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On the synthesis of a geared four-bar mechanism
Mechanism and Machine Theory 45 (2010) 1142–1152
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
On the synthesis of a geared four-bar mechanism Volkan Parlaktaş a,⁎, Eres Söylemez b, Engin Tanık a a b
Department of Mechanical Engineering, Hacettepe University, 06800 Ankara, Turkey Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 2 June 2009 Received in revised form 1 March 2010 Accepted 7 March 2010 Available online 28 April 2010 Keywords: Geared linkages Geared four-bar mechanisms
a b s t r a c t This paper presents an analysis and synthesis method for a certain type of geared four-bar mechanism (GFBM) for which the input and output shafts are collinear. A novel analysis method is devised, expressions for the transmission angle are derived and charts are prepared for the design of such mechanisms. It is observed that the GFBM considered is inherently a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. During the working stroke, approximately constant angular velocity at the output link is observed. For the type of GFBM analyzed, direction of rotation of the input link affects the force transmission characteristics. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Geared linkages are useful mechanisms, which can be formed by combining planar linkages with one or more pairs of gears [1,2]. A geared five link mechanism in general is a one degree of freedom planar mechanism with five revolute joints, one gear pair, and five links. Two different topologies are possible as shown in Fig. 1. In type A, there is a ternary joint between links 1, 2 and 3 whereas in type B all revolute joints are binary. Type A contains a four-bar loop whereas type B has a five-bar loop when the gear pairs are removed. The mechanism type B is extensively studied in literature [3–7]. The mechanism studied in this work has type A topology, which is named as GFBM in the literature. Geared four-bar mechanisms are generally investigated to obtain large swing angle, pilgrim-step motions [8,9], dwell motions and motion with approximately constant transmission ratio ranges. Neumann [10,11] classified the geared mechanisms and obtained their input– output motion functions. In Ref. [11] he searched for the type of mechanism which would produce the largest output oscillation and prepared design charts. Volmer et al. [12] studied GFBM and obtained a pilgrim-step motion by adding an internal gear train to the crank of the four-bar mechanisms. Volmer and Meiner obtained pilgrim-step motion by adding an internal gear train to the crank of slider-crank mechanisms [13,14]. Li and Cao [15] investigated the dwell characteristics of such mechanisms. The geared linkage mechanisms which are composed by adding the spur gears to each pair point of the planar four-bar mechanisms were classified and analyzed by Yokoyoma [16]. Geared linkages with non-circular gears are also studied in the literature [17,18]. This paper presents an analysis and synthesis method for a certain type of GFBM (type A1, input link 2, output link 3), which is different than the ones investigated in the previous works. It is observed that the GFBM considered is inherently a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. 2. Enumeration of the GFBM The mechanism studied in this work has a topology type A, which is simply a gear pair combined with a four-bar mechanism. Different mechanisms can be obtained by fixing links 1 to 5 respectively. These mechanisms can be summarized as in Fig. 2. Moreover, by changing the input and output links, different input–output functions can be achieved which is summarized for type ⁎ Corresponding author. E-mail address: [email protected] (V. Parlaktaş). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.03.007
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Fig. 1. Topology type A, and topology type B.
A1 in Table 1. Note that, instead of external gears internal gears can also be used in some of these mechanisms. Type A3 is nothing but a four-bar driven by another gear, and type A5 is a four-bar with a floating gear pair. The mechanism studied in this work is of topology type A1. One of the gears is rigidly connected to the coupler and the other gear has a fixed axis of rotation. This forms a planetary gear train with link 3 as the arm. The input torque is applied to link 2, and the output is obtained at the arm (link 3). Thus, the input and the output displacements are about the same rotation axis. 3. Motion analysis of the GFBM In order to obtain the relationship between the input (θ12) and output (θ13), an equation can be obtained from the loop closure equation of the four-bar mechanism (Fig. 3). iθ13
a3 e
iθ14
+ a4 e
iθ15
−a1 = a5 e
ð1Þ
There is another equation which is the velocity ratio of the planetary gear arrangement formed by links 2, 3 and 4: −
r4 ω −ω13 = 12 = −R r2 ω14 −ω13
ð2Þ
Integrating this equation: θ14 =
ðR + 1Þ θ θ13 − 12 R R
ð3Þ
+k
where, R = r4/r2 and k is a constant determined by the initial relative positions of links 2 and 4 with respect to link 3 (integration constant). Eq. (3) can be used in Eq. (1) to eliminate θ14 and a new equation can be obtained. Then, this obtained equation can be multiplied by its complex conjugate (eliminating θ15) to obtain a relationship between θ12 and θ13. However, obtaining an explicit relationship between the input and the output is very hard if not impossible by this method. Therefore, a novel approach is used to determine the input–output relationship in two steps.
Fig. 2. Enumeration of the GFBM.
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Table 1 Possible input–output links for type A1. Reference
−
[9]
[10]
Four-bar
[9,11]
Input link Output link
2 3
2 5
5 2
5, (3) 3, (5)
3 2
It is assumed that the input link of the mechanism is not link 2 but link 4. Hence, according to this assumption, an explicit relationship between θ14 and θ13 can be determined from the loop closure equation of the four-bar mechanism. Since, θ13 is determined as a function of θ14, then by using Eq. (3) a relationship between θ12 and θ14 can be obtained. Hence, for a given θ14 corresponding θ13 and θ12 can be determined explicitly. Therefore, a chart which gives the relationship between the input θ12 and output θ13 can be determined. With this approach, an iterative numerical solution is eliminated. For link 2 to have a complete rotation, the coupler link (link 4) must also have a complete rotation, while links 3 and 5 will have oscillations only. The full cycle rotation of link 3 imparts a full cycle rotation of the coupler link of the four-bar formed by links 1, 3, 4 and 5 (Fig. 3). This four-bar mechanism must then be of Grasshof type double-rocker mechanism (the sum of the longest and shortest link lengths is less than the sum of the two intermediate link lengths and the link opposite the shortest link is the frame). With this consideration, the rotation of link 3 can be determined in terms of the rotation of the coupler (link 4). Multiplying the loop closure Eq. (1) by its complex conjugate, the output θ13 can be determined as a function of θ14. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ13 = 2arctan −B∓ B2 −4AC = 2A
ð4Þ
where, A = K1 −K2 cos θ14 + K3 −K4 cos θ14 B = 2K2 sin θ14 C = K1 + K2 cos θ14 −K3 −K4 cosθ14 K1 = a21 + a23 + a24 −a25 K2 = 2a3 a4 K3 = 2a1 a3 K4 = 2a1 a4 : Since θ13 is determined for a given θ14, corresponding θ12 can be determined from Eq. (3). θ12 = ðR + 1Þθ13 −Rθ14 + k Therefore, the input–output relationship can be determined at two steps by solving Eqs. (4) and (5) respectively.
Fig. 3. The GFBM and the corresponding four-bar mechanism when the gears are removed.
ð5Þ
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k is a constant determined by the initial relative positions of links 2 and 4 with respect to link 3 (integration constant). It can be choosen arbitrarily. While obtaining input–output curves, in order to start θ12 values from 0° (when θ14 = 0°, θ13initial is calculated, so in order to obtain θ12 = 0°), k is chosen as: k = −ðR + 1Þθ13initial :
4. Transmission angle of the GFBM The transmission angle of a mechanism is defined as [19]: tanðμ Þ =
force component acting on the output link tending to produce output rotation : force component tending to apply pressure on the driven link bearings
Neglecting the mass of the links, the free-body diagrams of the links of the GFBM will be as shown in Fig. 4A. Where, Ti is the input torque applied to link 2, To is the output torque at link 3, and α is the pressure angle of the gears. The transmission angle of the mechanism can be obtained as: tanðμ Þ =
F43t : F43n
ð6Þ
From the free-body diagrams of the links, F43t and F43n can be obtained, and from Eq. (6), the transmission angle can be determined as: tanðμ Þ =
a r4
− 4 sinðθ15 −θ14 Þ−sinðθ15 −θ13 Þ a4 tanðαÞsinðθ15 −θ14 Þ r4
+ cosðθ15 −θ13 Þ
:
ð7Þ
If the direction of rotation of the input link is changed, then the transmission angle of the GFBM alters since the line of action of the force between links 2 and 4 changes (Fig. 4B). In this case, the transmission angle will be given by: tanðμ Þ =
a r4
− 4 sinðθ15 −θ14 Þ−sinðθ15 −θ13 Þ a r4
− 4 tanðαÞsinðθ15 −θ14 Þ + cosðθ15 −θ13 Þ
:
ð8Þ
Note that there is a minus sign in front of the term including α. Unlike all other mechanisms known up to now, for this mechanism the force transmission characteristics are also a function of rotation direction. A mechanism which has proper transmission angle values in one rotation direction can even lock if the direction of rotation of the input is reversed.
Fig. 4. A) The FBD of the links when link 2 is rotating counterclockwise, B) the FBD of link 4 when link 2 is rotating clockwise.
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Fig. 5. The position of the output link θ13 and the transmission angle µ wrt the input position θ12 for ccw and cw rotation direction.
Example 1. Consider the GFBM with link lengths: a1 = 1, a3 = 0.907, a4 = 0.306, a5 = 0.665, and R = 1. In Fig. 5, the position of the output link θ13 and the transmission angle µ are shown with respect to the input position θ12 for ccw and cw rotation direction. Note that, there are two different µ curves, according to the rotation of direction. Also note that, a large time ratio (2,27) is obtained. 5. Synthesis of the GFBM The problem is considered in two parts. The first part is the synthesis problem in which one must determine a four-bar mechanism of Grashof type with double-rocker proportions that have given swing angles ϕ and ψ for links 3 and 5 respectively. There is an infinite set of solutions for this part of the problem. The second part of the problem is concerned with the optimization. Out of the infinite possible set of solutions obtained in the first part, one must determine a particular mechanism whose maximum transmission angle deviation from 90° is a minimum. The extended and folded positions of the mechanism are shown in Fig. 6. At the extended position link 3 is at the forward position and at the folded position link 3 is at the fully withdrawn position. θ is the angle of link 3 and ξ is the angle of link 5 at the folded position. The loop closure equations for the folded and extended positions of the mechanism can be written as [20]: iθ
iξ
a3 e = a1 + ða5 −a4 Þe i ðθ + ϕ Þ
a3 e
ð9Þ iðξ + ψÞ
= a1 + ða5 + a4 Þe
:
ð10Þ
One can define Z1, Z2 and λ as: iθ
ð11Þ
iξ
ð12Þ
Z1 = a3 e Z2 = a5 e λ=
a4 : a5
ð13Þ
Without a loss of generality the fixed link can be chosen as unity; a1 = 1. Then, Eqs. (9) and (10) can be written in normalized form as: Z1 −Z2 ð1−λÞ = 1 iϕ
ð14Þ
iψ
Z1 e −Z2 e ð1 + λÞ = 1:
ð15Þ
Fig. 6. The dead-center positions of the GFBM.
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Fig. 7. Z1 and Z2 circles for the values of ϕ = 50° and ψ = 10°.
These complex equations are linear in terms of the unknowns Z1 and Z2. Then, Z1 and Z2 can be solved in terms of λ, ϕ, and ψ as: Z1 =
ð1−λÞ−ð1 + λÞeiψ ð1−λÞeiϕ −ð1 + λÞeiψ
ð16Þ
Z2 =
1−eiϕ : ð1−λÞeiϕ −ð1 + λÞeiψ
ð17Þ
Fig. 8. The design chart for cw input rotation and R = 1.
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As λ changes from −∞ to +∞, Z1 and Z2 describe a curve which is the loci of the tip of the vectors Ao Af and Bo Bf. These loci are two circles for any given value of ϕ and ψ. In Fig. 7, these two circles are shown for the values of ϕ = 50° and ψ = 10°. A line can be drawn from (0,0), which is Ao, at an angle θ with respect to A o Bo. Af is the intersection point of this line and the Z1 circle. Another line can be drawn from Bo at angle ξ with respect to A o Bo which intersects the circles at Af and Bf, respectively. This corresponds to the folded position of the mechanism, where the link lengths are, AoAf = a3, AfBf = a4, BoBf = a5 and AoBo = a1. Analytically, link lengths can be determined as: a3 =
qffiffiffiffiffiffiffiffiffiffi Z1 Z1
ð18Þ
a5 =
qffiffiffiffiffiffiffiffiffiffi Z2 Z2
ð19Þ
a4 =
qffiffiffiffiffiffiffiffiffiffiffi a25 λ2 :
ð20Þ
The link lengths are functions of the free parameter λ, and swing angles ϕ, and ψ. For a given ϕ and ψ, there is a set of solutions with respect to the free parameter λ. A necessary but not sufficient condition for double-rocker proportions is: 0 b jλ j b 1:
ð21Þ
In order to obtain double-rocker proportions there are limits on swing angles as: 0 b ϕ b 180˚ ϕ 2
−90˚ b ψ b
ð22Þ ϕ 2
+ 90˚:
ð23Þ
Fig. 9. The design chart for ccw input rotation and R = 1.
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5.1. Transmission angle optimization Among the set of solutions for a given ϕ and ψ, the one for which the maximum deviation of the transmission angle from 90° is a minimum must be selected (λopt will be determined). The design charts are obtained for a given output swing angle ϕ and the corresponding input link rotation β. Therefore, the optimum mechanism in terms of the transmission angle can be determined, for a given output swing angle ϕ and a time ratio, from the design charts. Using Eq. (5) at the folded and extended positions, β can be obtained for a gear ratio R, and according to the input direction of rotation (− sign for cw input) as: β = ðR + 1Þϕ∓Rπ−Rψ:
ð24Þ
A parametric optimization routine is developed and design charts for optimum GFBM are prepared by using MATLAB. According to the direction of rotation of the input, the transmission angle of the GFBM changes. That condition leads to two different optimum mechanisms which have the same output swing angle ϕ and corresponding input rotation β. Therefore, transmission angle optimization is performed for both of the input directions of rotation and two sets of design charts are prepared. The affect of gear ratio (R) is very clear, it affects the transmission angle; if R is increased, then the transmission angle improves. Therefore, transmission angle optimization is performed for several gear ratios and the corresponding design charts are also prepared. Some of those design charts are displayed in Figs. 8–11. The Y-axis represents the output swing angle ϕ, and the X-axis represents the corresponding input link rotation β. The full lines represent the optimum λ parameter, and the dotted lines represent the maximum deviation of the transmission angle from 90° (δµmax) for the corresponding mechanism. Since, ϕ and β are the given parameters, a value for ϕ and another value for β is chosen from Y and X axes respectively. Then, these values are intersected and the value of optimum λ parameter, and for that mechanism, the value of maximum deviation of the transmission angle from 90° can be obtained (since the value of optimum λ is determined the link lengths can easily be determined from Eqs. (18)–(20)). If the obtained δµmax is not preferred, then ϕ or β is changed, and another optimum solution can be obtained. Therefore, by the aid of these design charts, by specifying ϕ and β, the corresponding optimum mechanism and δµmax for that mechanism can be determined easily. It can be observed that transmission characteristics of the mechanisms deteriorate as the input rotation β approaches to R × 180° (R = 1,2,3,4). Therefore, for an acceptable transmission angle, centric mechanisms can be obtained for small output swing
Fig. 10. The design chart for cw input rotation and R = 2.
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Fig. 11. The design chart for cw input rotation and R = 4.
angles only. Hence, it can be concluded that this type of GFBM is useful for quick-return motions. This condition is explained as below. If a mechanism is to be designed from Fig. 8, which has an output swing angle ϕ = 50° and a large time ratio of 2 (β = 120°), from the intersection point of these values it is seen that the value of λ ≈ 0.7, and for that mechanism the maximum deviation of the transmission angle from 90° is ≈42°. If a smaller time ratio is desired, for example 210/150 (still ϕ = 50°) β is chosen as 150°. Then, from the intersection point of these values it is determined that λ ≈ 0.67 and δµmax ≈ 62° which lead to a mechanism with poor transmission characteristics. So, it can be observed that as the time ratio approaches to 1 (β approaches to 180°) for a fixed value of ϕ, the maximum deviation of the transmission angle from 90° (δµmax) of the corresponding mechanism increases (also note that in Fig. 10, since R = 2, as time ratio approaches to 1, β approaches to 360°). The gear ratio R significantly affects the transmission angle of the GFBM. As R increases, the transmission angle improves and mechanisms with larger output swing angles can be obtained (Figs. 10 and 11). Conversely, if R is decreased, transmission characteristics deteriorate. Therefore, transmission angle optimization is performed for several gear ratios and the corresponding design charts are also prepared.
Fig. 12. The transmission angle µ and the output link position θ13 wrt the input link position θ12.
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Fig. 13. The angular velocity ω13 and the angular acceleration α13 of the output link.
Example 2. For a given swing angle ϕ = 40°, and corresponding input rotation β = 100° (time ratio = 2,6) (cw input, R = 1), λopt can be determined from Fig. 8 as, λopt ≅ 0.2 (δµmax ≅ 30°). Then, the link lengths can be determined from Eqs. (16)–(20), and (24) as: a3 = 0.53, a4 = 0.181, a5 = 0.867. In Figs. 12 and 13 the transmission angle µ, the output link position θ13 and the angular velocity ω13 and acceleration α13 of the output link are shown with respect to the input link position θ12. The mechanism has quick-return motion characteristics and in the forward stroke approximately constant velocity of the output link is obtained (ω12 = 1 rad/s). 6. Conclusion In this work a new type of geared four-bar mechanism for which the input and output shafts are collinear has been investigated. A novel analysis method is devised, expressions for the transmission angle are derived and charts are prepared for the optimum design of such mechanisms. By the aid of these design charts, by specifying the output swing angle and the time ratio, the corresponding mechanism which has the best transmission characteristics can be determined easily. It is observed that, the gear ratio significantly affects the transmission angle; as the gear ratio (R) increases, the transmission angle improves. According to the direction of rotation of the input, there are two different optimum mechanisms which have the same output swing angle and corresponding input rotation. During the working stroke, approximately constant angular velocity at the output link is observed. This type of GFBM is suitable as a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory.2010.03.007. References [1] T.J. Li, W.Q. Cao, J.K. Chu, The topological representation and detection of isomorphism among geared linkage kinematic chains, Transactions of the ASME 25th Biennial Mechanisms, 1998, DETC98/MECH5812. [2] T. Li, W. Cao, Kinematic analysis of geared linkage mechanisms, Mechanism and Machine Theory 40 (2005) 1394–1413. [3] A.G. Erdman, G.N. Sandor, Kinematic synthesis of a geared five-bar function generator, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 11–16. [4] S.A. Oleska, D. Tesar, Multiply separated position design of the geared five-bar function generator, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 74–84. [5] A.V. Mohanrao, G.N. Sandor, Extension of Freudenstein's equation to geared linkages, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 201–210. [6] C.K. Lin, C.H. Chiang, Synthesis of planar and spherical geared five-bar function generators by the pole method, Mechanism and Machine Theory 27 (2) (1992) 131–141. [7] T.W. Lee, F. Freudenstein, Design of geared five-bar mechanisms for unlimited crank rotations and optimum transmission, Mechanism and Machine Theory 13 (2) (1978) 235–244. [8] K. Luck, K.H. Modler, Getriebetechnik: analyse, synthese, optimierung, Springer, Berlin, 1990. [9] J. Volmer, Getriebetechnik: koppelgetriebe, VEB Verlag Technik, 1, auflage, Berlin, 1979. [10] R. Neumann, Einstellbare fünfgliedrige raderkoppelgetriebe, Maschinenbautechnik 28 (1979) 211–215. [11] R. Neumann, Bauelemente auslegung von getrieben berechnung von getrieben mit grossem schwingwinkel, Maschinenbautechnik 29 (1980) 405–406. [12] J. Volmer, H. Riedel, E. Rissman, Bauelemente auslegung von getrieben berechnung von schritt und pilgerschrittgetrieben, Maschinenbautechnik 29 (1980) 121–124. [13] J. Volmer, Rationalisierungsmittel für den konstrukteurarbeitsblaetter zur auslegung von getrieben, Maschinenbautechnik 29 (1980) 125–127. [14] J. Volmer, C. Meiner, Bauelemente auslegung von getrieben berechnung von schritt und pilgerschrittgetrieben, Maschinenbautechnik 27 (1978) 219–220.
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[15] T. Li, W. Cao, Kinematic Synthesis of Planar Geared Four-bar Linkages with Prescribed Dwell Characteristics, Proceedings of the 11th World Congress in Mechanism and Machine Science (2004) 3 Tianjin China 1190–1193. [16] Y. Yokoyoma, Studies on the geared linkage mechanisms, Bulletin of the JSME 17 (112) (1974) 1332–1339. [17] K.H. Modler, E.C. Lovasz, G.F. Bär, R. Neumann, D. Perju, M. Perner, D. Margineanu, General method for the synthesis of geared linkages with non-circular gears, Mechanism and Machine Theory 44 (4) (2009) 726–738. [18] E.C. Lovasz, K.H. Modler, D. Perju, R. Neumann, D. Margineanu and M. Perner, Non-circular Gear Wheels in the Geared-linkages Mechanisms, 12th IFToMM World Congress, Besançon (France), 3 (2007) 284–289. [19] H. Alt, Der übertragungswinkel und seine bedeutung für das konstruıeren peridischer getriebe (Transmission angle and its importance for the design of periodic mechanisms), Werkstattstechnik 26 (1932) 61–64. [20] F. Freudenstein, F.P. Primrose, The Classical Transmission Angle Problem, The Institution of Mechanical Engineers, C96/72 Mechanisms, London, 1972.
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
On the synthesis of a geared four-bar mechanism Volkan Parlaktaş a,⁎, Eres Söylemez b, Engin Tanık a a b
Department of Mechanical Engineering, Hacettepe University, 06800 Ankara, Turkey Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 2 June 2009 Received in revised form 1 March 2010 Accepted 7 March 2010 Available online 28 April 2010 Keywords: Geared linkages Geared four-bar mechanisms
a b s t r a c t This paper presents an analysis and synthesis method for a certain type of geared four-bar mechanism (GFBM) for which the input and output shafts are collinear. A novel analysis method is devised, expressions for the transmission angle are derived and charts are prepared for the design of such mechanisms. It is observed that the GFBM considered is inherently a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. During the working stroke, approximately constant angular velocity at the output link is observed. For the type of GFBM analyzed, direction of rotation of the input link affects the force transmission characteristics. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Geared linkages are useful mechanisms, which can be formed by combining planar linkages with one or more pairs of gears [1,2]. A geared five link mechanism in general is a one degree of freedom planar mechanism with five revolute joints, one gear pair, and five links. Two different topologies are possible as shown in Fig. 1. In type A, there is a ternary joint between links 1, 2 and 3 whereas in type B all revolute joints are binary. Type A contains a four-bar loop whereas type B has a five-bar loop when the gear pairs are removed. The mechanism type B is extensively studied in literature [3–7]. The mechanism studied in this work has type A topology, which is named as GFBM in the literature. Geared four-bar mechanisms are generally investigated to obtain large swing angle, pilgrim-step motions [8,9], dwell motions and motion with approximately constant transmission ratio ranges. Neumann [10,11] classified the geared mechanisms and obtained their input– output motion functions. In Ref. [11] he searched for the type of mechanism which would produce the largest output oscillation and prepared design charts. Volmer et al. [12] studied GFBM and obtained a pilgrim-step motion by adding an internal gear train to the crank of the four-bar mechanisms. Volmer and Meiner obtained pilgrim-step motion by adding an internal gear train to the crank of slider-crank mechanisms [13,14]. Li and Cao [15] investigated the dwell characteristics of such mechanisms. The geared linkage mechanisms which are composed by adding the spur gears to each pair point of the planar four-bar mechanisms were classified and analyzed by Yokoyoma [16]. Geared linkages with non-circular gears are also studied in the literature [17,18]. This paper presents an analysis and synthesis method for a certain type of GFBM (type A1, input link 2, output link 3), which is different than the ones investigated in the previous works. It is observed that the GFBM considered is inherently a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. 2. Enumeration of the GFBM The mechanism studied in this work has a topology type A, which is simply a gear pair combined with a four-bar mechanism. Different mechanisms can be obtained by fixing links 1 to 5 respectively. These mechanisms can be summarized as in Fig. 2. Moreover, by changing the input and output links, different input–output functions can be achieved which is summarized for type ⁎ Corresponding author. E-mail address: [email protected] (V. Parlaktaş). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.03.007
V. Parlaktaş et al. / Mechanism and Machine Theory 45 (2010) 1142–1152
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Fig. 1. Topology type A, and topology type B.
A1 in Table 1. Note that, instead of external gears internal gears can also be used in some of these mechanisms. Type A3 is nothing but a four-bar driven by another gear, and type A5 is a four-bar with a floating gear pair. The mechanism studied in this work is of topology type A1. One of the gears is rigidly connected to the coupler and the other gear has a fixed axis of rotation. This forms a planetary gear train with link 3 as the arm. The input torque is applied to link 2, and the output is obtained at the arm (link 3). Thus, the input and the output displacements are about the same rotation axis. 3. Motion analysis of the GFBM In order to obtain the relationship between the input (θ12) and output (θ13), an equation can be obtained from the loop closure equation of the four-bar mechanism (Fig. 3). iθ13
a3 e
iθ14
+ a4 e
iθ15
−a1 = a5 e
ð1Þ
There is another equation which is the velocity ratio of the planetary gear arrangement formed by links 2, 3 and 4: −
r4 ω −ω13 = 12 = −R r2 ω14 −ω13
ð2Þ
Integrating this equation: θ14 =
ðR + 1Þ θ θ13 − 12 R R
ð3Þ
+k
where, R = r4/r2 and k is a constant determined by the initial relative positions of links 2 and 4 with respect to link 3 (integration constant). Eq. (3) can be used in Eq. (1) to eliminate θ14 and a new equation can be obtained. Then, this obtained equation can be multiplied by its complex conjugate (eliminating θ15) to obtain a relationship between θ12 and θ13. However, obtaining an explicit relationship between the input and the output is very hard if not impossible by this method. Therefore, a novel approach is used to determine the input–output relationship in two steps.
Fig. 2. Enumeration of the GFBM.
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Table 1 Possible input–output links for type A1. Reference
−
[9]
[10]
Four-bar
[9,11]
Input link Output link
2 3
2 5
5 2
5, (3) 3, (5)
3 2
It is assumed that the input link of the mechanism is not link 2 but link 4. Hence, according to this assumption, an explicit relationship between θ14 and θ13 can be determined from the loop closure equation of the four-bar mechanism. Since, θ13 is determined as a function of θ14, then by using Eq. (3) a relationship between θ12 and θ14 can be obtained. Hence, for a given θ14 corresponding θ13 and θ12 can be determined explicitly. Therefore, a chart which gives the relationship between the input θ12 and output θ13 can be determined. With this approach, an iterative numerical solution is eliminated. For link 2 to have a complete rotation, the coupler link (link 4) must also have a complete rotation, while links 3 and 5 will have oscillations only. The full cycle rotation of link 3 imparts a full cycle rotation of the coupler link of the four-bar formed by links 1, 3, 4 and 5 (Fig. 3). This four-bar mechanism must then be of Grasshof type double-rocker mechanism (the sum of the longest and shortest link lengths is less than the sum of the two intermediate link lengths and the link opposite the shortest link is the frame). With this consideration, the rotation of link 3 can be determined in terms of the rotation of the coupler (link 4). Multiplying the loop closure Eq. (1) by its complex conjugate, the output θ13 can be determined as a function of θ14. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ13 = 2arctan −B∓ B2 −4AC = 2A
ð4Þ
where, A = K1 −K2 cos θ14 + K3 −K4 cos θ14 B = 2K2 sin θ14 C = K1 + K2 cos θ14 −K3 −K4 cosθ14 K1 = a21 + a23 + a24 −a25 K2 = 2a3 a4 K3 = 2a1 a3 K4 = 2a1 a4 : Since θ13 is determined for a given θ14, corresponding θ12 can be determined from Eq. (3). θ12 = ðR + 1Þθ13 −Rθ14 + k Therefore, the input–output relationship can be determined at two steps by solving Eqs. (4) and (5) respectively.
Fig. 3. The GFBM and the corresponding four-bar mechanism when the gears are removed.
ð5Þ
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k is a constant determined by the initial relative positions of links 2 and 4 with respect to link 3 (integration constant). It can be choosen arbitrarily. While obtaining input–output curves, in order to start θ12 values from 0° (when θ14 = 0°, θ13initial is calculated, so in order to obtain θ12 = 0°), k is chosen as: k = −ðR + 1Þθ13initial :
4. Transmission angle of the GFBM The transmission angle of a mechanism is defined as [19]: tanðμ Þ =
force component acting on the output link tending to produce output rotation : force component tending to apply pressure on the driven link bearings
Neglecting the mass of the links, the free-body diagrams of the links of the GFBM will be as shown in Fig. 4A. Where, Ti is the input torque applied to link 2, To is the output torque at link 3, and α is the pressure angle of the gears. The transmission angle of the mechanism can be obtained as: tanðμ Þ =
F43t : F43n
ð6Þ
From the free-body diagrams of the links, F43t and F43n can be obtained, and from Eq. (6), the transmission angle can be determined as: tanðμ Þ =
a r4
− 4 sinðθ15 −θ14 Þ−sinðθ15 −θ13 Þ a4 tanðαÞsinðθ15 −θ14 Þ r4
+ cosðθ15 −θ13 Þ
:
ð7Þ
If the direction of rotation of the input link is changed, then the transmission angle of the GFBM alters since the line of action of the force between links 2 and 4 changes (Fig. 4B). In this case, the transmission angle will be given by: tanðμ Þ =
a r4
− 4 sinðθ15 −θ14 Þ−sinðθ15 −θ13 Þ a r4
− 4 tanðαÞsinðθ15 −θ14 Þ + cosðθ15 −θ13 Þ
:
ð8Þ
Note that there is a minus sign in front of the term including α. Unlike all other mechanisms known up to now, for this mechanism the force transmission characteristics are also a function of rotation direction. A mechanism which has proper transmission angle values in one rotation direction can even lock if the direction of rotation of the input is reversed.
Fig. 4. A) The FBD of the links when link 2 is rotating counterclockwise, B) the FBD of link 4 when link 2 is rotating clockwise.
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Fig. 5. The position of the output link θ13 and the transmission angle µ wrt the input position θ12 for ccw and cw rotation direction.
Example 1. Consider the GFBM with link lengths: a1 = 1, a3 = 0.907, a4 = 0.306, a5 = 0.665, and R = 1. In Fig. 5, the position of the output link θ13 and the transmission angle µ are shown with respect to the input position θ12 for ccw and cw rotation direction. Note that, there are two different µ curves, according to the rotation of direction. Also note that, a large time ratio (2,27) is obtained. 5. Synthesis of the GFBM The problem is considered in two parts. The first part is the synthesis problem in which one must determine a four-bar mechanism of Grashof type with double-rocker proportions that have given swing angles ϕ and ψ for links 3 and 5 respectively. There is an infinite set of solutions for this part of the problem. The second part of the problem is concerned with the optimization. Out of the infinite possible set of solutions obtained in the first part, one must determine a particular mechanism whose maximum transmission angle deviation from 90° is a minimum. The extended and folded positions of the mechanism are shown in Fig. 6. At the extended position link 3 is at the forward position and at the folded position link 3 is at the fully withdrawn position. θ is the angle of link 3 and ξ is the angle of link 5 at the folded position. The loop closure equations for the folded and extended positions of the mechanism can be written as [20]: iθ
iξ
a3 e = a1 + ða5 −a4 Þe i ðθ + ϕ Þ
a3 e
ð9Þ iðξ + ψÞ
= a1 + ða5 + a4 Þe
:
ð10Þ
One can define Z1, Z2 and λ as: iθ
ð11Þ
iξ
ð12Þ
Z1 = a3 e Z2 = a5 e λ=
a4 : a5
ð13Þ
Without a loss of generality the fixed link can be chosen as unity; a1 = 1. Then, Eqs. (9) and (10) can be written in normalized form as: Z1 −Z2 ð1−λÞ = 1 iϕ
ð14Þ
iψ
Z1 e −Z2 e ð1 + λÞ = 1:
ð15Þ
Fig. 6. The dead-center positions of the GFBM.
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Fig. 7. Z1 and Z2 circles for the values of ϕ = 50° and ψ = 10°.
These complex equations are linear in terms of the unknowns Z1 and Z2. Then, Z1 and Z2 can be solved in terms of λ, ϕ, and ψ as: Z1 =
ð1−λÞ−ð1 + λÞeiψ ð1−λÞeiϕ −ð1 + λÞeiψ
ð16Þ
Z2 =
1−eiϕ : ð1−λÞeiϕ −ð1 + λÞeiψ
ð17Þ
Fig. 8. The design chart for cw input rotation and R = 1.
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As λ changes from −∞ to +∞, Z1 and Z2 describe a curve which is the loci of the tip of the vectors Ao Af and Bo Bf. These loci are two circles for any given value of ϕ and ψ. In Fig. 7, these two circles are shown for the values of ϕ = 50° and ψ = 10°. A line can be drawn from (0,0), which is Ao, at an angle θ with respect to A o Bo. Af is the intersection point of this line and the Z1 circle. Another line can be drawn from Bo at angle ξ with respect to A o Bo which intersects the circles at Af and Bf, respectively. This corresponds to the folded position of the mechanism, where the link lengths are, AoAf = a3, AfBf = a4, BoBf = a5 and AoBo = a1. Analytically, link lengths can be determined as: a3 =
qffiffiffiffiffiffiffiffiffiffi Z1 Z1
ð18Þ
a5 =
qffiffiffiffiffiffiffiffiffiffi Z2 Z2
ð19Þ
a4 =
qffiffiffiffiffiffiffiffiffiffiffi a25 λ2 :
ð20Þ
The link lengths are functions of the free parameter λ, and swing angles ϕ, and ψ. For a given ϕ and ψ, there is a set of solutions with respect to the free parameter λ. A necessary but not sufficient condition for double-rocker proportions is: 0 b jλ j b 1:
ð21Þ
In order to obtain double-rocker proportions there are limits on swing angles as: 0 b ϕ b 180˚ ϕ 2
−90˚ b ψ b
ð22Þ ϕ 2
+ 90˚:
ð23Þ
Fig. 9. The design chart for ccw input rotation and R = 1.
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5.1. Transmission angle optimization Among the set of solutions for a given ϕ and ψ, the one for which the maximum deviation of the transmission angle from 90° is a minimum must be selected (λopt will be determined). The design charts are obtained for a given output swing angle ϕ and the corresponding input link rotation β. Therefore, the optimum mechanism in terms of the transmission angle can be determined, for a given output swing angle ϕ and a time ratio, from the design charts. Using Eq. (5) at the folded and extended positions, β can be obtained for a gear ratio R, and according to the input direction of rotation (− sign for cw input) as: β = ðR + 1Þϕ∓Rπ−Rψ:
ð24Þ
A parametric optimization routine is developed and design charts for optimum GFBM are prepared by using MATLAB. According to the direction of rotation of the input, the transmission angle of the GFBM changes. That condition leads to two different optimum mechanisms which have the same output swing angle ϕ and corresponding input rotation β. Therefore, transmission angle optimization is performed for both of the input directions of rotation and two sets of design charts are prepared. The affect of gear ratio (R) is very clear, it affects the transmission angle; if R is increased, then the transmission angle improves. Therefore, transmission angle optimization is performed for several gear ratios and the corresponding design charts are also prepared. Some of those design charts are displayed in Figs. 8–11. The Y-axis represents the output swing angle ϕ, and the X-axis represents the corresponding input link rotation β. The full lines represent the optimum λ parameter, and the dotted lines represent the maximum deviation of the transmission angle from 90° (δµmax) for the corresponding mechanism. Since, ϕ and β are the given parameters, a value for ϕ and another value for β is chosen from Y and X axes respectively. Then, these values are intersected and the value of optimum λ parameter, and for that mechanism, the value of maximum deviation of the transmission angle from 90° can be obtained (since the value of optimum λ is determined the link lengths can easily be determined from Eqs. (18)–(20)). If the obtained δµmax is not preferred, then ϕ or β is changed, and another optimum solution can be obtained. Therefore, by the aid of these design charts, by specifying ϕ and β, the corresponding optimum mechanism and δµmax for that mechanism can be determined easily. It can be observed that transmission characteristics of the mechanisms deteriorate as the input rotation β approaches to R × 180° (R = 1,2,3,4). Therefore, for an acceptable transmission angle, centric mechanisms can be obtained for small output swing
Fig. 10. The design chart for cw input rotation and R = 2.
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Fig. 11. The design chart for cw input rotation and R = 4.
angles only. Hence, it can be concluded that this type of GFBM is useful for quick-return motions. This condition is explained as below. If a mechanism is to be designed from Fig. 8, which has an output swing angle ϕ = 50° and a large time ratio of 2 (β = 120°), from the intersection point of these values it is seen that the value of λ ≈ 0.7, and for that mechanism the maximum deviation of the transmission angle from 90° is ≈42°. If a smaller time ratio is desired, for example 210/150 (still ϕ = 50°) β is chosen as 150°. Then, from the intersection point of these values it is determined that λ ≈ 0.67 and δµmax ≈ 62° which lead to a mechanism with poor transmission characteristics. So, it can be observed that as the time ratio approaches to 1 (β approaches to 180°) for a fixed value of ϕ, the maximum deviation of the transmission angle from 90° (δµmax) of the corresponding mechanism increases (also note that in Fig. 10, since R = 2, as time ratio approaches to 1, β approaches to 360°). The gear ratio R significantly affects the transmission angle of the GFBM. As R increases, the transmission angle improves and mechanisms with larger output swing angles can be obtained (Figs. 10 and 11). Conversely, if R is decreased, transmission characteristics deteriorate. Therefore, transmission angle optimization is performed for several gear ratios and the corresponding design charts are also prepared.
Fig. 12. The transmission angle µ and the output link position θ13 wrt the input link position θ12.
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Fig. 13. The angular velocity ω13 and the angular acceleration α13 of the output link.
Example 2. For a given swing angle ϕ = 40°, and corresponding input rotation β = 100° (time ratio = 2,6) (cw input, R = 1), λopt can be determined from Fig. 8 as, λopt ≅ 0.2 (δµmax ≅ 30°). Then, the link lengths can be determined from Eqs. (16)–(20), and (24) as: a3 = 0.53, a4 = 0.181, a5 = 0.867. In Figs. 12 and 13 the transmission angle µ, the output link position θ13 and the angular velocity ω13 and acceleration α13 of the output link are shown with respect to the input link position θ12. The mechanism has quick-return motion characteristics and in the forward stroke approximately constant velocity of the output link is obtained (ω12 = 1 rad/s). 6. Conclusion In this work a new type of geared four-bar mechanism for which the input and output shafts are collinear has been investigated. A novel analysis method is devised, expressions for the transmission angle are derived and charts are prepared for the optimum design of such mechanisms. By the aid of these design charts, by specifying the output swing angle and the time ratio, the corresponding mechanism which has the best transmission characteristics can be determined easily. It is observed that, the gear ratio significantly affects the transmission angle; as the gear ratio (R) increases, the transmission angle improves. According to the direction of rotation of the input, there are two different optimum mechanisms which have the same output swing angle and corresponding input rotation. During the working stroke, approximately constant angular velocity at the output link is observed. This type of GFBM is suitable as a quick-return mechanism. Large time ratios can be obtained with acceptable force transmission characteristics. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory.2010.03.007. References [1] T.J. Li, W.Q. Cao, J.K. Chu, The topological representation and detection of isomorphism among geared linkage kinematic chains, Transactions of the ASME 25th Biennial Mechanisms, 1998, DETC98/MECH5812. [2] T. Li, W. Cao, Kinematic analysis of geared linkage mechanisms, Mechanism and Machine Theory 40 (2005) 1394–1413. [3] A.G. Erdman, G.N. Sandor, Kinematic synthesis of a geared five-bar function generator, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 11–16. [4] S.A. Oleska, D. Tesar, Multiply separated position design of the geared five-bar function generator, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 74–84. [5] A.V. Mohanrao, G.N. Sandor, Extension of Freudenstein's equation to geared linkages, Transactions of the ASME-Journal of Engineering for Industry 93 (1971) 201–210. [6] C.K. Lin, C.H. Chiang, Synthesis of planar and spherical geared five-bar function generators by the pole method, Mechanism and Machine Theory 27 (2) (1992) 131–141. [7] T.W. Lee, F. Freudenstein, Design of geared five-bar mechanisms for unlimited crank rotations and optimum transmission, Mechanism and Machine Theory 13 (2) (1978) 235–244. [8] K. Luck, K.H. Modler, Getriebetechnik: analyse, synthese, optimierung, Springer, Berlin, 1990. [9] J. Volmer, Getriebetechnik: koppelgetriebe, VEB Verlag Technik, 1, auflage, Berlin, 1979. [10] R. Neumann, Einstellbare fünfgliedrige raderkoppelgetriebe, Maschinenbautechnik 28 (1979) 211–215. [11] R. Neumann, Bauelemente auslegung von getrieben berechnung von getrieben mit grossem schwingwinkel, Maschinenbautechnik 29 (1980) 405–406. [12] J. Volmer, H. Riedel, E. Rissman, Bauelemente auslegung von getrieben berechnung von schritt und pilgerschrittgetrieben, Maschinenbautechnik 29 (1980) 121–124. [13] J. Volmer, Rationalisierungsmittel für den konstrukteurarbeitsblaetter zur auslegung von getrieben, Maschinenbautechnik 29 (1980) 125–127. [14] J. Volmer, C. Meiner, Bauelemente auslegung von getrieben berechnung von schritt und pilgerschrittgetrieben, Maschinenbautechnik 27 (1978) 219–220.
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