## 3 Method Development
### Modal Amplitudes in Space State
Then  
\[\sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x)f_{1}(i,x_{i},t)dx= \sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x_{i})\delta_{l}(i) \delta(x_{a})f_{1}(t)dx=W_{l}^{j}(x_{a})f_{1}(t)\] \[\sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x)f_{2}(i,x _{i},t)dx=\sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x_{i})\delta_{r }(i)f_{2}(t)dx=f_{2}(t)\int\limits_{0}^{l_{m}}W_{r}^{j}(x_{m})dx\] \[\sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x)f_{3}(i,x _{i},t)dx=\sum\limits_{i=1}^{n}\int\limits_{0}^{l_{i}}W_{i}^{j}(x_{i})(-gA_{i} \rho_{i})dx=-g\sum\limits_{i=1}^{n}A_{i}\rho_{i}\int\limits_{0}^{l_{i}}W_{i}^{ j}(x_{i})dx\]  
Denoting, \(\forall j\in\mathbb{N}\),  
\[b_{i}^{j}=\int\limits_{0}^{l_{i}}W_{i}^{j}(x_{i})dx,\quad B^{j}= \sum\limits_{i=1}^{n}\rho_{i}A_{i}\int\limits_{0}^{l_{i}}W_{i}^{j}(x_{i})dx,\] \[\sum_{k=1}^{\infty}W_{i}^{k}(x)\left[E_{i}I_{i}(\beta_{i}^{k})^{4}\,T^{k}(t)+ \gamma\rho_{i}A_{i}\frac{dT^{k}(t)}{dt}+\rho_{i}A_{i}\frac{d^{2}T^{k}(t)}{dt^{ 2}}\right]=f(i,x_{i},t)\]0  
the expression (24) can be written as  
\[\sum_{k=1}^{\infty}W_{i}^{k}(x)\left[E_{i}I_{i}(\beta_{i}^{k})^{4}\,T^{k}(t)+ \gamma\rho_{i}A_{i}\frac{dT^{k}(t)}{dt}+\rho_{i}A_{i}\frac{d^{2}T^{k}(t)}{dt^{ 2}}\right]=f(i,x_{i},t)\]1  
The previous expression describes the dynamics of the \(j\)-th modal amplitude and its time derivative, i.e., \(T^{j}(t)\) and \(V^{j}(t)\). In this way, the space state model of the first \(m\) modal amplitudes is \[\frac{d\mathbb{T}(t)}{dt}=\left[ \begin{array}{cccccc}0&1&0&0&...&0&0\\ -(\omega^{1})^{2}&-2\eta^{1}\omega^{1}&0&0&...&0&0\\ 0&0&0&1&...&0&0\\ 0&0&-(\omega^{2})^{2}&-2\eta^{2}\omega^{2}&...&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&...&0&1\\ 0&0&0&0&...&-(\omega^{m})^{2}&-2\eta^{m}\omega^{m}\end{array} \right]\mathbb{T} (t)\] \[+\left[ \begin{array}{ccccc}0&0&0\\ \frac{W_{1}^{1}(x_{a})}{S^{j}}&\frac{b_{r}^{1}}{S^{j}}&\frac{B^{1}}{S^{j}}\\ 0&0&0\\ \frac{W_{1}^{2}(x_{a})}{S^{j}}&\frac{b_{r}^{2}}{S^{j}}&\frac{B^{2}}{S^{j}}\\ \vdots&0&0\\ \frac{W_{1}^{m}(x_{a})}{S^{j}}&\frac{b_{r}^{m}}{S^{j}}&\frac{B^{3}}{S^{j}} \end{array} \right]\mathbf{f}(t) \tag{26}\]  
where the state is given by  
\[\mathbb{T}(t)=\left[ \begin{array}{ccccc}T^{1}(t)&V^{1}(t)&T^{2}(t)&V^{2}(t) &...&T^{m}(t)&V^{m}(t)\end{array} \right]^{T} \tag{27}\]  
and the input vector is given by  
\[\mathbf{f}(t)=\left[ \begin{array}{ccccc}f_{1}(t)&f_{2}(t)&-g\end{array}  \right]^{T} \tag{28}\]  
Finally, based on (10), the displacement at \(x_{y}\) on a beam element \(e_{j}\) can be approximated by considering the first \(m\) modal components as  
\[w(x_{y},t)\simeq\left[ \begin{array}{ccccc}g(\beta_{j}^{1},x_{y})\mathbf{c} _{j}^{1}&0&g(\beta_{j}^{2},x_{y})\mathbf{c}_{j}^{2}&0&...&g(\beta_{j}^{m},x_{ y})\mathbf{c}_{j}^{m}&0\end{array} \right]\mathbb{T}(t) \tag{29}\]  
The previous equation can be replicated as many times as it is required to calculate displacements at different locations. Let us remark that there are no restrictions on the type of loads that can be applied (constant, periodic, punctual, distributed, etc.) Moreover, the simulation of the system (26)-(29) requires low computational efforts compared with numerical methods such as finite differences or \(FEM\).