## 3 Interpolation Algorithm
### Transfer Function Approximation of Bernoulli Beam  
At this point, we have a differential equation that represents the Bernoulli beam model of the structure. A finite element model is generated from that model in the following form [28]:  
\[\mathbf{M}\ddot{y}(t)+\mathbf{C}\dot{y}(t)+\mathbf{K}y(t)=0 \tag{10}\]  
* \(\mathbf{M},\mathbf{C},\mathbf{K}\in\mathbb{R}^{N\times N}\) are the mass, damping, and stiffness matrices, respectively;
* \(y(t)\in\mathbb{R}^{N}\) is the vector of generalized coordinates;
* \(t\in\mathbb{R}^{+}\) denotes time. It is important to note that this finite element model is a reduced version. Creating a finite element model from the Bernoulli beam model is a standard textbook exercise [31; 32; 33], so it is not detailed here. Note that for damping, we will use Rayleigh damping, but any diagonalizable damping is sufficient. The important factor here is that the new model can contain the interpolation parameter \(\rho\), which is not possible using ordinary FEM model. These parameters can be used to calculate the sensitivity of the natural frequency and, by extension, the sensitivity of the poles of the system. This is very important because if we can calculate the sensitivity of the poles regarding the parameters, then the sampling density of the interpolation parameter can be checked. For now, the  
Let the circle around \(p_{j1}\) when \(\zeta=\zeta_{i+1}\) be \(C_{j1}\) and the circle around \(p_{j2}\) when \(\zeta=\zeta_{i}\) be \(C_{j2}\). So,  
\[C_{j1}=\{p_{j1}+t|(\zeta_{i+1}-\zeta_{i})\lambda_{j,\zeta_{i}}|e^{ i\theta}:\quad\forall t\in[0,1]\;and\;\forall\theta\in[0,2\pi)\}\] \[C_{j2}=\{p_{j2}+t|(\zeta_{i+1}-\zeta_{i})\lambda_{j,\zeta_{i+1}}| e^{i\theta}:\quad\forall t\in[0,1]\;and\;\forall\theta\in[0,2\pi)\}\]  
Let  
\[\mathcal{D}=\{D_{j}=C_{j1}\cup C_{j2}:j=1,2,3\ldots n\}, \tag{14}\]  
where \(n\) is the number of poles. Each \(D_{j}\) can either be connected or disjoint, depending solely on the properties of \(C_{j1}\) and \(C_{j2}\). From now on, we say \(D_{j}\) is connected, meaning that \(C_{j1}\) and \(C_{j2}\) are overlapping. Let  
\[\mathcal{S}=\bigcup_{j=1}^{n}D_{j} \tag{15}\]  
Notice that if all \(D_{j}\) are connected, then \(\mathcal{S}\) can contain at most \(n\) disconnected regions. These regions are the \(D_{j}\), and they are formed by overlapping circles. Now, we describe the sampling criteria of the system. As it is stated before, the interpolated poles must lie on an artificial trajectory between the end positions that are the samples of the system under examination. But it has to be known which pole corresponds to which pole. In [20], pole matching was used based on their hyperbolic distance, but it has no guarantee that the poles matched correctly. This is primarily due to the absence of an estimation for the fineness of sampling across the parameter range. Assuming that every \(D_{j}\) is connected for any sampling, we can define the criterion:  
**Criterion 1**. : _If the set \(\mathcal{S}\) on the complex plane contains exactly \(n\) disjoint subsets, then the pole matching problem is perfectly solved and the interpolating parameter sampling is sufficient, where \(n\) is the number of poles._  
In other words, if every \(D_{j}\) is connected and the sets satisfy \(D_{j}\cap D_{i}=\emptyset\) for all \(i\
eq j\) (i.e., each compound set \(D_{j}\), formed by a pair of circles, is disjoint from all others), then the sampling is fine enough. In the following, we prove that if the circles \(C_{j1}\) and \(C_{j2}\) overlap for every sampling (i.e., all the \(D_{j}\) are connected), then there exists a sampling that generates exactly \(n\) disjoint regions on \(\mathcal{S}\). Proof. : Let \(p_{j1}\) and \(p_{j2}\) denote the \(j\)th pole at \(\zeta_{i}\) and \(\zeta_{i+1}\), respectively. Let \(\lambda_{j,1}\) and \(\lambda_{j,2}\) be the sensitivities of the \(j\)th pole at \(\zeta_{i}\) and \(\zeta_{i+1}\).