In order to achieve high reproduction accuracies, a high-quality mesh is required [5]. According to the literature, there is a strong preference for using linear hexagonal finite elements for mechanical problems [6], while tetrahedral elements are preferred for thermal problems. Many researchers addressed the problem of all-hexahedral-meshing (e.g. [7]), because of the absence of robust automatic hex-meshing algorithms. In aeronautic applications, for example, this is the standard meshing-strategy [8]. The elastic FE-model presented in this paper consists of hexahedral elements in order to achieve high calculation accuracy with low computing time. Hence, particular attention is paid to the pre-processing.
3.1 Geometry Preparation and Mesh Generation
Since only those parts of the geometry are of interest, which have a significant influence on the model’s behaviour, the geometry is simplified by removing filigree, geometrical elements such as small holes, chamfers, small radii, nuts and bolts etc. Furthermore, replacing parts of the model’s geometry, such as tie rods, with idealised FE-elements, such as link elements (see Fig. 2, b), simplifies the model and helps to avoid unwanted mesh refinements. The density of some components has been adjusted to maintain the overall mass distribution. This is important since the mass distribution of the system, represented by the mass
matrix MM, needs to be modelled accurately, to ensure that the obtained
eigenfrequencies and eigenvectors determined by Eq. 1, reflect the stiffness behaviour of the system corresponding to its stiffness matrix K. Therefore, the 327 tie rods (4 % of the overall mass) are considered (compare Fig. 2, b). Replacing the tie rods may result in some difficulties, though. Since the holes, through which the tie rods are mounted, were removed to avoid unwanted mesh refinements in those areas, the top and bottom plate of the model need to be segmented to create nodes at the former locations of the tie rods. This allows the creation and attachment of link elements to substitute the tie rods. Decomposition of the geometry into predominantly cuboid bodies, which the meshing algorithm can process more easily, finalises the geometry preparation (see Fig. 2, a).
Finite elements of various shape, size and polynomial order were investigated for meshing the model, including shell and solid elements. An assessment on the model’s reproduction accuracy concluded that linear hexahedrons, with the size of the models aluminium plate’s thickness (8 mm), yielded the most
accurate solution to eigenvalue problems with an acceptable time effort. The all-hexahedral mesh is shown in Fig. 2, a.
Fig. 2. Z-slide: (a) prepared model with an all-hexahedral mesh; (b) line elements to model the tie rods and the elastic suspension.
3.2 Contact Modelling
Due to the large number of contact areas between the aluminium sheets, homogenisation of the contact bahaviour is necessary. In addition, linear contact models are used to perform simulative modal analysis. This is reasonable, since preservation of the pretension can be assumed for all operating conditions. Due to roughness, measured contact stiffness values are significantly lower than theoretical values predicted from smooth surfaces in contact [9]. Since the aluminium plates and ribs are water jet cut, significant squareness deviations of the edges are unavoidable, which also influence the contact stiffness. Preceding investigations showed that a contact model is needed to map the numerous contacts between the aluminium sheets of the structure. Thereby, the stiffness distribution can be adjusted properly. A variety of contact models were created and compared to represent the tangential and normal contact stiffness between the sheets. Contact models using linear springs, a layer of thin material in the contact-area (transversely isotropic material behaviour), in conjunction with multipoint constraint based contacts and other ANSYS contact elements, were used to connect the aluminium plates of the structure. Adjustable ANSYS contact elements using the Pure Penalty- algorithm were used in this paper. Since ANSYS Workbench provides only one parameter to adjust the contact stiffness, additional APDL commands are included to parameterise the normal and tangential contact stiffness independently. In order to obtain reliable parameters of the contact stiffnesses, pre-investigations on a simpler but comparable test rig were carried out. The identified normal and tangential contact stiffness are found to be 765 N/mm³
and 400 N/mm³ respectively. These parameters were used in the presented model of the Z-slide for dynamical, elastic as well as for thermo-elastic analyses.
3.3 Comparison of the Measured and Simulated Elastic Behaviour
The stiffness of the elastic suspension, used for mounting the Z-slide during the experimental modal analyses, was measured to parameterise the corresponding springs in ANSYS (compare Fig. 2, b). The averaged modal damping ratio of the Z-slide yielded approximately 0.005. This value was used for the simulative modal analysis. From Fig. 3 it is evident, that the (global) system behaviour can be represented by the model accurately. However, by performing deformation measurements, it has to be verified whether the model can predict the local stiffness behaviour. Furthermore, the model needs to be improved by adjusting the model parameters in order to predict vibrations, dominated by the substructure of the linear motor components, as can be seen in Fig. 3 at approx. 265 Hz. Therefore, in particular, the contact parameters between the top plate and the substructure of the linear motor components need to be optimised. According to [10], the simulation-based approach for identifying the contact stiffness on the basis of experimental modal analyses can be applied.
Fig. 3. (top) frequency response function; (bottom, left) characteristic mode-shape: simulation; (bottom, right) characteristic mode-shape: measurement (60 positions investigated).