Simulation of fault and clearance induced effects in rolling

Transcription

1 Simulation of fault and clearance induced effects in rolling element earings Tahsin Doguer, Jens Strackeljan Otto-von-Guericke-Universität Magdeurg, Fakultät für Maschinenau Institut für Mechanik, Universitätsplatz 2, Magdeurg, 39106, Germany Phone: , Astract Rolling element earings are vital to many rotating machines. The main aspect in condition monitoring of rotating machinery is fault detection. Early detection of faults is essential for the life time estimation. Virational signals contain the response information of a earing due to the fact that they are directly related to contact forces, which arise due to static, dynamic loads and contact/impact interactions etween rolling elements and raceways. In this paper we concentrate on fault induced virations and virations due to an oversized clearance in earing, which can also e considered as a fault. For the investigations a rolling element earing is modelled as a multi-ody system consisting of outer ring, inner ring, rolling elements and cage. Faults can e introduced as simple spherical shapes on raceways and on rolling elements, which allow the application of non-linear Herzian contact theory. Clearance is investigated in two categories. First one is in radial direction etween rolling element and raceways and second one is in tangential direction etween rolling element and cage. Variation of clearance results in change of load distriution on rolling elements and raceways. In the simulation contact force, displacement, velocity and acceleration of odies are calculated for each step of the time integration, so that induced virational signals can e estalished in time and in frequency domain. Comparison of amplitudes from earings with different clearances gives a good indicator aout the effects on virations. Furthermore relationship etween amplitudes and clearance can e otained and results can e used to estimate the life time of the earing. 1. Introduction In order to estimate the life time of a rolling-element earing the applied load, speed, geometrical and material properties can e considered as the main factors (1,2,3). Load distriution, stress, deflection and load-carrying capacity are directly influenced y the geometrical properties, and hence diametral clearance (1). Inadequate or inappropriate lurication may also influence greatly the life time, resulting in friction and excessive heat emission especially at high rotational speeds. Also the geometry of cage pocket could e important for the life time, due to redistriution of luricant in the pocket and luricant supply for elastohydrodynamic contact, which is essential for the film thickness (4,5,6). In this paper main attension is given to diametral clearance and rolling element cage interaction.

2 The rolling element earing model in this work was a 2-dimensional multi-ody system, consisting of inner ring, outer ring, rolling elements and cage. A more detailed description of the model is given in (7). The cage was modeled as force elements with spring and damper components, acting etween adjacent rolling elements. In the present work also the cage was modeled as a rigid ody with 3 degrees of freedom, in x -, y -direction and rotation around the centre of mass. Cage possessed ody-fixed pockets, in which the corresponding rolling elements were placed. Mass moment of inertia, forces due to rolling element-pocket-contact and momentum around the rotational axis were considered (8,9,10). Figure 1 shows the modeled odies, springs, dampers and switches for contact force and momentum in one of the translational directions. Inner ring Cage Rolling element Outer ring Figure 1. Model of earing components and contact switches ased on Kelvin-Voigt-Formulation Formulation of contact force ased on the non-linear Herzian contact theory and elastohydrodynamic (EHD) damping, which was proposed y Dietl for point formed rolling contact. Further details were explained in (7,11). 2. Theory 2.1 Diametral clearance Based on the geometry of single-row all earings and angular-contact earings diametral clearance c d can e defined as the greatest possile distance due to the displacement of one raceway with respect to the other in radial direction until contact etween rolling element and raceways occurs, see Figure 2. Simple geometrical relationship can e written using raceway and rolling element diameters ( d, d d ) (1) : c o i, d d d...(1) d o i 2 According to this definition the greatest possile displacement for the cage is c d / 4 and for the inner ring is 2 (c d / 4). 2

3 d o c d d i Figure 2. Diametral clearance c d The contact of the rolling element with the cage pocket can e separated in two zones. In the load zone the cage is driven y the rolling element, thus contact etween rolling element and pocket occurs on the side where translational velocity vector of rolling element is directed to. In unloaded zone, rolling element is driven y cage and the contact occurs on the opposite side (12). As the rolling element rolls into and out of the load zone the rolling radius varies with respect to centre of inertia, thus its angular velocity changes respectively. 2.2 Tangential clearance (pocket clearance) Clearance etween rolling element and cage pocket is defined as pocket clearance (12). Pockets were fixed on the cage, similar to the fault elements explained in (7). pocket cage l all pocket l all y r p r c t O x Figure 3. Pocket clearance c t The contact force in the pocket is calculated y the indentation and the indentation velocity. In Figure 3 the distance l etween centre of mass of the rolling element and centre of the pocket can e written as in (2), where r is the vector from the centre of inertia to the rolling element and r p the vector to the pocket respectively. Pocket clearance is controlled y c t and the relationship for can e determined as shown in (2). Velocity of pocket centre r is calculated as shown in (3), where r p ca is the vector from centre of inertia to centre of mass of the cage, r cap is the vector from centre of mass of the cage to the centre of the pocket. Angular ca and translational r ca velocities of the cage are calculated y a MATLAB-intern integrator and taken from the state space for each time step (7). rcap l r r p, 0, l ct...(2) l ct, l ct r r, r p ca rcap r ca...(3) p ca 3

4 Indentation velocity is the scalar relative velocity etween centres of pocket r and p rolling element r, which is calculated y the projection of r rel on the unit vector n l as shown in Figure 3. If the projection of r rel and n l fall in same direction is positive, and negative for opposite directions, which is important for the sign of damping force. n l r rp T, r rel r r p, r rel n l r r...(4) 2 n p The contact force and the momentum are calculated depending on the variales given in (5) and (6) for each time step. Further detail was given in (7). Zero moment is exerted to the rolling element, hence contact force passes through the centre of mass of the rolling element. F M 2.3 Cage load due to improper pocket clearance p f (,, nl ), F Fp...(5) T rcap Fp, M (0,0,0)...(6) p Angular velocity of the rolling elements around the centre of inertia was calculated, in order to show the effect of the load, which is exerted on the cage due to improper adjustment of c t as shown in Figure 12, on the right. In Figure 4, r is the same vector as defined in Section 2.2. n and t are the unit vectors, which can e written as in (7). y r t r n, O x Figure 4. Angular velocity of the rolling element around the centre of inertia Velocity of the rolling element r is taken from the state space for each time step. Scalar velocity of the rolling element in peripheral direction r t is otained via projection of r on t as shown in (8). Finally the angular velocity is simply the division of scalar velocity and distance. r t l n r, t T n r, cos( ) sin( ) T sin( ) cos( ) 0...(7)

5 r t T r t, 2 t r t r... (8) 3. Applications 3.1 Influence of diametral clearance on load distriution The variation of load distriution in a non-rotating deep-groove all earing (DGBB) of type 6309 under a static external load was investigated as the c d was varied from -90 to 90 µm with a step size of 10 µm. External load of 1000 N was applied on inner ring in vertical direction at 270 with respect to the centre of inertia and was kept constant for all calculated c d steps. Bearing contained eight rolling elements with an offset angle of 45. First rolling element was positioned at 270, so that it was on the same plane as the external load, thus the most heavily loaded one. Adjacent rolling elements numer 8 and numer 2 were positioned at 225 and 315 respectively. Also numer 5 at 90 in the unloaded zone was oserved. The simulation time was 0.2 s with a step size of 2-17 s. In Figure 5 the variation of normal force etween rolling element and outer ring for each of the 19 simulated c d step was shown. The x-axis was normalised y the maximum c d of 90 µm. Besides positive values of c d, pretension in earing was also simulated via negative values of c d. F N [N] Ball5,90 Ball8,225 Ball1, c d /c d,max Figure 5. Load distriution according to diametral clearance c d Load on the most heavily loaded rolling element, represented y the red curve with triangles, rised linearly as the c d was increased and approximated the value of exerted force on inner ring. At c d, max almost the whole load was carried y the rolling element numer 1. Black curve with circles represented the load on the rolling elements numer 8 and numer 2 and decreased continuously as the c d was increased. Blue curve with rectangles represented numer 5 and was at zero due to unloaded zone. In the pretension region the load on each rolling element rised independently from the position in the earing as the pretension was increased. The possiility of plastic deformation of contact partners due to relatively high values in pretension region was not considered in 5

6 the calculation of normal forces, which should e oviously considered in real applications via relationship etween contact pressure and yield stress (9). 3.2 Influence of diametral clearance on intact earing The outer ring response of an intact earing was oserved, in order to investigate the influence of diametral clearance c d. A deep-groove all earing (DGBB) of type 6309 was simulated, for which the asic dimensions can e found in the catalogue of manufacturer (13). Operating conditions were a rotating inner ring at 1000 rpm, a non-rotating constant radial load of 1000 N at 270 with respect to the centre of inertia and a non-rotating outer ring. Figure 6 shows the acceleration signal of outer ring. Comparision of maximum values of oth time signals on the left and right hand side shows a clear growth of amplitudes depending on c d. y (2) [m/s 2 ] Time [s] Figure 6. Outer ring acceleration time signal at diametral clearance (left) and c 50m (right) d Time [s] c d 0m On the left c d was set to zero and maximum amplitude was read as 0.06 m/s 2. On the right c d was set to 50 µm and maximum amplitude was read to e 0.6 m/s 2. The increase of amplitudes due to c d was determined as factor 10. Also a range of c d from zero to 90 µm with a step size of 10 µm was investigated, thus with increasing c d the increase of amplitudes was oserved, which was not shown in further detail in this work. 3.3 Influence of diametral clearance on a earing with a defected outer ring Considered a rolling element which strikes the faulty region on the earing raceway, the response of the ody to the impact can e expressed as the measured acceleration. The direct relationship etween acceleration and contact force was shown in (7). Moreover as shown in Figure 5, with increasing diametral clearance the load carried y the most heavily loaded rolling element increases. Therefore also a relationship etween the intensity of impulse and the diametral clearance can e expected. In order to investigate this statement, a series of simulations was carried out. An approach was proposed, 6

7 which aimed to deliver a method for feature extraction. The results were found to e in good agreement with this statement. In the following, the extraction of one feature to represent one c d step was explained. In order to otain features for a range, extraction process was applied for each c d step. Only the parameter c d was varied from 0 to 80 µm with a step size of 10 µm. All other parameters such as load, speed, fault size and fault position were kept unchanged. A fault was introduced on the outer ring raceway. The fault or the faulty region consists of so called circular fault elements. Properties of the faulty region, such as the length and the steepness can e controlled through the distance etween the first and the last element f, radius R f and numer n of the elements. The assumption of circular shape allows the use of the contact algorithm etween two aritrary circular, rigid odies and the application of Herzian equations for the calculation of contact force. Fault elements are ody-fixed and can e introduced on each ody in multi-ody system. Further details can e seen in (7). The same DGBB with the same operating conditions was chosen as explained in Section 3.2. For the definition of faulty region, following setup of the three parameters was used: f 0. 5 mm, R f 5 mm and n 3. Then the distance etween two adjacent fault elements was 0.25 mm. Fault was placed in the load zone at 270 on the outer ring raceway and was on the same plane as the constant load applied on inner ring. Acceleration of the outer ring in the load direction was shown in Figure 7. The signal depicted typical characteristics of a small fault on outer ring raceway with repeated impact responses at outer ring all pass frequency. Simulated time was 0.2 s. At the eginning of the signal a short part with a duration of aout 0.05 s was left unconsidered, in which the non-relevant effects vanished, that were caused y integration process. The used signal was aout 0.15 s of duration and the numer of contained impact responses was sufficient to otain a mean value, which depicted the character of the signal Figure 7. Partition of signal. Outer ring fault, diametral clearance c 50µm d 7

8 Local maxima of each impact response was otained y using a peak find function written in MATLAB, red dots in Figure 7. The signal was separated into five parts with equal lengths. Each part egan at a point, for which the corresponding local maximum served as reference. This point was called trigger point. Figure 8 shows one of the parts, on which FFT was performed. Trigger point was determined y moving the window with respect to local maximum on the time axis, red dot in Figure 8. Sampling frequency of the signal was 2 17 Hz and the constant window length for FFT was 2 11 points, which was equivalent to s and resolution of 64 Hz. The window positions and corresponding FFT were shown in Figure 8 and Figure 9. FFT 2 FFT 1 FFT 3 Figure 8. Zoom in second partition, determination of region for FFT The trigger point for FFT 1 was located 100 data points to the left from the local maximum (red dot in Figure 8), 100 points to the right for FFT 2, and 500 points to the right for FFT 3. Each FFT showed two peaks at 1530 Hz and 4806 Hz. As the window was moved parallel to the time axis the ratio of the two peaks changed as shown in Figure 9. The peak at the higher frequency vanished, the wider the window was moved to the right on time axis. The reason was, due to the higher contact damping d EHL (7,11) at Ns/m etween elements of the earing high frequency part decayed faster than the low frequency part, which possessed a lower structural damping at 16 Ns/m. The time signal in Figure 8 was in agreement with this statement. In Figure 9 the peak at 4806 Hz decreased rapidly from FFT 1 to FFT 3, which stated a lost of information aout the system response. FFT 1 was determined to contain the most information, thus position of trigger point of FFT 1 was used for the rest of the impact responses in the time signal. As in Figure 9, FFT 1 shown, each impact response delivered two peaks, which were oth natural frequencies of the earing, hence important in order to reach a conclusion for the feature extraction. Each peak was treated separately, due to the fact that at the considered stage it could not e determined which one of the two peaks was more significant for the feature extraction. Mean values of oth peaks over five impact responses were calculated separately. 8

9 Figure 9. FFT of defined regions in Figure 8. Variation of amplitudes depending on the chosen section in time signal. Acceleration in m/s 2, Frequency in khz For one time signal and one diametral clearance two mean values were calculated, which were shown in Figure 10. As mentioned at the eginning of this section, the proposed method was applied on a range of c d from 0 to 80 µm with a step size of 10 µm and each delivered two mean values at 1530 Hz and at 4806 Hz. Figure 10 exhiited a significant difference etween oth curves. In the whole c d range the amplitudes of the high frequency curve varied etween µm, which was equivalent to factor On the other hand variation of low frequency curve was etween µm, equivalent to factor 10. According to manufacturer catalogue c d values for DGBB 6309 for industrial applications are in a range of 10-50µm. Higher values could e considered as a fault. It could e concluded, that low frequency curve was more sensitive to changes of c d and therefore could e used for feature extraction. y (2) [m/s 2 ] Hz Hz c d [µm] Figure 10. Mean acceleration value for each diametral clearance c d step, and for two larger defects at 50m 3.4 Influence of defect size The size of fault on the outer ring raceway was varied to investigate the influence on the outer ring acceleration. Simulations were carried out for only two sizes of faults, while c d 9

10 all other parameters such as position of fault, load and speed were kept constant and were setup as explained in Section 3.2. Diametral clearance was set to 50 µm and was also constant as the fault size was varied. Acceleration values of 120 and 130 m/s 2 were otained for the listed fault parameters in Tale 1 and shown in Figure 10. Each one of these values were the mean value of the peaks at 1530 Hz shown in Figure 9, FFT 1. The rise of the acceleration was ovious as the fault ecame wider and steeper. Tale 1. Setup of fault elements Section f [mm] R f [mm] At this stage it should e noted, optically the structural difference etween time signals due to variation of fault size could only e detected y the comparision in large scale. In Figure 11 the time signal of the third configuration in Tale 1 was shown in analogy to Figure 8. Compared to Figure 8 the difference of signal structure at the eginning of the urst could e seen. n Figure 11. Zoom in second partition of the time signal. Outer ring fault, f 0.8mm, R 1mm, c 50µm 3.5 Pocket clearance in cage (pheriperal clearance) f In several works in literatur the assumption was made that the angular velocity of one rolling element around the centre of inertia was equal to the angular velocity of the cage. In this work, these were calculated separately considering mass moment of inertia and centrifugal forces. It could e useful to show the effects, which are caused y the forces that act on the cage in unloaded zone especially for high rotational speeds. d 10

11 [rad/s] Time [s] Time [s] Figure 12. Angular velocity of one rolling element around the centre of inertia. ct c d (left) and t cd c (right) In the simulations shown in Figure 12 only the sizes of c d and c t were varied, for which the operating conditions were as explained in Section 3.2. On the left c t was 50 µm and c d was 20 µm. Angular velocity decreased in load zone and increased in unloaded zone, due to the variation of rolling radius as shown in (8) and under the influence of constant vertical load. Contact etween rolling element and outer ring was not roken during the whole simulation due to the load and centifugal force. On the right c t was 5 µm and c d was 20 µm. The same sine form as in the first case could e oserved. Noticeale difference was the high frequent viration in the unloaded zone. The reason was, due to the clearance setup ( ct cd ) rolling element lost contact to inner and outer ring in unloaded zone and was driven only y the cage, which caused a displacement not only in peripheral, ut also in radial direction, due to centrifugal force. 4. Conclusions Load distriution on the rolling elements was effected y the position of the rolling elements in the earing. The influence of diametral clearance on load distriution was shown under a constant vertical radial load. It was found that diametral clearance and load distriution are inversely propotional. In other terms, the wider the diametral clearance the higher the load carried y the most heavily loaded rolling element. Moderate diametral clearance could e advantageous to avoid the pretension in the earing. It can e concluded that diametral clearance should e kept in moderate rates to allow an adequate load distriution on rolling elements. Acceleration signal of intact outer ring showed remarkale increase of amplitudes due to variation of diametral clearace. The same statement was not confirmed for a defected outer ring. Proposed method however showed a clear dependency etween acceleration and diametral clearance and also different fault sizes could e recognised while diametral clearance was kept unchanged. On the other hand pocket clearance played a compensating role in the unloaded zone for the considered earing setup and operating conditions in this paper, allowing the rolling elements lean and roll on outer ring raceway due to the centrifugal force. Improper adjustment of pocket clearance size could e especially 11

12 critical in high speed rolling element earings due to the reason that centrifugal force rises with the square of angular velocity. It can e concluded that pocket clearance could play a preventive role against onset of excessive load on cage and excitation of cage virations. Due to the critical effect of geometrical properties on load distriution, clearance could play an essential role on earing life. References 1. B J Hamrock, Machine Elements, WCB/McGraw-Hill, J Strackeljan and S Lahdelma, Adaptive Features in Condition Monitoring Systems, The sixth International Conference on Condition Monitoring and Machinery Failure Prevention Technologies, 22 th 25 th Juni, Dulin, pp , J Strackeljan and S Lahdelma, Smart Adaptive Monitoring and Diagnostic Systems In Proceedings of the 2 nd International Seminar on Maintenance, Condition Monitoring and Diagnostics. 28 th -29 th Septemer, Oulu, Finland, POHTO Pulications, pp , T Doguer and J Strackeljan, Viration Analysis using Time Domain Methods for the Detection of small Roller Bearing Defects, SIRM th International Conference on Virations in Rotating Machines, 23 th 25 th Feruary, Vienna, Austria, 2009, Paper ID T Doguer and J Strackeljan, New Time Domain Method for the Detection of Roller Bearing Defects, International Conference on Condition Monitoring and Machinery Failure Prevention Technologies, Edinurgh, pp , T Doguer, J Strackeljan and C Daniel, Nutzung von Mehrkörperdynamik programmen zur Simulation von Wälzlagerschäden, 7. Aachener Kolloquium für Instandhaltung, Diagnose und Anlagenüerwachung - AKIDA 2008, Germany, 18 th - 19 th Novemer, Aachen, Germany, T Doguer, J Strackeljan and P Tkachuk, Using a dynamic roller earing model under varying fault parameters, The sixth International Conference on Condition Monitoring and Machinery Failure Prevention Technologies, 22 th 25 th Juni, Dulin, pp , K H Hunt and F Crossley, Coefficient of Restitution Interpreted as Damping in Viroimpact, Journal of Applied Mechanics, Vol 97, pp , K L Johnson, Contact Mechanics, Camridge University Press, W J Stronge, Impact Mechanics, Camridge University Press, P Dietl, Damping and Stiffness Characteristics of Rolling Element Bearings, PhD thesis, Technische Universität Wien, Austria, K Hahn, Dynamik-Simulation von Wälzlagerkäfigen, PhD thesis, Technische Universität Kaiserslautern, Germany, FAG Bearing Handook,