Elastic Waves of a Single Elasto-Hydrodynamically Lubricated Contact

Figure 1 shows a schematic drawing of the experimental setup of tests on a steel plate with the brand name Toolox44 that is considered to have infinite length and breadth. The plate used has the dimensions of 1 × 0.5 × 0.051 m and can be considered infinite for a measurement duration of \(84\,\upmu \hbox {s}\) because reflections in horizontal directions are not relevant. The shortest horizontal reflection time (based on width 0.5m) is \(84.75\,\upmu \hbox {s}\) which means the elastic wave does not return to the point of measurement within the recorded time. Hence, the plate can be considered infinite and boundaries do not need to be considered in the horizontal direction. However, across the plate (0.051 m) reflection times are shorter and therefore boundaries need to be considered in the vertical direction. Therefore, the plate was boundary restricted in one dimension and infinite in the two horizontal dimensions. Due to the two boundary free dimensions, the plate is considered to be an infinite plate throughout the article.

The plate was, during the experiments, resting on two rubber cylinders \(({\varnothing }10\,\hbox {mm})\) placed on each end of the 1-m side (Fig. 1). The AE sensor was mounted centred underneath the plate and was vertically aligned with a magnetic holder. In this way, the ball impact occurred centred on the sensor position only separated vertically by the plate thickness (epicentral setup). To improve the accuracy of the impact position, the magnetic holder was shaped as a sphere to obtain automatic alignment based on gravity.

Acoustic emission signals were recorded by a broadband flat-response transducer (\(\hbox {WS}\alpha \)) from Physical Acoustics in combination with a preamplifier (Physical Acoustic 2/4/6C). The sensor head of this sensor has a diameter of 19 mm. In addition to the amplifying circuit, the amplifier had an inbuilt bandpass filter with corner frequencies of 100 kHz and 1.2 MHz. In all experiments presented in this article, the AE sensor was attached with beeswax to the specimens. Therefore, a small amount of beeswax was applied onto the plate and heated to \(75\,^{\circ }\hbox {C}\), followed by pressing the AE sensor with dead weight onto the plate. Before starting the tests, a temperature measurement made certain that the plate was cooled to below \(25\,^{\circ }\hbox {C}\). All signals of the acoustic emission sensor presented in this investigation were compensated for the sensor function. This sensor function was obtained by a glass capillary burst experiment as suggested by McLaskey and Glaser [15] as a method for absolute calibration.

For experiments with the infinite plate setup, four different steel balls were used and each ball was dropped from three different heights (Table 1). Balls were released by a magnetic holder. This holder consists of a coil and a iron cylinder, which is partly insert into the coil. The cylinder has additionally the shape of a half sphere at one side. This half-spherical shape limits the contact area between holder and the ball to release, allowing a clean release without deflection. The penetration depth of the coil and the current were adjusted for each ball size to minimize the magnetic forces to a minimum, but still large enough to overcome gravity. Each combination (ball size and height) was repeated 8 times for the lubricated experiments. Hertzian reference measurements under dry conditions were repeated 4 times for each combination.

Tests for boundary restricted systems were executed on disc samples with a diameter of 103.8 mm. The disc samples only differed in height (Table 2). In this investigation, all disc materials were Steel2511. Due to the dimensions of the disc, reflection cannot be neglected and boundaries of all dimensions need to be considered. Therefore, the term boundary restricted system will be used for all experiments with discs. The balls were for the boundary restricted cases dropped from three different heights onto the discs. As previously shown [16], there are in boundary restricted systems two major cases; one where the exciting source and elastic wave are independent, and one where both interact with each other. The latter case (interaction between source and wave) is studied in this article for an EHL contact. The laser doppler vibrometer (LDV) used for acoustic emission measurement in the boundary restricted system is the Polytec OFV056. The Polytec OFV056 was used in combination with a control and preamplifying unit of the same manufacturer (Polytec OFV3001S). As shown in Fig. 2, the laser had to be redirected, which was done by an optical mirror (Thorlabs PF20-03-P01).

The sensitivity setting was chosen to be 1.25 mm/s, which resulted in a frequency range of 1 Hz to 1 MHz of the LDV system. As for the infinite plate setup, the measurement point and impact point were vertically aligned to measure an epicentral impact. The magnetic holder and sampling card were also identical in both measurement setups. However, the setups differ in terms of positioning of the plates and discs, respectively. While the plate was resting on rubber cylinders (Fig. 1), the discs needed to be clamped in between two rubber seals (Fig. 2), because the net weight of the discs was not enough to prohibit movement during impacts. The dimension of the rubber seals was \({\varnothing }100\,\hbox {mm}\) and the thickness was \({\varnothing } 2.5\,\hbox {mm}\). For the boundary restricted systems, both EHL contact and reference measurements with Hertzian contacts were repeated 8 times for each combination (ball size, disc height and drop height).

In Fig. 4 forces are generated in both a dry and a lubricated contact for an impact of a ball (RB2 G20) from a height of 350 mm. Differences in contact time, as well as amplitude, are observed. The EHL force reaches its maximum earlier than the Hertzian force function, which is in line with the investigation of Larsson and Höglund [18]. Therefore, the force functions are considered to be evaluated and are the main input for simulations presented in this investigation.

Results of simulated signals based on the force functions show decreased zero frequencies due to the increased contact time for EHL contacts and additional excitation of higher frequencies (Fig. 5) . The blue solid line represents the spectra of an impact of a ball (RB2 G20) dropped from a height of 350 mm. The spectra show distinct frequency minimums (zero frequencies, \(f_{{\text {Zero}}, N}\)) based on the contact time of the Hertzian contact as shown by McLaskey and Glaser [13]:

The red dashed line shows the spectra of the simulated EHL impact. Comparing the spectra to the Hertzian impact shows zero frequencies shift towards lower frequencies. Using Eq. (7) gives indication of expected positions of frequency minimums. While Hertzian zero frequencies agree with McLaskey’s and Glaser’s [13] suggested positions \((f_{{\text {norm}}, N} = [1.75, 2.75, 3.75])\), zero frequencies for EHL impacts shift according to the simulation to 1.61 and 2.53. The third zero frequency cannot be identified, which can be a result of several causes, such as cavitation, signal-to-noise ratio or an uncertain defined contact time. Even though Zong et al. [14] demonstrate cavitation excites high frequencies, which would be in line with covering zero frequencies such as in this case for third order and higher orders, the uncertain defined contact time in the EHL case is the most plausible explanation for the covering of the zero frequencies of third order and higher, which is observed in both simulation and measurement. Signal-to-noise ratio can be eliminated, due to the fact that covering of the zero frequencies of third and higher order is present in the simulations.

Figure 6 shows the comparison of simulation (blue dashed line) and measured (yellow solid line) signals for a Hertzian impact. As an earlier investigation [13] showed, agreement is good and could be replicated for this investigation. This demonstrates that Hertzian measurement is a good reference measurement. A slight overestimation of the simulated signal can be seen, which is especially interesting if compared to Fig. 7 (comparison of EHL impacts). For EHL impacts, the simulation (red dashed line) underestimates the measured signal (green solid line). The overestimation in the Hertzian case is known [13], but the underestimation in the EHL case was not previously known. This underestimation becomes even more contradictory when the lubricant layer is taken into account. Instead of having an interface of steel and air, the interface on the upper part of the plate consists of steel and lubricant. This affects the reflection coefficient (8), which decreases due to an increase in density (\(\rho _2\) increases). A change of reflection coefficient was not taking into account by using Hsu’s script [17] for Green’s function. The amplitude of EHL impacts is not only higher in comparison with the amplitude of the simulated signal (Fig. 7), but also higher in comparison with the amplitude of a Hertzian impact. Figure 8 shows the spectra of the measured signals of both Hertzian impact (yellow dashed line) and EHL impact (green dashed line). Both impacts are caused by a steel ball \(({\varnothing }2\,\hbox {mm})\) dropped from identical height (350 mm) with the paraffin lubricant layer as the only difference. The difference in amplitude indicates a bigger change in momentum for the EHL impacts in comparison with the Hertzian impacts, which is in line with higher ball rebounds for Hertzian impacts observed during the tests. However, integration of the force functions suggest otherwise. While the impulse for the EHL impact (RB2 G20 and 350 mm drop height) is \(-151.4*10^{-6}\,\hbox {N}\,\hbox {s}\), the integration of the Hertzian force results in \(-173.9*10^{-6}\,\hbox {N}\,\hbox {s}\).

While amplitudes deviate between simulation and measurement, the effect of covering third- and higher-order zero frequencies predicted by the simulation is also observed for measured signals (Fig. 7). The position of the zero frequencies, however, does differ between simulated and measured signals. Instead of a shift towards lower frequencies, as suggested by the simulation (Fig. 5), zero frequencies of an EHL impact shift towards higher frequencies when compared to the Hertzian reference impact (Fig. 8). Therefore, a moving average of 50 measurements points in the frequency domain (blue and red line in Fig. 8) was used to identify the position of the zero frequencies within the noisy measurements.

This shift of zero frequencies is even more apparent in Fig. 9 where normalized zero frequencies for all test conditions of the infinite plate are shown. All frequencies, of both Hertzian impact and EHL impacts, are normalized by multiplication with the contact time \(t_c\) of the Hertzian reference case. The error bars represent the standard deviation for the four and the eight repetitions of Hertzian and EHL measurements, respectively. The transparent marker represents the Hertzian reference measurements. As found in McLaskey and Glaser [13], normalized zero frequencies of Hertzian reference measurements fit Reed’s [19] contact model well (\(f_{{\text {Zero}}, 1}=1.75\) and \(f_{{\text {Zero}}, 2}=2.75\)). EHL impacts (markers with solid fill) in comparison are without exception at higher normalized zero frequencies. The deviation between different test conditions (ball size and drop height) as well as between repetitions is larger than for the Hertzian reference measurements. This is most probably a result of the way in which the lubricant layer is prepared, because any deviation in initial film thickness is expected to influence the zero frequencies by influencing impact velocity. A significant difference between the lubricants used could, however, not be observed, even though signals of EHL impacts with the lubricant 80W90 (grey marker) tend to have lower normalized frequencies as EHL impacts with paraffin (black marker) especially for the first zero frequency.

Both amplitude and the shift of zero frequencies behave in a contradictory manner when comparing simulations and measurements. The difference in amplitude could be caused by the lubricant layer acting as a couplant during excitation. However, this is not an explanation for the mismatch of zero frequencies. Another approach to provide an explanation of the difference between simulated and measured spectra could be solidification of the lubricant layer and the nature of the EHL contact itself. As Fig. 10 shows, the elastic deformations are substantially different as regards an EHL and a Hertzian impact. Note must also be taken that the y-axis in Fig. 10 is not to scale and deformations therefore are highly emphasized, in order to show basic differences with and without lubricant. While the Hertzian impact (Fig. 10a) results in a uniform deformation of both plate and ball, the EHL impact (Fig. 10b) results in a deformation that is more localized due to the partly solidified lubricant (indicated by the colour scheme). Furthermore, the maximum deformation is larger in the EHL case compared to Hertzian impact for the center position, as was previously observed by Larsson and Höglund [18].

Simulations in this investigation, are based on a point assumption. A global force function is applied at a single point assuming the contact area is so small that it can be considered to be a single point. However, this assumption also implies that deformations within the contact area are uniform so that the global force function may be used, which might be true for the Hertzian impact, but is definitely violated by the EHL impact. The difference on a local scale could be the cause of both amplitude (higher maximum deformation for EHL impacts) and frequency shift (localized faster deformation changes). A possible solution for future investigations could be to use the Green’s function approach to transfer deformations within the system instead of global forces coupled to deformations by fully elastic models.

A previous study [16] showed that an elastic wave excited by Hertzian contacts in boundary restricted systems reduces to a resonance problem once wave and source interact. As Fig. 11 shows the difference in excitation force between EHL impact and Hertz impact does not influence the measured signal. With both excitations, identical resonances are triggered and the spectra are almost identical, except for an additional spike in between resonances for the Hertzian excitation.

A change in impact energy by changing ball size only influenced the amplitudes (Fig. 12) of the resonances, which is also in line with previous Hertzian investigations [16]. However, a change in ball size does influence the resonance cut-off, due to the change of contact time. For Hertzian impact, the −20dB cut-off and the first zero frequency agree to a high extent [16]. In the case of EHL impacts, it was found that resonance cut-offs occur much earlier. As Fig. 13 shows the Hertzian contact (blue dashed line) excites to a higher degree higher resonances in comparison with an EHL contact (red solid line). A reason for the earlier cut-off could be the damping of the EHL film when compared with the dry Hertzian impact.