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On the design of slider-crank mechanisms. Part I:multi-phase motion generationKevin Russella, Raj S. Sodhib,*aArmaments Engineering and Technology Center, US Army Research, Development and Engineering Center,Picatinny Arsenal, NJ 07806-5000, USAbDepartment of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USAReceived 24 February 2003; received in revised form 12 July 2004; accepted 12 July 2004Available online 28 September 2004AbstractA method for designing slider-crank mechanisms to achieve multi-phase motion generation applicationstypically accomplished by adjustable planar four-bar motion generators is presented. The benefit of thismethod is twofold. First, multiple phases of prescribed rigid body positions are achievable using a mech-anism with fewer moving parts than the planar four-bar mechanism. Second, the slider-crank motion gen-erator can achieve phases of prescribed rigid body positions without any physical or automatedadjustments of its moving pivots between phases. A slider path that enables the slider-crank motion gen-erator to achieve two phases of prescribed rigid body positions is designed by using 7th order polynomialsto connect the moving pivot paths of the follower link of the adjustable planar four-bar motion generator.This polynomial generates smooth radial displacement, velocity, acceleration and jerk profiles with bound-ary conditions that can be prescribed. The example problem in this work considers a two-phase movingpivot adjustment of a planar four-bar mechanism.? 2004 Elsevier Ltd. All rights reserved.0094-114X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2004.07.009*Corresponding author. Tel.: +1 973 596 3362; fax: +1 973 642 4282.E-mail address: sodhi (R.S. Sodhi)./locate/mechmtMechanism and Machine Theory 40 (2005) 285299MechanismandMachine Theory1. IntroductionPlanar four-bar mechanisms are widely used in mechanical systems and devices. Due to the pla-nar kinematics, joint type and joint axis orientations of the planar four-bar mechanism, it can bepractical to design and implement these mechanisms (compared to most four-bar spatial mecha-nisms). In addition, an extensive array of graphical and analytical design and analysis methodsexists for planar four-bar mechanisms.Motion generation problems in mechanism synthesis require that a rigid body be guidedthrough a series of prescribed positions. The four-bar linkage shown in Fig. 1 can be used to pro-duce this motion by making the rigid body as a part of its coupler link. Fig. 2 shows motion gen-eration for the three positions in an assembly machine. An ideal motion of the coupler can only beapproximated by several discrete precision positions. Since a linkage has only a finite number ofsignificant dimensions, the designer may only prescribe a finite number of precision points. Afour-bar linkage can satisfy up to five prescribed positions for the motion generation problem.However, an adjustable four-bar linkage can satisfy more than five given positions with the sameCouplerFig. 1. Planar four-bar mechanism.Fig. 2. Planar four-bar loading mechanism.286K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299hardware. The moving pivots of a four-bar linkage can beadjusted in two different ways: withadjustable crank/follower lengths (Fig. 3) and with fixed crank/follower link adjustments (Fig. 4).The adjustable linkages can provide solution for two phases of general plane motion (Fig. 5). Ifa four-bar linkage is designed to reach positions 1, 2 and 3 in phase 1, after the adjustments, theFig. 3. Adjustable crank length.Fig. 4. Fixed crank length.PHASE ONEPHASE TWO412356Fig. 5. Two phases of prescribed rigid body positions.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299287same linkage can reach three new positions 4, 5 and 6 in the second phase 2. Both phases of mo-tion can be accomplished using the same hardware by adjusting one or more of the linkage param-eters. The linkage can create the motion precisely at these positions and will approximate themotion at other positions. The more precision positions are used, the closer to the ideal motionis the actual motion of the coupler.In the area of adjustable linkages for motion generation, published work is somewhat limited119. Previous work includes the work of Ahmad and Waldron 1 who developed a techniquefor synthesizing a four-bar linkage with adjustable driven fixed pivot. They solved two-phaseproblems with a maximum total number of five positions. Tao and Krishnamoorthy 2 developedgraphical synthesis procedures to generate variable coupler curves with cusps. McGovern andSandor 3,4 presented methods to synthesize adjustable mechanisms for function and path gen-eration using complex variables. Funabashi et al. 5 presented general methods to design planar,spherical and spatial mechanisms which can adjust inputoutput relationships. Shoup 6 designedadjustable spatial slider-crank mechanism to be used as a variable displacement pump. Cheun-chom and Kota 7 have presented general methods for the synthesis of adjustable mechanismsusing adjustable dyads. Wilhelm 8 developed synthesis techniques for two-phase motion gener-ation problems of adjustable four-bar linkages. Wang and Sodhi 9 developed solutions for thetwo-phase adjustable moving pivot problems with three positions in each of the two phases. Rus-sell and Sodhi 10,11 recently presented methods for synthesizing adjustable three-dimensionalmechanisms for multi-phase motion generation with tolerances. Using these methods, spatialRRSS mechanisms can be synthesized to achieve phases of prescribed precise rigid body positionsand rigid body positions with tolerances. Recently Chang 12 presented synthesis of adjustablefour-bar mechanisms generating circular arcs with specified tangential velocities.If there is any performance-related limitation to the adjustable planar four-bar mechanism, it isthat manual or automated adjustments are required to achieve all of the prescribed phases inmulti-phase applications. Manual adjustments can be time consumingespecially if the adjust-ment procedure is involved and the mechanism adjustments must be performed frequently. Imple-menting automated adjustment capabilities may make the mechanism impractical from a financialstandpointespecially when operations and maintenance expenditures are considered.For an adjustable planar four-bar motion generator that incorporates both moving pivot andlink length adjustments for the follower link and only moving pivot adjustments for the cranklink, an equivalent slider-crank motion generator can be designed to achieve multiple phases ofprescribed rigid body positions. The benefits of the method are that multiple phases of prescribedrigid body positions are achievable using a mechanism with fewer moving parts than the planarfour-bar mechanism and the slider-crank motion generator can achieve phases of prescribed rigidbody positions without any physical or automated adjustments of its moving pivots betweenphases.In this work, a method to design slider-crank motion generators to achieve multi-phase motiongeneration applications typically accomplished by adjustable planar four-bar motion generators ispresented. A slider path that enables the slider-crank motion generator to achieve two phases ofrigid body positions is designed by using 7th order polynomials to connect the moving pivot pathsof the follower link of the adjustable planar four-bar motion generator. The radial displacement,velocity, acceleration and jerk parameters of the moving pivot of the follower link are also pre-scribed using the boundary conditions of these polynomials.288K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 2852992. Rigid body guidance and multi-phase motion generationThe slider-crank motion generator design method presented in this work is adaptable to virtu-ally any multi-phase motion generation method available that incorporates moving pivot adjust-ments with fixed and adjustable crank and follower lengths respectively. The authors 10,11developed the multi-phase motion generation method utilized in this work.The planar four-bar motion generator is illustrated in Fig. 6. In this work, link a0a1is the des-ignated crank link and link b0b1is the designated follower link. Links a0a1and b0b1of the pla-nar four-bar mechanism must satisfy the constant length condition only since its fixed and movingpivot joint axes remain parallel. Given a fixed pivot b0and a moving pivot b1the constant lengthcondition in Eq. (1) 20,21 must be satisfied when synthesizing the crank and follower links of theplanar four-bar mechanism.bj? b0Tbj? b0 b1? b0Tb1? b0j 2;3;.;n1whereb0 b0x;b0y;1b1 b1x;b1y;1bj Dij?b1andDij? pjxqjxrjxpjyqjyrjy111264375pixqixrixpiyqiyriy111264375?12Eq. (1) can be rewritten as Eq. (3). In Eq. (3), the variable R represents the length of the crank orfollower link.One objective of this work is to design an equivalent slider-crank motion generator for anadjustable planar four-bar motion generator. Although the moving pivots of both the crankjajjpYbpXqrrqjjiii01a0a1bbFig. 6. The planar four-bar motion generator with rigid body points p, q and r.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299289and follower link of the planar four-bar mechanism are adjustable, only the length of the followerlink will be adjusted (not the crank link). By doing this, the equivalent slider-crank motion gen-erator to be designed will have a fixed crank link length and a slider path that accounts for theadjustment of the follower link.bj? b0Tbj? b0 R2j 2;3;.n3Eq. (2) is a rigid body displacement matrix. It is a derivative of the spatial rigid body displacementmatrix 20,21. Given the coordinates for a rigid body in position i and the subsequent j, ma-trix Dij is the transformation matrix required to transform coordinates from position i to posi-tion j. Variables p, q and r in Eq. (2) represent the position of the rigid body in two-dimensionalspace. Although the position of a rigid body in two-dimensional space is commonly described by asingle point and a displacement angle (p and h for example), the authors chose to describe the rigidbody using three points for computational purposes. If the user prefers to describe the rigid bodyusing conventional notation, the displacement matrix in Eq. (2) will be replaced with the conven-tional plane rigid body displacement matrix 20,21. Since there are four variables (b0x, b0y, b1xand b1y), a maximum of five rigid body positions can be prescribed, with no arbitrary choiceof parameter for one phase (see Table 1).Points p, q and r should not all lie on the same line in each rigid body position. Taking thisprecaution prevents the rows in the rigid body displacement matrix (Eq. (2) from becoming pro-portional. With proportional rows, this matrix cannot be inverted.In Table 1, the maximum numbers of prescribed rigid body positions for the adjustable planarfour-bar motion generator for several phases are given. The number of fixed and moving pivotcoordinates for the crank and follower links determine the maximum number of rigid body posi-tions. In the example problem in this work, an equivalent slider crank is designed to achieve atwo-phase moving pivot adjustment application for an adjustable planar four-bar motiongenerator.In the two-phase, adjustable moving pivot example problem in this work, the required un-knowns are a0, a1, a1n, b0, b1and b1n. The unknowns a0and b0represent the fixed pivots of theplanar four-bar mechanism. The unknowns a1, a1n, b1and b1nrepresent the moving pivots inphase 1 and phase 2 of the planar four-bar mechanism. Since each of these unknowns has twocomponents, there are a total of 12 variables to determine.a0 a0x;a0ya1 a1x;a1ya1n a1nx;a1nyb0 b0x;b0yb1 b1x;b1yb1n b1nx;b1nyTable 1Prescribed rigid body position and phase variations for the adjustable planar four-bar mechanismNumber of phasesMaximum number of rigid body positionsCrank or follower linksNumber of unknownsNumber of free choices1540286031180m5 + 3(m ? 1)2 + 2m0290K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Eqs. (4)(8), were used to calculate five of the six unknowns in a0, a1and a1n. The variable a0xandthe link length R1are specified.D12?a1? a0TD12?a1? a0 ? R21 04D13?a1? a0TD13?a1? a0 ? R21 05D14?a1? a0TD14?a1? a0 ? R21 06D56?a1n? a0TD56?a1n? a0 ? R21 07D57?a1n? a0TD57?a1n? a0 ? R21 08Eqs. (9)(13) were used to calculate five of the six unknowns in b0, b1and b1n. The variable b0xand the link lengths R1and R2are specified.D12?b1? b0TD12?b1? b0 ? R21 09D13?b1? b0TD13?b1? b0 ? R21 010D14?b1? b0TD14?b1? b0 ? R21 011D56?b1n? b0TD56?b1n? b0 ? R21 012D57?b1n? b0TD57?b1n? b0 ? R21 0133. Trajectory generationAfter incorporating the multi-phase motion generation method described in the previous sec-tion, the user can synthesize a planar four-bar motion generator and determine the paths of itsmoving pivots. The moving pivot paths of the follower link must be connected in a manner thatwill allow smooth displacement, velocity, acceleration and jerk transitions between the determinedmoving pivot paths. Abrupt or discontinuous transitions will ultimately result in excessive wearon the slider-crank mechanism. The slider path of the equivalent slider-crank motion generatorwill be comprised of the moving pivot paths of the follower link and the trajectories that connectthem.During the operation of an adjustable four-bar mechanism within a particular phase, the radialpositions of the moving pivots of the crank and follower links are constant and the radial veloc-ities, accelerations and jerks of these moving pivots are zero. The same holds true during movingpivot, constant link length adjustments of the adjustable planar four-bar mechanism. When mov-ing pivot and link length adjustments considered, the radial positions, velocities, accelerations andjerks of the moving pivots undergo a transition from the link parameters in the former phase tothe link parameters in the latter phase. If transition curves are generated for the follower link, andthese curves are connected piecewise to the moving pivot curves of the follower corresponding toK. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299291the phases before and after this transition, a single slider path is generated that will account for thetransition between phases (or follower link moving pivot adjustment).A 7th order polynomial 22,23 is required to specify the radial position, velocity, accelerationand jerk parameters of the moving pivot of the follower link of the adjustable planar four-bar mo-tion generator during the transition between phases.Rh a0 a1h a2h2 a3h3 a4h4 a5h5 a6h6 a7h714The radial displacement, velocity, acceleration and jerk boundary conditions for this polynomialareRh0 R015dRh0dh0_R016dR2h0dh20R017dR3h0dh30Rv018Rhf Rf19dRhfdhf_Rf20dR2hfdh2fRf21dR3hfdh3fRvf22In this work, the term R0is the length of the follower link in phase one (link b0b1) and the term Rfis the length of the follower link in phase two (link b0b1n). The constraints specify a linear set ofeight equations with eight unknowns whose solutions area0 R023a1_R024a2R0225a3Rv0626292K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299a435h4fRf? R0?_R0hf?12R0h2f?16Rv0h3f?15h3f?_R0?R0hf?12Rv0h2f?52h2f?R0?Rv0hf 16hfRv027a5?84h5fRf? R0?_R0hf?12R0h2f?16Rv0h3f?39h4f?_R0?R0hf?12Rv0h2f?72h3f?R0?Rv0hf 12h2fRv028a670h6fRf? R0?_R0hf?12R0h2f?16Rv0h3f?34h5f?_R0?R0hf?12Rv0h2f?132h4f?R0?Rv0hf 12h3fRv029a7?20h7fRf? R0?_R0hf?12R0h2f?16Rv0h3f?10h6f?_R0?R0hf?12Rv0h2f?2h5f?R0?Rv0hf 16h4fRv0304. Example problemTwo-phase moving pivot adjustments of the adjustable planar four-bar motion generator withfixed crank and adjustable follower lengths are exemplified in this section. Listed in Table 2 arethe X- and Y-coordinates of p, q and r for seven prescribed rigid body positions.Table 2Prescribed rigid body positions for the adjustable planar four-bar motion generatorpqrPhase 1Position 1?0.5175, 0.9640?0.2148, 1.50490.3551, 1.3103Position 2?0.4502, 1.0207?0.1413, 1.55810.4263, 1.3570Position 3?0.3786, 1.0720?0.0645, 1.60640.5011, 1.3997Position 4?0.3030, 1.11730.0152, 1.64920.5792, 1.4382Phase 2Position 5?0.0583, 1.20420.4515, 1.68340.9782, 1.3914Position 60.1449, 1.21550.5383, 1.69451.0648, 1.4023Position 70.2319, 1.21950.6249, 1.69881.1517, 1.4070K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299293Eqs. (4)(8) were used to calculate five of the six unknowns in a0, a1and a1n. The variable a0xand the link length R1were specified (a0x= 0 and R1= 1). Using the following initial guesses:a0y 0:1a1 ?0:5;0:5a1n ?0:5;0:5the planar four-bar mechanism solutions converged toa0y 0:0761a1 ?0:7049;0:7859a1n ?0:1739;1:0608:a0b0a1b1b1nXYa1nr1p1p2p3p4p5p6p7q1q2q3q4q5q6 q7r2r3r4r5r6r7Fig. 7. Adjustable planar four-bar motion generator and corresponding prescribed rigid body positions.ab41n57a01ab01bba1n4D57D1na1nbYXFig. 8. Moving pivot paths of the synthesized adjustable planar four-bar motion generator.294K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Eqs. (9)(13) were used to calculate five of the six unknowns in b0, b1and b1n. The variable b0xand the links length R1and R2were specified (b0x= 1.5, R1= 1.5 and R2= 1.3).Using the following initial guesses:b0y 0:1b1 0:6;1:2b1n 0:6;1:2the planar four-bar mechanism solutions converged tob0y ?0:1064b1 0:6821;1:1505b1n 1:2964;1:1775:Using the calculated fixed and moving pivot parameters, the resulting adjustable planar four-bar motion generator is illustrated in Fig. 7. An equivalent slider-crank motion generator was de-signed for the planar four-bar motion generator in this work.In Fig. 8, the starting and ending positions for the synthesized adjustable planar four-bar mo-tion generator in phases one and two are illustrated. Since the crank link underwent a constant0aa1b4b11nbYXq1r1p1phase 1phase 2transitionFig. 9. Equivalent slider-crank motion generator and initial rigid body position.1.251.301.351.401.451.501.550.000.000.250.300.35crank displacement angle radslider radial displacementFig. 10. Radial slider displacement versus crank angle for synthesized slider-crank motion generator.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299295link length moving pivot adjustment, all of the moving pivot positions (a1through a4and a1nthrough D57a1n) for this link lie on the same circle. The moving pivot positions (b1through b4and b1nthrough D57b1n) for the follower link lie on two different circles (one for each phase).To complete the slider path of the equivalent slider-crank motion generator (the path betweenb1and b1n), Eq. (14) was used calculate a path to link both of the follower moving pivot arcsin Fig. 8. Using Eq. (14) and the prescribed boundary conditions, the slider path in Fig. 9 wasdesigned. This slider path produces the radial displacement, velocity, acceleration and jerk profilesillustrated in the Figs. 1013.Listed in Table 3 are the X- and Y-coordinates of p, q and r for seven prescribed rigid bodypositions generated by the equivalent planar slider-crank mechanism. To achieve positions 2, 3and 4 in Table 3, link a0a1is rotated to 130?, 125? and 120? with respect to the X-axis. To achievepositions 5, 6 and 7 in Table 3, link a0a1nmust rotated to 100?, 95? and 90? with respect to theX-axis. In both phases, the crank angle is initially 135? with respect to the X-axis and the rigidbody point coordinates at this crank position are the coordinates in position 1 of Table 3.-1.2-1.0-0.8-0.6-0.4-0.20.00.000.000.250.300.35crank displacement angle radslider radial velocityFig. 11. Radial slider velocity versus crank angle for synthesized slider-crank motion generator.-15-10-50510150.000.000.250.300.35crank displacement angle radslider radial accelerationFig. 12. Radial slider acceleration versus crank angle for synthesized slider-crank motion generator.296K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Illustrated in Figs. 1013 are the radial displacement, velocity, acceleration and jerk profiles(with respect to crank displacement angles) during the transition from phase 1 to phase 2 forthe equivalent slider-crank motion generator. The radial velocity, acceleration and jerk profileboundary conditions (Eqs. (16)(18) and Eqs. (20)(22) were specified to zero in order to generatevelocity, acceleration and jerk profiles that are continuous with those profiles outside of the tran-sition. The radial displacement profile boundary conditions (Eqs. (15) and (19) were specified tothe phase 1 and phase 2 lengths of the follower link (R0= 1.5 and Rf= 1.3) in order to generatedisplacement profiles that are continuous with those profiles outside of the transition also.5. DiscussionThe slider path design method presented in this work is applicable for most of the existing mo-tion generation methods for adjustable planar four-bar mechanisms since only the fixed and mov-ing pivots for the synthesized mechanism are required. Although a two-phase moving pivotproblem was exemplified in this work, additional phases of prescribed rigid body positions can-250-200-150-100-500501001502002503000.000.000.250.300.35crank displacement angle radslider radial jerkFig. 13. Radial slider jerk versus crank angle for synthesized slider-crank motion generator.Table 3Rigid body positions generated by equivalent slider-crank motion generatorpqrPhase 1Position 1?0.5175, 0.9640?0.2148, 1.50490.3551, 1.3103Position 2?0.4508, 1.0205?0.1417, 1.55770.4258, 1.3564Position 3?0.3797, 1.0715?0.0654, 1.60570.5002, 1.3988Position 4?0.3046, 1.11670.0138, 1.64850.5778, 1.4372Phase 2Position 50.0794, 1.20840.4747, 1.68581.0001, 1.3915Position 60.1493, 1.25780.5510, 1.72981.0724, 1.4285Position 70.2232, 1.30290.6302, 1.77041.1482, 1.4632K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299297be incorporated. By calculating the fixed and moving pivots of the planar four-bar mechanism foreach additional phase, and additional transition paths for each additional phase (using Eqs. (14)(30) the equivalent slider-crank motion generator can be designed to achieve additional phases.Although the position of a rigid body in two-dimensional space is commonly described by a singlepoint and a displacement angle (p and h for example), the authors chose to describe the rigid bodyusing three points for computational purposes. If the user prefers to describe the rigid body usingconventional notation, the displacement matrix in Eq. (2) will be replaced with the conventionalplane rigid body displacement matrix 20,21. Computer aided design software was to prescribethe mechanism parameters in this work and mathematics software was used to compute the mech-anism solutions. This software enabled the tabulated prescribed and calculated parameters to beexpressed in four significant figures.6. ConclusionA design method for slider-crank mechanisms to achieve multi-phase motion generation appli-cations accomplished by planar four-bar mechanisms with adjustable moving pivots is presentedin this work. The benefit of the method is that using it, multiple phases of prescribed rigid bodypositions are achievable using a mechanism with fewer moving parts than the planar four-barmechanism. Another benefit of this method is that using it, slider-crank mechanisms can be de-signed to achieve phases of prescribed rigid body positions without any physical or automatedadjustments of its moving pivots between phases. A slider path that enables the slider-crank mech-anism to achieve two phases of prescribed rigid body positions is designed by using 7th order pol-ynomials to connect the moving pivot paths of the follower link of the adjustable planar four-barmechanism. The example problem in this work considers a two-phase moving pivot adjustment ofthe adjustable planar four-bar mechanism.References1 Ahmad, Anees, K.J. Waldron, Synthesis of adjustable planar 4-bar mechanisms, Mechanism and Machine Theory14 (1979) 405411.2 D.C. Tao, S. Krishnamoorthy, Likage mechanism adjustable for variable coupler curves with cusps, Mechanismsand Machine Theory 13 (6) (1978) 577583.3 J.F. McGroven, G.N. Sandor, Kinematic synthesis of adjustable mechanisms (Part 1: Function generation),ASME Journal of Engineering for Industry 95 (2) (1973) 417422.4 J.F. McGroven, G.N. Sandor, Kinematic synthesis of adjustable mechanisms (Part 2: Path generation), ASMEJournal of Engineering for Industry 95 (2) (1973) 423429.5 H. Funabashi, N. Iwatsuki, Y. Yoshiaki, A synthesis of crank length adjusting mechanisms, Bulletin of JSME 29(252) (1986) 19461951.6