The Response to Tacoma Narrows

After a disaster like Tacoma Narrows, it would be a brave person who did not over-react. Should a similar event occur, with a design that was not obviously different to that which failed, blame would be swiftly directed, but someone who had clearly made a great effort to forestall catastrophe would probably be safe.

The next picture shows a possible design that might resist the tendency to torsional oscillations. The main cables are pulled together and clamped somewhere near the antinodes of torsional oscillations. Would the design reduce the tendency to oscillate? What would be the difficulties in designing such a bridge? What would be the difficulties in construction?

The diagram below shows the ratio of depth of deck to span of deck for some of the larger suspension bridges (of their time) of the 20th century. We see clearly that after the very conservative Manhattan Bridge, designed around the deflection theory, depths become very shallow indeed. The crucial combination was shallow deck with flat vertical plate girders on each side. The Tacoma Narrows and the Bronx-Whitestone were the two which oscillated (the second not destructively). The George Washington escaped this fate, perhaps because pf its great weight. The truss added later was purely for the already foreseen second roadway.

In the diagram, AB means aerodynamic box girder, PG means plate girder, and TG means truss girder.

After the Tacoma Narrows event, the plate girder fell out of favour, and deep trusses were the norm. Only in 1966 did suspension bridges become slender again, using the aerodynamically streamlined deck. The slenderness of the first Tacoma Narrows bridge was approached, but never significantly exceeded. Although the George Washington span was also very shallow, the width of the deck was very much wider than that of Tacoma Narrows. The same was true, to a lesser extent, of the Bronx Whitestone.  The Tacoma Narrows crash probably required several contributory causes to be present – very shallow deck, very narrow deck, plate girders, broken stiffening stays, and possibly coincident resonant frequencies.

Flags demonstrate clearly the effects of wind on flexible objects. Metal is normally thought of as quite rigid, but rigidity, of course, depends on size and thickness. When aircraft first began to reach speeds of 500 mph (800 kph), flutter of control surfaces, or even of wings, became a phenomenon to be reckoned with.  

People sometimes wonder how a heavy airliner can even leave the ground. "Heaviness" depends on shape and speed. Anyone who has tried to cope with an umbrella in a high wind will have had a hint that aerodynamic forces increase rapidly as the speed increases. We could define aerodynamic heaviness in terms of minimum efficient flying speed. A Concorde is then "heavier" than a Boeing 747.

Tall metal chimneys used to oscillate in a  wind, until helical strakes were fitted, to spoil the airflow. Why helical? Because they work equally well for all wind directions.

Controlling Oscillations

The diagram above shows a typical streamlined bridge section, similar to that of the Severn suspension bridge. To see the section of the Humber bridge, click here. It is probably better if any aerodynamic force is downwards rather than upwards, "prestressing" the system, because the hangers can resist extra downward force, but are helpless against uplift. The roadway of the Severn suspension bridge has an aerodynamically streamlined cross-section, in the form of a closed box, and is suspended by inclined hangers, which provide some stiffening in the manner of a triangulated truss.

This box section corresponds to the use of monoplane construction for aircraft, with an inherently stiff wing, replacing biplanes, based on trusses, for most applications.

Each hanger, except for the very shortest ones, is provided with a small device near the bottom, which absorbs energy. This prevents the bridge from acting as a huge Aeolian harp. Near the halfway points between the low and high points of the main cables the hangers have dampers near their mid-points as well. Some of the pictures below show dampers and the inclined hangers. Others show the shape of the cross-section of the deck, which is designed to produce a slight downward force in a wind. The last picture shows one of the original dampers that were installed when the bridge was built. The later versions, added when the bridge was refurbished, are somewhat different in detail.

The essence of a damper is that there shall be relative motion involving friction.  It can be solid against solid, solid against liquid, or within a liquid. The disadvantage of a solid system is wear. A simple example of a damping system is the Stevenson strut used by some owners of caravans and glider-trailers. It absorbs energy, reducing the effects of oscillation between car and trailer. Energy is absorbed, too, by tyres, which go through a cycle of shear and tension at each point in each revolution. That is why they get hot, and why much less power is needed to pull a load on a steel railway than on a road.

Whatever the system, the effective viscosity must be suitable. If too low, not enough energy is absorbed: if too high, not enough relative motion occurs: in the case of a trailer, the response must be fast enough to allow bends and corners to be taken safely. The absorbed energy is proportional to relative distance times force, and the absorbed power is proportional to relative speed times force. Such a system always absorbs energy, whatever the direction of motion, and is analogous to a electrical resistor. Mass and elasticity can be likened to capacitance and inductance, and before the age of digital computers, people often built electrical analogues to study system behaviour without having to make expensive, and often large, parts.

To imagine how a damper might work, imagine a small metal cup resting on ice. If you place a stick in it and wiggle it about without touching the cup, you will experience no resistance. If the stick is frozen into the cup with ice, wiggling the stick will wiggle the cup. Again, no energy will be absorbed. But if the cup is full of water, when the stick is wiggled, it will do work against the viscosity of the water. The trick is to choose the optimal value of the viscosity to absorb the greatest possible energy. The pointer of an electrical meter is damped in order to bring it to rest quickly when the current changes.

These dampers are equivalent to mutes in musical instruments – for a cable-stayed bridge or a suspension bridge, the object is to avoid creating a giant harp. Pianofortes are equipped with dampers that stop the sound when a key is released, but if the sustaining pedal is pressed, the dampers are held off, and the strings continue to sound. With the pedal down, strings are free to vibrate in response to energy derived from the strings that are struck. You can also sing a note into a piano with the pedal down, and hear some of the strings responding. Mutes may be clipped to the bridges of members of the violin family, though they may work by changing the mass.  The instructions to the player are con sordino and senza sordino. The use of mutes for brass instruments is also very common.

Recently, some extra panels were added near the towers of the Severn bridge to influence the airflow near the towers. The panels are composed of many slats. The larger panels have electric motors which can be used to change the angle of the slats. These slats and motors can be seen in the pictures. The second picture is a close-up view of a few slats, with a distant view of the baffles on the other side of both road and tower. Triangular fairings are seen in front of the fins of many medium sized aircraft, where they delay stalling of the fin, which would result in loss of directional stability and control.

The Pont de Normandie has extra wires which connect the main wires to prevent oscillation. During the construction of the Pont de Normandie there was great concern that the unjoined cantilevers might vibrate too much in a high wind. There was a contingency plan to install active dampers. In these, sensors detect the motions of the bridge, and the amplified signals are used to move heavy masses in such a way as to cancel most of the motion. The principle is used in some tall buildings to control swaying. Should the power supply be lost, the control goes too.

Once you introduce a power supply, you have a continuous source of energy.  Many a designer of electronic amplifiers has discovered that he has built an oscillator.  If an amplifier has a gain of 1000, it only needs a little over 1/1000 the of the output energy to reach the input in the wrong phase, and you have an oscillator.  The older designer will remember terms like "motor-boating" and "squegging".  Microphones, amplifiers and loudspeakers can create the same effect.

Types of Energy

Before looking at the causes of oscillations, let’s look at energy in a structure. The graphs on the left below represent a structure which is four times as stiff as the one represented on the right, as shown from the graphs of force versus deflection.

The stored energy is proportional to both force and deflection, and is positive for deflections in both directions. It is proportional to the square of the deflection.

In the same way, kinetic energy built up when a structure moves is proportional to the square of the speed. It, too, is positive whatever the direction of motion.

We can see, then, that when a structure oscillates about its rest position, the peaks of energy will occur at twice the frequency of the oscillation.

Click here to run or download a program showing the exchange of stored energy and kinetic energy in an oscillating cantilever. Because the stored energy is a combination of compression energy and tension energy, which are portrayed as red and blue respectively in this web-site, the stored energy is shown as magenta, a combination of red and blue.

Here is a diagram showing stress versus energy.

What happens is we go on increasing the stress? This.

At a certain point, the material is over-stressed, and fails, and the energy falls to a very low value. The red line shows a possible maximum working stress, with a large safety factor. We can imagine an oscillatory stress that causes the energy to vary along the curve, while staying below the red line. Is this a satisfactory mode of operation? The answer depends on the material. For aluminium alloys, the answer is that if you run enough cycles, the material will fail. The lower the maximum stress, the more cycles you can run. For steel, the answer is different. Below a certain stress level, the material will never fail, however many cycles it is subjected to. What happens to the aluminium alloy is that every cycle of stress changes the internal structure of the material, very slightly. The effect is cumulative. Very much over simplified, the result is like the graph below, where the strength of the material is eroded with every stress cycle.

The material gets weaker and weaker, and eventually fails at a level where it would have been more than strong enough at the start. At some time before the failure, distress might be visible in the form of tiny cracks. Eventually, a crack will propagate with great speed, resulting in loss of the part.  For those who are old enough, the letters G-ALYP will be familiar. Many structures, such as aircraft, are designed to stop cracks at pre-determined lines.

What can generate oscillations in an elastic structure? The answer is any impulse. A vehicle moving on to a span generates a load that was not there before. As it moves, a wave of strain will travel in both directions along the span. It may reflect from the ends, forming a standing wave, which will appear as an oscillation. The oscillation will not persist for ever – energy will be lost by absorption in the material. One aim of the designer is to make the damping and stiffness so great that vibrations cannot build up to undesirable levels.

Impulsive Excitations

The diagram below shows the changes in position, velocity, and acceleration produced by an impulse acting on an elastic object. The result is a damped sinusoidal oscillation if the material behaves linearly. The strain energy is proportional to the square of the displacement, while the kinetic energy is proportional to the square of the velocity. The total energy is the sum of the two forms of energy, and it decays exponentially with time.  Here is a link to a real example.

Note that there can be times when the amplitude suddenly drops, when the new impulse is out of phase with the current oscillation. These graphs are a little unrealistic because the vibration produced by a truck travels in both directions as a wave. So the effect of a truck at a given begins somewhat gradually before the truck gets there.

If you place a mug, jug, or bottle near your ear, you may hear the noises in the room, as filtered by the container, giving a kind of slightly tuned noise. This is analogous to the way the bridge responds to impulses.

If you have access to a piano, open the lid, press and hold down one key, and shout into the piano. You will hear its response as it filters your noise and selects its own frequency. Many musical instruments are designed use this filtering. The pipe-organ, pan-pipes and whistle use a steady flow of air to create pure tones. The strings of a  violin use the steady movement of the bow to create the required tones.

A laser uses the steady input of energy to create a pure light signal. An exact number of wavelengths fits between the mirrors. Because the wavelength is minute, thousands of slightly different modes would be possible, but for the fact that the atoms in the laser select a narrow band of frequencies.

It is sometimes necessary to use a fast shutter speed to take photographs on a big bridge,, because of the vibration. The vibration can be felt before a heavy vehicle reaches you because the waves travel faster than the vehicles. If you look at a film of the Tacoma Narrows bridge you can work out roughly what the speed of the waves was. Knowing the length of the span and the frequency of the oscillation, and the fact that one wavelength fitted into the span, we can calculate the speed of the waves.

The speed is equal to the span multiplied by the frequency, or the span divided by the period. If the span were 1000 metres, and the period of oscillation were 5 seconds, the speed would be 200  metres per second, or 720 km/hr, or 450 mph. Making the span stiffer would make the waves faster, but only as the square-root of the stiffness. Making it heavier would slow them down, also proportionally to the square-root of the mass.

The next picture shows the effect of more frequent impulses with both polarities.  This could represent torsional oscillations set up by heavy traffic on opposite sides of the road.

Please click here if you want to see more graphs about superposition of waves, but come back here afterwards.

Periodic Excitations

The next simulation shows the case where each impulse comes exactly in step with a cycle of the previous oscillation. The amplitude of the individual impulses has been been reduced by a factor of ten from the value used in the first picture. Even so, the amplitude builds to a fairly high value.

Each impulse is assumed to add the same momentum to the system. Another way would be to add the same amount of energy each time. We see that the amplitude builds up until as much energy is lost during each cycle as is gained from each impulse. If the amplitude builds up too much there could be structural damage.

This is rather like the case of someone on a swing receiving a rhythmic push from someone else. It is a completely different mechanism from the case in which the person on the swing provides the energy themselves by moving up and down at the right times, which occur twice per cycle. One way to reduce the final amplitude is to increase the loss of energy through absorption in an inelastic material. This is difficult to do at the scale of a large bridge.

In the next picture the damping has been increased, resulting in smaller oscillations.

Effects of Increased Stiffness

Another way, shown below, to reduce the amplitude of the oscillations is to increase the stiffness of the structure, which will increase the natural frequency and reduce the effect of the impulses.

After the Tacoma Narrows disaster, some bridges, such as the Mackinac Straits bridge, were built with very deep trusses. These bridges were not affected by oscillations.

You can see how bandwidth is related to decay time by gently striking something which rings, such as a bell, a wine-glass, or a metal plate. If you grip more and more tightly, the tone becomes shorter and less clear in frequency. The behaviour of damped oscillations can be heard in another page – Damping

Click here for a bit of maths about sines and exponentials.

This happened to the Severn suspension bridge. Over many years the volume of traffic rose to levels that could not have been considered likely at the time the bridge was built. Eventually it was repaired and strengthened.  Even the towers were made stronger, by an ingenious method. Some steel tubes were placed inside the towers until they filled the entire height. Then by applying suitable forces they could be made to take some of the load.

Types of Oscillation

There have been discussions about the exact nature of the oscillations which affect big bridges. Are they caused by resonance, or by intermittent forces? Other pages in this web-site illustrate the difficulty of assigning bridge types to categories. The same phenomenon occurs in this topic also.

Wind-induced forces may be far from sinusoidal, but they can excite normal modes of structures. If two parts of the structure have similar natural frequencies, or if a natural frequency is excited by external forces, there is potential for trouble.

In electronics we can build sinusoidal oscillators, and also relaxation oscillators and square-wave oscillators. But we can also add a tuned circuit to a pulse type circuit to make good sines. A class C RF amplifier does just this. The behaviour analogous to the once-per-cycle push that you use to get a child’s swing going. The tiny push of the escapement of a pendulum clock or a hairspring clock is very similar in its effect. As in many other fields, the boundaries between defined categories can be difficult to find.

The steady motion of a violin bow or the wheel of a hurdy-gurdy interacts with the strings to produce the resonant frequency of the string. The steady flow of air on to the lip of an organ pipe or the hole of a flute has a similar effect. Subtle and varied are the ways of oscillators. Any electronic engineer knows that a high frequency amplifier is waiting for a chance to oscillate. Many circuits and components are specifically designed to produce oscillations from a steady flow of energy. In the days of valves (tubes), the transitron, magnetron and klystron were invented, and in fact the last two are still in use. A magnetron produces the oscillations in a microwave oven.

Yet another kind of oscillator is the laser, in which a steady flow of electrical or electromagnetic energy can produce an oscillatory output.

Oscillation of Cable-Stayed Bridges

Although the cable-stayed bridge is inherently stiffer than a suspension bridge, the relationship is reversed during construction. Construction of the deck of a suspension bridge does not begin until the cables are complete, and so all parts of the bridge are connected, however tenuously.  ut the cable-stayed span is built out in stages from each tower, and when the span is almost complete, the long cantilevers are at the mercy of the wind.

The diagrammatic plan view below, showing a part of a bridge, suggests what might happen. The amplitude is exaggerated. The deck could also oscillate in other modes with higher frequencies. In principle there could be horizontal oscillations allowed by torsion in the towers, and vertical ones allowed by bending of the towers. This calculation assumes that the deck is fixed at the tower. Without such a connection, the deck could oscillate in a horizontal pendulum mode.

The lower diagram suggests that when the two halves of the span have been joined, the resultant rigidity reduces the amplitude of any oscillations. It also increases the frequency – we can see this from the shorter wavelength, about equal to the span.

These pictures show a cantilever with and without vertical oscillation, caused by a gust of wind.

In principle an active damping system could be created using movable masses near the ends of the cantilevers during construction. Small signals from sensors on the deck would be amplified and used to control hydraulic or electric motors to move the masses. The system would require emergency power generators in case of a power supply failure. Such a system has been used in tall narrow buildings. Because the moving mass is much smaller than the effective mass of the structure it must move more quickly. The system must be unconditionally stable.

Lateral oscillation of a completed bridge cannot be eliminated: it can only be reduced to an acceptable level. What is acceptable? If the coefficient of friction of all vehicles and peoples’ shoes is at least f, and the acceleration due to gravity is g, then the lateral acceleration should be less than fg. In practice it will be much less than this. A more practical value is the level at which drivers and pedestrians begin to feel uncomfortable or alarmed. It would be especially dangerous if drivers tried to correct for perceived movement, as this might even build up the oscillations.  

A very unusual effect did occur on the millennium bridge in London, related to the walking of people on the bridge, which was closed after only two days of use, was only re-opened after a large number of dampers had been installed. The engineers did this without much effect on the appearance of the bridge.

These two pictures show a footbridge in the tree canopy in Sarawak, on the beautiful island of Borneo. The bridge was hung from a number of trees, and had little tendency to vertical oscillation. Laterally, however, it was a different story. It was difficult to avoid walking in step with the movements of the bridge induced by the previous person, producing the same phenomenon as in the London millennium bridge. One solution was to take longer and slower steps, and to put the pressure on the bridge gradually at each step. Keeping a greater distance from the previous person was also helpful. The deck was in part made from wooden spars, and in other from aluminium ladders with planks on top.

Oscillations Part 2 – Resonance

Links to Other Web-Sites About Oscillation and Wind

and about Tacoma Narrows Bridge

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