Abstract
Most of the major faults in gas turbines often originate from the rotor shaft. Some of these faults could result to misalignment, imbalance, crack, rubimpact and eccentricity. This study analyzes nonlinear factors such as lubefilm surge and thermalstress. Seal and bearing quality significantly affect the performance of a rotor system. The nonlinear dynamic characteristics of a hightemperature rotor–bearing–seal coupled system are investigated. Dynamic trajectories, Poincaré maps, frequency spectra diagrams are used to analyze the features of the rotorbearingseal coupled system in terms of various parameters. Several nonlinear phenomena in the hightemperature rotor–bearing–seal coupled system, such as periodic, doubleperiodic, multiperiodic, and quasiperiodic motion are investigated. Through these curves and the program, the dynamic characteristics of gas turbine rotors could be easily and accurately calculated. To validate this method, the critical speed of a real rotor of an actual gas turbine was calculated by means of the program and curves. The results are consistent with the measured data and may contribute to further understanding the nonlinear dynamics of the hightemperature rotorbearingseal coupled system.
1. Introduction
Gas turbines efficiently generate clean power and have been extensively applied because of its global market demand. As one of the key components of the gas turbine, the performance of a rotor directly affects the engine efficiency. With the recent technological advancements, rotors are being developed toward having towards large span, possessing more flexibility and carrying heavier load. However, rotors are constantly exposed to high temperature and harsh environment, thereby making them prone to failure. Rotor failure usually results to rubbing, imbalance, crack and other phenomena. Therefore, research on the mechanism of fluidsolid interaction in the rotorbearingseal system is crucial for improving modern rotating machines.
In the early stage of rotor dynamics, study on the rotorbearing sealing system used linear models. However, linear theory failed to explain a number of phenomena; thus, nonlinear theory gradually developed. For a better simulation of bearing nonlinearity, some nonlinear seal force models have been proposed. The present study adopts the Muszynska [1, 2] model to describe the sealing force considering that such model can describe the nonlinear characteristics of the sealing force and has a clear physical meaning.
Based on Muszynska nonlinear seal force model, Ding et al. [3, 4] investigated an unstable mechanism with the singledisc rotor seal system and then analyzed the direction of Hopf bifurcation and the influence of instability on the unbalance mass. In another paper, Ding et al. [5] analyzed the Hopf bifurcation behavior of a symmetric rotorseal system by using Muszynska nonlinear seal fluid dynamic force model. Moreover, Li et al. [68] studied both the dynamic stability and imbalance in the rotor system by adopting the aerodynamic force of the labyrinth seal in the Muszynska mode. Wang [9, 10] proposed and validate a hybrid numerical method to calculate pressure distribution of ultrashort gas bearing system and rotor orbits. In his another papers, Wang utilize this method to confirm the changes of rotor nonlinear dynamic behavior and bearing systems characterized as the rotor mass is increased, the results indicated that the method is useful for gas bearing systems. In addition, the rotor becomes unstable when a certain rotational speed is reached and when the whirl amplitude increases along with the increasing speed. At very high speeds, the seal and the rotor would rub. What is more, the seal force models have also been developed by many other researchers. In order to study the characteristics of the rotor motion, He et al. [11] presented a new nonlinear sealing force model based on the double control body model and rotor motion equation with drawing the trajectory of the rotor, Poincare and bifurcation diagram. Moreover, Yan et al. [12] proposed a transient CFD method for analyzing the complex annular seals of the rotorseal coupled system. Their findings show that the system exhibits complex dynamic behavior and possible chaotic motion as the rotor speed increases. The pressure ratio has a significant effect on the frequency response of the firstorder critical speed; however, it does not significantly affect the motion state and the frequency response of the rotating speed.
Some nonlinear oilfilm force models have been proposed to better simulate the nonlinearity of the sliding bearing. Jing et al. [13] proposed a continuum model that is based on the finite element method to analyze the oil whip phenomenon. For the oilwhirl and oilwhip phenomena, Castro et al. [14] employed a nonlinear force model to perform a detailed analysis to verify the effects of unbalance, journalbearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold. Leng et al. [15] simulated the rotor movement, the traditional eight stiffness, and the damping coefficient of the linear system response under the oil film force model when the speed increases by comparing the analysis made after considering the nonlinear oil film force characteristics of the rotational speed of the oil film when oscillation occurs in advance. Zhang et al. [16] developed a mathematical model and a computational methodology to simulate the complicated flow behaviors of the journal microbearing in the slip regime. Their findings indicated that the rotary motion was stable and exhibited halffrequency whirling when the system located in the lower stability region, whereas the rotary motion displayed highfrequency whirling when the system was located in the upper stability region. Riemann et al. [17] stated the rotor instabilities are related to the operational speed and structural parameters. They used the $\mu $synthesis control technique to reduce the oil whip instability. Furthermore, Yang et al. [18] proposed a new nonlinear dynamic analysis method mainly to analyze the effects of mass eccentricity on dynamic behaviors of rotor system supported by oilfilm journal bearings. Ma et al. [19] investigated the influence of angular acceleration, parallel misalignment, and angular misalignment on the oilfilm instability of a flexible rotorbearing system in the runup and rundown processes at high speeds. In the runup process, lower misalignment levels delay the onset of the second vibration mode instability; however, instability may occur in advance under high misalignment levels. Furthermore, the amplitudes of multiple frequency components under parallel misalignment conditions are more evident than amplitudes under the angular misalignment conditions. Sekhar et al. [20] introduced two different approach to identify the unbalanced fault in a rotor system: equivalent loads minimization method and vibration minimization method. The combination of equivalent loads minimization method with the modified theoretical fault model and vibration minimization method is more effective than using the equivalent load minimization method in identifying unbalance fault. Dufour et al. [21] proposed an onboard rotor model to investigate the nonlinear dynamics of a supportexcited flexible rotor with hydrodynamic journal bearings. Compared with the rigid bearings, linearized hydrodynamic bearings have greater influence on the size of the instability regions of the rotor.
Many researchers investigated the dynamic behaviors of the rotorbearingseal system in relation to the coupling effect of the nonlinear oil film force and seal force. Previous literature [2227] combined nonlinear oil film force model with Muszynska seal force model to study the nonlinear dynamics behavior and stability of rotor bearing sealing system. By comparing the response of the rotor system with the sealing force, it was found that the nonlinear sealing force can improve the stability region of the rotor system and the complex behavior was suppressed. The effect of changing the rotational speed, eccentricity, or sealing gap parameter on the rotor system was also studied. Shen et al. [28] conducted a theoretical and experimental study to analyze the nonlinear dynamic behavior of the rotorbearing seal system. Their investigation showed that the seal force would reduce the system threshold of instability and the vibration amplitude of the rotor system. Ma et al. [29] studied the influence of speed and eccentricity of the sealing force on the rotorbearing seal system under two kinds of load conditions. The nonlinear oilfilm force model was based on the assumption that short bearings are used. Research shows that the sealing force can improve the critical speed, whereas secondorder speed does not have a significant effect on critical speed. Moreover, Ma et al. [30] adopted a mathematical approach to investigate the influence of rotational speed on the dynamic characteristics of the seal with another circular bearing oil film force model of Capone. The time domain, frequency spectrum, phase and Poincare diagrams under different speeds are calculated. Results reveal the influence of rotational speed on the nonlinear vibration of the rotor.
Thermal stress can also significantly influence on the rotary system. Zhu et al. [3133] employed the finite element method to study the internal temperature distribution of steam turbine components which results in large thermal deformation and thermal vibration of the rotor. The influence of radial temperature difference, imbalance, heat bending and the combined action on the thermal vibration characteristics of the rotor bearing system is analyzed by using thermal structure dynamics coupling theory. Results showed that thermal bending alters the vibration characteristics of the rotor system. This study focuses on the rotorbearingseal system in gas turbines under thermal stress.
2. Mathematical model of a rotorbearingseal system with two discs
In order to study nonlinear dynamic behavior efficiently, a mathematical model of a rotorbearing seal system, which is depicted in Fig. 1.
Fig. 1Nonlinear rotorbearingseal system of the gas turbine
2.1. Equation of motion
${m}_{1}{Y}_{1}^{\text{'}\text{'}}+C{Y}_{1}^{\text{'}}+{K}_{1}\left({Y}_{1}{Y}_{2}\right)={f}_{Y}{m}_{1}g+{f}_{T},$
${m}_{2}{X}_{2}^{\text{'}\text{'}}+C{X}_{2}^{\text{'}}+{K}_{2}\left({X}_{2}{X}_{3}\right)+{K}_{1}\left({X}_{2}{X}_{1}\right)={F}_{X}+{m}_{2}r{\omega}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\omega t+{f}_{T},$
${m}_{2}{Y}_{2}^{\text{'}\text{'}}+C{Y}_{2}^{\text{'}}+{K}_{2}\left({Y}_{2}{Y}_{3}\right)+{K}_{1}\left({Y}_{2}{Y}_{1}\right)={F}_{Y}+{m}_{2}r{\omega}^{2}\mathrm{s}\mathrm{i}\mathrm{n}\omega t{m}_{2}g+{f}_{T},$
${m}_{3}{X}_{3}^{\text{'}\text{'}}+C{X}_{3}^{\text{'}}+{K}_{2}\left({X}_{3}{X}_{2}\right)+{K}_{3}\left({X}_{3}{X}_{4}\right)={F}_{X}+{m}_{3}r{\omega}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\omega t+{f}_{T},$
${m}_{3}{Y}_{3}^{\text{'}\text{'}}+C{Y}_{3}^{\text{'}}+{K}_{2}\left({Y}_{3}{Y}_{2}\right)+{K}_{3}\left({Y}_{3}{Y}_{4}\right)={F}_{Y}+{m}_{3}r{\omega}^{2}\mathrm{s}\mathrm{i}\mathrm{n}\omega t{m}_{3}g+{f}_{T},$
${m}_{4}{X}_{4}^{\text{'}\text{'}}+C{X}_{4}^{\text{'}}+{K}_{3}\left({X}_{4}{X}_{3}\right)={f}_{X}+{f}_{T},$
${m}_{4}{Y}_{4}^{\text{'}\text{'}}+C{Y}_{4}^{\text{'}}++{K}_{3}\left({Y}_{4}{Y}_{3}\right)={f}_{Y}{m}_{4}g+{f}_{T}.$
where, ${m}_{1}$ – the left journal mass; ${m}_{2}$ – mass of the left disc; ${m}_{1}$ – mass of the right disc; ${m}_{4}$ –the right journal mass; $C$ – system damping; ${K}_{1}$, ${K}_{2}$, ${K}_{3}$ – stiffness matrix of the system; ${X}_{1}$, ${Y}_{1}$ –the journal displacement; ${X}_{2}$, ${Y}_{2}$ – displacement of the left disc center; ${X}_{3}$, ${Y}_{3}$ – displacement of the right disc center; ${X}_{4}$, ${Y}_{4}$ – the right journal displacement; $r$ – eccentricity of the disc; ${f}_{X}$, ${f}_{Y}$ – the oilfilm forces in x and y directions; ${F}_{X}$, ${F}_{Y}$ – the seal forces in x and y directions; ${f}_{T}$ – thermal stress.
2.2. Nonlinear oilfilm force and thermal stress
Nonlinear oilfilm force [34] based on the assumption of short bearings can be calculated as:
where, $\sigma $ – Sommerfeld number; $\mu $ – lubricant viscosity; $L$ – the bearing length; ${C}_{b1}$ – the left bearing clearance; ${C}_{b4}$ – he right bearing clearance; ${C}_{s}$ – the seal clearance; $R$ – the bearing radius; $\overline{{f}_{x}}$, $\overline{{f}_{y}}$ – the dimensionless nonlinear oilfilm forces.
Assume the temperature field is steady, the heat conduction equation for steady temperature field [31]:
Introduce the thermal boundary conditions:
Thermal stress caused by the steady state temperature:
where, $\alpha $ – thermal expansion coefficient; $E$ – elastic modulus; $\upsilon $ – Poisson ratio; ${T}_{0}$ – environment temperature of rotor system; $\overline{T}$ – surface temperature of rotor.
2.3. Nonlinear seal force
Muszynska model is used to describe the nonlinear seal force because it not only reflects the nonlinear characteristics of seal force but also describes a clear physical meaning:
$\left[\begin{array}{cc}mf& 0\\ 0& mf\end{array}\right]\left[\begin{array}{l}{X}^{\text{'}\text{'}}\\ {Y}^{\text{'}\text{'}}\end{array}\right].$
In Eq. (12): ${K}_{f}$ – equivalent stiffness; ${m}_{f}$ – equivalent mass; ${D}_{f}$ – equivalent damping; ${\tau}_{f}$ – fluid average circumferential velocity ration; $\omega $ – rotational speed:
Furthermore, ${K}_{0}$, ${D}_{0}$ in Eq. (13) are given by Childs [35]:
${\mu}_{2}=\frac{\sigma}{1+\xi +2\sigma}\left(\frac{1}{6}+{E}_{f}\right),{\mu}_{3}=\frac{\pi {R}_{f}\mathrm{\Delta}P}{\lambda},$
In order to facilitate calculation and avoid excessive truncation errors, the dimensionless transformations are given as follows:
${x}_{3}=\frac{{X}_{3}}{{C}_{s}},{y}_{3}=\frac{{Y}_{3}}{{C}_{s}},{x}_{4}=\frac{{X}_{4}}{{C}_{b4}},{y}_{4}=\frac{{Y}_{4}}{{C}_{b4}},$
Taking the oilfilm force, thermal stress and seal force substitute into the Eq. (1), then the dimensionless equations are carried out:
${y}_{1}^{\text{'}\text{'}}+\frac{C}{{m}_{1}\omega}{y}_{1}^{\text{'}}+\frac{{K}_{1}}{{m}_{1}{\omega}^{2}}{y}_{1}\frac{{K}_{1}{C}_{s}}{{m}_{1}{C}_{b1}{\omega}^{2}}{y}_{2}=\frac{\sigma}{{m}_{1}{C}_{b1}{\omega}^{2}}\overline{{f}_{y}}\frac{g}{{C}_{b1}{\omega}^{2}}+\frac{{f}_{T}}{{m}_{1}{C}_{b1}{\omega}^{2}},$
${x}_{2}^{\text{'}\text{'}}+\frac{C}{({m}_{2}+{m}_{f})\omega}{x}_{1}^{\text{'}}+\frac{{K}_{1}}{\left({m}_{2}+{m}_{f}\right){\omega}^{2}}\left({x}_{2}\frac{{C}_{b1}}{{C}_{s}}{x}_{1}\right)+\frac{{K}_{2}}{({m}_{2}+{m}_{f}){\omega}^{2}}({x}_{2}{x}_{3})$
$=\frac{\overline{{F}_{x}}}{({m}_{2}+{m}_{f}){\omega}^{2}}+\frac{{m}_{2}r\mathrm{c}\mathrm{o}\mathrm{s}{\tau}_{f}}{({m}_{2}+{m}_{f}){C}_{s}}+\frac{{f}_{T}}{({m}_{2}+{m}_{f}){C}_{s}{\omega}^{2}},$
${y}_{2}^{\text{'}\text{'}}+\frac{C}{({m}_{2}+{m}_{f})\omega}{y}_{2}^{\text{'}}+\frac{{K}_{1}}{\left({m}_{2}+{m}_{f}\right){\omega}^{2}}\left({y}_{2}\frac{{C}_{b1}}{{C}_{s}}{y}_{1}\right)+\frac{{K}_{2}}{({m}_{2}+{m}_{f}){\omega}^{2}}({y}_{2}{y}_{3})$
${x}_{3}^{\text{'}\text{'}}+\frac{C}{({m}_{3}+{m}_{f})\omega}{x}_{3}^{\text{'}}+\frac{{K}_{2}}{\left({m}_{2}+{m}_{f}\right){\omega}^{2}}({x}_{3}{x}_{2})+\frac{{K}_{3}}{\left({m}_{2}+{m}_{f}\right){\omega}^{2}}\left({x}_{3}\frac{{C}_{b4}}{{C}_{s}}{x}_{4}\right)$
$=\frac{\overline{{F}_{x}}}{({m}_{3}+{m}_{f}){\omega}^{2}}+\frac{{m}_{2}r\mathrm{c}\mathrm{o}\mathrm{s}{\tau}_{f}}{({m}_{3}+{m}_{f}){C}_{s}}+\frac{{f}_{T}}{({m}_{3}+{m}_{f}){C}_{s}{\omega}^{2}},$
${y}_{3}^{\text{'}\text{'}}+\frac{C}{({m}_{3}+{m}_{f})\omega}{y}_{3}^{\text{'}}+\frac{{K}_{2}}{\left({m}_{3}+{m}_{f}\right){\omega}^{2}}({y}_{3}{y}_{2})+\frac{{K}_{3}}{\left({m}_{2}+{m}_{f}\right){\omega}^{2}}\left({y}_{3}\frac{{C}_{b4}}{{C}_{s}}{y}_{4}\right)$
$=\frac{\overline{{F}_{y}}}{({m}_{3}+{m}_{f}){\omega}^{2}}+\frac{{m}_{3}r\mathrm{s}\mathrm{i}\mathrm{n}{\tau}_{f}}{({m}_{3}+{m}_{f}){C}_{s}}\frac{{m}_{3}g}{\left({m}_{3}+{m}_{f}\right){C}_{s}{\omega}^{2}}+\frac{{f}_{T}}{\left({m}_{3}+{m}_{f}\right){C}_{s}{\omega}^{2}},$
${x}_{4}^{\text{'}\text{'}}+\frac{C}{{m}_{4}\omega}{x}_{4}^{\text{'}}+\frac{{K}_{3}}{{m}_{4}{\omega}^{2}}\left({x}_{4}\frac{{C}_{s}}{{C}_{b4}}{x}_{3}\right)=\frac{\sigma}{{m}_{4}{C}_{b4}{\omega}^{2}}\overline{{f}_{x}}+\frac{{f}_{T}}{{m}_{4}{C}_{b4}{\omega}^{2}},$
${y}_{4}^{\text{'}\text{'}}+\frac{C}{{m}_{4}\omega}{y}_{4}^{\text{'}}+\frac{{K}_{3}}{{m}_{4}{\omega}^{2}}\left({y}_{4}\frac{{C}_{s}}{{C}_{b4}}{y}_{3}\right)=\frac{\sigma}{{m}_{4}{C}_{b4}{\omega}^{2}}\overline{{f}_{y}}\frac{g}{{C}_{b4}{\omega}^{2}}+\frac{{f}_{T}}{{m}_{4}{C}_{b4}{\omega}^{2}}.$
$\overline{{F}_{x}}$, $\overline{{F}_{y}}$ are the dimensionless sealing force that written by:
3. Analysis of experimental results
In this study, the rotorbearing seal system of gas turbines is simplified into a doubledisc rotorbearing model, and two different sliding bearings with different diameters provide support at both ends. In a bearing with oil film force, sealing force is applied on the disc, which is a more common model.
Rotational speed has a significant influence on rotary movement. Rotors generally start with slow rotations under low rotational speed. As the speed gradually increases, the rotor will also increase its rotational speed, such as doubleperiodic motion, quasiperiodic motion and the resonance of firstorder critical speed. However, after the firstorder critical speed has been achieved and the speed continues to increase, rotary movement will change gradually from the nonstationary stage to the smooth stage. Analytical results presented in Fig. 2 to Fig. 4 are based on different speeds under the same temperature. Axis orbit, timedomain waveform, spectrogram, waterfall and Poincaré maps are showned in the Fig. 2. Poincaré maps can be interpreted as a discrete dynamical system. The stability of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding Poincaré map. The multiple frequency components of the spectrum and Poincaré map presented in Fig. 2 indicate that the system is in a quasiperiodic motion. However, with the speed continuing to increase as shown in Fig. 3 and Fig. 4, the movement of the rotor system changes gradually toward the transition to stable region, and harmonic components will continue to reduce. This condition reflects the principle by which the supercritical unit operates: when the speed exceeds the critical speed, the increase in a certain range of speed will stabilize the system further.
Figs. 7 to 9 present the original data and the data after the fivelayer and sixlayer wavelet transforms, respectively. The waveforms of the original data indicate that the system is in a quasiperiodic motion rather than in a periodic motion and that multiple frequency components are observed in the spectrum. Fig. 6 presents the analytical result after fivelayer wavelet processing which shows that only the fundamental frequency components (2X and 3X) were retained. Fig. 7 presents a more detailed analysis of the sixlayer wavelet transform where only a fundamental frequency was retained.
Fig. 2QuasiPeriodic motion
a) Axis orbit
b) Timedomain waveform
c) Spectrogram
d) Waterfall
e) Poincaré map
Fig. 3QuasiPeriodic motion
a) Axis orbit
b) Timedomain waveform
c) Spectrogram
d) Waterfall
e) Poincaré map
Fig. 4Period doubling motion
a) Axis orbit
b) Timedomain waveform
c) Spectrogram
d) Waterfall
e) Poincaré map
Fig. 5Analysis result map of original data
a) Timedomain waveform
b) Spectrogram
c) Waterfall
Fig. 6Analysis result map after fivelayer wavelet processing
a) Timedomain waveform
b) Spectrogram
c) Waterfall
Fig. 7Analysis result map after sixlayer wavelet processing
a) Timedomain waveform
b) Spectrogram
c) Waterfall
This study considers the thermal stress in the entire model. The rotorbearing component of a gas turbine is constantly exposed to high temperatures; thus, the rotorbearingseal system is influenced by thermal stress. Therefore, in this paper, thermal stress is added to the right of the motion equation to analyze the influence of high temperatures on the system and find out which force of the rotorbearing seal system is based on a steadystate temperature field. For simplification, the temperature field is assumed to be steady, and the size of the thermal stress is proportional to the temperature difference. Fig. 8 presents twotime domain waveform figures which only differ in temperatures. Comparing the contents of Fig. 8, results showed that vibration value increases as temperatures rise. Rising temperatures increase the rotor amplitude, pushing it to enter into the unstable region more quickly.
With the increased speed of the rotating shaft, the oil film vortex begins to appear and will remain until the first critical speed is increased. When the rotational speed is increased, the energy of the power frequency vibration is transferred to the oil film. When the speed becomes twice higher than the first critical speed, oil film oscillation occurs, and the vibration amplitude rapidly increases at this time.
After the analysis of the whole equation of motion, findings show that the system instability will occur in advance. The temperature rise affects the material performance, thereby reducing the stiffness of the rotor, bearing and other parts. Critical speed and shaft stiffness are proportional to the increase in temperature. Therefore, the instability of the rotorbearingseal system depends on the temperature increase. In the disc motion equation, the displacement of the rotor in the direction of $x$ and $y$ is larger than that of the sealing force when the sealing force is not considered. Therefore, the sealing force improves rotor system stability. Rotational speed, disc eccentricity, and seal clearance also significantly influence the rotorbearingseal system.
Fig. 8Timedomain waveform with different temperature
a) Low temperature
b) High temperature
4. Conclusions
1) The dynamic equations of the hightemperature rotorbearingseal coupled system are established and analyzed. The level of thermal force increases the instability region. The dynamic behavior of the system becomes more complicated when the thermal stress is considered. Moreover, nonlinear and experimental investigations are conducted intensively based on this model in this paper.
2) The observation of the nonlinear research demonstrates that in the Poincaré diagrams, the quasiperiodic motion occurs at a relatively high rotating speed, which is caused by the oil force. Moreover, the degree of confusion of the system decrease as the rotating speed increase, which shows that it contributes to reduce the oil whirl.
3) Based on wavelet, the quasi wavelet method is proposed. Quasi wavelet transform is a timefrequency analysis method based on convolution filter for similar wavelet analysis. The truncation error not appear in the analysis of the arbitrary waveform (except for the most extreme waveform) of the large capacity data.
4) The experimental analysis indicates that the response of rotorbearing system is influenced by the environment temperature. The vibration amplitude decreases when the environment temperature gets down and rises up when the rotating speed increases. According to the analysis results, the correctness of theoretical model was verified.
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About this article
This work is supported by the National Natural Science Foundation of China (Grant No. 11502140) and the “Innovation Action Program” for the Local Colleges and Universities of Capacity Building Projects in 2015 (Grant No. 15110501000).
Conceived and designed the experiments and Mathematical model: Rui Zhu. Analyzed the experimental data: Yanru Zhang. Contributed analysis tools: Jianxing Ren. Responsible for guiding and modifying the thesis: Hongguang Li and Qingkai.