Ettore Pennestrì Pier Paolo Valentini Dipartimento di Ingegneria Meccanica Università di Roma Tor Vergata Roma - Italy May 3, 2009

Transcription

1 A review of simple analytial methods for the kinemati synthesis of four-bar and slider-rank funtion generators for two and three presribed finite positions Ettore Pennestrì Pier Paolo Valentini Dipartimento di Ingegneria Meania Università di Roma Tor Vergata Roma - Italy May 3, Introdution Kinemati synthesis is an useful tool mainly used in the first stages of mehanisms design. The possibility to obtain through omputation the dimensions of a linkage whose parts reprodue a presribed motion allows to spare time and inrease auray. It is well known that time onsuming trial-and-error kinemati synthesis tehniques are often adopted at the industry level. This situation is aused by different reasons suh as: - Many useful kinemati synthesis methods are sattered in tehnial literature whih spans a time interval of deades. These methods are not always available on line and their retrieval often requires a manual library searh. - Most of the university textbooks onentrate on the theory of few lassial methods of kinemati synthesis. Considered their didati purposes, the desription of the methods is usually made with abundane of details. Hene, the style of desription is far to be onise and essential, as the one usually adopted in manuals dediated to industrial designers. The main purpose of this paper is to provide industrial designers with readyto-use kinemati synthesis methods. For this reason, the desription format adopted is onsistent with this purpose. In fat, the methods have been summarized in an algorithmi way and with expliative numerial examples provided. 1

2 Eah method has been also programmed in the sripting language Ch 1. This work has been written to provide the engineer with simple and readyto-use kinemati synthesis proedures whih an assist him in the early stages of mehanisms design. The transmission angle No design problem admits a unique solution. Therefore, quality indies must be adopted for sreening out non optimal solutions. Often a ompromise solution should be searhed in order to satisfy simultaneously more than one optimality riterion. It is of paramount importane the omparison of design alternatives on the basis of the quality of motion transmitted. One of the oldest performanes index is the transmission angle and has been introdued by H. Alt in t a µ t r µ Input link B Output link Figure 1: Definition of transmission angle The definition of the transmission angle is as follows (see Figure 1) Let B be the point where the fore is transmitted to the output link. The transmission angle µ is the angle between the diretion t a of the veloity of B of the output link and the diretion t r of the relative veloity of B with respet to the driving link. It an be observed that: - µ = 90 is the most favorable value of the transmission angle; - the transmission of motion is theoretially impossible when µ = 0 and µ = 180 (loking ondition); 1 Ch is a C/C++ interpreter for ross-platform sripting, shell programming, D/3D plotting, numerial omputing. Ch is a free and user-friendly alternative to C/C++ ompilers. Ch an be downloaded from The listings of the Ch programs developed are available for download at the following web page

3 - it is therefore signifiant the differene 90 µ instead of the absolute value of µ (in other words, aording to the disussed riterion, a mehanism with µ = 50 has the same merit as one with a value of µ = 130 ); - the loation of µ depends on the way in whih the mehanism is driven. With referene to Figure, let µ be the inner angle between the oupler and output lever. B Input link a A Coupler b µ Output link d Figure : Transmission angle in a four-bar linkage In the absene of frition, gravity and inertia: the oupler an transmit only a tensile fore F; the torque transmitted to the output link of length L is FL sinµ; a loking ondition ours when the oupler and the output lever are aligned. This suggest that, for satisfatory operation, the value of µ should not get values smaller than 30. In addition to fore transmission there is another reason to omply with the rule above. A poor transmission angle makes the position of the output lever very sensitive to learane in the joints, manufaturing toleranes on link lenghhs, and hanges in dimension due to thermal expansions [8, 9]. The expression ( b + ) ( a + d ) + adosφ osµ = (1) b is valid for all onfigurations of the mehanism. If the mehanism an reah the onfigurations shown in Figure 3, then the minimum and maximum values of the transmission angle an be omputed as follows: [ ] b + (1 a) µ min = aos, (a) b [ ] b + (1 + a) µ max = aos. (b) b 3

4 B 1 B µ max µ min b a A A 1 d Figure 3: Configurations of minimum and maximum transmission angle The transmission angle of a slider rank (see Figure 4): t a r Input link θ l µ t r t r µ µ ε µ t a Input link Figure 4: Loations of transmission angle in a slider rank - with the rank as the input link, is the angle between the oupler and the line perpendiular to the veloity of the slider; - with the slider as the input link (e.g. internal ombustion engine) is the angle between the oupler and the rank. Let us denote with r, the rank length, l the oupler length, ε, the slider-rank offset, θ the rank angle. In the first ase, the value of the transmission angle at a generi position is given by µ = artan ε + r sin θ π (3) l (ε + r sinθ) 4

5 The minimum value of the transmission angle, attained for θ=90, is ( ) l µ min = artan 1 (4) r + ε There is not a general onsensus on the most appropriate quality index of transmission motion in the presene of fores. The index should aount for the true fores ating on the mehanism. Here are some more general definitions of the transmission angle [7]: - the angle whose tangent is the ratio of the tangential fore to the normal fore on the output link (proposed by W. Rössner); - the angle between the diretion of the fore on the output link and the diretion of the absolute motion of the point of appliation of the fore on the output link (proposed by E. Lenk). 3 Quik-return rank-roker four-bar linkages In the design of mehanisms it is often neessary to design a linkage that will onvert a rotating input motion into an osillating output motion. The methods herein desribed will permit the design of four-bar linkages of this nature with a onsiderable auray. These methods, however, are limited to roker osillation angles less than 180. With the aronym Type FOURB-CR we will denote all the proedures for the design of a plane rank-roker mehanism for presribed values of the osillating angle ψ of the output link and angle 180 +θ of the input rank. (see Figure 6). The angle θ is positive when..w. The rank-roker an provide a quik-return motion. Assuming uniform angular speed of the rank, time ratio T R is defined as the ratio of the time taken for the forward stroke to the time taken for the return stroke. Its value is given by TR = Duration of forward stroke Duration of return stroke = θ 180 θ. (5) The reader may find useful the plot in Figure 5 to quikly obtain θ as a funtion of TR. The optimal design of these type of four-bar linkages an be stated as follows [5]: Determine the rank-roker proportions of a four-bar mehanism with a given swing angle ψ and orresponding rank rotation θ+180, or time ratio, suh that the maximum deviation of the transmission angle from 90 is minimum. 5

6 TR θ (deg) Figure 5: Angle θ.vs. Time ratio 3.1 Algorithm FOURB-CR-1 Purpose: Design of a plane rank-roker mehanism for presribed values of the osillating angle ψ of the output link, of angle π + θ of the input rank and initial rank angle θ 0. (see Figure 6 for the nomenlature). B θ b ψ B 1 A A 0 a A ψ 1 0 θ 0 d=a 0 B 0 B 0 Figure 6: Nomenlature Data: Angles ψ and θ Unknowns: Link lengths ratios a/d, b/d and /d Soure: [, 11] Computational proedure 1. Assume a onsistent value for the angle θ 0 6

7 . Let n 1 = sin θ 0 n = osθ 0 n 3 = sin (θ + θ 0 ) n 4 = sin (ψ θ θ 0 ) n 5 = os(ψ θ θ 0 ) 3. Compute [ ] n1 (n 3 + n 4 ) ψ 0 = atan n n 3 n 1 n 5 4. Let p 1 = sin ψ 0 sin θ 0 p = sin (ψ + ψ 0) sin (θ + θ 0 ) p 3 = sin (ψ 0 θ 0 ) 5. Compute d = 1 = sin θ 0 p 3 a = (p 1 p ) b = (p 1 + p ) The extremes values µ min e µ max of the transmission angle for the synthesized mehanism follow from the equations [ ] b + (1 a) µ min = aos, b [ ] b + (1 + a) µ max = aos. b The proedure an be repeated for different values of the angle θ 0 until satisfatory extremes values of the transmission angle are obtained. 7

8 Numerial example Let θ=10, ψ=50 and θ 0 =30. The previous formulas give ψ 0 =81.05 and a = b = = d = 1 The extreme values of the transmission angle are Program: FOURB-CR-1.h µ min = 47.0 µ max = Algorithm FOURB-CR- Purpose: Design of a plane rank-roker mehanism for presribed values of the osillating angle ψ of the output link, angle π + θ of the input rank and initial roker angle ψ 0 (see Figure 6 for the nomenlature). Data: Angles ψ and θ Unknowns: Link lengths ratios a/d, b/d and /d Soure: [, 11] Computational proedure 1. Assume a onsistent value for the angle ψ 0. Let θ f = θ y = ψ 0 + ψ θ f m = sinψ 0 sin θ f m 1 = sin y + osψ 0 sin θ f sin ψ 0 osθ f m 0 = osψ 0 osθ f osy 3. Solve w.r.t. x m x + m 1 x + m 0 = 0 4. If the roots are omplex the initial data are not onsistent. 5. For eah root x, let θ 0 = atan(x) 6. Let p 1 = sin ψ 0 sin θ 0 p = sin (ψ + ψ 0) sin (θ + θ 0 ) p 3 = sin (ψ 0 θ 0 ) 8

9 7. Compute link legths d = 1 = sin θ 0 p 3 a = (p 1 p ) b = (p 1 + p ) 8. Aept only the solution with all positive values of link lengths. The extremes values µ min e µ max of the transmission angle for the synthesized mehanism follow from the equations [ ] b + (1 a) µ min = aos, b [ ] b + (1 + a) µ max = aos. b The proedure an be repeated for different values of the angle θ 0 until satisfatory extremes values of the transmission angle are obtained. Numerial example Let θ=36.5, ψ=85 and ψ 0 =70. The previous formulas give θ 0 =33.89 and a = b = = d = The extreme values of the transmission angle are Program: FOURB-CR-.h µ min = 4.3 µ max = Algorithm FOURB-CR-3 Purpose: Determine the rank-roker proportions of a four-bar mehanism with a given swing angle ψ and orresponding rank rotation θ + 180, or time ratio, suh that the maximum deviation of the transmission angle from 90 is minimized (see Figure 6 for the nomenlature). Data: Angles ψ and θ 0 Unknowns: Link lengths ratios a/d, b/d, /d, maximum deviation of the transmission angle from 90. Soure: [5, 4] Computational proedure 9

10 1. Let θ f = θ Solve the ubi equation w.r.t. x t = tan θ f ( ) θf ψ u = tan x 3 + x t x t ( 1 + t ) u = 0 3. Let λ = t x and, if you want a rank-roker mehanism, aept only those solutions whih satisfy the inequality 1 λ u t 4. For all the feasible values of λ ompute a = sin ψ b = λa = sin θ f + λ os θ f d = sin θ f ψ + λ os θ f ψ 5. Among all the feasible values of λ hoose the solution with the least value of (a ± d) b sin = b where 0 90, + sign if θ < 0, sign if θ > 0. Numerial example Let θ=-10, ψ=45. The previous formulas give a = b = = d = 1 The extreme values of the transmission angle are : Use (), p.4 µ min = µ max =

11 Thus, the maximum deviation of the transmission angle from the optimal value of 90 is = For the ase of TR = 1 (i.e. θ = 0) the method does not give feasible results and algorithm FOURB-CR-4 should be used. Program: FOURB-CR-3.h 3.4 Algorithm FOURB-CR-4 Purpose: Design a rank-roker four-bar linkage with presribed unit time ratio TR = 1, roker osillating angle ψ and minimum value µ min of the transmission angle. A mehanism with unit time ratio is said entri four-bar linkage (see Figure 7). Data: Angles ψ and θ = 0, 0 < µ min < ( 90 ψ ) a b ψ d Figure 7: Centri four-bar linkage Unknowns: Link lengths ratios a/d, b/d and /d Soure: [13] Computational proedure 1. Compute b d = 1 osψ os µ min d = a d = 1 ( ) b d 1 ( b d) os µ min (b ) ( ) + 1 d d 11

12 Numerial example Let ψ=50 and µ min = 30. The previous formulas give a = b = = d = 1 The maximum value of the transmission angle is (use last of eqs. ()) = Program: FOURB-CR-4.h Design harts for this proedure have been prepared and inluded in another setion. Alternatively, the problem an be solved by means of the formulas dedued by J.G. Volmer 3. d a = os µ min ot ψ os µ min sin ψ, (6a) b a = os ψ os µ min sin ψ a = 1 sin ψ 4 Drag-link four-bar linkages, (6b). (6) One of the most ommon appliations of the drag-link four-bar mehanism is as a oupling between two parallel shafts. It provides 1 to 1 overall speed ratio while generating, over the yle, a nonuniform rotation of the output shaft. These mehanisms usually have favorable motion transmission harateristis. 4.1 Algorithm DRAGLINK-1 Purpose: Design a drag-link mehanism with design positions orresponding at input angles φ 1 =0 and φ =180 : - with the same minimum value (osµ min = osµ max ) of transmission angle at design positions; - with a presribed rotation ψ 0 of the output link. The presribed rotation of the input link is φ = φ φ 1 =180 (see Figure 8). Data: ψ 0, µ min. Unknowns: Link lengths ratios a/d, b/d and /d. Soure: [1] Computational proedure 3 A modern soure whih reports the formulas is [10], p

13 B µ max = b A A 1 d a A0 ψ 0 B 0 µ min µ min B Figure 8: Nomenlature 1. Let =1;. Compute 4 sin ψ 0 b = sin ( ψ 0 µ min ) ( ) b sinµmin λ = atan + b osµ min a 1 = b sin µ min sin λ b 1 = os ψ 0 + b os( ψ 0 µ min ) osλ a = a 1 + b 1 d = a 1 a Numerial example Design a drag-link four-bar for ψ 0 =10 and µ min =45. The formulas listed above give ( = 1) Therefore, the link ratios are b = λ = a 1 = b 1 = a = b = a/d = b/d= /d = If the term under square root is negative no solution exists. 13

14 Design harts for this problem are given in [1] and [1]. These harts have been also inluded in this paper. Program: DRALINK1.h 4. Algorithm DRAGLINK- A 1 b B 1 a φ 1 ψ φ A 0 B 0 d=a 0 B 0 A B Figure 9: Drag-link mehanism at design positions Purpose: Design a drag-link mehanism in its two unity veloity ratio positions for presribed rank rotations, φ and ψ, and minimized maximum deviation µ max of the transmission angle from 90 (see Figure 9). The method of drag-link design previously desribed often introdues exessive veloity and aeleration. This is due to the 180 input link rotation arbitrarily hosen for the design. The mehanisms so ahieved are not used for their maximum apaity. Therefore, this method approahes the problem with a different design riterion. Data: Input link rotation φ, Output link rotation ψ. Unknowns: Link lengths ratios a/d, b/d and /d and input rank angle φ 1 at the first design position. Soure: [15, 16] Computational proedure(without transmission angle optimization) 14

15 1. Let 5 L = φ ψ ( ) φ t = tan ( ) ψ u = tan ( ) L v = tan. Assume 3. Compute link lengths λ = Length of oupler Length of input rank v d = 1 + v u + λ a = 1 + u λ v b = 1 + v t + λ = 1 + t Computational proedure (with transmission angle optimization) 1. Let. Solve the ubi equation w.r.t. x L = φ ψ ( ) φ t = tan ( ) ψ u = tan ( ) L v = tan x 3 + x t x t ( 1 + t ) 5 For drag-link proportions the limits on φ and L are The algorithm fails when φ = L 180 u 90 + L φ 70 + L 15

16 3. Let λ = t x and aept only those solutions whih satisfy the inequality 1 λ u t 4. For eah real feasible solution of the ubi let ǫ = λ ut 5. Compute link lengths ratios a d = sin( ψ/) 1 + t ǫ sin ( L/) b d = utǫ d = sin ( φ/) sin( L/) 1 + u ǫ 6. Compute maximum deviation µ max of transmission angle from 90 sin µ 1 = sin a + d b + da b sin µ = sin a + d b da b µ max = max (µ 1, µ ) 7. Among all the feasible solutions hoose the one with the minimum value of µ max. 8. Compute the initial angular position of input rank r = b os( ψ/) a sin( L/) s = d sin ( ψ/) a sin ( L/) φ 1 = ATAN (r, s) φ F. Freudenstein 6 demonstrated, by means of a kinemati inversion, the orrespondene between the design of a drag-link mehanism and the one of a rank-roker mehanism. 6 See his disussion in the paper authored by L.W. Tsai [15]. 16

17 Numerial example Design a drag-link four-bar for φ=170 and ψ=130. The formulas listed above give λ opt = and a d = b d = d = The maximum deviation of the transmission angle is µ max = The input rank angle at first design position is φ 1 = This design proedure an also be arried out by means of design harts. Program: DRALINK.h 5 Funtion generator four-bar: positions In the previous setions have been presented speialized algorithms for the oordination of rotation of input and output links of rank-roker and drag-link mehanisms. Both input and output links are adjaent to the frame. In this setion a similar problem will be approahed, but for a general fourbar linkage. The type of four-bar linkage is not predetermined, as in the ases previously disussed. 5.1 Algorithm FOURB-FG-1 Purpose: Design a four-bar linkage suh that the absolute angular displaements φ j and ψ j of input and output links are oordinated. The initial angular positions of driving and driven links are denoted by φ 0 and ψ 0, respetively.(see Figure 10). Data: φ j and ψ j with φ j ψ j, φ 0, a/d, /d Unknowns:Link lengths ratio b/d and ψ 0. Soure: [3] Computational proedure 1. Let. Compute ξ = φ j ǫ j = ψ j t = tanφ 0 q = tanψ 0 d = 1 sin ǫ j r aj = sin (ξ j ǫ j ) sinξ j r bj = sin (ξ j ǫ j ) 17

18 B φ j A b ψ j a φ ψ 0 0 A 0 d = A 0 B 0 B 0 Figure 10: Nomenlature 3. Choose a value for a and φ 0 and ompute X = 1 a osφ 0 and t = tanφ 0 4. Compute a 1 = r aj osǫ j a osφ 0 t sin (ξ ǫ) + os(ξ ǫ) b 1 = r aj sin ǫ j a osφ 0 t os(ξ ǫ) sin (ξ ǫ) 1 = r bj (t osξ + sinξ) 5. Solve w.r.t.ψ 0 a 1 sin ψ 0 + b 1 osψ = 0 6. Compute x A = a osφ 0 y A = a sin φ 0 x B = d + osψ 0 y B = sin ψ 0 b = Numerial example Design a four-bar linkage suh that: Rotation of driving link: φ j =60 Rotation of driven link: ψ j =30 (x B x A ) + (y B y A ) 18

19 Initial position of driving link: φ 0 =50 Link length ratios: a/d = , /d = The proedure gives two different solutions: Solution n.1: b/d = and ψ 0 = Solution n.: b/d = and ψ 0 = Program: FOURB-FG-1.h 6 Funtion generator four-bar:3 positions 6.1 Algorithm FOURB-FG3-1 Purpose: Design a four-bar linkage so that the output angles φ 1, φ and φ 3 are oordinated with the input angles ψ 1, ψ and ψ 3, respetively. (see Figure 11 for the nomenlature). Data: φ 1, φ, φ 3 and ψ 1, ψ, ψ 3 φ 3 φ a b φ 1 d ψ 3 ψ 1 ψ Figure 11: Freudenstein s equation:nomenlature Unknowns: Link lengths ratios a/d, b/d and /d. The values of the transmission angles at eh position are also omputed. Soure: Reported in almost all books on kinematis synthesis published after The algorithm is due to F. Freudenstein. The book [6] reports a detailed algebrai treatment and an interesting analysis of errors. Computational proedure 19

20 1. For i = 1,, 3, ompute A i1 = osφ i A i = osψ i A i3 = 1 B i = os(φ i ψ i ). Solve w.r.t. k the system [A] {k} = {B} 3. Compute d = 1 d = 1 k 1 a d = 1 k b = a + + d ak 3 4. Compute transmission angle values at eah design position ( b + ( a + d ) ) + adosφ i µ i = aos b Numerial example The values of input-output angles are summarized in Table 1 i φ i ψ i Table 1: Four-bar funtion generator: Input-output angles. The set of equations [A] {k} = {B} is k 1 k k 3 = (7) 0

21 Solving the system we have k 1 = k = k 3 = Therefore, the proportions of the four-bar are a/d = b/d = /d = The values of the transmission angle in the design positions are µ 1 = µ = µ 3 = Program: FOURB-FG3-1.h 7 Funtion generator slider-rank: and 3 positions 7.1 Algorithm SLID-FG-1 Purpose: Design a slider-rank for presribed range of motion of the slider and of the driving rank. (see Figure 1 for nomenlature) r l θ 1 θ ε s s 1 Figure 1: Funtion generator slider-rank: Nomenlature Data: θ = θ θ 1 + π, range of rotation of the rank, s = s 1 s, slider displaement between design positions. Unknowns: ε, slider-rank offset, r, rank length and l, oupler length. Soure:[] Computational proedure 1. Assume ε 1

22 . Compute s 1 = 1 s + 1 s 4 (ε s ot θ + ε ) s = s 1 s 3. Compute ) θ 1 = artan ( εs1 θ = θ 1 + θ + π 4. Compute r = 1 ( s1 s ) osθ 1 osθ l = 1 ( s1 + s ) osθ 1 osθ The hoie of ε ould be tied to an optimum value of the transmission angle µ. For this purpose the values of µ and µ min an be respetively omputed as follows: µ = artan ε + r sin θ π l (ε + r sinθ) ( ) l µ min = artan 1 r + ε Numerial example Design a slider rank mehanism with θ =145 and s=3.30. If ε = 0.9 is assumed, then r = , l =.768, θ 1 = -1.9, and θ = 31.71, µ min = 8.5. Program: SLIDFG1.h Graphial synthesis proedures are reported by J. Volmer [17] 7 and D.C. Tao [14] 8 7. Algorithm SLID-FG- Purpose: Design a slider-rank for presribed dead-enter positions. (see Figure 1 for nomenlature). The formulas are those reported in the previous subsetion, but used with a different sequene. 7 see p see p.36-43

23 Data: θ 1, θ, rank angular positions at dead-enter onfigurations, s 1,s, slider position at dead-enter onfigurations. Unknowns: ε, slider-rank offset, r, rank length and l, oupler length, µ min, minimum value of transmission angle. Soure:[] Computational proedure 1. Compute. Compute 3. Compute r = 1 ( s1 s ) osθ 1 osθ l = 1 ( s1 + s ) osθ 1 osθ ε = s 1 tanθ 1 ( ) l µ min = artan 1 r + ε Numerial example Design a slider rank mehanism if the following dead-enter positions are presribed θ 1 = -16, θ = -54, s 1 = , s = The omputed sliderrank dimensions are: r = , l =.93150, ε = , µ min = Program: SLIDFG.h 3

24 REFERENCES 4 Referenes [1] C. Bagi. Synthesis of the Double-Crank Four-Bar Plane Mehanism with Most Favorable Transmission Via Pole Tehnique. In A.H. Soni, editor, Linkage Design Monograph. National Siene Foundation, [] F.Y. Chen. An analytial method for synthesizing the four-bar rank-roker mehanism. ASME Journal of Engineering for Industry, 91:45 54, [3] C.H. Chiang. Kinematis and Design of Planar Mehanisms. Krieger Publishing Company, Malabar, Florida, 000. [4] N.P. Chironis and Slater N. eds. Mehanisms and mehanial devies sourebook. MGraw-Hill Book Company, nd edition, [5] F. Freudenstein and Primrose E.J.F. The lassial transmission angle problem. In Proeedings of the Conferene on Mehanisms, pages , London, 197. The Institution of Mehanial Engineers. [6] J. Grosjean. Kinematis and Dynamis of Mehanisms. MGraw-Hill Book Compnay, [7] K. Hain. Applied Kinematis. MGraw-Hill Book Company, New York, nd edition, [8] A.S. Hall. Kinematis and Linkage Design. Waveland Press, In., Illinois, [9] R.S. Hartenberg and J. Denavit. Kinemati Synthesis of Linkages. MGraw-Hill, New York, [10] K. Luk and K.H. Modler. Getriebetehnik - Analyse, Synthese, Optimierung. Springer Verlag, [11] P.R. Pamidi. Gross Motion Synthesis of the Plane Crank-Roker Mehanism. In A.H. Soni, editor, Linkage Design Monographs, pages National Siene Foundation, [1] A.H. Soni. Mehanism Synthesis and Analysis. Krieger Publishing Co., Malabar, Florida, [13] A.H. Soni and R.J. Brodell. Design of the rank-roker mehanism with unit time ratio. Journal of Mehanisms, 5:1 4, [14] D.C. Tao. Applied Linkage Synthesis. Addison-Wesley, [15] L.W. Tsai. Design of Drag-Link mehanisms With Minimax Transmission Angle Deviation. ASME Journal of Mehanisms, Transmissions, and Automation in Design, 105: , See disussion by F. Freudenstein.

25 REFERENCES 5 [16] L.W. Tsai. Design of drag-link mehanisms with optimum transmission angle. ASME Journal of Mehanisms, Transmissiona, and Automation in Design, 105():54 59, June [17] J. Volmer. Getriebetehnik - Leherbuh. VEB Verlag Tehnik, Berlin, seond edition, 1976.