KinematicAnalysisandT_省略_erBeating_UpMec.pdf

Received date 2010 01 19 Foundation items National Natural Science Foundation of China No 50875243 Zhejiang Technique Innovation Group of Modern Textile Machinery China No 2009R50018 Foundation of Education Department of Zhejiang Province China No Y201019088 Foundation of New Textile R D Emphasised Laboratory of Zhejiang Province China No 2009FZD004 Correspondence should be addressed to CHEN Jian neng E mail jiannengchen zstu edu cn Kinematic Analysis and Test Study of Elliptic Gear and Crank Rocker Beating Up Mechanism ZHAO Xiong 赵雄 REN Gen yong 任根勇 CHEN Jian neng 陈建能 College of Mechanical Engineering and Automation Zhejiang Sci Tech University Hangzhou 310018 China Abstract In order to analyze the kinematic performances of elliptic gear and crank rocker EGCR beating up mechanism kinematic mathematic models of the mechanism were established and an aided analysis and simulation software were compiled This software can display the kinematic characteristics and simulation motion of the mechanism according to different parameters It also supplies a platform for human computer interaction A group of satisfactory parameters were selected by the software A test bed of EGCR beating up mechanism was developed according to these parameters The kinematic performances of the mechanism were verified by high speed video tape recorder Keywords beating upmechanism elliptic gear crank rocker kinematics CLC number TS103 135Document code A Article ID 1672 5220 2011 02 0222 04 Introduction Beating up mechanism is one of the key mechanisms of a loom It beats the weft which is inserted by weft insertion mechanism to form the fabric Its function is to transform the constant speed rotation of the loom s spindle to the non constant speed reciprocating swing of the sley In order to allow the weft insertion mechanism to finish inserting the wefts the sley of the beating up mechanism should have adequate dwell time or relative dwell time in the front position The performances of beating up mechanism determine the fabric squality andalsodecidethequalityand competitiveness of a loom 1 Nowadays there are three general kinds of beating up mechanisms four bar linkage beating up mechanism six bar linkage beating up mechanism and conjugated cam beating up mechanism Generally four barlinkagebeating up mechanism is the simplest mechanism with 65 75 relative dwell time Six bar linkage beating up mechanism has about 120 relative dwell time It has more hinges which produce larger cumulative errors The dwell time of conjugated cam mechanism is 220 240 but the conjugated cam mechanism needs veryhigh precision machining If there are some processing errors it will cause certain vibration 2 4 In this paper a new type of beating up mechanism based on elliptic gear and crank rocker EGCR was produced 5 and its kinematic mathematic models were established A test bed was developed and the kinematic performances of the mechanism were verified by high speed video tape recorder which demonstrated that this new mechanism could meet the requirements of weft beating up 1EGCR Beating Up Mechanism Figure 1 shows the EGCR beating up mechanism in its initial position O is one of the focuses of the active elliptic gear 1 and the rotation centre of the loom s spindle A is one of the focuses and the rotation centre of the driven elliptic gear 2 Crank AB l1 is fixed on the driven elliptic gear Rocker CD l3 is driven by BC l2 and swings reciprocally Sley DE l5 is fixed with CD by axis D and swings reciprocally together with CD By optimizing the eccentricity k the ratio of the elliptic gear s minor radius b to major radius a the angle the included angle between AD l4 and crank AB when crank AB and linkage BC are collinear that is the sley DE will be in the front position the included angle between the major axis of the active elliptic gear and AO the included angle between AD and x axis and the lengths oflinkagesincrank rockermechanism thekinematic performances of this novel beating up mechanism become excellent which are similar to those of the conjugated cam beating up mechanism 1 Active elliptic gear 2 Driven elliptic gear DE Sley Fig 1EGCR beating up mechanism with its initial position 2Kinematic Mathematic Models of EGCR Beating Up Mechanism 2 1Mathematical models of the driven elliptic gear In Fig 1 when the active gear 1 rotates anticlockwise at a constant speed gear 2 will rotate clockwise at a non constant speed Given the angular displacements of the active gear 1 1 and driven gear 2 2 the distance from mesh point P to axis O is r1and PA is r2 With calculation then r1 b2 a c cos 1 1 r2 b2 a c cos 2 2 where c is the distance between the elliptic gear center and the focus 1ranges from 0 to 2 and 2from 0 to 2 6 According to the transmission principles of elliptic gears r1 2a r2 3 222Journal of Donghua University Eng Ed Vol 28 No 2 2011 that is cos 2 a c cos 1 b 2 2a2 2ac cos 1 b2 c a c 4 From Eq 4 the relationship between 2and 1can be obtained According to the principles of gear transmission 2 1 r1 r2 5 Given that the velocity of active gear is constant 2 1 r 1r2 r1r 2 r2 2 2a r 2 r2 2 1 6 TakingthederivativeofEqs 1 and 3 r 1 b2 c sin 1 a c cos 1 2 1and r 2 r 1 so in Eq 6 r 2 b2 c sin 1 a c cos 1 2 1 7 2 2Kinematical models of rocker CD Since the crank is fixed on the driven elliptic gear its angular velocity and angular acceleration are the same as those of the driven elliptic gear That is to say j 1 2 j 1 2 and j1 2 From Fig 2 the following equations can be deduced 7 Fig 2Crank rocker mechanism j2 arctan yC yB xC xB 8 j4 arctan yB yD xB xD 9 j3 arccos l3 2 xD xB 2 yD yB 2 l2 2 2l3 xD xB 2 yD yB 槡 2 j4 10 j 2 Vx B cos j3 Vy B sin j3 l2 sin j2 j3 11 j 3 Vx B cos j2 Vy B sin j2 l3 sin j2 j3 12 j 2 c1 cos j3 c2 sin j3 l2 sin j2 j3 13 j 3 c1 cos j2 c2 sin j2 l3 sin j2 j3 14 In Eqs 13 and 14 c1 ax B l3 j 2 3 cos j3 l2 j 2 2 cos j2 15 c2 ay B l3 j 2 3 sin j3 l2 j 2 2 sin j2 16 2 3Kinematic models of E in x axis direction on the sley DE Displacement equation sx E xD lDE cos j3 17 Velocity equation Vx E lDE j 3 cos j3 2 18 Acceleration equation ax E lDE j 3 2 cos j 3 lDE j 3j cos j3 2 19 3AidedAnalysisandSimulation SoftwareofMechanismand Parameter Optimization 3 1Aided analytical software The visualization of the mechanism is analyzing and optimizing process that can display more information of the process to users Users can observe the whole process and find out the essential parameters ofthe mechanism Human computer interaction analysis and optimization combine the virtues of both human and computers Humans possess the capabilities of illegible illation judgment and innovation which can help to dispose random events as well Meanwhile computers are good at accurate calculation and repeative work Human andcomputercanfullydisplaytheirrespective advantages in human computer interaction optimization Thus satisfactory parameters can be easily achieved 8 10 Based on the above kinematic models of EGCR beating up mechanism an aided analysis and simulation software are compiled which is shown in Fig 3 It can be used to analyze the influences of different mechanism parameters and verify whether there exist interferences among the components of the mechanism Fig 3Aided analysis and simulation software of mechanism With this software users can input mechanism parameters such as a k l1 l2 l3 l4 and rotary speed of the loom s spindle The mechanism motion simulation will be shown on the left of the interface the displacement velocity and acceleration curves of point E will be shown respectively on the right of the interface the optimal value of will be shown on the left bottom of the interface The displacement curve shows that dwell time in the rear position decreases as k increases 322Journal of Donghua University Eng Ed Vol 28 No 2 2011 meanwhile the acceleration curve shows that the maximal acceleration also decreases as k increases For kinematic performances of the mechanism k needs to be optimized according to the requirements of the beating up weft l1 l2 l3 and l4also need to be modified by the user in company with k so as to achieve ideal kinematic performances 3 2Optimization results analysis One of the most important performances of beating up mechanism is the dwell time of sley Increasing mechanism parameter k can prolong dwell time meantime the maximal acceleration increases remarkably and the oversize fluctuation of acceleration will degrade mechanism dynamic performance Designer must therefore make a balance between prolonging dwell time and controlling acceleration fluctuation With the above aided analysis and simulation software a group of parameters was obtained 4 k 0 85 135 a 71 233 mm l1 40 mm l2 100 mm l3 180 mm l4 199 mm and lDE 189 5 mm Based on these parameters when the loom has a speed of 300 r min the kinematical curve of the beating point E is shown in Fig 4 When the beating up mechanism is on the rear position the displacement curve is almost flat The dwell time of sley is close to 200 ranging from 92 to 285 which will not lead to interference of the beating up mechanism and weft insertion mechanism In addition during this period the curves of the velocity and acceleration of the sley are almost close to 0 Therefore it will not cause vibration which will benefit the weft s entery and exit from shed At the end of the beating up the maximaldisplacementis 85mmandthemaximal acceleration is 615 843 4 m s2 which both can meet the requirements of the beating up weft Fig 4The kinematic curves of EGCR beating up mechanism 4Test Study Based on the above parameters a test bed of EGCR beating up mechanism is developed Fig 5 Using the high speed video tape recorder and video analysis software Blaster s MAS the displacement and velocity are obtained with loom spindle srotaryspeedat100r min Theoreticaland experimental displacements of beating point E are shown in Fig 6 and theoretical and experimental velocities of beating point E are shown in Fig 7 The actual measured displacement curve is consistent with the theoretical one but the actual measured velocity curve shows some fluctuation There are two reasons for this finding the gap among the components of the mechanism causes vibration the video analysis contains errors 422Journal of Donghua University Eng Ed Vol 28 No 2 2011 5Conclusions 1 In this paper the EGCR beating up mechanism had been produced Its kinematic mathematic models had been established and an aided analysis and simulation software had been compiled byvisual basic Agroup ofsatisfactory parameters had been got by this software 2 A test bed of the EGCR beating up mechanism was developed Withthevideotaperecorder itskinematic performances were verified This demonstrated the validity of the models and the feasibility of the mechanism References 1 Zhu S K Gao W D Weaving Machine M 2nd ed Beijing China Textile Apparel Press 2004 267 268 in Chinese 2 Liang H S Hu Q E Wang G C et al The Fuzzy Optimization Design ofthe Four Link Weft Beat Up Mechanism J Machine Design and Research 2005 21 2 72 75 in Chinese 3 Ma S P Optimal