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Improved mixed elastohydrodynamic lubrication of hypoid gears by the optimization of manufacture parameters
Wear 438-439 (2019) 102722
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Improved mixed elastohydrodynamic lubrication of hypoid gears by the optimization of manufacture parameters☆
T
Vilmos V. Simon Budapest University of Technology and Economics, Faculty of Mechanical Engineering, Department for Machine and Product Design, H-1111 Budapest, Műegyetem rkp. 3, Hungary
ARTICLE INFO
ABSTRACT
Keywords: Mixed EHL Hypoid Gears Pressure Temperature Efficiency Manufacture
Extensive wear appears in the case of dry contacts, or when the lubrication of the contacting surfaces is not appropriate. The aim of this research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth surface processing. A full numerical analysis of the thermal mixed EHL in hypoid gears is applied. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair based on the manufacturing procedure is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted. By using the developed method, the influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results recommendations are formulated to improve the mixed EHL and the efficiency of face-milled hypoid gears.
1. Introduction In no conformal contacts, such as is in gears, forces are transmitted through a thin film of lubricant, which separates the two solid mating components. However, under usual operating conditions, the thin film of lubricant is not sufficient to completely separate the surfaces, and the asperities on opposing surfaces come in contact. This is the regime of mixed elastohydrodynamic lubrication (mixed EHL) in which the applied load is shared by the asperities and the lubricant film. Predicting performance of gears operating in the mixed EHL regime is of significant importance because the asperity contacts give rise to high local pressures, which can significantly lower the fatigue life of the gears and lead to extensive wear. At the same time, the asperity contact friction and lubricant shearing in mixed lubricated contacts generate heat, which results in extreme local temperature raise. All these factors have big influence on the wear of contacting gear tooth surfaces. The early models of mixed lubrication were developed based on a
☆
stochastic approach [1]. In the last decades many theoretical and experimental research works were directed towards the more sophisticated deterministic model of mixed elastohydrodynamic lubrication. Only some of the related papers are referenced [2–16]. The first paper on the full thermo-elastohdrodynamic lubrication analysis of gears was published in 1981 [17]. In the last decades many research works are directed towards the EHL analysis in different types of gears [18–33]. The investigations have shown that in the case of full elastohydrodynamic lubrication, only a relatively small load (torque) can be transmitted. By applying a torque usually employed in gear pairs, mixed elastohydrodynamic lubrication appears. Recently, a considerable number of papers were published on mixed EHL in different types of gears. Some of them are referenced [34–50]. In the mixed EHL important role has the geometrical separation of the contacting tooth surfaces. Nowadays, in all the hypoid gears appropriate tooth surface modifications are introduced to reduce the sensitivity of the gear pair to tooth errors and misalignments inherent in
This paper was originally accepted for the International Conference on Wear of Materials 2019. E-mail address: [email protected].
https://doi.org/10.1016/j.wear.2019.01.053 Received 27 August 2018; Received in revised form 6 January 2019; Accepted 9 January 2019 0043-1648/ © 2019 Elsevier B.V. All rights reserved.
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Nomenclature
pinion and gear teeth temperatures, K T1, T2 U1, V1, W1 components of the velocity vector of the pinion tooth surface, m s 1 U2, V2, W2 components of the velocity vector of the gear tooth surface, m s 1 u,v,w components of the lubricant's velocity vector, m s 1 z exponent of the Roeland's pressure viscosity relation profile angle of the head-cutter, deg pressure viscosity exponent, Pa 1 lubricant thermal expansivity, K 1 T pressure density coefficients, Pa 1 1, 1 tilt angle, deg temperature viscosity exponent, K 1 swivel angle, deg roughness amplitudes, μm 1, 2 η lubricant viscosity, Pa s ambient lubricant viscosity, Pa s 0 efficiency eff µs coefficient of friction µ1 , µ 2 Poisson's ratios for the pinion and gear materials ρ lubricant density, kg m 3 ambient lubricant density, kg m 3 0 shear stress, Pa rotation angles of the pinion and the gear rolling through 1, 2 mesh, deg cradle angle, deg
c cp c1 ,c2 d E1, E2 e f g h h0 ig1
sliding base setting, m specific heat of lubricant, J kg 1 K 1 constants of the Roeland's pressure viscosity relation composite normal displacement of contacting surfaces, m modulii of elasticity of the pinion and gear materials, Pa basic radial, mm basic machine center to back increment, m basic offset, m oil film thickness, m minimum oil film thickness, m velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surface kp max , kT max , kTaver , k maximum pressure, maximum oil temperature, average temperature, efficiency ratios k0 thermal conductivity of lubricant, W m 1 K 1 p oil film pressure, Pa ph hydrodyanmic pressure, Pa asperity contact pressure, Pa pc Pe Péclet number Ploss power loss, kW rprof 1, rprof 2 radii of the circular arc head-cutter profile, m radius of the head-cutter, mm rt1 s separation due to the geometry of contacting surfaces, m T oil film temperature, K Taverage average temperature, K the gear box and to avoid the very dangerous edge contact. These modifications are realized by the appropriate manufacturing procedure. During the last decades many research works have been directed towards the manufacture of spiral bevel and hypoid gears with optimal tooth surface modifications [51–85]. In this paper the influence of the manufacturing parameters on the conditions of mixed EHL characteristics is investigated. On the basis of the obtained results recommendations are formulated to reduce the pressure and temperature in the oil film and to improve the efficiency of the gear pair. The full numerical solution for the mixed thermal elastohydrodynamic lubrication in hypoid gears is employed. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted.
temperature T(x,y,z,t), and shear τ(x,y,t) distributions along the contacting tooth surfaces. The load distribution model employed [87] considers all the essential components of the gear tooth compliance (tooth bending, shear, and Hertzian tooth contact deformations) as well as any intentional tooth surface modifications and manufacturing errors to predict the load distribution. The second component is described in the following sections. The governing equations of the mixed EHL analysis are the Reynolds, energy, and elasticity equations.
2. Model of mixed thermal elastohydrodynamic lubrication in hypoid gears
where
2.1. The Reynolds equation In the analysis of mixed lubrication of gears, the contact zone is divided into two different types of area: the hydrodynamic (EHL) regions where the two surfaces are separated by the lubricant film, and the asperity contact regions where the two surfaces are in direct contact. In the hydrodynamic region the pressure at the interface between the two tooth surfaces is governed by the transient generalized Reynolds equation for point contact. This type of equation takes into account the viscosity and density variations along and across the lubricant film for the purpose of thermal analysis. The general transient Reynolds equation for point contact is x
F0 =
The modeling methodology of mixed gear lubrication consists of two major components. They are (i) a gear load distribution model to determine the load sharing among the tooth pairs instantaneously in contact and the normal tooth force distribution along the tooth surface, (ii) the mixed EHL model to predict the transient pressure p(x,y,t),
p p + F2 = y x y
F2
x
F3 (U1 F0
U2)
(W1
W2) +
( h) t
h z dz ; F1 = dz; F2 = 0 * * h z dz = F1 0 * h
0
h 0
z z *
y
F3 (V1 F0
V2)
(1)
F1 dz; F3 = F0
U1, V1, W1,U2, V2, W2 are the tooth surface velocity components. The boundary conditions of the Reynolds equation are 2
+
(2)
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p (x in , y , t ) = p (x out , y , t ) = p (x , yin , t ) = p (x , yout , t ) = 0 p (x , y , t ) 0 for x in < x < x out , yin < y < yout
2.2. Film thickness equation
(3)
The instantaneous lubricant film thickness, h, or the gap between two rough surfaces, is calculated by the geometric equation given below
The condition p (x , y, t ) 0 means that during the process of pressure relaxation, once a negative nodal pressure was encountered, it was immediately forced to be zero. In such a way the cavitation conditions at the unknown edge of the film in the outlet region were satisfied automatically. The effective viscosity, * , describes the non-Newtonian lubricant properties which can be calculated as follows:
*=
( e / 0) ; sinh( e / 0)
e
2 x
=
+
2 y
h (x , y , t ) = h 0 (t ) + s (x , y, t )
0
e
p
(T T 0) ;
=
0
(4)
(1 +
1 p 1+ 1 p
)
ln(c1 p
0)
1+
p c2
(5)
U2)
h + (V1 x
V2)
h h + = 0 when h y t
where K d =
U2)
h + (V1 x
V2)
h =0 y
(9)
y , t ) + d (x , y , t )
2
cos
ycon Ly
(10)
µ12
1
1
ymin
+
E1
ph ( , ) + pc ( , )
ymax
x min
1
) 2 + (y
(x
µ22
)2
d d
(11)
; ph is the hydrodynamic pressure and pc
E2
is the asperity contact pressure. 2.3. Temperature rise equation
(6)
The tooth temperature is higher than that of the surrounding air-oil mixture because of the frictional heating in the oil film and at the asperity contacts. In mixed lubrication, the heat is generated from shear film and asperities contacts. 2.3.1. The temperature rise in the EHL oil film - the energy equation The thermal analysis in the oil film is based on the solution of the energy equation in the hydrodynamic films and heat conduction equation in solids, simultaneously, along with other governing equations. The temperature and the viscosity are varying in the direction of the film thickness. The energy equation for the lubricant film is
(7)
T T T +u +v t x y
cP
where is a constant with very small value pre-assigned in the computer program. At the border between the hydrodynamic and contact regions h = but h/ x , h/ y , and h/ t may not be zero. Within the contact regions, where h = , however, it is reasonable to subsequently turn off the squeeze term, h/ t , in Eq. (7). This will lead to a further reduced equation as follows: (U1
x con Lx
x max
d x, y, t = Kd
The unified lubrication-contact approach in mixed lubrication uses the same Reynolds equation in both hydrodynamic and contact regions. The idea is based on the belief that the solution of Reynolds equation under the constraint of h = 0 will give the same results as that from the contact equation. EHL practice has proven that Hertzian pressure is the asymptote of hydrodynamic pressure when film thickness is sufficiently small, although moleculardynamic simulation suggests that some flow may still penetrate through the tiny gaps between the surfaces As the film thickness approaches zero in a contact region, the third power of h in Eq. (1) makes the pressure-driven terms vanish, and the Reynolds equation is reduced to the following form: (U1
2 (x ,
where A denotes the amplitude of the sinusoidal wavy, L x and L y stand for its wavelengths in the x and y directions of the contact area. The tooth surface deformation, d, is calculated by
z
1
2
(x , y , t ) = A cos
It should be mentioned that the density is also temperature dependent, but the investigations have shown that in the EHD lubrication of gears it can be neglected. is constant in In the viscosity-pressure relationship the exponent the case of Barus equation and it is pressure dependent in the Roelands’ expression: =
y, t )
where h 0 (t ) is the minimal distance of the surfaces, s (x , y , t ) is the geometrical separation of the contacting teeth surfaces, 1 (x , y, t ) and 2 (x , y , t ) denote the roughness amplitude of surface 1 and 2, respectively. Sinusoidal wavy roughness is assumed
The stresses x , y , and e are functions of pressure and temperature, varying along all directions within the film. The characteristic shear stress of the Eyring fluid, 0 , however, is assumed to be constant. For a mineral oil: 0 = 8 MPa . The viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included: =
1 (x ,
k0
2T
x2
+
2T
y2
2T
+
z2
=
T
T
p p p +u +v + * t x y u z
2
v z
+
2
(12)
In the case of non-Newtonian fluid, the velocities are given as:
(8)
z
u=
With the assumption of a smooth transition between the fluid and asperity contact areas, this unified Reynolds equation system, defined by the above equations, governs the mixed EHL behaviour of the contact, considering both the hydrodynamic and the metal-metal contact pressures simultaneously. In this study, a threshold film thickness value was used to distinguish between the use of equation for a fluid film contact and the equation for an asperity contact. With consideration that physical gap between the mating surfaces cannot accommodate less than two layers of lubricant molecules for any hydrodynamic flow to happen, an asperity contact condition is assumed at any point if h is less than the threshold value of = 2.5 nm .
0 z 0
(V2
where
z dz *
z
h
0
e
1 dz + U1 v = * z
V1)
e
e
=
0
h h dz 0 *
;
1 dz * z
0
z dz *
p + e (U2 x h e e
h
z 0
U1) 1 dz *
1 dz + V1 * e
=
h2 h z 0 * dz
The boundary conditions for the above equations are
3
p + e y h
(13)
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T (x , y , z , t ) |x = xmin = T0 if u (x , y, z , t ) |x = xmin
0
T x x = xmin
=0
if u (x , y, z , t ) |x = xmin < 0
T x x = xmax
=0
if u (x , y, z , t ) |x = xmax > 0
T (x , y , z , t ) |x = xmax = T0 if u (x , y, z , t ) |x = xmax T (x , y , z , t ) |y = ymax = T0 if v (x , y, z, t ) |y = ymin T y T y
if v (x , y, z, t ) |y = ymin < 0
=0
if v (x , y, z, t ) |y = ymax > 0
y = ymax
T (x , y , z , t ) |y = ymax = T0 if v (x , y, z, t ) |y = ymax
0
T z
= k1 z=0
T1 z1
and k 0 z1 max
T z
= k2 z=h
T2 z2
z 2 max
2ql k
is the Péclet number, where
=
k0 cp
is the
The mechanical power loss at each gear tooth contact is caused by the viscous shear within the lubricated areas of the contact and by the contact friction due to any direct asperity interactions. The power loss due to viscous friction across the fluid film at position (x,y) and time t is determined by integrating the product of the local shear stress EHL and sliding velocity derivates along the film thickness as
(14)
PlossEHL (x , y , t ) = AEHL
(15)
h 0
EHLx (x ,
y, z, t )
u + z
EHLy (x ,
v z
y, z, t )
(17)
dz
Assuming no slips between the lubricant and the tooth surfaces and considering both the Poiseuille and Couette flows, the viscous shear stress components that varies along the film thickness direction are given as
1 (1 + Pe )
V2 )2
V l 2
2.4. Power losses
2.3.2. The temperature rise at asperity contacts The flash temperature rise at each contacting asperity spot can be calculated as Tf1 =
U2) 2 + (V1
thermal diffusivity of the solid, V is the velocity of the heat source relative to the contact area, l is the half length of the contact zone, k 0 is the thermal conductivity of the solid, is the density, and cp is the specific heat.
Furthermore, the following heat flux continuity conditions on the two interfaces between the lubricant and the bounding solids should be satisfied at any instant: k0
(U1
In Eq. (16) Pe =
0 0
=0 y = ymin
VS =
EHLx (x ,
(16)
y , z , t ) = * (x , y , z , t )
u z
EHLy (x ,
y, z , t ) = * (x , y, z, t )
v z
(18)
where q = µs p VS is the heat generated per unit area of contact, further µs is the coefficient of friction, p is the contact pressure at the contact VS area, and is the relative sliding velocity,
The velocities u and v are defined by Eq. (13). Within the asperity contact regions, the shear stress due to sliding is
Fig. 1. Machine tool setting for pinion teeth finishing. 4
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V.V. Simon
defined as asp (x , y, t ) = µs p (x , y , t ), and the corresponding friction power loss is Plossasp (x , y, t ) = Aasp µs p (x , y, t ) (U2
U1 + V2
r
(T 1)
PlossEHL (x , y, t ) + i=1
Plossasp (x , y , t )
(T 1,1)
v0
=
Power Ploss Power
(T 1,1)
v0
(T 1)
(T2)
tained by a simple coordinate transformation of vector r T2 from system KT2 (rigidly connected to head-cutter T2 ) into the to the stationary coordinate system K, as it follows
(21)
r
The mixed elastohydrodynamic lubrication conditions are strongly influenced by the real shape of the gap between the contacting teeth surfaces. This gap is determined by the manufacturing procedure. 3.1. The Geometrical separation of the contacting tooth surfaces
(1) P
y (2) P )2 + (z (1) P
z (2) P ) 2
(25)
The Reynolds and energy equations are coupled through changes in the lubricant rheological state (density and viscosity), dependent on the pressure and temperature. A full numerical solution using finite difference form of the Reynolds and energy equations is required to calculate the pressure and temperature distributions in the oil film. Lubricant film temperature alters three-dimensionally, thus a three-dimension grid mesh is applied in the fluid film. The (x,y,z) reference frame is attached to the contact zone and moves with it as the gears roll in mesh. The x and y axes denotes the directions along the oil film and the z axis is in the direction of the oil film thickness. The computational domain is torque load dependent and defined by maximum Hertzian half width amax as of 2.5 a max x 2.5 amax and by the length of the contact zone l as of 0 y l . It is discretized into Nx × Ny × Nz grid elements. The x and y increments are kept constant and sufficiently small to capture the measured surface roughness variation accurately. In the first step for any instantaneous meshing position, defined by the position angle of the pinion, 1, the load distribution is calculated to determine the load sharing among the tooth pairs instantaneously in contact and the normal tooth force distribution along the contact area. The method presented in Ref. [87] is applied.
(22)
(2) P
where r and r are the position vectors in the stationary coordinate system K(x,y,z), of the corresponding points on the pinion and (1) P (2) P gear teeth surfaces. The determination of r and r is based on fulfilling the following intersection conditions: x
(T 2)
= M2 ( 2) Mg1 r T 2
4. Computational procedure
The separation of teeth surfaces in a given mesh point P(x,y) is defined as the distance of the corresponding surface points that are the intersection-points of the gear tooth normal with the pinion and gear tooth surfaces: x (2) P )2 + (y (1) P
(2)
The details of the theory of manufacture and meshing of face-milled hypoid gears are presented in Refs. [84] and [86]. Eqs. (23–25) represent a system of equations with unknowns uP , P , P , t P , and P , where uP and P are the parameters of head-cutter surface for pinion tooth surface generation, P is the cradle angle (Fig. 1), t P and P are the parameters of the head-cutter surface for the gear teeth formation. The solution is obtained by iterations.
3. The influence of the manufacture parameters on the mixed elastohydrodynamic lubrication in hypoid gears
(x (1) P
is the relative velocity vector of the head-cutter to the pinion and
e 0 is the unit normal vector of the generator surface of the headcutter. The details of the generation of pinion tooth surface are presented in Ref. [86]. The position vector of the formed gear tooth surface points is ob-
where Power is the transmitted power by the gear pair.
s (x , y ) =
(24b)
=0
where r is the radius vector of head-cutter surface points, matrices M1, Mp1, Mp2 , Mp3, and Mp4 provide the coordinate transformations from system KT1 (rigidly connected to the cradle and head-cutter T1) to the stationary coordinate system K . Angle 1 is the rotational angle of the pinion in the gear mesh. The second equation describes mathematically the generation of the pinion tooth surface by the head-cutter;
where NEHL is the number of the discretized elements of the computational domain where the fluid film is maintained between the two surfaces and Nasp is the number of the discretized elements where actual asperity contacts take place. The efficiency of the gear pair is defined as eff
(T 1)
e0
(T1) T1
(20)
j=1
(24a)
r T 1 (rt1, rprof 1, rprof 2)
Nasp
NEHL
= M1 ( 1) Mp4 (i g1) Mp3 (c, f , g ) Mp2 (e ) Mp1 ( , )
(19)
V1)
In Eqs. (17) and (19), AEHL is the area of the hydrodynamic contact between the mating flanks and Aasp is the area of the asperity contact. The total instantaneous mechanical power loss over the entire contact area with both the fluid and asperity contact regions is then found as Ploss (t ) =
(1)
x (1) P y y (1) P z z (1) P x x (2) P y y (2) P z z (2) P = = = = ex(2) P ey(2) P ez(2) P ex(2) P e y(2) P ez(2) P
(23) 3.2. The pinion and gear tooth surface A hypoid gear pair with the generated pinion and the non-generated gear is considered. The pinion is the driving member. In order to reduce the sensitivity of the gear pair to errors in teeth surfaces and to the mutual position of the mating members appropriately chosen modifications are introduced into the teeth of the pinion. As a result of these modifications theoretically point contact of the meshed teeth surfaces appears instead of linear contact. The modifications are introduced by the variation in machine tool settings and in the profile of the headcutter's cutting edges. The machine tool settings used for pinion tooth finishing are specified in Fig. 1: sliding base setting (c), basic radial (e), basic machine center to back increment (f), basic offset (g), tilt angle ( ), and swivel angle ( ). The other manufacture parameters are the velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surface (ig1), the radius of the hadd-cutter (rt1), and the radii of the head-cutter profile (rprof 1, rprof 2 ). The tooth surface of the pinion is defined by the following system of equations:
Table 1 Pinion and gear design data.
5
Design parameters
Pinion
Gear
Number of teeth Module, mm Pinion offset, mm Pressure angle, deg Mean spiral angle, deg Face width, mm Pitch diameter, mm Outside diameter, mm Pitch angle, deg
10 3.4 35 20 52 37.8 70.944 77.131 24.5381
41
27.3 31 184.622 185.843 63.3212
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Table 2 Lubricant characteristics and operating parameters. Ambient lubricant viscosity, Pas
0.19361
−1
0.14504∙10−7 0.027 50 80 2000
Pressure viscosity exponent, Pa Temperature viscosity exponent, K−1 Supplied oil temperature, C Transmitted torque, Nm Pinion's revolution per minute, rpm
Fig. 2. The influence of the applied tilt angle for pinion manufacture on the geometrical separations of contacting tooth surfaces. Δ
Fig. 5. The influence of the sliding base setting variation in pinion finishing on the mixed-EHL characteristics.
Fig. 3. The influence of the applied tilt angle for pinion manufacture on the pressure distributions on the two tooth pairs instantaneously in contact.
Δ
Fig. 6. The influence of the basic radial variation in pinion finishing on the mixed-EHL characteristics.
satisfying the prescribed convergence criterions. 3. The new values of lubricant density and viscosity are calculated and step 2 is repeated. 4. After new converged thermo-elastohydrodynamic pressures are obtained, the pressure distribution is corrected due to the instantaneous balance between the lubricant reaction and the applied contact load (determined through LTCA). 5. When all the convergence criteria are met, the rotation angle of the pinion is advanced within the meshing cycle and the iterative process is repeated. To observe the mixed-EHL conditions through the whole mesh cycle, the meshing cycle is divided into suitable number discrete steps of the pinion rotation angle.
Δ
Fig. 4. The influence of the head cutter radius variation in pinion finishing on the mixed-EHL characteristics.
In second step the following iterative procedure is used to make the EHL calculation: 1. By assuming initial values for pressure and temperature distributions the density and the viscosity of the lubricant are calculated. 2. The pressure and temperature distributions are determined by solving the dicretized form of the Reynolds and energy equations
5. Results A computer program was developed to implement the formulation provided above. By using this program the influence of the tool 6
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Δ Δβ
Fig. 7. The influence of the basic machine center to back increment variation in pinion finishing on the mixed-EHL characteristics.
Fig. 9. The influence of the tilt angle in pinion finishing on the mixed-EHL characteristics.
Δ
Δδ
Fig. 8. The influence of the basic offset variation in pinion finishing on the mixed-EHL characteristics.
Fig. 10. The influence of the swivel angle variation in pinion finishing on the mixed-EHL characteristics.
geometry and machine tool setting parameters on the maximum pressure, maximum and average temperatures and on the efficiency in a face-milled (Gleason type) hypoid gear pair was investigated. The main design data of the example gear pair used in this study are given in Table 1. The lubricant characteristics and operating parameters of the gear pair are presented in Table 2. In the investigated gear pair tooth surface modifications are introduced. Therefore, theoretically point contact of the tooth surfaces occurs, causing 1 up to 3 tooth pairs instantaneously in contact through the mesh cycle. The number of tooth pairs instantaneously in contact depends on the instantaneous rotational position of the gear pair and on the transmitted torque (load). The geometrical separation of tooth surfaces depends on the manufactured tooth surface geometry and on instantaneous rotational position of the gear pair. The influence of the tilt angle ( ) in the manufacture procedure on the geometrical separation of contacting tooth surfaces is shown in Fig. 2 for the rotational position of the gear pair when the instantaneous contact point is in the middle point of the gear tooth. Fig. 3 shows the pressure
distributions for the two tooth pairs instantaneously in contact when the pinion teeth are manufactured by different values of the tilt angle. It can be seen that the tilt angle has a considerable influence on the geometrical separations of the instantaneously contacting tooth surfaces and on the pressure distributions, too. The influence of the machine tool settings for pinion manufacture on the maximum pressure, temperature and average temperature in the contact region and on the efficiency of the gear pair is shown in Figs. 4–11. In these figures factors kp max , k , kT max , kTaver represent the ratios of the values of the maximum pressure ( pmax ), efficiency ( ), maximum oil temperature (Tmax ), and average flash temperature (Taver ) obtained by applying varying head-cutter parameters and machine tool settings, and the values of pmax 0 , 0 , Tmax 0 , and Taver0 obtained by applying the basic values of manufacturing parameters calculated by the p commonly used formulas. It means that kp max = p max , k = , T
max 0
0
kTmax = T max , and kTaver = T aver . It can be observed that the radius of the max 0 aver0 head cutter and some of the machine tool settings have a considerably T
7
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References [1] N. Patir, H.S. Cheng, An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication, J. Lub. Tech. 100 (1978) 12–17. [2] N. Ren, D. Zhu, W.W. Chen, Y. Liu, Q.J. Wang, A three-dimensional deterministic model for rough surface line-contact EHL problems, J. Trib. 131 (Art. 011501) (2009) 1–9. [3] M. Kaneta, F. Guo, J. Wang, I. Krupka, M. Hartl, Pressure increase in elliptical impact elastohydrodynamic lubrication contacts with longitudinal asperities, J. Trib. 135 (1) (2013) 1–6 (Art. No. 011503). [4] N. Imai, T. Kato, Effects of texture patterns on hydrodynamic and mixed lubrication characteristics, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 227 (2013) 898–904. [5] X.L. Yan, X.L. Wang, Y.Y. Zhang, Influence of roughness parameters skewness and kurtosis on fatigue life under mixed elastohydrodynamic lubrication point contacts, J. Trib. 136 (Art. 031503) (2014) 1–7. [6] M. Masjedi, M.M. Khonsari, On the effect of surface roughness in point-contact EHL: formulas for film thickness and asperity load, Trib. Int. 82 (2015) 228–244. [7] W. Pu, J. Wang, D. Zhu, Friction and flash temperature prediction of mixed lubrication in elliptical contacts with arbitrary velocity vector, Trib. Int. 99 (2016) 38–48. [8] F.E. Kennedy, X. Tian, Modeling sliding contact temperatures, including effects of surface roughness and convection, J. Trib. 138 (Art. 042101) (2016) 1–9. [9] Z. Liu, A. Gangopadhyay, Friction reduction in lubricated rough contacts: numerical and experimental, Stud., J. Trib. 138 (Art. 021506) (2016) 1–12. [10] L. Chang, Y.R. Jeng, Q. Yu, A unified mixed-lubrication model of rolling-sliding contacts from elastohydrodynamic lubrication to boundary lubrication, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 230 (2016) 1056–1070. [11] X. Wang, Y. Liu, D. Zhu, Numerical solution of mixed thermal elastohydrodynamic lubrication in point contacts with three-dimensional surface roughness, J. Trib. 139 (Art. 011501) (2017) 1–12. [12] C. Gu, X. Meng, Y. Xie, X. Kong, Performance of surface texturing during start-up under starved and mixed, Lubr., J. Trib. 139 (Art. 011702) (2017) 1–11. [13] M. Ciavarella, On the significance of asperity models predictions of rough contact with respect to recent alternative theories, J. Trib. 139 (Art. 021402) (2017) 1–11. [14] C.J. Hooke, G.E. Morales-Espejel, Rapid analysis of low-amplitude sinusoidal roughness in rolling-sliding elastohydrodynamic contacts including thermal effects, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 232 (2018) 1690–1706. [15] E. Fatourehchi, M. Mohammadpour, S. Theodossiades, H. Rahnejat, Transient thermal analysis of mixed-elastohydrodynamic contact of high performance transmission in a dry sump environment, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. (2018) 1–13. [16] S. Li, U. Parmar, The effects of Microdimple texture on the friction and thermal behavior of a point, Contact, J. Trib. 140 (Art. 041503) (2018) 1–12. [17] V. Simon, Elastohydrodynamic lubrication of hypoid gears, J. Mech. Des. 103 (1981) 195–203. [18] V. Simon, Influence of Misalignments on EHD Lubrication in Hypoid Gears, Proceedings 10th ASME International Power Trans. Gearing Conference Las Vegas Paper No. DETC2007/PTG-34014 671-680. [19] V. Simon, Optimal tooth modifications in face-hobbed spiral bevel gears to reduce the influence of misalignments on elastohydrodynamic lubrication, J. Mech. Des. 136 (Art. 071007) (2014) 1–9. [20] V. Simon, Improvements in Gear Lubrication, Proceedings Lub., Maint. Trib. Conference, Bilbao Spain 646-650, 2016. [21] G. Koffel, F. Ville, C. Chengenet, P. Velex, Investigations on the power losses and thermal effects in gear transmissions, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 223 (2009) 469–479. [22] R.W. Snidle, H.P. Evans, Some aspects of gear tribology, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 223 (2009) 103–113. [23] S. Li, A. Kahraman, Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 225 (2011) 740–753. [24] M. Mohammadpour, S. Theodossiades, H. Rahnejat, Elastohydrodynamic lubrication of hypoid gear pairs at high loads, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 226 (2012) 183–198. [25] Y.Q. Wang, Z.C. He, W. Su, Effect of impact load on Transient Elastohydrodynamic lubrication of Involute spur gears, Trib. Trans. 55 (2012) 155–162. [26] L. Chang, Y.R. Jeng, P.Y. Huang, Modeling and analysis of the meshing losses of involute spur gears in high-speed and high-load conditions, J. Trib. 135 (Art. 011504) (2013) 1–11. [27] M. Liu, C. Zhu, H. Liu, H. Ding, Z. Sun, Effects of working conditions in TEHL performances of a Helical gear pair With non-Newtonian fluids, J. Trib. 136 (Art. 021502) (2014) 1–9. [28] S. Li, A thermal tribo-Dynamic mechanical power loss model for spur gear pairs, Trib. Int. 88 (2015) 170–178. [29] H.U. Jamali, K.J. Sharif, H.P. Evans, R.W. Snidle, The transient effects of profile modifications on elastohydrodynamic oil films in helical gears, Trib. Trans. 58 (2015) 119–130. [30] A. Diez-Ibarbia, A. Fernandez-Del-Rincon, P. Garcia, A. De-Juan, Assessment of Load Dependent Friction Coefficients and Their Influence on Spur Gear Efficiency, Meccanica 53 (2018) 425–445. [31] P.R. Sekar, E.V. Geo, L.J. Martin, A mixed finite element and analytical method to predict load, mechanical power loss and Improved efficiency in non-standard spur gear drives, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 231 (2017) 1408–1424. [32] A. Clarke, H.U. Jamali, K.J. Shariff, H.P. Evans, B. Shaw, Effects of profile errors on
Δ
Fig. 11. The influence of the velocity ratio variation in the kinematic scheme of the machine tool for the generation of the pinion tooth surface on the mixedEHL characteristics.
influence on the mixed EHL conditions. The investigations have shown that the influence of the radii of the head cutter profile on the mixed EHL characteristics is negligible. The maximum pressure can be reduced considerably by decreasing the head cutter radius (Δrt1, Fig. 4), the basic radial (e1, Fig. 6) and by increasing the swivel angle ( 1 > 3 deg , Fig. 10). Small reductions in the maximum pressure can be achieved by the variation in the sliding base setting (c1, Fig. 5). Only small improvements in the efficiency of the gear pair can be obtained by the variation in the sliding base setting (c1, Fig. 5), basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and basic offset (g1, Fig. 8). The swivel angle ( 1, Fig. 10) has a slightly bigger effect on the efficiency of the gear pair. Almost all the machine tool settings have a considerable influence on the maximum and average temperature. The biggest reduction in the temperatures can be achieved by the variation in the basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and tilt angle ( 1 , Fig. 9). 6. Conclusions The aim of the presented research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth finishing, and by this to reduce the wear of the contacting tooth surfaces. The influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results the following recommendations can be formulated to improve the mixed EHL in face-milled hypoid gears: 1. The maximum pressure can be reduced considerably by decreasing the head cutter radius (ΔrA1, Fig. 4), basic radial (e1, Fig. 6) and by increasing the swivel angle ( 1 > 3 deg , Fig. 10). 2. The biggest reduction in the temperatures can be achieved by the variation in the basic radial (e1, Fig. 6), in the basic machine center to back increment (f1, Fig. 7) and in the tilt angle ( 1 , Fig. 9). 3. Only small improvements in the efficiency of the gear pair can be obtained by the variation in the sliding base setting (c1, Fig. 5), basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and basic offset (g1, Fig. 8). The swivel angle ( 1, Fig. 10) has a slightly bigger positive effect on the efficiency of the gear pair. 8
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[58] S. Mo, Y. Zhang, Spiral bevel gear true tooth surface Precise Modeling and experiments studies based on machining adjustment parameters, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 229 (2015) 2524–2533. [59] I.G. Perez, A. Fuentes, R.R. Orzaez, An approach for determination of Basic machine-tool settings From blank data in face-Hobbed and face-milled hypoid gears, J. Mech. Des. 137 (Art. 093303) (2015) 1–10. [60] R. Tan, B. Chen, C. Peng, General mathematical model of spiral bevel gears of purerolling contact, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 229 (2015) 2810–2826. [61] W. Zhang, B. Cheng, X. Guo, M. Zhang, Y. Xing, A motion control method for face Hobbing on CNC hypoid Generator, Mech. Mach. Theory 92 (2015) 127–143. [62] G.F. Bär, On optimizing the basic geometry of hypoid gears, Mech. Mach. Theory 104 (2016) 274–286. [63] H. Chen, X. Zhang, X. Cai, Z. Ju, C. Qu, D. Shi, Computerized design, generation and simulation of meshing and Contact of Hyperboloidal-Type Normal Circular-Arc Gears, Mech. Mach. Theory 96 (2016) 127–143. [64] Y. Yang, S. Mao, W. Guo, Y. Kuang, Pinion development of face-milled spiral bevel and hypoid gears based on contact attributes, Int. J. Adv. Man. Tech. 84 (2016) 2347–2356. [65] Y. Zhang, H. Yan, New methodology for determining basic machine settings of spiral bevel and hypoid gears manufactured by duplex helical method, Mech. Mach. Theory 100 (2016) 283–295. [66] W. Guo, S. Mao, Y. Yang, Y. Kuang, Optimization of cutter Blade profile for faceHobbed spiral bevel gears, Int. J. Adv. Man. Tech. 85 (2016) 209–216. [67] G. Li, Z. Wang, A. Kubo, K. Yuan, Z. Geng, Error-sensitivity analysis of Ease-Off based for spiral bevel gears using a semi-analytical loaded contact model, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 230 (2016) 147–161. [68] M. Wasif, Z.C. Chen, S.M. Hasan, Determination of cutter-head geometry for the face-milling of hypoid gears, Int. J. Adv. Man. Tech. 86 (2016) 3081–3090. [69] P. Wang, Y. Zhang, M. Wan, Global synthesis for face milled spiral bevel gears with zero transmission errors, J. Mech. Des. 138 (Art. 033302) (2016) 1–9. [70] N. Park, A generalized hypoid gear synthesized with common crown rack positioned between pinion and gear blanks, J. Mech. Des. 139 (Art. 085001) (2017) 1–9. [71] H. Ding, J. Tang, Y. Zhou, J. Zhong, G. Wan, A multi-objective correction of machine settings considering loaded tooth contact performance in spiral bevel gears by nonlinear interval number optimization, Mech. Mach. Theory 113 (2017) 85–108. [72] M. Kolivand, G. Steyer, C. Krieger, M.F. Stroh, A study on hypoid gears NVH robustness, SAE Int. J. Veh. Dyn. Stab. NVH 1 (2017) 417–427. [73] G. Li, Z. Wang, A. Kubo, Error-sensitivity analysis for hypoid gears using a real tooth surface contact model, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 231 (2017) 507–521. [74] Y. Zhou, C.C. Chen, J. Tang, S. Liu, An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears, Comp. -Aided Des. 84 (2017) 15–24. [75] A. Marciniec, J. Pacana, J.M. Pisula, P. Fudali, Comparative analysis of Numerical methods for the determination of contact pattern of spiral bevel gears, Aircr. Eng. Aero. Tech. 90 (2018) 359–367. [76] J.J. Yang, Z.H. Shi, H. Zhang, T.X. Li, S.W. Nie, B.Y. Wei, Dynamic analysis of spiral bevel and hypoid gears with high-order transmission errors, J. Sound Vib. 417 (2018) 149–164. [77] Y. Zhuo, X. Xiang, X. Zhou, X. Ye, Quasi-static tooth contact analysis of hypoid gear drive with coaxiality deviations, J. Braz. Soc. Mech. Sc. Eng. (40) (2018) (Art. 328). [78] Z. Zhou, J. Tang, H. Ding, Accurate modification methodology of universal machine tool settings for spiral bevel and hypoid gears, Proc. Inst. Mech. Eng. Part B: J. Eng. Man. 232 (2018) 339–349. [79] V. Simon, The influence of misalignments on mesh performances of hypoid gears, Mech. Mach. Theory 33 (1998) 1277–1291. [80] V. Simon, Optimal tooth modifications in hypoid gears, J. Mech. Des. 127 (2005) 646–655. [81] V. Simon, Machine tool settings to reduce the sensitivity of spiral bevel gears to tooth errors and misalignments, J. Mech. Des. 130 (Art. 082603) (2008) 1–9. [82] V. Simon, Generation of hypoid gears on CNC hypoid generator, J. Mech. Des. 133 (Art. 121007) (2011) 1–9. [83] V. Simon, Optimization of face-hobbed hypoid gears, Mech. Mach. Theory 77 (2014) 164–181. [84] V. Simon, Micro tooth surface topography of face-milled hypoid gears, Mech. Mach. Theory 104 (2016) 370–381. [85] V. Simon, Optimal machine tool settings for face-Hobbed hypoid gears manufactured on CNC hypoid generator, Int. J. Adv. Man. Tech. 88 (2017) 1579–1594. [86] V. Simon, Tooth Contact Analysis for Modified Hypoid Gears, Proceedings Fourth World Cong. Theory Mach. Mech, Newcastle upon Tyne, 87-92, 1975. [87] V. Simon, Load distribution in hypoid gears, J. Mech. Des. 122 (2000) 529–535.
lubrication performance of Helical gears, Trib. Int. 111 (2017) 184–191. [33] S. Mo, T. Zhang, G. Jin, S. Zhu, J. Gong, J. Bian, Elastohydrodynamic lubrication characteristics of spiral bevel gear subjected to shot Penning treatment, Math. Prob. Eng. (2018) 1–12 (Article ID 3043712). [34] M.J.A. Holmes, H.P. Evans, R.W. Snidle, Analysis of mixed lubrication effects in simulated gear tooth contacts, J. Trib. 127 (2005) 61–69. [35] S. Li, A. Kahraman, A transient mixed elastohydrodynamic lubrication model for spur gear pairs, J. Trib. 132 (Art. 011501) (2010) 1–9. [36] M. Kolivand, S. Li, A. Kahraman, Prediction of mechanical gear mesh efficiency of hypoid gear pairs, Mech. Mach. Theory 45 (2010) 1568–1582. [37] H. Liu, K. Mao, C. Zhu, X. Xu, Mixed lubricated line contact analysis for spur gears using a deterministic model, J. Trib. 134 (Art. 021501) (2012) 1–7. [38] L. Han, D.W. Zhang, F.J. Wang, Predicting film parameter and friction coefficient for Helical gears considering surface roughness and load variation, Trib. Trans. 56 (2013) 49–57. [39] H.P. Evans, R.W. Snidle, K.J. Sharif, B.A. Shaw, J. Zhang, Analysis of micro-elastohydrodynamic lubrication and prediction of surface fatigue damage in micropitting tests on helical gears, J. Trib. 135 (Art. 011501) (2013) 1–9. [40] A.B. Ghahnavieh, S. Akbarzadeh, P. Mosaddegh, A numerical study on the performance of straight bevel gears operating Under mixed lubrication regime, Mech. Mach. Theory,. 75 (2014) 27–40. [41] H.L. Dong, J.B. Hu, X.Y. Li, Temperature analysis of involute gear based on mixed elastohydrodynamic lubrication theory considering tribo-dynamic behaviors, J. Trib. 136 (Art. 021504) (2014) 1–12. [42] A.E. Serest, S. Akbarzadeh, Mixed-Elastohydrodynamic analysis of Helical gears using load-sharing concept, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 228 (2014) 320–331. [43] M. Mohammadpour, S. Theodossiades, H. Rahnejat, P. Kelly, Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs, Proc. Inst. Mech. Eng. Part K: J. Multi-body Dyn. 228 (2014) 19–33. [44] W. Pu, J. Wang, R. Yang, D. Zhu, Mixed elastohydrodynamic lubrication with threedimensional machined roughness in spiral bevel and hypoid gears, J. Trib. 137 (Art. 041503) (2015) 1–11. [45] H. Moallem, S. Akbarzadeh, A. Ariael, Prediction of micropitting life in spur gears operating under mixed-lubrication regime using load-sharing concept, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 230 (2016) 591–599. [46] L.I. Paouris, S. Theodossiades, M. De la Cruz, H. Rahnejat, A. Kidson, G. Hunt, W. Barton, Lubrication analysis and sub-surface stress field of an automotive Differential hypoid gear pair under dynamic load, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 230 (2016) 1183–1197. [47] M. Mohammadpour, S. Theodossiades, H. Rahnejat, D. Dowson, Non-newtonian mixed thermo-elastohydrodynamics of hypoid gear pairs”, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. (2017) 1–21. [48] V. Elisaus, M. Mohammadpour, S. Theodossiades, H. Rahnejat, Effect of teeth micro-geometrical form modification on contact kinematics and efficiency of high performance transmissions, Proc. Inst. Mech. Eng. Part K: J. Multi-body Dyn. 231 (2017) 538–555. [49] E. Fatourehchi, H. Shahmohamed, M. Mohammadpour, R. Rahmani, S. Theodossiades, H. Rahnejat, Thermal analysis of an oil jet-dry sump transmission gear Under mixed-elastohydrodynamic conditions, J. Trib. 140 (Art. 051502) (2018) 1–11. [50] J. Moss, A. Kahraman, C. Wink, An experimental study of influence of lubrication methods on efficiency and contact fatigue life of spur gears, J. Trib. 140 (Art. 051103) (2018) 1–11. [51] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, 2nd edition, Cambridge University Press, New York, 2004. [52] Q. Fan, R.S. DaFoe, J.W. Swanger, Higher-order tooth flank form error correction for face-milled spiral bevel and hypoid gears, J. Mech. Des. 130 (Art. 072601) (2008) 1–7. [53] A. Artoni, M. Gabiccini, M. Kolivand, Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears: only the pinion needs corrections, Mech. Mach. Theory 61 (2013) 84–101. [54] J. Astoul, E. Mermoz, M. Sartor, J.M. Linares, A. Bernard, New methodology to reduce the transmission error of the spiral bevel gears, CIRP Annuals – Manuf, Tech 63 (2014) 165–168. [55] R. Takeda, M. Komori, Tatsuya Nishino, Y. Kimura, Takayuki Nishino, K. Okuda, S. Yamamoto, Performance analysis of generated hypoid gear based on measured tooth flank form data, Mech. Mach. Theory 72 (2014) 1–16. [56] Z.C. Chen, M. Wasif, A generic and theoretical approach to programming and postprocessing for hypoid gear machining on multi-axis CNC face-milling machines, Int. J. Adv. Manuf. Tech. 81 (2015) 135–148. [57] K. Kawasaki, T. Isamu, H. Gunbara, H. Houjoh, Method for remanufacturing largesized skew bevel gears using cnc machining center, Mech. Mach. Theory 92 (2015) 213–229.
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Improved mixed elastohydrodynamic lubrication of hypoid gears by the optimization of manufacture parameters☆
T
Vilmos V. Simon Budapest University of Technology and Economics, Faculty of Mechanical Engineering, Department for Machine and Product Design, H-1111 Budapest, Műegyetem rkp. 3, Hungary
ARTICLE INFO
ABSTRACT
Keywords: Mixed EHL Hypoid Gears Pressure Temperature Efficiency Manufacture
Extensive wear appears in the case of dry contacts, or when the lubrication of the contacting surfaces is not appropriate. The aim of this research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth surface processing. A full numerical analysis of the thermal mixed EHL in hypoid gears is applied. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair based on the manufacturing procedure is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted. By using the developed method, the influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results recommendations are formulated to improve the mixed EHL and the efficiency of face-milled hypoid gears.
1. Introduction In no conformal contacts, such as is in gears, forces are transmitted through a thin film of lubricant, which separates the two solid mating components. However, under usual operating conditions, the thin film of lubricant is not sufficient to completely separate the surfaces, and the asperities on opposing surfaces come in contact. This is the regime of mixed elastohydrodynamic lubrication (mixed EHL) in which the applied load is shared by the asperities and the lubricant film. Predicting performance of gears operating in the mixed EHL regime is of significant importance because the asperity contacts give rise to high local pressures, which can significantly lower the fatigue life of the gears and lead to extensive wear. At the same time, the asperity contact friction and lubricant shearing in mixed lubricated contacts generate heat, which results in extreme local temperature raise. All these factors have big influence on the wear of contacting gear tooth surfaces. The early models of mixed lubrication were developed based on a
☆
stochastic approach [1]. In the last decades many theoretical and experimental research works were directed towards the more sophisticated deterministic model of mixed elastohydrodynamic lubrication. Only some of the related papers are referenced [2–16]. The first paper on the full thermo-elastohdrodynamic lubrication analysis of gears was published in 1981 [17]. In the last decades many research works are directed towards the EHL analysis in different types of gears [18–33]. The investigations have shown that in the case of full elastohydrodynamic lubrication, only a relatively small load (torque) can be transmitted. By applying a torque usually employed in gear pairs, mixed elastohydrodynamic lubrication appears. Recently, a considerable number of papers were published on mixed EHL in different types of gears. Some of them are referenced [34–50]. In the mixed EHL important role has the geometrical separation of the contacting tooth surfaces. Nowadays, in all the hypoid gears appropriate tooth surface modifications are introduced to reduce the sensitivity of the gear pair to tooth errors and misalignments inherent in
This paper was originally accepted for the International Conference on Wear of Materials 2019. E-mail address: [email protected].
https://doi.org/10.1016/j.wear.2019.01.053 Received 27 August 2018; Received in revised form 6 January 2019; Accepted 9 January 2019 0043-1648/ © 2019 Elsevier B.V. All rights reserved.
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V.V. Simon
Nomenclature
pinion and gear teeth temperatures, K T1, T2 U1, V1, W1 components of the velocity vector of the pinion tooth surface, m s 1 U2, V2, W2 components of the velocity vector of the gear tooth surface, m s 1 u,v,w components of the lubricant's velocity vector, m s 1 z exponent of the Roeland's pressure viscosity relation profile angle of the head-cutter, deg pressure viscosity exponent, Pa 1 lubricant thermal expansivity, K 1 T pressure density coefficients, Pa 1 1, 1 tilt angle, deg temperature viscosity exponent, K 1 swivel angle, deg roughness amplitudes, μm 1, 2 η lubricant viscosity, Pa s ambient lubricant viscosity, Pa s 0 efficiency eff µs coefficient of friction µ1 , µ 2 Poisson's ratios for the pinion and gear materials ρ lubricant density, kg m 3 ambient lubricant density, kg m 3 0 shear stress, Pa rotation angles of the pinion and the gear rolling through 1, 2 mesh, deg cradle angle, deg
c cp c1 ,c2 d E1, E2 e f g h h0 ig1
sliding base setting, m specific heat of lubricant, J kg 1 K 1 constants of the Roeland's pressure viscosity relation composite normal displacement of contacting surfaces, m modulii of elasticity of the pinion and gear materials, Pa basic radial, mm basic machine center to back increment, m basic offset, m oil film thickness, m minimum oil film thickness, m velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surface kp max , kT max , kTaver , k maximum pressure, maximum oil temperature, average temperature, efficiency ratios k0 thermal conductivity of lubricant, W m 1 K 1 p oil film pressure, Pa ph hydrodyanmic pressure, Pa asperity contact pressure, Pa pc Pe Péclet number Ploss power loss, kW rprof 1, rprof 2 radii of the circular arc head-cutter profile, m radius of the head-cutter, mm rt1 s separation due to the geometry of contacting surfaces, m T oil film temperature, K Taverage average temperature, K the gear box and to avoid the very dangerous edge contact. These modifications are realized by the appropriate manufacturing procedure. During the last decades many research works have been directed towards the manufacture of spiral bevel and hypoid gears with optimal tooth surface modifications [51–85]. In this paper the influence of the manufacturing parameters on the conditions of mixed EHL characteristics is investigated. On the basis of the obtained results recommendations are formulated to reduce the pressure and temperature in the oil film and to improve the efficiency of the gear pair. The full numerical solution for the mixed thermal elastohydrodynamic lubrication in hypoid gears is employed. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted.
temperature T(x,y,z,t), and shear τ(x,y,t) distributions along the contacting tooth surfaces. The load distribution model employed [87] considers all the essential components of the gear tooth compliance (tooth bending, shear, and Hertzian tooth contact deformations) as well as any intentional tooth surface modifications and manufacturing errors to predict the load distribution. The second component is described in the following sections. The governing equations of the mixed EHL analysis are the Reynolds, energy, and elasticity equations.
2. Model of mixed thermal elastohydrodynamic lubrication in hypoid gears
where
2.1. The Reynolds equation In the analysis of mixed lubrication of gears, the contact zone is divided into two different types of area: the hydrodynamic (EHL) regions where the two surfaces are separated by the lubricant film, and the asperity contact regions where the two surfaces are in direct contact. In the hydrodynamic region the pressure at the interface between the two tooth surfaces is governed by the transient generalized Reynolds equation for point contact. This type of equation takes into account the viscosity and density variations along and across the lubricant film for the purpose of thermal analysis. The general transient Reynolds equation for point contact is x
F0 =
The modeling methodology of mixed gear lubrication consists of two major components. They are (i) a gear load distribution model to determine the load sharing among the tooth pairs instantaneously in contact and the normal tooth force distribution along the tooth surface, (ii) the mixed EHL model to predict the transient pressure p(x,y,t),
p p + F2 = y x y
F2
x
F3 (U1 F0
U2)
(W1
W2) +
( h) t
h z dz ; F1 = dz; F2 = 0 * * h z dz = F1 0 * h
0
h 0
z z *
y
F3 (V1 F0
V2)
(1)
F1 dz; F3 = F0
U1, V1, W1,U2, V2, W2 are the tooth surface velocity components. The boundary conditions of the Reynolds equation are 2
+
(2)
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p (x in , y , t ) = p (x out , y , t ) = p (x , yin , t ) = p (x , yout , t ) = 0 p (x , y , t ) 0 for x in < x < x out , yin < y < yout
2.2. Film thickness equation
(3)
The instantaneous lubricant film thickness, h, or the gap between two rough surfaces, is calculated by the geometric equation given below
The condition p (x , y, t ) 0 means that during the process of pressure relaxation, once a negative nodal pressure was encountered, it was immediately forced to be zero. In such a way the cavitation conditions at the unknown edge of the film in the outlet region were satisfied automatically. The effective viscosity, * , describes the non-Newtonian lubricant properties which can be calculated as follows:
*=
( e / 0) ; sinh( e / 0)
e
2 x
=
+
2 y
h (x , y , t ) = h 0 (t ) + s (x , y, t )
0
e
p
(T T 0) ;
=
0
(4)
(1 +
1 p 1+ 1 p
)
ln(c1 p
0)
1+
p c2
(5)
U2)
h + (V1 x
V2)
h h + = 0 when h y t
where K d =
U2)
h + (V1 x
V2)
h =0 y
(9)
y , t ) + d (x , y , t )
2
cos
ycon Ly
(10)
µ12
1
1
ymin
+
E1
ph ( , ) + pc ( , )
ymax
x min
1
) 2 + (y
(x
µ22
)2
d d
(11)
; ph is the hydrodynamic pressure and pc
E2
is the asperity contact pressure. 2.3. Temperature rise equation
(6)
The tooth temperature is higher than that of the surrounding air-oil mixture because of the frictional heating in the oil film and at the asperity contacts. In mixed lubrication, the heat is generated from shear film and asperities contacts. 2.3.1. The temperature rise in the EHL oil film - the energy equation The thermal analysis in the oil film is based on the solution of the energy equation in the hydrodynamic films and heat conduction equation in solids, simultaneously, along with other governing equations. The temperature and the viscosity are varying in the direction of the film thickness. The energy equation for the lubricant film is
(7)
T T T +u +v t x y
cP
where is a constant with very small value pre-assigned in the computer program. At the border between the hydrodynamic and contact regions h = but h/ x , h/ y , and h/ t may not be zero. Within the contact regions, where h = , however, it is reasonable to subsequently turn off the squeeze term, h/ t , in Eq. (7). This will lead to a further reduced equation as follows: (U1
x con Lx
x max
d x, y, t = Kd
The unified lubrication-contact approach in mixed lubrication uses the same Reynolds equation in both hydrodynamic and contact regions. The idea is based on the belief that the solution of Reynolds equation under the constraint of h = 0 will give the same results as that from the contact equation. EHL practice has proven that Hertzian pressure is the asymptote of hydrodynamic pressure when film thickness is sufficiently small, although moleculardynamic simulation suggests that some flow may still penetrate through the tiny gaps between the surfaces As the film thickness approaches zero in a contact region, the third power of h in Eq. (1) makes the pressure-driven terms vanish, and the Reynolds equation is reduced to the following form: (U1
2 (x ,
where A denotes the amplitude of the sinusoidal wavy, L x and L y stand for its wavelengths in the x and y directions of the contact area. The tooth surface deformation, d, is calculated by
z
1
2
(x , y , t ) = A cos
It should be mentioned that the density is also temperature dependent, but the investigations have shown that in the EHD lubrication of gears it can be neglected. is constant in In the viscosity-pressure relationship the exponent the case of Barus equation and it is pressure dependent in the Roelands’ expression: =
y, t )
where h 0 (t ) is the minimal distance of the surfaces, s (x , y , t ) is the geometrical separation of the contacting teeth surfaces, 1 (x , y, t ) and 2 (x , y , t ) denote the roughness amplitude of surface 1 and 2, respectively. Sinusoidal wavy roughness is assumed
The stresses x , y , and e are functions of pressure and temperature, varying along all directions within the film. The characteristic shear stress of the Eyring fluid, 0 , however, is assumed to be constant. For a mineral oil: 0 = 8 MPa . The viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included: =
1 (x ,
k0
2T
x2
+
2T
y2
2T
+
z2
=
T
T
p p p +u +v + * t x y u z
2
v z
+
2
(12)
In the case of non-Newtonian fluid, the velocities are given as:
(8)
z
u=
With the assumption of a smooth transition between the fluid and asperity contact areas, this unified Reynolds equation system, defined by the above equations, governs the mixed EHL behaviour of the contact, considering both the hydrodynamic and the metal-metal contact pressures simultaneously. In this study, a threshold film thickness value was used to distinguish between the use of equation for a fluid film contact and the equation for an asperity contact. With consideration that physical gap between the mating surfaces cannot accommodate less than two layers of lubricant molecules for any hydrodynamic flow to happen, an asperity contact condition is assumed at any point if h is less than the threshold value of = 2.5 nm .
0 z 0
(V2
where
z dz *
z
h
0
e
1 dz + U1 v = * z
V1)
e
e
=
0
h h dz 0 *
;
1 dz * z
0
z dz *
p + e (U2 x h e e
h
z 0
U1) 1 dz *
1 dz + V1 * e
=
h2 h z 0 * dz
The boundary conditions for the above equations are
3
p + e y h
(13)
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T (x , y , z , t ) |x = xmin = T0 if u (x , y, z , t ) |x = xmin
0
T x x = xmin
=0
if u (x , y, z , t ) |x = xmin < 0
T x x = xmax
=0
if u (x , y, z , t ) |x = xmax > 0
T (x , y , z , t ) |x = xmax = T0 if u (x , y, z , t ) |x = xmax T (x , y , z , t ) |y = ymax = T0 if v (x , y, z, t ) |y = ymin T y T y
if v (x , y, z, t ) |y = ymin < 0
=0
if v (x , y, z, t ) |y = ymax > 0
y = ymax
T (x , y , z , t ) |y = ymax = T0 if v (x , y, z, t ) |y = ymax
0
T z
= k1 z=0
T1 z1
and k 0 z1 max
T z
= k2 z=h
T2 z2
z 2 max
2ql k
is the Péclet number, where
=
k0 cp
is the
The mechanical power loss at each gear tooth contact is caused by the viscous shear within the lubricated areas of the contact and by the contact friction due to any direct asperity interactions. The power loss due to viscous friction across the fluid film at position (x,y) and time t is determined by integrating the product of the local shear stress EHL and sliding velocity derivates along the film thickness as
(14)
PlossEHL (x , y , t ) = AEHL
(15)
h 0
EHLx (x ,
y, z, t )
u + z
EHLy (x ,
v z
y, z, t )
(17)
dz
Assuming no slips between the lubricant and the tooth surfaces and considering both the Poiseuille and Couette flows, the viscous shear stress components that varies along the film thickness direction are given as
1 (1 + Pe )
V2 )2
V l 2
2.4. Power losses
2.3.2. The temperature rise at asperity contacts The flash temperature rise at each contacting asperity spot can be calculated as Tf1 =
U2) 2 + (V1
thermal diffusivity of the solid, V is the velocity of the heat source relative to the contact area, l is the half length of the contact zone, k 0 is the thermal conductivity of the solid, is the density, and cp is the specific heat.
Furthermore, the following heat flux continuity conditions on the two interfaces between the lubricant and the bounding solids should be satisfied at any instant: k0
(U1
In Eq. (16) Pe =
0 0
=0 y = ymin
VS =
EHLx (x ,
(16)
y , z , t ) = * (x , y , z , t )
u z
EHLy (x ,
y, z , t ) = * (x , y, z, t )
v z
(18)
where q = µs p VS is the heat generated per unit area of contact, further µs is the coefficient of friction, p is the contact pressure at the contact VS area, and is the relative sliding velocity,
The velocities u and v are defined by Eq. (13). Within the asperity contact regions, the shear stress due to sliding is
Fig. 1. Machine tool setting for pinion teeth finishing. 4
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defined as asp (x , y, t ) = µs p (x , y , t ), and the corresponding friction power loss is Plossasp (x , y, t ) = Aasp µs p (x , y, t ) (U2
U1 + V2
r
(T 1)
PlossEHL (x , y, t ) + i=1
Plossasp (x , y , t )
(T 1,1)
v0
=
Power Ploss Power
(T 1,1)
v0
(T 1)
(T2)
tained by a simple coordinate transformation of vector r T2 from system KT2 (rigidly connected to head-cutter T2 ) into the to the stationary coordinate system K, as it follows
(21)
r
The mixed elastohydrodynamic lubrication conditions are strongly influenced by the real shape of the gap between the contacting teeth surfaces. This gap is determined by the manufacturing procedure. 3.1. The Geometrical separation of the contacting tooth surfaces
(1) P
y (2) P )2 + (z (1) P
z (2) P ) 2
(25)
The Reynolds and energy equations are coupled through changes in the lubricant rheological state (density and viscosity), dependent on the pressure and temperature. A full numerical solution using finite difference form of the Reynolds and energy equations is required to calculate the pressure and temperature distributions in the oil film. Lubricant film temperature alters three-dimensionally, thus a three-dimension grid mesh is applied in the fluid film. The (x,y,z) reference frame is attached to the contact zone and moves with it as the gears roll in mesh. The x and y axes denotes the directions along the oil film and the z axis is in the direction of the oil film thickness. The computational domain is torque load dependent and defined by maximum Hertzian half width amax as of 2.5 a max x 2.5 amax and by the length of the contact zone l as of 0 y l . It is discretized into Nx × Ny × Nz grid elements. The x and y increments are kept constant and sufficiently small to capture the measured surface roughness variation accurately. In the first step for any instantaneous meshing position, defined by the position angle of the pinion, 1, the load distribution is calculated to determine the load sharing among the tooth pairs instantaneously in contact and the normal tooth force distribution along the contact area. The method presented in Ref. [87] is applied.
(22)
(2) P
where r and r are the position vectors in the stationary coordinate system K(x,y,z), of the corresponding points on the pinion and (1) P (2) P gear teeth surfaces. The determination of r and r is based on fulfilling the following intersection conditions: x
(T 2)
= M2 ( 2) Mg1 r T 2
4. Computational procedure
The separation of teeth surfaces in a given mesh point P(x,y) is defined as the distance of the corresponding surface points that are the intersection-points of the gear tooth normal with the pinion and gear tooth surfaces: x (2) P )2 + (y (1) P
(2)
The details of the theory of manufacture and meshing of face-milled hypoid gears are presented in Refs. [84] and [86]. Eqs. (23–25) represent a system of equations with unknowns uP , P , P , t P , and P , where uP and P are the parameters of head-cutter surface for pinion tooth surface generation, P is the cradle angle (Fig. 1), t P and P are the parameters of the head-cutter surface for the gear teeth formation. The solution is obtained by iterations.
3. The influence of the manufacture parameters on the mixed elastohydrodynamic lubrication in hypoid gears
(x (1) P
is the relative velocity vector of the head-cutter to the pinion and
e 0 is the unit normal vector of the generator surface of the headcutter. The details of the generation of pinion tooth surface are presented in Ref. [86]. The position vector of the formed gear tooth surface points is ob-
where Power is the transmitted power by the gear pair.
s (x , y ) =
(24b)
=0
where r is the radius vector of head-cutter surface points, matrices M1, Mp1, Mp2 , Mp3, and Mp4 provide the coordinate transformations from system KT1 (rigidly connected to the cradle and head-cutter T1) to the stationary coordinate system K . Angle 1 is the rotational angle of the pinion in the gear mesh. The second equation describes mathematically the generation of the pinion tooth surface by the head-cutter;
where NEHL is the number of the discretized elements of the computational domain where the fluid film is maintained between the two surfaces and Nasp is the number of the discretized elements where actual asperity contacts take place. The efficiency of the gear pair is defined as eff
(T 1)
e0
(T1) T1
(20)
j=1
(24a)
r T 1 (rt1, rprof 1, rprof 2)
Nasp
NEHL
= M1 ( 1) Mp4 (i g1) Mp3 (c, f , g ) Mp2 (e ) Mp1 ( , )
(19)
V1)
In Eqs. (17) and (19), AEHL is the area of the hydrodynamic contact between the mating flanks and Aasp is the area of the asperity contact. The total instantaneous mechanical power loss over the entire contact area with both the fluid and asperity contact regions is then found as Ploss (t ) =
(1)
x (1) P y y (1) P z z (1) P x x (2) P y y (2) P z z (2) P = = = = ex(2) P ey(2) P ez(2) P ex(2) P e y(2) P ez(2) P
(23) 3.2. The pinion and gear tooth surface A hypoid gear pair with the generated pinion and the non-generated gear is considered. The pinion is the driving member. In order to reduce the sensitivity of the gear pair to errors in teeth surfaces and to the mutual position of the mating members appropriately chosen modifications are introduced into the teeth of the pinion. As a result of these modifications theoretically point contact of the meshed teeth surfaces appears instead of linear contact. The modifications are introduced by the variation in machine tool settings and in the profile of the headcutter's cutting edges. The machine tool settings used for pinion tooth finishing are specified in Fig. 1: sliding base setting (c), basic radial (e), basic machine center to back increment (f), basic offset (g), tilt angle ( ), and swivel angle ( ). The other manufacture parameters are the velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surface (ig1), the radius of the hadd-cutter (rt1), and the radii of the head-cutter profile (rprof 1, rprof 2 ). The tooth surface of the pinion is defined by the following system of equations:
Table 1 Pinion and gear design data.
5
Design parameters
Pinion
Gear
Number of teeth Module, mm Pinion offset, mm Pressure angle, deg Mean spiral angle, deg Face width, mm Pitch diameter, mm Outside diameter, mm Pitch angle, deg
10 3.4 35 20 52 37.8 70.944 77.131 24.5381
41
27.3 31 184.622 185.843 63.3212
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Table 2 Lubricant characteristics and operating parameters. Ambient lubricant viscosity, Pas
0.19361
−1
0.14504∙10−7 0.027 50 80 2000
Pressure viscosity exponent, Pa Temperature viscosity exponent, K−1 Supplied oil temperature, C Transmitted torque, Nm Pinion's revolution per minute, rpm
Fig. 2. The influence of the applied tilt angle for pinion manufacture on the geometrical separations of contacting tooth surfaces. Δ
Fig. 5. The influence of the sliding base setting variation in pinion finishing on the mixed-EHL characteristics.
Fig. 3. The influence of the applied tilt angle for pinion manufacture on the pressure distributions on the two tooth pairs instantaneously in contact.
Δ
Fig. 6. The influence of the basic radial variation in pinion finishing on the mixed-EHL characteristics.
satisfying the prescribed convergence criterions. 3. The new values of lubricant density and viscosity are calculated and step 2 is repeated. 4. After new converged thermo-elastohydrodynamic pressures are obtained, the pressure distribution is corrected due to the instantaneous balance between the lubricant reaction and the applied contact load (determined through LTCA). 5. When all the convergence criteria are met, the rotation angle of the pinion is advanced within the meshing cycle and the iterative process is repeated. To observe the mixed-EHL conditions through the whole mesh cycle, the meshing cycle is divided into suitable number discrete steps of the pinion rotation angle.
Δ
Fig. 4. The influence of the head cutter radius variation in pinion finishing on the mixed-EHL characteristics.
In second step the following iterative procedure is used to make the EHL calculation: 1. By assuming initial values for pressure and temperature distributions the density and the viscosity of the lubricant are calculated. 2. The pressure and temperature distributions are determined by solving the dicretized form of the Reynolds and energy equations
5. Results A computer program was developed to implement the formulation provided above. By using this program the influence of the tool 6
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Δ Δβ
Fig. 7. The influence of the basic machine center to back increment variation in pinion finishing on the mixed-EHL characteristics.
Fig. 9. The influence of the tilt angle in pinion finishing on the mixed-EHL characteristics.
Δ
Δδ
Fig. 8. The influence of the basic offset variation in pinion finishing on the mixed-EHL characteristics.
Fig. 10. The influence of the swivel angle variation in pinion finishing on the mixed-EHL characteristics.
geometry and machine tool setting parameters on the maximum pressure, maximum and average temperatures and on the efficiency in a face-milled (Gleason type) hypoid gear pair was investigated. The main design data of the example gear pair used in this study are given in Table 1. The lubricant characteristics and operating parameters of the gear pair are presented in Table 2. In the investigated gear pair tooth surface modifications are introduced. Therefore, theoretically point contact of the tooth surfaces occurs, causing 1 up to 3 tooth pairs instantaneously in contact through the mesh cycle. The number of tooth pairs instantaneously in contact depends on the instantaneous rotational position of the gear pair and on the transmitted torque (load). The geometrical separation of tooth surfaces depends on the manufactured tooth surface geometry and on instantaneous rotational position of the gear pair. The influence of the tilt angle ( ) in the manufacture procedure on the geometrical separation of contacting tooth surfaces is shown in Fig. 2 for the rotational position of the gear pair when the instantaneous contact point is in the middle point of the gear tooth. Fig. 3 shows the pressure
distributions for the two tooth pairs instantaneously in contact when the pinion teeth are manufactured by different values of the tilt angle. It can be seen that the tilt angle has a considerable influence on the geometrical separations of the instantaneously contacting tooth surfaces and on the pressure distributions, too. The influence of the machine tool settings for pinion manufacture on the maximum pressure, temperature and average temperature in the contact region and on the efficiency of the gear pair is shown in Figs. 4–11. In these figures factors kp max , k , kT max , kTaver represent the ratios of the values of the maximum pressure ( pmax ), efficiency ( ), maximum oil temperature (Tmax ), and average flash temperature (Taver ) obtained by applying varying head-cutter parameters and machine tool settings, and the values of pmax 0 , 0 , Tmax 0 , and Taver0 obtained by applying the basic values of manufacturing parameters calculated by the p commonly used formulas. It means that kp max = p max , k = , T
max 0
0
kTmax = T max , and kTaver = T aver . It can be observed that the radius of the max 0 aver0 head cutter and some of the machine tool settings have a considerably T
7
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References [1] N. Patir, H.S. Cheng, An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication, J. Lub. Tech. 100 (1978) 12–17. [2] N. Ren, D. Zhu, W.W. Chen, Y. Liu, Q.J. Wang, A three-dimensional deterministic model for rough surface line-contact EHL problems, J. Trib. 131 (Art. 011501) (2009) 1–9. [3] M. Kaneta, F. Guo, J. Wang, I. Krupka, M. Hartl, Pressure increase in elliptical impact elastohydrodynamic lubrication contacts with longitudinal asperities, J. Trib. 135 (1) (2013) 1–6 (Art. No. 011503). [4] N. Imai, T. Kato, Effects of texture patterns on hydrodynamic and mixed lubrication characteristics, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 227 (2013) 898–904. [5] X.L. Yan, X.L. Wang, Y.Y. Zhang, Influence of roughness parameters skewness and kurtosis on fatigue life under mixed elastohydrodynamic lubrication point contacts, J. Trib. 136 (Art. 031503) (2014) 1–7. [6] M. Masjedi, M.M. Khonsari, On the effect of surface roughness in point-contact EHL: formulas for film thickness and asperity load, Trib. Int. 82 (2015) 228–244. [7] W. Pu, J. Wang, D. Zhu, Friction and flash temperature prediction of mixed lubrication in elliptical contacts with arbitrary velocity vector, Trib. Int. 99 (2016) 38–48. [8] F.E. Kennedy, X. Tian, Modeling sliding contact temperatures, including effects of surface roughness and convection, J. Trib. 138 (Art. 042101) (2016) 1–9. [9] Z. Liu, A. Gangopadhyay, Friction reduction in lubricated rough contacts: numerical and experimental, Stud., J. Trib. 138 (Art. 021506) (2016) 1–12. [10] L. Chang, Y.R. Jeng, Q. Yu, A unified mixed-lubrication model of rolling-sliding contacts from elastohydrodynamic lubrication to boundary lubrication, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 230 (2016) 1056–1070. [11] X. Wang, Y. Liu, D. Zhu, Numerical solution of mixed thermal elastohydrodynamic lubrication in point contacts with three-dimensional surface roughness, J. Trib. 139 (Art. 011501) (2017) 1–12. [12] C. Gu, X. Meng, Y. Xie, X. Kong, Performance of surface texturing during start-up under starved and mixed, Lubr., J. Trib. 139 (Art. 011702) (2017) 1–11. [13] M. Ciavarella, On the significance of asperity models predictions of rough contact with respect to recent alternative theories, J. Trib. 139 (Art. 021402) (2017) 1–11. [14] C.J. Hooke, G.E. Morales-Espejel, Rapid analysis of low-amplitude sinusoidal roughness in rolling-sliding elastohydrodynamic contacts including thermal effects, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 232 (2018) 1690–1706. [15] E. Fatourehchi, M. Mohammadpour, S. Theodossiades, H. Rahnejat, Transient thermal analysis of mixed-elastohydrodynamic contact of high performance transmission in a dry sump environment, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. (2018) 1–13. [16] S. Li, U. Parmar, The effects of Microdimple texture on the friction and thermal behavior of a point, Contact, J. Trib. 140 (Art. 041503) (2018) 1–12. [17] V. Simon, Elastohydrodynamic lubrication of hypoid gears, J. Mech. Des. 103 (1981) 195–203. [18] V. Simon, Influence of Misalignments on EHD Lubrication in Hypoid Gears, Proceedings 10th ASME International Power Trans. Gearing Conference Las Vegas Paper No. DETC2007/PTG-34014 671-680. [19] V. Simon, Optimal tooth modifications in face-hobbed spiral bevel gears to reduce the influence of misalignments on elastohydrodynamic lubrication, J. Mech. Des. 136 (Art. 071007) (2014) 1–9. [20] V. Simon, Improvements in Gear Lubrication, Proceedings Lub., Maint. Trib. Conference, Bilbao Spain 646-650, 2016. [21] G. Koffel, F. Ville, C. Chengenet, P. Velex, Investigations on the power losses and thermal effects in gear transmissions, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 223 (2009) 469–479. [22] R.W. Snidle, H.P. Evans, Some aspects of gear tribology, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sc. 223 (2009) 103–113. [23] S. Li, A. Kahraman, Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 225 (2011) 740–753. [24] M. Mohammadpour, S. Theodossiades, H. Rahnejat, Elastohydrodynamic lubrication of hypoid gear pairs at high loads, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 226 (2012) 183–198. [25] Y.Q. Wang, Z.C. He, W. Su, Effect of impact load on Transient Elastohydrodynamic lubrication of Involute spur gears, Trib. Trans. 55 (2012) 155–162. [26] L. Chang, Y.R. Jeng, P.Y. Huang, Modeling and analysis of the meshing losses of involute spur gears in high-speed and high-load conditions, J. Trib. 135 (Art. 011504) (2013) 1–11. [27] M. Liu, C. Zhu, H. Liu, H. Ding, Z. Sun, Effects of working conditions in TEHL performances of a Helical gear pair With non-Newtonian fluids, J. Trib. 136 (Art. 021502) (2014) 1–9. [28] S. Li, A thermal tribo-Dynamic mechanical power loss model for spur gear pairs, Trib. Int. 88 (2015) 170–178. [29] H.U. Jamali, K.J. Sharif, H.P. Evans, R.W. Snidle, The transient effects of profile modifications on elastohydrodynamic oil films in helical gears, Trib. Trans. 58 (2015) 119–130. [30] A. Diez-Ibarbia, A. Fernandez-Del-Rincon, P. Garcia, A. De-Juan, Assessment of Load Dependent Friction Coefficients and Their Influence on Spur Gear Efficiency, Meccanica 53 (2018) 425–445. [31] P.R. Sekar, E.V. Geo, L.J. Martin, A mixed finite element and analytical method to predict load, mechanical power loss and Improved efficiency in non-standard spur gear drives, Proc. Inst. Mech. Eng. Part J: J. Eng. Trib. 231 (2017) 1408–1424. [32] A. Clarke, H.U. Jamali, K.J. Shariff, H.P. Evans, B. Shaw, Effects of profile errors on
Δ
Fig. 11. The influence of the velocity ratio variation in the kinematic scheme of the machine tool for the generation of the pinion tooth surface on the mixedEHL characteristics.
influence on the mixed EHL conditions. The investigations have shown that the influence of the radii of the head cutter profile on the mixed EHL characteristics is negligible. The maximum pressure can be reduced considerably by decreasing the head cutter radius (Δrt1, Fig. 4), the basic radial (e1, Fig. 6) and by increasing the swivel angle ( 1 > 3 deg , Fig. 10). Small reductions in the maximum pressure can be achieved by the variation in the sliding base setting (c1, Fig. 5). Only small improvements in the efficiency of the gear pair can be obtained by the variation in the sliding base setting (c1, Fig. 5), basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and basic offset (g1, Fig. 8). The swivel angle ( 1, Fig. 10) has a slightly bigger effect on the efficiency of the gear pair. Almost all the machine tool settings have a considerable influence on the maximum and average temperature. The biggest reduction in the temperatures can be achieved by the variation in the basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and tilt angle ( 1 , Fig. 9). 6. Conclusions The aim of the presented research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth finishing, and by this to reduce the wear of the contacting tooth surfaces. The influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results the following recommendations can be formulated to improve the mixed EHL in face-milled hypoid gears: 1. The maximum pressure can be reduced considerably by decreasing the head cutter radius (ΔrA1, Fig. 4), basic radial (e1, Fig. 6) and by increasing the swivel angle ( 1 > 3 deg , Fig. 10). 2. The biggest reduction in the temperatures can be achieved by the variation in the basic radial (e1, Fig. 6), in the basic machine center to back increment (f1, Fig. 7) and in the tilt angle ( 1 , Fig. 9). 3. Only small improvements in the efficiency of the gear pair can be obtained by the variation in the sliding base setting (c1, Fig. 5), basic radial (e1, Fig. 6), basic machine center to back increment (f1, Fig. 7) and basic offset (g1, Fig. 8). The swivel angle ( 1, Fig. 10) has a slightly bigger positive effect on the efficiency of the gear pair. 8
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