dynamic behavior of ball bearings under axial vibration

Dynamic Behavior of Ball Bearings under Axial Vibration

Virgil Hinque* and René Seiler*

Abstract

The paper addresses the dynamics of ball bearings when exposed to vibration loads along their axis of rotation. Following common practice in space mechanisms design, the bearings are mounted in either hard preloaded or soft preloaded pairs. A computer-based model has been developed for the analysis and prediction of the load-deflection characteristics in bearing systems. Furthermore, the model may be used to quantify the maximum loads applied on the bearings and the resulting stresses during a vibration test or a spacecraft launch.

In parallel to the model development, an experimental test program has been carried out in order to get sufficient data for model correlation. In this context, the paper also elaborates on the post-processing of the acquired test signals and discusses specific effects, for instance nonlinearities due to the use of snubbers, in the time domain as well as in the frequency domain.

Introduction

Many space mechanisms use ball bearings for rotation functions. Therefore, assessing the bearing performance for the relevant environmental conditions is one of the typical challenges faced during the equipment design process. In this frame, it is common engineering practice to reduce the effect of a sine and random vibration environment to quasi-static equivalent loads and stresses. The relevant ball bearing systems often comprise two identical deep-groove or angular-contact bearings in an axially preloaded configuration. Several studies on the influence of the preload and other parameters on the structural behavior of such bearing assemblies have been done by the European Space Tribology Laboratory (ESTL). In a recent investigation, 25 ball bearing cartridges (“test units” or “bearing housings”) with different preload and snubber configurations were submitted to a series of sine and random vibration tests. The discussion of findings was mainly based on the analysis of frequency-domain data and bearing damage assessment via visual inspection [1].

The ESTL investigation inspired a number of ideas for continuation of the research, among others the development of a computer-based model that would be able to simulate the behavior of the bearing cartridges, especially those showing nonlinear features in their response. An adequate model should be able to predict the load transmission across the bearings in static and dynamic load situations. As the main sizing criterion for ball bearings is based on the allowable peak Hertzian contact pressure between the balls and the races [2], accurate knowledge of the maximum bearing loads is a key aspect for successful bearing selection and implementation in a space mechanism.

During the current investigation at the European Space Research and Technology Centre (ESTEC), a model was built using MATLAB®/Simulink®, with only the axial degree of freedom in a bearing taken into consideration. Because model correlation with real test results is of importance, a test program complementary to that reported in [1] has been conducted, with specific focus on the acquisition and interpretation of time-domain data. The following chapters describe the computer-based model, the design of the test units, as well as the details of the test campaign and corresponding results. The last part of the paper is dedicated to the comparison between the model output and the experimental test data.

* European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands

Proceedings of the 44th Aerospace Mechanisms Symposium, NASA Glenn Research Center, May 16-18, 2018

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Motivation and Background

Ball bearing systems in space mechanisms are usually composed of two angular contact or deep-groove bearings, which are preloaded along their axial direction in order to ensure a sufficiently high stiffness, the required precision of rotation, and a stable friction torque (within acceptable limits). The bearing preload can be applied using two different basic methods: hard or soft preload. In the hard preload method, a deflection is forced on the bearing pair by rigid mounting parts, corresponding to the desired preload magnitude (knowing the load-deflection characteristic of the bearings). The soft preload method relies on a compliant element, typically some spring-type component, to apply an axial force on the bearing pair corresponding to the controlled deflection of the compliant element (i.e., knowing its load-deflection curve). Both methods have their pros and cons [3] that will not be further detailed here for brevity. In fact, both methods are used in space mechanisms, and they imply different approaches for modelling and simulation. A schematic overview of both methods is given in Figure 1.

Figure 1. Hard and soft preload methods in ball bearing systems [3]

When an axial load is applied on a bearing pair (during launch or normal operation), it is generally shared between the two bearings, increasing the total load on one of them and gradually off-loading the other bearing. When a bearing gets completely off-loaded, the balls lose the controlled contact with the races, a phenomenon commonly known as gapping. This effect does not happen in the same way for the two different preload methods.

Hard preloaded bearings are pressed together with a displacement �� corresponding to the preload

magnitude on the load-deflection curve of an individual bearing. For reaching the onset of gapping in a bearing, it must be displaced by the same amount in the opposite direction, causing an axial load corresponding to 2 ∙ �� in the other bearing. Assuming a relationship between axial bearing load and

deflection according to Equation 1 (see e.g. [4]) and knowing that the entire load will then be carried by the remaining bearing, the off-loading or gapping force may be estimated with:

where � is a bearing specific stiffness coefficient. The static behavior of a hard-preloaded bearing pair can be visualized according to Figure 2. When gapping has occurred in either direction, the load-deflection curve follows that of an individual bearing. Both characteristics sum up in the operating range without gapping. Therefore, the linearized stiffness around the preload point (origin in Figure 2) can be approximated by twice the stiffness of a single bearing at the preload magnitude.

�� = � ∙ �2 ∙ �� = 2

�� ∙ � ∙ ��

�� = 2√2 �� ≈ 2.83 �� (1)

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Figure 2. Axial load-deflection characteristic and gapping points for hard preloaded bearings

Figure 3. Axial load-deflection characteristic and gapping points for soft preloaded bearings

Soft preloaded bearing pairs have a static and dynamic behavior very different from that of hard preloaded pairs. Their fundamental difference in stiffness for the two loading directions causes an asymmetric load-deflection characteristic. Furthermore, off-loaded races may experience considerable axial travel, independent from the rest of the bearing. Therefore, their motion should be taken into account as additional mass bodies in a dynamic model. In order to remain concise, the related equations of motion and further model details are not presented here. As a key effect, when a force equivalent to the preload magnitude is applied on the preload spring via the adjacent bearing, gapping occurs in the opposite bearing. If the force keeps increasing, the spring stiffness will dominate the load-deflection characteristic of the bearing system. Without gapping, the stiffness of the bearing opposite to the spring is dominating. Therefore, the linearized stiffness of the bearing system can be approximated by the stiffness of a single bearing at the preload magnitude. The load deflection curve of a soft preloaded bearing pair is presented in Figure 3. Hence, soft

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preloaded bearing pairs tend to be more compliant in general, and gapping occurs at load points lower than with equivalent bearing pairs under hard preload. Sometimes, snubbers are added to limit the stroke in the gapping range by adding a mechanical end-stop. The snubber effect can also be seen in Figure 3.

Modelling and Simulation Approach

In general, ball bearings have nonlinear load-deflection characteristics, with substantial asymmetries and possibly additional dynamic effects, e.g. due to snubbers, in the soft preloaded case. Therefore, predicting their dynamic behavior may be rather challenging. This has been the main reason to create a computer-based model for the investigation. The widely used software package MATLAB®/Simulink® was selected for modelling and simulation. At this stage of the research, the dynamic model only takes the axial degree of freedom of the bearings into account. In this context, a numerical solver integrates the equations of motion for the bearing system and generates results in the time domain. The load-deflection characteristics of the bearings are modelled on the basis of the bearing geometry, mounting and preload configuration as well as material parameters, referring to established bearing analysis, see for instance in [5]. Under quasi-static assumptions, they are pre-computed by an iterative numerical solver (based on the same algorithm as used in the ball bearing software tool CABARET [6]) before running a dynamic simulation, and they form the “backbone” of the model. Furthermore, other relevant phenomena, e.g. gapping and snubber contact, have been added to the dynamic model.

As part of the modelling and simulation process (see overall structure in Figure 4), the user can input any time-domain profile for housing acceleration (base excitation), equivalent to a shaker test. Alternatively, a force profile may be applied on the shaft, simulating a static tension/compression test. On this basis, the dynamic behavior of the individual bodies is computed, including the bearing loads. A separate post-processing module has been developed to extract additional values of interest like contact stresses or gapping distances.

Figure 4. Overall structure of the modelling and simulation approach

Experimental Test Set-up

In order to correlate the model results with real experimental data, a vibration test program has been carried out. Three different bearing cartridges have been used as test units. In fact, the same test articles had been used before in a related investigation led by ESTL [1]. All of them comprise two bearings preloaded in a back-to-back configuration. Their design is shown in Figure 5. On the top of the shaft, a dummy mass of 1.25 kg has been attached, with the option of increasing the total mass by an additional 0.625 kg. For the soft preloaded bearing cartridges, a set of Belleville washers (conical disc springs) at the bottom end of the shaft pushes on the inner ring of the lower bearing. The inner ring has a clearance fit, i.e. is allowed to slide along the shaft. Moreover, a snubber is added at the top of the bearing assembly. When touching a shoulder

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on the shaft, it prevents excessive displacements during gapping. The three test units differ from each other in terms of their preload characteristics, summarized in Table 1.

Figure 5. Cross section of the bearing cartridges (test units)

During the test program, the bearing cartridges were mounted on two different shakers, the first one for low amplitudes and the second for higher-level vibrations. The test units were only excited along the axial direction. One accelerometer (two for the stronger shaker) was used for input acceleration control and monitoring. Another accelerometer was placed on the dummy mass. Furthermore, three load cells were mounted in the adapter placed between the shaker and the test unit. Their purpose was to monitor the shaker input force by direct measurement. The phase of the force signals was also used to identify the direction of the structural modes observed, and to discriminate any non-axial modes. The test set-ups are depicted in Figure 6. In the left picture, a test unit with standard dummy mass is mounted on the small shaker. In the right picture, a test unit with additional dummy mass sits on the stronger shaker.

Table 1. Test unit preload parameters

Test ID Preload type Preload magnitude Preload stiffness

3 Soft 20 N 250 N/mm

6 Soft 20 N 900 N/mm

22 Hard 160 N (not applicable)

The test units have been exposed to a number of vibration tests: sine sweep, constant frequency sine with different amplitudes and random vibration.

Summary of Test Results

For all the tests, the time-domain signals of all sensors were recorded. The sine sweeps were carried out from 20 Hz to 2 kHz, with an input amplitude varying from 0.1 g to 1 g for the soft preload test units and from 1 g to 9 g for that with hard preload. The random tests ranged from 0.1 grms to 9 grms for all units. The constant frequency sines were applied at 5 different frequencies around the first main resonance of the test units (two below, one at the resonance frequency and two above). The amplitude ranged from 0.2 g to 5 g for the soft preload units and from 1 g to 58 g for the hard preload unit. Most of the tests were repeated using the additional dummy mass. A comparison of the low-level sine sweeps before and after the high-level runs confirmed that the dynamic response of the bearing cartridges was not affected by the tests.

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Figure 6. Test units mounted on the small and stronger shaker, including instrumentation

Discussion of Test Results

Frequency Domain A first overview of the results can be obtained by inspecting the estimated Power Spectral Densities (PSDs) of the output (response) acceleration, for random excitation (see Figure 7). For the hard preload test unit, the resonance frequency decreases slightly with increasing excitation magnitude. Some other features have been found in the PSD plots, but a closer analysis of the load cell phases indicated that those features resulted from a cross-coupling between a radial mode and the axial excitation. For the soft preload cases, it is found that the resonance frequency (and, hence, the apparent stiffness of the bearing system) starts decreasing with an increasing excitation magnitude. However, when the excitation becomes high enough to reach snubber contact, then the resonance frequency of the system starts rising again. Moreover, higher frequency content is introduced due to contact with the snubber.

Moreover, Figure 8 underlines that the quality factor and the resonance frequency are linked, as functions of the excitation level. Overall, the observations have confirmed the findings by ESTL derived from the previous test campaign [1].

Time Domain (for constant frequency sine vibration input) In the runs with constant frequency sine excitation, the increasing input levels were maintained over sufficiently long periods of time in order to reach steady state. Thanks to this, the measurement noise and other random components could be reduced by coherent averaging of the response signals. The averaged responses are shown for the hard preload case and one soft preload case (#3) in Figure 9. While the hard preload test unit behaves symmetrically, the acceleration asymmetry is clearly visible for the soft preload test unit.

At resonance, housing acceleration (input) and dummy mass and shaft acceleration (output) are shifted in phase. Therefore, the bearings undergo higher deflections. Double integration of input and output accelerations and subsequent evaluation of their difference allows to estimate the relative displacement between housing and dummy mass and shaft. Consequently, multiplying the dummy mass acceleration by the total moving mass (dummy mass plus shaft) results in a good estimate of the dynamic force applied on the shaft. This allows for a comparison with the theoretical (quasi-static) load-deflection curve presented in Figure 2. However, it should be kept in mind that the complete equation of motion comprises:

��̈�� = ��, �� − �{�̇��, �̇��, ��, ��, … } (2)

NASA/CP—2018-219887 88

where K is a generalized stiffness term considering the nonlinear load-deflection characteristic of a bearing system as described in Figure 2 and Figure 3, and L is a generalized loss term catering for viscous damping, contact interface friction and other dissipative effects. Thus, the loss term manifests itself in the form of hysteresis between the load-deflection curves for the two motion directions.

Figure 7. Output acceleration PSD for the hard preload unit (left) and soft preload unit #3 (right)

Figure 8. Quality factor and resonance frequency vs. excitation magnitude

The corresponding results are presented in Figure 10, including the modelled load-deflection curves. The hard preload unit reacts in an almost perfectly linearly form (also acknowledging the rather high preload). The hysteresis and, hence, the corresponding loss term is very small. The dummy mass and shaft load reaches ~1200 N, which is about 7.5 times the bearing preload. Therefore, considerable gapping is occurring in the bearings, however the transition to and from the gapping state appears entirely smooth.

Only one of the soft preload units (#3) is presented for brevity. As the main difference compared to the hard preload case, the load (and, hence, the stiffness) evolves in a much more nonlinear way over the entire deflection range, as expected. The bearing system enters the gapping state (= “compliant range” in spring compression direction) at approximately 50 N, which is around 2.5 times the preload. However, in the spring relaxation direction, the gapping state is left at around 20 N, i.e. at the preload. Such discrepancy may be explained by a combination of elastic spring force and viscous damping and interface friction forces at the Belleville washers. Evidently, a considerable amount of energy is dissipated in the gapping process and the interaction with the preload spring.

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Figure 9. Steady-state response of the hard preload unit (left) and soft preload unit #3 (right), constant frequency excitation with increasing amplitude

Figure 10. Quasi-static force vs. deflection for hard preload (left) and soft preload (#3, right)

Time Domain (for random vibration input) For the random vibration tests, the histograms of the input and output accelerations have been analyzed to infer on the underlying statistics. Figure 11 shows the histograms for the hard preload unit and increasing levels of excitation. The hard preload unit maintains a normally distributed output (no skewness) across all excitation levels applied, even when high peak amplitudes (and therefore gapping) are reached. The red curve represents a best-fit analytical model of a normal distribution.

Figure 12 shows the histograms for one of the soft preload test units. For low levels of excitation, any distortions (as identified via skewness) remain small. However, the response gets more and more skewed (asymmetric) with increasing input power. In this context, the left side of the histogram corresponds to the gapping state. For high-level excitation (see right histogram), two modes become evident as spiky features. They correspond to the compression and relaxation processes of the preload spring. Furthermore, when the shaft hits the snubber, the load increases rapidly, which may be noticed as the elevated left-side tail of the histogram. The right-side tail of the distribution corresponds to the stiff range of bearing compression. In fact, the highest bearing loads are reached in that region.

A common practice to approximate the response RMS level can be found with the Miles formula [7]. Using the resulting RMS value (standard deviation), a 3σ value can be calculated and used to size the bearings. Table 2 summarizes the number of dummy mass acceleration samples (and indirectly, the bearing forces)

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that are greater than the related 3σ value. For this purpose, the natural frequencies and quality factors for small levels of excitation have been applied in Miles formula. Because the 3σ rule is based on the fact that only 0.27% of the samples will be higher (normally distributed samples), it becomes evident that this rule is only valid for the hard preload unit, for the specific cases investigated.

The dummy mass acceleration can be related to the dummy mass load and, hence, to the bearing loads. On this basis, using the bearing design parameters, the model can deliver the Hertzian contact stress in the ball-race contacts (equal load distribution among all balls in axial direction). The maximum Hertzian contact pressure reached during in the tests (estimated via the maximum output acceleration) are compared to those obtained via the predicted 3σ values in Table 3. It can be seen that the extreme value events are significantly higher than what would be predicted with the Miles formula (even for the hard preload unit, although the predictions are closer).

Figure 11. Histograms of the output acceleration for random excitation 2.2, 4.5 and 8.9 grms

(hard preload unit)

Figure 12. Histograms of the output acceleration for random excitation 1.1, 2.2 and 4.5 grms

(soft preload unit #3)

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Table 2. Percentage of output acceleration samples exceeding the related 3σ value

Test unit ID

1.1 grms 1.6 grms 2.2 grms 4.5 grms 6.3 grms 8.9 grms

#3 (soft) 1.04% 0.62% 0.78% 1.54% 3.07% 5.51%

#6 (soft) 1.41% 1.92% 1.21% 1.70% 2.88% 2.64%

#22 (hard) 0.37% 0.37% 0.57% 0.26% 0.15% 0.29%

Table 3. Comparison of maximum contact pressures with 3σ equivalent values

Model Correlation

Ball Bearing Stiffness Around the preload point and for low levels of vibration, the stiffness of the bearing cartridges can be linearized. This “tangential” or “differential” stiffness (slope of the load-deflection curve around the origin) relates to the resonance frequency.

The load-deflection curves of a single bearing, as predicted using the model equations in [5] and derived from the test results, are compared in Figure 13 for the soft preload units (left) and for the hard preload unit (right). Several geometry parameters are used to compute the load-deflection characteristics. The one with the highest uncertainty (and the strongest effect on the stiffness) is the contact angle, linked to the radial clearance in deep groove bearings. A range from 5 to 20 degrees is indicated in Figure 13. As stated before, single bearings of the soft preload units can be represented and compared directly, just by plotting the non-gapping range and shifting it by the preload magnitude. Indeed, the stiffness of the single bearing dominates that range. The stiffness of a single bearing in a hard preload configuration is more difficult to extract from the test results. Nevertheless, a linearized approximation may be obtained by dividing the stiffness by two and shifting it by the preload magnitude and corresponding deflection.

Via above approach, the free contact angle of test unit #3 and #6 is estimated with approximately 13 degrees. The stiffness of the hard preload test unit is higher than expected with such free contact angle. At this stage of the investigation, two potential reasons are considered: The free contact angle might actually be higher than 13 degrees, or the preload magnitude might be higher than expected, keeping in mind that a hard preload is set by a small deflection with high precision, which is more sensitive to errors. Another interesting observation comprises the load-deflection characteristic for the soft preload cases: It can be fit with a function� = � ∙ ��, where the exponent � ≈ 2.1, which is higher than the 1.5 used to derive the rule of �� ≈ 2.83 ∙ �� . In order to correlate the hard preload results with the model, another

bearing model using a different contact angle and preload magnitude was assumed. Its characteristic corresponds to a model function with an exponent of 1.79, closer to 1.5.

Test unit ID

Max.Hertzian Pressure

[MPa]

1.1 grms 1.6 grms 2.2 grms 4.5 grms 6.3 grms 8.9 grms

#3 (soft)

3σ 940 1120 1188 1569 1693 1830 peak 1382 1486 1772 2240 2478 2595

#6 (soft)

3σ 975 1102 1299 1386 1605 1739 peak 1534 1661 1902 2049 2171 2563

#22 (hard)

3σ 1300 1483 1651 2104 2308 2543 peak 1541 1762 1949 2457 2613 2899

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Figure 13. Load-deflection characteristics of the test bearings vs. model output (left: hard preload unit, right: soft preload units)

Sine Sweeps Results A comparison between the Simulink® model output and the test results is made using the output of the sine sweeps runs. Figure 14 shows the resulting peak acceleration vs. excitation frequency. The peak amplitude is presented for both motion directions acknowledging the strong asymmetry in the soft preload cases.

Figure 14. Output acceleration magnitude vs. excitation frequency (left: hard preload unit, right: soft preload unit #3)

For both test units, the largest difference is found with the higher quality factor derived from the test data. This may be due the fact that the input acceleration was somewhat disturbed during the test runs because the shaker controller had some problems to keep the input level steady (possibly a sign of shaker saturation). For the hard preload unit, the system acts in a fairly linear way (left plot in Figure 14). Also for the soft preload unit, the overall evolution of the response is broadly similar. However, the nearly vertical rise at about 380 Hz and the pronounced asymmetry around the resonance for the highest excitation level (right plot in Figure 14) is typical of a non-linear transition in a vibrating system. In this particular case, it happens when the dummy mass acceleration is high enough to create gapping and to reach the spring compression range in the load-deflection characteristic.

In Figure 15, the deflection is plotted against the force on the shaft for the sine sweep with the highest amplitude. In the test results, the gapping range appears stiffer than predicted by the model. This was already found in Figure 10, and is probably caused by friction and other effects in the spring assembly. As

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a major element of the continued investigation, it will be attempted to measure the static load-deflection characteristic of the bearing cartridges with higher precision, for a more comprehensive model validation.

Figure 15. Load-deflection characteristics of the test units for the highest amplitude sine sweeps (left: hard preload unit; right: soft preload unit #3)

Constant Frequency Sine Vibration Input and Snubber Interaction In some of the constant frequency sine runs with the soft preload units, the deflection in the gapping range was large enough to hit the snubber. This interaction is also represented in the Simulink® model. Figure 16 shows the model output compared to the experimental results for test units #3 and #6.

Figure 16. Snubber interaction for the soft preload units (left: #3, right: #6)

The blue lines represent the model output while the green lines are the experimental results. Within the long and nearly horizontal stroke, it can be seen that the predicted stiffness of the spring matches the test data rather well. At the right end of the stroke, the snubber interaction is evident. However, the stiffness transition and evolution in the vibrations test results appears smoother than in the simulated output, an area for future model refinement.

In general, it can be noticed that the behavior of the bearing system is more complex when the snubber is hit. Concerning the maximum load reached for re-contact after gapping (left end of the graphs), the model shows good correlation. However, the load and stroke reached when hitting the snubber is not well predicted yet (right end of the graphs).

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Random Vibration Results Since the model will also accept a random time series as input, the signal acquired from the input accelerometer (on the shaker interface) can be directly used as model excitation for comparison of the results. This avoids any differences in the input due to shaker control, etc. The histograms of the resulting loads on the dummy mass are presented in Figure 17 and Figure 18.

For the hard preload case, the random vibration response of the model matches the test data rather closely, see Figure 17. Due to the skewness and relatively complex shape of the histograms for soft preload (Figure 18) a statistical interpretation tends to be more difficult than for the hard preload case. Nevertheless, theoverall shapes of the model response histograms follow those for the test response. For the highestexcitation (right plot in Figure 18), the two statistical modes, i.e. the two clearly separated spikes, arerepresented by only a single peak in the model response. This is explained by the fact that the hysteresisof the spring preload system is not fully represented in the model yet.

Figure 17. Histograms of the dummy mass loads for the hard preload unit (blue: derived from vibration test data, green: model output)

Figure 18. Histograms of the dummy mass loads for the soft preload unit #3 (blue: derived from vibration test data, green: model output)

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rms

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Conclusions

Relying on earlier work regarding the vibration response of preloaded ball bearings, the investigation has focused on the dynamic behavior of simplified bearing systems. In this conjunction, a dynamic model for the axial degree of freedom has been developed in MATLAB®/Simulink®. As a key capacity beyond predicting the load-deflection characteristics of bearing systems, it produces a numerical solution to the dynamic problem. On top of this, a series of vibration tests has been performed on three sample bearing cartridges. For validation of the computer-based model, special attention has been given to the post-processing of the test results in the frequency domain as well as in the time domain.

The main findings of the investigation encompass the following aspects:

• In general, the behavior of hard preloaded bearings appears more linear and predictable with higherconfidence than that of soft preloaded bearings. Snubber interactions in conjunction with gappingand hysteresis in soft preload systems are challenging details, which require further work.

• The model-based prediction of load-deflection characteristics relies on an accurate knowledge ofthe most relevant bearing design parameters (like the free contact angle), which appears difficultto obtain for individual bearing samples without dedicated measurements (among others, becauseof production tolerances).

• Performing static load tests to infer on the load-deflection characteristics of bearing systems maybe helpful, however getting reliable results from static tests appears more difficult than expected(potential problems with test rig stiffness, sensor precision and parasitic effects).

Acknowledgments

In the frame of the ongoing research project, the authors gratefully acknowledge the cooperation and support by Grant Munro and Simon Lewis of the European Space Tribology Laboratory (ESTL, ESR Technology). Furthermore, the authors would like to thank Ronan Le Letty and Lionel Gaillard (ESTEC Mechanisms Section) for their encouragement and help, specifically regarding the in-house tests and analysis work.

References

1. Munro, G., Checkley, M., Forshaw, T., Seiler, R. "Preloaded Bearing Characteristics under AxialVibration." Proceedings of the 15th European Space Mechanisms and Tribology Symposium(ESMATS 2013), September 2013.

2. European Cooperation of Space Standardization (ECSS) “Space Engineering: Mechanisms.”ECSS-E-ST-33-01C, Rev. 1, ESA, February 2017.

3. Videira, E., Lebreton, C., Gaillard, L., Lewis, S.D., “Guidelines for space mechanism ball bearingdesign assembly and preloading operations.” ADR, ESA and ESTL, September 2013.

4. Guay, P., Frikha, A. "Ball Bearing Stiffness. A New Approach Offering Analytical Expressions."Proceedings of the 16th European Space Mechanisms and Tribology Symposium (ESMATS2015), September 2015.

5. Harris, T. A. “Rolling Bearing Analysis.” Second edition, John Wiley and Sons, 1984.

6. Lewis, S. D. "Advanced bearing technology - Analysis and modelling report.” ESTL/TM/119,European Space Tribology Laboratory, March 1991.

7. Sarafin, T. P. “Spacecraft Structures and Mechanisms.” Springer, 1995.

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dynamic behavior of ball bearings under axial vibration

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