page_title: Tide-predicting machine
section_title: Principle
hierarchical_section_title: Tide-predicting machine / Principle
caption_attribution_description: English: Scan made by uploader from copy of Thomson's design sketch for the third tide-predicting machine, 1879-81, as published in Proceedings of the Institution of Civil Engineers, vol.65, 1881 --- too old to be in copyright.
context_page_description: A tide-predicting machine was a special-purpose mechanical analog computer of the late 19th and early 20th centuries, constructed and set up to predict the ebb and flow of sea tides and the irregular variations in their heights – which change in mixtures of rhythms, that never repeat themselves exactly. Its purpose was to shorten the laborious and error-prone computations of tide-prediction. Such machines usually provided predictions valid from hour to hour and day to day for a year or more ahead.
  The first tide-predicting machine, designed and built in 1872-3, and followed by two larger machines on similar principles in 1876 and 1879, was conceived by Sir William Thomson. Thomson had introduced the method of harmonic analysis of tidal patterns in the 1860s and the first machine was designed by Thomson with the collaboration of Edward Roberts, and of Alexander Légé, who constructed it.
  In the US, another tide-predicting machine on a different pattern was designed by William Ferrel and built in 1881-2. Developments and improvements continued in the UK, US and Germany through the first half of the 20th century.
context_section_description: Thomson conceived his aim as to construct a mechanism that would evaluate this trigonometrical sum physically, e.g. as the vertical position of a pen that could then plot a curve on a moving band of paper.
  There were several mechanisms available to him for converting rotary motion into sinusoidal motion.  One of them is shown in the schematic (right).  A rotating drive-wheel 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 can make the shaft move up and down within limits.  This arrangement shows that 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 , can then be expressed as
  , where  is the radial distance from the wheel's center to the peg,  is the rate at which the wheel turns (in radians per unit of time), and  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.
  This arrangement makes a physical analog of just one trigonometrical term.  Thomson needed to construct a physical sum of many such terms.
  At first he inclined to use gears.  Then he discussed the problem with engineer Beauchamp Tower before the British Association meeting in 1872, and Tower suggested the use of a device that (as he remembered) was once used by Wheatstone. It was a chain running alternately over and under a sequence of pulleys on movable shafts.  The chain was fixed at one end, and the other (free) end was weighted to keep it taut. As each shaft moved up or down it would take up or release a corresponding length of the chain. The movements in position of the free (movable) end of the chain represented the sum of the movements of the different shafts.  The movable end was kept taut, and fitted with a pen and a moving band of paper on which the pen plotted a tidal curve.  In some designs, the movable end of the line was connected instead to a dial and scale from which tidal heights could be read off.
  One of Thomson's designs for the calculating part of a tide-predicting machine is shown in the figure (right), closely similar to the third machine of 1879-81. A long cord, with one end held fixed, passed vertically upwards and over a first upper pulley, then vertically downwards and under the next, and so on. These pulleys were all moved up and down by cranks, and each pulley took in or let out cord according to the direction in which it moved. These cranks were all moved by trains of wheels gearing into the wheels fixed on a drive shaft. The greatest number of teeth on any wheel was 802 engaging with another of 423. All the other wheels had comparatively small numbers of teeth. A flywheel of great inertia enabled the operator to turn the machine fast, without jerking the pulleys, and so to run off a year's curve in about twenty-five minutes. The machine shown in the figure was arranged for fifteen constituents in all.
  Thomson acknowledged that the use of an over-and-under arrangement of the flexible line that summed the motion components was suggested to him in August 1872 by engineer Beauchamp Tower.
caption: Thomson's design for the third tide-predicting machine, 1879-81
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