Dynamic modelling of planetary gears of automatic transmissions

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DOI: 10.1243/14644193JMBD138

2008 222: 229Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body DynamicsM Inalpolat and A Kahraman

Dynamic Modelling of Planetary Gears of Automatic Transmissions

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229

Dynamic modelling of planetary gearsof automatic transmissionsM Inalpolat and A Kahraman∗

Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio, USA

The manuscript was received on 17 January 2008 and was accepted after revision for publication on 29 May 2008.

DOI: 10.1243/14644193JMBD138

Abstract: In this paper, a generalized dynamic model for multi-stage planetary gear trains ofautomotive automatic transmissions is proposed. The planetary gear train is formed by N numberof planetary gear sets of different types (single-planet, double-planet, or complex-compound),connected to each other in any given kinematic configuration. In addition, each planetary stagecan have any number of parallel planet branches. A generalized power flow formulation and a gearmesh load distribution model are used to determine the stiffnesses, displacement excitations, andfundamental frequencies at the gear mesh interfaces. The natural modes are computed by solvingthe corresponding eigenvalue problem. The forced vibration response to gear mesh excitations isobtained by applying the modal summation technique. At the end, the model is applied to a three-stage planetary gear train representative of an automotive automatic transmission applicationto demonstrate the influence of coupling stiffnesses and the kinematic configurations on thenatural modes and the dynamic response.

Keywords: automatic transmissions, planetary gear sets, gear dynamics

1 INTRODUCTION

Planetary gear trains are used extensively in auto-motive and heavy truck automatic transmissions androtorcraft drive trains. They present various advan-tages over their counter-shaft alternatives. First ofall, they transmit more power per volume since thepower is split into a number of identical parallel planetbranches. Their axisymmetric configuration requiresvery little radial bearing support since the radial gearmesh forces cancel out [1], allowing some of the centralmembers (typically the sun gear) to float so that sen-sitivity of the gear set to various manufacturing errorscan be minimized [2]. This also leads to reduced noiseand vibration levels [3]. Finally, their co-axial design iswell suited for them to be incorporated with wet andone-way clutches, and friction bands so that input,output, and reaction members of the gear train can bevaried conveniently to achieve multiple gear (speed)ratios from the same gear train.

∗Corresponding author: Department of Mechanical Engineering,

The Ohio State University, 201 W. 19th Avenue, Columbus, OH

43210, USA. email: [email protected]

Recent trends in automatic transmissions are to-wards more gear ratios (six or more) as well as higheroperating speeds. A higher number of gear ratios isachievable with multi-stage (three or more) plane-tary gear trains operated under elaborate kinematicplanetary gear configurations. These planetary stagesare connected either between their central membersthrough structures such as shafts or hollow tubes orthrough common planets and carriers between stages.In early phases of product design, several candidateconcept cross-sections must be evaluated for theirdynamic behaviour and compared with those of thetransmissions of the competitors. As each gear ratiorepresents a different kinematic configuration, andhence a unique dynamic behaviour, these new trans-mission concepts and competitive transmissions mustbe studied in each gear ratio separately. This is avery time consuming task when it is done manuallythrough customized models of each transmission ateach gear ratio. Development of a generalized dynamicmodel of an N -stage planetary gear train under anykinematic configuration (defined by input, output,and reaction members and connections) that wouldallow such analyses effectively is the main focus of thisstudy.

JMBD138 © IMechE 2008 Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics

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230 M Inalpolat and A Kahraman

Gear dynamics literature includes a number ofdynamic models for single-stage gear sets [4–8].These models vary in terms of the degrees of free-dom included (purely torsional, two-dimensional, orthree-dimensional) as well as the gear mesh modelsemployed (linear time-invariant models with con-stant gear mesh stiffnesses and no backlash, lineartime-varying models with fluctuating mesh stiffnessand no backlash, or non-linear time-varying mod-els with both mesh stiffness fluctuations and gearbacklash included). Meanwhile, published studieson dynamic modelling of compound and complex-compound planetary gear trains are very limited[9, 10]. Perhaps, the first extensive dynamic model oncomplex-compound planetary gear sets was proposedby Kahraman [9]. In this model, two most commontypes of complex-compound planetary gear arrange-ments used in automatic transmissions (Ravigneauxand another long planet arrangement) were mod-elled through sets of dynamic models representingeach power flow achievable from these gear trains.He used purely torsional models to predict the naturalmodes of each configuration at each power flow condi-tion. These power flow configurations included single-and double-planet arrangements as well. He alsoshowed that the asymmetric planet modes could beseparated from overall (axisymmetric) modes and pro-vided simplified closed-form expressions for naturalfrequencies and mode shapes.

To the best of the knowledge, no published study isavailable on generic modelling of planetary gear trainsformed by any number of stages as in automatic trans-missions. Therefore, the main goal of this study is todevelop a general-purpose dynamic model of an N -stage planetary gear train consisting of single-planet,double-planet, or complex-compound planetary gearsets. As the main goal of this study is to demon-strate the modelling approach and provide a tool thatcan be used in the early stages of design, torsionalmodels of Kahraman [9] will be used for each stageof the gear train. Meanwhile, if desired, expandedtwo-dimensional [4] or three-dimensional [5] modelscan also be used within the same general modellingframework proposed in this study.

Since each gear ratio represents a different powerflow condition, gear mesh frequencies and gear meshforces transmitted are different at each gear ratio. Inaddition, the average stiffness and the transmissionerror excitation of a gear mesh are dependent onforce transmitted by it. Therefore, the generality of thedynamic model to be developed depends heavily on itscapability of computing the static gear mesh forces aswell as the mesh stiffnesses for each gear ratio implic-itly. Therefore, this study aims at not only developinga lumped-parameter dynamic model of a multi-stageplanetary gear train consisting of any combinationof single- or double-planet planetary gear sets, but

also incorporating the gear dynamics model with ageneralized kinematics and power flow formulationsfor computation of gear mesh frequencies, gear meshforces transmitted as well as a gear contact modelfor an automated computation of stiffnesses and thetransmission error excitations of both internal andexternal meshes of each planetary gear set implicitly.

2 MODELLING METHODOLOGY

A dynamic model of an N -stage planetary gear trainwill be proposed in this section. This discrete modelwill use a simplified representation of each gear meshby considering a linear spring acting along the lineof action of the gear mesh. Generality of this modeldepends on its ability to compute its parameters thatare implicitly dependent on gear range, input speed,and load transmitted. It is known that the value ofthe gear mesh stiffness changes in a non-linear fash-ion with the transmitted force. Therefore, given theinput torque to the gear train, force transmitted byeach gear mesh must be determined before comput-ing the mesh stiffnesses. The same is true for the gearmesh transmission error excitations. Also consideringthat the different power flow configurations at differ-ent gear ratios result in different gear mesh forces,computations of the gear mesh forces and stiffnessesmust be repeated at each input torque value and eachgear ratio. In order to automate these computations tomake the model general, the modelling methodologyshown in Fig. 1 is utilized. This methodology combinesfour different models.

1. A kinematics formulation to compute gear rota-tional speeds and gear mesh frequencies.

2. A power flow computation formulation to deter-mine the forces transmitted by each gear mesh asa function of input torque and the gear range ofinterest.

3. A gear contact model to predict the stiffness andtransmission error of each internal and externalgear mesh at a given transmitted force level.

4. A multi-stage planetary gear dynamics model forpredicting free and forced vibration behaviour ofthe gear components of the transmission.

The following sections describe each of these fourmodels in the sequence they appear in Fig. 1.

2.1 Kinematics formulation

Here, the formulation proposed by Kahraman et al.[11] is followed for the calculation of gear speedsand gear mesh frequencies for any given transmissionconfiguration. There are two major planetary arrange-ments used in passenger car automatic transmissions,single-planet and double-planet planetary gear sets as

Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics JMBD138 © IMechE 2008



Modelling of planetary gears 231

Fig. 1 Flowchart of the automatic transmission planetary gear dynamic modelling methodology

shown in Figs 2(a) and (b). Each gear set is formedby an n number of identical planet branches (oneplanet p in Fig. 2(a) and two meshing planets b andd in Fig. 2(b)) that are simultaneously in mesh with acentral external gear s and/or a ring gear r. All of theplanets are held by a common carrier c. Each planetis free to rotate relative to its carrier. The single-planetplanetary gear set shown in Fig. 2(a) is formed by twobasic kinematic units: an external gear mesh (p−s−c)consisting of the sun gear, a planet gear, and the car-rier and an internal gear mesh (p−r−c) formed by aplanet gear, the ring gear, and the carrier. Kinematicequations governing these kinematic units are given as

�sZs + �pZp − �c(Zs + Zp) = 0 (1a)

�r Zr − �pZp − �c(Zr − Zp) = 0 (1b)

where �j is the absolute angular velocity of a com-ponent j and Zj is the number of teeth of gear j. Thekinematic equations for the double-planet planetarygear set as shown in Fig. 2(b) are given for the threebasic kinematic units, b−s−c, d−r−c, and b−d−c as

�sZs + �bZb − �c(Zs + Zb) = 0 (2a)

�r Zr − �dZd − �c(Zr − Zd) = 0 (2b)

�bZb + �dZd − �c(Zb + Zd) = 0 (2c)

Another class of planetary gear sets, complex-compound gear sets, can also be found incontemporary transmissions, such as the Ravigneauxarrangement shown in Fig. 2(c). Here, two double-planet gear sets are connected through their carriersand one of the planets (planet b, called the long

Fig. 2 (a) A single-planet planetary gear set, (b) a double-planet planetary gear set, and (c) acomplex-compound gear set





planet). For the first double-planet gear set, basickinematic units, a−s1−c, a−b−c, and b−r1−c canbe handled by equations (2a) to (2c). Likewise, thebasic kinematic units of the second double-planet gearare b−s2−c, d−r2−c, and b−d−c are represented byequations (2a) to (2c).

Equations (1a), (1b), and (2a) to (2c) yield a numberof relationships equal to the number of gear meshesthat exist in a given stage of the planetary gear train.Additional relationships are provided by any con-nection between two members i and j that makesthem rotate at the same angular velocity, i.e. �i −�j = 0. They include carrier-to-carrier (�c1 − �c2 = 0)and planet-to-planet (�b1 − �b2 = 0) connections ofthe complex-compound gear sets. Additionally, anymember k that is fixed (held stationary) must havezero velocity, �k = 0. Finally, the speed of the inputmember is known in advance. Using these kinematicrelationships, a matrix equation A� = B is formed andsolved for the vector of unknown absolute rotationalspeeds �. Once the absolute speeds of the componentsare known, the gear mesh (tooth passing) frequency ofthe jth planetary gear set is defined as

ωmj = Zsj(�sj − �cj) = Zrj(�rj − �cj) (3)

2.2 Power flow formulation

Before the quasi-static gear mesh forces can be calcu-lated, torque values acting on each component mustbe determined. For the jth (single- or double-planet)gear set connected to the other planetary gear setsthrough a number of connections, the total externalmoments and powers should balance [11], that is

Tsj + Trj + Tcj +N∑

i=1i �=j

∑k

∑m

Tki/mj = 0 (4a)

Tsj �sj + Trj �rj + Tcj �cj +N∑

i=1i �=j

∑k

∑m

Tki/mj�mj = 0

(4b)

where k, m = s, b, d, r, and c. Here, Tsj , Trj , and Tcj arethe external torque values applied on central mem-bers (sun, ring, and carrier) of the jth gear set, N isthe total number of gear sets (stages) in the systemand Tki/mj is the torque applied by member k of stagei on member m of stage j. Writing equations (4a) and(4b) for each stage provides 2N equations while thereare 3N + 2Nrc unknown torque values, where Nrc is thetotal number of connections among stages. Additionalequations come from other known conditions. First,any central member is subject to zero external torqueif it is not an input, output or reaction member. Sec-ond, Tkj/mi + Tmi/kj = 0 for each connection. Finally theinput torque value is known Tin = T .

With these additional equations, the system of 3N +2Nrc linear equations is solved for the unknown exter-nal and connection torque values. Given these torquevalues, the gear mesh forces of each stage can be cal-culated by using the static equilibrium equations. Fora single-planet gear stage j, each of the static momentbalance equations for members sj, rj, and cj about therotational axis of the gear set is given in vector formas [11]

T sj + T con/sj +n∑

i=1

[r sj × F pj/sj]i = 0 (5a)

T rj + T con/rj +n∑

i=1

[r rj × F pj/rj]i = 0 (5b)

T cj + T con/cj +n∑

i=1

[r cj × F pj/cj]i = 0 (5c)

Here T j (j = s, r, c) is the external torque vector onmember j, T con/j is the vector of total connectiontorques applying on member j, F k/j is the normal meshforce vector applied by gear k on gear j in the trans-verse plane of gears, and F p/c is the planet bearingforce applied on the carrier by the planet again in thetransverse plane of the gears. Here r sj and r rj are theradius vectors of s and r defined from the centre of thegear to the pitch point. Similarly, r cj is defined fromthe rotational centre to the centre of a planet. Finally,n is the total number of planet branches and i is theplanet index. Given the values of T j and T con/j from thetorque formulation, equations 5(a) to (c) can be solvedfor tangential gear mesh forces as

Fpj/sj = 1nrsj

⎛⎝Tsj +

∑k,m

Tkm/sj

⎞⎠ (6a)

Fpj/rj = 1nrrj

⎛⎝Trj +

∑k,m

Tkm/rj

⎞⎠ (6b)

The equations similar to equations (5a) to (5c),(6a), and (6b) can be written for double-planet andcomplex-compound gear sets as well to find thegear mesh forces, once the torque values are com-puted [11].

2.3 Gear contact model

For the computation of the stiffness values of bothinternal and external gear meshes in the planetarygear train, a load distribution model (LDP), initiallyproposed by Conry and Seireg [12] and later furtherdeveloped by Houser [13], is used in this study. Thismodel is designed to compute elastic deformations atany point of the gear surface, given the tooth compli-ance, applied torque, and the initial unloaded tooth





separations. For the solution of the gear contact prob-lem, conditions of compatibility and equilibrium areconsidered. The condition of compatibility states that,for any point within the contact zone, the sum ofelastic deformations of two bodies and the initial sep-aration must be greater than or equal to the rigid bodydisplacement. The condition of equilibrium impliesthat the sum of the moments applied on a gear bodymust be equal to zero. The load distribution problemis solved iteratively in this model by using a modifiedSimplex algorithm. The mesh stiffness componentsdue to tooth bending, base rotation, and shear defor-mations are computed using simplified finite elementsformulations and combined with the compliance atthe Hertzian contact to predict the overall gear meshstiffness as a function of gear rotation. In addition, themotion transmission error of the gear pair is predictedas a periodic function at the mesh frequency ωm of thegear mesh.

2.4 Dynamic model

In this study, it will be assumed that the mesh stiff-nesses are time-invariant and tooth separations donot take place, both being reasonable assumptions forplanetary gear meshes formed by helical gears [14].These main assumptions allow a linear time-invarianttorsional model and linear solution techniques in thefrequency domain. However, the modelling method-ology applies to cases where non-linear time-varyingbehaviour is expected as well. In such cases, directnumerical integration of the equations of motionwould be required for a time domain solution. Theoverall dynamic model considers any combination ofa number of single-planet, double-planet, or complex-compound planetary gear connected to each othertorsionally according to a user-defined kinematicconfiguration.

Figure 3 shows undamped torsional dynamic mod-els for single-planet, double-planet, and complex-compound planetary gear sets. As mentioned earlier,the methodology provided here is valid with moreadvanced models of each gear stage as well. In Fig. 3,gears and the carrier are modelled as lumped inertias.Mesh stiffnesses are modelled by using linear springsacting along the lines of action. Any connections tothe housing are modelled by using additional torsionalsprings, which are depicted in Fig. 3 as linear springsfor clarity purposes.

2.4.1 Equations of motion

With a user-defined number of identical planetbranches n(i ∈ [1, n]), equations of motion for asingle-planet planetary stage shown in Fig. 3(a) are

given, based on the unforced dynamic model formu-lations of Kahraman [9], as

Ic θc +n∑

i=1

Ip θpic −n∑

i=1

[ksprcpδspi + krprcpδrpi]

+ kcgθc = 0, (7a)

Ir θr +n∑

i=1

krprrδrpi + krgθr = 0 (7b)

Is θs +n∑

i=1

ksprsδspi + ksgθs = 0 (7c)

Ip θpic + Ip θc + ksprpδspi − krprpδrpi = 0 (7d)

where the relative gear mesh displacements aredefined as δspi = rsθs − rcpθc + rpθpic + espi and δrpi =rrθr − rcpθc − rpθpic + erpi. In equations (7a) to (7d), anoverdot denotes differentiation with respect to time,rcp = rs + rp, and Ij and rj are the polar moment ofinertia and the base circle radius and of gear j (j =r, s, p), respectively. The total inertia of the carrierassembly with n planets Ic in equation (7a) is definedas Ic = Ic + nIp + nMpr2

cp where Ic is the polar massmoment of inertia of the carrier without planets, andplanet bearings, but with planet pins, rcp is the radiusof the circle passing through the planet centres (pinhole circle), and Mp is the mass of a planet. Constantgear mesh springs, ksp and krp, and the transmis-sion error excitations, espi(t) and erpi(t), are appliedalong the lines of action of the gear pairs. Torsionalsprings kjg connect the jth member to the hous-ing, intended to hold this member stationary if it isassigned a large numerical value. In equation (7d), theplanet coordinate θpic is defined as the relative dis-placement of planet pi with respect to its carrier c, i.e.θpic = θpi − θc where θpi is the absolute displacement ofplanet pi.

Similarly, equations of motion of the double-planetsystem shown in Fig. 3(b) are given as

Ic θc +n∑

i=1

(Ib θbic + Id θdic)

−n∑

i=1

[ksbrcbδsbi + krdrcdδrdi] + kcgθc = 0 (8a)

Ir θr +n∑

i=1

krdrrδrdi + krgθr = 0 (8b)

Is θs +n∑

i=1

ksbrsδsbi + ksgθs = 0 (8c)

Ib θbic + Ib θc + ksbrbδsbi + kbdrbδbdi = 0 (8d)

Id θdic + Id θc − krdrdδrdi + kbdrdδbdi = 0 (8e)





Fig. 3 Dynamic models of (a) a single-planet gear set, (b) a double-planet gear set, and (c) acomplex-compound gear set

where i ∈ [1, n]. Here, relative gear mesh displacementterms are defined as

δsbi = rsθs − rcbθc + rbθbic + esbi (8f)

δrdi = rrθr − rcdθc − rdθdic + erdi (8g)

δbdi = rbθbic + rdθdic + ebdi (8h)

rcb = rs + rb, rcd = rr − rd , and Ic = Ic + n(Ib + Id) +n(Mbr2

cb + Mdr2cd) where Mb and Md are the masses of

planet b and d, Ib, and Id are the polar mass momentsof inertia, and rcb and rcd are the respective pin circleradii. The planet coordinates are again defined relativeto the carrier as θbic = θbi − θc and θdic = θdi − θc .

Likewise, the equations of motion of the dynamicmodel of a complex-compound gear set shown inFig. 3(c) are written in terms of the rotational coor-dinates θc , θr1, θr2, θs1, θs2, θaic = θai − θc , θbic = θbi − θc

and θdic = θdi − θc (i ∈ [1, n]), respectively, as

Ic θc +n∑

i=1

(Ia θaic + Ib θbic + Id θdic)

−n∑

i=1

[ks1arcaδs1ai + ks2brcbδs2bi − kr1brcbδr1bi

− kr2drcdδr2di] + kcgθc = 0 (9a)

Ir1θr1 −n∑

i=1

kr1brr1δr1bi + kr1gθr1 = 0 (9b)

Ir2θr2 −n∑

i=1

kr2drr2δr2di + kr2gθr2 = 0 (9c)

Is1θs1 +n∑

i=1

ks1ars1δs1ai + ks1gθs1 = 0 (9d)





Is2θs2 +n∑

i=1

ks2brs2δs2bi + ks2gθs2 = 0 (9e)

Ia θaic + Ia θc + ks1araδs1ai − kabraδabi = 0 (9f)

Ib θbic + Ib θc + kr1brbδr1bi + ks2brbδs2bi

− kabrbδabi − kbdrbδbdi = 0 (9g)

Id θdic + Id θc − kbdrdδbdi + kr2drdδr2di = 0 (9h)

where Ic = Ic + n(Ia + Ib + Id) + n(Mar2ca + Mbr2

cb +Mdr2

cd) and the relative gear mesh displacements aredefined as

δs1ai = rs1θs1 − rcaθc + raθaic + es1ai (9i)

δs2bi = rs2θs2 − rcbθc + rbθbic + es2bi (9j)

δr1bi = rcbθc − rr1θr1 + rbθbic + er1bi (9k)

δr2di = rcdθc − rr2θr2 + rdθdic + er2di (9l)

δabi = −raθaic − rbθbic + eabi (9m)

δbdi = −rbθbic − rdθdic + ebdi (9n)

Equations (7a) to (7d) till equations (9a) to (9n)can be written in matrix form to obtain the stiff-ness Kj and mass Mj matrices of gear stage j. Here,the dimension of these matrices is (n + 3) if the gearstage j is formed by a single-planet gear set with nplanets, whereas it is (2n + 3) and (3n + 5) for dou-ble and complex-compound gear stages with n planetbranches, respectively. The overall gear mesh stiffnessand mass matrices of an N -stage planetary gear train(j ∈ [1, N ]) are then assembled as

Kg = diag[K1 K2 · · · KN ] (10a)

M = diag[M1 M2 · · · MN ] (10b)

Next, a coupling stiffness matrix Kc that includestorsional stiffness values of connections among thestages of gear sets is obtained and added to the over-all gear mesh stiffness matrix to obtain the overallstiffness matrix of the N -stage planetary gear train asK = Kg + Kc. With this, the equations of motion of anN -stage planetary gear train is given as

Mq + Cq + Kq = F (11)

Here q is the overall displacement vector defined interms of displacement vectors of each gear stage as

q =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

q1

q2

...q

N

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(12a)

where qj (j ∈ [1, N ]) are defined for single-planet,double-planet, and complex-compound gear stages,

respectively, as

qj = [θc θr θs θp1c · · · θpnc]Tj (12b)

qj = [θc θr θs θb1c θd1c · · · θbnc θdnc]Tj (12c)

qj = [θc θr1 θr2 θs1 θs2 θa1c θb1c θd1c · · · θanc θbnc θdnc]Tj

(12d)

There is no practical or theoretical means of deter-mining the damping matrix C in equation (11). There-fore, a constant modal damping value will be usedin the forced response calculations, based on theexperience on damping of gear meshes.

2.4.2 Steady-state forced response

The forcing vector F in equation (11) consists of forcevectors F j of each gear stage j as

F =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

F 1

F 2

...F N

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(13a)

From equations (7a) to (7d) till equations (9a) to(9n), F j of single-planet, double-planet, and complex-compound gear stages are found as

F j =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

n∑i=1

ksprcpespi(t) +n∑

i=1

krprcperpi(t)

−n∑

i=1

krprr erpi(t)

−n∑

i=1

ksprsespi(t)

−ksprpesp1(t) + krprperp1(t)...

−ksprpespn(t) + krprperpn(t)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

j

(13b)

F j =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

n∑i=1

ksbrcbesbi(t) −n∑

i=1

krdrcderdi(t)

−n∑

i=1

krdrr erdi(t)

−n∑

i=1

ksbrsesbi(t)

−ksbrbesb1(t) + kbdrbebd1(t)

−krdrderd1(t) + kbdrdebd1(t)...

−ksbrbesbn(t) + kbdrbebdn(t)

−krdrderdn(t) + kbdrdebdn(t)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

j

(13c)





F j =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[ n∑i=1

ks1arcaes1ai(t) +n∑

i=1

ks2brcbes2bi(t)

−n∑

i=1

kr1brcber1bi(t) −n∑

i=1

kr2drcder2di(t)]

n∑i=1

kr1brr1er1bi(t)

n∑i=1

kr2drr2er2di(t)

−n∑

i=1

ks1ars1es1ai(t)

−n∑

i=1

ks2brs2es2bi(t)

−ks1araes1ai(t) + kabraeabi(t)[− kr1brber1bi(t) − ks2brbes2bi(t)

+kabrbeabi(t) + kbdrbebdi(t)]

kbdrdebdi(t) − kr2drder2di(t)

...

− ks1araes1an(t) + kabraeabn(t)[ − kr1brber1bn(t) − ks2brbes2bn(t)

+ kabrbeabn(t) + kbdrbebdn(t)]

kbdrdebdn(t) − kr2drder2dn(t)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

j

(13d)

respectively. In equations (13a) to (13d), each stage hasa distinct fundamental mesh frequency ωmj as definedby equation (3).

The gear contact model is used to determine time-varying transmission error of each external and inter-nal gear pair, and the harmonic amplitudes andthe corresponding phase angles are determined byFourier analysis. The phasing relationships betweenthe meshes of each stage and the phasing relation-ships between the stages must be included to obtainthe transmission error of each mesh to be used inequations (13a) to (13d). For this purpose, one of thesun–planet meshes of the first planetary gear stage(j = 1) is considered as the reference mesh. Withoutloss of generality, assume that the first stage is a single-planet gear set. The transmission error of the referencemesh formed by sun–planet 1 is defined in Fourierseries form as

[esp1(t)](j=1) =L∑

�=1

E (1)sp�

sin[�ωm1t + φ(1)sp�

] (14)

where E (1)sp�

and φ(1)sp�

are the amplitude and phase angleof the �th harmonic term, superscript indicates thegear stage, and L is the total number of harmonicterms considered. Defining, a phase angle betweenthis reference mesh on the first gear set and the

reference sun–planet 1 mesh on stage j as �( j) (bydefinition, �(1) = 0), the transmission error functionsof the single-planet gear at the jth stage are defined as

[espi(t)]( j) =L∑

�=1

E ( j)sp�

sin[�ωmjt + φ( j)sp�

+ �Z ( j)s

( j)i + ��( j)] (15a)

[erpi(t)]( j) =L∑

�=1

E ( j)rp�

sin[�ωmjt + φ( j)rp�

+ �Z ( j)r

( j)i

+ �γ ( j)sr + ��( j)] (15b)

where ( j)i is the position angle of planet i from the

reference planet on stage j and γ( j)

sr the phase anglebetween the sun–planet and ring–planet meshes ofplanet 1 on stage j [15, 16]. The terms Z ( j)

s ( j)i represent

the phase angles between the sun–planet i mesh andthe sun–planet 1 mesh on the same stage j [5, 17]. Like-wise, Z ( j)

r ( j)i is the phase angle between ring–planet i

and ring–planet 1 meshes.If the jth stage of the gear train is formed by a double-

planet gear set, the corresponding transmission errorexcitations in equation (13c) are defined as

[esbi(t)]( j) =L∑

�=1

E ( j)sb�

sin[�ωmjt + φ( j)sb�

+ �Z ( j)s

( j)i + ��( j)] (16a)

[erdi(t)]( j) =L∑

�=1

E ( j)rd�

sin[�ωmjt + φ( j)rd�

+ �Z ( j)r

( j)i

+ �γ ( j)sr + ��( j)] (16b)

[ebdi(t)]( j) =L∑

�=1

E ( j)bd�

sin[�ωmjt + φ( j)bd�

+ �Z ( j)s

( j)i

+ �γ( j)

sd + ��( j)] (16c)

Here, again, the coefficients E ( j)sb�

, E ( j)rd�

, and E ( j)bd�

and thecorresponding phase angles φ

( j)sb�

, φ( j)rd�

, and φ( j)bd�

asso-ciated with each mesh are computed by the contactmodel. In addition to the phase angle γ

( j)sr between

the sun and ring meshes of a planet branch, thephase angle γ

( j)sd between the sun–planet b and planet

b-planet d meshes is also required.Finally, if the jth stage is formed by a complex-

compound gear set as the one shown in Fig. 3(c), thereare six different gear meshes, each requiring a different





transmission error excitation given as

[es1ai(t)]( j) =L∑

�=1

E ( j)s1a�

sin[�ωmjt + φ( j)s1a�

+ �Z ( j)s1

( j)i + ��( j)] (17a)

[er1bi(t)]( j) =L∑

�=1

E ( j)r1b�

sin[�ωmjt + φ( j)r1b�

+ �Z ( j)r1

( j)i

+ �γ( j)

s1r1 + ��( j)] (17b)

[eabi(t)]( j) =L∑

�=1

E ( j)ab�

sin[�ωmjt + φ( j)ab�

+ �Z ( j)s1

( j)i

+ �γ( j)

s1b + ��( j)] (17c)

[ebdi(t)]( j) =L∑

�=1

E ( j)bd�

sin[�ωmjt + φ( j)bd�

+ �Z ( j)s2

( j)i

+ �γ( j)

s2d + ��( j)] (17d)

[es2bi(t)]( j) =L∑

�=1

E ( j)s2bi�

sin[�ωmjt + φ( j)s2bi�

+ �Z ( j)s2

( j)i + ��( j)] (17e)

[er2di(t)]( j) =L∑

�=1

E ( j)r2di�

sin[�ωmjt + φ( j)r2di�

+ �Z ( j)r2

( j)i

+ �γ( j)

s2r2 + ��( j)] (17f)

Here, with [es1a1(t)]( j) as the transmission error of thereference mesh (sun 1–planet a1) of this gear set,angles γ

( j)s1r1, γ ( j)

s1b, γ ( j)s2d , and γ

( j)s2r2 are the additional phase

angles needed to determine the complete relationshipbetween the reference mesh and the other meshesr1b1, ab1, b1d1, s2b1, and r2d1, respectively, all ofwhich are defined by the geometry of the gears andthe carrier.

With the above excitations, the forcing vector givenby equation (13a) is defined by N number of fun-damental gear mesh frequencies (and their integermultiples) and as many periodic transmission errorexcitations as the number of gear meshes in the geartrain, each having its own unique phase angle. Thesolution of this multi-excitation, linear, time-invariantproblem is done in several stages. First, the corre-sponding eigenvalue problem of the undamped sys-tem Ku = ω2Ku is solved to determine the undampednatural frequencies ωi and the corresponding modeshapes ui of the gear train. Then, the forced vibra-tion response due to each of the transmission errorexcitations is determined individually by using modalsummation technique with uniform modal damping[18]. Finally, the total steady-state response from theN -stage gear train is determined as a superposition ofindividual responses to each excitation.

3 AN EXAMPLE ANALYSIS

A five-speed, rear–wheel–drive automatic transmis-sion that consists of three single-planet planetary gearstages is used here as an application example. Here,for the sake of simplicity, the gear parameters werekept the same for all three stages except the numberof planet branches. Table 1 lists the basic gear designparameters of each planetary stage. The first stagehas four planets, whereas the other two stages havethree planets each. The stick diagram and the clutch-ing sequence of the kinematic configuration are shownin Fig. 4. Here, the five forward gear ratios are achievedby activating five clutches according to the scheduledefined in the table of Fig. 4 (clutches required forreverse gear ratio are not shown). In this application,ratios 1 to 3 are underdrive ratios, the fourth gear ratiois the direct drive (the output angular velocity is equalto the input angular velocity), and the fifth gear ratiorepresents the overdrive conditions. In direct driveconditions, no dynamic model is required since thevelocities of gears with respect to each other are zero,i.e. there is no meshing action and entire gear trainrotates as a single rigid body.

Table 1 Basic design parameters of the example gear train

Parameter Sun Planet Ring

Number of teeth 34 18 70Module (mm) 1.5 1.5 1.5Pressure angle (◦) 21.3 21.3 21.3Face width (mm) 30 30 30Root diameter (mm) 46 23.75 110Outside diameter (mm) 52.74 30.5 –Minor diameter (mm) – – 103.45Pin circle diameter (mm) 78Tooth thickness (mm) 1.895 2.585 1.884

Fig. 4 The stick diagram and the clutching schedule ofthe example five-speed automatic transmission





Table 2 shows the results of the power flow analysisat input torque levels of Tin = 300 Nm. It indicates thatthe static forces transmitted by gear meshes changewith gear ratio. It also shows that certain gear meshesare not loaded at certain gear ratios. For instance,meshes of the first gear stage are unloaded in secondand third gear ratios. According to the methodologydeveloped by Kahraman et al. [11], any unloaded gearstages are automatically excluded from the dynamicmodel by issuing very small gear mesh stiffnesses tothese unloaded gear meshes. Table 2 also lists the aver-age gear mesh stiffness values computed by the gearcontact module at these gear mesh force values. It isobserved from these results that the gear mesh stiff-ness values at fifth gear ratio are nearly 6–8 per cent

Table 2 Static gear mesh forces and average gear meshstiffness values at Tin = 300 Nm

Stages

1 2 3

Gear range s/p r/p s/p r/p s/p r/p

Gear mesh forces (N)1 1429 1429 3829 3829 2830 28302 0 0 2830 2830 1905 19053 0 0 1905 1905 0 05 235 235 465 465 465 465

Mesh stiffness (N/μm)1 437 526 451 546 447 5402 – – 447 540 441 5323 – – 441 532 – –5 413 492 422 504 422 504

lower than those at the first gear ratio, even though thevalue of Tin is the same.

The undamped natural frequencies ωi predicted bythe model are listed in Table 3. Here, the torsionalstiffness values of the coupling elements are definedas 108 Nm/rad and the ground connections (reactionelements) are assigned a very high torsional stiffnessvalue of 1010 Nm/rad. It is observed from Table 3 that ωi

vary significantly as a function of gear ratio. Since thegear train is semi-definite, a single rigid body modewith ω1 = 0 is predicted at the at all gear ratios asshown in Table 2. In first and fifth gear ratios, all gear

Table 3 Predicted natural frequencies of the examplesystem at Tin = 300 Nm

Natural frequencies, ωi (kHz)

First gear Second gear Third gear Fifth gear

0 0 0 01.09 1.42 1.55 0.771.79 2.18 10.38 1.922.88 10.54 10.59 (2) 2.25

10.54 (3) 10.59 (2) 11.95 9.6910.57 10.67 (2) 10.22 (3)10.62 10.72 10.2510.67(2) 18.36 10.3310.72(2) 10.3710.92 12.6918.28 18.3218.32

Numbers listed in parentheses next to the natural frequenciesindicate that these modes are repeated.

Fig. 5 Examples of mode shapes of the example gear set: (a) mode at ωi = 1.09 kHz at the first gearratio and (b) mode at ωi = 10.22 kHz at the fifth gear ratio





stages are loaded so that there is only one zero naturalfrequency representing the rigid body mode. Mean-while, additional zero-frequency modes are predictedin second and third ratios since there are unloadedgear stages. Figure 5 illustrates two of the naturalmodes. Here, dashed lines represent the shape of themode. In Fig. 5(a), a mode at 1.09 kHz at the first gearratio is shown. In this mode, planets move the sameway with respect to their central members indicatingthat this mode is an axisymmetric overall mode. Allthree stages contribute to this mode significantly. Sim-ilarly, Fig. 5(b) illustrates a repeated (planet) mode at10.22 kHz at the fifth gear range. None of the centralmembers rotate in this mode while each planet of thefirst stage has a different type of a motion. The othertwo stages do not contribute to this planet mode.

The amplitudes, E ( j)sp�

and E ( j)rp�

of the first five meshharmonics (� ∈ [1, L]) of the static transmission errorexcitations at each gear mesh, computed by the con-tact module at the static gear mesh force values

specified in Table 2, are given in Table 4 for the firstgear range. Here, phase angles, φ

( j)sp�

and φ( j)rp�

of eachharmonic are assumed to be zero. Further assumingzero phase angles between the stages �(2) = �(3) = 0and 4 per cent modal damping ratio, the steady-state response q of the gear train to the force vector

Table 4 The first five harmonic amplitudes E ( j)sp�

and

E ( j)rp�

(in μm) of the static transmission errorexcitations at Tin = 300 Nm

Stages

1 2 3

� Esp Erp Esp Erp Esp Erp

1 0.3 0.036 0.5 0.011 0.5 0.0672 0.022 0.018 0.06 0.012 0.049 0.0583 0.012 0.010 0.032 0.014 0.019 0.0214 0.004 0.008 0.021 0.006 0.012 0.0075 0.002 0.008 0.006 0.001 0.005 0.003

Fig. 6 Dynamic mesh forces at sun–planet 1 (—–) and ring–planet 1 (- - - - -) mesh interfaces for(a) stage 1, (b) stage 2, and (c) stage 3





F defined by equation (13a) is predicted. With allθi(t) known, the dynamic gear mesh forces can bedefined as

F ( j)spi (t) = [ksprsδspi(t)]( j) (18a)

F ( j)rpi = [krprrδrpi(t)]( j) (18b)

The dynamic mesh forces are important for twomain reasons. First of all, they are transmitted tothe transmission housing to generate structure-bornegear noise levels. Second, they cause additional cyclicgear tooth stresses accelerating the occurrence oftooth fatigue failures.

Figure 6 shows a set of example of the peak-to-peak gear mesh force amplitudes for this system asa function of input speed when the transmission isoperating at the first gear range. In Fig. 6(a) for themeshes of the first stage, peak dynamic mesh force

amplitudes are observed at a number of speed val-ues, the most significant ones are observed at 1880,3765, and 5540 r/min. Each of these resonance peakscan be linked to a certain natural mode and certaintransmission error harmonic amplitude exciting thatparticular mode. For instance, the peak at 1880 r/min(ωm1 = 717 Hz, ωm2 = 324 Hz, and ωm3 = 482 Hz) is dueto the natural mode at ωi = 2.88 kHz excited by fourthharmonic of transmission errors of the first stage, rep-resenting the condition ωm1 ≈ ωi/4. In Fig. 6(b), forthe meshes of the second stage, the largest resonancepeak at 3470 r/min (ωm1 = 1323 Hz, ωm2 = 600 Hz, andωm3 = 891 Hz) caused by the second harmonic of theexcitations of the third stage as it excites the modeat ωi = 1.79 kHz when ωm3 ≈ ωi/2 and simultaneouslyby the third harmonic of the excitations of the sec-ond stage as it excites the mode at ωi = 1.79 kHz whenωm2 ≈ ωi/3. Similarly, the largest peak in Fig. 6(c) for

Fig. 7 Dynamic displacements of the carrier (—–), ring gear (- - - - -), and sun gear (·-·-·-·-·) for (a)stage 1, (b) stage 2, and (c) stage 3





the third stage represents the condition ωm2 ≈ ωi/3where ωi = 1.79 kHz.

Figure 7 shows the peak-to-peak displacements θs,θr , and θc of each stage at the same condition as Fig. 6.In Figs 7(a) and (c), θ(1)

s = θ(3)s = 0, since the sun gears

of the first and third stages are both fixed in the firstgear ratio according to clutching schedule shown inFig. 4. The amplitudes of the sun gear of the secondstage are also small since it is connected to the sun gearof the third stage that is stationary. The frequencies(speeds) at which resonance peaks occur are the sameas those in Fig. 6, while the relative ranking of theseamplitudes are now different. For instance, the largestresonance peak for the ring gear of stage 1 (Fig. 7(a))occurs at 2110 r/min that corresponds to the reso-nance condition of ωm3 ≈ ωi/3 where ωi = 1.09 kHz isthe frequency of one of the axisymmetric modes listedin Table 3.

The phase angles among the planetary gear stages(�( j)) in an automatic transmission defined differentlyduring each gear shift as the final positions of twoconnected members relative to each other is deter-mined by the application of a clutch or band. Forthis reason, the vibration and noise measurementsfrom the same transmission at the same conditionsdiffer slightly from each other. Figure 8 aims at illus-trating the influence of these phase angles. Here,the analysis of the same example system is repeatedfor four different phase conditions: (a) �(2) = �(3) = 0,(b) �(2) = π/2, �(3) = π, (c) �(2) = π, �(3) = 0, and (d)�(2) = 0, �(3) = 3π/2. In Fig. 8, peak-to-peak F (1)

sp1(t) val-ues for these four cases are compared at the first gearratio to show that the phase angles between the stageshave a secondary effect on the forced response.

Fig. 8 Dynamic mesh force at the sun-p1 mesh ofthe first planetary gear stage for four differ-ent phasing conditions: �(2) = �(3) = 0 (—–),�(2) = π/2, �(3) = π (- - - - -), �(2) = π, �(3) = 0(·-·-·-·-·), and �(2) = 0, �(3) = 3π/2 (· · · · · · )

4 CONCLUSIONS

A generalized torsional dynamic model of a multi-stage planetary gear train was developed in this studyfor evaluation of the planetary gear sets of automatictransmissions in the early states of design for theirdynamic behaviour. The planetary gear train mod-elled here was formed by N number of planetary gearsets of different types, connected to each other in agiven kinematic configuration. A generalized powerflow formulation and a gear mesh LDP were coupledwith the dynamic model to automatically determinethe gear mesh compliances as well as the power flowpaths within any transmission. Phasing relationshipsbetween the meshes of each particular planetary gearstage as well as among different planetary gear stageswere defined. Free vibration analysis was performed topredict the natural frequencies and the correspond-ing mode shapes and the modal summation tech-nique was employed to determine the forced vibrationresponse due to transmission error excitations appliedat each gear mesh interface. A three-stage planetarygear train representative of a five-speed automatictransmission was used to demonstrate the capabilitiesof the model. The results indicate that the kinematicconfigurations, coupling conditions, gear ratios, andthe phasing among the planets substantially influencethe overall modal behaviour and the forced dynamicresponse of the automatic transmissions. They alsoshow that the phase angles among individual plan-etary gear stages have a secondary influence on thegear mesh force spectra.

REFERENCES

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9 Kahraman, A. Free torsional vibration characteristicsof compound planetary gear sets. Mech. Mach. Theory,2001, 36, 953–971.

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Dynamic modelling of planetary gears of automatic transmissions