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 12th IFToMM World Congress, Besanon, June 18-21, 2007 Review of Researches on Ring-plate Gear Reducers with Small Tooth Number Difference Ce Zhang * Yimin Song + Jun Zhang School of Mechanical Engineering, Tianjin University, Tianjin, 300072, P.R. China AbstractRing-plate gear reducer with small tooth number difference has drawn great attention from academic and industrial fields in China due to its special characteristics such as high transmission ratio, high load capacity, simple structure and low manufacturing cost. However, its application is limited because of the lack in systematic analysis method and design theory. Researches on ring-plate reducer are reviewed from aspects of force analysis, load capacity, balance and vibration reduction, mechanical efficiency and serial design of products. In addition, some questions to be solved are discussed. Key words: ring-plate gear reducer, force analysis, load capacity, balance, vibration reduction, efficiency I Introduction The predecessor of the ring-plate transmission with small tooth number difference is the K-H-V coaxial planetary gear train with small tooth number difference, which has two shortcomings: the limitation of the size of planetary bearing as well as its short service life; and absolute necessity of the output connection mechanism, as shown in Fig.1. Fig. 1. Sketch of K-H-V transmission In 1985, Chen Zongyuan, a Chinese engineer, invented the three-ring-plate transmission, which ingeniously overcame the above-mentioned problems of K-H-V transmission 1. *E-mail: ce_ E-mail: E-mail: zhang_ The three-ring-plate transmission is shown in Fig 2. Fig. 2. Three-ring-plate transmission Input shaft 1 and support shaft 2, which are the cranks in the parallelogram mechanism, share one eccentric part. The coupler 3 of the parallelogram mechanism, namely the ring-plate, is also an internal gear meshing with the external gear mounted on the output shaft 4 and rotating around a fixed axis. The ring-plate 3 driven by input shaft moves translationally and does not rotate. Three parallelograms are arranged to avoid the uncertainty when the crank and coupler are aligned. The phase angle is 120 degree. The transmission ratio is in the range of 11-99 for a single-stage drive, and can reach 9801 for a double-stage one, and even higher when needed. For the visualization purpose, a CAD solid model of the symmetrical three-ring gear reducer is shown in Fig. 3. Compared with other transmissions, the ring-plate transmission with such characteristics as higher transmission ratio, higher load capacity, simpler structure and lower production cost has drawn academic and industrial attentions as well. In China, this transmission has already been applied in metallurgical industry, mining, transportation, building and light industry. However, due to limitation in research and development and flaws in earlier design, some problems such as vibration, ring-plate break and short service life of the bearings arise. To some degree, its image has been undermined and further application has been limited. In this paper, a review is presented from aspects of force analysis, load capacity, balance and vibration reduction, mechanical efficiency and serial design of products. Fig. 3. Three-ring gear reducer（symmetrical） II Force Analysis The ring-plate transmission is a kind of over-constraint mechanism, and its force analysis is complicated due to its sensitivity to elastic deformations and manufacturing errors. The compatibility equations of deformations are necessary in the formulation. In 1990s, a few works discussed force analysis of this transmission, but the basic assumptions in the works may deviate from practical application. The results of the analysis, therefore, could be open to discussion. To improve the dynamic performance of the ring-plate reducers, an in-depth analysis on dynamics and vibration of the reducer is necessary. Yang and Zhang 2 established elastodynamic equations of this transmission, and analyzed the natural frequencies, vibration modes and dynamic response. But the influence of ring-plates deformation on dynamic behavior of the system has been ignored. By use of finite element modeling, the stresses, displacements and deformations of the ring-plates have been analyzed in reference 3,4. It has been revealed that tension and bending deformation of the plates have an unnegligible influence on the systems kinematic and dynamic performance. So these two kinds of deformation of the plates cannot be overlooked and are included as one of the parameters of compatibility equations in 4. A force analysis program of the transmission, in which many factors have been taken into consideration, could already be put into actual use 5. The time history of the loads in gears and bearings can be obtained through this program. III Load Capacity The researches on load capacity focus on the effect of so-called multi-teeth elastic meshing, which is a special phenomenon in internal gearing with small tooth number difference. As shown in Fig. 4, when the No.0 teeth pair is in conjugation, the clearances between working surfaces of the neighboring teeth pairs (No.1-4) are very small. When the teeth deform under load, the clearances may disappear and multi-teeth meshing takes place. Fig.4. Multi-teeth elastic meshing Early in 1965 6 predicted the existence of this effect and proposed a formula to calculate the clearances value. Later Sunaga 7 further proved its actual existence through an experiment. Based on tooth stiffness, clearances and manufacturing error, Shu 8, 9 deduced the number of simultaneously meshing teeth pairs and measured out the load distribution on the teeth pairs through photo-elastic experiment. According to Sunaga and Shus works, the number of simultaneously meshing teeth pairs cannot exceed 3-5. But in their experiments they failed to give the measured values of errors of the experimented gears. Since the base-pitch errors are random and have an enormous influence upon the load distribution, the results of their experiments cannot be taken as proofs of the actual effect of multi-teeth meshing. Later, the reference 10 made similar effort. Because the base-pitch errors are much greater than the clearances in No. 3 and No.4 teeth pairs, their approaches to the gear errors are open to discussion. The reference 11 precisely deduced the theoretical value of the clearances, calculated the elastic deformations and obtained the number of the simultaneously meshing teeth pair and the load distribution factors. The conclusion is that normally the number of working teeth pairs is only 3, cannot reach 9-18, as some product samples claimed. The most influential error is base-pitch error. According to references 11 base-pitch error may result in instantaneous single-tooth meshing. They advised that the load distribution factors be introduced into fatigue strength estimation of the teeth bending, and that the maximum bending stress be checked at the same time. It is necessary to estimate the influence of errors on load distribution more precisely. The first analysis shows that the positive effect of this phenomenon is too important to be neglected, even though the negative influence of errors should be taken into account. IV Balancing and Vibration Reduction The shaking forces of the three ring-plates counter- balance each other. But there exists a periodic shaking moment, which makes the reducer vibrate. In early design the unsymmetrical arrangement is employed, as shown in Fig. 2. This configuration makes the reducer vibrate strongly. Balancing is also necessary even for the symmetrical arrangement. The image of the transmission has been compromised because of the flaws in early design. Reference 12 has successfully resolved the balancing of the three-ring-plate reducers by mounting four counterweights on the input and support shafts, as shown in Fig. 6. The two-ring-plate reducers (shown in Fig.5) designed by Wang 13,14, which used a split path arrangement to avoid the uncertainty of motion, have a better dynamic performance. However, the structure is much complicated. Fig. 5. Two ring-plates reducer with split path arrangement The authors of this paper proposed a new type of two- ring-plate reducer, as shown in Fig.6, in which the phase angle between the two parallelograms is a little less than , and the split path arrangement can be removed. Four counterweights 7 are mounted on the high-speed shafts and the transmission is well balanced 15. 180 Fig. 6. Two-ring-plates reducer without split path arrangement To obtain load equilibrium among the three parallel mechanisms, the references 16-18 came up with elastic ring and oil-film floating respectively. V Mechanical Efficiency The relative sliding between the teeth of internal meshing with small tooth difference is slight. The friction between the gear surfaces is not the main reason of power loss. It differs from other transmissions in this aspect. Liu 19,20 pointed out that there was almost no room left between the convex and concave gear surfaces. The heat given out due to lubricant-film being squeezed out is dominant factor affecting the mechanical efficiency. Liu proposed that solid lubricants be employed. His experiment demonstrated that the efficiency could reach 97.4% by use of solid lubrication, while the efficiency by use of liquid lubrication only 90.6%. Granted Lius point is acceptable, the efficiency of the reducer cannot be correctly evaluated by using the traditional formula of mechanical efficiency evaluation. Therefore the efficiency of the ring-plate reducer is lower than that of the 2K-H planetary reducer. When in the case of higher transmission power and continuous operation, a heat balance analysis is indispensable. VI Serial Design of Products The majority of factories in China, which produce this kind of reducers, adopt the recommended standard stipulated in 1995 by Chinese Ministry of Metallurgical Industry 21,22. Both the theoretical analysis and the practical experiences revealed that the actual load capacity is much greater than that given in the standard. Obviously, the original design did not take the effect of multi-teeth elastic meshing into account and was very conservative in load capacity evaluation. Now, having fully accomplished the theoretical analysis of the ring-plate transmission, the conditions for developing more advanced production standard are mature: 1. Kineto-elastostatic analysis method with consideration of deformation of ring-plate 4. 2. Load capacity evaluation method with consideration of the effect of multi-teeth elastic meshing 11. 3. Program package of CAD, which can automatically determine geometrical parameters of internal meshing with small difference of teeth 23. 4. Optimization method of the parameters, which can keep a compromise between the bearing service life and the gears strength 5. Having summed up all the above-mentioned work, the division of mechanical transmission of Tianjin University has already completed a serial design of products, including 400 models (20 sizes each with 20 speed ratio). The load capacity of the newly designed products is much higher than that in the old standard, some even higher than the hardened tooth surface reducers 5. VII Suggestions and Prospect This transmission has a promising prospect. In less demanding applications, many general gear and worm reducers, even two-stage planetary gear reducers should be replaced by it. In order to make the best of its advantages, following work should be carried out. 1. A reliability design method for bending strength of teeth should be established by taking manufacturing errors into account. 2. A completely elastodynamic analysis should be worked out to verify whether it can be employed in the case of higher speed (more than 1500rpm). References 1 Chen Z.Y. and others. Three-ring-plate Reducer (amplifier) Transmission. Chinese Patent No.: CN85106692A, 1985 2 Yang J. M., Zhang C., et al. Elasto-Dynamic Analysis of Three-ring-plate Reducers. Chinese Journal of Mechanical Engineering, 36 (10): 54-58, October 2000. 3 Zhang G.H., Han J. L., Long H. Stress analysis of driving ring board of three-ring type gear reducer. Chinese Journal of Mechanical Engineering, 30 (2): 58-63, April 1994. 4 Zhang Y. X. Elasto-static analysis of three-ring gear reducer in consideration of gear-coupler deformations， Thesis, Tianjin University, January 2005 5 Zhang J. 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M., et al. Load equilibrium of three-ring-plate reducer and finite element analysis. Mechanical Design, 16(9): 25-27, 1999. 19 Liu W. Q., Zhang Q. X., Lei T. J. Experimental investigation on using solid lubrication in a planetary reducer with planet internal gears. Chinese Journal of Mechanical Engineering, 31(3): 39-43, 1995. 20 Liu W. Q., Zhang Q. X., Lei T. J. Analysis and investigation on thermgenesis in a reducer with planet internal gears. Transactions of Tsinghua University， 35(2): 54-57, 1995. 21 Three-ring-plate Reducer. Standard stipulated by Chinese Ministry of Metallurgical Industry, YB/T079-1995, 1995. 22 RH two-ring-plate reducer. Standard stipulated by Chinese Ministry of Mechanical Industry, JB/T 10299 2001, 2001. 23 Yu Y. H. Computer-aided geometric design and parameters optimization of internal gearing with small tooth number difference. Thesis, Tianjin University， January 2005