# Vibration Responses of a Coaxial Dual-Rotor System with Supporting Misalignment
## 2 Modeling of a Dual-Rotor System with Supporting Misalignment  
### Model of a Dual-Rotor System  
The structural diagram of a coaxial dual-rotor system is shown in Figure 1. It is a lumped parameter model [26] of an aeroengine (Figure 2). This coaxial dual-rotor system was obtained through the following three steps: First, the solid model of an aeroengine was built by measuring its size. Second, the solid model was imported into a finite element analysis software to obtain the dynamic parameters of the aeroengine. Finally, the dual-rotor system with similar parameters was designed according to the principle of dynamic similarity. The shaft of an inner rotor is solid, and the shaft of an outer rotor is hollow. Regarding the dual-rotor system, the two rotors are connected by an inter-shaft bearing (Bearing 5). The inner race of the inter shaft bearing is mounted on the inner rotor, while the outer race is mounted on the outer rotor. The outer and inner rotors rotate independently, and can rotate at different rotating speeds and in different directions [26]. Disks 1 and 2 represent a three-stage low pressure blade wheel. Disk 3, 4, and 5 represent a low-pressure turbine, a high-pressure compressor, and a high-pressure turbine, respectively. It is inconvenient for the aeroengine prototype to carry out the misalignment test because it is difficult to adjust the degree of misalignment, and the working speed of the aeroengine is also too high for the misalignment test. Therefore, the structural parameters of the lumped parameter model are optimized according to the principle of dynamic similarity. Ignoring the cross stiffness and cross damping coefficients, the bearing stiffness coefficients (\(\text{k}_{\text{xx}}\) and \(\text{k}_{\text{yy}}\)) and damping coefficients (\(\text{C}_{\text{xx}}\) and \(\text{C}_{\text{yy}}\)) in the horizontal and vertical directions are shown in Table 1. The stiffness of the bearing is much higher than that shown in the table. In order to obtain the bearing stiffness in the table, a special squirrel cage support seat should be designed in the test. The sections of the inner and outer rotor shafts are all circular. The section diameters are expressed as D1, D2, D3, D4, and D5, respectively. The diameters of the shafts are shown in Table 2.  
The five discs have the same size. The diameter, thickness, elastic modulus, and density of each disk are shown in Table 3.  
The dual-rotor system is modeled by the finite element method [36]. The inner rotor is modeled with 15 elements and 16 nodes. The outer rotor is modeled with 8 elements and 9 nodes. Thus, the model has 23 elements and 25 nodes in total. The nodes of the inner rotor are denoted by blue dots. The nodes of the outer rotor are denoted by red squares, which are shown in Figure 1. The node locations are listed in Table 4.  
Assembling all the damping matrices, stiffness matrices, and mass matrices of elements, the general dynamic equations of the dual-rotor system can be written as [36]:  
\[\mathbf{M}\ddot{\mathbf{q}}+(\mathbf{C}+\Omega_{1}\mathbf{G}_{1}-\Omega_{2}\mathbf{G}_{2})\dot{ \mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{F}_{g}+\mathbf{F}_{u}+\mathbf{F}_{b} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the dual-rotor system, respectively; \(\mathbf{q}\) is the displacement vector; \(\Omega_{1}\) and \(\Omega_{2}\) are the rotation speed of the inner rotor and the outer rotor, respectively; \(\mathbf{G_{1}}\) and \(\mathbf{G_{2}}\) are the gyroscopic matrixes of the inner and the outer rotor, respectively; \(\mathbf{F_{g}}\) is the gravity vector; \(\mathbf{F_{u}}\) is the vectors of the unbalance excitation of the rotor system; \(\mathbf{F_{b}}\) is the bearing load vector caused by misalignment of bearings, which will be derived in Section 2.2.