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Mustafa Umut Sarac EnduranceAbbrecher
Anmeldungsdatum: 19.09.2018 Beiträge: 82 Wohnort: Istanbul  Turkey

Verfasst am: 12.10.2018, 15:47 Titel: Mechanical Computer for Felix Hess Boomerang Trajectory 


Last night , I read about Nobel Prize winner Enrico Fermi and his invention Fermiac.
At 1946, neutron trajectory calculations are so difficult , hundreds of women were calculating such things day and night on mechanical calculators.
Fermi invented a notebook size trolley with a pen on its tip and when he changes the gears of the wheels , trolley was drawing the neutron path in nuclear reactor. Device is named as Fermiac or Fermi's Trolley.
I thought same can be made for boomerang trajectory.
I am an archaeologist and have no university mathematics and physics education but I can think such a analog mechanical computer for felix hess trajectory drawing, by if solidworks and matlab works together.
We would code matlab by felix hess boomerang trajectory computer program source code and matlab interacts with solidworks and evolutionary program to design such a mechanical computer.
When it is completed , you send the file to 3d printing house and final print and a pen draws boomerang trajectory on paper with changedvariables
I dont know if the differential would be needed to be calculated but there are many differential calculators made by MECCANO toy parts.
If you are smart and knowing what you are doing , you can go from meccano route.
I am after no electric and osciloscope using route but there are analog computers which uses electric and electronic.
I will post meccano publications list and important aircraft ww2 mechanical computer papers.
And I will upload analog electronic computers making design books. 

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Mustafa Umut Sarac EnduranceAbbrecher
Anmeldungsdatum: 19.09.2018 Beiträge: 82 Wohnort: Istanbul  Turkey

Verfasst am: 12.10.2018, 15:50 Titel: 


MECCANO IS A MECHANICAL TOY BRAND. STILL ACTIVE. THIS IS A LIST OF MECCANO ARTICLES DESCRIBES MECCANO DIFFERENTIAL ANALYZERS AND THEIR WORKING OR USE AT THE RESEARCH INSTITUTES
Adler, M., "Meccano Torque Amplifier", 2001 Online description of a demonstration torque amplifier using standard Meccano parts.
Amble, O., "On a principle of Connexion for Bush Integrators," Journal of Scientific Instruments, 23, 284287, 1946. A survey of regenerative connections of one or two integrators, by which it is possible to obtain functions such as the logarithm, square root, or any rational power.
Anon., "Sir William Thomson's Harmonic Analyser," Engineering, 30, 561, 1880. A description of a seven integrator version of William Thomson's harmonic analyser (see Thomson (1878)) constructed by R. W. Munro for the Meteorological Office. Contains a detailed engraving.
Anon., "With the Editor. Meccano Aids Scientific Research," Meccano Magazine, XIX, 6, 441, 1934. A one page introduction to a following article Anon. (1934B) discussing the Manchester Meccano differential analyzer and Bush's prototype at MIT.
Anon., "Machine Solves Mathematical Problems. A Wonderful Meccano Mechanism," Meccano Magazine, XIX, 6, 442444, 1934. A good article, with many excellent pictures of both the Manchester Meccano differential analyzer and Bush's prototype at MIT. A short extract from this is reproduced in Wright (1978).
Anon., "ServoMechanism Demonstrated Today by Professor Harold L. Hazen," The MIT Tech, LIV, 9, 1, 1934. Describes a servo mechanism designed to follow a curve on the differential analyzer input table. Can be found online here: P1, P4.
Anon., "A Diligent Machine," The Manchester Guardian, 25 Jan 1934, p 8. A curious short column commenting on the report Anon (1934E) later in the same issue.
Anon., "A Calculating Machine Working by Curves for Manchester University," The Manchester Guardian, 25 Jan 1934, pp 910. More than a full column devoted to the full scale analyzer being built for Manchester University, describing in layman's terms the kinds of applications it would be used for.
Anon. "Differential Analyser at Manchester University," Engineering, 140, 3268, 8892, 1935. A description of the MetropolitanVickers machine at Manchester. Includes good pictures including the digital revolution counters and time interval camera.
Anon., "Differential Analyser for the University of Manchester," Nature, 135, 535, 1935. A brief column reporting on the opening ceremony for the Manchester MetropolitanVickers machine.
Anon., "The Differential Analyser," The Engineer, 160, 4149, July 19 & 26, 1935. An excellent two part article describing the mechanical principles and construction of the differential analyzer.
Anon. "The Differential Analyser in Electrical Engineering," Nature, 143, 36, 1939. A review of the two papers Hartree (1938C) and Hartree (1938D).
Anon., "The ElectroMechanical Brain  MIT.'s Differential Analyzer Advances Science by Freeing it from PickandShovel Work of Mathematics," LIFE, Jan 14, 1946, pp 7376. A brief article with outstanding pictures describing the Rockefeller differential analyzer at MIT.
Anon., Annual Report of the University of Malaya 194950, pp 4445. Contains a brief mention of Prof. J. C. Cooke's Meccano differential analyzer; specifically that it then had three integrators, with a fourth planned.
Anon., "A Meccano Calculating Machine. Solving Complex Mathematical Equations," Meccano Magazine, XXXVI, 1, 11, 1951. A one page article describing the Meccano differential analyzer built by Prof. J. C. Cooke at the University of Malaya, Singapore.
Anon., "Mechanical Differential Analyser with SKF Bearings," The SKF Ball Bearing Journal, #3, 6772, 1953. Description of the Chalmers University DA, with excellent pictures. This machine is unusual in having the interconnect oriented vertically with integrators hanging below. The integrator design is also unconventional.
Anon., "Differential Analyser," The GMM Series of Modern Supermodels No. 4, London: The Chief Meccanoman, 1967. The first set of model building instructions published for a Meccano model of Bush's differential analyzer.
Anon., "Pioneer Computer Goes To Washington," USC University Bulletin, 26, 13, Jan 23, 1978.
Reports on the dismantling of the UCLA differential analyzer. It was sent to the Smithsonian, where it remains in storage.
Anon. "Among the Model Builders," Meccano Magazine, 58, 3, 5455, 1973. Reports on the rediscovery of the Cambridge Meccano differential analyzer in New Zealand at MOTAT. Includes a picture reproduced from the New Zealand Herald (Anon (1973B)). The text is somewhat confused as to the provenance of this machine.
Anon., "Toy Used to Build 'Brain Box' in 1930s," New Zealand Herald, 2 June, 1973. A brief report on the rediscovery of the Cambridge Meccano differential analyzer and its installation at MOTAT. Includes a good picture with Dr. Whale, which is reproduced in Anon (1973).
Anon., "Computer Display," Museum News, MOTAT, Sep. 1973. A brief report on the then new computer display at MOTAT, featuring the Cambridge Meccano differential analyzer.
Anon., "Among the Model Builders," Meccano Magazine, 59, 2, 36, 1974. A follow up article reproducing a letter from a Mr. Barcroft who was a laboratory assistant working under the direction of A Porter operating a Meccano differential analyzer at the Air Defence and Research Establishment, Malvern in 1942. It is unclear just which machine he refers to.
Anon. "New Computer Display,", Museum News, MOTAT, Sep. 1981. A brief report on the recently updated computer display at MOTAT. The article states that at this time the Meccano differential analyzer "is still capable of performing calculus."
Anon., "That Which was Lost has been Found," New Zealand Federation of Meccano Modellers Magazine, 17, 3, 1993. The editorial reproduces a picture from the New Zealand Herald of the Cambridge Meccano differential analyzer after it was rediscovered at MOTAT. There are brief notes plus a reproduction of another article from The Dominion of June 23, 1993.
Anon., "Meccano Differential Analyser No.2," New Zealand Federation of Meccano Modellers Magazine, 25, 1, 2001. Although no author is identified, these notes are edited from Tee (1993). See also Irwin (2001).
Ashurst, F. G., Pioneers of Computing. London: Frederick Muller, 1983. An excellent collection of short biographies. Chapter 7 is devoted to Vannevar Bush and the differential analyzers.
Ashdown, G. L., and Selig, K. L., "A General Purpose Differential Analyser Part 1  Description of Machine," Elliott Journal, 1, 2 4448, 1951. A six integrator machine using diskballcylinder integrators and magslip based servo followers. Incorporates photo electric curve followers on the input tables. See also Hersom (1952).
Asprey, W. (Editor), Computing Before Computers. Ames, Iowa: Iowa State University Press, 1990. Chapter five "Analog Computing Devices" contributed by Alan G. Bromley provides a good general survey of analog computation devices, from early planimiters, Kelvin's harmonic analyzer, differential analyzers, to modern electronic analog techniques. The full text is available online here.
Barton, J. C., Campbell, D. A., and Read, R. C., "An Analog Method for Studying Multiple Scattering," Proceedings of the Physical Society, LXX, 8A, 605614, 1957. This paper describes a simulation of multiple scattering conducted using a source of random motion "coupled to a differential analyser, built in Meccano, which is similar to one built by Hartree and Porter (1935)."
Beard, R. E. "The Differential Analyser" Royal College of Science Journal, 12, 127138, 1942. The text of a lecture delivered before the Society on 24 February 1942, describing the basic principles of the differential analyzer and including a picture of Beard's own machine.
Beard, R. E., "The Construction of a Small Scale Differential Analyser and its Application to the Calculation of Actuarial Functions," Journal of the Institute of Actuaries. LXXI (part II), 193227, 1942. A brief description of the machine, with a more extensive discussion on its application to actuarial work. A summary of the ensuing discussion after the paper was presented is also included.
Berends, T., "Historic Machine found by MOTAT," New Zealand Herald, 29 June, 1993. A report on the rediscovery of the Meccano differential analyzer at MOTAT. Pictured with the machine are Dr. H. Whale and R. Dearing, director of the museum, who states that it will be restored and displayed in a prime position in the museum.
Berry, T. M., "PolarizedLight Servo System," AIEE Transactions, 63, 4, 195198, 1944. Description of the photo electric follower system used on the General Electric differential analyzer (see Kuehni (1944)).
Blackett, P. M. S. and Williams, F. C., "An Automatic Curve Follower for the Differential Analyser," Proceedings of the Cambridge Philosophical Society, 35, 494505, 1939. This design uses a photoelectric slope detector, a mechanical arrangement to generate the tangent of an angle, plus an integrator to follow the slope of the curve more smoothly than with the simple servo arrangement of Hazen (1936).
Boerdijk, Ir. A. H., "Constructive Use of Friction in Torque Amplifiers and Constant Torque Devices," Constructor Quarterly, 4, 2425, June 1989. Describes the principles of the torque amplifier and related torque limiting devices.
Bowles, M. D., "U.S. Technological Enthusiasm and British Technological Skepticism in the Age of the Analog Brain," IEEE Annals of the History of Computing, 18, 4, 515, 1996.
This article is a comparative analysis of the British and U.S. differential analyzers from 1930 to 1945. The author examines the development of the Bush and Hartree analyzers in the context of the U.S. engineering community and the British scientific community. Includes many interesting details.
Bückner, H., The Differential Analyser, Gõttingen, undated.
This monograph develops a mathematical theory of setups with several free inputs both from a topological and an analytical point of view using the theory of Pfaffian systems. It was developed independently of Shannon (1941). It is very remote from the practicalities of actual machine operation.
Bush, V., "The Differential Analyzer. A New Machine for Solving Differential Equations," Journal of the Franklin Institute, 212, 447488, 1931. Bush's original paper giving a detailed account of the first differential analyzer built at MIT in 1930.
Bush, V., and Caldwell, S. H., "A New Type of Differential Analyzer," Journal of the Franklin Institute, 240, 255, 1945. A lengthy paper describing the very large scale second generation machine at MIT (the Rockefeller DA). This system made extensive use of shaft angle encoders and servo motors, allowing the mechanical integrators to be interconnected electrically through a matrix of telephone switching relays rather than though mechanical shafting, and programmed from punched paper tape.
Bush, V., Pieces of the Action, New York: William Morrow and Company, 1970. Bush's autobiography. Contains only brief mention of the differential analyzer work.
Cairns, W. J., Crank, J., and Lloyd, E. C., "Some Improvements in the Construction of a Small Scale Differential Analyser and a Review of Recent Applications," Armament Research Department Theoretical Research Memo. No. 27/44, 1944. UK National Archives reference DEFE 15/751 C20779. Describes improvements made to the Cambridge model to enhance reliability and usability: stronger output arms in the torque amplifiers, lighter integrator discs, clutches in the lead screw drives, and a double input table similar to Hartree's for time lag problems. Applications described include problems in heat flow, explosive detonations, and transmission line simulations.
Campbell, S. M. "Beatrice Helen Worsley: Canada's Female Computer Pioneer," IEEE Annals of the History of Computing, 25, 4, 5162, 2003. A short biography of Beatrice Worsley. Includes a brief mention of the Meccano differential analyzer she built over a 6 week period in the summer of 1948.
Cook, A. C. Differential Analyzer Manual, Schenectady: General Electric
A manual prepared for the use of General Electric Staff and prospective users of the analyzer. Chapter II is particularly useful as it sets out methods of assembling the integrators and other special units for a great variety of functions.
Cook, A. C., and Maginniss, F., J., "More Differential Analyzer Applications." General Electric Review, 52, 8, 1420, 1949. This paper is a follow up to Maginniss (1945), reporting on eight more engineering applications of the General Electric analyzer. The front cover of this issue has an excellent picture of the analyzer.
Cook, A. C., "Special Devices Aid Differential Analyzer Solution of Complex Problems," Transactions of AIEE, 69, 13651370, 1950.
Describes a range of special devices added to the GE differential analyzer. These include a curve follower, vector summation device, sinusoid generator, multiplier, and others.
Cossons, N. (ed.), Making of the Modern World, London: John Murray, 1992.
An illustrated volume highlighting 100 key inventions from the collections of the London Science Museum. Includes a beautiful picture of Hartree's full scale machine.
Cresswell, J., MOTAT: Museum of Transport and Technology of New Zealand (Inc.), Auckland, New Zealand: Hamlyn, 1976. Almost too late, a small group of enthusiasts joined together in an attempt to preserve the remaining relics of New Zealand's transport and engineering history. The result of their enterprise is the subject of this book. The Meccano differential analyzer is discussed on p102.
Croarken, M., Early Scientific Computing in Britain, Oxford: Oxford Science Publications, 1990. Chapter 5 is devoted to a discussion of the Manchester and Cambridge differential analyzers
Croarken, M., "The Emergence of Computing Science Research and Teaching at Cambridge, 19361949," IEEE Annals of the History of Computing, 14, 4, 1015, 1992.
This article describes the motivation behind the creation of the laboratory. It covers the period during which both the model and full scale differential analyzers were installed and operated.
Croarken, M., "Computing in Britain During World War II," IEE History of Technology Summer Meeting 6th July 2002, London, 2002. Includes a discussion of the use of the Manchester and Cambridge differential analyzers for military calculations. Includes a very extensive set of references.
Crank, J., The Differential Analyser, London: Longmans, 1947. An excellent introduction by the person in charge of operation of the full scale differential analyzer in the Mathematical Laboratory at the University of Cambridge. Includes many pictures and diagrams and a fairly detailed account of the construction of the Meccano machines.
Cundy, H. M., and Rollett, A. P,. Mathematical Models, Oxford: Oxford University Press, 1961. Brief description of the principles of Bush's differential analyzer. Mentions the possibility of Meccano construction.
Dalton, J., "Continuing the Saga of the Differential Analyser," Meccanoman's Newsmag, #68, 1994. Reports on an interview with Maurice Wilkes on the early history of the Cambridge Meccano differential analyzer.
Darwin, C. G., "Douglas Rayner Hartree 18971958," Biographical Memoirs of Fellows of the Royal Society, 4, 103116, 1958. An excellent short biography of Hartree. Includes a bibliography listing his published works.
Eames, C., and Eames, R. A., Computer Perspective, Cambridge, MA: Harvard University Press, 1973. This book is based on an exhibition conceived and assembled for IBM, displaying aspects of the intellectual and socioeconomic environments in the sixty years leading up to the modern computer. Page 119, titled "Meccano and Quantum Mechanics", is devoted to the Manchester Meccano differential analyzer.
Eyres, N. R., "Meccano in the Classroom," Mathematical Gazette, 54, 389, 282283, Oct 1970.
A short article describing a two integrator Meccano differential analyzer with no torque amplifiers, used for educational purposes.
Fail, R., "Electromechanical Servo for Differential Analyser," Meccanoman's Journal, #12, 310,1968. A rudimentary electromechanical servo design to replace a torque amplifier.
Fail, R., "Mini Differential Analyzer," Midlands Meccano Guild Gazette, No. 16, 49, April 1993. A small demonstration model from standard Meccano parts with full construction details. Two integrators and output table. Uses electromechanical servos instead of torque amplifiers.
Fischer, C. F., "Reminiscences at the end of the Century," Molecular Physics, 98, 10431050, 2000 A collection of autobiographical notes including an interesting discussion of the author's work as a research student with Hartree around the time he was building the first Meccano differential analyzer. Available online here.
Fischer, C. F., Douglas Rayner Hartree  His Life in Science and Computing, Singapore: World Scientific, 2003. This scientific biography of Douglas R. Hartree not only describes important events in his life but also outlines his contributions to a number of fields. Hartree was very interested in the process of computation. When he learned of a differential analyzer for solving differential equations, he first built a model using Meccano.
Ford, H. C., "Mechanical Movement" United States Patents 1,317,915, 1,317,916, 1919. The disk/ball/cylinder integrator used by Hannibal Ford in military fire control systems. I am not aware of this type of integrator ever being used in a differential analyzer, but it has the advantage of being able to transmit substantial torque without requiring a torque amplifier. The second patent has a derivative capable of even greater loading.
Gray, E., "The Torque Amplifier," New Zealand Federation of Meccano Modellers Magazine, Oct 1992. An unusual application of the torque amplifier to amplify the outputs of servo motors in a motor chassis.
Hartree, D. R., F.R.S., and Porter, A., "The Construction and Operation of a Model Differential Analyser," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 79, 5174, 1935. A detailed account of the construction and operation of the Meccano differential analyzer at Manchester University including a number of photographs of the machine.
Hartree, D. R., F.R.S., and Ingham, J., "Note on the Application of the Differential Analyser to the Calculation of Train Running times," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 83, 115, 1938. An interesting paper on a relatively simple second order equation. Although the work was done on the full scale machine, reference is made to the fact this would be a suitable problem for the Meccano differential analyzer described in Hartree (1935).
Hartree, D. R., "The Mechanical Integration of Differential Equations," Mathematical Gazette, 22, 342364, 1938. A comprehensive account of the construction and application of the Manchester differential analyzers with a good close up photograph of the integrators of the full scale machine.
Hartree, D. R, and Nuttall, A. K., "The Differential Analyser and its Application in Electrical Engineering," Journal of the Institution of Electrical Engineers, 83, 643647, 1938. A fairly detailed account of the MetropolitanVickers machine with good pictures, including of the special input table. A somewhat briefer section discusses actual applications.
Hartree, D. R., and Porter, A., "The Application of the Differential Analyzer to transients on a Distortionless Transmission Line," Journal of the Institution of Electrical Engineers, 83, 648656, 1938. Examines the behavior of transients on a finite distortionless transmission line. This problem required use of a special input table that can feed back a solution after a fixed delay. Application to lightening arresters is discussed.
Hartree, D. R., "The Bush Differential Analyser and its Applications," Nature, 146, 3697, 319323, 1940. A general description of the differential analyzer. Despite the title, the pictures are actually of the MetropolitanVickers machine at Manchester.
Hartree, D, R., "A Great Calculating Machine: The Bush Differential Analyser and its Applications in Science and Industry," Proceedings of the Royal Institution, 31, 151170, 1940. A paper presented at the Royal Institution weekly evening meeting on May 17, 1940. Describes the differential analyzer in general, the full scale Manchester machine specifically, and some of its applications, including the calculation of train running times. Mentions the Meccano differential analyzer.
Hartree, D. R., "The Thirtyfourth Kelvin Lecture: Mechanical Integration in Electrical Problems," Journal of the Institution of Electrical Engineers, 90, 435442, 1943. Lecture delivered before The Institution on 29, April, 1943. After a brief description of the differential analyzer (including mention of Kelvin's contribution) details solutions to several problems in electrical engineering obtained using the machine.
Hartree, D. R., "Differential Analyser," Ministry of Supply Permanent Records of Research and Development No. 17502, 1946/9. A detailed account of the use of the Manchester differential analyzer during the war. Probably the largest collection of applications gathered in a single document. Section 1.4 documents other differential analyzers in the UK, including Meccano models.
Hartree, D. R., F.R.S., Calculating Instruments and Machines, Urbana: University of Illinois Press, 1949. An extensive discussion of differential analyzers with many photographs including the Meccano differential analyzer described in Hartree (1935). Includes a chapter on its application to partial differential equations. There is a suggestion for the solution of a set of nonlinear simultaneous equations for a problem in spherical, sound waves remarkable in that it uses no integrators, just an intricate interconnection if input tables, output tables and multiplier/divider units. Extensive references are provided.
Hartree, J., and Tee, G., "Toy Story," New Scientist, 4 Mar 2000. Letters to the editor discussing the differential analyzer, in response to the question from a reader as to whether any significant inventions or principles owe their discovery to the use of Meccano. Available online here.
Hazen, H. L., Jaeger, J. J., and Brown, G. S., "An Automatic Curve Follower," Review of Scientific Instruments, 7, 353357, 1936. Describes an automatic curve follower using a photocell, servo motor, and torque amplifier, which can follow the black/white boundary of a curve on an input table. See also Blackett (1939).
Heffron, W. G., "Operation and Application of the Differential Analyzer," Product Engineering, 23, 4, 164170, 1952. General description of the operation of a differential analyzer with particular reference to the General Electric 14 integrator machine.
Hersom, S. E., and Selig, K. L., "A General Purpose Differential Analyser Part II  Application of Machine," Elliott Journal, 1, 3, 7680, 1952. Describes the application of the Elliott differential analyser to a thermionic microwave diode. Gives particular attention to scale factors. See also Ashdown (1951).
Hey, T., The Quantum Universe, Cambridge University Press, 1987. A brief reference to Hartree's atomic structure calculations, including a picture of Hartree and Porter with the Meccano differential analyzer.
Hogle, H., "Torque Amplifier," Canadian MeccaNotes, 6, 19, June 1997. An unusual torque amplifier design in Meccano parts. Probably not sensitive enough for use in a differential analyzer.
Holst, P. A., "Svein Rosseland and the Oslo Analyzer," IEEE Annals of the History of Computing, 18, 4, 1626, 1996. At one time the Oslo analyzer was the world's largest; technically advanced, highly accurate, and used by theoretical physicists from around the world. A discussion of the machine, and the man who created it.
Irwin, W., "Differential Analyser No. 2," New Zealand Federation of Meccano Modellers Magazine, 25, 2, 2001. Letter to the editor identifying the source of the article in the preceding issue (Anon (2001)) and reporting on more recent efforts to restore the Meccano differential analyzer at MOTAT.
Irwin, W., "Differential Analyser Myths," New Zealand Federation of Meccano Modellers Magazine, 25, 3, 2001. A short piece exposing a number of myths circulating about Meccano differential analyzers.
Irwin, W., "The Differential Analyzer Explained," New Zealand Federation of Meccano Modellers Magazine, 26, 3, 2002. Describes the principles of operation of a differential analyzer. Illustrated with a picture of an earlier Meccano machine by the current author and the Science Museum exhibit of an integrator from the Meccano differential analyzer described in Hartree (1935). Available online here.
Irwin, W., "Meccano Differential Analyser . . . and New Zealand's First Computer," The International Meccanoman, #46, IX.2005. An account of the differential analyzer exhibits at the 2005 NZFMM Easter Convention in Auckland, New Zealand, which included the restored section of the Cambridge Meccano machine and the author's version of the model in Fail (1993) .
Irwin, W., "Differential Analyzer  Adding Unit," New Zealand Federation of Meccano Modellers Magazine, 31, 6, 89, 2007. Details of the adding unit used in the original Cambridge Meccano differential analyzer together with a modern reconstruction.
Irwin, W., "Propagation of an Urban Legend, a Differential Analyser Myth," New Zealand Federation of Meccano Modellers Magazine, 31, 6, 1415, 2007. Debunks the myth that the Cambridge Meccano differential analyzer was used by Barnes Wallis for the design of the "bouncing bomb".
Irwin, W., "The Cambridge Meccano Differential Analyser No.2," The Driving Wheel, 5: The Museum of Transport and Technology Society, 2013. Reprinted (with fewer photos) in Resurrection. The Bulletin of the Computer Conservation Society, 64, Winter 2013/4. An updated account of the history of, and the full restoration of the Cambridge Meccano differential analyzer at MOTAT.
Jackson, A. S., Analog Computation, New York: McGrawHill, 1960.
Although primarily about electronic analog machines, this book includes a concise section on mechanical differential analyzers (pp. 573578).
Janssen, E., and Lebell, D., "Applications of the Mechanical Differential Analyzer to Electrical Engineering," Electrical Engineering, 70, 432435, 1951. Applications for the differential analyzer to magnetic amplifiers, pulse transformers and electron accelerators.
King, D., "Historic Computer Lost from MOTAT," New Zealand Herald, section 4, page 5, April 20, 1993. A half page report on the loss of the Meccano differential analyzer from MOTAT. Gives a history of the machine and a picture of it in the museum's former computer display.
King, D., "Heritage Rusts to Bits in Rain," New Zealand Herald, section 3, page 5, April 27, 1993. Another report on the events surrounding the loss of the Meccano differential analyzer from MOTAT.
Kuehni, H. P., and Peterson, H. A., "A New Differential Analyzer," AIEE Transactions, 63, 5, 221228 (discussion 429431), 1944. Detailed technical description of the General Electric 14 integrator differential analyzer which used a Polaroid optical follower system on the integrators. The follower system is described in Berry (1944).
Kryloff, A., "Sur un intégrateur des équations différentielles ordinaires," Bulletin de l'Académie Impériale des Sciences de St.Pétersbourg, Ser. V, T.XX, 1, Jan. 1904. In this little known work, Kryloff describes a machine based on the work of Thompson (Thompson (1876A)), but using radically different integrators. It is unclear if the machine was successful as the paper was written while it was still under construction. This paper is in French.
LennardJones, J. E., Wilkes, M. V., and Bratt, J. B., "The Design of a Small Differential Analyser," Proceedings of the Cambridge Philosophical Society, 35, 485, 1939. A detailed description of the construction and testing of the Meccano differential analyzer at Cambridge University. In an early test using four integrators, the wave equation of the hydrogen atom was solved giving the value of the ground state energy correct to one part in 500.
Lowe, I., "Ancient Computer Down and Out," New Scientist, 138, 1873, p.50, 15 May 1993. (May not appear in all editions.) A follow up to articles in the New Zealand Herald (King (1993A) and King (1993B)) reporting on the loss of the Cambridge Meccano differential analyzer from MOTAT.
Macauley, T., "Operating the Meccano Differential Analyser,", unpublished, MOTAT, Auckland, New Zealand, 1978. A brief operating manual for the Meccano differential analyzer at MOTAT which indicates that it was in operation in 1978 for demonstrations. One of the 5 integrators was not functional.
Maginniss, F.J., "Differential Analyzer Applications,", General Electric Review, 48, 5, 5459, 1945. A discussion of eight applications of the differential analyzer to engineering problems and the techniques used to handle them.
Marsh, P., "The Meccano Set Computer," New Scientist, 80, 1134, (supplement 2829), 1978. A popular article on the history of the Manchester differential analyzers.
Massey, H. S. W., Wylie, J., Buckingham, R. A., and Sullivan, R., "A Small Scale Differential Analyser  Its Construction and Operation," Proceedings of the Royal Irish Academy, 45A, 1, 121, 1938. A four integrator machine. All the spur gears used in this machine are of Meccano manufacture. Helical gears and sprockets are from Bond's. Everything else is of custom design. The paper contains some excellent pictures.
Michel, J. G. L., "Extensions in Differential Analyzer Technique," Journal of Scientific Instruments, 25, 10, 357361, 1948. In this paper, a constructive technique is developed for obtaining the results arrived at by Amble (1946) from analytical considerations. The technique is extended to include the integral of a quotient, and the inversion of functions. Mentions that one of these techniques was applied on the Cambridge Meccano differential analyzer.
Michel, J. G. L., "Errors of Friction Wheel Integrators," Journal of Scientific Instruments, 32, 2, 4344, 1955. Analyzes the error inherent in a wheel and disk integrator whenever there is sliding motion of the wheel. This error is approximately proportional to the torque which must be applied to the wheel.
Mindell, D. A., Between Human and Machine  Feedback, Control, and Computing Before Cybernetics, Baltimore: Johns Hopkins University Press, 2002. Mindell shows how the modern sciences of systems emerged from disparate engineering cultures and how they converged during W.W.II. Chapter 5, Analog Computing at MIT, is devoted to the work of Vannevar Bush's lab.
Myers, D. M., and Blunden, W. R., "The C.S.I.R.O. Differential Analyser," Proceedings of Conference on Automatic Computing Machines, Sydney Australia, 1951. A description of the construction of a 10 integrator differential analyzer at the University of Sydney. This machine uses electrical interconnections between the units using a system called "Mtype" transmission. Foe a more detailed account, see Myers (1952).
Myers, D. M., and Blunden, W. R., "The C.S.I.R.O. Differential Analyser," Journal of the Institution of Engineers, Australia, 24, 195204, OctNov, 1952. A more comprehensive description of the C.S.I.R.O. analyzer than given in Myers (1951). It includes a section on applications to which the instrument has been applied.
Nieman, C. W., "Bethlehem Torque Amplifier," American Machinist, 66, 21, 895897, 1927. Nieman's original description of the torque amplifier, with application to automobile power steering. The caption of one figure contains the phrase "adaptable to computing machines" though there is no other mention in the text, and this article considerably predates Bush's application. See also US patents 1751645, 1751647, and 1751652.
O'Neill, R., "Meccano 'Dam Busters' computer stars at MOTAT," Computerworld, 16 July, 2007. A review of the new computing exhibit at MOTAT which features the recently restored Cambridge Meccano differential analyzer.
Owens, L., "Vannevar Bush and the Differential Analyzer: The Text and Context of an Early Computer," Technology and Culture, 27, 1, 1986, 6395. Reprinted in Nyce, J. M. From Memex To Hypertext, Academic Press, 1991. Discusses the background to Bush's machines, from the early product integraphs through the mechanical analyzer to the mighty Rockefeller electromechanical analyzer.
Partridge, A., "Torque Amplifier," Midlands Meccano Guild Gazette, No 7, 5, September 1982. A simple servo based design using a motor and differential to operate electrical contacts.
Partridge, A., "Torque Amplifiers," Constructor Quarterly, 19, 4042, March 1993. Constructional details of two torque amplifier designs using Meccano parts.
Paynter, H. M., "The Differential Analyzer as an Active Mathematical Instrument," IEEE Control Systems Magazine, 9, 7, 38, 1989. This presentation features the essential role played by amplification and control in the successful development of the differential analyzer. Mentions the Meccano machines, but incorrectly states that their torque amplifiers were made entirely from Meccano parts.
Peierls, R., Bird of Passage, Princeton University Press, 1985. The highly readable autobiography of Rudolf Peierls. Contains a brief mention of Hartree and the Meccano differential analyzer (p104). Peierls was the supervisor of A. M. Wood when he was building a Meccano machine at Birmingham (see Wood (1942)) and on p137 he confirms this machine was never completed.
Porter, A., "An Approximate Determination of the Atomic Wave Functions of the Chromium Atom," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 79, 7581, 1934. Presents the results of approximate calculations of the wave functions of the chromium atom, carried out on the Meccano differential analyzer described in Hartree (1935).
Porter, A., Differential Analyser Log Book, MS474 London: Science Museum Library. Hand written day to day notes recorded from July to December 1935 as the full scale analyzer at Manchester was being commissioned. These notes make fascinating reading. It would appear that getting reliable operation from the machine was very challenging.
Porter, A., The Differential Analyser and Some Applications, University of Manchester PhD Thesis, 1936. Porter's PhD Thesis in which he describes both the model and full size machines, including modifications to handle timelag problems. He covers in detail numerous applications.
Porter, A., Introduction to Servomechanisms, London: Mehuen & Co., 1950. This is a general treatment of the theory of servomechanisms. However, the first chapter uses as examples both the mechanical torque amplifier of the original Bush differential analyzer, and the electromechanical servo system used on the later Rockefeller differential analyzer.
Porter, A., "Building the Manchester Differential Analyzers: A Personal Reflection," IEEE Annals of the History of Computing, 25, 2, 8692, 2003. A delightful memoir by one of the pioneers. Written in his 93rd year. This memoir is extracted from chapter 4 of Porter (2004).
Porter, A., So Many Hills to Climb: My Journey from Cumbria to North Carolina, Silver Spring: The Beckham Publications Group, 2004. Arthur Porter describes a remarkable life with eloquent sensitivity and charming candor. Chapter 4 covers his time at Manchester University and the building of the differential analyzers. This chapter was in large part reprinted in Porter (2003).
Robinson, T. B., "The Meccano Set Computers," IEEE Control Systems Magazine, 25, 4, 7483, 2005. A history of the Meccano based small scale differential analyzers.
Robinson, T. B., "A Reconstruction of the Differential Analyzer in Meccano," IEEE Control Systems Magazine, 25, 4, 8489, 2005. Description of the author's modern reconstruction in Meccano of a differential analyzer based on Bush's prototype.
Rose, H. E., "The Mechanical Differential Analyser: Its Principles, Development, and Applications," Proceedings of the Institute of Mechanical Engineers, 159, 1948, 4654 and 6280.
This is a review paper which contributes nothing original. It is followed by a record of the discussion which ensued when it was presented, in which A. Porter leads a blistering attack for both its lack of originality and for the arbitrary change in notation which Rose adopted. It is further followed by miscellaneous communications relating to mechanical and electrical analyzers.
Rosseland, von S., "Mechanische Integration von Differentialgleichungen," Die Naturwissenschaften, 27, 44, 729735, 1939. A description of the 12 integrator full scale machine at the Institut für Theoretische Astrophysik, Oslo, Norway. The paper is in German.
Shannon, C. E., "Mathematical Theory of the Differential Analyzer," Journal of Mathematics and Physics, XX, 4, 1941. Reprinted in Claude E. Shannon, Collected Papers (ed. Wyner A. D. and Sloane, N. J. A.) WileyIEEE Press, 1993. A heavy duty mathematical analysis of the class of problems that can be addressed by the differential analyzer, assuming only that the machine has an unlimited number of integrators and adders. An interesting paper, but somewhat remote from the practicalities of actual machine operation.
Small, J. S., The Analogue Alternative: The Electric Analogue Computer in Britain and the USA, 19301975 (Studies in the History of Science, Technology and Medicine), Routledge, 2001. While primarily concerned with electronic analog machines, chapter 2 provides a good introduction, including Hartree's development of differential analyzers at Manchester. Includes a picture of the Meccano differential analyzer. Extensive references.
Smillie, K., "People, Languages, and Computers: A Short Memoir," IEEE Annals of the History of Computing, 26, 2, 6274, 2004. A biographical memoir which briefly mentions work by Jim Howland to extend Beatrice Worsley's Meccano differential analyzer in 1951. Essentially the same material can be found online here and here.
Smillie, K., "A J Simulator for a Meccano Differential Analyzer," Vector, 23, 3 2008.
This paper describes a fiveintegrator differential analyzer simulator implemented in J and gives two examples of its use.
Smillie, K., "Simulation of Meccano Differential Analyzers", 2008.
A more extended version of Smillie (2008a) published on line. The simulator code is available for download via this page.
Smith, P., "Who sez it cannot be done!," Meccano Engineer, #11, March 1996. A short article with a picture of a one integrator Meccano DA constructed by T. Brooker, in which the torque amplifier is constructed entirely from standard Meccano parts. Probably the first person to have done it.
Soroka, W. W., Analog Methods in Computation and Simulation, New York: Mc GrawHill. A comprehensive survey a vast array of both mechanical and electrical methods current at the time of writing. Includes much material on the differential analyzer with extensive references.
Spackman, L., "A Meccano Differential Analyser," Meccanoman's Newsmag, #67, 1993. Discussion of the history of the Cambridge Meccano differential analyzer in New Zealand.
Strong, C. L., "The Amateur Scientist: A plan for an analogue computer that can be built for about $50," Scientific American, June 1968. A design for a home built two integrator differential analyzer with an example application. Construction of the machine would be greatly simplified by the use of Meccano!
Schultes, D., "On the Development and Use of Differential Analyzers," April 2004. A short essay presenting the development of the differential analyzer. In addition to discussing the mechanical machines it includes a section on the later electronic machines and a short comparison with present day technology.
Tee, G. J., "Meccano Differential Analyser No. 2," unpublished notes, 1993. Notes on the complex history of the Cambridge Meccano differential analyzer, shipped to New Zealand in 1950, almost dismantled in the 1960's, displayed at MOTAT in the 70's then lost in 1993 and finally found again in damaged condition.
Thomson, J., "An Integrating Machine having a new Kinematic Principle," Proceedings of the Royal Society, 24, 262, 1876. Description of the ball and disk integrator in which a ball rests under gravity between a horizontal cylinder and an inclined integrator disk thus avoiding the sliding necessary in a wheel and disk integrator. Inspired by the work of Prof. James Clark Maxwell who applied a similar approach to an improved planimeter design. Reprinted in Thomson (1879).
Thomson, Sir W., "An Instrument for Calculating the Integral of the Product of two Given Functions," Proceedings of the Royal Society, 24, 266, 1876. Application of the ball and disk integrator described in Thomson (1876A) to the evaluation of the integral of a product of functions. Reprinted in Thomson (1879).
Thomson, Sir W., "Mechanical Integration of Linear Differential Equations of the Second Order with Variable Coefficients," Proceedings of the Royal Society, 24, 269, 1876. Sir William Thomson first suggested that the integrators developed by his brother could be interconnected to produce solutions of differential equations. The idea was not practical at the time because of the lack of torque amplifiers. Reprinted in Thomson (1879).
Thomson, Sir W., "Mechanical Integration of the General Linear Differential Equation of any Order with Variable Coefficients," Proceedings of the Royal Society, 24, 271, 1876. A generalization of the scheme proposed in Thomson (1876C) to equations of arbitrary order. Reprinted in Thomson (1879).
Thomson, Sir W., "Harmonic Analyzer," Proceedings of the Royal Society, 27, 371, 1878. A description of a harmonic analyzer using up to eleven of the ball and disk integrators described in Thomson (1876A) and the principles detailed in Thomson (1876B) where one of the functions in the product is a sine or cosine. This machine was used to extract the coefficients of a Fourier series for the purposes of tide prediction. Reprinted with an addition dated April, 1879 in Thomson (1879).
Thomson, Sir W., and Tait, P. G., Treatise on Natural Philosophy, Vol. 1, Cambridge: Cambridge University Press, 1879. Although a textbook on physics, it contains an Appendix (curiously called Appendix B' even though it's the only one) which contains reprints of several papers on integrating devices previously published in the Proceedings of the Royal Society. See Thomson (1876A), Thomson (1876B), Thomson (1876C), Thomson (1876D), and Thomson (1878),
Travis, I., "Differential Analyzer Eliminates Brain Fag," Machine Design, 7, 7, 1518, 1935. Description of the ten integrator differential analyzer built at the Moore School of Engineering, University of Pennsylvania. This machine included two polar input tables.
Wildes, K. L. and Lindgren N. A., A Century of Electrical Engineering and Computer Science at MIT, 18821982, Cambridge: MIT Press, 1985.
Chapter 4 provides a nontechnical historical account of four generations of analog technology developed under Bush's guidance, from the early integraphs through to the mighty Rockefeller DA.
Wilkes, M. V., Memoirs of a Computer Pioneer, Cambridge, MA: The MIT Press, 1985. Describes his encounter with the Meccano differential analyzer at Cambridge and how he came to take responsibility for it. He also describes the addition of the fifth integrator by Miss E Monroe in 1937.
Williams, M. R., "UTEC and Ferut: The University of Toronto's Computation Centre," Annals of the History of Computing, 16, 2, 412, 1994. Contains a brief reference to Beatrice (Trixie) Worsley's Meccano differential analyzer at the Unversity of Toronto. There is a picture of the machine in an early stage of construction..
Williams, M. R., A History of Computing Technology, 2nd Edition, WileyIEEE Computer Society Press, 1997 Broad survey of computing history. Chapter 5 covers analog machines in general with a section on the differential analyzers. Mentions the Hartree Meccano differential analyzer and has an interesting reference to a Meccano machine built in Toronto by Beatrice (Trixie) Worsley.
Winston, B., Media Technology and Society: A History: From the Telegraph to the Internet, Routledge, 1998 Contains a brief reference to Hartree's construction of a differential analyser in Meccano after visiting MIT to see Bush's machine.
Wood, A. M., The Design and Construction of a Small Scale Differential Analyser and its Application to the Solution of a Differential Equation, University of Birmingham MSc. Thesis, 1942. The design of a small scale machine (6 integrators) built largely of Meccano parts is described. Torque amplifiers and integrator carriages are custom built. Because of war time material shortages, only two integrators were completed at the time the thesis was written. A second section describes the solution of a equation performed using the Meccano differential analyzer at Cambridge University. Peierls (1985) confirms it was never completed.
Worsley, B. H., "Construction of a Model Differential Analyzer," Worsley Archives, box 3, folder 10, Queen's University Archives, Ontario, 1948. A memo to Dr. B. A. Griffith, Toronto Computation Center, dated 10 September, 1948 describing the construction of a three integrator Meccano differential analyzer.
Worsley, B. H., "Differential Analyzer," Worsley Archives, box 3, folder 10, Queen's University Archives, Ontario, undated. A set of course notes from the University of Toronto Department of Physics, for a 4th year practical course using Worsley's model differential analyzer. Undated, but must be 1949 or later because of a reference to Hartree (1949).
Wilson, A. H., "The Binding Energies of the Hydrogen Isotopes" Proceedings of the Cambridge Philosophical Society, 34, 365374, 1938. Although fundamentally a paper about nuclear physics, it is interesting because this is the problem which resulted in the addition of a fifth integrator to the Cambridge Meccano differential analyzer.
Wright, G., The Meccano Super Models (The Hornby Companion Series, Vol. 2), London: New Cavendish, 1978. Reproduced on p.31 is a small extract from Anon (1934B).
Copyright (c) 2010 Tim Robinson
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What is a Differential Analyser?
The Differential Analyser is a mechanical analogue computer. What does this mean and where does it fit into the scheme of things?
There are two distinct branches of the computer family. One branch descends from the abacus, which is an extension of finger counting. The devices that stem from the abacus use digits to express numbers, and are called digital computers. These include calculators and electronic digital computers.
The other branch descends from the graphic solution of problems achieved by ancient surveyors. Analogies were assumed between the boundaries of a property and lines drawn on paper by the surveyor. The term “analogue” is derived from the Greek “analogikos” meaning by proportion. There have been many analogue devices down the ages, such as the nomogram, planimeter, integraph and slide rule. These devices usually perform one function only. When an analogue device can be “programmed” in some way to perform different functions at different times, it can be called an analogue computer. The Differential Analyser is such a computer as it can be set up in different configurations, i.e. “programmed”, to suit a particular problem.
In an analogue computer the process of calculation is replaced by the measurement and manipulation of some continuous physical quantity such as mechanical displacement or voltage, hence such devices are also called continuous computers. The analogue computer is a powerful tool for the modelling and investigation of dynamic systems, i.e. those in which some aspect of the system changes with time. Equations can be set up concerned with the rates of change of problem variables, e.g. velocity versus time. These equations are called Differential Equations, and they constitute the mathematical model of a dynamic system.
The Differential Analyser solves differential equations by integration. It makes use of one or more wheel and disc integrators (or Kelvindisc integrators), interconnected by shafts in various ways to suit the problem equations. The process of integration can be illustrated by the simple example of the acceleration of a car. This can be represented for input by a curved graph showing speed varying with time. Say one wanted to find out the distance travelled in a certain time, say five minutes. The period of 5 minutes can be divided into much smaller intervals of say 10 seconds each, and assuming a constant speed over each interval from the graph, a distance travelled for each interval is calculated. The sum of the distances travelled in successive intervals is then the total distance travelled. The smaller the interval taken the more accurate will be the result. This is called Integration, and is the function performed by the integrator in a continuous manner.
How does the Differential Analyser Work?
A typical mechanical Differential Analyser consists of the following components:
 Two or more Integrator units (including one Torque Amplifier per unit).
 adding and counting units.
 Input and Output tables.
 A gearing and shafting system to link everything together.
The Integrator:
The Integrator is in essence a variablespeed gear, and takes the form of a rotating horizontal disc on which a small knifeedged wheel rests. The principle used is shown in fig. 2. The vertical axis of the horizontal disc is supported in a movable carriage so that the distance of the point of contact of the wheel on the disc, from the centre of the disc, can be varied. The two inputs to the integrator are therefore the rotation of the disc x, and the carriage movement y. When the unit is integrating, the disc rotates and undergoes a translational movement simultaneously. The rotating disc drives the wheel by friction. The rotation of the wheel represents the output of the integrator. For proper operation the wheel must not slip on the disc so the rim of the wheel must be as sharp as possible, usually hardened steel on a polished glass disc.
The Torque Amplifier:
The torque on the shaft carrying the integrator wheel is very small due to the low frictional force between the wheel and disc. In order to drive other units of the machine, the shaft from the integrator passes into a Torque Amplifier, the output shaft of which rotates with the same velocity as the input shaft but with greatly increased torque. The principal of the torque amplifier is essentially that of the capstan, where a small force, applied at one end of a friction band wrapped around a rotating drum, produces considerably increased tension at the other end of the band. The principle is shown diagrammatically in fig. 3. The two drums are driven by a motor in opposite directions. Movement of the input shaft tightens the thread on the motor driven drum, which causes the output shaft to rotate with amplified torque. The construction of a Meccano demonstration model of a torque amplifier is described by Michael Adler in the International Meccanoman, no. 34, September 2001.
Adding and Counting Units:
These can be included in the shafting when required. Adding units consist of a differential gear system so arranged that the angular rotation of one shaft connected to it is the sum of the angular rotation of two other shafts. Counters are used when it is desired to know the number of revolutions made by a particular shaft.
Input and Output Tables:
The Input Table is a unit whereby information concerning the differential equation is transmitted to the machine. A stylus is manually made to follow a preplotted curve, representing the known functional relationship between the variables. The result is mechanically linked to the input drives. Not all problems require the use of the input table.
The Output Table is similar in construction to the input table. The solution to the equation is drawn by a pen on paper, in the form of a curve, thus plotting the results from the various mechanical linkages.
Operation:
Motors are required for each torque amplifier, and one motor is required for the main input drive. Secondary inputs are manual via the input table. Tim Robinson’s fine model above gives the reader a general idea of a typical setup.
I have purposely not gone into any mathematical background here. Suffice it to say that the Integrators, Input and Output tables, Adders and Counters if required, are all linked together by gearing and shafting to suit the particular differential equation solution required. This requires a lot of effort in setting up for each different problem and is in effect the “programming” of the machine which defines it as a computer. Of course a thorough knowledge of the mathematics of the problem is required by the user.
Examples of usage:
A practical example of the type of calculation for which the Differential Analyser has been used to predict results is the use by river control authorities in New Zealand for the calculation of soil erosion characteristics of various surface areas. The rate of erosion is among other things dependent upon the following:
1) the rate at which the water is falling on the surfaces
2) the resistance the surface cover offers to the flow of surface water
3) the speed of flow of surface water
4) the volume of water flowing
The Differential Analyser was used in many other fields of science and engineering before being displaced by faster digital computers.
It is rumoured that a differential analyser was used in the development of the "bouncing bomb" by Barnes Wallis for the "Dam Busters" attack on the Ruhr valley hydroelectric dams in WW2. This was first mentioned in MOTAT literature in 1973. However after extensive enquiries and literature searches over the last few years, no evidence can be found that the Differential Analyser no. 2, nor any other differential analyser, was used for this purpose. Considering the secrecy surrounding war time activities at the time it could still be possible, but most people from that era are now deceased. Two remaining personalities still alive from that era were consulted, namely Arthur Porter and Maurice Wilkes, but neither could substantiate the rumour.
References:
Analog Computation, by Albert S Jackson, McGraw Hill 1960
The Differential Analyser, by J Crank, Longmans Green & Co. 1947
Differential Analyser, GMM leaflet no. 4, The Meccanoman's Club, 1967
Torque Amplifiers, by Alan partridge, Constructor Quarterly no. 19, March 1993
Meccano Torque Amplifier, by Michael Adler, International Meccanoman no. 34, September 2001
The Meccano Set Computers, by Tim Robinson, IEEE Control Systems Magazine vol. 25 no. 3, June 2005
republished in Constructor Quarterly no. 72, June 200
APPENDIX
For those of a mathematical inclination, the simplest differential equation which can be solved by the Differential Analyser is d2y/dx2 = y
Since the equation has constant coefficients no manual controls via the input table are required. The output table records dy/dx as a function of y so the plotted output should be a circle. This is known as the "circle test" for accuracy. If there is slippage or any backlash the machine will draw a close spiral instead of a circle.
For other Differential equations the output might be a sine wave (damped or undamped), a plotted trajectory or other curve, depending on the problem being investigated. 

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