Computing linkages

Mechanical calculators based on hinged bars, instead of gears. And their applications.

Analog calculating machines usually contain lots of gears (differentials), cams, ball-and-disc integrators and rack-and-pinions. But would it be possible to construct such calculating machines only using hinged rods? In the first instance, one would think only linear functions could be represented by such a mechanism but that is not true. This presentation describes “computing linkages” and the work of Antonín Svoboda on their systematic development.

Introduction

Analog calculating machines can immediately present the results of a calculation on a continuous scale, in contrast to digital calculators that perform the calculation step by step, and present them in a “rounded-off” fashion. Unlike slide rules, some analog calculating machines can be integrated in a mechanical sensor-actuator system, so there is no need for a human hand to set up the calculation and read the result. Most mechanical analog calculating machines contain gears, cams and cam followers, rack and pinions, and ball-and-disk integrators. Some of them contain mechanisms purely consisting of linked rods, which we will call bar-linkage-mechanisms. Bar-linkage mechanisms can be used as “stand alone” calculators as well. Pure bar linkages mechanisms are only built from solid bodies that are hinged to each other or to a fixed base. In practical applications, these mechanisms also contain tracks or curved slots along which a pivot of a bar can slide.

Bar-linkage mechanisms have attracted the attention of mathematicians and engineers for centuries, mainly for their kinematic properties. People like James Watt, Charles-Nicolas Peaucellier and Pafnuty Chebyshev designed linkages for linear movements (figure 1). Watt’s mechanism approximated a straight line, but Peaucelliers en Chebyshevs mechanisms gave an exact straight line segment. Bar linkages have also been designed to draw a variety of curves, including ellipses.

The advantages of bar-linkage computers relative to “geared” computers are:[1]

- Bar-linkages require less space
- Bar-linkages have less friction
- Bar-linkages have small inertia
- Bar-linkages have a stable performance
- It is possible to approximate some complicated functions with a simple bar linkage
- Bar-linkages are easy to combine into complex systems
- Bar-linkages are cheap

Disadvantages are:

- Bar-linkages usually have a structural error
- Not all mathematical functions can be represented by bar-linkages
- Complexity of bar-linkages increases with increasing accuracy
- Bar-linkages are difficult to design
- The travel of the bar-linkage mechanism is limited, which affects its use as part of an automated system. Mechanical errors, like backlash and elasticity, should be reduced by careful construction

Adding

A bar-linkage mechanism can be used as an adder, for instance by a construction similar to the pole attachments used for a team of horses. This mechanism has been applied in a machine for measuring the surface area of leather (figure 2).

Adding, multiplying and dividing

In a balance computer patented by L.W. Imm[3] a large number of bar linkages is used to add the weights of loads in different compartments of an aircraft, and to calculate the combined center of mass. The weights are represented by sliding bars that are set through a rack and pinion by turning a knob (figure 3). This briefcase-size device was made by the Librascope Company for a.o. the Lockheed PV-1, Douglas DC-3 and Lockheed "14". Imm also developed a bar-linkage computer for aircraft power vs. fuel consumption.[4] A standalone bar-linkage mechanism to solve linear equations was patented[5] by Arnold Spilker. The mechanism shown in figure 4 solves two linear equations in 2 unknowns

a11•x1 + a12•x1 = ß1

a21•x2 + a22•x2 = ß2

but can be easily extended for a larger number of equations and unknowns. The coefficients aij of these equations are set by adjusting the position of the “central” pivot of a bar, thus dividing the bar in two sections. After the pivots are fixed on the bars, the pivots can move on a horizontal slide. The mechanism has to be adjusted for each new set of aij’s and it would be difficult to incorporate it in an automated system which varies aij’s.

Figure 1: Linear movement mechanisms of Watt, Peaucellier and Chebyshev.

(Click links for an animation)

Figure 2: Leather measuring machine[2]

Figure 3: The center-of-mass section of Imm's balance computer

Figure 4: Spilker's linear equation solver

Nomograms

If we allow the result scales of the computer to be non-linear, a bar-linkage can mechanise a nomogram. A pretty example is the Posograph (figure 5) patented[6] by Auguste-Robert Kaufmann in 1922. This device represents a relation between 6 inputs and 1 output variable, and was made in different versions for photographic and cinematographic calculations. Its mechanism, which was proudly displayed in its instruction book, is shown in figure 6. Note that most of the scales are categorical. The physics behind them is non-linear.

The Bloch Schnellrechner (figure 7), which also mechanizes a nomogram with non-linear scales,[7] is strictly speaking not a bar-linkage calculator because it uses rods provided with slots in which the connecting pins slide, instead of fixed hinges.

Consul, the Educated Monkey (figure 8), and similar educational toys, are pure linkages. The central slotted bar is not mathematically necessary.[8] For simplicity, the monkey has discrete “scales”, but the result could be represented by a grid-like graph if we discard the “square number” option at the far right.

Figure 7: Bloch's Schnellrechner

Figure 5: The Posograph

Figure 6: The Posograph mechanism

Figure 8: Consul, the Educated Monkey (photo Stefan Drechsler)

Antonín Svoboda

During the Second World War, Antonín Svoboda developed methods for designing bar linkage computers, which were published after the war in the ultimate book on linkage calculators: “Computing mechanisms and linkages”. Svoboda was born on October 14, 1907 in Prague.[9] He earned a degree in electrical engineering from the Czech Institute of Technology in 1931, and a PhD in 1936. During his service in the Czech army the following two years, he helped to design an analog anti-aircraft gun fire control system[10] based on a differential analyser.[9] At the beginning of the Second World War he moved to Paris with his colleague Vladimir Vand[11] and worked for SAGEM (Société d'Application General d'Electricité Mécanique). Later, in 1941, he went to the USA, at first working for the ABAX Corporation in New York, designing an anti-aircraft control for the 40 mm Bofors gun.[10,12] In 1943 he started working at the Radiation Laboratory of the Massachusetts Institute of Technology. The Radiation Laboratory was established in 1940 to develop radar systems. By the end of the war, it had 3900 employees.[13]

Svoboda was again involved in the development of gun fire control systems, and contributed to the Mark 42 ballistics computer of the Mark 56 anti-aircraft defense system.[10] The Mark 56 consisted of a radar set and a huge amount of electronic and electromechanical controlling and computing units, among which two Mark 42 ballistics computers, to allow for two types of guns to be aimed at the same target. The Mark 42 had five mechani-cal inputs, one by hand (initial velocity) and the others by servo’s. The results, projectile time of flight, superelevation, drift, range rate, and fuze time, were converted into electrical analog form by rotating potentiometers. The primary ballistics unit weighed 290 kg, the secondary one 250 kg.[14,15] The unit was produced by the Librascope Company. When the British adapted the Mark 56 for their own MRS 3 system, they replaced the Mark 42 computer by a “geared” electromechanical one.[15] After the war the Radiation Laboratory closed but some of the scientists were invited to contribute a volume to the “M.I.T. Radiation Laboratory Series”. Most books dealt with electronics, but the Svoboda’s contribution was purely mechanical: “Computing mechanisms and linkages”. In 1946 he returned to Czechoslovakia and worked on relay computers. He went back to the USA in 1964 and became professor of computer sciences at the University of California in Los Angeles. Svoboda died on May 18, 1980.

Figure 9: Mark 56 fire control system with two Mark 42 ballistics computers[14]

Figure 10: Svoboda and the ballistics computer[9]

Function generators

In his book, Svoboda describes various kinds of elementary bar-linkages: The ideal harmonic transformer has as input an angle Xi and as output a displacement Xk, with Xk = R sin Xi (figure 11). The non-ideal harmonic transformer has as input an angle Xi and as output a displacement Xk, with Xk = R sin Xi + E(Xi), where E(Xi) is a deviation from the harmonic transformation (figure 12). It is this deviation which can be used to make the harmonic transformer approximate another function in a limited domain, for instance Xk = R tan Xi for 0°<Xi<50°. Svoboda gives extensive tables to help fitting a harmonic transformer to the desired function. Note that in the ideal harmonic transformer (figure 11), a half-Peaucellier movement is used to achieve a pure parallel motion, an “infinite bar”, linked to the rotating bar whereas in the non-ideal harmonic transformer (figure 12) a short bar links the rotating bar to the slider.

Having different kinds of input and output (angle vs. displacement) can complicate the integration of this mechanism into a complex calculator. The solution is to add another linkage. The double harmonic transformer (figure 13) transforms a displacement into another displacement. Again, one can distinguish ideal and non-ideal harmonic transformers. For the design of such a transformer, the tables for the single harmonic are used in combination with graphs to find matching pairs of transformers for a given function. The design process is iterative, and convergence is not guaranteed. A three-bar linkage (figure 14) has an angular input and output. Svoboda presents two methods to design such linkages: a nomographic one, using a single nomogram for all purposes, and a geometrical one, using two charts that have to be drawn from scratch for each problem. Using a double three-bar linkage, Svoboda was able to make a logarithm generator with evenly spaced input and output scales (figure 15) for 1 = Xi = 50 with a maximum error of 0.003.[16]

Figure 11: Ideal harmonic transformer

Figure 12: Non-ideal harmonic transformer

Figure 13: Ideal double harmonic transformer

Figure 14: Three-bar linkage (aka four-bar linkage)

Figure 15: Double 3-bar linkage for log(x)

Figure 16: Multiplier: Xi = Xj x Xk

Bar linkages with two inputs

A common example of a bar linkage with two inputs is the multiplier. Pure bar linkage mechanisms cannot perform exact multiplications with two variable multiplicands, but Svoboda presents a multiplier that is pretty accurate and can handle positive and negative multiplicands (figure 16). In his design method, Svoboda starts with a contour-graph of the output vs. the inputs. Then he performs a geometrical transformation that approximates the two-dimensional contours by a single output scale. Usually, the resulting input and output scales are curved. Another transformation, mapping the two input scales upon each other, could result in the grid of the Educated Monkey. As an example of another two-input bar linkage Svoboda discusses the step-by-step design of a simple ballistics computer that calculates gun elevation from ground range and relative altitude of the target without aerodynamic corrections.

Tweaking

There are several methods to improve bar linkages by adding new types of constructions. One of them is replacing hinges by eccentric hinges (figure 17), which is typically done for the output link to minimize the structural error. Another method is the use of slots in which pivots slide. Figure 18 shows a multiplier in which the output scale is curved. This is an obvious disadvantage, as is the fact that one input scale is a line segment and the other a circle segment. By adding transformer linkages to the inputs and outputs one can get a calculator with linear scales, like the multiplier previously shown in figure 16.

Fire control computers

The use of bar-linkage mechanisms in complex calculators took off during the First World War and continued during the interbellum.[17] These calculators were part of gun fire control systems, especially for naval and anti-aircraft guns. Mechanical gun fire computers were still used in the 1970’s.[17] An early example is a torpedo director patented in 1893 by Walter Gordon Wilson[18] (figure 19). Wilson is well known as the designer of early British tanks.[19]

In 1918 he patented[20] a far more complicated fire control computer which combines a cam follower with bar linkages (figure 20). The cam follower and the upper linkages are used to calculate gun elevation from observed range and angle of sight. The three “square” linkages in the lower right part of the computer were used to calculate corrections for muzzle velocity, wind and air density. These corrections were added to the estimated elevation using the vertical bars at the far right. Linkages can also be found in the Vickers Predictor[17,21] and in the artillery calculator of Kurt Pannke[22] (figure 21).

Figure 17: Three-bar linkage with an excentric

Figure 18: Multiplier with curved scale

Figure 19: Wilson's torpedo director

Figure 20: Wilson's fire control calculator

Figure 21: Pannke's fire control calculator

Other Applications

Although mechanical analog computers were mainly used for military purposes,[17] there were some civilian uses. A bar linkage has been designed for measuring electrical resistance. It divides the measured voltage and current values.[23] A bar linkage has been proposed for the geometric summation of measured real and reactive electrical power to get apparent electrical power.[24] Bar linkages can be found in a protractor-like device to evaluate synchronous alternating current motors.[25] A differential flow meter with computing linkages has been produced by the Hagan Corp. of Pittsburgh[26] (figure 22).

Final remarks

Kinematic linkage design has a long history and still receives much attention, for instance in robotics. The design of computing linkages, however, once part of a “basic text and reference book” for the US Navy,[27] is now a forgotten art.

Figure 22: Hagan flow calculator

References

1. Antonín Svoboda, “Computing Mechanisms and Linkages”, Dover Publications, New York, 1965; originally published as: M.I.T. Radiation Laboratory Series Volume 27 (1946).
2. Picture: German Shoe-Museum Hauenstein, see also Robert W. Strout , “Machine for measuring the area of surfaces”, US Patent 738580, Sep. 8, 1903 (assigned to the Vaughn Machine Company) A variant of this mechanism in “polar coordinates” is the “fractal vise” patented bij Paulin K. Kunze and produced by Mantle & Co, New York
3. L.W. Imm, “Balance Computer”, US Patent 2179822, Nov. 14, 1939; US Patent 2373566, Apr 10, 1945 (assigned to Librascope Inc.); E. Churchill, “Automatic Brains”, Popular Aviation, Feb 1940.
4. L.W. Imm, “Power Computer”, US Patent 2394180, Feb 5, 1946; “Dress Rehearsals for Danger Aloft”, Popular Mechanics, Aug. 1940.
5. Arnold Spilker, “Vorrichtung zum Auflösen linearer Gleichungen”, German Patent 415286, Jun 17, 1925. The better known equation solver by Josef Nowak consisted of gears.
6. Auguste-Robert Kaufmann, “Appareil indiquant le temps de pose pour l'obtention de clichés photographiques”, French Patent 542107, Aug. 5, 1922.
7. Georg Bloch, “Rechentafel”, German Patent 347926, Jan. 27, 1922; Georg Bloch, “Rechentafel nach Patent 347926 mit in Schlitzen beweglichen Schiebern”, German Patent 361708, Oct. 18, 1922.
8. William Robertson, “Toy”, US Patent 1188490, June 27, 1916 and “Calculating device”, US Patent 1286112, Nov. 26, 1918 (assigned to the Educational Novelty Company, Dayton).
9. “Antonin Svoboda”, Engineering and Technology History Wiki.
10. Antonín Svoboda, OH 35. Oral history interview by Robina Mapstone, Nov. 15, 1979. Charles Babbage Institute, University of Minnesota, Minneapolis.
11. Vladimir Vand went to Glasgow, became an expert in crystallographic calculations, and worked a.o. with Francis Crick.
12. The Swedish Bofors 40 mm guns could also be equipped with English Kerrision Predictors, American Sperry and even the Dutch (well... German) Hazemeyer fire control units.
13. Wesley W. Stout, “The great detective”, Chrysler Corp, Detroit, 1946, p.66.
14. "Naval Ordnance and Gunnery, Volume 2, Fire Control", NavPers 10798-A, U. S. Government Printing Office, 1958, Chapter 26.
15. Norman Friedman, "The Naval Institute guide to world naval weapons systems", 1997-1998, page 59 and 350.
16. Antonín Svoboda, “Mechanism for use in computing apparatus”, US Patent 2340350, Feb. 1, 1944.
17. Allan G. Bromley, “British mechanical gunnery computers of world war II“, manuscript, Jan. 1984.
18. Walter Gordon Wilson, “Improvements in Directors for ‘Whitehead’ Torpedoes and the like”, British Patent 17656/1893, Nov. 11, 1893.
19. “Walter Gorden Wilson”, Wikipedia.
20. Walter Gordon Wilson, “Improved means for computing the elevation or deflection of guns and the setting of fuses”, British Patent 113136, Feb. 7, 1918.
21. Percy Willis Gray, “Improvements in or relating to apparatus for use in the fire control of anti-aircraft guns”, British Patent 236250, July 3, 1925 (assigned to Vickers Ltd.).
22. Kurt Pannke, “Delostrelecké pocitadlo denních vlivu” [artillery calculator], Czech Patent 24349, Feb. 25, 1928. It is unclear if this is part of the 50 kg T4 device mentioned in The Field Artillery Journal, Nov-Dec 1931. At the end of the First World War, Pannke designed an “Artillerie-Plan-Rechenmaschine” for Mercedes Büromaschinen- und Waffen-Werke. In the 1950’s Pannke manufactured a Fourier analyzer, especially for crystallography (W. Hoppe, K. Pannke, Z. Krist. 107 (1956) p.451).
23. “Elektrisches Messgerät zur Anzeige des Quotienten aus zwei elektrischen Grössen, insbesondere Quotientenrelais für den Schutz von elektrischen Leitungen”, Swiss Patent 144893, May 16, 1931 (assigned to Siemens-Schuckertwerke AG, Berlin)
24. Wilhelm Stäblein, “Einrichtung zur geometrischen Summierung von Meßgrössen, insbesondere zum Zwecke der Fernmessung von Scheinleistungen”, German Patent 566565, Dec. 19, 1932 (assigned to Allgemeine Elektricitäts-Gesellschaft, Berlin)
25. Vladimir Karapetoff, “Calculator”, US Patent 1648733, Nov. 8 ,1927 (assigned to General Electric)
26. Otto B. Vetter, “Calculating linkage”, US Patent 2527282, Oct. 24, 1950 (assigned to Hagan Corp.); Canadian department of Trade and Commerce, “Approval of pressure and temperature compensated recording ring balance flow meters model 3000, 3001, 3002, 3003, 3004”, Ottawa, 1960
27. “Basic Fire Control Mechanisms”, Ordnance Pamphlet 1140, 1944 (section 7)

This paper appeared in the Proceedings of the 16th International Meeting of Collectors of Calculating Instruments, Leiden, Sept. 17-18, 2010.

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