# Lubrication Condition Monitoring in EHD Line Contacts of Thrust Needle Roller Bearing Using the Electrical Impedance Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurement Principle

#### 2.1. Outline

#### 2.2. Geometrical Model

_{1}is the oil film thickness in the oil film-forming part of the contact area, b denotes the contact half-width, r denotes the roller radius, L denotes the straight part of the roller’s length, S

_{1}denotes the apparent contact area, S

_{2}denotes the roller surface area surrounding the contact area (note that S

_{2}in Figure 1 represents the projected area on the xy plane), and α denotes the breakdown ratio (i.e., 0 ≤ α ≤ 1). Consequently, the surface upon which the breakdown (that is, h = 0) occurs in the EHD line contact is represented by αS

_{1}, as shown in Figure 1.

_{2}in Figure 1 is a function representing the height of a roller surface around a contact area (that is, b ≤ x ≤ b + r), and we use argument $\phi $, as shown in the figure, to express it in Equation $\left(1\right)$, as follows:

#### 2.3. Electrical Model

_{1}in Figure 2 denotes the electrical resistance in areas wherein oil film is breaking down, C

_{1}denotes the capacitance in the oil film formed area within an EHD line contact, and C

_{2}denotes the capacitance in the straight part of the roller around an EHD line contact, respectively. As shown in Figure 2, in this study, it is assumed that all contact areas are represented by the same equivalent circuit. The contact areas in the upper and lower races are connected in their respective series circuits, and these are joined by parallel circuits, which are equal in number to the rolling elements. This implies that this method determines the average value of the oil film thickness and breakdown ratio in all contact areas. In Figure 2, R

_{1}, C

_{1}, and C

_{2}are each represented by their respective equations, as follows:

_{10}denotes the electrical resistance under a stationary contact (i.e., α = 1). Generally, h

_{1}≪ r; hence, Equation $\left(4\right)$ can be approximated as shown in Equation $\left(5\right)$ below.

#### 2.4. Complex Impedance Analysis

_{1}can be obtained from the following equation.

_{1}.

## 3. Experimental Details

#### 3.1. Apparatus

_{a}. To apply the electrical impedance method to this bearing, a carbon brush pressed against the rotating shaft was used to introduce an AC voltage from a LCR meter between the upper and lower races. Additionally, by using a rubber timing belt for the rotating shaft, power was transmitted in an insulated state from the motor. A ceramic ball was set between the spring and the lower race, thus applying the axial load to the test bearing. The use of a ceramic ball allowed the AC voltage to be applied only to the test bearing, and it also prevented the lower race from tilting against the rollers. This tester was not only capable of simultaneously measuring the average oil film thickness $\overline{h}$ and oil film breakdown ratio α, but also the lower race temperature T and bearing torque M. T was measured by attaching a thermocouple to the lower race, whereas M was measured using a torque meter.

#### 3.2. Materials

_{t}= 5.6 mm, and roller end face chamfer curvature r

_{c}= 0.5 mm. Additionally, the roller material was SUJ2 (Young’s modulus: 208 GPa, Poisson’s ratio: 0.3). Furthermore, the material of the upper and lower races used in this test (inner diameter: 25 mm, outer diameter: 42 mm, thickness: 1.0 mm) was SK5 (Young’s modulus: 206 GPa, Poisson’s ratio: 0.3), and the material of the cage was SPCC (Young’s modulus: 205 GPa, Poisson’s ratio: 0.3). Additionally, the root-mean-square roughness, R

_{q1}and R

_{q2}, of the rollers and upper and lower races are R

_{q1}= 0.15 μm and R

_{q2}= 0.12 μm, respectively. As the test bearing is an actual product and the manufacturing errors of the rollers and races (roundness, waviness, etc.) vary within their respective tolerances, the experimental results are affected by these factors; however, the developed method measures the average value of the total contact area when an axial load is applied, which mitigates the effects of the variation.

^{2}/s), and the test was conducted using oil lubrication. The relative permittivity ε

_{oil}of the mineral oil was ε

_{oil}≈ 2.1 in the AC frequency range f = 20 Hz to 1.0 MHz, so the oil film thickness was calculated by assuming that ε = ε

_{oil}ε

_{0}= 2.1ε

_{0}F/m. Note that the dielectric constant of vacuum ε

_{0}= 8.85 × 10

^{−12}F/m.

#### 3.3. Procedure

_{e}= 1.1 V, AC frequency: f = 10 kHz) while loading an axial load F

_{a}= 1.5 kN onto the test bearing. Before rotating the bearing, we first measured the initial complex impedance Z

_{0}using stationary contacts (i.e., α = 1). Based on the value of Z

_{0}, the electrical resistance R

_{10}in Equation $\left(14\right)$ can be obtained using the following equation.

_{0}| and θ

_{0}in Equation $\left(18\right)$ are the magnitude and phase of Z

_{0}, we found that by substituting Equation $\left(18\right)$ into Equation $\left(14\right)$, α is not affected by n. Subsequently, the upper race of the test bearing was rotated at a speed of N = 3000 min

^{−1}for 5 h, and |Z|, θ, T, and M, using dynamic contacts, were simultaneously measured (sampling rate: 0.5 Hz). The average oil film thickness $\overline{h}$ obtained using the electrical impedance method was compared with the theoretical central oil film thickness measurement that was derived using the Ertel–Grubin equation [37] in order to verify the oil film measurement accuracy. Here, the theoretical oil film thickness was calculated using the viscosity of the lubricant obtained from the lower race temperature T; this varied over time. The wear track formed on the lower race after the test was observed using an optical microscope, and the surface roughness was measured using an optical interference microscope.

## 4. Experimental Results

#### 4.1. Measurements of |Z| and θ

_{a}= 1.5 kN, in a practical thrust needle roller bearing, Figure 5 shows the measured values of the modulus |Z| (upper) and phase θ (lower) of the complex impedance Z at a rotational speed of N = 3000 min

^{−1}. |Z| varied from 0.03 kΩ to 30 kΩ, approximately, and θ varied from 0° to –85°, approximately, under experimental conditions. In the following section, the average oil film thickness $\overline{h}$ and breakdown ratio α will be quantified using |Z| and θ.

#### 4.2. Measurements of h, α, T, and M

#### 4.3. Observations of Tested Bearing

_{q2}= 0.12 μm to 0.08 μm). Figure 6 shows that the value of $\overline{h}$ after one hour was also approximately 100 nm with no breakdown area; thus, it is thought that the asperities that were greater than the oil film thickness were worn, and running-in wear was completed.

## 5. Discussion

#### 5.1. Validation of Measured h-Values

_{2c}generated in the chamfer of the roller end face was ignored, and oil film thickness was determined using the capacitance C

_{2}generated only in the straight part of the roller. However, to be precise, the oil film thickness should be calculated using the total capacitance C

_{2t}, which reflects the actual roller shape, including the chamfer of the roller end face. Therefore, we calculated C

_{2t}using the finite element method (FEM) [38], and we compared it with the C

_{2}obtained from Equation $\left(5\right)$. The simulations were performed using COMSOL

^{®}Multiphysics (version 6.0), employing the AC/DC module. More specifically, the simulations were performed by assuming that the roller and race were made of steel and filled with test oil (mineral oil). The race and analytical boundary were grounded, and a potential of 1 V was applied to the roller. Figure 8 shows the triangular mesh that was automatically assigned by the software to the roller and race geometry that was used in the simulations. Furthermore, in this simulation, we did not consider elastic deformation (i.e., F

_{a}= 0 N). Hence, given that this FEM analysis does not consider C

_{1}occurring in the EHD line contact, it implies that it can simply be compared with C

_{2}, as obtained from Equation $\left(5\right)$.

_{2}and C

_{2t}when changing the oil film thickness h

_{1}(see Figure 1) under hydrodynamic lubrication (i.e., α = 0) conditions, assuming that the roller’s surroundings are sufficiently filled with the mineral oil used in this test. In accordance with Figure 9, it was found that C

_{2}closely agrees with C

_{2t}in the range of h

_{1}< 1 μm, even though C

_{2}only calculates the capacitance in the straight part of the roller. However, in the range where h

_{1}> 1 μm, C

_{2t}is slightly larger than C

_{2}. Furthermore, we found that this difference gradually widens as h

_{1}thickens. For reference, C

_{1}at F

_{a}= 1.5 kN, obtained using Equation $\left(3\right)$, is also shown in Figure 9. In the range of h

_{1}< 30 nm, C

_{1}is larger than C

_{2}; despite this, C

_{2}significantly affects the accuracy of the oil film measurement.

_{1}= 100 μm. In Figure 10, it is determined that the electric field generated at the chamfer of the roller end face cannot be ignored when h

_{1}is thick. In other words, the thicker h

_{1}is, the more C

_{2c}cannot be ignored, which is thought to have led to the results shown in Figure 9. However, the oil film thickness in EHD line contacts is generally less than 1 μm, thus suggesting that ignoring C

_{2c}does not significantly affect oil film measurement accuracy (see Figure 6).

_{2c}cannot be ignored. Figure 11 shows the results of the calculating relationship between h

_{1}and C

_{2}/C

_{2t}for varying L/L

_{t}at r = 1.0 mm and r

_{c}= 0.5 mm. From Figure 11, we found that as L/L

_{t}increases, C

_{2}/C

_{2t}increases (i.e., the effects of C

_{2c}become smaller). However, we also found that even if L/L

_{t}is relatively small, and the lower h

_{1}is, the more C

_{2c}can be ignored. Given that L/L

_{t}≈ 0.82 for the rollers used in the bearing test, it is thought that the electrical impedance method developed in this study has a sufficient oil film measurement accuracy within the range of h

_{1}< 1 μm, as shown in Figure 11.

#### 5.2. Validation of Measured α-Values

_{q1}and R

_{q2}denote the root-mean-square roughness of the roller and lower race before the test, Figure 12 shows the relationship between Λ and α. From Figure 12, we can determine that when Λ < 3, α increases as Λ decreases, although the variation is large. Johnson et al. [39] predicted that the number of asperity contacts within the EHD contact was given by a Poisson distribution, and they pointed out that theoretically, the oil film breaks at Λ < 3. Hence, Figure 12 suggests that this method can quantitatively evaluate $\overline{h}$ and α simultaneously. Furthermore, the reason for the large variation in the test results shown in Figure 12 is that the test conditions are accompanied by running-in wear, as shown in Figure 7. Additionally, Figure 6 shows that α, as well as $\overline{h}$, decreased with time, except immediately after starting the test, finally reaching α ≈ 0 and $\overline{h}$ ≈ 100 nm. This result is also expected to support the idea of progressive running-in wear in EHD line contacts.

## 6. Conclusions

- We theoretically demonstrated that the thickness and breakdown ratio of oil films can be simultaneously measured using the complex impedance generated when a sinusoidal voltage is applied to EHD line contacts.
- We applied the developed method to an actual thrust needle bearing, and we simultaneously measured the oil film thickness and breakdown ratio. The oil film thickness and breakdown ratio fluctuated significantly immediately after starting the test, but the breakdown ratio decreased over time. Furthermore, the oil film thickness after one hour was thought to be completed when the running-in wear was found to almost match the theoretical value. It was also confirmed that the lower race temperature and bearing torque were both constant values at that time, thus indicating that the lubricated condition was stable.
- Based on the results of measuring the surface roughness of the lower race after the test, asperities of 100 nm or more, observed in the undamaged part, were not found in the wear track. One hour after the start of the test, the oil film thickness was also approximately 100 nm, and there was no breakdown area; it is thought that running-in wear was almost completed at this time.
- When using this method, the oil film thickness in the EHD line contacts is determined by ignoring the capacitance generated in the chamfer of the roller end face. As a result of calculating the capacitance in the actual roller shape using FEM, we confirmed that there is generally no problem in ignoring the capacitance that occurs in the chamfer of the roller end face. However, this suggests that when the length of the roller’s straight part is relatively short compared with the total length of roller, the capacitance occurring in the chamfer part of the roller end face cannot be ignored.
- We clarified the relationship between the oil film parameter Λ and breakdown ratio α in EHD line contacts. When Λ < 3, we found that α tended to increase as Λ decreased. The reason for the large variation in the obtained test results is that running-in wear occurred in this study.

## 7. Patents

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$b$ | Hertzian halfwidth | [m] |

f | AC frequency | [Hz] |

f(y) | height of surface roughness in y-axis direction | [m] |

h | coordinate perpendicular to the xy plane | [m] |

h_{1} | oil film thickness in lubricated area | [m] |

h_{2} | oil film thickness in surrounding area expressed as ${h}_{2}={h}_{1}+r\left(1-\mathrm{cos}\phi \right)$ | [m] |

$\overline{h}$ | mean oil film thickness expressed as $\overline{h}=\left(1-\alpha \right){h}_{1}$ | [m] |

j | imaginary unit | [–] |

n | number of rollers per bearing | [–] |

$r$ | radius of roller | [m] |

${r}_{\mathrm{c}}$ | chamfer curvature at roller edge | [m] |

t | time | [s] |

x | coordinate in rolling direction | [m] |

y | coordinate across rolling direction | [m] |

C_{1} | capacitance in lubricated area within EHD line contact | [F] |

C_{2} | capacitance in roller straight area surrounding EHD line contact | [F] |

C_{2c} | capacitance in chamfer area | [F] |

C_{2t} | total capacitance between roller and race reflecting actual geometry | [F] |

E | electric field strength | [V/m] |

F_{a} | axial load | [N] |

I | alternating current expressed as $I=\left|I\right|\mathrm{exp}\left(j\left(\omega t-\theta \right)\right)$ | [A] |

|I| | amplitude of alternating current | [A] |

L | length of roller straight area | [m] |

L_{t} | total length of roller | [m] |

M | bearing torque | [N·m] |

N | rotational speed of upper race | [s^{−1}] |

R_{1} | resistance in breakdown area under a dynamic contact | [Ω] |

R_{10} | resistance of the breakdown area under a stationary contact (i.e., α = 1) | [Ω] |

${R}_{\mathrm{q}1}$ | root mean square roughness of roller | [m] |

${R}_{\mathrm{q}2}$ | root mean square roughness of lower race | [m] |

S_{1} | Hertzian contact area | [m^{2}] |

S_{2} | roller surface area surrounding EHD line contact | [m^{2}] |

T | lower race temperature | [°C] |

V | sinusoidal voltage expressed as $V=\left|V\right|\mathrm{exp}\left(j\omega t\right)$ | [V] |

|V| | amplitude of sinusoidal voltage | [V] |

V_{e} | RMS amplitude of sinusoidal voltage expressed as ${V}_{\mathrm{e}}=\frac{1}{\sqrt{2}}\left|V\right|$ | [V] |

Z | complex impedance expressed as $Z=V/I=\left|Z\right|\mathrm{exp}\left(j\theta \right)$ | [Ω] |

|Z| | modulus of complex impedance under dynamic contacts | [Ω] |

|Z_{0}| | modulus of complex impedance under stationary contacts | [Ω] |

α | breakdown ratio of oil films | [–] |

ε | dielectric constant of lubricant expressed as $\epsilon ={\epsilon}_{\mathrm{oil}}{\epsilon}_{0}$ | [F/m] |

ε_{0} | dielectric constant of vacuum | [F/m] |

ε_{oil} | relative permittivity of lubricant | [–] |

θ | phase of complex impedance under dynamic contacts | [deg] |

θ_{0} | phase of complex impedance under stationary contacts | [deg] |

ν | kinematic viscosity of lubricant | [m^{2}/s] |

φ | polar angle | [rad] |

ω | angular frequency of AC voltage expressed as $\omega =2\pi f$ | [rad/s] |

Λ | film parameter expressed as $\Lambda =\overline{h}/\sqrt{{R}_{\mathrm{q}1}{}^{2}+{R}_{\mathrm{q}2}{}^{2}}$ | [–] |

$\Psi $ | dimensionless number expressed as $\Psi =-\frac{8\left(1-\alpha \right)b\mathrm{sin}\theta}{{\pi}^{2}\epsilon nLr\omega \left|Z\right|}$ | [–] |

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**Figure 2.**Electrical model of a thrust needle roller bearing. R

_{1}: resistance in breakdown area; C

_{1}: capacitance in lubricated area within EHD line contact; and C

_{2}: capacitance in the straight area of the roller surrounding the EHD line contact.

**Figure 5.**Time evolution of the measured modulus |Z| (top) and phase θ (bottom); bearing: FNTA2542, lubricant: mineral oil, rotational speed: N = 3000 min

^{−1}, axial load: F

_{a}= 1.5 kN, RMS amplitude: V

_{e}= 1.1 V and frequency: f = 10 kHz; red plots in graphs: measured values by LCR meter.

**Figure 6.**Time evolution of the measured oil film thickness h (top left), breakdown ratio α (top right), lower race temperature T (bottom left), and bearing torque M (bottom right); bearing: FNTA2542, lubricant: mineral oil, rotational speed: N = 3000 min

^{−1}and axial load: F

_{a}= 1.5 kN; red plots in graphs: measured values using the electrical impedance method; black broken line in top left graph: theoretical prediction at T using the Ertel–Grubin equation [37].

**Figure 7.**Observations of the lower race surface after the experiment; (

**a**) photograph of the wear track and (

**b**) surface roughness profile around the boundary of the wear track; black vertical line at y = 0 in (

**b**): boundary of the wear track; red broken line in (

**b**): f(y) = 100 nm.

**Figure 8.**Triangular mesh used for simulations; radius of roller: r = 1.0 mm, length of roller straight area: L = 4.6 mm, chamfer curvature at roller edge: r

_{c}= 0.5 mm and total length of roller: L

_{t}= 5.6 mm.

**Figure 9.**Effect of a chamfer at the roller edge on capacitance between the roller and race for varying oil film thickness h

_{1}with no breakdown area (i.e., α = 0); lubricant: mineral oil and F

_{a}= 0 N; red line in graph: theoretical prediction C

_{2}using Equation $\left(5\right)$; black open circles in graph: simulated values C

_{2t}using COMSOL

^{®}Multiphysics; red broken line in graph: theoretical prediction C

_{1}at F

_{a}= 1.5 kN using Equation $\left(3\right)$.

**Figure 10.**Electric field distribution in the yh plane at x = 0; (

**a**) the entire view of simulation and (

**b**) the enlarged view of the chamfer at the roller edge; oil film thickness: h

_{1}= 100 μm, lubricant: mineral oil and F

_{a}= 0 N; color bar in simulated results: electric field strength: E [V/m].

**Figure 11.**Relationship between the oil film thickness h

_{1}and ratio of C

_{2}/C

_{2t}for the varying ratio of L/L

_{t}with no breakdown area (i.e., α = 0); lubricant: mineral oil, F

_{a}= 0 N, radius of roller: r = 1.0 mm and chamfer curvature at roller edge: r

_{c}= 0.5 mm; red lines in graph: theoretical prediction using Equation $\left(5\right)$ and COMSOL

^{®}Multiphysics; black broken line in graph: h

_{1}= 1 μm.

**Figure 12.**Relationship between film parameter Λ and breakdown ratio α; bearing: FNTA2542, lubricant: mineral oil, rotational speed: N = 3000 min

^{−1}and axial load: F

_{a}= 1.5 kN; red plots in graph: measured values using the electrical impedance method; black broken line in graph: Λ = 3.

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## Share and Cite

**MDPI and ACS Style**

Maruyama, T.; Radzi, F.; Sato, T.; Iwase, S.; Maeda, M.; Nakano, K.
Lubrication Condition Monitoring in EHD Line Contacts of Thrust Needle Roller Bearing Using the Electrical Impedance Method. *Lubricants* **2023**, *11*, 223.
https://doi.org/10.3390/lubricants11050223

**AMA Style**

Maruyama T, Radzi F, Sato T, Iwase S, Maeda M, Nakano K.
Lubrication Condition Monitoring in EHD Line Contacts of Thrust Needle Roller Bearing Using the Electrical Impedance Method. *Lubricants*. 2023; 11(5):223.
https://doi.org/10.3390/lubricants11050223

**Chicago/Turabian Style**

Maruyama, Taisuke, Faidhi Radzi, Tsutomu Sato, Shunsuke Iwase, Masayuki Maeda, and Ken Nakano.
2023. "Lubrication Condition Monitoring in EHD Line Contacts of Thrust Needle Roller Bearing Using the Electrical Impedance Method" *Lubricants* 11, no. 5: 223.
https://doi.org/10.3390/lubricants11050223