The influence of elastohydrodynamic lubrication on the stiffness of deep groove ball bearing

where, δ c e n is the mutual approach (the relative motion between two undeformed points within the contact bodies in the direction normal to the lubricated contact surface). Note the displacement of the load point δ d is the mutual approach for the dry contact.

The schematic diagrams of dry contact and EHL contact are shown in Fig. 1. For the dry contact, the displacement of the load point δ d is equal to the elastic deformation D d at the contact center. However, for a lubricated contact, due to the hydrodynamic lift effect of fluid films, the displacement of the load point δ c e n is the difference of the elastic deformation D c e n and film thickness h c e n at the contact center:

2.2. Contact force and mutual approach relationship

Before incorporating the effect of thin film lubrication on the analysis of bearing equilibrium model and hence the bearing stiffness, an explicit relationship between the EHL contact force and the mutual approach is developed. Here, the method proposed by Nonato is adopted in this study. Nonato [10] assumed that the relation for lubricated ball element and raceway contact follows a unified behavior for a given constant bearing speed, similar as that for the dry contact, but shifted by a certain constant amount. Then, the following formulation was given [10]:

2

KEHLδn+ΔQEHL=Q,

where δ and Q are mutual approach and contact force at the ball element-raceway contact, respectively. The stiffness coefficient KEHL, the exponent n, the contact force constant ΔQEHL can be obtained via a curve fitting of the data of contact forces and mutual approaches. This contact assumption has been validated indirectly by the results from a laboratory rotor-bearing test rig [10].

The geometry and material parameters of deep groove ball bearing and lubricant parameters are listed in Table 1. The chosen inner ring rotational speeds of bearing are 6308 r/min and 18924 r/min and the outer ring is fixed. Ignoring the gross slip at the rolling element-raceway contact, the linear speed um at the center of ball element and the rotational speed of ball element nb about its own axis can be calculated as [25]:

3

um=πDm120ni1-DbDm,

4

nb=Dmni2Db1-DbDm2,

where, ni is the rotational speed of bearing inner ring, Db is the diameter of ball element, and Dm is the average of Di and Do which are inner and outer raceway diameter, respectively. The rolling speed ur at the ball element-raceway contact in bearing and the centrifugal inertia force Fr applied on ball element can be obtained by:

5

ur=2πnb120Db,

6

Fr=2mbum2Dm,

where, mb is the mass of ball element and its value is 0.02 kg. Based on the above Eqs. (3)-(6), the rolling speeds are ur= 10 m/s and ur= 30 m/s, respectively, corresponding to inner ring speeds of 6308 r/min and 18924 r/min, and the inertia forces applied on ball element are 40 N and 354 N, respectively.

For the bearing with a high speed, the centrifugal inertia force applied on ball element must be considered for the calculation of EHL behavior. The contact force range at the ball element and inner raceway contact varies from 600 N to 2000 N, and the outer raceway contact force is the summation of the inner raceway contact force and inertia force on ball element. Following the similar analysis by Nonato [10], the multilevel technique [22] is adopted to model the EHL contact, which is introduced to numerically evaluate the Reynolds equation governing the gap film flow. In order to achieve a better numerical convergence and reduce the discretization errors, the asymmetric integrated control volume scheme [25] is used to discretize the second-order partial differential Reynolds equation. The Roelands equation [26]is used for the viscosity-pressure relation, and the lubricant compressibility is modelled with the Downson and Higginson [27]. The discrete convolution and fast Fourier Transform (DC-FFT) scheme [28] is applied here to reduce the computation time of elastic deformation at EHL contact domain. A complete description of discretization and solution process of EHL problem can be found in [22, 24], and the application of DC-FFT scheme for solving elastic deformation is discussed in detail in [27].

Table 1Parameters for deep groove ball bearing and lubricant oil

Parameters

Value

Outer raceway diameter (mm)

82

Inner raceway diameter (mm)

48

Ball diameter (mm)

17

Number of rolling elements

8

Inner/Outer raceway groove curvature radius coefficient

0.54

Radial clearance (mm)

6×10-3

Elastic modulus (Pa)

2.07×1011

Poisson’s ratio

0.3

Atmosphere viscosity (Pa∙s)

9.6×10-2

Pressure viscosity coefficient (Pa-1)

1.82×10-8

The elastic deformations and film thicknesses at the inner and outer raceway contacts are obtained by multilevel technology. The mutual approach for each contact is calculated by the Eq. (1). Based on the Eq. (2) proposed by Nonato [10], the contact force at each contact can be expressed as shown in Eq. (7) as the function of the mutual approach:

7

Ki,o δi,o ni,o +ΔQi,o =Qi,o ,

where, the index i and o represent the inner and outer raceways contact, respectively.

It is worth noting here that a combine loaded deep groove ball bearing at high speeds acts as an angular contact ball bearing due to the effects of gyroscopic motion of the ball element, which can result in a little difference for the inner and outer contact angles. In this paper, the zero gyroscopic moment is assumed for each ball element, therefore, the inner and outer contact angles remain the same. Based on the assumption, the total mutual approach between inner and outer raceways at the position of a ball element can be obtained as follow:

8

δ=δi +δo =Qi -ΔQi Ki 1/ni +Qo -ΔQo Ko 1/no .

Due to the centrifugal inertia force applied on the ball elements, the inner and outer raceways contact forces satisfy Qo =Qi +Fr. Then, Eq. (8) can be rewritten as:

9

δ =δi +δo =Qi -ΔQi Ki 1/ni +Qi -ΔQo*Ko 1/no ,

here, ΔQo*=ΔQo -Fr. Nonato assumed that the total mutual approach and inner raceways contact force follow the similar behavior to each individual contact. Therefore, the Eq. (9) is rewritten as:

10

δ=Qi -ΔQK1/n,

11

Kδn+ΔQ=Qi.

The inner raceway contact force at the ball element position is related to the total mutual approach between the raceways by the above Eq. (11). Actually, for the dry contact, the same expression as Eq. (11) can also be derived when the centrifugal inertia force acting on the ball element is considered. Therefore, Eq. (11) is used here to describe the contact relation at the ball element position between the raceways for both lubricated and dry contacts.

For the above given rolling speeds (or bearing speeds), the total mutual approaches due to different inner raceway contact forces are shown in Fig. 2 for both lubricated and dry contacts. The unknown coefficients in Eq. (11) can be obtained by fitting the inner raceway contact forces and the total mutual approaches. A least square method is employed to find the unknown coefficients as shown in Tables 2. The coefficients of determination for these fits are almost 1, which represent that the perfect matching is achieved between the fitted formula and calculated data.

As can be seen from Table 2, the contact force constant ΔQ is negative for the dry contact. Here, the negative value implies that the there is no inner raceway contact force until the mutual approach exceeds a certain value. The reason might be due to the fact that the elastic deformation at the outer raceway contact resulting from the centrifugal inertia force on the ball element leads to a gap between the ball element and inner raceway. Therefore, the inner raceway comes into contact with the ball element resulting in the generation of contact force only if the mutual approach between raceways is large enough to eliminate the gap. For a larger inertia force on the ball element due to higher rolling speed, the gap becomes larger and a bigger mutual approach is required for the generation of the inner raceway contact force.

For the EHL contact, the combined effect of the lubricated film and the centrifugal inertia force determines the value of the contact force constant. On one hand, the lubricated film forms at the inner and outer raceways contacts at near to zero contact force due to rolling speed. On the other hand, the centrifugal inertia force results in the generation of the elastic deformation at the outer raceway contact and the gap at the inner raceway contact. For a high bearing speed, the lubricated film thickness is not enough to fill the gap, which is similar to the dry case and results in the negative contact force constant. For a low bearing speed, however, the hydrodynamic lift effect is dominant compared to the effect of the centrifugal inertia force. The inner raceway moves away from the outer raceway. In order to keep the mutual approach between raceways at zero value a positive load, i.e. contact force constant, should be applied to the inner raceway contact.

Fig. 2Contact force and mutual approach relation for both dry and EHL contacts between inner and outer raceways at two different rolling speeds

Table 2Related coefficients for inner raceway contact force and total mutual approach relation between inner and outer raceways at ball element position

Rolling speed (m/s)

K (N/mn)

n

ΔQ (N)

Adjusted-R

10 (DRY)

1.078×1010

1.500

–19.214

1

30 (DRY)

1.180×1010

1.510

–155.465

1

10 (EHL)

8.562×109

1.475

30.207

1

30 (EHL)

7.281×109

1.457

–57.050

1