Coupling vibration analysis of turbine shared support rotor-bearing system with squeeze lm dampers

The turbine shared support structure is used widely in aeroengines, but theoretical and experimental research on a rotor - bearing system containing a shared turbine support structure is lacking. This paper reports research into the coupling vibration response of a squeeze - film - damper rotor - bearing system that has two spools with different rotation speeds and is supported by a turbine shared support structure. The problem is addressed by means of rotor - bearing system tests and the finite - element method. Based on the features of a turboshaft engine with a turbine shared support structure, a rotor - bearing test system with a shared support structure is designed, and a dynamic model of the test system is built based on Timoshenko beam elements. The experimental and simulation results indicate that the unbalanced response of the rotor - bearing system with a shared support structure may involve either the sum or difference of the fundamental frequencies of the rotors of the gas generator and power turbine. The simulations show that the imbalance of the power turbine rotor, the radial and bending stiffnesses of the shared support structure, and the radial clearances of squeeze film dampers at the rear of the rotor - bearing system all affect the coupling response. The amplitude of the coupling response can be suppressed effectively by (i) selecting reasonable parameter values for the turbine shared support structure and (ii) exerting strict control over the spool imbalance.


Introduction
The core components of an aeroengine are its rotor-bearing systems, and the vibration characteristics of the latter determine directly whether the engine can work in harsh environments with high temperatures, pressures, speeds, and high loads, meanwhile, the vibration condition which has a critical influence on the overall performance of the engine. With improvements in aeroengine performance and increased reliability requirements, modern aeroengine rotor-bearing systems are mostly dual-rotor ones.
The support scheme at the turbine involves rotor-bearing systems with either intermediate bearings or a turbine shared support structure (SSS), and the different supporting structures result in different coupling vibrations between the rotor-bearing systems.
Of the two support schemes, the turbine SSS is effective in large turbofan engines at coordinating the larger transition ducts between the high-pressure and low-pressure turbines, such as in the GE90 and CF6-50 aeroengines, and it has a better impact on the clearances of high-pressure-ratio components in the hot section of mid-size turbofan engines compared to when intermediate bearing support is used [1,2]. The SSS is also effective at controlling the power turbine (PT) tip clearance in turboshaft engines and improving the turbine efficiency and maintenance performance. The PT can be replaced without exposing the rear bearing cavity, such as in the MTR390 aeroengine.
In summary, the turbine SSS is used widely in large, medium, and small engines.
However, it contains nonlinear components in the form of squeeze film dampers (SFDs), and it carries two spools that rotate simultaneously with different speeds. Therefore, the nonlinear characteristics of the oil films in the SFDs and the mutual vibration between the two spools have adverse effects on the dynamic responses of the rotor-bearing systems in aeroengines.
In the past few decades, much work has been done on coupling vibration analysis of rotor-bearing and whole-engine systems in aeroengines, resulting in many achievements. However, that literature concerns mainly the coupling vibrations of rotor-bearing and whole-engine systems containing intermediate bearings, with the main modeling approaches involving Newton's second law, the Lagrange equations, and the finite-element (FE) method. Lu et al. [3,4], Thiery and Aidanpäaä [5], and Yang et al. [6] used Newton's second law to establish an analysis model of a rotor-bearing system containing nonlinear components. Gao et al. [7,8] and Hou et al. [9] used the Lagrange equations to establish a dynamic analysis model of a dual-rotor-bearing system with nonlinear elements and studied its response. Lu et al.
[10], Wang et al. [11], Chen [12], Yu et al. [13], and Yang et al. [14] established a dynamic analysis model of dual-rotor-bearing and whole-engine systems using the FE method. Meanwhile, SFDs are used extensively for vibration suppression in aeroengines [22][23][24][25]. However, the severe nonlinear characteristics of the oil-film force in an SFD may result in a dramatic change in the rotor response; more seriously, they may cause instability and damage to the rotor system [26][27][28]. Many scholars remain concerned about SFD-rotor coupling vibrations. Inayat-Hussain et al. [29] used numerical simulation to investigate the unbalanced response of a single-rotor system containing SFDs without centering springs; with changing shaft imbalance, the response of the rotor system experienced a series of period-doubling bifurcations that caused the system to enter a state of chaotic motion after a period of time, which introduced cyclic stresses and may have rapidly reduced the fatigue limit of the shaft. Qin et al. [30] suggested that the support stiffness has considerable influence on the SFDs; excessive support stiffness causes oil whirl and leads to SFD failure. Chen et al. [31] studied the coupling relationship between rigid-body translation and precession of SFD-unsymmetrical rotor-bearing systems; the results showed that the bifurcation The turbine SSS supports the rear three bearings of the two rotors simultaneously.  (1) The mass matrix e m , stiffness matrix e k , and gyroscope matrix e g of the shaft segment are given in Eqs.
(2)-(5), the derivations of which are given elsewhere [21]: where ρe, Ae, and le are the material density, cross-sectional area, and axial length of the shaft section, respectively, and we write m1-m10 as where Ee is the modulus of elasticity and b1-b4 are formulated as g g g g g g g g g g g I g g g g lg g g g g g g g g All disk is simulated as a concentrated mass unit that accounts for the gyroscopic where md, Id, and Ip are the mass, diametral moment, and polar moment of inertia of the disk, respectively.

Equations of motion for rotor-bearing system
The nonlinear dynamic equations of a rotor-bearing system can be expressed as ( ) e f

Mu + C + ΩG u + Ku
where M, C, G, and K are the mass, damping, gyroscope, and stiffness matrices of the rotor-bearing system, respectively, which can be obtained according to the method in Sect. 3.1. Here, u is the displacement vector of the rotor-bearing system, F e is the unbalanced force, and F f is the nonlinear force introduced by an SFD. The unbalanced force F e is expressed as where m is the unbalanced mass, e is its eccentricity, 0  is its initial phase, and ω is the angular velocity of the spool.
where x and y are the horizontal and vertical displacements, respectively, of the shaft diameter, L is the axial width of the SFD, R is the average radius of the SFD, c is the SFD radius clearance, μ is the dynamic viscosity of the oil, and I1, I2, and I3 are Sommerfeld integrals.
Herein, we treat the squirrel cage as an isotropic linear spring whose stiffness is set directly, and we ignore the stiffness and damping of the rolling bearings. Having established the dynamic analysis model of the rotor-bearing system, we use the implicit Newmark-β method to solve for its response.

Finite-element model
The analysis model of the test system is shown schematically in Fig. 3, and the structural parameter values for the shaft, disk, squirrel cage, and SFD are given in Tables 1-4, respectively. In Table 3, supports 1-6 are shaft supports with only their radial stiffness (kxx, kyy) considered. Support 7 is the support of the turbine SSS, with both radial stiffness (kxx, kyy) and bending stiffness (kxz, kyz) considered. In the model established herein, we suppose that kxx = kyy and kxz = kyz.     For safety reasons, the dual-rotor speeds in the tests were increased in two steps:

Fig. 3 Finite-element model
in step 1, the GG spool speed was increased from zero to 6000 rpm while the PT spool remained stationary; in step 2, the GG spool speed remained at 6000 rpm while the PT spool speed was increased from zero to 6000 rpm. In the theoretical analysis, to avoid encountering harmonics such as multiples of 0.5, 1.5, or 2, we set the GG rotor speed to be 1.3 times the PT one.
In both the tests and the FE analysis, the results show that the turbine-

Coupling vibration analysis of rotor-bearing system
Based on the validated analysis model in Sect. 3.4, we subject the turbine-SSS dual-rotor system to coupling vibration research. The parameters studied in this section are the imbalance of the PT rotor, the radial and bending stiffnesses of the SSS (kxx and kxz), and the radial clearances of SFDs 4 and 6. We also analyze the regular influence patterns of each parameter on the amplitudes of the four frequency components of ω1, ω2, ω1−ω2, and ω1+ω2.

Influence of imbalance of power-turbine rotor
The imbalance of the PT rotor is given in Table 5 Figures 6 and 7 show that the imbalance of the PT rotor has little influence on the amplitude corresponding to ω1 at the measuring points of the PT and GG rotors but a greater influence on the responses corresponding to ω2, ω1−ω2, and ω1+ω2. The responses corresponding to the latter three frequency components all increase with increasing imbalance of the PT rotor. When the imbalance of the PT rotor is small, the responses corresponding to ω1−ω2, and ω1+ω2 are extremely small and almost negligible. Therefore, we reason that exerting strict control over the spool imbalance is an effective way to control the coupling vibration response of the SSS rotor-bearing system. The radial stiffness of the SSS is given in Table 6, and the other calculation parameters of the dynamic analysis model are consistent with those of the model in Sect. 3.3.  The bending stiffness of the SSS is given in Table 7, and the other calculation parameters of the dynamic analysis model are consistent with those of the model in Sect. 3.3. Figures 10 and 11 show the responses of the SSS rotor-bearing system at the measuring points of the PT and GG rotors, respectively, for different values of the SSS bending stiffness.  The radius clearance of SFD 4 is given in Table 8 and corresponds to 1-3‰ of the journal radius. The other calculation parameters of the dynamic analysis model are consistent with those of the model in Sect. 3.3. Figures 12 and 13 show the response of the SSS rotor-bearing system at the measuring points of the PT and GG rotors, respectively, for different values of the radius clearance of SFD 4.  increasing radius clearance, and the peak response decreases by more than 50%. After 140 Hz, the response of ω1+ω2 for each radius gap is at a low level. Figure 13b and d show that the responses of ω2 and ω1+ω2 at the measuring point of the GG rotor decrease with increasing radius clearance of SFD 4. The response of ω1−ω2 at the measuring point of the GG rotor decreases with increasing radius clearance before 140 Hz; after 140 Hz, the response of ω1−ω2 does not change regularly with the radius clearance of SFD 4, but in general the peak response for large radius clearance is smaller than that for small radius clearance. As radius clearance of SFD 4 increases, the response of ω1 at the measuring point of the GG rotor increases before 90Hz, and then reduces after 90 Hz, but judging from the response amplitude, the response amplitude for large oil-film thickness has a smaller range of change and the maximum amplitude in the full frequency range is smaller than the response amplitude for small radius clearance of SFD 4.
Therefore, increasing the oil-film clearance of SFD 4 within a reasonable range would help to (i) suppress the coupling vibration response of the SSS rotor-bearing system and (ii) reduce the responses corresponding to the fundamental frequencies of the PT and GG rotors. The radius clearance of SFD 6 is given in Table 9 and corresponds to 1-3‰ of the journal radius. The other calculation parameters of the dynamic analysis model are consistent with those of the model in Sect. 3.3. Figures 14 and 15 show the responses of the SSS rotor-bearing system for different values of the radius clearance of SFD 6. Comparing the response amplitudes of each frequency in Figs. 14 and 15 shows that the oil-film thickness of SFD 6 has a greater impact on the response amplitudes of frequencies ω1 and ω1−ω2 at the measuring points of both the PT and GG rotors but little influence on those of ω2 and ω1+ω2. From the perspective of suppressing the maximum amplitude, a reasonable increase in the oil-film clearance of SFD 6 would help to reduce the response amplitudes of ω1 and ω1−ω2 at the measuring points of the PT and GG rotors of the SSS rotor-bearing system.

Conclusions
Based on the features of a turboshaft engine with a turbine SSS, a test system with a turbine SSS was designed, and an FE model of the test system was established. We studied how the imbalance of the PT rotor, the radial and bending stiffnesses of the SSS, and the radius clearances of the SFDs at the SSS affected the coupling vibration response of the rotor-bearing system, and the main conclusions are as follows.
Both the experimental and theoretical research showed that an SSS rotor-bearing system with SFDs may exhibit coupling vibration responses with frequencies ω1−ω2 and ω1+ω2.
The experimental verification showed that the dynamic analysis model of the rotor-bearing system with an SSS established by FE modeling is accurate and reliable.
Theoretical research using this dynamic analysis model can better reflect the coupling vibration of a rotor-bearing system with an SSS.
The spool imbalance in an aeroengine has a greater impact on the coupling vibration response of a rotor-bearing system with an SSS. Exerting strict control over the spool imbalance would help to suppress the coupling vibration response of a rotor-bearing system with an SSS.
The SSS radial stiffness (kxx) should avoid certain specific frequency components to reduce the probability of coupling vibration arising in an SSS rotor-bearing system.
Furthermore, a larger SSS bending stiffness (kxz) has a positive impact on suppressing the coupling vibration response of a rotor-bearing system with an SSS.
Finally, in a reasonable range, increasing the radial clearances of the SFDs at the SSS would help to reduce the response amplitude of a rotor-bearing system with an SSS.