40 M. Khan et al. force increases, the second bending mode comprises increasing measures of the total system energy, increasing to 35% of the total system energy, while that for the axial mode decreases to 55%. Therefore, as the axial mode softens, the second bending mode in Y does become more excited and contributes more to the overall response. For a purely linear system, these ratios would not change as the relative excitation between the two modes would remain constant. However, due to the nonlinearity introduced by the bolted joint, there appears to be coupling between these two modes that results in a decrease in apparent damping and increased excitation of the second bending mode in Y. Therefore, at high force levels, the response of the structure is no longer mostly axial, but has significant lateral motion due to the activation of the second bending mode. From a fatigue testing standpoint, this means that a multiaxial load will be applied to the bolt, which limits the viability of the structure to consistently test bolts to failure. There may be experimental methods to reduce lateral motion using another shaker, but further research would be needed. Although traditional joint models would not be able to account for this nonlinear modal coupling, calibration to modal test data can still be performed to develop useful approximations. The following sections detail the linear joint models developed and model calibration performed for the structure. 4 Structural Dynamic Modeling Four different FE models were created, each with a distinct representation of the bolted joint. All meshes were created using the CUBIT meshing software and consisted mainly of ten-noded tetrahedral (TET10) elements. The KB, washers, and AP were modeled using these TET10 elements, with the bolt representation dependent on the modeling technique used. For fixed base analysis, the bottom surface of the AP was fixed. An image of the model is shown in Fig. 10. 4.1 Model 1: Solid Bolt and Tied Joint Interface The first model uses a solid bolt with washers and a nut. The bolt head, washers, and nut are connected to the KB and AP using Tied Data functionality, which constrains nodes from different meshes, effectively “gluing” them together. For the models presented here, preload is not included and instead the load transfer between the bolt and members is represented with spring elements. In Model 1, the interface between the KB and AP is connected using the Tied Joint capability in SIERRA Structural Dynamics (SIERRA/SD) [1]. The Tied Joint model creates a virtual spring element with six user-defined stiffness parameters (three translational and three rotational), which are used to connect two surfaces. The contact radius was adjusted based on later model calibration steps. Figure 11 shows the Model 1 configuration. Fig. 10 FE model of structure
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