A dynamic model of a double-helical planetary gear set

A dynamic model of a double-helical planetary gear set

Prashant Sondkar ?,Ahmet Kahraman

Department of Mechanical and Aerospace Engineering,The Ohio State University,201W 19th Ave.,Columbus,OH 43210,USA

a r t i c l e i n f o a

b s t r a

c t

Article history:

Received 26July 2012

Received in revised form 11July 2013Accepted 14July 2013

Available online 8August 2013A linear,time-invariant model of a double-helical planetary gear set is proposed in this paper.The model formulation allows the analysis of a gear set with any number of planets,any planet phasing and spacing configurations and any support conditions.Torsional,transverse,axial and rotational (rocking)motions of gears and the planet carrier are included in this three-dimensional model.Planets are allowed to be positioned equally or unequally spaced around the sun gear.The natural modes are computed by solving the corresponding Eigen value problem.The modal summation technique is used to find the forced response due to periodic gear mesh excitations under constant or proportional modal damping conditions.An example gear set consisting of four equally-spaced planets is considered at the end to demonstrate the influence of key design parameters including right-to-left stagger angle,and support conditions on the dynamic response of the double-helical planetary gear system.Numerous modes with dominant tilting motions are predicted to underline the essence of using a 3D formulation for double helical gears.Staggering of gear teeth is shown to impact the dynamic response of the gear system substantially.Different values of stagger are shown excite different types of modes,resulting in different dynamic response curves.It is also found that left and right sides of double helical gears can carry equal or different dynamic load amplitudes based on the right-to-left stagger values.Likewise the influence of the sun support conditions is shown to be more pronounced for the case of a non-zero stagger.

?2013Elsevier Ltd.All rights reserved.

Keywords:

Gear dynamics

Double-helical planetary gear sets Planet mesh phasing

Double-helical gear stagger

1.Introduction

Planetary gear sets are used commonly in many automotive,industrial,aerospace and wind turbine gearbox applications.The typical applications include jet propulsion systems,rotorcraft transmissions and passenger vehicle automatic transmission and transfer cases.A simple planetary gear set consists of a central sun gear (s ),three or more identical planet gears (p )in mesh with the sun gear as well as an internal ring gear (r ).The planets are held by a rigid carrier (c )through rolling element bearings in between.One of the main advantages of planetary gear sets is that the input torque is split into number of parallel planet paths,providing increased power density (power-to-weight ratio).Planetary gear sets consist of either spur,(single)helical or double-helical gears.Spur planetary gear sets can be commonly found in heavy machinery and off-highway gearboxes and transmissions,the helical planetary gear sets are the norm for all automotive applications,while double-helical planetary gear sets are often found in high-speed aerospace and rotorcraft applications.Their high load carrying capacity and ability to cancel out the axial thrust forces make double helical planetary gear sets suitable for such applications.

Dynamic analysis of a double-helical planetary gear set,as any other gear system,is required for two main reasons.One reason is associated with the resultant noise outcome.High-frequency dynamic forces created at the gear meshes are transmitted to gearbox housing and the surrounding support structures to generate structure-borne noise.Therefore,reduction of vibrations necessitates a

Mechanism and Machine Theory 70(2013)157–174

?Corresponding author.Tel.:+16143124253;fax:+16142923163.E-mail address:sondkar.1@https://www.360docs.net/doc/4f2205910.html, (P.

Sondkar).0094-114X/$–see front matter ?2013Elsevier Ltd.All rights reserved.

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j ou r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /m e c h mt

158P.Sondkar,A.Kahraman/Mechanism and Machine Theory70(2013)157–174 Nomenclature

C Damping matrix

D Diameter

e Transmission error

b e Amplitude of l-th harmonics of transmission error

F Force acting on planet i due to coupling between carrier and planet i

F spi Dynamic mesh force for sun–planet i gear mesh

F rpi Dynamic mesh force for ring–planet i gear mesh

F Force vector

FW Face width

I Diametral mass moment of inertia

J Polar mass moment of inertia

k Planet bearing stiffness

k sp Mean value of sun–planet i gear mesh stiffness

k rp Mean value of ring–planet i gear mesh stiffness

k x?Support bearing stiffness in x direction(?=s,r,c)

k y?Support bearing stiffness in y direction(?=s,r,c)

k z?Support bearing stiffness in z direction(?=s,r,c)

kθx?Support bearing stiffness inθx direction(?=s,r,c)

kθy?Support bearing stiffness inθy direction(?=s,r,c)

m Mass

M Moment acting on planet i due to coupling between carrier and planet i

M Mass matrix

N Number of planets

p Relative mesh displacement

q Forced vibration response vector

Q Mode shape

r?Base radii of gear?(?=s,r,pi)

r c Planet pin hole radius

T Torque

x Forced vibration response in x direction

y Forced vibration response in y direction

z Forced vibration response in z direction

Z Number of teeth on gear

αpi Position angle for planet i

βHelix angle

γspi phase angle between the s-pi mesh on the left side and the reference s-p1mesh

γrpi phase difference between the r-pi mesh and the r-p1mesh on the left side

γstg phase difference between the left and right side due to intentional nominal stagger of the teeth γrs phase difference between the reference s-p1mesh and the r-p1mesh on left side

εStiffness multiplier for proportional damping

ζDamping factor

ηMass multiplier for proportional damping

θx Forced vibration response inθx direction

θy Forced vibration response inθy direction

θz Forced vibration response inθz direction

ΘDynamic compliance

σPhase angle of l-th harmonics of transmission error excitation

κPhase difference

?Transverse pressure angle

ψAngle made by plane of action with vertical y axis

ωNatural frequency

ωm Mesh frequency

Subscripts

b Bore

c Carrier

better understanding of the dynamic behavior is crucial.The other main reason stems from the durability and reliability of the planetary gear set.Dynamic gear mesh and bearing forces that alternate about the static forces transmitted,cause dynamic stresses and dynamic factors to potentially impact the fatigue lives of the gears and bearings adversely.Hence,dynamic analysis of a double helical planetary gear set becomes a critical step in the design process which aims at developing quiet and reliable transmissions.

Numerous analytical models on planetary gear dynamics consisting of spur or helical gears can be found in the literature while research on double-helical gears is quiet limited.Most of these models are focused on analytical work comprising of discrete-parameter models with gears bodies assumed to be rigid and tooth flexibilities are modeled as a springs.In their review paper,Yang and Dai [1]summarize these torsional,two-dimensional (2D)and three-dimensional (3D)models employing linear or nonlinear formulations as well as constant or periodic gear mesh stiffnesses.Types of analyses performed in these published studies also vary as some focused solely on prediction of natural modes [2–4],some others emphasized the excitation mechanism and cancelation/minimization of these excitations by properly phasing the planets [5–7],while a number of studies predicted forced vibration response and dynamic tooth load [8–12].More recently,Parker et al.[13]and Kahraman et al.[14]used a deformable-body contact model developed by Vijayakar [15]to study the dynamic behavior of planetary gear sets with flexible rims.

Literature on double-helical gears is quiet sparse and limited to single gear pair.Thomas [16]developed an analytical model for investigating load distribution and transmission error of a double helical gear pair under quasi-static conditions.The model was later used by Clapper and Houser [17]to investigate the root stresses of double helical gears and comparisons were made to experiments performed.Zhang et al.[18]carried out noise optimization of double helical parallel shaft gearbox by developing a 3D finite elements model.Jauregui and Gonzalez [19]developed single degree-of-freedom model to study axial vibrations of double helical gear pair due to manufacturing errors.Quasi-static and dynamic analyses of a double-helical gear pair was carried out by Ajmi and Velex [20]by combing a 12degree-of-freedom (DOF)model of the left side of gear pair with another 12-DOF model of the companion right side pair using Euler beam elements.The model is then analyzed to study the effect of floating pinion and staggering of teeth on the quasi-static and dynamic behavior of a gear pair.Anderson et al.[21]conducted experiments on double-helical planetary gear set to measure efficiency,vibration amplitudes and stress levels.

To the best knowledge of the authors,no published study is available on the dynamic modeling of double-helical planetary gear sets.Accordingly,overall aim of this study is to investigate the dynamic behavior of double-helical planetary gear system through the development of general analytical model.The proposed model will be used to investigate the free and forced vibration characteristics of double-helical planetary gear sets within ranges of several key design parameters.

2.Modeling methodology

A three-dimensional discrete model of an N -planet double-helical planetary gear set is proposed.A deformable-body dynamic i Index for number of planets l Harmonics number m Mean value mesh Gear mesh

n Number of beam element r Ring gear

pb Planet bearing pi Planet i gear s Sun gear

sup Support bearing x x direction y y direction z z direction θx θx direction θy θy direction θz θz direction

λNumber of degrees of freedom

Superscript L Left side of double helical gear R Right side of double helical gear

159

P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

6(N +3)degree-of-freedom (DOF)model of a helical planetary gear set according to Kahraman [11]using the method proposed by Ajmi and Velex [20].The following assumptions are made in the model formulation:

(i)The bodies representing the sun,planet and ring gears and the carrier are assumed to be rigid.

(ii)Each gear mesh flexibility is represented by a linear spring acting on the plane of action normal to gear tooth surfaces

(inclined by helix angle β).

(iii)Time-varying component of mesh stiffness due to fluctuation of number of tooth pairs in contact is neglected as this was

shown to be a good assumption for a helical gears [5,22].

(iv)Planets are assumed to be identical to each other such that each planet-sun and planet-ring mesh can be assumed to have

the same properties.However,if needed the stiffness and displacement excitation at each mesh can be defined individually through minor revision to the proposed model.

(v)Frictional forces arising from tooth sliding are considered to be negligible in accordance with the off-line-of-action gear

vibration measurements of Kang and Kahraman [23].

(vi)Left and right sides of double-helical gears are assumed identical in geometry expect the hand of the teeth is opposite,but

staggered by a certain angle with respect to each other.

(vii)Damping of the system is represented by either a constant (uniform)modal damping or proportional damping.

Fig.1shows the overall dynamic model of an entire double-helical planetary gear set with only one planet shown for simplicity purposes.Under the assumptions listed above,the model formulations will be done first for three basic sub-systems:(i)a sun –planet i pair (left or right side),(ii)a ring –planet i pair (left or right side),and (iii)a carrier –planet i pair (left or right side).A beam formulation will then be introduced to combine left and right sides of double helical gear set.A general assembly process will then be employed to obtain overall mass and stiffness matrices including support bearing conditions.2.1.Formulation of sub-systems

2.1.1.A sun –planet i pair

Fig.2illustrate an dynamic model of an external gear pair,which represents the one side (either left or right side)of the sun gear (subscript s )meshing with planet-i (subscript pi )located at arbitrary position angle αpi .As shown in Fig.2,the plane of action of the gear pair makes an angle Ψspi with the vertical y axis.This angle can be defined in terms of the transverse pressure angle ?sp of the sun –planet pair and αpi as:

ψspi ?

?sp ?αpi ;??sp ?αpi ;

T s :Counterclockwise ;T s :Clockwise ;

(e1T

where T s is external torque acting on the sun gear.The undamped equations of motion for the s-pi pair are derived using the helical gear pair formulations of Kahraman [11].These equations for sun gear motions are

m s y

€s t eTtk sp cos βcos ψspi p spi t eT?0;m s x €s t eTtk sp cos βsin ψspi p spi t eT?0;m s z €

s t eT?k sp sin βp spi t eT?0;I s θ€

ys

t eTtk sp r s

sin βcos ψspi p spi

t eT?0;I s θ€

xs t eTtk sp r s sin βsin ψspi p spi t eT?0;J s θ

€zs t eTtk sp r s cos βp spi t eT?T s =2N eT:e2a –f T

The corresponding equations of motion for the degrees of freedom of planet i are

m p y

€pi t eT?k sp cos βcos ψspi p spi t eT?0;m p x €pi t eT?k sp cos βsin ψspi p spi t eT?0;m p z €

pi t eTtk sp sin βp spi t eT?0;I p θ

€ypi t eTtk sp r p sin βcos ψspi

p spi t eT?0;I p θ€

xpi t eTtk sp r p sin βsin ψspi p spi t eT?0;e3a –f T

160P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

Fig.1.

Dynamic model of a double-helical planetary gear set.

161

P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

In these equations,m ?,I ?and J ?are mass,the diametral mass moment of inertia and the polar mass moment of inertia of one side (left or right),and r ?is base circle radius of gear ?(?=s,pi ).k sp is the average gear mesh stiffness for sun planet pair.P spi (t )represents the relative displacement of the s-pi mesh in the direction normal to tooth surface such that

p spi t eT?

y s ?y pi cos ψspi tx s ?x pi sin ψspi tr s θzs tr p θzpi h i

cos βtr s θys tr p θypi cos ψspi tr s θxs tr p θxpi sin ψspi t?z s tz pi h i

sin β?e spi t eT:

e4T

where e spi (t )is loaded static transmission error excitation at the s-pi mesh and βis the helix angle.

Eqs.(2)and (3)are written in matrix form as

M s

00

M p

!q €s t eTq €pi t eT()tk sp K 11spi K 12spi sym :K 22spi

2435q s t eTq pi t eT()?f sm tf si t eT

f spi t eT():e5T

2.1.2.A ring –planet i pair

Next consider the meshing of the same planet i with the ring gear (subscript r )on one side of the double-helical system as shown in Fig.3.With pi being at the same angular position αpi ,the plane of action of this internal mesh makes an angle Ψrpi with the vertical y axis,which is defined in terms of the transverse pressure angle ?rp of the ring –planet pair and αpi as

ψrpi ?

?rp tαpi ;??rp ?αpi ;

T s :Counterclockwise ;T s :Clockwise

:(e6T

pi

y pi

z 162P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

The undamped equations of motion for the r-pi pair shown in Fig.3are derived for the motions of the ring and planet i ,respectively,as

m r y

€r t eTtk rp cos βcos ψrpi p rpi t eT?0;m r x €

r t eT?k rp cos βsin ψrpi p rpi t eT?0;m r z €

r t eTtk rp sin βp rpi t eT?0I r θ

€yr

t eT?k rp r r

sin βcos ψrpi p rpi

t eT?0;I r θ€xr t eTtk rp r r sin βsin ψrpi p rpi t eT?0;J r θ€zr t eTtk rp r r cos βp rpi t eT?T r =2N eT;e7a –f T

m p y

€pi t eT?k rp cos βcos ψrpi p rpi t eT?0;m p x €

pi t eTtk rp cos βsin ψrpi p rpi t eT?0m p z €

pi t eT?k rp sin βp rpi t eT?0I p θ€

ypi t eTtk rp r p sin βcos ψrpi

p rpi t eT?0;I p θ€xpi t eT?k rp r p sin βsin ψrpi p rpi t eT?0;J p θ€zpi t eT?k rp r p cos βp rpi t eT?0:

e8a –f T

In Eq.(7a –f),m r ,I r and J r are mass,the diametral mass moment of inertia and the polar mass moment of inertia of one side of

the ring gear,respectively,and r r is base circle radius of the ring gear,respectively,and k rp is the average gear mesh stiffness of the ring –planet pair.Similar to the s-pi formulation,p rpi (t )represents the relative displacement of the r-pi mesh along the plane of action normal to the tooth surfaces

p rpi t eT?

y r ?y pi cos ψrpi t?x r tx pi sin ψrpi tr r θzr ?r p θzpi h i

cos βt?r r θyr tr p θypi cos ψrpi tr r θxr ?r p θxpi sin ψrpi tz r ?z pi h i

sin βte rpi t eT

e9T

zc

163

P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

with e rpi (t )being the loaded static transmission error excitation at the r-pi mesh.Eqs.(7)and (8)can be put into the matrix form as

M r 00

M p

!q €r t eTq €pi t eT

&'tk rp

K 11

rpi

K 12

rpi

sym :K 22

rpi

"#q r t eT

q pi t eT

&

'?

f rm tf ri t eT

f rpi t eT

&

'

:

e10T

2.1.

3.A carrier –planet i pair

Fig.4shows a one side of a carrier –planet i pair,with planet i positioned at an angle αpi and attached to the carrier via a bearing.Rotation center of the planet i represented by the z pi axis is at a distance r c from the rotational axis z c of the carrier.The planet bearing

is modeled as a diagonal stiffness matrix K bp ?Diag k y k x k z k θy k θx 0??

that couples planet pi to the carrier c along a circle of radius r c .Bearing forces and moments due to any arbitrary motions of pi and c of this particular side are defined as

F y t eT?k y

y c ?y pi tr c θzc cos αpi h i ;F x t eT?k x x c ?x pi ?r c θzc sin αpi h i

;

F z t eT?k z z c ?z pi ?r c θyc cos αpi tr c θxc sin αpi h i

;

M y t eT?k θy θyc ?θypi

;

M x t eT?k θx θxc ?θxpi

:e11a –e T

Here,pi can rotate in the θz direction with no resistance such that M z (t )=0.With these bearing forces and moments,the Eqs.of motion of the c-pi pair are written as

m c y €c t eTtF y t eT?0;m c x €c t eTtF x t eT?0;m c z €c t eTtF z t eT?0;

I c θ€yc t eTtM y t eT?r c cos αpi F z t eT?0;I c θ€xc

t eTtM x

t eTtr c

sin αpi

F z

t eT?0;

J c θ€

zc t eT?r c sin αpi F x t eTtr c cos αpi F y t eT?0;e12a –f T

m p y €pi t eT?F y t eT?0;m p x €pi t eT?F x t eT?0;m p z €

pi t eT?F z t eT?0;I p θ€

ypi t eT?M y t eT?0;I p θ€

xpi t eT?M x t eT?0:e13a –e T

Eqs.(12)and (13)are written in matrix form as

M c 00

M p

!q €c t eTq

€pi t eT&'tK 11

cpi K 12

cpi sym :

K 22

cpi

"#q c t eT

q pi t eT

&

'

?00

&':

e14T

2.2.Connection of the left and right sides

The sub-system models shown in Figs.2to 4consist of only one side of the double helical gears.In an actual double-helical system,left and right sides of a gears and the carrier are either one-piece (for planet and sun gears)or connected rigidly (for the ring gear and the carrier).While they might be perceived to be rigid,these connecting structures have certain flexibility as measured in comparison to gear mesh stiffnesses.As done earlier by Ajmi and Velex [20],left and right sides of gear (s ,r ,pi )are connected by finite beam elements to capture such compliances accurately.Also for practical reasons,use beam elements between left and right sides creates additional finite element nodes to be used for connecting this model to a given drivetrain in a more accurate way.

In Fig.5,a double-helical gear is divided into three pieces:left side at inner diameter D b ,the right side at the same diameter,and connecting structure of outside diameter D g .The connecting structure spans from gear face mid-point of the right side to the gear face mid-point of the left side.Here,partitioning is done such that the total mass of the gear equals the sum of individual masses of the left 164P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

are also handled the same way by using a two-element finite element model,thus representing each gear by three nodes.While not obvious here,the additional node of the beam elements created in middle is useful in interfacing this model with larger driveline model where gear set is employed.With this,the sub-matrices for the connecting structures are given as

K?e?

K11?e1K12?e10

K22?e1tK11?e2K12?e2

sym:K22?e2

2

64

3

75;M

?e

?

M11?e1M12?e10

M22?e1tM11?e2M12?e2

sym:M22?e2

2

64

3

75:e15a;bT

where?=s,r,pi,c and subscript e represents beam elements for each component.

These sub-matrices corresponds to the displacement sub-vector

q?e?

q?

àá

L

q?

àá

M

q?

àá

R

8

<

:

9

=

;:e16T

where subscripts L,R and M indicate left side,right side and middle nodes of each components respectively.

2.3.The overall system equations

These sub system matrices defined by Eqs.(5),(10)and(14)are assembled systematically along with right-to-left coupling matrices defined by Eq.(15)to obtain the overall equations of motion of a double-helical planetary gear set consisting of N planets (a total of18(N+3)degrees of freedom)as

Mq€teTtC q_teTtK mtK b

eTq teT?F teTe17aT

where the overall displacement vector q(t),the mass matrix M,gear stiffness matrix K m and the force array F(t).Arrays q(t)and F(t)are defined below while M and K m are given in the Appendix A.

q teT?

q se teT

q re teT

q ce teT

q p1e teT

?

8

>>>

>>>

<

>>>

>>>

:

9

>>>

>>>

=

>>>

>>>

;

;F teT?

F s teT

F r teT

F p1teT

?

8

>>>

>>>

<

>>>

>>>

:

9

>>>

>>>

=

>>>

>>>

;

:e17b;cT

11

element 1

beam

element 2

right side

gear

segment

Fig.5.Beam elements devised to connect left and right sides of a gear.

165 P.Sondkar,A.Kahraman/Mechanism and Machine Theory70(2013)157–174

The sub-arrays included in F (t )are defined as

F s t eT?

f sm tX N i ?1f si !L 0f sm

tX N i ?1

f si !R

8>>>>>>><>>>>>>>:9

>>>>>>>=

>>>>>>>;;F r t eT?f rm tX N i ?1f ri !L 0f rm

tX N i ?1

f ri !R

8>>>>>>><>>>>>>>:9

>>>>>>>=

>>>>>>>;;

F pi t eT?f spi tf rpi

L

0f spi tf rpi

R

8><>:9>=>;

:e17d –f T

A support stiffness matrix K b is defined to include the matrices for the sun (K bs ),ring (K br ),and carrier (K bc )supports.

K b ?Diag 0

K bs

K br

K bc

0…0? :

e17g T

Here,K bs ,K br and K bc are 6×6matrices that are applied to the middle nodes of the respective connecting structures which

can be defined as

K b ζ?Diag k y ζ

k x ζ

k z ζ

k θy ζ

k θx ζ

k θz ζ?

?:

e17h T

where ?=s ,r ,c ..C is the proportional damping matrix that is given as C =εK +ηM where εand ηare constants.In Eq.(17g,h)the sub-stiffness matrix K b ζrepresenting fixed member has very high k θz ζvalue while other two central members would have k θz ζ=0.

In Eq.(17c),individual forcing vectors forming F (t )include the transmission error excitations that are given as part of the relative gear mesh displacements in Eqs.(4)and (9).These periodic excitations (a total of 4N of them for an N -planet gear set)at individual gear meshes have the same fundamental frequency that is equal to the gear mesh frequency ωm .The transmission error excitation at a given s-pi or r-pi mesh can be computed by using a gear load distribution model [24]as well as the average gear mesh stiffnesses k sp and k rp .While each of these periodic excitations e spi (t )and e rpi (t )along the meshes of the right and left sides of the gear set have the same waveforms at the same harmonic amplitudes,they possess a unique phasing relationship defined by N ,αpi (αp 1=0and i [1,N ]),and the numbers of teeth on the sun,ring and planet gears.

The s -p 1mesh on the left side of the gear set is chosen here as the reference mesh.The transmission error excitation at this reference mesh is given in Fourier series form as

e L eT

sp 1t eT?

X L l ?1

b e spl cos l ωm t tσspl ;

e18a T

where b e spl and σspl are the amplitude and phase angle of the l -th harmonic of this excitation as predicted by the gear load

distribution analysis [24].The transmission error functions on other s-pi meshes on the left side can be defined relative to the reference s -p 1mesh as

e L eT

spi t eT

?

X L l ?1

b e spl cos l ωm t tσspl tl γspi

;

i ∈2;N ? :e18b T

Here γspi is the phase angle between the s -pi mesh on the left side and the reference s -p 1mesh [11,25]

γspi ?

Z s αpi ;?Z s αpi ;

for CW planet rotation ;for CCW planet rotation :

(e18c T

where Z s is the number of teeth on the sun gear.With γstg defined as phase difference between the left and right side due to

intentional nominal stagger of the teeth (e.g.γstg =πfor 50%stagger and γstg =0for no stagger),the excitation on the s -pi on the right side of the gear set is defined as

e R eT

t eT

?

X L b e cos l ωt tσ

166P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

Meanwhile,the transmission error excitation at the r -p 1mesh on the left side is given as

e L eT

rp 1t eT?

X L l ?1

b e rpl cos l ωm t tσrpl tl γrs ;

e19a T

where γrs is phase difference between the reference s -p 1mesh and the r -p 1mesh,both on the left side as defined in Ref.[25].b

e rpl and σrpl are the amplitude and phase angle o

f the l -th harmonic of the excitation.Similarly,excitations on any other r -pi meshes on the left side are given as

e L eT

rpi t eT

?

X L l ?1

b e rpl cos l ωm t tσrpl tl γrpi tl γrs

;

i ∈2;N ? :e19b T

Here γrpi is the phase difference between the r -pi mesh and the r -p 1mesh on the left side:

γrpi ?

?Z r αpi ;Z r αpi ;

for CW planet rotation ;for CCW planet rotation

(e19c T

where Z r denotes number of teeth on ring gear.With the same stagger phase angle γstg defined between the left and right sides,

excitations of the ring –planet meshes of the right side of the gear set are defined as

e R eTrpi t eT

?

X L l ?1

b e rpl cos l ωm t tσrpl tl γrpi tl γrs tl γstg ;

i ∈1;N ? :e19d T

It is noted here that the phase angles γstg ,γspi ,γrpi and γrs define the relative phasing amongst all 4N gear meshes of planetary

gear set in terms of number of teeth of the gears and planet position angle.2.4.Solution methodology

The mass and stiffness matrices,M and K =K m +K b ,in Eq.(17a)is considered for free vibration analysis of a given system.The corresponding Eigen value problem for a undamped system yields natural frequencies ωλand normalized mode shapes Q λ(λ∈[1,N dof ]).The response of the given system to transmission error excitations can be obtained using Modal Summation Technique.Based on phasing information the forcing vector can be expressed as sum of 4N vectors (each representing the excitation at one gear mesh)as

F t eT?

X 4N k ?1

F k t eT:

e20T

Response to each individual forcing vector F k (t )is obtained by modal summation as

q k t eT?e

F k k ?p X L l ?1X N dof λ?1

Θλl j ωm eTb e ?pl cos l ωm t tσ?pl tκ

;

e21a T

where ?=s ,r ,j ?????????

?1p and κrepresents appropriate phasing terms defined by Eq.(18)or Eq.(19).e F k is the amplitude of F k (t ).Θλl (j ωm )is dynamic compliance matrix given by

Θλl j ωm eT?Q λQ Τλω2

λ?l 2ω2m à

átj 2l ζλωλωm eT

;e21b T

where Q λis the λ-th normalized mode shape.Modal or proportional damping can be employed in this formulation.For modal damping ζλ=ζand for the proportional damping ratio ζλof the λ-th mode is

ηtεω2λ

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P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

where ηand εare the proportional damping constants.The overall displacement vector is given as the of steady-state responses to each of these excitations

q t eT?

X 4N k ?1

q k t eT:

e22T

With q (t )known,dynamic mesh forces at each of the 4N gear meshes can be obtained as (F ?pi )L /R (t )=k ?p p ?pi (t ),?=s ,r .

Likewise the components of the support bearing forces are computed by using user defined support stiffness matrices and the corresponding gear (or carrier)displacements.With the linear system and periodic excitation,displacements and corresponding dynamics mesh forces will be periodic and amplitudes of these periodic forces (F ?pi )L /R ,?=s ,r can be calculated at different mesh frequencies to study the dynamic behavior of the system under transmission error excitations.3.Example results

A double helical planetary gear set consisting of four equally spaced planets (N =4)with a stationary (non-rotating)carrier is used here as an example system.Table 1lists its basic gear design parameters.Total number of degree of freedom for the model is N dof =18(N +3)=126.Individual contact analyses of one sun –planet and one ring –planet mesh were carried out by using a gear pair load distributions model [24]under specified loading condition to determine the average mesh stiffnesses values (k sp and k rp ),and the harmonic components of the transmission error excitations (amplitudes b e spl and b e rpl ,and phase angles σspl and σrpl in Eqs.(18)–(19)where l ∈[1,3](Table 2)).Connecting structures between left and right sides have the dimensions listed in Table 1according to Fig.5.Table 1also lists the support stiffness values for the sun and ring gears and the carrier as well as the planet bearing stiffness values.

Eigen value solution is carried out for the system defined in Table 1to predict the undamped natural frequencies ωλas listed in Table 3.The corresponding modes are classified as in-phase,sequentially phased and counter-phased adapting the planetary gear set mode classification of Refs.[7,11]to describe the 2D (planar)modes from 2D models.Such classifications are not fully descriptive here,however,since the most modes exhibit 3D motions.The modal summation procedure described in Section 2was employed to predict the forced response vector q (t )to excitations given in Eqs.(18)and (19).Fig.6plots the maximum dynamic mesh force amplitudes (F spi )L and (F rpi )L for sun –planet and ring –planet meshes of the left side,respectively,as a function of gear mesh frequency (ωm =Z s Ωs =Z r Ωr for the case of fixed carrier where Ωs and Ωr are the rotational speeds of the sun and ring gears in Hz,

Table 2

Transmission error excitations.Harmonics b e spl (μm)b e rpl (μm)σspl (rad)σrpl (rad)10.4050.483?0.769?0.70920.0880.135?1.299?1.543Table 1

Basic design parameters of the example gear system.

Sun

Planet Ring Carrier Number of teeth

4739125

–Normal module (mm) 1.81 1.81–Helix angle (°)

21.521.5–Normal pressure angle (°)22.522.5–Base radius (mm)41.834.7111.2–Mass (kg)

2.40.73 6.814.4Mesh stiffness (N/μm)564531–k y ζ,k x ζ(N/μm)

1001001000–k θy ζ,k θx ζ(1e6Nm/rad)5510–D g (mm)7525064–D b /D o (mm)30

263.8

42

–

ε 1.35e ?6η

50

168P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

ωm [Hz]

600

1200

1800

2400

3000

0200040006000800010000

(b)

9451

2994

(F rpi )L

(F spi )L

Table 3

Predicted natural frequencies and mode types.Sr.no.Mode no.Frequency (kHz)Mode type

110.987In phase

22(2) 1.262Sequentially phased 33 1.902In phase

44(2) 1.965Sequentially phased 55(2) 2.076Sequentially phased 66 2.506In phase 77 2.518In phased

88(2) 2.522Sequentially phased 99(2) 2.994Counter phased 1010 3.166Planet mode ???

?

21228.398Planet mode ??

?

?

2325(2)8.852Sequentially phased 24269.037In phase

2527(2)9.451Sequentially phased ???

?

282911.047Planet mode ?

?

?

?

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P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

respectively).Fig.7shows the corresponding dynamic gear mesh forces (F spi )R and (F rpi )R for the right side meshes.In both figures,results for three right-to-left gear tooth stagger phase angles of γstg ?0;π2and πare compared.Several observations can be made from Figs.6and 7:

?Maximum dynamic sun –planet mesh force amplitudes at all planet meshes at a given (left or right side)are equal (i.e.F sp 1=F sp 2=F sp 3=F sp 4).The same is true for the ring –planet mesh forces as well.

?Each of the resonance peaks is associated with a particular mode excited by a certain harmonic amplitude l of the excitations.The frequencies of these modes as well as the harmonic exciting the mode are specified as labels for each resonance peak.Shapes of some of the excited modes are illustrated graphically in Fig.8.

?Maximum dynamic mesh forces vary considerably with γstg considered.Not only amplitudes of response but also resonance frequencies change with γstg .Different values of stagger excite different types of modes.For γstg =0,excited modes exhibit motion with both left and right sides of double helical gears move together as one piece as in the case of modes at natural frequencies of ωλ=1262and 8852Hz,both of which are classified as sequentially phased modes in Table 3.If the modal displacements for these modes are applied to relative gear mesh displacement expressions of Eqs.(4)and (9)with transmission error terms discarded,one would arrive at templates for these modes in the form

e23T

ωm [Hz]

(F rpi )R

(F spi )R

600

1200

1800

2400

3000

0200040006000800010000

9451

2994

(b)

Fig.7.Maximum dynamic mesh force amplitudes at the right side (a)s-pi and (b)r-pi meshes.

170P.Sondkar,A.Kahraman /Mechanism and Machine Theory 70(2013)157–174

It is noted in these modes that the right and left side gear meshes are in unison,and hence get excited for γstg =0.On the other hand the sequentially phased modes at ωλ=2076and 3757Hz exhibit equal but opposite motions on the right and left sides such that

e24T

As a result,they are excited by the excitations for γstg =π.Also it is evident from Figs.6and 7that second harmonic of excitation for γstg =πexcites the modes with left and right sides moving in unison exhibiting relative mesh displacements similar to Eq.(23)(ωλ=2ωm =8398Hz).Any stager angle other than 0and πexcites both types of modes but with less amplitudes,as illustrated in curves for γstg =π/2.It is noted here that connecting left and right sides of double helical gears rigidly instead of by Euler beam elements would not be able to capture the dynamics associated with these models.

?For the cases when γstg =0or π,maximum gear mesh forces at the left and right side meshes are equal:(F spi )L =(F spi )R and (F rpi )L =(F rpi )R while (F spi )L ≠(F spi )R and (F rpi )L ≠(F rpi )R for any other γstg .

In Fig.9,the influence of floating the sun gear radially on the forced response curves is demonstrated for both γstg =0and π.In the model,radial floating condition for sun gear is achieved by reducing the radial sun support stiffnesses significantly.Modes

Fig.8.Representative mode shapes at (a)ω22=8398Hz and (b)ω27=9451Hz.

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of the gear set with the stagger condition of γstg =πexhibit changes at lower frequency ranges (say ωm b 5000Hz)when the sun gear is switched from a radially fixed (piloted)condition to a radially floating condition as shown in Fig.9(b).Very little influence of sun gear support is evident in Fig.9(a)for γstg =0.4.Conclusions

In this study,a linear,time-invariant model of a double-helical planetary gear set was proposed.The model allows the analysis of a double-helical planetary gear set with any number of planets,any planet phasing and spacing configurations and any support conditions.The model included all rigid body degrees of freedom of gears and the carrier in a three-dimensional manner as well as the coupling amongst them.The model allows planets to be positioned equally or unequally spaced around the sun gear with the assumption that any unbalance issues associated with unequal planet spacing are addressed independently.The model captures the phasing relationships between the planet meshes as well as the right-to-left phase differences associated with the staggering the teeth of each double helical gear.Free and forced vibration analyses of the model were performed by solving the governing Eigen value problem to the corresponding undamped system and applying the modal summation technique with proportional damping.

An example set of analyses were performed to point various unique features of double-helical planetary gear sets.Numerous modes with dominant tilting motions were demonstrated to underline the essence of using a 3D formulation for double helical gears.Results also indicated that staggering of gear teeth influences the dynamic behavior of the gear system substantially.Different values of stagger excite different types of modes,resulting in different dynamic response curves.It is also noted that left and right sides of double helical gears can carry equal or different dynamic load amplitudes based on the right-to-left stagger

ωm [Hz]

(F rpi )L (F spi )L Fig.9.Effect of radially floating sun gear on dynamic mesh force amplitudes for (a)γstg =0and (b)γstg =π.

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Acknowledgments

Authors thank Pratt &Whitney for supporting of this research activity.

Appendix

A

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