Under the flight maneuvering of an aircraft, the maneuvering load on the rotor is generated, which may induce the change of dynamic behavior of aeroengine rotor system. To study the influence on the rotor dynamic behavior of constant maneuvering overload, a nonlinear dynamic model of bearing-rotor system under arbitrary maneuver flight conditions is presented by finite element method. The numerical integral method is used to investigate the dynamic characteristics of the rotor model under constant maneuvering overload, and the simulation results are verified by experimental works. Based on this, the dynamic characteristics of a complex intermediate bearing-squeeze film dampers- (SFD-) rotor system during maneuvering flight are analyzed. The simulation results indicate that the subharmonic components are amplified under constant maneuvering overload. The amplitude of the combined frequency components induced by the coupling of the inner and outer rotors is weakened. The static displacements of the rotor caused by the additional excitation force are observed. Besides, the period stability of the movement of the rotor deteriorates during maneuver flight. The design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic under constant maneuvering overload.

The maneuverability is an essential indicator in evaluating the performance of military aircraft. For military drones, it is unnecessary to consider the pilot’s ability to withstand maneuvering loads, and the maneuvering overload will be significantly increased to improve the maneuverability in the future. Additional excitation load will be generated acting on the aeroengine rotor system under the condition of maneuver flight, which may affect the dynamic behavior of the rotor [

In recent years, significant attention has been paid to the dynamic behavior of the rotor system during maneuvering flight. Xu and Liao [

Furthermore, Brentner et al. [

Relative simple models, such as Jeffcott and lumped parameter rotor, are widely adopted in the above mentioned literature, which can efficiently disclose the nonlinear characteristics of rotor system. However, the analysis cannot precisely reflect the vibration behavior of the actual aeroengine complex rotor system. Using the finite element theory, the transient response of a rotor system with SFD during maneuver flight was investigated by Han and Ding [

This paper aims to establish a nonlinear dynamic model of bearing-rotor system under arbitrary maneuver flight conditions using FEM. The added damping, stiffness, and maneuvering load generated by maneuver flight are obtained. The numerical integral method is used to investigate the vibration behavior of the rotor under constant maneuvering overload, and the simulation results are verified by experiments. Based on this, the vibration characteristics of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload are analyzed through the above methods.

To obtain the dynamic differential equations of a disk during maneuvering flight, a coordinate system is presented in Figure

Coordinate system of disk.

And the kinetic energy of a disk of rotor system under arbitrary maneuvering flight actions can be written as

The rotational kinetic energy of the disk during the movement of the rotor is [

Substitute equations (

The schematic diagram of a Timoshenko beam element is shown in Figure

Schematic diagram of beam element.

Similar to the derivation process of a disk, the dynamic differential equations of the beam element under arbitrary maneuver flight conditions can be expressed as

Rearranging the differential equations of motion of the disks and beam elements, the dynamic differential equations of the rotor system under arbitrary maneuver flight actions can be expressed as

In the process of the whirling of the rotor, the nonlinear forces of rolling ball bearing can be written as follows [

Substituting the nonlinear forces of bearing into the generalized external force

To validate the correctness of the dynamic model presented in this paper, a test rig for simulating the maneuvering flight of a bearing-rotor system is proposed and designed, as shown in Figure

Test rig for simulating maneuver flight of a bearing-rotor system.

Diagrams of pitch angle. (a) 0°. (b) 30°. (c) 90°.

Figure

Bearing-rotor system.

Inertial properties and unbalance configuration.

Name | Disk 1 | Disk 2 | Disk 3 |
---|---|---|---|

Mass (kg) | 0.4 | 0.4 | 0.5 |

Polar moment of inertia (g·m^{2}) | 0.00038 | 0.00038 | 0.00041 |

Unbalance (10^{−5} kg·m) | 4 | 4 | 5 |

Parameters of the bearings.

Name | Bearing 1 | Bearing 2 |
---|---|---|

Radius of outer ring (mm) | 30 | 30 |

Radius of inner ring (mm) | 10 | 10 |

No. of rollers | 8 | 8 |

Radial clearance ( | 6 | 6 |

To validate the correctness of the dynamic model, nine kinds of maneuver flight conditions are investigated by numerical simulation and experimental methods, respectively, as shown in Table

Maneuver flight parameters.

Flight maneuvers | ||||||
---|---|---|---|---|---|---|

1 | Hovering | 10 | 12.12 | 8.08 | 0 | 0 |

2 | Hovering | 20 | 17.15 | 11.43 | 0 | 0 |

3 | Hovering | 30 | 21 | 14 | 0 | 0 |

4 | Rolling | 10 | 0 | 0 | 12.12 | 8.08 |

5 | Rolling | 20 | 0 | 0 | 17.15 | 11.43 |

6 | Rolling | 30 | 0 | 0 | 21 | 14 |

7 | Synthetic | 10 | 10.5 | 7 | 6.06 | 4.03 |

8 | Synthetic | 20 | 14.85 | 9.90 | 8.57 | 5.72 |

9 | Synthetic | 30 | 18.19 | 12.12 | 10.5 | 7 |

The finite element model is established, which consists of 40 elements, including 35 beam elements, 3 disk elements, and 2 supporting elements, as shown in Figure

The mesh diagram of the bearing-rotor system.

Dimension and elements information of the bearing-rotor system.

Node number | Axial location (m) | Bearing/disk | Element number | Outer diameter (m) | Inner diameter (m) |
---|---|---|---|---|---|

1 | 0.001 | Disk 3 | 1 | 0.010 | 0 |

2 | 0.015 | 2 | 0.010 | 0 | |

3 | 0.030 | 3 | 0.010 | 0 | |

4 | 0.051 | 4 | 0.010 | 0 | |

5 | 0.066 | 5 | 0.010 | 0 | |

6 | 0.081 | 6 | 0.010 | 0 | |

7 | 0.096 | Bearing 2 | 7 | 0.010 | 0 |

8 | 0.101 | 8 | 0.012 | 0 | |

9 | 0.116 | 9 | 0.012 | 0 | |

10 | 0.133 | 10 | 0.012 | 0 | |

11 | 0.148 | 11 | 0.012 | 0 | |

12 | 0.163 | 12 | 0.012 | 0 | |

13 | 0.178 | 13 | 0.012 | 0 | |

14 | 0.193 | 14 | 0.012 | 0 | |

15 | 0.208 | Disk 2 | 15 | 0.012 | 0 |

16 | 0.225 | 16 | 0.012 | 0 | |

17 | 0.240 | 17 | 0.012 | 0 | |

18 | 0.255 | 18 | 0.012 | 0 | |

19 | 0.270 | 19 | 0.012 | 0 | |

20 | 0.285 | 20 | 0.012 | 0 | |

21 | 0.300 | 21 | 0.012 | 0 | |

22 | 0.315 | 22 | 0.012 | 0 | |

23 | 0.330 | Disk 1 | 23 | 0.012 | 0 |

24 | 0.341 | 24 | 0.012 | 0 | |

25 | 0.356 | 25 | 0.012 | 0 | |

26 | 0.371 | 26 | 0.012 | 0 | |

27 | 0.386 | 27 | 0.012 | 0 | |

28 | 0.401 | 28 | 0.010 | 0 | |

29 | 0.406 | Bearing 1 | 29 | 0.010 | 0 |

30 | 0.419 | 30 | 0.010 | 0 | |

31 | 0.436 | 31 | 0.010 | 0 | |

32 | 0.451 | 32 | 0.010 | 0 | |

33 | 0.466 | 33 | 0.010 | 0 | |

34 | 0.480 | 34 | 0.010 | 0 | |

35 | 0.490 | 35 | 0.010 | 0 | |

36 | 0.500 |

The Newmark-

Waterfall spectra plots for the vibration of disk 3 without maneuver flight. (a) Numerical simulation. (b) Experimental results.

The waterfall spectra plots for the vibration of disk 3 with the rotor operating in 20–600 rad/s under the maneuver flight conditions of hovering, rolling, and the synthetic maneuver of hovering and rolling with the overload of 20 g are shown in Figures

Waterfall spectra plots for the vibration of disk 3 during hovering. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.

Waterfall spectra plots for the response of disk 3 during rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.

Waterfall spectra plots for the vibration of disk 3 during the synthetic maneuver of hovering and rolling. (a) Horizontal response, numerical simulation. (b) Horizontal response, experimental results. (c) Vertical response, numerical simulation. (d) Vertical response, experimental results.

With comparison of the influence of maneuver flight actions on the magnitude of the static displacements in numerical simulation and experimental works, Figure

Variation result of the static displacements of disk 3 with overload under maneuver flight conditions. (a) Horizontal magnitude, numerical simulation. (b) Horizontal magnitude, experimental results. (c) Vertical magnitude, numerical simulation. (d) Vertical magnitude, experimental results.

According to the above result analysis, the dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions presented in this paper has the superior ability to predict the dynamic behavior of the rotor system under constant maneuvering overload.

The dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions has been established and verified by experimental works. Based on this, the dynamic behavior of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload is investigated using the finite element model proposed in this paper.

The nonlinear oil film forces of SFD in the horizontal and vertical directions can be obtained by Reynolds equation [

Similar to the nonlinear forces of rolling ball bearing as equations (

The intermediate bearing-SFD-rotor system is presented in Figure ^{3}, Young’s modulus is 196 GPa, and the shear modulus is 75.5 GPa. The speed ratio of the rotor system is defined as

Structural diagram of the dual rotor system.

Inertial properties and unbalance configuration of disks.

Name | Inner rotor | Outer rotor | ||
---|---|---|---|---|

Disk 1 | Disk 4 | Disk 2 | Disk 3 | |

Mass (kg) | 2.41 | 2.38 | 3.17 | 1.57 |

Polar moment of inertia (kg·m^{2}) | 0.0082 | 0.0082 | 0.0146 | 0.0059 |

Unbalance (10^{−5} kg·m) | 2 | 2 | 1.5 | 3 |

Radial stiffness of elastic supports.

Name | I | II | III |
---|---|---|---|

Stiffness (N/m) | 2.9 × 10^{7} | 4.4 × 10^{7} | 2 × 10^{7} |

Parameters of SFDs.

Name | Inner rotor | Outer rotor | |
---|---|---|---|

SFD I | SFD II | SFD III | |

Radius (mm) | 25 | 18 | 35 |

Length (mm) | 13 | 13 | 14 |

Radial clearance (mm) | 0.25 | 0.18 | 0.35 |

Dynamic viscosity (10^{−2} Pa·s) | 1.0752 | — | — |

Parameters of the intermediate bearing.

Name | Radius of inner ring (mm) | Radius of outer ring (mm) | No. of rollers | Radial clearance ( |
---|---|---|---|---|

Value | 9.35 | 14.15 | 9 | 6 |

The finite element model of the dual rotor system is established, and the geometric dimensions and information of each element are listed in Table

Dimension and elements information of the dual rotor system.

Node number | Axial location (m) | Bearing/disk | Element number | Outer diameter (m) | Inner diameter (m) |
---|---|---|---|---|---|

1 | 0.100 | Disk 1 | 1 | 0.018 | 0 |

2 | 0.130 | 2 | 0.018 | 0 | |

3 | 0.160 | 3 | 0.018 | 0 | |

4 | 0.190 | 4 | 0.018 | 0 | |

5 | 0.220 | 5 | 0.018 | 0 | |

6 | 0.255 | 6 | 0.018 | 0 | |

7 | 0.289 | 7 | 0.018 | 0 | |

8 | 0.323 | 8 | 0.018 | 0 | |

9 | 0.359 | 9 | 0.018 | 0 | |

10 | 0.384 | 10 | 0.018 | 0 | |

11 | 0.409 | 11 | 0.018 | 0 | |

12 | 0.433 | 12 | 0.018 | 0 | |

13 | 0.445 | Bearing 1 | 13 | 0.018 | 0 |

14 | 0.490 | 14 | 0.022 | 0 | |

15 | 0.560 | 15 | 0.022 | 0 | |

16 | 0.650 | 16 | 0.022 | 0 | |

17 | 0.753 | 17 | 0.022 | 0 | |

18 | 0.820 | 18 | 0.022 | 0 | |

19 | 0.958 | 19 | 0.022 | 0 | |

20 | 1.030 | 20 | 0.022 | 0 | |

21 | 1.065 | Bearing 4 | 21 | 0.022 | 0 |

22 | 1.087 | 22 | 0.022 | 0 | |

23 | 1.100 | 23 | 0.022 | 0 | |

24 | 1.140 | Disk 4 | 24 | 0.022 | 0 |

25 | 1.170 | 25 | 0.022 | 0 | |

26 | 1.190 | 26 | 0.022 | 0 | |

27 | 1.211 | 27 | 0.017 | 0 | |

28 | 1.215 | Bearing 2 | 28 | 0.014 | 0 |

29 | 1.220 | 29 | 0.014 | 0 | |

30 | 1.230 | End of the inner rotor | |||

31 | 0.642 | 30 | 0.035 | 0.03 | |

32 | 0.655 | 31 | 0.035 | 0.03 | |

33 | 0.679 | Bearing 3 | 32 | 0.035 | 0.03 |

34 | 0.687 | 33 | 0.038 | 0.03 | |

35 | 0.711 | 34 | 0.038 | 0.03 | |

36 | 0.737 | 35 | 0.038 | 0.03 | |

37 | 0.762 | Disk 2 | 36 | 0.038 | 0.03 |

38 | 0.808 | 37 | 0.038 | 0.03 | |

39 | 0.854 | 38 | 0.038 | 0.03 | |

40 | 0.899 | 39 | 0.038 | 0.03 | |

41 | 0.945 | 40 | 0.038 | 0.03 | |

42 | 0.960 | Disk 3 | 41 | 0.038 | 0.03 |

43 | 0.980 | 42 | 0.038 | 0.03 | |

44 | 1.000 | 43 | 0.07 | 0.03 | |

45 | 1.020 | 44 | 0.06 | 0.03 | |

46 | 1.040 | 45 | 0.06 | 0.03 | |

47 | 1.064 | Bearing 4 | 46 | 0.06 | 0.03 |

48 | 1.074 |

Taking hovering as the research object, the dynamic behavior of the complex intermediate bearing-SFD-rotor system under constant maneuvering overload is analyzed. The maneuvering parameters of the aircraft are shown in Table

Parameters of maneuver flight.

Maneuvers | Overload (g) | ||
---|---|---|---|

Horizontal turn | 98 | 1 | 10 |

Solving the dynamic differential equations under the conditions of straight flight and maneuver flight, respectively, Figures

Waterfall spectra plots for the vibration of disk 3. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.

Waterfall spectra plots for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.

Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.

The bifurcation diagrams of the vibration response varying with the rotational speed of disk 4 are shown in Figure

Bifurcation diagrams for the vibration of disk 4. (a) Horizontal response, straight flight. (b) Horizontal response, hovering. (c) Vertical response, straight flight. (d) Vertical response, hovering.

Orbit of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.

Poincare maps of disk 4. (a) 300 rad/s, straight flight. (b) 300 rad/s, hovering. (c) 400 rad/s, straight flight. (d) 400 rad/s, hovering. (e) 500 rad/s, straight flight. (f) 500 rad/s, hovering. (g) 550 rad/s, straight flight. (h) 550 rad/s, hovering.

The coaxial co- or counterrotating rotors exhibit different vibration characteristics [

Vertical response of disk 3. (a)

Vertical response of disk 4. (a)

The spectrograms for the vertical vibration of disks 3 and 4 are shown in Figures

Spectrograms for the vertical vibration of disk 3. (a) 300 rad/s,

Spectrograms for the vertical vibration of disk 4. (a) 300 rad/s,

A nonlinear dynamic model of the bearing-rotor system under arbitrary maneuver flight conditions is modeled in this paper, and the added damping, stiffness, and maneuvering load generated by maneuver flight are obtained. The numerical integral method is used to investigate the vibration behavior of the rotor under constant maneuvering overload. Moreover, the simulation results are compared and verified by the experimental results. Based on this, the vibration characteristics of a complex intermediate bearing-SFD-rotor system under constant maneuvering overload are analyzed. Conclusions are as follows:

The maneuver flight action can lead to some static displacements of the rotor. It is found that the static displacements in the horizontal direction caused by the additional centrifugal force do not change with the rotating speed of the rotor. And the static displacements in the vertical direction induced by the additional gyro moment increase with the rise in the rotating speed. For the future aircraft with high maneuverability and overload, the clearance of rotors and stators needs to be specially designed to prevent rub-impact fault induced by the static displacements.

Compared with the condition of straight flight, the combined frequency components induced by the coupling of the inner and outer rotors through the intermediate bearing are weakened. In contrast, the subharmonic vibration component caused by the nonlinear characteristics of SFDs is amplified under constant maneuvering overload. The amplitude of subharmonic vibration may far exceed the fundamental frequency vibration under certain maneuver flight conditions, which may do great harm to the bearings of rotor system, and it needs to be paid great attention in the design of aeroengine.

The periodicity of motion of the rotor deteriorates during maneuver flight. For the dual rotor system under the maneuver action of hovering studied in this paper, when the rotating speed exceeds the 3rd critical speed of the rotor system, the movement of the rotor system develops from pseudo-period to chaos gradually as the rotational speed continues to increase.

The co- and counterrotating dual rotor systems have different vibration characteristics under constant maneuvering overload. The design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic induced by large maneuvering overload.

The data that support the findings of this study are available upon request from the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (grant number 51775266) and the National Science and Technology Major Project (grant number 2017-IV-0008-0045).