OPTIMISATION OF A DRIVE SYSTEM AND ITS EPICYCLIC GEAR SET

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1 OPTIISATION OF A DRIVE SYSTE AND ITS EPICYCLIC GEAR SET Nicolas Bellegade, Philippe Dessante, Piee Vidal, Jean-Claude Vannie To cite this vesion: Nicolas Bellegade, Philippe Dessante, Piee Vidal, Jean-Claude Vannie. OPTIISATION OF A DRIVE SYSTE AND ITS EPICYCLIC GEAR SET. ISEF XIII Intenational Symposium on Electomagnetic Fields in echatonics, Electical and Electonic Engineeing, Sep 007, Pague, Czech Republic. pp.1-6, 007. <hal-00777> HAL Id: hal Submitted on 9 Jan 008 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.

2 ISEF XIII Intenational Symposium on Electomagnetic Fields in echatonics, Electical and Electonic Engineeing Pague, Czech Republic, Septembe 13-15, 007 OPTIISATION OF A DRIVE SYSTE AND ITS EPICYCLIC GEAR SET Nicolas BELLEGARDE, Philippe DESSANTE, Piee VIDAL, Jean-Claude VANNIER Dépatement Énegie Supélec, 1 ue Joliot-Cuie 9119 Gif-su-Yvette, Fance, nicolas.bellegade@supelec.f Abstact This pape descibes the design of a dive consisting of a DC moto, a speed educe, a lead scew tansfomation system, a powe convete and its associated DC souce. The objective is to educe the mass of the system. Indeed, the volume and weight optimisation of an electical dive is an impotant issue fo embedded applications. Hee, we pesent an analytical model of the system in a specific application and aftewads an optimisation of the moto and speed educe main dimensions and the battey voltage in ode to educe the weight. I. INTRODUCTION The system studied in this pape is a linea electical dive system ealized with a Ni-H battey bank, a DC/DC convete, a DC moto, a speed educe and a lead-scew device. The aim of the system is to move a load along a linea displacement. Regading the load, we can define mainly two specifications. Fistly, it has to apply a athe high static foce to ovecome some static fiction foce. This has to be done at constant speed o at standstill. Secondly, it has to be diven fom one point to anothe in a given time. This specification implies a dynamic foce, a given acceleation and a maximum speed depending on the kind of displacement. In ode to optimise the weight of the system, and mainly the battey, DC moto and speed educe weights, geometical and physical elations have to be witten fo each component. These elations ae then linked with the othes to make a global optimisation of the system. The constaints ae based upon the load specifications. The mathematical optimisation is pefomed with the help of vaious numeical methods like Genetic Algoithm, Random Seach, Diffeential Evolution and Nelde ead. II. SYSTE ODEL 1. Powe convesion system A geneal epesentation of the system is given in figue 1. U bat ibat DC/DC Convete i u DC oto LOAD C em γ v x Ω θ Figue 1: Powe convesion system. C Speed Reduce ρ ρ t Ω θ Lead Scew Device C em : toque developed by the moto [ N ] C : esistive toque [ N ]. ρ : tansfomation atio of the speed educe. ρ t : tansfomation atio of the lead-scew device [ ad / m ]. θ m : moto shaft angle [ ad ]. Ω m : moto shaft speed [ ad / s ]. x : linea displacement [ m ]. v : linea speed [ m / s ].

3 Concening the mechanical pat, the lead-scew is epesented by its tansfomation atio deduced fom the scew pitch while the speed eduction system intoduces a speed tansfomation atio. The elation between the moto shaft angle θ m and the linea displacement x can be witten as follows: Whee ρ is the global tansfomation atio.. Speed educe volume A epesentation of the speed educe is given in figue. θ = ρ ρ x = ρ x (1) m t α1 b α R b 3 R 1 α3 b R α b Ω m Ω R Figue : Speed educe stuctue. The speed educe is constitued by two epicyclic geas (chaacteized by R 1,R ) and one cylindical gea with staght oute teeth (chaacteized by R 3,R ). We can expess the volume V of this speed educe as follows: [ R1 + R ] [ α1 + α ] + α 3 + α ] V = π b R () 3. Load specifications Hee, all load specifications ae expessed on the moto shaft. In the consideed application, the moto has to geneate two sots of toques, imposed by the load. A static toque which is necessay to each the beakaway foce on the load just befoe it stats to move o to maintain the speed at a constant value. The static toque C is given by: em s C em s = K1 (3) R 1 + R R1 Whee 1 K is a constant depending on the speed, the lead-scew tansfomation atio, the diffeent esistive foces due to the load. The moto must also geneate a dynamic toque which is equied when the diffeent esistive foces ae at thei maximum values.

4 The dynamic toque C em d can be expessed as follows: Cem d = K () R 1 + R R1 Whee K is a constant depending on the same paametes as K 1. An othe toque can be also consideed. Indeed, the RS toque Cem RS is inteesting because it is diectly linked to the coppe losses in the moto. Its expession is given by: C em RS R + 1 R R1 = K 3 J + K + K 5 R + 1 R R1 J (5) J is the total inetia of the system. The constants K 3, K, K 5 ae functions of the aveage and the ms values of the load speed and acceleation, the lead-scew tansfomation atio and the aveage and the ms values of the diffeent esistive foces.. oto specifications Among the limits concening the moto, thee is a maximal oto speed Ω, function of the maximal linea speed v, defined by: Ω = ρ ρ v (6) The maximum peak powe P given by the moto to the load has also to be consideed, whee K 6 is a constant depending on the diffeent esistive foces: P v t = K 6 (7) Fo the moto s design, it is possible to define the main dimensions by the peak toque nominal toque C em n and the oto inetia J m as seen in pevious papes [1-6]: C em, the C C J emn em m = π A = p H π = μ m L R B γ B R L E L p R L (8) These thee elationships intoduce thee main dimensions paametes fo the design : the oto adius R, the oto length L and the pemanent magnet thickness E. The emaining paametes ae constants and have the following significations: p = 1 : pole s numbe. H = 80 ka / m : magnet s peak magnetic field. B = 0, T : ai gap flux density. A L = 7850 A / m : ms cuent linea density. γ = 0, 75 : pole s ovelapping facto. p 3 μ 5070 kg / m : moto mass density m =

5 III. SYSTE OPTIISATION The established elationships ae used to define the constaints duing the optimisation pocedue. Two types of constaints ae consideed: the physical constaints which pemit to ensue that the moto can supply load equiements and the geometical constaints which pemit to define a feasible moto and speed educe (see [7] fo futhe details). Concening the physical constaints, the moto peak toque has to be geate than the static and dynamic toques. The nominal toque has also to be geate than the equied ms toque. C C C em em emn > C > C > C emd ems emrs (9) The maximum powe supplied by the battey bank has to be geate than the maximum powe consumption. [ U bat i bat ] > P (10) The maximum speed equied by the load has to be kept lowe than the nominal speed of the moto Ω. n Ω n > ρ ρ v (11) t Concening the geometical constaints, the following will be used in ode to have feasible moto and speed educe: R mm R L 5 R R E 3mm R R mm 1 R 6mm (1) Depending on the application, diffeent cost functions can be minimized. Hee, we pesent an optimisation whee the weight of the system is the cost function. We sum the moto, speed educe and battey bank masses: bat + m + The moto and speed educe masses ae defined as follows: m = (13) = μ m π R L (1) [ R1 + R ] [ α1 + α ] + α 3 + α ] = π μ b R (15) The battey bank mass is a function of the voltage if we take a constant capacity. ( ) = (16) bat f U bat

6 The total weight can be expessed by the following expession: [ α 3 + α ] ( U ) + μ π R L + π μ b [ R + R ] [ α + α ] = f bat m R (17) The seven optimisation paametes ae: R, L, E, R1, R,, R. IV. RESULTS We pesent hee the esults concening the definition of the battey, the speed educe and the moto whose chaacteistics ae optimised fo a given load. In this example, the load chaacteistics ae pesented in the table I. γ v eff = 1,mm / s eff = 30,9mm / s γ = 60 mm / s v = 15 mm / s K1 = 0, 3N K =, 7N K 3 = 396N / kg m K = 3, 6N K 5 = 10,N / kg m K 6 = 1995W s / m Table I : Load chaacteistics. Figue 3: oto and speed educe weights vesus ρ max. Befoe optimisation, the battey voltage is 1 V and the weight is equal to 1180 g. The optimisation pocedues uses the constaints (9-1) and seaches a set of values fo R, L, E, R1, R,, R and U bat which minimises the total mass of the system in equation (17). The optimisation is made with athematica and vefied with atlab and an home made genetic algoithm. Figue : oto dimensions vesus ρ max. Figue 5: Speed educe dimensions vesus ρ max.

7 Figues 3-5 show the esults obtained afte the optimisation pocedue. The battey voltage is equal to 1 V and its weight is 60 g. As seen in a pevious pape [-5], boundaies ae used to limit the vaiation of the paametes to feasible values. It appeas that the esults ae contolled by the uppe bounday value fo the tansfomation atio ρ = ρ ρt. Figue 3 shows the moto and speed educe weights of the system in function of the constaint ρ max. The system mass deceases until ρ max = 95ad / mm, whee it emains constant. Befoe the pevious value, the minimum is obtained fo ρ = ρmax. Thee is no need to have a tansfomation atio geate than 95 ad / mm because the system weight will not decease anymoe. At this point, the decease of the moto weight is compensated by the incease of the speed educe weight. The esulting active mass is 830 g. Figues and 5 show the dimensions of the moto and the speed educe vesus the uppe constaint ρ max. Concening the moto, we can see in Figue that the adius deceases until ρ max = 95ad / mm and the two othes paametes ae independant of ρ max. In figue 5, we can note that two adius incease until ρ 95ad / mm while the two othes stay constant. max = V. CONCLUSION In this pape, the weight of an electomechanical convesion system has been optimised. Fistly, a model of the moto and the speed educe has been done. This model links the moto and speed educe main dimensions to thei pefomances. Then, the battey bank, moto and speed educe weights have been witten in function of the optimisation paametes. Secondly, an optimisation pocedue was executed in ode to minimise the objective function which is the weight of the pevious components. Diffeent numeical optimisation methods wee used to valid the methodology. All gave the same esults. The initial total weight was 1180g and the obtained total weight afte optimisation is equal to 830g. We can note that the weight is educed by 30%. VI. ACKNOWLEDGENT The authos would like to thank Viax company, and paticulaly. Fédéic BERNIER fo his help and his financing. Refeences [1] acua E., Ripoll C., Vannie J-C, Design and Simulation of a Linea Actuato fo Diect Dive, PCI001, pp 317-3, june 19-1, 001, Nünbeg (Gemany). [] acua E., Ripoll C., Vannie J-C, Design, Simulation and Testing of a P Linea Actuato fo a Vaiable Load, PCI00, pp 55-60, may 1-16, 00, Nünbeg (Gemany). [3] acua E., Ripoll C., Vannie J-C, Optimization of a Bushless DC oto Load Association, EPE003, Sept 0-0, 003, Toulouse (Fance). [] Ph. Dessante, J-C Vannie, Ch. Ripoll, Optimisation of a Linea Bushless DC oto Dive, ICE 00, Septembe 00. [5] Ph. Dessante, J-C Vannie, P. Vidal, Optimisation of a Linea Bushless DC oto Dive and the Associated Powe Supply, AES 005 Civil o ilitay All-Electic Ship Confeence, 13 1 Octobe 005 Pais- Vesailles, Fance. [6] Rioux C., Théoie généale compaative des machines électiques établie à pati des équations du champ électomagnétique, Revue généale de l Electicité (RGE), mai [7] Nudin., Poloujadoff., Faue A., Synthesis of Squiel Cage oto : A Key to Optimisation, IEEE Tans on Enegy Convesion, Feb 199.