<p style="text-align: center;"><img src="/ueditor/php/upload/image/20251015/1760510921108078.png" title="1760510921108078.png" alt="6.png"/></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Conventions In the figures below, blue indicates the designed gear, red the manufactured or measured gear and green the digital twin. Introduction to the Problem Loaded tooth contact analysis of gears, considering or neglecting gear misalignment, may be performed by gear design software such as KISSsoft or other similar tools. Approximately a dozen such commercial software programs are used in an international environment, most of them based on analytical approaches. Very few are based on FEM approaches. With this, the performance characteristics of a gear design are assessed, considering: Transmission error TE and its spectral content. Peak to peak transmission error PPTE. Contact stress, its maximum value and distribution. Gear mesh force variation over a meshing cycle. Micropitting and scuffing safety, flank and root strength, considering the calculated load distribution.&nbsp;</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">In principle, it is possible to measure the above performance characteristics also on manufactured gears. Costs in time, equipment and money typically do not allow for this approach in a manufacturing environment. This means that deviations in the geometry of the manufactured gear compared to the designed gear are assessed only in the sense of whether the manufactured gear meets the quality grade stipulated on the manufacturing drawing, e.g., along ISO 1328. But a question “… we manufactured a batch of gears, we measured a form error of xyz μm, is the resulting degradation of gear strength acceptable or do we need to scrap the batch?” cannot be answered. Closed Loop as Solution If, however, the manufactured and measured gear geometry could be looped back into the original KISSsoft gear design and combined with the designed gear to create a digital twin of the manufactured gear(s), then this twin—or these twins—could be subjected to the same calculations. And the resulting performance characteristics, e.g., contact pressure, may be compared to those of the designed gear. Strength rating methods, e.g., ISO 6336, would then allow for the calculation of gear strength of the manufactured gear and a degradation may be assessed by a gear designer familiar with the requirements. A decision, whether a batch of gears having a geometric deviation needs to be scrapped, is then based on relevant performance characteristics (strength, lifetime, reliability, scuffing safety, etc.), not only on a gear quality number.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">The calculation of the above-mentioned performance parameters of the designed gear as the reference and of several gears or batches of gears as manufactured and measured is done in KISSsoft through variants of the geometry. This means that simultaneously, several gears are defined in one calculation. All have the same gear macro and micro geometry, but each has a different amount of deviation from the reference gear (of course, the reference gear has zero deviation from itself and is variant No. 1), as measured. This means that in a single calculation file, the reference and as many digital twins as required are managed. The reference design and the digital twins are then subjected to an LTCA. Transmission error, contact stress levels or contact pattern shape and position of the digital twins are then compared to the results of the designed gear. If deviations in the performance parameters are within acceptable limits, the gear or batch of gears is approved. If deviations are too high, the gear or batch of gears is rejected. With this approach, quality control is far more target-oriented compared to using only a gear quality number. It also requires more experience to assess the performance characteristics.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Creating the Digital Twin Measuring a Grid of Points on the Flank as State of the Art</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">To create a digital twin, it is state of the art to measure a grid of points on the flank, compare their measured coordinates to the coordinates calculated based on the gear macro and micro geometry, and feed the deviations back to the designed gear to convert it to a digital twin. This approach has been available for some time. The gear macro geometry, as designed in the gear design software, is exported as a Gleason GAMA software-compatible file or through GDE format, along with VDI/VDE 2610 guideline. Furthermore, direct export and import of a grid of points (rather than their coordinates and normal vector) from KISSsoft to GAMA is available. With this approach, the gear measurement machine knows the grid, the grid point designed coordinates and the grid point normal vectors. The measurement of the as-is coordinates of these points is then performed using a tactile probe. In Figure 7, the grid indicated consists of 8 × 8 points (green dots). The output is then an 8 × 8 table showing the deviation of the measured grid point coordinates from the designed grid point coordinates or rather the deviation as a scalar, to be understood in the direction of the normal vector of the corresponding point. This table is then imported into the design software, where the designed gear is modified by the imported deviation table to generate the digital twin. Such a digital twin is suitable for the assessment of tooth contact patterns and load distribution on the flank and will already be most helpful to assess manufacturing deviations like errors in pressure angle, helix angle, crowning or twist.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Measuring Flank Waviness as Next Step In the above process, only 8 × 8 = 64 points are measured and the shape of the flank between the measured points is not known. A more refined approach is to not only transfer individual points of information from the measurement but whole traces in lead and profile directions (which is just a much higher number of points). Traces are measured and displayed in Gleason GAMA software, see the yellow lines on the left side of Figure 8. These traces show a waviness; they may also be visible in printouts, as shown in the top right of Figure 8. These traces are now to be represented in the gear design software so that the gear model there also includes the waviness as it is present in the manufactured gear. At the time of writing this paper, this was not yet automatically possible but required a manual approximation of the shape of the trace in the gear design software, using functions to approximate it there. Note the comment in the section on “Future Work” where this point is addressed as a subject for improvement of the software interface. With this approach, the digital twin contains far more information on the manufactured gear since a much higher number of data points is considered. At the time of writing this paper, however, it is only possible to import one trace in lead and one trace in profile direction (or one trace with an inclination as explained in the following section). This is again a shortcoming of the current software version to be addressed in the future. The underlying assumption is that waviness is a systematic result of the manufacturing process, manufacturing machine and tool properties. Hence, it is reasonable to assume, in a first approximation, that the waviness in profile direction is constant along the face width or the waviness in lead direction is constant in profile direction. In the next section, definitions and descriptions of how the waviness is defined in the gear design software are given.</span></p><p style="text-align: justify;"><span style="font-family: arial, helvetica, sans-serif; font-size: 12px;">Definitions and Description of Waviness On the left side of Figure 9, the waviness is shown as a three-dimensional image. The vertical axis shows the deviation of the manufactured and measured geometry from the designed geometry. Note the waviness in one direction. Note that the plane is twisted due to the natural twist from manufacturing. Note that this waviness is concerning one flank. In a mesh, several flanks are in contact at one point in time, and the waviness in one contact and the waviness in the next contact (for contact ratio above unity) overlap. On the right side of Figure 9, the definition of waviness on the flank, inclined by an angle, is shown. The amplitude, wavelength, inclination angle and phase shift are needed to define the waviness where the shape is a sinusoidal form. If the angle is set to zero, a waviness in profile direction results; if set to 180 degrees, a waviness in lead direction results. Several such waviness definitions may be superimposed. On the left of Figure 10, waviness patterns in lead and profile direction are superimposed, creating a “rough” gear flank. Depending on the amplitude of this “roughness”, different maximum peak stresses result. Such an approach may be helpful to explain the formation of micropitting in a pattern resembling grinding marks. In the right of Figure 10, several waviness definitions having different phase shift, amplitude and wavelength are superimposed in the profile direction. This results in a composite waviness in profile direction that is not intuitively recognized as a superposition of sinusoidal shapes but looks realistic. To represent the measured waviness, e.g., in profile direction, the user must come up with several sinusoidal curves and superimpose them. By adjusting phase shift, amplitude and wavelength for each sinusoidal curve, different patterns for the resulting waviness are achieved. With this approach, by increasing the amplitude, wear of a tool and an increase in waviness in production between dressing of the tool may be considered. Currently, the software lacks the functionality to create the individual sinusoidal shapes, again, a function to be added.</span></p><p><br/></p>