JH36-400機(jī)械壓力機(jī)機(jī)身部分及其上橫梁加工工藝的設(shè)計(jì)【說明書+CAD】,說明書+CAD,JH36-400機(jī)械壓力機(jī)機(jī)身部分及其上橫梁加工工藝的設(shè)計(jì)【說明書+CAD】,jh36,機(jī)械,壓力機(jī),機(jī)身,部分,部份,及其,橫梁,加工,工藝,設(shè)計(jì),說明書,仿單,cad On the design of slider-crank mechanisms. Part I: method is twofold. First, multiple phases of prescribed rigid body positions are achievable using a mech- * Corresponding author. Tel.: +1 973 596 3362; fax: +1 973 642 4282. E-mail address: sodhiadm.njit.edu (R.S. Sodhi). Mechanism and Machine Theory 40 (2005) 285299 Mechanism and Machine Theory 0094-114X/$ - see front matter C211 2004 Elsevier Ltd. All rights reserved. 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 automated adjustments 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 polynomials to 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 proles with bound- ary conditions that can be prescribed. The example problem in this work considers a two-phase moving pivot adjustment of a planar four-bar mechanism. C211 2004 Elsevier Ltd. All rights reserved. multi-phase motion generation Kevin Russell a , Raj S. Sodhi b, * a Armaments Engineering and Technology Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USA b Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA Received 24 February 2003; received in revised form 12 July 2004; accepted 12 July 2004 Available online 28 September 2004 Abstract A method for designing slider-crank mechanisms to achieve multi-phase motion generation applications typically accomplished by adjustable planar four-bar motion generators is presented. The benet of this doi:10.1016/j.mechmachtheory.2004.07.009 1. Introduction Planarfour-barmechanismsarewidelyusedinmechanicalsystemsanddevices.Duetothepla- nar kinematics, joint type and joint axis orientations of the planar four-bar mechanism, it can be practical 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 methods exists for planar four-bar mechanisms. Motion generation problems in mechanism synthesis require that a rigid body be guided through 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- erationforthethreepositionsinanassemblymachine.Anidealmotionofthecouplercanonlybe approximated by several discrete precision positions. Since a linkage has only a nite number of signicant dimensions, the designer may only prescribe a nite number of precision points. A four-bar linkage can satisfy up to ve prescribed positions for the motion generation problem. However, an adjustable four-bar linkage can satisfy more than ve given positions with the same Coupler 286 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 Fig. 1. Planar four-bar mechanism. Fig. 2. Planar four-bar loading mechanism. K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 287 hardware. The moving pivots of a four-bar linkage can beadjusted in two dierent ways: with adjustable crank/follower lengths (Fig. 3) and with xed crank/follower link adjustments (Fig. 4). The adjustable linkages can provide solution for two phases of general plane motion (Fig. 5). If a four-bar linkage is designed to reach positions 1, 2 and 3 in phase 1, after the adjustments, the Fig. 3. Adjustable crank length. Fig. 4. Fixed crank length. PHASE ONE PHASE TWO 4 1 2 3 5 6 Fig. 5. Two phases of prescribed rigid body positions. same linkage can reach three new positions 4, 5 and 6 in the second phase 2. Both phases of mo- tioncanbeaccomplishedusingthesamehardwarebyadjustingoneormoreofthelinkageparam- eters. The linkage can create the motion precisely at these positions and will approximate the motion at other positions. The more precision positions are used, the closer to the ideal motion is the actual motion of the coupler. In the area of adjustable linkages for motion generation, published work is somewhat limited 119. Previous work includes the work of Ahmad and Waldron 1 who developed a technique for synthesizing a four-bar linkage with adjustable driven xed pivot. They solved two-phase problemswith a maximumtotal numberofvepositions. Tao andKrishnamoorthy 2 developed graphical synthesis procedures to generate variable coupler curves with cusps. McGovern and Sandor 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, sphericalandspatialmechanismswhichcanadjustinputoutputrelationships.Shoup6designed adjustable 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 mechanisms using 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 the two-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-dimensional mechanisms for multi-phase motion generation with tolerances. Using these methods, spatial RRSS mechanisms can be synthesized to achieve phases of prescribed precise rigid body positions and rigid body positions with tolerances. Recently Chang 12 presented synthesis of adjustable four-bar mechanisms generating circular arcs with specied tangential velocities. If there is any performance-related limitation to the adjustable planar four-bar mechanism, it is that manual or automated adjustments are required to achieve all of the prescribed phases in multi-phase applications. Manual adjustments can be time consumingespecially if the adjust- mentprocedureis involvedandthemechanismadjustmentsmustbeperformedfrequently.Imple- mentingautomatedadjustmentcapabilitiesmaymakethemechanismimpracticalfromanancial standpointespecially when operations and maintenance expenditures are considered. For an adjustable planar four-bar motion generator that incorporates both moving pivot and link length adjustments for the follower link and only moving pivot adjustments for the crank link, an equivalent slider-crank motion generator can be designed to achieve multiple phases of prescribed rigid body positions. The benets of the method are that multiple phases of prescribed rigid body positions are achievable using a mechanism with fewer moving parts than the planar four-bar mechanism and the slider-crank motion generator can achieve phases of prescribed rigid body positions without any physical or automated adjustments of its moving pivots between phases. In this work, a method to design slider-crank motion generators to achieve multi-phase motion generationapplicationstypicallyaccomplishedbyadjustableplanarfour-barmotiongeneratorsis presented. A slider path that enables the slider-crank motion generator to achieve two phases of rigidbodypositionsisdesignedbyusing7thorderpolynomialstoconnectthemovingpivotpaths of 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- 288 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 scribed using the boundary conditions of these polynomials. 2. Rigid body guidance and multi-phase motion generation The 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 xed and adjustable crank and follower lengths respectively. The authors 10,11 developed the multi-phase motion generation method utilized in this work. The planar four-bar motion generator is illustrated in Fig. 6. In this work, linka 0 a 1 is the des- ignated crank link and linkb 0 b 1 is the designated follower link. Links a 0 a 1 andb 0 b 1 of the pla- narfour-barmechanismmustsatisfytheconstantlengthconditiononlysinceitsxedandmoving pivot joint axes remain parallel. Given a xed pivot b 0 and a moving pivot b 1 the constant length condition in Eq. (1) 20,21 must be satised whensynthesizing the crank and follower links of the planar four-bar mechanism. b j C0b 0 T b j C0b 0 b 1 C0b 0 T b 1 C0b 0 j 2;3; .; n 1 where b 0 b 0 x ; b 0y ;1 b 1 b 1x ; b 1y ;1 b j D ij C138b 1 Eq. (1) follower K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 289 j a j j p Y b p X r j j i 0 1 a 0 a 1 b b q r q i i One objective of this work is to design an equivalent slider-crank motion generator for an adjustable planar four-bar motion generator. Although the moving pivots of both the crank 111 111 can be rewritten as Eq. (3). In Eq. (3), the variable R represents the length of the crank or link. and D ij C138 p jx q jx r jx p jy q jy r jy 2 6 4 3 7 5 p ix q ix r ix p iy q iy r iy 2 6 4 3 7 5 C01 2 Fig. 6. The planar four-bar motion generator with rigid body points p, q and r. and follower link of the planar four-bar mechanism are adjustable, only the length of the follower link will be adjusted (not the crank link). By doing this, the equivalent slider-crank motion gen- erator to be designed will have a xed crank link length and a slider path that accounts for the adjustment of the follower link. b j C0b 0 T b j C0b 0 R 2 j 2;3; .n 3 Eq.(2)isarigidbodydisplacementmatrix.Itisaderivativeofthespatialrigidbodydisplacement matrix 20,21. Given the coordinates for a rigid body in position i and the subsequent j, ma- trixD ij is the transformationmatrixrequiredtotransformcoordinatesfrom positioni toposi- tion space. singlepointandadisplacementangle(pand h forexample),theauthorschosetodescribetherigid Prescribed Number 290 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 Number of unknowns Number of free choices 15 4 0 28 6 311 8 m rigid body position and phase variations for the adjustable planar four-bar mechanism of phases Maximum number of rigid body positions Crank or follower links body using three points for computational purposes. If the user prefers to describe the rigid body using 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 (b 0 x ,b 0y ,b 1x and b 1y ), a maximum of ve rigid body positions can be prescribed, with no arbitrary choice of 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 this precaution 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 planar four-bar motion generator for several phases are given. The number of xed and moving pivot coordinates 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 a two-phase moving pivot adjustment application for an adjustable planar four-bar motion generator. In the two-phase, adjustable moving pivot example problem in this work, the required un- knowns are a 0 , a 1 , a 1n , b 0 , b 1 and b 1n . The unknowns a 0 and b 0 represent the xed pivots of the planar four-bar mechanism. The unknowns a 1 , a 1n , b 1 and b 1n represent the moving pivots in phase 1 and phase 2 of the planar four-bar mechanism. Since each of these unknowns has two components, there are a total of 12 variables to determine. a 0 a 0 x ; a 0y a 1 a 1x ; a 1y a 1n a 1nx ; a 1ny b 0 b 0 x ; b 0y b 1 b 1x ; b 1y b 1n b 1nx ; b 1ny Table 1 j. Variablesp,qandrinEq. (2)representthepositionoftherigidbodyintwo-dimensional Althoughthepositionofarigidbodyintwo-dimensionalspaceiscommonlydescribedbya 5+3(m C0 1) 2 + 2m 0 13 1 0 13 1 0 1 D 14 C138b 1 C0b 0 T D 14 C138b 1 C0b 0 C0R 2 0 11 3. Trajectory After willallow moving on the will be them. During positions ities, pivot, ing pivot K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 291 jerks of the moving pivots undergo a transition from the link parameters in the former phase to the link parameters inthe latter phase. Iftransition curvesare generatedfor the follower link, and these smoothdisplacement,velocity,accelerationandjerktransitionsbetweenthedetermined pivot paths. Abrupt or discontinuous transitions will ultimately result in excessive wear slider-crank mechanism. The slider path of the equivalent slider-crank motion generator comprised of the moving pivot paths of the follower link and the trajectories that connect the operation of an adjustable four-bar mechanism within a particular phase, the radial of the moving pivots of the crank and follower links are constant and the radial veloc- accelerations and jerks of these moving pivots are zero. The same holds true during moving constant link length adjustments of the adjustable planar four-bar mechanism. When mov- and linklengthadjustments considered,the radialpositions, velocities, accelerations and tion, the user can synthesize a planar four-bar motion generator and determine the paths of its moving pivots. The moving pivot paths of the follower link must be connected in a manner that 1 D 56 C138b 1n C0b 0 T D 56 C138b 1n C0b 0 C0R 2 1 0 12 D 57 C138b 1n C0b 0 T D 57 C138b 1n C0b 0 C0R 2 1 0 13 generation incorporating the multi-phase motion generation method described in the previous sec- Eqs. (4)(8), wereused to calculateveof the six unknownsina 0 ,a 1 anda 1n . The variablea 0 x and the link length R 1 are specied. D 12 C138a 1 C0a 0 T D 12 C138a 1 C0a 0 C0R 2 1 0 4 D 13 C138a 1 C0a 0 T D 13 C138a 1 C0a 0 C0R 2 1 0 5 D 14 C138a 1 C0a 0 T D 14 C138a 1 C0a 0 C0R 2 1 0 6 D 56 C138a 1n C0a 0 T D 56 C138a 1n C0a 0 C0R 2 1 0 7 D 57 C138a 1n C0a 0 T D 57 C138a 1n C0a 0 C0R 2 1 0 8 Eqs. (9)(13) were used to calculate ve of the six unknowns in b 0 , b 1 and b 1n . The variable b 0 x and the link lengths R 1 and R 2 are specied. D 12 C138b 1 C0b 0 T D 12 C138b 1 C0b 0 C0R 2 1 0 9 D C138b C0b T D C138b C0b C0R 2 0 10 curves are connected piecewise to the moving pivot curves of the follower corresponding to thephases transition A andjerk tion The are Inthis is the eight equations with eight unknowns whose solutions are 292 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 a 2 R 0 2 25 a 3 R v 0 26 a 1 _ R 0 24 a 0 R 0 23 dh 0 _ R 0 16 dR 2 h 0 dh 2 0 R 0 17 dR 3 h 0 dh 3 0 R v 0 18 Rh f R f 19 dRh f dh f _ R f 20 dR 2 h f dh 2 f R f 21 dR 3 h f dh 3 f R v f 22 work,thetermR 0 isthelengthofthefollowerlinkinphaseone(linkb 0 b 1 )andthetermR f length of the follower link in phase two (link b 0 b 1n ). The constraints specify a linear set of Rh 0 R 0 15 dRh 0 beforeandafterthistransition,asinglesliderpathisgeneratedthatwillaccountforthe between phases (or follower link moving pivot adjustment). 7th order polynomial 22,23 is required to specify the radial position, velocity, acceleration parametersofthemovingpivotofthefollowerlinkoftheadjustableplanarfour-barmo- generator during the transition between phases. Rha 0 a 1 h a 2 h 2 a 3 h 3 a 4 h 4 a 5 h 5 a 6 h 6 a 7 h 7 14 radial displacement, velocity, acceleration and jerk boundary conditions for this polynomial 6 a 4 35 h 4 f R f C0 R 0 C0 _ R 0 h f C0 1 2 R 0 h 2 f C0 1 6 R v 0 h 3 f C18C19 C0 15 h 3 f C0 _ R 0 C0 R 0 h f C0 1 2 R v 0 h 2 f C18C19 5 2h 2 f C0 R 0 C0R v 0 h f 1 6h f R v 0 27 a 5 C084 h 5 f R f C0 R 0 C0 _ R 0 h f C0 1 2 R 0 h 2 f C0 1 6 R v 0 h 3 f C18C19 39 h 4 f C0 _ R 0 C0 R 0 h f C0 1 2 R v 0 h 2 f C18C19 C0 7 2h 3 f C0 R 0 C0R v 0 h f 1 2h 2 f R v 0 28 a 6 70 6 R f C0 R 0 C0 _ R 0 h f C0 1 R 0 h 2 f C0 1 R v 0 h 3 f C18C19 34 5 C0 _ R 0 C0 R 0 h f C0 1 R v 0 h 2 f C18C19 h 7 f 2 f 6 f h 6 f 2 f 2 v 1 v 4. Example Two-phase xed the X Table Prescribed Phase Position Position Position Position Phase Position Position Position K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 293 C0 h 5 f C0 R 0 C0R 0 h f 6h 4 f R 0 30 problem moving pivot adjustments of the adjustable planar four-bar motion generator with crank and adjustable follower lengths are exemplied in this section. Listed in Table 2 are - and Y-coordinates of p, q and r for seven prescribed rigid body positions. 2 rigid body positions for the adjustable planar four-bar motion generator pqr 1 1 C00.5175, 0.9640 C00.2148, 1.5049 0.3551, 1.3103 2 C00.4502, 1.0207 C00.1413, 1.5581 0.4263, 1.3570 3 C00.3786, 1.0720 C00.0645, 1.6064 0.5011, 1.3997 4 C00.3030, 1.1173 0.0152, 1.6492 0.5792, 1.4382 2 5 C00.0583, 1.2042 0.4515, 1.6834 0.9782, 1.3914 6 0.1449, 1.2155 0.5383, 1.6945 1.0648, 1.4023 h f 2 6 h f 2 13 2h 4 f C0 R 0 C0R v 0 h f 1 2h 3 f R v 0 29 a 7 C020 R f C0 R 0 C0 _ R 0 h f C0 1 R 0 h 2 C0 1 R v 0 h 3 C18C19 10 C0 _ R 0 C0 R 0 h f C0 1 R v 0 h 2 C18C19 7 0.2319, 1.2195 0.6249, 1.6988 1.1517, 1.4070 Eqs. (4)(8) were used to calculate ve of the six unknowns in a 0 , a 1 and a 1n . The variable a 0 x and the link length R 1 were specied (a 0 x =0 and R 1 =1). Using the following initial guesses: a 0y 0:1 a 1 C00:5;0:5 a 1n C00:5;0:5 the planar four-bar mechanism solutions converged to a 0y 0:0761 a 1 C00:7049;0:7859 a 1n C00:1739;1:0608: a0 b0 a1 b1 b1n X Y a1n r1 p1 p2 p3 p4 p5 p6 p7 q1 q2 q3 q4 q5 q6 q7 r2 r3 r4 r5 r6r7 Fig. 7. Adjustable planar four-bar motion generator and corresponding prescribed rigid body positions. a b 4 1n 57 1 a 1 b b a 1n 4 D 57 D 1n a 1n b Y 294 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 b 0 X a 0 Fig. 8. Moving pivot paths of the synthesized adjustable planar four-bar motion generator. Eqs. (9)(13) were used to calculate ve of the six unknowns in b 0 , b 1 and b 1n . The variable b 0 x and the links length R 1 and R 2 were specied (b 0 x =1.5, R 1 =1.5 and R 2 =1.3). Using the following initial guesses: b 0y 0:1 b 1 0:6;1:2 b 1n 0:6;1:2 the planar four-bar mechanism solutions converged to b 0y C00:1064 b 1 0:6821;1:1505 b 1n 1:2964;1:1775: Using the calculated xed 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 constant 0 a a 1 b 4 b 1 1n b Y X q1 r1 p1 phase 1 phase 2transition Fig. 9. Equivalent slider-crank motion generator and initial rigid body position. K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 295 1.25 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 crank displacement angle rad 1.30 slider radial displacement 1.35 1.40 1.45 1.50 1.55 Fig. 10. Radial slider displacement versus crank angle for synthesized slider-crank motion generator. link length moving pivot adjustment, all of the moving pivot positions (a 1 through a 4 and a 1n through D 57 a 1n ) for this link lie on the same circle. The moving pivot positions (b 1 through b 4 and b 1n through D 57 b 1n ) for the follower link lie on two dierent circles (one for each phase). To complete the slider path of the equivalent slider-crank motion generator (the path between b 1 and b 1n ), Eq. (14) was used calculate a path to link both of the follower moving pivot arcs in Fig. 8. Using Eq. (14) and the prescri