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/Engineering ScienceEngineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical /content/225/1/194The online version of this article can be found at: DOI: 10.1243/09544062JMES2321194 2011 225:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering ScienceH-S Yan and C-C YehIntegrated kinematic and dynamic designs for variable-speed plate cam mechanisms Published by: On behalf of: Institution of Mechanical Engineers can be found at:ScienceProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringAdditional services and information for /cgi/alertsEmail Alerts: /subscriptionsSubscriptions: /journalsReprints.navReprints: /journalsPermissions.navPermissions: /content/225/1/194.refs.htmlCitations: What is This? - Jan 1, 2011Version of Record at Northwestern Polytechnical University on March 10, 2014Downloaded from at Northwestern Polytechnical University on March 10, 2014Downloaded from 194Integrated kinematic and dynamic designsfor variable-speed plate cam mechanismsH-SYanand C-CYehDepartment of Mechanical Engineering, National Cheng Kung University,Tainan, Republic of ChinaThe manuscript was received on 21 February 2010 and was accepted after revision for publication on 26 April 2010.DOI: 10.1243/09544062JMES2321Abstract: Thisworkpresentsasystematicapproachtoimprovethekinematicanddynamicchar-acteristics of plate cam mechanisms by using the Bezier curve as the input speed trajectory. Theanalytical models of kinematic and dynamic designs for a variable-speed plate cam mechanismare derived first. Speed trajectories of the cam are obtained by employing the Bezier curve.Then,an approach to integrate kinematic and dynamic designs for motion adaptation and to reducetheinputtorqueatthesametimebasedonmulti-objectiveoptimizationisproposed.Twoexam-ples are provided for the design models to illustrate the design process. Finally, an experimentalset-up is established, and the data of related characteristics is measured to verify the proposeddesign for variable-speed plate cam mechanisms.Keywords: variable-speed input, Bezier curve, cam mechanism, kinematic design, dynamicdesign, multi-objective optimization1INTRODUCTIONCammechanismsarewidelyusedinvariousmachines, since a specific design process can be usedto generate better motion characteristics. Further-more, followers can be driven in various kinds ofmotion by simple cam profiles. Traditionally, a cammechanism is usually designed to operate at a con-stant input speed. Its follower is driven by the camdirectly to achieve a desired motion.The power source of a cam mechanism is the inputtorque of an electric motor, which highly depends ontheinertialforceandexternalloads.Iftheinputtorqueincreases, the output dynamic characteristics will beaffected.Traditionalapproachestoimprovekinematicanddynamiccharacteristicsofcammechanismsaretoamend cam contours or add other mechanisms 14.Nowadays, servomotors can flexibly adapt the driv-ing speeds. If the motor can run with a variable speed,the motion characteristics can be improved withoutchanging the cam contour. With such a concept, theinput speed function has to be provided with enoughCorresponding author: Department of Mechanical Engineering,National Cheng Kung University, No. 1, Ta-Hsueh Road, Tainan7010,Republic of China.email:hsyan.twflexibility for various mechanisms. A Bezier curve,a high-order differentiable continuous function, isdetermined by a set of control points. Its shape canbe realized intuitively by the disposition of the controlpoints. In addition, this curve allows the motion to becontinuous.Sankyo Seisakusho Co. 5 announced a pick-and-place and an index device with the servomotor tosmooth the shape of output motion and to reduce thepeak value of the acceleration. Wu et al. 6 proposedthe design approach for a translating cam with vary-ing input velocity provided by an offset slider-crankmechanism. Yan and Tsai 7 developed an approachwith variable input speed to improve the kinematiccharacteristics of plate cam mechanisms. Yan andYeh 8 improved the dynamic characteristics of platecam mechanisms by using Fourier series for the inputspeed trajectory. This work intends to integrate kine-matic and dynamic designs by using the Bezier curveas the input speed function of the cam.2ANALYSIS OF KINEMATIC AND DYNAMICDESIGNSConsider a plate cam mechanism with a reciprocityroller follower as shown in Fig. 1, where d is the diam-eter of the follower shaft, s the follower displacement,Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from Integrated kinematic and dynamic designs for variable-speed plate cam mechanisms195Fig.1A cam-follower systemL1 s the distance between the bearing and the roller,L2the distance between the bearings, rfthe rollerradius,rbthebasecircleradiusofthecam,e theoffset,and the pressure angle between the cam and the fol-lower.Thefollowerdisplacements(t)canbeexpressedasafunctiong()ofthecamangulardisplacement(t)and time t ass(t) = g(t)(1)The follower speed v(t) and acceleration a(t) can beobtained by differentiating equation (1) with respectto time t consecutively asv(t) = g?(t)(t)(2)a(t) = g?(t)2(t) + g?(t)(t)(3)where functions g?() and g?() are the first- andsecond-order derivatives of the follower displacementfunction g() with respect to cam angular displace-ment (t), respectively. The cam angular velocity(t)and acceleration(t) are the first- and second-orderderivatives of cam angular displacement (t) withrespect to time t, respectively, i.e.(t) = d/dt and(t) = d2/dt2.To simplify the analytical process of the cam mech-anism, the follower displacement, velocity, and accel-eration are normalized. When the follower completesa rise or fall motion of the total stroke h with a kind ofgeometrycurveinatimeperiod,thecamalsorotatesanangle inthesameperiod.NormalizedtimeT,nor-malized follower displacement S(T), and normalizedcam rotation angle ?(T) can be defined asT =t(4)S =sh(5)? =(6)for all T 0,1, t 0, S 0,1, s 0,h, ? 0,1, and 0,. On the basis of equations (1) to(3), normalized follower displacement S(T), velocityV(T), and acceleration A(T) can be expressed asS(T) = G?(T)(7)V(T) = G?(T)?(T)(8)A(T) = G?(T)?2(T) + G?(T)?(T)(9)where G(?) is the normalized follower displacementfunction. G?(?) and G?(?) are the first- and second-order derivatives of the normalized follower displace-ment function G(?) with respect to normalized camangular displacement ?(T), respectively. Normalizedcam angular velocity?(T) and acceleration?(T) arethe first- and second-order derivatives of the normal-ized cam angular displacement ?(T) with respect tonormalizedtimeT,respectively,i.e.?(t) = d?/dt and?(t) = d2?/dt2.For the analysis of dynamic pressure, angle between the cam and the follower can be derived bythe theory of envelope as 9 = arctan?s? es +?(rb+ rf)2+ e2?(10)wheres?isthefirst-orderderivativeofthefollowerdis-placement function g() with respect to cam angulardisplacement (t), rbthe base circle of the cam, rftherollerradiusofthefollower,ande theoffsetofthecam.On the basis of free body diagram of the followerand the cam as shown in Fig. 2, the force and torqueequilibrium equations can be expressed asFncos F0+ ks mfg (Fa+ Fb) Fe= mf s(11)Fa Fnsin Fb= 0(12)Fa(L1 s) Fb(L1+ L2 s) 0.5d(Fa Fb) = 0(13)where Faand Fbare the bearing forces acting on thefollower, s?the follower velocity, Fnthe contact forcebetweenthecamandthefollower,F0thepreloadofthespring,k thespringratio,mfthemassofthefollower,thefrictioncoefficient,andFetheexternalloadexertedon the follower.Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from 196H-SYan and C-CYehFig.2Free body diagram of the follower and camOn the basis of equations (11) to (13), the contactforce Fncan be derived asFn=mf s + F0+ ks + mfg + Fecos (2L1+ L2 2s d)/L2sin(14)Finally, by considering the torque equilibrium of thecam, the input torque Q from the motor can beexpressed asQ = Fne cos + (rb+ rf+ s)sin + Ic (15)3BEZIER CURVEThe shape of a Bezier curve is controlled by a setof control points. It is a polynomial function and itsorder depends on the number of points. An nth-orderBezier curve can be defined by n + 1 control points.The function of Bezier curve P(u) can be expressed asP(u) =n?i=0PiBi,n(u)(16)where Piis the coordinate vector of the ith controlpoint and u the position parameter at which the valueis between 0 and 1. Bi,n(u) is the weighting function ofthe control points and can be expressed asBi,n(u) =n!i!(n i)! ui (1 u)ni,u 0,1(17)Here, the Bezier curve is used to define the rota-tional movement of the plate cam mechanism. Thenormalized cam angular velocity?(T) in a period canbe defined by an nth-order Bezier curve as?(T) =n?i=0PiBi,n(T)(18)Bi,n(T) =n!i!(n i)! Ti (1 T)ni,T 0,1(19)An important characteristic of the Bezier curveis its nth-order differentiable function. This char-acteristic can ensure that the kinematic propertiesof the plate cam system are continuous. Normal-izedcamangulardisplacement?(T)andacceleration?(T) can be derived by integrating and differentiatingequation (18), respectively, as?(T) =?(T)dT + c =?n?i=0PiBi,n(T)dT + c=n?i=0Pi?Bi,n(T)dT + c,T 0,1(20)?Bi,n(u)dT = n!i?j=01(i j)!(n i + j + 1)! Tij(1 T)ni+j+1(21)?(T) =d?(T)dT=n?i=0PidBi,n(T)dT,T 0,1(22)dBi,n(u)dT=n!i!(n i)!iTi1(1 T)ni (n i)Ti(1 T)ni1(23)The continuous feature also allows the curve shapeto be flexibly altered without changing the boundaryconditions.Inviewoftheaboveadvantagesformotiondesign, Bezier curves with simple and intuitive natureare employed to represent the cam speed trajectories.For the normalized cam angular displacement, theinitial normalized cam angular displacement is set tozero,i.e.?(0) = 0,andthetotalnormalizedcamangu-lardisplacementisrequiredtobe1,i.e.?(1) = 1.Sub-stituting these two requirements into equation (20)yieldsn?i=0Pi= n + 1(24)c = 1(25)For the continuous movement of the cam, the camangular velocity, acceleration, and jerk must beequal at the beginning and ending, i.e.?(0) =?(1)and?(0) =?(1), and the two requirements can beexpressed asP0 Pn= 0(26)P0 P1 Pn1+ Pn= 0(27)Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from Integrated kinematic and dynamic designs for variable-speed plate cam mechanisms197The degree of Bezier curve affects the complicationof the curve. If the degree is larger, the function candescribe a more complicated curve. Due to the com-puting speed of the experimental devices, a Beziercurve with the order higher than 16 is not feasible.4INTEGRATED KINEMATIC AND DYNAMICDESIGNSThe integration of kinematic and dynamic designs isto optimize the kinematic and dynamic characteris-ticsatthesametime.Formulti-objectiveoptimization,the constraint method and the weighting method areapplied. The constraint method uses the kinematiccharacteristic as an inequality constraint. Only whenthekinematiccharacteristicisbetterthantheconstantinput speed, the dynamic design of the input speedfunctioncanbeobtained.Theweightingmethodspec-ifies the kinematic and dynamic characteristics withdifferent weightings in the optimal object function.Here, the kinematic design is to decrease the peakvalue of the follower acceleration, and the dynamicdesignistodecreasethepeakvalueoftheinputtorqueof motor.4.1The constraint methodBy comparing the constraint method for integratingkinematic and dynamic designs, the objective func-tions are the same, but the constraint method has anadditionalconstraint,whichisthecharacteristicvaluehas to be better than the use of constant input speed.The optimization problem can be expressed asObjective function: minf (P0,P1,.,Pn) = peak|Q|(28)subject to the constraintsn + 1 n?i=0Pi= 0(29)P0 Pn= 0(30)P0 P1 Pn1+ Pn= 0(31)|?(T1) ?1| 0(32)|?(T2) ?2| 0(33)|?(T3) ?3| 0(34)0 ? P0,P1,.,Pn? 2(35)0.8 ? ? 1.2(36) min(Fn) + 0.005 0(37)peak|A| peak|AC| 0(38)Equations (29) to (36) are derived from the anal-ysis of kinematic, equation (37) is derived from thedynamic design, and equation (38) is an additionalconstraint of the constraint method.4.2The weighting methodThe weighting method has to determine the weight-ings for the objects. In this work, the optimized resultsare compared with the results of constant speed, andthe weightings can be determined from the results ofconstant speed. The constraints are the same as thedynamic design. The optimization problem can bewritten asObjective function:minf (P0,P1,.,Pn) =peak|A|peak|AC|+peak|Q|peak|QC|(39)subject to the constraint equations (29) to (37).In what follows, examples are provided for eachdesign method to illustrate the design process. Thecam-follower system with the motion program isshowninTable1,andrelatedspecificationsofthecammechanism are listed inTable 2.Example 1 demonstrates for the constraint method.The average cam speed is 200r/min. A 16th-orderBeziercurveisusedtorepresentthenormalizedspeedtrajectory. There are 17 control points, which can beused as the design variables. The optimal model forthe constraint method of integrated kinematic andTable 1Motion program of the cam-follower systemMotion periodCam rotationangle ()Followerstroke (mm)Dwell 10600Rise (MS)6018025Dwell 21802400Fall (MS)24036025Table 2Specifications of the cam mechanismItemsSpecificationsCamRadius rb(mm)40Cam thickness tc(mm)14Cam mass (cam and cam shaft) mc(kg)1.803Cam moment of inertia aboutrotating axis (including cam, camshaft, torque transducer, and twocouplings) Ic(kgmm2)0.002306kg-mm2FollowerRoller radius rf(mm)12Roller thickness tf(mm)11Follower shaft diameter d (mm)30Initial distance from roller to bearing l1(mm)96Span between two bearings l2(mm)58Follower mass mf(kg)1.075SpringSpring ratio k (N/mm)2.79N/mmTotal length l0(mm)60Preload length lp(mm)5Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from 198H-SYan and C-CYehFig.3Input torque of Example 1dynamic designs is formulated asObjective function:minf (P1,P2,.,Pn) = peak|Q|(40a)subject to the constraintsn + 1 n?i=0Pi= 0(40b)P0 Pn= 0(40c)P0 P1 Pn1+ Pn= 0(40d)|?(T1) ?1| 0(40e)|?(T2) ?2| 0(40f)|?(T3) ?3| 0(40g)Fig.4Motion characteristics of Example 1Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from Integrated kinematic and dynamic designs for variable-speed plate cam mechanisms199Fig.5Input torque of Example 20 ? P0,P1,.,Pn? 2(40h)0.8 ? ? 1.2(40i) min(Fn) + 0.005 0(40j)peak|A| peak|AC| 0(40k)The optimized trajectory of normalized cam angu-lar speed has the following control points: 1.0877,1.2209, 1.2139, 0.0030, 1.5445, 2.0000, 0.1722, 0.0693,1.4878, 1.3018, 1.5582, 0.4812, 0.1356, 2.0000, 0.6814,0.9546, and 1.0877. Figure 3 shows the input torque.The trajectories of normalized cam angular displace-ment, velocity, and acceleration, which are deter-mined by the design variables, are shown in Fig. 4(a).Thecorrespondingfollowermotioncharacteristicsareshown in Fig. 4(b). Figure 4 also displays the motioncharacteristics for the constant cam speed, whichFig.6Motion characteristics of Example 2Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from 200H-SYan and C-CYehFig.7Experimental set-upcan be contrasted with the design results. When thevariable-speed trajectory is applied, the peak value ofthe input torque is decreased from 1.29 to 1.17Nmby 9.24%, and the peak value of the normalized fol-lower acceleration is the same as that in the constantcam speed.Example2demonstratestheweightingmethod.Theaverage cam speed is also 200r/min. A 16th-orderBeziercurveisusedtorepresentthenormalizedspeedtrajectory. There are 17 control points as the designvariables. The optimal model for constraint methodof integrated kinematic and dynamic designs is thenformulated asminf (P1,P2,.,Pn) =peak|A|peak|AC|+peak|Q|peak|QC|(41a)subject to the constraint equations (40b) to (40j).Fig.8Experimental results of input torque (Example 1)Theoptimizednormalizedcamspeedtrajectoryhasthe following control points: 1.0479, 1.2097, 1.4334,0.0000,1.0128,2.0000,1.0416,0.0000,0.2368,2.0000,2.0000, 0.1220, 0.1754, 2.0000, 0.7865, 0.8862, and1.0479. Figure 5 shows the input torque. The trajecto-riesofnormalizedcamangulardisplacement,velocity,and acceleration, which are determined by the designvariables, are shown in Fig. 6(a). The correspondingfollower motion characteristics are shown in Fig. 6(b).Figure6alsodisplaysthemotioncharacteristicsfortheconstant cam speed, which can be contrasted to thedesign results. When the variable-speed trajectory isapplied,thepeakvalueoftheinputtorqueisdecreasedfrom 1.29 to 1.18Nm by 8.7%, and the peak valueof the normalized follower acceleration is decreasedfrom 49.75 to 47.52 by 4.5%.Table 3Experimental devicesInstrumentsSpecificationsTypeProviderController DSP boardand connector panel1. TITMS320C31 DSP board2. Four A/D and four D/A converters3. Two incremental sensor connectorsController board: DS1102dSpaceConnector panel: CP1102Servomotor and driver1. Rated output: 2kW2. Rated torque: 9.54Nm3. Rated speed: 2000r/min4. Rotormomentofinertia:0.00152kgm2Motor: MDMD202P1GPanasonicDriver: MEDDT7364Optical encoder1. 2500pulse/round2. 10000 resolution/roundIncrementalPanasonicOptical scale1. Accuracy: 3m2. Resolution: 0.08m3. Maximalmeasuringlength:70mmLIF471HeidenhainTorque transducer1. Capacity: 50Nm2. Output: 1.499mV/VTP-5KMCBKyowaLoad cell1. Capacity: 200N2. Output: 0.985mV/VLMA-A-200NKyowaStrain amplifiers1. Excitation: 10V2. Magnification: 1000 timesWGA-100B-01 A110KyowaPersonal computerCeleron 1.0G1.0GHzSynnexProc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from Integrated kinematic and dynamic designs for variable-speed plate cam mechanisms201Fig.9Experimental results of motion characteristic (Example 1)Fig.10Experimentalresultsofinputtorque(Example2)5EXPERIMENTAL RESULTS AND DISCUSSIONInadditiontotheplatecammechanism,thehardwarefor the experimental system also includes a controllerboard and connecter panel, a servomotor and motordrive, an optical encoder, an optical linear scale, atorque transducer, a load cell, two strain amplifiers,and a personal computer, as shown in Fig. 7. Therelated specifications are tabulated inTable 3.The experimental results prove the feasibility inthe realistic applications with the cam rotating atthe design speed trajectory. Figures 8 to 11 showthe resulting experimental data, including cam angu-lar displacement, cam angular velocity, cam angularacceleration, follower displacement, follower velocity,follower acceleration, and input torque. Among themeasurement results, the displacement and velocityagree with the corresponding theoretical predictions.Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science at Northwestern Polytechnical University on March 10, 2014Downloaded from 202H-SYan and C-CYehFig.11Experimental results of motion characteristic (Example 2)Therearesomerelativelylargerdeviationsintheaccel-eration and input torque. It may be caused due to thereasonthattheaccelerationiscomputedbynumericaldifferentiation. And, the experimental set-up demon-strates the feasibility of the variable-speed plate cammechanisms.6CONCLUSIONSThis work targets on the kinematic and dynamic anal-yses of variable-input speed plate cam mechanismsfirst. The speed trajectories of the cam are derived byemploying Bezier curve, and then motion continuityconditions are developed. Multi-objective optimiza-tion methods are used, for the integrated kinematicand dynamic designs. Examples are provided for eachdesignmethodtoillustratethedesignprocess,andtheresults are discussed. Finally, an experimental set-upverifies the proposed design models