# Nonlinear Dynamics of an Elastic Stop System and Its Application in a Rotor System
## 4 Application to a Flexible Rotor
### Model of the Rotor  
A flexible rotor-support system is shown in Figure 18. There exist three bearings: 1\(\#\) bearing, 2\(\#\) bearing, and 3\(\#\) bearing. The 1\(\#\) bearing is ball bearing, 2\(\#\) and 3\(\#\) bearing are roller bearing. The material is assumed to be 45\(\#\) steel, of which the density is 7800 kg/mm\({}^{3}\) and the elastic modulus is 210 Gpa. The stiffness of the 1\(\#\) bearing-support is 1 \(\times\) 10\({}^{8}\) N/m, and the stiffness of the 2\(\#\) and 3\(\#\) bearing-support is 3 \(\times\) 10\({}^{6}\) N/m. The mass of the 1\(\#\) pedestal is 2 kg (m\({}_{\rm bl}\)) and the masses of the 2\(\#\) and 3\(\#\) pedestal are 1 kg (m\({}_{\rm b2}\)) and 1.5 kg (m\({}_{\rm b3}\)), respectively. In Ansys19.0, the beam element Beam188 is used to model the rotor shaft, the Combin14 element is used to simulate the support, and the Mass21 element is used to simulate the pedestal mass. The first and second order natural vibration modes of the derived linear rotor system are shown in Figure 19, and the corresponding natural frequencies are 57.6 Hz and 159.7 Hz. In engineering, the rotor is often operating above the first critical speed; thus, the excitation frequencies near the first resonance peak are focused on in the following analysis.  
Figure 18: The flexible rotor–support system with three bearings and an overhanging disk: (**a**) structure and parameters; (**b**) nonlinear finite element model.  
To analyze the nonlinear behavior, the model of elastic stop is introduced into the governing equations at the node of 3# bearing. The governing equations can be derived by using the method in Appendix B and are as follows:  
\[\mathbf{M\ddot{q}+}(\mathbf{C}+\omega\mathbf{G})\mathbf{\dot{q}}+\mathbf{Kq}+ \mathbf{F}_{nl}\mathbf{=}\mathbf{F}_{ex} \tag{21}\]  
in which \(\mathbf{M}\), \(\omega\mathbf{G}\), and \(\mathbf{K}\) are the matrix of mass, gyroscopic moment, and stiffness, and are obtained by the finite element method; \(\mathbf{C}\) is the damping matrix and is obtained by  
\[\mathbf{C}=c_{\mathrm{a}}\mathbf{M}+c_{\mathrm{\beta}}\mathbf{K} \tag{22}\]  
\[c_{\mathrm{a}}=\frac{2\zeta\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}} \tag{23}\]  
\[c_{\mathrm{\beta}}=\frac{2\zeta}{\omega_{1}+\omega_{2}} \tag{24}\]  
\(\zeta\) is damping ratio, which is 0.03 in this paper; \(\omega_{1}\) and \(\omega_{2}\) are the lower and upper bounds of the considered frequency, which are 62.8 rad/s(10 Hz) and 6283.2 rad/s(1000 Hz) in this paper.  
\(\mathbf{F}_{\mathrm{ex}}\) is the unbalance excitation force; \(\mathbf{F}_{nl}\) is the nonlinear force caused by elastic stop model and is expressed as  
\[\mathbf{F}_{\mathrm{nl}}=k_{p}\bigg{(}1-\frac{\delta}{r}\bigg{)}f_{c}(r- \delta)\bigg{(}\mathbf{w}_{1}\bigg{(}\mathbf{w}_{1}^{T}\mathbf{q}\bigg{)}+ \mathbf{w}_{2}\bigg{(}\mathbf{w}_{2}^{T}\mathbf{q}\bigg{)}\bigg{)} \tag{25}\]  
\(\mathbf{w}_{1}\) and \(\mathbf{w}_{2}\) are the vectors indicating the direction of the nonlinear force in the system; \(r\) is the radial displacement amplitude and is calculated by  
\[r=\sqrt{\left(\mathbf{w}_{1}[x]\right)^{2}+\left(\mathbf{w}_{2}[x]\right)^{2}} \tag{26}\]  
The harmonic responses of the linear system without pedestal looseness are obtained by Ansys19.0 and MatlabR2017b (the program of the HBM method in Section 2.2), respec  
Figure 19: Natural vibration modes of the derived linear rotor system: (a) the first mode; (b) the second mode.  
tively. In the analysis, the unbalance forces are located at the disk and the middle of the shaft, and the unbalanced forces are 10 g:mm. The response of the disk and the support of 3# bearing are shown in Figure 20. The frequency of the resonance peak is 59.7 Hz due to the gyroscopic effect. The amplitudes obtained by Matlab are higher than the one obtained by Ansys by the ratio of 6%, which verifies the solving method.