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KHOA HỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 EVALUATE ERROR IN TRANSMISSION RATION OF ENVELOPE CHAIN AND INDEXING CHAIN IN SPIRAL-BEVEL-GEAR MANUFACTURING MACHINE GLEASON ZFTKK250X5 ĐÁNH GIÁ SAI SỐ TỶ SỐ TRUYỀN CỦA XÍCH BAO HÌNH VÀ XÍCH PHÂN ĐỘ CỦA MÁY GIA CÔNG BÁNH RĂNG CÔN RĂNG CONG GLEASON ZFTKK250X5 Nguyen Tho Son1,*, Nguyen Van Mui1, Le Thanh Son2 ABSTRACT The paper presents the method which evaluates the gear ratio error of envelope chain and indexing chain spiral-bevel-gear generator ZFTKK250X5 in the Gleason system. The elements in power transmission chains are gear transmitters, which are subject to kinetic errors, pitch errors, cumulative pitch errors, etc... Therefore, the errors of these chains will be the sum of the errors of the elements in the chains during machining, so when we process the Gleason gear system, its quality is directly affected by these errors. To evaluate these above errors, it is necessary to determine how many degrees of the gear pairs in the chains during the geometry and the gradation process, thereby calculating the errors of each pair of gears in the chain and the gear ratio of the elements. Finally, multiplying them together gives us the gear ratios of the chain when it comes to the errors. From there, the authors get the gear ratio error of the chains. Keywords: Envelope chain, indexing chain, transmission ratio, pitch error. TÓM TẮT Bài báo trình bày về xây dựng phương pháp đánh giá sai số tỷ số truyền của xích bao hình và xích phân độ của máy gia công bánh răng côn răng cong Gleason ZFTKK250X5. Các phần tử trong các chuỗi xích là các cặp bánh răng ăn khớp với nhau, các cặp bánh răng mắc phải những sai số động học, sai số bước răng, sai số tích lũy bước răng,… vì vậy sai số của những chuỗi xích này sẽ là tổ hợp các sai số của các phần tử ăn khớp trong quá trình gia công, nên khi gia công bánh răng Gleason chất lượng của nó bị ảnh hưởng trực tiếp từ các sai số này. Để đánh giá được sai số trên cần xác định xem các cặp bánh răng ăn khớp trong quá trình bao hình và phân độ quay được một góc bao nhiêu độ, từ đó tính được các sai số của từng cặp bánh răng trong các xích, tính các tỷ số truyền của các phần tử, nhân chúng lại với nhau ta được tỷ số truyền của xích khi kể đến các sai số. Từ đó ta thu được sai số tỷ số truyền giữa các xích. Từ khóa: Xích bao hình, xích phân độ, tỷ số truyền động, sai số bước. 1 Faculty of Mechanical Engineering, University of Economic and Technical Industries Hanoi University of Science and Technology * Email: ntson2012@uneti.edu.vn Ngày nhận bài: 10/5/2020 Ngày nhận bài sửa sau phản biện: 21/7/2020 Ngày chấp nhận đăng: 18/8/2020 2 48 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 56 - Số 4 (8/2020) 1. INTRODUCTION Spiral bevel gears have advantages of large coincidence degree, smooth transmission and high strength. Moreover, they are widely applied in fields of ships, aerospace, and defense technology, etc. Therefore, forming the tooth surface of the Spiral bevel gears is of great interest to many scientists and has solved many basic problems to be applied in production. Scientists have come up with a variety of research and problem-solving solutions such as the gear surface theory, the solutions to improve the working quality of Spiral bevel gear [2, 3]. In Viet Nam, there are not many companies producing this type of gear, so the machinery and equipment for processing this type of gear are not many. Therefore, the study of this type of machine is necessary, especially the final machine for this type of gear. For this study, the author studied the ZFTKK250X5 machine manufactured by Modul [1]. In machining ZFTKK250X5, there are two important power transmission chains: envelope chain and indexing chain. These two chains are crucial to the quality of the Spiral bevel gear of the Gleason system. Therefore, it is necessary to evaluate the transmission errors of these two chains, because when knowing how much the error is and depending on what factors, it is possible to find a method to minimize the error. Gear ratio error of envelope chain and indexing chain spiral-bevel-gear generator ZFTKK250X5 in the Gleason system directly affects the working quality of this gear pair, specifically the two biggest factors affecting the Gleason gear system pair was: i) When there is a gear ratio error of envelope chain, the gear ratio of the generating crown gear and the generated gear (gear workpiece) is no longer in the correct gear ratio, causing the tooth height profile and the spiral direction profile of the tooth is no longer correct, thus causing tooth profile error and affecting the bearing pattern of Gleason's gears. Website: https://tapchikhcn.haui.edu.vn SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 ii) When there is a gear ratio error of an indexing chain that makes indexing inaccurate when machining a tooth, this makes the gear workpiece carry tooth pitch error. Therefore, the determination of these errors is necessary to determine how much these errors are theoretical, then find a way to minimize these errors. 2. METHODS OF EVALUATING THE GEAR RATIO ERROR To evaluate the the gear ratio error, the authors followed these steps: Step 1: Draw the kinematic schematic of the machine Sample directly taken from the gear forming chain and intermittent chain on ZFTKK250x5. Since the documents came with the machine do not include schematic so the authors take samples directly from the machine. After disassembling, measuring, the authors archived the kinematic schematic of the machine as Figure 1. cycle) pass through the gears pairs the gear pairs replacing the intermittent drive pass through the worm drive a2 c 2 x (X) b 2 d2 1 360  → cycle through the 96 zp workpiece. Where a2, b2; c2, d2 are the gears pairs replacing the indexing chain corresponding to the number of tooth in a c 16 the manufactured gear zp. 2 x 2  b2 d2 zp Equation for the indexing chains: 1 cycle of hydraulic clutch L2 rack L2 (corresponding to 2 cycle of the gear pair Z17 and Z72) x 72 a2 c 2 1 360 x x x  24 b 2 d2 96 zp ii) Equation for the envelope chains 1 cycle of shaking stand → pass through the 120 worm drive (I) → 1 pass through the bevel 25 gears pair (II) → pass 24 through the gears replacing gear forming d b drive 1 x 1 (III) → pass c1 a1 through the bevel gears 24 pair (IV) → pass 25 through the bevel gears 29 (V) → pass pair 29 through the bevel gears 29 pair (VI) → pass 29 Figure 1. Kinematic Schematic of ZFTKK250X5 [1] Step 2: Write the equations of the chains in the machine i) Equation for the indexing chains Intermittent mechanism: After the hydraulic clutch L2 cut the transmission of the envelope chain by pushing the planetary gears away from the envelope chain, at the same time closed the pin for gear Z72 and Z17 to fit. The hydraulic clutch L2 has the rack to perform the intermittent mechanism. The chain is intermitted as followed: 1 cycle of hydraulic clutch L2 rack L2 → pass through the gear pairs Z17 and Z72 make the gears to rotate 720 (2 Website: https://tapchikhcn.haui.edu.vn 72 (IX) → pass through 24 through the bevel gears pair 25 (VII) → pass through the 20 71 (VIII) pass through the planetary gear 71 30 17 72 17 (IX) → pass through the gear pairs x x x pairs 17 72 17 30 a c replacing the intermittent drive 2 x 2 (X) → pass through b2 d2 bevel gears pair the worm drive z 1 → d cycle through the workpiece. 96 zp Vol. 56 - No. 4 (Aug 2020) ● Journal of SCIENCE & TECHNOLOGY 49 KHOA HỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 Equation for the envelope chain: (1 cycle of shaking stand) 120 25 d1 b1 24 29 25 71 30 17 72 17 a2 c2 1 zd x x x x x x x x x x x x x x  1 24 c1 a1 25 29 20 71 17 72 17 30 b2 d2 96 zp x (cycle of the workpiece) More specific dimensions are in the below table: Table 1. Specification of the gears in the chains No Specification 1 Worm wheel z120 2 Worm screw k = 1 (shaft I) m Z   5.6 120 4.4 20 3 Straight bevel gears Z25 (Shaft I) 5.6 1 4.4 20 2.5 25 0 20 4 Straight bevel gears Z24 (Shaft I) 2.5 24 0 20 5 Change gears of the envelope chain a1; b1 ; c 1 ; d1 6 Straight bevel gears Z24 (Shaft III) 2 0 0 Thus to determine the rotation error of the gear in the chain, one needs to: i) Look up the errors of the gears in [4, 5] ii) If the gears finished a whole number of cycle (1 cycle, 2 cycle,…) then the erros will be calculated by fi’, if not the errors would consist of the cumulative gear pitch error Fpr plus the kinematic error of gears fi’. 3. DETERMINE THE TRANSMISSION ERRORS OF THE CHAINS Consider the spiral bevel gear pairs that need to be manufactured have the specifications: Table 2. Specifications of the helical bevel gear pairs need to be manufactured [1, 2, 3] No Specification Symbol Pinion Gear 1 Spiral angle BET 35 35 Number of teeth Z 15 30 Average module Hand of Spiral mn Right = 1; left = 0 3.5 1 3.5 0 2.5 24 0 20 2 7 Straight bevel gears Z25 (Shaft IV) 8 Straight bevel gears Z29 (Shaft IV) 2.5 25 0 20 2.5 29 0 20 3 4 9 Straight bevel gears Z29 (Shaft V) 2.5 29 0 20 - Number 2 1 2 2 of teeth 2 of generating crown is 2 10 Straight bevel gears Z29 (Shaft V) 2.5 29 0 20 zd  z  z  15  30  33.54101966 11 Straight bevel gears Z29 (Shaft VI) 2.5 29 0 20 12 Straight bevel gears Z25 (Shaft VI) 13 Straight bevel gears Z24 (ShaftVII) 2.5 25 0 20 - Gear replacing the indexing chain when machining pinion z1: 2.5 20 0 20 14 Spur gear Z71 (Shaft VII) 15 Spur gear Z71 (Shaft VIII) 2 71 0 20 2 71 0 20 16 Spur gear Z30 (Shaft VIII) 2 30 0 20 17 Spur gear Z17 18 Spur gear Z72 2 17 0 20 72 0 20 30 0 20 0 20 1 4.57 20 a1 c1 25 25 x   b1 d1 z d 33.54101966 4.57 20  0.745356  0 20 2 19 Spur gear Z30 (Shaft IX) 2 20 Change gears of the indexing chain a2; 2 b2 ; c 2 ; d2 21 Worm screw k = 1 (Shaft X) 3.15 22 Worm wheel z96 3.15 23 Spur gear Z17 (indexing chain) in clutch L2 24 Rack in clutch L2 3 96 17 3 - 0 20 25 Spur gear Z72 (indexing chain) 3 72 0 20 26 Spur gear Z72 (indexing chain) 3 24 0 20 Where m: Module; z: Number of teeth; : Spiral angle; : Profile angle. Step 3: Determine the errors in rotation of the components in the kinematic chain There are a lot of errors in gears [4, 5], however the ones that affects the number of cycles of the gear transmission are: kinematic error of gears fi, cumulative gear pitch error Fpr,… In the scope of this study, the article only considers the kinematic errors fi’ and cumulative gear pitch error Fpr. 50 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 56 - Số 4 (8/2020) a2 c2 16 16 48 60 x    x b2 d2 z1 15 45 60 - Gear replacing the indexing chain when machining gear z2: a2 c 2 16 16 40 48 x    x b2 d2 z2 30 60 60 - Gear replacing the envelope chain when machining pinion and gear: 40 62 x  0.745193 64 52 Transmission errors of the envelope chain when calculating the alternative gear i  0.00016 Through the sampling, all the spur gears in the kinematic chain of the machine have the module of m = 2 and the precision degree of 6, the straight bevel gear m = 2.5, precision degree 7, the worm wheel on the shaking stand z = 120 and precision degree 7, the worm wheel on the workpiece shaft has the precision degree of 6. 3.1. Transmission errors of the indexing chain Calculating the transmission errors of the manufactured a c 48 60 gear z1 based on the replacing gears 2 x 2  x b2 d2 45 60 was shown in table 3, 4. Website: https://tapchikhcn.haui.edu.vn SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 Table 3. Calculating the transmission errors of the indexing chain in gear z1 No 1 2 3 4 5 6 7 8 9 10 Specs L2 z = 17 z = 72 z = 24 a2 = 48 b2 = 45 c2 = 60 d2 = 60 k=1 z = 96 n 2 2 6 6 6.4 6.4 6.4 6.4 0.06 zk 0 0 0 0 0 4.5 4.5 4.5 0.4 6.4 Fpr 80 25 45 25 32 32 45 45 63 63 fi 20 18 20 18 18 18 18 18 12 20 fp F Δφ nmax nmin 4.706 20 0.067 1.471 18 0.061 720.13 719.87 0.313 20 0.016 720.14 719.86 0.521 18 0.043 2160.04 2159.96 0.333 18 0.021 2160.06 2159.94 0.356 19.6 0.025 2304.02 2303.98 0.375 19.69 0.019 2304.04 2303.96 0.375 19.69 0.019 2304.02 2303.98 1.313 12.53 0.005 2304.02 2303.98 1.313 28.4 0.011 24.01 23.99 Table 4. Results of the transmission errors in the indexing chain z1 Specs Z72/Z24 a2/b2 c2/d2 1/Z96 Largest transmission ratio 3.0006 1.0667 1.00003 0.01042 Smallest transmission ratio 2.9993 1.0666 0.99997 0.01041 Theoretical Transmission 3.0000 1.0666 1.00000 0.01042 ratio Error in largest transmission ratio Error in smallest transmission ratio Error in transmission ratio Error in angle when machining gear z1 Error in normal surface when machining gear z1 (mm) iz1i 0.0334 0.0333 0.0333 2.564E-05 -2.56E-05 5.126E-05 0.0369052 20.640953 fp: Cumulative gear pitch error for one tooth: fp  Fpr 45 25 32 32 45 45 36 63 20 18 18 18 18 18 12 20 0.313 0.521 0.333 0.356 0.375 0.375 1.125 1.313 20.00 18.00 18.00 18.00 18.00 22.50 12.23 24.20 0.0159 0.0430 0.0258 0.0172 0.0215 0.0215 0.0049 0.0096 720.14 719.86 2160.04 2159.96 2160.07 2159.93 1440.02 1439.98 1440.04 1439.96 1152.02 1151.98 1152.03 1151.97 12.01 11.99 Table 6. Results of the transmission errors in the indexing chain z2 Specs iz2i Z72/Z24 a2/b2 c2/d2 1/Z96 Largest transmission ratio 3.0007 0.6667 0.8000 0.0104 0.0167 Smallest transmission ratio 2.9993 0.6666 0.8000 0.0104 0.0166 Theoretical Transmission ratio 3.0000 0.6667 0.8000 0.0104 0.0167 Error in largest transmission ratio 1.89E-05 Error in smallest transmission ratio -1.89E-05 Error in transmission ratio 3.78E-05 Error in angle when machining gear z2 0.027226 Error in normal surface when machining gear z2 (μm) 30.45528 iz2i: Transmission errors in the indexing chain z2 Transmission error in percent ipz2d (%)  3.78x105 x100%  0.23% 0.0167 Error in normal surface when machining gear z2: Fp  30 .46mm 3.2. Transmission error of the envelope chain Angle of shaking stand, we can caculate in the material… With the machine ZFTKK250X5 and ZFTKKR250X5 to ensure the width of the tooth B, the angle of shaking stand has to be choosen as d = 55. - Look up the table and do the similar calculation like the indexing chain, the transmission errors of the envelope chain based on the replacing gears a1 c1 25 25 40 62 x    0.745356  x  0.745193 . b1 d1 zd 33.54101966 64 52 When machining the bevel gear z1 with the replacing gear a2 c2 48 60 x  x b2 d2 45 60 5.126x105 x100%  0.154% 0.03333333 Error in normal surface when machining gear z1: Table 7. Calculating the transmission error of the envelope chain when machining gear z1 Fp  20 .64mm Calculating the transmission error when machining gear a c 40 48 z2 based on the replacing gears 2 x 2  x was b2 d2 60 60 shown in the table 5, 6. Table 5. Calculating the transmission errors of the indexing chain in gear z2 No Specs n zk Fpr fi fp F Δφ nmax nmin 1 L2 0 80 20 4.706 20.00 0.0674 2 z=17 2 0 25 18 1.471 18.00 0.0607 720.13 719.87 Website: https://tapchikhcn.haui.edu.vn z=72 2 0 z=24 6 0 a2=40 6 0 b2=60 4 0 c2=48 4 0 d2=60 3.2 12 k=1 3.2 0.2 z=96 0.033 3.2 z n: Rotation cycle (cycle) zk: Number of odd teeth: zk = [n - int(n)].z F: Sum of errors: if zk = 0 then F = fi else F = fi + zk.fp D: Pitch diameter Δφ = 2F/D: Error of rotation angle nmax : Maximum Rotation angle: nmax = n.360 + Δφ nmin : Minimum Rotation angle: nmin = n.360 - Δφ iz1i: Transmission errors in the indexing chain z1 Transmission error in percent ipz1d (%)  3 4 5 6 7 8 9 10 No 1 2 3 4 5 6 7 Specs z120 k=1 z25 z24 d1 = 52 c1 = 62 b1 = 64 F 82.78 15.78 70.00 40.25 26.75 19.53 19.53 Δφ 0.01 0.00 0.13 0.08 0.03 0.02 0.02 nmax 55.01 6600.00 6600.13 6875.08 6875.11 5766.15 5766.16 nmin 54.99 6600.00 6599.87 6874.92 6874.89 5766.11 5766.09 Vol. 56 - No. 4 (Aug 2020) ● Journal of SCIENCE & TECHNOLOGY 51 KHOA HỌC CÔNG NGHỆ 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 a1 = 40 z24 z25 z29 z29 z29 z29 z25 z20 z71 z71 z30 z = 17 z = 72 z = 17 z = 30 a2 = 48 b2 = 45 c2 = 60 d2 = 60 k=1 z = 96 42.14 44.03 40.87 42.93 42.93 42.93 42.93 40.87 46.19 84.59 84.59 30.13 24.74 21.43 24.74 18.07 18.04 18.85 18.89 18.89 13.05 63.05 0.06 0.08 0.07 0.07 0.07 0.07 0.07 0.07 0.11 0.07 0.07 0.06 0.08 0.02 0.08 0.03 0.66 0.79 0.59 0.59 0.17 0.03 P-ISSN 1859-3585 E-ISSN 2615-9619 9225.87 9225.95 8856.85 8856.92 8856.84 8856.91 8856.84 8856.92 11071.07 11071.14 11071.04 11071.09 19537.09 4612.92 19537.09 11071.00 11071.66 11809.82 11810.41 11809.62 11809.79 123.04 9225.75 9225.66 8856.70 8856.63 8856.71 8856.64 8856.71 8856.63 11070.86 11070.79 11070.90 11070.84 19536.92 4612.89 19536.92 11070.93 11070.27 11808.25 11807.65 11808.44 11808.27 122.99 Table 8. Results of the transmission error of the envelope chain when machining gear z1 Specs Z120/1 Z25/Z24 d1/c1 b1/a1 Z24/Z25Z29/Z29Z29/Z29 Largest 120.02880 1.04170 0.83873 1.60002 0.96002 1.00002 1.00002 transmission ratio Smallest 119.97122 1.04163 0.83869 1.59998 0.95998 0.99998 0.99998 transmission ratio Theoretical 120.00000 1.04167 0.83871 1.60000 0.96000 1.00000 1.00000 transmission ratio Z25/Z20 Z71/Z71 Z30/Z17 Z17/Z72 Z72/Z17 Z17/Z30 a2/b2 c2/d2 Z1/Z96 1.25003 1.00002 1.76473 0.23611 4.23533 0.56667 1.06680 1.00017 0.01042 1.24997 0.99998 1.76468 0.23611 4.23526 0.56666 1.06653 0.99983 0.01041 1.25000 1.00000 1.76471 0.23611 4.23529 0.56667 1.06667 1.00000 0.01042 Specs Largest transmission ratio Smallest transmission ratio Theoretical transmission ratio Error in largest transmission ratio Error in smallest transmission ratio Error in transmission ratio Error in transmission ratio in percentage envelope chain when machining gear z1 2.23885194 2.23426876 2.23655914 0.002292804 -0.00229038 0.004583187 0.204921353 Table 9. Results of the transmission error of the envelope chain when machining gear z2 Largest transmission ratio envelope chain when machining gear z2 1.119138766 Smallest transmission ratio 1.117421019 Theoretical transmission ratio 1.11827957 Error in largest transmission ratio 0.000859197 Error in smallest transmission ratio -0.00085855 Error in transmission ratio 0.001717748 Error in transmission ratio in percentage 0.153606273 Specs Error in transmission error of the envelope chain in z2 percentage ibh (%)  0.154% 4. CONCLUSION The results of the study show that: the transmission error of the envelope chain and indexing chain of the machine ZFTKK 250x5 of Modul appears when machining the Gleason spiral-bevel-gear. The results when calculating the kinematic chain when machining gear pairs z15/z30 yield: Specs z1 = 15 z2 = 30 Transmission error in the indexing chain (%) 0.154% 0.23% Error in tooth pitch calculated at the normal surface due to error in intermittent transmission error 20.64μm 30.46μm Transmission error in the envelope chain (%) 0.205% 0.154% With this result, to increase the fitting quality of the gear, one need to shorten the kinematic chain of the envelope chain and the indexing chain. REFERENCES [1]. Modul, 1966. Form-walzm berechnungsanleitung KFW fur kreibogenzahn - kegelrader (verjungte zhanhohe). Karl-Mark - Germany. [2]. Faydor L. Litvin. Development of Gear technology and Theory of Gearing. NASA RP 1406. [3]. Faydor L. Litvin, Alfonso Fuentes, Kenichi Hayasaka, 2006. Design, manufacture, stress analysis, andexperimental tests of low-no ise high endurance spiral bevel gears. Mechanism and Machine Theory. [4]. Ninh Duc Ton, Nguyen Thi Xuan Bay, 1999. Dung sai - Đo lường. Education Publishing House. [5]. Vietnam Standard TCVN 1067:1984, 1984. Truyền động bánh răng trụ Dung sai. Vietnam National Committee of Science and Technology. Error in transmission error of the envelope chain in z1 percentage ibh (%)  0.205% - Similarly, the error in the transmission ratio of the envelope chain when machining z2 (with the replacing a c 40 48 gear 2 x 2  x ) can be calculated as table 9. b2 d2 60 60 52 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 56 - Số 4 (8/2020) THÔNG TIN TÁC GIẢ Nguyễn Thọ Sơn1, Nguyễn Văn Mùi1, Lê Thanh Sơn2 1 Khoa Cơ khí, Trường Đại học Kinh tế - Kỹ thuật Công nghiệp 2 Trường Đại học Bách khoa Hà Nội Website: https://tapchikhcn.haui.edu.vn