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1、Dynamic Analysis of Four -Bar Beating -up M echanismwith Joint ClearanceXUE Wen -liang(薛文良)1*, CHEN Guo -zhen(陈国珍)2, CHEN Ge(陈 革)3, ZHU Zhang -chen(祝章琛)1, CHENG Long -di(程隆棣)11 Key Laboratory of Textile Science & T echnology , M inistry of Education, D onghua University , Shanghai 201620, China2 Sha
2、nghai ShenA n T ex tile Co. , Ltd. , Shanghai 200020 , China3 College of Mechanical Engineering, Donghua University , Shanghai 201620, ChinaAbstract: For the four -bar beating -up mechanism of air -jetloom,the plain bearing of linkage is the bearing withdynamicload, and is immersed in the lubricant
3、-box. If thejoint clearance isconsidered,the research on linkagemovement could be very complicated.In this paper, thekinematic characteristics of four -bar beating -up mechanismwith joint clearance were studied by analyzing the trace ofjournal cente r and the balance of radial, tangential forces,and
4、bearing load. The region of principal vibration and itsforming cause swere discussed.And the results couldinterpret themeasuringcurvesoffour -barbeating -upmechanism completely.Key words: dynamic analysis; four-bar linkage; beating -upmechanism; joint clearanceCLCnumber: TS193. 3 Document code: AArt
5、icle ID: 1672 -5220(2010) 04 -0559 -07IntroductionFor the four -bar beating -up mechanism,the crankshaft bearing and rockshaft bearing are the ball bearing,and the couple journal bearing and crank journal bearingare the journal bearing. The crank, coupler, rocker, andtwo journal bearings are all imm
6、erged in the lubricant, andthere is the radial clearance between the ring and pin ofjournal bearing. The clearance is helpful to the relativemotion of links by providing the space of moving andlubricating, but it also leads to the discontinuous motion ofkinematic pair,which is the main reason to aff
7、ect themechanismdynamiccharacteristics 1 -3.Forthemechanisms transferring the force and moment applied inthe industry manufacturing, asthe joint clearance enlargingthe dynamic characteristics will be destructive under thecondition of high speed and high load.The joint clearance of four -bar beating
8、-up mechanismis within the range of tolerance. If running smoothly someclearance in the joint of mechanism is always necessary.But the clearance is often increased by wearing, then theimpact forces could cause the vibration,noise,bearingdeterioration,andthewearcould alsobeincreased.Therefore, the vi
9、brations become one basic study contentabout mechanism kinetics. Xi et al. studied the relationbetween the dynamic balance and the vibration4,andSoong et al. studied the dynamic characteristics of crank -slider with joint clearance 5.The dynamic alternating load is endured by the journalbearing of f
10、our -bar beating -up mechanism. The forces oflinks are transferred by the oil film, and the pin center ofcouple journal moves on the trace. When the eccentricity islittle, the motion of rocker is unstable, which should leadto the instantaneous impact between the ring and the pinand more serious wear
11、, so the strong vibration will happenmore frequently.1 Four -Bar Linkage with Joint ClearanceIfthejoint clearance ofmechanism is zero,thecenters of bearing and journal are in the same position. Butif the clearance appears, two centers will be separated,which can result in the eccentricity, and there
12、 is the extradegree of freedom for the mechanism.The beating -up mechanism withjointclearanceisshown in Fig. 1, which is the crank -rocker inversion. Thebearing B is made up of crank and coupler, the bearing Cis made up of coupler and rocker, the rocker shaft bearingDis madeup ofrockerandground link
13、,and thecrankshaft bearing A is made up of ground link and crank.For the beating - up mechanism,the connecting arm pinbearing C is easiest to be worn, so it is assumed that thereis only the clearance between the coupler pin and ring, andno clearance in other three bearings.In this paper, themeasurin
14、g loom is the ZA 203 air - jet loom and its speed is450 r/min. XY reference coordinate frame of mechanismshown in Fig. 1 is set according to the practical workingposition. The origin is the bearing center of crankshaft A ,and the coordinate of rocker shaft bearing D center arexD= 8 mmandyD= 159. 8mm
15、. The other parameters are559Journal of Donghua University (Eng. Ed .) Vol. 27, No .4 ( 2010) Received date: 2009 - 01 -15* Correspondence should be addressed to XU E Wen - liang, E -mail: xwl dhu. edu. cnshown in Table 1.Fig. 1 Four -bar linkage used for beating weft up into clothTable 1 Main param
16、eters of four -bar mechanismLinkageLength/ mmMass/ kgMo ment of inertial/( kg # cm2)Crank AB3535. 80155Coupler BC752. 27114R ocker CD14510. 42528Sley and sley sword DE)64. 953 728G round link AD160) The sley and sley sword DE is the working link offour -bar mechanism, which is integrated with the ro
17、cker.When the sley reciprocates, the reed pushes the weft intothe fell of cloth for the inertia of sley and sley sword drivenby the rocker and elbow position of mechanism. For thelinks of four -bar beating -up mechanism, the mass of sleyand sley sword including the rocker is much heavier thanother l
18、inks. The main body of rocking shaft is the steeltube of large diameter. The motion of rocker is stable andits property of ant- i jamming is well, so the sley sword ischosen as the measuring objective of beating -up mechanismacceleration curve.A piezoelectric accelerometer is adsorbed on the sleyswo
19、rd directly by the magnetic seat, which is shown inFig. 2. The sley sword is short and strong, which outputsignal could reflect sensitively the dynamic characteristicsof four -bar linkage.The measuring acceleration curve ofsley sword isshown inFig. 3,whichaccords withthetheoreticalacceleration curve
20、, and characterized in.(1) The curve is not smooth, which means that thetheoretical smooth curve has been added by the oscillationswith different frequencies and amplitudes (see Fig. 4).Fig. 2 Sketch of measuring positionFig. 3 Measuring curveFig. 4 Contrast between measuring curve andtheoretical sm
21、ooth curve(2) The oscillations with different frequencies expressstrict periodical, and the difference is only the amplitudes(see Fig. 3) .For the four -barbeating -up mechanism with fixedgeometrical and kinematic parameters, in the first two or560 Journal of Donghua University ( Eng. Ed. ) Vol.27,
22、No. 4 (2010)three years the output acceleration signal of mechanism willappear again correctly in the next period, and the vibrationpulse with fixed frequency and amplitude will also appearagaineveryperiodofcrankangle.Sotheoutputacceleration signal of four -bar beating -up mechanism withjoint cleara
23、nce is not the random signal, but the main bodyof signal is the signal with the strict periodicity. The outputcurve of mechanism is mainly made up of signals with thestrictperiodicity,andalsoincludessomerandomcomponent.(3) The random component of vibration will increaseas the working time of loom. T
24、hat is to say as the workingtime of mechanism increases, the random component ofoutput curve will be gradually improved.Shown in Fig. 5, B is the included angle of coupler orits extended line and crank. WhenB= 0b or 180b, the crankand thecouplerare in the same line,and when theacceleration of crank
25、is zero,the mechanism is in theposition of elbow link. When B =90 b or 270b, the crankandthecouplerareverticaleachother,andtheacceleration of crank is large. According to the four specialpositions above, the swing process of sley could be dividedinto four regions.Fig. 5 B angle of four - bar mechani
26、sm2 Relative Trace of Coupler Pin Centerw ith Joint Clearance at Static BalancePosition According to the measuring curve, there is the impact oflinks in the joint clearance. By measuring the clearance ofcrank bearing is2Lm, and the clearance of coupler pin journalis 42Lm. So it could be concluded th
27、at the impact in the jointclearance will happen mainly in the coupler pin journal.2. 1 Coupler pin journal with clearanceThere are two cylinders of pin hole and journal shownin Fig. 6. The radius of journal bearing hole is R , theradiusof journal is r , and the radius clearance isc= R- r.The distanc
28、e between two central lines of cylinders is theeccentricitye =C1C2.Thecouplerandtherockerconnected by the coupler journal bearing are both movableparts, so the bearing and the journal are also both turning.There is the oil between two cylinders, and the thickness hof extrusion film ish = c+ esinH= c
29、 (1+ EsinH ) ,(1)where E=e/c is the eccentricity; H is the angle includedbetween h and C1C2in the vertical direction.Fig. 6 Geometric dimension of journal for radial loadF is the total load of journal bearing, Ubcis the linearvelocity of pin along the pin surface, and Ucdis the linearvelocity of jou
30、rnal along the ring surface( see Table 2) .Table 2 Main parameters of coupler pin bearingNominal dimensionof journal bearinghole and journal2R/ mmWidth ofplain journalbearing L / mmV iscosityco efficient oflubricantG/( N # s# m- 1)Radius clearancec = R - r/ mm40420. 01750. 0212. 2 Journal center tra
31、ce of static balance positionThe crank and the coupler are both immersed in thelubricant -box.The lubricantin thejoint clearance ofjournal and bearing will turn with them.There is thehigherhydrodynamicpressureintheregionoflessclearance, and the lower hydrodynamic pressure in theregion of larger clea
32、rance, which leads to the deliveryhead. The bearing load is endured by the hydrodynamicpressure of lubricant for the effect of oil wedge. The staticbalance position of journal center is the balance position ofoil film force and bearing force. There is the differencebetween coupler pin journal and or
33、dinary radial bearing.(1) The direction and intensity of load on the couplerpin journal alternate as the time.(2) The swing velocity of journal and bearing changesas the time.The relation between bearing load Fanddimensionless oil film force is shown in Eq. ( 2) 6.5 =F# ( c/ R)2G# ( Ubc+ Ucd) # L,(2
34、)561Journal of Donghua University (Eng. Ed .) Vol. 27, No .4 ( 2010) where F is the bearing load, G is the dynamic viscosity,and Ubcand Ucdare shown in Fig. 6. According to the datain Table 3, the balance position 7 = N(F , C1C2) could besolved by the interpolation.Table 3 Bearing parameters of circ
35、ular cylinder(ratio of width and radiusL/2R = 1. 0 )EccentricityE0.100. 200.300. 400. 500.600.70Eccentric angle 7 / (b) 75.11 63. 59 55.55 50. 08 45. 32 40.54 36.02Oil film forceX0.197 0. 415 0.680 1. 029 1. 524 2.296 3.654 The journal center trace could be drawn according to theeccentric angle and
36、the eccentricity. The journal center trace ofbearing with the static load usually is semicircle. But the load ofcoupler pin journal will vary in the size and direction as theturning of crank, so the journal center will move along the traceof static balance position. The relative trace of journal cen
37、terstatic balance relative to the coupler pin journal hole shown inFig. 7 is 8 -shaped.The polar angle shown in Fig. 7 is thedirection angle of hole, and the polar radius is the eccentricity.According to the figure, there are three times for the journalcenter coming into the circle of E = 0. 1. When
38、 the crank angle isat 72 b -114 b, it is the first time for the journal center coming intothe circle, and the least eccentricity is 0. 001. Less eccentricity, Fig. 7 Relative trace of journal center static balance relativeto hole( the margent angle is the crank angle)the system stability is worse an
39、d worse. When the crank angle isat 178 b -192 b, it is the second time for the journal center cominginto the circle, and the least eccentricity is 0. 098. When thecrank angle is at 302 b -338 b, it is the third time for the journalcenter coming into the circle, and the least eccentricity is 0.001.An
40、d there are two times for the center touching the outer circleof E=1, where the angular velocities of coupler and rockerrelative to the bearing load F are the same, but their directionsare reverse, and the impact will happen between two links.2. 3 Balance between oil film force and bearing loadFnis
41、the reaction force component in the direction oflink BC length, and Ftis the reaction force component inthe vertical direction of link BC. Between 80 b and 150 b, Ftis close to zero, and at 92 b Fnis the least. The bearingreaction force is endured mainly by the oil film thicknessh, and if a vertical
42、 load was placed on the journal, the oilwould squeeze and there is the sufficient pressure to supportthe load.The pressure could be divided into the radialforce F0nin the direction of tie line and the tangential forceF0tin the vertical direction of tie line.F0n=G(Ubc+ Ucd) L5c2#E2(1- E2)2, ( M C1C2)
43、F0t=14#G(Ubc+ Ucd) L5c2#PE( 1- E2)5/ 2, ( L C1C2)(3)where Ubcis the relative linear velocity of hole surface,Ucdis the relative linear velocity of journal surface. Thereaction force of journal for the oil film will change as theturning of crank. The bearing load is the concentratedforce, and the rea
44、ction force is the distribution force. Thefunction of radial force F0nis to support the journal. In thedirection of turning, the journal center will move for theaction of tangential force F0t, and the eccentricity willincrease. The tangential force will destroy the balance ofjournal. When the main b
45、earing load is in the direction ofcoupler length, the component of force Ftin the verticaldirection is very little. The pressure curves of F0nin thedirection of couple and F0tin the vertical direction areshown in Fig. 8. Because the oil film force F0tis not like thebearing force Ftwhich is around ze
46、ro, in this condition F0tis easy to destroy the stability of journal motion.562 Journal of Donghua University ( Eng. Ed. ) Vol.27, No. 4 (2010)Fig. 8 Curves of couple pin bearing load and oil film reaction f orce The rigidity of oil film leads to the elastic force of oilfilm,so the rigidity coeffici
47、ent influences the dynamiccharacteristics of bearing and the journal center will movefor the disturbance. The cross rigidity coefficient of oilfilm is large, which indicates that the coupling effect of oilfilm force in the vertical direction of radial and tangentialis also large. The cross rigidity
48、coefficient changes with theeccentricity. The eccentricity is less, the absolute value ofcross rigidity coefficient is larger. There is a vortex motionof center for the imbalance of oil film force F0tand bearingload component Ft. So the cross coupling rigidity of oilfilm will excite the vibration of
49、 system.The firsthump in theacceleration curve ofsleybetween 88 b and 156b shown in Fig. 4 is the strong vibrationregion. There are two reasons.Firstly,when the sleyleaves the front dean center, the radial reaction force Ftofcouple pin journal is near zero. Secondly, between 72 b and114b the journal
50、 center crosses the inner circle, where theeccentricity is little, but it is clear that according to thetrace of journal center the stability in this region is theworst when the crank turning one rotation. In this region,there is a breaking point of bearing force direction for thecouple pin journal.
51、 The smallest force is 20 N at 92 b, andthere is a motion of two links with same velocity butreverse direction in the couple pin journal.3 Stability Judgment of Journal Motion inBearing with Joint Clearance One of basic research contents for the mechanismkinetics of bearing with joint clearance is t
52、he impact oflinkages. The kinematic pair with a large clearance is easyto lead to the vibration of mechanism. When the contactbetween journal and bearing of revolving joints losing, theimpact will happen. The experiential Formula( 4) whetherthe impact will happen could be concluded according to thee
53、xperiments6, 7.When the ratio between the directionchanging ratio B c of the nominal bearing force and theminimum magnitude of bearing force Fmincalculated from azero -clearance analysis is less than the unity, the contactbetween journal and bearing could be kept on.B c| Fmin| 1.(4)The impactbetween
54、the journal and bearing willhappen when one of following two conditions appears.(1) There is the smallest bearing load, or even zerobearing load.For the four -bar linkage immersed in the lubricant,the bearing load is endured by the oil film. For the bearingwith static load, the eccentricity is the f
55、unction of bearingload. When the load is zero, the journal and bearing are inthe concentric position. As the load increasing, the journalcenter will be going to the bearing center. And when theload increases, the journal will contact the bearing innerhole. The hydrodynamic force of journal bearing h
56、as greatinfluence for the stability of linkage. The bearing load isless, the eccentricity is less,and the stability of journalmotion is worse, which accords with Formula( 4) .The journal bearing of beating -up mechanism is thebearing with dynamic load. For the tangential force of oilfilm, there is t
57、he disturbance of journal center motion. Asthere is no enough input load in the tangential direction, itis clear that the smallest bearing load is one condition ofjournal motion imbalance.( 2)When the direction of bearingload changes tooquickly, the journal ring and the pin could not keep the contac
58、t.When the crank angle shown in Fig. 9 is at 92 b and 322 b,the direction angle of couple pin bearing load changes quicklyandthesmallestbearingforce alsoappears there.Thehydrodynamic pressure of oil film could support the quickchange of bearing force direction, and the contact betweenring and pin wi
59、ll not lose, if the clearance is in the normalallowable range. According to Formula( 4), in every rotationof crank there are two angles where the probability of four -barlinkage imbalanceis thehighest.Butaccordingtothemeasuring curve shown in Fig. 4, the vibration intensities attwo angles are differ
60、ent. At 92 b the angular velocity is just563Journal of Donghua University (Eng. Ed .) Vol. 27, No .4 ( 2010) dropping from the maximal value, and the vibration in theregion ofmaximal angularvelocityis strongand itsamplitude is large.The vibrating amplitude at 322b issmall. There are two reasons for
61、the different vibrations.(1) At 92b, the force Ftof couple bearing fluctuatesthe zero within the period. But at 322b, the angle includedbetween the crank and the couple is close to 90b, and thebearing force Ftis close to the zero then goes upwardsimmediately.(2) At 92b, the journal center is in the
62、inner circle,where E= 0. 001, and at 322 b E= 0. 004. The eccentricity isless, then the journal motion is easier to be unstable.Fig. 9 Direction angle of bearing f orce According to the above discussion, Formula(4) could notdescribe the impact in bearing with joint clearance completely,and the appli
63、cability of Formula(4) is discussed in Ref. 7 .When the load -carrying of couple journal bearing issmall, the eccentricity ratio of journal center is small. Andwhen the load is large,the eccentricity ratio is large.When the angle velocities of coupler and crank relative tothe bearing load are the sa
64、me, the pin will contact the ringfor the load. The couple journal bearing will usually be intheconditionofstartingupandrelativestillnessalternately, and for the oil clings to the journal, the oilfilm will build up the sufficient pressure to support theload, and will turn or stop with the surfaces of
65、 bearing andjournal, because the angle velocities of coupler and rockerchange frequently. When starting up there is no effectiveoil film, and in the interim of stop the oil film will break.The bearing is always in the condition of mixed friction.The joint clearance of kinematic pair not only leads t
66、o thevibration with high frequency by itself, but also is the inducedfactor, which leads to the impact exciting the steady -statevibration of mechanism. The natural frequency of linkage isthe most important factor influencing the dynamic response ofmechanism. The measuring curve of rocker is the sup
67、erpositionof vibration with stable low frequency curve and vibration withmuch higher frequency.The clearance will enlarge as theworking time of machine. So there is the impact in the jointclearance , and the frequency and amplitude of vibration willalso change. Such an impact could cause the vibrati
68、on andnoise, and increase the wear.4 Conclusions(1) For the four -bar linkage with joint clearance, theoutput time signal of rocker acceleration in the period willappear again correctly in the next period, and the vibrationimpulse with certain frequency and amplitude also willappear at the fixed cra
69、nk angle. The output curve is mainlymade up of the periodical signals, also includes a little ofrandom signals.( 2)The plain bearingof four -barlinkageis thebearing with dynamic load, and the relative trace of journalcenter in the connecting arm pin bearing is the basis ofjudging the journal motion
70、stability in the bearing with jointclearance. The journal center is closer to the center ofbearing hole, the motion stability is worse. If the time ofjournal center staying in the most inner circle is long, thedisturbance of journal will be excited and develop to theimpact of members.( 3)Thelinkagem
71、echanismis immersedinthelubricant -box, and the oil film is enough to endure thebearing load, especially endured by the radial componentof the oil film force. For the action of tangential force, ifthere is no enough force to endure the input load in thetangential direction, it is easy to lead to the
72、 disturbance ofjournal center.(4) The first hump in the acceleration curve is theregion with the strongest vibration. The stability of journalmotion in the couple pin bearing with joint clearance is themost important factor of mechanism vibration.References 1 Flores P, Ambr sio J. Revolute Joints wi
73、th Clearance inMultibody Systems J . Computers and Structures, 2004,82( 17/ 18/ 19) : 1359 -1369. 2 Flores P, Ambr sio J, Claro J C P, et al . A Study on564 Journal of Donghua University ( Eng. Ed. ) Vol.27, No. 4 (2010)Dynamics ofMechanical Systems Including Joints withClearance and Lubrication J .
74、 Mechanism and MachineTheory , 2006, 41( 3) : 247 -261. 3 Schwab A L , Meijaard J P, Meijers P. A Comparison ofRev olute Joint Clearance Models in the Dynamic Analysisof Rigid and Elastic Mechanical Systems J . Mechanismand Machine Theory, 2002, 37( 9) : 895 - 913. 4 Xi F F, Sinatra R. Effect of Dyn
75、amic Balancing on Four -Bar Linkage Vibrations J .Mechanismand MachineTheory , 1997, 32 ( 6) : 715 - 728. 5 Soong K,Thompson B S.A Theoretical & ExperimentalInvestigation ofthe Dynamic Response of a Slider -CrankMechanism with Radial Clearance in the Gudgeon -Pin Joint J.Journal of Mechanical Design
76、, 1990, 112( 2) : 183 -189. 6 Li Z , Li L, Bai S X. A New Method of Predicting theOccurrence of Contact Loss Between Pairing Elements inPlanar Linkages with Clearances J .MechanismandMachine Theory , 1992, 27 ( 3) : 295 -301. 7 SeneviratneLD,Earles SWE.ChaoticBehaviorExhibited during Contact Loss in a Clearance Joint of aFour -Bar M echanism J . Mechanism and Machine Theory ,1992, 27 ( 3) : 307 -321.565Journal of Donghua University (Eng. Ed .) Vol. 27, No .4 ( 2010)
