[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】一种新型双环齿轮减速器的设计与运动仿真-外文文献

Transactions of Tianjin UniversityISSN100624982 pp1632168Vol. 13 No. 3 Jun. 2007Design and Dynamics Simulation of a NovelDouble2Ring2Plate Gear Reducer 3ZHANGJun(张 俊 ) , SONG Yimin(宋轶民 ) , ZHANG Ce (张 策 )(School of Mechanical Engineering , Tianjin University , Tianjin 300072 , China)Abstract: A p atente d double2ring2plate gear re ducer was designe d and its dynamic p erformancewas simulate d. One unique characteristic of this novel drive is that the p has e angle difference be2tween two p arallelogram mechanis ms is a little less than 180 de gree and four counterweights on twocranks hafts are designe d to balance inertia forces and inertia moments of the mechanis ms. Its op er2ating principle , a dvanta ges , and design iss ues were dis cuss e d. An elasto2dynamics model was pr2es ente d to ac quire its dynamic resp ons e by considering the elastic deformations of ring2plates ,gears , bearings , etc . The simulation res ults reveal that comp are d with housing bearings , planetarybearings work in more s evere conditions . The fluctuation of loa ds on gears and bearings indicatesthat the main reas on for re ducer vibration is elastic deformations of the system rather than inertiaforces and inertia moments of the mechanis ms .Keywords : double2ring2plate gear re ducer ; planetary trans mission ; elasto2dynamicsAccepted date : 2006211228.ZHANGJun , born in 1981 , male , doctorate student.3 Supported by the Key Project of Ministry of Education of China ( No. 106050 ) , National Natural Science Foundation of China(No. 50205019) , and Doctoral Foundation of Ministry of Education of China(No. 20040056018) .Correspondence to ZHANGJun , E2mail : zhang- jun tju. edu. cn.Three2ring gear reducer , an internal gear planetarytransmission , claims many advantages , including largetransmission ratio , high loading capacity , and compactvolume1 . However , there still exist some disadvantages inits application. One is the unbalanced inertia moments ex2erted on the housing bearings of crankshafts during its work2ing process. The unbalanced moments , named the shakingmoments , may produce negative vibration and noise 2 .And with the increase of input speed , the vibration getsmore severe. Another one is the fretting wear of eccentricsleeves3 . The six eccentric sleeves on crankshafts bringnot only assembling difficulties but also premature fatigue ofplanetary bearings.To eliminate the above disadvantages , Xin et al 4 proposed a fully2balanced three2ring gear reducer. Thephase angle differences between middle ring2plate and twoside ones are both 180 degree , and the thickness of middlering2plate is twice the side ones. Thus , the inertia forcesand inertia moments of three phases of mechanisms are fullybalanced. However , from the viewpoint of mechanism ,such an internal gear planetary transmission is a combina2tion of three parallelogram linkages and internal gear trans2missions juxtaposed , whose motion will be uncertain whenthe coupler is collinear with the cranks. And this position isoften named“ dead point” . As the phase angle differencebetween two adjacent mechanisms is 180 degree , threephases of parallelogram mechanisms will reach the“ deadpoint” at the same time. Therefore , to overcome the“ deadpoint” , this device needs two additional timing belts todrive the crankshafts simultaneously , which makes thestructure of the reducer more complicated. Besides thecomplexity of structure , the fretting wear of eccentricsleeves still remains. To solve the problems concerning ec2centric sleeves , Tang et al presented a similar double2ring2plate gear reducer5 . Four counterweights are fixed on twocrankshafts to balance the inertia moments. Similarly , thephase angle difference between the juxtaposed parallelogrammechanisms is 180 degree. Because of the problem of“ dead point” , an additional bridge gear pair is needed todivert the input into two crankshafts. The both drives men2tioned above need a first stage transmission to overcome the“ dead point” , which makes the structure of the drive com2paratively incompact.In this paper , a novel double2ring2plate gear reduc2 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. er6 is designed and the afore2mentioned disadvantages ofinternal gear planetary transmission are eliminated. By con2sidering elastic deformations of the parts , a systematic elas2to2dynamic model is developed to reveal its dynamic perfor2mance.1 Mechanism design111 Structure and operating principleThe basic structure of the double2ring2plate gear re2ducer is shown in Fig. 1.Fig. 1 Schematic of a double2ring2plate gear reducerThe two corresponding eccentrics on each crankshaftand one ring2plate form a parallelogram mechanism. Whenthe input shaft is driven , the ring2plate will perform a trans2lational motion. Through the meshing of internal gear on thering2plate with external gear on the output shaft , the poweris output with a large transmission ratio.One unique feature of this drive is that the phase angledifference between two parallelogram mechanisms is a littleless than 180 degree. When one parallelogram mechanismis at the“ dead point” where the coupler is collinear withthe cranks , the other one is at“ regular position” . Throughthe gear meshing , the mechanism at“ regular position” willcarry the other mechanism through the“ dead point” suc2cessfully. Thus the reducer can rotate continuously with asingle power input. Compared with previous internal gearplanetary transmissions1 ,4 ,5 , this drive needs no first stagetransmission. Therefore , the volume is much more com2pact. Moreover , the cancellation of eccentric sleeves helpsto eliminate the fretting wear.112 Calculation for counter weightsAs mentioned above , the phase angle difference be2tween two parallelogram mechanisms is a little less than 180degree. So when the reducer works , the inertial forces andinertial moments of two parallelogram mechanisms cannot bebalanced ,which will produce both shaking forces and shak2ing moments on housing bearings of two crankshafts ,leadingto vibration and noise. Hence , four counterweights are de2signed to eliminate the shaking forces and moments.Let the inertia force produced by each parallelogrammechanism be Fi ( i = 1 ,2) , and we haveFi = (015 mb + mH) e 2 (1)where mb and mH represent the mass of the ring2plate andthe tumbler , and e , stand for eccentric of the sleeves (orthe crank length) and angular velocity of input shaft , re2spectively.According to the dynamic balance theory for rigid ro2tor , we can choose two balance planes , named I and II , inwhich the shaking forces and moments are balanced bycounterweights. The balance condition is described as fol2lows :F1 + F2 + Fe1 = 0F1 + F2 + Fe2 = 0 (2)where Fi , Fi are the components of Fi in balanceplanes I and II respectively , while Fe1 , Fe2 are the in2ertia forces yielded by counterweights in related balanceplanes.By solving Eq. (2) , we can obtain the mass of coun2terweight me i and the eccentric radius rp i using the follow2ing formula :Fe i = me i rp i 2 (3)2 Elasto2dynamics analysisTo evaluate the performance of the drive , an elasto2dynamics model is developed to simulate its dynamic re2sponse.A prototype of the double2ring2plate gear reducer isdesigned for case study. Its main parameters are listed inTab. 1 , where the nomenclatures are explained as follows :A Distance between input shaft and output shaft ,mm ;z1 , z2 Tooth number of external gear and internalgear , respectively ;m Module of gear pair , mm ;e Eccentric of the sleeves , mm ; Phase angle difference between two parallelogrammechanisms , degree ;461Transactions of Tianjin University Vol . 13 No. 3 2007 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. x1 , x2 Modification coefficient of gear pair respec2tively ; Meshing angle of gear pair , degree ;di Radius of crankshafts , mm ;n Input speed of crank , rPmin ;mb Mass of ring2plate , kg ;mH Mass of tumbler , kg ;mo Mass of output shaft , kg ;me Mass of counterweight , kg ;J xb , Jzb Inertia moments of ring2plate about x and zaxis respectively , kg m2 ;J xo , Jzo Inertia moments of output shaft about x ,and z axis respectively , kg m2 ;J s Inertia moments of support shaft about z axis , kgm2 ;kp Stiffness of planetary bearing , 108 NPm ;km Stiffness of gear meshing , 108 NPm ;ko Stiffness of housing bearing on output shaft , 108NPm ;To Rated output torque , kN m.Tab. 1 Main parameters of prototypeA z1 z2 m e 200 51 53 4 512 1761603 8x1 x2 di n mb01662 11164 431985 2 40 1 500 2918mH mo me J xb J zb J xo3152 38176 1152 01235 01973 01191J zo J s kp km ko To01194 01000 5 5 14 019 410It is necessary to point out that the phase angle differ2ence is 31396 2 degree less than 180 degree , whichequals a half of tooth angle of the internal gear. As men2tioned before , to overcome“ dead point” , the phase angledifference between two parallelogram mechanisms must notequal 180 degree. Theoretically , any unequal to 180 de2gree can make the mechanisms get through the “ deadpoint” . But if is too close to 180 degree , the errors ofmanufacturing and assembling may bring unpredictable trou2bles. On the other hand , if is much less than 180 de2gree , the inertia forces and inertia moments of the mecha2nisms will increase and a larger mass of counterweights isneeded , which may probably bring difficulty to structure de2sign issues.211 ModelingThe double2ring2plate gear reducer is an over2con2strained mechanism , which requires some coordinate rela2tions when a dynamics analysis is made. Therefore , someelastic deformations are taken into consideration to derivethe coordinate relations. The deformations include those ofcrankshafts , gearings , bearings , ring2plates and errors ofeccentrics.Fig. 2 shows the elastic deformations of one phase ofparallelogram mechanism. The dashed lines represent theactual position of the mechanism while the solid lines indi2cate the theoretical position. Here , OI A and OS B are the2oretical lengths of input eccentric and support eccentric ;OI OI and OS OS are bending deformations of input shaftand support shaft ; OI A1 and OS B1 are positions of eccen2trics on input shaft and support shaft without any errors ;A1 A2 and B1 B2 are run2out error and indexing error of ec2centrics on input shaft and support shaft , respectively ;A2 A3 and B2 B3 denote elastic deformations of input plane2tary bearing and support planetary bearing , respectively ,and i is the crank angle of the ith parallelogram mecha2nism.Fig. 2 Deformations of one phase of parallelogram mechanismTo simplify the analysis , the overall transmission is di2vided into several subsystems. The dynamic model for eachsubsystem is developed separately and then assembled withthe coordinate relations to get the global dynamics equation.The process is a little similar to Yangps derivation1 .However , in Yangps model the ring2plates were merely con2sidered as rigid bodies while in this model the elastic defor2mations of ring2plates are taken into account. Previous re2searches revealed that the elasticities of ring2plate played asignificant role in ring2plate type gear reducerps dynamicperformance7 ,8 . By considering all these deformations , wederive some alternative coordinate relations as follows.From the vector loops of OI OI A1 A2 A3 A andOS OS B1 B2 B3 B , we can derive the following equations :Ua x = G1 Xn1 + G2 Xn2 - XI - G3 + G4561ZHANG Jun et al : Design and Dynamics Simulation of a Novel Double2Ring2Plate Gear Reducer 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. Ua y = H1 Yn1 + H2 Yn2 - YI - H3 - H4 (4)Ub x = G5 Xn1 + G6 Xn2 + Es - XS - G7 + G8Ub y = H5 Yn1 + H6 Yn2 - Ec - YS - H7 - H8(5)where Xn i , Yn i are the tensile compression and bending de2formations of the ith ring2plate , respectively ; and XI , YI ,XS , YS denote the bending deformations of input shaft andsupport shaft in x and y directions , respectively ; Ua x ,Ua y , Ub x , Ub y represent the elastic deformations of plane2tary bearings on input shaft and support shaft in x and y di2rections respectively ; is the elastic angular displacementof support shaft with respect to its initiative position.As to the deformations of gear meshing , we name pithe relative displacement of meshing pair along the actionline for the ith ring2plate and we finally have :pi = L1 i Xn i + L2 i Yn i + L3 i Xo (6)where Xo is the displacement of output shaft. In Eqs.(4) (6) , the variant vectors have the same meaning as inYangps1 , but the compositions of coefficient matrixes suchas G , H , L are different. The detailed elements of thosematrixes and vectors will not be listed here for content limi2tation.By inserting the coordinate relations into the dynamicequations of subsystems , we obtain the global dynamicequation for this novel drive as follows :MX + KX = Q (7)where M , K , X , Q represent global mass matrix , globalstiffness matrix , vector of global generalized coordinates ,and vector of global excitations , respectively. Their compo2sitions are as follows :M = diag( Mi ) i = 1 , ,10K =K11 K16 K17K22 K28 K29K33 K35 K36 K37K44 K45 K48 K49K55 K56 K57 K58 K59K66 K68 K6 ,10sym. K77 K79 K7 ,10K88 K8 ,10K99 K9 ,10K10 ,10X = XI , YI , XS , YS , , Xn1 , Xn2 , Yn1 , Yn2 ,Xo TQ = Qi Ti = 1 , ,10The global dynamic equation consists of 32 second2order differential equations with periodically time2variantstiffness matrix and excitation vector.It is worthy to point out that the elements in mass ma2trix and stiffness matrix are different from those in Yangpsmodel1 because of the different coordinate relations wehave deduced.212 SimulationBy solving the global dynamic equations , we can ob2tain the dynamic responses of the double2ring2plate gear re2ducer.The dynamic forces of gear meshing are shown in Fig.3 where the horizontal axis represents the rotation angle ofthe first crank. Obviously , the dynamic responses betweentwo parallelogram mechanisms are similar but with a littledifference in amplitude and phase angle , which indicatesthat the load distribution is unequal between two phases ofmechanisms. The unequal load sharing may attribute toelastic deformations of the parts in this over2constrainedtransmission system. And this unequal load distribution ismuch more apparent when the mechanisms reach the regionsaround“ dead point” .Fig. 3 Dynamic forces acting on gearsThe dynamic reactions on planetary bearings of the in2put shaft are shown in Fig. 4. It can be seen that two paral2lelogram mechanisms share similar dynamic responses. Thepeaks indicate that the reactions on planetary bearings varyquite severely. The varying of planetary bearing reactions isdue to the elastic deformations of the transmission systemunder the external load.Fig. 5 shows the dynamic reactions on two housingbearings of the input shaft. Herein , the solid curve denotesthe forward housing bearing of the shaft while the dashedone stands for the backward housing bearing.By comparing Figs. 4 and 5 , we can find that the re2actions of planetary bearings are much greater than those ofhousing bearings. This accounts for the premature fatigue ofplanetary bearings in this type of planetary gearing. There2fore , roller bearing is strongly recommended as the plane2tary bearing because of its high loading capacity.Though the counterweights are designed to balance the661Transactions of Tianjin University Vol . 13 No. 3 2007 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. Fig. 4 Dynamic reactions on planetary bearings of input shaftFig. 5 Dynamic reactions on housing bearings of input shaftinertia forces and inertia moments of the mechanisms , thereactions on housing bearings still fluctuate greatly. Notic2ing that all the reactions of housing bearings are finallytransferred to the reducer case and may cause negative vi2bration , we need to calculate the shaking moments producedby those reactions.The shaking moments of the reducer produced by reac2tions of all housing bearings are demonstrated in Fig. 6.Herein, Figs. 6 ( a) and ( b) are the shaking momentsabout x axis and y axis , respectively. And the solid curvesrepresent the shaking moments of the reducer produced bydynamic reactions of housing bearings while the dashed onesstand for the inertia moments of the mechanisms before thecounterweights are fixed.Apparently , the mechanismsp inertia moments aremuch smaller than the reducerps shaking moments in bothdirections. Therefore , it can be further predicated that themain reason for reducer vibration is not the inertia forces orinertia moments of the mechanisms but the elastic deforma2tions of the parts produced by the external load.Even though the inertia forces and inertia moments arefully balanced by the well2designed counterweights , whichmeans the dashed curves in Fig. 6 become straight lineswith zero amplitude , the fluctuation of reducerps shakingFig. 6 Shaking moments of the reducermoments still remains noticeable. In other words , instead ofeliminating the vibration of reducer case , the design ofcounterweights can only suppress it to some extent.3 ConclusionsA novel double2ring2plate gear reducer is designed andits dynamic performance is simulated with an elasto2dynam2ics model. This drive is featured by non2180 degree phasedifference. This provides the feasibility of single power in2put and makes a compact volume for the reducer. To bal2ance the inertia forces and inertia moments of the mecha2nisms , four counterweights are designed on the crankshafts.The dynamic simulation results indicate t