## 3 Method Development
### Boundary Conditions
\] (13)
Simple supported endpoint: there is no vertical displacement, there is no bending moment: \[\text{General:} w_{i}(x_{i}^{e},t)=0, \frac{\partial^{2}w_{i}(x_{i},t)}{\partial x_{i}^{2}}\Bigg{|}_{x_{i }^{e}}=0,\] Modal: \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]
* Additional boundary conditions are required for endpoints at which two or more beams are coupled. These conditions depend on how many beams are coupled at the same point, if these are aligned or not, and if torsion is allowed or not. Four configurations can occur, as shown in Figure 2. The most general case involves four beams \(e_{h},e_{i},e_{j}\) and \(e_{l}\), coupled at the same point, where \(e_{h}\) and \(e_{i}\) are aligned between them, \(e_{j}\) and \(e_{l}\) are perpendicular to them, and torsion is allowed. In the following, take Figure 2d) as reference. For this, the following conditions must be defined:
* The following conditions are required to guarantee displacements continuity \[\text{General:} w_{h}(x_{h}^{e},t)=w_{i}(x_{i}^{e},t)=w_{j}(x_{j}^{e},t)=w_{l}(x_{l }^{e},t)\] Modal: \[g(\beta_{h},x_{h}^{e})\mathbf{c}_{h}=g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}= g(\beta_{j},x_{j}^{e})\mathbf{c}_{j}=g(\beta_{l},x_{l}^{e})\mathbf{c}_{l}\] where \(x_{h}^{e}\), \(x_{i}^{e}\), \(x_{j}^{e}\), \(x_{j}^{e}\) denote the endpoints coordinates where the beams \(e_{h},e_{i},e_{j},e_{l}\) are coupled. * The aligned beams require slope continuity, i.e., \[\text{General:} \frac{\partial w_{h}(x_{h},t)}{\partial x_{h}}\Bigg{|}_{x_{h}^{e}} =\frac{\partial w_{i}(x_{i},t)}{\partial x_{i}}\Bigg{|}_{x_{i}^{e}},\] \[\frac{\partial w_{j}(x_{j},t)}{\partial x_{j}}\Bigg{|}_{x_{j}^{e}} =\frac{\partial w_{l}(x_{l},t)}{\partial x_{l}}\Bigg{|}_{x_{i}^{e}}\] Modal: \[g_{x}(\beta_{h},x_{h}^{e})\mathbf{c}_{h}=g_{x}(\beta_{i},x_{i}^{e}) \mathbf{c}_{i},\] \[g_{j}(\beta_{j},x_{j}^{e})\mathbf{c}_{j}=g_{x}(\beta_{l},x_{l}^{e}) \mathbf{c}_{l}\]
* Another condition is due to a balance of vertical forces, in detail, shear forces must be coherent with the acceleration of the mass attached at the coupling point, which is generally formulated as \[E_{h}I_{h}\left.\frac{\partial^{3}w_{h}(x_{h},t)}{\partial x_{h}^{3}} \right|_{x_{h}^{e}}-E_{i}I_{i}\left.\frac{\partial^{3}w_{i}(x_{i},t)}{\partial x _{i}^{3}}\right|_{x_{i}^{e}}+E_{j}I_{j}\left.\frac{\partial^{3}w_{j}(x_{j},t)}{ \partial x_{j}^{3}}\right|_{x_{j}^{e}}\] \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]0 where \(v\) is the vertical displacement of the attached mass \(m\); if there is no mass attached at this point then the right term is substituted by \(0\). By using (1) and (4), the mass acceleration can be expressed in terms of the displacement at the endpoint of any of the coupled beams, e.g., \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]1 \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]2 The corresponding modal formulation becomes \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]3 (17) \[g(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0, gxx(\beta_{i},x_{i}^{e})\mathbf{c}_{i}=0.\]4
By considering the balance of moments along the longitudinal axis of beams \(e_{h}-e_{i}\) and \(e_{j}-e_{l}\), and the aforementioned torsional spring effects, two boundary conditions are imposed as follows  
\[ \begin{split} E_{h}I_{h}\left.\frac{\partial^{2}w_{h}(x_{h},t)}{ \partial x_{h}^{2}}\right|_{x_{h}^{e}}-E_{i}I_{i}\frac{\partial^{2}w_{i}(x_{i},t)}{\partial x_{i}^{2}}\right|_{x_{h}^{e}}-k^{j}\left(\phi_{j}^{e}-\phi_{j}^{ f}\right)\\ -k^{l}\left(\phi_{l}^{e}-\phi_{l}^{f}\right)=mr^{2}\frac{d^{2}\phi _{j}^{e}}{dt^{2}}\\ E_{j}I_{j}\left.\frac{\partial^{2}w_{j}(x_{j},t)}{\partial x_{j}^ {2}}\right|_{x_{j}^{e}}-E_{i}I_{i}\frac{\partial^{2}w_{l}(x_{i},t)}{\partial x _{l}^{2}}\right|_{x_{h}^{e}}-k^{h}\left(\phi_{h}^{e}-\phi_{h}^{f}\right)\\ -k^{i}\left(\phi_{i}^{e}-\phi_{i}^{f}\right)=mr^{2}\frac{d^{2} \phi_{h}^{e}}{dt^{2}}\end{split}