## 2 Methodology of the Analysis Method
### Reference Test Case  
A lumped parameter model of a rotating bladed disk assembly is assumed as a test case for the verification of the method. This type of model was extensively used for investigating the effects of mistuning on the forced response of bladed disks [24; 25; 26; 27]. Similar models [15; 28] and models based on finite element method [10] were also employed by the researchers as numerical simulators for generating the tip timing data in order to test their analysis methods. Figure 3 shows a schematic representation of a lumped parameter model. In the model two degrees of freedom are considered for each bladed disk sector (one for the blade and one for the disk sector). Each sector is attached to its neighboring sectors by the coupling springs, \(k_{c}\). To avoid rigid body motion, an extra spring, \(k_{d}\), connects each sector to the ground. The masses of the blade and the disk in each sector are respectively denoted by \(m_{b}\) and \(m_{d}\), and \(k_{b}\) represents the blade stiffness. The equations of motion for the bladed disk system can be written as:  
\[\mathbf{M\ddot{q}}(t)+\mathbf{C\ddot{q}}(t)+\mathbf{Kq}(t)=\mathbf{F}(t) \tag{8}\]  
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) represent the mass, damping and stiffness matrices, while \(\mathbf{q}\) and \(\mathbf{F}\) denote the displacement and external force vectors, respectively. Very similar models with the same matrices of Equation (8) can be found in [24; 29]. Damping matrix is constructed according to the proportional damping model (\(\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}\)) in which \(\alpha\) and \(\beta\) are known constants. The excitation force is assumed as a traveling wave excitation force acting on each blade. The force can be written as:  
\[f_{i}\left(t\right)=f_{0}\cos\left(\omega t-\frac{2\pi EQ}{N}\left(i-1\right) \right)\text{; }i=1,2,3,\ldots,N \tag{9}\]  
where \(N\) is the total number of blades and \(f_{0}\) is the amplitude of excitation. The excitation frequency is given by \(\omega=EO.\Omega\), where \(\Omega\) denotes the rotating speed. Figure 3: Lumped parameter model of a bladed disk assembly used for simulation. Figure 2: A typical blade response and BTT data during a resonance passage. A typical case of dual mode resonance, is the splitting frequency phenomenon [25; 30] which takes place in a mistuned bladed disk assembly. Mistuning is caused by small unavoidable imperfections in manufacturing process or by deviation of the components from their ideal design shape due to machine operation which eventually breaks the cyclic symmetric features of the system. One major consequence of the mistuning is the splitting of the frequencies of two orthogonal modes. In the tuned system the two orthogonal modes have the same natural frequency. When the symmetry of the system is lost due to mistuning, two frequencies are separated from each other and the corresponding modes appear as two distinct modes with the same nodal diameter. In Figure 4, it is displayed how this phenomenon appears in a Campbell diagram where there are two close lines instead of one for a given mode shape (same nodal diameter). This will cause the excitation of two close resonances by the same EO at two different rotational speeds. Here a bladed disk assembly is assumed as a test case consisting of 48 blades equally spaced around the disk. The parameters for the bladed disk model are given in Table 1. The BTT sample signals are acquired by two virtual sensors located at the angles of 0 and \(\pi/4\). A pattern of imperfections with respect to the tuned system is applied in order to induce the mistuning. Indeed, the mistuning is applied to the assembly based on a random distribution of the blade mass with the standard deviation (SD) of 0.05 in one case and 0.10 in the other case.