## 2 Model
Moreover, \(m\) and \(I_{v}\) are the mass and second inertia moments per unit-length, evaluated with respect to the centroid (\(\
u=x,y,z\)), and a dot denotes time-differentiation. When no external load and no lumped masses are present at the ends of the beam, the mechanical boundary conditions read:  
* Clamped at \(A\) and free at \(B\): \(N_{B}=T_{yB}=T_{zB}=M_{xB}=M_{yB}=M_{zB}=0\);
* Simply supported and torsionally restrained in \(A\), \(B\): \(M_{yA}=M_{yB}=M_{zA}=M_{zB}=0\). It is assumed that the beam material is hyperelastic, so that the existence of a strain energy function, quadratic in the strains, is postulated, namely:  
\[\phi=\frac{1}{2}\left(c_{11}\,\varepsilon^{2}+c_{22}\,\gamma_{y}^{2}+c_{33}\, \gamma_{z}^{2}+c_{44}\,\kappa_{x}^{2}+c_{55}\,\kappa_{y}^{2}+c_{66}\,\kappa_{ z}^{2}\right), \tag{3}\]where the elastic constants \(c_{ii}\) (\(i=1,\ldots,6\)) must be identified according to the micro-structure of the grid beam. The constitutive law follows from the Green law as:  
\[\left( \begin{array}{c}N\\ T_{y}\\ T_{z}\\ M_{x}\\ M_{y}\\ M_{z}\end{array} \right)=\left( \begin{array}{ccccc}c_{11}&0&0&0&0&0\\ &c_{22}&0&0&0&0\\ &&c_{33}&0&0&0\\ &&&c_{44}&0&0\\ &&&c_{55}&0\\ &&&c_{66}\end{array} \right)\left( \begin{array}{c}\varepsilon\\ \gamma_{y}\\ \gamma_{z}\\ \kappa_{x}\\ \kappa_{y}\\ \kappa_{z}\end{array} \right). \tag{4}\]  
Then, equilibrium (2) is written in terms of displacements, and by using Equations (4) and (1), the elasto-dynamic problem is obtained:  
\[ \begin{split}&-\mathbf{M}\bar{\mathbf{u}}+\mathbf{K}_{2}\mathbf{u} ^{\prime\prime}+\mathbf{K}_{1}\mathbf{u}^{\prime}+\mathbf{K}_{0}\mathbf{u}+ \mathbf{p}=\mathbf{0},\\ &\mathbf{A}_{1}\mathbf{u}_{A}^{\prime}+\mathbf{A}_{0}\mathbf{u}_{A}= \mathbf{0},\\ &\mathbf{B}_{1}\mathbf{u}_{B}^{\prime}+\mathbf{B}_{0}\mathbf{u}_{B}= \mathbf{0}.\end{split}  \tag{5}\]  
Equation (5)-a is a system of partial differential equations, sided by geometric and mechanical boundary conditions at \(A\) and \(B\), Equation (5)-b,c, and by initial conditions, prescribing that the system is initially at rest (not reported here). Moreover,  
\[ \begin{split}\mathbf{u}:&=\left(u,v,w,\theta_{x},\theta_{y}, \theta_{z}\right)^{T},\\ \mathbf{M}:&=\text{diag}\left(m,m,m,I_{x},I_{y},I_{z} \right),\\ \mathbf{K}_{1}:&=\left(\begin{array}{ccccc}0&0&0&0&0&0\\ &0&0&0&-c_{22}\\ &&0&0&c_{33}&0\\ &&&0&0&0\\ &&&0&0\\ \text{SKW}&&&0\end{array}\right),\quad\mathbf{K}_{0}:=\left(\begin{array}{ cccccc}0&0&0&0&0&0\\ &0&0&0&0&0\\ &&0&0&0&0\\ &&&0&0&0\\ &&&-c_{33}&0\\ \text{SYM}&&&-c_{22}\end{array}\right),\end{split}  \tag{6}\]  
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{K}_{i}\) (\(i=0,1,2\)) are symmetric or skew-symmetric stiffness matrices, and \(\mathbf{u},\mathbf{p}\) are displacement and load vectors, respectively. Boundary operators, involved in Equation (5)-b,c, are defined as:  
* Clamped at \(A\) and free at \(B\): \[ \begin{split}\mathbf{A}_{1}:=&\mathbf{0},\quad\mathbf{A}_{0}:=& \mathbf{I},\\ &\mathbf{B}_{1}:=\mathbf{K}_{2},\quad\quad\mathbf{B}_{0}:=& \left(\begin{array}{ccccc}0&0&0&0&0&0\\ 0&0&0&0&0&-c_{22}\\ 0&0&0&0&c_{33}&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array}\right);\end{split} \] (7)
* Simply-supported and torsionally restrained in \(A\), \(B\): \[ \begin{split}\mathbf{A}_{1}=\mathbf{B}_{1}:=&\text{diag} \left(0,0,0,0,c_{55},c_{66}\right),\quad\mathbf{A}_{0}=\mathbf{B}_{0}:=& \text{diag}\left(1,1,1,1,0,0\right).\end{split} \] (8)  
It is worth noticing that in the static case, that is, when inertial effects are disregarded, the elastic problem (5) consists in a system of ordinary differential equations and boundary conditions in the unknown displacement field. They can be easily solved in closed-form due to the uncoupled constitutive law, entailing that the equations of the equivalent model are the classical (partially uncoupled) equations of the Timoshenko beam model [43]. Moreover, the free dynamics problem associated with Equation (5) also admits a closed-form solution, since the mass matrix is diagonal; see [43; 44; 45].