Oscillation

The behaviour of a bridge is not fully specified by the static forces within it. Any bridge can move. All the parts have both mass and elasticity, and can exchange energy between kinetic energy of the motion. and the strain energy of bending, stretching or torsion.

The bridge does not sit in a vacuum, doing nothing. It experiences the wind, and it experiences the live loads caused by traffic. These two facts have a profound effect on bridge design. Even the steps of pedestrians can affect a light, flexible foot-bridge.

The picture to the left represents a side view of one tower of a suspension bridge before the suspended parts have been added.

The left hand bar represents the static situation. The right hand shape (exaggerated) reminds us that the tower can oscillate. In principle there could be higher modes, the next mode having a node near the top, but these modes have higher frequencies and smaller amplitudes. The oscillations can be a serious problem before the bridge has been completed, especially when the two legs have not yet been joined by a cross member at the top, or lower down.

Like any other massive elastic object, the tower will have a natural resonant frequency. Energy transferred from the wind will tend to excite this resonance.

Furthermore, because the towers are not streamlined, it is possible for them to shed vortices downstream. These tend to occur on alternate sides of an obstacle, making for an oscillatory situation. This can be observed with a flapping flag or even with the rope slapping against the flag-pole.

Tall metal chimneys are usually provided with helical strakes, which affect the flow in such a way that the vortices do not occur. The helical shape makes the system work well whatever the direction of the wind.

Flutter has been known to affect the wings of high-speed aircraft, though this is now unusual, as the phenomenon is well understood and the technology is mature.

The picture below represents the first five modes of oscillation of a hanging cable, such as an empty clothes line or a cable of an incomplete suspension bridge.

In the fundamental mode at the top, the oscillation would require large changes in the length of the cable, so this mode is strongly suppressed.

The main mode is therefore the second harmonic, with a node in the middle. This has important consequences for the behaviour of bridges.

The millennium bridge in London has eight parallel cables, spaced away from the deck. Had some been curved inwards, so that they were close to the deck at the centre, the fundamental horizontal mode might have been somewhat suppressed. Having little depth, it has little resistance to vertical movement also.

The next picture represents the lowest modes of oscillation of a beam which is fixed only at the ends.

Because the cables of a suspension bridge cannot support the fundamental oscillation with an antinode in the middle, this mode is suppressed in the deck also. In fact the slight upward curvature of the deck, found in most suspension bridges, will also tend to suppress the fundamental mode . The main mode of the deck will be the second harmonic, with a central node. The oscillation can be a simple vertical oscillation, or a twisting motion with opposite sides of the roadway moving in opposite directions, and opposite ends of the bridge tilting in opposite directions.

Suppose we were to connect a huge hydraulic ram to the deck of a bridge, and we slowly swept the frequency of oscillation from zero to perhaps 1 Hz. The response of the bridge might look rather like the diagram below, though the diagram is not intended to be at all realistic in detail.

The horizontal scale denotes the harmonics, whose responses are coloured in red, green, blue and black. The result is shown in white.

In practice the frequencies might not be so simply related, for example because the side spans are often not equal to half the main span, and because the cable is curved and not straight. Although the details are not realistic, we see that the fundamental is missing, and that the harmonics are lower and wider as the frequency increases. The fourth harmonic is barely visible as a slight bump. A wider resonance means a faster dying away of the oscillation after the excitation has been removed.

This relationship between widths of structures in the time domain and in the frequency domain is fundamental to waves and oscillations. It means, for example, that the position and frequency of a wave cannot both be determined with absolute accuracy. The product of the two uncertainties can be calculated. The behaviour of damped oscillations can be heard in another page - Damping - as if the oscillations had been speeded up by a factor of several hundred.

We have looked at three modes of oscillation - towers, cables and deck. Should any two of these have similar frequencies there could be a serious problem. What could excite a resonance? Any source of energy of about the right frequency could be dangerous. This is why soldiers break step on a small bridge.

But in normal conditions a bridge is not apparently subjected to periodic forces. So thought the designers of the Tacoma narrows bridge. The diagrams below represent the cross-section of the long, narrow and shallow deck of the Tacoma bridge.

The top picture shows the static situation. If there is a wind it can slightly tend to tilt the deck, especially if gusty. If the tilt happens to increase the lift near the leading edge, the tilt may increase. In a strong enough wind the tilt may increase until the airflow breaks away, and a stall occurs. The energy stored in the torsional strain now starts to transform into kinetic energy of rotation. When the bridge reaches the normal position it does not stop because of its inertia, and it will tilt in the opposite direction. In certain conditions a steady wind can sustain such an oscillation. A falling leaf or sheet of card may oscillate in a similar way.

Near the bottom the circles represent a train of vortices downstream of an obstacle in a wind.

The frequency of generation is equal to the speed of the wind divided by the distance between the vortices. If this frequency of a pair of vortices is related to the natural torsional resonance frequency then there could be a problem. The Tacoma narrows bridge was reduced to wreckage in a short time, by a moderate wind. Henry Ford is reputed to have said that history is bunk. Without knowing the context, and what he meant, we cannot evaluate this statement. But if you are building aircraft, bridges, or any other critical structure, it is a good idea to know what has happened before, and to find the right balance between over-cautiousness and rashness. Informed boldness is perhaps a good way to make progress.

The immediate response to the Tacoma Narrows crash was to build deep trusses under decks, much as the first aircraft used biplane construction to achieve rigidity.

It may seem strange that a steady wind can produce oscillations from in simple object, but, as we see from the picture above, simple causes do not always have simple effects. This picture shows the tear in a KitKat wrapper after the back of a knife has been pulled through it.

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 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. 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?

At the bottom of the big diagram above, and repeated here, a typical streamlined bridge section is shown. It is probably better if any aerodynamic force is downwards rather than upwards, "prestressing" the system. The roadway of the Severn suspension bridge has an aerodynamically streamlined cross-section, suggested by the work of Fritz Leonhardt, 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 for most applications.

Each hanger, except for the very shortest ones, is provided with a small device near the bottom, which absorbs energy. 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.

Recently, some extra panels were added near the towers to influence the airflow near the towers.

The panels are composed of many slats. The larger panels have electric motors which can be sued 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.

Before looking at the causes of oscillations, let's look at energy in a structure. The graphs on the left 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 as the peaks of position or speed.

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.

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.

The next diagram shows the deflection and energy resulting from a series of impulses of random energy, occurring at random times. This crude simulation is not very realistic, and includes impulses only in one direction. The picture could represent vertical oscillations resulting from the passage of vehicles past a point on a span.

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.

You can feel these vibrations quite easily when a large vehicle passes on a big bridge. You then realise that the phrase "live load" really means something. The Severn suspension bridge has footways, enabling people to walk from Aust to Chepstow. The footways are in fact substantial enough to take small vehicles used by the maintenance crews. On the footways you can easily feel the vibrations of the bridge.

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 and pan-pipes 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 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 in this situation, 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 proportional to the square-root.

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

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. 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.

Another way, shown below, is 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.

The next picture shows the onset and the end of a short train of impulses. The build-up and the decay of the displacements occur on the same time-scale. The greater the damping the quicker the system follows the input. This is analogous to the requirement for wide bandwidth in an AM radio receiver in order to follow the modulations imposed by the audio signal.

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.

There can be other types of oscillation which are not resonant. Suppose a bridge is near a big city, and the flow of traffic is mainly into the city in the morning, and out of the city in the evening. Then the weight of traffic will twist the bridge one way in the morning, and the other way in the evening. Even worse, if all the toll-booths are all at one end of the bridge, then traffic which is queuing will build up on the bridge in one direction, and on the land in the other direction. During peak periods one side could be solid with traffic while the other is lightly loaded.

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.

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. But 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

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

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.