1、93FTM10High Speed, Heavily Loaded and Precision AircraftType Epicyclic Gear System Dynamic Analysis byUsing AGMA Gear Design Guidelines Enhanced byExact Definition of Dynamic Loadsby: K. Buyukataman, United Technologies Pratt /_.% !n_ 400_ 200- Xl.ss P Y2.00Ya.ooco Xa.ooX2.0 YI.85x. VariationintheMa
2、gnitudeofStressFigure 7 Variation of Bending Stresses Along the ContactSurface, as Investigated by Peterson and BaudHelical Gear Coordinate SystemFigure 5 Schematic of High Contact Ratio Version of The cartesian y, z coordinates shown in Figure 8 defineSuccessful Test Gears the beginning of contact
3、on the faces of a pair of teeth. This isalso the deflection of the beginning of the motion, andspecified as:S=x-jIM I At the centerline of the contact zone, we have S = 0, and_ -_ _ thus, x=j. This is shown in Figure 9. This figure also definesthe center of the y coordinate, y=0, which is also calle
4、d theK K instantaneous center of rotation. This location coincides with2 Nthejth tooth. Thus, at time zero, S=0, y= 0, z=0, which defines? / / / / f/ /1 the contact position. The load at time zero by definition isi r/ / / / r/ /i“ ( “ always set equal to zero within the GENESIS code.(a) K = k = spri
5、ng rate orstiffness of systemt = time required to yinsert wedge /_(error) or to (FTANtpIo-L)/2reach maximum force F/2 -/-,-_,= displacement of ,/ /i, , masses during load N2 _ /transfer _ Aa = accelerationof -(FTAN Smass during load TA/ I/ (F qJb+L)/2transferu= 1/2at 2 F = applied F = Facei/_._./_._
6、 tpb= Basehelix(tangential load) -F/2 0 = PressureangleFd = F + K(e-u) Fd total dynamic= (FTAN_b-L)/2load of motion(b) (a)Figure 8 Helical Gears: (a) Coordinate System, and (b)Figure 6 Basie LogieofLowand High Contact Ratio Gear Gear Mesh Motion, as Investigated by W. D.Impact Load, UsingTuplins Wed
7、ge Terminology MarkZ 8Z 0 Z 87-.0 0 carder and ring gear, as well as some other components)i - I p I Z 12 require tailoring of their stiffness and damping characteristics.L _. L,.I . i r J I _L using time dependent equations of motion. This, of course,yi_ _._ ._J_ I_:_i i-_ y requires Design Team ex
8、perience and judgement in system- design and simulation. Team efforts to reach the very best areI I I I essential in the frontiers of technology advancements.L-_A .J JPitch Plane I I I t I In this study, the term “stiffness“ refers to a function_b,r / _/-ToothNo.j I i. A I I which expresses the rate
9、 of application of load and the-/_ _ qJ F _ _ : L/2 following changes of TE in attached masses. “Stiffness“ is a/, , _ _ I I _ nonlinear system of differential equations. Similarly, the termr-_-_-I- _- ,_-_, “damping“ refers to the outcome of the relative motion, “ “sF-0-5“_-s-“t-T- _ “_P _ H-J- _ b
10、etween load transfer components and/or to the combinedL-_-/_ ! effect of the system of masses. “Damping“ in this model is also F- - greatly affected by the shear forces of the fluid films whichy , exist within the gear meshes and bearing surfaces.I q Lq_oooo, b- F/2 -I_ F/2 -.-4 Contact System Model
11、The dynamic model of the gear system is set to satisfy(b) design considerations. A detailed description of the model isFigure 8 Helical Gears: (a) Coordinate System, and (b) beyond the scope of this paper. However, a partial schematicGear Mesh Motion, as Investigated by W.D. illustrating the system
12、model is shown in Figure 10. TheMark (Continued) number of planets in this model can be any number greaterthan or equal to one. Figure 10 shows a two-dimensional crossy ToothPairLI2 IJ2 section of one planet and its immediate surroundings. Eachi No.j=O I1 , 21 3 4 5 6- ,- - -/- -/r-t-_- r- -/- -7 -
13、“7“ component in the system is represented by rotational and/ / translational masses, Mr and Mt.AI I / / / /-“ q / I / rZoneof /1=/2 t/ /l A I/ Contact / / / /,p./J_Y/ / / / / / Ir_“ “/ / , / - x Planetn Planet geart/o ,/z_,/_y3z_H/4_,/5A / /I/ / _b/I / / / PlanetcarrierF;/ / / “_/ ! / / / / / _Pn/
14、/ / ,/ / ./ / /“/ /_.z _ V Z-_ / L_ Sungear-_ e_nRinggearxi- I_, KZ-L+l_-. r, O=Pressureangle_ 9Figure 9 Transfer Functions, as Investigated by W.D. %P_.aMark _lane_ :LDetail tooth load analysis is beyond the scope of this _-paper. However, the above defines the beginning of all /analysis and system
15、 simulation. Analysis performed by W. D. Mark shows that the random component of thetransmission error (TE) is the sum of random components of _“the system which may be expressed using Fourier series. This Planet i+1 _ dcomponent has a direct contribution to random fluctuations of _o rP_dynamic load
16、 and is also time dependent. The peak value K rdefinition requires activation of data banks stored in Darklinesrepresent: rzGENESIS. - Sungear and splineconnection- Planet and journal bearing connectionCritical Issue Note:One can study and simulate different effects of a real K = Spring rate represe
17、ntationsystem consisting of given masses, rotations, time, and time C = Damping rate representationdependent forces using a coordinate system as previously 0 = Phaserepresentationdefined. However, it should not be forgotten that industryproven design features (e.g., floating sun, flexible supported
18、Figure 10 Partial Schematic of System Model5The gear mesh analysis is the most complex input to the important to distinguish between the higher harmonics ofsystem dynamic analysis. The advanced code used in the mesh higher modes. Both consist of high frequency vibrations butanalysis is based on the
19、various special studies of NASA and resultant motions are physically different.consortiums. This code, called GENESIS, in its final formcontains all that is known by the writers in the art of Spur and To understand the higher harmonics of vibrations, weHelical gears. GENESIS uses various experimenta
20、l tables of must examine the number of amplitude nodal cylindersNASA,AVSCOM,and other sources duringitsanalysisthrough concentric with the axis of each gear. The first harmonics aresystematic referral. GENESIS is an artificial intelligence the simplest vibration for which reflection occurs. Therouti
21、ne developed to be used for design and optimization of reflected waves combine to produce one nodal cylinder.the aircraft main drive gear meshes. GENESISis used today Higher harmonics are of increasing complexities, the waveswithin the first coordinate system which is the center block of combining t
22、o give two, three, etc., nodal cylinders.system simulation.Brief explanations on the subject can be found in Refs. 31The carder needs to be modeled as a subsystem which and 32. The motion acrossthe cross section of a gearvibratingconsists of a central mass with various de-central masses, in the form
23、s of the firstfive lower modes is shown in Figure 11.based on planet locations and torque flow. All supports withinand outside the carder need to be simulated using the classical rapproach of time proven damperand spring systems which are _-_)- _0functions of time. O_-_ _. /f-The equations of motion
24、 were based on the selectedquadruplet coordinate system and were a direct application ofthe concepts of mass, distance, time, and force. Stress wavesand their interaction carded Newtons laws and the Lagrangeequations one step further. However, the basic equation (a) “_/ r (b)remained in the form def
25、ined decades ago. f_-I J ue=0; iJz=_-O/-F=ma+kx+ex O_ Ur=Ue=Uz=Owhere: F= Fd*sin(wt) or F = Ft + Fd*Cos(wt) ue_0;Ur=Uz=O-_p/ I /_ UyStrained _._._-_- _ _p4-. :.-_-r-_i“ - -Laxisin _ I -%,Tneutral Strained“neutralaxisEnhancements and explanations: elementarytheory-_ _. i _j/_-in p_c theoryx = f(TEi,
26、DT, t) r _() rDT = distance travelled O_- Nodalplaneof 0 “_TEi instantaneous transmission error _,_ _radialmotion _TEi = TEx (load dependent) + TEy (random)Ct = f(t), instantaneous compliance of meshF, Fi = instantaneous load x,._ _ _ _wt = phase angleV1, V2 = instantaneous gear deflections in direc
27、tionparallel to plane of contact (d) “_-_-_- )El, E2 = components of static transmission error on theteeth (a) Longitudinalmode; n=0;Uo=0;Ur=0;Uz=0(b) Torsionalmode;n=0; Ur=Uz=0; Uo=0O1, 02 = componentsof relativedisplacement (e) Transversemode;n=lR1, R2 = base radii of gears (d) Screwmode;n=2TEx, T
28、Ey = componentsof transmissionerror (e) Nodesignation; n=3Ft = static loadFd = vibratory or dynamic load. Figure 11 Motion Across Cross Sections of CylindersVibrating in the Fundamentals in the First FiveModal Analysis Modes (Schematic). “u“ is the directionaldisplacement, “n“ is the number of nodal
29、Analysis of the systems basic dynamic characteristics diameters.requires accurate determination of its natural frequencies andmode shapes. A dynamic analysis of the given system of planet gearsrequires that the system should be converted to its equivalentA detailed example of the basic principles of
30、 gear matrix form using stif_ess and mass properties. The use ofdynamics is given in Ref. 31. We must be very cautious and damping characteristics in the matrix is optional to definealert during examination of high frequency waves, since it is nodal shapes and natural frequencies. However, once theI
31、above isknown, Refs. 31 and 32 define what needs to be done Major Model Componentsif the excitation occurs within the operating envelope andbecomes destructive to safe operation. Gears and their mesh stiffness havethe greatest influenceon the outcome of natural frequencies, modes, andSeveral program
32、 packages exist to solve the matrix of amplitudes. A proper mesh analysis is the key to real lifeequations. However, interpretation of the results is usually matching results. High contact ratio (HCR) gears, byvery difficult. Interpretation requires experience and definition, come with flexible toot
33、h structures and withbetterknow-how in planetary gear system dynamics. Figure 12 is a load sharing characteristics than conventional gears. Thisgraphical representation of a typical matrixsolution showing provides increased damping and stabilization throughout thethebasiemodeshapeofthesystemwhichmay
34、beconsidered operating range. Ref. 34 states that the dynamic energyequivalent to Figure ll(e) of the basic single gear mode. exchange of HCR gears is less than 10 percent of standardtooth proportion gears. This is confirmed by the systemsimulation.-_“N_, Journal bearings are an essential part of th
35、e system_- . dynamic analysis. A system dynamic analysis cannot be doneI without considering the dynamic film thickness of the ovalized:_ _._“ bearings Figure Displacement gear massshown in 14. ofcenters is greatly affected by the eccentricity of bearing mass“ -_“ _i centers. Displacement of the bea
36、ring mass centers is also ai,.“_ “_ ./ direct result of loads imposed on them by the gears. In a _,_7 c- dynamic system, the engineer takes these loads and first determines the fluid film pressure at a particular time, thenapplies the equations of motion to find the position of the_. _/ planet mass.
37、 A closed-loop computer simulation nsuallyhomes in on the final location at that particular instant of time. “ and defines mode shapes, natural frequencies and theiramplitudes.Figure 12 Equal Mass Displacements , ,.,., _ ,. ,_It is emphasized here that all planets in the system do not “. “ _ “ ;.,ne
38、cessarily displace in the same rotational direction. A typical j_ _ _ _,_,:example of this is shown in Figure 13, which similarly could beconsidered as an equivalent to the single system of Figuresll(b) and 1l(e). The mode shown in Figure 13 not only affects _,_.“_/; _- Zeroloadjournalsurface _ h_,“
39、the value of each tooth mesh load, it also has a great influence _ _5/,:,)/ _._.-on whirl and load sharing among the planets. An example of _ _l_l _“the above is shown in Ref. 33. _g_,_c, _. . _ “ -t “ Resultantload and )_l_:6 ovahzatlon directionFigure 14 Actual Operating Conditions RepresentationD
40、esign CharacteristicsThe system design characteristics (e.g., special featuresto improve load sharing) affect the resonating frequencies andFigure 13 Unstable Load Distribution their amplitudes. System parameters need to be set to considerthe floating sun, the flexible ring gear, the journal bearing
41、s, theDetermination of the exact position of each planet double helieals, etc., so that dynamic loading of the geardependsonthedeterminationoftheexaetpositionofitsmass system will simulate reality and/or mateh with thecenter. Here, the planet mass center happens to coincide with experimental data (R
42、ef. 27). Variations induced by speeialthe mass center of the journal bearing on which it rides, design features and manufacturing errors always havenoticeable influence on dynamic loading, mode shapes, Systems-l), Ref. 26, is used to simulate the dynamics of afrequencies, vibration amplitudes, and u
43、ltimately, sizing of the single-stage planetary gear with five planets and a fixed ring.final design. The gears are double helical gears, and the input and outputare the sun gear and carrier, respectively.Thus, the precise modeling of gear system dynamics asflexible solid links interacting with mult
44、i-contact lower and Figures 15 through 26 present simulation results for thehigher pairs requires complete analysis of each flexible link third planet and sun and first planet and rign mesh. These(gears, carriers, etc.) and of their contact points at each instant results include plots of the ratio o
45、f dynamic load over staticin time, and developing the corresponding non-holonomie load, PV index, Hertz stress, flash temperature, and contactconstraints. This is a formidable task given the present stress on both its meshes with the sun and ring gears. Otherknowledge of dynamic modeling of flexible
46、 bodies and with planets have rather similar plots phased accordingly as thetodays computational facilities, mesh proceeds. These plots illustrate trends in the stress andtemperature that are relatively accurate in narrow strips ofA reasonable alternative is to approximate the system time; a) Figure
47、s represent simulation of Helical gears withwith lumped masses connected by springs and dash pots. Spur gear type of representation; b) Figures use the trueDetermination of the degrees of freedom, the number of Helical gearrepresentation. However, further examination oflumped masses, spring and dash
48、 pots, and evaluation of the the simulation results leads to some concerns as explainedcorresponding compliance and damping values are then further.central to any such modeling.Figures 27 through 36 contain the individual tooth loadThe mass-spring model is the main building block of all transfer and
49、 also the gear load transmitted between the sun,existing gear dynamic simulation packages (Ref. 24). The planets, and ring. Changes in the load transmission at theprogram GRDYNMULT (Ref. 25) is recognized as the most beginning and end of each tooth mesh, as seen in these plots,versatile in system size, type and analysis capabilities, indicate a large jerk (third time derivative of motion)
