Influence of support stiffness on aero-engine coupling vibration quantitative analysis

3.1. Modal test of the whole tester

3.1.1. Test scheme

The single-point excitation and multi-point measurement methods are adopted in the experiment. 13 test points are selected on the tester (Fig. 2 shows the positions of the measuring points), 6 of the 13 points are located on the surface of the rotor and the remaining 7 points are located on the surface of the casing. In the test, the accelerations of the test points are measured using B&K4508ICP acceleration sensors. The sine excitation method is adopted in the experiment. Point 1 is the exciting point.

Fig. 2Test points positions schematic diagram of the whole tester modal experiment

3.1.2. Test results and analysis

The first three modal parameters of the tester (listed in Table 1.) can be obtained using the vibration modal identification software. The genetic algorithm is used to identify the modal parameters in Table 1. The reciprocal of the difference between the theoretical expression values and the measured values (obtained by the modal test) of frequency response is set to the objective function. The first order natural frequency is 38.2 Hz, and the modal shape of the rotor and stator coupling exhibits a rigid body vibration, wherein both the rotor and stator exhibit pitching behavior. The second order natural frequency is 46.6 Hz, and the modal shape of the rotor exhibits a rigid body movement. The third order natural frequency is 113.3 Hz, and the modal shape of the rotor exhibits bending.

Table 1Experimental modal results

Order	1st	2nd	3rd
Natural frequency (Hz)	38.20	46.57	113.42
Damping ratio	0.0183	0.0167	0.0134

Fig. 3The first three order modal shapes of the rotor tester

a) The 1st order

b) The 2ndorder

c) The 3rd order

Fig. 3 shows the first, second, and third order modal shapes. The modal shape of the stator is presented on the upper half of Fig. 4, and the lower half of the figure shows the modal shape of the rotor. Clearly, in the first three orders of tester modal shapes of the tester, only the modal shape of the first order shows the coupling vibration of the stator and rotor. In the second and third modal shapes, the vibration in the stator casing is considerably lower than the vibration in the rotor; hence, the coupling degree of the rotor and stator is low.

The stator-casing thickness of the rotor tester is 4 mm, which implies that the stiffness of the stator is high. The phenomenon of coupling vibration of the stator and rotor is not caused by their structures alone. The vibration in the stator in the horizontal direction is due to the mounting method. The rear mounting of the tester is a hanging hinge that lifts the tail of the tester, implying that the rear of the tester is not constrained in the horizontal direction. Hence, the rigid modal shape of the first order stator and rotor coupling is due to the mounting method of the tester.

3.2. Finite element modeling of the tester

The rotor tester is a complex structure comprising observation holes, incentive holes, bolts, keyways, and other small-size structures. These structures have little effect on the dynamic characteristics of the whole tester. However, these structures will increase the scale of the finite element model when meshing the geometrical model, which increases the computation steps. Hence, the structure is simplified for the modeling process, and some small-size structures are neglected. The rotor material is 30CrMnSiA, and the casing material is 1Cr18Ni9Ti. Table 2 lists the material parameters.

Solid185 is selected to mesh the rotor and stator systems of the rotor tester, and Conbin14 is selected to simulate the bearings of the rotor tester. The stiffness values of the supports of the front and rear bearings are consistent with the true bearing stiffness. The bearings are isotropic in nature. The finite element model is updated based on the experimental results. Table 3 lists the stiffness values of the supports of the model, where $k_{1x}$ and $k_{1y}$ are the front support-stiffness values in the horizontal and vertical directions, respectively. $k_{2x}$ and $k_{2y}$ are the rear support stiffness values in the horizontal and vertical directions, respectively. The tester has designed the elastic support structure in two supporting positions. The $k_{1x}$, $k_{1y}$, $k_{2x}$ and $k_{2y}$ in the Table 3 are the series total stiffness of the elastic support structure and the bearing structure. $k_{3x}$ and $k_{3y}$ are the stiffness values of the front mounting in the horizontal and vertical directions, respectively. $k_{4x}$ and $k_{4y}$ are the stiffness values of the rear mounting in the horizontal and vertical directions, respectively.

Fig. 4 shows the supports location of the whole tester. Fig. 5 shows the finite element model of the tester.

Fig. 4Location of the supports

Fig. 5Finite element model of the rotor tester

3.3. Verification of the finite element model

The finite element model is analyzed with the ANSYS software. Fig. 6 shows the first three order modes. Every order modal shape obtained by simulation is similar to the corresponding experimental modal shape.

Fig. 6The first three order modal shapes in the mounting condition in test room

a) The 1st order

b) The 2nd order

c) The 3rd order

The simulation results are compared with the experimental first three order natural frequencies. Table 4 lists the results. In the table, the relative error in the calculation is based on the test results.

The harmonic response analysis of the simulation model is conducted. The points corresponding to the test points are selected in the finite element model. The acceleration frequency response functions of each selected point are calculated, and compared with the test results. Fig. 7 shows the frequency response curves of the test and the simulation for points 1, 3, 4, 5, 7, and 13, in which points 1, 3, 4, 5 are at the rotor, and points 7 and 13 are at the stator.

Fig. 7Comparisons between the frequency response functions of the experiment and the simulation

a) Test point 1 (rotor)

b) Test point 3 (rotor)

c) Test point 4 (rotor)

d) Test point 5 (rotor)

e) Test point 7 (stator)

f) Test point 13 (stator)

It can be seen from Fig. 3 and Fig. 6 that the first three order modal shapes obtained from the simulation and the experiment are similar. The first-order modal shapes both exhibit the rotor and stator coupling vibration, wherein both the rotor and stator exhibit pitching behavior. The second modal shapes of the rotor both exhibit rigid-body vibration. The third order modal shapes of the rotor both exhibit bending.

The Table 4 shows that the first three order natural frequencies of simulation are in good agreement with the experimental results. The maximal relative error between simulation and the test of the first three order natural frequencies is only –0.49 %.

It can be clearly seen form the Fig. 7 that the simulation acceleration-frequency response functions strongly coincide with the experimental ones at the points 1, 3, 4, 5, 7, and 13.

In summary, the finite element model developed in this study can reflect the dynamic characteristics of actual tester. The modeling method is appropriate, and the model is accurate. The model can be used to calculate and predict the characteristics of the real tester.

4.1. Influence analysis of support stiffness on natural frequencies

Taking the front support stiffness ($k_{1x}$) as an example, to study the influence of the single support stiffness value on the natural frequencies, the rear support and mountings stiffness values are fixed at their original values, and $k_{1x}$ is changed from 2×10⁵N/m to 1×10⁷N/m, the first three natural frequencies of the rotor tester are calculated under different rear bearing stiffness. Fig. 8 shows the results.

Fig. 8The relationship between the first 3 orders natural frequencies and front support stiffness values

It can be seen from the Fig. 8 that the $k_{1x}$ has a great effective to the first two order rigid modal natural frequencies, and has almost no influence on the third order bending modal natural frequencies. The first-order natural frequency of the tester increases gradually with $k_{1x}$, and when $k_{1x}$ increases to 1×10⁶N/m, the first-order natural frequency is stable at 38.7 Hz. The second-order natural frequency increases greatly with $k_{1x}$. With $k_{1x}$ increases, the third-order natural frequency has little changes.

4.2. Influence analysis of the supports stiffness on natural frequencies and mode shapes

The first three natural frequencies of the tester change with $k_{1x}$ changes, and what about the modal shapes? The first three order modal shapes under different $k_{1x}$ are shown in Fig. 9. In order to compare the different modal shapes, the modal shapes are normalized. In Fig. 9 the first rotor point displacements of each modal shape are set to 1, i.e. all the displacements of each modal shape are divided by the corresponding compressor rotor displacements. And in every modal shape, the left is the compressor and the right is the turbo.

We can read from Fig. 9 that the first two modal shapes change greatly with $k_{1x}$, and the third ones just has little changes.

(1) In the first-order modal shapes, with $k_{1x}$ increases, the turbo displacements both rotor and stator increase. The rotor pitches intensified, and also the stator. The coupling relationship between rotor and stator is changed. It’s mainly because the compressor vibration is limited with the front support stiffness enhancing, therefore the relative displacement of the turbine increases.

(2) In the second-order mode shapes, with $k_{1x}$ increases, the second-order modal shape of rotor is transformed from pitch to translational motion. The turbo vibration displacements both of rotor and stator are decrease. When $k_{1x}$ is more than 1×10⁶ N/m, the modal shapes are little changes, the rotor vibration turns to translational motion, and also the stator.

(3) The third-order modal shapes changes slightly with the increase of $k_{1x}$. In the modal shape, the rotor is bending vibration and the stator is almost not move.

In summary, natural frequencies and modal shapes of the rigid mode (the first two mode of the tester) are significant affected by $k_{1x}$, and the bending mode ones are little. For the rotor rigid mode, the transformation mainly focuses on the supports, so the rigid mode is affected by support stiffness greatly. The transformation of the rotor bending mode mainly focuses on rotor itself, so the bending mode is affected by support stiffness slightly.

Fig. 9The first 3 orders modal shapes in different front support stiffness values

a) The first order

b) The second order

c) The third order

5.1. Rotor-stator coupling factor

In order to quantitatively measure the degree of coupling vibration between the rotor and stator system, a new evaluation index is defined. The rotor-stator coupling factor $C_i$ of the $i$th order mode is defined as:

Among them, ${\left|R_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${\left|S_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the maximum absolute values of the whole rotor and stator modal displacements in the $i$th order mode, respectively. The closer the $C_i$ is to 1, the higher rotor-stator coupling degree, when $C_i$ equals to 1, it means the coupling degree is highest.

5.2. Influence analysis of the supports stiffness on the rotor-stator coupling factor

Taking $k_{1x}$ as an example, the rear support and mountings stiffness values are fixed at their original values, and $k_{1x}$ is changed from 2×10⁵N/m to 1×10⁷N/m, the rotor-stator coupling factor of each mode shapes are computed under different $k_{1x}$ values. The first-order mode results are shown in Fig. 10 and Fig. 11.

Fig. 10The relationship between the first modal maximum absolute displacements of the rotor and the stator system and the fore support stiffness values

Fig. 11The relationship between the first modal coupling factors and the fore support stiffness values

We can read those conclusions from Fig. 10 and Fig. 11:

The first-order rotor-stator coupling factor ($C_1$) first increases and then stabilizes, with $k_{1x}$. When $k_{1x}$ is less than 1×10⁶ N/m, the maximum absolute rotor displacements reduce and the stator one’s increase, the rotor-stator coupling degree increases, the factor increases. It’s mainly because that the first-order mode shapes mainly includes rotor pitches and stator pitches, stator structure and rotor structure in-phase pitch, with the $k_{1x}$ increases the stator and rotor pitch degree increase, so $C_1$ increases.

Under different supports stiffness, the degree of rotor-stator coupling of the first order mode of the tester changes to some extent. The influence of support stiffness on the degree of rotor-stator coupling is non-linear, and only affects in a certain range.

6.1. The section rubbing risk coefficient between rotor and stator

In order to study the coordination problem of cross-section vibration modal shape between rotor and stator, and do quantitative comparison of the rubbing risk of cross-section, the coefficient of cross-section rub-impact risk between rotor and stator at the section $j$ under the $i$th order mode is defined as:

where $R_{ij}$ and $S_{ij}$ are the rotor and stator modal displacements at section $j$ of the $i$th order modal shapes, respectively; $\underset{j}{Max}\left|R_{ij}\right|$ is the maximum absolute value of $R_{ij}$; $\underset{j}{Max}\left|S_{ij}\right|$ is the maximum absolute value of $S_{ij}$.

Eq. (2) can be used to compare the risk level of blade-casing rub-impact in different conditions, including different sections, different stiffness values in the same section, and so on. When we design a structure, maybe we have several plans, we can calculate the sections rotor-stator rubbing risk coefficients of different projects, which can give some help to compare the plans in the rub-impact characteristics.

6.2. Influence analysis of the supports stiffness on the section rubbing risk coefficient

Taking the first-order vibration mode as an example, the influence of $k_{1x}$ on the first-order mode rotor-stator rubbing risks at the compressor and turbo are studied. Fig. 12 shows the rotor and stator modal displacements at the compressor and turbo sections, which clearly describes each sections displacements variation with $k_{1x}$. Fig. 13 shows $T_{1C}$ and $T_{1T}$ under different $k_{1x}$, in which $T_{1C}$ and $T_{1T}$ are the compressor and turbine section rubbing risk coefficients in the first-order mode, respectively.

Fig. 12The relationship between the first modal absolute displacements of the rotor and the stator system in compressor section and turbo section and the fore support stiffness values

Fig. 13The relationship between the first modal rotor-stator rubbing risk coefficient of compressor and turbo sections and the fore support stiffness values

We can read those conclusions from Fig. 12 and Fig. 13:

(1) In the first-order vibration mode, $T_{1C}$ first decreases and then stabilizes, with $k_{1x}$ increases. When $k_{1x}$ is less than 2×10⁶ N/m, the rotor displacements at the compressor section reduce and the stator displacements slightly increase, so the rubbing risk at this section is less, $T_{1C}$ reduces. When $k_{1x}$ is more than 2×10⁶N/m, the rotor and stator displacements at compressor section tends to be stable, so also $T_{1C}$ also tends to be stable.

(2) In the first-order vibration mode, $T_{1T}$ first increases and then stabilizes, with $k_{1x}$ increases. When $k_{1x}$ is less than 1.4×10⁶N/m, the rotor and stator displacements at the turbine section increase, but the stator displacement increases less than the rotor displacement increases. The rubbing risk between rotor and stator at the section increases, $T_{1T}$ increases. When $k_{1x}$ is more than 1×10⁶ N/m, the rotor and stator displacements at compressor section tends to be stable, so also $T_{1T}$ also tends to be stable.

(3) In the first-order vibration mode, when $k_{1x}$ is less than 1×10⁶ N/m, the compressor section has a higher rubbing risk than the turbine section.

The change of the supports stiffness causes the change of the modal shapes, which causes the change of the relative position of the rotor and the stator, so that the clearance characteristics of rotor and stator in cross-section will be changed, which can change the rubbing risk between rotor and stator. The influence of the supports stiffness on the clearance characteristics of rotor and stator in cross-section is non-linear, which will tend to be stable with the increase of the supports stiffness. Through analysis, it can be seen that the section rubbing risk coefficient proposed in this study can quantitatively reflect the risk of rub-impact between rotor and stator, the changes trend of the coefficient can reflect the risk changes of cross-section rub-impact.