## 3 Method Development
### Boundary Conditions  
Computing the modes for the beam sections is a coupled problem, i.e., the modes shapes of a beam depend on the modes of the other beams in the structure. This behavior results from the beam coupling, which can be described by boundary conditions. Thus, the first step in the modes computation is to establish boundary conditions for all the beams in the structure that consider the beams coupling. The beam couplings in the CNC _router_ are mechanical elements, such as bearings, ballscrews and masses, which may have their own elastic properties and behavior. For this reason, masses can be attached at the coupling points in the model. However, elasticity at couplings is not considered in the model, since the elastic contribution of these components will be small because these are built with rigid geometries and materials. The relative positions of the beams studied in this work are those that a CNC _router_ observes during operation, i.e., beam elements being either collinear or perpendicular. In detail, two boundary conditions must be established for each endpoint of each beam, otherwise stated, each function \(w_{i}(x_{i},t)\) requires a total of four boundary conditions (two for each endpoint). These conditions depend on whether the beam is coupled to another beam at the endpoint or not. Figure 2 shows four possibilities of beams coupling. An interesting feature will appear when two or more beams are perpendicularly coupled, as shown in Figure 2d). In this figure, beams \(e_{h}\) and \(e_{i}\) act as torsional springs for beams \(e_{j}\) and \(e_{l}\), and vice versa. To compute the torsion of such springs, variables \(\phi_{h}^{e}\), \(\phi_{i}^{e}\), \(\phi_{j}^{e}\) and \(\phi_{l}^{e}\) (resp. \(\phi_{h}^{f}\), \(\phi_{i}^{f}\), \(\phi_{j}^{f}\), and \(\phi_{l}^{f}\)) are defined to express the torsional deformation of beam \(e_{h}\), \(e_{i}\), \(e_{j}\) and \(e_{l}\), respectively, at the coupling end-point (resp. at the other end-point). The sign of these angles is defined according to the right-hand rule. These torsional deformations are null at fixed and simple supported endpoints. On the other hand, they can be expressed as slopes of perpendicular beams at coupling endpoints, e.g.,  
\[\phi_{j}^{e}\simeq tan(\phi_{j}^{e})=\left.\frac{\partial w_{h}(x_{h},t)}{ \partial x_{h}}\right|_{x_{h}^{e}} \tag{11}\]  
In the following, boundary conditions for different cases are described as both, its general formulation in terms of \(w_{i}(x_{i},t)\) and its modal formulation in terms of \(W_{i}(x_{i})=g_{x}(\beta_{i},x_{i})\mathbf{c}_{i}\). For this, let us denote the first, second, and third derivatives of \(g(\beta_{i},x_{i})\) with respect to \(x_{i}\) as \(g_{x}(\beta_{i},x_{i})\), \(g_{xx}(\beta_{i},x_{i})\) and \(g_{xxx}(\beta_{i},x_{i})\), respectively. * In the case of uncoupled endpoints (e.g., \(x_{1}=0\) at \(e_{1}\) of Figure 1), the boundary conditions can be defined as one of the following cases:
* Free endpoint (cantilever): there are neither bending moments nor shear forces: \[ \begin{aligned} \text{General:}&\frac{ \partial^{2}w_{i}(x_{i},t)}{\partial x_{i}^{2}}\bigg{|}_{x_{i}^{e}}=0,& \frac{\partial^{3}w_{i}(x_{i},t)}{\partial x_{i}^{3}}\bigg{|}_{x_{i}^{e}}=0, \\ \text{Modal:}& g_{xx}(\beta_{i},x_{i}^{e})\mathbf{c}_{i} =0,& g_{xxx}(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0,\end{aligned} \] (12) where \(x_{i}^{e}\) denotes the coordinate of the endpoint (thus \(x_{i}^{e}=0\) for one endpoint or \(x_{i}^{e}=l_{i}\) for the other endpoint). * Fixed endpoint: there is no vertical displacement, there is no slope: \[ \begin{aligned} \text{General:}& w_{i}(x_{i}^{e},t)=0,& \frac{\partial w_{i}(x_{i},t)}{\partial x_{i}}\bigg{|}_{x_{i}^{e}}=0,\\ \text{Modal:}& g(\beta_{i},x_{i}^{e})\mathbf{c}_{i} =0,& g_{x}(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\end{aligned}