Their applications, physics, and math. -- Peter Ceperley
There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.
Brunelleschi's Dome - Its structure and construction
The dome that was added to the top of the Santa Maria del Fiore cathedral in Florence in 1446-1461 is rich in history and interesting detail. Two of the more touted features of the dome are the structural design and the novel method of construction which allowed the dome to be constructed without the usual temporary internal bracing.
The web and literature is rich with great overviews of all the features and historical events of the dome's construction. However while reading about this remarkable engineering feat, I was frustrated by the lack of mechanistic explanations about how the dome was actually assembled and exactly how its touted features combine to make it work. In this posting I present what I have inferred from combing through many articles and making educated guesses from an engineer's point of view.
1. The Dome and its structural features
The geometry: This dome was a massive undertaking. It was to be the largest in the world at the time, 144 feet across and with a base 171 feet above the cathedral floor and rising to 375 feet. The dome would weigh about 37,000 tons with over 4 million bricks.
The geometry of the dome was not a simple hemisphere but rather an eight sided shape, similar to the shape of an eight sided felt hat, each side made of a bendable flat piece, with the pieces laced together at their edges. At each level, the shape of this dome is octagonal. The vertical curve is circular with its center of curvature at the base of the dome, one-fifth the dome's span from the edge (see the figure at the right) as prescribed by its designer. While the designer Arnolfo di Cambio 124 years earlier had laid out the shape of the dome, most of the engineering details to make it possible were left up to the builder, i.e. to Brunelleschi.
Exaggerated potential deformation caused by the weight of the upper dome. In reality with a stone or brick structure vertical cracks would appear on the lower sidewalls and the dome would collapse. The chains are designed to act like strong belts, resisting the lower deformation and holding the structure together.
Chain made of 80 sandstone beams placed end to end and interconnected with galvanized iron links to form an octagon. Each sandstone beam was approximately 17inch by 17 inch in cross section by 7.5 feet long and weighed about 1700 lbs. Chains for the upper part of the dome were progressively shorter in circumference. Below each chain was a series of shorter stone beams laid perpendicular to the chain to connect the chain to both inner and outer domes.
Some writers say that two of these stone chains were used at each level with the perpendicular ties interconnecting them and the assembly resembling railroad tracks. Two parallel chains at each level would better resist the stresses in the middle of the flat octagonal sides. The ribs and double dome construction (covered next) also help resist these forces.
Of central importance was the lack of internal or external bracing, such as tie beams and flying buttresses. The key to making the structure stable without bracing was the novel use of massive sandstone and wooden tension rings, called "chains", embedded in the dome at regular intervals. One sandstone chain was at the base of the dome, the wooden chain was next, followed by three more sandstone chains. The sandstone chains were at 35 foot intervals with the wood chain between the first and second stone chains. The original design called for complete iron chains on top of each sandstone chain for added strength; however, it's not clear that these were ever added. Today, builders would use rebar to provide strength in tension.
Drawing showing the ribs that separated the inner and outer domes.
Brunelleschi's sketch showing ribs and brick orientation on inner and outer domes. This sketch greatly exaggerates the size of the bricks.
Double dome construction: Also adding to the strength-to-weight ratio was the design which used two nested domes, and inner one and outer one, interconnected with a lattice of brick ribs. The drawing and sketch at the right show the ribs. This design of two shells with spacers between them is used extensively today whenever a large strength to weight ratio is desired, e.g. modern aircraft design.
The outer dome was 2 feet thick at the base and tapered to 1 foot thickness at the top. The inner shell was 7 feet thick at the base and tapered to 5 feet thick at the top.
The ribs also transfer the weight of the outer dome onto the inner dome. The inner dome was designed to be very strong being thicker and with its bricks oriented for strength. The outer dome could then be optimized for resistance to the weather. A nicely done cutaway drawing of the dome and its parts is at the National Geographic website.
Lightweight bricks: Also important was the use of bricks for most of the construction of the dome in place of the heavy stone which was usually used for construction of cathedrals. In fact except for the dome, this cathedral (Santa Maria del Fiore) was otherwise made of stone. To further cut the weight of the dome, the bricks used for the inner dome were special low weight bricks, while those used for the outer dome were the more normal weight brick (still lighter than stone) for better resistance to the weather.
As shown in Brunelleschi's sketch at the right, the bricks in the inner dome were inclined at steeper and steeper angles sloping inwards as one progresses up the dome, to best resist the weight of the structure above. The bricks of the outer dome were laid at a fixed angle to slope away from the interior to best shed rain water.
No centering: The ceilings and arches of most cathedrals were built using temporary wooden structures called "centering" that would support the stone work until the complete arch and supporting structures were completed. It was a huge event to remove this temporary structure and hope the completed arch or vaulted ceiling would not collapse killing all the workmen doing the removal. In the case of Brunelleshi's dome there was no obvious way to provide the massive amount of timber required for the centering, at least at a reasonable cost. Furthermore, the danger of removing the centering on such a massive structure was close to unthinkable. Brunelleshi won the contract to build the dome by convincing the town fathers that he could do the job without centering.
Photo of the herringbone brick work visible in the space between the domes.
Brunelleschi's sketch of the herringbone pattern looking down at the dome from above. The brick size is greatly exaggerated. This sketch also falsely suggests that the dome is curved all the way around, when in fact it was octagonal with eight straight sides.
Top edge of bricks being laid in a herringbone pattern. The upright bricks from the previous course stick up and provide secure points for the new course of bricks (shown in white). This upright bricks also breaks up the circumference so that each section could be laid and stabilized independently. This illustration does not show the other layers of bricks required for the 5 to 7 foot thick wall. These other layers would be laid along with the inner layer (shown here), with perhaps the inner layer being several courses ahead of the other layers (which perhaps were also staggered as per the masons' convenience).
Herringbone brick pattern: Brunelleschi's novel herringbone brick pattern made possible the laying of the upper level bricks without centering or temporary bracing. At these higher levels the dome curved inwards making it hard to keep the newly laid bricks with their wet mortar from slipping off and falling to the cathedral floor 300 feet below. At each level the vertical bricks in the herringbone pattern stuck up and provided secure points between which to set the new bricks. The workers, perhaps with wooden clamps, could hold the bricks between each set of vertical bricks until the mortar began to set up. The mortar was allowed to thoroughly dry for about a week to secure the newly laid bricks before the next course of brick was laid. The upright bricks also separated the circumference into short 3 foot sections allowing bricklayers at different parts of the circumference to progress at slightly different rates. These breaks made the whole, layer-by-layer laying much more forgiving.
This herringbone pattern also bonded the layers of bricks together making a more stable interior plastering surface.
↑ Possible dome layout in the riverbank sand. The dome's span (from octagonal corner to opposite octagonal corner) would be laid out by a line (in blue) from point a to point b. The perpendicular bisector d is drawn along crossing arcs c. This perpendicular bisector is the axis of the dome. Points e are laid out along the span line 1/5 the span's distance to either end (points a or b). Points e are the center of the dome's vertical arcs f (in green). The x and y coordinates of the red reference points can be measured along the green arcs by measuring the distance from each to the span line (to get the y coordinate) and the distance each is from the axis d (to give the x coordinate).
↑ One method of determining the correct template's position that would have been available to the workmen at the time. "dtc" means distance-to-center. The height measurement would be the distance from the template tip to the cathedral floor with the distance from the dome's base to the floor subtracted off.
Maintaining the correct curve: The workmen used several methods to keep their structure in agreement with the design as they were laying the bricks. First of all, one of the advantages of this octagon shape over a hemisphere is that only the eight corners (or "seams") of the octagonal dome are difficult to keep in line. The bricks in between each pair of corners lay along a straight line and can be kept true with a simple taut string (the same method brick layers use today). Then (and now) the corners are built up a few courses higher than the sides, then a string is stretched between the corners and the bricks are carefully laid along this string. It is also possible that the workmen instead stretched the string from the templates (discussed in the next paragraph) carefully positioned and secured at the corners, instead of from brick corners.
How to keep the corner arches true? Brunelleshi had a half mile of riverbank smoothed so he could lay out details of his design. From this work on the riverbank, a number of 8.5 feet long by 2 foot wide curved wooden and metal templates were fashioned. Records could have been made of the x and y coordinates of the corner arches at a number of reference points. The templates were secured between the inner and outer domes as they began to be laid, next to the corners and moved up as the domes progressed to serve as guides to allow workers to keep the correct corner curvature. In setting a template a workmen could hang a plumb bob from the end of the template to the cathedral floor to measure the height of the end and the distance the template was from the desired center of the dome and compare this with the plan. They could also check that the plumb bob pointed to the correct radial line on the cathedral floor. Most likely Brunelleschi used specialists (surveyors of old) to do the above measurements. Their job would include keeping the templates accurately positioned and secured each time they were moved so that the masons would lay bricks on the proper curve. Their professional reputation would hang on their keeping the cathedral's dome true to the design.
The stairs: Four sets of stairways were built into the design of the dome. These mostly went through the space between the domes and occasionally cut through the ribs as shown here. They provided a very functional way to get men and materials up to the building site. Two of the stairways were used for workers climbing up and two for going down. Today they are used for tourists wanting to climb to the top of the dome and see Florence below them. Youtube has videos of the climb, for example this one.
Taccola's drawing of Brunelleschi's lifting device, shown here with a single horse instead of an oxen team. The screw at the base raises and lowers the vertical shaft to select raising or lowering for the crane direction without needing to reverse the animals.
The lifting machines: One of Brunelleschi's achievements was the successful invention of several lifting machines, needed to lift the thousands of tons of material hundreds of feet upwards to be incorporated into the dome. One of the cranes was powered by a pair of oxen turning a horizontal gear (much like a capstan on a ship), replacing men who had powered earlier similar machines. Another machine invented by Brunelleschi was located high on the construction site and was used to precisely position heavy loads such as the blocks for the chains. It resembles modern day tower cranes used to build most of today's skyscrapers. This website has drawings of two of the lifting machines.
2. The construction
Today, workmen would probably construct a steel skeleton and then add the outer and inner skins. Because of the lack of modern steel and the large dimensions of the dome, Brunelleschi used a different approach. Brunelleschi had the workmen lay the chains, the ribs and both domes all at the same time, progressing vertically layer-by-layer upwards. Workmen were able to add about 1 course of bricks (one brick thickness across the width of the inner and outer domes) a week and progress about a foot in height every month. The dome was finished in 16 years, interrupted at times by wars and the plague. Much of the labor went into getting the material up to the height of the raw edge of the dome. With the inner dome being of a comfortable thickness, 5 to 7 feet thick, workmen could uses the raw upper edge as a platform on which to stand and work. They may have used planks and platforms to bridge the gap to the outer dome where necessary. The illustrations below show how the dome may have progressed at various stages of the construction. The "lantern" at the top was added last and finished after Brunelleschi's death. An outside covering of tile was added centuries later.
Cross sections of the dome at various stages of completion. The dome progressed vertical layer by vertical layer including the embedded chains, ribs, stairways and other details. The completed dome is shown in a very light gray.
I used Ross King's well written book Brunelleschi's Dome: How a Renaissance Genius Reinvented Architecture as the starting reference. Also, the internet is full of articles written about Brunelleschi and his dome. In addition to the internet links in the text above, here I list two more that I found to most useful to understanding the structural details.
Nice animation about the building of the dome: here.
Lots of images of structural details, on pages 22-38: here.
Waves, Berkeley Physics Course - vol. 3, Frank S. Crawford, Jr. McGraw-Hill 1965. This book is suitable for an add-on to an introductory course on college or university physics. It discusses all sorts of aspects of waves and has a multitude of home experiments. One could probably make a great science fair project from one of them. As to its math level, it mostly uses algebra, with some calculus in the mix.
Physics of waves, by Elmore and Heald, originally published by McGraw-Hill in 1969, but currently published by Dover. This book covers many different wave systems, such as waves on a string, on a membrane, in solids, in fluids, on a liquid surface, and electromagnetic waves. It also covers the many aspects of waves. It has an excellent chapter on diffraction.
The Feynman lectures on physics, Feynman, Leighton, and Sands, Addison-Wesley 1963. Three volumes. These cover many aspects of physics. They are perhaps best suited for someone who has made it through an introductory sequence in college or university physics, and wants to read about the subject from a more sophisticated point of view. They are not particularly math intensive, more just into discussing concepts with some math as required. These are books you read to understand a physicist's mind. Perhaps 10% to 20% of the chapters are about waves and resonances.
Electromagnetic books that I use:
Engineering Electromagnetics, Hayt (with Buck on more recent editions), McGraw-Hill. An easy to read, compact junior-level text for electrical engineering students.
Fields and waves in communication electronics, Ramo, Whinnery, Van Duzer, Wiley. A upper level/graduate level text for electrical engineering student. Covers practically every aspect of applied electromagnetic fields in some depth. Is not a book to sit down and read for philosophy, but rather to look up the rational behind certain devices or design methods.