Vibration characteristics of rotor system with disk-shaft clearance-eccentric coupling faults

Abstract

Clearance between the disk and shaft can lead to the aggravation of the system fault. To solve this problem, a disk-shaft dynamic model with the clearance-eccentric coupling faults is established by the finite element method. Based upon the model, the dynamic characteristics of the system with different influence factors such as clearance and rotating speed are acquired by comprehensive analysis. Meanwhile, the accuracy of the dynamic model is verified by experiment investigations. The results show that frequency components in the spectrum include the shaft’s rotating frequency and high frequency multiplication components. And the sawtooth phenomenon from the trajectory diagram of disk center can be observed. With the increase of rotating speed, the disk-shaft contact stress and strain energy first decrease and then increase, while the speed difference of disk to shaft and vibration amplitude increase, and the number of high frequency multiplication components in the spectrum decreases gradually. Furthermore, with the increase of clearance, the stable values of the disk-shaft contact stress and strain energy decrease, while the vibration amplitude, the number of high frequency multiplication components and speed difference increase gradually, and the trajectory shape of disk center changes from closed circle to hollow ring.

Keywords

Disk-shaft system clearance-eccentric coupling faults dynamic characteristics sawtooth phenomenon

Introduction

Rotating machinery can be widely used in various industrial applications, such as energy, electric power, transportation, petrochemical industry, and astronautics, and plays an irreplaceable important role.^1–6 Due to the weak installation quality and long-term vibration, looseness of rotating machinery components is common, which will affect the dynamic characteristics of system and even results in structure failure. The research of looseness fault has attracted the attention of many researchers at the moment and many achievements has been achieved.^7–9 To solve the problem of the misdiagnosis of the bearing looseness fault and other faults such as misalignment and rub-impact, Dong et al.¹⁰ proposed a fault signal recognition method. According to the numerical simulation and experimental investigation on the single span rotor system model with looseness at one end, they analyzed the spectrum characteristics of fault signal and summed up the changing regularity under different faults. Based on the lumped mass model, Yu et al.¹¹ built the failure models of three fasteners, and further discussed the dynamics response of the three kinds of rotor system. Using the numerical analysis method, a rotor-bearing-pedestal systems are constructed by Cao et al.¹² for analyzing the dynamic characteristics under different fit clearances. They investigated the dynamic characteristics of rotor system by employing the time displacement responses and Fourier spectra and given a description of the stability changes for the rotor system by the phase trajectory. Mercier et al.¹³ did a non-linear dynamic analysis on the disk-shaft system subjected to the friction at the disk-shaft contact interface and discussed the effect of the relative velocity deviations at each node of the contact interface on the dynamic characteristics of the system. Wang et al.¹⁴ investigated the non-linear dynamic response of a rod fastening rotor system considering the effect of the residual unbalance. Aimed at the problem of the bolt looseness, the finite element method was employed by Qin et al.¹⁵ to simulate the severity of bolt looseness under different preloads, and the change rule of local stiffness for the rotor was analyzed. Based on augmented Lagrange method, the rotor-sliding bearing system featured with looseness-rubbing coupling faults was introduced by Liu et al.¹⁶ to discuss the non-linear dynamic characteristics. In order to identify the pedestal looseness of the disk-shaft system, a non-linear evaluation method was proposed by Jiang et al.^17,18 and the dynamic characteristics under different pedestal looseness clearances were investigated. Yang et al.¹⁹ proposed the geometrical non-linear disk-shaft system featured with imbalance-rub-pedestal looseness coupling faults, and the variation of the resonant performance and rub region under different loose stiffness was revealed by the investigation on the effect of the disk-shaft rub, geometrical nonlinearity, and the pedestal looseness on the dynamic performance for the system. For the fit looseness fault of the outer ring for bearing, the numerical integration method was employed by Wang et al.²⁰ to discuss the response performance of the rotor-rolling bearing system with coupling faults.

However, these studies mainly focus on the looseness fault of non-rotating parts, and the looseness fault of rotating parts is neglected. As the core work component of rotor systems, the rotating disk is operated with harsh operation condition long time, and the looseness fault of the mating surface between the disk and shaft is easy to occur. For instance, the rotating speed of the rotor similar to the turbocharger compressor will higher than 30,000 r/min and the inlet temperature can reach 700°C. Besides, in this extreme circumstance, the integral impeller of the turbocharger compressor is connected with the bushing and the spindle by means of the gland with taper, the disk-shaft mating interface is more vulnerable to the looseness fault, thereby the strong nonlinearity of the disk-shaft connection and the uncertainty of the location for the unbalanced mass can be caused. Hence, the looseness fault of the rotating parts should be deserved an intensive attention, otherwise, it will create a serious safety problem in engineering practice and cause enormous economic damages.

Based on the deficiency, some preliminary exploration on the looseness fault of the rotating parts has been provided by researchers. Considering the looseness fault of the connecting bolts between rotating turntable and rotary drum used in large rotating machines, Qin et al.²¹ analyzed the time varying stiffness of joint interface with bolt looseness by constructing a nonlinear finite element model. The finite element code was developed by Behzad and Asayesh ²² for exploring the response performance of the rotor-bearing system under the looseness fault of the rotating disk. However, the developed model only analyzes the dynamic feature when the disk-shaft clearance was large and assumed that the disk-shaft was always in contact, which was inconsistent with the clearance or interference fit of the disk-shaft contact interface in practical running. Through the investigation of the numerical simulation and experimental study, a novel contact model of looseness fault for the rotating part was developed by Wei et al.²³ to analyze the dynamic characteristics of the rotating part under the looseness fault. Based on the Hertzian impact-contact theory, the disk-shaft rubbing model and the disk-shaft dynamic model with looseness fault were established by Liu et al.²⁴ to investigate the non-linear dynamic properties. According to model and simulate the looseness fault of the disk-shaft system, Li et al.²⁵ analyzed the effects of structure parameters for the motion state of rotor system, and further discussed the influence of oil film force on the vibration characteristics of the fault system.

Only the single looseness fault of the disk-shaft system is considered at the moment. However, the occurrence of looseness fault will inevitably produce twin faults. Compared with the single fault, the nonlinear vibration characteristics of the coupling faults is highly complicated. Additionally, the tiny clearance will be appeared on the disk-shaft connecting surface when the failure of interference connection. At this time, the motion state between the disk and shaft will be changed and the motion state can be divided into four parts, i.e., contact, separation, collision, and re-contact. Moreover, with the disk-shaft vibration intensified, the more complex dynamic characteristics may be caused by the clearance-eccentric coupling faults. Hence, aimed at the problem of the disk-shaft clearance-eccentric coupling faults, the finite element method is employed to build the disk-shaft system model. Based on the model, the change law of the disk-shaft contact stress, strain energy, disk-shaft speed difference, time-frequency characteristics, and the trajectory of disk center under different disk-shaft clearances and rotating speeds are discussed. Furthermore, according to construct the rotor system test-bed, the simulation results can be verified through experiments. Moreover, the nonlinear characteristics of the system under the disk-shaft clearance-eccentric coupling faults can be further revealed by analyzing the vibration characteristics of the system.

The disk-shaft system model under the clearance-eccentric coupling faults

The disk-shaft system model

As shown in Figure 1, the model diagram of the disk-shaft system under the disk-shaft clearance-eccentric coupling faults can be expressed, where both ends are supported by bearings. Moreover, the cooperation style of disk-shaft contact interface is defined as interference fit. In order to better analyze the dynamic characteristics of the disk-shaft system with clearance-eccentric coupling faults, the rotating disk is assumed to install at the center part of the shaft and the whirl effect of the rotor caused by other parts is ignored.

Figure 1.

Disk-shaft system model.

As the clearance between the disk and shaft appeared, the disk and shaft are no longer considered as a whole, and the motion between the disk and shaft is maintained by contact friction. Hence, the motion formula is employed to express the motion relationship between the disk and shaft. In Figure 1, the rotating shaft is equivalent to a concentrated mass block with mass m₁, and its center position is O₁. The rotating disk can be equivalent to a concentrated mass block with a mass of m₂, and its center position is O₂. The position of the concentrated mass block is the position of the shaft of the disk-shaft contact section. The shaft and disk can vibrate in a plane space and act relative to each other.

Dynamic equation

During vibration of the rotating shaft, it is assumed that m₁, k₁ and c₁ are the concentrated mass, the concentrated stiffness coefficient, and the concentrated damping coefficient of the rotating shaft, respectively, x₁ and y₁ are the vibration displacement of the centroid for the rotating shaft, respectively. Hence, the rigid restoring force of the rotating shaft can be expressed by -k₁x₁ and –k₁y₁, and the damping force can be defined by –c₁ẋ₁ and –c₁ẏ₁. Additionally, the rotating shaft is also affected by the gravity -m₁g, and there are relative forces F_x, F_y between the rotating shaft and the rotating disk. Hence, the motion equation of the rotating shaft can be defined as

\begin{array}{l} m_{1} x_{1}^{″} + c_{1} x_{1}^{'} + k_{1} x_{1} + F_{x} = m_{1} e {φ^{'}}^{2} \cos φ + m_{1} e φ^{″} \sin φ \\ m_{1} y_{1}^{″} + c_{1} y_{1}^{'} + k_{1} y_{1} + F_{y} = m_{1} e {φ^{'}}^{2} \sin φ + m_{1} e φ^{″} \cos φ - m_{1} g \end{array}

(1)

in which e is the eccentricity of the rotating shaft and φ express the rotating angle of the mass center for the rotating shaft.

The motion equation of the mass centre for the rotating disk can be expressed as

\begin{array}{l} m_{2} x_{2}^{″} + c_{2} x_{2}^{'} - F_{x} = m_{2} u {θ^{'}}^{2} \cos θ + m_{2} u θ^{″} \sin θ - F_{o x} \\ m_{2} y_{2}^{″} + c_{2} y_{2}^{'} - F_{y} = m_{2} u {θ^{'}}^{2} \sin θ - m_{2} u θ^{″} \cos θ - m_{2} g - F_{o y} \\ J_{p} θ^{″} + c_{θ} θ^{'} = M_{c} - m_{2} g u \cos θ \end{array}

(2)

in which m₂, c₂ and u are the equivalent mass, damping coefficient, and eccentricity of the rotating disk, respectively, x₂ and y₂ are the vibration displacement of the centroid of the rotating disk, respectively, and θ is the rotating angle of the rotating disk. Besides, the components of centrifugal force in x direction and y direction can be expressed by F_ox and F_oy, respectively, the moment of inertia for the rotating disk can be represented by J_p, and c_θ, M_c is the rotational damping moment coefficient of the rotating disk and the rotational torque provided by the friction between the disk and shaft.

Additionally, the unbalanced force F₀ can be produced due to the eccentric fault of the rotating disk occurs, and the unbalanced force can be denoted by

F_{0} = U ω^{2} = m_{2} l ω^{2}

(3)

in which U is the unbalance of the rotating disk, l is the distance from the centroid to the rotor axis position, and ω is the rotating speed of the disk.

Modeling of the disk-shaft system under the clearance-eccentric coupling faults

To better simulate the nonlinear vibration characteristic of the disk-shaft system under the clearance-eccentric coupling faults, a finite element model of the system is established by the finite element analysis software ANSYS which has been demonstrated to as a powerful method in earlier research.^26–28 As shown in Figure 2, the structure parameters of the rotating disk can be described by the internal diameter with 19 mm and external diameter with 76.2 mm, and those of the rotating shaft can be described by the diameter with 19 mm and length with 320 mm. Besides, the radius of the eccentric mass block is assumed as 5 mm for simulating the eccentric fault of the rotating disk.

Figure 2.

Finite element model of the disk-shaft system.

The material properties of the shaft and disk for the system is same. The physical prosperities of the material during modeling can be denoted as that the density is 7890 kg/m³, the elastic modulus is 209 GPa, and the Poisson’s ratio is 0.269. Additionally, the material properties are assumed to be isotropic within the calculation.

In modeling, the cooperation style between the disk and shaft is defined as friction contact and interference fit. Additionally, the different interference fit can be simulated by the adjustment of the interference. In order to simulate the vibration characteristics, the material models of the rotor system are made of flexible materials. For simulating the support function of bearing, spring parts are employed to define the stiffness and damping. Besides, the rotating of the shaft can be achieved by the addition of the rotating pair. Gravitational acceleration g is added along the -y direction for all components. At the same time, a certain rotating speed is applied to the rotating shaft.

Due to the cooperation style between the disk and shaft is defined as the interference fit, the disk-shaft contact state can be reflected by the variation of contact stress. Moreover, the disk-shaft contact state can be changed by the variation of clearance. With the eccentricity fault occurs, different rotating speeds will make the rotating disk produce different centrifugal force. Consequently, the disk-shaft clearance, and the shaft’s rotating speed are important factors affecting the running state of disk and shaft. The effects of the disk-shaft clearance and the shaft’s rotating speed on the dynamic characteristics between the disk and shaft are further discussed within the following research.

Simulation

In the previous sections, the finite element model of the disk-shaft system has been constructed. Moreover, the disk-shaft clearance and the rotating speed of the shaft have been found to be key factors for the determination of the disk-shaft running state. Hence, based on the finite element model, this section mainly discusses the changes of the disk-shaft contact stress, the strain energy, the time-frequency characteristics, the trajectory of the disk center, and the disk-shaft speed difference under different speeds or clearances.

Variation of the contact stress

In this section, finite element simulations are employed to discuss the variations in the contact stress of the disk-shaft contact interface with the rotating speed and clearance, as shown in Figures 3 and 4. The solid yellow line in the figure represents the mean value under steady state.

Figure 3.

Variation of the disk-shaft contact stress under different rotating speeds.

Figure 4.

Variation of the disk-shaft contact stress under different clearances.

From Figure 3, the evolution process of contact stress along with the rotating speeds can be observed. Besides, the disk-shaft clearance is defined as 0.001 mm and remains unchanged. As presented in Figure 3, we can observe that the contact stress present periodic variation with the increase of time. Moreover, with the increase of rotating speed, the amplitude of the contact stress decreases first and then increases. It is widely known that the collision between the disk and the shaft can be produced when the clearance-eccentric coupling faults occurs. Furthermore, the collision phenomenon between the disk and shaft will result in the increase of the contact force, that is, the contact stress increases.

As shown in Figure 4, the evolution process of contact stress between the disk and shaft along with the clearance can be observed, where the rotating speed is 1800 rpm. Consider the practical engineering application, the clearance between the disk and shaft can not be too large. Consequently, the clearances in Figure 4 with 0.001 mm, 0.002 mm, 0.003 mm, and 0.005 mm are investigated. From Figure 4, it can be seen that the final stable value of the disk-shaft contact stress gradually reduce with the increase of clearance. However, it is generally known that the variation of the eccentric force can be achieved by the adjustment of the rotating speed, and the eccentric force is constant when the rotating speed is constant. In other words, with the increase of clearance, the rotating disk and the rotating shaft are not always in contact, so the contact stress between the disk and shaft decrease.

Variation of the strain energy

Actually, the strain energy of the system can be determined by the stress and strain. Besides, the change of stress and strain can be reflected by the variation of the strain energy. Hence, we focus on the variation of the strain energy in this section, and the results are illustrated in Figures 5 and 6.

Figure 5.

Variation of the strain energy under different rotating speeds.

Figure 6.

Variation of the strain energy under different clearances.

Figure 5 shows the variation of strain energy at different speeds under a clearance of 0.001 mm. It can be seen from the Figure 5 that the amplitude of strain energy first decreases slightly and then increases with the increase of rotating speed, and the number of periods for the strain energy also increases. The reason for this phenomenon is that the eccentric force of the disk increases with the rotating speed increases, the contact stress also increases, and the collision between the disk and the shaft is intensified, so the strain energy increases.

The change law of the strain energy as a function of the clearance between the disk and shaft under the rotating speed with 1800 rpm is given in Figure 6. From Figure 6, we can see clearly that the amplitude of strain energy decreases with the increase of disk-shaft clearance. Besides, the initial value of the strain energy has bigger change without obvious rule. Additionally, compared with the simulation results in Figure 4, the change law of the strain energy is similar to the contact stress, and therefore the relationship between the strain energy and the contact stress can be verified.

Analysis of the time frequency characteristics for the disk-shaft system

As shown in Figure 7, the time-frequency diagram of the disk-shaft system under the speed with 900 rpm can be observed when the clearance between the disk and shaft is defined as 0.001 mm. The data used in the time-frequency diagrams are all from the x direction. It can be seen from Figure 7(a) that the vibration amplitude varies regularly with time. From Figure 7(b), it can be observed that the frequency component is composed of the rotating frequency of the shaft and high frequency multiplication components, in which the generation of high frequency multiplication components is caused by the rubbing and eccentricity between disk and shaft.

Figure 7.

Time frequency diagram of the disk-shaft system under the speed with 900 rpm.

Similarly, the time-frequency characteristic of disk-shaft system under the speed with 1200 rpm and 1800 rpm is also discussed, as illustrated in Figures 8 and 9. It can be also seen that the regular variation of the vibration amplitude with time. Compared with the change law shown in Figure 7(a), it can be found that the vibration amplitude increases with the increase of rotating speed. In other words, with the increase of rotating speed, the intensification of the rubbing phenomena between the disk and the shaft can be found, and the increase of the centrifugal force, thereby the vibration amplitude becomes larger. In spectrum diagram, the rotating shaft’s rotating frequency and the high frequency multiplication components can be also found within the spectrum.

Figure 8.

Time frequency diagram of the disk-shaft system under the speed with 1200 rpm.

Figure 9.

Time frequency diagram of the disk-shaft system under the speed with 1800 rpm.

As shown in Figure 10, the time-frequency characteristics of the disk-shaft system under the rotating speed with 2400 rpm can be observed. According to Figure 10(a), it can be also observed that the regular variation of the vibration amplitude with time. Compared with the waveform under the rotating speed is 1800 rpm, it can be observed that with the increase of speed, that is, the increase of eccentric force, the vibration amplitude increases. Similarly, the rotating frequency of the shaft and the high frequency multiplication components can be found in the spectrum. Compared with the spectrum under the speed is 1800 rpm, it can be found that the number of high frequency multiplication components decreases with the increase of the rotating speed, while the amplitude of rotating frequency becomes larger.

Figure 10.

Time frequency diagram of the disk-shaft system under the speed with 2400 rpm.

As shown in Figure 11, the time-frequency characteristics of the disk-shaft system under the clearance with 0.002 mm when the rotating speed is 1800 rpm can be observed. In Figure 11(b), the frequency components in the spectrum include the shaft’s rotating frequency and high frequency multiplication components. Similarly, the time-frequency characteristic of the system under the clearance with 0.003 mm is also discussed, as illustrated in Figure 12. Also, the vibration amplitude changes regularly with time. With the increase of disk-shaft clearance, the vibration amplitude is also increasing. In other words, with the increase of the clearance, the vibration will be intensified, thereby the vibration amplitude increases. Moreover, in Figure 12(b), as the clearance increase, the number of high frequency multiplication components is also increased.

Figure 11.

Time frequency diagram of the disk-shaft system under the clearance with 0.002 mm.

Figure 12.

Time frequency diagram of the disk-shaft system under the clearance with 0.003 mm.

As shown in Figure 13, the time-frequency characteristic of the disk-shaft system under the clearance with 0.005 mm is also displayed. Compared with the results shown in Figures 11(a) and 12(a), it can be known that the vibration amplitude increases with the increase of clearance. Additionally, according to compare with the results shown in Figures 11(b), 12(b) and 13(b), it can be seen that the amplitude of the frequency components increases with the increase of clearance.

Figure 13.

Time frequency diagram of the disk-shaft system under the clearance with 0.005 mm.

Analysis of the trajectory for the disk center

As shown in Figure 14, the trajectory of the disk center under different rotating speeds when the clearance between the disk and shaft is 0.001 mm is given. In Figure 14(a), the trajectory shape of the disk center can be found to present a circle shape with radioactive. Additionally, the variation of trajectory shape for the disk center is regular, and the sawtooth phenomenon of the trajectory shape can be observed. In Figure 14(d), the trajectory of the disk center can be found to present a hollow ring structure, and the certain sawtooth phenomenon is also existed. In other words, with the increase of rotating speed, the eccentric force also increases, and the deviation phenomenon of the disk from the normal position is more severe. According to compare the trajectory shape of the disk center under different rotating speed, it can be observed that with the increase of rotating speed, the amplitude of the trajectory is also increasing, and the track shape changes from closed circle to hollow ring.

Figure 14.

Trajectory of the disk center under different rotating speeds.

The trajectory of the disk center under different clearances when the rotating speed is 1800 rpm is given in Figure 15. In Figure 15, the trajectory shape of the disk center presents a hollow ring shape. With the increase of clearance, the trajectory of the disk center is gradually moved closer to the center and the inner diameter of the hollow ring decreases. In other words, with the increase of clearance, the vibration of disk will be intensified, thereby the increase of the amplitude for the trajectory can be created.

Figure 15.

Trajectory of the disk center under different clearances.

Analysis of the speed difference between the disk and the shaft

The variation of speed difference between the disk and shaft of the disk-shaft system with rotating speeds is studied when the clearance between the disk and shaft is 0.001 mm, as given in Figure 16. In Figure 16, with the increase of the speed, the speed difference between the disk and shaft during the start-up phase increases. In other words, with the increase of the rotating speed, the centrifugal force of the eccentric disk also increases and the looseness between the eccentric disk and the rotating shaft is appeared, thereby an increasing speed difference between the eccentric disk and the rotating shaft. At the same time, the increase of rotating speed leads to a shortening of the start-up process, which results in a faster stabilization of the speed difference.

Figure 16.

Speed difference between the disk and the shaft under different rotating speeds.

Similarly, the variation of the speed difference between the disk and shaft under different clearance when the speed is 1800 rpm is also given, as shown in Figure 17. In Figure 17, it can be seen that with the increase of clearance between that disk and shaft, the speed difference during the start-up phase also increases. In other words, as the increase of the clearance, the rotating speed of the disk decreases and the shaft is kept steady, thereby the speed difference increases. At the same time, the increase in clearance leads to an ever-lengthening start-up process, and the speed difference of the disk shaft takes longer to reach stability.

Figure 17.

Speed difference between the disk and the shaft under different clearances.

Experiment

In order to perform the demonstration on the accuracy of the proposed dynamic model, the experiment investigation is carried out on the dynamic properties of disk-shaft system for the rotor test-platform subjected to the clearance-eccentric coupling faults. The test-platform is shown in Figure 18. The experimental equipment is divided into three main parts, i.e., ZT-3 rotor test-platform, sensors, and the signal acquisition system. The structure parameters of the rotating disk used in experiment can be described as that the inner diameter is 19 mm, the outer diameter is 76.2 mm, and the thickness is 25 mm, as given in Figure 19. In addition, the sensors used in this experiment include the eddy current displacement sensor and the photoelectric sensor.

Figure 18.

Rotor test rig with ZT-3.

Figure 19.

The structure of rotating disk.

In Figure 19, the conical structure with thread shape is employed to keep the connection of the disk and shaft. Hence, according to adjust the conical structure, and the clearance between the disk and shaft can be simulated. From Figure 19, the six balance weights with uniformly distributed can be observed on the rotating disk. Hence, the eccentric fault of the rotating disk can be simulated by the adjustment of the number of balance weights in experiment.

Analysis of the time frequency characteristics

In this section, the time frequency characteristics of the disk-shaft system under different clearances between the disk and shaft is experimental studied when the speed is 1800 rpm, as illustrated in Figures 20–23. Because of the inconvenience of operation, the clearance is larger than the simulation. As given in Figure 20, the time-frequency characteristic of the disk-shaft system is analyzed. Meanwhile, the disk-shaft clearance is defined as 0.01 mm. From Figure 20(a), it can be clearly seen that the variation of the vibration amplitude of the system is basically stable, and the average value remains around 0 mm. Furthermore, from Figure 20(b), we can see that the frequency component includes two components: the rotating frequency of rotating shaft and the high frequency doubling component. Besides, the amplitude of rotating frequency of the rotating shaft is the largest.

Figure 20.

Time frequency diagram of the disk-shaft system under the clearance with 0.01 mm.

Figure 21.

Time frequency diagram of the disk-shaft system under the clearance with 0.02 mm.

Figure 22.

Time frequency diagram of the disk-shaft system under the clearance with 0.03 mm.

Figure 23.

Time frequency diagram of the disk-shaft system under the clearance with 0.05 mm.

Similarly, from Figures 21(a), 22(a) and 23(a), it can be found that the variation of the vibration amplitude is also held steady. Compared with the result under the clearance with 0.01 mm given in Figure 20(a), it can be seen that the vibration amplitude of the rotor system increases with the increase of the clearance. In other words, with the clearance increases, the vibration of the disk will be intensified and the eccentric force is also increase, thereby the vibration amplitude of the rotor system will also increase. The frequency component also includes the rotating frequency component of the rotating shaft and the high frequency multiplication components. Compared with the result shown in Figure 20(b), with the increase of the clearance, the amplitude of rotating frequency component of rotating shaft also increases.

As given in Figures 24 and 25, the time-frequency characteristics of the disk-shaft system under different rotating speeds are studied when the clearance is 0.01 mm. From Figures 22(a) and 23(a), it can be clearly seen that with the increase of rotating speed, the vibration between disk and shaft will be intensified. Besides, the stable value of vibration amplitude is not around 0 mm due to the existence of eccentric fault for the rotating disk. Also, from Figures 22(b) and 23(b), we can see that there are always the rotating shaft’s rotating frequency and high frequency multiplication components within the spectrum. Moreover, with the increase of rotating speed, the amplitude of the rotating frequency of the rotating shaft increases. By comparing the changeable rule of time frequency characteristics under different influencing factors from the Analysis of the time frequency characteristics for the disk-shaft system, the similar laws between the simulation results and experimental results can be observed.

Figure 24.

Time frequency diagram of the disk-shaft system under the rotating speed with 900 rpm.

Figure 25.

Time frequency diagram of the disk-shaft system under the rotating speed with 1200 rpm.

Analysis of the trajectory for the disk center

The trajectory of the disk center under different clearances at a rotating speed of 1800 rpm is given in Figure 26. With the increase of the clearance, the amplitude of the trajectory increases, and the trajectory will also change. Besides, the trajectory are hollow, and the sawtooth phenomenon can be found.

Figure 26.

Trajectory of the disk center under different clearances.

The trajectory of the disk center under different rotating speeds at a clearance of 0.01 mm is given in Figure 27. With the increase of rotating speed, it can be found that the amplitude of the trajectory increases, but the basic shape of the trajectory is kept roughly constant, and sawtooth phenomenon of the trajectory can also be found.

Figure 27.

Trajectory of the disk center under different rotating speeds.

According to compare the simulation results illustrated in Simulation with the experimental results, it can be noticed that the similar phenomena between the simulation results and the experimental results can be observed. Thus the accuracy of the proposed dynamic model can be verified by the experimental research.

Conclusions

In this paper, the main objective is to analyze the nonlinear vibration characteristics of the disk-shaft dynamic model with the clearance-eccentric coupling faults by the combination of the numerical simulation and experimental study. The phenomena of coupling faults can be observed in both the simulation and experiments.

(1) With the rotating speed increase, the disk-shaft contact stress and strain energy is reduced firstly then is increased, when the disk-shaft clearance as a fixed value. Furthermore, when the rotating speed as a fixed value, both the disk-shaft contact stress and strain energy decrease as the disk-shaft clearance increase.

(2) The variation of the waveform for the disk-shaft system has periodical feature, and both the amplitude of the waveform and the number of cycles increase with the disk-shaft clearance or rotating speed increase. Moreover, the frequency components from the spectrum diagram are composed of the rotating frequency component of the shaft and the high frequency multiplication components. Additionally, the disk-shaft speed difference changes stably when the disk-shaft system steadily operates, and the speed difference also increase with the clearance or rotating speed increase.

(3) The variation of the trajectory shape the disk center is multifarious and complex and the sawtooth and mutation phenomenon of the trajectory shape can be observed, resulted by the disk-shaft rubbing and eccentric. Also, the trajectory shape changes from closed circle to hollow ring with the increase of the rotating speed.

Footnotes

Author contributions

Zhinong LI was in charge of the whole trial and is the corresponding author of this paper. Fang QIAO and Xingfu MA wrote the manuscript, Yunlong Li and Fei Wang were assisted with resources. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the grant from National Natural Science Foundation of China (Grant No. 52075236), Laboratory of Science and Technology on Integrated Logistics Support, National University of Defense Technology (Grant No. 6142003190210), Aero Engine Corporation of China (Grant No. KY-1003-2021-0017), and Key projects of Natural Science Foundation of Jiangxi Province (Grant No. 20212ACB202005).

ORCID iDs

Zhinong Li

Yunlong Li

References

Costello

JJA

West

McArthur

SDJ

. Machine learning model for event-based prognostics in gas circulator condition monitoring. IEEE Trans Reliab 2017; 66(4): 1048–1057.

Song

Choi

, et al. A novel fault diagnosis method for high-temperature superconducting field coil of superconducting rotating machine. Appl Sci 2020; 10(1): 223.

Ferrari

Mariscotti

Motta

, et al. Electromagnetic emissions from electrical rotating machinery. IEEE Trans Energy Convers 2001; 16(1): 68–73.

Wan

, et al. Rotating machinery fault diagnosis based on convolutional neural network and infrared thermal imaging. Chinese J Aeronaut 2020; 33(2): 427–438.

Xiong

Liang

Wan

, et al. The order statistics correlation coefficient and PPMCC fuse non-dimension in fault diagnosis of rotating petrochemical unit. IEEE Sens J 2018; 18(11): 4704–4714.

Zhang

Harne

, et al. Statistical quantification of DC power generated by bistable piezoelectric energy harvesters when driven by random excitations. J Sound Vib 2019; 442: 770–786.

Lee

Choi

. Fault diagnosis of partial rub and looseness in rotating machinery using Hilbert-Huang transform. J Mech Sci Technol 2008; 22(11): 2151–5162.

Chu

Tang

. Stability and non-linear responses of a rotor-bearing system with pedestal looseness. J Sound Vib 2001; 241(5): 879–893.

Karlberg

. Investigation of a saddle node bifurcation due to loss of contact in preloaded spherical roller thrust bearings. Chaos Solitons Fractals 2009; 41(4): 1750–1759.

10.

Dong

Zhang

Yan

, et al. Research on dynamic feature of the sliding bearing lubricating film of pedestal looseness for rotating machinery. J Vib Measurement Diagnosis 2014; 34(6): 1050–1056.

11.

Zhang

Zhou

. Effects of fastener failure at different parts on rotor systems outputs. IEEE Access 2019; 7: 108521–108529.

12.

Cao

Shi

, et al. Vibration and stability analysis of rotor-bearing-pedestal system due to clearance fit. Mech Syst Signal Process 2019; 133: 106275.

13.

Mercier

Jezequel

Besset

, et al. Nonlinear analysis of the friction-induced vibrations of a rotor-stator system. J Sound Vib 2018; 443: 483–501.

14.

Wang

Jin

, et al. Nonlinear effects of induced unbalance in the rod fastening rotor-bearing system considering nonlinear contact. Arch Appl Mech 2020; 90(5): 917–943.

15.

Qin

Han

Chu

. Bolt loosening at rotating joint interface and its influence on rotor dynamics. Eng Fail Anal 2016; 59: 456–466.

16.

Liu

Xue

Jia

, et al. Response characteristics of looseness-rubbing coupling fault in rotor-sliding bearing system. Math Probl Eng 2017; 2017: 8742468–8742510.

17.

Jiang

Peng

, et al. Nonlinearity measure based assessment method for pedestal looseness of bearing-rotor systems. J Sound Vib 2017; 411: 232–246.

18.

Jiang

Kuang

, et al. Rub-impact detection in rotor systems with pedestal looseness using a nonlinearity evaluation. Shock Vib 2018; 2018: 7928164–7928215.

19.

Yang

Cao

, et al. Response evaluation of imbalance-rub-pedestal looseness coupling fault on a geometrically nonlinear rotor system. Mech Syst Signal Process 2019; 118: 423–442.

20.

Wang

Guan

Chen

, et al. Characteristics analysis of rotor-rolling bearing coupled system with fit looseness fault and its verification. J Mech Sci Technol 2019; 33(1): 29–40.

21.

Qin

Han

Chu

. Bolt loosening at rotating joint interface and its influence on rotor dynamics. Eng Fail Anal 2016; 59: 456–466.

22.

Behzad

Asayesh

. Numerical and experimental investigation of the vibration of rotors with loose disks. Proc Inst Mech Eng C J Mech Eng Sci 2010; 224(C1): 85–94.

23.

Wei

Chu

. Speed characteristics of disk-shaft system with rotating part looseness. J Sound Vib 2020; 469: 115127.

24.

Liu

. Effects of unsteady oil film force on rotor system with loose disk and shaft. J Vib Shock 2019; 38(17): 268–275.

25.

Liu

, et al. Research on dynamic modeling and simulation of rotors with loose disk. J Mech Eng 2020; 56(7): 60–71.

26.

Zeng

Zhao

, et al. Dynamic response characteristics of the shaft-blisk-casing system with blade-tip rubbing fault. Eng Fail Anal 2021; 125: 105406.

27.

Zhu

, et al. Rotor stress analysis for high-speed permanent magnet machines considering assembly gap and temperature gradient. IEEE Trans Energy Convers 2019; 34(4): 2276–2285.

28.

CholUK

Qiang

ZhunHyok

, et al. Nonlinear dynamic simulation analysis of rotor-disk-bearing system with transverse crack. J Vib Eng Tech 2021; 9(7): 1433–1445.