Effects of misalignment on the nonlinear dynamics of a twoshaft rotorbearinggear coupling system with rubimpact fault
Xin Lu^{1} , Junhong Zhang^{2} , Liang Ma^{3} , Jiewei Lin^{4} , Jun Wang^{5} , Jie Wang^{6} , Huwei Dai^{7}
^{1, 2, 3, 4, 6, 7}State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China
^{2, 5}Department of Mechanical Engineering, Tianjin University Renai College, Tianjin 301636, China
^{4}Corresponding author
Journal of Vibroengineering, Vol. 19, Issue 8, 2017, p. 59605977.
https://doi.org/10.21595/jve.2017.18476
Received 13 April 2017; received in revised form 8 September 2017; accepted 3 October 2017; published 31 December 2017
JVE Conferences
In rotorbearing system, the misalignment can lead to secondary faults, such as the rubimpact fault. How the development of misalignment could affect the rubimpact force and other dynamic responses is important for the vibration control and fault diagnosis of rotor system. In this paper, a mathematical model of a twoshaft rotorbearing gear coupling system is established. The model is validated through the misaligned force of gear coupling, the supporting force of the lubricated ball bearing and the whole rotor response. To overcome the limit of traditional approaches and further dig out fault characteristics, the timefrequency method is introduced to analyze the dynamic response from both the time and frequency domains points of view. It is found the misalignment effect on rotor dynamic responses is mainly focused in high rotating speed range. The development of misalignment has an obvious adverse effect on the rotor stability. Some intermittent components in low frequency range due to the occurrence and development of misalignment can be found affecting the rotor stability at low speed range, and the high frequency rubimpact components can be found responsible for the unstable state of rotor.
Keywords: rotor system, ball bearing, gear coupling, meshing force, parallel misalignment, rubimpact, timefrequency analysis.
1. Introduction
For high efficiency, the clearance between the rotor and the stator is decreasing in rotating machineries. In this circumstance, the rotor and the stator are prone to rubimpact fault which can be considered as a secondary fault caused by unbalance, misalignment and other faults. The rubimpact fault can lead to abnormal vibration, overheat and component wearing [1].
In terms of the influence factors of the rubimpact fault, the effects of stiffness, damping, and mass imbalance on the radial rubimpact fault are most discussed. Patel et al. [2] investigated the nonlinear lateraltorsional vibration characteristics of a rotor contacting a viscoselastically suspended stator, and found that the friction coefficient, the rotorstator relative inertia and the contact damping have nonlinear complex effects on the motion characteristics. Liu et al. [3] established a rubimpact model considering the influence of the speed whirling and the radialtorsional coupling, and analyzed the dynamics characteristics of the radial, the whirl and torsional vibration under no rub, full annual rub and partial rub conditions. Yuanet et al. [4] studied the influence of a radial rubimpact on the imbalance response of a rotor and found that the magnitude of the lateral response decreased while that of the torsional response increased with a growing radial rubimpact fault. Although the effects of rotor properties on the rubimpact fault has been studied, the influence of other faults on the rubimpact has not been well discussed, such as how the misalignment could affect the dynamic response of a rubimpact rotor system.
However, the rubbing fault always occurs along with another fault, and the misalignment is a common twinborn fault resulting in the rubbing fault. Many researchers have studied the dynamic characteristics of coupling with misalignment. A twoshaft Jeffcott rotor with rigid coupling was developed to investigate the effect of misalignment on the lateral and torsional responses of the rotor system [5]. In order to study the effects of coupling, Zhao [6] derived a dynamic meshing force model for the gearspline coupling according to the classic deformation theory of gear tooth and obtained the characteristics of the coupling meshing force. It is found that the meshing forces in the $x$ and $y$ directions are identical with no misalignment, change nonlinearly when the misalignment is small, and vary linearly when the misalignment gets larger. Zhao [7] and AlHussain [5] investigated the effect of the parallel misalignment on the lateral and torsional responses of two rotating shafts. Paolo [8] proposed a simulation method to describe the behavior of a real shaft supported by several oil film bearings, and analyzed the nonlinear effects caused by the rigid coupling misalignment.
About the coupling fault of the rubimpact and the misalignment, Huang and Zhou [9] studied the vibration characteristics of the rotor system with the parallel misalignment and the rubimpact, and analyzed the effect of the parallel misalignment, the bearing stiffness and the mass eccentricity on the system dynamic behavior. It should be noticed that the meshing force of the coupling was ignored in that study. Fu [10] used a series of methods which including experiment and simulation to study the nonlinear characteristics of a rotor system with misalignment and rub fault. Zhang and Ma [11] studied dynamic responses of a rotorballbearing system with a misalignment and rubimpact coupling fault, in which the effect of the ElastoHydrodynamic Lubrication (EHL) on system dynamics was the main content. They also established a twostage rotor system connected by a gear coupling and supported on ball bearings and study the nonlinear dynamics of the system under misalignment fault [12]. But they only studied the dynamic characteristics of misalignmentrubbing fault, but less studied on the influence factors on the rub characteristics. There still needs a further study on the connection between the rubimpact faults and the induced failure.
Most of the previous researches were focused on numerical simulations and vibration characteristics of faults system. However, it has been proved that vibration signals are nonstationary and timevarying due to the change of rotational speed and the nonlinear vibration of a faulty rotor system [13]. In order to improve the ability of fault diagnosis, numbers of methods have been proposed for signal processing, condition monitoring and pattern identification [1417]. Gu [1820] proposed a series of novel algorithms benefiting the vibration signal processing and fault diagnosis. Timefrequency analysis is an effective tool to extract machinery health information because it can identify the signal frequency components to reveal their time variant features. Wang [21] investigated the effect of signal decomposition methods on timefrequency analysis and found that only using the empirical mode decomposition (EMD) method is not appropriate for realistically decomposing the vibration signal. Zhang and Ma [22] proposed an improved EMD method on timefrequency analysis and applied it on rotating machine fault diagnosis.
In this paper, a rotorbearinggear coupling system model with rubimpact and misalignment faults is developed. The improved ball bearing model is built up considering the EHL condition, and the meshing force is taken into account in the gear coupling model for connecting the two rotor sections. The effect of misalignment on the rubimpact rotor dynamic responses is analyzed using bifurcation diagram, Pointcaré map, spectrum diagram and timefrequency analysis.
2. Model descriptions
2.1. Misaligned gear coupling
When the gear coupling is misaligned, the male and female couplings are not concentric (as shown in Fig. 1). The gear teeth on one side tightly mesh with the sleeve while those on the other side contact loosely. In this condition, the meshing force can be divided into: (a) the meshing force associated with torque and (b) the meshing force associated with dynamic vibration.
a) Meshing force associated with torque.
When the gear teeth are in contact with the sleeve to transmit a torque, the meshing force is affected by three factors: the torqueinduced deformation, the meshing distance and the meshing stiffness. According to [7], the meshing force caused by the torque can be obtained:
where ${F}_{g}^{T}$ is the meshing force associated with torque, ${\varphi}_{g}$ is the deformation angle of gear tooth, ${l}_{g}$ is the equivalent meshing distance of gear tooth, and ${k}_{g}$ is the meshing stiffness of gear tooth. The ${\varphi}_{g}$ of all the gear teeth are identical under a static driven torque.
b) Meshing force associated with vibration.
During the torque transmission, the vibration of the rotor system leads to a relative movement between the meshing tooth and sleeve. The meshing force associated with vibration can be expressed as:
where ${F}_{g}^{D}$ is the meshing force caused by vibration, ${\phi}_{g}$ is the position angle of gear tooth, and ${e}_{g}$ is the misalignment of gear coupling.
Without torque the meshing force is zero and independent to the vibration displacement. Since the meshing force cannot be negative, the total meshing force of the $i$th tooth can be expressed as:
The meshing forces in the $y$ and $z$ directions are as follows:
c) Misalignment force
In this study, the misalignment takes place at the gear coupling and produces two forces affecting the dynamic characteristics of the rotor system: the meshing force and the misalignment force. The misalignment forces in the $y$ and $z$ directions of the rotor system are expressed as [11]:
where ${y}_{s}$ and ${z}_{s}$ are the movements of the intermediate tooth of gear coupling, ${M}_{g}$ is the mass of the intermediate tooth of gear coupling, $\omega $ is the angle velocity, $\psi $ is the initial phase. Since the intermediate tooth has to satisfy both couplings at the same time that misalignment occurs, its rotating angle velocity is related with the rotor speed and the orbit is a circle.
2.2. Ball bearing
The schematic diagram of the force applied on a ball bearing is shown in Fig. 2. Based on the Hertzian contact [23], the contact stiffness of the ball bearing can be determined as follows [24]:
where ${k}_{b}^{H}$ is the contact stiffness of ball bearing. ${Q}_{b}$ is the load of ball, ${k}_{c}$ is the contact deformation coefficient of the ball, and ${F}_{b,c}$ is the centrifugal force of the ball. The subscript $j$ means the $j$th rolling element.
Fig. 1. The Schematic diagram of gear coupling
Fig. 2. The Schematic diagram of ball bearing
Due to the existence of oil film, the lubrication between rolling element and races should be considered. Under the EHL condition, the oil film pressure is close to Hertzian contact stress in most of the contact zone, but quite different at the entering and departing ends.
According to the pervious study [12], it is found that most of the oil film clearances in the contact zone are nearly unchanged, so the central oil film clearance can be used as equivalent. In the case of pointcontact, the oil film thickness equation is suitable for the ball bearing:
where ${H}_{b}$ is the nondimensional central oil film clearance, $U={\mu}_{b}u/2E\text{'}{R}_{b}$ is a nondimensional velocity parameter, $G=\alpha {E}^{\text{'}}$ is a nondimensional material parameter, $W=\frac{{Q}_{b}}{{E}^{\text{'}}{R}_{b}^{2}}$ is a nondimensional load parameter, ${e}_{b}$ is the ellipticity of the rolling element, ${\mu}_{b}$ is the dynamic viscosity, $u$ is the linear velocity, ${E}^{\text{'}}=\frac{E}{{\left(1v\right)}^{2}}$ is the equivalent Young’s modulus, $\upsilon $ is the Poisson’s ratio, $\alpha $ is the pressureviscosity coefficient, and ${R}_{b}$ is the curvature radius.
Then the oil film stiffness between the rolling element and the inner race can be calculated:
$=\frac{1}{0.18023}{\stackrel{}{U}}^{0.67}{\stackrel{}{G}}^{0.53}{{Q}_{j}}^{1.067}{{E}^{\text{'}}}^{0.067}{{R}_{b}^{i}}^{1.134}{\left(10.61{e}^{0.73{K}_{eli}}\right)}^{1}.$
The oil film stiffness between the rolling element and the outer race ${k}_{EHL}^{o}$ can be calculated in the same way. Thus, the overall oil film stiffness can be obtained:
So, the global stiffness and the radial contact load of the $j$th rolling element are as follows:
2.3. Rubimpact fault
As shown in Fig. 3, the rubimpact force can be decomposed into a radial force ${F}_{N}$ and a tangential force ${F}_{T}$:
where $e={\left({y}_{d2}^{2}+{z}_{d2}^{2}\right)}^{\frac{1}{2}}$, $\sigma $ is the radial clearance between the rotor and the stator, ${k}_{r}$ is the radial stiffness coefficient of the stator wall, and $f$ is the radial friction coefficient between the rotor and the stator.
Fig. 3. The schematic of rubimpact forces
Then the rubimpact forces in the $y$ and $z$ directions can be given as:
2.4. Equations of motion
In this study, the rotor system (shown in Fig. 4) consists of two shafts, two disks, three ball bearings and one gear coupling. The rotor is supported in a m pressure is Disk 1 is mounted on the left end of Shaft 1 which is supported by Ball bearings 1 and 2, and Disk 2 is associated with Shaft 2 which is supported by Ball bearing 3 at the right end. The two shafts are connected by the gear coupling.
Fig. 4. The model of the rotorbearingSFD system
Two assumptions are made as below:
1) The deformations of the journals, the disks, the gear coupling, and the ball bearings are ignored. So, the rotor system is dispersed into 11 lumped mass points connected by mass less shaft sections with axial stiffness;
2) The displacements of the rotor in the axial and torsional directions are negligible so that each mass point has four degrees of freedom including two translations in the lateral and vertical directions and two rotations about the lateral and vertical directions.
The equations of motion of the rotor system are derived as:
where ${M}_{d1}$ and ${M}_{d2}$ are the masses of Disk 1 and Disk 2, ${M}_{r1}$ to ${M}_{r6}$ are the masses of shaft sections, ${C}_{d}$ and ${C}_{b}$ are the viscous damping factors of disks and ball bearings, ${K}_{r}$ is the stiffness of shaft sections, ${F}_{P,y}$ and ${F}_{P,z}$ are the rubimpact forces in the lateral and vertical directions, ${F}_{b,y}$ and ${F}_{b,z}$ are the supported forces of ball bearings in the lateral and vertical directions, ${F}_{m,y}$ and ${F}_{m,z}$ are the misalignment forces in the lateral and vertical directions, ${F}_{g,y}$ and ${F}_{g,z}$ are the meshing forces in the lateral and vertical directions, and ${e}_{d1}$ and ${e}_{d2}$ are the eccentricities of Disk1 and Disk 2.
2.5. Numerical method
In this paper, equations of motion are solved by the fourth order RungeKutta method, and the time step in the iterative procedure is set as $\u2206n=$ 1×10^{5} s. The time varying data corresponding to the first 500 periods generated by numerical integration are deliberately excluded in order to discard the transient solutions. The parameters used are listed in Table 1.
2.6. Timefrequency method
Due to the nonlinearity of the system response and the frequency overlap of fault characteristic, traditional features obtained from bifurcation diagram, Pointcaré map and frequency spectrum are not enough for analyzing the faulty rotor system. In this paper, the timefrequency method is employed to dig out more fault information that hidden from time and frequency domains. To this end a wavelet packet decomposition (WPD) improved EMD technique is applied to solving the problem [22]. Although a vibration signal can be decomposed into several IMFs by EMD, some highfrequency smallamplitude harmonics cannot be separated from one modulated IMF. So, the WPD is used to further decompose an IMF into narrow bands. In this way, some discontinuous high frequency contents can be found to illustrate the rubimpact fault. The timefrequency analysis can be operated in four steps:
1) Decompose the vibration signal by EMD method;
2) Further decompose the obtained IMFs using WPD technique and reform the WP coefficients;
3) Choose reformed IMFs that of high concerns as the first IMFs according to the node energy and the correlation between the reformed and original IMFs, and add other IMFs into the residual;
4) Apply Hilbert transform to the newgenerated IMFs to obtain the timefrequency distribution of the vibration signal.
Table 1. Parameters of the rotorbearing system
Parameter

Value

Parameter

Value

${M}_{d1}$

15.78 kg

${M}_{r6}$

2.25 kg

${M}_{d2}$

5.23 kg

${M}_{r7}$

1.61 kg

${M}_{r1}$

3.7 kg

${K}_{r}$

2.5×10^{7} N/m

${M}_{r2}$

2.3 kg

${C}_{d}$

800 Ns/m

${M}_{r3}$

1.1 kg

${C}_{b}$

1500 Ns/m

${M}_{r4}$

2.25 kg

${k}_{rub}$

2×10^{7} N/m

${M}_{r5}$

2.25 kg

$\sigma $

3.5×10^{5}m

3. Model validation
3.1. Ball bearing model
To validate the ball bearing model developed in this paper, the calculation results are compared with the results in [25]. The comparison is conducted using the information in frequency domain. Fig. 5(a) shows the results from [25] and Fig. 5(b) is the results from the model.
From both results, it is found that when the speed is far from the critical speeds the motion of the rotor mainly represents at the ball passage frequency and relevant harmonic components. The amplitude of vibration at the VC frequency is far higher than the following harmonics and decreases with increasing VC frequencies. Although the rotor speed is different, the vibration amplitude distribution along the frequency range in this paper is exactly in the same pattern as Ref [25], and both results show the motions are periodic. As a conclusion, the ball bearing model developed in this paper is validated.
3.2. Gear coupling model
In order to verify the misalignment model of the gear coupling, the simulation results of the proposed model are compared with the results of [7]. In the simulation, the vibration displacements in the $z$ and $y$ directions are both 0.001 mm, the static misalignment changes from 0 to 0.2 mm, and the deformation angle of gear tooth is 0°. The meshing forces in the $y$ and $z$ directions are shown in Fig. 6. From the comparison, it is found that the two components of meshing force of the proposed model are in the same pattern as the results of [7]. The difference in the amplitude of meshing force is because the transmitted torque is much smaller than that in the reference due to the rotor structure. When the misalignment is zero, the meshing forces in the two directions are the same. With increasing misalignment, the magnitudes of meshing forces in the two directions begin to increase sharply within the misalignment range from 0 to 1×10^{‒4} m. As the misalignment further increases, the two components of meshing force increase very slow and reach stable levels until the end. Obviously, the model developed in this paper is able to reflect the dynamic behavior of gear coupling correctly.
Fig. 5. Rotor responses in the $x$ direction: a) in this paper, b) in [25]
a)
b)
Fig. 6. Meshing forces under different misalignments: a) in this paper, b) in [7]
a)
b)
3.3. Rotor system model
To examine the rotor model, a group of experimental results [12] under different misalignments are used. The comparison between the simulation and the experiments are given in Table 2 and Fig. 7.
Fig. 7. Vibration amplitudes of: a) simulation, b) experiment with increasing misalignment
a)
b)
Table 2. Comparison between simulation and experimental results with different misalignments
Misalignment (m)

Amplitude at fundamental frequency and harmonics


1×

2×

3×

4×

5×


sim

test

sim

test

sim

test

sim

test

sim

test


0.7×10^{−4}

0.95

–5.62

–19.28

–16.14

–21.86

–24.85

–29.32

–32.25

–35.82

–36.81

1.4×10^{−4}

0.95

–5.96

–10.98

–16.07

–13.71

–20.62

–18.50

–28.14

–21.92

–21.43

2.1×10^{−4}

1.10

0.25

–6.56

–13.02

–9.01

–14.27

–11.00

–24.48

–18.71

–27.81

2.8×10^{−4}

1.37

–2.16

–3.54

–11.16

–5.81

–19.94

–6.68

–35.87

–16.61

–25.89

As the misalignment changes, both simulation and experimental amplitudes varies obviously. For the simulation, the amplitudes at all frequencies increase with increasing misalignment. For the experiment, the amplitudes at the fundamental frequency increases monotonously with increasing misalignment, while the amplitudes at other harmonic frequencies increase first then decrease at the end. The reason of this different trend may be the increase of misalignment in the gear coupling leads to a misalignment between the drive motor and the rotor which can bring unpredictable effect on the vibration of rotor system. This conclusion can be found in [26] as well. So, the mathematical model of the twoshaft rotorbearing system with a misaligned gear coupling can be validated.
4. Results and discussion
4.1. Rubimpact analysis under misalignments
For the rotor system under rubimpact fault but without misalignment, the bifurcation diagram and Pointcaré maps are shown in Fig. 8 and Fig. 9 respectively. The motion of the rotor system is periodic in the rotating speed range of 1000 rad/s to 1650 rad/s and becomes quasiperiodic in the range of 1650 rad/s to 1800 rad/s. Two following jump points appear at 1810 rad/s and 1815 rad/s. Then the motion returns to periodic until 1930 rad/s and goes into quasiperiodic again from 1935 rad/s to 2005 rad/s. Between 2005 rad/s and 2200 rad/s, the motion alternates between periodone and periodtwo. After that, the motion of the rotor loses its regularity and falls into quasiperiod.
Fig. 8. Bifurcation diagram of the rotor system under rubimpact fault without misalignment
Fig. 9. Pointcaré maps of the rotor system under rubimpact fault without misalignment
Under 1×10^{‒5} m misalignment, the bifurcation diagram of the rubimpact rotor system is shown in Fig. 10 and the corresponding Pointcaré maps are shown in Fig. 11. The motion of the rotor is periodone from 900 rad/s to 1450 rad/s. In the range of 1450 rad/s and 1800 rad/s, the motion of the rotor is quasiperiod and jumps at 1805 rad/s, 1810 rad/s, 1815 rad/s and so on. From the last jump point to 2270 rad/s, the motion of the rotor is unstable and goes into chaos. Then the motion becomes periodic from 2275 rad/s to 2345 rad/s and finally returns to quasiperiodic.
Fig. 10. Bifurcation diagrams of the rubimpact rotor system with 1×10^{‒5}^{}m misalignment
Fig. 11. Pointcaré maps of the rubimpact rotor system with 1×10^{‒5} m misalignment
When the misalignment is increased to 1.4×10^{‒1} m, the bifurcation diagram of the rubimpact rotor system is shown in Fig. 12 and the corresponding Pointcaré maps are shown in Fig. 13. The dynamic orbit shows periodone motion when the rotating speed is below 1375 rad/s. As the rotor speed increases to 1590 rad/s, the motion of rotor loses its stability and falls into chaos. After that the motion returns to periodone from 1595 rad/s to 1645 rad/s and goes into quasiperiod from 1650 rad/s to 1805 rad/s. After a series jump points, the motion of rotor goes back to periodone in a short range of 1810 rad/s to 1840 rad/s, becomes quasiperiod from 1845 rad/s to 2240 rad/s, and then falls into chaos. At the end, the system stability improves but the motion of rotor is still in quasiperiod.
Fig. 12. Bifurcation diagrams of the rubimpact rotor system with 1.4×10^{‒4} m misalignment
It can be concluded the rubimpact rotor system is always stable at lower speed regardless misalignment. But increasing misalignment can bring the motion of rotor into quasiperiod earlier and enlarge the speed range of quasiperiod/chaos motion. In other words, the stability of rotor system decreases with increasing misalignment. However, the motions of rotor are unusual that returns to periodic from quasiperiodic in some highspeed ranges, which is further discussed in the next section.
Fig. 13. Pointcaré maps of the rubimpact rotor system with 1.4×10^{‒4} m misalignment
4.2. Effect of misalignment on rotor dynamics
Fig. 14 shows the bifurcation diagram of the rubimpact rotor system with varies of misalignments under 1500 rad/s. It can be found that the dynamic orbit shows periodone with increasing misalignment from 0 to 3×10^{‒5} m. In this misalignment range, the rubimpact force is periodic and vibration amplitudes only can be found at fundamental and 2× fundamental frequencies in the spectrum diagram. With increasing misalignment, the peak rubimpact force increases from 1000 N to 6757 N, the variation of rubimpact force is irregular, the motion of rotor system is more unstable, and vibration at some low frequencies can be found around the fundamental frequency.
Fig. 14. Bifurcation diagram of the rubimpact rotor with different misalignments under 1500 rad/s
Fig. 16 and Fig. 17 show the dynamic responses of the rotor system with varies of misalignment at 2320 rad/s. It can be found that the motion of the rotor is quasiperiodic when the misalignment develops from 0 to 7.5×10^{‒5} m.
Without misalignment, the rubimpact force is irregular, and the peak amplitude is 1960 N, and low frequency vibration around the fundamental frequency can be found on the spectrum. As the misalignment grows from 7.5×10^{‒5} m to 9.3×10^{‒5} m, the motion of rotor goes into periodone, the peak amplitude of rubimpact force reduces to 1250 N, the force time history becomes more regular, and vibration around the fundamental frequency mitigates a lot. With further increased misalignment, the motion of rotor returns to quasiperiod, the rubimpact force becomes irregular again in time domain, the amplitude of rubimpact force is almost doubled, and a considerable vibration peak appears at low frequency.
Fig. 15. Pointcaré map, rub force, rotor orbit and spectrum diagram of rubimpact rotor at 1500 rad/s under misalignment of: a) 0 m, b) 5×10^{‒5} m, c) 1.85×10^{‒4} m, d) 3.3×10^{‒4} m
a)
b)
c)
d)
Fig. 16. Bifurcation diagram of the rotor system with different misalignments at 2320 rad/s
Fig. 17. Pointcaré map, rub force, rotor orbit and spectrum diagram of rubimpact rotor at 2320 rad/s under misalignment of: a) 0 m, b) 9×10^{‒5} m, c) 1.2×10^{‒4} m, d) 2.2×10^{‒4} m
a)
b)
c)
d)
4.3. Timefrequency analysis
Fig. 18 gives the timefrequency maps of the rubimpact force of rotor under different misalignments. Without misalignment (as Fig. 18(a)), the rubimpact force is nearly continuous with few weak intermittent components at high frequency, and the energy distributes around the fundamental frequency evenly. When the misalignment appears at a relatively low level (as Fig. 18(b)), numbers of intermittent components can be found at low frequency range, the energy distribution is messy, and the magnitude of energy starts to increase. As the misalignment further develops (as Fig. 18(c), (d) and (f)), the intermittent components reduce a lot, the energy distribution of the signal becomes more regular, and the magnitude of energy stops increasing. It can be seen that the occurrence of misalignment has the biggest impact on the rubimpact force of rotor system, which is greatly deteriorated due to a small increment of misalignment from nothing to a very low level. In other words, the stability of rubimpact rotor decreases a lot at the moment that misalignment starts, but stops to further decrease with further increasing misalignment.
Fig. 18. Timefrequency maps of rubimpact force at 1500 rad/s under misalignment of: a) 0 m, b) 5×10^{‒5} m, c) 1.85×10^{‒4} m, d) 2×10^{‒4} m, e) 2.4×10^{‒4} m, f) 3.3×10^{‒3} m
a)
b)
c)
d)
e)
f)
Fig. 19 shows the timefrequency maps of the rubimpact force of rotor under different misalignments at higher rotating speed. When there is no misalignment, it is found that the energy distribution of rubimpact force is not as evenly as that at lower speed (shown in Fig. 18 (a)) due to some low frequency components. When the misalignment is 5×10^{‒5} m, lots of intermittent components appear at low frequencies which make the energy distribution more irregular, but the magnitude of energy decreases. As the misalignment increases to 9×10^{‒5} m, the energy of rubimpact force is focused on the fundamental frequency, and more intermittent disturbances spread in low frequency range. But some high frequency components start to show periodic distribution. With increasing misalignment, the magnitude of energy increases, and the distribution is more irregular. Combined with Fig. 17, it can find that the motion of rotor turns from quasiperiod to periodone due to the increase of misalignment, but this state is unstable because of the periodic intermittent high frequency rubimpact component. With further development of misalignment, the rubimpact force becomes more irregular and results in chaos of the rotor system.
Fig. 19. Timefrequency maps of rubimpact force at 1940 rad/s under misalignment of: a) 0 m, b) 5×10^{‒5} m, c) 9×10^{‒5} m, d) 1.2×10^{‒4} m, e) 2.2×10^{‒4} m, f) 3×10^{‒4} m
a)
b)
c)
d)
e)
f)
5. Conclusions
In this paper, a model of a rotorball bearinggear coupling system with rubimpact and misalignment faults is established. The lubrication of ball bearing, and the meshing forces of gear coupling are considered. The effect of misalignment on the rubimpact rotor system is analyzed. Some conclusions are as follows:
1) Misalignment can strongly affect the stability of the rubimpact rotor system at high rotating speed. Both the starting point of quasiperiod motion and the range of quasiperiod/chaos motion are changed due to the development of misalignment.
2) The motion, orbit and vibration spectrum of the rotor and the rubimpact force are highly related to the change of misalignment. Generally, the development of misalignment brings adverse effect on rotor stability, and the speed range of the unstable motion increases with increasing misalignment. However, the effect pattern still depends on the rotating speed.
3) Unusual effects of misalignment on the rubimpact rotor dynamics can be found at high frequencies which cannot be well explained by bifurcation diagram, Pointcaré map, rotor orbit and frequency spectrum. Using the timefrequency method, some intermittent high frequency rubimpact components can be found responsible for the unstable state of rotor.
4) Some intermittent components in low frequency range due to the occurrence and development of misalignment can be found affecting the rotor stability at low speed range. Since the intermittent disturbances reduce quickly with increasing misalignment, the motion of rotor returns to regular at a constant speed even the misalignment goes worse.
5) Using the timefrequency analysis, some fault characteristics once unable to be observed become visible, in particular for the misalignmentrubimpact coupled fault. It indicates that a qualitative or even quantitative assessment of the fault might be possible on the basis of measurable quantities.
Acknowledgements
We are grateful to the Joint Funds of National Science Foundation of China and Civil Aviation Administration Foundation of China (No. U1233201), Major projects of CAAC (MHRD20160106) and National Science Foundation of China (51501222) for providing financial support for this work. The authors declare that there is no conflict of interests regarding the publication of this paper.
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