## 2 Dynamic Equilibrium Equation for Structures with Interacting Diaphrags  
For simplicity, diaphrags are assumed to have only three degrees of freedom, namely two in-plane translations, along x- and y-directions, and the rotation around the z-direction.  
Referring to the \(i\)-th diaphragm, one can define \(m_{i}\) as the mass, \(J_{0,i}\) as the polar moment of inertia and \(m_{i}^{x\gamma}\) and \(m_{i}^{y\gamma}\)as static moments, \(\overline{k}_{i}^{x}\) and \(\overline{k}_{i}^{y}\) as, respectively, the translational stiffnesses in the x-direction and in the y-direction, \(\overline{k}_{i}^{y}\) as the torsional stiffness, \(\overline{k}_{i}^{x\gamma}\)and \(\overline{k}_{i}^{y\gamma}\) mixed stiffness terms that regulate the coupling between the translational and rotational degree of freedom, and \(u_{i}\), \(v_{i}\) and \(\gamma_{i}\) as the displacements in the x-direction, in the y-direction, and the rotation, respectively.  
In free undamped vibration conditions, the dynamic equilibrium of the \(i\)-th diaphragm, if connected only to the ground, writes:  
\[\left[\begin{array}{ccc}m_{i}&0&m_{i}^{x\gamma}\\ 0&m_{i}&m_{i}^{y\gamma}\\ m_{i}^{x\gamma}&m_{i}^{y\gamma}&J_{0,i}\end{array}\right]\left\{\begin{array}[ ]{ccc}\ddot{u}_{i}\\ \ddot{v}_{i}\\ \ddot{\gamma}_{i}\end{array}\right\}+\left[\begin{array}{ccc}\overline{k}_ {i}^{x}&0&\overline{k}_{i}^{x\gamma}\\ 0&\overline{k}_{i}^{y\gamma}&\overline{k}_{i}^{y\gamma}\\ \overline{k}_{i}^{x\gamma}&\overline{k}_{i}^{y\gamma}&\overline{k}_{i}^{x }\end{array}\right]\left\{\begin{array}{c}u_{i}\\ v_{i}\\ \gamma_{i}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\} \tag{1}\]  
Now assume that the generic \(i\)-th diaphragm is part of a system of \(n\) interacting diaphrags. The interaction is assumed to be chain-like, i.e., only between adjacent diaphrags, and it is described by means of linear springs.  
In analogy with Equation (1), it is possible to define the mass matrices of the system \(\mathbf{M_{xx}}\) and \(\mathbf{M_{yy}}\), the matrix of polar moments of inertia \(\mathbf{M_{YY}}\) and the matrices of the static moments \(\mathbf{M_{xy}}\) and \(\mathbf{M_{yy}}\), as well as the stiffness matrices along the three directions \(\mathbf{K_{xx}}\), \(\mathbf{K_{yy}}\) and \(\mathbf{K_{YY}}\), and the mixed terms stiffness matrices \(\mathbf{K_{xy}}\) and \(\mathbf{K_{YY}}\), so that the equilibrium equation in compact form writes in terms of \(3n\times 3n\) matrices:  
\[\left[\begin{array}{ccc}\mathbf{M_{xx}}&0&\mathbf{M_{xy}}\\ 0&\mathbf{M_{yy}}&\mathbf{M_{yy}}\\ \mathbf{M_{xy}}&\mathbf{M_{yy}}&\mathbf{M_{yy}}\\ \end{array}\right]\left\{\begin{array}{ccc}\bar{\mathbf{u}}\\ \bar{\mathbf{v}}\\ \bar{\mathbf{v}}\end{array}\right\}+\left[\begin{array}{ccc}\mathbf{K_{xx} }&0&\mathbf{K_{xy}}\\ 0&\mathbf{K_{yy}}&\mathbf{K_{yy}}\\ \mathbf{K_{xy}}&\mathbf{K_{yy}}&\mathbf{K_{yy}}\\ \end{array}\right]\left\{\begin{array}{c}\mathbf{u}\\ \mathbf{v}\\ \gamma\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\} \tag{2}\]  
Defining then the translational stiffness of the springs connecting the \(i\)-th diaphragm with two adjacent diaphrags in the x-direction as \(k_{i}^{x}\) and \(k_{i+1}^{x}\), the stiffness matrix along the x-direction \(\mathbf{K_{xx}}\) writes:  
\[\mathbf{K_{xx}}=\left[\begin{array}{ccccc}k_{1}^{x}+k_{2}^{x}+\overline{k }_{1}^{x}&-k_{2}^{x}&\ldots&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ \ldots&-k_{i}^{x}&k_{i}^{x}+k_{i+1}^{x}+\overline{k}_{i}^{x}&-k_{i+1}^{x}& \ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&\ldots&\ldots&-k_{n}^{x}&k_{n}^{x}+\overline{k}_{n}^{x}\end{array}\right] \tag{3}\]  
Similarly to Equation (3), also the stiffness matrix along the y-direction, \(\mathbf{K_{yy}}\), and rotation \(\gamma\), \(\mathbf{K_{YY}}\), can be formulated.  
The interaction between the \(i\)-th diaphragm and the adjacent ones by means of linear springs is described in Figure 1.