1 Introduction

The vibration of the whole engine mainly comes from the rotor system, in which the most common cause of the excessive vibration is the rotor mass unbalance [1]. Because of longer shafting and relatively lower support stiffness, the vibration of dual-rotor bearing system is much more sensitive to mass-unbalance [2]. It is easy to reduce the performance of the engine, like causing the failure of bearing, blade damage, rub-impact of rotor-to-stator et al., eventually leading to shortening the service time because of existence of unbalance mass in rotor system. According to the statistics of an engine maintenance factory of China, the number of returning to the factory in advance caused by serious unbalance fault accounts for more than 60% of the total every year, which will seriously reduce the aircraft attendance rate [3]. Therefore, the dynamics balancing of rotor system is a necessary prerequisite for smooth high-speed operation of aero-engine, but vibration analysis is the first necessary premise.

Vibration analysis, as one of the most important criteria, is always used to evaluate the quality of an engine. The fault analysis of aeroengine rotor system mainly focuses on misalignment [4,5,6,7,8], cracked rotor [9,10,11], rub-impact [12,13,14] and multiple faults [15,16,17,18,19], and fan blade off (FBO) is the hot topic in the field of unbalance fault [20,21,22] at present. However, the studies for dynamic characteristics and laws of the rotor-bearing system simultaneously only considering the single unbalance factor have not gained sufficient attention in available investigations, namely that the analysis of unbalance fault caused by system operation or assembly is relatively less, and the research on unbalanced vibration of rotor system mainly focuses on the influence of structural parameters on unbalanced vibration and the influence of unbalanced parameters on the inherent characteristics, for instance, Al-Solihat and Behdinan analyzed the influence of unbalance mass on the critical speed region [23], Wang presented that the response amplitudes decrease with increasing rotating damping coefficients and increase with increasing unbalance for fan rotor system [24], Yang et al. [25] studied the sensitive frequency ranges and effect factors of unbalanced vibration for different sides of blade-disk rotor, etc.

Design of experiment method is adopted in this paper to study the vibration response of the dual-rotor system with the unbalanced disk(s) in different quantities and positions at multiple measurement points, so as to explore the global distribution law of the unbalance vibration response among working speeds and the correlation between the unbalance parameters and the vibration response of the dual-rotor system, which would be helpful to recognize the dynamic characteristics of dual-rotor system unbalance and provide theoretical guidance for the maneuver balance of aircraft.

2 Method and object

The method of Design of experiment (DOE) is to research the correlation between multiple factors and response variables [26]. Latin Hypercube Sampling (LHS), as one of common DOE methods, is a random sampling method with the feature “space filling”, which can ensure that the entire variable space is covered by sample points, so adopted to determine unbalance m·e (where m is the unbalance mass and e is eccentric distance at which unbalance is placed) and phase angle of unbalance disks. The sample space can be determined by Eq. (1) [27],

$$X_{j}^{(i)} = \frac{{\pi_{j}^{(i)} - U_{j}^{(i)} }}{N},1 \le i \le N,1 \le j \le d$$
(1)

where N is the data size, d is the number of variables, π is random number from 1 to N, U is a random number independent of π between [0,1] and X is the sample point, which can be converted to the scale ranges of unbalance parameters ([200,1000] g mm and [0,360]°) by simple algebraic iteration. The Pearson correlation coefficient is applied to study linear correlation between amplitude and unbalance parameters in this study, which can be calculated by Eq. (2) [28],

$$\rho_{X,Y} = \frac{{\text{cov} \left( {X,Y} \right)}}{{\sigma_{X} \sigma_{Y} }}$$
(2)

where X is the unbalance parameter, Y is the corresponding vibration amplitude in the gravity direction, cov is the covariance, and σX and σY is the standard deviation of X and Y; ρX,Y ranges from +1 to − 1, where 1 is total positive linear correlation, 0 is no linear correlation, and − 1 is total negative linear correlation.

Figure 1 shows the structure diagram and the basic size parameters of a dual-rotor system, where M1–M5 are the measuring points at the low-pressure (LP) part, in which M1 is located at the centroid of 1# supporting, M2 at the center of mass of the low-pressure compressor disc (LPC), M3 at the centroid of 2# supporting, M4 at the center of mass of the low-pressure turbine disc (LPT), and M5 at the centroid of 3# supporting; M6–M9 are the measuring points of the high-pressure (HP) part, in which M6 is located at the centroid of 4# supporting, M7 at the center of mass of the high-pressure compressor disc (HPC), M8 at the center of mass of the high-pressure turbine disc (HPT), and M9 at the centroid of the 5# supporting. The horizontal and vertical time-domain signals at each measuring point are collected.

Fig. 1
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The design drawing of the simplified dual-rotor system

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The 3D model of the dual-rotor system is built by Inventor and saved in “x_t” format, which is then imported into ADAMS software, as shown in Fig. 2. The material used in ADAMS is structural steel (E = 2.07 × 1011N/m2, ρ = 7801.0 kg/m3 and Poisson’s ratio υ 0.29). After these settings, the mass and moment of inertia of each part will be calculated and generated automatically according to the part size.

Fig. 2
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Simulation model in ADAMS

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In order to truly reflect the connection states between various parts of the model, the parts that were originally bolted together are connected with fixed pairs. The bearing at each supporting point is represented by BUSHING module, which radial stiffnesses are shown as in Table 1; since there are no blades, the aerodynamic effect of the rotor system can be negligible, so axial stiffness is not considered. Damping of all the bearing supports is Rayleigh damping, which can be determined by mass matrix and stiffness matrix of the system. The high- and low-pressure rotors are driven by STEP functions respectively and N1 and N2 represent the rotational frequency of low-pressure and high-pressure rotors respectively in this paper. The three steady-state operating speeds of the dual-rotor system are shown in Table 2. The sampling frequency is set to 20,000. The X-axis direction is the axial direction, the Y-axis direction is the gravity direction, the Z-axis direction is the horizontal direction, and the direction that coincides with the positive direction of the Z-axis is defined as the 0° phase angle.

Table 1 The radial stiffness values at the supporting points
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Table 2 The three steady-state operating speeds of the dual-rotor system
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ADAMS developed by MDI, as one of the most authoritative and widely used dynamic analysis software of mechanical system as of now, applies Lagrange method to calculate multi-body dynamics and its equation form is \(A\left( {q,t} \right)\ddot{q} = B\left( {q,\dot{q},t} \right)\). It was first proposed to solve the problem in spacecraft field and has been widely used until now. Its advantages are that the number of equations is the least, the number of coordinates of the tree topology system is equal to the degree of freedom of the system, and the dynamics equation is easily converted into ordinary differential equations [29].

3 Results

3.1 Single unbalanced disk

Based on the LHS sampling method and the principle of “the sample size is at least 10 times the number of variables” [30], thirty different groups of unbalance m·e and phase angle in LPC, LPT, HPC and HPT are obtained respectively as shown in Fig. 3.

Fig. 3
figure 3

The unbalanced vector distribution of the single disk, a LPC, b LPT, c HPC, d HPT

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The unbalance sets are based on each sample condition so that the vibration responses of all selected unbalance conditions under three speeds are obtained via ADAMS simulations, and the results are shown in Fig. 4 and Figs. 18, 19 and 20 of “Appendix”. It can be readily found that, (1) N1 is the forced vibration frequency of the rotor system caused by the unbalanced LP disk (LPC or LPT) while N2 by the unbalanced HP disk (HPC or HPT), which agree with the literatures [31,32,33]; (2) because of the existence of damping, the slight frequency shift appears compared with Table 2 in the amplitude-frequency diagrams, which agrees with the literature [34]; (3) moreover, the axis trajectory is a regular circle for single unbalanced disk.

Fig. 4
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Time- and frequency- domain diagrams and axis trajectory of vibration response for unbalanced LPC

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The response amplitudes and the ranges in the Y direction are aggregated and shown in Fig. 5 and “Appendix” Figs. 21 and 22, in which each line corresponds to the unbalanced parameter in Fig. 3. It can be found from these that, (1) the largest response amplitude and most sensitive place to unbalance vibration both is at LPC for unbalanced LPC while at LPT for unbalanced LPT; (2) the response amplitude of the unbalance vibration at 4# bearing supporting is the largest for unbalanced HPC, in which the sensitivity to the unbalanced vibration is the highest, the reason of which is that the mass of HPC is quite large while the 4# support is closest to HPC and its supporting stiffness is relatively lower; (3) and the same situation happens for unbalanced HPT that the largest amplitude and the most sensitive position to unbalance both is the 5# support. According to dual-rotor structure features and the variation law of vibration amplitudes among the different measuring points, the vibration transmission routes are shown in Figs. 6, 7, 8 and 9, along which vibration amplitudes gradually decreases roughly and vibration energy is transferred from the rotors to the outside.

Fig. 5
figure 5

The vibration amplitude of single unbalanced disk at idle speed, a LPC, b LPT, c HPC, d HPT

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Fig. 6
figure 6

Vibration transmission route for unbalanced LPC

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Fig. 7
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Vibration transmission route for unbalanced LPT

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Fig. 8
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Vibration transmission route for unbalanced HPC

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Fig. 9
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Vibration transmission route for unbalanced HPT

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The amplitude of vibration response at each measuring point also varies with the increase of rotating speed. It is the next work to verify whether if there have fixed relations between the vibration amplitudes at different rotating speeds under the same unbalance condition. M *idle (subscript “idle” refers to idle speed, and superscript “*” can refer to cruising speed and full speed, the same below) is used to represent the proportion of vibration amplitudes at other speeds to those at idle speed under the same unbalance condition, so the ratios are calculated through the least square fitting as shown in Fig. 10 and Figs. 23, 24 and 25 of “Appendix” wherein it can be clearly seen that the vibration amplitudes among rotating speeds show the constant proportionality relationship.

Fig. 10
figure 10

The proportional relationship of vibration amplitudes among three rotating speeds for unbalanced LPC

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The Pearson correlation coefficient R*,Y between the unbalance parameters and the corresponding vibration amplitude is shown in Table 3 (the results are nearly the same under the three rotational speeds); Y represents the vibration amplitude, and * represents the unbalance parameters where A represents the m·e and B represents its phase angle. It can be seen from Table 3 that the vibration response amplitude is strongly correlated (linearly correlated) with A, and extremely weakly negatively correlated or extremely weakly correlated or weakly correlated with B.

Table 3 Pearson correlation coefficient R*,Y for single unbalanced disk
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3.2 Double unbalanced disks

Similarly based on the same sampling method and principle, forty different groups of the unbalance m·e and phase angle of the double disks severally in HP and LP rotors (combinations with (LPC, HPC), (LPC, HPT), (HPC, LPT) and (HPT, LPT)) are obtained respectively as shown in Fig. 11 and Figs. 26, 27, and 28 of “Appendix” (four experimental variables at this time). The vibration amplitudes at the measuring points under 40 unbalanced conditions are obtained via ADAMS simulations, and the results at cruising speed are shown in Fig. 12 and Figs. 12, 13 and 14, from which it can be found that, (1) N1 and N2 are the forced vibration frequencies for unbalanced dual rotors system, and the same law can also be obtained at idle and full speeds; (2) the axis trajectory is an enveloping ring.

Fig. 11
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Distribution of unbalanced vectors of double disks, a LPC, b HPC

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Fig. 12
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Time- and frequency-domain diagrams and axis trajectory of vibration response for unbalanced LPC and HPC

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Fig. 13
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Vibration amplitudes excited by N1 of idle speed, a (LPC, HPC), b (LPC, HPT), c (HPC, LPT), d (HPT, LPT)

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Fig. 14
figure 14

Vibration amplitudes excited by N2 of idle speed, a (LPC, HPC), b (LPC, HPT), c (HPC, LPT), d (HPT, LPT)

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The response amplitudes at measuring points and the ranges in the Y direction are shown in Figs. 13 and 14 and Figs. 31, 32, 33 and 34 of “Appendix”. It can be found from the comparison of Figs. 13a, b versus Fig. 5a that the amplitude ranges excited by N1 are almost equal to those for unbalanced LPC and the vibration amplitude at LPC is also the largest with the highest sensitivity to unbalance when the LP unbalance disk is LPC; it can be found from the comparison of Figs. 13c, d versus Fig. 5b that the amplitude ranges excited by N1 are almost equal to those for unbalanced LPT and the vibration amplitude at LPT is also the largest with the highest sensitivity to unbalance when the LP unbalance disk is LPT; it can be found from the comparison of Figs. 14a, b versus 5c that the amplitude ranges excited by N2 are almost equal to those for unbalanced HPC and the vibration amplitude at 4# bearing point is also the largest with the highest sensitivity to unbalance when the HP unbalance disk is HPC; it can be found from the comparison of Figs. 14c, d versus 5d that the amplitude ranges excited by N2 are almost equal to those for unbalanced HPT and the vibration amplitude at 5# bearing point is also the largest with the highest sensitivity to unbalance when the HP unbalance disk is HPT. The above rules can also be found in the comparisons of “Appendix” Figs. 31, 32 and 4 at cruising speed and “Appendix” Figs. 33, 34 and 5 at full speed.

It is similar to the case of a single unbalanced disk that the vibration amplitudes at different rotating speeds also have constant proportionality relationship under the same unbalanced condition for unbalanced LPC and HPC as shown in Fig. 15 (the comparison results for unbalanced (LPC,HPT), (HPC,LPT) and (HPT,LPT) are shown in Figs. 35, 36 and 37 of “Appendix” respectively), which ratios are calculated through the least square fitting. It can be seen from the data comparison of Figs. 15 versus 10 and “Appendix” Fig. 24 that the ratios for unbalanced LPC and HPC are basically consistent with those of separate unbalanced LPC and HPC respectively, and the error rates are no more than 9.6%; it can be seen from the data comparison of “Appendix” Fig. 35 versus Figure 10 and “Appendix” Fig. 25 that the ratios of unbalanced LPC and HPT are basically consistent with those of separate unbalanced LPC and HPT respectively, and the error rates are no more than 5.4%; it can be seen from the data comparison of “Appendix” Fig. 36 versus “Appendix” Figs. 23 and 24 that the ratios of unbalanced HPC and LPT are basically consistent with those of separate unbalanced HPC and LPT respectively, and the error rates are no more than 8.5%; it can be seen from the data comparison of “Appendix” Fig. 37 versus “Appendix” Figs. 23 and 25 that the ratios of unbalanced HPT and LPT are basically consistent with those of separate unbalanced HPT and LPT respectively, and the error rates are no more than 9.33%.

Fig. 15
figure 15

The proportional relationship of amplitudes among three speeds for unbalanced LPC and HPC

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The Pearson correlation coefficients R*,Y between the unbalance parameters and the corresponding vibration amplitude are shown in Figs. 16 and 17 (the results are nearly the same under the three rotational speeds); Y represents the vibration amplitude, and * represents the unbalance parameters, where A and B represent the m·e and phase angle of the unbalanced plate in LP rotor respectively, and C and D represent the m·e and phase angle of the unbalanced plate in HP rotor respectively. It can be found from Fig. 16 that the vibration response amplitude under N1 is strongly correlated (linearly correlated) with parameter A, extremely weakly (negatively) correlated with parameter B (except for unbalanced LPC and HPC that it is moderate correlation), and (extremely) weakly (negatively) correlated with parameters C and D. It can be found from Fig. 17 that the vibration response amplitude under N2 is strongly correlated (linearly correlated) with parameter C, extremely weakly (negatively) correlated with parameters A, B and D.

Fig. 16
figure 16

Pearson correlation coefficient R*,Y under N1

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Fig. 17
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Pearson correlation coefficient R*,Y under N2

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4 Discussion

The research contents and direction of this article different from the previous scholars’ studies only on quite few unbalanced conditions [23, 24, 35,36,37,38,39,40,41,42,43] is new and relatively lacking, and there are very few researches applying DOE in rotor-dynamics unbalance vibration field in which these two studies are representative, (1) Mahadeshwar applied response surface method to sample unbalanced weight, radius, speed and position of the rotor disc on the unbalance in a lathe motorized spindle with one-disc and two-support in order to explore the effect for the vibration responses experimentally [44], (2) Ahmed did DOE for the factors (including unbalance mass) affecting the vibration characteristics and used the response surface method and forced rotordynamic analyses together with Finite-Element-Analysis to evaluate the sensitivities of these factors and their contributions to the imbalance vibration [45]. Hence, the current work is an important and necessary effort in the printed literatures to illustrate and analyze the use of DOE upon the unbalance vibration of a dual-rotor system.

The vibration response amplitude is linearly correlated with m·e in rotor system from the Pearson correlation coefficients, so the linear expression can be determined between amplitude and unbalance (note: the equations for single unbalanced disk are showed in the Tables 4, 5, 6 and 7 of “Appendix”, and as for double unbalanced disks the expressions could be roughly seen as the combinations of two one-unbalanced-disk situations. The accurate prediction for vibration response amplitude based on unbalanced parameters of double unbalanced disks has been basically completed by Chen [30] from the simulation way, but the reverse research that prediction of unbalanced parameters for double unbalanced disks in the dual-rotor system still needs more endeavor to be accomplished in simulation and experiment, which will be one of the focuses in the next work). It can be seen that the vibration amplitude at each measuring point is stable in a certain range within the given ranges of unbalanced parameters and the vibration amplitudes among different rotating speeds have the clear proportional relations so that we can determine the amplitude results at the maximum operating speed according to those at the minimum, and then judge whether the allowable tolerance will be exceeded by it or not. In this way, we can firstly determine the unbalanced disc(s) according to the unbalanced vibration amplitude curve; secondly, unbalance m·e and amplitudes at other operating speed can be diagnosed, so that the aim is achieved that the real-time monitoring for the unbalanced fault of the engine.

Table 4 The fitting equations between amplitude and unbalanced parameters for unbalanced LPC
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Table 5 The fitting equations between amplitude and unbalanced parameters for unbalanced LPT
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Table 6 The fitting equations between amplitude and unbalanced parameters for unbalanced HPC
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Table 7 The fitting equations between amplitude and unbalanced parameters for unbalanced HPT
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5 Conclusion

The unbalance parameters in a dual-rotor system were sampled based on LHS method in order that the vibration response amplitudes under the whole unbalance scales were obtained through dynamic simulation and, consequently, the global law of the unbalance vibration response could be analyzed. The main conclusions are as follows,

  1. (1)

    The vibration amplitude at each measuring point is stable in a certain range within the given ranges of unbalance parameters. The vibration transmission routes of the rotor system can be inferred from dual-rotor structure features and the variation of vibration amplitudes among the measuring points.

  2. (2)

    Single unbalanced disk: the vibration response amplitude at LP unbalanced disc is the largest, while the largest amplitude for HP unbalanced disc happened at the nearest support to the unbalance position, and the response amplitude gradually decreases along the vibration transmission route.

  3. (3)

    HP and LP double unbalanced disks: the vibration amplitude ranges excited by N1 are basically the same as those of the corresponding single unbalanced LP disk (LPC or LPT) respectively, while the ranges excited by N2 are basically the same as those of the corresponding single unbalanced HP disk (HPC or HPT) respectively.

  4. (4)

    The vibration response amplitudes for single unbalanced disk under the same unbalance condition are proportional among three rotational speeds, while the amplitudes for high- and low-pressure double unbalanced disks excited by N1 and N2 are also proportional among three rotational speeds, and the ratios are basically consistent with those of the corresponding single unbalanced disk, which error rates are no more than 9.6%.

  5. (5)

    The calculation of the Pearson correlation coefficients unveiled that the vibration amplitude is linearly related to unbalance m·e and nearly uncorrelated with the distribution of unbalance phase angle for the situations of single unbalanced disk and high- and low-pressure double unbalanced disks (the correlation with phase angle increases slightly compared with that for the single unbalanced).

The results of this study need to be further qualitatively verified on the dual-rotor tester, but how to select the right unbalanced trial weights is a challenge because the counterweight mass should be subject to Latin hypercubic sampling, which requires accurate manufacturing technology. In addition, the selection of metal materials with higher density to manufacture trial-weight bolts that matched with the bolt hole in the disks, which has less impact on the rotor system because it takes up less space.