Design charts based methods for the kinematic

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May 12, 2009 - Freudenstein and Primrose for the optimal design of crank-rocker mechanisms. 1 .... the use of the following equations a d. = − sin ψ0. 2 cos(φ0.
Design charts based methods for the kinematic synthesis of four-bar function generators Ettore Pennestr`ı Pier Paolo Valentini Dipartimento di Ingegneria Meccanica Universit`a di Roma Tor Vergata 00133 Roma - Italy May 12, 2009

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Introduction

This paper describes kinematic synthesis procedures by means of design charts. The use of such tools are traditionally well accepted by industrial designers. In fact, they usually are reported in manuals and catalogs for the selection of mechanical components on the basis of input requirements. The main advantages of design charts are - visualisation of the entire design space; - immediate computation of link dimensions and of their variations as a function of design parameters; - avoidance of software and/or computing devices; - possibility to use them through different design paths. For many design problems, charts are ideally suited to optimum design whereby the designer can visually survey the range and relationships of dimensional parameters. The methods herein presented are limited to crank-rocker and drag-link fourbar linkages. For each method the required design charts are included and a simple numerical example on how to use the charts is provided. For the first time the classical method of Freudenstein and Primrose [3, 2] is available in form of design charts and this should increase its use and popularity. Moreover it is shown the equivalency of results provided by the semi-analytical method of Volmer, also embodied in the VDI norms [6, 8], and the method of Freudenstein and Primrose for the optimal design of crank-rocker mechanisms

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Design of crank-rocker linkage with optimum transmission angle

Purpose: Design a crank-rocker four-bar linkage with a given rocker swing angle Ψ and corresponding crank rotation 180◦ + θ, such that the maximum deviation of the transmission angle from 90◦ is minimized. The charts reported in the following pages have been computed on the basis of the algorithm originally proposed by Freudenstein and Primrose [3, 2]. The computational steps of this algorithm have been already reported in a companion paper1 published in this Bulletin . The nomenclature adopted for

B1

B2 b θ

A0

θ0

ψ

c ψ0

A1 d=A0 B0

B0

a A2 Figure 1: Nomenclature. θ is positive when c.c.w. this problem is shown in Figure 1. One can compute the link length ratios using the charts shown in • Figures 2, 4, 6, when θ < 0; • Figures 3, 5, 7, when θ > 0. The extreme values µmin and µmax of the transmission angle can be obtained from the charts shown in: • Figure 8, when θ < 0; • Figure 9, when θ > 0. Example Determine the crank-rocker optimal proportions of a four-bar linkage with a 1 E. Pennestr` ı, P.P. Valentini, A review of simple analytical methods for the kinematic synthesis of four-bar and slider-crank function generators for two and three prescribed finite positions.

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swing angle of ψ = 40◦ and a crank rotation of 180◦ + θ = 160◦ . Solution In this case θ = −20◦ . From the charts reported in Figures 2, 4 and 6 one readily obtains a ≈ 0.25 , d b ≈ 0.52 , d c ≈ 0.78 . d The minimum value of the transmission angle (see Figure 8 ) is µmin ≈ 66◦ .

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(1a) (1b) (1c)

0.9 −20° θ=−10°

0.8

−30°

0.7

−40°

0.6 −50°

0.5 a d

0.4

−60°

0.3 −70°

0.2

0.1

0 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 2: Design of a crank rocker four-bar linkage. Case:θ < 0

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1 0.9 +70°

0.8 +60° +50°

0.7 +40°

0.6 +30° a d 0.5

+20°

0.4 θ=+10°

0.3 0.2 0.1 0 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 3: Design of a crank rocker four-bar linkage. Case:θ > 0

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1 θ=−10° −20°

0.9 −30°

0.8 0.7 −40°

0.6 −50° b d

0.5 0.4 −60°

0.3 −70°

0.2 0.1 0 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 4: Design of a crank rocker four-bar linkage. Case:θ < 0

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1.2 1.1 1 0.9

+20°

θ=+10°

0.8 +50° +40° +30°

0.7 b d

+60°

0.6 +70°

0.5 0.4 0.3 0.2 0.1 0 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 5: Design of a crank rocker four-bar linkage. Case:θ > 0

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1

−70°

−60°

−50° −40°

−30° −20°

0.9

θ=−10°

0.8 c d

0.7

0.6

0.5 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 6: Design of a crank rocker four-bar linkage. Case:θ < 0

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1.3

1.2

1.1 +70°

1 +60° c d

+50°

0.9 +40° +30°

0.8

+20°

0.7 θ=+10°

0.6

0.5 20

40

60

80 100 120 140 160 180 ψ (deg)

Figure 7: Design of a crank rocker four-bar linkage. Case:θ > 0

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180

µmax (deg)

170

−70° −60° −50°

160

−40°

150

−30°

140

−20°

130 θ=−10°

120 110 100 90

µmin (deg)

80 70 60 50 40 30 −70° −60°

20

−50°

−40°

10

−30°

−20°

θ=−10°

0 0

20 40

60 80 100 120 140 160 180 ψ (deg)

Figure 8: Design of a crank rocker four-bar linkage. Case:θ < 0

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180 170 160 µmax (deg)

+70°

150 +50°

140

+60°

+40°

130 +20°

120

+30°

θ=+10°

110 100 90 80 70 θ=+10°

60

+20°

+30°

µmin (deg)

50

+40°

40 +50°

30

+60° +70°

20 10 0 0

20 40

60 80 100 120 140 160 180 ψ (deg)

Figure 9: Design of a crank rocker four-bar linkage. Case:θ > 0

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It is interesting to compare the above results with an alternative approach based on design charts often presented in German textbooks2 and design manuals 3 . These charts, commonly known as Alt charts in memory of the first person who suggested a solution to the problem, were published by VDI4 in the version prepared in 1958 by J.G. Volmer.

b

c ψ0

φ0 a φ1

d

Figure 10: Nomenclature The method, with reference to the nomenclature of Figure 10, is based on the use of the following equations   sin ψ20 cos φ20 + φ1 a   , (2a) =− d sin φ20 − ψ20   ψ0 φ0 sin + φ sin 1 2 2 b   , (2b) = ψ0 φ0 d cos 2 − 2 s  2   a c b b a cos φ1 . (2c) −2 = 1+ + + d d d d d where the value of φ1 is obtained from the first of the Alt charts together with max µmin (see Figure 11). 2 See

[6] p.195, [8] p.477, [9], [4], p.22 [5]p.397 4 Verein Deutscher Ingenieure, Society of German Engineers. 3 See

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max

m = 0° min

70°

80°

60°

50°

40°

30°

20°

10°

270° 90° 10°

240°

100°

20°

30°

210°

40°

φ0

50° 60° 70°

180° 60° 50°

10° 40°

20°

30°

150°

30°

20°

40° 10°

50°

120°

f1 =80° 0

30°

70°

60°

60°

90°

120°

150°

ψ0 Figure 11: VDI chart for the design of four-bar linkage with optimal transmission angle (adapted from [8]) Such a chart, for the case ψ0 = 40◦ and φ0 = 160◦ , returns the values of φ1 ≈ 50◦ and µmin ≈ 32◦ . It must be observed that if one substitute in (2) this value of φ1 the same link length ratios (1) are obtained. However, the minimum and maximum value of the transmission angle estimated by the Alt chart is different from µmin ≈ 66◦ , µmax =≈ 148◦ . returned by the chart reported in Figure 8. The apparent discrepancy is solved considering that, when θ < 0, i.e. φ0 > 180◦, the minimum and maximum transmission angles can also be defined as shown in Figure 12.

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µmax

µmin

Figure 12: Definition of extreme values of transmission angle according to Volmer’s method when φ0 > 0. In conclusion, the optimality criterion adopted by F. Freudenstein and E.J.F. Primrose [3] is the same originally proposed by J.G. Volmer for the compilation of the Alt charts and the numerical results of the two methods are thus coincident. To improve the accuracy, in the charts herein presented the link length ratios and minimum values of transmission angles can be read directly on the ordinate axes.

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Design of drag-link four-bar linkage

Purpose: Design a drag-link mechanism with design positions corresponding at input angles φ1 =0◦ and φ2 =180◦ and a prescribed value µmin of the minimum transmission angle. The nomenclature is shown in Figure 13. The numerical

B1

180 - µ max= µ min

b

c

A2

d

A1 a

A0

B0 ∆ψ

µ min

0

B2

Figure 13: Design of a drag-ling:Nomenclature procedure for this problem has been already presented in a previous paper con-

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tained in this Bulletin 5 . Example Design a drag-link four-bar linkage such that whan the input crank rotates of 180◦ degrees the opposite crank rotates of ∆ψ0 = 110◦ . The prescribed value of the minimum transmission angle is µmin = 45◦ . Solution One enter the charts shown in Figures 14, 15, 16, for ∆ψ0 = 110◦, on the horizontal scale, and draw a vertical line to read from the curves of µmin = 45◦ on the vertical the link length ratios ad , db , dc , respectively. For the prescribed numerical data one approximately reads a ≈ 2.85 , d b ≈ 2.75 , d c ≈ 1.55 . d C. Bagci [1] suggests to investigate the degree of nonuniformity of the output velocity ratio as a function of ∆ψ0 for a given value of µmin . 5 see E. Pennestr` ı, P.P. Valentini, A review of simple analytical methods for the kinematic synthesis of four-bar and slider-crank function generators for two and three prescribed finite positions

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6 5.5 5 4.5 55° 60°

4 45° 50°

a d 3.5 35°

3 25°

2.5

40°

30°

20°

2

15°

1.5 µmin=10°

1 20

40

60

80

100 120 140 160 180 ∆ψ0 (deg)

Figure 14: Design of a drag-link for prescribed rotation ∆ψ0 of output link and minimum value of the transmission angle 16

6 5.5 5 4.5 60°

4 55°

b d

50°

3.5

45° 40°

3 35° 30°

2.5 25° 20°

2

15°

1.5

µmin=10°

1 20

40

60

80

100 120 140 160 180 ∆ψ0 (deg)

Figure 15: Design of a drag-link for prescribed rotation ∆ψ0 of output link and minimum value of the transmission angle 17

6 5.5 5 4.5 4 c d 3.5

3

60° 55°

2.5

50° 40° 45°

2 15° 20°

35° 25° 30°

1.5 µmin=10°

1 20

40

60

80

100 120 140 160 180 ∆ψ0 (deg)

Figure 16: Design of a drag-link for prescribed rotation ∆ψ0 of output link and minimum value of the transmission angle 18

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Design a drag-link four-bar linkage with minmax transmission angle deviation

Purpose: Design a drag-link mechanism in its two unity velocity ratio positions for prescribed crank rotations, ∆φ and ∆ψ, and minimized maximum deviation ∆µmax of the transmission angle from 90◦ (see Figure 17). For best performances

A1

B1

b

c a

φ1

∆ψ

A0

B0

d=A0 B0

∆φ

A2 B2

Figure 17: Drag-link mechanism at design positions it is necessary that at initial and final design positions (see Figure 17) the lag is minimum and maximum, respectively. Hence, at design positions, the angular velocity ratio of the output link to the input link equals to one [7]. The design charts presented in this section have been prepared making use of an analytical procedure already described and originally due to L.W. Tsai. These charts allow a quick kinematic synthesis of a drag-link four-bar linkage. The input data are the angular displacement ∆ψ of the output link and the corresponding displacement ∆φ of the input link. The resulting four-bar linkage is optimal in the sense that the maximum deviation ∆µmax of the transmission angle from 90◦ is minimized. The link length ratios are computed using the charts shown in Figures 18, 19, 18. The corresponding value of ∆µmax is readily obtained by means of the chart shown in Figure 21. Example Design a drag-link four-bar linkage with optimal transmission angle variation

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such that the rotations of input and output cranks between design positions are ∆φ = 170◦ , ∆ψ130◦ . At design positions the input-output velocity ratio is unity. Solution The value ∆L needs to be computed first ∆L = ∆φ − ∆Ψ = 40◦ . Then, entering the charts reported in the Figures 18, 19 and 20, from the abscissa value ∆φ = 170◦ , and reading the ordinate of the intersection point of a vertical line with the curve ∆L = 40◦ , one obtains a ≈ 4.3 , d b ≈ 2.7 , d c ≈ 3.0 . d The maximum deviation of transmission angle from 90◦ is ∆µmax ≈ 45◦ .

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10 9.5 9 8.5

∆L=20°

∆ L=20°

8 7.5 7 6.5 a d

30°

6

30°

40°

5.5 5 4.5 4

70°

40° 50°

50° 60°

60° 80°

70° 80°

3.5 3 2.5

2 160 165 170 175 180 185 190 195 200 ∆φ (deg) Figure 18: Design of a drag-link four-bar linkage with minimax transmission angle deviation from 90◦ 21

8 7.5 7 6.5 ∆ L=20°

6 ∆ L=20°

5.5

30° 40°

5 b d

4.5

30° 40° 50°

50°

4

60° 70°

60°

3.5 3

80°

70° 80°

2.5 2 1.5 1 160 165 170 175 180 185 190 195 200 ∆φ (deg) Figure 19: Design of a drag-link four-bar linkage with minimax transmission angle deviation from 90◦ 22

6.5 ∆L=20°

6 5.5 5 4.5 c d

4

30°

30°

40°

40°

50°

50°

60°

60°

70° 80°

70° 80°

3.5 3 2.5 2

1.5 160 165 170 175 180 185 190 195 200 ∆φ (deg) Figure 20: Design of a drag-link four-bar linkage with minimax transmission angle deviation from 90◦ 23

80 ∆L=80°

∆L=80°

70 70°

70°

60°

60

60°

50°

50°

∆µmax (deg)

40° 30°

30°

50

40°

20° 20°

40

30

20

10 160 165 170 175 180 185 190 195 200 ∆φ (deg) Figure 21: Design of a drag-link four-bar linkage with minimax transmission angle deviation from 90◦ 24

REFERENCES

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References [1] C. Bagci. Synthesis of the Double-Crank Four-Bar Plane Mechanism with Most Favorable Transmission Via Pole Technique. In A.H. Soni, editor, Linkage Design Monograph. National Science Foundation, 1976. [2] N.P. Chironis and Sclater N. eds. Mechanisms and mechanical devices sourcebook. McGraw-Hill Book Company, 2nd edition, 1996. [3] F. Freudenstein and Primrose E.J.F. The classical transmission angle problem. In Proceedings of the Conference on Mechanisms, pages 105–110, London, 1972. The Institution of Mechanical Engineers. [4] P.W Jensen. Classical and Modern Mechanisms for Engineers and Inventors. Marcel Dekker, Inc., New York, 1991. [5] H. Krzenciessa, K. Luck, K.H. Modler, and G. Nerge. Mechanismen (getriebetechnik). In K. Luck, S. Fronius, and J. Klose, editors, Taschenbuch Maschinenbau, volume Band 3, pages 331–440. VEB Verlag Technik, Berlin. [6] K. Luck and K.H. Modler. Getriebetechnik - Analyse, Synthese, Optimierung. Springer Verlag, 1995. [7] L.W. Tsai. Design of Drag-Link mechanisms With Minimax Transmission Angle Deviation. ASME Journal of Mechanisms, Transmissions, and Automation in Design, 105:686–691, 1983. See discussion by F. Freudenstein. [8] J. Volmer. Getriebetechnik - Leherbuch. VEB Verlag Technik, Berlin, second edition, 1976. [9] J.G. Volmer. Four-bar linkages. Product Engineering, pages 71–76, November 12 1962.