A 3D-ball bearing model for simulation of axial load variations Petro Tkachuk and Jens Strackeljan Otto-von-Guericke-Universität Magdeburg, Fakultät für Maschinenbau Institut für Mechanik Universitätsplatz 2, 39106 Magdeburg, Germany phone: +49 0391 67 {12885}{18438} e-mail: {petro.tkachuk}{jens.strackeljan}@ovgu.de Abstract Ball bearings are widely applicable elements in various machineries and thus their performance and operation reliability are of very great importance. The vibration analysis is commonly used for diagnostic and condition monitoring of rotating systems and ball bearings separately. This paper represents a 3-dimensional dynamic model of a one or double-row ball bearing for fault simulation, which was created to investigate the influence of axial load variations on the vibration signal. The model should help to understand the effects that appear in a ball bearing under axial load. We know from experimental results that a slight variation in the contact conditions could result in a significant change of the exiting force signal and the measured acceleration. In some cases the change of the signal structure could lead to a wrong classification with standard techniques like envelope analysis. The paper shows these dependencies. 1. Introduction Ball bearings, as a potential source of vibration in rotating machinery systems are an object of numerous experimental researches and theoretical studies. Different effects that appear in ball bearings fault detection, such as false brinelling, nonlinear effects caused by force deformation relationships, effects caused by axial load variations, fault diagnostic in the case of combined axial and radial loads, the problem of early fault detection, determination of limits like smallest detectable fault and so on, are of a very great importance for the development of advanced condition monitoring systems. The conditions of permanent technical progress and constantly growing safety and reliability requirements on ball bearings create a necessity and interest in simulation of ball bearings. Simulation results may be very helpful in understanding the processes that are happening inside the ball bearing, because frequently experimental results appear to be time-consuming and expensive. A lot of works and researches have been devoted to modelling of bearings, particularly to bearings with localized defects and it s diagnostic. The main concepts of bearing and fault bearing simulation can be found in literature (1, 2, 3, 4, 5).The following paper is devoted to investigations of axial load influence on the behaviour and the vibration signal of ball bearings with localized defects using a 3-dimensional dynamic model of ball bearing. The represented model was developed on the base of the 2-dimensional ball bearing model shown in (1). The idea of fault simulation consists in description of
contact between a rolling element and two (or more) so-called fault elements. Those elements are circles, defined on one or more components (rings, rolling elements) of the system. A radius, amount and distance between fault elements determine the geometry (length, depth, width) of the defect. The algorithm of the model is realized in a self written multi body program in MATLAB. 2. Theory 2.1 Ball bearing model The represented model regards a ball bearing as a solid multi body system, which consists of rolling elements (balls), outer and inner rings, and cage. Every body of the system is connected with the beginning of the inertial system via spring-damperelements. The basic idea of the model for one of the ball bearings is shown in Figure 1 (4). The cage is modeled as a point-to-point force element. The condition of equal and constant distance between the centers of masses of neighbouring rolling elements determines this force. Outer ring Contact force Ball Contact force Inner ring co m a, J a c H m k, J k c H m i, J i c i d o d i Figure 1. The idea of a ball bearing model for one of the balls In order to investigate the dynamic properties of a ball bearing, considering the influence of axial load, each body of the represented model has 4 degrees of freedom: translational in x, y, z directions and rotational direction (rotation around z-axis). The differential equation of motion of the system is given through a second order differential equation. Initial conditions are given with a vector of displacement and velocity for each mass. For the numerical integration this equation is formulated in state space (see details in (1) ). 2.2 Contact model As a rule a contact model for dynamic simulation can be divided into two main parts: contact detection and calculation of contact forces. Requirements for the contact detection for dynamic simulation are very high, because of its considerable influence on the stability of solution and computing time. In the represented ball bearing model a socalled slicing technique (according to DIN ISO 281 (6) ) is used for contact calculation. The main idea of this technique is the subdivision of rolling elements into a number of independent slices. Though slicing technique is basically oriented for calculation of line contacts in roller bearings, it can be also applied for ball bearings. Point contacts have just to be substituted for equivalent line contacts (3). 2
2.2.1 Contact detection Each body of the system is divided into a number of slices n along the z-axis (Figure 2). Therefore contact detection and, if the contact has occurred, calculation of penetration Δs i [mm], takes place in x, y plane for each slice, correspondingly. The problem reduces to the contact detection between two circular bodies. The principal approach for contact detection between two circular bodies that was also used for this model is shown in (1). This approach has the advantage of high stability and allows avoiding time-consuming spatial contact detection. The examples of ball bearings models that can be simulated in the present stage of the software development are shown in Figure 3. y b sl z i-th slice of a ball D w α sl z s Δs i y F ri z F ai Figure 2. The main approach of slicing technique z x y Figure 3. a) radial ball bearing, b) double-row angular ball bearing Moreover, the described algorithm allows simulating diverse geometries of races, which are given through functions f (z, x) or f (z, y) and is compatible with the surface defect model (see 2.4). For most of the ball bearings the geometry of their races is a part of a torus or can be approximately given as it. 3
2.2.2 Contact force For a 3-dimensional ball bearing model a radial and axial component of the contact force have to be determined. The radial component of contact force F r [N] (scalar value) is calculated according to DIN ISO 281 (6), as the sum of contact forces for each slice F ri [N], correspondingly. n F F (1) r i1 A contact force F ri for each slice is being calculated as a sum of a non-linear elastic part F e [N] and a dissipative part F d [N]. 10 9 ri e d s i Ni ri F F F c s d v (2) c s 35948 Lel Dw Lel (3) n 2 Here c s is a contact stiffness of one slice [N/mm^10/9], D w diameter of a ball [mm], si penetration of one slice [mm], L el theoretical length of equivalent line contact [mm], d damping coefficient [N (s/m)], vni - relative velocity in contact normal direction [m/s]. The damping coefficient d is being used to describe the impact process of two solid bodies and is calculated according to (7). This coefficent depends on penetration si, contact stiffness c s, and material coefficient. 10 3 9 d cs si (4) 2 The axial components of contact force separately. Those depend on corresponding radial components F ri and angle between centers of mass of a ball and a slice (Figure 2). F ai [N] are calculated for each slice [rad] sl F F tan (4) ai ri sl Total axial force F a [N] is determined, as given in equation: n F F (5) a i1 In the next step the vector of radial and axial components has to be calculated via multiplication by unit vectors in contact normal direction n i, j and axial direction z 0, correspondingly. The unit vectors in contact normal direction n i, j are determined for each contact partner during contact detection algorithm (see (1) ). FR 1 Fr n1, FR2 Fr n2, n1 n2 (6) F F z, F F z (7) A1 a 0 A2 a 0 Here indexes 1 and 2 are associated with contact partners. ai 4
2.3 Friction force A scalar value of friction force F fi [N] is calculated according to the Coulomb friction law, for each slice separately. This value depends on contact force in contact normal direction F ri and friction coefficient R, which is a function of relative tangential velocity v [m/s]. T F v F i1 fi R T ri F f n F fi (8) Here F f is a total friction force [N]. The direction of friction force coincides with the direction of the relative tangential velocity v T. Therefore friction force vector is determined as: vt FF FF (9) v The total force vector could include contact, gravity, friction forces and external loads for each body of the system. 2.4 Surface defect simulation As already mentioned, the idea of fault simulation consists in the description of contact between a rolling element and two (or more) so-called fault elements. Those elements are circles, defined for each slice of one or more components (outer, inner rings and rolling elements) of the system. The fault model is body-fixed, thus it is applicable for each body of the system. The main approach of fault simulation on outer ring is shown in Figure 4.. T Figure 4. Surface defect model on outer ring 5
The main parameters which describe the fault geometry are angular position of the simulated fault f [rad], radius of fault elements R f [mm], fault width f bi [mm] and fault length in z-direction f lz [mm]. For each slice two fault elements are attached tangentially to the outer race. Fault width, which is a distance between those elements, can be defined as a function fbi F( zsl ) on the interval f lz. Varying those parameters allows modeling different types of defects, for example micro and macro defects, raw surfaces etc. 3. Results and discussion Figures 5a), b) show some experimental vibration signals of a vehicle wheel bearing with a small defect located on the outer ring. From this investigation acceleration measurements from different combinations of axial and radial load conditions were used as reference signals. The bearing was loaded for case a) with radial load of 5 kn and for case b) with radial load of 5 kn and axial load of 3 kn. Once the bearing is loaded with axial components, amplitudes of vibration signal are getting higher and the signal gets a typical fault structure. Simulation results may help to understand this effect Figure 5. Experimental vibration signal of ball bearing with outer ring defect (left) radial load 5 kn, (right) radial load 5 kn axial load 3 kn Figure 6. Simulated vibration signal of outer ring with defect (left) radial load 7 kn, (right) radial load 7 kn axial load 3 kn 6
For the verification of ball bearing model a vehicle wheel bearing with defect on outer ring was simulated. It is an angular double-raw ball bearing, which has 13 balls for each of the outer ring races with a diameter of 11.9 mm. A series of simulations were made in order to investigate the influence of axial load on the vibration signal. Corresponding simulated vibration signals are shown in Figure 6. In both cases periodical peaks are induced in the vibration signal of the system due to the contact between rolling element and faulty region. The first parts of both time signals are dominated by the influence of the initial conditions and have no physical background. However, the response of the axial loaded bearing model differs from the pure radial loaded model (the difference of acceleration amplitudes averages 30 %). The analysis of contact forces distribution and contact conditions can help to explain this effect. Figure 7. Distribution of normal contact forces in a ball bearing model under radial load of 7 kn Figure 8. Distribution of normal contact forces in a ball bearing model under radial load of 7 kn and axial load of 3 kn The distributions of contact forces in normal contact direction for the ball bearing model with defect on outer ring are shown in Figures 7 and 8. The inner ring is loaded with 7
constant external load of 7 kn in radial direction, and is driven with constant frequency of 30 Hz (1800 rpm). The defect is defined on one of the races of outer ring and is specified with the following values of geometrical parameters: radius of fault elements R is set to 0.3 mm, fault width f and fault length f are both set to 1.5 mm. Angular f bi position of the simulated fault f is set to -90, which means that fault is located in the middle of the loading zone. Those figures illustrate the influence of axial load on contact conditions between rolling element and outer ring in defected region. Under axial load the contact area shifts significantly relatively to pure radial loading. In addition, axial load have considerable influence on load distribution in a bearing. lz mm mm mm mm Displacement of the centre of mass of the ball z b [mm] Figure 9. Distribution of normal component of contact forces left) radial load of 7 kn, right) radial load 7 kn axial load 3 kn Figure 9 represents the distribution of the normal contact forces of each active slice for one of the balls to that point of time, when the rolling element is located in the middle of the loading zone (t = 0.017992 s). In spite of a slight displacement of the ball s center of mass, which is caused by the axial load (Δz b = 0.05 mm), the displacement of the contact zone is of the same order as fault dimensions (f bi =1.5 mm) and is equal to 0.9 mm. Besides, axial load results in growth of the normal component of the contact force (Figure 9). This leads to considerable variations of contact conditions and as a result to the different system responses in vibration signals. However, it has to be mentioned that axial loads do not always lead to an increase of amplitudes. It depends on contact conditions between rolling element and faulty region. The following example demonstrates this effect very good. Figure 10 shows the distribution of normal components of contact force for the ball bearing model, which is under radial load of 7 kn and axial load of -3 kn. The contact forces, as well as the contact area are considerably smaller in comparison to cases that are shown in the Figures 7 and 8. Figure 11 demonstrates the simulated vibration signal of the system. While the first part of the time signal, until 0.02 s is dominated by the initial condition, the following time signal shows no clear fault structure. Due to the unstable contact in faulty region one could not see periodical peaks in the induced vibrations. The value of normal 8
components of contact force, as well as the position and the size of contact area varies significantly with the simulation time, which leads to the varying of amplitudes in the vibration signal. Figure 10. Distribution of normal contact forces in a ball bearing model under radial load of 7 kn and axial load of -3 kn y (2) [m/s 2 ] 90 60 30 0-30 -60-90 0 0.02 0.04 0.06 Time [s] Figure 11. Simulated vibration signal of outer ring with defect radial load 7 kn and axial load -3 kn 4. Conclusion A 3D dynamic ball bearing model for fault simulation is presented in this paper. It allows to model ball bearings with different geometries of raceways. The fault model is applicable for each body of the system, so that defects of inner ring, outer ring and rolling elements can be simulated. The model shows the necessity of bearing modeling in three dimensions, especially of vehicle wheel bearings, which are frequently loaded with axial load forces. Experimentally obtained results, as well as simulation results, show the considerable influence of axial load on amplitudes in vibration signals, which is caused by varying contact conditions and load distribution in a ball bearing. The next 9
step is to improve the damping models by including the advanced lubrication algorithms, in order to get more realistic simulation data. Further research in this area seems to be of great importance for a better understanding of the signal structure of bearing faults and the definition of fault related features. Therefore the bearing diagnostic could benefit from this simulation. References 1. T Doguer, J Strackeljan and P Tkachuk, Using a dynamic roller bearing model under varying fault parameters, The sixth International Conference on Condition Monitoring and Machinery Failure Prevention-CM2009, Dublin, pp 907-918, June, 2009. 2. J Strackeljan, T Doguer and C Daniel, Nutzung von Mehrkörperdynamikprogrammen zur Simulation von Wälzlagerschäden, Proceedings AKIDA Konferenz, Aachen, Germany, 2008. 3. R Teutsch, Kontaktmodelle und Strategien zur Simulation von Wälzlagern und Wälzführungen,Dissertation, Technische Universität Kaiserslautern, 2005. 4. S Sassi, B Badri and M Thomas, A Numerical Model to Predict Damaged Bearing Vibrations, Journal of Vibration and Control, Vol 13, pp 1603-1628, 2007. 5. N Sawalhi, Diagnostics, prognostics and fault simulation for rolling element bearings, dissertation, University of New South Wales, 2007. 6. DIN ISO 281, Wälzlager, Beiblatt 4: Dynamische Tragzahlen und nominelle Lebensdauer, Deutsches Institut für Normung, 2003. 7. K Hunt and F Crossley, Coefficient of Restitution Interpreted as Damping in Vibroimpact, Transaction of the ASME, Journal of Applied Mechanics, pp 440-445, June, 1975. 10