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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 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 40 637 647639 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 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 b 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 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 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 Z14 0 and Z24 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 Znl 5 3 7 Znl 2 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 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 3 5 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 40 637 647 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 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 Z2 Z1 Np an integer 10 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 integer 11 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 sHP 12 sF sFP 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 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 Y 15 The values of HPand FPare given by sHP sHlim ZN ZL ZR ZV ZW ZX 16 sFP sFlim YST YNT YdrelT YRrelT YX 17 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 40 637 647641 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 Va p 4 b14d1 2d4 2 18 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 2d40 2 19 and for the constructional solution of Fig 2d it is expressed as Vd p 4 b14 b240 max d1 2d4 d2 2 20 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 40 637 647 The same objective function for the constructional solutions of Fig 2c d is expressed as follows KEcd 1 2 I1w2 1 Np 2 m4 m40 v2 4 Np 2 I4 I40 w2 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 40 637 647643 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 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 644Int J Adv Manuf Technol 2009 40 637 647 minimum kinetic energy solutions For example for the c