Beams and Gyroscopes

Beams and gyroscopes? What kind of a title is that? What have beams to do with gyroscopes? The answer is - nothing at all. So why write this page? Read on and find out. The diagram below shows a possible cross section for a beam. If we take each element of the cross section, and multiply its area by the square of its distance from the neutral axis of the beam, we get a quantity that is proportional to the contribution of that element to the stiffness of the beam. The neutral axis of a beam is the plane along which there is neither tension or compression. If we add up the effects of all the calculated elements and divide by the total area, we get a sort of average of the squared distances from the neutral axis. We can then take the square root of this value to get a length rather than a length squared. The stiffness of a beam is proportional to the square of this length. We see that the the greater this length, the stiffer the beam, in other words, moving material away from the neutral axis makes a beam stiffer. So I-beams, boxes and tubes are good shapes to use. This representative length could be called the root-mean-square (rms) distance from the neutral axis, rms being a commonly used concept. Electric power, for example, is proportional to squared voltage or squared current, so specifying alternating supplies by the rms values allows AC and DC calculations to use the same formulae. Since rms is a sort of average, the peak AC voltage is bigger than the rms. With a 110 volts supply, the peak is about 156 volts, and with a 240 volt supply, the peak is about 339 volts. The scatter of a set of measurements is also sometimes specified by a quantity like rms, called the standard deviation. Nevertheless, with beams, although the representative distance could be called rms, it isn't: it is called the radius of gyration, even though nothing is gyrating. Why do we call it that? The reason is that the root-mean-squared radius comes into play in a completely unrelated situation, namely, the rotation of an object, where it is indeed called the radius of gyration, and the use of the name has been extended outside that field. The resistance of an object to a change in its rate of rotation is called its moment of inertia. In the same way that linear acceleration requires a force, angular acceleration requires a torque. For a mass M, an acceleration A and a force F we have F = MA. For a moment of inertia I, an angular acceleration A and a torque T we have T = IA. The moment of inertia is equal to the mass of an object multiplied by the square of a representative length called the radius of gyration, which is in fact the root-mean-square distance of its elements from the axis of rotation. Thus moving the mass away from the axis increases the resistance to rotation. Somehow, the name "radius of gyration" has been carried over to beams, even though they don't rotate. So that is the tenuous connection between beams and gyroscopes. Because beams are used in an asymmetric environment, a gravitational field, to be exact, their symmetry generally reflects this fact. Beams are usually possess horizontal reflection symmetry, and they may possess vertical reflection symmetry, as in the case of I-beams. Pre-stressed beams, at least internally are usually vertically asymmetrical/ What they seldom have is four-fold rotation symmetry, because they usually have depth greater than width. Laying a beam on its side will almost always reduce its resistance to sagging, for this reason. Four-fold symmetry means that there are four different angles through which the shape can be rotated, with a result that looks the same as the initial condition. There is an exception to the requirement for a beam to be stiffer in one direction than another. Where can you see a structure that is equally resistant to bending in any direction, because forces can come from anywhere? The answer is a tower, which has to resist wind pressure and buckling in every direction. Thus we might expect very tall towers to have a circular - specifically tubular - cross section, because a tube has all the mass as far as possible from any bending axis. In practice, trussed towers often have four-fold symmetry, while concrete ones may be polygonal or "asterisk" shaped. By asterisk we mean a shape like the one shown below, but not necessarily with six sectors. The CN tower, for example, has three sectors. Why not tubular? A tube is very stiff, but as all the material is around the edge, placing and supporting services such as lifts and cable ducts may present problems, especially as the diameter will usually vary with height above the ground. The three-fold asterisk, or Y shape means that there is much material at the centre to contain and support services. We may well ask whether a polygon (especially a triangle) or an asterisk (especially a Y) is equally stiff in all directions. It turns out that for any cross section with N-fold rotational symmetry, the radius of gyration is the same about any axis in the section, as long as N is greater than 2. The value 2 is typical of horizontal beams for which rotational symmetry is not required. Nevertheless, this is not the whole story, because buckling has not been mentioned: tubes, polygons and asterisks may need stiffening flanges at intervals along their length. The tubes of the Forth railway bridge and the Royal Albert Saltash bridge have these flanges. In Slovenia, you can see racks, called kozolecs, for drying hay. These have many horizontal poles with square cross section. Sometimes a pole becomes twisted, so that the two ends differ in orientation by 90 degrees. Because the stiffness of a rotationally symmetric beam with N > 2 is independent of orientation, the twisting does not matter. The picture at left shows an example near Stara Fuzina. It was this that motivated the calculation to check rotational invariance of a ploygonal beam. The second picture shows the distribution of tensile and compressive stresses for two orientations of a horizontal square pole. Symmetry Invariance and symmetries are powerful ideas in physics. The laws of conservation of energy, momentum and angular momentum may seem somewhat arbitrary, but they are in fact equivalent to invariance of the laws of physics with changes in time, position and rotation. In other words, the world possesses certain symmetries. Note that what is invariant is a law of physics, not the measurements. Thus measured lengths and times depend on the speed of motion of the measured objects relative to the frame of reference of the measuring instruments, though the same laws are true in all frames. And standing up does not feel like lying down, because the locality is not perfectly symmetrical, as gravity defines a direction that is special. A more abstract principle is gauge invariance, which does not correspond to any everyday observable variable. Music and poetry both possess symmetry, though it is broken symmetry, otherwise every stanza of a poem or song would be the same. The repetition of rhythm and metre are symmetries, as is the repetition of rhyming patterns. The practitioners of both disciplines are happy to break the symmetry in the interests of variety - that is one difference between art and science. Some physical symmetries are broken also - otherwise the three families of quarks and leptons would be identical, and presumably physicists would have discovered only one family of basic particles. Here are some asterisks and polygons. Pillars, columns and piers inside buildings are often circular or polygonal in cross section, sometimes with subsidiary columns or other decoration, still with rotational symmetry. The case N = 2 is different from the others, because it denotes not a column but a wall, which is far more stable in one direction than the other, not only against bending, but against toppling. Click here for a discussion of stability. Gyroscopes Since we mentioned the subject of rotating objects, perhaps we may reconsider them now. The invariance under angle mentioned above means that for N > 2, any N-fold symmetrical shape will do as a flywheel or a gyroscope, though from the point of view of energy storage or resistance to break-up caused by huge accelerations, a circular shape may be best. Calculation shows that for N = 2, the precession of a gyroscope is not the same for all axes through the plane of the object, because the moment of inertia varies with the angle, and so the use of such a shape in an a gyrocompass, auto-pilot or inertial navigator would result in varying forces as the object rotated. A low-pass filter might be employed, but it is much simpler to use a shape with N > 2. Sampling and aliases Having made a tenuous connection between gyroscopes and beams, how about an even more fanciful one? Consider a gyroscope rotor comprising N (> 2) compact masses on a circle. Looking at the circle edge-on, we see the N masses in a straight line. From their positions we can calculate the diameter of the circle, and the phases of the objects relative to our line of sight, even if none of the objects is at an extremity of its motion, provided we have been told that they are equally spaced on a circle. If there are only two objects, we cannot do this. What has this to do with anything? The circle is closely related to a sinusoid, and our objects can represent the samples in a digitizer. The inability to reconstruct the situation with N = 2 is the analogue of Nyquist's theorem for sampling. If we take a sine of infinite duration (and there is no other kind of sine), and we take N samples per cycle, we can reconstruct the phase and amplitude of the sine, as long as N is greater than 2, even by a minute amount. But we cannot do it with N equal to 2 or N less than 2, even slightly less. As so often, it is all too easy to make a statement that is not correct. Having said that we can reconstruct the correct sinusoid through a suitable set of points, we have to admit that if we have a set of uniformly spaced points that lie on a sinusoid, it is possible to draw an infinite number of different sinusoids through them. Which is the right one? And what is different when we obey the Nyquist rule? In order to be sure we are getting the right answer, we have to know the analogue bandwidth of whatever precedes our digitizer - then we know an upper limit for the possible signal frequency. The Nyquist sampling frequency is then twice this frequency, and we have to choose the lowest frequency of sinusoid that fits the data. If the sampling rate is too low, the lowest frequency sinusoid is not the right one, but if we have no idea what the right frequency is, we are lost. If, however, we do have a good idea what the signal frequency is, we can measure tiny changes in it by using a very slow and very accurate sample rate, because the small changes in the signal will make large changes in the measured frequency. See also the page about aliases for more information. A discussion about the stability of polygonal towers can be found by clicking here. Again, the number 2 is exceptional, because few towers are thin in one dimension. We have looked at several topics where the value 2 causes behaviour different from that produced by other numbers, even though the topics are totally unrelated. The 2 comes in simply because of mathematical or physical properties. So we must not assume that mathematical similarity always implies a physical connection. Can you think of any more such topics involving the number 2? In another page, you can read about the Fibonacci numbers that clearly appear in nature, giving a completely false impression that they are fundamental to plant growth. This page has described several unrelated topics, starting with the fact that "radius of gyration" is used in different contexts. On the other hand, it sometimes happens that apparently unrelated topics have deep connections. The wave mechanics and matrix mechanics of early non-relativistic quantum mechanics illustrate this point - they were capable of producing the same results, while appearing quite different. Taniyama and Shimura conjectured that elliptic curves and modular forms were somehow equivalent. This was proved by Wiles, Taylor, Breuil, Conrad and Diamond, and the work of the first two authors led to a proof of Fermat's last theorem. Thus the conjecture was very fruitful. |