A dynamic model of a doubledisk rubimpact rotorbearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotorbearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the perioddoubling bifurcation. With the increase of the eccentricity, the quasiperiodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.
In the previous study of rotor dynamics, the singledisc rotor system is the main research object. The single disk rotor system is a real simplification of the actual rotor system. Its structure is simple, and it can well reflect the dynamic characteristics of the actual system. Scholars at home and abroad have used this model to discover nonlinear phenomena in many rotor systems. Luo et al. [
However, in actual industrial production, there is often more than one rotor, and there may be a plurality of turntables or a plurality of rotor systems connected to each other by a coupling to form a multispan large rotor system. There are many studies on multiple turntables or multiple rotors at home and abroad. Luo et al. [
In this paper, a singlespan doubledisc rotorbearing system model is established by considering the nonlinear bearing oil film force. The numerical analysis method is used to solve the differential equations of the system, and the nonlinear dynamic characteristics of the rotor system are analyzed. This provides a theoretical basis for the rotor system.
As shown in Figure
(a) Mechanical model of the rotor system. (b) Model of rub impact.
The local friction model of the rotating stator is shown in Figure
Decompose the rubbing force into the
Comprehending (
Introducing dimensionless transformations
Substitute it into equation (
Introducing dimensionless transformations
The parameters of the dualdisk rotor system are shown in Table
Parameters of the rotor system.
Parameter  Value 


4.0 kg 

32.0 kg 

32.0 kg 

2.5 × 10^{7} N·m^{−1} 

1 × 10^{7} N·m^{−1} 

1050 N·s·m^{−1} 

2100 N·s·m^{−1} 

0.05 mm 

0.05 mm 

0.018 Pa·s 

25 mm 

12 mm 

0.11 mm 

0.1 
Figure
Bifurcation diagram of the rotor system in the
In order to show in detail the response characteristics of the system in the speed range 200–2500 rad/s, in this section, multiple speeds are selected and the corresponding time histories, axes, phase diagrams, Poincaré sections, etc., are plotted to illustrate the system’s motion.
When
When the rotational speed increases to a certain value, the film force will generate a whirl to change the rotor system’s motion state, and the system will produce doublecycled bifurcation from cycle one to cycle two. Figure
Response of the rotor system when
As the rotation speed continues to increase, the effect of oil film force begins to increase further. At this time, the oil film force is still the main factor affecting the stability of the rotor system. As the rotation speed increases, the whirlpool of the oil film increases and the rotor system becomes unstable. Figure
Response of the rotor system when
Figure
Response of the rotor system when
When the speed increases to 1800 rad/s, the system is a threecycle motion. As the speed increases, the rotor system again enters the quasiperiodic motion after a brief period of three response intervals. Figure
Response of the rotor system when
Eccentricity often exists in the rotor system, and its main causes are structural design irrational, manufacturing or installation process errors, and material inhomogeneity. Eccentricity is the main source of unbalanced forces in the rotor system. From the differential equations, it can be seen that unbalanced forces have a great impact on the system. Under the same parameters, the eccentricity is different, and the magnitude of the imbalance force is different. The response of the system will also be very different. Therefore, it is of great significance to study the influence of eccentricity on rotor system response. In this section, the selected subrigidity is 1 × 10^{7} N/m, and the eccentricity is the changing parameter, and the other parameters remain unchanged.
Figure
Bifurcation diagram of the rotor system in the
When the rotation speed is low, the rotor system performs a stable quasiperiodic motion with an interval of 200–650 rad/s, where the system is in cycle one at 200–500 rad/s. When the rotational speed reaches 500 rad/s, the oil film vortex starts to occur, the system produces a perioddoubling bifurcation and a period two movement, and there are even peaks in the time history diagram. The Poincaré section is the two isolated points, as shown in Figure
Response of the rotor system at different rotation speeds when
When the speed is greater than 650 rad/s, the oil film whirl continues to increase, the system motion state continues to change and enters almost periodic motion state, as shown in Figure
When the rotational speed is greater than 900 rad/s, the rotor system leaves the quasiperiodicity region through the reverse bifurcation process and becomes a quasiperiodicity one. When the rotational speed increases to 1525 rad/s, the rotor system again branches into a wide range of quasiperiodic motion. Figure
Figure
Bifurcation diagram of the rotor system in the
When the rotor rotates at a low speed, the system performs a stable quasiperiodic motion in the range of 200–525 rad/s, and in the speed range 200–500 rad/s, it is a cycle one movement. When the rotational speed exceeds 500 rad/s, the system is affected by the whirl of the oil film. The doublecycle bifurcation becomes a quasiperiodicity second movement. As shown in Figure
Response of the rotor system at different rotation speeds when
With the increase of the rotational speed, the whirl of the oil film at the bearing begins to gradually intensify and finally the oil film destabilizes, and the system continues to branch into the quasiperiodic motion. The rubbing force during the system’s quasiperiodicity movement is also increasing, but the film force is still the main factor affecting the stability of the system. Figure
As the rubbing force increases, the effect of oil film whirl at the bearing is weakened, and the rotor system leaves the quasiperiodicity cycle and enters the cycle two movement with an interval of 650–925 rad/s. As the rotational speed continues to increase, the rubbing force will continue to increase and gradually become the main factor affecting the operation of the system. The system will once again produce a bifurcation to enter the quasiperiodic motion state. Figure
The speed continues to increase, and the rotor system moves backwards into cycle three. When the speed exceeds 1550 rad/s, the rotor system will enter the quasiperiodic motion, as shown in Figure
The response bifurcation diagram with the change of rotation speed is compared when the eccentricity is 0.01 mm, 0.03 mm, and 0.05 mm. It can be seen that when the rotation speed is low, the rotor system motion is not affected by the eccentricity, and it is a stable cycle one and a cycle two movement. The amplitude at the bearing is small, and there is a sharp corner. As the amount of eccentricity increases, the rotor system begins to appear in more abundant forms of movement. In the range of 500–1000 rad/s, as the amount of eccentricity increases, the system’s quasiperiodic motion interval decreases and the period two motion intervals appear. When
According to the calculation formula of the rotorimpact frictional force model, it can be seen that the stator stiffness
Figure
Bifurcation diagram of the rotor system in the
Response of the rotor system at different rotation speeds when
With the intensification of the oil film vortex, the system continues to branch and enters the PK cycle. When the system at a speed of 775 rad/s, the system makes a period 7 movement. When the rotor speed reaches 925 rad/s, the system oil film is in chaos. Figure
Due to the presence of eccentricity in the system, there is always an imbalance force in the rotor system. According to the unbalanced force term in the system dynamics differential equation group, it can be known that when the rotation speed is low, the unbalanced force is small and the system stability is less affected. At this time, the film force is the main factor affecting the stability of the system. With the increase of the rotational speed, the unbalanced force in the system will gradually increase and gradually become the main factor affecting the stability of the system. When the rotation speed exceeds 1050 rad/s, the rotor system goes backwards into the cycle two movement. Figure
Figure
Bifurcation diagram of the rotor system in the
The system performs a stable quasiperiodic motion at low speeds with an interval of 200–675 rad/s. When the rotation speed is less than 525 rad/s, the system is in a cycle one movement. When the rotation speed exceeds 525 rpm, the system will cycle twice as much as the whirl of the oil film. As shown in Figure
Response of the rotor system at different rotation speeds when
The oil film vortex aggravating system continues to occur twice the period into the chaotic movement. As shown in Figure
With the further increase of the rotational speed, the unbalanced force due to eccentricity in the system becomes larger. When the rotor speed exceeds 925 rad/s, the system under the action of the unbalanced force goes through the backward bifurcation and enters the cycle two movement. In this process, the system instability caused by the oil film force is gradually offset, and the system movement is gradually stabilized. When the rotational speed increases to 1100 rad/s, the system enters a short chaotic motion range 1050–1060 rad/s and then goes backwards again into cycle two.
When the speed exceeds 1200 rad/s, the system goes into quasiperiodic motion. Figure
Figure
Comparing the response of the rotor system with different stator stiffness at
In this paper, the dynamic model of the doubledisc rotorbearing system with rubbing fault is established. The numerical integration method is combined with MATLAB to solve the dynamic differential equation of the system. The effects of the speed of the rotor, the disc eccentricity, and the stator stiffness on the response of the rotorbearing system are analyzed. The following conclusions are obtained:
The rotor system moves periodically in the lowspeed range 200–767 rad/s. As the speed of the rotor continues to increase, the influence of the oil film force begins to further increase. When the speed is greater than 767 rad/s, the system loses stability and performs chaotic motion. When the speed is
The rotor system is sensitive to changes in the eccentricity of the disc. When the speed is low, the motion of the system is not affected by the amount of eccentricity, and it is a stable period one and two periods of motion. As the amount of eccentricity increases, the system begins to appear in more abundant forms of motion. In the range of 500–1000 rad/s, as the eccentricity increases, the system’s periodic motion interval decreases and the period two motion interval appears. When
The increase of stator stiffness in the rotor system can suppress the chaotic vibration caused by the oil film oscillation in the system to some extent. However, when the stiffness exceeds a certain value, the chaotic area will appear again. In the interval of more than 1500 rad/s, the change of stator stiffness has little effect on the system response, and the rotor system is in the almost periodic motion state. During this period, there is also a small range of periodic three movements, and the corresponding interval range has not changed significantly.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was financially supported by the National Science Foundation (11272190) and the Key Research and Development project of Shandong Province (Public Welfare Projects): Research on Key Technologies of Vehicle Flywheel Battery Vibration Suppression in Complex Environment (2019GGX103024).