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1、Available online at ScienceDirectMechanism and Machine Theory 43 (2008) 115-122MechanismandMachine TheoryMechanism analysis of a trisectorLyndon O. Barton*Delaware State University, Dover, Delaware, United States Received 20 July 2006; received in revised form 27 April 2007; accepted 17 October 2007
2、AbstractThis paper presents a graphical procedure for analyzing a trisector - a mechanism used for trisecting an arbitrary acute angle. The trisector employed was a working model designed and built for this purpose. The procedure, when applied to the mechanism at the 60 angle, which has been proven
3、to be not trisectable as well as the 45 angle for benchmarking (since this angle is known to be trisectable), produced results that compared remarkably in both precision and accuracy.For example, in both cases, the trisection angles found were 20.00000, and 15.00000, respectively, as determined by T
4、he Geometers Sketch Pad software. Considering the degree of accuracy of these results (i.e. five decimal places) and the fact that it represents the highest level of precision attainable by the software, it is felt that the achievement is noteworthy, notwithstanding the theoretical proofs of Wantzel
5、, Dudley, and others Underwood Dudley, A Budget of Trisections, Springer-Verlag, New York, 1987; Clarence E. Hall, The equilateral triangle, Engineering Design Graphics Journal 57 (2) (1993); Howard Eves, An Introduction to The History of Mathematics, sixth ed., Saunders College Publishing, Fort Wor
6、th, 1990; Henrich Tietze, Famous Problems of Mathematics, Graylock Press, New York, 1965. 2007 Elsevier Ltd. All rights reserved.Keywords: Mechanism analysis; Angle trisector; Four bar mechanism; Slider crank mechanism; Slider-coupler mechanism; Famous problems in mathematics1. IntroductionThe probl
7、em of the trisection of an angle has been for centuries one of the most intriguing geometric challenges for mathematicians 1-4. According to Underwood Dudley 1 author of A Budget of Trisections,Certain angles can be trisected without difficulty. For example, a right angle can be trisected, since an
8、angle of 30 can be constructed. However, there is no procedure, using only an unmarked straight edge and compasses, to construct one-third of an arbitrary angle.Dudley then proceeded to lay out a proof of this statement by showing that a 60 angle cannot be trisected. Also, in the same text, he refer
9、enced the work of Pierre Laurent, Wantzel, who in 1837 first proved that such trisection was impossible. Yet in a more recent paper by Hall 2, a proof was presented to show that a three toTel.: +1 302 366 8879.E-mail address: lynobar0094-114X/S - see front matter 2007 Elsevier Ltd. All rights reserv
10、ed. doi:10.1016/j.mechmachtheory.2007.10.005116L.O. Barton/Mechanism and Machine Theory 43 (2008) 115122one relationship between certain angles can exist for acute angles. However, Hall 2 did not develop or present a procedure for the trisection.The purpose of this paper is not to contradict or deba
11、te the established proofs alluded to, but to present a summary of the results obtained from my study of a trisector mechanism, which I designed and built as part of the study 5. It is hoped that these results as well as the approach used in developing them will provide others in the mathematics and
12、science community valuable physical insights into the nature of the problem.The results presented are based on an analysis, using unmarked ruler and compasses only (aided by The Geometers Sketch Pad software), of the 60 angle that has been proven to be not trisectable as well as the 45 angle that is
13、 known to be trisectable.2. TheoryThe proposed method is based on the general theorem relating arcs and angles.Let ZECG (or 30) be the required angle to be trisected. With center at C and radius CE describe a circle. Given that a line from point E can be drawn to cut the circle at S and intersect th
14、e extended side GC at some point M such that the distance SM is equal to the radius SC, then from the general theorem relating to arcs and angles,ZEMG = 1/2(ZECG ZSCM)2ZEMG + ZSCM = ZECG Since DCSM is an isosceles DZSCM = ZEMG = 0 Therefore 3ZEMG = ZECG or 30 = ZECG.3. Trisector design and analysisT
15、he trisector mechanism illustrated in Fig. 1 is a compound mechanism consisting of a four-bar linkage, CEDA, where CE is the crank, ED is the coupler, and DA is the follower, and a slider-crank linkage 6, CFVE, where CV is the crank, FE the connecting rod, and F the slider. The links for the four-ba
16、r and slider-crank are designed so that the pin joints are all located at equal distances apart, and both linkages are mounted on a common base and connected at fixed axis C, where the two cranks CE and CV meet, as well as at crank pin E, via a pivoting slot through which the connecting rod slides.
17、Thus, as the four-bar crank CE is rotated in one direction or the other, between the 90 and 0 positions, the connecting rod FE of the slider-crank divides the angle formed by said crank and coupler into a 2-1 ratio. Also, if crank CE is rotated, say in a clockwise manner, the connecting rod FE would
18、 be forced to undergo combined motion of translation, where sliding would occur at both ends, and rotation only at the pivoting slot end. Meanwhile crank CV of the slider-crank would be forced to rotate, but in a counterclockwise manner.By assuming slider F is not constrained (or not restricted by t
19、he slot), while other parts of the mechanism are in motion, the mechanism behaves like a sliding coupler mechanism 6 where, it was possible to show thatL.O. Barton/Mechanism and Machine Theory 43 (2008) 115122117m/ ECG = 90. OOP 00 mZETA = 30.00000CE = 6.25437 cm FOLLOWER VC = 6.25437 cm VF = 6.2543
20、8 cmA FIXED AXISFIXED HORIZON SLOT WITH SLIDERC FIXED AXIS90X30FIGURE1Fig. 1. Trisector shown at 90 angle, where cade is A four-bar mechanism, coupled to A slider crank mechanism CVFE at fixed axis C and at E, where connecting rod FE slides as well as Pivots as its slider end F moves within a fixed
21、slot along the ground AC.the path point T or joint at slider F is practically a smooth circular path (See Appendix Fig. A1). This path intercepts the straight path that F would normally describe, when constrained within the slot, at a unique point. Further, the analysis will show that this point loc
22、ates the vertex of the angle formed by the connecting rod and said slot, which represents the required trisection angle.4. Procedure for Figs. 1A or 2AReferring to Figs. 1A and 2A let CEDA and CVFE represent the four bar and slider crank components, respectively of the trisector, as shown in the 90
23、position.EmZECG = 60.00000HFH = 1 4.97404 cm FE = 1 4.97404 cmTV =6.25438 cm = 6.25437 cmDECVDEDE = 6.25437 cm/FE= 1 6.00374 cm0- / .TE =wv16.00374 cm = 1.02970 cmV/ FT =1.02970 cm/ HE= 1.02970 cmVT*f W/ T60X20FIG1AFig. 1A. Trisector at 60 angle with sliding end of the connecting rod FE is unconstra
24、ined while crank CV moves to an intermediate position, CV.118L.O. Barton/Mechanism and Machine Theory 43 (2008) 115122mECG = 45.00000FW = 6.25437 cm TV = 6.25437 cm TE = 17.21238 cm FE = 17.21238 cm VW = 1.32886 cm FT = 1.32886 cm HE = 1.32886 cm FH = 15.88351 cm FE = 15.88351 cm DE = 6.25437 cm45X1
25、5FIGURE2AFig. 2A. Trisector at 45 angle with slider end of connecting rod FE unconstrained, while crank CV is assumes an intermediate position, CV.1. With center at C and radius CE, construct an arc from point E to point G on the ground to represent the path of E as crank CE of the four-bar is rotat
26、ed between its normal position at 90 and the ground at 0.1. Draw the four-bar in its 60 or 45 position where ZECG is equal to 60 or 45.1. Assuming point F to be a fixed axis, temporarily, join points F and E with a segment FE to represent the connecting rod segment FH when rotated about F to its new
27、 position dictated by crank CE.2. Still assuming point F to be fixed, but crank CV disconnected from the connecting rod, construct an arc cutting FE at W. The arc VW then represents the path of V as the rod segment FH is rotated to FE.3. Now, assume that F is not fixed, and also not constrained by t
28、he fixed slot, then the rod FE will change position to become TE, while crank CV will be rotated accordingly to a new position. Therefore,(a) extend segment FE from point F to point T to represent HE (i.e. the portion of the connecting rod that slides within the pivoting slot at E),(b) with center a
29、t C and radius CV, describe an arc from V cutting TE at point V. The arc VV will represent the path of V between its original position on FE to its final position on TE.Note that in the new configuration of the linkage, because F is assumed to be unconstrained or free to move, point T falls below th
30、e ground, and TE represents the new position of the connecting rod FE. Therefore TE = HE = VW, where VW represents the change in positions of point V along the relocated connecting rod TE.5. Procedure for Figs. 1B or 2B1. With the linkage in any given acute angular position (for example, 60 or 45 be
31、ing considered in this paper), join points A and V with a segment AV.2. From point T, construct a line TX parallel to AV and from the same point another line TY perpendicular to ground AC.3. Bisect the angle formed by lines TX and TY above and define the point where the bisector intersects ground AC
32、 as point P.4. Connect point E to point P with segment EP and define intersection of this segment with the circular path AV as point R.5. With center at R and radius RC, construct an arc to cut RP at N, where N defines the unconstrained end of the connecting rod in some intermediate position.L.O. Ba
33、rton/Mechanism and Machine Theory 43 (2008) 11512211960X20FIGURE1BFig. 1B. Same as Fig. 1A with crank CV positioned between points V0 and V, namely, at R.v fmECG = 45.00000RC = 6.25437 cmzXDEYV i*0000/ / N- /N/- TA CG45X15FIGURE2BFig. 2B. Same as Fig. 2A, with crank CV positioned between points V0 a
34、nd V, namely, at R.6. Procedure for Figs. 1C or 2C1. Referring to Figs. 1C or 2C, join points N and T with segment NT and construct a bisector of this segment to cut link extension E0C at point O, thereby establishing the center of the circular path of the unconstrained end of the connecting rod.120
35、L.O. Barton/Mechanism and Machine Theory 43 (2008) 115122mZECG = 60.00000 mZEMA = 20.00000Fig. 1C. Same as Fig. 1B, with circular path through T and N added to define point M the vertex of the required angle E0 MA.mZECG = 45.00000mZEMA = 15.0000045X15 FIGURE 2CFig. 2C. Same as Fig. 2B with circular
36、path through points T, and N added to define point M the vertex of the required angle, E0MA.2. With center at O and radius TO, construct an arc to intersect ground AC and define the point of intersection as point M. This point locates the vertex of the angle formed by the connecting rod and the grou
37、nd AC.3. Join points E0 and M to form segment E0M and angle E0MA, which represents the required trisection angle (note that E0M cuts arc AV at S).L.O. Barton/Mechanism and Machine Theory 43 (2008) 115122mECG = 60.00000 mEMA = 20.00000121D, EFAG60X20FIGURE1DFig. 1D. Final position of linkage showing
38、resultant angle. E0MA same as LCG.mZECG = 45.00000 mZEMA = 15.00000DECG45X15FIGURE2DFig. 2D. Final position of linkage showing resultant angle ZEMA same as ZLCG.Figs. 1D or 2D show the final positions of the linkages with their connecting rods forming their respective trisection angles.7. SummaryA g
39、raphical procedure for analyzing a trisector mechanism has been presented. The procedure, when applied to the mechanism at the 60 angle, which has been proven to be not trisectable, as well as the 45 angle, which is known to be trisectable, has yielded results that compared remarkably in both precis
40、ion and accuracy.For example, the results, as determined by The Geometers Sketch Pad software, for the angles 60 and 45 were, respectively 20.00000 and 15.00000 (i.e. five decimal places) which represented the highest level of precision attainable by the software. Thus, it is felt that this degree o
41、f accuracy is not only significant, but noteworthy, notwithstanding the theoretical proofs by Wantzel, Dudley, and others 1-4.122L.O. Barton/Mechanism and Machine Theory 43 (2008) 115122Fig. A1. Figure shows path of point T when it is not constrained to move within the fixed horizontal slot.Appendix
42、See Fig. A1. References1 Underwood Dudley, A Budget of Trisections, Springer-Verlag, New York, 1987.2 Clarence E. Hall, The equilateral triangle, Engineering Design Graphics Journal 57 (2) (1993).3 Howard Eves, An Introduction to The History of Mathematics, sixth ed., Saunders College Publishing, Fort Worth, 1990.4 Henrich Tietze, Famous Problems of Mathematics, Graylock Press, New York, 1965.5 Lyndon O. Barton, Another device for trisecting an angle, Mathematical Spectrum 38 (January) (2006).6 Lyndon O. Barton, Mechanism Analysis, second ed., Marcel Dekker, NY, 1993.
