diagonal ties combined to cause a fatal combination of transverse and torsional (twisting) oscillations.
But what caused the eventual collapse of the bridge?
Most A level text books will attribute the collapse of the bridge to resonance caused by the frequency of the driving force (supplied by the wind) being similar to the natural frequency of the bridge causing a drastic increase in the amplitude of oscillation (see resonance curve below) and the eventual collapse of the bridge due to the stress exerted by the increased amplitude.
This has several fatal assumptions, the most obvious of which is that the gusts of wind would occur with any defined or regimented period (which is clearly ridiculous).
A more credible explanation is that that collapse was triggered by a phenomenon called Vortex Shedding.
A vortex is a region in fluid medium that flows around an axis. Vortex shedding is where a fluid flows past a buff (non streamlined) object and results in an oscillating flow of vortexes that detach periodically from either side of the body. It is believed that these periodic vortexes exerted a periodic force alternately on the top and bottom causing the torsional oscillation around the central line of the bridge.
It can also be inferred that the frequency at which vortexes were produced must have been similar to the natural frequency of the bridge. This would have caused resonance and thus would explain why the amplitude of the torsional oscillation was large enough (and able to overcome frictionaltension forces that would have reduced the amplitude) to exert enough strain on the bridge to cause it to collapse.
However, there is more to this argument than originally appears.
For the mathematically inclined, the frequency of shredding vortexes is defined by,
Where St is a constant called the Strouhal number (for the Tacoma Bridge this constant is 0.11), f is the frequency of the detachment of vortexes, D is frontal dimension (in this case, the depth of the bridge platform was 8 ft (2.44 m)) and V is the Velocity of the wind.
From this we can calculate the the frequency of the vortex shredding is close to 1 Hz (1 vortex being produced per second). The observed frequency of the bridges oscillations was close to 0.2 Hz.
This means that the Shredding Vortexes cannot have caused resonance and there must have been another phenomenon in action.
Whilst the shredding vortexes may have caused the initial torsional oscillation, the increasing amplitude was a self induced.
When an object changes direction in a fluid stream, it causes new vortexes to form behind it. This is known as a Flutter Wake. This is caused when the air flow is disconnected from the surface and vortexes flow into the newly formed low pressure region. This can be seen with a schematic of a plane wing changing direction.
Similar to the shredding vortexes, the flutter wake exerted a force on the bridge increasing the amplitude of oscillations. As the amplitude of the oscillations increased, so did the difference in air pressure between the surface of the bridge and the undisturbed air flow causing more vorticity. This in turn increased the force exerted by the flutter wake and as a result increased the amplitude until the bridge reached breaking point.
The mean net force experienced by the bridge can be described as negative damping, rather than the amplitude decreasing over time, the amplitude increases until breaking point.
This is a perhaps similar to “Which came first, the chicken or the egg?”.
The (shredding) vortexes causes motion, and the motion causes more (flutter) vortexes.
The wind supplies the power, and the motion supplies the power tapping mechanism.