Author Contributions
Conceptualization, S.Z. and F.W.; methodology, S.L.; software, F.W.; validation, F.W., X.M. and Y.L.; formal analysis, S.L.; investigation, S.Z. and F.W.; resources, F.W.; data curation, X.L.; writing—original draft preparation, S.L.; writing—review and editing, S.Z. and S.L.; visualization, X.L.; project administration, X.M. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Model diagram: (a) structural diagram of the industrial turbine engine; (b) cross-section of the industrial turbine engine; (c) structure diagram of the SFD.
Figure 1.
Model diagram: (a) structural diagram of the industrial turbine engine; (b) cross-section of the industrial turbine engine; (c) structure diagram of the SFD.
Figure 2.
Campbell diagram of the rotor system.
Figure 2.
Campbell diagram of the rotor system.
Figure 3.
Mode shapes of the first two critical speeds: (a) mode shape for critical speed 1588 rad/s; (b) mode shape for critical speed 3276 rad/s.
Figure 3.
Mode shapes of the first two critical speeds: (a) mode shape for critical speed 1588 rad/s; (b) mode shape for critical speed 3276 rad/s.
Figure 4.
Bifurcation diagram of support I in 1000−4500 rad/s: (a) bifurcation diagram − ; (b) bifurcation diagram − ; (c) bifurcation diagram − ; (d) bifurcation diagram − .
Figure 4.
Bifurcation diagram of support I in 1000−4500 rad/s: (a) bifurcation diagram − ; (b) bifurcation diagram − ; (c) bifurcation diagram − ; (d) bifurcation diagram − .
Figure 5.
Waterfall diagram of support I in 1000–4500 rad/s: (a) waterfall diagram − ; (b) waterfall diagram − ; (c) waterfall diagram − ; (d) waterfall diagram − .
Figure 5.
Waterfall diagram of support I in 1000–4500 rad/s: (a) waterfall diagram − ; (b) waterfall diagram − ; (c) waterfall diagram − ; (d) waterfall diagram − .
Figure 6.
Horizontal response of the rotor in 1000−4500 rad/s: (a) ; (b) (c) ; (d) ; (e) ; (f) ; (g) ; (h) .
Figure 6.
Horizontal response of the rotor in 1000−4500 rad/s: (a) ; (b) (c) ; (d) ; (e) ; (f) ; (g) ; (h) .
Figure 7.
Horizontal response of support I in 1000–4500 rad/s under different unbalanced configuration: (a) ; (b) .
Figure 7.
Horizontal response of support I in 1000–4500 rad/s under different unbalanced configuration: (a) ; (b) .
Figure 8.
Bifurcation diagram of support I in 1000–4500 rad/s: (a) , ; (b) , ; (c) , ; (d) ; (e) , ; (f) , ; (g) , ; (h) ,.
Figure 8.
Bifurcation diagram of support I in 1000–4500 rad/s: (a) , ; (b) , ; (c) , ; (d) ; (e) , ; (f) , ; (g) , ; (h) ,.
Figure 9.
Waterfall diagram of support I in 1000–4500 rad/s: (a) , ; (b) ,; (c) , ; (d) ; (e) , ; (f) , ; (g) , ; (h) ,.
Figure 9.
Waterfall diagram of support I in 1000–4500 rad/s: (a) , ; (b) ,; (c) , ; (d) ; (e) , ; (f) , ; (g) , ; (h) ,.
Figure 10.
Horizontal response of support I in 1000–4500 rad/s with , : (a) 0X; (b) 1X.
Figure 10.
Horizontal response of support I in 1000–4500 rad/s with , : (a) 0X; (b) 1X.
Figure 11.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 11.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 12.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 12.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 13.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 13.
Horizontal response of support I in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 14.
Horizontal response of the rotor system in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 14.
Horizontal response of the rotor system in 1000−4500 rad/s with , : (a) 0X; (b) 1X.
Figure 15.
Orbit and spectrum of support I with , and : (a) orbit; (b) spectrum.
Figure 15.
Orbit and spectrum of support I with , and : (a) orbit; (b) spectrum.
Figure 16.
Dual-rotor test rig.
Figure 16.
Dual-rotor test rig.
Figure 17.
Structural diagram of the coaxial rotor system.
Figure 17.
Structural diagram of the coaxial rotor system.
Figure 18.
Experimental results from Reference [
22] (Chapter 5, Figure 5.25): (
a) orbit of the inner rotor disk −
; (
b) orbit of the outer rotor disk −
; (
c) orbit of the inner rotor disk −
; (
d) orbit of the outer rotor disk −
.
Figure 18.
Experimental results from Reference [
22] (Chapter 5, Figure 5.25): (
a) orbit of the inner rotor disk −
; (
b) orbit of the outer rotor disk −
; (
c) orbit of the inner rotor disk −
; (
d) orbit of the outer rotor disk −
.
Figure 19.
Results obtained with method in this work: (a) orbit of the inner rotor disk − ; (b) orbit of the outer rotor disk − ; (c) orbit of the inner rotor disk − ; (d) orbit of the outer rotor disk − .
Figure 19.
Results obtained with method in this work: (a) orbit of the inner rotor disk − ; (b) orbit of the outer rotor disk − ; (c) orbit of the inner rotor disk − ; (d) orbit of the outer rotor disk − .
Table 1.
Stiffness of elastic supports (squirrel cage) in industrial turbine engine.
Table 1.
Stiffness of elastic supports (squirrel cage) in industrial turbine engine.
| Support I | Support II |
---|
Stiffness (N/m) | 1.92 × 107 | 2.75 × 107 |
Table 2.
Parameters of SFDs in industrial turbine engine.
Table 2.
Parameters of SFDs in industrial turbine engine.
| SFD I | SFD II |
---|
Radius R/mm | 39.00 | 38.75 |
Length L/mm | 20.30 | 7.20 |
Radial clearance c/mm | 0.07 | 0.09 |
Dynamic viscosity μs/10−3Pa·s | 5.12 |
Table 3.
Inertia properties of blades for each disk.
Table 3.
Inertia properties of blades for each disk.
| Compressor I | Compressor II | Compressor III | Compressor IV | Turbine |
---|
Mass (kg) | 0.17 | 0.36 | 0.28 | 1.269 | 2.12 |
Polar moment of inertia (×10–3 kg·m2) | 4.35 | 1.15 | 0.79 | 9.03 | 10.00 |
Table 4.
Dimension and elements information of the rotor system.
Table 4.
Dimension and elements information of the rotor system.
Node No. | Axial Location (m) | Bearing/Disk | Element No. | Outer Diameter (m) | Inner Diameter (m) |
---|
1 | 0 | | 1 | 0.018 | 0.00 |
2 | 0.08143 | | 2 | 0.018 | 0.00 |
3 | 0.16286 | | 3 | 0.018 | 0.00 |
4 | 0.24429 | | 4 | 0.018 | 0.00 |
5 | 0.24909 | | 5 | 0.018 | 0.00 |
6 | 0.25479 | | 6 | 0.018 | 0.00 |
7 | 0.28879 | | 7 | 0.018 | 0.00 |
8 | 0.32279 | | 8 | 0.018 | 0.00 |
9 | 0.35879 | Disk no.1 | 9 | 0.018 | 0.00 |
10 | 0.38369 | | 10 | 0.018 | 0.00 |
11 | 0.40859 | | 11 | 0.018 | 0.00 |
12 | 0.43349 | | 12 | 0.018 | 0.00 |
13 | 0.43869 | Bearing no.1 | 13 | 0.018 | 0.00 |
14 | 0.44479 | | 14 | 0.022 | 0.00 |
15 | 0.54752 | | 15 | 0.022 | 0.00 |
16 | 0.65025 | | 16 | 0.022 | 0.00 |
17 | 0.75298 | | 17 | 0.022 | 0.00 |
18 | 0.85571 | | 18 | 0.022 | 0.00 |
19 | 0.95844 | | 19 | 0.022 | 0.00 |
20 | 1.06117 | | 20 | 0.022 | 0.00 |
21 | 1.06517 | Bearing no.4 | 21 | 0.022 | 0.00 |
22 | 1.06867 | | 22 | 0.022 | 0.00 |
23 | 1.08867 | | 23 | 0.022 | 0.00 |
24 | 1.10867 | Disk no.2 | 24 | 0.022 | 0.00 |
25 | 1.14274 | | 25 | 0.022 | 0.00 |
26 | 1.17681 | | 26 | 0.022 | 0.00 |
27 | 1.21088 | | 27 | 0.017 | 0.00 |
28 | 1.21488 | Bearing no.2 | 28 | 0.014 | 0.00 |
29 | 1.21838 | | 29 | 0.014 | 0.00 |
30 | 1.23038 | End of inner rotor |
31 | 0.64200 | | 30 | 0.035 | 0.03 |
32 | 0.66065 | | 31 | 0.035 | 0.03 |
33 | 0.67930 | Bearing no.3 | 32 | 0.035 | 0.03 |
34 | 0.68650 | | 33 | 0.038 | 0.03 |
35 | 0.71170 | | 34 | 0.038 | 0.03 |
36 | 0.73690 | | 35 | 0.038 | 0.03 |
37 | 0.76210 | Disk no.3 | 36 | 0.038 | 0.03 |
38 | 0.80784 | | 37 | 0.038 | 0.03 |
39 | 0.85358 | | 38 | 0.038 | 0.03 |
40 | 0.89932 | | 39 | 0.038 | 0.03 |
41 | 0.94506 | | 40 | 0.038 | 0.03 |
42 | 0.99080 | Disk no.4 | 41 | 0.038 | 0.03 |
43 | 1.01430 | | 42 | 0.038 | 0.03 |
44 | 1.02030 | | 43 | 0.070 | 0.03 |
45 | 1.03330 | | 44 | 0.060 | 0.03 |
46 | 1.06030 | | 45 | 0.060 | 0.03 |
47 | 1.06430 | Bearing no.4 | 46 | 0.060 | 0.03 |
48 | 1.07380 | | | | |
Table 5.
Stiffness of elastic supports (squirrel cage) in the test rotor system.
Table 5.
Stiffness of elastic supports (squirrel cage) in the test rotor system.
| Support I | Support II | Support III | Support IV |
---|
Stiffness (N/m) | 1.45 × 106 | 2.21 × 105 | 9.29 × 105 | |
Table 6.
Parameters of the intermediate bearing in the test rotor system.
Table 6.
Parameters of the intermediate bearing in the test rotor system.
Radius of Inner Ring (mm) | Radius of Outer Ring (mm) | No. of Rollers | Contact Stiffness (N/m3/2) | Radial Clearance (μm) |
---|
9.37 | 14.13 | 9 | 7.055 × 109 | 6 |
Table 7.
Parameters of SFDs in the test rotor system.
Table 7.
Parameters of SFDs in the test rotor system.
| Inner Rotor | Outer Rotor |
---|
Support I | Support II | Support III |
---|
Radius R/mm | 25 | 18 | 35 |
Length L/mm | 15 | 15 | 20 |
Radial clearance c/mm | 0.1 | 0.1 | 0.08 |
Dynamic viscosity μs/10−2Pa·s | 1.0752 |
Table 8.
Unbalanced configuration and inertia properties of disks.
Table 8.
Unbalanced configuration and inertia properties of disks.
| Inner Rotor | Outer Rotor |
---|
Disk 1 | Disk 2 | Disk 3 | Disk 4 |
---|
Unbalance (×10−5 kg·m2) | 2 | 4 | 1 | 2 |
Mass (kg) | 2.3386 | 2.3386 | 3.2590 | 1.6303 |
Polar moment of inertia (kg·m2) | 0.00815 | 0.00815 | 0.01561 | 0.00661 |