減速器-圓錐圓柱齒輪減速器設(shè)計(jì)【鏈?zhǔn)捷斔蜋C(jī)傳動(dòng)裝置】【F=2500M V=0.67ms D=445 L=800mm】,鏈?zhǔn)捷斔蜋C(jī)傳動(dòng)裝置,F=2500M V=0.67ms D=445 L=800mm,減速器-圓錐圓柱齒輪減速器設(shè)計(jì)【鏈?zhǔn)捷斔蜋C(jī)傳動(dòng)裝置】【F=2500M,V=0.67ms,D=445,L=800mm】,減速器 Transactions of Tianjin University ISSN100624982 pp1632168 Vol. 13 No. 3 Jun. 2007 Design and Dynamics Simulation of a Novel Double2Ring2Plate Gear Reducer 3 ZHANGJun(張 俊 ) , SONG Yimin(宋軼民 ) , ZHANG Ce (張 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 erformance was simulate d. One unique characteristic of this novel drive is that the p has e angle difference be2 tween two p arallelogram mechanis ms is a little less than 180 de gree and four counterweights on two cranks hafts are designe d to balance inertia forces and inertia moments of the mechanis ms. Its op er2 ating principle , a dvanta ges , and design iss ues were dis cuss e d. An elasto2dynamics model was pr2 es 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 , planetary bearings work in more s evere conditions . The fluctuation of loa ds on gears and bearings indicates that the main reas on for re ducer vibration is elastic deformations of the system rather than inertia forces and inertia moments of the mechanis ms . Keywords : double2ring2plate gear re ducer ; planetary trans mission ; elasto2dynamics Accepted 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 planetary transmission , claims many advantages , including large transmission ratio , high loading capacity , and compact volume1 . However , there still exist some disadvantages in its application. One is the unbalanced inertia moments ex2 erted on the housing bearings of crankshafts during its work2 ing process. The unbalanced moments , named the shaking moments , may produce negative vibration and noise 2 . And with the increase of input speed , the vibration gets more severe. Another one is the fretting wear of eccentric sleeves3 . The six eccentric sleeves on crankshafts bring not only assembling difficulties but also premature fatigue of planetary bearings. To eliminate the above disadvantages , Xin et al 4 proposed a fully2balanced three2ring gear reducer. The phase angle differences between middle ring2plate and two side ones are both 180 degree , and the thickness of middle ring2plate is twice the side ones. Thus , the inertia forces and inertia moments of three phases of mechanisms are fully balanced. However , from the viewpoint of mechanism , such an internal gear planetary transmission is a combina2 tion of three parallelogram linkages and internal gear trans2 missions juxtaposed , whose motion will be uncertain when the coupler is collinear with the cranks. And this position is often named“ dead point” . As the phase angle difference between two adjacent mechanisms is 180 degree , three phases of parallelogram mechanisms will reach the“ dead point” at the same time. Therefore , to overcome the“ dead point” , this device needs two additional timing belts to drive the crankshafts simultaneously , which makes the structure of the reducer more complicated. Besides the complexity of structure , the fretting wear of eccentric sleeves still remains. To solve the problems concerning ec2 centric sleeves , Tang et al presented a similar double2ring2 plate gear reducer5 . Four counterweights are fixed on two crankshafts to balance the inertia moments. Similarly , the phase angle difference between the juxtaposed parallelogram mechanisms is 180 degree. Because of the problem of “ dead point” , an additional bridge gear pair is needed to divert the input into two crankshafts. The both drives men2 tioned above need a first stage transmission to overcome the “ dead point” , which makes the structure of the drive com2 paratively incompact. In this paper , a novel double2ring2plate gear reduc2 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. http:/ er6 is designed and the afore2mentioned disadvantages of internal gear planetary transmission are eliminated. By con2 sidering elastic deformations of the parts , a systematic elas2 to2dynamic model is developed to reveal its dynamic perfor2 mance. 1 Mechanism design 111 Structure and operating principle The basic structure of the double2ring2plate gear re2 ducer is shown in Fig. 1. Fig. 1 Schematic of a double2ring2plate gear reducer The two corresponding eccentrics on each crankshaft and one ring2plate form a parallelogram mechanism. When the input shaft is driven , the ring2plate will perform a trans2 lational motion. Through the meshing of internal gear on the ring2plate with external gear on the output shaft , the power is output with a large transmission ratio. One unique feature of this drive is that the phase angle difference between two parallelogram mechanisms is a little less than 180 degree. When one parallelogram mechanism is at the“ dead point” where the coupler is collinear with the cranks , the other one is at“ regular position” . Through the gear meshing , the mechanism at“ regular position” will carry the other mechanism through the“ dead point” suc2 cessfully. Thus the reducer can rotate continuously with a single power input. Compared with previous internal gear planetary transmissions1 ,4 ,5 , this drive needs no first stage transmission. Therefore , the volume is much more com2 pact. Moreover , the cancellation of eccentric sleeves helps to eliminate the fretting wear. 112 Calculation for counter weights As mentioned above , the phase angle difference be2 tween two parallelogram mechanisms is a little less than 180 degree. So when the reducer works , the inertial forces and inertial moments of two parallelogram mechanisms cannot be balanced ,which will produce both shaking forces and shak2 ing moments on housing bearings of two crankshafts ,leading to vibration and noise. Hence , four counterweights are de2 signed to eliminate the shaking forces and moments. Let the inertia force produced by each parallelogram mechanism be Fi ( i = 1 ,2) , and we have Fi = (015 mb + mH) e 2 (1) where mb and mH represent the mass of the ring2plate and the tumbler , and e , stand for eccentric of the sleeves (or the crank length) and angular velocity of input shaft , re2 spectively. According to the dynamic balance theory for rigid ro2 tor , we can choose two balance planes , named I and II , in which the shaking forces and moments are balanced by counterweights. The balance condition is described as fol2 lows : F1 + F2 + Fe1 = 0 F1 + F2 + Fe2 = 0 (2) where Fi , Fi are the components of Fi in balance planes I and II respectively , while Fe1 , Fe2 are the in2 ertia forces yielded by counterweights in related balance planes. By solving Eq. (2) , we can obtain the mass of coun2 terweight me i and the eccentric radius rp i using the follow2 ing formula : Fe i = me i rp i 2 (3) 2 Elasto2dynamics analysis To evaluate the performance of the drive , an elasto2 dynamics model is developed to simulate its dynamic re2 sponse. A prototype of the double2ring2plate gear reducer is designed for case study. Its main parameters are listed in Tab. 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 internal gear , respectively ; m Module of gear pair , mm ; e Eccentric of the sleeves , mm ; Phase angle difference between two parallelogram mechanisms , degree ; 461 Transactions of Tianjin University Vol . 13 No. 3 2007 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. http:/ x1 , x2 Modification coefficient of gear pair respec2 tively ; 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 z axis 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 , kg m2 ; kp Stiffness of planetary bearing , 108 NPm ; km Stiffness of gear meshing , 108 NPm ; ko Stiffness of housing bearing on output shaft , 108 NPm ; To Rated output torque , kN m. Tab. 1 Main parameters of prototype A z1 z2 m e 200 51 53 4 512 1761603 8 x1 x2 di n mb 01662 11164 431985 2 40 1 500 2918 mH mo me J xb J zb J xo 3152 38176 1152 01235 01973 01191 J zo J s kp km ko To 01194 01000 5 5 14 019 410 It is necessary to point out that the phase angle differ2 ence is 31396 2 degree less than 180 degree , which equals a half of tooth angle of the internal gear. As men2 tioned before , to overcome“ dead point” , the phase angle difference between two parallelogram mechanisms must not equal 180 degree. Theoretically , any unequal to 180 de2 gree can make the mechanisms get through the “ dead point” . But if is too close to 180 degree , the errors of manufacturing and assembling may bring unpredictable trou2 bles. On the other hand , if is much less than 180 de2 gree , the inertia forces and inertia moments of the mecha2 nisms will increase and a larger mass of counterweights is needed , which may probably bring difficulty to structure de2 sign issues. 211 Modeling The double2ring2plate gear reducer is an over2con2 strained mechanism , which requires some coordinate rela2 tions when a dynamics analysis is made. Therefore , some elastic deformations are taken into consideration to derive the coordinate relations. The deformations include those of crankshafts , gearings , bearings , ring2plates and errors of eccentrics. Fig. 2 shows the elastic deformations of one phase of parallelogram mechanism. The dashed lines represent the actual position of the mechanism while the solid lines indi2 cate the theoretical position. Here , OI A and OS B are the2 oretical lengths of input eccentric and support eccentric ; OI OI and OS OS are bending deformations of input shaft and support shaft ; OI A1 and OS B1 are positions of eccen2 trics on input shaft and support shaft without any errors ; A1 A2 and B1 B2 are run2out error and indexing error of ec2 centrics on input shaft and support shaft , respectively ; A2 A3 and B2 B3 denote elastic deformations of input plane2 tary bearing and support planetary bearing , respectively , and i is the crank angle of the ith parallelogram mecha2 nism. Fig. 2 Deformations of one phase of parallelogram mechanism To simplify the analysis , the overall transmission is di2 vided into several subsystems. The dynamic model for each subsystem is developed separately and then assembled with the 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 con2 sidered as rigid bodies while in this model the elastic defor2 mations of ring2plates are taken into account. Previous re2 searches revealed that the elasticities of ring2plate played a significant role in ring2plate type gear reducerps dynamic performance7 ,8 . By considering all these deformations , we derive some alternative coordinate relations as follows. From the vector loops of OI OI A1 A2 A3 A and OS OS B1 B2 B3 B , we can derive the following equations : Ua x = G1 Xn1 + G2 Xn2 - XI - G3 + G4 561 ZHANG Jun et al : Design and Dynamics Simulation of a Novel Double2Ring2Plate Gear Reducer 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. http:/ Ua y = H1 Yn1 + H2 Yn2 - YI - H3 - H4 (4) Ub x = G5 Xn1 + G6 Xn2 + Es - XS - G7 + G8 Ub y = H5 Yn1 + H6 Yn2 - Ec - YS - H7 - H8 (5) where Xn i , Yn i are the tensile compression and bending de2 formations of the ith ring2plate , respectively ; and XI , YI , XS , YS denote the bending deformations of input shaft and support shaft in x and y directions , respectively ; Ua x , Ua y , Ub x , Ub y represent the elastic deformations of plane2 tary bearings on input shaft and support shaft in x and y di2 rections respectively ; is the elastic angular displacement of support shaft with respect to its initiative position. As to the deformations of gear meshing , we name pi the relative displacement of meshing pair along the action line 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 in Yangps1 , but the compositions of coefficient matrixes such as G , H , L are different. The detailed elements of those matrixes and vectors will not be listed here for content limi2 tation. By inserting the coordinate relations into the dynamic equations of subsystems , we obtain the global dynamic equation for this novel drive as follows : MX + KX = Q (7) where M , K , X , Q represent global mass matrix , global stiffness matrix , vector of global generalized coordinates , and vector of global excitations , respectively. Their compo2 sitions are as follows : M = diag( Mi ) i = 1 , ,10 K = K11 K16 K17 K22 K28 K29 K33 K35 K36 K37 K44 K45 K48 K49 K55 K56 K57 K58 K59 K66 K68 K6 ,10 sym. K77 K79 K7 ,10 K88 K8 ,10 K99 K9 ,10 K10 ,10 X = XI , YI , XS , YS , , Xn1 , Xn2 , Yn1 , Yn2 , Xo T Q = Qi Ti = 1 , ,10 The global dynamic equation consists of 32 second2 order differential equations with periodically time2variant stiffness matrix and excitation vector. It is worthy to point out that the elements in mass ma2 trix and stiffness matrix are different from those in Yangps model1 because of the different coordinate relations we have deduced. 212 Simulation By solving the global dynamic equations , we can ob2 tain the dynamic responses of the double2ring2plate gear re2 ducer. The dynamic forces of gear meshing are shown in Fig. 3 where the horizontal axis represents the rotation angle of the first crank. Obviously , the dynamic responses between two parallelogram mechanisms are similar but with a little difference in amplitude and phase angle , which indicates that the load distribution is unequal between two phases of mechanisms. The unequal load sharing may attribute to elastic deformations of the parts in this over2constrained transmission system. And this unequal load distribution is much more apparent when the mechanisms reach the regions around“ dead point” . Fig. 3 Dynamic forces acting on gears The dynamic reactions on planetary bearings of the in2 put shaft are shown in Fig. 4. It can be seen that two paral2 lelogram mechanisms share similar dynamic responses. The peaks indicate that the reactions on planetary bearings vary quite severely. The varying of planetary bearing reactions is due to the elastic deformations of the transmission system under the external load. Fig. 5 shows the dynamic reactions on two housing bearings of the input shaft. Herein , the solid curve denotes the forward housing bearing of the shaft while the dashed one stands for the backward housing bearing. By comparing Figs. 4 and 5 , we can find that the re2 actions of planetary bearings are much greater than those of housing bearings. This accounts for the premature fatigue of planetary bearings in this type of planetary gearing. There2 fore , roller bearing is strongly recommended as the plane2 tary bearing because of its high loading capacity. Though the counterweights are designed to balance the 661 Transactions of Tianjin University Vol . 13 No. 3 2007 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. http:/ Fig. 4 Dynamic reactions on planetary bearings of input shaft Fig. 5 Dynamic reactions on housing bearings of input shaft inertia forces and inertia moments of the mechanisms , the reactions on housing bearings still fluctuate greatly. Notic2 ing that all the reactions of housing bearings are finally transferred to the reducer case and may cause negative vi2 bration , we need to calculate the shaking moments produced by those reactions. The shaking moments of the reducer produced by reac2 tions of all housing bearings are demonstrated in Fig. 6. Herein, Figs. 6 ( a) and ( b) are the shaking moments about x axis and y axis , respectively. And the solid curves represent the shaking moments of the reducer produced by dynamic reactions of housing bearings while the dashed ones stand for the inertia moments of the mechanisms before the counterweights are fixed. Apparently , the mechanismsp inertia moments are much smaller than the reducerps shaking moments in both directions. Therefore , it can be further predicated that the main reason for reducer vibration is not the inertia forces or inertia moments of the mechanisms but the elastic deforma2 tions of the parts produced by the external load. Even though the inertia forces and inertia moments are fully balanced by the well2designed counterweights , which means the dashed curves in Fig. 6 become straight lines with zero amplitude , the fluctuation of reducerps shaking Fig. 6 Shaking moments of the reducer moments still remains noticeable. In other words , instead of eliminating the vibration of reducer case , the design of counterweights can only suppress it to some extent. 3 Conclusions A novel double2ring2plate gear reducer is designed and its dynamic performance is simulated with an elasto2dynam2 ics model. This drive is featured by non2180 degree phase difference. This provides the feasibility of single power in2 put and makes a compact volume for the reducer. To bal2 ance the inertia forces and inertia moments of the mecha2 nisms , four counterweights are designed on the crankshafts. The dynamic simulation results indicate that compared with the housing bearings , the planetary ones work in more severe conditions. Different from the prevailing viewpoint that the inertia forces and inertia moments are main vibra2 tion resource of ring2plate type gear reducers2 , this inves2 tigation reveals that the vibration of this novel double2ring2 plate gear reducer is mostly caused by elastic deformations of the parts rather than by inertia forces and inertia mo2 ments. References 1 Yang Jianmin , Zhang Ce , Qin Datong et al. Elasto2dy2 761 ZHANG Jun et al : Design and Dynamics Simulation of a Novel Double2Ring2Plate Gear Reducer 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved. http:/ namic analysis of three2ring reducersJ . Chinese Journal of Mechanical Engineering , 2000 , 36 ( 10) : 54 58 ( in Chinese) . 2 Cui Jiankun , Zeng Zhong. Calculation and balance for shaking force and shaking moment of three2ring gear reduc2 erJ . Machine Design and Research , 1996 (3) : 39 40 (in Chinese) . 3 Cui Jiankun , Zhang Guanghui. Study on fretting wear of eccentric sleeves in three2ring2plate gear reducer J . Journal of Machine Design , 1996 (10) : 31 33 (in Chi2 nese) . 4 Xin Shaojie , Li Huamin , Liang Yongsheng. Study on equilibrating load mechanism of oil film floating on a new type of three2ring gear reducerJ .Mechanical Science and Technology , 2000 , 19(4) : 581 583(in Chinese) . 5 Tang Guoliang , Wang Shuhao. Double Crank Double2Ring2 Plate Gear Reducer with Few Tooth Number Difference P . CN : 2118208U , 1992210207(in Chinese) . 6 Zhang Ce , Song Yimin , Zhang Jun. Double2Ring2Plate Gear Reducer P . CN : ZL 20042008568610 , 2005212207 (in Chinese) . 7 Zhang Yongxin. Elasto2Static Analysis of Three2Ring Gear Reducer in Consideration of Gear2Coupler Deformations D . Tianjin : School of Mechanical Engineering , Tianjin University , 2005(in Chinese) . 8 Zhang Guanghui , Han Jielin , Long Hui. Stress analysis of driving ring board of three2ring type gear reducerJ .Chi2 nese Journal of M