## 3 Kinematics and Dynamics of the Elastic Part
Recalling (7)-(9) in the kinetic energy formula, one can write the total kinetic energy, \(E_{e}\), of the element, as well as the total energy of its deformation, \(V\!e\), as [24; 25; 26; 29; 30; 31]:  
\[E_{e}=\frac{1}{2}\int_{m_{e}}\hat{\mathbf{v}}^{T}\cdot\hat{\mathbf{v}}\cdot dm; \tag{10b}\] \[V_{e}=\frac{1}{2}\int_{V_{e}}\hat{\mathbf{\epsilon}}^{T}\cdot\hat{\mathbf{v}}\cdot dV, \tag{10b}\]  
and rewritten to [24; 25; 26; 29; 30; 31]:  
\[E_{e}=\frac{1}{2}\hat{\mathbf{\mu}}_{e}^{T}\cdot\hat{\mathbf{M}}_{e}\cdot\dot{\hat\mathbf{{q}}}_{e}; \tag{11a}\] \[\hat{\mathbf{M}}_{e}=\frac{1}{2}\rho\cdot\int_{x_{e}}\int_{y_{e}}\int_{z_{e}}\hat{ \mathbf{N}}_{e}^{T}\cdot\hat{\mathbf{N}}_{e}\cdot dx\cdot dy\cdot dz;\] (11b) \[V_{e}=\frac{1}{2}\hat{\mathbf{q}}_{e}^{T}\cdot\hat{\mathbf{K}}_{e}\cdot\hat{\mathbf{q}}_{e};\] (12a) \[\hat{\mathbf{K}}_{e}=\frac{1}{2}\int_{x_{e}}\int_{y_{e}}\int_{z_{e}}\hat{\mathbf{B}}_{ l}^{T}\cdot\hat{\mathbf{D}}_{e}\cdot\hat{\mathbf{B}}_{l}\cdot dx\cdot dy\cdot dz, \tag{12b}\]  
where the obtained matrices \(\hat{\mathbf{M}}_{e}\) and \(\hat{\mathbf{K}}_{e}\) are the mass matrix and the elasticity/stiffness matrix, respectively, of the element expressed in the local coordinate system. We can omit the subsequent step of classic development now since if we focus on the presently investigated case, the local systems are collinear to the global system, and therefore rotational transformations are unnecessary. Accordingly, the local coordinates are identical to their absolute complements. If one intends to obtain the global vector of the displacements, \(\mathbf{q}_{\mathbf{c}}^{*}\), and the global vector of loads, \(\mathbf{P}_{\mathbf{c}}^{*}\), all the elements' matrices of displacements and loads should be grouped and written as:  
\[\mathbf{q}_{\mathbf{c}}^{*}=col(\mathbf{q}_{i}); \tag{13a}\] \[\mathbf{P}_{\mathbf{c}}^{*}=col(\mathbf{P}_{i}), \tag{13b}\]  
where \(\mathbf{q}_{i}\) is the vector of displacements of the \(i\)th node; and \(\mathbf{P}_{i}\) is the vector of loads of the \(i\)th node. Furthermore, one can write the global matrices for the shape functions, masses, and elasticity components. With a correct allocation of blocs of the local matrices:  
\[N\mathbf{e}q=[\ NiNj\ ]ij[qiq]ji\ ;\quad\mathbf{A}\mathbf{e}q=[AiiAijAji]ij[qiq]ij\, \tag{14}\]  
one can write the global matrices as [24, 25, 26, 29, 30, 31]:  
\[\mathbf{N}_{\mathbf{c}}^{*}\mathbf{q}_{\mathbf{c}}^{*}= \begin{bmatrix}0&N_{i}&0&N_{j}&0\\ \cdots&i&\cdots&j&\cdots\\ &&&&&\ddots\\ &&&&&&\ddots\\ \end{bmatrix}\begin{bmatrix}q_{1-i}\\ q_{i}\\ \vdots\\ q_{i-j}\\ \vdots\\ \vdots\\ \end{bmatrix} ;\quad\mathbf{A}_{\mathbf{c}}^{*}\mathbf{q}_{\mathbf{c}}^{*}= \begin{bmatrix}0&0&0&0 &0\\ 0&A_{\mathbf{i}}&0&A_{\mathbf{i}}&0\\ 0&0&0&0&0\\ 0&A_{\mathbf{i}}&0&A_{\mathbf{i}}&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \vdots\\ \end{bmatrix}\begin{bmatrix}\mathbf{q}_{1-i}\\ \vdots\\ i\\ i\\ \mathbf{q}_{i-j}\\ \vdots\\ \end{bmatrix}\begin{bmatrix}\mathbf{q}_{1}\\ \vdots\\ i\\ \vdots\\ \vdots\\ \end{bmatrix} . \tag{15}\]  
Then, thanks to the cumulative property of the kinetic and potential energy, the global matrices of the subpart are obtained as sums of all the matrices of elements [24, 25, 26, 29, 30, 31]:  
\[\mathbf{M}_{\mathbf{c}}^{*}=\sum_{e=1}^{n_{\mathbf{c}}}\mathbf{M}_{\mathbf{c}}^{*}; \tag{16a}\] \[\mathbf{K}_{\mathbf{c}}^{*}=\sum_{e=1}^{n_{\mathbf{c}}}\mathbf{K}_{\mathbf{c}}^{*}, \tag{16b}\]  
where \(n_{\mathbf{c}}\) is a number of the finite elements considered in the system.