1、14FTM20 AGMA Technical Paper Influence of Central Members Radial Support Stiffness on Load Sharing Characteristics of Compound Planetary Gearsets By Z. Peng and S. Wu, Wuhan University2 14FTM20 Influence of Central Members Radial Support Stiffness on Load Sharing Characteristics of Compound Planetar
2、y Gearsets Zeming Peng and Shijing Wu, Wuhan University The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract Abstract: In this study, a non-linear dynamics model of R
3、avigneaux compound planetary gearsets which adopts the intermediate floating component is set up based on concentration parameter. By considering the position errors and eccentric errors, the dynamic load sharing factors of the gearsets are calculated. The relationship between central members radial
4、 support stiffness and the dynamic load sharing factors is obtained and the influence of central members radial support stiffness on load sharing characteristic is analyzed. The research results show that central members radial support stiffness affects the gear pairs which are directly contacted to
5、 the central members, while the effect is rather small in the gear pairs which are not directly connected. Reducing the radial support stiffness of the central members helps improve the load sharing performance of the system. Copyright 2014 American Gear Manufacturers Association 1001 N. Fairfax Str
6、eet, Suite 500 Alexandria, Virginia 22314 October 2014 ISBN: 978-1-61481-112-1 3 14FTM20 Influence of Central Members Radial Support Stiffness on Load Sharing Characteristics of Compound Planetary Gearsets Zeming Peng and Shijing Wu, Wuhan University Introduction Planetary gearsets, a classification
7、 of epicyclic gears, have several advantages over fixed-center counter-shaft gear systems, including higher power density (transmitted power to gearsets volume ratio), compactness, ability to achieve multiple speed ratios through different power flow arrangements, and lower gear noise. In addition,
8、axi-symmetric orientation of the planet gears in the gearsets creates negligible radial bearing forces and provides a self-centering capability. This relieves the requirement for bearing support. Based on the above advantages, planetary gearsets have been widely used in transportation, aerospace and
9、 energy development areas. However, these advantages of planetary transmissions rely heavily on the assumption that each pinion carries an equal share of the total torque applied. In the production process, gear manufacturing and assembly variations, as well as design parameters may prevent such equ
10、al load sharing characteristics affecting the transmission performance. The majority of published studies on dynamic load sharing focus on one-stage planetary arrangements. Hidaka, et al., 1-3 studied the planet load sharing of three-planet gearsets to show, both experimentally and theoretically, th
11、at perfect load sharing in a three-planet gearsets is achievable only if at least one central member (ring gear, sun gear, or carrier) is allowed to float. The same conclusion was confirmed by Muller 4. Kahraman 5, in his 1994 paper, constructed a dynamic mathematical model of a planetary gear stage
12、, which could be set to an arbitrary number of planets and corresponding possible gear sizes and tolerance variations, and fixity or not of the sun gear. In another paper 6, Kahraman considered load sharing of planetary gearsets again, both in a mathematical model and in experimental work, and showe
13、d reasonable agreement between his experimental results and mathematical model for a four-planet system. Ligata et al., 7 did further work along the lines of the paper of Kahraman 6 but added the numbers of planets and torque as parametric variables in their experimental study and obtained reasonabl
14、e agreement with the theory. In Ligatas work, they demonstrated in an experiment that three-planet systems show excellent load sharing and that four-planet systems with the planets opposite each other show good load sharing between opposed planets, but not so good otherwise. They also mentioned, and
15、 one can see in the data plots, that for constant error and other variables, with the torque held constant, load sharing gets better for higher torques. Bodas and Kahraman 8 used a two-dimensional (2D) deformable-body model of a planetary gearsets and demonstrated theoretically that adding more plan
16、ets makes the system more sensitive to certain gear and carrier manufacturing errors and assembly variations. They showed that different types of errors acting on each planet could be combined into a total planet error eirepresenting the effective tangential (in the circumferential direction on the
17、circle formed by planet centers) error of planet i. Singh 9-10 used a three-dimensional (3D) model of the same configuration to obtain similar conclusions. He showed that the directions of the pinhole position errors are important, with the errors in tangential direction having the most critical imp
18、act on planet load sharing. He concluded that increasing the number of planets in the system without appropriately tightening the pinhole position tolerances fails to deliver expected planet load reductions. His predictions clearly showed the maximum planet loads for an n-planet (n 4) system can bec
19、ome higher than the corresponding loads for a planetary gearsets with a smaller number of planets, unless the error magnitudes are appropriately controlled. All of the models cited above only focus on one-stage planetary gearsets. The demand for fuel economy and more ratios for different speed and t
20、orque make vehicle automatic transmissions with compound planetary gearsets very desirable, while few scholars studied the load sharing characteristics of compound planetary gearsets. In this study, a dynamics model of Ravigneaux compound planetary gearsets which adopts the intermediate floating com
21、ponent is set up based on concentration parameter. By considering the influence factors, including central member radial support stiffness, gear eccentric errors, gear position errors and backlash, the load sharing factors of the gearsets are calculated. The curves of the relationships between centr
22、al members radial support stiffness and the load sharing factors of the 4 14FTM20 gearsets are obtained and the influence of central members radial support stiffness on load sharing characteristic is analyzed. Dynamic models Ravigneaux compound planetary gear transmissions are based on a simple plan
23、etary gear train with a clever combination, whose structure is more complex than a simple planetary gear train. The Ravigneaux compound planetary gear system studied in the paper is illustrated in Figure 1. A long planet b connects two planes of double-planet gearsets s1-a-b-r and s2-b-r. Here s1and
24、 s2are the sun gears, r is the ring gear, and a is the short gear. All planets, a and b, are supported by a single carrier c. The compound arrangements shown in Figure 1 have four central members, s1, s2, r and c that can be used as input, output or reaction members. At any given power flow conditio
25、n, only three of these four members will have assignments. Therefore, by applying different clutching arrangements, the input, output and reaction members can be selected in different ways from the four central members of the gearsets. This way, it is possible to obtain permutation P (4, 3) = 24 dis
26、tinct power flow configurations. With such inherent ratio flexibility, it is feasible to achieve up to five desirable forward and at least one reverse gear ration using the same compound gearsets under different clutching schemes. This allows significant reductions in transmission size and weight ma
27、king compound planetary units very desirable for such applications. The dynamic model of this system employs a number of simplifying assumptions. 1. Each gear body is assumed to be rigid and the flexibilities of the gear teeth at each gear mesh interface are modelled by a spring having periodically
28、time-varying stiffness acting along the gear line of action. This mesh stiffness is subject to a clearance element representing gear backlash. 2. Each central member was assumed to move in the torsional direction and radial x, y direction, while planets a, b were assumed to move in the torsional dir
29、ection only. 3. As the damping mechanisms at the gear meshes and bearings of a planetary gearsets are not easy to model, viscous gear mesh damping elements are introduced representing energy dissipation of the system. In Figure 2, the central members s1, s2, r and c, which are mounted on linear elas
30、tic bearings with the stiffnesses ks1, ks2, kr, kc, are constrained by torsional linear springs of stiffness magnitude ks1t, ks2t, krt, kct. The magnitudes of torsional stiffness constrains can be chosen accordingly to simulate different power flow arrangements with different fixed central members.
31、Each gear body l (l = s1, s2, r, c, anand bn) is modelled as a rigid disk with a polar mass moment of inertia Il, radius rland torsional displacement l. Here lis the vibrational component of the displacement defined from the nominal rotation of the gear. External torques Ti(I = s1, s2, r and c) are
32、applied to the central members to represent input, output and reaction torque values. The carrier inertia, Ice, is defined in equation 1. Figure 1. Ravigneaux compound planetary gearsets arrangements considered in this study 5 14FTM20 Figure 2. Dynamic model of Ravigneaux compound planetary gearsets
33、 22ce c a ca b cbIINmrmr (1)where Ic is polar mass moment of inertia of the carrier alone without planets; N is total number of planet sets a-b in the gearsets, ma is mass of planets a; mb is mass of planets b; rca, rcbare radii of circles passing through the centers of planets a and b, and are defi
34、ned as: ,coscos coss1 acarbA s2bBcbrrrrr r rr(2) The mesh of gear pair j (j = s1-an, s2-bn, r-bn and an-bn) is represented by a periodically time-varying stiffness element kjsubjected to a piecewise linear backlash function f (j) that includes a clearance of amplitude bj. A time-varying displacement
35、 function of ej (t) is applied along the line of action to account for position error and eccentric error. Loses of lubricated gear contacts are represented by constant viscous damper coefficient cj. Mesh errors analysis Pinion position error Figure 3 illustrates how the position of gear is changed
36、as a function of gear position error. O is the ideal location of s1 center. The sun gear position error of magnitude As1makes the ideal location point O move to O. The tangential magnitude of the position error of planet s1, in Figure 3, can be represented by equations 3 and 5. 6 14FTM20 Figure 3. P
37、osition error of As1sins1-s1an s1 c s1 nAA t (3) where c, s1, nare the angular velocity of carrier, the initial angle of position error and the planet spacing angles. 12nnN (4) Similarly, the tangential magnitude of the position error of planet s2, r, an and bn can be represented by: sinsinsinsinsin
38、sinsins2-s2bn s2 c s2 nr-rbn r c r 3 nan-s1an an anan-anbn an 1 anbn-anbn bn 2 bnbn-s2bn bn bnbn-rbn bn bnAA tAA tAAAAAAAAAA (5) 1, 2, 3and an, bn, rare illustrated in Figure 4. Figure 4. Geometric diagram of position errors 7 14FTM20 Eccentric error Figure 5 illustrates how the position of rotation
39、 center is changed as a function of eccentric error. The eccentric error s1 of magnitude Es1moves the gear rotation center from O to O. Similar to position error, the tangential magnitude of the position error of gears can be represented by equation 6, where is the initial angle of eccentric error.
40、sinsinsinsinsinsins1-s1an s1 s1 c s1 ns2-s2bn s2 s2 c s2 nr-rbn r c r 3an-s1an an a c anan-anbn an a c 1 anbn-anbn bn b c 2 bnnEE tEE tEE tEE tEE tEE t sinsinbn-s2bn bn b c bnbn-rbn bn b c bnEE tEE t (6) Dynamic transmission error By considering both position errors and eccentric errors, the tangent
41、ial dynamic transmission errors of gear pairs can be represented by ej(t) as s1an s1-s1an an-s1an s1-s1an an-s1ans2bn s2-s2bn bn-s2bn s2-s2bn bn-s2bnrbn r-rbn bn-rbn r-rbn bn-rbnanbn an-anbn bn-anbn an-anbn bn-anbneA A E EeA A E EeA A E EeA A E E (7) Figure 5. Eccentric error of Es18 14FTM20 Equatio
42、ns of motion The relative gear mesh displacements for the s1-an, s2-bn, r-bn, and an-bn are expressed in equations 8 and 9: sin cos cossin cos cossin cos coss1an s1 c s1an s1 c s1an s1 s1 c a an c b s1ans2bn s2 c s2bn s2 c s2bn s2 s2 c bB bn c b s2bnrbn r c rbn r c rbn bA bn c r r c brxx yy r r exx
43、yy r r exx yy r cosrbnanbn a an c bA bn c b anbnerr e (8) where s1an an s1s2bn bn s2rbn bn r (9) an,bnare the angle between the line which cross the ideal geometric center of an, bn and the coordinate origin O and the x axis of coordinate OXY. Define Pjand Djas the elastic and damp meshing force of
44、the gear pairs respectively, which can be expressed as follows. jj jjjjPkfDc(10) The piecewise-linear displacement functions are defined as 22() 0222jjjjjjbbbfbb (11) The equations of motion shown in Figure 2 are written as 112 sin cos sin cos 02 cos cos cos cos 0cos cos2s1 s1 c s1 c s1 s1an s1an b
45、s1an s1an b s1 s12s1 s1 c s1 c s1 s1an s1an b s1an s1an b s1 s1s1 s1 s1 s1an b s1 s1an b s1NNnnmx y x P D kxmy x y P D kyIrP rD k 11ts1 s1NNnnT(12) 112 sin cos sin cos 02 cos cos cos cos 0cos cos2s2 s2 c s2 c s2 s2bn s2bn b s2bn s2bn b s2 s22s2 s2 c s2 c s2 s2bn s2bn b s2bn s2bn b s2 s2s2 s2 s2 s2bn
46、 b s2 s2bn b s2NNnnmx y x P D kxmy x y P D kyIrP rD k 11ts2 s2NNnnT(13) 9 14FTM20 11 11112 sin cos sin cos sin cossin cos sin cos sin cos 02cos2ce c c c c c s1an s1an b s1an s1an b s2bn s2bn bs2bn s2bn b rbn rbn b rbn rbn b c c2ce c c c c c s1an s1aNN Nnn nNNNnnmx y x P D PDP kxmy x y P 11 111111cos
47、 cos cos cos coscos cos cos cos cos cos 0cos cos cosn b s1an s1an b s2bn s2bn bs2bn s2bn b rbn rbn b rbn rbn b c cce c ca s1an b ca s1an b cb s2bn b cb s2NN Nnn nNNNnnnNNnnDPDPDkyIrPrDrPrD 11 111 1cos coscos cos cosbn b cb rbn bcb rbn b ab anbn b ab anbn b ct c cNN Nnn nNN Nnn nrPrD rP rD k T (14) 1111112 sin cos sin
