Integrated Non-linear FE Module for Rolling Bearing Analysis: Reprint ...
Integrated Non-linear FE Module for Rolling Bearing Analysis Dipl.-Ing. Hermann Golbach
INA reprint “Proceedings of NAFEMS WORLD CONGRESS ’99 on Effective Engineering Analysis”, Volume 2 Newport, Rhode Island, USA 25 – 28 April, 1999
Integrated Non-linear FE Module for Rolling Bearing Analysis Dipl.-Ing. Hermann Golbach
Rolling bearings are commonly used machine elements which the engineer encounters in the design of all kinds of machine systems. Due to their properties as nonlinear, statically indeterminate systems, rolling bearings place high demands on calculation methods. A model for representing the non-linear mechanical behaviour of rolling elements and rolling bearings has been developed for static finite element analyses and converted into the form of a user-defined element for use in the ABAQUS/Standard system. This user element determines, as a kind of structural
element, the non-linear contact stiffness in the Hertzian contact area between the rolling element and raceway on the basis of analytical geometrical and elastic considerations. Unlike the representation of the rolling element using conventional continuum elements and the resulting unavoidable fine mesh, this user element manages with the minimum degrees of freedom. At the same time, the accuracy achieved is very satisfactory, as has been indicated by a comparison with the results of a continuum finite element model of the particular rolling element type. In practical
Figure 1 Needle roller bearing and linear guidance system
2
application, only the essential geometric parameters of the rolling element or rolling bearing are required for element definition. The user element, integrated as a type of module into the FE structure of a machine system, enables realistic analyses both of the load distribution of the rolling bearing under the influence of the elastic environment and of the stress and deformation of the components adjacent to the bearing.
External force
Housing: – undeformed – deformed
Load reactions
Ball bearing
Shaft
Roller bearing
Contact pressure
Figure 2 Deformed system comprising of bearings, shaft and housing
1 Introduction INA Wälzlager Schaeffler oHG is an international manufacturer of rolling bearings and a supplier to the automotive industry, employing over 24 000 people. The central calculation department at the company headquarters in Herzogenaurach (Germany) develops calculation methods and tools for various products, particularly rolling bearings. Rolling bearings are some of the most important basic components in machine, vehicle and plant construction. They are used wherever machine elements moving in relation to each other require mutual support. As their name suggests, the rolling elements roll between the raceways of the components moving in relation to each other. Among other things, rolling bearings are differentiated according to the type of movement relations in rotating and linear motion bearings and the shape of the rolling element in ball and cylindrical roller bearings. Figure 1 shows two examples from INA’s extensive range of products: a rotating bearing with needle rollers as the rolling elements and a linear motion bearing with balls as the rolling elements.
The external load is transmitted from one raceway to the other, distributed over several rolling elements. There is a very high local stress created at the contact point between the rolling elements and the raceways, which occurs cyclically when the bearing is in motion. In order to ensure that this extreme stress can be endured reliably and for the required fatigue life, a rolling bearing made from excellent quality, designed under consideration of various environmental conditions, is essential. Analysis of the load distribution of the bearing is a key task, and is the first step in determining the fatigue life of the bearing.
2 Calculation of the load distribution in rolling bearings Even the best rolling bearing will fail if its system behaviour is not adapted to the machine into which it is integrated. It is therefore necessary to analyse and determine in advance the intensive interactions between bearings and adjacent components. For example, in a system comprising of bearings, shaft and housing, the bending of the shaft and the deformation of the housing influence the
bearing reactions and therefore the load distribution within the bearing. In turn the stiffness of the bearing influences the elastic line of the shaft (Figure 2). In mechanical terms, the system components, bearing, shaft and housing, represent spring elements which form a statically indeterminate spring system. The bearing, which generally has several load-transmitting rolling elements, is itself also a high-grade statically indeterminate spring system, which is also characterised by the strongly non-linear spring behaviour of the rolling elements: • rolling elements behave in a unidirectional fashion, i. e. they only transmit compressive and not tensile forces • the compressive load-deflection relationship within the rolling contact is non-linear • the behaviour of ball bearings (compared to roller bearings) is more complicated due to the fact the pointshaped contact areas may shift under load and the direction of force transmission also changes under load.
3
Roller (undeformed) drawn)
F: Roller force D: Roller diameter b: Semi-contact-width b = b (F), b
INA reprint “Proceedings of NAFEMS WORLD CONGRESS ’99 on Effective Engineering Analysis”, Volume 2 Newport, Rhode Island, USA 25 – 28 April, 1999
Integrated Non-linear FE Module for Rolling Bearing Analysis Dipl.-Ing. Hermann Golbach
Rolling bearings are commonly used machine elements which the engineer encounters in the design of all kinds of machine systems. Due to their properties as nonlinear, statically indeterminate systems, rolling bearings place high demands on calculation methods. A model for representing the non-linear mechanical behaviour of rolling elements and rolling bearings has been developed for static finite element analyses and converted into the form of a user-defined element for use in the ABAQUS/Standard system. This user element determines, as a kind of structural
element, the non-linear contact stiffness in the Hertzian contact area between the rolling element and raceway on the basis of analytical geometrical and elastic considerations. Unlike the representation of the rolling element using conventional continuum elements and the resulting unavoidable fine mesh, this user element manages with the minimum degrees of freedom. At the same time, the accuracy achieved is very satisfactory, as has been indicated by a comparison with the results of a continuum finite element model of the particular rolling element type. In practical
Figure 1 Needle roller bearing and linear guidance system
2
application, only the essential geometric parameters of the rolling element or rolling bearing are required for element definition. The user element, integrated as a type of module into the FE structure of a machine system, enables realistic analyses both of the load distribution of the rolling bearing under the influence of the elastic environment and of the stress and deformation of the components adjacent to the bearing.
External force
Housing: – undeformed – deformed
Load reactions
Ball bearing
Shaft
Roller bearing
Contact pressure
Figure 2 Deformed system comprising of bearings, shaft and housing
1 Introduction INA Wälzlager Schaeffler oHG is an international manufacturer of rolling bearings and a supplier to the automotive industry, employing over 24 000 people. The central calculation department at the company headquarters in Herzogenaurach (Germany) develops calculation methods and tools for various products, particularly rolling bearings. Rolling bearings are some of the most important basic components in machine, vehicle and plant construction. They are used wherever machine elements moving in relation to each other require mutual support. As their name suggests, the rolling elements roll between the raceways of the components moving in relation to each other. Among other things, rolling bearings are differentiated according to the type of movement relations in rotating and linear motion bearings and the shape of the rolling element in ball and cylindrical roller bearings. Figure 1 shows two examples from INA’s extensive range of products: a rotating bearing with needle rollers as the rolling elements and a linear motion bearing with balls as the rolling elements.
The external load is transmitted from one raceway to the other, distributed over several rolling elements. There is a very high local stress created at the contact point between the rolling elements and the raceways, which occurs cyclically when the bearing is in motion. In order to ensure that this extreme stress can be endured reliably and for the required fatigue life, a rolling bearing made from excellent quality, designed under consideration of various environmental conditions, is essential. Analysis of the load distribution of the bearing is a key task, and is the first step in determining the fatigue life of the bearing.
2 Calculation of the load distribution in rolling bearings Even the best rolling bearing will fail if its system behaviour is not adapted to the machine into which it is integrated. It is therefore necessary to analyse and determine in advance the intensive interactions between bearings and adjacent components. For example, in a system comprising of bearings, shaft and housing, the bending of the shaft and the deformation of the housing influence the
bearing reactions and therefore the load distribution within the bearing. In turn the stiffness of the bearing influences the elastic line of the shaft (Figure 2). In mechanical terms, the system components, bearing, shaft and housing, represent spring elements which form a statically indeterminate spring system. The bearing, which generally has several load-transmitting rolling elements, is itself also a high-grade statically indeterminate spring system, which is also characterised by the strongly non-linear spring behaviour of the rolling elements: • rolling elements behave in a unidirectional fashion, i. e. they only transmit compressive and not tensile forces • the compressive load-deflection relationship within the rolling contact is non-linear • the behaviour of ball bearings (compared to roller bearings) is more complicated due to the fact the pointshaped contact areas may shift under load and the direction of force transmission also changes under load.
3
Roller (undeformed) drawn)
F: Roller force D: Roller diameter b: Semi-contact-width b = b (F), b
