Transmission angle in mechanisms (Triangle in mech)

Mechanism and Machine Theory 37 (2002) 175±195

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Transmission angle in mechanisms (Triangle in mech) Shrinivas S. Balli a

a,b

, Satish Chand

a,*

Department of Mechanical Engineering, M.N. Regional Engineering College, Allahabad, 211 004 UP, India b Basaveshwar Engineering College, Bagalkot 587 102, Karnataka, India Received 17 May 2000; received in revised form 16 February 2001; accepted 20 August 2001

Abstract The transmission angle is an important criterion for the design of mechanisms by means of which the quality of motion transmission in a mechanism, at its design stage can be judged. It helps to decide the ``Best'' among a family of possible mechanisms for most e€ective force transmission. Literature on transmission angle in a planar 4-, 5-, 6- and 7-bar linkages and spatial linkages is organized. A survey of synthesis of mechanism with transmission angle control is made. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Transmission angle; Mechanism; Linkage; Quality of motion; Synthesis; Optimum

1. Introduction Mechanisms for non-uniform transmission of motion such as linkages are characterized by continuously changing transmission ratios. Ideally a smooth motion throughout the whole range of operation is expected. For designing such mechanisms it is important to utilize fully all possibilities known from theory and practical experiences [1]. The criteria for the design of mechanism are low ¯uctuation of in put torque, compact in size and links proportion, good in force transmission, low periodic bearing loads, less vibrations, less wear, optimum transmission angle and higher harmonics. The transmission angle is an important criterion for the design of mechanism as was pointed out by Alt [3]. It is very dicult to say that a particular design is the only solution for a given problem. For example, there are number of alternate possibilities for the motion relationship between the input

*

Corresponding author. Tel.: +91-532-445103x1110; fax: +91-532-445106, -445107. E-mail addresses: ssballi@redi€mail.com, [email protected] (S.S. Balli), satishchand@redi€mail.com, [email protected] (S. Chand). 0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 6 7 - 2

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and output links of a simple four bar linkage which has a very few parameters. When the number of links and number of degrees of freedom increase the design becomes more complicated. Because of the many parameters, the possible solutions may also increase. This calls for an extensive research on this topic alone [1]. Therefore, it is better to have a measure of how well a mechanism might ``run'' when it is still in design stage, i.e. the e€ectiveness with which motion is imparted to the output link. It implies a smooth operation in which a desired force component is available to produce a torque or a force in an output member [2]. A good transmission angle is the solution to most of the problems in mechanisms. In this paper, the primary concentration is on transmission angle and the other aspects of synthesis are attended only to the extent to which they overlap. The transmission angle in planar 4-bar mechanism is discussed followed by 5-, 6-, 7-link and spatial mechanisms. A brief touch of dwell mechanisms, adjustable mechanisms and dead center position mechanisms is also made. 2. Transmission angle in planar 4-bar mechanism 2.1. De®nition Transmission angle is a smaller angle between the direction of velocity di€erence vector VBA of driving link and the direction of absolute velocity vector VB of output link both taken at the point of connection (Fig. 1) [2]. It is the angle between the follower link and coupler of a 4-bar linkage [12]. The de®nitions are related to a joint variable and depend on the choice of driver and driven links. It appears to be an acute angle l and an obtuse angle (180° l). It varies throughout the range of operation and is most favorable when it is 90°. The recommended transmission angle is 90°  50° [3]. In mechanism having a reversal of motion, i.e. if roles of i/p and o/p links are reversed during the cycle, transmission angle must be investigated for both directions of motion transmission [1]. Referring to the Fig. 1, transmission angle in the form of equation is given by, c2 2ad cos h2 : …1† 2bc Transmission of motion is impossible when transmission angle is 0° or 180°. If transmission angle is zero, no torque can be realized on output link, i.e. mechanism is at its dead center pol ˆ cos

1

d 2 ‡ a2

b2

Fig. 1. Showing transmission angle.

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sition. A large transmission angle does not necessarily guarantee the low ¯uctuation of torque. Very small or very large transmission angle results in large error of motion, high sensitivity to manufacturing error, noisy and unacceptable mechanism [8]. It is not the absolute value of transmission angle but its deviation from 90° that is signi®cant. Di€erent limits for transmission angle suggested are 35±145°; 40±140°; 45±135° [10]. 2.2. Maximum and minimum transmission angles The transmission angles at the extreme positions of a double rocker linkage will also be the minimum and maximum values of transmission angle for the entire motion of mechanism (Fig. 2) [7]. In case of crank-rocker and drag mechanisms, the transmission angle will be minimum when input crank angle is zero and maximum when input crank angle is 180° (Figs. 3 and 4).These occur twice in each revolution of the driving crank. They do not occur at the extreme positions of the linkage. Hain [9] recommends that a drag mechanism is dynamically ecient when it experiences the same lower limit of the minimum transmission angle at its two positions where they occur and when this lower limit, …lmin †max is as large as possible. Maximum and minimum transmission angles depend greatly on rocker angle. The concept of pseudo-mini±max transmission angle is discussed by Gupta [17].

Fig. 2. Double-rocker mechanism.

Fig. 3. The crank-rocker mechanism.

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Fig. 4. Double-crank mechanism.

2.3. Optimum transmission angle The deviation of transmission angle from 90° is the measure of reduction in e€ectiveness of force transmission. So the aim in linkage design is to proportion the links so that these deviations are as small as possible, especially in the presence of appreciable joint friction [12]. If the range of operation is suciently small, it seems as if we could obtain a linkage with optimum variation of transmission angle if it is set equal to 90° in the designed position [8]. Among the family of possible 4-bar linkages, there is one linkage that has a minimum transmission angle, which is greater than the minimum transmission angles of all the others. This is called optimum transmission angle and this particular linkage has the best dimensions for most e€ective force transmission [9]. If the designer tries to optimize the linkage with respect to its force transmission characteristics simultaneously with optimum transmission angle synthesis, it increases the diculty of problem extensively. Therefore combined force transmission and synthesis studies have been restricted to relatively simple linkages. On the other hand, the problem simpli®es signi®cantly if the transmission angle is restricted such that Dl < d where Dl ˆ maximum deviation of transmission angle l from optimum one and d ˆ a speci®c bound on Dl [11]. 2.4. Constant transmission angle In some of the applications like direct contact mechanism (cam and follower mechanism) and trammel mechanism, the transmission angle is kept constant. In such cases, the transmission angle depends on the shape of the curved surface of the output link. If the curve is of the form of a logarithmic spiral having a constant rise, then the transmission angle remains constant for some part of motion. Hain [1] has shown how this can be done in 6-link coupler mechanism. 2.5. Importance of transmission angle Alt and Volmer [3] used the transmission angle to isolate better chains for various linkage applications. The mechanism designed with maximal transmission angle criterion will have minimum force acting along the coupler and on the bearings resulting in small friction torque at

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the shafts. Though a good transmission angle is not a cure-all for every design problem, however, for many mechanical applications it can guarantee for the performance of linkage at higher speed without unfavorable vibrations [13]. When l ˆ 90°, most e€ective force transmission takes place and the accuracy of output motion is less sensitive to manufacturing tolerances of link lengths and clearance between joints and change of dimensions due to thermal expansion [12]. Mechanisms having transmission angle too much deviated from 90°, exhibit poor operational characteristics like noise and jerk at high speeds [1,2,10]. If it is 0°, self-locking takes place. Thus the transmission angle of a mechanism provides a very good indication of the quality of motion, the accuracy of its performance, expected noise output and its costs in general [8]. In other words, it is a simple and useful coecient of performance for mechanisms for non-uniform transmission of motion. It serves as a basis for comparing mechanisms. 2.6. Transmission angle and other parameters The e€ect of transmission angle on the other parameters of mechanism such as pressure angle, force transmission, velocity, acceleration, input crank angle, friction, mechanical advantage tolerance and clearance and performance sensitivity is discussed in the following sections. 2.6.1. Transmission angle and pressure angle Pressure angle (k) is the complimentary indicator to the transmission angle [20] and is given by k ˆ j90 lj. It will reach two extreme values during the cycle when crank and ®xed links are collinear. If these values are equal, then desirable chain characteristic results [1], i.e. kmax ˆ kmin < 45°. Small pressure angles lead to reasonable mechanical advantages and a high quality of motion transmission [1]. Pressure angle is signi®cant not only as an indicator of good force and motion transmission but also as a prime factor in the linkage sensitivity to small design parameter errors. Larger the maximum pressure angle, more sensitive the linkage will be [21]. Small pressure angles are consistent with minimal mechanical errors [2]. Therefore, it is reasonable to attempt to minimize the maximum pressure angle in a linkage design. 2.6.2. Transmission angle and force transmission Alt [3] de®nes transmission angle as: l ˆ tan

1

Ft ; Fb

…2†

where Ft is the force tending to move the driven link and Fb is the force tending to apply pressure on the driven link. Rossner [1] considered the forces acting at the pivots to evaluate the quality of transmission more accurately and de®ned transmission as, tan l ˆ

Tangential force : Normal force on o=p link

…3†

According to Link, it is the angle between the direction of force and the direction motion [1]. When the force along the coupler drives the output link at 90°, i.e. when transmission angle is as close to 90° as possible for entire motion of mechanism, the bending in the linkage will be reduced

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and will produce the most favorable force transmission condition [27]. The minimum permissible transmission angle of a linkage mechanism depends on magnitude of transmitted forces including d'-Alembert's inertia forces [10]. The transmission angle does not consider the dynamic forces due to velocity and accelerations. Because of this reason, it is widely used in kinematic synthesis stage during which the lengths and mass properties of the links are still unknown [11], i.e. kinematically determined transmission angle does not re¯ect the action of gravity or dynamic forces. For the determination of transmission characteristics of the linkage only, it is not necessary to analyze the forces and torque acting at each joint of the whole mechanism. It is suggested to ®nd the nature not the magnitude of the force system acting at joint by ¯oating link and to see how this force system can a€ect the rotation of output link. This force system can be conceived by simple conceptions of statics [16]. Liu and Angeles [100] resort to transmission angle to measure the goodness of force or torque transmission to output link of 4-bar mechanism. To evaluate the performance of a linkage in terms of its transmission angle within the mobility of i/p link, the transmission quality is introduced. 2.6.3. Transmission angle, velocity and acceleration Higher the value of a° =x20 , the poorer will be the acceleration characteristics where a0 is the angular acceleration of output link and x angular velocity of input link. Though the cam driven and geared linkages have better acceleration characteristics (lesser value of a0 =x2 ) pure linkages are used where acceleration characteristics are not of major importance. It is because of ease of manufacture, cheaper cost and compactness. The design technique that correlates the input and output link positions though permits the designer to select the transmission angle at the ends of cycle, provides little control over the acceleration characteristics of the mechanism [14]. Best acceleration characteristics occur when initial transmission angle is greater than 90° and they are worst when initial transmission angle is smaller than 90°. Variation in the ®nal transmission angle has negligible e€ect on acceleration. In crank rocker mechanism when angle of oscillation reaches 180° mechanism will be in its dead center position. An auxilliary force is needed for operation leading to extremely poor acceleration characteristics. The drag mechanism with optimum transmission angle has less acceleration and hence a smooth motion. With lower values of transmission angle the mechanism is subjected to high acceleration, objectionable noise and jerk at high speed. Therefore high speed mechanism should have lmin > 40°. All these lead to design a mechanism with optimum transmission angle for good acceleration characteristics [15]. 2.6.4. Transmission angle and input (crank) angle The choice of initial input angle h2 depends on following design control factors ± smooth in force transmission, low in fatigue and wear loads, compact in size and link proportions. Among these the most important is the force transmission characteristics [7]. With decrease in input angle h2 ; lmin increases and crank length decreases. As h2 tends to 0°; lmin approaches maximum value and crank length approaches zero which is impractical. Practical considerations limit the minimum length of crank. A compromise is to be made between reduction in crank length and increase in lmin when Dh ˆ 180° [18]. Higher the value of initial input angle more optimistic the transmission angle will be.

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Transmission angle as a function of input angle for various Grashof classes is dealt with and categories of devices best suited for a particular type of transmission behavior is indicated by Shoup in discussion of [17]. A functional relationship l ˆ f …h2 † is produced by variation of the motion variable h2 . An in®nite number of 4-bars exist that produce the same functional relationship [19]. 2.6.5. Transmission angle and friction Friction can change the basic manner of operation of a mechanism and impair its usefulness. In functioning of mechanism friction forces can hardly be distinguished from bending forces. These in¯uence the mode of operation of mechanism. In cases involving transmission of forces through a mechanism, frictional forces may be neglected. In other cases, it is not allowable and many investigations on the e€ect of friction have been made [1]. When transmission angle deviates signi®cantly from 90°, the torque on the output bar falls o€ and may not be sucient to overcome friction in the system. To avoid this situation, Dl ˆ j90° lj should not be too great. There is no de®nite upper limit for Dl because in practice the existence of inertia forces may eliminate the undesirable force relationships that prevail under static conditions [26]. Lenk and Bock take the frictional characteristics of a mechanism into consideration in evaluating the quality of motion transmission. Bock suggested using the cosine of the angle of force application as a better measure for the quality of transmission of motion [1]. 2.6.6. Transmission angle and mechanical advantage Mechanical advantage is the instantaneous ratio of output force (F4 ) or torque (T4 ) to the input force (F2 ) or torque (T2 ) of a mechanism, i.e. Mechanical advantage ˆ

F4 T4 RC O4 sin l ˆ ˆ ; F2 T2 RB O2 sin b

…4†

where RC O4 and RB O2 are the segments into which the common instant center cuts the line of centers [16] and if the e€ect of inertia is neglected. As the transmission angle becomes small the mechanical advantage decreases and even a small amount of friction will cause the mechanism to lock. The mechanical advantage of a 4-bar mechanism is directly proportional to sin l. A linkage with a good mechanical advantage may have an unacceptable transmission angle and a linkage with an excellent transmission angle in a particular position may not have a sucient mechanical advantage. Since both the transmission angle and mechanical advantage vary with linkage position, either parameter can be critical to the designer in certain positions [54]. 2.6.7. E€ect of tolerance, clearance and play on transmission angle It is necessary to have good transmission angle to have low sensitivity to manufacturing error. If transmission angle is large enough and tolerances are small enough, the e€ect of tolerance on output, is negligible [1,23]. The design allowing for eventual mechanical errors is the most favorable from standpoint of linkage length ratio and transmission angle [24]. For small l the e€ects of tolerances in the link dimensions are much more unfavorable than for a l near 90°. To know the e€ect of clearances, each member must be thoroughly investigated. As long as the members of the joint remain in contact on one side, the clearances will have no harmful e€ects. But greater the clearances, greater the impact becomes due to change of direction. The problem of

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avoiding directional changes in oscillating mechanisms appears to be solvable. Thus it is possible to decide how much the tolerances can be increased without impairing the performance of the mechanism [1]. Maximum error in o/p of a 4-bar mechanism for a given range of i/p occurs when jp=2 lj is the largest. Good transmission properties generally imply smaller error due to clearance. Error is least when l ˆ p=2. As l moves away from 90°, the error increases rapidly [25]. 2.6.8. Transmission angle and performance sensitivity In linkages used for instrumentation and control, sensitivity should be optimum so that output link responds well, even for very small changes in the input. Also it is most desirable to incorporate good transmission characteristics in the linkage. Rao [87] proposes a method to synthesize a 4-bar linkage for three precision points having good transmission characteristics. A mechanism with a poor transmission angle also subjects to large mechanical errors. Faik and Erdman [86] derived the relationship between the proportions of a mechanism and its sensitivity. Sensitivity is related to 4-bar solution space concept developed by Barker [101]. Transmission angle plays a crucial role in the upper bound sensitivity control during synthesis. A sensitivity coecient contains the smallest value of transmission angle. If transmission angle is zero or p, the sensitivity value for all linkages will tend to in®nity. Transmission angle should be controlled to be greater than a desired value and should be kept as close as possible to p=2 which makes output variables less sensitive to the transmission angle [85]. 2.7. Synthesis of planar 4-bar mechanism with optimum transmission angle ± a survey One important task of kinematic synthesis is to design a linkage in such a way that the transmission angle does not fall below a prescribed minimum value. Using transmission angle for rating the suitability of 4-bar linkages, a number of design procedures are proposed. Following is the brief survey of such works. These graphical and analytical approaches include motion, path and function generation. The references provided are representative rather than comprehensive. Closed form equations are presented [7,27] for coordinating the prescribed extreme positions of a double rocker mechanism. Design is reduced to the solution of algebraic quadric equation. A crank-rocker mechanism is designed [13] by an approximate graphical method by Eschenbach and Tesar by applying the knowledge of plane algebraic curve tracing. They established a geometric bounded region to limit the link lengths of chain on the basis of inequality constraint 45° < l < 135°. To represent the overall quality of quick returning crank rocker mechanism with respect to its transmission variation in a complete input crank rotation, a suitable norm of cosl in the range of the input crank rotation (0; 2p) is to be set. The mechanism is considered to be optimal when such particular norm is minimized as suggested by Gupta [6]. An approximate analytical procedure to synthesize crank-rocker mechanism with a better transmission angle is presented for four di€erent situations. Closed form equations are developed by Khare and Dave for i/p crank angle h1 > 180°, h1 < 180°, h1 ˆ 180° and h1 ˆ 180° ‡ w0 (w0 ˆ o=p angle) and minimum transmission angle for the complete rotation of crank. The upper and lower bounds on crank angle correspond to a dead centre position of rocker are obtained by using Grashof's criterion and the design is optimized by maximizing the lmin [18]. Cleghorn and Fenton [34] and Lu [35] developed analytical synthesis methods of crank-rocker mechanism with optimum transmission angle.

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For unit time ratio, a crank-rocker mechanism with transmission angle criterion is designed by Hain [1], Brodell and Soni [33], Hall [12], Harrisberger, Volmer and Chen [5]. Such iterative method is proved to be ecient for other types of 4-bar linkages too. It is interesting to note that satisfying condition lmin ˆ 180° lmax , satis®es the condition that the time-ratio for the designed crank rocker mechanism is unity [22]. By writing vector loop equations based on the geometry of constraint, if the parameters are allowed to vary, the solutions to loop equations provide a family of possible designs. A ®nal design is determined by introducing an appropriate initial angle of the crank on the basis of a design criterion [5]. For certain ranges of speci®cation such as time-ratio and rocker swing angle, there is an unique optimal Chebyshev solution for quick return crank rocker mechanism which can be found by solving a cubic equation in terms of a design parameter [36]. The feasible ranges of design parameters are enlarged when transmission angle over the working stroke is taken as design criterion and slow stroke of the rocker is taken as the working stroke. Formulae are presented to facilitate the design with speci®ed optimum options [37]. Several numerical design procedures based on the mathematical programming technique presented incorporate various checks in the computation to ensure that the input link of the designed 4-bar linkage is a crank and transmission angle variation is within reasonable range [37±39]. Some other methods [1,40] require the analysis of a large number of points on the circle point curve for the four position synthesis to insure input link rotability and to optimize transmission angle. Graphical iteration schemes to identify crank type mechanism as well as to resolve problem of crank ordering and branching in the precision position synthesis problem are proposed by Filemon [41,42] and Waldron [43±45]. Gupta has designed precision point and least square 4-bar function generator with fully rotable input links and desired transmission angle variation. This algebraic-geometrical method can also be used e€ectively in the position synthesis problem. This identi®es signi®cant portions of the design regions for crank type 4-bar mechanism in which transmission angle variation is speci®ed [47]. Gupta [48] extends this theory for designing crank type 4-bar mechanism for position, path and function generation in an uni®ed way. The increased use of mechanism for high-speed applications requires the knowledge of inertia forces and harmonic analysis of the motion involved. Freudenstain determined the dimensions of linkage minimizing higher harmonics and optimizing the transmission angle [20]. To determine the dimensions of a drag linkage, minimizing the higher harmonics and optimizing the transmission angle, Hain [46] presented an analysis, which enables the designer to ®nd whether a given mechanism possesses sizable overtones or not. A drag mechanism with more emphasis on control on transmission angle rather than i/p±o/p motion coordination is designed and error in coordination is optimized by least square method by Kulkarni and Khan [15]. Crank type mechanisms for exact generation of a set of FSP of a moving plane with transmission angle control are given by Kazerounian and Gupta [105]. Synthesis of 4-bar path generating mechanism with optimum transmission characteristics is presented by Roth and Freudenstain [22]. A numerical method for generators of function, path and rigid motions is given by Cossaltor et al. [49] giving wide variability ranges for parameters chosen randomly. Mechanism assembly is managed by introducing a proper penalty function. Kacachios [51] used augmented Lagrangian penalty function coupled with a variable metric algorithm for the dimensional synthesis of 4-bar mechanism.

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The performance characteristics such as acceleration, transmission angle and design requirements of mechanism which ®nd the widest practical use belong to broad rather than those that are highly specialized in nature were dealt with by Schnarbach [53] and Aronson [14]. Mini±max velocity ratio chart for determination of the positions of the dynamically ecient drag linkage for minimum velocity ratios are developed by Cemil Bagci [4]. To design a 4-bar function generator of mini±max transmission angle type, an analogous to geometric entities (i.e. circle point curve) in the precision point theory, loci of optimum transmission angle is derived by Gupta [17]. The points on these curves or the intersections of these with other curves give the desired and practical optimum solutions. Chiang et al. [52] provided the inaccessible previous works of higher order synthesis techniques of 4-bar generators in a concise manner. Rao [50] tried the application of geometric programming to synthesis. Dijksman [19] gives a functional relation between input crank angle and transmission angle l ˆ f …h† which is produced by variation of motion variable h. Resit Soylu [11] determines the dimensions of a mechanism such that the constraint (maximum) deviation of transmission angle from optimum one is less than a speci®ed bound on it is satis®ed. Sun and Waldron [88,91] suggest a graphical solution to satisfy branch, order and transmission angle requirements. In a motion generation problem, it is not, however, possible to control the actual values of the maximum and minimum transmission angles without trial and error. They recti®ed the segments on circle point curve giving transmission angle variations in design range [47]. Chuang and Chiang [89] present a computer-aided method minimizing the maximum error and the maximum deviation of transmission angle from 90° by varying the scale factors. Using the mean values of the optimizing scale factors obtained in both cases, it is made possible to synthesize a 4-bar function generator with a compromise between optimum structural error and optimum transmission angle. Shoup and Pelan [39] tried to optimize transmission angle and structural error simultaneously for the design of function generating mechanism over some prescribed portion of total range of motion by iteration based on solution of nonlinear algebraic equations. Barker and Shu [90] give the three position equations, which are combined with the equal deviation of transmission angle condition on the non-dimensional link lengths to produce a thirdorder polynomial. The roots of this polynomial produce potential solutions, which must be evaluated for defects. Once the defects are removed, the remaining solutions yield defect-free GCRR and GCCC mechanisms, which have equal deviation on transmission angle. The selective precision synthesis (SPS) is a design method that uses optimization techniques which can be applied to the path, motion and function generating planar mechanism with or without velocity and acceleration speci®cations for which exact accuracy is neither possible to obtain nor needed due to link manufacturing tolerance and need for pin joint clearances. Arbitrary limits of precision (accuracy neighborhoods) are speci®ed around precision points. These limits de®ne the error envelope and optimized mechanism is determined without branch, order or locking defects. The designer can arbitrarily select large or small allowable deviations for any and all of speci®ed precision conditions. If an unreasonable set of allowable deviations was selected, it is likely that no solution would be found [55]. Bajpai and Kramer [99] states that a mechanism constructed from dyads that satis®es all of the constraints speci®ed may have locking defect. This happens when neither of its cranks is able to rotate fully. At each

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Fig. 5. Slider-crank mechanism.

extreme point in the path (motion/function) generation, l becomes 0° or 180° and no motion is possible beyond that point. 2.8. Transmission angle of slider-crank mechanism The transmission angle in a slider crank mechanism is de®ned as the angle between the coupler rod and the normal to the straight line path of the slider [32]. It is maximum when the input crank angle is 270° and minimum when the input crank angle is 90° (Fig. 5). In an inverted slider crank mechanism, the slider pair at the o/p link always maintains the transmission angle to be 90°. Naik and Amarnath [84] provide a method of synthesis for adjustable slider crank mechanism for di€erent time ratios and constant strokes and constant time ratio and di€erent strokes. The method yields a design zone in which mechanism selected for adjustability experiences desired minimum transmission angle leading to a practical solution to the problem. Vadasz [83] developed design charts providing the designer with the links and eccentricity dimensions for a prescribed minimum transmission angle and time ratio. The user of these charts can select the dimensions and con®gurations of slider crank mechanism when minimum transmission angle lying between 5° and 175° is prescribed. 3. Transmission angle in planar 5-, 6- and 7-bar mechanisms As the number of links becomes more than four and the driver is a ¯oating link, the de®nition of transmission angle has peculiar diculties [2] and even does not have a common agreement for all mechanisms. In some cases transmission angle is not de®ned so far and those mechanisms with accepted transmission angle may fail to predict the quality of motion transmission [76]. A compound mechanism may have more than two transmission angles. In this section, the de®nition and the role of transmission angle in planar 5-, 6- and 7-bar mechanisms are discussed (Fig. 6). 3.1. Transmission angle in planar 5-bar mechanism A 5-bar linkage (Fig. 7(a)) is a 2-DOF mechanism having two independent driving links O2 A and O5 C. The transmission angle in this case is de®ned as the angle between the two couplers AB and BC.

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Fig. 6. 7-link, 2-DOF linkage.

Fig. 7. (a) 5-bar mechanism: h2 ˆ 180°, h5 ˆ 0°; (b) 5-bar mechanism: h2 ˆ h5 ˆ 0°.

Rose [70] called these couplers secondary links. The transmission angle should be continuous, i.e. it must not become 0° or 180° at any position in motion. If transmission angle is assigned a constant value, secondary links AB and BC (Fig. 7(a)) form a rigid coupler and the 5-bar loop reduces to a 4-bar linkage. Then depending upon the input link, either the angle O2 AB or BCO5 represents the new transmission angle. A ®ve bar linkage is called a coupler crank linkage if both input links can be arbitrarily rotated without reaching a dead position, i.e. 0 < l < 180° or lmin > 0° and lmax < 180° must hold true [72]. This condition does not have any singular position for any input angle meaning that the linkage has at least two input cranks [73]. The positions at which the maximum (when input angles h2 ˆ 180° and h5 ˆ 0°) and minimum values of transmission angle occur (when input angles h2 ˆ 0° and h5 ˆ 0°) are shown in Figs. 7(a) and (b), respectively. Minimum values of transmission angle will also occur when input angles h2 ˆ h5 ˆ 0°; h2 ˆ 0° and h5 ˆ 180°; h2 ˆ h5 ˆ 180° [72]. Inverted mechanism of 5-bar has two transmission angles. With the two transmission angles as guides, the second link set is designed completing 5-bar loop [70,73]. Balli and Chand [74] proposed a synthesis method for 5-bar variable topology mechanism with transmission angle control. They used allowable lmin and lmax to design 5-bar variable topology mechanism for motion between extreme positions [71].

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3.2. Transmission angle in planar 6-bar mechanism Hain [1] has identi®ed the di€erent cases of 6-bar linkage such as: (i) 6-bar linkage consisting of two 4-bar linkages in series having two transmission angles (Fig. 8). In this case, there may be more than one transmission angle and the positions of the mechanism in which the extreme values of l occur are dependent only on the dimensions of the basic 4-bar linkage and on the position of the ®xed pivot [1]. (ii) 6-bar linkage with coupler drive (a) with one transmission angle; (b) with two transmission angles. In a two-phase synthesis of 6-bar linkage design, the designer makes a suitable choice for the location of moving pivot in a coupler plane [75]. The designer can freely vary moving point so as to optimize his design with respect to transmission angle or other criteria. The method also provides a computer program output, which consists of a selected group of 6-bar mechanisms capable of meeting the performance speci®cations. Selection of mechanism is made after passing through optimal transmission test for further consideration. Both dimensional variations and transmission angle of 6-bar linkage are closely related to the magnitude of mechanical error, i.e. transmission angle is very sensitive to the variations of linklengths. Yan and Wu [76] proved that mechanisms with accepted transmission angle might fail to predict the quality of motion. They dealt with the counter examples of l of in 6-bar linkages. 3.3. Transmission angle in planar 7-bar mechanism The transmission angle in 7-bar mechanism is dealt in [78]. It is the angle between the direction of F6 and the component along Z6 (Fig. 6). It is necessary to control the deviation of transmission angle from 90° for which the frictional resistance does not become excessive. To obtain a practically useful mechanism, it is necessary to prescribe limits on the transmission angle and link length ratio. With a view to enable to investigate the e€ectiveness of the chosen minimization technique, Lakshminarayana and Narayanmurthi [79] control the transmission angle alone by limiting its values in the design position by the method of transformation of variables and their maximum rate of change by simply rejecting the point that violate the constraints as failures.

Fig. 8. 6-bar mechanism.

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The reasonable values of transmission angles can be located by considering one of the two inputs ®xed at a time each. The transmission angles thus obtained are named intermediate transmission angle and output transmission angle [80]. A graphical method of 7-bar linkage synthesis to generate functions of two variables is presented by Mrutyunjaya [77] yielding an in®nite number of solutions and enabling subsequent selection on the basis of transmission angles.

4. Transmission angle in other mechanisms The transmission angle in cam-follower mechanism, dwell linkage, adjustable mechanisms, dead center position mechanism and spatial mechanism is discussed in the following sections. 4.1. Transmission angle in cam and follower mechanism Cams having shape of constant rise will have a constant transmission angle for some part of the motion. Beyer [56,57] discussed this for disc and cylindrical cams. In a roller follower, the roller diameter has no in¯uence on the transmission angle though it can in¯uence the frictional forces. Pressure angle in cam mechanism is the complimentary angle to transmission angle [1]. The de®nitions of pressure angle and transmission angles are applicable to followers driven by only a single cam. For followers driven by multiple inputs, these de®nitions are not applicable. Dresner and Bunton [65] draw attention to this fact and presented a precise de®nition of pressure angle along with mathematical consequences that properly characterizes the performance of either single input or multi input cam follower mechanism. For single input systems this definition is shown to be equivalent to the de®nitions for pressure angle found in the literature. They also discussed the applicability of this de®nition to determine the transmission angle for planar and spatial linkages with multiple input. 4.2. Transmission angle in dwell mechanism Smith and Tesar [67] provided a number of comprehensive contour charts to enable the designer to select the linkage lengths and orientations of four bar dwell mechanism to satisfy the design criteria, i.e. minimum transmission angle, establishing the region of optimum orientation. Choudhari and Tesar's [68] design charts supply working data to enable the quick design of 6-bar cycloidal dwell linkage design based on four ISP having transmission angle within speci®ed limits. Raju et al. [69] provide design charts for 6-link double dwell linkage based on transmission angle. 4.3. Transmission angle in adjustable mechanisms In adjustable mechanisms, since there are reactions on the adjustable link, that link is also an output link. To keep the reactions small and avoid them completely if possible the transmission angle must be very small or if possible, zero [1]. In the construction of a 4-bar linkage adjustable for o/p velocity while holding i/p and o/p displacements constant, the o/p crank-pin at its terminal position is selected based on a favorable transmission angle and suitable crank length [81].

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Krishnamoorthy and Tao [82] have given a synthesis procedure for an adjustable linkage mechanism for generating intersecting and non-intersecting circular arcs of speci®ed radii. Usually one gets better results by having one or two free choices of parameters for an adjustable 4-bar linkage, than solution without any free choice of parameter. This is because the design of linkage has to satisfy not only the basic equations but also the transmission angle, etc. [92]. 4.4. Transmission angle in dead center position mechanism The dead center position of a planar 4-bar mechanism is de®ned as the position when transmission angle becomes zero [1,32]. In case of 5-bar if the two coupler links lie on a common straight line, then 5-bar loses one-degree of freedom and becomes a temporarily 4-bar linkage. At this position the linkage is at a dead position or at an indeterminate position [72]. The dead center position of 6-bar linkage and transmission angle are discussed in [66,76]. For symmetric 2-DOF, 7-link mechanism in a position of symmetry the transmission angle is equal to 0° or 180°. Then the mechanism will be in a locking position. If such a linkage is required, it is necessary to provide an auxiliary drive [78]. A second driving link is used in dead center position, then there is generally another transmission angle at another point of linkage so that it is possible to overcome the dead lock. This driving link may be a link, which gives up some of its kinetic energy to overcome dead center position. Spring, ¯ywheel, lever may be used for the purpose [1]. 4.5. Transmission angle in spatial mechanism The de®nition of transmission angle is justi®ed in spherical 4-bar linkage too, [58]. Transmission index (TI) in case of spatial 4-bar linkage is given by sinl [59]. A general index of the quality of motion transmission for spatial mechanism based on transmission angle is developed using the theory of screw by Sutherland and Roth [59]. A method for synthesis of spatial linkages with desirable motion transmission and mechanical error characteristics is presented which is similar to the method given by Denavit and Hartenberg [2]. Alizade et al. [63] used the same algorithm (that of planar 7-links, 2-DOF, function generating mechanism) for optimal synthesis of 2-DOF spatial RSSRP mechanism using transmission angle in equality design constraint for force transmission. The simply skewed 4-bar spatial mechanism, centric simply skewed 4-bar mechanism and a skew slider crank mechanism are designed based on transmission angle criteria. Soylemez and Freudenstein [64] also provide the design charts, which give optimum transmission angle and corresponding initial crank angle. Soni and Dikkpati synthesized a spherical 4-link function generator for three position points maintaining the deviation of transmission angle from 90° to a minimum through out the range of rotation of the crank. The method is same as that of Shoup and Pelan [39] for planar version of the problem. The transmission angle of a spherical 4-bar linkage throughout the whole range of input angle can be optimized in a Chebyshev sense, i.e. the deviation of it from 90° can be minimized if its maximum absolute values at the middle and at both ends of the range are equal and opposite [58]. Equations in simple trigonometric form are presented for optimization method of transmission angle of spherical crank-rocker and double-rocker function generator by Chiang [60,61].

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McLarnan [102] used replacing technique and optimized both transmission angle and structural error for three position points. Chen [62], after designing RRPR mechanism for coordinating two extreme positions, allowed position angle of the in put crank (measured from horizontal axis in ccw) to vary within a reasonable range and suggested to obtain an optimal design based on transmission angle. 5. Transmission angle and design curves for optimum linkage design Design charts re®ne and simplify the design procedure. The suitable dimensions of mechanisms can be read o€ directly, saving much of time. They, however, cannot displace all other methods [1]. In early years, the charts for designing crank-rocker mechanism with optimum transmission angle were provided by Alt [28,29]. Improved versions of the same are provided by Volmer and Jensen [30]. They are again modi®ed by Leu and Lun [31]. Jensen [9] developed a chart to single out the linkage with optimum transmission angle from a family of possible crank-rocker linkages. Design charts are available for ®nding optimum solutions for planar crank mechanism when time ratio and rocker swing angles are speci®ed, in the works of Hain [1], Hall [12] and Soni [32]. Design charts of crank rocker mechanism with optimum transmission angle for output link completing the forward stroke in 180° rotation of the input crank are evolved by Brodell and Soni [33]. To facilitate the design of crank rocker mechanism with optimum transmission angle over its working stroke for a given time ratio and other parameters as rocker swing angle or allowable transmission angle for both working and return stroke, a set of design charts are devised by Lun and Leu [37]. Bagci [4] developed two types of design charts for the selection of dynamically ecient drag mechanisms with most favorable transmission angle when designing them as driving loops of other mechanism loops. They are (i) mini±max velocity ratio chart (ii) input±output displacement chart.

6. Transmission angle as an index of merit Since the ease with which it can be visually inspected, the transmission angle has become a commonly accepted measure of quality of kinematic design of a linkage. If transmission angle becomes too small, mechanical advantage becomes small and even a very small amount of friction will cause mechanism to jam [16]. The most widely accepted design bounds imposed on l are 45° < l < 135°, over a full cycle of motion. Hall [12] puts in other words, j90° ljmax is to be considered one ``®gure of merit'' in comparing alternate possible designs for a particular application. 7. Role of computers in optimum mechanism design Synthesis of 4-bar linkage has been reduced to the solution of algebraic equations. The use of computers in solving these equations is a simple standard process. From among many possible

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solutions generated and tabulated by computer, an optimal design can quickly be obtained based on transmission angle [7]. Graphical methods of dimensional synthesis are desirable because, visualization of solution space yields an intuitive knowledge that can be easily be lost when using only equations to solve a synthesis problem. Computers can eliminate the tedium and inaccuracy inherent in graphical design. By using CAD package the graphical methods are rendered to speed up while maintaining reasonable accuracy and allowing the designer to concentrate on developing a more intuitive understanding of the synthesis process [55,103,104]. SYNTRA ± SYNthesis using TRAces is a software package, which can be used to design mechanisms with prescribed transmission angle. It can handle three-position synthesis problem for function, motion and path generation. It has two versions-Apollo and IBM-PC. It is able to ®nd all the solutions, which have equal deviations of transmission angle from the ideal one [93]. KINSYN-III (KINematic SYNthesis), a revised version of KINSYN has ``analysis during synthesis'' messages about Grashof type, transmission angle and branching. There are ¯ashing signals that appear on screen when de®ned-conditions are satis®ed. The user can request a ¯ashing signal when a 4-bar being designed is found to have the speci®ed transmission angle [96]. LINCAGES-6 (Linkage Interactive Computer Analysis and Graphically Enhanced Synthesis). By applying non-linear programming (complex number) methods and dyadic construction of mechanism general solutions through computer can be readily determined. LINCAGES-6 deals with recti®cation method that has signi®cantly enhanced the software over the years [55]. RECSYN-(RECti®ed SYNthesis) [94] is the improved version of [95,97,98]. It incorporates the control of maximum and minimum transmission angle in the design positions to enable the design optimization [95]. A rectifying synthesis method of 6-bar linkage (Watt-I crank driven) with wellde®ned transmission angles is guided to pass through 4 FSP. It eliminates circuit, branch and order defects [97]. The synthesis of motion generation of crank driven 4-bar mechanism designed to go through 2, 3 or 4 FSP, based on optimum theory and recti®ed synthesis. The technique for automatic synthesis presented has been implemented in the software package [94]. 8. Conclusions In the initial stage of kinematic design of any mechanism, whether it is a geometrical or an analytical method of synthesis, the knowledge of transmission angle is necessary. It helps the designer to compare his design with other similar mechanisms and to select the best among the family of mechanisms. Though the transmission angle is not the only solution for every design problem, however, for many mechanical applications it can guarantee for the performance of linkage at higher speed without unfavorable vibrations. The solution space for the feasible mechanisms is also reduced. Therefore, an attempt is made to bring the entire information about transmission angle in spatial 4-bar, planar 4-, 5-, 6- and 7-link mechanisms under one umbrella.

References [1] K. Hain, Applied Kinematics, McGraw-Hill, New York, 1967 (translated by Adams and others). [2] R.S. Hartenberg, J. Denavit, in: Kinematic synthesis of Linkages, McGraw-Hill, New York, 1964, pp. 46±47.

192

S.S. Balli, S. Chand / Mechanism and Machine Theory 37 (2002) 175±195

[3] Von H. Alt, Der ubertragungswinkel und seine bedeutung fur dar konstruieren periodischer getriebe, Werksstattstechnik 26 (Heft4) (1932) 61±65. [4] B. Cemil, Synthesis of double crank (drag link) driven mechanism with adjustable motion and dwell time ratios, Mechanism and Machine Theory 12 (1977) 619±638. [5] F.Y. Chen, An analytical method for synthesizing four bar crank rocker mechanism, J. Eng. Ind. Trans. ASME Ser. B 91 (1) (1969) 45±54. [6] K.C. Gupta, A note on the optimum design of four bar crank rocker mechanism, Mech. Mach. Theory 12 (1977) 247±254. [7] F.Y. Chen, Position synthesis of the double rocker mechanism, J. Mech. 4 (1969) 303±310. [8] J.T. Kimbrella, in: Kinematics Analysis and Synthesis, McGraw-Hill, New York, 1991, pp. 14±15. [9] P.W. Jensen, Selecting the dimensions of four bar mechanisms, Mach. Design (June 22) (1961) 173±176. [10] D.C. Tao, in: Applied Linkage Synthesis, Addison-Wesley, Reading, MA, 1964, pp. 7±12. [11] S. Resit, Analytical synthesis of mechanism part-1, Transmission angle synthesis, Mech. Mach. Theory 28 (1993) 825±833. [12] A.S. Hall, in: Kinematics and Linkage Design, Prentice-Hall, Englewood Cli€s, NJ, 1961, p. 41. [13] P.W. Eschenbach, D. Tesar, Link length bounds on the four bar chain, J. Eng. Ind. Trans. ASME Ser. B 93 (1) (1971) 287±293. [14] A. Raphael, Large oscillation mechanisms, Mach. Design (Nov.) (1960) 190±196. [15] S.V. Kulkarni, R.A. Khan, Synthesis of a four bar mechanism with optimum transmission angle, in: Proceedings of 6th World Congress on Theory of M/c and Mechanisms, 1983, pp. 94±97. [16] S. Joseph, U. John Jr., in: Theory of Machines and Mechanism, second ed., McGraw-Hill, New York, 1995, p. 132. [17] K.C. Gupta, Design of four bar function generators with mini±max transmission angle, J. Eng. Ind. Trans. ASME (May) (1977) 360±366. [18] A.K. Khare, R.K. Dave, Optimizing four bar crank-rocker mechanism, Mech. Mach. Theory (1979) 319±325. [19] E.A. Dijksman, How to design four bar function cognetes, Inst. Mech. Eng. 847- (1975) 853. [20] F. Freudenstain, Harmonic analysis of crank rocker mechanism with applications, J. Appl. Mech. Trans. ASME, Ser. E 26 (1959) 673±675. [21] Sutherland, Siddal, Dimensional synthesis of linkages by multifactor optimization, Mech. Mach. Theory (1974) 81±95. [22] B. Roth, F. Freudenstain, G. Sondor, Synthesis of a four bar link path generating mechanisms with optimum transmission characteristics, in: Transactions of Seventh Conference on Mechanisms, Purdue University, Lafayette, Ind., 1962, pp. 44±48. [23] K. Lakshminarayana, R.G. Narayanmurthi, On the analysis of e€ect of tolerance in linkage, J. Mech. 6 (1971) 59±67. [24] R.E. Garrett, S.H. Allen, E€ect of tolerance and clearance in linkage design, J. Eng. Ind. (1969) 198±202. [25] S.A. Kolhatker, K.S. Yajnik, The e€ects of pay in the joints of a function generating mechanism, Mech. Mach. Theory 5 (1970) 521±532. [26] P. Burtan, in: Kinematics and Dynamics of Planar Machinery, Prentice-Hall, Englewood Cli€s, NJ, 1931, p. 17. [27] A.K. Khare, R.K. Dave, Synthesis of the double rocker mechanism with optimum transmission characteristics, Mech. Mach. Theory 15 (1980) 77±80. [28] H. Alt, VDI Z-85 (3) (1941) S69±S72. [29] H. Alt, Getriebechnik 10 (1942) S173. [30] J. Volmer, P.W. Jensen, Product Eng. 23 (1962) 71±76. [31] Y. Leu, M. Lun, J. Waxi Inst. Light Ind. 11 (1992) 142. [32] A.H. Soni, Mechanism Synthesis and Analysis, McGraw-Hill, New York, 1974. [33] R.J. Brodell, A.H. Soni, Design of the crank rocker mechanism with unit time ratio, J. Mech. 5 (1970) 1±4. [34] W.L. Cleghorn, R.G. Fenton, Mech. Mach. Theory 19 (1984) 319±324. [35] K. Lu, J. Tianjin Univ. Mech. Mach. Theory (1986) 68±76. [36] F. Freudenstain, E.J.F. Primose, The transmission angle problem, in: Proceedings of Conference on Mechanisms, institute of Mechanical Engineers, London, 1973, pp. 105±110.

S.S. Balli, S. Chand / Mechanism and Machine Theory 37 (2002) 175±195

193

[37] L. Ming, L. Yonghou, Design of crank rocker mechanism with optimum transmission angle over working stroke, Mech. Mach. Theory 31 (4) (1996) 501±511. [38] R.L. Fox, K.C. Gupta, Optimization technology as applied to mechanism design, J. Eng. Ind. Trans. ASME (May) (1973) 657±663. [39] T.E. Shoup, B.J. Pelan, Design of a four bar mechanism for optimum transmission angle and optimum structural error, in: Proceedings of Second OSU applied Mechanism Conference, Stilwater, Okla, 1971, pp. 4.1±4.9. [40] R. Beyer, The Kinematic Synthesis of Mechanisms, McGraw-Hill, New York, 1963. [41] E. Filemon, In addition to Burmester theory, in: Proceedings of 3rd World Congress on Theory of M/c and Mechanisms,Yugoslavia, 1971, pp. 63±78. [42] E. Filemon, Useful ranges of centre point curves for design of crank rocker linkages, Mech. Mach. Theory 7 (1972) 47±53. [43] K.J. Waldron, Elimination of branch problem in graphical burmester mechanism synthesis for four ®nitely separated positions, J. Eng. Ind. Trans. ASME (1976) 176±182. [44] K.J. Waldron, Graphical solution of the branch and order problem of linkage synthesis for multiple separated positions, Trans. of ASME paper no.76 DET 16. [45] K.J. Waldron, Location of burmester synthesis solution with fully rotable cranks, Mech. Mach. Theory (1977). [46] K. Hain, Drag link mechanisms, Mach. Design (June) (1958) 104±113. [47] K.C. Gupta, A general theory for synthesizing crank type four bar function generators with transmission angle control, J. Appl. Mech. Trans. ASME (1978) 415±421. [48] K.C. Gupta, Synthesis of position, path and function generating four bar mechanism with completely rotable driving links, Mech. Mach. Theory (1980) 93±101. [49] V. Cossalter et al., A simple numerical approach for optimum synthesis of a class of planar mechanisms, Mech. Mach. Theory 27 (1992) 357±366. [50] A.C. Rao, Synthesis of four bar function generators using geometric programming, Mech. Mach. Theory 14 (1979) 141±149. [51] A.J. Kacachios, S.J. Tricamo, Optimal kinematic synthesis of a four bar mechanism with mini±max structural error, in: Proceedings of 6th World Congress on Theory of M/c and Mechanisms, 1983, pp. 81±83. [52] C.H. Chiang et al., On a technique for higher order synthesis of four bar function generators, Mech. Mach. Theory 24 (1989) 195±205. [53] K. Schnarbach, VDI Z-96 (9) (1954) S279±81. [54] A.G. Erdman, G.N. Sondor, in: Mechanism Design: Analysis and Synthesis, 1, Prentice-Hall, NJ, 1984, p. 124. [55] A.G. Erdman, Modern Kinematics-Developments in Last 40 Years, Wiley, New York, 1993. [56] R. Beyer, Der ubetragungswinkel bei kurvenscheiben, Reuleaux-Mitteilungen 1 (1933) S67. [57] R. Beyer, Der ubetragungswinkel am zylinderkurventrieb, Reuleaux-Mitteilungen 3 (Heft 3) (1935) S.167±S.169. [58] C.H. Chiang, Kinematics of Spherical Mechanisms, Cambridge University Press, Cambridge, 1988. [59] G. Sutherland, B. Roth, A transmission index for spatial mechanism, J. Eng. Ind. Trans. ASME (1973) 589±597. [60] C.H. Chiang, Design of spherical and planar crank-rockers and double rockers as function generators-I, crankrockers, Mech. Mach. Theory 21 (1986) 287±296. [61] C.H. Chiang, Design of spherical and planar crank-rockers and double rockers as function generators-II, doublerockers, Mech. Mach. Theory 21 (1986) 297±305. [62] F.Y. Chen, Algebraic synthesis of a class four-bar linkages and their equivalent mechanisms, J. Mech. (1971) 213±230. [63] R.I. Alizade, A.V. Mohan Rao, G.N. Sandor, Optimum synthesis of two degree of freedom planar and spatial function generating mechanism using the penalty function approach, J. Eng. Ind. Trans. ASME (May) (1975) 629±634. [64] E. Soylemez, F. Freudenstein, Transmission optimization of spatial 4-link mechanics, Mech. Mach. Theory 17 (1982) 263±283. [65] T.L. Dresner, K.W. Bunton, De®nitions of pressure and transmission angles applicable to multi-input mechanisms, Trans. ASME, J. Mech. Design Dec. (1991) 495±499. [66] H.S. Yan, L.I. Wu, On the dead-center positions of planar linkage mechanisms, Trans. ASME, J. Mech. Design 111 (1989) 40±46.

194

S.S. Balli, S. Chand / Mechanism and Machine Theory 37 (2002) 175±195

[67] J.A. Smith, D. Tesar, Use of contour charts for dwell mechanism design, in: Second Applied Mechanisms Conference, Oklahoma, State University, Oklahoma, 1969, pp. 11-1±11-13. [68] A. Chaudhari, D. Tesar, Cycloidal dwell linkage design charts based on four in®nitesimally separated positions, in: Second Applied Mechanisms Conference, Oklahoma, State University, Oklahoma, 1969, pp. 12-1±12-11. [69] D.L.N. Raju, Y.R. Patole, C. Amarnath, Design charts for double dwell linkage mechanisms, in: Proceedings of the Eighth NaCoMM Conference, IIT, Kanpur, 1997, pp. 15±20. [70] R Stanley, Five-bar loop synthesis, Mach. Design (1961) 189±195. [71] S.S. Balli, S. Chand, Synthesis of a ®ve-bar mechanism with variable topology for motion between extreme positions, Mech. Mach. Theory 36 (10) (2001) 1147±1156. [72] K.L. Ting, G.H. Tsai, Mobility and synthesis of ®ve-bar programmable linkages, in: Proceedings of nineth OSU Applied Mechanisms Conference, Kanas City, MD, October 1985, pp. III-1±III-8. [73] K.L. Ting, Five-bar grash of criteria, Trans. ASME, J. Mech. Transmiss. Automat. Design 108 (1986) 533±537. [74] S.S. Balli, S. Chand, Synthesis of a ®ve-bar mechanism of variable topology type with transmission angle control, Trans. ASME, Mech. Design (2000) Manuscript No. 010103 (communicated). [75] R.E. Kaufman, Synthesis of the Watt-I six-bar linkage form ®ve position higher coupler motion correlated with input crank rotations, in: Proceedings of Second OSU Applied Mechanisms Conference, Oklahoma State University, Stillwater, 1971, pp. 28-1±28-5. [76] H.S. Yan, L.I. Wu, Counter examples in transmission angle, in: Proceedings of 10th OSU Applied Mechanisms Conference, Oklahoma State University, Stillwater, Oklahoma, 1987, pp. 1±4. [77] T.S. Mruthyunjaya, A note on synthesis of the general, two-degree-of-freedom linkage, Mech. Mach. Theory 10 (1975) 77±80. [78] R.E. Philipp, F. Freudenstein, Synthesis of two-degree-of-freedom linkages-a feasibility study of numerical methods of synthesis of bivariate function generators, J. Mech. 1 (1966) 9±21. [79] K. Lakshminarayana, R.G. Narayanmurthi, Derivate synthesis of plane mechanisms to generate functions of two variables, J. Mech. 5 (1970) 249±271. [80] K. Lakshminarayana, A simpli®ed approach to two degree of freedom linkages, J. Eng. Ind. Trans. ASME (1973) 584±588. [81] D.C. Tao, Adjustable four-bar linkages, Mach. Design (December 20) (1962) 127±129. [82] S. Krishnamoorthy, D.C. Tao, Adjustable linkage mechanisms using coupler points on pole tangent for variable circular arcs, in: Proceedings of OSU Conference on Applied Mechanisms, pp. 1-1±1-20. [83] A.F. Vadasz, Design charts for slider crank mechanisms with prescribed minimum transmission angle and time ratio, in: Proceedings of Proceedings of 6th Annual Applied Mechanisms Conference,Colorado, October 1979, pp. XXXI-1±XXXI-8. [84] D.P. Naik, C. Amarnath, Planar slider-crank mechanisms adjustable for prescribed dead-center positions, J. Indian Inst. Sci. 66 (1986) 247±252. [85] M.Y. Lee, A.G. Erdman, S. Faik, A generalized performance sensitivity synthesis methodology for four-bar mechanisms, Mech. Mach. Theory 34 (1999) 1127±1139. [86] S. Faik, A.G. Erdman, Sensitivity distribution in the synthesis solution space of four-bar linkages, Trans. ASME, J. Mech. Design 113 (1991) 3±9. [87] A.C. Rao, Kinematic design of four-bar function generators with optimum sensitivity, Mech. Mach. Theory 10 (1975) 531±535. [88] J.W.H. Sun, K.J. Waldron, Graphical transmission angle control in planar linkage synthesis, Mech. Mach. Theory 16 (1981) 385±397. [89] S.H. Chuang, C.H. Chiang, Fifth order synthesis of plane four-bar function generators optimized by varying scale factors, Mech. Mach. Theory 22 (1987) 53±63. [90] C.R. Barker, G.H. Shu, Three-Position function generation of planar four-bar mechanisms with equal deviation transmission angle control, Trans. ASME, J. Transmiss. Autom. Design 110 (1988) 435±439. [91] J.W.H. Sun, K.J. Waldron, Graphical transmission angle control in planar linkage synthesis, in: Proceedings of Sixth Annual Applied Mechanisms Conference, Colorado, October 1979, pp. XXXIV-1±XXXIV-8. [92] S.J. Wang, R.S. Sodhi, Kinematic synthesis of adjustable moving pivot four-bar mechanisms for multi-phase motion generation, Mech. Mach. Theory 31 (1996) 459±474.

S.S. Balli, S. Chand / Mechanism and Machine Theory 37 (2002) 175±195

195

[93] C.R. Barker, P.L. Tso, An enhanced version of the synthesis package SYNTRA, in: 10th OSU Applied Mechanisms Conference, New Orleans, LA, December 7±9, 1987. [94] S. Bawab, S. Sabada, U. Srinivasan, G.L. Kinzel, K.J. Waldron, Automatic synthesis of crank driven four-bar mechanisms for two three or four-position motion generation, Trans. ASME, J. Mech. Design (1997) 225±231. [95] S.M. Song, K.J. Waldron, Theoretical and numerical improvements to an interactive linkage design programRECSYN, in: Proceedings of Sixth Annual Applied Mechanisms Conference, Colorado, October 1979, pp. VIII-1± VIII-8. [96] A.J. Rubel, R.E. Kaufman, KINSYN III: A new human-engineered system for interactive computer aided design of planar linkages, J. Eng. Ind. Trans. ASME (May) (1977) 440±448. [97] S. Bawab, G.L. Kinzel, K.J. Waldron, Recti®ed synthesis of six-bar mechanism with well de®ned transmission angles for four-position motion generation, Trans. ASME, J. Mech. Design 118 (1996) 377±383. [98] J.C. Chuang, R.T. Strong, K.J. Waldron, Implementation of solution recti®cation techniques in an interactive linkage synthesis program, Trans. ASME, J. Mech. Design 103 (1981) 657±664. [99] A. Bajpai, S. Kramer, Detection and elimination of mechanism defects in the selective precision synthesis of planar mechanisms, Mech. Mach. Theory 20 (1985) 521±534. [100] L. Zheng, J. Angeles, The properties of constant branch for-bar linkages and their applications, Trans. ASME, J. Mech. Design 114 (December) (1992) 574±579. [101] C.R. Barker, A complete classi®cation of planar four-bar linkages, Mech. Mach. Theory (6) (1985) 535±554. [102] C.W. McLarnan, On linkage synthesis with minimum error, J. Mech. 3 (2) (1968) 101±105. [103] A.G. Erdman, Computer-aided design of mechanism: 1984 and beyond, Mech. Mach. Theory (1985) 245±249. [104] A.G. Erdman, Computer-aided mechanism design: now and the future, Trans. ASME, Special 50th Anniver. Design Issue 117 (June) (1995) 93±100. [105] S.M.K. Kazerounian, K.C. Gupta, Synthesis of position generating crank-rocker or drag-link mechanisms, Mech. Mach. Theory (4) (1982) 243±247.