## 3 Contact Model
### Reduction Technique  
The search for the values of the \(k_{n}\) and \(k_{t}\) was carried out by comparing the frequencies of the first and second flexural modes of the FEM model with those obtained from the measurements. This procedure required the use of a global method varying \(k_{n}\) and \(k_{t}\). In this regard, the degrees of freedom of the beam and support were reduced. In particular, the technique proposed by [PERSON] [24] was adopted in the present study, already implemented in commercial finite element software. The method implements a change of coordinates of the equations of motion to obtain a description of the system with a reduced number of generalized coordinates. The first step is to break down the degrees of freedom of the model into master DOFs (m), which will be also present in the final reduced model, and slave DOFs (s). In a general case, the system of equations of the motion is of the form:  
\[[M]\left\{\ddot{u}\right\}+[K]\left\{u\right\}=\left\{0\right\} \tag{1}\]  
where the matrices \([M]\) and \([K]\) and the vector \(\left\{u\right\}\) can be considered in the following form:  
\[\left\{u\right\}=\left\{ \begin{array}{c}u_{m}\\ u_{s}\end{array} \right\} \tag{2}\]  
\[[M]=\left[ \begin{array}{cc}M_{mm}&0\\ 0&M_{ss}\end{array} \right]\]  
\[[K]=\left[ \begin{array}{cc}K_{mm}&K_{ms}\\ K_{sm}&K_{ss}\end{array} \right]\]  
Defining the static modes \(\left[\overline{\Phi}_{C}\right]\) as the displacements of the slave DOFs due to the displacements of master DOFs, one can extract:  
\[[K_{sm}]\{u_{m}\}+[K_{ss}]\{u_{s}\}=\{0\}\ \ \rightarrow\ \ \{u_{s}\}=\left[\overline{\Phi}_{C}\right]\{u_{m}\} \tag{3}\]  
with  
\[[\overline{\Phi}_{C}]=-[K_{ss}]^{-1}[K_{sm}] \tag{4}\]  
The internal modes \([\Phi_{N}]\) are defined as:  
\[\omega^{2}[M_{ss}][\Phi_{N}]=[K_{ss}][\Phi_{N}] \tag{5}\]  
When all the \"master DOFs\" are bound, it is now assumed that the movements of the \"slave DOFs\" can be approximated by a combination of a subset of the internal modes \([\Phi_{N}]\) as:  
\[\{u_{s}\}=[\Phi_{N}]\{p_{N}\}=[\overline{\Phi}_{N}][\Phi_{N}]\bigg{\{} \begin{array} []{c}\overline{p}_{N}\\ \ p_{N}\end{array} \bigg{\}}\simeq[\overline{\Phi}_{N}]\{\overline{p}_{N}\} \tag{6}\]  
Figure 13: Detail of dovetail joint in FEM model with nodes at contact interfaces. A: SlotContact_Xp_Zp_Nodes. B: SlotContact_Xp_Zn_Nodes. Defining a new generalized coordinate system consisting of the master DOFs and the coordinates \(\{p_{N}\}\) as:  
\[\{p\}=\left\{ \begin{array}{c}u_{m}\\ \overline{p}_{N}\end{array} \right\} \tag{7}\]  
Thus, the movements of the slave DOFs can be written according to the new coordinates using the static modes and internal modes in the following form:  
\[\{u_{s}\}=[\Phi_{\mathbb{C}}]\{u_{m}\}+[\Phi_{N}]\{p_{N}\} \tag{8}\]  
The above relation implies that the initial degrees of freedom and the new generalized coordinate system are related by the following transformation matrix:  
\[\left\{ \begin{array}{c}u_{m}\\ u_{s}\end{array} \right\}=\left[ \begin{array}{cc}I&0\\ \overline{\Phi}_{\mathbb{C}}&\overline{\Phi}_{N}\end{array} \right]\left\{ \begin{array}{c}u_{m}\\ \overline{p}_{N}\end{array} \right\} \tag{9}\]  
Defining \([\alpha]=\left[ \begin{array}{cc}I&0\\ \overline{\Phi}_{\mathbb{C}}&\overline{\Phi}_{N}\end{array} \right]\) and \(\{p\}=\left\{ \begin{array}{c}u_{m}\\ \overline{p}_{N}\end{array} \right\}\), the equations of motion of the structure in the new coordinates are therefore:  
\[[\alpha]^{T}[M][\alpha]\{\breve{p}\}+[\alpha]^{T}[K][\alpha]\{p\}=\{0\}\ \ \rightarrow\ \ [\overline{M}]\{\breve{p}\}+[\overline{K}]\{p\}=\{0\} \tag{10}\]  
where  
\[[\overline{K}]=[\alpha]^{T}[K][\alpha] \tag{11}\]  
\[[\overline{M}]=[\alpha]^{T}[M][\alpha]\]  
To analyze the problem of interest, the master nodes chosen for the beam are:  
* 33 nodes for each of the four contact interfaces
* 21 nodes along the longitudinal axis of the beam on each side, see Figure 14b, to be capable of viewing the mode shapes of the reduced model.