Rolling Bearing Calculations with Considerations of Geometric Deviations

<p style="text-align: center;"><img src="/ueditor/php/upload/image/20230912/1694528231281212.png" title="1694528231281212.png" alt="4.png"/></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Rolling bearing calculations are usually based on the assumption of ideal nominal geometries. However, actual components and assemblies are always subject to statically distributed geometric deviations resulting from the manufacturing and assembly processes. This leads to changes in the internal geometric conditions which have an effect on bearing characteristics such as the service life. The&nbsp;FVA-Workbench&nbsp;makes it possible for users to consider these geometric deviations in bearing calculations for more reliable results.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Problem Statement</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Rolling bearings are indispensable components in all fields of mechanical engineering which allow for precise, low-friction, and cost-effective bearing arrangements for rotating components. Rolling bearings are typically calculated according to ISO/TS 16281. This generally assumes an ideal geometry for the actual raceways with no misalignment of the bearing axes.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">However, this ideal situation can never be achieved in practice. The manufacturing and assembly processes always lead to geometric deviations which are expressed as a static distribution (Ref. 2). Furthermore, the loads during operation can cause deformations which influence the operating behavior of the bearings. For example, Fingerle (Ref. 3) has shown that meshing forces acting on the planetary gears cause ovalization of the wheel body. This leads to an increased or reduced bearing life compared to an ideally stiff wheel, depending on the conditions.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The integration of geometric deviations and their statically distributed characteristics in rolling bearing calculations results in a more accurate representation, which leads to improved results.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The&nbsp;FVA-Workbench&nbsp;(Ref. 4) software allows users to consider these form and position deviations in rolling bearing designs. The scripting interface also makes it possible to integrate user-defined routines. The following will show how these options can be used to perform bearing calculations, taking these statically distributed geometric deviations into consideration.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Consideration of Geometric Deviations in RollingBearing Design</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The&nbsp;FVA-Workbench&nbsp;can be used for the modeling, configuration, and calculation of transmission systems. When modeling the entire system, an STP file of the casing can be imported, which can be then coupled with the bearing outer ring via a simple click workflow. This makes it possible to calculate the stiffness matrix and determine the equivalent stiffness according to Guyan (Ref. 5) without the use of external FE software. In addition to using bearing data according to manufacturer specifications (e.g., from rolling bearing catalogs), the rolling bearing geometry can also be directly specified. This includes profiling of the rolling elements and raceways, which makes it easy to consider form deviations.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Deviations on the outer ring raceway can be defined as radial deviations from the ideal circular form. For example, this can be used to define ovalization of the outer bearing ring.&nbsp;Figure 1 shows an example of form deviation of the outer ring, enhanced one-hundred-fold for clarity.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">For rolling bearings with line contact, any kind of profiling can be defined in the axial direction for the rolling elements on the inner and outer ring.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Generally, manufacturers use specifically designed profile functions to improve the service life. In theory, however, any kind of form deviation of the raceways can be represented.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Figure 2 shows some example deviations in the axial direction with eight interpolation points. The rolling elements correspond to the suggested profile according to DIN 26281—with a crowned form deviation on the inner ring and a tapered form deviation on the outer ring. These values are also enhanced one-hundred-fold for clarity.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">When using the slice model, DIN 26281 prescribes dividing the bearing into at least 30 axial slices. The&nbsp;FVA-Workbench&nbsp;supports any resolution, as long as this lower limit is not exceeded. For optimal resolution of the desired profiling in the slice model, a number of profiling interpolation points at least equal to the number of slices should be selected.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">On the outer ring, additive deviations can be overlaid in both the longitudinal and circumferential directions.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">In addition to these deviations on the raceways, an offset and misalignment of the casing-bearing seat can also be defined by specifying values for the vertical (v) and transverse (w) axes (for identification of the axes, see Figure 4). In the calculation, these values represent boundary conditions that are considered accordingly in solving the quasi-static equilibrium in the shaft-bearing-casing system.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The Scripting Module (Ref. 6) can be used to automate complex processes in the&nbsp;FVA-Workbench. Simple commands can be used to load additional data, run calculations, and create custom output reports. Possible output formats include simple text files, clearly prepared HTML reports or ready-configured&nbsp;Excel&nbsp;files.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">For example, the Scripting Module can be used to load previously generated samples and process them in sequence. In this case, a sample describes a possible configuration of deviations as they can occur in reality.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Figure 3 shows the sequence for the statistical tolerance analysis in this example. The calculation is configured in an XML file in which the components and the form and position deviations are defined. The samples are then automatically created based on the input file and the configured sampling strategy (randomly or systematically according to an experimental plan). The input file is processed by a MATLAB script, which saves the geometry data of the individual samples in an Excel file that can be read by the&nbsp;FVA-Workbench. A model file for which the calculations are to be performed is also required as an input.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The&nbsp;FVA-Workbench&nbsp;is then started in Batch Mode via a command line call and automatically processes the generated samples and saves the calculation results in an Excel file.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The statistical analysis of the results is then performed using suitable MATLAB scripts. A results report can also be generated for individual samples in the&nbsp;FVA-Workbench. This report is output in HTML format and includes detailed, graphical calculation reports for the individual system components (e.g., shaft bending, flank pressure in tooth meshes, or lifetime).</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Calculation Example</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The following example will demonstrate the sequence using a simple system consisting of a casing and shaft with a fixed-floating bearing arrangement with a radial force F acting on the center of the shaft.</span></p><p><br/></p>