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ORIGINAL ARTICLE Optimal mechanical spindle speeder gearbox design for high-speed machining D. R. Salgado and the turning pair is the link between the arm (member 3) and the planet. In the present work, the expression “simple planet” will be used for a planet constructed with a single gear, such as the planet of Fig. 2a,b, and “double planet” for one constructed with two gears, such as the planets of Fig. 2cf. A more detailed explanation of the structure of PGTs may be found in 9, 11. 2.1 Efficiency considerations It is possible to prove that the efficiency of the multiplier based on the four-member PGT is higher if it is designed with an input by the arm (member 3). This is the reason why all mechanical spindle speeders are designed as multiplier four-member PGTs with an input by the arm member. 2.2 Economic and operating considerations Of the solutions with a double-planet configuration (Fig. 2cf), that of Fig. 2d is more interesting from an economic point of view, since it offers the advantage of not using a ring gear. The reason for this is that spindle speeder gears must be hardened, tempered and ground to avoid high heating, and a ground ring gear is more expensive than a ground non-ring gear. Also, if the ring gear is not ground, heat buildup will occur in a shorter period of time, and this heating limits and reduces the input speed and torque. The constructional solution of Fig. 2a presents the advantage over the other solution constructed with simple abcdef Fig. 2 The six constructional solutions of the four-member PGT Fig. 1 a Members of a plane- tary gear train (PGT). b A mechanical spindle speeder Int J Adv Manuf Technol (2009) 40:637647639 planets (Fig. 2b) in that the ring gear is the fixed member. For this reason, the constructional solution of Fig. 2b is not used for mechanical spindle speeder design, since it increases the kinetic energy of the spindle speeder considerably. Following this same reasoning, the construc- tional solutions of Fig. 2e,f are not appropriate config- urations from the solutions constructed with double planets for mechanical spindle speeder design. 2.3 Planet member considerations In spindle speeder design, it is quite important to choose an optimal number of planets for the required power and speed ratio. The number of planet members (Np) can vary from two to three, four or even more, depending on the application for which it is designed. For example, the mechanical spindle speeder of Fig. 1a has three planet members (Np=3). This number must be as small as possible in order to reduce the weight and the kinetic energy of the transmission, while ensuring a good distribution of the load to each of the planet gears. Whichever the case, the planets must always be arranged concentrically around the PGTs principal axis to balance the mass distribution. In short, for mechanical spindle speeders, only the constructional solutions of Fig. 2a,c,d must be considered for an optimal spindle speeder design. In particular, these constructional solutions are the ones that are most often used by manufacturers. 3 Constraints on mechanical spindle speeder design In this section, the constraints for the mechanical spindle speeder design are described. They are grouped into three sets, according to the type of constraint. These are: Constraints involving gear size and geometry PGT meshing requirements Contact and bending stresses 3.1 Constraints involving gear size and geometry The first constraint is a practical limitation of the range for the acceptable face width b. This constraint is as follows: 9m ? b ? 14m1 where m is the module. The module indicates the tooth size and is the ratio of the pitch diameter to the number of teeth in the gear. For gears to mesh, their modules must be equal. Gear ISO standards and design methods are based on the module. All of the kinematic and dynamic parameters of the transmission depend on the values of the tooth ratios Znl, where Znlis the tooth ratio of the gear pair formed by the linking members n and l. In particular, Znlis defined as: Znl Zn Zl 2 For the definition of the tooth ratios to satisfy the Willis equations, Znlmust be positive if the gear is external (meshing geargear) and negative if it is internal (meshing ring geargear) 10, 11. For the train of Fig. 2a, one would have to take Z140 and Z240. In theory, the tooth ratios can take any value, but in practice, they are limited mainly for technical reasons because of the difficulty in assembling gears outside of a certain range of tooth ratios. In this work, the tooth ratio for the design of mechanical spindle speeders are quite close to the recommendations of Mller 12 and the American Gear Manufacturers Association (AGMA) norm 13, and are: 0:2 Znl 53 ?7 Znl ?2:24 with the constraint given by Eq. 3 being for external gears and that by Eq. 4 for internal gears. It is important to note that these constraints are valid for designs with different numbers of planets (Np). In respecting these values, one achieves mechanical spindle speeder designs that are smaller, lighter and cheaper. Another constraint that will be imposed on the design of spindle speeders with double planets is that the ratio of the diameters of the gears constituting a double planet is: 1 3 d4 d0 4 35 where d0 4is the diameter of the planet gear that meshes with member 2 and d4is the diameter of the planet gear that meshes with member 1 (see Fig. 2). In the constructional mechanical spindle speeders based on the PGT of Fig. 2c,d, the tooth ratios Z14and Z240are related to the radii of the gears constituting the planet. In particular, the following geometric relationship must be satisfied in the spindle speeder configuration of Fig. 2c: 1 2 d1 d4 1 2 d2? d0 4 ? 6 Expressing the above equation in terms of the module of the gears, it is straightforward to find that the ratio of the diameters of gears 4 and 4 conditions the value of Z14and Z240: This ratio is: d0 4 d4 Z14 1 Z240jj ? 1 7 640Int J Adv Manuf Technol (2009) 40:637647 Likewise, one obtains for the case of the configuration in Fig. 2d the expression: d0 4 d4 Z14 1 Z240 1 8 Lastly, one assumes a minimum pinion tooth number of: Zmin? 189 3.2 Planetary gear train meshing requirements The meshing requirements are given by the AGMA norm 13. The following constraint (Eq. 10) is for the design of Fig. 2a: Z2? Z1 Np an integer10 where Z1is the number of teeth on the sun gear (member 1) and Z2is the number of teeth on the ring gear (member 2). The sign in Eq. 10 depends on the turning direction of the sun and ring gear with the arm fixed. The negative sign must be used when the sun and ring gear turn in the same direction with the arm member fixed. Planetary systems with double planets must, either of which, factorise with the number of planets in the sense of Eq. 11 below (see AGMA norm 13): Z2P2? Z1P1 Np an integer11 where P1and P2are the numerator and denominator of the irreducible fraction equivalent to the fraction Z04 Z4 ; where Z0 4 is the number of teeth of the planet gear that meshes with member 2 and Z4is the number of teeth of the planet gear that meshes with member 1 (see Fig. 2): Z0 4 Z4 P1 P2 3.3 Contact and bending stresses The torques on each gear of the proposed spindle speeder designs were calculated taking power losses into account. This aspect allows one to really optimise the mechanical spindle speeder design, unlike the optimisation studies in which these losses are not considered 14, 15. The procedure for obtaining torques and the overall efficiency of the spindle speeder is that described by Castillo 11. For each of the gears of the spindle speeder configura- tion, the following constraints relative to the Hertz contact and bending stresses must be satisfied: sH sHP12 sF sFP13 For the calculation of the gears, the ISO norm was followed. The values of the stresses of Eqs. 12 and 13 are defined by this norm as: H ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi KA? KV? KH? KH p ? ZH? ZE? Z? Z ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi Ft b ? d ? u 1 u r 14 F KA? KV? KF? KF? Ft b ? m ? YF? YS? Y? Y15 The values of HPand FPare given by: sHP sHlim? ZN? ZL? ZR? ZV? ZW? ZX16 sFP sFlim? YST? YNT? YdrelT? YRrelT? YX17 It is important to emphasise that the tangential force Ft was obtained from the calculation of the torques taking the power losses into account. To include power losses in the overall efficiency calculation, we used the concept of ordinary efficiency 10, 11, which is what the efficiency of the gear pair would be if the arm linked to the planet were fixed. By means of this efficiency, one introduces into the overall efficiency calculation of the PGT the friction losses that take place in each gear pair. For this, we took a value of 0=0.98 for the ordinary efficiencies, i.e. 2% of the power passing through each gear pair is lost by friction between these gears. In studies that do not take this power loss into account, the value of the tangential forces is only approximate and may be quite different in the case of PGTs because of the possibility of power recirculation 10. Given the start-up characteristics of machine tools in general, we took an application factor of KA=1. The pressure angle is =20. The material chosen for the gears is a steel with Hlim 1;360 N?mm2and Flim 350 N?mm2: Lastly, the distribution of the loads to which each of the planet gears is subjected was determined using the distribution factors recommended in the AGMA 6123 A- 88 norm 13 as a function of the number of planets (Np). Int J Adv Manuf Technol (2009) 40:637647641 4 Objective functions and design variables Various works have presented methods for the optimisation of a conventional transmission 1423, but only a few studies have proposed optimisation techniques for the design of PGTs 20, 21. In addition, none of these studies on PGTs 24, 25 calculate exactly the torques to which each of the gears is subjected, since they do not consider the power losses in the different gear pairs of the PGT. Nevertheless, it is known that power losses in these transmissions may be considerably greater than in an ordinary gear train 10, 11, and, therefore, an optimal design must take this factor into account. Indeed, not considering power losses, as well as not ensuring an optimal mechanical spindle speeder design, impedes one from knowing its overall efficiency with certainty. In this section, we describe the objective functions and the design variables. The objective functions are the volume function and the kinetic energy function. It is important to bear in mind that these functions have different expressions, depending on the constructional solution adopted for the spindle speeder design. In particular, the volume function for the constructional solution with simple planets (Fig. 2a) is expressed as follows: Va p 4 b14d1 2d4218 where Varepresents the total volume of the gears. The same objective function for the constructional solution of Fig. 2c takes another form, and is expressed as follows: Vc p 4 b14 b240 ? max d1 2d4; d2 2d40219 and for the constructional solution of Fig. 2d, it is expressed as: Vd p 4 b14 b240 ? max d1 2d4; d2220 where b14is the face width of gears 1 and 4, and b240is the face width of gears 2 and 4. The kinetic energy function is also different for the constructional solutions with simple and double planets, as can easily be deduced. The function for the constructional solution of Fig. 2a is expressed in the following form: KEa 1 2 I1w2 1 Np 1 2 m4v2 4 1 2 I4w2 4 ? 21 where I4, w4and m4are the moment of inertia, the rotational speed and the mass of the planet gear, respectively, and v4is the translation speed of the centre of the planet gear. In the above expression, I1is the moment of inertia of the sun member and Npis the number of planet gears. Table 1 Optimal designs of spindle speeders based on the constructional solution of Fig. 2a Spindle designPin(kW) n (rpm) m (mm)b (mm) (mm)Vol. (mm3) KE210?6 mm 5 s2 ? KE310?6 mm 5 s2 ? T(mm) 1:3.5, Z1=24 Z4=18 Z2=6010 kW1.2514.841469,850860,9051,057,74177.30 8,000 rpm1.2511.912564,285908,1521,115,79182.75 16 kW1.2517.031883,4481,672,5292,054,93378.86 10,000 rpm1.2515.232582,1421,812,9702,227,48582.75 1:4, Z1=18 Z4=18 Z2=5420 kW2.530.7515471,7181,754,2732,280,555139.76 3,000 rpm2.525.3225441,2781,864,0762,423,300148.96 30 kW2.526.2216406,1004,235,9375,506,718140.44 5,000 rpm2.523.6221387,8914,289,5045,576,355144.60 45 kW2.532.40463,76911,443,06014,875,978135.00 8,000 rpm2.522.7118359,4119,804,36112,745,669141.95 1:5, Z1=18 Z4=27 Z2=721.7 kW0.66.2609,181166,090230,17343.20 24,000 rpm0.65.4588,150104,760145,18143.62 2 kW0.79.751721,27069,27195,98852.70 10,000 rpm0.78.482520,59874,688103,50655.61 3.5 kW0.79.651520,640213,482295,85152.18 18,000 rpm0.77.772719,545237,579329,24456.56 5 kW0.911.681440,934361,818501,42066.78 13,000 rpm0.99.652538,754392,580544,05171.50 6.4 kW111.921552,045573,010794,09574.54 13,000 rpm19.932549,223615,591853,10679.44 7 kW113.921762,011593,508822,50375.30 12,000 rpm111.212858,557657,453911,12081.54 8 kW1.2512.001187,770865,0871,198,86591.68 10,000 rpm1.2511.252081,077872,0341,208,49295.78 642Int J Adv Manuf Technol (2009) 40:637647 The same objective function for the constructional solutions of Fig. 2c,d is expressed as follows: KEcd 1 2 I1w2 1 Np 2 m4 m40v2 4 Np 2 I4 I40w2 4 22 In Eqs. 21 and 22, the energy of the arm has been neglected because this member can be designed in different and variable forms, and because it is considerably less than that of the planetary system. The design variables are of the constructional solution chosen from those of Fig. 2a,c,d, the number of planet gears (Np), the module of the gears (mi), the number of teeth on each gear (Zi), the face width (bi) and the helix angle (i). When these design parameters are determined by minimising the above objective functions, the PGT is perfectly defined. Table 2 Optimal designs of spindle speeders based on the constructional solution of Fig. 2a (cont.) Spindle designPin(kW) n (rpm)m (mm)b (mm)Vol. (mm3)KE210?6 mm 5 s2 ? KE310?6 mm 5 s2 ? T(mm) 1:6, Z1=18 Z4=36 Z2=902.5 kW0.76.302022,247248,709355,29867.04 18,000 rpm0.68.502222,653191,109273,01358.24 5.3 kW0.910.571558,355708,7681,012,52683.86 15,000 rpm0.98.762554,946758,0541,082,93489.37 7 kW1.512.2125212,852667,212953,160148.95 5,000 rpm1.2517.6727221,326498,477712,111126.26 7 kW1.2512.1115129,047798,7861,141,124116.47 9,000 rpm1.2511.2520126,682828,5431,183,633119.72 9.3 kW1.2512.2914129,7601,928,2152,754,593115.94 12,000 rpm1.2511.2519126,6822,007,1002,867,285119.72 10 kW1.2515.7714166,4841,718,6982,455,284115.94 10,000 rpm1.2511.4330151,5081,963,4092,804,871129.90 1:7, Z1=18 Z4=45 Z2=1083 kW113.7019140,453251,865365,659114.22 5,000 rpm110.6030129,475276,759401,801124.70 5 kW0.811.112376,852835,9801,213,68293.86 15,000 rpm0.89.313072,790894,5461,298,70999.76 7 kW0.810.831467,4661,834,0272,662 65389.05 25,000 rpm0.87.653059,7922,040,3602,962,21899.76 1:8, Z1=18 Z4=54 Z2=1263 kW0.68.241439,271615,788902,41577.91 25,000 rpm0.66.672536,468655,435960,51683.42 4 kW0.68.061840,0121,069,9581,567,98579.49 32,000 rpm0.66.912537,7701,112,2171,629,91483.42 1:10, Z1=18 Z4=72 Z2=1623 kW0.65.711947,4031,339,6931,982,746102.80 32,000 rpm0.65.432146,2791,341,9151,986,034104.12 4 kW0.66.251851,2382,236,3353,309,776102.20 40,000 rpm0.65.482549,5202,380,0453,522,466107.25 Table 3 Optimal designs of spindle speeders based on the constructional solution of Fig. 2c Spindle design 14=240m14=m240(mm)b14=b240(mm)d1=d4(mm)d1=d40 (mm) Vol. (mm3) KE210?6 mm 5 s2 ? T (mm) 1:5, 5 kW, 13,000 rpm240.911.0819.7564.6478,475668,15369.13 80.89.9824.6920.20 Z1=20Z2=80Z4=25 Z40 25 1:6, 5.3 kW, 15,000 rpm260.910.1218.0272.1789488865,89678.10 40.88.5630.0424.05 Z1=18Z2=90Z4=30 Z40 30 1:8, 3 kW, 25,000 rpm40.67.3612.0365.5358,743719,21172.17 160.97.0030.0723.40 Z1=20Z2=70Z4=50 Z40 25 1:10, 4 kW, 40,000 rpm130.66.1412.3059.5849,4221,271,83373.78 250.65.4230.7416.55 Z1=20Z2=90Z4=50 Z40 25 Int J Adv Manuf Technol (2009) 40:637647643 5 Results and discussion The optimisation problem of mechanical spindle speeders described in this paper was applied to a set of different designs of spindle speeders, i.e. different speed ratios and powers covering the entire marketed range. Tables 1 and 2 summarise all of the cases studied for the design based on the constructional solution of Fig. 2a and show the optimal designs. In these tables, the first and second columns list the speed ratio, the input power and the maximum output speed for each design. The first column also indicates the tooth number of each member for the minimum volume and Table 4 Optimal designs of spindle speeders based on the constructional solution of Fig. 2d Spindle design 14=240m14=m240 (mm) b14=b240 (wmm) d1=d4 (mm) d1=d40 (mm) Vol. (mm3) KE210?6 mm 5 s2 ? T (mm) 1:5, 5 kW, 13,000 rpm171.12510.1521.1747.66182,9474,964,871105.85 24.50.810.6442.3415.88 Z1=18Z2=54Z4=36 Z40 18 1:6, 5.3 kW, 15,000 rpm28.31.12510.1522.9953.63221,4368,157,084114.97 200.811.1845.9915.32 Z1=18Z2=63Z4=36 Z40 18 1:8, 3 kW, 25,000 rpm300.67.3512.4739.31104,9204,136,54595.59 170.77.2741.5614.55 Z1=18Z2=54Z4=60 Z40 20 1:10, 4 kW, 40,000 rpm260.66.6212.0139.9891,8896,682,16692.11 80.67.1740.0512.11 Z1=18Z2=66Z4=60 Z40 20 1:51:61:71:81:91:10 2 4 6 8 10 12 14 Speed ratio Ratio between the volume and kinetic energy of the spindle speeder gearbox based on the constructional solucion of Fig. 2(c) and Fig. 2(d), and the volume and kinetic energy of that based on the constructional solution of Fig. 2(a) Vc/Va KEc/KEa Vd/Va KEd/KEa volume kinetic energy Fig. 3 Ratio between the volume and kin