Mechanism analysis of a trisector1

1、mechanism analysis of a trisectorlyndon o. barton*delaware state university, dover, delaware, united statesabstractthis 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 b

2、uilt for this purpose. 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 for benchmarking (since this angle is known to be trisectable), produced results that compared remarkably in both precision and accuracy.for exampl

3、e, in both cases, the trisection angles found were 20.00000, and 15.00000, respectively, as determined by the 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

4、 software, it is felt that the achievement is noteworthy, notwithstanding the theoretical proofs of wantzel, 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); h

5、oward eves, an introduction to the history of mathematics, sixth ed., saunders college publishing, fort worth, 1990; henrich tietze, famous problems of mathematics, graylock press, new york, 1965.keywords: mechanism analysis; angle trisector; four bar mechanism; slider crank mechanism; slider-couple

6、r mechanism; famous problems in mathematics1. introductionthe problem 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

7、 difficulty. for example, a right angle can be trisected, since an 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 th

8、at a 60 angle cannot be trisected. also, in the same text, he referenced 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 to. one relationship between certain angles can

9、 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 debate 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

10、 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 science community valuable physical insights into the nature of the problem.the results presented are based on an analysis, using unmarked ruler an

11、d 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 known to be trisectable.2. theory the proposed method is based on the general theorem relating arcs and angles.let zecg (or 30) be the required an

12、gle 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 the 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,

13、 3. trisector design and analysisthe 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

14、 slider. the links for the four-bar 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

15、 which the connecting rod slides. 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 m

16、anner, the connecting rod fe would 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 c

17、onstrained (or not restricted by the 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 that the path point t or joint at slider f is practically a smooth circular path (see appendix fig. a1). this path int

18、ercepts 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 locates the vertex of the angle formed by the connecting rod and said slot, which represents the required trisection angle.4. procedure for f

19、igs. 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 position 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

20、of the four-bar is rotated between its normal position at 90 and the ground at 0.2. draw the four-bar in its 60 or 45 position where zecg is equal to 60 or 45.3. 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 ro

21、tated about f to its new position dictated by crank ce.4. 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.5. now, assume that f is not fixed, and a

22、lso not constrained by the 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

23、 at e),(b) with center at 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

24、, point t falls below the 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 (

25、for example, 60 or 45 being 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 bisec

26、tor intersects ground ac 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.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 intermed

27、iate position. 6. procedure for figs. 1c or 2creferring 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. 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 forme