1、 STD-AGMA S4FTM2-ENGL 1994 = Ob87575 0004543 352 94FTM2 An Analytical Method for the Calculation of the Efficiency of Planetary Gears by: Michel Pasquier, Gear Department - CETIM, France and Pierre Foucher, Ecole Suprieure de 1Energie et de Matriaux ORLEANS, France American Gear (VVVVVVVVV Manufactu
2、rers Association TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesAn Analytical Method for the Calculation of the Efficiency of Planetary gears Michel Pasquier, Gear Department - CETIM, France and Pierre Foucher, Ecole Suprieure de IEner
3、gie et de Matriaux ORLEANS, France me statements and opinions contained herein are those of the author and should not be conshued as an official action or opinion of the American Gear Manufacturers Association. ABSTRACT This paper presents a synthesis of an analytical method for the calculation of t
4、he efficiency of simple or compound planetary gear aainS. It is based on fundamentals formulae. It is intended to improve the accuracy of the rating of the efficiency of pianetary gears to be included in a caicuiation of the thermal capacity. Copyright O 1994 American Gear Manufacturers Association
5、1500 King Street, Suite 201 Alexandria, Vrrginia, 223 14 October, 1994 ISBN 1-555894367 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGMA SLiFTM2-ENGL 1774 b87575 0004543 125 W AN ANALYTICAL METHOD FOR THE CALCULATION OF THE EFFICIENCY OF PLANET
6、ARY GEARS MICHEL PASQUIER - Dr PIERRE FOUCHER“ +GEAR DEPARTMENT - CETIM (FRANCE) *+ Ecole Suprieure de IEnergie et de Matriaux ORLEANS (FRANCE) 1 INTRODUCTION In the industrial field, planetary (or epicyclic) gears are generally designed for high speed applications such as turbo gear drives. They ar
7、e also used when very high torque must be transmitted at medium and low speed (vertical cement mill . ). Sometimes planetary gears are appropriated when dimensions of a gear unit with an equivalent gear ratio or transmitted torque is excessive (manufacturing, transportation.), for example marine gea
8、r for high torque Diesel engine. Planetary gears are very often applied in automotive industry, such as trucks gear boxes, automatic automotive gear boxes and axle drives. In the industrial field the arrangements of epicyclic gears is generally quite the same. In special applications, specific arran
9、gements are designed. But in automotive industry, arrangements are more complex and very different. These arrangements are intended to realise many ratio, to make compatible the dynamic of the engine (torque vs speed) with respect to the dynamic of vehicle. That is the reason why the efficiency of p
10、lanetary gears has been studied by gear designers of automotive industry. 111, 121. The efficiency of industrial planetary gears is rarely one of the main problems. This is quite different in automotive industry and for special applications such as spatial application where a very high ratio is requ
11、ired in conjunction with a low number of meshes and a minimum amount of soacai The above mentioned studies proposes graphical methods, which sometimes may be difficult to use, even for an experienced gear designer. More recently to improve the knowledge, automotive gear designers have developed more
12、 accurate and more user friendly methods i31. This paper presents a synthesis of an analytical method, which may be included in the design stage of a planetary gear, because it is based on very well known fundamental formulae of planetary gears. 2 FUNDAMENTAL FORMULAE OF THE MOTION OF A PLANETARY GE
13、AR TRAIN 2.1 KINEMATICS FORMULAE In order to simplify the following explanations, let us consider a simple planetary gear like those drawn on figure 1. It is made of two kinds of gears: One external gear (1)-(4) and one internal gear (4142) It is obvious that the results may be applied to other plan
14、etary gear trains including more than two types of gears. Let us consider two coordinate systems - I?, (O,Z0 ,yo ,io) related to the frame (Reference coordinate system). - R, (O,?, ,i3 ,f3) related to the planet carrier. (relative coordinate system) The motion of R, with respect to R, may be written
15、 as follows: COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesIn the relative coordinate system R3 (O, ji3 , p3 ,I3 ) , an observer located on the planet carrier can see that this one has an equivalent function to a housing for the gears. (The planet ca
16、rrier is fixed in this coordinate system R3). So, in this coordinate system R3 the motion of a planetary gear is the same as a gear unit with two gear sets connected together. I Figure 1 - example of simple planetary gear It is assumed that: - w : is the rotation speed of the component i (in rdls) w
17、ith respect to the cornponentj . - The number of teeth are always positive. In the coordinate system R3 , the gear ratio ub of the first gear set (1 1- (4 may be written as follows: . (1) O13 - O10 -O30 “4 =bZE - - O43 O40 21 where Z; = number of teeth of the lSt planet gear 2, = number of teeth of
18、the sun gear E = -I if (1 - (4) is an externat gear set E = +1 if (1 - (4) is an internal gear set The gear ratio of the second gear i 4) - (2) may be written as follows: . (1“) =zq *, =a10 -030 vaio =a20 -030 aio =o. . A training motion with a rotation speed equal to the rotation speed of the plane
19、t carrier O 3o 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services2.2 STATIC FORMULA 3 CALCULATION OF THE EFFICIENCY It is assumed that the rotation speed of each component is constant, and so the dynamic moment of the planetary gear is equal to O. So t
20、he dynamic moment theorem applied to the planetary gear may be written as fol1ows:ialgebraic formula) E, +E2 +E, =o=.c, =-(cl +C2) 2.3 ENERGETIC FORMULAE , . . (6) Assume that: eo =lOlo = Input power (O) or output power (O) or output power i O) or output power i O1 E = Output power on driven shaft (
21、O) Applying the principle of conservation of energy in the reference coordinate system Ro, we obtain:(algebraic formula) this without any consideration on the arrangement. 3.3.3 Star power is lower than input power If the input star power 6, or pZ3 is lower than the actual input power Fm, it exists
22、a “sharing of power“. That means the efficiency of the planetary gear is higher than those of a gear unit that have the same ratio. 3.4 APPLICATION TO COMPOUND PLANETARY GEARS The above method may be applied to compound planetary gears. Additional formula have to be written especially kinematics and
23、 static. In the simple case of a coupled planetary gear, the efficiency should be calculated for each of the simple planetary gears of the coupled planetary gear. The total efficieiicy is equal to the product of the efficiency of each simple planetary gears. To couple each simple planetary gears the
24、 dynamic moment theorem has to be applied on the common shaft which connects each planetary gears. Sometime this is not very easy to write the formula especially when a coupling or a joint is used because its efficiency has to be introduced like the “equivalent meshing efficiency“ of each gearing. 3
25、.5 SIMPLIFIED METHOD A simplified method may be developed when the following exists: - the star ratio of the planetary gear (simple or compound) is superior to 1. -The star efficiency qb is close to 1. The star power 6, may be easily written as a function of the actual power 6, or pz0 The efficiency
26、 may be written as follows: n = number of meshes 4 EXAMPLE OF CALCULATION . (17) A simple fixed differential gear is shown in the following figure 4. (3) is the input shaft and (2) is the output shaft. (1 ) is the fixed gear. A differential gear enables a very high ratio with only few gears and a mi
27、nimum amount of space. The fundamental equations of the simple four differential gear are as follows: 4.1 Hypothesis and assumptions. input = 131 output = 121 fixed = i11 4.2 Equations The star ratio is: 5 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servic
28、es STD-AGHA SqFTflZ-ENGL 1774 W Ob87575 0004546 707 D end the cmputation of the gear ratio gives: “ =%=I- I O30 b cornputation of the star efficiency The computation of the efficiency using detailed method gives: (l), (7), (9), (1 1) and (1 5) llb -l) hbUb -I) ?= Figure 4- Fixed differential planeta
29、ry gear 4.3 Numerical application: First, assume that the number of teeth of each annulus is respectively equal to z1 =78andz2 =73. Let us assume that: - the output torque is always considered equal to c2 = -94915.25rnN(-840,000/b.in) - the input rotation speed is always considered equal to 030 =157
30、rd.s-+0 The ratio is equal to: 65 I 60 I 1,014 10.0138 22 I 19 I 1,08 I 0,074 The star power p13 and Pz3 are equal to: -14,283,021 14,871,950 -14.1 13,991 14,695,951 22 -1 3,251,470 13,797.865 The efficiency is equal to: 0,048 1 ,O4 0.26 1 ,O4 0,66 5 CONCLUSION Since this method is an analytical one
31、, it can be easily included as a part of a software for the design of planetary gears. That means the calculations which are sometimes complex, depending on the arrangement, becomes more easily to undertake. This method gives an accurate value of the efficiency. It is quite obvious that it depends o
32、n the accuracy of the value of the efficiency determined for all individual components of the planetary gear train. This accuracy becomes now very important with one of the mains topics of research and standardisation: The thermal capacity of industrial gears. In the case of planetary gears, the mai
33、n source of heat is the planetary gear train itself. REFERENCES l I Le train plantaire: De la cinmatique au rendement - Tomes 1 et 2 G FLEISCHEL - S.1.A 21 Les trains picycloidaux P RAVIGNEAUX i31 MECANIQUE DES TRAINS PLANETAIRES DES BOITES DE VITESSE - 1 partie: Etude cinematique, dynamique et energetique des trains plantaires engrenages P FOUCHER P LEPELLETIER i41 DUDLEYS GEAR HANDBOOK second edition Dennis P TOWSEND Mc GRAW HIIL -C.L.E.S.I.A - 1962 S.1.A - C.L.E.S.1.A . . 6 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services
