A novel collaborative optimization assembly process method for multi-performance of aeroengine rotors

In the assembly process of aeroengine, the aeroengines have been rejected at pass-off test for high vibration of high-pressure rotor under the condition of meeting the geometric accuracy. Repeated disassembly and trial assembly will lead to interface damage, which will affect the safety and reliability of the engine. Therefore, a multi-objective collaborative optimization process method for geometric accuracy and dynamic performance is put forward based on the practical engineering problems. In the proposed method, the primal problem firstly reveals the transfer mechanism of coaxiality and unbalance of rotor system, and the prediction model of the coaxiality and unbalance of the rotor system was established. Then, combined with the high-precision reduced-order model of aeroengine high-pressure combined rotor, the system vibration responses under different assembly states are obtained. Finally, a multi-objective collaborative optimization assembly process method considering coaxiality, unbalance and vibration response is obtained. The results show that the collaborative optimization process method considering assembly accuracy and dynamic performance is the optimal assembly strategy, which can reduce the vibration amplitude of the key nodes of the rotor system while ensuring high assembly accuracy and achieve the low-level vibration of the rotor system.


Introduction
As the final link of product production, the assembly performance of aeroengine directly determines the dynamic performance of rotor system. Due to the manufacturing deviation of rotor parts, the deviation is amplified step by step in the stacking assembly process of each rotor part, resulting in the change of characteristic parameters such as center of gravity offset and moment of inertia, which affects the dynamic performance of the rotor system. Under the influence of uncertain factors such as the decline of bolt preload and additional unbalance, the rotor system will have engineering problems such as interface slip, wear, and vibration out of tolerance, which seriously affect the safe and reliable operation of the engine. At present, it is difficult for aeroengine manufacturers to accurately predict rotor vibration in assembly, and some engines failed the test due to out-of-tolerance vibration. Therefore, it is urgent to establish a multi-objective assembly process method for dynamic performance and assembly accuracy in order to meet the technical requirements of high assembly accuracy and lowlevel vibration of the rotor system [1,2].
In recent years, scholars have carried out extensive research on the deviation accumulation and transmission mechanism of parts in the assembly process. Bourdet et al. used the small displacement torsor (SDT) method to characterize the offset of the ideal surface of parts and applied it to the comprehensive tolerance analysis of assembly system [3]. In order to realize the cumulative analysis of deviations in the assembly process, Whitney et al. introduced the homogeneous coordinate transformation in robot kinematics to characterize the relative position offset caused by geometric deviation caused by machining deviation [4,5]. Based on SDT method and the homogeneous coordinate transformation theory [6][7][8], many scholars have proposed a series of improved prediction models and methods for the assembly accuracy of mechanical systems [9][10][11][12][13]. These modeling and analysis methods provide a rich theoretical basis for the prediction of system assembly accuracy and the comprehensive analysis of tolerance. Based on the SDT theory and homogeneous coordinate transformation theory, Sun et al. revealed the influence mechanism of assembly deformation on the overall assembly accuracy of mechanical system [14][15][16]. Some scholars have proposed the theoretical model and method of deviation accumulation control on the assembly path of rotor system [17][18][19].
In order to improve the assembly accuracy of mechanical system, scholars deeply study the assembly process optimization method of rotating machinery based on the assembly accuracy prediction model. Rao and Wu proposed a multilevel component assembly tolerance optimization method by sequential quadratic programming, comprehensively considering the tolerance design requirements and manufacturing cost [20]. Yang et al. proposed two assembly methods of component stacking and collimation by controlling deviation accumulation for the assembly of rotating machinery such as turbine rotor [21,22]. Wang et al. studied the spatial projection characteristics of positioning deviation and orientation deviation of single-stage rotor in the whole rotor system based on the typical bolt connection form of aeroengine and then predicted the coaxiality deviation of the whole rotor system [23]. Liu et al. analyzed the propagation mechanism of unbalance and eccentricity deviation in the assembly process and proposed a method to minimize the initial unbalance in the assembly stage of multistage rotor aeroengine based on genetic algorithm [24]. Sun et al. proposed a new method of unbalance propagation control in precision rotating parts assembly considering machining deviation, measurement deviation, and assembly deviation. The optimal assembly phase angle of each stage rotor is obtained by genetic algorithm, and the minimum unbalance of the system is obtained [25]. Zhou and Gao proposed new general models to indicate the geometric deviations and assembly accuracy in typical rotor parts by simulating large quantities of rotor assemblies under different docking conditions [26].
Scholars have made a series of achievements in the research of assembly accuracy prediction and assembly process of the rotor system. At present, scholars mainly focus on the control of assembly accuracy and initial imbalance, which cannot fully reflect the vibration level of the engine. What's more, there is a complex coupling relationship among initial unbalance, coaxiality, and vibration response in the assembly process of rotor system. The single objective process optimization method is difficult to meet the problems of assembly accuracy and dynamic performance at the same time. At present, there are few reports on the optimization method of aircraft generator rotor assembly process considering the system assembly progress and dynamic performance.
The main purpose of this paper is to propose a multiobjective assembly process optimization method considering the coaxiality, initial unbalance, and vibration response of rotor system. Firstly, the deviation transfer accumulation model of multi-stage rotor assembly accuracy is established to obtain the coaxiality and initial unbalance of the assembled rotor system. Then, the reduced order model of rotor dynamics under the coupling of mechanical characteristics of joint and unbalance is established. Finally, a novel collaborative optimization assembly process method for dynamic performance is proposed to ensure high assembly accuracy and low-level vibration of the rotor system.

Model and method
The high-pressure rotor of aeroengine is composed of front shaft, compressor, and turbine and rear shaft. The rotor parts are connected by short bolts, and the connection diagram of high-pressure simulated rotor is shown in Fig. 1.

Deviation feature recognition
In the assembly process of rotor system, there is a certain manufacturing deviation on the assembly surface. The deviation of assembly will continue to accumulate and transfer along the assembly path, which will affect the assembly accuracy of the whole rotor system. For the matching surface, the least square method is used to fit the matching surface morphology data [27], as shown in Fig. 2a. For the rabbet surface, the offset of centroid position is fitted by the least square method, as shown in Fig. 2b.
The mathematical equations of the fitted plane and circle are: where dX and dY are the displacement characteristics of the fitting circle, and dθxp and dθyp are rotation feature of matching surface. (1) The deviation vector matrix of a single part obtained based on STD is

Assembly accuracy model
For the stacked assembly of multiple rotating parts in the aeroengine rotor system, different axial dimensions will lead to different position deviations in the axial direction of the assembly path (Z direction). Therefore, the structure size on the Z axis has a great influence on deviation transmission. The variable dz is a very small amount relative to Z, and Z + dz is represented by Z. Since the characteristic variation of the rotation vector around the Z axis is very small, the characteristic matrix ignores the manufacturing deviation of rotating features. Therefore, the expression of the modified homogeneous transformation matrix M is The assembly process of rotor system can be regarded as two-level assembly between the latter part and the former part. Taking the assembly of two sub-parts as an example, the assembly deviation accumulation model is shown in Fig. 3.
From Eq. (4), the deviation characteristic matrices of Part 1 and Part 2 can be expressed as The transfer deviation matrix of the assembly of two parts can be expressed as Since the rotation vector and translation vector are small quantities, ignoring the high-order small quantities, Eq. (7) can be simplified as follows: According to the homogeneous coordinate transformation theory, the each part deviation matrix is multiplied through the assembly of a matching surface, until the end of the assembly system [28]. The deviation propagation matrix for the n parts is where R n is the angle cumulative deviation matrix, P n is the displacement cumulative deviation matrix, and P n = [x n , y n , z n ] T . The cumulative deviation of the rotor system is calculated according to the displacement deviation matrix P n in Eq. (9). Therefore, the amplitude and phase of eccentricity for rotor system are

Unbalance prediction model
In the assembly process of multistage rotor, the rotor unbalance includes the unbalance caused by manufacturing deviation and the additional unbalance caused by assembly accuracy. The manufacturing deviations of rotor parts are continuously accumulated and transmitted through the assembly interface, which affects the position of the center of mass relative to the actual rotation axis and then changes the distribution state of initial unbalance of rotors.
The assembly diagram of three-stage rotor system is shown in Fig. 4. O 1 , O 2 , and O 3 are the centroids of rotors tested on the ideal matching plane, C 1 , C 2 , and C 3 are the centroid positions of rotors on assembled components, and the coordinate system XYZO is the global absolute coordinate system. Fig. 3 Assembly diagram of the two-stage rotor system The centroid coordinates of each part tested in the ideal plane are written in the form of homogeneous coordinate matrix.
The centroid coordinates of the rotor part are projected into the global coordinate system XYZO.
The coordinate of unbalance of each stage rotor under the actual working rotation axis is where H is the change matrix between the actual rotation axis and the ideal rotation axis.
where R a and P are the rotation change matrix and translation transformation matrix of the actual rotation axis relative to the ideal rotation axis, respectively.
The coordinates of unbalance of the ith stage rotor under the actual working rotation axis is where R Gi is the coordinates of unbalance of ith stage rotor under the actual working rotation axis. The unbalance vector sum U of the rotor system is where m i is the unbalance of the ith part.

Reduced order model of rotor dynamics
According to the structure, support form, and working principle of the aeroengine high-pressure rotor system in Fig. 1, the discrete rotor dynamics model considering the assembly characteristics of the joint interface is established, in which the setting of each element and node is shown in Fig. 5. The aeroengine rotor system is bolted with multiple joints. In order to accurately study the influence of the mechanical characteristics of the joint on rotor dynamics, it is necessary to fully characterize the dynamic parameters of the joint into the rotor dynamics model. The combination of lumped parameter method and finite element method is used for hierarchical modeling of the high-pressure rotor system. The lumped parameter method is used at the joint to concentrate the dynamic parameters of the joint on the two coupling nodes of the joint interface. The drum, front axle, and rear axle are modeled by the finite element method. The motion equation of the rotor system considering the assembly characteristics of the joint surface is where q(t) is the generalized displacement, M is the mass matrix and inertia matrix, K is the generalized stiffness matrix, C is the Rayleigh damping matrix, Ω is the angular velocity of the rotor, G is the gyro effect matrix, F(t) is the generalized load of the rotor system, and f[q(t), q(t)] is the nonlinear contact load of the joint interface.
The rotor system considers Rayleigh damping, which is expressed as where α and β are the Rayleigh damping parameters. The value of those two parameters are calculated by the first and second natural frequencies and damping coefficients of the rotor system [29,30].
where ω 1 and ω 2 are the first two natural frequencies of the system; ξ 1 and ξ 2 are the damping ratio for the corresponding to natural frequency. The unbalance excitation of ith node is obtained by where me is the mass unbalance at the ith node, and φ is the phase of unbalance at the ith node. where there is no unbalance excitation, the excitation is set to 0 vectors. So the unbalance excitation of the rotor system is The detailed modeling and solution process of rotor dynamics with mechanical characteristics of bolted joint are shown in references [31,32].

Collaborative optimization process method
In order to control the assembly accuracy and initial unbalance of rotor system, the assembly process method of adjusting the assembly phase angle of rotor parts is adopted. By adjusting the assembly phase of each part, the unbalance distribution can be improved while ensuring the assembly accuracy, so as to realize the low-level vibration of rotor system.
For the convenience of description, five different assembly strategies are defined: Strategy 1: multi-objective optimal assembly method with minimum coaxiality and vibration amplitude, Strategy 2: double objective optimal assembly method with minimum coaxiality and unbalance, Strategy 3: minimum coaxiality assembly method, Strategy 4: minimum unbalance vector sum assembly method, and Strategy 5: direct assembly method.
Optimization model of Strategy 1: where F cd (θ) is a multi-objective function, c(θ) is the coaxiality of the rotor system, d(θ) is the vibration amplitude of key nodes, v is the number of key nodes, and θ i is the assembly phase of ith stage rotor. Due to the connection characteristics of the high-pressure rotor system, the change of its phase during assembly is constrained by the bolt mounting hole. The number of bolt holes in the flange plate simulating the high-pressure rotor system is 36, and each change of the installation angle is an integral multiple of 10°. Subject to: where c max is the maximum value of coaxiality of kth stage rotor system, d max is the maximum value of vibration amplitude of kth stage rotor system, and t i is the number of bolts on the matching interface of stage I rotor.
With the same method, the optimization models of the other four assembly strategies are established. Optimization model of Strategy 2: where u max is the maximum value of unbalance of the kth stage rotor system.
Optimization model of Strategy 3: Optimization model of Strategy 4: Strategy 5 is a direct assembly method, without considering the adjustment of assembly phase angle. All parts are assembled as 0 phase.
Due to the large difference in the numerical order of coaxiality, unbalance, and vibration response of the rotor system, each objective variable is dimensionless. The calculation Fig. 6 Assembly process collaborative optimization flow chart of rotor system flow of multi-objective rotor assembly phase optimization model is shown in Fig. 6.

Deviation test results
In order to accurately calculate the coaxiality and unbalance of the assembly system, the manufacturing deviation of all parts of rotor system must be obtained firstly. In term of manufacturing of aeroengine parts, the tolerance requirements are very high. The deviation of matching surface of each part is one of the key factors affecting its assembly accuracy. Accurate feature extraction of manufacturing deviation of connecting joint is the key factor of rotor assembly accuracy and unbalance prediction. The high-precision rotation measuring instrument is used to measure the end face and rabbet runout of the joint interface. The comprehensive deviation test platform of single-stage and multi-stage rotor system is shown in Fig. 7a and b. The rotor unbalance test platform is shown in Fig. 7c. The unbalance of single-stage rotor is tested by vertical dynamic balancing machine. The unbalance data of single parts are shown in Table 1. The rotation characteristics and displacement characteristics by fitting the end face runout are shown in Tables 2 and 3, respectively.

Deviation propagation model verification
In order to verify the accuracy of the deviation transfer model, the runout of parts and assembly of rotor measured respectively are shown in Tables 4 and 5 respectively.
For the assembly accuracy of two-stage assembly and multi-stage assembly, the experimental results and prediction results for assembly are shown in Fig. 8.   Fig. 7 Assembly accuracy and unbalance test platform: a single-stage rotor accuracy test, b multistage rotor accuracy test, and c unbalance test platform As can be seen from Fig. 8, the monomer model has high accuracy in two-stage assembly deviation prediction. For the eccentricity value, the maximum calculation deviation is 5.49%; for eccentric phase, the maximum calculation deviation is 2.20%. The calculation deviation of the monomer model gradually accumulates with the assembly number, and the maximum deviation of the eccentricity value is 14.81%; for the eccentric phase, the maximum calculation deviation is 2.58%.

Assembly strategy optimization results
According to the structural parameters and manufacturing deviation parameters of aeroengine rotor system, combined with the assembly accuracy and unbalance deviation propagation model, the coaxiality and unbalance of the rotor system are gained, and then, the vibration amplitude of the key nodes is obtained through the rotor dynamics reduced-order model (). The assembly phase, coaxiality, and initial unbalance of assembly strategies 1 ~ 5 are acquired by optimization algorithms respectively, as shown in Table 6. The vibration response of the key nodes of the rotor system obtained by various assembly strategies is shown in Fig. 9.
The optimization results of different strategies are shown in Table 7. v c , v u , and v d represent the ratio of coaxiality(c), unbalance(u), and maximum vibration amplitude (d max ) of other strategies to Strategy 5 respectively.

Discussions
The monomer model has high accuracy in two-stage assembly. For the eccentricity value, the maximum calculation deviation is 5.49%; for the eccentric phase, the maximum calculation deviation is 2.20%. It can be seen from Fig. 8 that with the continuous increase of assembly stages, the calculation deviation of monomer model gradually accumulates, and the maximum deviation of eccentricity value is 14.81%, indicating that the influence of assembly deformation on eccentricity value cannot be ignored. For the eccentric phase, the maximum deviation is 2.58%, indicating that the assembly deformation has little effect on the eccentric phase, and the monomer model still has high calculation accuracy. In general, although there are some deviations between the predicted eccentricity and the measured results, the eccentricity accumulation trend is basically the same, indicating that the monomer model can be applied to the deviation prediction of multistage assembly. The high prediction accuracy of the assembly deviation accumulation model provides a theoretical basis for the accurate calculation of the unbalance of the rotor system considering the assembly and manufacturing deviation of the joint.
As shown in Table 7, whether the assembly process method adopts Strategy 3 or Strategy 4, the vibration amplitude of the rotor system can be reduced. Strategy 4 can greatly reduce the  vibration amplitude and keep the unbalance vector sum at a low level, but it is difficult to ensure the accuracy of coaxiality. Therefore, in the assembly process method of single objective optimization, Strategy 3 is the better assembly process method, which not only improves the assembly accuracy of coaxiality but also reduces the vibration of the whole rotor system to a certain level. The assembly process methods of Strategy 1 and Strategy 2 can effectively reduce the vibration level of the rotor system while ensuring the coaxiality requirements. The assembly process method of Strategy 1 has greater advantages in improving the dynamic performance of the rotor system. The maximum vibration amplitude of the system using the assembly process method of Strategy 1 is 50% lower than that of Strategy 2, which fully shows that the objective of minimizing the unbalance vector of rotors parts cannot fully reflect the dynamic performance of rotors. Therefore, the multi-objective optimal assembly method based on assembly accuracy and dynamic performance is a better assembly process strategy, which can not only ensure high assembly accuracy but also more comprehensively reflect the vibration level of the whole rotor system.

Conclusions
Based on the typical connection structures of aeroengine rotor system, this paper studies the assembly accuracy and unbalances transmission mechanism considering Fig. 8 The measured and predicted results of two-stage and multi-stage assembly: a eccentricity value and b eccentricity phase the manufacturing deviation and assembly error and then establishes the prediction model of coaxiality and unbalance of rotor system. On this basis, combined with the reduction dynamics model of rotor connection structure, this paper analyzes the vibration response under different assembly states. The collaborative optimization assembly process method for coaxiality, unbalance, and vibration response is obtained to achieve high-quality and high-efficiency assembly of aeroengine rotor. The main conclusions are as follows: 1. An assembly accuracy prediction model considering manufacturing deviation and assembly error of parts is established. The accuracy prediction method is suitable for the systematic assembly deviation prediction of multistage rotor assembly and accurately identifies the weak links of connection, so as to provide theoretical guidance for the improvement of assembly accuracy in the assembly stage. 2. A reduced-order dynamics model of aeroengine rotor system considering the influence of bolted joint characteristics is established, based on the lumped parameter method and finite element method [31,32]. The reduced order model integrates the assembly factors of aeroengine rotor, meets the requirements of calculation efficiency and accuracy, and provides a theoretical basis for the assembly process optimization of rotor system. 3. An aeroengine rotor assembly process method for dynamic performance is proposed. Taking the highpressure simulated rotor of aeroengine as the research object, the assembly accuracy and vibration performance of different assembly processes are analyzed respectively. The optimization results show that the assembly coaxiality and maximum vibration amplitude for Strategy 1 are decreased by 40.9% and 76.2% respectively, compared with Strategy 5. So the collaborative optimization process method for dynamic performance not only reduces the vibration amplitude of the rotor but also reduces the technical requirements of coaxiality and unbalance. The novel optimization process method considering dynamic performance is a more efficient and economical assembly process method, compared with the traditional geometry centered process strategy.  The basic parameters of the simulation high-pressure rotor are shown in Table 8. According to reasonable equivalence, the relevant parameters of the shaft and disk of the high-pressure rotor system are shown in Tables 9 and 10, respectively.     The runout deviation of the assembly surface of the rotor system is shown in Fig. 10.
Author contribution All authors contributed to the study conception and design. Tao Li: established the dynamic model of rotor connection considering assembly error, proposed a multi-objective collaborative optimization model for geometric accuracy and dynamic performance, and drafted the manuscript.
Zhenhua Wen: planned and coordinated the research project and its funding and summarized the current situation and the existing problems about assembly of rotating combined rotor system.
Binbin Zhao: proposed the accuracy prediction model of error and unbalance of typical bolted combined rotor and participated in the verification of accuracy prediction model.
Qingchao Sun: carried out the experimental verification of the theoretical model and analyzed and discussed the experimental results.
Funding The work was supported by Foundation of He'nan Educational Committee, China (23B460001), the National Natural Science Foundation of China (51975539) and partly by the Key Science and Technique R&D Program of Henan Province, China (212102210275). The authors also wish to thank them for their financial support.

Data Availability
The datasets obtained during the current work are available from the corresponding authors upon request.
Code availability Not applicable.

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Disclaimer The authors declare that the paper is original and has been written based on the own finding. It is confirmed that all the authors are aware and satisfied of the authorship order and correspondence of the paper.
The submitted work has not been published elsewhere in any form or language.