5.1 Nonlinear thermal-dynamic coupling effect
The dynamic responses of the dual-rotor system considering the thermal effect of the intershaft bearing can be obtained by the solving procedure shown in Fig. 6. The root-mean-square (RMS) of the horizontal and vertical dynamic responses of a rotor is used to represent the vibration amplitude of this rotor. The amplitude-frequency curves of the LP rotor for the dynamic model alone and the thermal-dynamic coupling model are shown in Fig. 7(a) and Fig. 7(b), respectively. The solid blue line represents the speed-up process, i.e., the start-up process in which the rotation speed slowly increases from low to high, and the rotation speed ratio is kept constant. The dotted red line represents the speed-down process, i.e., the shutdown process in which the rotation speed slowly decreases from high to low, and the rotation speed ratio is kept constant. The initial dynamic parameters of the two models are the same: the initial radial clearance of the intershaft bearing is \({\delta _0}=20{\text{ \varvec{\mu}m}}\), and the speed ratio is \(\lambda =1.2\).
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(a) For the dynamic model alone. |
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(b) For the thermal-dynamic coupling model. |
Figure 7 Amplitude-frequency curve of the LP rotor in the dual-rotor system. |
Comparing the similarities between Fig. 7(a) and Fig. 7(b), it can be found that both have two resonance regions, i.e., A and B. The dual-rotor system is subjected to dual-frequency unbalanced excitation from the HP and LP rotors. When the rotation speed of the HP and LP rotors increases or decreases at a constant rotational speed ratio, the unbalanced excitations of the HP and LP rotors successively pass through the first-order critical speed of the dual-rotor system. Therefore, the two interrelated primary resonances A and B are formed. Moreover, the amplitude-frequency curves of both show nonlinear dynamic characteristics, such as the jump phenomenon and the bistable phenomenon, because of the nonlinear factors of the intershaft bearing.
Comparing the differences between Fig. 7(a) and Fig. 7(b), when the thermal effect of the intershaft bearing is considered, the resonance regions A and B obviously move to the right, i.e., the resonance frequencies ωA and ωB increase, which indicates that the critical speed of the system increases. The resonance peaks rA and rB obviously decrease, and rB decreases more than rA. Figure 7(a) has three bistable intervals A, B and C, while Fig. 7(b) has only two bistable intervals, A and B. The bistable intervals ΔωA and ΔωB become significantly narrower, and \(\Delta {\omega _{\text{A}}}<\Delta {\omega _{\text{B}}}\) in Fig. 7(a) while \(\Delta {\omega _{\text{A}}}>\Delta {\omega _{\text{B}}}\) in Fig. 7(b). The jump amplitudes of the jump phenomenon in the speed-up process ΔA1 and ΔB1 decrease obviously, and ΔB1 decreases more than ΔA1. The jump amplitudes of the jump phenomenon in the speed-down process ΔA2 and ΔB2 increase obviously, and ΔB2 increases more than ΔA2.
To further explore the influence of the thermal effect of the intershaft bearing on the nonlinear dynamic characteristics of the dual-rotor system, Table 1 shows the resonance frequency, resonance peak, bistable interval, and jump amplitude in the speed-up and speed-down processes for the dynamic model and thermal-dynamic coupling model. In other words, the thermal effect of the intershaft bearing can increase the resonance frequency of the system, reduce the resonance peak, shorten the bistable interval, lower the jump amplitude in the speed-up process, and increase the jump amplitude in the speed-down process. The thermal effect of the intershaft bearing helps weaken the nonlinear dynamic characteristics of the dual-rotor system, and its influence on resonance region B is more significant than that on resonance region A. The rotation speed of resonance region B is greater than that of resonance region A. Thus, the thermal effect of the intershaft bearing in resonance region B is more significant than that in resonance region A.
Table 1
Resonance frequency, resonance peak, bistable interval, and jump amplitude in speed-up and speed-down processes for the dynamic model and thermal-dynamic coupling model.
Parameter | Dynamic model | Thermal-dynamic coupling model | Rate of change |
Resonance frequency ωA (rad/s) | 681 | 698 | + 2.5% |
Resonance frequency ωB (rad/s) | 817 | 831 | + 1.7% |
Resonance peak rA (µm) | 115.3 | 67.4 | -41.5% |
Resonance peak rB (µm) | 108.4 | 49.6 | -54.2% |
Bistable interval ΔωA (rad/s) | 27 | 15 | -44.4% |
Bistable interval ΔωB (rad/s) | 38 | 11 | -71.1% |
Jump amplitude ΔA1 (µm) | 105.3 | 58.0 | -44.9% |
Jump amplitude ΔB1 (µm) | 102.0 | 43.2 | -57.7% |
Jump amplitude ΔA2 (µm) | 26.1 | 29.0 | + 11.1% |
Jump amplitude ΔB2 (µm) | 23.5 | 28.6 | + 21.7% |
To provide insight into the nonlinear dynamic responses for the two resonant peaks in Fig. 7(b), the nonlinear dynamic response analysis of the two resonant peaks \({\omega _{\text{A}}}\) and \({\omega _{\text{B}}}\) in the speed-up and speed-down processes for the thermal-dynamic coupled model are displayed in Fig. 8-Fig. 11, including the time history diagram of the vertical and horizontal dynamic responses, the orbit diagram of the LP rotor, and the spectrum diagram. Herein, ω1 and ω2 are the rotation frequencies of the LP and HP rotors, \(3{\omega _1} - 2{\omega _2}\), \(2{\omega _1} - {\omega _2}\), \(2{\omega _2} - {\omega _1}\) and \({\omega _1}+{\omega _2}\) are their combination frequencies, and \(2{\omega _1}\) and \(2{\omega _2}\) are their double frequencies.
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(a) Vertical response. | (b) Horizontal response. | (c) Orbit diagram. | (d) Spectrum diagram. |
Figure 8 Nonlinear dynamic responses for the thermal-dynamic coupling model at \({\omega _{\text{A}}}=698{\text{ rad/s}}\) in the speed-up process. |
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(a) Vertical response. | (b) Horizontal response. | (c) Orbit diagram. | (d) Spectrum diagram. |
Figure 9 Nonlinear dynamic responses for the thermal-dynamic coupling model at \({\omega _{\text{A}}}=698{\text{ rad/s}}\) in the speed-down process. |
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(a) Vertical response. | (b) Horizontal response. | (c) Orbit diagram. | (d) Spectrum diagram. |
Figure 10 Nonlinear dynamic responses for the thermal-dynamic coupling model at \({\omega _{\text{B}}}=831{\text{ rad/s}}\) in the speed-up process. |
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(a) Vertical response. | (b) Horizontal response. | (c) Orbit diagram. | (d) Spectrum diagram. |
Figure 11 Nonlinear dynamic responses for the thermal-dynamic coupling model at \({\omega _{\text{B}}}=831{\text{ rad/s}}\) in the speed-down process. |
Both Fig. 8 and Fig. 10 depict the speed-up process; Fig. 8 shows resonance region A and Fig. 10 shows resonance region B. The dynamic signals of both are periodic, Fig. 8 is almost a sinusoidal signal, and Fig. 10 seems to be a beat vibration. The orbits of the LP rotor center for both look like a ring. The dominant frequency in Fig. 8 is the rotation frequency of the HP rotor ω2, which means that resonance region A is mainly excited by the unbalanced excitation of the HP rotor. The dominant frequency in Fig. 10 is the rotation frequency of the LP rotor ω1, which means the resonance region B is mainly excited by the unbalanced excitation of the LP rotor. Both Fig. 9 and Fig. 11 depict the speed-down process, while the rotation speeds of Fig. 9 and Fig. 8 are the same, and the rotation speeds of Fig. 11 and Fig. 10 are the same. The amplitude of the dynamic signals in the speed-down process is much smaller than that in the speed-up process. However, the waveforms, orbits of the LP rotor center, and frequency components are much more complex. The motions of the system are almost periodic motions or perhaps even chaotic motions.
The nonlinear thermal effect of the intershaft bearing considering the dynamic characteristics of the dual-rotor system can also be obtained by the solving procedure shown in Fig. 6, i.e., the thermal-dynamic coupling model. The thermal effect of the intershaft bearing without considering the dynamic characteristics of the dual-rotor system can be obtained by solving the heat transfer model in Eq. (30) alone, where the intershaft bearing provides a support force for the dual-rotor system to counteract the gravity as a static load [21]. The relationship curves of the steady-state temperature of the intershaft bearing rollers with the rotation speed for the heat transfer model alone and the thermal-dynamic coupling model are shown in Fig. 12(a) and Fig. 12(b), respectively. The solid blue line represents the speed-up process, and the dotted red line represents the speed-down process. The thermal environment of the two models is the same, as the ambient temperature is \({T_\infty }=20{\text{ }}^\circ {\text{C}}\).
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(a) For the heat transfer model alone. | (b) For the thermal-dynamic coupling model. |
Figure 12 Relationship curve between the steady-state temperature of rollers and the rotation speed. |
In Fig. 12(a), for the thermal effect of the intershaft bearing without considering the dynamic characteristics of the dual-rotor system, the speed-up and speed-down processes coincide with each other. The higher the rotational speed is, the higher the steady-state temperature of the intershaft bearing, i.e., the more significant the thermal effect. In Fig. 12(b), for the thermal effect of the intershaft bearing considering the dynamic characteristics of the dual-rotor system, it can be found that the thermal effect of the intershaft bearing is consistent with Fig. 12(a) in the nonresonance regions, i.e., outside the resonance regions A and B. The speed-up and speed-down processes coincide with each other. The higher the rotation speed is, the more significant the thermal effect of the intershaft bearing. However, in resonance regions A and B, the thermal effect of the intershaft bearing also shows complex nonlinear phenomena such as the jump phenomenon and the bistable phenomenon, similar to the nonlinear dynamic characteristics of the dual-rotor system. Moreover, the jump amplitude and width of the bistable interval in resonance region A are slightly larger than those in resonance region B. This makes the thermal effect of the intershaft bearing extremely significant in the resonance region, especially in the speed-up process, and the temperature is much higher than that in the adjacent nonresonant regions.
To provide insight into the nonlinear thermal effect of the intershaft bearing for the thermal-dynamic coupling model in Fig. 12(b), the transient temperature of the intershaft bearing in the speed-up and speed-down processes when the rotation speed is \({\omega _{\text{A}}}=698{\text{ rad/s}}\) and \({\omega _{\text{B}}}=831{\text{ rad/s}}\) are illustrated in Fig. 13. When the temperature no longer rises, the system transforms from a transient heat transfer state into a steady-state heat transfer state, and the intershaft bearing is in a dynamic thermal equilibrium state.
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(a) At \({\omega _{\text{A}}}=698{\text{ rad/s}}\). | (b) At \({\omega _{\text{B}}}=831{\text{ rad/s}}\). |
Figure 13 Transient temperature of the rollers in the speed-up and speed-down processes. |
Figure 13(a) is in resonance region A, while Fig. 13(b) is in resonance region B. The transient temperature curves in the speed-up and speed-down processes do not coincide. Over time, the temperature increases gradually, but the rate of increase slowly decreases to zero, and the final temperature remains unchanged to reach the steady-state heat transfer state. The rotation speed of resonance region B is much higher than that of resonance region A. Therefore, the temperature of the resonance region B in the speed-up and speed-down processes is higher than that of the resonance region A. However, the jump amplitude of the steady-state temperature ΔTA of resonance region A is significantly larger than that of the steady-state temperature ΔTB of resonance region B.
Overall, the nonlinear thermal-dynamic coupling effect means that the nonlinear dynamic characteristics of the dual-rotor system are deeply coupled with the thermal effect of the intershaft bearing, and both have significant influences on each other. The thermal effect of the intershaft bearing can increase the resonance frequencies and reduce the resonance peaks of the dual-rotor system. Moreover, it helps weaken the nonlinear dynamic characteristics of the dual-rotor system, and its influence on resonance region B is more significant than that on resonance region A. In contrast, the nonlinear dynamic characteristics of the dual-rotor system have a significant influence on the thermal effect of the intershaft bearing in the resonance regions. This makes the steady-state temperature of the intershaft bearing exhibit nonlinear thermal effects, such as the jump phenomenon and bistable phenomenon, and the nonlinear thermal effect in resonance region A is more significant than that in resonance region B.
5.2 Operating radial clearance
The operating radial clearance of the intershaft bearing is related to its initial radial clearance and is closely related to the temperatures of the intershaft bearing rollers and the `inner and outer raceways. Its formula is shown in Eq. (6). Therefore, the rotation speed has an important influence on the operating radial clearance of the intershaft bearing. For initial radial clearance of the intershaft bearing is \({\delta _0}=5{\text{ \varvec{\mu}m}}\), \({\delta _0}=10{\text{ \varvec{\mu}m}}\), \({\delta _0}=15{\text{ \varvec{\mu}m}}\) and \({\delta _0}=20{\text{ \varvec{\mu}m}}\), the relationship curves between the operating radial clearance of the intershaft bearing and the rotation speed are shown in Fig. 14.
For Fig. 14, in the nonresonance regions, i.e., outside the resonance regions A and B, the operating radial clearance of the intershaft bearing \({\delta _{\text{t}}}\) gradually decreases with increasing rotation speed. The relationship curves are straightforward because in the nonresonance regions, the higher the rotation speed is, the higher the temperature of the intershaft bearing, the greater the thermal expansion, and the smaller the operating radial clearance. Moreover, the operating radial clearance decreases significantly with decreasing initial radial clearance \({\delta _{\text{0}}}\). When the initial radial clearance decreases to \({\delta _0}=10{\text{ \varvec{\mu}m}}\), the operating radial clearance even becomes a “negative clearance” at some rotation speeds. When the initial radial clearance continues to decrease to \({\delta _0}=5{\text{ \varvec{\mu}m}}\), the operating radial clearance becomes “negative clearance” at all rotation speeds.
In resonance regions A and B, the relationship curves between the operating radial clearance of the intershaft bearing and the rotation speed become highly complex due to the thermal-dynamic coupling effect. The operating radial clearance decreases sharply in resonance regions A and B during the speed-up process. When the initial radial clearance of the intershaft bearing is \({\delta _0}=20{\text{ \varvec{\mu}m}}\) and \({\delta _0}=15{\text{ \varvec{\mu}m}}\), the operating radial clearance shows nonlinear characteristics, such as the jump phenomenon and bistable phenomenon. The jump amplitude of the speed-up process and the bistable interval in resonance region B are both smaller than those in resonance region A. When the initial radial clearance decreases to \({\delta _0}=10{\text{ \varvec{\mu}m}}\) and \({\delta _0}=5{\text{ \varvec{\mu}m}}\), the nonlinear characteristics, such as the jump phenomenon and the bistable phenomenon, gradually disappear, and are replaced by linear characteristics. This means that with the decrease in the initial radial clearance, the nonlinear characteristics of the operating radial clearance gradually disappear and turn into linear characteristics.
In other words, the thermal-dynamic coupling effect of the dual-rotor system has a significant influence on the operating radial clearance of the intershaft bearing. The operating radial clearance of the intershaft bearing gradually decreases with increasing rotation speed in the nonresonance regions, decreases sharply in the resonance regions, and shows nonlinear characteristics such as the jump phenomenon and the bistable phenomenon. With the decrease in the initial radial clearance of the intershaft bearing, the nonlinear characteristics of the operating radial clearance gradually disappear and turn into linear characteristics. However, when the initial radial clearance of the intershaft bearing is small enough, “negative clearance” occurs in the operating radial clearance of the intershaft bearing, which may lead to shaft clamping and endanger the operation of the rotor system.
5.3 Effect of the initial radial clearance
The initial radial clearance of the intershaft bearing has an important influence on its operating radial clearance and has a significant impact on the dynamic characteristics of the dual-rotor system and the temperature of the intershaft bearing. For initial radial clearances of the intershaft bearing of \({\delta _0}=5{\text{ \varvec{\mu}m}}\), \({\delta _0}=10{\text{ \varvec{\mu}m}}\), \({\delta _0}=15{\text{ \varvec{\mu}m}}\) and \({\delta _0}=20{\text{ \varvec{\mu}m}}\), the amplitude-frequency curves of the LP rotor and relationship curves between the steady-state temperature of the intershaft bearing’s rollers and the rotation speed for the thermal-dynamic coupling model are displayed in Fig. 15, where the solid lines indicate the speed-up process and the dotted lines indicate the speed-down process.
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(a) Amplitude-frequency curves of the LP rotor |
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(b) Relationship curves between the steady-state temperature of rollers and the rotation speed. |
Figure 15 Nonlinear thermal-dynamic coupling effect of the system under different initial radial clearances. Solid lines indicate speed-up processes and dotted lines indicate speed-down processes.
Figure 15 (a) shows that the initial radial clearance of the intershaft bearing \({\delta _{\text{0}}}\) has a crucial influence on the dynamic characteristics of the dual-rotor system. As the initial radial clearance decreases, the resonance peaks of A and B obviously move to the right, i.e., the resonance frequencies ωA and ωB increase obviously. However,the resonance peaks rA and rB increase significantly, and rB increases more and gradually exceeds rA. Moreover, nonlinear characteristics such as the jump phenomenon and bistable phenomenon in resonance regions A and B gradually become weak until they disappear entirely and then turn into linear characteristics.
In Fig. 15 (b), it can be seen that the initial radial clearance of the intershaft bearing \({\delta _{\text{0}}}\) has a significant influence on the thermal effect of the intershaft bearing in both the resonance and nonresonance regions. In the nonresonance regions, the steady-state temperature of the intershaft bearing increases gradually with decreasing initial radial clearance. In resonance regions A and B, the steady-state temperature increases sharply, and the magnitude of the increase rises significantly with the decrease in the initial radial clearance. Moreover, nonlinear characteristics such as the jump phenomenon and bistable phenomenon gradually become weak until they disappear entirely and then turn into linear characteristics.
In summary, the initial radial clearance of the intershaft bearing has an essential influence on the thermal-dynamic coupling effect of the dual-rotor system. With the decrease in the initial radial clearance, the resonance frequencies and peaks of the dual-rotor system both increase significantly. The steady-state temperature of the intershaft bearing increases gradually in the nonresonance regions and increases sharply in the resonance regions. Furthermore, the nonlinear characteristics, such as the jump phenomenon and bistable phenomenon, for both the amplitude-frequency curves of the dual-rotor system and the thermal effect of the intershaft bearing gradually become weak until they disappear entirely and then turn into linear characteristics.
5.4 Effect of the ambient temperature
The ambient temperature has an essential influence on the nonlinear thermal-dynamic coupling effect of the dual-rotor system as well as a significant impact on the operating radial clearance of the intershaft bearing. First, we focus on the influence of ambient temperature on the nonlinear thermal-dynamic coupling effect of the dual-rotor system. For ambient temperatures of \({T_\infty }=20{\text{ }}^\circ {\text{C}}\), \({T_\infty }=30{\text{ }}^\circ {\text{C}}\), \({T_\infty }=40{\text{ }}^\circ {\text{C}}\) and \({T_\infty }=50{\text{ }}^\circ {\text{C}}\), the nonlinear thermal-dynamic coupling effect of the dual-rotor system is displayed in Fig. 16, where the solid lines indicate the speed-up process and the dotted lines indicate the speed-down process.
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(a) Amplitude-frequency curves of the LP rotor |
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(b) Relationship curves between the steady-state temperature of the rollers and the rotation speed. |
Figure 16 Nonlinear thermal-dynamic coupling effect of the dual-rotor system under different ambient temperatures. Solid lines indicate speed-up processes and dotted lines indicate speed-down processes.
In Fig. 16 (a), it can be seen that the effect of ambient temperature on resonance region B is more significant than that on resonance region A. With increasing ambient temperature, A moves to the right slightly, while B shifts to the right significantly, i.e., the resonance frequency ωA is almost still, while ωB increases significantly; resonance peak rA increases slightly, and rB increases obviously. Furthermore, the bistable intervals ΔωA and ΔωB become significantly wider; the jump amplitudes during the speed-up process ΔA1 and ΔB1 increase obviously, while the jump amplitudes during the speed-down process ΔA2 and ΔB2 decrease significantly. This indicates that nonlinear characteristics such as the jump phenomenon and the bistable phenomenon in resonance regions A and B also gradually become stronger with increasing ambient temperature.
Figure 16 (b) shows that the ambient temperature has an important influence on the thermal effect of the intershaft bearing in both the resonance and nonresonance regions. In the nonresonance regions, the steady-state temperature of the intershaft bearing increases gradually with increasing ambient temperature, while its increase is obviously less than that of the ambient temperature. In resonance regions A and B, the steady-state temperature of the intershaft bearing increases sharply, and the magnitude of the increase rises significantly with increasing ambient temperature. Meanwhile, nonlinear characteristics such as the jump phenomenon and the bistable phenomenon gradually become stronger.
Finally, we examined the influence of the ambient temperature on the operating radial clearance of the intershaft bearing. For ambient temperatures of \({T_\infty }=20{\text{ }}^\circ {\text{C}}\), \({T_\infty }=30{\text{ }}^\circ {\text{C}}\), \({T_\infty }=40{\text{ }}^\circ {\text{C}}\) and \({T_\infty }=50{\text{ }}^\circ {\text{C}}\), the relationship curves between the operating radial clearance of the intershaft bearing and the rotation speed are shown in Fig. 17, where the solid lines indicate the speed-up process and the dotted lines indicate the speed-down process.
In Fig. 17, the ambient temperature has an important influence on the operating radial clearance of the intershaft bearing in both the resonance and nonresonance regions. In the nonresonance regions, the operating radial clearance increases gradually with increasing ambient temperature. In resonance regions A and B, the operating radial clearance decreases sharply, and the magnitude of the decrease rises significantly with increasing ambient temperature. Meanwhile, nonlinear characteristics such as the jump phenomenon and bistable phenomenon gradually become stronger.
In conclusion, the ambient temperature has a crucial influence on the nonlinear thermal-dynamic coupling effect of the dual-rotor system and the operating radial clearance of the intershaft bearing. As the ambient temperature increases, the frequency and peak of the second resonance peak of the system increase significantly; in the non-resonant regions, the steady-state temperature and operating radial clearance of the intershaft bearing increase; in the resonance regions, the nonlinear characteristics, such as the jump phenomenon and bistable phenomenon, become stronger for both the thermal-dynamic coupling effect of the dual-rotor system and the operating radial clearance of the intershaft bearing.