Fig. 1 shows the global coordinate system, loading, and displacements of a D-ACBB. The inner ring load and displacement vectors, {F} and {d}, are respectively represented by

Fig. 1 
Global coordinate system, loading, and displacements

Fig. 2 shows the loads and displacements of the inner ring and ball defined at the inner ring cross-section of a typical ball. Because a bearing consists of two rows, without loss of generality, two local coordinate systems are introduced for a specific ball in accordance to (r_m, f_m, z), where m = L, and R represent the left and right rows, respectively. The displacement vectors of the inner ring on the left and right cross-sections are defined at the reference points, which coincide with the inner raceway groove centers (L and R), as

Fig. 2 
Local coordinate system, loading, and displacements

The inner ring contact load vectors of left and right rows are written by

The corresponding displacement vectors of balls at the left and right rows are

The relationship between the global and local displacements of the inner ring is described by

where [R_ϕm] represents the transformation matrix for the left and right rows, which can be expressed as follows:

where ϕ_m is the location angle of the ball. x_L and x_R are the distances from the bearing’s center to the reference points of the left and right rows, respectively.

Fig. 3 shows the free-body diagram of a ball, in which the contact forces are represented by Q_a. Herein, a = i, e represents the inner and outer rings. Q_a can be determined by using the well-known Hertzian contact formula.¹⁴ F_c and M_g are the centrifugal force and gyroscopic moment acting on the ball, respectively, which can be determined as described in.¹⁴ The equilibrium equations of the ball are then expressed as

Fig. 3 
Free-Body diagram of one of the balls on the left row

where l_i and l_e indicate the distribution parameters for the gyroscopic moment, which, in this study, are assumed to be identical on both raceways, l_i = l_e = 1. D_a and a indicate the ball diameter and contact angle, respectively. The determination of α was described by de Mul et al..¹⁵ The above equations are subsequently solved to estimate the ball displacement vector, {v_m}, and the ball-ring contact forces. Because these equations are nonlinear, this study uses the iterative Newton–Raphson method as the computational algorithm. The inner ring contact load is found as

Having obtained the inner ring contact loads for all the balls, the global equilibrium equations of the inner ring can be found by taking the summation of the external loads and all inner ring contact loads, as follows:

where j indicates the ball index. Z represents the number of balls for each row. The Newton–Raphson method is employed to solve the global equations of the inner ring required to estimate the global displacements of the bearing, {d}. The bearing stiffness matrix can be derived analytically via the Jacobian matrix of the global equilibrium equations for double-row cylindrical roller bearings, as described in.¹⁴

The fatigue life of a rolling bearing is conventionally defined as the number of revolutions of a bearing until the first appearance of flaking on one of its rings or rolling elements. The bearing’s fatigue life has been predicted statistically based on the theory proposed by Palmgren.¹⁶ It is well known that even identical bearings operating under identical conditions may have different fatigue lives.¹⁷ The fatigue life of D-ACBBs presented in this study is the basic reference rating life (L_10r) that represents an estimated value of fatigue life with a probability of 90%.

To consider more general loading conditions—including the effects of angular misalignment, combined loads, and axial clearance—this study determined the bearing fatigue life based on the ISO16-281 standard.¹⁸ In this case, the contact forces of all the rolling elements are required, which are found by using the presented five-DOF quasi-static model. First, the basic dynamic radial load rating of bearing Q_c should be determined, which represents the influence of the internal dimensions of the bearing. Details of calculating Q_c are presented in.¹⁸ Next, the dynamic equivalent rolling element load Q_e is determined. Q_e reflects the effect of loading conditions on the bearing fatigue life in accordance to

Finally, the basic reference rating life of D-ACBB can be found as follows:

The calculated stiffness, displacement, and fatigue life of D-ACBB using the proposed model were compared with those elicited using commercially available codes.¹³ The bearing clearance e and rotational speed n were selected as 0 μm and 1000 rpm, respectively.

Fig. 4(a) shows the displacement of the bearing as a function of the radial load F_y. Subject to the radial loads imposed on the bearing, only radial displacements are induced, which gradually increased as the radial load increased. Fig. 4(a) demonstrates that the calculated radial displacement agrees well with that estimated based on the commercial code. Figs. 4(b)-4(d) depicts the stiffness values of the D-ACBB as a function of radial load. The diagonal stiffness elements are displayed for comparison purposes. Figs. 4(b)-4(d) clearly indicates that the stiffness values of the presented model and commercial code are in a very good agreement for all radial loads.

Fig. 4 
Stiffness and radial displacement of the D-ACBB as a function of radial load F_y (rotational speed n = 1000 rpm)

Fig. 5 illustrates the basic reference rating life of the D-ACBB as a function of the radial load. It is shown that the fatigue life of bearings is gradually reduced at increasing radial loads. As expected, the calculated bearing fatigue lives are well matched with those obtained using the commercial code, thus successfully validating the proposed model.

Fig. 5 
Fatigue life of the D-ACBB as a function of radial load

D-ACBBs are often used in the presence of small internal clearance under various operating conditions. Such clearance allows the inner and outer rings to displace axially with respect to each other. Positive axial clearance is mainly assigned for avoiding bearing seizure caused by thermal expansion. However, D-ACBBs may be installed with a negative axial clearance. This implies that the D-ACBBs are initially preloaded before operation. Such a preload leads to the development of constant elastic compressive forces at the contact points between the rolling element and raceway surfaces. This leads to an increase in the bearing rigidity to prevent or suppress shaft run-out, vibration, and noise, as well as to improve the running and locating accuracies. This also reduces smearing, and regulates rolling element rotation.⁴

This section discusses the effect of axial clearance on the bearing stiffness and fatigue life. The axial clearance was chosen to range from -25 to 25 μm investigation purposes, and the rotational speed was kept constant at 1000 rpm.

Figs. 6 and 7 show the effect of axial clearance on the stiffness and fatigue life ratios, which are defined as the ratios between the stiffness and fatigue life at an arbitrary axial clearance, and the corresponding values at zero axial clearance. The stiffness values of D-ACBB at negative axial clearance are obviously higher than those of the D-ACBBs with positive clearance because of the increased initial preload. However, as shown in Fig. 7, increasing the magnitude of the negative axial clearance reduces the bearing fatigue life, especially when an increased radial load is applied. This is mainly caused by the increase of the ball and race contact loads, as shown in Fig. 8. It should be noted that the positive axial clearance has a minor effect on the radial stiffness values (k_yy and k_zz) of the D-ACBB (Fig. 6).

Fig. 6 
Stiffness ratio as a function of axial clearance (F_y = 1000 N, n = 1000 rpm)

Fig. 7 
Fatigue ratio as a function of axial clearance (n = 1000 rpm)

Fig. 8 
Contact force between the ball and inner race (n = 1000 rpm)

This section presents the effects of rotational speed and radial load on the stiffness and fatigue life of D-ACBB. The axial clearance was set to 0 μm. The radial and moment stiffness values show opposing behavior in the low rotational speed range, as shown in Fig. 9. However, under increased rotational speeds, both stiffness values increase. Additionally, the increase of rotational speed would lessen the effect of radial load on the bearing stiffness values. This could be because the effect of inertial loading under increased speeds prevails over the radial load acting on the bearing.

Fig. 9 
Stiffness values of the D-ACBB as a function of rotational speeds (ε = 0 μm)

Fig. 10 shows the fatigue life of the D-ACBB as a function of rotational speed for various radial loads. It is observed that the bearing fatigue life decreases as the rotational speed increases. This means that increasing the rotational speed leads to the increase of the contact load between the ball and bearing races, as demonstrated in Fig. 11. The effect of the rotational speed on the bearing’s fatigue life is significant at low bearing loads.

Fig. 10 
Fatigue life of the D-ACBB as a function of rotational speed (e = 0 μm)

Fig. 11 
Maximum contact force between ball and races

This section presents the effect of angular misalignment on the bearing stiffness and fatigue life of D-ACBBs at different axial clearances. The bearing load and rotational speed are assumed constant at F_y = 1000 N and n = 1000 rpm, respectively. Fig. 12 depicts the bearing radial stiffness and fatigue life ratios for various axial clearances. Only the radial stiffness k_yy is shown because the other stiffness elements will have the same behavior. Fig. 13 evaluates the ball and inner race contact load at two values of axial clearance, namely, ε = -7.5 μm and ε = 7.5 μm. Fig. 12 shows that the stiffness of D-ACBB increases with increase in angular misalignment. This behavior may be explained by the increase in the number of balls that sustain the applied load, as shown in Fig. 13(a).

Fig. 12 
Stiffness and fatigue life ratios a function of misalignment angle

Fig. 13 
Contact force between ball and inner ring of aligned and misaligned D-ACBBs

Contrary to the observed variation for the bearing stiffness, the fatigue of the bearing decreases with misalignment increases. This is due to the increase of the contact load in the rolling elements (Fig. 13(a)). However, the variations in both stiffness and fatigue lives are negligible for the D-ACBB that has significantly large axial clearance, as shown in Fig. 12. This is because the internal contact load distribution of the ball and race are unaffected by angular misalignments in the case of sufficiently high axial clearances (such as μ = 7.5 μm), as shown in Fig. 13(b). It is clear from Fig. 13(b) that the internal load distributions are almost identical for both aligned and misaligned bearings.

The bearing arrangement is known to affect the characteristics of the implemented bearings.^2,19 The effect of the bearing arrangement on the bearing stiffness and fatigue life is discussed in this section. Two types of arrangements are considered: back-to-back, or “O” arrangements, and face-to-face, or “X” arrangements, as shown in Fig. 14. For comparison, the stiffness and fatigue life ratios are introduced, which are defined as the ratios between the stiffness and fatigue lives of the bearings in “X” and “O” arrangement.

Fig. 14 
D-ACBB arrangement and effective load center distance

Figs. 15 and 16 respectively show the stiffness and fatigue life ratios of D-ACBB as a function of the applied radial load. The axial clearance and rotational speed of the bearing are set to 0 μm and 1000 rpm, respectively. One can see that the bearing in the “O” arrangement is beneficial in terms of moment stiffness. It is obvious that the distance between the effective load centers of the two bearing rows L_e is larger when the bearing is in this “O” arrangement, as shown in Fig. 14. Consequently, the capability of the bending resistance for D-ACBB in the “O” arrangement is higher than that of the “X” arrangement. On the other hand, the bearing arrangement does not affect the radial stiffness, or the fatigue life of the bearing. As demonstrated in Figs. 15(a) and 16, the radial stiffness and fatigue life ratios are almost equal to 1.0, regardless of the radial load. It is noted that the D-ACBBs in the “X” arrangement may have a higher fatigue life under low radial load and misalignment conditions compared to that of the bearing in the “O” arrangement, as shown in Fig. 16.

Fig. 15 
Stiffness ratio as a function of radial load

Fig. 16 
Fatigue ratio as a function of radial load under angular misalignment