## 2 Mathematical Model
### The Motion Governing Equation of Gas Turbine Rotor with Blade off  
Considering the complexity of the rotor structure investigated, the varying cross-section structures are simulated with a series of cylindrical and conical beam elements; after that, the system's DOFs are reduced by Guyan theory. On the basis of this, a typical high-speed gas generator rotor with complex structures shown in Figure 1 is discretized based on the FE method and Guyan reduction technology, as shown in Figure 3a. To facilitate the understanding of the modeling process, the final equivalent mechanical model is presented in Figure 3b, in which the rotatory inertia, shear effect, and gyroscopic moment are considered. Each node of the main element has four DOFs (two translational and two rotational DOFs). In view of the fact that the left and right bearings are rolling bearings, the supports are modeled as linear spring-damping models, where the supporting stiffness and damping are \(2\times 10^{4}\) N/mm and 0.5 N\(\cdot\)s/mm, respectively. The blades at each stage are regarded as lumped mass [33] applied to the corresponding nodes, such as nodes 4, 7, 8, and so on. For a cylindrical or conical beam element shown in Figure 3a, the (8 \(\times\) 8) inertia or stiffness matrixes can be expressed in the general form  
\[A^{\text{e}}=\left[ \begin{array}{cccccc}a_{11}&&&&\\ 0&a_{22}&&&&sym&&\\ 0&a_{32}&a_{33}&&&&\\ a_{41}&0&0&a_{44}&&&&\\ a_{51}&0&0&a_{54}&a_{55}&&&\\ 0&a_{62}&a_{63}&0&0&a_{66}&&\\ 0&a_{72}&a_{73}&0&0&a_{76}&a_{77}&\\ a_{81}&0&0&a_{84}&a_{85}&0&0&a_{88}\end{array} \right] \tag{12}\]  
Corresponding to Equation (12), the (4 \(\times\) 4) stiffness and inertia matrixes in _XOZ_ plane are well documented in any book [34] or reference [35; 36] about rotor dynamics. Once the 4-by-4 matrix is known, the 8-by-8 matrix (Equation (12)) can be obtainedaccording to the relationships between the \(XOZ\) and \(YOZ\) planes [37]. The gyroscopic matrix can be written as  
\[G^{\rm e}=\left[ \begin{array}{cccccc}0&&&&\\ G_{21}&0&&\text{skew}&\text{sym}\\ G_{31}&0&0&&\\ 0&G_{42}&G_{43}&0&&\\ 0&G_{52}&G_{53}&0&0&&\\ G_{61}&0&0&G_{64}&G_{65}&0&\\ G_{71}&0&0&G_{74}&G_{75}&0&0\\ 0&G_{82}&G_{83}&0&0&G_{86}&G_{87}&0\end{array} \right] \tag{13}\]  
\[G_{21}=\frac{-\rho I}{15L(1+\Phi)^{3}}(36),\ G_{31}=\frac{\rho I}{15L(1+\Phi)^ {3}}(3-15\Phi)L,\]  
\[G_{43}=\frac{-\rho I}{15L(1+\Phi)^{3}}(4+5\Phi+10\Phi^{2})L^{2},\ G_{83}=\frac{ -\rho I}{15L(1+\Phi)^{3}}(-1-5\Phi+5\Phi^{2})L^{2},\]  
\[G_{61}=-G_{21},G_{71}=G_{31},\ G_{42}=G_{31},G_{52}=G_{21},G_{82}=G_{31},\]  
\[G_{53}=G_{31},G_{64}=G_{31},\ G_{74}=-G_{83},G_{65}=G_{21},G_{75}=-G_{31},G_{ 86}=-G_{31},\ G_{87}=G_{43}.\]  
where \(\rho\), \(I\), \(L\), \(E\), and \(G\) represent the density, area moment, length, elastic modulus, and shear modulus, respectively. \(\Phi=12 EI/(0.886 AGL^{2})\). Once all of the sub-element beam equations in each main element are deduced into the 8 \(\times\) 8 main matrix on the basis of Guyan theory, the mass matrix (\(\mathbf{M}_{\rm r}\)), the stiffness matrix (\(\mathbf{K}_{\rm r}\)), and the gyroscopic matrix (\(\mathbf{G}_{\rm r}\)) for the whole rotor system can be obtained by assembling all of the main element matrices, and the assembly procedure is depicted in  
Figure 3: Dynamic model of the rotor-support system with 18 master nodes. (**a**) FE model with master- and sub-elements. (**b**) The mechanical model equal to (**a**). (**c**) Supporting model. Figure 4. Then, based on the rotor dynamics theory, the dynamic governing equation with 72 DOFs describing the rotor motion induced by blade off is deduced as  
\[(M+M_{\text{b}}(t))\ddot{\mathbf{q}}+(\mathbf{C}+\mathbf{G}+\mathbf{G}_{\text{b}}(t))\dot{\mathbf{q}} +\mathbf{K}\mathbf{q}=\mathbf{F}(t) \tag{14}\]  
where \(\mathbf{q}=[x_{1},y_{1},\theta_{x1},\theta_{y1},\dots,x_{18},y_{18},\theta_{x18}, \theta_{y18}]^{\text{T}}\) is the generalized displacement vector.