Theoretical and Experimental Research on the Floating-Sleeve Bearing Applied in Roller-Cone Bit

In oil and gas drilling, the roller-cone bit and the hybrid bit configured with roller cones are two commonly used rock-breaking tools; however, service lives of the bits are limited by the sliding bearings therein. In order to improve the performance and reliability of the bearing in the roller cone, a novel bearing with a floating sleeve is researched in this paper. Firstly, motion characteristics of the bearing are researched, and motion relations between the elements in the bearing are analyzed, respectively.*en, on the basis of the motion relations, threemotion statuses of the float sleeve are clarified, including the sleeve start to rotate, the sleeve rotates stably with the pushing force exerted by the rollers, and the sleeve rotates stably with the opposing force exerted by the rollers. At last, bearing performances in three different conditions are respectively tested in the experiment.*e results, with the specific speed value being in the range of 0.375∼0.833 and the average value in 0.613∼0.618, show that the sleeves with 6∼12 rollers are able to start and rotate stably under 5 kN radial load, indicating that the sleeves can be started to rotate and work stably, whereas, for the sleeve with less than 4 rollers, the specific speed value varies in the large range of 0∼1.176, indicating that this sleeve cannot rotate stably in the bearing.*is research revealed that the amount of rollers is an important factor that determines whether the sleeve is able to float stably in the bearing; therefore, roller quantity should be focused in the bearing design to optimize its performance.


Introduction
As an effective rock-breaking tool, the roller-cone bit once was the most commonly used in oil and gas exploitation.Since the service life of the roller-cone bit is limited by its vulnerable bearings, it is gradually replaced by the polycrystalline diamond compact bit and other new drill bits.However, considering that the roller-cone bit still has advantages in drilling in hard and brittle formation, in this paper, a bearing with the floating sleeve is introduced into the roller-cone bit to improve the performance and prolong the service life of the bit.
In the situation with high speed and light load, many research achievements on performance of the bearing with the floating sleeve were conducted by researchers in this field.Porzig et al. conducted a thermal analysis on the high-speed floating-sleeve bearing, discussing the influence of thermal boundary condition on the feature parameters of the bearing, and revealing the importance of temperature for the bearing performance [1].Koutsovasilis et al. researched the subsynchronous vibration of the turbine rotor configured with a floating-sleeve bearing and discussed the influence of vibration on the rotor and the bearing in high-speed condition [2].Tamunodukobipi et al. researched the dynamic performance of the floatingsleeve bearing.ey found that, with the rotation control mechanism configured with a special lubrication system, damping capacity of the bearing and dynamic performance of the rotor were significantly improved in the rotation speed of 10000 r/min [3].Chasalevris conducted a research on the operating characteristics of the bearing under high-speed condition and put forward a precise analytic method for the bearing with cylindrical floating sleeve of finite length, of which the result was in good accordance with the numerical solution [4].Guo et al. researched the lubrication of the floating-sleeve bearing in a turbocharger under a high-speed and light-load condition and put forward a new method to lubricate the bearing by applying an external pressure, which successfully reduced the friction and improved the stability of the bearing [5].Kang et al. researched the operation mechanism of the floating sleeve in the bearing system.On basis of the Hirs bulk flow theory and Moody friction factor, with numerical simulation in the rotation speed of 5000 r/min, they found that the floating sleeve could significantly diminish the nonlinearity of oil film stiffness and damping factor that caused by the large amount of unbalance [6].Zhang et al. conducted a theoretical analysis on the bearing in poor lubricated condition.ey found that, when the angular speed of the shaft was lower than 19098 r/ min, leakage rate of the outer oil film increased as the shaft rotated faster, while when higher than 19098 r/min, the rate remained stable.e phenomenon indicates that the floatingsleeve bearing is suitable for high-speed situation [7].Zhang et al. researched the floating-sleeve bearing in a rotor charger and discussed the influence of system parameters (including the eccentric mass, lubricating oil viscosity, and structure parameters of the sleeve) on the speed ratio and whirl ratio of the sleeve in the speed range of 20000 r/min∼150000 r/min [8].Pei et al. conducted a research on the lubricating property of the floating sleeve.ey found that relation between the speed ratio of the sleeve and the rotation speed of the bearing was notably nonlinear.Specifically, as the speed increased, the speed ratio would firstly rise rapidly and then fall gradually, indicating that there was an optimum speed range for the sleeve [9].Li et al. researched the lubricating property of the floating sleeve in high-speed and light-load condition and mainly discussed the influence of thermal elastic deformation on the lubricating property of the floating sleeve [10].
rough the survey and analysis on related research results, the authors find that the bearing with the floating sleeve is already widely used in many high-speed and lightload areas and that a lot of impressive breakthroughs have been achieved.However, since the roller-cone bit works under a typical low-speed and heavy-load situation (with the speed in 60 r/min∼120 r/min and the load over 40 kN) where the floating sleeve can hardly be floated, the bearing with the floating sleeve cannot be directly utilized in the bit.Besides, since seldom research has been conducted in this condition, the service life of the bearing in the roller-cone bit is still limited by its bearings.
In order to solve the problem that the floating sleeve cannot stably float under low-speed and heavy-load situation, a novel bearing with a force-to-float sleeve (hereinafter referred to as the floating sleeve or simply the sleeve) is put forward, wherein the floating sleeve is motivated to stably rotate by the rollers inside of it.Generally, as illustrated in Figure 1, the bearing comprises an inner rotating body (i.e., the shaft journal), an outer rotating body (i.e., the bushing), a floating sleeve configured in-between, and a set of cylindrical rollers configured inside of the sleeve.It should be noted that diameter of each roller is slightly larger than the thickness of the sleeve, so that when the inner and outer rotating bodies rotate relative to each other, the rollers will be driven by the friction force and rotate between the cylindrical surfaces of the inner and outer rotating bodies; accordingly, the sleeve will be pushed by the rollers to rotate along the axis of the bearing.us, the sleeve is forced to rotate stably between the two rotating bodies.

Motion Relations between the Elements of the Bearing.
According to the actual motion status of the bearing in a roller-cone bit, the journal is set as fixed, and the rotating direction is ordered as the clockwise being positive while otherwise negative.Accordingly, the motion of the elements is illustrated in Figure 2.
As shown in Figure 2, as some loads are applied on the bearing, the bushing will rotate along the bearing axis with the speed n 1 .Taking geometric parameters of all the elements as known, the linear speed v 1 of the contact point between the roller and the bushing should be where n 1 is the rotating speed of the bushing and D 1 is the inner diameter of the bushing.Apparently, angular rotation speed of the roller can be derived as follows: where D 3 is the mean roller diameter.erefore, rotating speed of the roller is According to the motion relations of the elements, motion of the roller in one of its rotation period is illustrated as Figure 3. Specifically, the roller's rotation period is T � 1/n 3 ′ , and circumference of the roller cylinder is C � πD 3 .Since there is no relative slip between the bushing and the roller, the arc length of bushing rotation in one of the roller's rotation period is twice the circumference of the roller cylinder; as a result, the rotation angle of the bushing can be derived as θ 1 � 2π((2πD 3 )/(πD 1 )) � 4πD 3 /D 1 .Accordingly, the revolution angle of the roller in one of its rotation period should be On the basis of the above equation, revolving (along the bearing axis) speed of the roller is Since rotation of the sleeve is motivated by the rollers, angular speed of the sleeve should be the same as the angular speed of the roller revolution, that is, where ω R ′ is the angular speed of revolution of the roller.So, rotating speed of the floating sleeve is 2 Shock and Vibration erefore, linear speed at the medium diameter of the sleeve should be where D 2 is the medium diameter of the oating sleeve.Further, when both the bushing and the shaft journal rotate along the bearing axis, angular speed of the oating sleeve is where ω i is the angular speed of the journal shaft and ω o is the angular speed of the bushing.
And angular speed of rotation (along its own axis) of the roller is

Analysis on the Motion of the Floating Sleeve.
As the key element in the novel bearing, stable rotation of the sleeve is of great importance for the performance of the bearing.In the situation that the bushing is xed, motion of the sleeve can be divided into three statuses, which are (a) the sleeve start to rotate, (b) the sleeve rotates stably with the pushing  force exerted by the rollers, and (c) the sleeve rotates stably with the opposing force exerted by the rollers.
(a) When the sleeve starts to rotate, the bushing, which is motivated by some applied loads, will rotate along the bearing axis with the angular speed ω o .On the other hand, once the bushing starts to rotate, it will contact with the sleeve and exert a torque on the sleeve as the driving moment M 1 .Meanwhile, the sleeve will inevitably contact with the shaft journal during its motion, so that it will suffer a resisting moment M 2 .At the same time, the roller will revolve along the bearing axis with an angular speed of ω R ′ and will exert another driving moment M g on the sleeve.It should be noted that, when the roller contacts with the sleeve body during the motion, friction will be generated between the roller and the roller slot in the sleeve, but it is not considered in this paper since the friction is quite slight.On the basis of the above analysis, whether rotation of the sleeve can be started or not is determined by the resultant moment of M 1 , M 2 , and the moment exerted by all rollers  M g , that is, where M f is the resultant moment on the floating sleeve, M 1 is the moment on the floating sleeve exerted by the bushing, M 2 is the moment on the floating sleeve exerted by the shaft journal, and M g is the moment on the floating sleeve exerted by the roller.
Apparently, if M f > 0, sleeve rotation will be started or accelerated and if M f ≤ 0, the rotation cannot be started.(b) e sleeve rotates stably with the pushing force exerted by the rollers.Having been stated, the rotation of the sleeve will be gradually accelerated with the moment M f , and accordingly, the angular speed ω f of it will be increased.As the angular speed of the sleeve grows, driving moment  M g exerted by the rollers, as well as the moment M 1 exerted by the bushing, will decrease, while the resisting moment M 2 from the journal will increase, which means the resultant driving moment M f as well as the angular acceleration will gradually decrease.Until the angular speed ω f of the sleeve reaches the roller's angular revolving speed ω R ′ , the resultant moment M f will nearly be equal to 0; thus, rotation of the sleeve is stabilized.(c) e sleeve rotates stably with the opposing force exerted by the rollers.With the motivation of the resultant moment M f , rotation of the sleeve will accelerate.If the angular speed ω f of the sleeve exceeds the roller's angular revolving speed ω R ′ , moment  M g exerted by the rollers will turn from a driving moment into a resisting one; thus, equation (11) should be changed as follows: According to equation ( 12), with resisting moments M 2 and  M g , the resultant moment of the sleeve will be negative, which means the sleeve will be decelerated.Until ω f approaches ω R ′ , M f will be approximately equal to 0; then, rotation of the sleeve is stabilized.

Experimental Study on Motion of the Floating Sleeve
3.1.Mechanism of the Experiment Platform.On basis of the bearing structure in an 8 1/2 in.tricone roller bit, an experiment platform with a simplified bearing (as the test sample) is designed and manufactured, of which the structure diagram is shown in Figure 4.When the experiment is started, the radial loading module will firstly exert radial load on the bearing, and then, the power module will input a driving load (a torque to motivate the rotation) on the shaft.During the operation, sensor 1 will record the rotation speed of the floating sleeve and sensor 2 will record shaft speed; meanwhile, sensor 3 will record the radial load on the bearing.Considering the actual working condition of the bearing in a roller-cone bit, operating parameters in the experiment are determined as thickness of the sleeve (represented as the inner and outer diameter), diameter of the roller cylinder, radial load on the bearing, and numbers of the rollers.Accordingly, the specific parameters are listed in Table 1, wherein three sets of experiments are planned, and three groups of parameters are set, respectively.

Apparatus and Samples.
e main apparatus used in this experiment is the universal material tester, which is hydraulic driven and is able to supply a maximum pressure of 60t, as illustrated in Figure 5. e tester mainly consists of two parts, i.e., the electrohydraulic part (as illustrated in the left of Figure 5) and the tension-compression part (as illustrated in the right of Figure 5).Specifically, the electrohydraulic part comprises a power unit, a throttle circuit, an electronic control unit, and an oil storage tank, and the tension-compression part comprises a hydraulic circuit, a hydraulic cylinder, a set of material fixture, and a tensioncompression gear.
According to the actual structure parameters of the bearing in an 8 1/2 in.tricone bit, the apparatus that needs to be designed and manufactured in this experiment including the bearing samples and its auxiliary holder is shown in Figure 6, wherein the holder consists of a bearing support, two rolling bearings, and a crank handle, as shown in Figure 7.
e bearing sample as shown in Figure 8 comprises a bushing (Figure 8 In this research, experiments of the bearing samples are respectively conducted in three different operating conditions, and structure parameters of each bearing are listed in Table 2.

4
Shock and Vibration

Scale Plates and Scale
Reading.Since neither of the sensors 1 and 2 can record the rotation data in real time, a camera is used to record the rotation in the experiment, and three scale plates are designed and applied on the bearing samples.
(1) Scale plate on the bushing: circumference of the bushing is uniformly divided into 72 sectors and marked, as shown in Figure 9.
(2) Scale plate on the shaft: in accordance with the circumference of the shaft (the smaller end), a scale plate marked with scale lines every 10 mm is designed.To scale the shaft, the scale plate is pasted on the cylindrical surface of the shaft end, as shown in Figure 10.When the experiment is finished, the rotation angle of the bushing will be read according to the angular position recorded in the video.
(3) Scale plate on the sleeve: since rotation speed of the sleeve is the most important parameter, scale plate and reading of the sleeve must be carefully treated.Specifically, circumference of the sleeve is equally divided into 8 sectors and is marked with three colors, as shown in Figure 11.When the experiment is finished, the rotation angle of the sleeve will be read according to the angular position of the scale lines by reference to the scale lines on the bushing.

Results and Analysis.
In this experiment, the input torque is supplied by human hands, while the radial load comes from the universal material tester.During the rotating process of the bearing, a professional camera is utilized and is set at 30 fps to record the angular positions of the rotatable elements.Until the experiment is finished, the video will be exploded into a series of high quality images with each image showing the bearing status in one frame.By comparing the angular positions of the journal and the sleeve in different frames, the angular speeds ω i (of the shaft) and ω f (of the sleeve) can be calculated.Some of the images extracted from the video are shown in Figure 12.
According to the operating condition of the bearing, the bushing stays stationary, i.e., ω o � 0 rad/s.On the other hand, medium diameter of the sleeve D 2 measures 55.099 mm, and average outer diameter of the rollers D 3 measures 4.03 mm.Taking these two diameters into equation ( 9), angular speed of the sleeve is calculated as ω f ≈ 0.463ω i .
In one group of the experiment where the radial load is 5 kN and 12 rollers are configured, experimental data in 10 seconds of the rotation are extracted and processed;  6 Shock and Vibration consequently, angular speed variations of both the journal and the sleeve are achieved, as shown in Figure 13.e curves in Figure 13 show that angular speed variation trend of the shaft (shaft journal) is in good accordance with that of the sleeve (the similarity degree is more than 90%), indicating that rotation of the sleeve can be successfully started driven by the rotating shaft and that the sleeve is able to rotate stably with the pushing force (driving moment) exerted by the 12 rollers.
On the basis of the angular speeds of the two elements, the speci c angular speed value between the sleeve and the shaft ω f /ω i is calculated, as shown in Figure 14, ω f /ω i is in the range of 0.375∼0.833.Notably, the average value of ω f /ω i substantially stays around 0.6, and precise calculation shows that the value (i.e., the rotation speed ratio) is 0.613.On the other hand, theoretical calculation before the experiment shows that this ratio is 0.463, indicating that, with 12 rollers and under the radial load of 5 kN, the di erence between the experimental and theoretical value is around 24%.
In another group of the experiment where the radial load is 5 kN while 6 rollers are con gured, similarly, experimental data in 10 seconds of the rotation are extracted and processed, and the angular speed variations of both the shaft journal and the sleeve are achieved, as shown in Figure 15.
Similarly, the curves in Figure 15 show that the angular speed variation trend of the shaft (shaft journal) is also in good accordance with that of the sleeve (the similarity degree is more than 90%), indicating that rotation of the sleeve can

Shock and Vibration 7
be successfully started driven by the rotating shaft and that the sleeve is able to rotate stably with the pushing force (driving moment) exerted by the 6 rollers.
Similarly, the specific angular speed value between the sleeve and the shaft ω f /ω i is calculated, as shown in Figure 16, ω f /ω i is in the range of 0.400∼0.800.Notably, the average value of ω f /ω i substantially stays around 0.6, and precise calculation shows that the value (i.e., the rotation speed ratio) is 0.618.By reference to the theoretical value 0.463, the difference between the experimental and theoretical value is around 25%.
In the third group of the experiment where the radial load is 5 kN while 4 rollers are configured, experimental data in 10 seconds of the rotation are extracted and processed, and the angular speed variations of both the shaft journal and the sleeve are achieved, as shown in Figure 17. e curves in Figure 17 show that the angular speed variation trend of the shaft (shaft journal) is substantially in accordance with that of the sleeve, but at some of the time points (such as 1.83 s, 3.50 s, and 8.8 s), the speed trend of the two elements is quite different.Besides, in most of the time period, speed of the sleeve nearly equals to that of the shaft, meaning that a stable speed difference between these two elements cannot be achieved, so that the sleeve is not able to float stably in the bearing.Apparently, the sleeve with 4 rollers can neither function effectively to reduce the friction nor prolong the service life of the bearing.
Based on the angular speeds of the two elements, ω f /ω i is calculated as shown in Figure 18, wherein ω f /ω i lies in a large range of 0.0∼1.176.Calculation result shows that the specific value ω f /ω i is quite unstable and that speed fluctuation of the sleeve is too dramatic to keep a stable rotation.

Conclusions
In this paper, performance of the floating-sleeve bearing is researched with the indoor experiment, and results and conclusions of this research are as follows: (a) eoretical analysis on the floating-sleeve bearing is conducted, and the theoretical calculating equations for both the sleeve and the shaft are derived.Besides,   results, with the speci c speed value being in the range of 0.375∼0.833and the average value in 0.613∼0.618,show that both the sleeves with 12 and 6 rollers are able to start and rotate stably under 5 kN radial load, indicating that the sleeves can be started to rotate and work stably in these operating conditions.(c) Results comparison of the three groups show that both the sleeves with 12 and 6 rollers are able to start and rotate stably, but for the sleeve with 4 rollers, the speci c speed value varies in a large range of 0∼1.176, indicating that this sleeve cannot rotate stably in the bearing.erefore, it can be concluded that the amount of rollers is an important factor that determines whether the sleeve is able to oat stably in the bearing; therefore, roller quantity should be focused in the design of the bearing to optimize the bearing performance.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no con icts of interest.

Figure 2 :Figure 3 :
Figure 2: Motion relations of the bearing elements.

Figure 4 :
Figure 4: Diagram of the experiment platform for the floating-sleeve bearing.e numbered parts are as follows: 1: rotation speed sensor for the floating sleeve, 2: rotation speed sensor for the power input shaft, and 3: radial load sensor.

Figure 9 :
Figure 9: Scale plate on the bushing.

Figure 11 :
Figure 11: Scale plate on the sleeve.

Figure 10 :
Figure 10: Scale plate on the shaft.

Figure 12 :Figure 13 :Figure 14 :
Figure 12: Some of the images extracted from the video.