Figure 1A shows a schematic of a solid-oil-air interface, that exists in the meniscus region of a contact, forming two boundaries at x = 0, h. Here, h is equivalent to the thickness of the oil film. It is assumed here that the boundaries are perfectly bonded and any wave travel is perfectly perpendicular to the boundaries. The acoustic impedance z of a material is the product of its density ρ and speed of sound c. When an ultrasonic wave strikes the boundary between two acoustically different materials, a portion transmits into the second material, and the remainder reflects. The reflection coefficient R quantifies the reflected proportion and is calculated in Eq. 1: R = z 2 − z 1 z 2 + z 1 (1)

From this equation it is clear that greater acoustic mismatches lead to a greater proportion of the wave being reflected. In Figure 1A, p _i represents the initial, incident wave pressure which has all of the wave energy. This then splits at the boundary between steel and oil where x = 0 and a transmitted wave p _a and reflected wave p _r form. The transmitted wave continues to the boundary at x = h and splits again at the oil boundary with air, into the transmitted p _t and reflected p _b . Where c ₂ is the speed of sound through the second material, the oil layer, and λ is the ultrasonic wavelength, if h is sufficiently large, to the point were 2h/c ₂ > λ, the reflections p _r and p _b will be observably distinct from one another. The time between reflections Δt can then be measured, and equates to: Δ t = 2 h c 2 (2)

Equation 2 can be rearranged to make h the subject, and so the thickness of the second body can be calculated. This is known as the Time-Of-flight method (TOF).

However, if h < λ the reflected signals p _r and p _b superimpose and are not distinct. The case of the complex reflection coefficient, from a layered material, in terms of pressure was given by Brekhovskikh (1960): R = R 23 + R 23 2 i α 2 h 1 + R 12 R 23 2 i α 2 h (3)

Where R ₁₂ and R ₂₃ are the reflection coefficients at the media boundaries at x = 0, h. α ₂ is a coefficient referring to the second medium, defined as: α 2 = k 2 + i β (4)

Where β is the attenuation coefficient, k ₂ is the wave number calculated as: k 2 = 2 π f c 2 (5)

where f is frequency.

Although Eq. 3 is complex, incorporating amplitude and phase, Haines et al. (1978) gives an equation for the real, amplitude portion: | R | = R 23 + R 12 e − 2 β h 2 − 4 R 12 R 23 e − 2 β h ⁡ sin 2 k 2 h 1 + R 12 R 23 e − 2 β h 2 − 4 R 12 R 23 e − 2 β h ⁡ sin 2 k 2 h 1 / 2 (6)

Kinsler et al. (2000) explains when a wave travelling through an acoustically soft material reaches the boundary with an acoustically harder material z ₁ < z ₂ the reflected portion of the wave is perfectly in phase with the incident wave. When z ₁ > z ₂ the reflected wave is 180° out of phase with the incident wave. Transmitted and incident waves across a boundary are always in phase regardless of acoustic impedances at a boundary. These rules still hold true at x = 0, h in a 3-body system. However, the phase between the transmitted wave p _a from the first boundary and the reflected wave p _b from the second boundary changes based on the distance between boundaries i.e. the thickness of the middle layer. Haines et al. (1978) showed that when z ₂ > z ₃ the middle layer resonates if the phase between p _a and p _b is ± (2n + 1) π/2 where n is any integer value (n = 0, 1, 2, 3, …). At these particular phases, p _a and p _b destructively interfere, reducing the amplitude of p _r. This is then observed by a drop in the reflection coefficient, as in Figure 1B, which is a model plot of Eq. 6. The first of these resonances is the fundamental frequency, f ₀. To find the relationship between fundamental frequency f ₀ and film thickness, Eq. 5 is used with f = f ₀, multiplied by the thickness of the middle body h, and equalled to the phase change when n = 0: k 2 h = 2 n + 1 π 2 ∴ 2 π f 0 h c 2 = π 2 (7) And so: h = c 2 4 f 0 (8)

Equation 8 shows that h ∝ 1/f ₀ and so the thicker films have smaller fundamental frequencies and thinner films have larger fundamental frequencies. This is shown in Figure 1B, which compares model data for an increasing film thickness. The 1 μm film shows no detectable reduction in R across the presented frequency range, demonstrating that there is a lower limitation of the resonance method. In reality, a resonance would occur if the layer could be excited by a high enough frequency(f > 200 MHz), but an increased frequency leads to more attenuation, and thus a diminishing reflection amplitude. The 20 μm film does show a resonant dip, but above a typical bandwidth of a 10 MHz sensor. This would therefore be measurable with a higher frequency sensor. The thicker films, of 100 μm and 200 μm, show several dips in R, occurring at the odd integer multiples of f ₀.

Swapping c ₂ for terms in the general wave equation c = λf, which holds true for ultrasonic waves, in Eq. 8 shows that the middle layer resonates at its fundamental frequency when its thickness is a quarter of that of the ultrasonic wave: h = λ 4 (9)

This resonance occurring in media at a quarter of the interacting wavelength is seen in other wave interactions, such as coupling piezoelectric elements to water and air Gomez Alvarez-Arenas (2004), and coupling ultrasonic matching-layers with metals in ultrasonic viscometers Schirru et al. (2015). A simple rearrangement of Eq. 8 can make the frequency the subject: f 0 = c 2 4 h (10)

Haines et al. (1978) reports that when z ₂ > z ₃ further resonances occur at odd integer multiples of the fundamental frequency (f ₀, 3f ₀, 5f ₀, etc.). These resonances are clear from Figure 1B and the frequencies are calculated by the following: 2 n + 1 f 0 = 2 n + 1 c 2 4 h (11)

Again n = 0, 1, 2, 3, … Higher order odd harmonics occur at higher integer multiples of a quarter of the wavelength. For example, where n = 3, 2 × 3 + 1 f 0 = 2 × 3 + 1 c 2 / 4 h 7 f 0 = 7 c 2 / 4 h

The resonance approach has been used to measure stationary lubricant film between shims Dwyer-Joyce et al. (2003) and hydrodynamic films in rolling bearings Zhang et al. (2019). Chen et al. (2005) used the resonance approach to measure condensing liquid films showing that the method is reliable for a film of varying thickness. Kanja et al. (2021) used a sensor pair in a pitch-catch formation to create a continuous standing wave in a free-film layer. The results show a high level of accuracy, but the approach requires more instrumentation space for the additional sensors. In the work presented in this paper, the resonance method is validated for thinning oil films of varying thickness. The resonance method is then used to assess whether a meniscus is present, and then quantify meniscus thickness and length in the inlet and outlet of an EHL line contact. These values then give an indication of how bearing operating conditions affect this meniscus formation.

Figure 2, taken from Nicholas (2021) shows the bespoke rig designed and manufactured by Ricardo United Kingdom Ltd, housed at the University of Sheffield described in Howard (2016). The rig houses a shaft mounted NU2244 cylindrical roller bearing, the inner raceway being stationary during testing, the outer raceway being belt driven up to a maximum of 100 revolutions-per-minute (rpm). A hall effect sensor monitored the rotation of the outer raceway of the test bearing whilst a purely radial load was applied to the shaft via two linkages. Load cells within each linkage gave load feedback. During oil testing a hydraulic pump fed lubricant between the raceways, at 60° from the bottom-dead-centre. An oil reservoir and scavenger pump were used to collect and recirculate this oil feed so that the bearing had a constant lubricant supply. A pressure transducer in the inlet to the oil pump ensured there was lubricant flow during testing. For the grease tests, the oil pump was removed and 1/3 of the free space was filled with grease. The load, rotational speed and oil pump were all controlled through a LabVIEW interface.

The instrumented inner raceway, seen in Figure 3A is mounted onto a bearing carrier before being shaft mounted. The bearing carrier has a machined instrumentation slot ≈ 10 mm wide and ≈ 5 mm deep where all sensors for testing must fit. This limited testing to bare ultrasonic piezo elements. 7 longitudinal sensors, bonded along the axis of the bearing with their centre positions 11 mm away from each other, were all active during testing. When excited, a wave transmits from the ultrasonic element, through the raceway material and reflects from the inner raceway-roller contact face. A guiding bolt ensures the position of the sensors is at the bottom of the bearing, at the point of maximum load when a roller passes the sensor location.

Two oils were used for testing, Alpha SP 320 and the less viscous Hyspin VG 32. Additionally, Mobil SHC 460 WT grease was tested. By using lubricants of different viscosities, densities and type (oil and grease) it was possible to investigate the potential to detect meniscus resonances and the effects on meniscus dimensions. Lubricant properties can be found in Table 1.

The ultrasonic test kit used was a personal computer with an in-built ultrasonic pulser receiver (UPR) to excite ultrasonic sensors, using a top-hat wave, and to record the reflections. The UPR can excite elements at a rate of 80 kHz known as the global pulse rate (GPR). The GPR is split over a possible 8 active channels and so if all channels are used, each channel pulse rate can be a maximum of 10 kHz. A digitiser records the reflected voltages at a rate of 100 MHz. Due to the high capture rate, a buffer is used to temporarily store recorded reflections before being permanently saved onto a hard drive. The UPR was controlled through a LabVIEW programme which allowed different pulse widths, delays, ranges and filters to be applied to each channel. This allows the user to optimise the excitation of the signal for the largest amplitude with the widest bandwidth. It also enables just the first reflection to be recorded which reduces data storage demands. Figure 3B shows a schematic of the UPR system linked up with the test bearing.

Oil tests comprised of three 10 s captures at loads from unloaded to 500 kN in 100 kN increments, and five incremental speeds of 20–100 revolutions per minute, bearing speed n.dm of 6,200 to 31,000. By keeping test times short, the UPR can record at much higher pulse rates without filling the internal buffer and losing data. K-type thermocouples monitored oil supply and inner raceway temperature.

For grease testing, the added complexity of the churning phase Lugt (2013) was considered. Under a constant load of 400kN, 5 break-in stages were identified for the different bearing speeds. During these stages, lasting between 2 h and 5.5 h, the sensors were active (transmitting and recording reflections) for 5s followed by a 25s lay-off period. This allowed the UPR pulse rate to be high and comparable to the oil tests, without creating too much data which would limit the overall test time.

The limits and precision of the resonance method are difficult to define as from Eq. 13 there is dependency not only on the frequency measurement, but also through the speed of sound value. The speed of sound is affected by several things such as temperature, pressure and aeration and so calculation of the measurement uncertainty of these parameters, which can be very difficult to measure in situ, should be estimated on a case-by-case basis. However, resonance detection is universal, regardless of the application.

For very thin films, f ₀ becomes increasingly large, as too does the required bandwidth to capture multiple resonances, and so only one resonance may be recorded. In this situation Chen et al. (2005) suggests that as no other resonances are detectable, the single resonance can be assumed to be f ₀. However, this does not hold true for all Bandwidths (BW). Take for example a typical bandwidth of a 10 MHz sensor between 5 MHz and 15 MHz. If f ₀ = 4 MHz, this resonance would not be detected, and only 3f ₀ = 12 MHz is within the bandwidth. If this was incorrectly assumed to be f ₀, the thickness measurement would be out by a factor of 1/3.

To address this issue, the authors propose two conditions in which either must be met to ensure a single observed resonance f _r is the fundamental frequency and not a higher order odd harmonic: f r ≤ B W max − B W min 2 (14) f r 3 ≥ B W min (15)

If either condition is met, then it is assumed f _r = f ₀. However, if the conditions cannot be satisfied, the assumption cannot be made, and at least 2 resonances must be recorded and Δ f ̄ calculated and used in Eq. 13 to make a thickness calculation. The 100 μm film in Figure 1B is an example of when the single resonance within the measured bandwidth cannot be assumed to be f ₀.

To validate that bare piezoelectric sensors could observe resonances in an oil film, the phenomenon was investigated on a simple spreading oil droplet. An aluminium plate, instrumented with a 10 MHz ultrasonic sensor and thermocouple, had droplets of different viscosity oils placed on the plate centre, corresponding with the sensor position, and were allowed to flow unhindered. Two Newtonian calibration oils were used for the experiment, Cannon S3 and the more viscous Cannon N10. Figure 7 shows a schematic of the test set-up. As the oil runs, the film thickness reduces, and so the layer measured becomes thinner over time. During the test, the sensor is continuously excited, to produce ultrasonic waves which interact with the oil layer, and then ‘listens’ for the reflections, in a pulse-echo configuration. The ultrasonic test equipment used was a compact low voltage version of the UPR kit used for the bearing test.

Figure 8 plots (a) and (c) show reflection coefficient at a single frequency of ≈ 11.5 MHz . As the oil thins, the film thickness passes through different integer multiples of the quarter wavelength, which cause a resonance in the oil layer and thus a dip in reflection coefficient. The resonances occur closer together in (a) as the oil is less viscous and so thins very quickly. In (c) the resonances occur over a long time period as the oil thins slower, and also have a more variable dip pattern. This is attributed to shearing in the lubricant by Pialucha et al. (1989). Plots (b) and (d) show the spectrogram analysis of the same oils thinning. By analysing simultaneously in the time and frequency domains, the thinning oil can be visualised as horizontal resonant fringes. As an oil spreads, h decreases, and so Δ f ̄ increases over time. Plots (a) and (c) are essentially horizontal slices through these spectrograms, and the frequency at which they were taken is observed by the dashed line.

Figure 9 shows the ultrasonic measurement of the thinning oil droplets for both viscosity oils. In the initial reflections, the oil droplet is thick and therefore measurable only by the well established TOF method. As the films continues to thin, reflections from the front and back face of the oil layer overlap and TOF becomes unusable, but the resonance technique is still applicable. for the less viscous s3 oil, the transition occurs at ≈ 20 s when the film is 300 μm thick. The oil film is measured to 112 μm after which the film becomes too thin for detection with this particular sensor. For the more viscous n10 oil, the transition does not occur until ≈ 80 s , and the oil film is measured for the full duration of the test. Therefore, it is shown the fundamental resonance is related to the film thickness, and the resonance approach is capable of measuring surface oil films.

Figure 4 shows the reflection coefficient, calculated from Eq. 12, change as 4 rollers pass over the piezo element location. Highlighted in blue is the contact region, and either side is the inlet and outlet meniscus, where there is a clear change from R = 1 to R ≈ 0.95 due to the presence of lubricant menisci. This type of plot shows the presence of a meniscus, but as sensitivity is limited between 1⪆R⪆0.95, the shape, length and thickness cannot be visualised.

Instead, a spectrogram similar to Figures 8B,D can be made by stacking together all the reflection plots, like those of Figure 4, at different bandwidth frequencies. Figure 10A shows an example spectrogram of a single roller contact with both an inlet and outlet menisci present, where the intensity of the plot is R. The intensity of the plot has been limited between 0.95 ≤ R ≤ 1.00 to improve the contrast around the resonant R values. 4 distinct areas are highlighted on the plot:

1. Red: outside the contact and meniscus regions where very little oil is present, and so no resonances can be seen here

Figure 10B has drawn by eye resonances, and for single reflections Δ f ̄ has been measured across the frequency range, and from that film thickness calculated using Eq. 13. It is clear here that thickening takes place in the contact entry and thinning takes place at the exit.

Of note is that the frequency jump between resonant minima is not always exact, as it is in theory, which is why it is important to calculate Δ f ̄ . There are two likely reasons for this deviation. The first is the digitisation rate of the ultrasonic test equipment being too low, meaning the signal padding increases the resolution “near” to the true resonance, instead of at the resonance. The second is an uneven film thickness within the sensor measurement area, meaning the resonances from multiple films are being detected. If the films are of a very similar amplitude, such as a meniscus inlet film growing into the contact, the resonance minimas could be blurred and/or shifted. If this was the case, taking the mean frequency jump would still be an appropriate way to quantify the mean film thickness. Regardless, both of these factors should be explored in future work.

Applying the detection approach of Section 3.2.3 to the resonant plots, Figure 11 shows the comparison between a single meniscus for VG 320 oil at Figure 11A 100 rpm and Figure 11B 40 rpm. For the majority of the measurement at both inlets there is a clear, consistent meniscus pattern. However, further away from the contact the pattern becomes much noisier, indicating the oil film in this region is below the minimum detectable threshold, but some noise peaks are still being detected causing the script to output a thickness value. Therefore, it can be assumed in these areas that the film is very thin in comparison with the measured menisci. Comparing cases a & b, the thickness values appear very similar, however the detectable menisci length at both inlet and outlet are shorter for the higher bearing speed case. Between the tests, there is a temperature change < 1 ° and the load is the same. Therefore, bearing speed is the dominating factor in this shortening effect. Likely, at the higher bearing speed, the oil cannot circulate around the rollers as quickly, and more oil is swept away from the track, meaning less oil availability at the inlet and outlet of the contacts.

Figure 12 shows a comparison of 2 inlet menisci recorded over a single 10s period. They seem to show two conflicting patterns, with a) being very thin and then growing into the inlet, and b) being thick and thinning into the inlet. Figure 12B is more representative of the norm for the test case. These results are taken from VG 320 oil at 20rpm and 500kN. Although the potential impact of either meniscus shape on the lubricating film is beyond the scope of this paper, clearly ultrasound has potential for detecting these changes if such an association were to be defined in the future.

Figure 13 shows a comparison of a single roller pass under the same bearing speed and load conditions, lubricated with VG 320 and VG 32 oil. The first order polynomial has been plotted for easier comparison. Although this is a comparison of just a single roller, it can be seen that the lower viscosity oil does not develop as thick a film at the contact inlet as the higher viscosity oil. Svoboda et al. (2013) found that thicker inlet films led to greater contact separation, and Crook (1961) found that contact film thickness increased with an increasing inlet viscosity. Figure 13 therefore suggests high viscosity oils create more separation in a contact by generating thicker films at the contact inlet.

Ultrasonic measurements were taken for a grease lubricated case during the initial churning phase. During churning the fresh grease fibres are severely broken and the bearing begins to starve Lugt (2013). Figure 14 shows the meniscus shape change under constant load and speed, but captured at four different times after start-up during the churning phase. Observing from (a) to (d), as the grease is worked, there is a shortening of the meniscus length towards the contact inlet, which is associated with starvation, suggesting ultrasonic sensors can observe grease churn. In plot (d), which is at 58% of the total churn time, no meniscus is detected at the inlet and instead only noise is present. Again, this does not necessarily mean a meniscus is not present, only that if there, it is thinner than the minimum detectable value, around 290 μm for the grease tests.

The outlet meniscus shows a similar pattern of shortening, and in (c) after 22% of the churn time, the outlet meniscus is not observed. However, in (d) at 58% of the churn time, an outlet meniscus is detected, suggesting that the meniscus recovery during churn is observable.