## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes  
The PDM shown in Figure 3 were derived purely through kinematic concepts and was not derived through structural interpretation. Thus, a procedure was developed to assemble PDM into three main families: (i) The classic modes, which are consistent with those of the [PERSON] theory; (ii) Primary modes, which are independent with Classic modes in certain vibration modes; and (iii) Secondary modes, which accompany Classic or Primary modes in vibration modes with relatively smaller participation. Among them, the priority is in a descending order in influencing the structural performance. This is just the basis to form a reduced thin-walled beam model: Classic and Primary modes are qualified to derive a higher order model being applicable for most cases, and a selective number of Secondary modes can be supplemented into the model in special cases for a higher accuracy.  
The initial governing equations are a coupled system, and it is difficult to have a deep sight in the relations of PDM in time domain. However, the amplitude functions can be uncoupled by solving the eigenvalue problem related to the homogenous governing equations, namely  
\[\int\limits_{L}\int\limits_{A}\mathbf{H}^{\mathrm{T}}\mathbf{C}^{\mathrm{T}} \rho\mathbf{CH}\frac{\partial^{2}\mathbf{x}}{\partial t^{2}}\mathrm{d}A \mathrm{d}z+\int\limits_{L}\int\limits_{A}\mathbf{H}^{\mathrm{T}}\mathbf{C}^{ \mathrm{T}}\mathbf{ECHx}A\mathrm{d}z=0. \tag{10}\]  
In the process, the finite element implementation adopts the quadratic Lagrange interpolation and leads to \(\mathbf{x}=\mathbf{N}\mathbf{X}\). As a result, the eigenvectors are derived and grouped in a matrix form \(\mathbf{X}\) as  
\[\mathbf{X}=\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)},\cdots,\mathbf{x}^{(k)}, \cdots,\mathbf{x}^{(N)}\right] \tag{11}\]  
where the superscript \((k)\) is the order of an eigenvector. Notice that \(\mathbf{x}^{(k)}\) is grouped with \(N\) amplitude functions as \(\mathbf{x}^{(k)}=[\chi_{1}(z),\chi_{2}(z),\)\(\ldots,\)\(\chi_{N}(z)]^{\mathrm{T}}\), and the amplitude function \(\chi_{i}(z)\) can be expressed in a discretization form as  
\[\mathbf{x}_{i}=\left[\chi_{i}(1),\chi_{i}(2),\cdots,\chi_{i}(N_{\mathrm{e}}) \right]^{\mathrm{T}} \tag{12}\]  
where \(N_{\mathrm{e}}\) is the number of interpolation nodes along \(z\). In other words, an eigenvector \(\mathbf{x}^{(k)}\) is actually spanned with \(N\) amplitude function vectors \(\mathbf{x}_{i}^{(k)}\) which further contain \(N_{\mathrm{e}}\) node values. Thus, it can be rewritten in a matrix form as  
\[\mathbf{x}^{(k)}=\left[\mathbf{x}_{1}{}^{(k)},\mathbf{x}_{2}{}^{(k)},\cdots, \mathbf{x}_{N}{}^{(k)}\right]\in\mathbb{R}^{N_{\mathrm{e}}\times N} \tag{13}\]  
where each column represents the axial variation of an amplitude function. More precisely, the eigenvector \(\mathbf{x}^{(k)}\) is now reformed as an \"eigen matrix\".  
The eigen matrix is deemed to inherit the structural behaviours and possess the information to assemble PDM. Within each eigen matrix, the collinear amplitude functions can be viewed as the quantities describing the same structural behaviour. In this sense, the corresponding set of PDM can be assembled together considering their participation. Therefore, one has to address the issues including determining the collinear amplitude functions (DCAF) in an executable way, assigning the weights of the deformation modes (AWDM) based on their participation and decomposing an amplitude function into components (DAFC) when related to multiple structural behaviours.