## 5 Dynamic Simulation and Real Application
### Dynamic Simulation  
The rub-impact model discussed is a Jeffcott rotor system, as shown in Figure 7.  
Figure 5: The waveform of the ideal IF between 0 and 0.6 s.  
Figure 6: The waveform of the estimated IF between 0 and 0.6 s. (**a**) The estimated IF using Hilbert transform; (**b**) The STFT of \(q(t)\) (The length of the window is 128); (**c**) The STFT of \(q(t)\) (The length of the window is 1280). The system can be described by the following equations:  
\[\left( \begin{array}{cc}m&\\ &m\end{array} \right)\left(\begin{array}{c} \begin{array}{c}\cdot\cdot\\ x(t)\\ \cdot\cdot\\ y(t)\end{array} \right)+\left( \begin{array}{c}c\\ &c\end{array} \right)\left( \begin{array}{c}x(t)\\ y(t)\end{array} \right)+\left( \begin{array}{c}k\\ &k\end{array} \right)\left( \begin{array}{c}x(t)\\ y(t)\end{array} \right)= \tag{31}\] \[me\omega^{2}\left( \begin{array}{c}\cos(\omega t)\\ \sin(\omega t)\end{array} \right)-\left( \begin{array}{c}0\\ mg\end{array} \right)+\left( \begin{array}{c}F_{x}(x(t),y(t))\\ F_{y}(x(t),y(t))\end{array} \right)\]  
in which \(m\) is the mass of the rotor system, the gravity of the rotor is \(mg\), \(c\) is the damping coefficient, \(k\) is the stiffness coefficient, \(x(t)\) is the lateral displacement, \(y(t)\) is the vertical displacement, \(e\) is the imbalance, \(\omega\) is the rotation speed, \(F_{x}(x(t),y(t))\) and \(F_{y}(x(t),y(t))\) are the rub-impact forces. The rub-impact forces are shown in the following equation  
\[\left( \begin{array}{c}F_{x}(x(t),y(t))\\ F_{y}(x(t),y(t))\end{array} \right)=-H(r(t)-\delta_{0})\frac{(r(t)-\delta_{0})k_ {c}}{r(t)}\left( \begin{array}{cc}1&-f_{c}\\ f_{c}&1\end{array} \right)\left( \begin{array}{c}x(t)\\ y(t)\end{array} \right) \tag{32}\]  
in which \(r(t)\) is the radial displacement of the disk center, \(\delta_{0}\) is the clearance between the rotor and the stator, \(k_{c}\) is the contacting stiffness for the stator, \(f_{c}\) is the friction coefficient and the Heaviside function \(H(. )\) is defined as  
\[H(u)=\left\{ \begin{array}{c}0,u<0\\ 1,u\geq 0\end{array} \right. \tag{33}\]  
The parameters in the above equations are shown in Table 2.  
Using the fourth order Runge-Kutta method to solve for \(x(t)\). The second derivative of \(x(t)\), which is \(x(t)\), using numerical method can be solved. Additionally, the step size of the numerical derivative is \(2.44140625\times 10^{-6}\)s.  
The waveform and FFT of \(x(t)\) is revealed in Figure 8, and \(\omega\) is set to 5616 r/min. It is illustrated that the amplitudes of \(1\times\) and \(2\times\) are relatively high, and the amplitude of \(3\times\) is lower, as revealed in Figure 8b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(m\) & 44 kg \\ \(c\) & \(7950\) N \(\cdot\) s/m \\ \(k\) & \(5.85\times 10^{4}\) N/m \\ \(e\) & \(4.1\times 10^{-5}\) m \\ \(k_{c}\) & \(3.8\times 10^{7}\) N/m \\ \(f_{c}\) & 0.2 \\ \(\delta_{0}\) & \(10^{-5}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters for the simulated dynamic system. Figure 7: Schematic of the rub-impact rotor system. (**a**) Overview; (**b**) Sectional view. FFT is applied to the IF of 1\(\times\), and the FFT of the IF is revealed in Figure 10a. We can see that the main fluctuating frequencies of the IF of 1\(\times\) are 468.0 Hz, 374.4 Hz, 280.8 Hz, 187.2 Hz, and 93.6 Hz. Additionally, 468.0 Hz is five-times the rotation frequency, 374.4 Hz is four-times the rotation frequency, 280.8 Hz is three-times the rotation frequency, 187.2 Hz is two-times the rotation frequency, 93.6 Hz is the rotation frequency. The envelope spectrum for 1\(\times\) is revealed in Figure 10b. The main fluctuating frequencies of the envelope of 1\(\times\) are 280.8 Hz, 1187.2 Hz and 93.6 Hz. Additionally, 280.8 Hz is three-times the rotation frequency, 187.2 Hz is two-times the rotation frequency, 93.6 Hz is the rotation frequency. Figure 8: Simulated vibration signal when the rotation speed is 5616 r/min. (**a**) The waveform of \(x(t)\); (**b**) Fourier spectrum of \(x(\overset{..}{t})\).