A rigid-body (RB) mode is defined as the free translation or rotation of a body without undergoing any internal deformation. In theory, there will be no internal deformation in a free-free normal mode analysis, where there are no loads or constraints, there should be 6 rigid body modes, three translation modes (TX, TY, TZ) and three rotation modes (RX, RY, RZ). These first 6 modes have a modal frequency of zero or very close to zero. If the system has an artificial internal constraint in some way that prevents it from moving freely, even though it contains no constraint to the world, it means that there is a loose constraint behind the scenes. This can be considered a side effect of modeling problems or formulations used in finite element simulation. Common causes of non zero RB mode frequencies in finite element analysis include contacts with relatively large or uneven gaps, overlapping contact areas, some contact formulation options and very stiff or compliant elements in the model. In a perfectly prepared model, the RB mode frequency is expected to be very close to zero, but this is not always the case. If grounding is present, it means that the modal results of the system may not be accurate because grounding can affect the stiffness matrix of the structure and change the load paths. Grounding may affect the load transfer in the model and the results may not be reliable. In some cases however, the grounding may be so weak that it does not affect the load transfer in the model and therefore can be ignored. The questions addressed in this study are: When the effect of RB modes that have a frequency > 0 can be ignored and when does it affect the simulation results? To ensure the reliability of the model, some criteria for RB mode frequencies have been established in the industry. A common practice is to keep RB modes below 1E-4 Hz. In the literature, no study has been found on the effects of the RB modes on the simulation results if they are significantly non zero. In some cases, the models do not meet this criterion, and a large effort is required to correct them. In this study, hypothesis of virtual grounding springs is put forward. Several FE models are tested to determine the stiffness of grounding springs due to non-zero RB modes. These components are connected to the ground, and the spring stiffness was varied (e.g. 1E2 to 1E9) to cover a range of values. Reaction forces are checked at each level to determine whether or not all applied forces are reacting on the supports. The deviation between the applied force and the reacting force starts at a value comparable to the calculated stiffness of the grounding spring, and this confirms the hypothesis. The frequency response of the models simulated on weaker foundation springs than grounding springs shows a deviation especially in the lower frequency range. Since the load always flows through the stiff regions, models with high foundation or neighborhood stiffnesses compared to the grounding spring show no deviation due to non-zero RB modes. It is recommended to compare the stiffness of the grounding spring with the foundation or neighborhood stiffness to decide whether it will effect further simulation results.
Reference | NWC23-0057-presentation |
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Authors | Dirik. H Starreveld. J Hyderabadwala. M |
Language | English |
Type | Presentation |
Date | 16th May 2023 |
Organisation | ASML |
Region | Global |
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