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Proceedings of the 11th World Congress in and Science August 18–21, 2003, Tianjin, China China Machinery Press, edited by Tian Huang

Kinematic Synthesis of Planar Geared Four-Bar Linkages with Prescribed Dwell Characteristics

Tuanjie Li College of Mechanical and Electrical Engineering, Xidian University, Xi’an 710071, China e-mail: [email protected]

Weiqing Cao Department of Mechanical Engineering, Xi'an University of Technology, Xi'an 710048, China

Abstract: Geared linkages are useful mechanisms, which can be y formed by combining planar linkages with one or more pairs of . As a result output motions can be obtained with reversing, nonreversing without dwell or nonreversing with dwell. As an A x2 example consider the design of the regressive or irregressive r two- four-bar linkages with prescribed dwell characteristics. 2 x B First of all, a simplified algorithm is given, including relevant 1 x analytical solutions and the crank existing conditions. Then the 3 r ϕ 1 ψ homotopy iteration method is introduced to solve the non-linear 1 x equations, all solutions can be found easily without choosing k A0 proper initial values. Finally the numerical examples illustrate B0 the synthesis procedures. (a) Regressive mechanism Keywords: Geared four-bar linkages, Instantaneous dwell, Homotopy iteration method y

1 Introduction B Dwell mechanisms ]1[ have numerous applications in x2 A r2 industry, especially in machine tools, packaging, textile machinery, process , and automation. The x3 ϕ intermittent-motion could be produced by Geneva, ratchet x ]4,3,2[ 1 1 k ψ mechanisms or other linkages , but their common x weakness is the bigger shock. With the increasing trend B0 towards higher speeds, mechanisms which are capable of A0 r1 producing shockless dwell or generating smooth and continuous acceleration have been the subject of many (b) Irregressive mechanism investigations. Geared linkages are useful mechanisms, Figure 1 Geared four-bar linkages which can be formed by combining planar linkages with one or more pairs of gears ]5[ . As a result output motions can be obtained with reversing, nonreversing without 2 Nomenclature dwell or nonreversing with dwell. As an example we Referring to Figure 1, unless otherwise specified, all consider the design of the two-gear four-bar linkages in angles are expressed in degrees, and all angular quantities this paper, which consists of a four-bar as its basic are positive in counter-clockwise direction. mechanism and two meshing gears centered on two pivots r2 of one of the links, and is a class of geared momentary i ±= =gear ratio, positive or negative r dwell mechanisms. There are two types of the two-gear 1 four-bar linkages, one is regressive, shown in Figure 1(a), according to whether the gear mesh is internal or in which the output gear is coaxial with the input, the external, respectively other is irregressive, shown in Figure 1(b), in which the , rr 21 =pitch radii of gears 1 and 2, respectively output gear is not coaxial with the input. The mechanisms xi =link length of the four-bar linkage convert uniform rotary motion into nonuniform or (j=1,2,3) when the length of the frame is equal reciprocating motion, they provide better motion to one transmission and lesser gear inertias, therefore, offer better ϕ =rotating angle of the crank AA with kinematic as well as dynamic characteristics. In addition, 0 they have advantages, such as simplicity and low cost. respect to the frame BA 00 ψ =rotating angle of the follower 0 BB relative to the 转载 中国科技论文在线 http://www.paper.edu.cn

frame A0 B0 of the crank A0 A with respect to the frame A0 B0 , and

k =rotation of the output gear 1 the follower B0 B respectively, while the output gear 1 is α, β =rotating angles of the crank A0 A and at rest with an instantaneous dwell.

the follower B0 B respectively while the output It can be seen from above, the link lengths — x1, x2

gear 1 is at rest with an instantaneous dwell and x3 —can be calculated from equation (5) or (6) with prescribed dwell characteristics, α, β and i. Conversely, 3 General Dwell Conditions given each length of links, the gear ratio i and dwell Technically speaking, dwell is defined to exist when one angles α and β can be calculated. or more derivatives of the output displacement with respect to the input displacement are equal to zero at an instant. The degree of dwell, therefore, refers to the 4 Methods of Solution number of consecutive derivatives which are zero at that 4.1 A simplified algorithm When it does not make strict demands on the transmitting instant [1] . A second degree dwell exists when the first and property, the simplified algorithm as follows may be taken second derivatives —velocity and acceleration—are both for the instantaneous dwell geared four-bar linkages. zero at the same instant. This might be called an 0 instantaneous or momentary dwell and is the type of Setting β = 90 and from equation (5), we have, motion generally sought for indexing in cams or 2 2 2 intermittent motion devices where the transfer function x1 − x2 + x3 +1− 2x1 cosα − 2x1x3 sinα = 0 joins an exact dwell function, since the moving mass is  x cosα − (1+ i) = 0 (7) brought to rest with zero inertia.  1  Referring to Figure 1, the zero-order transfer function sinα(x sinα − x ) = 0  1 3 is 0 k = k(ϕ) (1) Setting α = 90 and from equation (6), we have,

2 2 2 The zero-order transfer function of the four-bar x1 − x2 + x3 +1+ 2x3 cos β − 2x1x3 sin β = 0 linkage is  x3 cos β + (1+ i) = 0 (8) F(ϕ,ψ ) = 0 (2)  sin β (x sin β − x ) = 0  3 1 By differentiating equation (1), we have： All kinds of the relevant analytical solutions of the G(ϕ,ψ ) = 0 (3) equations (7) and (8) are derived shown in Table 1 and Table 2. By differentiating equation (3), we have： Prescribed Analytical Crank existing H (ϕ,ψ ) = 0 (4) (α,i ) solution conditions

Then the nonlinear equations may be expressed in 0 x (1 i) α = 180 1 = − + i < −2 terms of α, β , x1, x2 , x3 and i due to the equations (2), (3) i < −1 2 2 x3 > 2(1+ i) (2 + i) and (4). x2 = x3 + i Referring to the regressive mechanism shown in Figure 1(a), the design equations may be given as follows. 0 i > 0 , x > 1+ i or α = 0 x1 = 1+ i 2 i < x < 1+ i and  2 2 2 i > 0 2 2 2 F(α, β ) = x1 − x2 + x3 +1− 2x1 cosα + x2 = x3 + i  x2 > 1  2x3 cos β − 2x1 x3 cos(α − β ) = 0  0 x = 1+ i α = 0 1 −1 < i < 0 , x2 > 1 or G(α, β ) = x1 sin(α − β ) + (1+ i) sin β = 0 (5)  −1 < i < 0 2 2 x > 2(1+ i) (2 + i) x2 = x3 + i 3 H (α, β ) = sinα[x1 x3 sin(α − β ) + x1 sinα]−  sin β cos(α − β )[x x sin(α − β ) + x sin β ] = 0 0 0 x = (1+ i) / cosα  1 3 3 90 < α < 180 1 According to the 2 2 x2 = i value of α Referring to the irregressive mechanism shown in i < −1 2 2 Figure 1(b), the design equations may be written as, x3 = x1 − (1+ i) 2 2 2 F(α, β ) = x1 − x2 + x3 +1− 2x1 cosα +  00 < α < 900  2x cos β − 2x x cos(α − β ) = 0 Idem Idem 3 1 3  i > 0 G(α, β ) = x3 sin(α − β ) + (1+ i) sinα = 0 (6)  00 < α < 900 Idem No crank H (α, β ) = sin β[x x sin(α − β ) + x sin β ]−  1 3 3 −1 < i < 0  sinα cos(α − β )[x x sin(α − β ) + x sinα] = 0  1 3 1 Table 1 Regressive mechanism when β = 900 where α, β are dwell angles, namely, the rotating angles 中国科技论文在线 http://www.paper.edu.cn

4.2 Continuation method 7 Conclusion When dwell angles α and β are arbitrary values, we In a word, the dwell characteristics of planar two-gear have to solve the nonlinear equations (5) and (6). The four-bar linkages have been investigated in this paper. A simple solution to the design of the mechanisms is offered. continuation method [6] and the homotopy iteration [7] The simplicity of the design procedures presented here method , which rose in recent years, are brought to may be an aid to the practicing engineers for their designers’ attention, because all solutions can be found consideration and use of these versatile mechanisms on easily without choosing proper initial values. the design of high-speed machinery involving dwell. Here, the homotopy iteration method can be used to solve the nonlinear equations (5) and (6). Prescribed dwell Numerical Sketch condition solution Prescribed Analytical Crank existing 0 (β , i ) solution conditions α = 45 x1 = 5.6569

0 x2 = 3 0 β = 90 β = 0 x3 = −(1+ i) i < −2 x = 4 x1 > 2(1+ i) (2 + i) i = 3 3 i < −1 2 2 x2 = x1 + i 0 α = 135 x1 = 2.8284 0 x = 1+ i i > 0 , x > 1+ i or β =180 3 2 0 x = 3 β = 90 2 2 2 i x 1 i and i > 0 x = x + i < 2 < + 2 1 i = −3 x3 = 2 x2 > 1 0 x = 2.8794 0 α =120 1 β =180 x3 = 1+ i No crank 0 x = 2.0749 −1 < i < 0 2 2 β =100 2 x2 = x1 + i i = −2 x3 = 2.2267 x = −(1+ i) / cos β 0 0 3 According to the 0 0 < β < 90 α = 40 x1 = 4.5963 x 2 i 2 value of β i < −1 2 = 0 x2 = 2.3322 2 2 β = 80 x1 = x3 − (1+ i) i = 2 x3 = 4.5963

0 0 Idem Idem 90 < β < 180 Table 3 Examples of the regressive mechanisms i > 0 Prescribed dwell Numerical Sketch condition solution 900 < β < 1800 Idem No crank 0 −1 < i < 0 α = 90 x1 = 2.3835

0 x2 = 3 0 β = 50 Table 2 Irregressive mechanism when α = 90 x = 3.1114 i = −3 3

0 5 Geometrical Constraints α = 90 x1 = 3.3564 For complete rotation of cranks, Grashof criterion must be 0 x = 3 satisfied, that is, β =140 2 x = 5.2216 ' " i = 3 3 lmax + lmin < l + l (9) 0 x = 2.2267 x1, for a crank - rocker mechanism α = 80 1 where l =   min 0 x = 2.0749 1, for a drag link mechanism β = 60  2  The crank existing conditions can be determined due x = 2.8794 i = −2  3 to equation (9), which are shown in Table 1 and 2.

6 Examples 0 x = 7.7084 We can use the preceding results to synthesize the α = 80 1  proportions of geared four-bar linkages with prescribed β =1000 x2 = 2.0539 dwell condition. The numerical examples of the regressive  x = 8.6382 and irregressive mechanisms are described in Table 3 and i = 2  3

Table 4, respectively. Table 4 Examples of the irregressive mechanisms 中国科技论文在线 http://www.paper.edu.cn

Acknowledgements This research is sponsored by the NSFC(Grant No.59575509) and supported by the youth scientific research workstation of xidian university.

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