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Nonlinear dynamic characteristics of a rotor-bearing system with cubic nonlinearity are investigated. The comprehensive effects of the unbalanced excitation, the internal clearance, the nonlinear Hertzian contact force, the varying compliance vibration, and the nonlinear stiffness of support material are considered. The expression with the linear and the cubic nonlinear terms is adopted to characterize the synthetical nonlinearity of the rotor-bearing system. The effects of nonlinear stiffness, rotating speed, and mass eccentricity on the dynamic behaviors of the system are studied using the rotor trajectory diagrams, bifurcation diagrams, and Poincaré map. The complicated dynamic behaviors and types of routes to chaos are found, including the periodic doubling bifurcation, sudden transition, and quasiperiodic from periodic motion to chaos. The research results show that the system has complex nonlinear dynamic behaviors such as multiple period, paroxysmal bifurcation, inverse bifurcation, jumping phenomena, and chaos; the nonlinear characteristics of the system are significantly enhanced with the increase of the nonlinear stiffness, and the material with lower nonlinear stiffness is more conducive to the stable operation of the system. The research will contribute to a comprehensive understanding of the nonlinear dynamics of the rotor-bearing system.

In rotational machinery, the rotor is commonly supported by the rolling element bearing. The inner race of the bearing is affixed to a shaft and housing. Accordingly, there exists a coupling effect between the bearings and the rotor. It is essential to consider the rotor and bearings as an integrated part. Dynamic analysis of the rotor-bearing system is of great significance to exactly diagnose the fault of a rotor system. Therefore, much research on this topic has been carried out to analyze the dynamic characteristics of the rotor-bearing system.

Jeffcott rotor has been widely applied to study the fundamental problems of rotor dynamics [

Furthermore, other scientists have carried out research on the dynamics of the cubic nonlinear rotors. In the second section of mathematical modeling in [

It is concluded that the cubic nonlinear stiffness of the elastic support will cause the vibration and the hysteresis of the resonance curve when the rotor is running in the critical speed region, and the excessive amplitude of the rotor will also aggravate its unstable behavior. In summary, the research of this subject is of great significance and is a concrete problem in practical engineering, which is worthy of further and systematic research.

In the rotor-bearing system, there are many nonlinear factors such as bearing clearance, the Hertzian contact force, imbalance, and support materials. The nonlinear factors from the rolling bearing and the rotor may affect the operation of the system together. At present, there are rarely articles to study the synthetical effect of these two types of nonlinear factors. The research objective of this paper is to analyze the dynamic response of the rotor-bearing system under consideration of the cubic nonlinear factors.

The nonlinear dynamic in this research focuses on the global dynamics of the rotor-bearing system instead of the local characteristics. Therefore, a simplified approximate model is developed to research the complicated system of the rotor bearing. It is important to establish the analytical model of the rotor-bearing system. The first step is to build the model of the rolling element bearing. Figure _{1}) and the rotation around the center of the ball. The initial position of the mass center for the rotor at the bearing is _{1}. During the operation of the rotor, the position of the mass center for the rotor is _{2}. As shown in Figure _{1} and the outer race radius is _{2}. The outer race is affixed to the bearing pedestal and its vibration is ignored, and the inner race is connected to the shaft; then, the speed of the outer race for the bearing is

The schematic diagram of the rolling element bearing.

The angular velocity of the ball revolution is given by

Therefore, the position angle of the

Therefore, the radial contact deformation between the

The resultant force of the contact force between each element and the inner race is the force of the rolling bearing on the shaft, consequently,

Another important model for this study is the rotor system. Based on the rolling element bearing, the rotor model will be built in this section.

Among the nonlinear sources of the rotor-bearing system, one is caused by materials of the bearing seat, such as the rubber, the copper, and the aluminum alloys and the other is the radial clearance of rolling bearing, the VC vibration, the ball surface ripple, and the defect, etc. Some scholars have carried out the related experiments and fitted the elastic force and the bearing deformation. There is a mathematical relationship of the cubic nonlinearity, and the fitting results are compared with the traditional analytical model. It is proved that the cubic nonlinear fitting method is more suitable [

Schematic diagram of the rotor-bearing system.

In this model, the rotor is simplified into three lumped masses. The positions of points _{R}, and _{p} is the damping coefficients of the rotor at point _{b} is the damping coefficients of the rotor at both ends of the bearing. The horizontal and the vertical relative displacements at the points _{P}, _{P}, _{R}, _{R}, _{L}, and _{L}, respectively. The horizontal and the vertical relative displacement differences between the point

The horizontal and the vertical relative displacement differences between the point

According to Newton’s second theorem, the dynamic equation of the system can be obtained as follows:_{XR} and _{YR} are the horizontal and the vertical components of the reaction force for the rolling bearing on the rotor at the right end of the rotor, respectively; and _{XL} and _{YL} are the horizontal and the vertical components of the reaction force of the rolling bearing on the rotor at the left end of the rotor, respectively.

For the convenience of the research, equation (_{2} is a dimension parameter; and

The dimensionless governing equations are as follows:

The dynamics of the rotor-bearing system with the cubic nonlinearity are digitally simulated in Matlab. In order to calculate the nonlinear response of the system, the fourth-order Runge–Kutta method is applied to solve the steady-state response of the governing equations. The relevant calculation parameters of the rotor system are shown in Table _{2} is 4 × 10^{−5} mm. The system is analyzed by means of the bifurcation diagram, spectrum diagram, axis track diagram, Poincaré map, and the waterfall chart.

Calculation parameters of the rotor system.

Parameter | Value |
---|---|

Equivalent mass at disk, | 32.1 |

Equivalent mass at the left bearing, | 4.0 |

Equivalent mass at the right bearing, | 4.0 |

Damping coefficient at the disk, | 2100 |

Damping coefficient at the bearing | 1050 |

Linear stiffness coefficient of shaft, | 2.5 × 10^{7} |

A novelty of this study lies in consideration of the cubic nonlinearity of the shaft. Firstly, the influence of the cubic nonlinear stiffness on the dynamic behaviors will be focused on in this section. The model of the rolling element bearing selected in this paper is JIS6306. The relevant parameters are as follows: the inner radius _{1} = 40.1 mm, the outer radius _{2} = 63.9 mm, the number of balls _{1} versus the rotational speed Ω_{1} rad/s when the bearing clearance ^{3}, 3 × 10^{15} N/m^{3}, 3 × 10^{16} N/m^{3}. and 3 × 10^{17} N/m^{3}, respectively.

Bifurcation diagrams of the nondimensional displacement in _{1} versus the rotational speed Ω_{1} rad/s for the clearance ^{3}; (b) ^{15} N/m^{3}; (c) ^{16} N/m^{3}; (d) ^{17} N/m^{3}.

Bifurcation diagrams of the nondimensional displacement in _{1} versus the rotational speed Ω_{1} rad/s for the clearance ^{3}; (b) ^{15} N/m^{3}; (c) ^{16} N/m^{3}; (d) ^{17} N/m^{3}.

It can be seen from Figures ^{15} N/m^{3}_{,} the speed scope of the jumping phenomenon expands to 775–885 rad/s, as shown in Figure ^{17} N/m^{3}, only one jumping phenomenon exists at the beginning at the rotating speed of 792 rad/s and the chaos is more chaotic than that of smaller stiffness, as shown in Figure

The clearance between the element and races has a certain effect on the nonlinear behaviors of the responses. In order to study the influence of the clearance, the bifurcation diagram for the clearance _{0} = 20

The rotating speed is one of the key parameters affecting the dynamic behavior of the rotor system. In this section, the bifurcation diagrams, the waterfall plots, the time domain diagrams, the phase diagrams, and the Poincaré maps of the rotor system with nonlinear stiffness are drawn to exactly analyze the dynamics.

Figure _{1} versus the rotational speed Ω_{1} rad/s for ^{16} N/m^{3}; Figure _{n}) with the changing rotating speed Ω_{1} under the same parameters ^{16} N/m^{3}. Figure _{1} (0–1380 rad/s) because the varying compliance vibration is weaker when the rotating speed is lower, and the vibration induced by the eccentric force dominates the vibration; when the rotating speed increases, the spectrum diagram shows the combined frequencies including the 0.5

(a) Bifurcation diagram of the nondimensional displacement in _{1} versus the rotational speed Ω_{1} rad/s for ^{16} N/m^{3}.

Dynamic behaviors at the typical rotating speed. (a) P-1 motion: (a1) time history, (a2) spectrum, chart of axis track, (a3) phase trajectory, and (a4) Poincaré section when

Because of the uneven distribution of rotor mass, the rotor will produce eccentric force during the operation of the system. The existence of eccentric force leads to a big risk for the safety of the system. The exciting forces induced by the mass imbalance are the major source of rotor vibrations in the field, so it is necessary to analyze the effect of eccentricity on the motion characteristics of the rotor-bearing system. In order to study the influence of mass eccentricity on the rotor-bearing system, the nonlinear stiffness system is analyzed in this section. The imbalance is represented by the eccentricity of the rotor

Figure _{1} as the control parameter. Figure _{1} = 3 × 10^{16} N/m^{3}. Applying the dynamic analytical method similar to the previous section, it can be known that the orange circles in Figure

The bifurcation and waterfall chart when (a) ^{16} N/m^{3} and (b) ^{16} N/m^{3}.

A rolling rotor-bearing system with cubic nonlinear stiffness is established; in the model, the effects of the unbalanced excitation, the bearing clearance, the nonlinear Hertzian contact force, the varying compliance vibration, and the nonlinear stiffness of shaft material on the system are fully considered. The influences of the bearing clearance, the shaft nonlinear stiffness, and the rotating speed on the dynamic behaviors of the system are studied, and the main conclusions are as follows:

In engineering applications, the bearing clearance should be decreased as much as possible to reduce the risk of the system in chaos and jumping; the critical speed and the response peak of the system can be controlled by adjusting the stiffness of the shaft under specific scopes. The nonlinear characteristics of the system are significantly enhanced with the increase of the nonlinear stiffness. Therefore, the material with lower nonlinear stiffness is more conducive to the stable operation of the system.

The rotor system supported by the rolling bearing not only has the frequency components of varying compliance vibration and the working rotating speed, but also has the frequency components of the frequency division, the frequency multiplication, and frequency combination. When the speed increases, the frequency components corresponding to the speed gradually become the dominant frequency components. The larger the imbalance in the low-speed area and the high-speed area, the more unstable the system response.

The motion of the system will be in different motion states with the increase of the rotating speed under the condition of the same eccentricity. Under the condition of changing the eccentricity, the system usually maintains the same motion state, and only a few motion states will be deviated, usually at the rotating speed of the junction range for the chaotic motion and the periodic motion.

The mass eccentricity has an obvious effect on the dynamic behavior, with the increase of the eccentricity, the range and extent of the chaos has significant change, the amplitude and the rotational speed for the occurrence of the 1

All data, models, and code generated or used during the study appear in the submitted article.

The authors declare that they have no conflicts of interest.

The authors are grateful for the financial support provided by the National Natural Science Foundation of China under Grant number 51305267.