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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. 2c–f. 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. 2c–f), 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 a bcdef 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) 40:637–647 639 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 (N p ) 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 (N p =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 PGT’s 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 C20 b C20 14m e1T 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 Z nl , where Z nl is the tooth ratio of the gear pair formed by the linking members n and l. In particular, Z nl is defined as: Z nl ? Z n Z l e2T For the definition of the tooth ratios to satisfy the Willis equations, Z nl must be positive if the gear is external (meshing gear–gear) and negative if it is internal (meshing ring gear–gear) [10, 11]. For the train of Fig. 2a, one would have to take Z 14 >0 and Z 24 <0. 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 Müller [12] and the American Gear Manufacturers Association (AGMA) norm [13], and are: 0:2 < Z nl < 5 e3T C07 < Z nl < C02:2 e4T 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 (N p ). 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 < d 4 d 0 4 < 3 e5T where d 0 4 is the diameter of the planet gear that meshes with member 2 and d 4 is 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 Z 14 and Z 24 0 are 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 d 1 t d 4 eT? 1 2 d 2 C0 d 0 4 C0C1 e6T 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 Z 14 and Z 24 0: This ratio is: d 0 4 d 4 ? Z 14 t 1 Z 24 0jjC0 1 e7T 640 Int J Adv Manuf Technol (2009) 40:637–647 Likewise, one obtains for the case of the configuration in Fig. 2d the expression: d 0 4 d 4 ? Z 14 t 1 Z 24 0 t 1 e8T Lastly, one assumes a minimum pinion tooth number of: Z min C21 18 e9T 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: Z 2 C6 Z 1 N p ? an integer e10T where Z 1 is the number of teeth on the sun gear (member 1) and Z 2 is 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]): Z 2 P 2 C6 Z 1 P 1 N p ? an integer e11T where P 1 and P 2 are the numerator and denominator of the irreducible fraction equivalent to the fraction Z 0 4 Z 4 ; where Z 0 4 is the number of teeth of the planet gear that meshes with member 2 and Z 4 is the number of teeth of the planet gear that meshes with member 1 (see Fig. 2): Z 0 4 Z 4 ? P 1 P 2 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: s H < s HP e12T s F < s FP e13T 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: σ H ? ?????????????????????????????????????? K A C1 K V C1 K Hβ C1 K Hα p C1 Z H C1 Z E C1 Z " C1 Z β ???????????????????????? F t b C1 d C1 u t 1 u r e14T σ F ? K A C1 K V C1 K Fβ C1 K Fα C1 F t b C1 m C1 Y Fα C1 Y Sα C1 Y " C1 Y β e15T The values of σ HP and σ FP are given by: s HP ? s Hlim C1 Z N C1 Z L C1 Z R C1 Z V C1 Z W C1 Z X e16T s FP ? s Flim C1 Y ST C1 Y NT C1 Y drelT C1 Y RrelT C1 Y X e17T It is important to emphasise that the tangential force F t 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 η 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 K A =1. The pressure angle is α=20°. The material chosen for the gears is a steel with σ Hlim ? 1;360 N C14 mm 2 and σ Flim ? 350 N C14 mm 2 : 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 (N p ). Int J Adv Manuf Technol (2009) 40:637–647 641 4 Objective functions and design variables Various works have presented methods for the optimisation of a conventional transmission [14–23], 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: V a ? p 4 b 14 d 1 t 2d 4 eT 2 e18T where V a represents 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: V c ? p 4 b 14 t b 24 0eTC1max d 1 t 2d 4 ; d 2 t 2d 4 0eT 2 e19T and for the constructional solution of Fig. 2d, it is expressed as: V d ? p 4 b 14 t b 24 0eTC1max d 1 t 2d 4 ; d 2 eT 2 e20T where b 14 is the face width of gears 1 and 4, and b 24 0 is 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: KE a ? 1 2 I 1 w 2 1 t N p 1 2 m 4 v 2 4 t 1 2 I 4 w 2 4 C18C19 e21T where I 4 , w 4 and m 4 are the moment of inertia, the rotational speed and the mass of the planet gear, respectively, and v 4 is the translation speed of the centre of the planet gear. In the above expression, I 1 is the moment of inertia of the sun member and N p is the number of planet gears. Table 1 Optimal designs of spindle speeders based on the constructional solution of Fig. 2a Spindle design P in (kW) n (rpm) m (mm) b (mm) β (mm) Vol. (mm 3 ) KE 2 10 C06 mm 5 s 2 C16C17 KE 3 10 C06 mm 5 s 2 C16C17 φ T (mm) 1:3.5, Z 1 =24 Z 4 =18 Z 2 =60 10 kW 1.25 14.84 14 69,850 860,905 1,057,741 77.30 8,000 rpm 1.25 11.91 25 64,285 908,152 1,115,791 82.75 16 kW 1.25 17.03 18 83,448 1,672,529 2,054,933 78.86 10,000 rpm 1.25 15.23 25 82,142 1,812,970 2,227,485 82.75 1:4, Z 1 =18 Z 4 =18 Z 2 =54 20 kW 2.5 30.75 15 471,718 1,754,273 2,280,555 139.76 3,000 rpm 2.5 25.32 25 441,278 1,864,076 2,423,300 148.96 30 kW 2.5 26.22 16 406,100 4,235,937 5,506,718 140.44 5,000 rpm 2.5 23.62 21 387,891 4,289,504 5,576,355 144.60 45 kW 2.5 32.4 0 463,769 11,443,060 14,875,978 135.00 8,000 rpm 2.5 22.71 18 359,411 9,804,361 12,745,669 141.95 1:5, Z 1 =18 Z 4 =27 Z 2 =72 1.7 kW 0.6 6.26 0 9,181 166,090 230,173 43.20 24,000 rpm 0.6 5.45 8 8,150 104,760 145,181 43.62 2 kW 0.7 9.75 17 21,270 69,271 95,988 52.70 10,000 rpm 0.7 8.48 25 20,598 74,688 103,506 55.61 3.5 kW 0.7 9.65 15 20,640 213,482 295,851 52.18 18,000 rpm 0.7 7.77 27 19,545 237,579 329,244 56.56 5 kW 0.9 11.68 14 40,934 361,818 501,420 66.78 13,000 rpm 0.9 9.65 25 38,754 392,580 544,051 71.50 6.4 kW 1 11.92 15 52,045 573,010 794,095 74.54 13,000 rpm 1 9.93 25 49,223 615,591 853,106 79.44 7 kW 1 13.92 17 62,011 593,508 822,503 75.30 12,000 rpm 1 11.21 28 58,557 657,453 911,120 81.54 8 kW 1.25 12.00 11 87,770 865,087 1,198,865 91.68 10,000 rpm 1.25 11.25 20 81,077 872,034 1,208,492 95.78 642 Int J Adv Manuf Technol (2009) 40:637–647 The same objective function for the constructional solutions of Fig. 2c,d is expressed as follows: KE cd ? 1 2 I 1 w 2 1 t N p 2 m 4 t m 4 0eTv 2 4 t N p 2 I 4 t I 4 0eTw 2 4 e22T 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 (N p ), the module of the gears (m i ), the number of teeth on each gear (Z i ), the face width (b i ) and the helix angle (β 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 design P in (kW) n (rpm) m (mm) b (mm) β Vol. (mm 3 ) KE 2 10 C06 mm 5 s 2 C16C17 KE 3 10 C06 mm 5 s 2 C16C17 φ T (mm) 1:6, Z 1 =18 Z 4 =36 Z 2 =90 2.5 kW 0.7 6.30 20 22,247 248,709 355,298 67.04 18,000 rpm 0.6 8.50 22 22,653 191,109 273,013 58.24 5.3 kW 0.9 10.57 15 58,355 708,768 1,012,526 83.86 15,000 rpm 0.9 8.76 25 54,946 758,054 1,082,934 89.37 7 kW 1.5 12.21 25 212,852 667,212 953,160 148.95 5,000 rpm 1.25 17.67 27 221,326 498,477 712,111 126.26 7 kW 1.25 12.11 15 129,047 798,786 1,141,124 116.47 9,000 rpm 1.25 11.25 20 126,682 828,543 1,183,633 119.72 9.3 kW 1.25 12.29 14 129,760 1,928,215 2,754,593 115.94 12,000 rpm 1.25 11.25 19 126,682 2,007,100 2,867,285 119.72 10 kW 1.25 15.77 14 166,484 1,718,698 2,455,284 115.94 10,000 rpm 1.25 11.43 30 151,508 1,963,409 2,804,871 129.90 1:7, Z 1 =18 Z 4 =45 Z 2 =108 3 kW 1 13.70 19 140,453 251,865 365,659 114.22 5,000 rpm 1 10.60 30 129,475 276,759 401,801 124.70 5 kW 0.8 11.11 23 76,852 835,980 1,213,682 93.86 15,000 rpm 0.8 9.31 30 72,790 894,546 1,298,709 99.76 7 kW 0.8 10.83 14 67,466 1,834,027 2,662 653 89.05 25,000 rpm 0.8 7.65 30 59,792 2,040,360 2,962,218 99.76 1:8, Z 1 =18 Z 4 =54 Z 2 =126 3 kW 0.6 8.24 14 39,271 615,788 902,415 77.91 25,000 rpm 0.6 6.67 25 36,468 655,435 960,516 83.42 4 kW 0.6 8.06 18 40,012 1,069,958 1,567,985 79.49 32,000 rpm 0.6 6.91 25 37,770 1,112,217 1,629,914 83.42 1:10, Z 1 =18 Z 4 =72 Z 2 =162 3 kW 0.6 5.71 19 47,403 1,339,693 1,982,746 102.80 32,000 rpm 0.6 5.43 21 46,279 1,341,915 1,986,034 104.12 4 kW 0.6 6.25 18 51,238 2,236,335 3,309,776 102.20 40,000 rpm 0.6 5.48 25 49,520 2,380,045 3,522,466 107.25 Table 3 Optimal designs of spindle speeders based on the constructional solution of Fig. 2c Spindle design β 14 =β 24 0 m 14 =m 24 0 (mm) b 14 =b 24 0 (mm) d 1 =d 4 (mm) d 1 =d 4 0 (mm) Vol. (mm 3 ) KE 2 10 C06 mm 5 s 2 C16C17 φ T (mm) 1:5, 5 kW, 13,000 rpm 24 0.9 11.08 19.75 64.64 78,475 668,153 69.13 8 0.8 9.98 24.69 20.20 Z 1 =20 Z 2 =80 Z 4 =25 Z 4 0 ? 25 1:6, 5.3 kW, 15,000 rpm 26 0.9 10.12 18.02 72.17 89488 865,896 78.10 4 0.8 8.56 30.04 24.05 Z 1 =18 Z 2 =90 Z 4 =30 Z 4 0 ? 30 1:8, 3 kW, 25,000 rpm 4 0.6 7.36 12.03 65.53 58,743 719,211 72.17 16 0.9 7.00 30.07 23.40 Z 1 =20 Z 2 =70 Z 4 =50 Z 4 0 ? 25 1:10, 4 kW, 40,000 rpm 13 0.6 6.14 12.30 59.58 49,422 1,271,833 73.78 25 0.6 5.42 30.74 16.55 Z 1 =20 Z 2 =90 Z 4 =50 Z 4 0 ? 25 Int J Adv Manuf Technol (2009) 40:637–647 643 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 β 14 =β 24 0 m 14 =m 24 0 (mm) b 14 =b 24 0 (wmm) d 1 =d 4 (mm) d 1 =d 4 0 (mm) Vol. (mm 3 ) KE 2 10 C06 mm 5 s 2 C16C17 φ T (mm) 1:5, 5 kW, 13,000 rpm 17 1.125 10.15 21.17 47.66 182,947 4,964,871 105.85 24.5 0.8 10.64 42.34 15.88 Z 1 =18 Z 2 =54 Z 4 =36 Z 4 0 ? 18 1:6, 5.3 kW, 15,000 rpm 28.3 1.125 10.15 22.99 53.63 221,436 8,157,084 114.97 20 0.8 11.18 45.99 15.32 Z 1 =18 Z 2 =63 Z 4 =36 Z 4 0 ? 18 1:8, 3 kW, 25,000 rpm 30 0.6 7.35 12.47 39.31 104,920 4,136,545 95.59 17 0.7 7.27 41.56 14.55 Z 1 =18 Z 2 =54 Z 4 =60 Z 4 0 ? 20 1:10, 4 kW, 40,000 rpm 26 0.6 6.62 12.01 39.98 91,889 6,682,166 92.11 8 0.6 7.17 40.05 12.11 Z 1 =18 Z 2 =66 Z 4 =60 Z 4 0 ? 20 1:5 1:6 1:7 1:8 1:9 1:10 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) V c /V a KE c /KE a V d /V a KE d /KE a ? volume ? kinetic energy Fig. 3 Ratio between the volume and kinetic energy of the optimal spindle speeder gearbox designs based on the constructional solutions of Fig. 2c and Fig. 2d, and the corresponding gearbox designs based on the constructional solution of Fig. 2a for different speed ratios. The dots represent the ratio between the volumes and the open diamonds show the ratio between the kinetic energies. The dashed line represents the comparison between the design based on the construc- tional solutions of Fig. 2c,a, and the continuous line for the comparison between Fig. 2d,a. 644 Int J Adv Manuf Technol (2009) 40:637–647 minimum kinetic energy solutions. For example, for the case of speed ratio 1:3.5, we chose two multiplier designs, one for a power of 10 kW and another for 16 kW, with different maximum output speeds, which are 8,000 rpm and 10,000 rpm, respectively. For this design, the optimal number of teeth according to the objective functions are: for the output member Z 1 =24, for the planet gear Z 4 =18 and for the ring gear Z 2 =60. The two rows corresponding to the same power and maximum output speed correspond