Exploring Machine Motion Design
by Daina Taimina

This learning module is based on experience working with 7th and 8th graders in Lehmans Alternative Community School in Ithaca, NY as a 9-week project for technology class. The choice of actual kinematic models to explore was determined by the possibility to take actual Reuleaux models to school. Digitized Reuleaux kinematic model collection allows technology teachers to make his/her own choice of what models to explore and in which order.

Project schedule - one class a week

  1. Introduction - little overview of history of engineering and role of machines in it. History of F. Reuleaux kinematic model collection. class notes
  2. Exploring models of cams and planetary gears. Reuleaux triangle. class notes
  3. Exploring straight-line motion, pumps and the universal joint. class notes
  4. Exploring old machine design. class notes
  5. What is a creative design? Building a windmill. class notes
  6. History of steam engine. Different use of linkages. class notes
  7. Field trip. class notes
  8. Making your own machine design. class notes
  9. Student presentations. class notes

Class notes

Week 1: History of Engineering and F. Reuleaux kinematic model collection.

From the earliest days of recorded history the people now known as engineers - civil, chemical and mechanical, mining and metallurgical - have provided for man's unfolding material needs and wants. Engineering not only played an ever-increasing role in material life but also has had a profound effect on human relations. According to [Finch] a development of engineering can be historically determined as:

  • Food producing revolution (ca. 6000-3000 BC)
  • Appearance of urban society (ca. 3000-2000 BC)
  • Birth of Greek science (600-300 BC)
  • Revolution in power (Middle Ages)
  • Rise of modern science (17th century)
  • Steam and Industrial Revolution (18th century)
  • Electricity and beginnings of applied science (19th century)
  • Age of automatic control (20th century)

Discussion with students - what simple machines do you know? How do they work? Where are they used? What examples of machine motion do you know? Why it is necessary to change motion in machines?

Machines are mentioned in history texts long time ago. But what was considered a machine two thousand years ago differs considerably from our present ideas. Vitruvius a military engineer writing about 28 B.C. defined a machine as "a combination of timber fasten together, chiefly efficacious in moving great weights". About a century later Hero of Alexandria summarized the practice of his day by naming the "five simple machines" for "moving a given weight by a given force" as the lever, windlass, screw for power, wedge, and tackle block (pulley).

Around 1875 German professor Franz Reuleaux (1829-1905) identified six basic mechanical components from which machines are built:

  1. the eye-bar type of link, called crank in kinematics;
  2. the wheel, including gears;
  3. the cam in its many forms;
  4. the screw for communicating motion and force;
  5. the intermittent-motion devices called ratchets;
  6. the tension-compression organs, or parts having "one way rigidity", as belts (chains) and hydraulic lines.
Each of these components was invented and put in use already in antiquity.

Kinematics flourished in the 19th century as machine inventors learned to transmit information and forces (power) from one element in the machine to another. Scientific American featured a new invention in every issue. Steam-and water-based machines revolutionized the l9th century, but both of those energy sources generate circular motions, creating the need to convert these steady circular motions into nonsteady linear and curvilinear motion for machine applications. The challenge to create input-output kinematic devices that could convert circular motion into noncircular, complex, three-dimensional, intermittent motions attracted both practical inventors as well as mathematicians. Thousands of mechanisms were invented, designed, and built, nurturing the widespread use and manufacture of machines as process analogous to the plethora of electronic circuits in the early 20th century and software in the late 20th century.

Franz Reuleaux's mission was to codify, analyze, and synthesize kinematic mechanisms so that engineers could approach machine design in a rational way. He laid the foundation for a systematic study of machines by defining clearly the machine and mechanism, determining the basic building blocks, and developing a system for classifying known mechanism types. Reuleaux created in Berlin a collection of over 800 models of mechanisms and authorized a German company, Gustav Voigt, Mechanische Werkstatt, in Berlin, to manufacture these models so that technical schools could use them for teaching about machines to engineers. Cornell University, with the initiative of its first president A. D. White, acquired a large kinematic model collection in 1882.


Week 2: Exploring Reuleaux kinematic models - gears and cams

Q-1, Q-3, Q-4, L-1, L-2, L-4, L-5

Gears are one of man's oldest mechanical devices. The gear has been a basic element of machinery throughout all time from the earliest beginnings of machinery. The earliest known relic of gearing from ancient times is the "South Pointing Chariot" (about 2600 BC) A miniature replica of this chariot is on exhibit at the Smithsonian Institute.

In view of the intricacy of the South Pointing Chariot, it seems obvious that there must have been earlier use of gearing going back to at least 3000 BC. The earliest written records about gearing are dated from about 330 BC in the writings of Aristotle. He explained gear wheel drives in windlasses, pointed out that the direction of rotation is reversed when one gear wheel drives another gear wheel. From many writings it seems probable that both the Egyptians and Babylonians were using gear devices as far back as 1000 BC. The most probable uses were in clocks, temple devices and water lifting equipment. About 250 BC Ctesibius wrote about making water clocks and water organs using gears. About 230 BC Phylo of Byzantium made rack and pinion device to raise water, but Archimedes about 220 BC. Made devices to multiply force or torque many times. By 100 BC the gear art included both metal and wooden gears. Triangular teeth, buttressed teeth, and pin teeth were all in use. Spur gears, racks and pinions and worm gears had all arrived on the scene. Right-angle pin-tooth drives were in use and perhaps the first primitive bevel gear. The Romans and Greeks made wide use of gearing in clocks and astronomical devices. Gears were also used to measure distance or speed. One of the most interesting relics of antiquity is the "Antikythera machine" which is an astronomical computer. It had many gears in it - some of which were planetary.
Timeline of modern history of gearing:
Nicholas of Cusa Studied cycloidal curve.
Albrecht Durer Discovered epicycloids.
Girolamo Cardano First mathematics of gears in print.
Philip de la Hire Full mathematical analysis of epicycloids. Recommended involute curve for gearing. (Involute not used in practice until about 150 years later.)
Charles Camus Expanded of work of la Hire, developed theories of mechanisms, studied lantern pinion and gear, crown gears and beveled gears.
Leonard Euler Euler worked out design principles, worked out rules for conjugate action. Some consider him "the father of involute gearing".
Abraham Kaestner Wrote up practical methods for computing tooth shapes of epicycloid and involute gear teeth. Considered 15 to be a minimum pressure angle.
Robert Willis Wrote and taught extensively in gear field. A pioneer in gear engineering.
Edward Sand General theory of gear teeth. Provided theoretical basis on which all gear tooth-generating machines are based.

For a classroom discussion about gears good model can also be a toy "Spirograph", if it is available.


Week 3: Exploring Reuleaux kinematic models - straight-line mechanisms, pumps, universal joint.

S-01, S-02, S-03, S-04, S-09, S-14, S-16, S-17, S-18, S-19, S-22, S-24, S-30, S-32, S-33, S-34, S-35, S-36, S-37, S-38, S-39, F-02, F-03, F-06, P-01, P-02, P-03

Students were asked to observe mechanisms (in class we had just few of mentioned above), and then tell which of them are producing the same movement (straight-line mechanisms). More about importance of producing straight-line motion see "How to draw a straight line"

Discussion about pumps - what are the function of pumps? What motion is used in pumps? Can these models be used not only for pumping?

Discussion about universal joints - how motion has been transferred through it? Why it is useful? Where have you seen the universal joint? Historical description of the universal joint

Class activity: Choose one of the mechanisms we explored in class, make a rough sketch of a mechanism and then put that mechanism in some kind of context. For example - try to draw pipes coming to a pump or to show some machine where straight line mechanism would work (steam locomotive, oil refineries). While they were drawing we got into talk about history of machines and I promised to send web links to them about history of machines. They were particularly interested about Archimedes and Leonardo da Vinci machines.


Week 4: Exploring old machine designs.

I had prepared a handout with 13 pictures from old books and students were figuring out how those machines worked. After that we read a description of those pictures from a book. Students were having a lot of fun trying to understand those pictures. I was also asking them to find in pictures simple mechanisms we have seen from Reuleaux collection.


This class can explore other online resources that are given below.


Week 5: What is creative design? Originals and copies.

We talked about problem solving strategies and how similar they are for math problems and engineering or any other problems where you need to do steps in your thinking. Then we made three teams of students and made a little competition which team will find more differences in originals and copyists versions of pictures. I let them to look for that about 10 minutes and then each team were showing one of the pictures and explaining what differences they found.


Next task for teams was "Building a windmill": build a windmill out of index cards, tape, string, pencil, pushpins and a cup. Use the windmill to lift as many pennies as possible from the floor to the top of a table using hairdryer. This turned out to be a real fun and it involved a lot of discussions about the most efficient windmill design.


  • Buhl, Harold R. Creative Engineering Design, The Iowa State University Press, Ames, Iowa, 1960.
  • Ramelli, Agostino, The various and ingenious machines of Agostino Ramelli (1588), Baltimore : Johns Hopkins University Press, 1976.
  • Florman Samuel C. The Existential Pleasures of Engineering, St. Martin's Griffin, New York, 1976, 1994
  • Harrisberger, Lee Engineermanship: A Philosophy of Design. Brooks/Cole Publishing Company, Belmont, CA, 1966
  • Inventors' handbook http://web.mit.edu/afs/athena.mit.edu/org/i/invent/h-main.html
  • Design Process http://www.bergen.org/technology/despro.html

Week 6: History of steam engine. Linkages.

I started with telling history about inventing steam engine and biography of James Watt and Matthew Boulton. Prepared handout "Steps toward steam engine".

Playing with linkages - how to draw straight line, how to use linkages for calculating. Calculation appeared to be too complicated with this class. But we did a panthograph that they were interested very much. They also tried to connect links making their own design of linkages.

Useful are Didax "Geo Strips" (can order from Didax for $36, but they have limited use because their holes are set in particular distances). But if you want to construct linkages for computation then it is better to cut them out from cardboard or plastic file folders. For fastening them together it is convenient to use eyelets and a tool for putting them in (available in stores like JoAnn Fabrics, look for section where there are tools for sewing jeans).


Learning Module


Week 7: Field trip.

For Ithaca area and nearby schools it is possible to organize a field trip to Cornell University to see actual Reuleaux kinematic model collection and to visit Sibley School of Mechanical and Aerospace Engineering. But it can be another field trip to some site with interesting engineering where students can explore working machines and try to locate, which parts of machine design they have explored and how they are incorporated in that particular machine. For example, it could be field trip to nearby mills or oil refineries, or some construction place.


Week 8: Making your own machine design.

I was letting students to do free drawings of their ideas. For inventing their own machine it was suggested to use some of Reuleaux kinematic models and incorporate them in design. They first had to decide what do they want their machine to do. Then analyze what kind of motions will be in this machine, then which models can do those motions and how to connect separate parts. Macaylay's book is a good way to start a discussion and its informal illustrations are freeing students for drawing.


Week 9: Presenting design of a machine.

This was a fun class of presentations. Students admitted that, of course, it would be much more fun if they could present not only drawings but also some models. It is possible to make models out of cardboard or some other easy accessible materials, like Lego. It would be good to have a lab for that.