Figure 2 presents a schematic of the testing apparatus used in this study. The apparatus employed an open-type angular contact ball bearing as the test bearing, specified by the bearing dimension series 70. The cage’s dimensions include an outer diameter of 110 mm, an inner diameter of 70 mm, and a width of 20 mm. The cage material is phenolic resin with a density of 1,250 kg/m³ and Young’s modulus of 9.61 GPa. The cage is a cylindrical type guided by the outer ring, with pocket clearance c p of 0.555 mm and guide clearance c g of 0.427 mm. The lubrication was provided by extreme pressure grease, comprising 2.6 g of barium complex soap thickener and mineral oil with a kinematic viscosity of 105 mm²/s at 313 K and 12 mm²/s at 373 K.

The inner ring was mounted on a shaft supported by an angular contact ball bearing of the same series. The outer ring was secured in a fixed housing to restrict its movement. Noise measurement was conducted using a microphone positioned 40 mm from the housing’s end face, while a high-speed camera coaxially aligned with the shaft visualized the cage’s rigid body motion. High-intensity LED lighting ensured adequate exposure for the camera. An air cylinder attached to the housing exerted a constant axial load ( F a ), and a motor provided a constant angular velocity ( ω z ).

After a 2 h break-in period, sound pressure levels recorded by the microphone and optical images captured by the high-speed camera were synchronously documented. The recording spanned at least 50 shaft rotations, with the camera operating at up to 12,800 frames per second. The recorded area was a 93.75 mm square, equivalent to the diameter of the outer ring shoulder, with an image resolution of 1,024 × 1,024 pixels. This study analyzes results under test conditions of F a = 687 N and ω z ranging from 50 to 8,000 rpm.

It should be noted that, given that the tests were conducted with grease lubrication, the results may be influenced by the characteristics of the grease, such as its viscosity, consistency, and distribution. Therefore, approximately five trials per condition were performed to obtain the data reported in this paper. Within the same test, the occurrence and disappearance of cage noise were repeated, leading to some variability in the frequency of these occurrences. However, it was confirmed that the sound pressure levels during the occurrence of cage noise and the cage whirl velocities, as detailed in Section 2.3, remained relatively consistent under the same conditions. Consequently, this paper presents the most reliable data, including average values and ranges, for comparison with the computational results.

To determine the trajectory of the cage center from the test results, image-processing techniques were applied to multiple optical images captured by the high-speed camera. The cage’s center of gravity, calculated from its contour in each image, was used to establish the cage’s center. This method provided the trajectory data depicted in Figure 3. For each cage rotation, between 200 and 660 optical images were processed. The contour extraction involved binarization, setting a constant brightness threshold for each pixel to ensure the area of the cage contour remained consistent across different frames.

In Figure 3, the radial directions of the cage center’s inertial coordinates are represented on the vertical and horizontal axes. This presentation shows the results over a single rotation of the cage, including a circle representing the guide clearance diameter. In the present tests, where the rotational speed ( ω z ) was incrementally increased in steps, no cage noise was observed at speeds below 100 rpm, whereas cage noise was observed at speeds above 500 rpm. Therefore, the trajectories for conditions at rotational speeds of 100, 500, and 3,000 rpm are shown as representative samples. Notably, the reference origin for each condition is the center of gravity coordinates of the cage trajectory rather than the center of the shaft, considering the convenience of the image-processing technique used.

The observed trajectories suggest that the cage center can exceed the guide clearance owing to the elastic deformation caused by centrifugal forces because of the orbital motion of the cage. At ω z = 100 rpm, the cage center exhibits a wobbling motion, forming a crescent-shaped trajectory without generating any detectable noise. At higher speeds, ω z = 500 and 3,000 rpm, the cage performs a circular motion within the guide clearance. During one cage rotation, multiple revolutions of orbital motion occurred, with the orbital direction opposite to the rotation direction, indicating negative whirl. Noise associated with the cage was detected under these conditions.

The forces acting on the cage include contact and friction from the rolling elements and outer ring, as well as gravity. In particular, when the relationship between pocket clearance c p and guide clearance c g is c p > c g , the interaction with the outer ring becomes dominant. At ω z = 100 rpm, the cage’s wobbling is interpreted to center around an equilibrium position where the friction force from the outer ring counterbalances gravity acting in the negative y-axis direction. The wobble radius closely aligns with half of the guide clearance c g / 2 , suggesting it wobbles along the outer ring shoulder. At ω z = 500 and 3,000 rpm, similar forces act, but increased centrifugal force at higher rotational speeds increases the contact and friction forces from the outer ring, thereby making the friction force predominant over gravity, resulting in negative whirl.

Analysis of Figure 3 involved calculating the angle between the time-varying cage center coordinates and the origin. The derivative of this angle with respect to time was then used to compute the cage’s whirl velocity. These calculations allowed us to determine the time-averaged whirl velocity and time-averaged sound pressure during a single rotation of the cage, as shown in Figure 4. The graph plots time-averaged whirl velocity on the horizontal axis against time-averaged sound pressure on the vertical axis. Positive whirl velocity values indicate the cage orbiting in the same direction as its rotation, whereas negative values indicate an orbit in the opposite direction. Data points are marked with an “x” for cases where cage noise was audibly detected and with an “o” where noise was absent. Given the intermittent nature of the cage noise observed, data points at the beginning, middle, and end of each test condition were plotted.

From the data represented in Figure 4, test results can be categorized into two distinct types of cage behavior, detailed in Table 1. This categorization has allowed for the clear identification that negative whirl significantly contributes to the generation of cage noise. For a representative condition where negative whirl was observed, we further analyzed the time variations of whirl velocity and sound pressure. These findings are illustrated in Figure 5, where the upper panel shows the time variation of the whirl velocity, and the lower panel shows the corresponding sound pressure variations. The timing of fluctuations in these two measurements generally aligns, corroborating that negative whirl is a principal factor in the generation of cage noise.

This section outlines the analytical model developed to replicate the negative whirl observed in experimental settings, effectively capturing the essence of the phenomenon. This approach is instrumental in fundamentally addressing cage noise. To simplify the understanding of the phenomenon and reduce computational costs, a streamlined mechanical model comprising minimal elements is employed. For dynamic analysis of bearing focusing on cage behavior, Kannel and Bupara (1978) and Boesiger et al. (1992) investigated the in-plane motion of a rigid cage, validating this assumption through experimental data. According to Nogi et al. (2018), negative whirl primarily results from sliding friction between the cage and the outer ring. Therefore, this study focuses on the in-plane motion of a rigid cage, disregarding the interaction between the rolling elements and the raceways, with each rolling element theoretically orbiting at equal intervals. Thus, the equations of motion are confined to those of the cage.

The motion of a rigid cage, with two degrees of freedom for translation and one for rotation, is governed by the following equations based on Newton’s laws: m x ¨ = F x , (1) m y ¨ = F y , (2) I z θ ¨ = M z , (3) where m is the mass of the cage, I z is the moment of inertia about its rotational axis, F x and F y are the resultant forces in the x and y directions, respectively, and M z is the resultant moment about the axis of rotation.

A moving coordinate system, fixed at the cage center, facilitates analysis of the forces acting on the cage. The direction of cage eccentricity is denoted as the ε -axis, while the orthogonal direction is termed the φ -axis. The transformation from the moving coordinate system to the inertial coordinate system is expressed by the following Equation 4 for the rotation matrix: F x F y = cos   φ − sin   φ sin   φ cos   φ F ε F φ , (4) where φ is the angle between the ε -axis and the y-axis ( 0 < φ < 2 π ).

Figure 6 shows the forces acting on the cage, encompassing contact and friction forces from the rolling elements and outer ring, as well as gravity, as outlined in Section 2.2.

The contact state between the cage and rolling elements is effectively modeled by a series connection of the contact stiffness, as determined by Hertzian contact theory, and the deflection stiffness of the cage (Boesiger et al., 1992). Given that the contact stiffness substantially exceeds the deflection stiffness, only the latter is considered significant. A linear Voigt model, consisting of a spring and a dashpot in parallel, is employed as the contact model. The Coulomb friction model is used for the friction calculations.

When examining the interaction between the cage and the outer ring, relying solely on solid contact and Coulomb friction was found to predict a negative whirl velocity approximately ten times greater than what was observed experimentally. Furthermore, to induce negative whirl using a solid friction model, an unrealistically high Coulomb friction coefficient of at least 0.6 was required. Thus, the impact of fluid friction in this interaction cannot be ignored. As shown in Figure 7, the case where the cage center is eccentrically displaced by ε is considered. Given the study’s focus on grease lubrication, it is not assumed that the clearance is consistently filled with grease. It is assumed that a uniform thickness, h L , of lubricant exists on the shoulder of the outer ring, with h L treated as a constant input value. This model does not account for changes in film thickness due to side leakage or scooping. The presence of lubricant subjects the cage to film pressure and shear force, assumed to act within the geometrically contactable region between the cage and the lubricant (from ∠ P O Q to ∠ P O Q ′ ). To simplify the representation of cavitation, the film pressure is considered zero in regions of negative pressure.

The analysis also necessitates addressing collision phenomena. The pressure dependency of viscosity in the fluid lubrication model is described by the following equation (Barus, 1893): η = η 0   exp α P , (5) where η 0 is the base oil viscosity, and α is the pressure-viscosity coefficient. Furthermore, owing to potential interference between the cage and the outer ring, contact and friction models are adapted based on the cage’s eccentricity, ε . Given that both solid contact and fluid lubrication can occur owing to the surface roughness of the cage, the metal contact ratio λ r , as defined by Aihara (1987), is employed in the following Equation 6: λ r = 1 − exp 1.8 Λ 1.2 , (6) where Λ represents the film thickness ratio, calculated as the minimum film thickness between the cage and the outer ring divided by the surface roughness.

Table 2 outlines the friction model between the cage and the outer ring, considering the following three contact states:

1. When c g / 2 − h L ≥ ε :

In this pattern, the cage and lubricant do not geometrically contact, resulting in no interaction between the cage and the outer ring.

2. When ε > c g / 2 − h L and Λ > 0 :

This scenario involves geometric contact between the cage and lubricant where the film thickness ratio Λ > 0 . Here, both solid contact and fluid lubrication occur owing to the surface roughness of the cage, linked via the metal contact ratio λ r .

3. When ε > c g / 2 − h L and Λ ≤ 0 :

This signifies geometric contact with a film thickness ratio Λ ≤ 0 , indicating interference between the rigid cage and the outer ring. This is considered a transient state where only solid contact is modeled.

The forces acting on the cage, as delineated in Table 2, are formulated under various conditions. Under solid contact, the Voigt model and Coulomb friction, as outlined in Section 3.2, are applied. For fluid lubrication scenarios, the forces derive from Reynolds’ equation, considering the dominance of axial flow due to the cage width being typically less than one-fourth of the outer diameter. This scenario applies the theory of short-width journal bearings. The validity of this approach was confirmed through point contact elastohydrodynamic lubrication analysis, focusing on the interaction between the cage and the outer ring, which demonstrated that oil film pressure and tangential forces due to shear do not significantly affect the results.

The Reynolds equation is derived under this assumption, based on Cameron (1971), in the following Equations 7 and 8: ∂ ∂ y h 3 η ∂ p ∂ y = 6 u 1 + u 2 ∂ h ∂ x + 12 ∂ h ∂ t , (7) h = C 1 + ε ⁡ cos ⁡ θ , (8) where h is the oil film thickness, u 1 and u 2 are the surface velocities of the cage and outer ring, C is half the guide clearance, and ε is the eccentricity, respectively.

Assuming η as defined in Equation 5, Equation 7 cannot be solved algebraically using the approach for short-width journal bearings. Based on Cameron (1971), we apply Equation 9 to transform Equation 7. This transformation necessitates the case distinctions described in Equation 10, but the resulting modified Reynolds Equation 11 can then be solved algebraically. q = 1 − e − α p / α . (9) p = − 1 α ln 1 − α q  at  q < 1 / α   q  at  q ≥ 1 / α (10) ∂ 2 q ∂ y 2 = 6 η 0 h 3 u 1 + u 2 ∂ h ∂ x + 2 ∂ h ∂ t . (11)

Integrating Equation 11 with boundary conditions from Equation 12 yields Equation 13. p L / 2 = p − L / 2 = 0 (12) q = − 3 η 0 h 3 u 1 + u 2 ∂ h ∂ x + 2 ∂ h ∂ t L 2 4 − y 2 (13)

To simplify the representation of cavitation, the pressure is set to zero in regions of negative pressure. From Equations 10, 13, the oil film pressure is derived as p = max 0 , − 1 α ln 1 + 3 α η 0 h 3 u 1 + u 2 ∂ h ∂ x + 2 ∂ h ∂ t L 2 4 − y 2  at  q < 1 / α max 0 , − 3 η 0 h 3 u 1 + u 2 ∂ h ∂ x + 2 ∂ h ∂ t L 2 4 − y 2  at  q ≥ 1 / α . (14)

Pressure-induced forces F P − ε and F P − φ acting on the cage center are calculated from the following Equations 15 and 16: F P − ε = R ∫ − L / 2 L / 2 ∫ π − θ c π + θ c p ⁡ cos ⁡ θ d θ d y , (15) F P − φ = R ∫ − L / 2 L / 2 ∫ π − θ c π + θ c p ⁡ sin ⁡ θ d θ d y , (16) where θ c is ∠ POQ in Figure 7.

The shear force acting on the cage is calculated based on journal bearing theory (Cameron, 1971), as shown by Equations 17–20. Here, F S and M S represent the forces and moments acting on the cage center, respectively. It is important to note that the parameter p , which appears in Equation 17, is defined in Equation 14. Consequently, the calculations for Equations 18–20 necessarily require numerical integration. τ = ∂ p ∂ x h 2 + η u 1 − u 2 h (17) F S − ε = R ∫ − L / 2 L / 2 ∫ π − θ c π + θ c τ ⁡ sin ⁡ θ d θ d y (18) F S − φ = R ∫ − L / 2 L / 2 ∫ π − θ c π + θ c τ ⁡ cos ⁡ θ d θ d y (19) M S = R 2 ∫ − L / 2 L / 2 ∫ π − θ c π + θ c τ d θ d y (20)

Thus, the resultant forces acting on the cage from the interaction with the outer ring are detailed in Equations 21, 22. F ε = λ r F P − ε + F S − ε + 1 − λ r F D C (21) F φ = λ r F P − φ + F S − φ + 1 − λ r F D F (22)

Here, F D C and F D F represent the contact and friction forces under solid contact, respectively.

Table 3 lists the analytical conditions, which correspond to those under which negative whirl was observed in the experiments described in Section 2. Here, the angular velocities of ball rotation and orbit are set to values corresponding to an inner ring rotational speed of ω z = 3,000 rpm. The initial conditions are set with the cage center at coordinates x = 0 , y = 0 and the cage’s angular velocity at 142.5 rad/s. The analysis period is defined as 100 ms, within which the cage’s behavior is expected to stabilize. During this interval, the cage completes approximately 4.5 rotations.

The simulation addresses the dynamic problem of the cage’s contact or non-contact with the rolling elements and the outer ring, which varies over time. This variability renders Equations 1–3 as stiff differential equations. To effectively manage these, the LSODA (Petzold, 1983) numerical integration algorithm was utilized. LSODA is an algorithm that automatically switches between the Adams method and the BDF method depending on the stiffness of the ODE, allowing for efficient and accurate solutions to ODEs. It is important to note that although the BDF method selects the optimal scheme to ensure the accuracy and stability of the solution, higher-order schemes are known to increase numerical errors. Therefore, we compared the analysis results obtained using LSODA with those obtained using the Radau method, which is considered less susceptible to errors under similar conditions, and confirmed their consistency. In this study, we adopted LSODA, which has a proven track record in similar bearing dynamic analyses (e.g., Boesiger et al., 1992).

Figure 8 illustrates the trajectory of the cage center and the temporal changes in cage whirl velocity. It also shows the maximum and minimum whirl velocities observed during negative whirl events, as detailed in the visualization tests described in Section 2. The trajectory of the cage center reveals orbital motion, with trends broadly matching the experimental results shown in Figure 3 (right). The orbital radius remains consistently within the guide clearance, suggesting that the cage stays in contact with the outer ring via the lubricant. It is important to note that this analysis assumes a rigid cage; thus, unlike the experimental results, the cage center does not exceed the guide clearance due to elastic deformation caused by centrifugal force. From 25 ms onward, the whirl velocity remains generally constant and negative, signifying a steady-state condition characterized by continuous negative whirl. A comparative analysis between experimental and analytical whirl velocities indicates a discrepancy ranging from approximately 1.2–2.3 times. In contrast, an analysis based solely on solid contact and Coulomb friction between the cage and the outer ring resulted in a whirl velocity approximately ten times higher than the experimental value, indicating significant enhancements achieved with the analytical model introduced in Section 3.

The model expansion to include more degrees of freedom provides a nuanced representation of the forces driving the cage, encompassing contact, friction, and inertia. While the model discussed in Section 3 initially considered only the radial plane motion of a two-dimensional cage, extending this model to three dimensions allows for the resolution of the driving forces in the axial direction as well. This extension is expected to reduce the absolute value of the whirl velocity, which has been primarily determined within the radial plane, thereby aligning it more closely with the experimental findings.

Figure 9 shows the time variation of the tangential force F φ acting on the cage center. Under the current analytical conditions, there was no solid contact between the cage and the outer ring; thus, only fluid forces were involved. The force F φ is critical in inducing cage whirl, leading to positive whirl when positive and negative whirl when negative. Specifically, the oil film pressure force F P − φ tends to induce positive whirl, whereas the shear force F S − φ tends to induce negative whirl. The mechanism by which F S − φ induces negative whirl is linked to the direction of the shear force, which opposes the sliding velocity of the cage. This sliding velocity, determined by the rotational and translational speeds of the cage, predominantly aligns with the cage’s rotation direction. F S − φ acts in the opposite direction thereby inducing negative whirl.

The positive whirl induced by F P − φ can be elucidated by Figure 7, where the steady-state oil film pressure is influenced by the wedge effect. This effect generates positive pressure on the entry side and negative pressure on the exit side of the cage. Given the consideration of cavitation, the oil film pressure on the exit side is effectively zero. By integrating the oil film pressure along the arc from Q to P, the resultant force aligns with the direction of cage rotation, causing F P − φ to induce positive whirl.

Figure 8 reveals that for t > 25 ms, a consistent negative whirl velocity occurs, and the forces acting on the cage reach equilibrium. However, focusing on t ≤ 25 ms in Figure 9, F S − φ (inducing negative whirl) rises earlier than F P − φ (which counters negative whirl), indicating the initial cause of negative whirl.

In the steady state at t > 25 ms, the average value of the resultant force, as shown in Figure 9, is marginally negative. This suggests the presence of a counteracting force that moderates an increase in negative whirl. Figure 10 shows the time variation of the tangential force resulting from the interactions between the outer ring and the cage and between the rolling elements and the cage. The interaction with the rolling elements generates a tangential force that induces positive whirl, effectively balancing the tangential force that induces negative whirl owing to the interaction with the outer ring for t > 25 ms.

In the content up to Section 4.2, we constructed and validated an analytical model capable of reproducing the negative whirl observed in the 3,000 rpm experiment. In this section, to examine the applicability of this model to other rotational speeds, we evaluated the rotational speed dependency of cage whirl. The analytical conditions were based on those shown in Table 3, with only the angular velocity of ball rotation and orbit adjusted to match the inner ring rotational speed. In the experiment described in Section 2, the cooling effect was not expected due to the influence of grease lubrication, and it is considered that the guide surface experienced heating as the rotational speed increased, potentially leading to a decrease in the viscosity of the base oil on the guide surface. Here, two analytical conditions regarding the base oil temperature were set to identify the parameters needed to reproduce the cage behavior observed in the experiment. Specifically, one condition set a constant temperature regardless of rotational speed, and the other allowed the temperature to vary depending on the rotational speed. It should be noted that the base oil temperatures used in the analysis are reference values, as the guide surface temperatures were not measured in the experiments described in Section 2. The relationship between temperature and viscosity was considered using Walther’s equation (ASTM International, 1993).

The analytical results are shown in Figure 11. The left side of the figure shows the rotational speed dependency of the whirl velocity, with the experimental results also included. For each condition, the average values are plotted, and error bars indicating the upper and lower bounds are shown. Regarding the temperature settings in the analysis, the condition of constant temperature used 313 K, as in the content up to Section 4.2, while for the condition where temperature varies with rotational speed, the temperature was explored through trial and error to reproduce the cage behavior observed in the experiment. The right side of the figure shows the base oil temperatures determined for each rotational speed.

The analysis with the base oil temperature fixed at 313 K showed that the cage exhibited stable behavior at rotational speeds below N < 1,100 rpm, while negative whirl occurred at N ≥ 1,100 rpm. The absolute value of the whirl velocity also tended to increase with increasing rotational speed. However, this result qualitatively differs from the experimental findings, suggesting that the analytical model requires further refinement.

On the other hand, the analysis with the base oil temperature adjusted for each rotational speed indicated that the cage behavior qualitatively matched the experimental results when the temperature increase was determined through trial-and-error adjustments. This observation suggests that the base oil temperature had a significant impact on the experimental outcomes described in Section 2, underscoring the necessity of considering this factor when assessing cage instability in grease-lubricated systems. Additionally, while shear thinning is often considered in the modeling of grease lubrication, it was concluded that this effect could be neglected in this analysis due to the oil film thickness being on the order of tens of micrometers. However, this conclusion may lack sufficient supporting evidence. Therefore, future research should focus on measuring the guide surface temperature, visualizing the lubrication state during testing, and measuring the torque exclusively on the guide surface. These approaches are expected to provide a deeper understanding of cage behavior.