A spur gear mesh interface damping model based on ...cecs.wright.edu/~sheng.li/papers/A spur gear...

4 Int. J. Powertrains, Vol. 1, No. 1, 2011

Copyright © 2011 Inderscience Enterprises Ltd.

A spur gear mesh interface damping model based on elastohydrodynamic contact behaviour

S. Li and A. Kahraman* Department of Mechanical Engineering, The Ohio State University, 201 W. 19th Avenue, Columbus, OH 43210, USA Fax: 614-292-3163 E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: In this study, the damping mechanism at the interface of the mating spur gear teeth is investigated. The dynamic behaviour of a spur gear pair is represented by a two Degree-of-Freedom (DOF) nonlinear model with the damping provided by the instantaneous tribological behaviour of the tooth contacts. The shear stress distributions along the tooth surfaces are incorporated with the dynamic model to formulate the periodic gear mesh viscous damping. With the assumption the radii of curvature of the contact surfaces can be represented by the ones at the pitch point, a simplified expression for the damping ratio is formulated.

Keywords: gear dynamics; spur gears; gear lubrication; gear mesh damping.

Reference to this paper should be made as follows: Li, S. and Kahraman, A. (2011) ‘A spur gear mesh interface damping model based on elastohydrodynamic contact behaviour’, Int. J. Powertrains, Vol. 1, No. 1, pp.4–21.

Biographical notes: Sheng Li achieved his PhD degree in Mechanical Engineering at The Ohio State University in 2009. He continued his research in the fields of gear tribology, power loss and efficiency, contact fatigue including pitting and scuffing and gear dynamics as a post-doctoral fellow in 2009 and as a senior research engineer since 2010 at The Ohio State University.

Ahmet Kahraman is a Professor of Mechanical Engineering at the Ohio State University. He is the Director of Gleason Gear and Power Transmission Research Laboratory. He also directs Pratt & Whitney Center of Excellence in Gearbox Technology. His research focuses on several areas of power transmission and gearing including gear system design and analysis, gear and transmission dynamics, gear lubrication and efficiency, wear and fatigue life prediction, and test methodologies. He is a fellow of ASME and member of STLE.

A spur gear mesh interface damping model based on elastohydrodynamic 5

1 Introduction

Dynamic modelling of gear pairs and gear trains has been an active research field in power train engineering for two main reasons. One is directly related to the noise and vibration requirements that the gear system must meet. A vibratory model of the gear system predicts the natural modes and resonance frequencies as well as the dynamic gear mesh and bearing forces transmitted to the surrounding structures. Such a model allows one to check whether any resonance peak is present within the frequency range of operation. It also allows the determination of the optimal tooth profile modifications which minimise the gear mesh and bearing forces. The other main reason for gear dynamic modelling is the durability concern. Dynamic gear tooth forces are typically larger than those predicted under the quasi-static condition, potentially impacting the tooth bending and contact fatigue lives of the gear system.

A very large number of the dynamic models have been proposed over the last 50 years, as reviewed extensively by Ozguven and Houser (1988a) and more recently by Wang et al. (2003). These models vary in many aspects. Focusing on spur gears, single or multi-degree-of-freedom discrete models and Finite Element (FE) based deformable body models have been proposed. The published experimental data clearly indicate (Blankenship and Kahraman, 1995; Kahraman and Blankenship, 1996, 1997, 1999a, 1999b; Kubo et al., 1972; Munro, 1962; Umezawa et al., 1986) that spur gear pairs act as nonlinear time-varying systems due to the gear backlash and periodically varying gear mesh stiffness. Therefore, several discrete, nonlinear, time varying (NTV) models were shown to correlate to spur gear experiments well (Blankenship and Kahraman, 1995; Kahraman and Singh, 1990; Kahraman and Blankenship, 1996; Ozguven and Houser, 1988b; Theodossiades and Natsiavas, 2000). These models typically employed two rigid disks to represent the inertias of the gear bodies that are connected with each other through a gear mesh interface model along the line of action. The gear mesh interface formulation of these models consists of four main components. The first component is a time-varying stiffness employed to represent the overall gear mesh flexibility including the compliances associated with the contact, tooth bending, shear and base rotation of the gear teeth as well as the change of the number of tooth pairs in contact with rotation. This mesh stiffness is typically subjected to a piecewise-linear clearance with a dead zone (backlash) allowing tooth separations to take place. Modelling of this clearance component is required to capture the change in stiffness when such nonlinear motions occur. The third component is a periodic displacement excitation, often called the Transmission Error (TE) excitation, again applied along the line of action to account for the disturbances caused by intentional tooth profile modifications such as tip and root relief or profile crown as well as unintentional manufacturing deviations from the actual tooth profile. The last gear mesh interface component is a viscous damper to account for the energy loss at the gear mesh interface. Among those four components, the stiffness and transmission error functions can be predicted by using any gear contact or load distribution model (Conry and Seireg, 1973) under quasi-static condition; and the backlash magnitude can be determined from the effective center distance and tooth thickness values. However, the derivation of the damping element is not as straightforward as the others. Almost all of the previous gear dynamic models employed a user-defined constant viscous damping coefficient, stating that it must be determined empirically. A wide range of gear mesh damping ratio value can be cited in the experimental gear dynamics literature, ranging from the values as low as 1–2%

6 S. Li and A. Kahraman

(Blankenship and Kahraman, 1995; Kahraman and Blankenship, 1996, 1997, 1999a, 1999b; Munro, 1962) to those as high as 10% (Kubo et al., 1972; Umezawa et al., 1986). Although the accuracy of the natural frequencies and resonance frequencies would be influenced slightly by the actual value of the damping ratio within this range, its impact on the resonance peak amplitudes is significant. In addition, the nonlinear behaviour induced by backlash and parametric resonances associated with the time-varying gear mesh stiffness are both very sensitive to the damping value of the gear mesh (Kahraman and Blankenship, 1996; Ma and Kahraman, 2005, 2006). Higher values of damping typically suppress both the nonlinear behaviour and parametric sub-harmonic resonance, hence, dictating the response characteristics in terms of the degree of nonlinearity and sub-harmonic motions.

Another group of dynamic models employed FE based approach (Parker et al., 2000; Tamminana et al., 2007) where the deformations of the gear bodies were included in addition to the compliances of the gear mesh. In these models, the gear mesh stiffness and the TE excitation were included implicitly through a deformable multi-body formulation as well as the backlash condition. These models were shown to compare with the experiments even better (Tamminana et al., 2007), provided the level of the gear mesh damping is known. These deformable body models often employed a damping matrix that is proportional to the stiffness and mass matrices of the system with the two proportionality constants determined empirically.

Several recent studies including Kahraman et al. (2007) and He et al. (2007) investigated the off line of action vibrations of gear pairs that are caused by the friction at the gear mesh interface. In these models, either a simple dry friction model or sliding friction formulae obtained from experiments or Elastohydrodynamic Lubrication (EHL) models were used to couple the motions along the line of action with those along the off line of action. These models still retained a user-defined gear mesh damping to predict the line of action motions. On the other hand, gear EHL models were proposed to predict the gear mesh mechanical power losses mainly for efficiency prediction purposes (Li et al., 2009; Li and Kahraman, 2010b; Xu et al., 2007). The relation between the gear mesh interface mechanical power loss and damping has drawn very little attention.

It is clear from the review of the published work that very little is known about the level and mechanism of gear mesh damping. The question of how the EHL behaviour of gear contacts impacts the dynamic response remains mostly unanswered. The fidelity of using a constant damping value at every speed, every gear mesh instant and every lubricant type and temperature has not been investigated. There is no published formulation to allow an estimation of the gear mesh damping as a function of the operating conditions (speed and torque), surface conditions as well as lubricant parameters. This paper aims at developing a viscous gear mesh damping formulation from the principle of EHL theory to fill this void. A full film lubrication version of the gear EHL model (Li and Kahraman, 2010a) is incorporated with a purely torsional spur gear dynamics model to obtain an approximate equivalent damper in the direction of line of action. The variation of the damping value with the gear mesh position (gear rotation) is investigated and the influences of various contact conditions including speed, torque and lubricant temperature on the resultant gear mesh damping value are demonstrated.


2 Gear mesh viscous damper formulation

2.1 Discrete gear dynamics model

A conventional single DOF NTV dynamic model for spur gear pairs, which was shown to correlate well with various spur gear dynamics experiments (Tamminana et al., 2007), is used here to investigate the gear mesh damping mechanism. As shown in Figure 1(a), the mating gear pair with base radii of r1 and r2 is represented by two rigid wheels of polar mass moments of inertia J1 and J2, connected via a time-varying mesh spring element k(t) subject to a backlash of 2b and a viscous damper c, both applied along the line of action of the gears. As stated earlier, any manufacturing errors as well as any intentional tooth profile modifications are also considered and modelled as the external displacement excitation e(t). In the attempt to capture the actual gear mesh damping mechanism instead of assuming any constant damper, the power dissipation at the interface of the mating tooth pair (Figure 1(b)) is used to model the gear mesh damping in place of the constant c. It is noted in this figure that both the traction forces exerted on the pinion and gear tooth surfaces by the lubricant film, F1 and F2, and the tangential dynamic tooth surface velocities, u1 and u2, are in the off line of action direction of the gears.

Figure 1 Discrete dynamic model of a spur gear pair

With the positive directions of the alternating rotational displacements θ1 and θ2 and the applied torque T1 and T2 defined in Figure 1(a), the equations of motion of the spur gear pair can be written as

1 1 1 1 1 11

( ) ( ) ( ) ,N

n nn

J t r k t t T F Rθ δ=

+ = +∑ (1a)

2 2 2 2 2 21

( ) ( ) ( ) ,N

n nn

J t r k t t T F Rθ δ=

− = − −∑ (1b)


where the additional subscript n denotes the nth loaded tooth pair such that F1n and F2n represent the pinion and gear traction forces of the nth tooth pair in contact, and R1n and R2n are the corresponding contact radii as shown in Figure 2 for a tooth pair contacting at point C (The subscript n is omitted in the figure for simplicity purposes). Here, N denotes the number of tooth pairs in contact at a certain mesh position. For most spur gear pairs, the maximum N is 2, such that the contact ratio of the gear pair is between 1 and 2. A gear load distribution model similar to that of Conry and Seireg (1973) is used to determine the static transmission error under unloaded (e(t)) and loaded ( ( ))e t conditions. The difference between e(t) and ( )e t is used to estimate the mesh stiffness as

1 1( ) ( ) [ ( ) ( )]k t T r e t e t= − (Tamminana et al., 2007). The nonlinear restoring function δ(t) with the backlash nonlinearity is given in a piecewise linear form as

1 1 2 2 1 1 2 2

1 1 2 2

1 1 2 2 1 1 2 2

( ) ( ) ( ) if ( ) ( ) ( ) ;( ) 0, if ( ) ( ) ( ) ;

( ) ( ) ( ) if ( ) ( ) ( ) .

r t r t e t b r t r t e t bt r t r t e t b

r t r t e t b r t r t e t b

θ θ θ θδ θ θ

θ θ θ θ

− − − − − >= − − ≤ − − + − − < −

(2)

In this equation, the first condition represents the linear motions with no tooth separations, while the second and third conditions represent the tooth separations (single-sided impacts) and the back side contacts (double-sided impacts). Although tooth separations were demonstrated to occur commonly in spur gears, there has been no experimental data showing back contacts under loaded steady state conditions. With the friction induced damping acting in the off line of action direction, this discrete dynamic system has two degrees of freedom, one of which represents the rigid body motions.

Figure 2 Basic geometric parameters of a spur gear mesh


2.2 An EHL based gear mesh viscous damping formulation

The tooth surface velocities consist of a kinematic component due to the nominal rotation of the gears and an alternating (dynamic) component due to the vibratory motions of θ1 and θ2. For any arbitrary contacting tooth pair, the instantaneous tangential surface velocities of the pinion and gear are

1 1 1 1 2 2 2 22( ) ( ) ( ), ( ) ( ) ( )u t u R t t u t u R t tθ θ= + = + (3a,b)

where 1 1 1( )u R t ω= and 2 2 2( )u R t ω= are the kinematic surface velocities of the pinion and the gear that are rotating at the speeds of ω1 and 2 1 2 1( )Z Zω ω= (Z1 and Z2 are the numbers of teeth of the pinion and the gear) as shown in Figure 2. The movement of the surfaces of the tooth pair in contact entrains the lubricant into the contact zone, establishing a hydrodynamic fluid film that separates the contacting surfaces. The viscous shear that the lubricant film endures transforms certain amount of kinetic energy into frictional heat, dissipating the power to constrain the motion amplitudes, which is also referred as viscous damping effect.

As illustrated in Figure 3, the shear stress within the lubricant varies linearly along the film thickness direction z. Considering both Poiseuille and Couette flows, the lubricant shear stress is expressed as (Li and Kahraman, 2010b)

2 1( ) ( )1( , ) ( ) ( )

2( )x p xx z u u z h x

h x xητ

∗ ∂ = − + − ∂ (4)

where p and h denote the hydrodynamic pressure and film thickness, both of which vary along the rolling direction x (the off line of action direction) within the contact zone. Since the gear contact experiences large sliding motions as it moves away from the pitch line, the resultant high shear rates in the addendum and dedendum regions make inclusion of the non-Newtonian behaviour of the fluid necessary. Here, the Eyring fluid effective viscosity 0cosh ( )mη η τ τ∗ = is used to approximate the non-Newtonian effects, where τm is the mean viscous shear stress given as [ ]1

0 1 2 0sinh ( ) ( )m u u hτ τ η τ−= − (Li et al., 2009; Li and Kahraman, 2010a, 2010b), τ0 is the lubricant reference shear stress, and η is the lubricant viscosity that is dependent on the hydrodynamic pressure.

Figure 3 Viscous shear distribution across the film thickness

In this work, in order to avoid the overestimation of the viscosity in the high pressure range, the two-slope model of Allen in its modified form (Goglia et al., 1984) is used


with a typical automatic transmission fluid whose properties are listed in Table 1. This two-slope viscosity-pressure relation is given as

[ ]

0 12 3

0 0 1 2 3

0 1 2 .

exp( ), ;exp( ), ;exp ( ) ,

a

a b

t t b

p p pc c p c p c p p p p

p p p p p

η αη η

η α α

<= + + + ≤ ≤ + − >

(5)

This relationship divides the pressure into three ranges, low (p < pa), high (p > pb) and the transition regime in between, defining two pressure-viscosity coefficients α1 and α2 for the low and high pressure ranges, respectively. The coefficients c0, c1, c2 and c3 are determined in such a way that both η and dη/dp are continuous at the transition points. It is also noted here that any measured pressure-viscosity function can be used in place of equation (5).

Table 1 Basic parameters of the automatic transmission fluid used in this study

Temperature [°C] 25 50 75

Ambient viscosity η0 [Pa⋅s] 0.0479 0.0178 0.0086

Density ρ0 [kg/m3] 864.80 849.14 833.48

Pressure-viscosity parameters: α1 [GPa–1] 18.7 15.6 13.5

α2 [GPa–1] 2.9 2.4 2.2 Pa [GPa] 0.406 0.406 0.406 Pb [GPa] 0.812 0.812 0.812 Pt [GPa] 0.580 0.580 0.580

From equation (4), 1( ) ( ,0)x xτ τ= and 2 ( ) ( , )x x hτ τ= are obtained as the viscous shear

stresses acting on the pinion and gear tooth surfaces at z= 0 and z = h, respectively. For the computation of τ1(x) and τ2(x)

as the contact point C moves along the line of action

B1B2 as sown in Figure 2, the continuous gear EHL formulation of Li and Kahraman (2010a) is employed. Instead of treating the contact at each mesh position as an independent steady-state EHL event with constant contact parameters, this model follows the tooth contact from the Start of Active Profile (SAP) all the way to the tip of the tooth, in the process including the variations of the contact radii, surface velocities and the normal tooth force along the mesh position. The squeezing and pumping effects caused primarily by the sudden tooth load changes due to the fluctuation of the number of tooth pairs in contact are also captured in this model (Li and Kahraman, 2010a). Defining a computational domain of max max2.5 1.5a x a− ≤ ≤ in the direction of rolling where amax is the maximum half Hertzian width of the contacts through the entire line of action and applying a refined mesh of I grid elements to avoid any discretisation errors, the shear stresses at each grid node, 1 1( )i ixτ τ= and 2 2( )i ixτ τ= ( [1, ])i I∈ are then determined by using the above procedure.

Denoting the area of the uniform tooth surface grid element as A, the total traction forces exerted on the surfaces of the nth tooth pair ( [1, ])n N∈ are written as


1 1 2 21 1

, .I I

n i n ii in n

F A F Aτ τ= =

= = ∑ ∑ (6a,b)

Substituting equations (3) and (4) into equation (6), the following expressions of the traction forces are arrived

2 11 1 1 2 2

1 1

( ),

2

I I

n n ni i ii nn

u u h pF R R A Ah h x

ηθ θ η∗

∗

= =

− ∂ = − − + − ∂ ∑ ∑ (7a)

2 12 1 1 2 2

1 1

( ).

2

I I

n n ni i ii nn

u u h pF R R A Ah h x

ηθ θ η∗

∗

= =

− ∂ = − − + + ∂ ∑ ∑ (7b)

From these expressions, the viscous damping associated with the dynamic components of the velocities, 1 1R θ and 2 2 ,R θ for the nth contacting tooth pair is extracted as

1

.I

ni i n

D Ah

η∗

=

=

∑ (8)

It is seen that this viscous damping term is directly proportional to the lubricant viscosity and inversely proportional to the film thickness. Since the lubricant viscosity varies exponentially with the pressure according to equation (5) and the contact pressure changes with the mesh position as the gears roll, the viscous damper defined above becomes periodically time-varying. In addition, the surface velocities and the lubricant temperature (through its influence on the oil viscosity) dictate the film thickness, causing Dn to be impacted by the operating speed and temperature conditions as well.

Defining the mean sliding friction force and the rolling friction force as

2 1

1 1

( ), ,

2

I I

sn rni i ii nn

u u h pF A F Ah x

η∗

= =

− ∂ = = ∂ ∑ ∑ (9a,b)

the equations of motion (equation (1)) of the gear pair are rewritten with the above damping formulation as

21 1 1 1 1 2 2 1 1 1

1 1 1

( ) ( ) ( ) ( ) ( ) ( ),N N N

n n n n n n sn rnn n n

J t R D t R R D t r k t t T R F Fθ θ θ δ= = =

+ − + = + − ∑ ∑ ∑ (10a)

22 2 1 2 1 2 2 2

1 1

2 21

( ) ( ) ( ) ( ) ( )

( ).

N N

n n n n nn n

N

n sn rnn

J t R R D t R D t r k t t

T R F F

θ θ θ δ= =

=

− + −

= − − +

∑ ∑

∑ (10b)

These equations define a damping matrix in the form of

21 1 2

1 1mesh

21 2 2

1 1

.

N N

n n n n nn n

N N

n n n n nn n

R D R R D

R R D R D

= =

= =

− = −

∑ ∑

∑ ∑C (10c)


This viscous mesh damping matrix is periodically time-varying since both Dn and the radii of curvature vary periodically at the gear mesh frequency.

2.3 Definition of an equivalent viscous damper along the line of action direction

Most of the dynamic models reviewed earlier used a constant, user-defined viscous damper in the direction of line of action to represent the power losses taking place at the gear mesh. The formulation proposed in the previous section indicates that this is indeed a gross simplification of the actual meshing condition. However, the use of the proposed formulation requires one to couple a transient EHL model with a gear dynamics model, involving a certain level of complexity especially when applied to larger, multi-mesh gear systems. Therefore, a simplified formulation of an equivalent damper along the line of action direction has certain practical value. One such formulation is possible when the variations of the tooth contact radii of curvature along the mesh cycle are ignored. Considering the radii of curvature when the contact point C is at the pitch point as the representative radii of curvature, 1 1 tanR r φ= and 2 2 tanR r φ= as shown in Figure 2 where φ is the pressure angle. Substituting these radii into equation (10), one obtains

2 2 21 1 1 1 1 2 2 1

1 1

1 11

( ) tan ( ) tan ( ) ( ) ( )

tan ( ),

N N

n nn n

N

sn rnn

J t r D t r r D t r k t t

T r F F

θ φ θ φ θ δ

φ

= =

=

+ − +

= + −

∑ ∑

∑2 2 2

2 2 1 2 1 2 2 21 1

2 21

( ) tan ( ) tan ( ) ( ) ( )

tan ( )

N N

n nn n

N

sn rnn

J t r r D t r D t r k t t

T r F F

θ φ θ φ θ δ

φ

= =

=

− + −

= − − +

∑ ∑

∑ (11a,b)

which reduces the damping matrix of equation (10c) to 2

2 1 1 2mesh 2

1 1 2 2

tan .N

nn

r r rD

r r rφ

=

−= −

∑C (11c)

This form of the damping matrix is still periodic due to the 1

Nnn

D=∑ term.

Defining the relative gear mesh displacement along the line of action 1 1 2 2( ) ( ) ( )t r t r tθ θΘ = − as a new coordinate, equation (11) can be reduced to that of a

single DOF system as

2

1

( ) tan ( ) ( ) ( ) ( ) ( )N

e n en

m t D t k t t e t F tφ δ=

Θ + Θ + Θ − = ∑ (12a)

where 2 21 2 1 2 2 1( )em J J J r J r= + is the equivalent mass and ( )eF t is the excitation force

that


211 1 2 2

11 2 1

22

12

tan( ) ( )

tan( ).

Ne

e e sn rnn

Ne

sn rnn

m rrT r TF t m F F

J J J

m rF F

J

φ

φ=

=

= + + −

+ −

∑

∑ (12b)

With the dimensionless time and displacement parameters defined as natˆ ,t tω= ˆ bΘ = Θ

and ˆ ,e e b= equation (12a) can be further written into its non-dimensional form of

21

ˆ ˆtan ( ) ( )ˆ ˆ ˆˆ ˆ ˆ ˆˆ( ) ( ) 1 ( ) ( )N

nn a e

m mm e

D k t F tt t t e t

k k bk m

φδ= ′′ ′Θ + Θ + + Θ − =

∑ (13)

where km and ˆ( )ak t are the mean and alternating components of the mesh stiffness, .i.e., ˆ ˆ( ) ( ),m ak t k k t= + and ˆ ˆ ˆd d .t′Θ = Θ With this, an equivalent damping ratio corresponding

to the viscous damping at the mesh in the direction of the line of action is obtained as 2

1

tan .2

N

nnm e

Dk m

φζ=

= ∑ (14)

3 Example results

The gear mesh viscous damping term D in equation (8) (for any given tooth pair n) is a time-varying parameter rather than being constant as the contact point moves along the tooth profile from the SAP to the tip. An example spur gear pair having the parameters listed in Table 2 is considered in this work. In addition, an example automotive lubricant is used whose properties at various temperature levels are given in Table 1. Here, the ambient viscosity and density, and the first pressure-viscosity coefficient are measured values, while the second slope of the pressure-viscosity relationship and the transition pressures are estimated. Based on a smooth surface EHL analysis, the predicted D values are given in Figure 4 for T1 = 300 Nm, a lubricant temperature of 50°C and four different rotational speed values of Ω1 = 500, 1000, 1500 and 2000 rpm. It is observed in Figure 4 that D increases as the contact moves closer towards the pitch point, where the maximum damping value is reached. This is because of two primary reasons:

• the effective viscosity of the lubricant increases as the contact pressure increases near the pitch point

• the non-Newtonian behaviour of the lubricant is reduced as the relative sliding diminishes near the pitch point.

Two separate discontinuities are observed at the lowest and highest points of single tooth contact (LPSTC and HPSTC), where sudden changes of the normal tooth force occur due to the change in the number of loaded tooth pairs. It is also observed that the damping value reduces with increased speed, primarily due to increased film thicknesses. The corresponding viscous damping ratio values along the line of action under the same contact conditions are given in Figure 5. It is clear that ζ is not constant, but varies periodically at the gear mesh frequency. It reaches its maximum value when the contact is at the pitch point (for instance, for Ω1 = 500 rpm, ζ = 5% at the pitch point) and is


significantly lower in the double-tooth contact regions (as low as ζ = 0.1% for Ω1 = 500 rpm). The first three Fourier harmonic amplitudes of ζ at Ω1 = 500 rpm are 0.3, 0.23 and 0.2% and the mean value is 0.45%. The influence of the rotational speed of gears on the gear mesh damping is also evident in this figure. The ζ value at Ω1 = 2,000 rpm fluctuates between 2% and 0.03% while the first three harmonic amplitudes are reduced to 0.09, 0.07 and 0.06% with a mean value of 0.12%.

Considering the baseline operating conditions as T1 = 300 Nm, Ω1 = 2000 rpm and oil temperature of 50°C, the influence of the oil temperature (through the change in viscosity it causes) on damping is demonstrated in Figure 6. Here, the ζ values for 25, 50 and 75°C are quite different. At the lowest temperature value, the mean value of ζ is 0.17% while it is only 0.08% at 75°C. This can be directly attributed to the reduction in the fluid shear defined in equation (4) with reduced oil viscosity as listed in Table 1 for these three temperature values.

Table 2 Basic design parameters of the example spur gears considered

Number of teeth 50 Module [mm] 3.0 Pressure angle [deg] 20.0 Pitch diameter [mm] 150.0 Base diameter [mm] 140.954 Outside diameter [mm] 156.0 Root diameter [mm] 140.0 Circular tooth thickness [mm] 4.64 Face width [mm] 20.0 Center distance [mm] 150.0

Figure 4 Variation of the viscous damping D as a function of the roll angle of gear 1 at T1 = 300 Nm, 50°C and various rotational speed values


Figure 5 Variation of the viscous damping ratio ζ as a function of mesh position at T1 = 300 Nm, 50°C and various rotational speed values

Figure 6 Variation of the viscous damping ratio ζ as a function of mesh position at T1 = 300 Nm, Ω1 = 2000 rpm and various oil temperature speed values

In Figure 7, the influence of the input torque on the mesh damping ratio is illustrated through comparison of the ζ values of the gear pair operated at 2000 rpm, 50°C and three input torque values of 100, 200 and 300 Nm. These torque values correspond to the pitch line contact pressures of 0.45, 0.6 and 0.8 GPa, respectively, for this example gear pair. Here, it is found that the ζ values of the gear pair are higher for higher torque levels. The main reason for this is the strong dependence of oil viscosity on the pressure value as captured in equation (5).


Figure 7 Variation of the viscous damping ratio ζ as a function of mesh position at Ω1 = 2000 rpm, 50°C and various input torque values

Figure 8 Instantaneous P (left column solid line), h (left column dashed line) and effective viscosity (right column) distributions of the example spur gear pair at various contact positions defined on the top figure (a) Point A at SAP; (b) a point B between SAP and LPSTC; (c) point C at LPSTC; (d) point D at the pitch line and (e) point E at HPSTC T1 = 300 Nm, Ω1 = 2,000 rpm and the lubricant is 50°C


The transient gear EHL contact behaviour at several contact positions under the baseline condition are shown in Figure 8. In the top figure, the variation of the Hertzian pressure ph and the absolute value of the slide-to-roll ratio 1 2 1 22( ) ( )SR u u u u= − + are plotted against the roll angle of gear 1. In this figure, a number of representative contact instances are identified by points A to E. The corresponding hydrodynamic pressure and fluid film thickness distributions at each of these five instances are shown in the first column of Figure 8, starting from point A at the SAP, moving towards point C (LPSTC) all the way to point E (HPSTC). The corresponding effective viscosity distributions are presented in the second column of Figure 8. As the contact point moves upward from the SAP (point A), the increase in ph enlarges the contact zone and increases the effective viscosity values. This gradual increment in viscosity results in a relatively slow increase of D in Figure 4 and ζ in Figure 5–7 before the sudden increase at the LPSTC (point C). At the LPSTC, the contact pressure increases sharply due to the sudden increase in gear mesh force, elevating the effective viscosity to the order of 100 Pas. Additionally, pressure ripple is observed at both the inlet and outlet of the contact zone resulting from the substantial squeezing effects (Li and Kahraman, 2010a), causing η* ripples at the corresponding locations. Moving on to the pitch point (point D), the pressure level is only slightly increased, while the viscosity reaches to the order of 1,000 Pas. At the pitch point with pure rolling (no sliding), the non-Newtonian effects do not exist such that η* = η. At this instant, the relationship between η* and p becomes exponential, resulting in this significant increase of the η* value. This sliding effect on η* can be confirmed at HPSTC (point E), where the pressure distribution is about the same as that at the pitch point while the effective viscosity decreases to almost 10% of that at the pitch point due to the non-Newtonian effects. This explains the relatively sharp increases of D in Figure 4 and ζ in Figures 5–7 as the contact moves toward the pitch line from the LPSTC or HPSTC. Since the fluid film thickness varies limitedly as shown in Figure 8 due to the constant surface rolling velocity of this unity ratio example gear pair, the gear mesh damping variation is primarily determined by the change in lubricant viscosity according to equation (8).

4 Conclusion

In this study, a new viscous damping formulation for a spur gear pair was proposed based on a transient gear elastohydrodynamic lubrication model and a torsional dynamic model. The shear stress distributions along the tooth surfaces were computed by an elastohydrodynamic lubrication model of a spur gear pair, reduced to instantaneous surface traction forces, and incorporated into the dynamic model to formulate a periodically time-varying gear mesh viscous damping. With the assumption that the radii of curvature of the contact surfaces can be represented by those at the pitch point, a simplified expression for the damping ratio of the gear pair was derived along the line of action direction. The example results were presented at the end to state that the gear mesh damping is not constant. At a given set of operating conditions, the gear mesh damping is a periodically time-varying function with a fundamental frequency equal to the gear mesh frequency. In addition, the mean and alternating values of the gear mesh damping increase with increased torque while decrease with increased rotational speed and


lubricant temperature. These results clearly point to the shortcomings of the previous gear dynamic models in using a constant viscous gear mesh damping within the wide ranges of torque, speed and lubricant temperature.

Our current research on this topic includes an investigation of the impact of the gear surface roughness under mixed EHL conditions when asperity contacts are possible. We are also conducting companion investigations on the impact of dynamic tooth forces predicted by the gear dynamics model on the lubrication conditions of the gear contacts as well as the influences of the contact friction forces on the off line of action vibrations of the gear pairs. A sensitivity study of the impact of various gear design parameters on the gear mesh damping is also underway. The results of these studies will be reported in separate papers.

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Nomenclature

amax Maximum half Hertzian contact width b Half backlash c Constant viscous damping value Dn Gear mesh damping for the nth contacting tooth pair e Manufacturing errors and/or intentional tooth modifications F1, F2 Friction traction forces exerted on gears 1 and 2, respectively Fe Excitation force

snF Mean sliding friction force for the nth contacting tooth pair Frn Rolling friction force for the nth contacting tooth pair HPSTC Highest point of single tooth contact h Film thickness J1, J2 Polar mass moments of inertia of gears 1 and 2, respectively k Mesh stiffness km, ka Mean and alternating components of mesh stiffness LPSTC Lowest point of single tooth contact


me Equivalent mass n Index for loaded tooth pairs, [1, ]n N∈

N Number of tooth pairs in contact p Hydrodynamic pressure pa, pb Low and high pressure range thresholds ph Hertzian pressure pt Transition pressure r1, r2 Base radii of gears 1 and 2, respectively R1, R2 Contact radii of curvature of gears 1 and 2, respectively SAP Start of active profile

SR Slide-to-roll ratio, 1 2

1 2

2( )( )

u uSRu u

−=+

t Time

t Dimensionless time

T1, T2 Torque applied on gears 1 and 2, respectively u1, u2 Dynamic surface velocities in the direction of rolling of gears 1 and 2,

respectively

1 2,u u Mean surface velocities in the direction of rolling of gears 1 and 2, respectively

x Coordinate along the rolling direction z Coordinate along the film thickness direction Z1, Z2 Number of teeth of gears 1 and 2, respectively

α1, α2 Pressure viscosity coefficients at low and high pressure ranges, respectively

δ Nonlinear restoring function

φ Pressure angle

η Lubricant viscosity

η* Effective lubricant viscosity

η0 Lubricant viscosity at ambient pressure

θ1, θ2 Alternating rotational displacements of gears 1 and 2, respectively

Θ Displacement coordinate 1 1 2 2r rθ θΘ = −

Θ d dtΘ

ˆ ′Θ ˆ ˆd dtΘ

1 2,ϑ ϑ Roll angles of gears 1 and 2, respectively

ρ0 Lubricant density at ambient pressure

ζ Damping ratio

τ Lubricant shear stress

τ0 Reference shear stress of the lubricant

1 2,τ τ Shear stresses acting on tooth surfaces of gears 1 and 2, respectively


1 2,ω ω Angular velocities of gears 1 and 2, respectively

natω Undamped natural frequency

1Ω Gear 1 rotational speed

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