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ORIGINAL ARTICLEOptimal mechanical spindle speeder gearbox designfor high-speed machiningD. R. Salgado&F. J. AlonsoReceived: 7 May 2007 /Accepted: 3 January 2008 /Published online: 9 February 2008#Springer-Verlag London Limited 2008Abstract There are different solutions to upgrade a conven-tional machine tool to high-speed machining (HSM). One ofthe cheapest solutions is the use of mechanical spindlespeeders. Mechanical spindle speeders allows the increase ofthe speed of a machine tool by means of a multiplier gearbox,and they have been successfully used in a variety ofmachining processes, such as drilling, milling, tapping andeven grinding. They are mainly used in the mould and dieindustry, since they provide an effective solution for upgrad-ing an existing lower speed machine tool. In this work, thedesign of the planetary gear trains (PGTs) used in all of themechanical spindle speeders marketed nowadays is optimisedby minimising the volume and the kinetic energy of thesegearboxes, since their functionality depends directly on thesetwo criteria. In the authors opinion, the results can be of greatinterest for spindle speeder manufacturers.Keywords Spindlespeedergearboxdesign.Planetarygeartrain.High-speedmachining.OptimisationNomenclatureHHertz contact stressFbending stressHPallowable Hertz contact stressFPallowable bending stressHOnominal Hertz contact stressFOnominal bending stressHlimmaximum allowed Hertz contact stressFlimmaximum allowed bending stresspressure angleFttangential gear forcebface widthhelix angledpitch diametermmoduleKAapplication factorKHtrans. load sharing factor for pitting resistanceKHlong. load sharing factor for pitting resistanceKFtrans. load sharing factor for bending strengthKFlong. load sharing factor for bending strengthYFaform factor for bending strengthYNTlife factor for bending strengthYRrelTrelative rugosity factorYSastress concentration factorYSTstress concentration factorYXsize factor for bending strengthuratio between the diameters of the gears mesh-ing, being greater than 1rugositymeasurement of the small-scale variations in theheight of a physical surfaceYrelTnotch relative sensitivity factorYcontact ratio factor for bending strengthYhelix angle factor for bending strengthZEmaterial factorZHgeometry factor for pitting resistanceZLviscosity factorZNlife factor for pitting resistanceZRrugosity factor for pitting resistanceInt J Adv Manuf Technol (2009) 40:637647DOI 10.1007/s00170-008-1378-8D. R. Salgado (*)Department of Mechanical, Energetic and Materials Engineering,University of Extremadura,Sta. Teresa de Jornet 38,06800 Mrida, Spaine-mail: drsunex.esF. J. AlonsoDepartment of Mechanical, Energetic and Materials Engineering,University of Extremadura,Avda. Elvas s/n,06071 Badajoz, SpainZVvelocity factorZWhardness ratio factorZXsize factor for pitting resistanceZhelix angle factor for pitting resistanceZcontact ratio factor for pitting resistanceKVdynamic factorNpnumber of planet gearsKEkinetic energy of the planetary systemiangular speed of gear iv4speed of the planet gearmimass of gear iIimoment of inertia of gear iZinumber of teeth on gear iZnltooth ratio of the gear pair formed by links nand lefficiency of the planetary gear train (spindlespeeder)0ordinary or stationary efficiency of the gear pair1 IntroductionAll of the current trends in machiningfrom high-speedmachining (HSM) to knowledge-based systemsaregeared towards maximising production capabilities. HSMis growing rapidly, and it is providing several advantagesover conventional machining, such as reduced machiningtime, reduced mechanical stresses, reduced heating of theworkpieces, high surface quality, the use of smaller toolsetc. This necessity of HSM in industry has increasedsubstantially the amount of research in this field 14.Moreover, HSM represents a good solution for machininglight metals (aluminium and magnesium for automotive andaerospace applications), machining cast iron with ceramicinserts, machining composite materials and other materials,including kovar, titanium, inconel, etc.There are different solutions to approximate a conven-tional machine tool to HSM, providing an outstanding andcost-effective opportunity for upgrading an existing lowerspeed machine tool, and can save substantial investment innew capital equipment. One of the cheapest solutions is theuse of mechanical spindle speeders. Spindle speeders havebeen developed with proven performance in a variety ofmachining processes, such as drilling, milling, tapping andeven grinding. They are used particularly in finishingoperations, and are ideal for applications such as thoseencountered in the mould and die industry. In summation,mechanical spindle speeders are a low-cost option thatallows the increase of the speed of a conventional machinetool to the speed of HSM.The spindle is one of the main mechanical componentsin machining centres, since its design directly affects themachining productivity and finish quality of the work-pieces. Consequently, spindle design (static and dynamicstiffness, dimensions of the shaft, bearings, the designconfiguration etc.) has been studied in depth 58. Thefunctionality of mechanical spindle speeders dependsmainly on the optimal design of the transmission for therequired speed ratio and power. In particular, two factorsmust be taken into account because of their importance foran optimal design of the spindle speeder: minimum volumeand minimum kinetic energy of the transmission.The volume of the spindle speeder must be minimal forlow weight and to not reduce the operating space of themachine tool. But, also, mechanical spindle speeders mustbe designed for a long working life and, so, the kineticenergy of the transmission must be minimal to ensureoptimal functionality.These requirements for spindle speeder design lead tothe use of transmissions based on planetary gear trains(PGTs), since PGTs offer a very compact and efficientsolution (reduced weight and size in comparison withordinary gear trains), combined with high speed ratios andhigh efficiency. PGTs are also used in many machine toolsequipped with a motor gearbox to extend the constantpower range of the machine tool spindle drive motor at lowspeeds. Recently, their designs have been optimised by thecurrent authors 9.The common range of speed ratios for which mechanicalspindle speeders are designed is a multiplication factor from3.5 to 8, depending on the manufacturer. Only onemanufacturer offers a mechanical spindle speeder thatincreases the machine speed up to 10 times, with amaximum speeder output of 40,000 rpm and with 2 kWof power. Figure 1 shows a mechanical spindle speeder.The objective of this paper is to give a set of optimaldesigns of mechanical spindle speeders for different powersand speed ratios. In particular, the spindle speeder config-urations currently used by manufacturers are studied for allof the marketed range of powers and speed ratios, and theoptimal designs of these configurations (for each power andspeed ratio) are given and compared for all of that range.2 Considerations on the design of mechanical spindlespeedersIn this section, we explain some important considerationsthat must be taken into account for spindle speeder design.Mechanical spindle speeders are designed with a four-member PGT, which is the most widely used configurationcommercially because it covers almost the entire range ofspeed ratios employed in industrial applications, whilebeing the simplest PGT construction. This PGT has sixdifferent constructional solutions, depending on how the638Int J Adv Manuf Technol (2009) 40:637647members are designed. These six solutions are shown inFig. 2. A priori, the six constructional solutions of Fig. 2could be considered for the spindle speeder design.The members of PGTs are of different types, which willbe called suns, rings, arms and planets in the present work(see Fig. 1a).In Fig. 2, member 3 is the arm and members 4 and 4 arethe planets. Members 1 and 2 are different members,depending on the constructional solution, so member 1 is asun gear in the constructional solutions of Fig. 2a,c,d, and itis a ring gear in the rest of the constructional solutions. In thesame way, member 2 is a sun gear in the constructionalsolutions of Fig. 2b,d,e, and it is a ring gear in the rest of theconstructional solutions of Fig. 2. The gear pairs are the linksbetween members 1 and 4, and between members 2 and 4;and the turning pair is the link between the arm (member 3)and the planet. In the present work, the expression “simpleplanet” will be used for a planet constructed with a singlegear, such as the planet of Fig. 2a,b, and “double planet” forone constructed with two gears, such as the planets of Fig.2cf. A more detailed explanation of the structure of PGTsmay be found in 9, 11.2.1 Efficiency considerationsIt is possible to prove that the efficiency of the multiplierbased on the four-member PGT is higher if it is designedwith an input by the arm (member 3). This is the reasonwhy all mechanical spindle speeders are designed asmultiplier four-member PGTs with an input by the armmember.2.2 Economic and operating considerationsOf the solutions with a double-planet configuration (Fig.2cf), that of Fig. 2d is more interesting from an economicpoint of view, since it offers the advantage of not using aring gear. The reason for this is that spindle speeder gearsmust be hardened, tempered and ground to avoid highheating, and a ground ring gear is more expensive than aground non-ring gear. Also, if the ring gear is not ground,heat buildup will occur in a shorter period of time, and thisheating limits and reduces the input speed and torque.The constructional solution of Fig. 2a presents theadvantage over the other solution constructed with simple abcdefFig. 2 The six constructional solutions of the four-member PGTFig. 1 a Members of a plane-tary gear train (PGT). b Amechanical spindle speederInt J Adv Manuf Technol (2009) 40:637647639planets (Fig. 2b) in that the ring gear is the fixed member.For this reason, the constructional solution of Fig. 2b is notused for mechanical spindle speeder design, since itincreases the kinetic energy of the spindle speederconsiderably. Following this same reasoning, the construc-tional solutions of Fig. 2e,f are not appropriate config-urations from the solutions constructed with double planetsfor mechanical spindle speeder design.2.3 Planet member considerationsIn spindle speeder design, it is quite important to choose anoptimal number of planets for the required power and speedratio. The number of planet members (Np) can vary from twoto three, four or even more, depending on the application forwhich it is designed. For example, the mechanical spindlespeeder of Fig. 1a has three planet members (Np=3). Thisnumber must be as small as possible in order to reduce theweight and the kinetic energy of the transmission, whileensuring a good distribution of the load to each of the planetgears. Whichever the case, the planets must always bearranged concentrically around the PGTs principal axis tobalance the mass distribution.In short, for mechanical spindle speeders, only theconstructional solutions of Fig. 2a,c,d must be consideredfor an optimal spindle speeder design. In particular, theseconstructional solutions are the ones that are most oftenused by manufacturers.3 Constraints on mechanical spindle speeder designIn this section, the constraints for the mechanical spindlespeeder design are described. They are grouped into threesets, according to the type of constraint. These are:Constraints involving gear size and geometryPGT meshing requirementsContact and bending stresses3.1 Constraints involving gear size and geometryThe first constraint is a practical limitation of the range forthe acceptable face width b. This constraint is as follows:9m ? b ? 14m1where m is the module. The module indicates the tooth sizeand is the ratio of the pitch diameter to the number of teethin the gear. For gears to mesh, their modules must be equal.Gear ISO standards and design methods are based on themodule.All of the kinematic and dynamic parameters of thetransmission depend on the values of the tooth ratios Znl,where Znlis the tooth ratio of the gear pair formed by thelinking members n and l. In particular, Znlis defined as:ZnlZnZl2For the definition of the tooth ratios to satisfy the Willisequations, Znlmust be positive if the gear is external(meshing geargear) and negative if it is internal (meshingring geargear) 10, 11. For the train of Fig. 2a, one wouldhave to take Z140 and Z240.In theory, the tooth ratios can take any value, but inpractice, they are limited mainly for technical reasonsbecause of the difficulty in assembling gears outside of acertain range of tooth ratios. In this work, the tooth ratio forthe design of mechanical spindle speeders are quite close tothe recommendations of Mller 12 and the AmericanGear Manufacturers Association (AGMA) norm 13, andare:0:2 Znl 53?7 Znl ?2:24with the constraint given by Eq. 3 being for external gearsand that by Eq. 4 for internal gears. It is important to notethat these constraints are valid for designs with differentnumbers of planets (Np). In respecting these values, oneachieves mechanical spindle speeder designs that aresmaller, lighter and cheaper.Another constraint that will be imposed on the design ofspindle speeders with double planets is that the ratio of thediameters of the gears constituting a double planet is:13d4d04 35where d04is the diameter of the planet gear that meshes withmember 2 and d4is the diameter of the planet gear thatmeshes with member 1 (see Fig. 2).In the constructional mechanical spindle speeders basedon the PGT of Fig. 2c,d, the tooth ratios Z14and Z240arerelated to the radii of the gears constituting the planet. Inparticular, the following geometric relationship must besatisfied in the spindle speeder configuration of Fig. 2c:12d1 d4 12d2? d04?6Expressing the above equation in terms of the module ofthe gears, it is straightforward to find that the ratio of thediameters of gears 4 and 4 conditions the value of Z14andZ240: This ratio is:d04d4Z14 1Z240jj ? 17640Int J Adv Manuf Technol (2009) 40:637647Likewise, one obtains for the case of the configuration inFig. 2d the expression:d04d4Z14 1Z240 18Lastly, one assumes a minimum pinion tooth number of:Zmin? 1893.2 Planetary gear train meshing requirementsThe meshing requirements are given by the AGMA norm13. The following constraint (Eq. 10) is for the design ofFig. 2a:Z2? Z1Np an integer10where 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 thesun and ring gear with the arm fixed. The negative signmust be used when the sun and ring gear turn in the samedirection with the arm member fixed.Planetary systems with double planets must, either ofwhich, factorise with the number of planets in the sense ofEq. 11 below (see AGMA norm 13):Z2P2? Z1P1Np an integer11where P1and P2are the numerator and denominator of theirreducible fraction equivalent to the fractionZ04Z4; where Z04is the number of teeth of the planet gear that meshes withmember 2 and Z4is the number of teeth of the planet gearthat meshes with member 1 (see Fig. 2):Z04Z4P1P23.3 Contact and bending stressesThe torques on each gear of the proposed spindle speederdesigns were calculated taking power losses into account.This aspect allows one to really optimise the mechanicalspindle speeder design, unlike the optimisation studies inwhich these losses are not considered 14, 15. Theprocedure for obtaining torques and the overall efficiencyof 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 contactand bending stresses must be satisfied:sH sHP12sF sFP13For the calculation of the gears, the ISO norm wasfollowed. The values of the stresses of Eqs. 12 and 13 aredefined by this norm as:HffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKA? KV? KH? KHp? ZH? ZE? Z? ZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFtb ? d?u 1ur14F KA? KV? KF? KF?Ftb ? m? YF? YS? Y? Y15The values of HPand FPare given by:sHP sHlim? ZN? ZL? ZR? ZV? ZW? ZX16sFP sFlim? YST? YNT? YdrelT? YRrelT? YX17It is important to emphasise that the tangential force Ftwas obtained from the calculation of the torques taking thepower losses into account. To include power losses in theoverall efficiency calculation, we used the concept ofordinary efficiency 10, 11, which is what the efficiencyof the gear pair would be if the arm linked to the planetwere fixed. By means of this efficiency, one introducesinto the overall efficiency calculation of the PGT thefriction 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 lostby friction between these gears. In studies that do not takethis power loss into account, the value of the tangentialforces is only approximate and may be quite different inthe case of PGTs because of the possibility of powerrecirculation 10.Given the start-up characteristics of machine tools ingeneral, we took an application factor of KA=1. The pressureangle is =20. The material chosen for the gears is a steelwith Hlim 1;360 N?mm2and Flim 350 N?mm2:Lastly, the distribution of the loads to which each of theplanet gears is subjected was determined using thedistribution 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) 40:6376476414 Objective functions and design variablesVarious works have presented methods for the optimisationof a conventional transmission 1423, but only a fewstudies have proposed optimisation techniques for thedesign of PGTs 20, 21. In addition, none of these studieson PGTs 24, 25 calculate exactly the torques to whicheach of the gears is subjected, since they do not considerthe power losses in the different gear pairs of the PGT.Nevertheless, it is known that power losses in thesetransmissions may be considerably greater than in anordinary gear train 10, 11, and, therefore, an optimaldesign must take this factor into account. Indeed, notconsidering power losses, as well as not ensuring anoptimal mechanical spindle speeder design, impedes onefrom knowing its overall efficiency with certainty.In this section, we describe the objective functions andthe design variables. The objective functions are the volumefunction and the kinetic energy function. It is important tobear in mind that these functions have different expressions,depending on the constructional solution adopted for thespindle speeder design. In particular, the volume functionfor the constructional solution with simple planets (Fig. 2a)is expressed as follows:Vap4b14d1 2d4218where Varepresents the total volume of the gears. The sameobjective function for the constructional solution of Fig. 2ctakes another form, and is expressed as follows:Vcp4b14 b240 ? max d1 2d4; d2 2d40219and for the constructional solution of Fig. 2d, it is expressedas:Vdp4b14 b240 ? max d1 2d4; d2220where b14is the face width of gears 1 and 4, and b240is theface width of gears 2 and 4.The kinetic energy function is also different for theconstructional solutions with simple and double planets, ascan easily be deduced. The function for the constructionalsolution of Fig. 2a is expressed in the following form:KEa12I1w21 Np12m4v2412I4w24?21where I4, w4and m4are the moment of inertia, the rotationalspeed and the mass of the planet gear, respectively, and v4isthe translation speed of the centre of the planet gear. In theabove expression, I1is the moment of inertia of the sunmember and Npis the number of planet gears.Table 1 Optimal designs of spindle speeders based on the constructional solution of Fig. 2aSpindle designPin(kW)n (rpm)m (mm)b (mm) (mm)Vol.(mm3)KE210?6 mm5s2?KE310?6 mm5s2?T(mm)1:3.5, Z1=24 Z4=18 Z2=6010 kW1.2514.841469,850860,9051,057,74177.308,000 rpm1.2511.912564,285908,1521,115,79182.7516 kW1.2517.031883,4481,672,5292,054,93378.8610,000 rpm1.2515.232582,1421,812,9702,227,48582.751:4, Z1=18 Z4=18 Z2=5420 kW2.530.7515471,7181,754,2732,280,555139.763,000 rpm2.525.3225441,2781,864,0762,423,300148.9630 kW2.526.2216406,1004,235,9375,506,718140.445,000 rpm2.523.6221387,8914,289,5045,576,355144.6045 kW2.532.40463,76911,443,06014,875,978135.008,000 rpm2.522.7118359,4119,804,36112,745,669141.951:5, Z1=18 Z4=27 Z2=721.7 kW0.66.2609,181166,090230,17343.2024,000 rpm0.65.4588,150104,760145,18143.622 kW0.79.751721,27069,27195,98852.7010,000 rpm0.78.482520,59874,688103,50655.613.5 kW0.79.651520,640213,482295,85152.1818,000 rpm0.77.772719,545237,579329,24456.565 kW0.911.681440,934361,818501,42066.7813,000 rpm0.99.652538,754392,580544,05171.506.4 kW111.921552,045573,010794,09574.5413,000 rpm19.932549,223615,591853,10679.447 kW113.921762,011593,508822,50375.3012,000 rpm111.212858,557657,453911,12081.548 kW1.2512.001187,770865,0871,198,86591.6810,000 rpm1.2511.252081,077872,0341,208,49295.78642Int J Adv Manuf Technol (2009) 40:637647The same objective function for the constructionalsolutions of Fig. 2c,d is expressed as follows:KEcd12I1w21Np2m4 m40v24Np2I4 I40w2422In Eqs. 21 and 22, the energy of the arm has beenneglected because this member can be designed in differentand variable forms, and because it is considerably less thanthat of the planetary system.The design variables are of the constructional solutionchosen from those of Fig. 2a,c,d, the number of planet gears(Np), the module of the gears (mi), the number of teeth oneach gear (Zi), the face width (bi) and the helix angle (i).When these design parameters are determined by minimisingthe 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 mm5s2?KE310?6 mm5s2?T(mm)1:6, Z1=18 Z4=36 Z2=902.5 kW0.76.302022,247248,709355,29867.0418,000 rpm0.68.502222,653191,109273,01358.245.3 kW0.910.571558,355708,7681,012,52683.8615,000 rpm0.98.762554,946758,0541,082,93489.377 kW1.512.2125212,852667,212953,160148.955,000 rpm1.2517.6727221,326498,477712,111126.267 kW1.2512.1115129,047798,7861,141,124116.479,000 rpm1.2511.2520126,682828,5431,183,633119.729.3 kW1.2512.2914129,7601,928,2152,754,593115.9412,000 rpm1.2511.2519126,6822,007,1002,867,285119.7210 kW1.2515.7714166,4841,718,6982,455,284115.9410,000 rpm1.2511.4330151,5081,963,4092,804,871129.901:7, Z1=18 Z4=45 Z2=1083 kW113.7019140,453251,865365,659114.225,000 rpm110.6030129,475276,759401,801124.705 kW0.811.112376,852835,9801,213,68293.8615,000 rpm0.89.313072,790894,5461,298,70999.767 kW0.810.831467,4661,834,0272,662 65389.0525,000 rpm0.87.653059,7922,040,3602,962,21899.761:8, Z1=18 Z4=54 Z2=1263 kW0.68.241439,271615,788902,41577.9125,000 rpm0.66.672536,468655,435960,51683.424 kW0.68.061840,0121,069,9581,567,98579.4932,000 rpm0.66.912537,7701,112,2171,629,91483.421:10, Z1=18 Z4=72 Z2=1623 kW0.65.711947,4031,339,6931,982,746102.8032,000 rpm0.65.432146,2791,341,9151,986,034104.124 kW0.66.251851,2382,236,3353,309,776102.2040,000 rpm0.65.482549,5202,380,0453,522,466107.25Table 3 Optimal designs of spindle speeders based on the constructional solution of Fig. 2cSpindle design14=240m14=m240(mm)b14=b240(mm)d1=d4(mm)d1=d40(mm)Vol.(mm3)KE210?6 mm5s2?T(mm)1:5, 5 kW, 13,000 rpm240.911.0819.7564.6478,475668,15369.1380.89.9824.6920.20Z1=20Z2=80Z4=25Z40 251:6, 5.3 kW, 15,000 rpm260.910.1218.0272.1789488865,89678.1040.88.5630.0424.05Z1=18Z2=90Z4=30Z40 301:8, 3 kW, 25,000 rpm40.67.3612.0365.5358,743719,21172.17160.97.0030.0723.40Z1=20Z2=70Z4=50Z40 251:10, 4 kW, 40,000 rpm130.66.1412.3059.5849,4221,271,83373.78250.65.4230.7416.55Z1=20Z2=90Z4=50Z40 25Int J Adv Manuf Technol (2009) 40:6376476435 Results and discussionThe optimisation problem of mechanical spindle speedersdescribed in this paper was applied to a set of differentdesigns of spindle speeders, i.e. different speed ratios andpowers covering the entire marketed range. Tables 1 and 2summarise all of the cases studied for the design based onthe constructional solution of Fig. 2a and show the optimaldesigns. In these tables, the first and second columns listthe speed ratio, the input power and the maximum outputspeed for each design. The first column also indicates thetooth number of each member for the minimum volume andTable 4 Optimal designs of spindle speeders based on the constructional solution of Fig. 2dSpindle design14=240m14=m240(mm)b14=b240(wmm)d1=d4(mm)d1=d40(mm)Vol.(mm3)KE210?6 mm5s2?T(mm)1:5, 5 kW, 13,000 rpm171.12510.1521.1747.66182,9474,964,871105.8524.50.810.6442.3415.88Z1=18Z2=54Z4=36Z40 181:6, 5.3 kW, 15,000 rpm28.31.12510.1522.9953.63221,4368,157,084114.97200.811.1845.9915.32Z1=18Z2=63Z4=36Z40 181:8, 3 kW, 25,000 rpm300.67.3512.4739.31104,9204,136,54595.59170.77.2741.5614.55Z1=18Z2=54Z4=60Z40 201:10, 4 kW, 40,000 rpm260.66.6212.0139.9891,8896,682,16692.1180.67.1740.0512.11Z1=18Z2=66Z4=60Z40 201:51:61:71:81:91:102 4 6 8 101214Speed ratioRatio 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 energyFig. 3 Ratio between the volume and kinetic energy of the optimalspindle speeder gearbox designs based on the constructional solutionsof Fig. 2c and Fig. 2d, and the corresponding gearbox designs basedon the constructional solution of Fig. 2a for different speed ratios. Thedots represent the ratio between the volumes and the open diamondsshow the ratio between the kinetic energies. The dashed linerepresents the comparison between the design based on the construc-tional solutions of Fig. 2c,a, and the continuous line for thecomparison between Fig. 2d,a.644Int J Adv Manuf Technol (2009) 40:637647minimum kinetic energy solutions. For example, for thecase of speed ratio 1:3.5, we chose two multiplier designs,one for a power of 10 kW and another for 16 kW, withdifferent maximum output speeds, which are 8,000 rpm and10,000 rpm, respectively. For this design, the optimalnumber of teeth according to the objective functions are:for the output member Z1=24, for the planet gear Z4=18 andfor the ring gear Z2=60. The two rows corresponding to thesame power and maximum output speed correspond to theminimum volume and minimum kinetic energy solutions.The third, fourth and fifth columns give the module, theface width and the helix angle, respectively. The sixthcolumn lists the volume occupied by the gears and theseventh and eighth columns are the kinetic energies of thegear system when it is designed with two (KE2) or withthree (KE3) planet gears. The kinetic energy is expressedindependently of the specific value of the density of thesteel used in the gears. The units are, therefore, mm5/s2.Finally, the ninth column gives the total diameter of theplanetary transmission.Continuing with the case of speed ratio 1:3.5 and, inparticular, for 10 kW and 8,000 rpm, it can be seen that, forboth the minimum volume and minimum kinetic energydesigns, the module of the gears is 1.25. For the minimumvolume design, all gears must have a 11.91-mm face widthand must be constructed with a 25 helix angle. On thecontrary, if one wants a design in which the kinetic energy isminimum, the face width must be 14.84 mm and the helixangle 14. From this data, the diameter of any of the gearscan be straightforwardly deduced. For example, the diameterof member 1 is 33.10 mm, the diameter of the planets is24.82 mm and the ring gear diameter is 82.75 mm in theminimum volume solution. For this specific design of thespindle speeder (1:3.5, 10 kW and 8,000 rpm), it can beverified that the minimum kinetic energy solution has 8.6%more volume than the minimum volume solution. Theminimum volume design, however, only has 5.5% morekinetic energy than the optimal kinetic energy design.From Tables 1 and 2, several conclusions can beobtained. For example, it is important to bear in mind that,if the minimum volume and minimum kinetic energydesigns for a specific spindle speeder design correspondto gears with different modules, the differences betweenthese two designs are greater. In the set of spindle speedersstudied in this work, this occurs in two cases with speedratio 1:6. In particular, these are the designs: 2.5 kW and18,000 rpm, and 7 kW and 5,000 rpm. In the first case, i.e.1:6, 2.5 kW and 18,000 rpm, it can be verified that the ratiobetween the kinetic energies of the minimum volumedesign and the minimum kinetic energy design is 30.14%.However, the ratio between the volumes of those twodesigns is only 1.82%. In the second case, i.e. 1:6, 7 kWand 5,000 rpm, the results are similar: these percentages areonly slightly greater, being 33.85% and 4.00%, respective-ly. In summation, when the gears of these two optimaldesigns have different modules, the difference in the kineticenergies of the two designs is greater than in the cases inwhich gears of the two proposed designs have the samemodule. In these cases, the mechanical spindle design mustbe designed based on the minimum kinetic energy solution.Another important result is that the designs 1:4, 45 kWand 8,000 rpm, and 1:5, 7 kW and 24,000 rpm have onlyone solution, i.e. the minimum volume and minimumkinetic energy designs are one and the same. They are theonly cases in which the two designs coincide. In Table 1,for these spindle designs, as well as the optimal design, thedesign corresponding to =0 is given.Finally, the results of the optimal designs based on theconstructional solution of Fig. 2a (Tables 1 and 2) showthat the total diameter of the minimum kinetic energydesign is always smaller than the diameter of the minimumvolume spindle speeder design, as was to be expected apriori.Tables 3 and 4 summarise the spindle speeder designsbased on the constructional solutions of Fig. 2c,d, respec-tively. With these constructional solutions, only a fewdesigns of those proposed in Tables 1 and 2 are analysed.The main reason for this is that these solutions only presentadvantages with speed ratios greater than 1:12 10, and thisis not in the range of mechanical spindle speeders. In thesecases also, for each spindle speeder design (speed ratio,power and maximum output speed), only one solution isgiven, since for any design of the spindle speeder, theseconstructional solutions represent worse results than thoseobtained based on Fig. 2a from the point of view of thefunctionality of the multiplier. This design corresponds to acompromise solution between the minimum volume and theminimum kinetic energy design. For example, for amechanical spindle speeder with 1:5, 5 kW and 13,000rpm, an appropriate design based on the constructionalsolution of Fig. 2c is given in Table 3. The first column ofthis table lists the speed ratio, power and maximum outputTable 5 Efficiency of spindle speeders for different ordinaryefficiencies and speed ratiosSpeed ratioOrdinary efficiency0.980.9750.970.9650.961:3.50.9710.9640.9570.9490.9431:40.9700.9620.9550.9470.9401:50.9680.9600.9520.9440.9361:60.9670.9580.9500.9420.9341:70.9660.9570.9480.9400.9321:80.9650.9560.9470.9390.9301:100.9640.9550.9460.9370.928Int J Adv Manuf Technol (2009) 40:637647645speed. The second to sixth columns give the helix angles,modules, face widths and diameters of all the gears. In thisspindle speeder design, the data corresponding to the gearpair formed by gears 1 and 4 are: 24 helix angle, module0.9, face width 11.08 mm, and diameters of 19.75 mm and24.69 mm, respectively. The gear pair formed by gears 2 and4 has 8 helix angle, module 0.8, face width of 9.98 mm,and diameters of 64.64 mm and 20.20 mm, respectively. Thenumber of teeth of each gear is shown in the bottom row ofthis spindle speeder design. They are: Z1=20, Z2=80, Z4=25and Z40 25: The seventh and eighth columns give thevolume and the kinetic energy of the gear system. Finally,the ninth column gives the total diameter of the mechanicalspindle speeder. In the same way, the information about themechanical spindle speeder based on the constructionalsolution of Fig. 2d is given in Table 4.Analysing the information given in Tables 3 and 4 andcomparing it with the information of Tables 1 and 2, someinteresting conclusions about the optimal design of me-chanical spindle speeders can be drawn. This comparison issynthesised in Fig. 3.From Fig. 3 it can be deduced that spindle speeders mustbe designed based on the constructional solution of Fig. 2a,since the other two possibly interesting constructionalsolutions have greater volumes and kinetic energies. Also,the constructional solution of Fig. 2d is a poorer solutionthan that of Fig. 2c, as is easily deduced from Fig. 3. This ismainly because the rotation speed of the planet is higher inthe design of Fig. 2d than that in the design proposed in Fig.2c for the same speed ratio. This constructional solution (Fig.2d) only presents the advantage of not using a ring gear, andthis is more an economic than a functional consideration.This is the reason why this constructional solution is used inother gearboxes for machine tools (see Fig. 2b in 9), as isthe case in which the gearbox is designed to extend theconstant power range of a machine tool spindle drive motor.Another noteworthy result is that the ratios between thevolume and kinetic energy fall as the speed ratio increases.Indeed, in the 1:10 case, the volume and kinetic energy ofthe spindle speeder based on the design of Fig. 2c are lessthan the volume and kinetic energy obtained with thedesign based on the constructional solution of Fig. 2a (seeTables 2 and 3).It is also worth noting that the total diameter of thespindle speeder based on the constructional solution of Fig.2c is less for speed ratios 1:6, 1:8 and 1:10 than that basedon Fig. 2a (see Tables 2 and 3), and is similar to thatobtained for the speed ratio 1:5 (see Tables 1 and 3).Finally, the overall efficiencies of all of the proposedmechanical spindle speeder designs have been calculatedtaking into account power losses 11 for different ordinaryefficiencies. The results are summarised in Table 5.6 ConclusionsAccording to the results obtained in this work, it can beconcluded that, in general, the best design of a mechanicalspindle speeder is based on the constructional solution ofFig. 2a, which is that most often used by mechanicalspindle speeder manufacturers. Among all of the possiblespindle speeder designs based on the constructionalsolution of Fig. 2a for each speed ratio, power andmaximum output speed, the results given in Tables 1 and2 offer the most appropriate solutions, i.e. the minimumvolume and minimum kinetic energy solutions. In theauthors opinion, these results could be of great interest formanufacturers and engineers involved with the marketingand design of mechanical spindle speeders.Moreover, it is important to note that the constructi