钢锥锥轮式的无级变速器的传动与设计英语原文.pdf

On the design of slider-crank mechanisms. Part I: multi-phase motion generation Kevin Russell a, Raj S. Sodhib,* 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 benefi t of this method 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 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 profi les 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. ? 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: (R.S. Sodhi). Mechanism and Machine Theory 40 (2005) 285299 Mechanism and Machine Theory 1. Introduction Planar 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 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- eration for the three positions in an assembly machine. An ideal motion of the coupler can only be approximated by several discrete precision positions. Since a linkage has only a fi nite number of signifi cant dimensions, the designer may only prescribe a fi nite number of precision points. A four-bar linkage can satisfy up to fi ve prescribed positions for the motion generation problem. However, an adjustable four-bar linkage can satisfy more than fi ve given positions with the same Coupler Fig. 1. Planar four-bar mechanism. Fig. 2. Planar four-bar loading mechanism. 286K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 hardware. The moving pivots of a four-bar linkage can beadjusted in two diff erent ways: with adjustable crank/follower lengths (Fig. 3) and with fi 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. K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299287 same 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 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 fi xed pivot. They solved two-phase problems with a maximum total number of fi ve positions. Tao and Krishnamoorthy 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, spherical and spatial mechanisms which can adjust inputoutput relationships. Shoup 6 designed 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 specifi ed 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- ment procedure is involved and the mechanism adjustments must be performed frequently. Imple- menting automated adjustment capabilities may make the mechanism impractical from a fi nancial 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 benefi ts 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 generation applications typically accomplished by adjustable planar four-bar motion generators is presented. A slider path that enables the slider-crank motion generator to achieve two phases of 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. 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) 285299 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 fi 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, 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 fi xed and moving pivot joint axes remain parallel. Given a fi xed pivot b0and a moving pivot b1the constant length condition in Eq. (1) 20,21 must be satisfi ed when synthesizing the crank and follower links of the planar four-bar mechanism. bj? b0Tbj? b0 b1? b0Tb1? b0j 2;3;.;n1 where b0 b0x;b0y;1b1 b1x;b1y;1bj Dij?b1 and Dij? pjxqjxrjx pjyqjyrjy 111 2 6 4 3 7 5 pixqixrix piyqiyriy 111 2 6 4 3 7 5 ?1 2 Eq. (1) can be rewritten as Eq. (3). In Eq. (3), the variable R represents the length of the crank or follower link. 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 j aj j p Y b p X q r r qj j i i i 0 1 a0 a1 b b Fig. 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) 285299289 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 fi xed crank link length and a slider path that accounts for the adjustment of the follower link. bj? b0Tbj? b0 R2j 2;3;.n3 Eq. (2) is a rigid body displacement matrix. It is a derivative of the spatial rigid body displacement matrix 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-dimensional space. Although the position of a rigid body in two-dimensional space is commonly described by a single point and a displacement angle (p and h for example), the authors chose to describe the rigid 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 (b0x, b0y, b1x and b1y ), a maximum of fi 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 fi 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 a0, a1, a1n, b0, b1and b1n. The unknowns a0and b0 represent the fi xed pivots of the planar four-bar mechanism. The unknowns a1, a1n, b1and b1nrepresent 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. a0 a0x;a0ya1 a1x;a1ya1n a1nx;a1ny b0 b0x;b0yb1 b1x;b1yb1n b1nx;b1ny Table 1 Prescribed rigid body position and phase variations for the adjustable planar four-bar mechanism Number of phasesMaximum number of rigid body positionsCrank or follower links Number of unknownsNumber of free choices 1540 2860 31180 m5 + 3(m ? 1)2 + 2m0 290K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 Eqs. (4)(8), were used to calculate fi ve of the six unknowns in a0, a1and a1n. The variable a0xand the link length R1 are specifi ed. D12?a1? a0TD12?a1? a0 ? R2 1 04 D13?a1? a0TD13?a1? a0 ? R2 1 05 D14?a1? a0TD14?a1? a0 ? R2 1 06 D56?a1n? a0TD56?a1n? a0 ? R2 1 07 D57?a1n? a0TD57?a1n? a0 ? R2 1 08 Eqs. (9)(13) were used to calculate fi ve of the six unknowns in b0, b1and b1n. The variable b0x and the link lengths R1and R2 are specifi ed. D12?b1? b0TD12?b1? b0 ? R2 1 09 D13?b1? b0TD13?b1? b0 ? R2 1 010 D14?b1? b0TD14?b1? b0 ? R2 1 011 D56?b1n? b0TD56?b1n? b0 ? R2 1 012 D57?b1n? b0TD57?b1n? b0 ? R2 1 013 3. Trajectory generation After 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 its moving pivots. The moving pivot paths of the follower link must be connected in a manner that will allow smooth displacement, velocity, acceleration and jerk transitions between the determined moving pivot paths. Abrupt or discontinuous transitions will ultimately result in excessive wear on the slider-crank mechanism. The slider path of the equivalent slider-crank motion generator will be comprised of the moving pivot paths of the follower link and the trajectories that connect them. During the operation of an adjustable four-bar mechanism within a particular phase, the radial positions 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 moving pivot, 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 and jerks of the moving pivots undergo a transition from the link parameters in the former phase to the link parameters in the latter phase. If transition curves are generated for the follower link, and these curves are connected piecewise to the moving pivot curves of the follower corresponding to K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299291 the phases before and after this transition, a single slider path is generated that will account for the transition between phases (or follower link moving pivot adjustment). A 7th order polynomial 22,23 is required to specify the radial position, velocity, acceleration and 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 a7h714 The radial displacement, velocity, acceleration and jerk boundary conditions for this polynomial are Rh0 R015 dRh0 dh0 _ R016 dR2h0 dh2 0 R017 dR3h0 dh3 0 R v 0 18 Rhf Rf19 dRhf dhf _ Rf20 dR2hf dh2 f Rf21 dR3hf dh3 f R v f 22 In this work, the term R0is the length of the follower link in phase one (link b0b1) and the term Rf is the length of the follower link in phase two (link b0b1n). The constraints specify a linear set of eight equations with eight unknowns whose solutions are a0 R023 a1 _ R024 a2 R0 2 25 a3R v 0 6 26 292K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 a4 35 h4 f Rf? R0? _ R0hf? 1 2 R0h2 f ? 1 6R v 0h 3 f ? ? 15 h3 f ?_R0?R0hf? 1 2R v 0h 2 f ? 5 2h2 f ?R0 ?R v 0hf 1 6hf R v 0 27 a5 ?84 h5 f Rf? R0? _ R0hf? 1 2 R0h2 f ? 1 6R v 0h 3 f ? 39 h4 f ?_R0?R0hf? 1 2R v 0h 2 f ? ? 7 2h3 f ?R0 ?R v 0hf 1 2h2 f R v 0 28 a6 70 h6 f Rf? R0? _ R0hf? 1 2 R0h2 f ? 1 6R v 0h 3 f ? 34 h5 f ?_R0?R0hf? 1 2R v 0h 2 f ? 13 2h4 f ?R0 ?R v 0hf 1 2h3 f R v 0 29 a7 ?20 h7 f Rf? R0? _ R0hf? 1 2 R0h2 f ? 1 6R v 0h 3 f ? 10 h6 f ?_R0?R0hf? 1 2R v 0h 2 f ? ? 2 h5 f ?R0 ?R v 0hf 1 6h4 f R v 0 30 4. Example problem Two-phase moving pivot adjustments of the adjustable planar four-bar motion generator with fi xed crank and adjustable follower lengths are exemplifi ed in this section. Listed in Table 2 are the X- and Y-coordinates of p, q and r for seven prescribed rigid body positions. Table 2 Prescribed rigid body positions for the adjustable planar four-bar motion generator pqr Phase 1 Position 1?0.5175, 0.9640?0.2148, 1.50490.3551, 1.3103 Position 2?0.4502, 1.0207?0.1413, 1.55810.4263, 1.3570 Position 3?0.3786, 1.0720?0.0645, 1.60640.5011, 1.3997 Position 4?0.3030, 1.11730.0152, 1.64920.5792, 1.4382 Phase 2 Position 5?0.0583, 1.20420.4515, 1.68340.9782, 1.3914 Position 60.1449, 1.21550.5383, 1.69451.0648, 1.4023 Position 70.2319, 1.21950.6249, 1.69881.1517, 1.4070 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299293 Eqs. (4)(8) were used to calculate fi ve of the six unknowns in a0, a1and a1n. The variable a0x and the link length R1 were specifi ed (a0x= 0 and R1= 1). Using the following initial guesses: a0y 0:1a1 ?0:5;0:5a1n ?0:5;0:5 the planar four-bar mechanism solutions converged to a0y 0:0761a1 ?0:7049;0:7859a1n ?0:1739;1:0608: a0 b0 a1 b1 b1n X Y a1n r1 p1 p2 p3 p4 p5 p6p7 q1 q2 q3 q4 q5q6 q7 r2 r3r4 r5 r6r7 Fig. 7. Adjustable planar four-bar motion generator and corresponding prescribed rigid body positions. a b4 1n 57 a0 1 a b0 1 b b a1n 4 D 57 D 1n