1. ATM Conference 2006, Ormskirk Mark Dabbs (mark@mfdabbs.com)mark@mfdabbs.com

2. Watt’s Parallel Motion The best known of the approximations to straight line motion is that of Scottish inventor and engineer James Watt’s 4 Bar Linkage, invented in 1784. It took Watt several years to design a straight-line linkage that would change straight-line motion into circular motion. Years later Watt told his son: "Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other mechanical invention I have ever made." [Fergusson 1962]

3. Watt’s Parallel Motion When in mid-position the bars AB and CD are parallel, and BC is perpendicular to both. Point E is chosen at any point within the bar BC, whose displacement during a small rotation of the bars AB and CD is will be an approximate straight line. Since the diagram shows the mechanism at its mid-position we can assume that BC represents the direction of the line generated by point E.

4. Watt’s Parallel Motion Ranges of motion of bars AB and CD are shown.

5. The exact position of E’ is found by first assuming a small rotation of the bars so that B moves to B’ and C to C’. Let B’C’ cut BC or BC (produced) in E’. Then E’ will be the generating point for the straight line. To prove this we therefore need to show that the ratio is constant for all small rotations of AB and CD about their mid-positions. Watt’s Parallel Motion

6. At these rotation angles of AB and CD we see that point E’ is just about to diverge from its approximate straight line path. However, the angle has been greatly enlarged for for clarity in this and subsequent diagrams.

7. Watt’s Parallel Motion Proof: Draw B’H perpendicular to CB produced and C’K to BC. Draw B’F perpendicular to AB and C’G to CD.

8. Watt’s Parallel Motion Proof: Draw B’H perpendicular to CB produced and C’K to BC. Draw B’F perpendicular to AB and C’G to CD. Therefore, triangles B’E’H and C’E’K are similar. Hence

9. Watt’s Parallel Motion By Pythagoras:

10. Watt’s Parallel Motion For small rotations of the bars AB and CD it may be assumed that B’F be equal to C’G, despite the differing lengths of AB and CD. Hence, dividing (2) by (3) gives: Therefore, Substituting (4) into (1) gives as required, since AB and CD are the fixed lengths of the rotating bars.

11. Watt’s Parallel Motion In addition to the above work proving the point E generates a straight line approximation for small angles of rotation of the bars AB and CD, we also have a condition for the best possible position of point E within the link BC. Namely: That is, E divides the bar BC into the same ratio as the rotating bar lengths

12. Watt’s Beam Engine Watt’s Linkage Pantograph

13. The Double-Acting, Triple Expansion Steam Engine. The steam travels through the engine from left to right The engine shown in the picture below is a Double-Acting steam engine because the valve allows high-pressure steam to act alternately on both faces of the piston. The animation below it shows the engine in action: High-pressure steam inExhaust steam out

14. Hyperlinks to Euclidraw Dynamic Linkage Files Straight Line Approximation EUC Files Background Theory EUC Files Exact Line Generator Mechanism EUC Files Exact Translator Mechanism EUC Files Link 1 to EUC FileLink 10 to EUC FileLink 14 to EUC FileLink 17 to EUC File Link 2 to EUC FileLink 11 to EUC FileLink 15 to EUC FileLink 18 to EUC File Link 3 to EUC FileLink 12 to EUC FileLink 16 to EUC FileLink 19 to EUC File Link 4 to EUC FileLink 13 to EUC FileLink 20 to EUC File Link 5 to EUC FileLink 21 to EUC File Link 6 to EUC FileLink 22 to EUC File Link 7 to EUC FileLink 23 to EUC File Link 8 to EUC FileLink 24 to EUC File Link 9 to EUC FileLink 25 to EUC File

15. The Harmonic Range If A, B be two points on a straight line, and C, D two other points on the line, placed such that then the points A, C, B and D form a Harmonic Range; and C and D are Harmonic Conjugates with respect to A and B. The lengths AC, AB and AD are said to be in Harmonic Progression; and AB is said to be the Harmonic Mean between AC and AD. AC BD (Notice: All segments are directed. Thus, etc) --(1) Eq. (1) can be written in the form: Therefore, in order that the above proportion hold, it can be seen that one, and only one, of the points C, D must lie between A and B. For example:

16. AC BDO rr Extension: For the Harmonic Range A, C, B and D; set point O as the midpoint of AB. Let |OA| = |OB| = r. Then can be rewritten in terms of r as: which simplifies to the important result:

17. Ceva’s and Menelaus’ Theorems Ceva’s Theorem:

18. Proof of Ceva’s Theorem

20. Ceva’s Theorem (The Converse):

21. Menelaus’ Theorem:

22. Proof of Menelaus’ Theorem

24. The Complete Quadrangle In triangle ABQ Ceva’s Theorem states: In triangle ABQ Menelaus’ Theorem states: Dividing these results gives: Therefore, the “Complete Quadrangle” is a simple geometrical construction that creates the Harmonic Range A, C, B and D. Link 26 to EUC File

25. Software All of my geometry files were created with EuclidrawEuclidraw www.euclidraw.com Created by: Prof. Paris Pamfilos, University of Crete, Department of Mathematics pamfilos@math.uoc.gr