## 5 Computational Simulation Results and Discussion
### The Nonsynchronous Response  
The simulations show that the rotor system with SFDs can exhibit not only a synchronous response with a circular orbit but also a nonsynchronous response. The direct time integration method of the equation of motion (1) must be used to compute this type of response. In Figure 7, the blue curve denotes the nonsynchronous response computed at the nondimensional angular speed of a value of 8.7. The black curve represents the contour of the damper clearance, and the red curves illustrate the circular orbits.  
The behaviour of the computed response is assessed with the Poincare section, which consists of intersection points of the trajectory with a section plane at the time period. These points are determined at constant time intervals, corresponding to the period of excitation force. In Figure 7, the points of the Poincare section are drawn with a green colour. The Poincare section is constructed for the time history of displacements over 6000 times the period of the excitation force and the region of the plotted trajectory is filled with a curve of blue colour. The resulting set of intersection points in the Poincare section appears as a continuous closed curve. Therefore, it can be assumed that the nonsynchronous motion has a quasiperiodic character.  
The time history of the displacement in the horizontal direction of the nonsynchronous motion at a nondimensional angular speed of 8.7 is shown in Figure 8. The dependence in  
Figure 6: Frequency response curves for six values of the dimensionless unbalance (**a**) and for the expanded range of the dimensionless speed (**b**).  
Figure 8a is plotted for the time of 40 excitation periods and Figure 8b shows a detailed view. The time dependences exhibit a typical quasiperiodic motion.  
The frequency components calculated from the time history using the discrete Fourier transform (DFT) are shown in Figure 9. The response includes the rotational frequency \(\mathrm{f_{1}}=1\) (in the nondimensional form), a subharmonic component \(\mathrm{f_{2}}=0.2974\), and several of their difference frequencies. The response is a combination of two independent motions, with the ratio of their frequencies being an irrational number.  
The trajectory in the state space with the Poincare section is presented in Figure 10. In Figure 10a, a case with a nondimensional speed of 8.5328 is shown, when the nonsynchronous vibrations occur. Figure 10b depicts the case for a speed value of 10, which is at the end of the investigated region. The kinematic parameters are plotted in the state space for the duration 6000 times the excitation period long. The individual trajectories cannot be clearly distinguished because the envelope of the trajectory is filled with blue curves in Figure 10.  
Figure 8: Time history of the displacement in the horizontal direction of the nonsynchronous motion (**a**) and the detailed view (**b**).  
Figure 7: Nonsynchronous response (**a**) and the detail of its structure (**b**).  
The frequency response curve and the bifurcation diagram of the nonsynchronous response are drawn in Figure 11. The points of the bifurcation diagram are plotted with the blue circle symbols.  
The direct time integration with zero initial conditions was used for computing a nonsynchronous response. The integration was performed with a very fine nondimensional speed step in the vicinity of the origin of the nonsynchronous response. It was found that the origin of the nonsynchronous response occurs in the interval 8.5322-8.5327 of nondimensional speed and has a quasiperiodic character.  
Contrary to the results of [PERSON] and [PERSON] [33], the occurrence of the nonsynchronous vibrations was detected only in the D region of the frequency response curve (see Figure 3) and was not detected in the C region.  
Figure 10: The state space with the Poincaré section (denoted with green points) for the nondimensional speed of 8.5327 (**a**) and 10 (**b**).  
Figure 9: The frequency spectrum of the dimensionless horizontal displacement (**a**) and velocity (**b**).