More About Oscillations - Bridges and Otherwise

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In the DVD "Toward the Within" by Dead Can Dance, Brendan Perry explains that the name of the group can be taken to represent the ability of musicians to create animate music from inanimate objects. Objects are perhaps not quite as inanimate as we might think. Every single atom or molecule has an intrinsic average energy that is proportional to the absolute temperature in which it finds itself, though the energy at any moment varies in a random way. This thermal noise constitutes a lower level to the size of any signal that we can measure in a given frequency band. In electronic circuits this noise is called Johnson noise, and also white noise, because it is frequency independent.  

In addition to this noise, which is consistent with classical physics theory, quantum mechanics predicts the existence of vibrations and rotations which behave differently from classical motions. The almost spherical molecules of buckminster-fullerene rotate at very high speed at normal temperatures because the first rotational energy level corresponds to a much lower temperature. Cooling the substance to a low enough temperature makes the rotation stop completely.  

So we see that inanimate objects are never completely devoid of motion, though in normal life we are unaware of this.

In this page we will try to see how inanimate objects can be made to produce a huge variety of sounds, and how they can sometimes oscillate, or vibrate, when we definitely don't require it. Many is the designer who has built a radio frequency amplifier that oscillated, a servo-mechanism that hunted - and even in some forms of software got into never ending oscillating loops.

In all inanimate objects, there are two qualities that contribute to resonant oscillations - mass and elasticity. Mass is the quality that affects acceleration: elasticity is the quality that affects change of shape. These imprecise statements can be expressed more accurately by simple equations.

Newton's second law says that the force F required to produce an acceleration A in a mass M is given by this formula -

F = MA.

And Hooke's law says that the force F needed to produce a change in length E in an elastic object is given by this formula, in which K is a constant peculiar to that particular object -

F = KE.

Both of these formulas produce straight line graphs, and are called linear equations.

If we connect a spring to a mass, then by Newton's third law, the force that the mass exerts on the spring is equal and opposite to the force that the spring exerts on the mass. This means that we can construct a new equation, of the form -

MA = KE.

The equation contains two variables, the acceleration and the position, which are not obviously related, though in fact they are. We will consider only one very simple case, which will turn out to be the one that describes resonant oscillations.

Because the force is proportional to the displacement and the acceleration, we can draw graphs like this -

In order to make progress, we need to find a way of relating displacement and acceleration. One way to do this is to calculate the velocity in each case.  We will do this for an oscillatory motion. Firstly we look at the variables we already have, for a spring and a mass.

Note that the vertical scales are arbitrary here, because force, displacement and acceleration are all measured in different units. To get to velocity, we need to know that velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. This results in the graphs below.

What we see is that if we act on a mass and a spring with a force, the velocities are in opposite directions. The next point to see is that the displacement, velocity and acceleration are related to the frequency of the oscillation. If we oscillate a mass with a given amplitude, and we increase the frequency, the velocity must increase, and in the same way, if we keep the same velocity and increase the frequency, the acceleration must increase. That means, for a given acceleration, the velocity must decrease. The diagram below shows this.

At one frequency, for a given mass and elasticity, the velocities are equal and opposite, or, putting it another way, if we make the velocities go the same way, the forces are equal and opposite. This means that if you connect a mass and a spring, they should be able to oscillate with no input force, and therefore no input of energy. In practice, this does not happen, because some energy is lost in each cycle.  The special frequency is called the resonant frequency. You can check this behaviour by singing down a wide tube or in a bathroom, and varying the pitch of the sound.

This explanation is ridiculously long: that is what happens if you try to avoid mathematics, the appropriate language for so many things. Some of the power of mathematics lies in its ability to be concise and accurate. Not only that - it can be predictive in ways that you probably wouldn't foresee. Maxwell's equations for electromagnetic behaviour predicted the existence of electromagnetic waves. Einstein was delighted when his general relativity theory "predicted" the known perihelion of Mercury, but very upset indeed when it predicted the expansion of the entire universe, so upset, in fact, that he fudged the theory to allow a static universe. Then Hubble made his measurements. The theory also predicted the possibility of gravitational waves, which have not yet been detected. Dirac created a very neat equation to describe the behaviour of an electron: he got anti-matter out of it as well. Perhaps this is a scientific version of the Dead Can Dance comment - from simple ideas, we can derive amazing and unexpected predictions. And Nature can still provide phenomena that are not understood, such as the rings of Saturn. Science is not a matter of recording data into inanimate lists: it is a living thing that evolves.

This problem is better solved by the techniques of differential equations.

Returning to our mass and spring, if we measure the amplitude of oscillation versus frequency we get curves like these -

It looks fairly harmless, but when the amplitude can represent a massive suspension bridge thrashing in the wind with huge waves travelling along the deck at several hundred miles per hour, or the vibrating shaft of a huge generator set rotating at 3000 revolutions per minute, the reality is alarming. It is indeed almost as if the materials have come alive: they are certainly out of control.

Resonant oscillations can be avoided in several ways. One is to make sure that all parts of a structure have different resonant frequencies and to ensure that no input of energy occurs at these frequencies. But if the input energy comes from wind or tide, all frequencies can be excited. In some cases, the running frequency may be above the resonant frequency: in these cases it is well to accelerate through the resonant frequency as quickly as possible, and to remember to do the same when running down.

Another method is to increase rigidity, which is why many suspension bridges have such deep and well braced trusses.

DamperOld.JPG (46368 bytes)DampersBot.jpg (30553 bytes)Yet another way is to use devices which absorb energy, usually containing some kind of viscous material. In music, they are usually called mutes.  These two pictures show two types of damper which have been used on the Severn suspension bridge. The cables of any bridge are like the strings of a musical instrument, ready to vibrate with any input of energy. In some cable-stayed bridges, the main cables are connected together by subsidiary wires. Because the main cables have different lengths, they have different resonant frequencies, and so the cross-connections will tend to take energy out of resonating cables and pass it to others at frequencies that do not support resonance in those other cables.

Here is a link to the web-site of Jodi Rose, who is creating a sonic sculpture from the sounds of bridge cables around the world.

Some bridges, especially small ones, are sufficiently flexible that people can generate easily detectable oscillations. It is not a good idea to try to make a bridge move, because this can be alarming for other people.

Strangely enough, resonant devices can be used to prevent regular oscillations from reaching equipment that needs to be kept quiet. A carefully designed spring and mass arrangement between the floor and the equipment can be made to absorb energy quite strongly at one particular frequency..

Although we generally don't want structures to vibrate, musical instruments are exactly the opposite - we want them to vibrate, generally in a controlled manner. Energy can be transmitted into musical instruments by blowing into them or across them, by striking (pianoforte, xylophone, dulcimer, drum), by plucking (finger, plectrum or harpsichord mechanism), and by rubbing (bow or hurdy-gurdy wheel). The vibrating part may be a string or wire (one dimension), a membrane (two dimensions), or a bar (two or three dimensions), a column of air (one dimension), or a cavity containing air (three dimensions). In addition, many instruments, especially stringed ones, include coupling to a large object that can act as an acoustic transformer to better couple the energy to the air. These many methods contrast with the few ways of getting energy into a bridge (wind, traffic movement, pedestrian steps), and yes - bridges have been broken by soldiers marching in step - it is not a myth.

A large number of stringed instruments consist of a box with many strings stretched across, with series of posts, called bridges, which raise the strings from the box, and transmit the vibrations to it. From China across to eastern Europe, members of the family of dulcimers can be found, for example, in Hungary the cembalom, in China the yang ch'in. The general arrangement of the yang ch'in is like this: the interleaving of the strings is very convenient for the use of two hammers. The strings are struck using flexible bamboo hammers. The Hungarian cembalom is rather bigger, and sometimes provided with legs to make it self-supporting, and like all dulcimers, it is played with a pair of hammers. Unlike xylophone players, who can use up to four hammers at once, dulcimer players are content with one in each hand.

YangChin.gif (2085 bytes)

The stresses in dulcimers are not particularly high, but in the larger instruments such as pianofortes and harps, the frames are often massive, because the total pull of the strings is very large, often the equivalent of several tonnes. Even in the relatively small violin family, the stresses imposed by modern playing methods, and techniques like the Bartok pizzicato, have necessitated the rebuilding of many instruments. The force and energy now put into instruments to fill a large hall with music that was written for playing in small rooms (chamber music), together with 20th century styles of composition, has forced these changes.

Resonance can occur in electrical circuits, where capacitance and inductance play the roles of spring and mass. This is how we can tune a radio receiver to select the signals from one radio transmitter while rejecting others. The tuning must not be too narrow, because the signal modulations that carry the information occupy a finite bandwidth. Much ingenuity has been devoted to maximising the amount of information that can be carried with a given bandwidth.

On a much smaller scale, the spectra of the masses of particle groups emitted during the collision of so-called elementary particles show very similar effects. This is the result of energy E (mass) being associated with frequency F, the ratio being Planck's constant (h).  The equation is E = hF. The spectrum below was a result of kaons colliding with protons.  It shows two resonances between kaons and pions, and one between two pions. The mean lives of these very transient particles is of the order 10-23 seconds, and cannot be measured by timing, but the width of the peaks is inversely proportional to the mean lives, and are used to calculate these lives. The relationship again involves Planck's constant.

MassSpectra.JPG (63469 bytes)

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