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816:" approach for mechanically computing cosines as Thomson used, shown in the schematic (right). A rotating drive-wheel ("crank") is fitted with an off-center peg. A shaft with a horizontally-slotted section is free to move vertically up and down. The wheel's off-center peg is located in the slot. As a result, when the peg moves around with the wheel, it makes the shaft move up and down within limits. As a result of this arrangement when the drive-wheel rotates uniformly, say clockwise, the shaft moves sinusoidally up and down. The vertical position of the center of the slot, at any time
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traces a sinusoid. Each slotted yoke crank is connected by a shaft to a pulley, which causes the pulley to follow the sinusoidal motion. A chain going over and under pulleys sums each of their deflections to compute the tide. Along the top of the photo, connecting shafts drive slotted yoke cranks on both sides of the machine. One side of the machine computes the height of tides while the other determines the times of high and low tides.
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for the 1912 and 1913 tide tables. Then the machine was disassembled, polished, plated, lacquered, and reassembled in time to provide predictions for the 1914 tide tables. Comparisons of the accuracy of the mechanical predictions of tides compared to hand calculations for two challenging locations demonstrated errors in heights of 0.72 inches (1.83 cm) or less.
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Assuming the factors for a location are known, configuring the machine to compute tides for the location requires 2.5–4 hours. Predictions for a year’s tides at that location can then be produced in 8–15 hours. The calculations Tide-Predicting
Machine No. 2 can perform in 1 day would require a person
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A tide formula component crank on Tide-Predicting
Machine No. 2. The mechanical arrangement (a slotted crank yoke) converts circular motion to a vertical motion that traces a sinusoid. The operator adjusts the position of the pin on the crank to represent a component of the tide formula at a specific
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Tide-Predicting
Machine No. 2 was the first tide-predicting machine to incorporate both a paper graph of the tides–the approach used by earlier British machines–and dials and scales that showed the tide height and corresponding date and time–used by Tide-Predicting Machine No. 1. The dials and scales
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The slotted yoke cranks at the top and bottom (with the triangular pieces) move vertically in a sinusoidal pattern. The locations of their pins determine their amplitudes and phases, representing factors in the tide equation. The pulleys across the center move with the attached cranks. The summation
233:
Other work in the Coast and
Geodetic Survey took precedence over construction of the new machine, and a reduction in staff levels precluded all work on the new machine for three years. As a result, Tide-Predicting Machine No. 2 was not functional until 1910. It was first applied to predicting values
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Much consideration was given to the mechanical characteristics of the components to ensure reliability and accuracy. For instance, some components that were hard to replace were designed with a 50-year lifetime. Also, the summation chains were moved across gears under tension for a year of work days
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One side of Old Brass Brains is used to compute the height of the tide. A similar arrangement of components on the other side, but with cranks 90 degrees out of phase, represents the derivative with respect to time of the tide height formula. When the derivative is zero the time of high or low tide
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is the starting phase angle of the peg, measured in radians from the 12 o'clock position to the angular position where the peg was at time zero. An operator adjusted the location of each pin based on the empirically computed parameters for a port’s tides. This arrangement makes a physical analog of
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In 1895 the Coast and
Geodetic Survey grew concerned because Tide-Predicting Machine No. 1 had developed considerable wear from almost constant use over 12 years. The office decided to construct a new machine that was faster, more accurate, and more reliable. This became Tide-Predicting Machine No.
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started printing annual tide tables to support safe and effective maritime, coastal, and defense activities. Before long, these tables showed the times and heights of high and low tides to the nearest minute and tenth of a foot, respectively. Tables were printed for a year at a time and distributed
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This photo shows both the top and one side of Tide
Predicting Machine No. 2. The tide prediction formula implemented by the machine includes the addition of a series of cosine terms. The triangular metal pieces are part of slotted yoke cranks which convert circular motion to a vertical motion that
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The Coast and
Geodetic Survey designers also adopted from the earlier British machines the approach of summing terms by passing a chain over and under pulleys attached to the vertically oscillating yokes. The amount of chain remaining after passing over and under pulleys indicated the sum of the
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To compute tides for a coastal location, the operator has to configure the machine for that location. This is done by adjusting physical settings on the machine based on up to 37 factors. Those factors are determined empirically by harmonic analysis of a time series of tides at the location, and
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Around 1960 Old Brass Brains was modified to replace the hand crank with an electric motor and to add an automatic readout of heights and times. In 1965, the Coast and
Geodetic Survey retired the Tide-Predicting Machine No. 2, 55 years after it entered service, and started performing its tide
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and incorporated their best attributes in the design of the new machine. The machine, also known as “Old Brass Brains”, used an intricate arrangement of gears, pulleys, slides, and other components. The design of the new machine was approved in 1895, and construction began in 1896.
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made it much easier for an operator to precisely determine the height and time of high and low tides. The paper graph, referred to as a tide curve, was very useful as a record of the computation that could be checked later to confirm the calculations were performed correctly.
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the Coast and
Geodetic Survey produced annual tide tables for major ports four years in advance in case Old Brass Brains broke down or was sabotaged. The Coast and Geodetic Survey also provided tide predictions for a number of additional locations in the
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This photo shows the largest of the three sections of Tide-Predicting
Machine No. 2. The gears on the left transmit power from the hand crank. The components on the right contribute to the computation of the time of high and low
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of tide heights. This analysis can be performed with a record as short as two weeks but a 369-day sample is standard. The longer sample minimizes the errors introduced by wind storms, freshets, and other non-regular influences.
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represent the influence of the moon, sun, depth of bay, offshore islands, etc. Once computed the factors for a location can be applied to past and future years. and are shared widely so anyone can perform tide calculations.
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of the Coast and Geodetic Survey designed a tide-predicting machine. Fauth & Co. Instrument Makers built Tide-Predicting Machine No. 1 and delivered it in 1882. The Survey started using the machine routinely in 1883.
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attempt to account for these factors but lead to complex calculations. Originally, calculations were performed by hand, which was very labor-intensive and error-prone. The burden became even larger when the
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considerations, which were determined by Thomson and Darwin and have been almost universally used. For instance one speed represents the speed of a theoretical moon with a uniform speed in a circular
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terms. For example, a large value for a term would move its pulley further from a neutral position, deflecting the chain, and reducing the amount of excess chain remaining in system.
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The pulleys move up and down in a sinusoidal motion, representing components of the tide equation. The summation chain that passes above and below the pulleys sums their influences.
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are used to mark the start of hours and days on the paper graph and to stop the machine when high and low tides were reached so the operator can note the height and time.
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The prediction of tides is very challenging as it depends on multiple factors–including the alignment of the Sun and Moon, the shape of the coastline, and near-shore
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Old Brass Brains is 10.8 feet (3.3 m) long, 6.2 feet (1.9 m) high, 2 feet (0.6 m) wide and weighs approximately 2,500 pounds (1,134 kg).
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is the relative phase of the term. This is the equation computed by most tide-predicting machines, including Old Brass Brains which handles 37 such terms.
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Internals: the machine uses an intricate arrangement of gears, pulleys, chains, slides, and other mechanical components to perform the computations.
282:. Obtaining tide observations for those locations to support computation of the factors required for predictions was often a significant challenge.
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has been reached. An electrical circuit detects this condition and stops the machine so the operator can record the date, time, and tide height.
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508:{\displaystyle H_{0}+A_{1}\cos(\omega _{1}t+\varphi _{1})+A_{2}\cos(\omega _{2}t+\varphi _{2})+A_{3}\cos(\omega _{3}t+\varphi _{3})+\cdots }
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Tide Predicting Machine No. 2 is based on the first accurate mathematical approach for predicting tides, which was developed around 1867 by
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Rolin Harris and E. G. Fischer of the Coast and Geodetic Survey led the effort. The design team studied previous British and U.S.
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for specific locations. The machine can perform tide calculations much faster than a person could do with pencil and paper. The
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before being installed in the machine to ensure they were sufficiently flexible and their length would remain constant.
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1187:"A Great Brass Brain: A Unique Engine, on the Accuracy of Which Depend Millions of Dollars and Thousands of Lives"
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Tides are the rise and fall of sea levels caused by the combined effects of gravitational forces exerted by the
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coastal port before starting computations. The pin position affects the amplitude and phase of the sinusoid.
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298:(NOAA) maintains Tide-Predicting Machine No. 2 in working condition. The machine is at NOAA’s facility in
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The Power of the Sea: Tsunamis, Storm Surges, Rogue Waves, and Our Quest to Predict Disasters
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put the machine into operation in 1910. It was used until 1965, when it was replaced by an
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The operator reads the height and time of high and low tides from these dials and scales.
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To compute those terms the Coast and Geodetic Survey designers incorporated the same "
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turned by the operator provides the power for the machine’s mechanical calculations.
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Description of the U.S. Coast and geodetic survey tide-predicting machine, Issue 2
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10.8 feet (3.3 m) long, 6.2 feet (1.9 m) high, 2 feet (0.6 m) wide
49:, and other mechanical components to compute the height and time of high and low
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169:(as the U.S. Coast Survey was renamed in 1878) started using the more accurate
1147:"The Tide Prediction Centenary of the United States Coast and Geodetic Survey"
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just one term in the tide equation. Old Brass Brains computes 37 such terms.
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Around 1915, the machine was used to produce annual tide tables for 70 major
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To significantly reduce the work required to predict tides, in 1881
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of the term’s contribution to tide height above mean sea level,
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1277:"The Coast and Geodetic Survey Tide Predicting Machine No. 2"
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is the radial distance from the wheel's center to the peg,
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chain above and below the pulleys sums their influences.
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worldwide. Additional ports were added in later years.
1314:"Tide Predicting Machines - NOAA Tides & Currents"
1423:
American Mathematical Society/Bill Casselman (2009),
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Feature Column: Monthly Essays on Mathematical Topics
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mechanism for generating sinusoidal motion component
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894:{\displaystyle A_{1}\cos(\omega _{1}t+\phi _{1})}
1490:National Oceanic and Atmospheric Administration
1113:National Oceanic and Atmospheric Administration
549:is the height of mean sea level. For each term
296:National Oceanic and Atmospheric Administration
1416:American Mathematical Society/Tony Phillips,
8:
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1431:Operator with Tide-Predicting Machine No. 2
286:calculations with an electronic computer.
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19:For broader coverage of this topic, see
1241:. U.S. Coast and Geodetic Survey. 1915.
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167:United States Coast and Geodetic Survey
93:United States Coast and Geodetic Survey
55:United States Coast and Geodetic Survey
1203:10.1038/scientificamerican03071914-197
645:determines the frequency of the term,
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1385:"Fourier Analysis of Ocean Tides II"
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1077:"The fall of 'Old Brass Brains'".
173:for predictions of tides in 1884.
132:Early U.S. tide-prediction efforts
14:
1418:Fourier Analysis of Ocean Tides I
1151:International Hydrographic Review
326:for sea height is represented as
1439:"The tide predictions for D-Day"
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153:prior to the start of the year.
1185:Claudy, C. H. (March 7, 1914).
518:containing 10, 20 or even more
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1500:Silver Spring, Maryland (CDP)
1393:American Mathematical Society
580:{\displaystyle i=1,\ldots ,n}
259:125 days to perform by hand.
76:Tide Predicting Machine No. 2
65:Tide-Predicting Machine No. 2
27:Tide-Predicting Machine No. 2
117:2,500 pounds (1,134 kg)
1275:Fischer, E. G. (May 1912).
948:{\displaystyle \omega _{1}}
836:, can then be expressed as
715:{\displaystyle \omega _{i}}
638:{\displaystyle \omega _{i}}
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150:United States Coast Survey
18:
16:Mechanical analog computer
1341:. Macmillan. p. 39.
979:{\displaystyle \phi _{1}}
785:{\displaystyle \phi _{i}}
685:{\displaystyle \phi _{i}}
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195:tide-predicting machines
33:, was a special-purpose
1318:www.co-ops.nos.noaa.gov
300:Silver Spring, Maryland
21:Tide-predicting machine
1515:Mechanical calculators
1437:Parker, Bruce (2011).
1337:Parker, Bruce (2012).
1145:Hicks, Steacy (1967).
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1455:2011PhT....64i..35P
1293:1912PA.....20..269F
1191:Scientific American
1079:Product Engineering
738:. The coefficients
312:Sir William Thomson
215:electrical circuits
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59:electronic computer
35:mechanical computer
1106:"Old Brass Brains"
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1363:"NOAA Open House"
1348:978-0-230-12074-7
1281:Popular Astronomy
829:{\displaystyle t}
665:is the time, and
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316:Sir George Darwin
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106:Discontinued
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1197:: 197–198.
798:time series
1510:Navigation
1484:Categories
1323:2016-05-01
1064:References
158:bathymetry
122:Dimensions
98:Introduced
37:that uses
1473:0031-9228
1157:: 121–131
968:ϕ
937:ω
880:ϕ
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627:ω
616:amplitude
569:…
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488:φ
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320:frequency
213:-powered
901:, where
1451:Bibcode
1289:Bibcode
1008:Gallery
957:radians
730:in the
614:is the
522:terms.
324:formula
276:Pacific
269:During
211:Battery
205:A hand
43:pulleys
1471:
1398:May 8,
1368:May 1,
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1298:May 1,
1208:May 1,
1161:May 1,
1119:May 1,
322:. The
225:tides.
144:, and
114:Weight
47:chains
1505:Tides
1109:(PDF)
796:of a
728:orbit
264:ports
207:crank
51:tides
39:gears
1469:ISSN
1400:2016
1370:2016
1343:ISBN
1300:2016
1210:2016
1163:2016
1155:XLIV
1121:2016
765:and
736:Moon
294:The
138:Moon
109:1965
101:1910
85:1895
1459:doi
1199:doi
1195:110
854:cos
462:cos
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241:Use
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142:Sun
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