TRANSPORT PROBLEMS PROBLEMY TRANSPORTU
2008 Tom 3 Zeszyt 3
Bogusław ŁAZARZ, Grzegorz PERUŃ*, Sławomir BUCKI Silesian University of Technology, Faculty of Transport Krasińskiego St. 8, 40-019 Katowice, Poland *Corresponding author. E-mail:
[email protected]
APPLICATION OF THE FINITE-ELEMENT METHOD FOR DETERMINING THE STIFFNESS OF ROLLING BEARINGS Summary. The paper presents the results of numerical tests performed with use of the FEM method, the aim of which was determining the stiffness of the outer raceway rolling element - inner raceway system of bearing 6307. The characterization obtained has been compared with a characterization determined with analytical methods and in the next stage, it will be used to determine the total stiffness of the bearing, variable in working time. Correct modelling of bearing stiffness is one of important conditions for obtaining correct results of simulation calculations. Obtained results will allow the determination of possibilities of limiting vibroactivity of toothed gears, commonly used in transport.
ZASTOSOWANIE METODY ELEMENTÓW SKOŃCZONYCH DO WYZNACZANIA SZTYWNOŚCI ŁOśYSK TOCZNYCH Streszczenie. W artykule przedstawiono wyniki przeprowadzonych badań numerycznych z uŜyciem metody MES, których celem było określenie sztywności układu bieŜnia zewnętrzna - element toczny - bieŜnia wewnętrzna łoŜyska 6307. Otrzymana charakterystyka została porównana z charakterystyką wyznaczoną metodami analitycznymi i w kolejnym etapie zostanie uŜyta do wyznaczenia całkowitej, zmiennej w czasie pracy, sztywności łoŜyska. Prawidłowe zamodelowanie sztywności łoŜysk, jest jednym z warunków uzyskania poprawnych wyników obliczeń symulacyjnych, prowadzonych w celu znalezienia moŜliwości ograniczenia wibroaktywności przekładni zębatych, powszechnie stosowanych w transporcie.
1. INTRODUCTION Applying the FEM method for modelling a system comprised of an inner and outer raceway of the bearing and the rolling element between them is the next step of works on the method of calculating stiffness of bearings. This method is being developed for the needs of a model stand for testing toothed gears operating in a circulating power system [1]. Correct modelling of bearing stiffness is one of important conditions for obtaining correct results of simulation calculations and so it will allow reduction of the number of laboratory tests. It is expected that the results obtained with use of the model developed will allow the determination of possibilities of limiting vibroactivity of toothed gears, being the major element of power transmission systems.
34
B. Łazarz, G. Peruń, S. Bucki
In the discussed calculation method, it was assumed that the bearing stiffness depends on the stiffness of the outer raceway - rolling element – inner raceway systems being under load [7]. These stiffnesses are nonlinear functions of load imposed on the rolling element, thereby dependent on its position in relation to the direction of the force. Such approach allows taking into consideration disfunctions of the vibration signal resulting from the changeable bearing stiffness caused by both, changes in the position of rolling elements in relation to the direction of the force and damage and wear and tear of the interacting bearing elements.
2. METHODS OF DETERMINING THE STIFFNESS OF BEARINGS Descriptions of many methods of calculating bearing stiffness can be found in professional literature. The ones more precise require much data concerning the elements of a bearing [2], which most often are not available in manufacturers’ catalogues. For this reason, approximate methods are used most frequently, which allow determining the values of bearing deformation, depending on the value of load imposed on the bearing. On the modelled stand, eight single-row ordinary ball bearings 6307 are mounted. Radial dislocation of the journal of such bearing, determined as a function of maximal load of the rolling part and of the rolling element’s diameter, is described by dependence (1) [2, 4]. If, additionally, the number of rolling elements is known, dependence (2) can be used [2, 3]. 2
δr =
3 0,44 ⋅ Qmax 1 3 t
D ⋅ cos α
,
[µm]
(1)
where: Qmax − maximal load of the rolling part [N], Dt – rolling element diameter [mm], α – bearing operation angle [rad].
δ r = 0,96 ⋅ 3
Q2 R , [µm] ; Q = 0,1 ⋅ d k 2e
(2)
where: dk – diameter of the bearing ball [mm], R − radial load of bearing [N], e – number of rolling elements in the bearing. For small values of the bearing operation angle, whose value is used in formula (1), considerable occurrence of stiffness characteristics can be obtained for a number of rolling elements equal to 5, however, the 6307 bearings installed on the stand, depending on the manufacturer, have 7 or 8 rolling elements. The method suggested in this paper [7] requires determining the number of rolling elements under load as well as the stiffness characterization of one system: outer raceway – rolling element - inner raceway. In order to determine the number of rolling elements under load, the knowledge is necessary concerning the size of radial clearance of the bearing (for the 6307, the adequate values were taken from [5]) and the load distribution angle, ψε – Table 1. In order to determine the load distribution on individual rolling elements and the maximal load δmax of the rolling part, a notion of the load distribution angle coefficient is introduced [4]:
εd =
δ max
2δ max
1 1 g = (1 − cosψ ε ) = 1 − + g 2 2δ max + g 2
(3)
Application of the finite element method for determining the stiffness of rolling bearings
35 Tab. 1
ψε [°] δmax 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
Load distribution angle ψε for various values of radial clearance g and deformation δmax of bearing 6307 C2 clearance [µ µm] normal clearance [µ µm] C3 clearance [µ µm] min max min max min max 1 11 6 20 15 33 87,3 69,2 76,7 60,0 64,6 51,5 88,6 77,5 82,5 70,5 74,2 63,1 89,1 81,1 84,8 75,5 78,5 69,2 89,3 83,1 86,0 78,5 80,9 73,0 89,4 84,3 86,8 80,4 82,5 75,6 85,2 87,3 81,8 83,6 77,5 89,5 89,6 85,8 87,6 82,8 84,4 79,0 89,6 86,3 87,9 83,6 85,1 80,2 89,7 86,7 88,2 84,3 85,6 81,1 87,0 88,3 84,8 86,0 81,9 89,7 89,7 87,3 88,5 85,2 86,3 82,5 89,8 87,5 88,6 85,6 86,6 83,1 89,8 87,7 88,7 85,9 86,9 83,5 89,8 87,8 88,8 86,2 87,1 83,9 89,8 88,0 88,9 86,4 87,3 84,3 89,8 88,1 88,9 86,6 87,4 84,6
It arises from here that the load distribution area, ψε , depends on the maximal deformation δmax and clearance g in the bearing [4]:
ψ ε = arccos(
g 2δ max + g
)
(4)
Load distribution on the rolling elements in the bearing, depending on the value of the load distribution angle is shown in Fig. 1 [4]. εd=1 ψε=180°
εd=0,5 ψε=90°
0
