卷紙機(jī)的設(shè)計(jì)

卷紙機(jī)的設(shè)計(jì),卷紙,設(shè)計(jì) An extension of mechanism design optimization for motion generationQiong Shena, Yahia M. Al-Smadib, Peter J. Martinc, Kevin Russellc,*, Raj S. SodhiaaDepartment of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USAbAECOM, Special Structures Group, New York, NY 10005, USAcArmaments Engineering and Technology Center, US Army Research, Development and Engineering Center, Picatinny, NJ 07806-5000, USAa r t i c l ei n f oArticle history:Received 27 May 2008Received in revised form 19 January 2009Accepted 2 March 2009Available online 2 April 2009Keywords:Motion generationCoupler loadPlanar mechanismFour-bar mechanismOptimizationSelection algorithma b s t r a c tAs an extension of the authors published work on a search algorithm for motion generationwith Grashof, transmission angle and linkage perimeter conditions P.J. Martin, K. Russell,R.S. Sodhi, On mechanism design optimization for motion generation, Mechanism andMachine Theory 42 (10) (2007) 12511263, this work formulates a goal program to gen-erate four-bar mechanism fixed and moving pivot loci that considers prescribed couplerposes, a coupler load and maximum driver static torques.Published by Elsevier Ltd.1. IntroductionIn planar four-bar motion generation, the objective is to calculate the mechanism parameters required to achieve orapproximate a set of prescribed coupler poses. This mechanism design objective is particularly useful when the coupler mustachieve a specific displacement sequence for effective operation (e.g., specific end-effector orientations for accurate taskcompletion). In Fig. 1, five prescribed coupler poses are defined by the x and y-coordinates of variables p, q and r (Fig. 1a)and the calculated mechanism parameters to achieve the prescribed poses are the x and y-coordinates of fixed pivot vari-ables a0and b0and moving pivot variables a1and b1(Fig. 1b). The latter figure illustrates a planar four-bar motion generator.Motion generation for planar four-bar mechanisms is a well-established field. Recent contributions include the work ofYao and Angeles 3 who applied the contour method in the approximate synthesis of planar linkages for rigid-body guid-ance. By deriving a set of two bivariate polynomial equations and plotting these equations, the real solution to the optimi-zations-which correspond to the intersections of the contour plots are determined. Hong and Erdman 4 introduced a newapplication Burmester curves for adjustable planar four-bar linkages. Their method is applicable for the synthesis of four-barmechanisms is planar and spherical form and their work shows that nonadjustable mechanism solutions are special cases ofadjustable mechanism solutions. Zhou and Cheung 5 introduced an optimal synthesis method of adjustable four-bar link-ages for multi-phase motion generation. A modified genetic algorithm is used to seek the global optimal solution of an equa-tion set that includes constraints for fixed pivot positions, no branch defect, crank existence and link length ratios. Al-Widyan et al. 6 considered the robust synthesis of planar four-bar linkages for motion generation. Danieli et al. 7 appliedBurmester theory in the design of planar four-bar motion generators to reproduce tibia-femur relative motion. Goehler et al.0094-114X/$ - see front matter Published by Elsevier Ltd.doi:10.1016/j.mechmachtheory.2009.03.001* Corresponding author. Tel.: +1 07 806 5000; fax: +1 973 724 6027.E-mail address: kevin.russell1us.army.mil (K. Russell).Mechanism and Machine Theory 44 (2009) 17591767Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: applied parameterized T1motion theory to the synthesis of planar four-bar motion generators. This T1motion theory isgeneral and not limited to the second order parameterization that is associated with prior development of T1motion theory.Caracciolo and Trevisani 9 considered rigid-body motion control of flexible four-bar linkages. A discrete finite elementmodel of the four-bar mechanism accounts for geometric and inertial nonlinearities. Zhixing et al. 10 presented a guid-ance-line rotation method of rigid-body guidance for the synthesis of planar four-bar mechanisms. The method effectivelysolves the rigid-body guidance synthesis problem crankrocker mechanisms, double-rocker mechanisms and double-crankmechanisms for four, five or more than five rigid-body positions. Lin and Modler 11 presented a method to avoid branchdefects, order defects and ensure link rotatability in three-point path generation. The method considers (but is not limited to)planar four-bar mechanisms.Using a conventional motion generation model 12 the user can only calculate the planar four-bar mechanism parame-ters required to achieve or approximate a limited set of prescribed coupler poses. Although such solutions are effective forkinematic analyses, other design factors (e.g., static loads, deflections, stresses, strains, etc.) must be considered for a com-prehensive engineering analysis. This work considers static driving link torque for a given coupler load for motion genera-tion. Unlike previously published work on analytical motion generation with prescribed coupler loads, where the number ofprescribed coupler poses is limited and the mechanism solution is exact 2, the numerical model formulated in this workcan accommodate an indefinite number of prescribed coupler poses that are approximated by the mechanism solution. Byincorporating the calculated coupler load and driver static torque-based fixed and moving pivot loci in the search algorithm,planar four-bar motion generator solutions are sought that also satisfy specified Grashof conditions, transmission angle con-ditions and mechanism perimeter conditions.2. Conventional planar four-bar motion generationEqs. (1) and (2) encompass the numerical planar four-bar motion generation model presented by Suh and Radcliffe 12.Eqs. (1) and (2) ensure the constant lengths of the crank and follower links. Eq. (3) is a planar rigid-body displacementmatrix.When using this conventional mechanism synthesis model to calculate the components of the fixed pivots a0and b0(where a0= a0 x,a0y,1T, and b0= b0 x,b0y,1T) and moving pivots a1and b1(where a1= a1x,a1y,1T, and b1= b1x,b1y,1T) ofa planar four-bar motion generator (Fig. 1b), the user can specify a maximum of five coupler poses 12:D1j?a1? a0?TD1j?a1? a0? a1? a0Ta1? a0 0;1D1j?b1? b0TD1j?b1? b0 ? b1? b0Tb1? b0 0;2whereD1j? pjxqjxrjxpjyqjyrjy111264375p1xq1xr1xp1yq1yr1y111264375?1j 2;3;4;5:3XYa0b0b1a1XYp1q1r1p2q2r2p3q3r3p5q5r5p4q4r4a b Fig. 1. Prescribed coupler poses (a) and calculated planar four-bar mechanism (b).1760Q. Shen et al./Mechanism and Machine Theory 44 (2009) 175917673. Static torque constraint derivation for a planar four-bar mechanism with coupler loadGiven a virtual driver link displacement dh for the planar four-bar mechanism in Fig. 2, the virtual work done by drivertorque isdWT Tdh;4while the virtual work done by external coupler load F!isdWF F!? dq:5According to Suh and Radcliffe 12dq daPua?q ? a dhPu0?q ? a0;6whereda ?dhb ? b0TPu0?b ? a0?b ? b0TPua?b ? afg:7For the planar four-bar mechanism to reach static equilibrium at a given position, it must follow thatdW dWT dWF Tdh F!? dq 0:8Eq. (8) can be rewritten asTdh F!?dhb ? b0TPu0?b ? a0?b ? b0TPua?b ? afgPua?q ? a dh Pu0?q ? a0() 0:9Canceling out the dh term gives the expression of driver torque T in terms of unknowns to be synthesizedT ? F!?b ? b0TPu0?b ? a0?b ? b0TPua?b ? afgPua?q ? a Pu0?q ? a0();10wherePu0? Pua? 0?10100000264375:11Eq. (10) calculates the driver static torque for a planar four-bar mechanism under a given coupler load (also a = D1ja1,b = D1jb1and q = D1jq1).4. Planar four-bar motion generation goal program with driver link static torque constraintWhen Eq. (10) is incorporated into the conventional planar four-bar motion generation model included in Section 2, theresulting model is useful for calculating the planar four-bar mechanism fixed and moving pivot parameters required toapproximate a set of prescribed coupler poses and driver static torque requirements for a given coupler load.XYFpqra0b1a1b0TFig. 2. Planar four-bar mechanism with a coupler load.Q. Shen et al./Mechanism and Machine Theory 44 (2009) 175917671761Formulating Eqs. (1) and (2) into a single objective function (that accommodates an indefinite number of N prescribedcoupler poses) to be minimized yieldsfXXNj2D1j?a1?a0TD1j?a1?a0?a1?a0Ta1?a0hi2 D1j?b1?b0TD1j?b1?b0?b1?b0Tb1?b0hi2?;12where X a0 x;a0y;a1x;a1y;b0 x;b0y;b1x;b1yT.Substituting Eq. (10) into (13) gives a nonlinear inequality constraint which will keep the magnitude of the driver torqueless thansmax.jTij ?smax? 0;i 1;2;3;.;N:13Eq. (12) and inequality constraint (13) constitute a goal program from which mechanism solutions that approximate the pre-scribed coupler poses and maximum driver static torque conditions are calculated.The algorithm employed for solving this goal program (a nonlinear constraints problem) is SQP (Sequential Quadratic Pro-gramming) which uses quasi-Newton approach to solve its QP (Quadratic Programming) subproblem and line search ap-proach to determine iteration step. The merit function used by Han 13 and Powell 14 is used in the following form:WX fX Xmei1rigiX Xmime1rimax0;giX?;14where m is the total number of constraints, meis the number of equality constraints (which is zero in this case), and thepenalty parameter isri rk1i maxiki;12rki ki?;15where kiare estimates of the Lagrange multipliers.5. Motion generator selection algorithmMartin et al. 1 presented a selection algorithm for planar four-bar motion generators that considers Grashof criteria,transmission angle criteria and mechanism perimeter conditions. The algorithm input are the fixed and moving pivot locicalculated for a set of prescribed coupler poses. The algorithm then down-selects planar four-bar mechanism solutions giventhe specified Grashof mechanism type (or non-Grashof), minimum and maximum transmission angles, and mechanismperimeter (e.g., minimum perimeter) conditions.When the fixed and moving pivot loci calculated from the goal program in Section 4 are used as input for the four-barmotion generator selection algorithm, the down-selected mechanism will satisfy all of the prescribed selection algorithmconditions (blocks 1, 2 and 4 in Fig. 3) as well as approximate the prescribed coupler poses and the driver static torque con-straints for a given coupler load.In its original codified form, the algorithm presented by Martin et al. 1 requires a single fixed pivot locus and a singlemoving pivot locus as input. The goal program on the other hand calculates a locus for each of the variables a0, a1, b0and b1.When the loci for the fixed pivots are read into the macro as one continuous locus and likewise for the moving pivot loci, theoriginal codified algorithm will produce a combination of valid and invalid solution output. A filter loop (block 2 in Fig. 3)will exclude the invalid solutions prior to the mechanism perimeter loop (block 4 in Fig. 3). The filter loop codified in Math-cad is included in the Appendix. The user should insert the filter loop immediately after the Grashof criteria loop as shown inFig. 3 to update the original selection algorithm code.6. ExampleTable 1 includes the x and y-coordinates (in feet) of seven prescribed coupler poses. This corresponds to twice the max-imum number of prescribed coupler displacements available with the conventional motion generation method included inthis work 7 when a0 xand b0 xare prescribed. The maximum allowed driver torque will besmax= 60ft lbs and the couplerload will be F! 0;?100;0 lbs.With a prescribed range of a0 x= ?2.00, ?1.95, . , 2.00 and initial guesses of a0y= 0, a1= (0,1), b0= (2.5,0) andb1= (2.5,1.5), the fixed and moving pivot loci calculated from the goal program are illustrated in Fig. 4. After incorporatingthe fixed and moving pivot loci in to the updated motion generator selection algorithm with a transmission angle (trans.)condition of 40? 6 trans. 6 140?, a Grashof condition of crank-rocker”, and the minimum mechanism perimeter condition,the fixed and moving pivot coordinates of the selected planar four-bar motion (Fig. 5) generator are a0= (0.5500,0.2944),a1= (0.5906,0.9494), b0= (2.6993,0.0015), b1= (2.5945,1.4970). This mechanism has crank, coupler, follower and groundlengths of 0.6563, 2.0774, 1.4992 and 2.1692 respectively (satisfying Grashof crank-rocker conditions) and produces thetransmission angle profile illustrated in Fig. 6. This mechanism is also the most compact of the available solutions (satisfyingminimum perimeter condition).1762Q. Shen et al./Mechanism and Machine Theory 44 (2009) 175917671. Transmission Angle Criteria2. Grashof Criteria4. Mechanism Perimeter CriteriaSelection Algorithmdown-selected mechanism fixed/moving pivot loci3. Filter LoopFig. 3. Planar four-bar motion generator selection algorithm diagram.Table 1Prescribed coupler poses.pqrPos 10.5293, 1.33461.0587, 1.91921.8267, 1.7077Pos 2?0.0668, 1.08550.4410, 1.68901.2162, 1.5055Pos 3?0.3286, 0.54340.0816, 1.21700.8757, 1.1536Pos 4?0.1340, 0.10140.1072, 0.85230.8940, 0.9764Pos 50.3804, 0.07410.5044, 0.85291.2632, 1.0954Pos 60.9439, 0.49501.1831, 1.24651.9696, 1.3727Pos 71.1025, 1.02061.5555, 1.66612.3438, 1.5512-0.3-0.10.10.30.50.70.91.11.31.5-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0a0a1b0b1Fig. 4. Calculated motion generator (with a coupler load and torque) fixed and moving pivot loci.Q. Shen et al./Mechanism and Machine Theory 44 (2009) 175917671763The achieved rigid-body positions and driver static torques are listed in Table 2. Using the SimMechanics module in MAT-LAB, the achieved coupler poses and the driver static torques in Table 2 were measured for the planar four-bar motion gen-erator (Fig. 7). Fig. 8 includes plots of the scalar differences (jpointprescibed? pointachievedj) between the prescribed and achievedcoupler poses of the synthesized motion generator.7. DiscussionEq. (10) becomes invalid when the pivots a1, b1and b0are collinear. Such a state is possible when the four-bar mechanismreaches a lock-up” or binding position. When pivots a1, b1and b0are collinear, the denominator in Eq. (10) becomes zero(making the equation invalid). The Optimization Toolbox offered by mathematical analysis software MATLAB was used tocodify and solve the formulated goal program. The optimized solution was then fed to the MATLAB SimMechanics module(Fig. 7) to independently confirm the achieved coupler poses and driver static torques (Table 2) of the synthesized planarFig. 5. Down-selected planar four-bar motion generator (in SimMechanics).405060708090100110120130140050100150200250300350400crank disp. degtrans. angle deg.Fig. 6. Motion generator transmission angles.Table 2Coupler poses and driver torques achieved by the synthesized motion generator.pqrsPos 10.5293, 1.33461.0587, 1.91921.8267, 1.70772.0774Pos 2?0.0596, 1.06340.4410, 1.67281.2183, 1.4986?47.1020Pos 3?0.2399, 0.52850.1447, 1.21700.9406, 1.1833?57.6307Pos 4?0.0780, 0.10140.1493, 0.85660.9337, 0.9953?40.5240Pos 50.4407, ?0.07020.5044, 0.71591.2423, 1.016015.6433Pos 60.9790, 0.49501.2189, 1.24632.0056, 1.371763.3721Pos 70.9926, 1.01191.4332, 1.66612.2235, 1.566446.71641764Q. Shen et al./Mechanism and Machine Theory 44 (2009) 17591767four-bar motion generator. The filter loop should only be used in the original selection algorithm when individual loci forvariables a0, a1, b0and b1are used as input.Fig. 7. Prescribed and achieved coupler poses of down-selected mechanism (in SimMechanics).Q. Shen et al./Mechanism and Machine Theory 44 (2009) 1759176717658. ConclusionsBy calculating planar four-bar mechanism fixed and moving pivot loci data using a goal program that considers pre-scribed coupler poses, a coupler load and driver static torque constraints and using the loci as input for an updated versionof the selection algorithm introduced by Martin et al. 1, this work demonstrates the synthesis of planar four-bar motionsgenerators with respect to Grashof, transmission angle, mechanism perimeter and driver static torque criteria for a givencoupler load.Appendix.Filter loop (codified in Mathcad) for planar four-bar motion generator selection algorithmCell:= Mstack( ) for i 1rows(Cell) -1 M1(Celli,0 Celli,1 Celli,2 Celli,3) for j 0(end/2) -1 continue if (Celli,0 CRANKj) M1( ) if Celli,2 CRANKj+(end/2) Mstack(M, M1) M References1 P.J. Martin, K. Russell, R.S. Sodhi, On mechanism design optimization for motion generation, Mechanism and Machine Theory 42 (10) (2007) 12511263.2 C. Huang, B. Roth, Dimensional synthesis of closed-loop linkages to match force and position specifications, Journal of Mechanical Design 115 (1993)194198.3 J. Yao, J. Angeles, Computation of all optimum dyads in the approximate synthesis of planar linkages for rigid-body guidance, Mechanism and MachineTheory 35 (8) (2000) 10651078.4 B. Hong, A.G. Erdman, A method for adjustable planar and spherical four-bar linkage synthesis, ASME Journal of Mechanical Design 127 (3) (2005) 456463.5 H. Zhou, E.H.M. Cheung, Adjustable four-bar linkages for multi-phase motion generation, Mechanism and Machine Theory 39 (3) (2004) 261279.6 K. Al-Widyan, J. Angeles, J. Jess Cervantes-Snchez, The robust synthesis of planar four-bar linkages for motion generation, in: Proceedings of theASME Design Engineering Technical Conference, vol. 5A, 2002, pp. 627633.7 G.A. Danieli, D. Mundo, V. 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Modler, Path generation with emphasis on desired mechanism type and characteristics, in: Proceedings of the 11th World Congress inMechanism and Machine Science, vol. 3, China Machine Press, Tianjin, China, 2004, pp. 12641269.12 C.H. Suh, C.W. Radcliffe, Kinematics and Mechanism Design, John Wiley and Sons, New York, 1978.13 S.P. Han, A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications 22 (1977) 297.14 M.J.D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in: G.A. Watson (Ed.), Numerical Analysis, Lecture Notes inMathematics, vol. 630, Springer-Verlag, 1978.Q. Shen et al./Mechanism and Machine Theory 44 (2009) 175917671767