Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 12, N o 1, 2014, pp. 51 - 60 HIGH EFFICIENCY GEARS  UDC (621.8) Florian Ion Petrescu, Relly Victoria Petrescu Polytechnic University of Bucharest, Romania Abstract: The paper presents an original method for determining gear efficiency, gearing forces, velocities and powers. It analyzes the way in which certain parameters affect gear efficiency. Furthermore, an original method for determining geared transmissions efficiency as a function of the contact ratio is concisely presented. With the presented relations, one can make a dynamic synthesis of geared transmissions with the aim of increasing gearing mechanisms efficiency. Key Words: Gear, Geared Transmission, Contact Ratio, Dynamic Synthesis, Gear Efficiency 1. INTRODUCTION Today gears are present in all possible fields. Their key advantage lies in their providing for a very high working efficiency. Additionally, gears can transmit large loads [3, 13]. Regardless of their size, gears must be synthesized carefully considering the specific conditions. This paper intends to present the main conditions that must be met for a correct synthesis of the gear [8-11]. The pinnacle of using sprocket mechanisms is to be sought in ancient Egypt at least some thousand years before Christ. For the first time in human history, transmission wheels "spurred" to irrigate crops were used and so were worm gears for cotton processing [1-2]. In the year of 230 B.C., in the city of Alexandria in Egypt, multi-lever wheels with gear rack were used. Such gears were constructed and used from primeval times, above all for lifting heavy anchors of vessels and for winding catapult arms on the battlefields. They were later introduced in the vehicles with wind and water in order to reduce or enlarge the pump originating from windmills or water (see Fig. 1). The Antikythera Mechanism is the name given to an astronomical calculating device, measuring about 32 by 16 by 10 cm, which was discovered in 1900 in a sunken ship just off the coast of Antikythera, an island between Crete and the Greek mainland. Several kinds of evidence point unquestionably to around 80 B.C. as the date of the shipwreck. The device, made of bronze gears fitted in a wooden case, was crushed in the wreck, and parts of the faces Received March 02, 2014 / Accepted March 31, 2014  Corresponding author: Florian Ion Petrescu Bucharest Polytechnic University, Faculty of Transport, Theory of Mechanisms and Robots, Bucharest, Romania E-mail: petrescuflorian@yahoo.com Original scientific paper 52 F.I. PETRESCU, R.V. PETRESCU were lost, "the rest then being coated with a hard calcareous deposit at the same time as the metal corroded away to a thin core coated with hard metallic salts preserving much of the former shape of the bronze" during almost 2000 years of its underwater existence (Fig. 2). Fig. 1 Transmissions wheeled "spurred" to irrigate crops and worm gears used for cotton processing Fig. 2 The "Antikythera" mechanism – an astronomical calculating device Modern adventure began with the spur gear wheel spurred created by Leonardo da Vinci, in the 15 th century. He founded the new kinematics and dynamics stating inter alia the principle of superposition of independent movements (Fig. 3). Fig. 3 The spur gear wheel spurred created by Leonardo da Vinci, 15 th century High Efficiency Gears 53 Benz had engine with transmissions sprocket gearing and gear chain (patented in 1882, Fig. 4), [4-6] but the first gear patent (the drawings of the first gear transmission patented) and gearing wheels with chain were made in 1870 by the British Starley & Hillman [12]. Fig. 4 The Benz patent It is in 1912, in Cleveland (USA), that the production of industrial specialized wheels and gears (cylindrical, worm, conical, with straight teeth, inclined or curved; see Fig. 5) started. Fig. 5 In Cleveland, the production of industrial specialized wheels started in 1912 54 F.I. PETRESCU, R.V. PETRESCU Today, the gears are present everywhere in the world of mechanics, such as the car industry, electronics and electro-technique equipments, power industry, etc. (Fig. 6). Fig. 6 Gear today 2. DETERMINING GEAR EFFICIENCY AS A FUNCTION OF THE CONTACT RATIO One calculates the efficiency of a geared transmission, having in mind the fact that, at one moment, there are several couples of teeth in contact, and not just one [3]. Hence, the initial model incorporates four pairs of teeth in contact (4 couples) concomitantly [7-11]. The first couple of teeth in contact has the contact point i, defined by ray ri1, and pressure angle i1. The forces which act at this point are: motor force Fmi, perpendicular to position vector ri1 at i and the force transmitted from wheel 1 to wheel 2 through point i, Fti, parallel to the path of action and with the direction from wheel 1 to wheel 2. The transmitted force is practically a projection of the motor force onto the path of action. The defined velocities are similar to the forces (having in mind the original kinematics, or the precise kinematics adopted). The same parameters are defined for the other three points of contact – j, k and l (Fig. 7). The quantities of interest are: rb1 – the base radius of drive wheel 1; 1 – the circular velocity of wheel 1; z1 – the number of teeth of wheel 1;  – the pressure angle, so that 1 is the pressure angle for wheel 1 and 0 is the normal pressure angle on the pitch circle. High Efficiency Gears 55 Fig. 7 Four pairs of teeth in contact concomitantly As a starting point, we write the following relations between the velocities as [8-11]: 1 1 1 1 1 1 1 1 1 1 1 1 cos cos cos cos cos cos cos cos i mi i i i b j mj j j j b k mk k k k b l ml l l l b v v r r v v r r v v r r v v r r                                                 (1) From Eqs. (1), one obtains the equality of the tangential velocities (Eq. (2)), and, furthermore, the motor velocities are explicitly obtained (Eq. (3)): 1 1i j k l b v v v v r          (2) 1 1 1 1 1 1 1 1; ; ; cos cos cos cos b b b b mi mj mk ml i j k l r r r r v v v v                 (3) The forces transmitted concomitantly at the four points must be equal [8-11]: i j k l F F F F F          (4) The motor forces are given as: 56 F.I. PETRESCU, R.V. PETRESCU ; ; ; cos cos cos cos mi mj mk ml i j k l F F F F F F F F            (5) The momentary efficiency can be written in the following form: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 cos cos cos cos 4 1 1 1 1 cos cos cos cos 4 4 i i j j k k l lu i c m mi mi mj mj mk mk ml ml b b b b b i j k l i j k l i F v F v F v F vP P P P F v F v F v F v F r F r F r F r F r tg tg                                                                        2 2 j k l tg tg   (6) Eqs. (7) and (8) are auxiliary relations [7-11]: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ; ; ; ( ); 2 2 ( ); 2 2 2 2 ( ); b i b j b k b l b j i b j i b k i b k i b l i K i r tg K j r tg K k r tg K l r tg K j K i r tg tg K j K i r tg tg z z K k K i r tg tg K k K i r tg tg z z K l K i r tg tg K l K i                                                          1 1 1 2 2 3 3 b l i r tg tg z z                              (7) 111 2 3; 2 2; 2 z tgtg z tgtg z tgtg ilikij       (8) One keeps Eq. (8), with the sign plus (+) for the gearing where drive wheel 1 has external teeth (regardless if it is external or internal gearing), and with the sign (-) for the gearing where drive wheel 1 has internal teeth (the drive wheel has a ring form only for the internal gearing). The relation of momentary efficiency (Eq. (6)) uses auxiliary Eq. (8) and takes the form of Eq. (9). In Eq. (9), one starts with Eq. (6) where four pairs are in contact concomitantly, but then one generalizes the expression by replacing number 4 (four pairs) by variable E, which represents the whole number of the contact ratio +1, and after restricting the sum expressions, variable E is replaced by contact ratio 12, as well. High Efficiency Gears 57 The mechanical efficiency offers more advantages than the momentary efficiency, and will be calculated approximately, by replacing pressure angle α1 in Eq. (9) with normal pressure angle α0, so that Eq. (10) is obtained, where 12 represents the contact ratio of the gearing, and it will be calculated by Eq. (11) for the external gearing, and by Eq. (12) for the internal gearing: 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 11 2 2 2 2 1 111 4 4 4 2 2 2 4 ( ) ( 2 ) ( 3 ) 4 4 2 4 4 (0 1 2 3 ) 2 (0 1 2 3) 1 4 2 1 ( 1) 2 ( 1) 1 1 i i j k l i i i i i i E E i i i i tg tg tg tg tg tg tg tg z z z tg tg zz tg i tg i E zE z tg                                                                        2 2 1 1 2 11 2 2 1 1 2 11 2 2 1 1 12 12 122 11 44 ( 1) (2 1) ( 1) 6 2 1 2 ( 1)2 ( 1) (2 1) 1 3 1 22 1 ( 1) (2 1) ( 1) 3 i tgE E E E E E zE z tg EE E tg zz tg tg zz                                                                              (9) 2 2 0 0 12 12 122 11 1 22 1 ( 1) (2 1) ( 1) 3 m tg tg zz                     (10) 2 2 2 2 1 0 1 2 0 2 1 2 0. . 12 0 sin 4 4 sin 4 4 ( ) sin 2 cos a e z z z z z z                     (11) 2 2 2 2 0 0 0. . 12 0 sin 4 4 sin 4 4 ( ) sin 2 cos e e i i i ea i z z z z z z                     (12) The results of the performed calculations for various gear transmission parameters are summarized in Table 1 [8-11]. 58 F.I. PETRESCU, R.V. PETRESCU Table 1 Gear efficiency for different sets of gear transmission parameters The summarized results z1 0 [°] z2 12 ae 12 ae 21 ae 12 ai 12 ai 21 ai 42 20 126 1.79 0.844 0.871 1.92 0.838 0.895 46 19 138 1.87 0.856 0.882 2.00 0.850 0.905 52 18 156 1.96 0.869 0.893 2.09 0.864 0.915 58 17 174 2.06 0.880 0.904 2.20 0.876 0.925 65 16 195 2.17 0.892 0.914 2.32 0.887 0.933 74 15 222 2.30 0.903 0.923 2.46 0.899 0.942 85 14 255 2.44 0.914 0.933 2.62 0.910 0.949 98 13 294 2.62 0.924 0.941 2.81 0.920 0.956 115 12 345 2.82 0.934 0.949 3.02 0.931 0.963 137 11 411 3.06 0.943 0.957 3.28 0.941 0.969 165 10 495 3.35 0.952 0.964 3.59 0.950 0.974 204 9 510 3.68 0.960 0.970 4.02 0.958 0.980 257 8 514 4.09 0.968 0.975 4.57 0.966 0.985 336 7 672 4.66 0.975 0.980 5.21 0.973 0.989 457 6 914 5.42 0.981 0.985 6.06 0.980 0.992 657 5 1314 6.49 0.986 0.989 7.26 0.986 0.994 3. EFFICIENCY OF HELICAL GEARS AS A FUNCTION OF THE CONTACT RATIO In praxis, helical gears are used very often. For helical gears, the calculations show a decrease in yield with increasing tooth inclination angle (β). For angles not exceeding 25°, the efficiency of gears is rather good. However, when the inclination angle exceeds 25°, the gears will suffer a significant drop in yield. New calculation relationships can be given in Eqs. (13-15): 2 2 1 2 2 2 2 4 2 1 0 0 1 cos 2 ( cos ) cos ( 1) (2 1) 2 cos ( 1) 3 m z z tg tg z                             (13)   2 . . 2 3 1 0 1 2 3 2 0 2 1 2 0 1 [( 2 cos ) ] 4 cos ( cos ) 2 [( 2 cos ) ] 4 cos ( cos ) ( ) a e tg z tg z z tg z z z tg                                    (14)   2 . . 2 3 0 2 3 0 0 1 [( 2 cos ) ] 4 cos ( cos ) 2 [( 2 cos ) ] 4 cos ( cos ) ( ) a i e e i i e i tg z tg z z tg z z z tg                                    (15) 4. VALIDATION All the presented relationships have been validated by using the "Inventor" software package; a very good compliance is confirmed. Several examples have been computed for both external and internal gearing. The corresponding gear pairs have been drawn automatically by the "Inventor". Figs. 8 and 9 depict the gear pairs together with the main High Efficiency Gears 59 parameters of the considered gears for external and internal gearing, respectively. For angle 0 ranging between 10° and 20°, the contact line is perfect. For values greater than 25°, the software confirms that such a design is no longer safe, and for 0 taking values lower than 10°, the software has no details necessary for verification because it relies on the experimental values that no longer exist for such small values of angle 0. Fig. 8 Validated examples of external gearing Fig. 9 Validated examples of internal gearing 60 F.I. PETRESCU, R.V. PETRESCU 5. CONCLUSIONS The best efficiency is obtained with the internal gearing when drive wheel 1 is a ring. The minimum efficiency will be obtained when drive wheel 1 of the internal gearing has external teeth. For the external gearing, the best efficiency is obtained when the bigger wheel is the drive wheel. With decreasing normal angle 0, the contact ratio increases and efficiency increases as well. Efficiency increases too, when the number of teeth of drive wheel 1 (z1) increases. REFERENCES 1. Lei, X. M., Ge, Y. Z., Zhang, Y. C., Liu, P., 2011, Design and Analysis for High-Speed Gear Coupling, Applied Mechanics and Materials, 86, pp. 658-661. 2. Lin C., Hou, Y., J., Zeng, Q. L., Gong, H., Nie. L., Qiu, H., 2011, The Design and Experiment of Oval Bevel Gear, Applied Mechanics and Materials, 86, pp. 297-300. 3. Maros, D., 1958, Cinematica roţilor dinţate. Editura Tehnică, Bucureşti. 4. Matos, L., Marinho, B., 2011, A Comparison of the Delay Spread Obtained with Different Power Delay Profiles De-Noising Techniques, ENGEVISTA, 13(2), pp. 129-133. 5. Nogueira, O. C., Real, M. V., 2011, Estudo Comparativo De Motores Diesel Maritimos Atraves Da Analise De Lubrificantes Usados e Engehharia De Confiabilidade, ENGEVISTA, 13(3), pp. 244-254. 6. Oliveira, M. M. P., Lima, F. R., 2003, Analogias Entre o Desenho Instrumental e o Dedenho Computacional, ENGEVISTA, 5(9). 7. Pelecudi, C. H. R.., Maros, D., Merticaru, V., Pandrea, N., Simionescu, I., 1985, Mecanisme, Editura Didactică şi Pedagogică, Bucureşti. 8. Petrescu, V., Petrescu, I., 2002, Randamentul cuplei superioare de la angrenajele cu roţi dinţate cu axe fixe, Proceedings of the 7th National Symposium PRASIC, Braşov, vol. I, pp. 333-338. 9. Petrescu, R., Petrescu, F., 2003., The gear synthesis with the best efficiency, Proceedings of ESFA’03, Bucharest, vol. 2, pp. 63-70. 10. Petrescu, R. V., Petrescu, F. I., Popescu, N., 2007, Determining Gear Efficiency, Gear Solutions, pp. 19-28. 11. Petrescu, F.I., 2012, Teoria mecanismelor – Curs si aplicatii (editia a doua), Create Space publisher, USA, p. 284. 12. Rey G. G., 2013, Influencia de la lubricacion en la eficiencia de engranajes de tornillo sinfin , Ingineria Mecanica, 16(1), pp. 13-21. 13. Stoica, I. A., 1977, Interferenţa roţilor dinţate, Editura DACIA, Cluj-Napoca. ZUPČANICI VISOKE EFIKASNOSTI Rad predstavlja originalnu metodu za određivanje efikasnosti zupčastih prenosnika, kao i sila, brzina i snaga u prenosniku. Analizira se način na koji određeni parametri utiču na efikasnost prenosnika. Takođe, ukratko je predstavljena originalna metoda za određivanje efikanosti zupčastih prenosnika u funkciji stepena sprezanja. Pomoću predstavljenih relacija moguće je sprovesti dinamičku sintezu zupčastih prenosnika u cilju postizanja veće efikasnosti mehanizama. Ključne reči: zupčanik, zupčasti prenos, stepen sprezanja, dinamička sinteza, efikasnost zupčanika