Variation Law and Experimental Study on Multi-stage Critical Speed of an Aero-engine Power Turbine Rotor

Finite element method was used to establish the calculation model of the dynamic characteristics of the simulated power turbine rotor, and the dynamic characteristics of the rotor were systematically calculated and analysed. The variation law of the first three stages critical speeds of the rotor with the supporting stiffness and the mass of each disk were revealed. The dynamic characteristics test of the simulated power turbine rotor in the full speed range was completed, and the calculation results and test results were compared and analysed. The results show that the calculation model reflects the dynamic characteristics of the rotor well, and the rationality of the rotor structure design, supporting layout and critical speed distribution were verified. The study provides a theoretical basis for the adjustment of critical speed based on the supporting stiffness and disk’s mass, it also provides technical reference for the real rotor structure design and test.


Introduction
Rotor is an important rotating part of turbine machineries such as motors, pumps, compressors, gas turbines and aero-engines. It is of great significance to study rotor dynamics. Many scholars like Zhou [1], He [2], Zhang [3] , Tai [4], Du[5] and other researcher [6][7][8][9] have studied the dynamics of turbo machinery rotor and achieved notable results. Compared with the rotor of other turbo machinery, the rotor of aeroengine was characteristic of high speed, complex structure and bad working condition, there were great risks and difficulties in the dynamic research of the real rotor directly. Therefore, the dynamic analysis and research of the simulated rotor (connection structure, supporting layout, mass inertia, etc. were basically consistent with the real rotor) were usually carried out firstly. Deng [10,11] , Liu [12] and Wu [13], conducted dynamic analysis and experimental research on the simulated rotor of the aeroengine , which provided a reference for the structure design and test of the real rotor. Critical speed design is the core content of the rotor dynamics design of small and medium-sized aeroengine, which plays an important role in the development of small and medium-sized aeroengine. In order to meeting the margin requirements of the rotor critical speed design, it can be achieved by selecting appropriate supporting stiffness, changing the mass distribution, optimizing the rotor structure and other methods. Deng [14,20], Nie [15,16], Hong [17], Bai [18], Mei [19] and other scholars have carried out a systematic study on the influencing factors of the critical speed of the aeroengine, providing a theoretical basis for the adjustment of the critical speed. Whether the structure design, supporting layout and critical speed distribution of the rotor are reasonable or not needs to be verified by experiments. In this paper, dynamic analysis and experimental research were carried out on the flexible rotor with complex structure of aeroengine. The dynamic characteristic calculation model of the simulated power turbine rotor was established by finite element method. The dynamic characteristic of the rotor was systematically calculated and analysed. The laws of the first three critical speeds of the rotor change with supporting stiffness and each disk ' s mass were revealed, and the dynamic characteristic in the full speed range was carried out. The rationality of rotor structure design and dynamic design was verified by experimental research, which laid a solid foundation for the dynamic research of real rotor.

Rotor Structure
The schematic diagram of the simulated power turbine rotor is shown in figure 1. The whole rotor was mainly composed of power turbine shaft, first stage simulated power turbine disk, second stage simulated power turbine disk and other parts. The two stages simulated power turbine disks were connected by end teeth, the first stage simulated power turbine disk and power turbine shaft were connected by splines for transmitting torque with cylindrical centering at both ends, and the second stage simulated power turbine disk and power turbine shaft were connected by circle cylinder interference fit. The length of the rotor's power turbine shaft was nearly 1.5 meters, and the hollow structure was adopted. The mass center position, mass and moment of inertia of the simulated power turbine disks were in good agreement with the real disks. There were four bearings in the rotor, among which the No.1 bearing was ball bearing, the No.2, No.6.5 and No.7 bearings were all roller bearing. The simulated power turbine rotor was a slender, hollow structure flexible rotor with large aspect ratio.

Concentrated Mass
In order to facilitating the model, parts of the body of the two stages simulated power turbine disks were simulated with the concentrated mass, which characteristics are shown in table 1.

Finite Element Model
During the pre-processing of the model, the structure of the rotor has been simplified, and some small local structures such as chamfering and threaded holes have been neglected. The finite element software PATRAN was used to establish the model based on beam element, and then the analysis software SAMCEF/ROTOR was used to improve the calculation model (mainly including the establishment of concentrated mass elements and bearing elements). The model has 708 beam elements, 717 nodes, 2 concentrated mass elements and 4 bearing elements. The finite element calculation model of the rotor established is shown in figure 2.

Results of Critical Speeds and Mode Shapes
Under the condition of supporting stiffness in table 2, the critical speeds and mode shapes of the first three stages of the simulated power turbine rotor were calculated. The results of the first three stages critical speed and their margins are shown in table 3, and the first three stages mode shape are shown in figure 3.  It can be seen that the simulated power turbine rotor worked above the two stages critical speed, the second stage critical speed margin was slightly less than 20%, and the other two stages critical speed meet the design requirement. The first three stages mode shapes of the simulated power turbine rotor were all bending mode shapes, and the slender power turbine shaft was the main reason for the bending deformation of the rotor.

Law of Critical Speeds Change with Supporting Stiffness
Without changing the structure and mass distribution of the simulated power turbine rotor, the stiffness of the rotor in the table 2 was taken as the reference stiffness, and the law of the first three stages critical speeds of the rotor change with the supporting stiffness was obtained through calculation, which provides a theoretical basis for the adjustment of critical speeds based on the supporting stiffness.   to 20E+7N/m, curve of the first three stages critical speeds of the simulated power turbine rotor change with No.6.5 supporting stiffness is shown in figure 6, and the change rates of the first three stages critical speed are shown in table 6.When the No.6.5 supporting stiffness increased from 2E+6N/m to 5E+7N/m, the first stage critical speeds of the rotor increased by 131.28%, with the continuous increase to 20E+7N/m of the No.6.5 supporting stiffness, the first stage critical speeds increased by 8.23%. The second stage critical speeds of the rotor increased by 35.40% when the No.6.5 supporting stiffness increased from 2E+6N/m to 20E+7N/m. There was no substantial change in the third stage critical speed of the rotor when the No.6.5 supporting stiffness increased from 2E+6N/m to 20E+7N/m.

Law of Critical Speeds Change with Mass of The First Stage Simulated Power Turbine Disk.
When the density of first stage simulated power turbine disk varied from 1780 kg m -3 to 8400 kg m -3 , the curve of the first three stages critical speeds of the rotor change with density of the first stage simulated power turbine disk is shown in figure8, and the change rates of the first three stages critical speeds are shown in table 9. When the density of the first stage simulated disk increased from 1780 kg m -3 to 8400 kg m -3 , the change rates of the first three stages critical speeds of the simulated power turbine rotor were all less than 5%, indicating that the mass of the first stage disk has little effect on the first three stages critical speeds of the simulated power turbine rotor.

Law of Critical Speeds Change with Mass of The Second Stage Simulated Power Turbine Disk.
When the density of second stage simulated power turbine disk varied from 1780kg m -3 to 8400 kg m -3 , the curve of the first three stages critical speeds of the rotor change with density of the second stage simulated power turbine disk is shown in figure9, and the change rates of the first three stages critical speeds are shown in table 10. When the density of the second stage simulated disk increased from 1780 kg m -3 to 8400 kg m -3 , the change rates of the first two stages critical speed of the rotor were reduced 30.67% and 8.57% respectively, and there was no substantial change in the third critical speed. It shown that the mass of the second stage simulated disk has a certain influence on the first two critical speeds of the rotor, but has no substantial influence on the third critical speed.

Dynamic Characteristics Test and Result Analysis
Dynamic characteristics test of the rotor was carried out on the high-speed rotating rig. In figure 1, a displacement sensor D1 was arranged in the vertical direction of position A, a displacement sensor D2 was arranged in the vertical direction of position C, and two displacement sensors D3 and D4 were respectively arranged in the vertical and horizontal directions of position B. The rotor deflection at positions A, B and C were measured respectively. The curves of rotor deflection change with relative speed the rotor in the full speed range measured by the for sensors are shown in figure 10. In figure 10, the relative speed is defined as: relative speed = (actual speed / rated working speed) × 100%.   The calculation error of the second critical speed was 7.08-9.38%, less than 10%. The difference between rotor calculation model and actual structure is the main cause of error, and the measurement error is another source of the error. In fact, if the actual error of oil film stiffness, the measurement error of supporting stiffness and the complex structure of the rotor were considered, the calculation results still have good consistency with the test results. The rotor has obvious resonance peak at the critical speed, indicating that the rotor has been bending change, it is also consistent with the calculated vibration modes. From the test results, the margin of the second critical speed of the rotor was 23.71~25.31%, greater than 20%, which meets the critical speed margin requirement [21].

Conclusion
In this paper, the finite element calculation model of the dynamic characteristics of the slender simulated power turbine rotor was established, and the dynamic characteristics of the rotor were systematically calculated and analysed. The laws of the first three critical speeds of the rotor change with the supporting stiffness and the mass of each disk were revealed, which provides the theoretical basis for the adjustment of the critical speeds of the rotor, and the dynamic characteristics test verification of the rotor in the full speed range was completed. The main conclusions are as follows.
1.No.1 supporting stiffness has obvious influence on the third critical speed, No.2 supporting stiffness has obvious influence on the first three critical speeds, No.6.5 supporting stiffness has obvious influence on the first and second critical speeds, and No.7 supporting stiffness has obvious influence on the first and second critical speeds.
2. In a certain range of supporting stiffness, the first three critical speeds of the simulated rotor can be adjusted by adjusting the supporting stiffness.
3. The mass of the first stage simulated power turbine disk has no substantial influence on the first three critical speeds of the rotor. The mass of the second stage simulated power turbine disk has a significant influence on the first stage critical speed of the rotor, a certain influence on the second stage critical speed, and no substantial influence on the third stage critical speed.
4. Mode shapes of the first three stages mode shapes of the simulated power turbine rotor are all flexural mode shapes. The test results verified the rationality of the rotor structure design, supporting layout and critical speed distribution.
5.The calculated error of the finite element model is less than 10%, which is in good agreement with the experimental results.