Small geometric imbalance leaf disc equivalent calculation method based on main node

Abstract

The invention relates to the technical field of equivalent calculation methods for small geometric misalignment bladed discs, in particular to a small geometric misalignment bladed disc equivalent calculation method based on a main node, which comprises the following specific steps: S1, establishing a harmonic bladed disc structure dynamic model; s2, establishing an equivalent dynamic model of the maladjustment leaf disc main node; s3, a main node response prediction method of the imbalance blisk; and S4, based on the analysis, performing cyclic symmetry analysis on the harmonic bladed disc structure single-sector finite element model, and selecting a group of harmonic bladed disc structure modals under each pitch diameter to perform linear combination so as to represent the basic dynamic characteristics of the imbalance bladed disc. The method has the following beneficial effects that on the premise that the calculation precision is guaranteed, the calculation efficiency can be improved, and technical support is provided for imbalance vibration suppression and optimization design of the engine bladed disc.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of equivalent calculation method of small-geometrically-misaligned bladed disk, in particular to an equivalent calculation method of small-geometrically-misaligned bladed disk based on master node. BACKGROUND

[0002] The bladed disk system of an aero-engine usually adopts a tuned structure in the design, and each sector has the same physical and geometric parameters. However, in the actual processing process, due to the inevitability of tolerances, the bladed disk structure will produce random geometric small misalignment, which destroys its ideal periodic symmetry. This misalignment can cause the excitation energy to be highly concentrated on a few blades, increase the local amplitude, and trigger high-cycle fatigue failure, thereby affecting the safety and reliability of the engine. Therefore, accurately modeling and analyzing the vibration characteristics of the misaligned bladed disk is crucial for optimizing the design of the bladed disk and improving the stability of the system.

[0003] The existing misaligned bladed disk vibration analysis method mainly relies on high-precision full-size finite element modeling to obtain the vibration response of the bladed disk under the misaligned working condition. The main disadvantages of this method include:

[0004] 1. Large calculation scale: The complete finite element modeling needs to accurately describe the geometric shape and physical characteristics of the bladed disk structure, resulting in a huge size of the calculation matrix and high calculation cost.

[0005] 2. Low analysis efficiency: Due to the large amount of calculation, the conventional finite element method is difficult to quickly obtain results in parameter analysis and optimization design, affecting the efficiency of engineering application.

[0006] 3. Difficult to adapt to rapid optimization requirements: In actual engineering, the bladed disk system needs to be designed iteratively multiple times, and the traditional method is difficult to realize efficient vibration characteristic prediction and optimization under the premise of ensuring accuracy. SUMMARY

[0007] The purpose of the present application is to provide an equivalent calculation method of small-geometrically-misaligned bladed disk based on master node to solve the problems raised in the background.

[0008] To achieve the above purpose, the present application provides the following technical solution: an equivalent calculation method of small-geometrically-misaligned bladed disk based on master node, the specific steps are as follows:

[0009] S1, establish a tuned bladed disk structure dynamics model

[0010] Since the tuned bladed disk structure has cyclic symmetry characteristics, only the single-sector model of the bladed disk can obtain the dynamics characteristics of the bladed disk structure, and the motion equation (1.1) of the single sector is:

[0011] (K S -ω 2 MS +iD S )q S =f S +Q S ;

[0012] where K S is the stiffness matrix, M S is the mass matrix, D S is the damping matrix; q S is the sector nodal displacement vector, f S is the sector nodal load vector, Q S is the nodal force vector of adjacent sectors acting on the interface of the sector;

[0013] In order to make full use of the circumferential symmetry of the sector, the sector nodal displacement vector q S is divided into blocks as follows:

[0014] q S ={q l , q i , q r} T ;

[0015] where q l and q r are the left and right boundary nodal displacement vectors of the sector, respectively; q i is the internal nodal displacement vector of the sector;

[0016] Similarly, the nodal force vector Q S of adjacent sectors acting on the interface of the sector can be expressed as:

[0017] Q S ={Q l , 0, Q r} T ;

[0018] where Q l and Q r are the forces of adjacent sectors on the left and right boundary nodes, respectively;

[0019] When the blade disk structure is subjected to a force of a certain order excitation type, the following implicit boundary conditions exist at the left and right boundaries of the single sector:

[0020] The displacement boundary is as follows (1.2): q r =e iak q l ;

[0021] The force boundary is as follows (1.3): Q r =-e iαk Q l ;

[0022] where: i is imaginary unit, k is excitation order (harmonic number); a = 2p / N B is sector angle (N B is number of sectors of bladed disk);

[0023] Considering the balance relation between force and displacement in the force boundary formula and the displacement boundary formula, the node displacement can be expressed as the following formula (1.4):

[0024]

[0025] Substituting formula (1.2), (1.3) and (1.4) into (1.1), the following formula (1.5) can be obtained:

[0026]

[0027] where: The superscript * is Hermitian conjugate;

[0028] Analyzing the form of formula (1.5), the modal characteristic expression (1.6) of a single sector in a circular bladed disk structure can be obtained:

[0029]

[0030] In order to obtain the complete modal set of a tuned bladed disk, k must take all integers from 0 to N / 2, and the modal characteristic expression (1.6) can be obtained by substituting k = 0, 1, 2, …, N / 2 into formula (1.6). and It is known from the properties of and that they are Hermite matrices when k≠0 and N / 2, and the corresponding modes are two-fold frequency at this time, and the two modal shapes are complex conjugate to each other.

[0031] Up to now, the mode shape of the entire tuned bladed disk structure Φ(N×nN B ) can be obtained by expanding the mode shape of a single sector in the local coordinate system.

[0032] where: N is the number of degrees of freedom of the entire bladed disk structure, and n is the number of modes corresponding to each wave frequency (order).

[0033] When the number of sectors is odd, the mode shape of the entire tuned bladed disk structure can be expressed in the following form (1.7):

[0034]

[0035] When the number of sectors is even, the mode shape of the entire tuned bladed disk structure can be expressed in the following form (1.8):

[0036]

[0037] S2, Establishing an equivalent dynamic model of a misaligned bladed disk main node

[0038] After the mass of each blade is mismatched, the overall mismatch matrix of the bladed disk can be expressed as the following equation (1.10):

[0039]

[0040] Where: the misalignment quality matrix for each sector It can also be expressed in the following form (1.11):

[0041]

[0042] The sequential quadratic programming algorithm is used to analyze the misalignment matrix δM of the blade principal nodes. S The specific values ​​at corresponding positions are optimized iteratively; with the goal of ensuring that the errors of the first 6 natural frequencies after blade mass misalignment are all less than 0.01%, a set of optimal master node misalignment values ​​μ is finally output. m To achieve high-precision fitting of different mass misalignments for each blade; the misalignment matrix of blade m can be expressed as equation (1.12):

[0043]

[0044] In modal space, the misalignment correction matrix M Φ =Φ T δMΦ can be solved in blocks, yielding the following equation (1.13):

[0045]

[0046] Where: j and k represent M Φ The row and column indices of each sub-block after partitioning, Φ m j and Φ mk This refers to the block matrix at the corresponding position in the block matrix obtained in equation (1.7) or (1.8);

[0047] S3. Master Node Response Prediction Method for Off-Track Bladed Disks

[0048] The modal characteristics of the harmonic bladed disk structure can be expressed as follows (1.14):

[0049] KΦ=MΦΛ;

[0050] Where: K and M are the stiffness and mass matrices of the harmonic adjustable bladed disk structure; Φ and Λ are its corresponding eigenvectors and eigenvalue matrices;

[0051] Similarly, the modal characteristics of the misaligned bladed disk structure can also be expressed as equation (1.15):

[0052]

[0053] Wherein: delta K and delta M represent the change of the main node stiffness matrix and the mass matrix caused by the blade mistuning respectively;

[0054] The vibration mode of the mistuned bladed disk is expressed as a linear combination of the vibration mode of the tuned bladed disk, and expression (1.16) is obtained:

[0055]

[0056] Wherein: c is a combination coefficient matrix;

[0057] By substituting expression (1.12) into expression (1.11), expression (1.17) is obtained after simplification:

[0058]

[0059] Through the vibration mode Phi of the tuned bladed disk and the mistuning amount delta K and delta M of the main node, the eigenvalue of the mistuned bladed disk structure and the eigenvector (obtained through c) are obtained.

[0060] After mass normalization processing is performed on expression (1.17), expression (1.18) is obtained:

[0061] c T [I+Phi T delta M Phi] c = I.

[0062] Then the vibration mode of the mistuned bladed disk structure is also mass-normalized, and expression (1.19) is obtained:

[0063]

[0064] S4, based on the above analysis, a cyclic symmetry analysis is performed on a single sector finite element model of a tuned bladed disk structure, a group of tuned bladed disk structure modes under each pitch diameter are selected to perform linear combination to represent the basic dynamic characteristics of the mistuned bladed disk.

[0065] The application provides an equivalent calculation method for a small-geometric-mistuned bladed disk based on a main node, and has the following beneficial effects:

[0066] The cyclic symmetry of the whole bladed disc structure is destroyed by the blade mistuning, so the whole mistuned bladed disc model must be established for the bladed disc response prediction; the application provides an equivalent calculation method for the small geometric mistuned bladed disc based on a master node, so as to solve the problems of large scale of the mistuned whole bladed disc model and low calculation efficiency; the core idea of the method is to project the modal shape of the mistuned bladed disc structure to the modal shape space of the tuned bladed disc structure, that is, the tuned bladed disc structure mode is used to construct the mistuned bladed disc structure mode; since the position and size of the mistuned unit are difficult to determine for the specific actual blade, the method uses the concept of "master node" to construct the mistuned matrix combined with the sequential quadratic programming method (SQP), so as to simulate the small geometric mistuning on the blade; by reasonably selecting the master node, the random small geometric mistuning is equivalent to the mass mistuning form, and combined with the dimension reduction technology, the calculation scale is effectively reduced; at the same time, the optimization algorithm is used to fit the resonance frequency under the actual working condition, so as to realize the fast dynamic response prediction of the small geometric mistuned bladed disc. On the premise of ensuring the calculation accuracy, the application can improve the calculation efficiency, and provide technical support for the mistuned vibration suppression and the optimization design of the engine bladed disc. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 It is a local coordinate system diagram of the bladed disc in the application;

[0068] Figure 2 It is a modal shape diagram of the bladed disc mass mistuned sector finite element model in the application;

[0069] Figure 3 It is a master node selection diagram in the application;

[0070] Figure 4 It is a front 12-order mistuned frequency fitting convergence process diagram in the application. DETAILED DESCRIPTION

[0071] The technical solutions in the embodiments of the application will be clearly and completely described below with reference to the drawings in the embodiments of the application.

[0072] In the embodiment, the equivalent calculation method for the small geometric mistuned bladed disc based on the master node has the following specific steps:

[0073] S1, a tuned bladed disc structure dynamic model is established

[0074] Since the tuned bladed disc structure has the cyclic symmetry, only the single sector model of the bladed disc can obtain the dynamic characteristics of the bladed disc structure, and the motion equation (1.1) of the single sector is:

[0075] (K S -ω 2 M S +iD S )q S =fS +Q S ;

[0076] where: K S is the stiffness matrix, M S is the mass matrix, D S is the damping matrix; q S is the sector node displacement vector, f S is the sector node load vector, Q S is the node force vector of adjacent sector acting on the interface of this sector;

[0077] In order to make full use of the circumferential symmetry characteristics of the sector, the sector node displacement vector q S is divided into blocks as follows:

[0078] q S = {q l , q i , q r} T ;

[0079] where: q l and q r are the left and right boundary node displacement vectors of the sector, respectively; q i is the internal node displacement vector of the sector;

[0080] Similarly, the node force vector Q S of adjacent sector acting on the interface of this sector can be expressed as:

[0081] Q S = {Q l , 0, Q r} T ;

[0082] where: Q l and Q r are the forces of adjacent sectors on the left and right boundary nodes, respectively;

[0083] When the blade disk structure is subjected to the action of the order excitation type force, the following implicit boundary conditions exist at the left and right boundaries of the single sector:

[0084] The displacement boundary is as follows (1.2):

[0085] q r = e iak q l ;

[0086] The force boundary is as follows (1.3):

[0087] Q r = -e iαk Q l ;

[0088] where: i is imaginary unit, k is excitation order (harmonic number); a = 2p / N B is sector angle (N B is number of sectors of bladed disk);

[0089] Considering the balance relation between force and displacement in the force boundary formula and the displacement boundary formula, the node displacement can be expressed as the following formula (1.4):

[0090]

[0091] Substituting formula (1.2), (1.3) and (1.4) into (1.1), the following formula (1.5) can be obtained:

[0092]

[0093] where: The superscript * is Hermitian conjugate;

[0094] Analyzing the form of formula (1.5), the modal characteristic expression (1.6) of a single sector in a circular bladed disk structure can be obtained:

[0095]

[0096] In order to obtain the complete modal set of a harmonic bladed disk, k must take all integers within 0 to N / 2. From the properties of and , it can be known that they are Hermite matrices when k≠0 and N / 2, and the corresponding modes are two-fold frequency, and the two modal shapes are complex conjugate to each other;

[0097] Up to now, the mode shape Φ(N×nN B ) of the entire harmonic bladed disk structure can be obtained through the mode shape of a single sector, which is expanded in the local coordinate system;

[0098] where: N is the number of degrees of freedom of the entire bladed disk structure, and n is the number of modes corresponding to each wave frequency (order);

[0099] When the number of sectors is odd, the mode shape of the entire harmonic bladed disk structure can be expressed in the following form (1.7):

[0100]

[0101] When the number of sectors is even, the mode shape of the entire harmonic bladed disk structure can be expressed in the following form (1.8):

[0102]

[0103] It should be noted that please refer to Figure 1Each of the above-mentioned sector mode shapes is obtained in a respective local coordinate system, and the local coordinate system of the jth sector is obtained by rotating the first sector coordinate system by an angle of a(j-1).

[0104] S2, establishing an equivalent dynamic model of the imbalance bladed disk master node

[0105] After the imbalance of each blade, the imbalance matrix of the whole bladed disk can be expressed as the following formula (1.10):

[0106]

[0107] Wherein: the imbalance mass matrix of each sector And can be expressed in the following form (1.11):

[0108]

[0109] In the specific implementation, the present application uses the sequential quadratic programming algorithm to perform high-precision fitting on the imbalance of each blade by means of the idea of applying concentrated masses to a plurality of necessary master nodes. S The specific values of the corresponding positions are optimized and iterated; the final output is a set of optimal master node imbalance values μ m , so as to realize high-precision fitting of the imbalance of each blade; the imbalance matrix of the m blades can be expressed as the following formula (1.12):

[0110]

[0111] Therefore, in the modal space, the imbalance correction matrix M Φ = Φ T δMΦ can be solved in blocks, and the following formula (1.13) is obtained:

[0112]

[0113] Wherein: j and k represent the row index and column index of each sub-block of the block matrix M Φ After the block, Φ mj and Φ mk are the block matrices at the corresponding positions in the block matrix obtained in formula (1.7) or (1.8);

[0114] Subsequently, only a small number of nodes are subjected to modal analysis, thereby greatly reducing the matrix dimension while obtaining accurate imbalance modal characteristics;

[0115] In the dynamic response prediction analysis, the basic characteristics of the response at the imbalance natural frequency points are used, and only the dangerous nodes on the dangerous blades in the dangerous frequency band are selected for response analysis and calculation, which greatly improves the operation efficiency and accurately obtains the maximum forced response amplitude of the bladed disk.

[0116] S3. Master Node Response Prediction Method for Off-Track Bladed Disks

[0117] The modal characteristics of the harmonic bladed disk structure can be expressed as follows (1.14):

[0118] KΦ=MΦΛ;

[0119] Where: K and M are the stiffness and mass matrices of the harmonic adjustable bladed disk structure; Φ and Λ are its corresponding eigenvectors and eigenvalue matrices;

[0120] Similarly, the modal characteristics of the misaligned bladed disk structure can also be expressed as equation (1.15):

[0121]

[0122] Where: δK and δM represent the changes in the principal node stiffness matrix and mass matrix caused by blade misalignment, respectively;

[0123] In practice, directly solving equation (1.15) requires a complex inversion process, resulting in low computational efficiency. Therefore, the mode shape of the misaligned bladed disk is represented as a linear combination of the mode shapes of the harmonic bladed disk, as shown in expression (1.16) below:

[0124]

[0125] Where: c is the combination coefficient matrix; substituting equation (1.12) into equation (1.11), we can simplify to obtain equation (1.17):

[0126]

[0127] The characteristic values ​​of the misaligned bladed disk structure can be obtained by tuning the mode shape Φ and the misalignment values ​​δK and δM at the principal nodes. And eigenvectors (obtained via c); at this point, the dimension of the inverse matrix depends only on the number of mode shapes of the selected harmonic bladed disk structure, thus greatly reducing the size of the inverse matrix; furthermore, it can be proven that if equation (1.17) is normalized by "mass", equation (1.18) can be obtained:

[0128] c T [I+Φ T δMΦ]c=I;

[0129] The mode shapes of the misaligned bladed disk structure were also normalized accordingly, resulting in equation (1.19):

[0130]

[0131] Take the compressor whole disk blade with 41 blades as an example, Table 1 lists the modal frequencies calculated by the modal reduction method based on the master node and the accuracy of the finite element software ANSYS APDL after fitting the blades:

[0132] Table 1: Accuracy comparison of the blade frequencies calculated by the modal reduction method and the finite element software ANSYS APDL

[0133] Blade number \ modal order 1 2 3 4 5 6 max(error) 1 -0.003％ 0.011％ 0.016％ 0.090％ 0.157％ 0.172％ 0.172％ 2 0.008％ 0.006％ 0.012％ 0.088％ 0.086％ 0.128％ 0.128％ 3 -0.002％ 0.006％ 0.007％ 0.079％ 0.088％ 0.122％ 0.122％ 4 0.002％ 0.012％ 0.010％ 0.056％ 0.052％ 0.065％ 0.065％ 5 -0.002％ 0.015％ 0.027％ 0.070％ 0.140％ 0.118％ 0.140％ 6 0.001％ 0.001％ 0.008％ 0.039％ 0.043％ 0.059％ 0.059％ 7 0.006％ 0.004％ 0.006％ 0.002％ 0.004％ 0.010％ 0.010％ 8 0.005％ -0.006％ -0.009％ 0.018％ 0.072％ 0.052％ 0.072％ 9 -0.004％ 0.000％ 0.001％ 0.012％ 0.054％ 0.034％ 0.054％ 10 -0.001％ 0.005％ 0.004％ 0.078％ 0.054％ 0.084％ 0.084％ 11 -0.002％ 0.003％ 0.008％ 0.041％ 0.082％ 0.083％ 0.083％ 12 -0.014％ 0.009％ 0.018％ 0.025％ 0.064％ 0.081％ 0.081％ 13 0.006％ 0.002％ 0.016％ 0.041％ 0.138％ 0.124％ 0.138％ 14 0.000％ 0.000％ 0.007％ 0.032％ 0.079％ 0.048％ 0.079％ 15 0.000％ 0.004％ 0.012％ 0.042％ 0.133％ 0.101％ 0.133％ 16 0.005％ 0.006％ 0.005％ 0.135％ 0.063％ 0.092％ 0.135％ 17 -0.002％ 0.013％ 0.012％ 0.094％ 0.102％ 0.140％ 0.140％ 18 -0.002％ 0.009％ 0.013％ 0.077％ 0.143％ 0.151％ 0.151％ 19 0.004％ 0.008％ 0.018％ 0.047％ 0.162％ 0.126％ 0.162％ 20 -0.002％ 0.008％ 0.005％ 0.019％ 0.084％ 0.077％ 0.084％ 21 -0.002％ 0.015％ 0.012％ 0.148％ 0.114％ 0.129％ 0.148％ 22 0.000％ 0.016％ 0.020％ 0.056％ 0.226％ 0.163％ 0.226％ 23 -0.010％ 0.000％ 0.011％ 0.053％ 0.060％ 0.089％ 0.089％ 24 -0.002％ 0.012％ 0.018％ 0.044％ 0.196％ 0.165％ 0.196％ 25 0.009％ 0.016％ 0.018％ 0.073％ 0.212％ 0.208％ 0.212％ 26 -0.003％ -0.003％ 0.009％ 0.016％ 0.072％ 0.037％ 0.072％ 27 -0.006％ 0.002％ 0.013％ 0.125％ 0.133％ 0.199％ 0.199％ 28 0.000％ 0.003％ 0.008％ 0.032％ 0.073％ 0.044％ 0.073％ 29 -0.001％ 0.007％ 0.011％ 0.039％ 0.131％ 0.120％ 0.131％ 30 -0.001％ 0.005％ 0.010％ 0.053％ 0.085％ 0.060％ 0.085％ 31 -0.004％ 0.019％ 0.014％ 0.222％ 0.121％ 0.191％ 0.222％ 32 -0.001％ 0.009％ 0.018％ 0.028％ 0.178％ 0.090％ 0.178％ 33 -0.005％ 0.017％ 0.025％ 0.056％ 0.253％ 0.229％ 0.253％ 34 0.001％ 0.009％ 0.010％ 0.025％ 0.118％ 0.101％ 0.118％ 35 -0.005％ 0.012％ 0.014％ 0.067％ 0.199％ 0.172％ 0.199％ 36 0.000％ 0.009％ 0.004％ 0.083％ 0.057％ 0.076％ 0.083％ 37 0.000％ 0.006％ 0.009％ 0.048％ 0.084％ 0.060％ 0.084％ 38 -0.004％ 0.000％ 0.008％ 0.069％ 0.043％ 0.049％ 0.069％ 39 -0.004％ 0.003％ 0.009％ 0.052％ 0.084％ 0.077％ 0.084％ 40 -0.001％ 0.002％ 0.005％ 0.043％ 0.041％ 0.053％ 0.053％ 41 -0.002％ 0.015％ 0.027％ 0.070％ 0.140％ 0.118％ 0.140％ max(error) 0.009％ 0.019％ 0.027％ 0.222％ 0.253％ 0.229％ /

[0134] As shown in Table 1, based on the above analysis, by performing cyclic symmetry analysis on the finite element model of the single sector of the tuned bladed disk structure, a group of tuned bladed disk structure modes at each pitch radius is selected to represent the basic dynamic characteristics of the mistuned bladed disk by linear combination, which greatly reduces the matrix dimension of the inverse process in modal solution while ensuring accuracy. Thus, for the small geometric misalignment caused in the machining process, from the establishment of the mistuned bladed disk dynamic model to the calculation of its forced response, the calculation efficiency is greatly improved while ensuring the calculation accuracy.

[0135] Although embodiments of the present application have been shown and described, it is to be understood that various modifications, substitutions, replacements and changes can be made to these embodiments without departing from the principles and spirit of the present application, and the scope of the present application is defined by the appended claims and their equivalents.

Claims

1. A method for calculating the equivalent of a small geometrically misaligned bladed disk based on a master node, characterized in that, The specific steps are as follows: S1. Establish a dynamic model of the harmonic bladed disk structure; S2. Establish an equivalent dynamic model of the master node of the misaligned bladed disk; S3. A method for predicting the master node response of a misaligned rotary bladed disk; S4. Based on the above analysis, by performing cyclic symmetric analysis on the single-sector finite element model of the harmonic bladed disk structure, a set of harmonic bladed disk structure modes under each pitch diameter are selected for linear combination to characterize the basic dynamic characteristics of the misaligned bladed disk.

2. The method for equivalent calculation of a small geometrically misaligned bladed disk based on a master node according to claim 1, characterized in that, The specific steps for establishing the dynamic model of the harmonic bladed disk structure as described in S1 are as follows: Because the harmonic integrated bladed disk structure has cyclic symmetry characteristics, the dynamic characteristics of the bladed disk structure can be obtained only by a single sector model. The motion equation (1.1) for a single sector is: (K S -ω 2 M S +iD S )q S =f S +Q S ; Where: K S M is the stiffness matrix. S Let D be the mass matrix. S q is the damping matrix; S f is the sector node displacement vector. S Q is the sector node load vector. S This represents the nodal force vector exerted on the interface of this sector by adjacent sectors; To fully utilize the circular symmetry of the sector, the sector node displacement vector q S Divide into blocks: q S ={q l ,q i ,q r } T ; Where: q l and q r These are the displacement vectors of the left and right boundary nodes of the sector, respectively; q i The displacement vector of the nodes within the sector; Similarly, the nodal force vector Q acting on the interface of the adjacent sector is... S It can be represented as: Q S ={Q l ,0,Q r } T ; Q l and Q r These represent the forces acting on the left and right boundary nodes of adjacent sectors, respectively. When the bladed disk structure is subjected to forces of an order-type excitation, the following implicit boundary conditions exist on the left and right boundaries of a single sector: The displacement boundary is given by equation (1.2): q r =e iak q l ; The force boundary is given by equation (1.3): Q r =-e iαk Q l ; Where: i is the imaginary unit, k is the excitation order (harmonic number); α = 2π / N B Sector angle (N) B (Number of sectors on the bladed disk); Considering the equilibrium relationship between force and displacement in the displacement boundary formula and the force boundary formula, the nodal displacement can be expressed as equation (1.4): Substituting equations (1.2), (1.3), and (1.4) into (1.1), we obtain equation (1.5): in: The superscript * indicates Hermitian conjugation; Analyzing the form of equation (1.5), we can obtain the modal characteristic expression (1.6) for a single sector in a circumferential bladed disk structure: To obtain the complete modal set of the harmonic bladed disk, k must take all integers from 0 to N / 2. and According to its properties, it is a Hermite matrix when k≠0 and N / 2, and the corresponding mode is a double frequency, with the two mode shapes being complex conjugates of each other. At this point, the mode shape Φ(N×nN) of the entire harmonic bladed disk structure is determined. B ), then we can use the vibration mode of a single sector Obtained by extending in the local coordinate system; Where: N is the number of degrees of freedom of the entire bladed disk structure, and n is the number of modes corresponding to each wave frequency (order); When the number of sectors is odd, the mode shape of the entire harmonic bladed disk structure can be expressed in the following form (1.7): When the number of sectors is even, the mode shape of the entire harmonic bladed disk structure can be expressed in the following form (1.8):

3. The method for equivalent calculation of a small geometrically misaligned bladed disk based on a master node according to claim 2, characterized in that, The equivalent dynamic model of the master node of the misaligned bladed disk is established in S2. The specific steps are as follows: After the mass of each blade is mismatched, the overall mismatch matrix of the bladed disk can be expressed as the following equation (1.10): Where: the misalignment quality matrix for each sector It can also be expressed in the following form (1.11): The sequential quadratic programming algorithm is used to analyze the misalignment matrix δM of the blade principal nodes. S The specific values ​​at corresponding positions are optimized iteratively; with the goal of ensuring that the errors of the first 6 natural frequencies after blade mass misalignment are all less than 0.01%, a set of optimal master node misalignment values ​​μ is finally output. m To achieve high-precision fitting of different mass misalignments for each blade; the misalignment matrix of blade m can be expressed as equation (1.12): In modal space, the misalignment correction matrix M Φ =Φ T δMΦ can be solved in blocks, yielding the following equation (1.13): Where: j and k represent M Φ The row and column indices of each sub-block after partitioning, Φ mj and Φ mk It is the block matrix at the corresponding position in the block matrix in equation (1.7) or (1.8).

4. The method for equivalent calculation of a small geometrically misaligned bladed disk based on a master node according to claim 3, characterized in that, The master node response prediction method for the misaligned rotary bladed disk described in S3 includes the following specific steps: The modal characteristics of the harmonic bladed disk structure can be expressed as follows (1.14): KΦ=MΦΛ; Where: K and M are the stiffness and mass matrices of the harmonic adjustable bladed disk structure; Φ and Λ are its corresponding eigenvectors and eigenvalue matrices; Similarly, the modal characteristics of the misaligned bladed disk structure can also be expressed as equation (1.15): Where: δK and δM represent the changes in the principal node stiffness matrix and mass matrix caused by blade misalignment, respectively; The mode shape of the misaligned bladed disk can be expressed as a linear combination of the mode shapes of the harmonic bladed disk, resulting in expression (1.16): Where: c is the combination coefficient matrix; Substituting equation (1.12) into equation (1.11), we can simplify to obtain equation (1.17): The characteristic values ​​of the misaligned bladed disk structure can be obtained by tuning the mode shape Φ and the misalignment values ​​δK and δM at the principal nodes. and eigenvectors (obtained via c); After normalizing the mass of equation (1.17), we obtain equation (1.18): c T [I+Φ T δMΦ]c=I; The mode shapes of the misaligned bladed disk structure were also normalized accordingly, resulting in equation (1.19):

Images