Influence of dynamic behaviour on elastohydrodynamic ...cecs.wright.edu/~sheng.li/papers/Influence of dynamic behavior on... · elastohydrodynamic lubrication of spur gears ... Influence

Influence of dynamic behaviour onelastohydrodynamic lubrication of spur gearsS Li* and A Kahraman

Gleason Gear and Power Transmission Research Laboratory, The Ohio State University, Columbus, OH, USA

The manuscript was received on 1 November 2010 and was accepted after revision for publication on 14 April 2011.

DOI: 10.1177/1350650111409517

Abstract: In this study, the elastohydrodynamic lubrication (EHL) behaviour of high-speedspur gear contacts is investigated under dynamic conditions. A non-linear time-varying vibratorymodel of spur gear pairs is introduced to predict the instantaneous tooth forces under dynamicconditions within both the linear and non-linear operating regimes. In this model, the periodi-cally time-varying gear mesh stiffness and the motion transmission error are used as excitationsand a constant damping ratio is employed. This model allows the prediction of steady-state non-linear response in the form of tooth separation (contact loss). An earlier gear mixed EHL model [1]is adapted to simulate the lubrication behaviour of spur gear contacts under these dynamicloading conditions, considering the variations of the basic contact parameters, such as radii ofcurvature, sliding and rolling velocities, and measured roughness profiles, as the contact movesalong the tooth from the start of active profile to the tip. The EHL predictions under dynamicloading conditions are compared to those assuming quasi-static contact loads for gear setshaving smooth and rough surfaces to demonstrate the important influence of dynamic loadingon gear lubrication. The unique, transient EHL behaviour under the non-linear (intermittentcontact loss) condition is also illustrated.

Keywords: dynamics, non-linear, gear, mixed EHL, roughness

1 INTRODUCTION

Dynamic behaviour of gear systems has been studied

extensively for two primary reasons. One is that the

noise generated by a gear system is a direct conse-

quence of its dynamic behaviour. Any effort to reduce

gear noise of a transmission must focus on the reduc-

tion of the vibration levels of the gears. The second

reason is the durability concern. As the gear and bear-

ing force and stress amplitudes are often amplified

under dynamic conditions, such dynamic effects

must be taken into account in the design of gear

pairs. A large number of dynamics models have

been developed over the past 50 years. Most of

these models are for spur or helical gears as summar-

ized in the review papers by Ozguven and Houser [2]

and Wang et al. [3]. Based on the measured non-

linear dynamic behaviour documented in the

literature for spur gears [4–9], several non-linear

time-varying models of spur gears were proposed

with reasonable success in describing the published

experiments. These mostly torsional spur gear

dynamics models [10–15] used a clearance type

non-linear gear mesh function to take into account

the tooth separations in the presence of gear backlash

and considered the periodic time variation of the gear

mesh stiffness due to the fluctuation of the number of

loaded tooth pairs as the gears rotate. Likewise, in

line with the linear behaviour observed experimen-

tally [16], most helical gear pair models have

*Corresponding author: Gleason Gear and Power Transmission

Research Laboratory, The Ohio State University, 201 West 19th

Avenue, Columbus, OH 43210, USA.

email: [email protected]

740

Proc. IMechE Vol. 225 Part J: J. Engineering Tribology

neglected backlash while including additional rota-

tional, translational, and axial motions [16–20].

The tribology literature contains a wide spectrum

of elastohydrodynamic lubrication (EHL) models.

Sophisticated isothermal or thermal, Newtonian or

non-Newtonian, and line or point EHL models con-

sidering smooth or rough surface contacts have been

proposed to simulate the very complicated lubrica-

tion behaviour in a more comprehensive and accu-

rate way [21–26]. However, the contact parameters of

a gear pair, including the normal tooth force, radii of

curvature, and surface velocities, all vary as the gears

roll in mesh. Such a transient contact behaviour

might not be fully represented through the analyses

of certain discrete gear mesh positions of interest

using these EHL models with constant contact

parameters.

There are a limited number of published studies

that attempted to include the geometric, kinematic,

and load variations of gear contacts. Approximating

the elastic deformation by that of a smooth dry

Hertzian contact, Wang and Cheng [27, 28] predicted

the minimum film thickness and thermal character-

istics of spur gears having smooth tooth surfaces.

Larsson [29] and Wang et al. [30] proposed involute

spur gear EHL models for isothermal non-Newtonian

and thermal Newtonian fluids, respectively, employ-

ing assumed time-varying normal tooth force as the

contact moves along the line of action. These three

studies, while establishing the need for a specialized

EHL model for spur gears, lacked the ability to handle

the rough surface condition. In addition, these stu-

dies limited their treatments of the gear mesh defor-

mation to Hertzian effect. However, other effects due

to tooth bending, base rotation, and shear deforma-

tion were shown to be equally important in defining

gear mesh compliance [31]. Incorporating a gear load

distribution model considering all these essential

components of the gear tooth compliance, Li and

Kahraman [1] proposed a transient mixed EHL

model for spur gear pairs that is capable of handling

extreme asperity contact condition robustly by also

including the gear contact transient effects. Any man-

ufacturing errors or intentional tooth profile modifi-

cations were also taken into account.

In view of the above review of the gear dynamics

and gear tribology literature, one can point to an

apparent gap between these two disciplines. Tribo-

dynamic models for gear systems are not readily

available. Although there have been a few studies on

the gear mesh damping mechanism (induced by the

gear EHL contacts) [32] and on the influence of fric-

tion of lubricated contacts on gear dynamics [33, 34],

the impact of dynamic tooth force on gear EHL con-

tact is yet to be fully understood. The research by

Wang and Cheng [27, 28] incorporated a torsional

vibratory model into the EHL analysis to solve for

the minimum film thickness. However, the transient

gear parametric effects were not fully included.

Neither was the surface roughness. Accordingly, the

main objective of this study is to investigate the influ-

ence of gear dynamics on gear EHL behaviour.

In this study, a non-linear, time-varying dynamic

model of a spur gear pair is incorporated with a gear

mixed EHL model to predict the gear contact beha-

viour under both the linear and non-linear dynamic

conditions. The dynamic model includes the gear

mesh stiffness fluctuation, displacement excitation

due to the manufacturing errors and intentional

tooth corrections (also known as the transmission

error), and gear backlash non-linearity in a manner

similar to the model of Tamminana et al. [15]. The

dynamic model uses the quasi-static gear mesh stiff-

ness and the transmission error excitation predicted

using a gear load distribution model [31] to deter-

mine the instantaneous tooth contact force. The

EHL model used here is based on an earlier spur

gear EHL model [1] where the instantaneous contact

radii and surface velocities are defined using the invo-

lute geometry. With the dynamic tooth force provided

by the dynamic model, instantaneous film thickness

and normal pressure distributions of the tooth con-

tact are predicted by the EHL model as the contact

moves along the tooth surface.

In this study, the effect of the lubrication character-

istics on the dynamic response in terms of lubricant

stiffness and damping is not considered, such that the

dynamic solver and the EHL solver are uncoupled.

The investigation is also kept limited to spur gear

pairs with no significant lead modifications to allow

a line contact formulation. Other types of gear pairs

such as helical and hypoid gears were beyond the

scope of this study.

2 MODEL FORMULATIONS

2.1 Prediction of dynamic gear tooth

contact forces

In order to predict the dynamic loading on the teeth

of a spur gear pair, a single-degree-of-freedom (DOF)

discrete model similar to that of Tamminana et al.

[15] is employed. This torsional dynamic model,

shown in Fig. 1, consists of two rigid discs of radii

rb1 and rb2 (to represent the base circle radii of gears

1 and 2) and polar mass moments of inertia I1 and I2,

respectively. The gear mesh interface model consists

of (a) a parametrically time-varying gear mesh stiff-

ness kðt Þ, (b) a constant viscous damper c, and (c) an

externally applied gear mesh displacement excitation

Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 741


eðt Þ, all of which are applied along the line of action

(line tangent to the base circles of the gears). Here, the

least known parameter is c since there are several

power loss mechanisms present, including the vis-

cous losses at the gear mesh interface, viscous

losses on the bearings, and windage, churning, and

pocketing losses due to fluid–gear interactions within

the gear box. The damping element c is intended to

represent all these mechanisms. However, some of

them are not feasible to easily quantify. The time var-

iance of kðt Þ is mainly due to the fluctuation of the

number of loaded tooth pairs between two integers

(typically 1 and 2) as the gears roll in mesh. The dis-

placement excitation eðt Þ represents the motion

transmission deviation caused by intentional tooth

profile modifications as well as the manufacturing

errors under unloaded condition.

Bulk of the non-linear behaviour observed in spur

gear pairs occurs at instances when the dynamic force

amplitude exceeds the static load (preload) trans-

mitted by the gear pair. With the presence of gear

backlash, the gear teeth lose contact at such instances

and the gear mesh stiffness drops to zero instanta-

neously. As proposed earlier [12, 15], the mesh stiff-

ness kðt Þ is subjected to a piecewise linear clearance

function, d as illustrated in Fig. 1. This function is

composed of a dead zone (backlash) of size 2b

bounded by two unity slope regions representing

the linear and back contact conditions (no contact

loss).

With the positive directions of the alternating rota-

tional displacements, #1 and #2, and the applied

torque, T1 and T2, defined as shown in Fig. 1, the

dynamic equations read

I1€#1ðt Þ þ rb1kðt Þ�ðt Þ þ crb1½_sðt Þ � _eðt Þ� ¼ T1 ð1aÞ

I2€#2ðt Þ � rb2kðt Þ�ðt Þ � crb2½_sðt Þ � _eðt Þ� ¼ �T2 ð1bÞ

where sðt Þ ¼ rb1#1ðt Þ � rb2#2ðt Þ is the dynamic trans-

mission error and

�ðt Þ ¼sðt Þ � eðt Þ � b, sðt Þ � eðt Þ4b0, sðt Þ � eðt Þ

�� bsðt Þ � eðt Þ þ b, sðt Þ � eðt Þ5�b

8<: ð1cÞ

Since the generalized parameters of the two-DOF

model of Fig. 1 are semi-definite with a rigid body

mode at zero natural frequency, a new relative dis-

placement parameter is defined as �ðt Þ ¼ sðt Þ � eðt Þ.

With this, the equation of motion of the resultant

definite single-DOF model is derived as

me€�ðt Þ þ c _�ðt Þ þ kðt Þ�ðt Þ ¼ �F �me €eðt Þ ð2aÞ

where me is the equivalent mass defined as

me ¼ I1I2

�ðI1r2

b2 þ I2r2b1Þ. The constant force trans-

mitted by the gear mesh �F ¼ me rb1T1=I1ð þrb2T2=I2Þ,

where T1 and T2 are the constant external torques

applied to gears 1 and 2, respectively. The overdot

denotes the differentiation with respect to time t. The

non-linear restoring function � in Fig. 1 has the form of

�ðt Þ ¼�ðt Þ � b, �ðt Þ4b0, �ðt Þ

�� b�ðt Þ þ b, �ðt Þ5�b

8<: ð2bÞ

It is noted that the mesh stiffness kðt Þ consists of a

mean component, �k, and an alternating component,

kaðt Þ, i.e. kðt Þ ¼ �k þ kaðt Þ. With this, a set of dimen-

sionless parameters can be defined as �ðt Þ ¼ �ðt Þ�

b,

eðt Þ ¼ eðt Þ�

b, �ðt Þ ¼ �ðt Þt=b, � ¼ c=ð2

ffiffiffiffiffiffiffiffiffiffime

�k

qÞ and

F ¼ �F=ðb �kÞ, leading to a dimensionless form of equa-

tion (2a) as

d2�ðt Þ

dt 2þ 2�!n

d�ðt Þ

dtþ !2

n 1þkaðt Þ

�k

� ��ðt Þ

¼ !2nF �

d2eðt Þ

dt 2ð3Þ

where !n ¼

ffiffiffiffiffiffiffiffiffiffiffiffi�k=me

qis the undamped natural fre-

quency. The mesh stiffness kðt Þ of a gear pair can be

determined using a gear load distribution model [31]

or any finite element based gear contact model. In

the latter case, the difference of the gear static trans-

mission error between the loaded (~eðt Þ) and unloaded

(eðt Þ) conditions can be used to estimate the

mesh stiffness as kðt Þ ¼ ðT1

�rb1Þ

�½~eðt Þ � eðt Þ� [15].

Fig. 1 A purely torsional dynamic model of a spur gearpair

742 S Li and A Kahraman


Defining the dynamic mesh force from equation (3)

as

DMFðt Þ ¼ b �k2�

!n

d�ðt Þ

dtþ 1þ

kaðt Þ�k

� ��ðt Þ

( )ð4Þ

The individual dynamic tooth force, Wdðt Þ, can be

obtained approximately from the quasi-static tooth

force, Wsðt Þ, as [15]

Wdðt Þ ¼Wsðt ÞDMFðt Þ

T1=rb1

� �ð5Þ

2.2 Time-varying gear tooth contact parameters

Unlike the simple contact of cylinder pair, the gear

contact parameters, including the radii of curvature,

tangential surface velocities, and normal tooth force,

all vary as the contact moves along the line of action

when the gears rotate. According to the basic involute

gear geometry and kinematics, the contact radii of

curvature r1 and r2 are given as

r1ðt Þ ¼ rb1�1ðt Þ ð6aÞ

r2ðt Þ ¼ rb2�2ðt Þ ð6bÞ

where �1ðt Þ and �2ðt Þ are the roll angles of gears 1 and

2, respectively. As a result, the corresponding surface

velocities tangent to the tooth surfaces become vari-

able as well such that

u1ðt Þ ¼ !1r1ðt Þ ð7aÞ

u2ðt Þ ¼N1

N2!1r2ðt Þ ð7bÞ

where N1 and N2 are the number of teeth of gears 1

and 2, respectively, and !1 the angular velocity of

gear 1. Here, u1ðt Þ5u2ðt Þwhen the contact point tra-

vels long the dedendum of the driving gear (gear 1)

from the start of active profile (SAP) to the pitch

point such that the sliding velocity is negative, i.e.

usðt Þ ¼ u1ðt Þ � u2ðt Þ5 0. When the contact reaches

the pitch point, usðt Þ ¼ 0 (pure rolling condition).

Within the addendum range from the pitch point to

the tooth tip of the driving gear, usðt Þ4 0. In the pro-

cess, the rolling velocity urðt Þ ¼12 u1ðt Þ þ u2ðt Þ½ � varies

with the gear rotation as well. The resultant slide-

to-roll ratio of the tooth contact, defined as

SRðt Þ ¼ usðt Þ�

ur ðt Þ, varies from a negative limit to a

positive one with SRðt Þ ¼ 0 at the pitch point.

As the driving and driven gears roll in mesh, the

number of loaded tooth pairs alternates between

two integers that bound the profile contact ratio of

the gear pair (average number of tooth pairs in con-

tact), resulting in the periodic quasi-static tooth force

Ws ¼Wsðt Þ.

The gear load distribution model [31] referred to

earlier for the predictions of kðt Þ and eðt Þ is also

used to predict Wsðt Þ. This model includes the flex-

ibilities associated with tooth bending, shear defor-

mation, base rotation, as well as the contact

deformation. It also considers any deviation of the

tooth profile from the perfect involute primarily due

to the intentional modifications such as tip relief and

profile crown. In the absence of such modifications,

Wsðt Þ experiences sudden and drastic changes at the

instants when the number of loaded tooth pairs

changes from two to one (the lowest point of single

tooth contact, LPSTC) and from one to two (the high-

est point of single tooth contact, HPSTC).

2.3 Transient mixed EHL model for spur

gear contacts

Assuming negligible shaft misalignment and a rea-

sonable level of lead modification, the contact of a

spur gear pair can be characterized as a line contact.

The one-dimensional transient Reynolds equation

governs the fluid flow in the contact areas where suf-

ficient lubricant film is established

@

@xf ðt Þ

@pðx, t Þ

@x

� �¼@ ur ðt Þ�ðx, t Þhðx, t Þ½ �

@x

þ@ �ðx, t Þhðx, t Þ½ �

@tð8aÞ

where pðx, t Þ, hðx, t Þ, and �ðx, t Þ are the transient pres-

sure, film thickness, and density distributions along

the rolling direction x. The Ree–Eyring flow coeffi-

cient, f ðt Þ, is approximated as f ðt Þ ¼ ½�h3�ð12�Þ� cos

h½�mðt Þ��0� [1], where � is the lubricant viscosity, �0

the lubricant reference stress, and �m the mean vis-

cous shear stress; �mðt Þ ¼ �0 sinh�1 �usðt Þ�ð�0hÞ

. For

any local area where the fluid film is extremely thin,

say less than two layers of lubricant molecules,

hydrodynamic lubrication becomes impossible and

the reduced Reynolds equation [1, 25, 26] is used to

describe the contact

@ ur ðt Þ�ðx, t Þhðx, t Þ½ �

@xþ@ �ðx, t Þhðx, t Þ½ �

@t¼ 0 ð8bÞ

Once a smooth transition from the fluid film region

to the asperity contact region is assumed, this unified

Reynolds equation system of equations (8a) and (8b)

govern the EHL behaviour of the contact, considering

the hydrodynamic and asperity contact pressures

simultaneously.

Assuming only elastic deformation, the local film

thickness can be defined as

hðx, t Þ ¼ h0 tð Þ þ g0ðx, t Þ þ V ðx, t Þ

� R1ðx, t Þ � R2ðx, t Þ ð9Þ



where h0 is the reference film thickness, V the surface

elastic deflection, and g0 the unloaded geometric gap

between the mating tooth surfaces, which is time

dependent through the variable equivalent radius

of curvature reqðt Þ ¼ r1ðt Þr2ðt Þ�½r1ðt Þ þ r2ðt Þ� for

g0ðt Þ ¼ x2�½2reqðt Þ�. The terms R1ðx, t Þ and R2ðx, t Þ

represent the tooth surface roughness profiles mea-

sured in the profile (rolling and sliding) direction x.

Here, it is assumed that these roughness profiles

remain constant along the tooth face direction to

allow the use of this line contact formulation. This

is a reasonable assumption for certain finishing pro-

cesses such as grinding and shaving.

The next equation of interest is the load balance

equation, which states that the total contact force

due to the instantaneous pressure distribution over

the entire contact zone must balance the dynamic

tooth force at the same instant, i.e.

W 0dðt Þ ¼

Zpðx, t Þ dx ð10Þ

where W 0dðt Þ is the dynamic tooth force intensity

along the contact line (dynamic tooth force per unit

face width). The value of h0 in equation (9) must be

adjusted iteratively until the pressure distribution

pðx, t Þ satisfies equation (10).

Various forms of viscosity–pressure relationships

have been used in the past including the Barus’ expo-

nential relationship, the Roeland’s equation, as well

as the two-slope exponential relationship. However,

these relationships might not be accurate within very

wide pressure ranges experienced in rough gear con-

tacts [35]. The Doolittle–Tait relationship was pro-

posed as a potential remedy [35]. While any of the

other viscosity–pressure relationships can be used

as long as they represent the measured behaviour of

the lubricant accurately within the operating pressure

range, the Doolittle equation [36] used in this study is

� ¼ �0 exp B�Voccð1� �V Þ

ð �V � �VoccÞð1� �VoccÞ

� �ð11aÞ

where �0 the ambient viscosity, B the Doolittle para-

meter, and �Vocc the normalized occupied volume. The

normalized volume �V is a pressure-dependent para-

meter that can be modelled through the empirical

Tait equation of state [37] as

�V ¼ 1�1

ð1þ K 00Þln 1þ

p

K0ð1þ K 00Þ

� �ð11bÞ

where K0 and K 00 are the bulk modulus and the deri-

vative of bulk modulus with respect to pressure,

respectively, when p ¼ 0. The density–pressure rela-

tionship of the lubricant is modelled the same way as

described in Li and Kahraman [1], in which the details

of the discretization and linearization of the govern-

ing equations can be found.

3 RESULTS OF AN EXAMPLE ANALYSIS

In this section, the transient contact pressure and film

thickness distributions between the mating teeth of

an example spur gear pair under the dynamic loading

condition are presented at different mesh frequencies

to demonstrate the substantial impact of the dynamic

response on the EHL behaviour. The design para-

meters of the example spur gear pair are listed in

Table 1. The mass and inertia values of this gear

pair are also listed. This gear pair design was used

in several earlier experimental studies on the non-

linear dynamic behaviour of spur gear pairs [7–9,

12, 13]. The lubricant used in this study is Mil-

L23699, whose viscosity–pressure relationship at an

inlet temperature of 100�C is plotted in Fig. 2. The

black dots in the figure denote the measured viscosity

values at different pressures extracted from Fig. 2

of Bair et al. [35]. The solid line represents the

viscosity computed from equation (11) with the

required parameters regressed from the measured

data as �0 = 0.004 573 Pa s, B = 3.3054, �Vocc ¼ 0:6132,

K0 = 1.0747 GPa, and K 00 ¼ 10:076. The viscosity esti-

mates of equation (11) with these parameter values

are in good agreement with the measurement of Bair

et al. [35]. The density of this lubricant at the same

temperature and ambient pressure is �0 = 947.8 kg/

m3. The computational domain with the dimension

of �2:5amax � x � 1:5amax (amax is the maximum

Hertzian half-width of the contacts along the line of

action) is discretized into 512 elements with the grid

size �x ¼ 1:3 mm, which reasonably represents the

measured roughness resolution. The entire analysis

from the SAP to the tip is discretized into 1000 time

steps, resulting in the time resolution of 8� 10�5 s at

fm = 1250 Hz, 5� 10�5 s at fm = 2085 Hz and 4� 10�5 s at

fm = 2375 Hz. This represent a more refined time incre-

ment compared to those used in Larsson [29] and

Wang et al. [30]. To start the simulation at the SAP,

the Hertzian pressure is used as the initial guess and

iterated until the converged stationary EHL solution is

obtained. During the analysis from the SAP to the tip,

whenever the tooth force reaches zero (due to the

dynamic effect), a very small loading value of 5 N was

applied artificially, that is negligibly small compared to

typical tooth force levels at several kilo-Newtons.

Employing the experimentally determined damp-

ing ratio of � ¼ 0:01 (1 per cent) in equation (3) [7–9],

the dynamic model proposed above is used to predict

the steady-state dynamic response of the gear pair at

T1 = 250 Nm within the gear mesh (tooth passing) fre-

quency range of fm = 500–3500 Hz (fm ¼1

2�N1!1 where



!1 is in rad/s). The gear load distribution model is

implemented to determine kðt Þ of this gear pair at

T1 = 250 Nm, as shown in Fig. 3(a). Although the

load distribution model is capable of including any

profile modification, the example gear pair used here

has perfect involute profile such that eðt Þ ¼ 0. The

Fourier spectrum of kðt Þ, shown in Fig. 3(b), indicates

that �k ¼ 3:26� 108 N/m with the first three harmonic

amplitudes of k1 = 3.46� 107, k2 = 2.16� 107, and

k3 = 9.583� 106 N/m. With this, the natural frequency

of the corresponding linear, undamped system is !n

¼

ffiffiffiffiffiffiffiffiffiffiffiffi�k=me

q¼ 20 375 rad/s (fn ¼ !n=ð2�Þ ¼ 3242 Hz).

Here, the lubricant stiffness is not included sinceits magnitude, which ranges 2.5� 1010 to5.1� 1011 N/m for the operating conditions consid-ered (estimated from the Hamrock–Dowson for-mula), is several orders larger than that of thegear mesh stiffness. The predicted root-mean-squared (r.m.s) value of sðt Þ versus fm plot ofFig. 3(c) reveals one primary resonance peak at

fm ¼ fn � 3240 Hz caused by the first harmonic ofthe excitation as well as two super-harmonic reso-nances at fm ¼

12 fn � 1620 Hz and fm ¼

13 fn � 1080

Hz caused by the second and third harmonicterms of the excitation. In the vicinity of these reso-nance peaks, two stable motions coexist: the lowerbranch linear motion without tooth separation andthe upper branch non-linear motion with toothseparation which exhibits a softening type non-linear behaviour. It is the initial condition that dic-tates which motion should be exhibited by the gearpair. This predicted response of Fig. 3(c) agrees wellwith the published measurement using the sameexample gear pair [7–9].

Three representative operating conditions marked

in Fig. 3(c) are considered here to demonstrate the

impact of the dynamic response on the EHL beha-

viour of the gear set. They include two off-resonance

conditions at fm = 1250 and 2085 Hz (marked as points

I and II in Fig. 3(c)) and a non-linear resonant condi-

tion at fm = 2375 Hz (marked as point III in Fig. 3(c)).

The variations of the contact geometry parameters

r1, r2, and req, and the speed parameters u1, u2, us, ur,

and SR of the example gear pair operating at

fm = 1250 Hz as the contact moves from the SAP at

�1 ¼ 14:5� to the tip at �1 ¼ 27:2� are shown in Fig. 4.

As seen, the variations of the radii of curvature, sur-

face velocities, and sliding velocity are evident, while

ur is constant in this case since the gear pair has unity

ratio (i.e. the driving and driven gears are identical).

In Fig. 5(a), the quasi-static tooth force Ws predicted

by the gear load distribution model at T1 = 250 Nm is

compared to its dynamic counterpart Wd when the

gears are operated under the condition of point I

(Fig. 3(c)). Here, Ws is shown to increase linearly

from the SAP to the LPSTC at �1 ¼ 20:0�, and then

almost double at the LPSTC where one of the two

loaded tooth pairs loses contact. After staying rela-

tively constant till the HPSTC at �1 ¼ 21:7�, Ws experi-

ences a sudden drop as the gear mesh transmits to

two loaded tooth pairs. As shown in Li and Kahraman

[1], these drastic changes in Ws impact the EHL beha-

viour significantly. The corresponding dynamic tooth

force Wd (in the same figure) has a substantially dif-

ferent shape from that of Ws. As this frequency is near

the resonance peak of fm ¼13fn caused by the third

harmonic of the excitation, nearly three cycles of fluc-

tuation are observed in Fig. 5(a) for the Wd curve

within a base pitch. In Figs 5(b) and (c), the Wd

curves for points II and III (Fig. 3(c)) are compared

to the same Ws. It is seen that both the amplitude and

shape of Wd change significantly with fm and show

little resemblance to those of Ws, suggesting that per-

forming any EHL analysis using Ws is inaccurate in

the case of high-speed gearing where dynamic effects

Table 1 Design parameters of the example unity-ratio

spur gear pair used in this study

Number of teeth 50Module (mm) 3.0Pressure angle (degrees) 20.0Outside diameter (mm) 156.0Pitch diameter (mm) 150.0Root diameter (mm) 140.0Centre distance (mm) 150.0Face width (mm) 20.0Backlash (mm) 0.14Polar mass moment of Inertia (kg �m2) 0.0078Equivalent mass, me (kg) 0.785

Fig. 2 Viscosity–pressure relationship of lubricantMil-L23699 at 100�C



are prominent. It is also noted in Fig. 5(c) that the

dynamic tooth force becomes intermittent in the

non-linear region of operation. The Wd amplitude

for the upper branch motion of point III is amplified

to more than twice the maximum Ws value, while

sizable portions of the mesh cycle are spent under

zero contact load.

Starting with the case of perfectly smooth surfaces

(i.e. R1ðx, t Þ ¼ R2ðx, t Þ ¼ 0 in equation (8)), the mini-

mum film thickness hmin predictions of the EHL

simulations under Wdðt Þ and Wsðt Þ are compared in

Fig. 6 at the points I–III defined in Fig. 3(c). The hmin

curves that correspond to Ws (dashed lines) oscillate

between the LPSTC and somewhere after the HPSTC,

which is due to the strong squeezing effect at the

LPSTC induced by the sudden load jump-up and

the pumping effect right after the HPSTC, induced

by the sudden load jump-down as it was described

in Li and Kahraman [1]. Such film thickness spikes

were also reported in Larsson [29] and Wang et al.

[30] while the severity of these jumps varied, perhaps

due to the different operating conditions and coarser

time resolution used in those studies. Such quasi-

static behaviour, however, becomes irrelevant as the

Fig. 3 Gear mesh stiffness: time history kðt Þ (a) and the corresponding frequency spectrum ki ofthe example spur gear pair (b), and the r.m.s DTE amplitudes as a function of gear meshfrequency fm (c). Note: T1 = 250 Nm

Fig. 4 The variations of contact radii (a) and surfacevelocities (b) with �1 during a single toothengagement cycle of the example gear pair atthe mesh frequency of fmesh = 1250 Hz (point Iin Fig. 3)



hmin behaviour corresponding to Wd (solid lines) is

drastically different from that for Ws at all three fm

values of operation. The wavy shape of this dynamic

condition hmin with �1 is observed to be dictated by

the shape of Wd that itself is defined by the harmonic

term(s) of the excitation effective at that fm value. For

instance, at point I where fm is near the third super-

harmonic resonance, three cycles of the fluctuation of

Wd (in Fig. 5(a)) result in a similar qualitative shape of

hmin as shown in Fig. 6(a). Similarly, the effect of the

second harmonic of the excitation is evident in Fig.

6(b) at point II. Finally, under the non-linear condi-

tion which exhibits the loss of contact (point III), it is

difficult to point to the formation of fluid film as large

portions of the mesh cycle are spent with the contact-

ing surfaces away from each other. The resultant

dynamic film thickness variation in Fig. 6(c) is

accordingly very different from that using Ws as the

normal contact load.

The lubrication behaviour summarized in the form

of hmin predicted by the EHL model in Fig. 6 is com-

plemented by Figs 7 to 9 that provide instantaneous

pressure and film thickness distributions at various �1

values along the mesh cycle. In Fig. 7, the hðx, t Þ and

pðx, t Þ are shown at ten �1 values (denoted by points A

to J) under the quasi-static loading of Ws and rota-

tional speed of 1500 r/min (fm = 1250 Hz). The effects

of the sudden change in Ws at the LPSTC and the

variations of the other contact parameters (radii and

sliding velocity) are evident in this figure. When the

dynamic load Wd is considered (at point I,

fm = 1250 Hz), completely different hðx, t Þ and pðx, t Þ

are obtained in Fig. 8 at the same �1 values as in Fig. 7.

At the SAP (mesh position A) where Wd peaks, typical

smooth surface EHL pressure and film thickness

Fig. 6 Comparisons of the predicted EHL minimumfilm thickness along the driving gear tooth sur-face with Wsðt Þ and Wdðt Þ as the tooth force:(a) fm = 1250 Hz (point I in Fig. 3(c)); (b)fm = 2085 Hz (point II in Fig. 3(c)); and (c)fmesh = 2375 Hz (point III in Fig. 3(c)). Note:T1 = 250 Nm

Fig. 5 Comparisons of the tooth force variations alongthe driving gear roll angle between the quasi-static loading condition and the dynamic load-ing condition: (a) fm = 1250 Hz (point I in Fig.3(c)); (b) fm = 2085 Hz (point II in Fig. 3(c));and (c) fmesh = 2375 Hz (point III in Fig. 3(c)).Note: T1 = 250 Nm



distributions are observed. As Wd decreases with the

increasing �1, the hydrodynamic fluid film becomes

less pressurized and thicker. Beyond the mesh posi-

tion B, Wd starts to increase with the consequence of

wider fluid film and larger thickness, as shown for the

mesh position C. The resultant hðx, t Þ and pðx, t Þ,

shown in Fig. 8, point to the influence of Wd. A com-

parison between Figs 7 and 8 indicates that the hðx, t Þ

and pðx, t Þ solutions obtained using Ws or Wd are sub-

stantially different. It is also noted that the pressure

ripples at both the inlet and outlet zones introduced

by the sudden load change at the LPSTC (position G)

under the quasi-static condition are absent in Fig. 8

with Wd as the normal load. Next, the hðx, t Þ and

pðx, t Þ at ten different �1 values are shown in Fig. 9

for point III. It is evident that the EHL behaviour

under such non-linear dynamic condition has abso-

lutely no resemblance to the corresponding quasi-

static condition shown in Fig. 7. The fluid film is

formed between the mesh positions A and E to certain

extent during the first loaded segment of the contact

that dissolves completely during the period between

E and F where there is no tooth load. Afterwards, the

transient effort to form the fluid film is observed start-

ing from the mesh position F where the tooth contact

is reestablished. The hðx, t Þ and pðx, t Þ distributions

Fig. 7 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the quasi-static loading condi-tion with a rotational speed of 1500 r/min and T1 = 250 Nm



are highly transient for the rest of the load cycle (from

F to J).

At the end, an analysis using the ground tooth

surface profiles, shown in Fig. 10, is presented to

illustrate the EHL contacts of rough spur gear sur-

faces under the dynamic condition. These mea-

sured R1 and R2 profiles have the Rq values of

0.54 and 0.53 mm, respectively. Considering these

roughness profiles, the transient hðx, t Þ and pðx, t Þ

distributions of the example gear pair at the same

mesh positions and under the same dynamic con-

tact condition as in Fig. 8 are shown in Fig. 11.

Comparing Fig. 11 with Fig. 8, it can be seen that

the size of the contact zone of the dynamic rough

contact varies in the same way as that under the

dynamic smooth condition when the Wd fluctu-

ates. Due to the surface irregularities, however,

the local contact pressures in Fig. 11 can easily

exceed 1 GPa. At various instantaneous local con-

tact points, hðx, t Þ is zero, indicating actual asperity

contacts with the corresponding spikes displaying

in the pressure distributions. Meanwhile, the deep

roughness valleys introduce much larger local

film thickness values. Neither the pressure distri-

bution nor the film thickness distribution is

smooth and continuous. The influences of the vari-

able load Wd on hðx, t Þ and pðx, t Þ are also evident

in Fig. 11.

Fig. 8 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the dynamic loading conditionwith fmesh = 1250 Hz and T1 = 250 Nm



Fig. 9 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the dynamic loading conditionwith fmesh = 2375 Hz and T1 = 250 Nm

Fig. 10 Measured tooth surface roughness profiles in the direction of relative sliding and rolling



4 CONCLUSIONS

In this study, an EHL model of a gear pair was used in

conjunction with a dynamic model to predict the

lubrication behaviour under various dynamic speed

conditions. The non-linear time-varying dynamic

model of a spur gear pair was used to predict the

instantaneous tooth contact force under the dynamic

condition within both the linear and non-linear oper-

ating regimes. The predicted instantaneous tooth

force was fed into the gear EHL model to simulate

the lubrication behaviour of the spur gear contact

under the dynamic loading condition. The EHL

model included any asperity interaction activity as

well as the variations of radii of curvature, sliding,

and rolling velocities and measured roughness pro-

files as the contact moves along the gear tooth from

the SAP to the tip. The EHL results presented under

various dynamic loading conditions were shown to

differ from those under static tooth load conditions,

suggesting that the dynamic behaviour of the gear

pair must be included in describing the high-speed

gear tribology. The tooth separation that takes place

due to the backlash non-linearity was also shown to

impact the transient EHL behaviour in a unique

manner.

� Authors 2011

REFERENCES

1 Li, S. and Kahraman, A. A transient mixed elastohy-drodynamic lubrication model for spur gear pairs.J. Tribol., 2010, 132, 0115019.

2 Ozguven, H. N. and Houser, D. R. Mathematicalmodels used in gear dynamics – a review. J. SoundVib., 1988, 121, 383–411.

3 Wang, J., Li, R., and Peng, X. Survey of nonlinearvibration of gear transmission systems. Appl. Mech.Rev., 2003, 56, 309–329.

4 Umezawa, K., Ajima, T., and Houjoh, H. Vibration ofthree axis gear system. Bull. JSME, 1986, 29, 950–957.

5 Munro, R. G. Dynamic behavior of spur gears. PhDThesis, Cambridge University, 1962.

6 Kubo, A., Yamada, K., Aida, T., and Sato, S. Researchon ultra high speed gear devices. Trans. Jpn. Soc.Mech. Eng., 1972, 38, 2692–2715.

7 Kahraman, A. and Blankenship, G. W. Experimentson nonlinear dynamic behavior of an oscillator withclearance and time-varying parameters. J. Appl.Mech., 1997, 64, 217–226.

8 Kahraman, A. and Blankenship, G. W. Effectof involute contact ratio on spur gear dynamics.J. Mech. Des., 1999, 121, 112–118.

9 Kahraman, A. and Blankenship, G. W. Effect ofinvolute tip relief on dynamic response of spur gearpairs. J. Mech. Des., 1999, 121, 313–315.

10 Umezawa, K., Sata, T., and Ishikawa, J. Simulationof rotational vibration of spur gears. Bull. JSME,1984, 38, 102–109.

Fig. 11 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pairat the same mesh positions as in Fig. 8 under the dynamic loading condition withfm = 1250 Hz, T1 = 250 Nm, and the surface roughness profiles shown in Fig. 10



11 Kahraman, A. and Singh, R. Nonlinear dynamics ofa spur gear pair. J. Sound Vib., 1990, 142, 49–75.

12 Blankenship, G. W. and Kahraman, A. Steady stateforced response of a mechanical oscillator with com-bined parametric excitation and clearance type non-linearity. J. Sound Vib., 1995, 185, 743–765.

13 Kahraman, A. and Blankenship, G. W. Interactionbetween external and parametric excitations in sys-tems with clearance. J. Sound Vib., 1996, 194,317–336.

14 Ozguven, H. N. and Houser, D. R. Dynamic analysisof high speed gears by using loaded static transmis-sion error. J. Sound Vib., 1988, 125, 71–83.

15 Tamminana, V. K., Kahraman, A., and Vijayakar, S.A study of the relationship between the dynamic fac-tors and the dynamic transmission error of spur gearpairs. J. Mech. Des., 2007, 129, 75–84.

16 Kubur, M., Kahraman, A., Zini, D., and Kienzle, K.Dynamic analysis of a multi-shaft helical gear trans-mission by finite elements: model and experiment.J Vib. Acoust., 2004, 126, 398–406.

17 Kubo, A. Stress condition, vibrational excitationforce and contact pattern of helical gears with man-ufacturing and alignment errors. J. Mech. Des., 1978,100, 77–84.

18 Kahraman, A. Effect of axial vibrations on thedynamics of a helical gear pair. J Vib. Acoust., 1993,115, 33–39.

19 Kahraman, A. Dynamic analysis of a multi-mesh helical gear train. J. Mech. Des., 1994, 116,706–712.

20 Maatar, M. and Velex, V. Quasi-static and dynamicanalysis of narrow-faced helical gears with profileand lead modifications. J. Mech. Des., 1997, 119,474–480.

21 Venner, C. H. Higher-order multilevel solvers for theEHL line and point contact problem. J. Tribol., 1994,116, 741–750.

22 Holmes, M. J. A., Evans, H. P., Hughes, T. G., andSnidle, R. W. Transient elastohydrodynamic pointcontact analysis using a new coupled differentialdeflection method. Part 1: theory and validation.Proc. IMechE, Part J: J. Engineering Tribology, 2003,217, 289–303.

23 Kim, H. J., Ehret, P., Dowson, D., and Taylor, C. M.Thermal elastohydrodynamic analysis of circularcontacts. Part 1: Newtonian model. Proc.IMechE, Part J: J. Engineering Tribology, 2001, 215,339–352.

24 Zhao, J. and Sadeghi, F. Analysis of EHL circularcontact start up. Part 1: mixed contact model withpressure and film thickness results. J. Tribol., 2001,123, 67–74.

25 Zhu, D. On some aspects of numerical solutions ofthin-film and mixed elastohydrodynamic lubrica-tion. Proc. IMechE, Part J: J. Engineering Tribology,2007, 221, 561–579.

26 Li, S. and Kahraman, A. A mixed EHL model withasymmetric integrated control volume discretiza-tion. Tribol. Int., 2009, 42, 1163–1172.

27 Wang, K. L. and Cheng, H. S. A numerical solution tothe dynamic load film thickness, and surface

temperatures in spur gears. Part I: analysis. J. Mech.Des., 1981, 103, 177–187.

28 Wang, K. L. and Cheng, H. S. A numerical solution tothe dynamic load film thickness, and surface tem-peratures in spur gears. Part II: results. J. Mech.Des., 1981, 103, 188–194.

29 Larsson, R. Transient non-Newtonian elastohydro-dynamic lubrication analysis of an involute spurgear. Wear, 1997, 207, 67–73.

30 Wang, Y., Li, H., Tong, J., and Yang, P. Transientthermoelastohydrodynamic lubrication analysis ofan involute spur gear. Tribol. Int., 2004, 37, 773–782.

31 Conry, T. F. and Seireg, A. A mathematical program-ming technique for the evaluation of load distribu-tion and optimal modifications for gear systems.J. Eng. Ind., 1973, 95, 1115–1122.

32 Li, S. and Kahraman, A. A spur gear mesh interfacedamping model based on elastohydrodynamic con-tact behavior. Int. J. Powertrains, 2011, 1 (in press).

33 He, S., Gunda, R., and Singh, R. Effect of slidingfriction on the dynamics of spur gear pair with rea-listic time-varying stiffness. J. Sound Vib., 2007,301(N3), 927–949.

34 Kahraman, A., Lim, J., and Ding, H. A dynamicmodel of a spur gear pair with friction. InProceedings of the 12th IFToMM World Congress,Besancon, France, 18–21 June 2007.

35 Bair, S., Jarzynski, J., and Winer, W. O. The tem-perature, pressure and time dependence of lubricantviscosity. Tribol. Int., 2001, 34, 461–468.

36 Doolittle, A. K. Studies in Newtonian flow. II. Thedependence of the viscosity of liquids on free-space. J. Appl. Phys., 1951, 22, 1471–1475.

37 Cook, R. L., King, H. E., Herbst, C. A., andHerschbach, D. R. Pressure and temperature depen-dent viscosity of two glass forming liquids: glyceroland dibutyl phthalate. J. Chem. Phys., 1994, 100,5178–5189.

Appendix

Notation

amax maximum Hertzian half-width along the

line of action

b half backlash

B Doolittle parameter

c viscous damping

e, ~e gear static transmission error under

unloaded and loaded conditions,

respectively

f flow coefficient

fm gear mesh frequency (Hz)

fn natural frequency (Hz)

g0 geometry gap before deformation

h film thickness

h0 reference film thickness

I1, I2 polar mass moments of inertia of gears 1

and 2, respectively



k gear mesh stiffness�k, ka mean and alternating components of gear

mesh stiffness

K0 bulk modulus at p ¼ 0

K 00 derivative of bulk modulus with respect to

pressure at p ¼ 0

me equivalent mass

N1, N2 number of teeth of gears 1 and 2,

respectively

p pressure

rb1, rb2 base circle radii of gears 1 and 2, respectively

req equivalent radius of curvature,

req ¼ r1r2

�ðr1 þ r2Þ

r1, r2 contact radii of curvature of gears 1 and 2,

respectively

R1, R2 surface roughness profiles of gears 1 and 2,

respectively

s dynamic transmission error

SR slide-to-roll ratio, SR ¼ us=ur

t time

T1, T2 torques applied to gears 1 and 2,

respectively

u1, u2 surface velocities in the direction of rolling

of gears 1 and 2, respectively

ur rolling velocity, ur ¼12ðu1 þ u2Þ

us sliding velocity, us ¼ u1 � u2

V surface elastic deformation�V normalized volume�Vocc normalized occupied volume

Wd dynamic tooth force

W 0d dynamic tooth force per unit face width

Ws quasi-static tooth force

x coordinate along the rolling direction

� non-linear restoring function

� damping ratio

� lubricant viscosity

�0 lubricant viscosity at ambient pressure

�1, �2 roll angles of gears 1 and 2, respectively

#1, #2 alternating rotational displacements of

gears 1 and 2, respectively

1, 2 Poisson’s ratios of gear 1 and 2, respectively

� lubricant density

�0 lubricant density at ambient pressure

�0 reference shear stress of the lubricant

!n natural frequency (rad/s)

!1, !2 angular velocities of gear 1 and 2,

respectively



Documents

Influence of dynamic behaviour on elastohydrodynamic ...cecs.wright.edu/~sheng.li/papers/Influence of dynamic behavior on... · elastohydrodynamic lubrication of spur gears ... Influence