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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 1 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 2 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 M ller 12 and the American Gear Manufacturers Association AGMA norm 13 and are 0 2 Z nl 5 3 C07 Z nl C02 2 4 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 5 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 d 4 1 2 d 2 C0 d 0 4 C0C1 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 Z 14 and Z 24 0 This ratio is d 0 4 d 4 Z 14 1 Z 24 0jjC0 1 7 640 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 1 Z 24 0 1 8 Lastly one assumes a minimum pinion tooth number of Z min C21 18 9 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 10 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 11 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 12 s F s FP 13 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 K A C1 K V C1 K H C1 K H p C1 Z H C1 Z E C1 Z C1 Z F t b C1 d C1 u 1 u r 14 F K A C1 K V C1 K F C1 K F C1 F t b C1 m C1 Y F C1 Y S C1 Y C1 Y 15 The values of HP and 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 16 s FP s Flim C1 Y ST C1 Y NT C1 Y drelT C1 Y RrelT C1 Y X 17 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 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 20 The material chosen for the gears is a steel with Hlim 1 360 N C14 mm 2 and 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 2d 4 2 18 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 b 24 0 C1max d 1 2d 4 d 2 2d 4 0 2 19 and for the constructional solution of Fig 2d it is expressed as V d p 4 b 14 b 24 0 C1max d 1 2d 4 d 2 2 20 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 N p 1 2 m 4 v 2 4 1 2 I 4 w 2 4 C18C19 21 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 mm Vol mm 3 KE 2 10 C06 mm 5 s 2 C16C17 KE 3 10 C06 mm 5 s 2 C16C17 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 N p 2 m 4 m 4 0 v 2 4 N p 2 I 4 I 4 0 w 2 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 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 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 Vol mm 3 KE 2 10 C06 mm 5 s 2 C16C17 KE 3 10 C06 mm 5 s 2 C16C17 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 14 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 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 14 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 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 to the minimum volume and minimum kinetic energy solutions The third fourth and fifth columns give the module the face width and the helix angle respectively The sixth column lists the volume occupied by the gears and the seventh and eighth columns are the kinetic energies of the gear system when it is designed with two KE 2 or with three KE 3 planet gears The kinetic energy is expressed independently of the specific value of the density of the steel used in the gears The units are therefore mm 5 s 2 Finally the ninth column gives the total diameter of the planetary transmission Continuing with the case of speed ratio 1 3 5 and in particular for 10 kW and 8 000 rpm it can be seen that for both the minimum volume and minimum kinetic energy designs the module of the gears is 1 25 For the minimum volume desig