An Optimization Method for Bolt Tightening Process in Low-Pressure Rotor Assembly

Abstract

This invention proposes an optimization method for bolt tightening process in low-pressure rotor assembly. The method uses bolt preload as the optimization variable, coaxiality and stiffness as optimization objectives, and the uniform distribution of tightening torque and preload of already installed bolts during actual assembly as constraints. A dual-objective optimization model for coaxiality and stiffness is established. The NSGA-II algorithm is used to obtain a non-dominated solution set of initial preload for subsequent bolts to be installed, achieving simultaneous optimization of coaxiality and stiffness.

Description

Technical Field

[0001] This invention belongs to the field of aero-engine assembly and relates to an optimized bolt tightening process method for low-pressure rotor assembly. Background Technology

[0002] The low-pressure rotor is a crucial component of an aero-engine, primarily composed of a multi-stage stacked discs with flanges and bolted connections. Bolting is the most common fastening method. Bolting ensures both the fit accuracy of the discs and the tightness between them, transmitting torque during operation. Therefore, it is essential to maintain a uniform bolt preload as much as possible to avoid stress concentration. However, due to the elastic interaction between bolts, later-tightened bolts can affect earlier-tightened bolts, leading to a significant discrepancy between the actual and expected preload. An overly dispersed bolt preload distribution further impacts the entire aero-engine. Excessive preload can cause thread yielding, reducing the load-bearing capacity of the bolted connection structure, while insufficient preload can reduce bolt stiffness, leading to loosening and serious rotor failure. Furthermore, each tightening operation causes rotor deflection, and the cumulative effect of these deflections in space can cause relative deflection between upper and lower stage rotors, resulting in excessive rotor coaxiality.

[0003] The paper "Optimization of Tightening Process Parameters for Wheel Rim Bolts of a Loader" uses finite element method and UG software to cut the complex solid structure and divide it into high-quality hexahedral and pentahedral elements. After meshing, corresponding material properties are applied to different components of the model, and appropriate contact is set between different parts. Finally, the bolt preload is applied by controlling the one-dimensional spring elements connecting the two bolt parts through a corresponding loading curve. By controlling the loading time, the bolt stress variation curve can be obtained. The optimal tightening sequence is determined by comparing the mean and variance of bolt stress after tightening under different tightening sequences. However, this paper only optimizes the bolt tightening sequence, while the bolt tightening torque, which has a significant impact on the assembly in actual processes, is not addressed. In addition, the method in this paper is relatively complex and has limitations in practical applications. Summary of the Invention

[0004] Bolted connections are the primary method for connecting various components of low-pressure rotors in aero-engines. Due to the influence of elastic interactions, uneven distribution of bolt preload can easily occur. Although this may meet stiffness requirements, it can lead to spatial misalignment of components, potentially causing coaxiality deviations. To address the optimization conflict between coaxiality and stiffness in rotor assembly, this invention proposes a bolt tightening process optimization method for low-pressure rotor assembly. Using bolt preload as the optimization variable and coaxiality and stiffness as optimization objectives, and constrained by the uniform distribution of tightening torque and preload of already installed bolts during actual assembly, a dual-objective optimization model for coaxiality and stiffness is established. An optimization algorithm is then used to obtain a non-dominated solution set of initial preload for subsequent bolts to be installed, achieving simultaneous optimization of coaxiality and stiffness.

[0005] The technical solution of this invention is as follows:

[0006] The method for optimizing bolt tightening process for low-pressure rotor assembly includes the following steps:

[0007] Step 1: Establish a low-pressure rotor coaxiality prediction model c for the assembly process; establish a low-pressure rotor bolt connection stiffness model K;

[0008] Step 2: Construct a dual-objective optimization model for coaxiality and stiffness:

[0009] The objective function for the dual-objective optimization of coaxiality and stiffness after low-pressure rotor assembly is as follows:

[0010]

[0011] In the formula, λ1 is the weighting parameter of the coaxiality target, λ2 is the weighting parameter of the stiffness target; c is the coaxiality of the low-pressure rotor after assembly; K is the bolt connection stiffness of the low-pressure rotor; L is the bolt preload vector; l i Let be the preload force of the i-th bolt;

[0012] The optimization variable is: bolt preload vector L;

[0013] The constraints are:

[0014]

[0015] In the formula, c max K represents the maximum permissible coaxiality after the bolts are tightened. min This refers to the minimum allowable stiffness after the bolt is tightened; max The maximum preload force for a single bolt; min The minimum preload force for a single bolt;

[0016] Step 3: Solve the bi-objective optimization model established in Step 2 to obtain the optimal solution for bolt preload.

[0017] Furthermore, in step 3, the NSGA-II algorithm is used to solve the bi-objective optimization model established in step 2 to obtain a set of non-dominated solutions for bolt preload, and the optimal solution is selected from the set of non-dominated solutions for bolt preload.

[0018] Furthermore, in the constraints of step 2, the minimum preload force l of a single bolt... min The maximum preload is determined based on the minimum precision of the torque wrench. max According to the formula for the preload of steel connecting bolts, F ≤ (0.6~0.8)×δ s ×A is determined, where δ s Where A is the yield strength and A is the cross-sectional area of ​​the minor diameter of the thread.

[0019] Furthermore, in step 1, the process of establishing the low-pressure rotor coaxiality prediction model is as follows:

[0020] Step 1.1: During the assembly of the low-pressure rotor, a numerical simulation of the multi-bolt tightening process under a specific assembly process is performed. An elastic interaction model is established by fitting the preload change value, and the elastic interaction matrix under the current assembly process is obtained [A].

[0021] Step 1.2: Establish a low-pressure rotor coaxiality prediction model based on GRU network, and train the model to obtain the trained low-pressure rotor coaxiality prediction model based on GRU network.

[0022] The inputs for training the coaxiality prediction model include: the time step T of the assembly process. step 1. Rotor assembly influencing factors F within this time step; 2. Coaxiality of the low-pressure rotor Coa within this time step;

[0023] The rotor assembly influencing factors F include the assembly process F. gy And assembly process F gy The elastic interaction matrix [A] obtained in step 1.1 is converted into bolt preload during the assembly process and input into the coaxiality prediction model.

[0024] Furthermore, in step 1.1, the elastic interaction model is as follows:

[0025] Prlod=A(T lj ,T sx ,T lc ,T qd )

[0026] Where Prlod is the bolt preload; A is the elastic interaction matrix; T lj T is the bolt tightening torque; sx For the bolt tightening sequence; Tbs For the bolt tightening cycle; T qd This marks the starting point for tightening the bolt.

[0027] Furthermore, in step 1.1, based on the bolt tightening torque T... lj Bolt tightening sequence T sx Bolt tightening cycle T bs Bolt tightening starting point T qd The bolt tightening process under this assembly process was simulated, and based on the simulation results, the formula was used.

[0028]

[0029]

[0030] The elastic interaction matrix [A] under the current assembly process is obtained by solving; where a ij f represents the coefficients of the elastic interaction matrix. ij Δf represents the remaining preload of the i-th bolt after bolt j has been tightened; j This represents the change in preload when the j-th bolt is tightened.

[0031] Furthermore, in step 1.2, the assembly process F gy This includes bolt tightening torque, tightening sequence, tightening rounds, tightening start point, and drum installation phase; wherein the bolt tightening torque, tightening sequence, tightening rounds, and tightening start point are converted into bolt preload during the assembly process through the elastic interaction matrix [A] obtained in step 1, and are input into the coaxiality prediction model along with the drum installation phase.

[0032] Furthermore, by utilizing the bolt tightening torque, tightening sequence, tightening rounds, and tightening starting point, based on the elastic interaction formula...

[0033] {F0}+[A]{ΔF}={F f}

[0034] The preload of each bolt after each loading step is obtained; where {F0} is the initial preload of the bolt before each loading step (n×1); [A] is the elastic interaction matrix (n×n); {ΔF} is the bolt preload increment for each loading step (n×1); {F f} is the remaining preload of the bolt after each loading step (n×1); n is the number of bolts in the bolt group; each time a bolt is tightened is a loading step.

[0035] Furthermore, factors influencing the rotor assembly process also include machining quality F. jg Material properties F d Assembly accuracy F jd and environmental parameter Fhj Furthermore, the z-score standardization method was used to standardize the data of influencing factors during the rotor assembly process, so that all influencing factor data were of the same order of magnitude.

[0036] The processing quality F jg The vector form expression is:

[0037] F jg =(Q pm Q td Q cc Q zx Q y Q cz Q tz )

[0038] In the formula Q pm For the flatness of the upper and lower stop surfaces of the disc drum; Q td Q is the runout of the mating stop relative to the rotation axis; cc Q is the surface roughness at the mating point of the disc drum. zx For the straightness of the center axis of the drum; Q y Q represents the roundness of the outer and inner circumference of the disc drum. cz The perpendicularity of the drum end face to the axis; Q tz To ensure the coaxiality of the drum itself;

[0039] The material property F d The vector form expression is:

[0040] F cl =(E,τ,ρ,μ)

[0041] In the formula, E is Young's modulus; τ is Poisson's ratio; ρ is density; and μ is the coefficient of friction.

[0042] The assembly precision F jd The vector form expression is:

[0043] F jd =(Paral,Intf)

[0044] In the formula, Paral represents the parallelism of the upper and lower end faces of the disc drum stop; Intf represents the interference fit of the disc drum stop.

[0045] The environmental parameter F hj The vector form expression is:

[0046] F hj =(T,RH)

[0047] In the formula, T is the ambient temperature during rotor assembly; RH is the ambient humidity during rotor assembly.

[0048] Furthermore, in step 1, the connection stiffness model K is:

[0049]

[0050] In the formula, K is the bolt connection stiffness; K b For bolt stiffness; K c K represents the stiffness of the connected components. d For the contact stiffness of the mating surface.

[0051] Beneficial effects

[0052] To address the optimization contradiction between coaxiality and stiffness in rotor assembly, this invention proposes a bolt tightening process optimization method for low-pressure rotor assembly. Using bolt preload as the optimization variable and coaxiality and stiffness as optimization objectives, and constrained by the uniform distribution of tightening torque and preload of already installed bolts during actual assembly, a dual-objective optimization model for coaxiality and stiffness is established. The NSGA-II algorithm is then used to obtain a non-dominated solution set of initial preload for subsequent bolts to be installed, achieving simultaneous optimization of coaxiality and stiffness.

[0053] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0054] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0055] Figure 1 Implementation process for predicting the coaxiality of low-pressure rotors in the assembly process;

[0056] Figure 2 Comparison of prediction performance of coaxiality prediction models;

[0057] Figure 3 Axial variation of the area of ​​the pressure-bearing layer of the connected component;

[0058] Figure 4 Optimization of bolt tightening process;

[0059] Figure 5 Bi-objective optimization of coaxiality and stiffness distribution of non-dominated solutions. Detailed Implementation

[0060] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0061] To overcome the shortcomings of existing methods in resolving the optimization contradiction between coaxiality and stiffness in rotor assembly, and considering that bolted connections are the most representative connection method in aero-engine assembly, this invention provides a bolt tightening process optimization method for low-pressure rotor assembly. This method balances coaxiality and stiffness by adjusting bolt tightening process parameters during assembly. The method mainly includes three aspects: first, establishing a coaxiality prediction model during bolt tightening; second, establishing a stiffness calculation model during bolt tightening; and finally, addressing the optimization contradiction between coaxiality and stiffness in rotor assembly, using bolt preload as the optimization variable, coaxiality and stiffness as optimization objectives, and constrained by the uniform distribution of tightening torque and preload of already installed bolts during actual assembly, establishing a dual-objective optimization model for coaxiality and stiffness; and using the NSGA-II algorithm to obtain a non-dominated solution set of initial preload for subsequent bolts to be installed, achieving simultaneous optimization of coaxiality and stiffness. The basic process is as follows:

[0062] I. Unified Expression of Influencing Factors

[0063] There are many factors affecting coaxiality during rotor assembly, which can be divided into five categories. These five categories of factors are expressed in a unified way, as shown in Equation 1.

[0064] F = (F gy ,F jg ,F cl ,F jd ,F hj (1)

[0065] In the formula: F represents the influencing factors; F gy For assembly process; F jg For processing quality; F d For material properties; F jd For assembly accuracy; F hj These are environmental parameters.

[0066] II. Data Preprocessing

[0067] Because the data on influencing factors are inconsistent in magnitude and unit, in order to avoid excessive errors in subsequent prediction processes, it is necessary to standardize the data so that all influencing factor data are at the same magnitude.

[0068] III. Constructing an Elastic Interaction Model

[0069] During the assembly of a low-pressure fan rotor, there is an elastic interaction between the bolt groups. The tightening torque, tightening sequence, tightening rounds, and tightening start point in the assembly process are not easily input into the model. Therefore, this invention establishes an elastic interaction model to uniformly transform these assembly processes into changes in bolt preload, which can be represented by Equation 2. This invention establishes a three-dimensional model with the same structure as the rotor, uses the finite element method to simulate the changes in bolt preload under a specific assembly process, processes the obtained data to calculate the elastic interaction matrix, and establishes the elastic interaction model.

[0070] Prlod=A(T lj ,T sx ,T lc ,T qd (2)

[0071] In the formula: Prlod is the preload of the bolt group; A is the elastic interaction matrix; T lj T is the bolt tightening torque; sx For the bolt tightening sequence; T bs For the bolt tightening cycle; T qd This marks the starting point for tightening the bolt.

[0072] IV. Constructing a coaxiality prediction model

[0073] Based on the above steps, a coaxiality prediction model y is constructed. tz The number of loading steps included in the assembly process is used as the model's time step T. step The assembly process F within this time step T gy Processing quality F zp Material properties F cl Assembly accuracy F jd Environmental parameter F hj As input, the coaxiality Coa within this time step is used as the target output and fed into the established prediction model for training, resulting in the coaxiality prediction model. The establishment of the coaxiality prediction model is shown in Equation 3:

[0074]

[0075] In the formula: y tz A model for predicting the coaxiality of rotor assembly; T represents the training parameters for the prediction model. step θ is the input time step length; Coa is the coaxiality within that time step; θ is the parameter set obtained during model training.

[0076] V. Constructing a Stiffness Calculation Model

[0077] Stiffness is one of the important indicators for evaluating the performance of bolted connections. Especially for high-speed rotating components such as low-pressure rotors, greater stiffness means greater load capacity, which can effectively improve the reliability of the rotor. In addition to bolt stiffness and the stiffness of the connected components, this application incorporates the contact stiffness of the bonding surface into the bolted connection stiffness calculation model.

[0078] The stiffness of a bolted connection is mainly composed of three parts connected in series: bolt stiffness, stiffness of the connected parts, and contact stiffness of the mating surfaces.

[0079]

[0080] In the formula, K is the bolt connection stiffness; K b For bolt stiffness; K c K represents the stiffness of the connected components. d For the contact stiffness of the mating surface.

[0081] VI. Constructing a dual-objective optimization model for coaxiality and stiffness

[0082] Combining the coaxiality prediction model and the stiffness calculation model, the objective function for the dual-objective optimization of coaxiality and stiffness after low-pressure rotor assembly is shown below.

[0083]

[0084] In the formula, λ i The weights of the i-th objective function are: c = c, K = K, L = L ... i Let be the preload force of the i-th bolt.

[0085] The following is a detailed description of the above six parts:

[0086] I. Unified Expression of Influencing Factors

[0087] (1) Assembly process

[0088] Different assembly process parameters during rotor assembly will cause changes in bolt preload, which in turn will cause changes in coaxiality. The main process parameters in this process include bolt tightening torque, tightening sequence, number of tightening passes, tightening start point, and installation phase. The above five assembly process parameters are expressed in vector form as shown in Equation 6:

[0089] F gy =(T lj ,T sx ,T lc ,T qd ,P az (6)

[0090] In the formula: F gy For assembly process; Tlj T is the bolt tightening torque; sx For the bolt tightening sequence; T bs For the bolt tightening cycle; T qd P is the starting point for tightening the bolt. az Install phases for the disc drum.

[0091] (2) Processing quality

[0092] The difference in initial coaxiality caused by the different machining quality of the two-stage drum assembly leads to changes in the coaxiality after assembly. These machining qualities mainly include: flatness, runout, roughness, straightness, roundness, perpendicularity, and coaxiality. The coaxiality described here refers to the coaxiality of a single-stage drum itself, not the overall coaxiality after assembly. The above seven machining qualities are expressed in vector form as shown in Equation 7:

[0093] F jg =(Q pm Q td Q cc Q zx Q y Q cz Q tz (7)

[0094] In the formula: F jg For processing quality; Q pm For the flatness of the upper and lower stop surfaces of the disc drum; Q td Q is the runout of the mating stop relative to the rotation axis; cc Q is the surface roughness at the mating point of the disc drum. zx For the straightness of the center axis of the drum; Q y Q represents the roundness of the outer and inner circumference of the disc drum. cz The perpendicularity of the drum end face to the axis; Q tz This refers to the coaxiality of the drum itself.

[0095] (3) Material properties

[0096] The material properties of the drum itself affect the frictional force and the degree of deformation, thus influencing the change in coaxiality. The main material properties are: Young's modulus, Poisson's ratio, density, and coefficient of friction. These four material properties are expressed in vector form as shown in Equation 8:

[0097] F cl =(E,τ,ρ,μ) (8)

[0098] In the formula: F cl ρ is the material property; E is Young's modulus; τ is Poisson's ratio; ρ is density; μ is the coefficient of friction.

[0099] (4) Assembly accuracy

[0100] When there is a parallelism error between the upper and lower level discs / drums, due to the presence of a stop fit, uneven preload after the two contact surfaces are in contact will cause the upper and lower level discs / drums to deflect, ultimately affecting the coaxiality of the rotor after assembly. The assembly accuracy parameters include parallelism and interference fit. The above two assembly accuracy parameters are expressed in vector form as shown in Equation 9:

[0101] F jd =(Paral,Intf) (9)

[0102] In the formula: F jd For assembly accuracy; Parallel is the parallelism of the upper and lower end faces of the disc drum stop; Intf is the interference fit of the disc drum stop assembly.

[0103] (5) Environmental parameters

[0104] Environmental parameters during assembly can also affect assembly coaxiality to some extent. The main environmental parameters include temperature and humidity. These two main parameters are expressed in vector form as shown in Figure 10.

[0105] F hj =(T,RH) (10)

[0106] In the formula: F hj For environmental parameters; T is the ambient temperature during rotor assembly; RH is the ambient humidity during rotor assembly.

[0107] II. Data Preprocessing

[0108] Factors affecting coaxiality include data such as the assembly process, processing quality, and material properties of the parts. These data have different dimensions and orders of magnitude. When the levels of the data differ significantly, it will affect the accuracy and reliability of the subsequent prediction model. Therefore, it is necessary to standardize the original data.

[0109] The factors affecting coaxiality have design requirements during rotor design and process planning, causing them to fluctuate within a theoretical range, and their distribution follows a Gaussian distribution to a certain extent. Therefore, the z-score standardization method can be used to standardize the assembly data, as follows:

[0110]

[0111] In the formula: x' is the standardized data; μ is the mean of the influencing factor series; σ is the standard deviation of the influencing factor series.

[0112] 3: Constructing an elastic interaction model

[0113] The bolt tightening process was simulated using ABAQUS software, and an elastic interaction matrix was established based on the residual bolt preload. This patent uses an 8-bolt connection with a cross-tightening mechanism (2500 N) as an example to establish the elastic interaction matrix.

[0114] The elastic interaction of the bolt group under load is shown in Equation 12:

[0115] {F0}+[A]{ΔF}={F f} (12)

[0116] Where: {F0} is the initial preload of the bolt before each loading step (n×1); [A] is the elastic interaction matrix (n×n); {ΔF} is the bolt preload increment for each loading step (n×1); {F f} is the remaining preload of the bolt after each loading step (n×1); n is the number of bolts in the bolt group.

[0117] Each vector in the above formula is represented as follows:

[0118]

[0119] In the formula: f i 0 , Δf i and f i f These represent the remaining preload at the end of the previous loading, the preload increment of bolt i, and the final remaining preload after loading, respectively.

[0120] The elastic interaction matrix [A] is shown in Equation 14:

[0121]

[0122]

[0123] In the formula: a ij f represents the coefficients of the elastic interaction matrix. ij Δf represents the remaining preload of the i-th bolt after bolt j has been tightened; j This represents the change in preload when the j-th bolt is tightened.

[0124] The changes in bolt preload under cross-loading conditions simulated in finite element software are shown in Table 1:

[0125] Table 1. Remaining bolt preload (N) under different loading steps

[0126]

[0127] The elastic interaction matrix [A] can be obtained by using Equation 15, as shown below:

[0128]

[0129] IV. Constructing a coaxiality prediction model

[0130] (1) Calculation of elastic interaction matrix

[0131] In this embodiment, an 8-bolt connection is used, and the bolt tightening torque is 1.5 N·m. The preload variation is calculated. First, the tightening torque is converted into preload using Formula 16.

[0132] T = 0.2dF f (16)

[0133] In the formula, T is the tightening torque; F f d is the preload; d is the bolt outer diameter. This patent uses an M4 bolt with an outer diameter of 4mm, so F f =1875N.

[0134] Using Formula 12 and the previously established elastic interaction matrix to process and calculate the bolt preload variation, the actual preload during the tightening process can be obtained. The calculation process is as follows:

[0135] Step 1: When tightening the first bolt, F0 = [0,0,0,0,0,0,0,0] T ΔF = [1875,0,0,0,0,0,0,0] T After tightening the first bolt, the preload of the other bolts can be calculated using formula 12, as shown in the first column of Table 2.

[0136] Step 2: When tightening the second bolt, F0 is the preload force when tightening the first bolt, i.e. F0=[1875,78.8,97.5,97.5,0,91.9,91.9,0] T ΔF = [0, 1875, 0, 0, 0, 0, 0] T The preload of other bolts can be calculated using formula 12, as shown in the second column of Table 2.

[0137] Step 3: Following this process, the actual preload of all bolts after tightening can be calculated. See Table 2 for details.

[0138] Table 2 Actual bolt preload (N) during assembly.

[0139]

[0140] (2) Training of coaxiality prediction model

[0141] The entire assembly process of the low-pressure fan rotor is divided into n loading steps based on the number of bolts. Each loading step includes assembly factors, bolt preload values, and coaxiality values ​​within that time period. Network training mainly includes the forward computation part of the prediction model and the backward tuning part.

[0142] Forward calculation part of the prediction model: This step mainly takes the assembly factors, bolt preload and coaxiality value within the time period as input, and then calculates according to the forward calculation formula of GRU network. After the forward calculation of GRU network, the predicted value of rotor coaxiality at the next moment is output.

[0143] The forward computation formula for the GRU network is as follows:

[0144] r t =σ(W r [S t-1 ,x t (17)

[0145] z t =σ(W z [S t-1 ,x t (18)

[0146] S t =tanh(W h [r t *S t-1 ,x t (19)

[0147] S t =(1-z) t )*S t-1 +z t *S t (20)

[0148] y t =σ(W o S t ) (twenty one)

[0149] Among them, W r W represents resetting the weight of the gate. z W represents the weight of the updated gate. h S represents the hidden layer information at time t-1. t-1 W is the weight matrix used as input. o Indicates the output layer weights, y t This represents the predicted value of the GRU network at time t.

[0150] From the forward calculation formula, we can know W r W zW h W o These are the weight parameters that the GRU network needs to train and learn. The first three parameters can be divided into the following components:

[0151] W r =[W rx W rs W z =[W zx W zs W h =[W hx W hs ] (twenty two)

[0152] If the actual value of the sample data at time t is y t The output value at time t obtained using the GRU network is y. t Then the propagation loss of the network at time t (using the root mean square error MSE as the loss function) is:

[0153]

[0154] Therefore, the transmission loss of the network over the entire time period T is:

[0155]

[0156] Then, the intermediate error term in the network at time t can be obtained as follows:

[0157] δ y,t =(y' t -y t )·σ' (25)

[0158] δ S,t =δ y,t W o +δ z,t+1 W zS +δ t+1 W hS ·r t+1 +δ S,t+1 W rs +δ· S,t+1 ·(1-z t+1 (26)

[0159] δ z,t =δ t,S ·(S t -S t-1 )·σ' (27)

[0160] δ t =δ S,t ·z t ·tanh' (28)

[0161] δ r,t =S t-1 ·((δ S,t ·z t ·tanh')W hS )·σ' (29)

[0162] Since the derivative of the loss with respect to the weight matrix is ​​the gradient of the weight matrix, the sum of the gradients at each time step is the final gradient over the total time. Therefore, the weight parameter matrices W within time period T... o W zx W zs W hx W hs W rx W rs The gradient calculation formula is:

[0163]

[0164] Finally, by using the above formula and an appropriate learning rate η, the parameters of the GRU network model can be iteratively updated.

[0165] The backpropagation optimization of the prediction model: This part mainly uses the principle of error backpropagation to iteratively update the weights and biases of each neuron in the GRU network, thereby improving the accuracy of the prediction model. This patent uses the mean squared error as the loss function for this part, and then uses the Adam algorithm to optimize the gradient descent.

[0166] The Adam method calculates a dynamically changing coefficient τ using the moment of the current gradient, and then dynamically adjusts the learning rate. The formula for calculating τ is as follows:

[0167]

[0168] m t =p×m t-1 +(1-p)×g t (32)

[0169] n t =q×n t-1 +(1-q)×g t 2 (33)

[0170] Among them, g t This is the current gradient value for this parameter; m t n t They represent g respectively t The first-order moment estimate and the second-order moment estimate; p and q are real numbers between 0 and 1; c is a non-zero constant.

[0171] (3) Coaxiality prediction model test

[0172] Twelve randomly selected experimental samples were used to test the coaxiality prediction model trained with 4-fold cross-validation. The test results are attached. Figure 2 As shown.

[0173] To better evaluate the predictive performance of the coaxiality prediction model, the Nash efficiency coefficient (NSEC) values ​​for 12 test samples were calculated, and the calculation formula is shown in Equation 34. Generally, the smaller the error between the predicted and actual coaxiality values, the closer the NSEC value is to 1.

[0174]

[0175] The NSEC calculation results for the 12 test samples are shown in Table 3 below.

[0176] Table 3 Comparison of predicted and actual values ​​of the test samples.

[0177]

[0178]

[0179] Depend on Figure 2 It can be seen from the test results of the coaxiality prediction model in the 12 test samples that the predicted coaxiality values ​​are not significantly different from the actual values, and the prediction accuracy is relatively good. From the NSEC values ​​of the 12 test samples shown in Table 3, it can be seen that the prediction model can generally achieve coaxiality prediction of low-pressure fan rotors under multi-bolt tightening conditions.

[0180] V. Constructing a Stiffness Calculation Model

[0181] The stiffness of a bolted connection is mainly composed of three parts connected in series: bolt stiffness, stiffness of the connected parts, and contact stiffness of the mating surfaces.

[0182]

[0183] In the formula, K is the bolt connection stiffness; K b For bolt stiffness; K c K represents the stiffness of the connected components. d For the contact stiffness of the mating surface.

[0184] According to the German (VDI) standard, bolt stiffness is generally calculated by treating the bolt as a uniform cylinder.

[0185]

[0186] In the formula, E b d is the elastic modulus of the bolt material; d is the diameter of the bolt; T is the total thickness of the connected parts.

[0187] If the stress of the connected parts under bolt tightening is uniformly distributed radially, then the shape of the compressed layer will be a cone, such as... Figure 3 .

[0188] The stiffness of the connected components varies with the area of ​​the pressure layer, which is mainly affected by the elastic modulus and thickness of the material, as well as the cone angle θ. m It was decided that, based on the formula fitted in the literature "Improved Bolt Connection Stiffness Model Based on Finite Element Method", the elastic modulus and the cone angle θ caused by thickness would be established. m The changing relationship is shown in equation (37):

[0189] K c / E m d = -0.058E A / E B +1.309 (37)

[0190] In the formula, E m It represents the equivalent elastic modulus of the materials of the upper and lower connected components.

[0191] E m =E A E B / (E A +E B (38)

[0192] K c / E m d = -0.031E A / E B +1.047 (39)

[0193]

[0194] The area of ​​the pressure layer of the connected component is shown in Equation (41).

[0195]

[0196] In the formula, d w Indicates the clamping outer diameter of the bolt, d h Indicates the inner diameter of the threaded hole.

[0197] According to Hooke's Law, the stiffness of the connected parts is:

[0198]

[0199] Considering that the stiffness of the mating surface is mainly affected by the contact pressure, the stiffness value can only be calculated after obtaining the contact pressure at the mating surface. For connected parts of the same thickness, the outer diameter d m≥d w +Ltanθ, If there is no gasket between the connected parts, the distribution of contact pressure can be considered as a fourth-order polynomial of radius r, where θ is the assumed half-apex angle of the cylinder.

[0200] p n (r)=c4r 4 +c3r 3 +c3r 3 +c2r 2 +c1r+c0 (43)

[0201] In the formula, c i (i = 0, 1, ..., 4) are undetermined coefficients.

[0202] According to the literature "Contact of nominally flat surfaces", the contact stiffness of the mating surface is:

[0203]

[0204] In the formula, F m The applied preload; This shows the pressure distribution after a unit load is applied.

[0205] make but

[0206]

[0207] Combined (35)-(45), the stiffness of the bolted connection is:

[0208]

[0209] VI. Constructing a dual-objective optimization model for coaxiality and stiffness

[0210] (1) Bi-objective optimization function

[0211] Combining the above coaxiality prediction model and stiffness calculation model, the objective function for the dual-objective optimization of coaxiality and stiffness after low-pressure rotor assembly is shown in the following equation.

[0212]

[0213] In the formula, λ i The weights of the i-th objective function are: c = c, K = K, L = L ... i Let be the preload force of the i-th bolt.

[0214] (2) Optimize variables

[0215] The preload of bolts cannot be arbitrarily selected. To ensure the stability of the rotor connection and to withstand certain tension, torsion, and bending, the preload formula for general steel connection bolts is used:

[0216] F≤(0.6~0.8)×δ s ×A (48)

[0217] In the formula, δ s The yield strength (δ in this patent) s =320MPa); A is the cross-sectional area of ​​the minor diameter of the thread.

[0218] Substituting this into the context of this patent, F is calculated. max = 0.8 × 3.14 × 1.8 2 ×320=2604N.

[0219] Considering that the minimum accuracy of the torque wrench in this patent example verification is 1.5 N·m, and the minimum preload is calculated to be 1500 N, the range of bolt preload is between 1500 N and 2604 N.

[0220] (3) Constraints

[0221] To avoid stress concentration in the rotor after bolt tightening, the difference between the maximum and minimum preload of a single bolt should not exceed 20%. Therefore, the dual-objective optimization constraints for the multi-stage rotor are shown in equation (49):

[0222]

[0223] In the formula, c max K represents the maximum permissible coaxiality after the bolts are tightened. min This refers to the minimum allowable stiffness after the bolt is tightened; max The maximum preload force for a single bolt; min This is the minimum preload force for a single bolt.

[0224] (4) Solving the bi-objective optimization model based on the NSGA-II algorithm

[0225] Based on the objective function and constraints of the dual-objective optimization model of coaxiality and stiffness, the bolt preload is encoded, the fitness function is designed, and operators and mutation operators are selected. The NSGA-II algorithm is used to solve the model and find the non-dominated solution of bolt preload.

[0226] Fitness function

[0227] The goal of assembling the low-pressure rotor of an aero-engine is to minimize coaxiality while maintaining good rigidity, while ensuring uniform bolt preload. Therefore, this patent uses equation (47) as the fitness function of the NSGA-II algorithm.

[0228] Selection Operator

[0229] When optimizing bolt preload, the combination with optimal coaxiality and maximum stiffness is selected as the parent algorithm, and a new population is generated from it. Generally, two main methods are used for operator selection: tournament method and roulette wheel method.

[0230] In optimizing bolt preload, the tournament method typically selects a certain number of bolt preload individuals from the generated population, choosing the one with the best coaxiality and highest stiffness to participate in the next generation of bolt preload population. The above process consists of three steps:

[0231] Step 1: First, select the amount of bolt preload M for each tournament.

[0232] Step 2: Then, randomly select M groups of bolt preload forces from the bolt preload force population, and select the group with the best coaxiality and the greatest stiffness to be included in the next generation of bolt preload force population.

[0233] Step 3: If the existing population contains a total of N sets of bolt preload, then repeat steps 1 and 2 N times.

[0234] In the optimization of bolt preload, the compass betting method uses the fitness function of each preload as the basis for determining whether the preload should be included in the next generation of samples. This method consists of the following four steps:

[0235] Step 1: Calculate the preload L of the j-th bolt group j The probability of being selected P(L) j ):

[0236]

[0237] In the formula, M represents the population size for the dual-objective optimization of coaxiality and stiffness; F(L j ) represents the adaptability of the preload of the j-th group of bolts.

[0238] Step 2: Calculate the preload L of the j-th bolt group j Corresponding cumulative probability q(L) j ):

[0239]

[0240] Step 3: After comparing probabilities, determine whether the bolt preload is the parent force.

[0241] q(θ j-1 )<b<q(θ j (52)

[0242] In the formula, b represents a number randomly selected between [0,1]. If b happens to fall within the cumulative probability range of the bolt preload of the j-th group, then the bolt preload of the j-th group is regarded as the parent.

[0243] Step 4: If there are N groups in the next generation of bolt preload population, then repeat steps 2 and 3 N times.

[0244] By comparing the two methods, the first algorithm has fewer steps and higher efficiency. Therefore, the selection operator of the optimization model in this patent adopts the tournament method, selecting 10 individuals to participate in the tournament each time.

[0245] Cross operator

[0246] For bolt preload, a linear cross operator is used to generate the next generation of bolt preload individuals (child):

[0247] child=parent2+0.5(parent1-parent2) (53)

[0248] In the formula, parent1 is the parent of the first bolt preload; parent2 is the parent of the second bolt preload.

[0249] The next generation of bolt preload is obtained by linear cross-pollination of the parent bolt preload, thus giving the next generation of bolt preload individuals genetic diversity.

[0250] Mutation operator

[0251] Common mutation operators mainly include two categories: Gaussian mutation operator and adaptive mutation operator. Gaussian mutation operator is more suitable for handling unconstrained programming problems, while adaptive mutation algorithm has better applicability. Since the bolt preload between rotors is constrained, to avoid local optima in the bolt preload and to ensure uniform distribution of the bolt preload, this patent selects the adaptive mutation operator. This operator, based on sufficient mutation of the bolt preload, better suits the actual situation of low-pressure rotor bolt connections.

[0252] Algorithm Flow

[0253] The overall process of using the NSGA-II algorithm to optimize the bolt tightening process for coaxiality and stiffness is as follows: Figure 4 .

[0254] Simulation results and analysis of bi-objective optimization based on NSGA-II algorithm

[0255] To verify the feasibility and effectiveness of the coaxiality and stiffness optimization model proposed in this patent, simulation analysis was performed using a three-dimensional rotor model.

[0256] Based on the three-dimensional model of the rotor, the coaxiality of the low-pressure rotor after assembly is obtained by using the coaxiality prediction model. The bolt connection stiffness of the rotor can be calculated by using equation (46). The NSGA-Ⅱ algorithm is used to solve the calculation and obtain the non-inferior solution of the bolt group preload. The coaxiality and stiffness of each bolt group after assembly are shown in Table 4.

[0257] Table 4 Non-inferior solutions for preload of bolt groups optimized based on coaxiality and stiffness dual objectives.

[0258]

[0259] To more conveniently represent the coaxiality and stiffness of the bolt group in the non-dominated solution of the coaxiality and stiffness optimization model, the values ​​in Table 1 are mapped to obtain... Figure 5 In the figure, the left vertical axis represents coaxiality, the right vertical axis represents stiffness, and the horizontal axis represents the non-dominated solution group.

[0260] according to Figure 5 It can be seen that, firstly, under the constraints of rotor coaxiality extreme value being less than 75μm and stiffness extreme value being greater than 1.5×10⁸ N / m, the optimization model of this patent can obtain 5 sets of non-dominated solutions, corresponding to 5 sets of bolt preload forces. Secondly, under the action of the second set of bolt preload forces, the coaxiality accuracy is the highest at 50.9μm, and the stiffness also reaches the maximum of 1.93×10⁸ N / m. Selecting this set of bolt preload forces can bring both coaxiality and stiffness to the best state.

[0261] In summary, this patent establishes a dual-objective optimization model for coaxiality and stiffness, with coaxiality and stiffness as optimization objectives, preload as optimization variables, and uniform distribution of bolt preload as a constraint. Based on the model's characteristics, the NSGA-II algorithm is used to obtain a non-dominated set of solutions for bolt preload. Appropriate initial preloads are selected according to different practical requirements. Finally, simulation experiments demonstrate the feasibility of the dual-objective optimization method.

[0262] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for optimizing bolt tightening process in low-pressure rotor assembly, characterized in that: Includes the following steps: Step 1: Establish a low-pressure rotor coaxiality prediction model for the assembly process Establish a stiffness model for bolted connections of low-pressure rotors. K ; The process of establishing the low-pressure rotor coaxiality prediction model is as follows: Step 1.1: During the low-pressure rotor assembly process, a numerical simulation of the multi-bolt tightening process under a specific assembly technology is performed. An elastic interaction model is established by fitting the preload variation value, and the elastic interaction matrix under the current assembly technology is obtained by solving the model. ; The elastic interaction model is as follows: in Preload on the bolt assembly; The elastic interaction matrix; This refers to the bolt tightening torque; The order of bolt tightening; For the number of times the bolts are tightened; This marks the starting point for tightening the bolt. Step 1.2: Establish a low-pressure rotor coaxiality prediction model based on GRU network, and train the model to obtain the trained low-pressure rotor coaxiality prediction model based on GRU network. The inputs for training the coaxiality prediction model include: the time steps of the assembly process. Factors affecting rotor assembly during this time step F The coaxiality of the low-pressure rotor Coa within this time step; Factors affecting rotor assembly F Including assembly process And assembly process The elastic interaction matrix obtained in step 1.1 Convert it into bolt preload during the assembly process and input it into the coaxiality prediction model; Step 2: Construct a dual-objective optimization model for coaxiality and stiffness: The objective function for the dual-objective optimization of coaxiality and stiffness after low-pressure rotor assembly is as follows: In the formula, The weighting parameters for the coaxiality target. The weighting parameters for the stiffness objective; To ensure the coaxiality of the low-pressure rotor after assembly; For low-pressure rotor bolt connection stiffness; This is the bolt preload vector; For the first The preload force of each bolt; The optimization variable is: bolt preload vector. ; The constraints are: In the formula, This represents the maximum permissible coaxiality after the bolts are tightened. This represents the minimum allowable stiffness after the bolt is tightened. This represents the maximum preload force for a single bolt. The minimum preload force for a single bolt; Step 3: Solve the bi-objective optimization model established in Step 2 to obtain the optimal solution for bolt preload.

2. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 1, characterized in that: In step 3, the NSGA-II algorithm is used to solve the bi-objective optimization model established in step 2 to obtain a set of non-dominated solutions for bolt preload, and the optimal solution is selected from the set of non-dominated solutions for bolt preload.

3. The method for optimizing bolt tightening process for low-pressure rotor assembly according to claim 1 or 2, characterized in that: In the constraints of step 2, the minimum preload of a single bolt is... The maximum preload is determined based on the minimum precision of the torque wrench. According to the formula for the preload of steel connecting bolts Confirmed, among which For yield limit, This is the cross-sectional area of ​​the minor diameter of the thread.

4. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 1, characterized in that: In step 1.1, based on the bolt tightening torque... Bolt tightening sequence Bolt tightening cycle Bolt tightening start point The bolt tightening process under this assembly process was simulated, and based on the simulation results, the formula was used. The elastic interaction matrix under the current assembly process is obtained by solving the problem. ;in Represents the coefficients of the elastic interaction matrix. Indicates bolt After tightening, the first The remaining preload of each bolt; Indicates the first The change in preload when a bolt is tightened.

5. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 4, characterized in that: In step 1.2, the assembly process This includes bolt tightening torque, tightening sequence, tightening rounds, tightening start point, and drum installation phase; wherein the bolt tightening torque, tightening sequence, tightening rounds, and tightening start point are obtained from the elastic interaction matrix in step 1. It is converted into bolt preload during the assembly process and input into the coaxiality prediction model along with the installation phase of the disc drum.

6. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 5, characterized in that: By utilizing bolt tightening torque, tightening sequence, tightening cycles, and tightening start point, and based on the elastic interaction formula... The preload of each bolt after each loading step is obtained; where Initial preload of bolts before each loading step ; The elastic interaction matrix ; Bolt preload increment for each loading step ; The remaining preload of the bolt after each loading step ; The number of bolts in the bolt group; each tightening of one bolt constitutes one loading step.

7. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 1, characterized in that: Factors affecting the rotor assembly process also include machining quality. Material properties Assembly accuracy and environmental parameters Furthermore, the z-score standardization method was used to standardize the data of influencing factors during the rotor assembly process, so that all influencing factor data were of the same order of magnitude. The processing quality The vector form expression is: In the formula To ensure the flatness of the upper and lower stop surfaces of the disc drum; The runout of the stop relative to the rotation axis at the mating stop of the disc drum; The surface roughness of the mating joint between the disc and the drum; The straightness of the central axis of the drum; The roundness of the outer and inner circles of the disc drum; The perpendicularity of the drum end face to the axis; To ensure the coaxiality of the drum itself; The material properties The vector form expression is: In the formula Young's modulus; Poisson's ratio; Density; The coefficient of friction; Assembly accuracy The vector form expression is: In the formula To ensure the parallelism of the upper and lower end faces of the drum stop; The interference fit is applied to the stop of the disc drum. The environmental parameters The vector form expression is: In the formula The ambient temperature during rotor assembly; Ambient humidity during rotor assembly.

8. The bolt tightening process optimization method for low-pressure rotor assembly according to claim 1, characterized in that: In step 1, the connection stiffness model K for: In the formula, For bolted connection stiffness; For bolt stiffness; For the stiffness of the connected parts; For the contact stiffness of the mating surface.

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