The finite element model of a dual-rotor system was established by Timoshenko beam element. The dual-rotor system is a coaxial rotor whose supporting structure is similar to that of an aero-engine rotor system. The inner rotor is supported by three bearings, which makes it a redundantly supported rotor. The outer rotor connects the inner rotor by an intershaft bearing. The spectrum characteristics of the dual-rotor system under unbalanced excitation and misalignment excitation were analysed in order to study the influence of coupling misalignment of the inner rotor on the spectral characteristics of the rotor system. The results indicate that the vibration caused by the misaligned coupling of the inner rotor will be transmitted to the outer rotor through the intershaft bearing. Multiple harmonic frequency components, mainly 1
Rotor misalignment is one of the most common difficulties in the operation of rotating machinery. A misalignment rotor negatively influences the rolling, sealing, and coupling parts and can also produce eccentricity in the air gap. Therefore, a misaligned machine is more prone to failure due to the increased loads on bearings and couplings. In general, the misalignment in a rotor system is unavoidable and cannot be completely eliminated. Furthermore, it can disguise itself very well on industrial rotating machinery, which makes it not easy to be detected. Even if the alignment looks good when you do an offline check, a running misalignment may occur. What we witness is the secondary effects of misalignment as it slowly damages the machinery over long periods of time. Hence, the ability to clearly diagnose the presence of misalignment using vibration analysis can be vital in reducing costly machine unscheduled downtime. Vibration based identification of faults, such as rolling bearing fault [
In the fault diagnosis of misalignment, Lees et al. [
In terms of the dynamic characteristics of rotor misalignment, Hili et al. [
The research object of the previous studies is the multispan rotor system. However, little information has been done on the misalignment of the coaxial rotor. A coaxial rotor of an aero-engine is different from a series connection rotor of steam turbine in twofold. For one thing, the coaxial shafts corotate or count-rotate at different speeds, for another, the inner rotor which is supported by more than two bearings is a redundant structure. These special structures deserve to be paid more attention. Lu et al. [
Spectrum analysis is one of the basic techniques used to conduct misalignment diagnosis. Therefore, the spectrum analysis of an aero-engine multibearing dual-rotor system with coupling misalignment and disk unbalance has been studied by the finite element method (FEM) in the present study. A test rig has a similar bearing supporting structure like a real aero-engine was built to verify the theoretical studies. Experiments were performed on the test rig under different intentional misalignment conditions. The study could be useful in the detection of misalignment in the coaxial dual-rotor system diagnosis and in the design of an aero-engine rotor system.
A structural diagram of a typical twin-spool jet engine is shown in Figure
Structural diagram of a typical twin-spool jet engine [
Schematic of the dual-rotor system, nodes, and elements.
The dual-rotor system shown in Figure
Schematic diagram of the coordinate system of the shaft unit.
A rotation of a right-handed screw from
Using the FEM, the rotor is divided into a number of shaft elements with nodes at both ends of each element. The finite shaft element of the dual-rotor system is accomplished by means of the Timoshenko beam. Each shaft element has two nodes, and the node location is shown in Table
Node location.
Node | Position (m) | Remark |
---|---|---|
1 | 0.000 | Bearing 1 |
2 | 0.043 | |
3 | 0.083 | Disk 1 |
4 | 0.123 | |
5 | 0.164 | Disk 2 |
6 | 0.204 | |
7 | 0.260 | Bearing 2 |
8 | 0.380 | |
9 | 0.500 | |
10 | 0.620 | |
11 | 0.740 | |
12 | 0.860 | Bearing 5 |
13 | 0.925 | Disk 3 |
14 | 0.965 | |
15 | 1.010 | Bearing 3 |
16 | 1.098 | |
17 | 0.415 | |
18 | 0.440 | Bearing 4 |
19 | 0.490 | |
20 | 0.548 | Disk 4 |
21 | 0.636 | |
22 | 0.724 | Disk 5 |
23 | 0.792 | |
24 | 0.860 | Bearing 5 |
25 | 0.885 |
The rotors and disks are made from steel. The rotor structure is simplified, and the change of some shaft sections is ignored. The diameters of the shafts are shown in Table
Diameters of the shafts.
Parameters | |||||
---|---|---|---|---|---|
Values (m) | 0.07 | 0.056 | 0.035 | 0.07 | 0.042 |
All the five disks are identical. Disk 1 and disk 2 represent a low pressure compressor. Disk 3, disk 4, and disk 5 represent a low pressure turbine, a high pressure compressor, and a high pressure turbine, respectively. Disk 1, disk 2, disk 3, disk 4, and disk 5 are located at nodes 3, 5, 13, 20, and 22, respectively. The disk properties are listed in Table
Disk properties for the rotor system.
Parameters | Diameter (m) | Thickness (m) | Elastic modulus (MPa) | Density (kg/m^3) |
---|---|---|---|---|
Values | 0.24 | 0.038 | 2.06e5 | 7850 |
There are five bearings in the system, namely, bearing 1, bearing 2, bearing 3, bearing 4, and bearing 5. Bearing 5 is an intershaft bearing. All bearings are assumed as to be linear and isotropic. Bearing 1, bearing 2, bearing 3, and bearing 4 are located at nodes 1, 7, 15, and 18, respectively. The intershaft bearing, bearing 5, is located at node 12 and node 24. The bearings properties are listed in Table
Bearing properties for the rotor system.
Bearing | Node | ||
---|---|---|---|
Bearing 1 | 2.21 | 2.21 | 1 |
Bearing 2 | 14.5 | 14.5 | 7 |
Bearing 3 | 2.21 | 2.21 | 15 |
Bearing 4 | 9.29 | 9.29 | 18 |
Bearing 5 | 25.1 | 25.1 | 12 and 24 |
According to the Lagrange equation, the dynamic equations of disk element and shaft element can be derived as follows:
The dynamic equations of intershaft bearing and an ordinary bearing are the same, which can be expressed as
Force analysis diagram of the intershaft bearing.
Considering all the damping, stiffness, and mass matrices of shaft, disk, and bearing, the general dynamic equations of the dual-rotor system can be written as
The effects of misalignment on a rotor system depend on how the reaction forces accommodate the misalignment. There are many factors that affect the reacting forces and moments of a misaligned coupling, and the structure of the coupling is one of the important factors. [
The coupling of an aero-engine dual-rotor-bearing system [
According to the structural characteristics of the spline coupling, it can be regarded as a flexure coupling. The reaction forces and moments caused by the spline coupling misalignment can be calculated by the Gibbons–Sekhar formula [
Coupling coordinate system: (a) parallel misalignment and (b) angular misalignment [
For parallel misalignment,
For angular misalignment,
It can be observed that the reaction forces
As already mentioned in the introduction section, the reaction forces and moments calculated from the Gibbons–Sekhar formula are static loads. For a rotating shaft, the static loads can be decomposed as series dynamic periodic forces which cause vibrations. The dynamic misalignment forces and moments are treated as excitations at the coupling node of the finite element model, and only the 1
For the dual-rotor system shown in Figure
Angular misalignment of coupling of the inner rotor.
In this section, two case studies are performed to investigate the effects of coupling misalignment on the spectral characteristics of the dual-rotor system: (1) the dual-rotor system with only unbalanced excitations; (2) the system with both unbalanced excitations and misaligned excitations. The frequency responses of the two conditions are compared with each other to understand the mechanism of misalignment in the dual-rotor system. Assume that the inner and outer rotors rotate in the reverse direction; the rotation speed of the outer rotor is 1.6 times that of the inner rotor. Regardless of the unbalance moment, assume that only the out-of-balance forces exist at both node 3 (disk 1) of the inner rotor and node 20 (disk 4) of the outer rotor; all reacting forces and moments caused by the coupling misalignment are located at node 6. And the time domain dynamic response of the system can be obtained by solving the dynamic equation of the dual-rotor system with the Newmark-
A Campbell diagram is a map of natural frequency against rotor rotation speed. It is useful in identifying the forward and backward precession natural frequency components of the rotor system during spectral analysis. The Campbell diagram of the dual-rotor system, plotted against the inner rotor speed, is shown in Figure
Campbell diagram for the dual-rotor- bearing system.
The spectrum diagrams of the dual-rotor system with only unbalance forces are shown in Figures The dominant frequencies of the dual-rotor system are 1 The vibration amplitudes in horizontal direction and vertical direction at the same bearing (Figures Resonance occurs at bearing 1 when 1.6
Spectrum of out-of-balance at bearing 1 (inner rotor): (a) horizontal direction and (b) vertical direction.
Spectrum of out-of-balance at bearing 4 (outer rotor): (a) horizontal direction and (b) vertical direction.
In order to analyse the influence of misalignment on the spectrum characteristics of the dual-rotor system, it is assumed that the working conditions of the rotor system are exactly the same as the unbalanced conditions (as described in Section
Spectrum of misalignment at bearing 1 (inner rotor): (a) horizontal direction and (b) vertical direction.
Spectrum of misalignment at bearing 4 (outer rotor): (a) horizontal direction and (b) vertical direction.
Comparing Figures The vibration spectrum of misalignment is more complicated than that of unbalance. Both the inner and outer rotors appear frequency components dominated by 1 Misalignment intensifies the vibration in the misalignment direction. Comparing Figures Note that the further away from the misalignment coupling, the less affected by misalignment. Misalignment increases the vibration level of both the inner and outer rotor, but the most obvious change is in the direction of misalignment near the misaligned coupling. The maximum vibration amplitude of bearing 1 is higher than that of bearing 4 under the condition of misalignment (Figures
The radar chart shown in Figure Although bearing 2 is closest to the misaligned coupling, the maximum amplitude of the four bearings does not occur at the misalignment direction of bearing 2. Instead, the maximum amplitude occurs in the vertical direction of bearing 1 (B1V) because the inner rotor is a redundantly support rotor. The vibration of bearing 2 is suppressed because bearing 2 and bearing 3 make the inner rotor a stable structure. Although the stiffness of the four bearings is isotropic. The vibration amplitudes in the vertical direction are greater than that of the horizontal direction in both bearing 1 and bearing 2, which are near the misaligned coupling. However, bearing 3 and bearing 4, away from the coupling, show the opposite vibration level. Although the amplitudes of the four harmonic components of the additional force are the same, the amplitudes of the 2
Radar chart of harmonic frequencies at the four bearings.
The effects of misalignment on rotor frequency characteristics have been qualitatively analysed. The quantitative influence of different degrees of misalignment on the harmonic frequencies of the system will be analysed too. Assuming that bearing 1 is raised by 0.1 mm, 0.2 mm, 0.3 mm, 0.4 mm, and 0.5 mm, respectively, the variation trend of the vertical amplitude of bearing 1 with the misalignment value is obtained, which is shown in Figure
Amplitude of harmonic frequencies altering with misalignment.
As described in Section
Resonance of 3
Resonance of 2
Resonance of 1
The spectrum of bearing 1 is shown in Figure
From the above analysis, it can be known that all harmonic frequencies excited by misaligned coupling, 1
An orbit is a powerful diagnostic tool which can provide important and useful information about the dynamic motion of the rotor. The orbits of the dual-rotor system with different degree misalignment are shown in Figures The orbits of bearing 1 and bearing 4 are both in the shape of rings when only unbalanced excitation exists, as shown in Figure The orbits of bearing 1 tend to collapse toward a straight line for the angular misalignment along the Figure
Orbits of bearings without misalignment (Δ = 0 mm, Ω1 = 19 Hz, Δ2 = 30.4 Hz): (a) bearing 1 and (b) bearing 4.
Orbits of bearings with misalignment (Δ = 0.1 mm, Ω1 = 19 Hz, Ω2 = 30.4 Hz): (a) bearing 1 and (b) bearing 4.
Orbits of bearing 1 at different misalignment (Ω1 = 11 Hz, Ω2 = 17.6 Hz): (a) Δ = 0.1 mm and (b) Δ = 0.3 mm.
A dual-rotor test rig is built in order to verify the simulation results. Experiments were performed on the test rig under different intentional misalignment conditions.
The configuration of the test rig is shown in Figure
Experimental setup of the dual-rotor system with five supports.
Bearing type of the test rig.
Bearing | Types |
---|---|
Bearing 1 | NU1013 |
Bearing 2 | 7013AC |
Bearing 3 | NU1013 |
Bearing 4 | 7013AC |
Bearing 5 | NU1013 |
Gaskets of different thickness.
Vibration data are acquired from the four bearing pedestals by a B & K data collector. Both horizontal and vertical vibrations of the bearing pedestals are measured by accelerometers. The acceleration signal collected by the sensor is firstly converted into a displacement signal by double integration, and then the FFT algorithm is applied on the displacement signal to obtain the spectrum. Because the low-frequency noise will be generated in this process, a high-pass filter is adopted to filter low-frequency noise signals below 10 Hz.
Three test plans are performed to verify the simulation results.
The first experiment aims to confirm the spectral components of the misaligned dual-rotor system. When the spacer thickness is 0.3 mm and the rotation speed of the inner rotor is 21 Hz, the spectra of bearing 1 and bearing 4 are shown in Figures The misalignment excited the frequencies of 1 Comparing the vibration amplitude of both directions (Figures The vibration level of bearing 1 is higher than that of bearing 4 in both horizontal and vertical directions, which can be illustrated by comparing Figures
Spectrum of misalignment and unbalance at bearing 1, Ω1 = 21 Hz: (a) horizontal direction and (b) vertical direction.
Spectrum of misalignment and unbalance in bearing 4, Ω1 = 21 Hz: (a) horizontal direction and (b) vertical direction.
The second experiment is conducted to check the effects of different degrees of misalignment on the amplitude of different frequency components. Gaskets with thickness of 0.2 mm, 0.3 mm, 0.4 mm, 0.5 mm, and 0.6 mm, shown in Figure
The spectra corresponding to different spacer thicknesses are tested at the same speed (Ω1 = 21 Hz, Ω2 = 33.6 Hz). Two sets of tests are performed on each gasket, recording the amplitude of each frequency component. Then the amplitude of each frequency is averaged, and the experimental results are finally obtained as shown in Figure
Amplitude of harmonic frequencies altering with misalignment value (test results).
The third experiment is designed to verify the harmonic resonance caused by misalignment. The vibration of bearing 1 and bearing 4 at two rotation speeds is measured, as shown in Figures
Spectrum of misalignment and unbalance at bearing 1: (a) horizontal (Ω1 = 18 Hz, Ω2 = 28.8 Hz) and (b) horizontal (Ω1 = 11 Hz, Ω2 = 17.6 Hz).
Spectrum of misalignment and unbalance in bearing 4. (a) horizontal (Ω1 = 18 Hz, Ω2 = 28.8 Hz) and (b) horizontal (Ω1 = 11 Hz, Ω2 = 17.6 Hz).
Due to the limitation of experimental conditions, the orbits of the rotor cannot be measured directly. The orbits of the rotor are replaced by the orbits of bearings to analyse the influence of misalignment. The orbits of bearings are plotted by the vibration displacement of bearing pedestals in horizontal and vertical directions. Figure
Orbits of bearing without misalignment (Δ = 0 mm, Ω1 = 21.6 Hz, Ω2 = 34.6 Hz): (a) bearing 1 and (b) bearing 4.
Orbits of bearings with misalignment (Δ = 0.2 mm, Ω1 = 21.6 Hz, Ω2 = 34.6 Hz): (a) bearing 1 and (b) bearing 4.
Orbits of bearing 1 at different misalignment (Ω1 = 11.6 Hz, Ω2 = 18.6 Hz): (a) Δ = 0.2 mm and (b) Δ = 0.4 mm.
In this paper, a finite element model for a dual-rotor system with a misaligned coupling is developed using the Timoshenko beam elements. The response of the system is generated using the Newmark- The inner rotor with coupling misalignment makes both the inner and outer rotor appear 1 Although the vibration caused by misalignment of the coupling in the inner rotor can be transmitted to the outer rotor, the closer the coupling is to the bearing, the more severe the vibration is. And the vibration amplitude of the misalignment direction is larger than that of the direction without misalignment. The amplitudes of each harmonic frequency increase with the increase of the misalignment. Harmonic resonance occurs when any harmonic frequencies of the misalignment response coincide with a natural frequency of the system. And the vibration will be intensified. The orbits of bearings tend to collapse toward a straight line for the angular misalignment. And the orbit diverges into a reticulated diamond shape from a straight line shape as the increase of misalignment.
The data supporting this research article are available from the corresponding author or first author on reasonable request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to thank Dr. Qing Xiao for his valuable suggestions. The authors would also thank graduate students Zihao Liao and Liman Chen for their help in experiments. In addition, the authors would like to thank Linlin Song for her help in translation work. This work was supported by the Natural Science Foundation of China through the Grants 11672106, 11702091, and 51875196, the Natural Science Foundation of Hunan Province of China (2018JJ3140), and the Open Fund Project of Key Laboratory Hunan Province of Health Maintenance Mechanical Equipment (201702).