An interface reduction method for high-fidelity dynamic modeling of blades and disks

Abstract

The present invention relates to the technical field of finite element dynamic response analysis, and specifically discloses an interface reduction method for high-fidelity dynamic modeling of blades and disks, comprising the following steps: Step 1, dividing the blade disk model into two substructure models, blades and disks, and obtaining corresponding mass and stiffness matrices; Step 2, selecting nodes from the interface between blades and disks, generating a conversion matrix, and performing a reduction on the blades and disks to obtain a primary reduction model; Step 3, performing a secondary reduction on the primary reduction model using a substructure method to obtain a secondary reduction model used for response analysis calculations. The method improves the traditional substructure method to solve the technical problem that the traditional substructure method retains all degrees of freedom of the interface, resulting in insufficient improvement in calculation efficiency compared with the calculation efficiency of the complete model.

Description

Technical Field

[0001] The invention relates to an interface reduction method for high-fidelity dynamic modeling of a blade disk, and belongs to the technical field of finite element dynamic response analysis of blade disks. Background Art

[0002] As the demand for improving the structural performance of aircraft engines becomes increasingly urgent, higher requirements are placed on the blade disk structure. The blade disk structure has gradually developed from individual molding processes to integrated molding processes. While the development of the process has brought advantages such as reducing the number of structural parts, greatly reducing the structural mass, and improving the aerodynamic performance of the structure, the problems generated in the process cannot be ignored. In particular, there will be a certain geometric deviation between the integrated molded blades, which is called the detuning phenomenon. Detuning will lead to a serious decline in structural performance and life. Therefore, in order to calculate the vibration characteristics of the detuned blade disk, a large number of random detuning samples need to be analyzed.

[0003] For large-scale blade disk structures, dynamic characteristics analysis and vibration response calculations are usually completed directly through commercial finite element software, which will consume a lot of computing costs. Especially when performing dynamic characteristics analysis on multiple detuned blade disk samples, the computing cost increases exponentially. At this time, the substructure method is usually used to divide the model into several substructures to reduce the calculation scale and increase the calculation efficiency. However, the interface between the blade and the disk is often more complex, and the number of divided grids is large. At this time, the traditional substructure method will retain the full degree of freedom of the interface, resulting in the calculation efficiency not being significantly improved compared to the calculation efficiency of the complete model. Therefore, how to improve the calculation efficiency of the traditional substructure method has become an engineering problem that needs to be solved urgently. Summary of the invention

[0004] The present invention aims to provide an interface reduction method for high-fidelity dynamic modeling of blade disks, and to improve the traditional substructuring method to solve the technical problem that the traditional substructuring method retains all degrees of freedom of the interface, resulting in insufficient improvement in calculation efficiency compared with the calculation efficiency of the complete model.

[0005] To achieve the purpose of the present invention, the technical solution adopted by the present invention is:

[0006] An interface reduction method for high-fidelity dynamic modeling of a blade disk comprises the following steps:

[0007] Step 1: Divide the blade disk model into two substructure models: blade and disk, and obtain the corresponding mass and stiffness matrices;

[0008] Step 2: Select nodes on the interface between blades and disks, generate a transformation matrix, and perform a reduction on the blades and disks to obtain a reduced model;

[0009] Step 3: Use the substructure method to perform secondary reduction on the primary reduction model to obtain the secondary reduction model used in response analysis calculation.

[0010] Furthermore, the interface point selection strategy in step 2 includes, based on the premise of uniform selection, selecting points at the interface between the blade and the disk at a ratio of 50% of the total number of interface nodes to obtain an interface reduced node set.

[0011] Furthermore, in step 2, the interface node is selected according to one of the following three strategies:

[0012] (1) The strategy of uniformly selecting points in both horizontal and vertical directions, that is, first selecting interface nodes in horizontal intervals, and then selecting these nodes in vertical intervals;

[0013] (2) The longitudinal uniform interval point selection strategy, that is, the longitudinal interval selection of interface nodes;

[0014] (3) The strategy of selecting points evenly with staggered intervals in the horizontal and vertical directions is to first select points with staggered intervals in the vertical direction for each row of interface nodes.

[0015] Furthermore, in step 2, points are selected according to the three strategies, respectively, to obtain three different interface reduction node sets, and corresponding primary reduction models are obtained respectively;

[0016] In step 3, three different first-order reduction models are respectively subjected to second-order reduction to obtain three different second-order reduction models;

[0017] It also includes using three different quadratic reduction models to perform modal analysis and taking the one with the smallest relative error in natural frequency for response analysis calculation.

[0018] Furthermore, three different local quadratic reduction models are obtained by using a local wheel disc model that includes at least one blade substructure and the partial wheel disc mechanism in which it is located for modal analysis, and the interface reduction strategy corresponding to the quadratic reduction model with the smallest relative error of the natural frequency is taken as the interface point selection strategy for completing the model.

[0019] Furthermore, step one includes:

[0020] Assume that the interface force on the substructure is F Γ , the free vibration equations of the blade and the impeller plate structure are established as follows:

[0021]

[0022] Where X is the coordinate matrix, M is the mass matrix, K is the stiffness matrix, and the superscripts b and d represent the blade and impeller structure respectively. are the blade interface force and the disk interface force.

[0023] Furthermore, the first reduction in step 2 includes generating a unit matrix of corresponding scale according to the degree of freedom of the substructure, and converting it into a transformation matrix for reduction according to the interface node selection result. When reducing a certain degree of freedom, the transformation matrix of the blade substructure and the wheel plate structure is generated by redefining the row of the corresponding degree of freedom and deleting the column of the corresponding degree of freedom. Then the first reduction matrix of the overall structure is obtained as

[0024]

[0025] According to the following coordinate transformation relationship:

[0026]

[0027] Will Substituting into the free vibration equation of each substructure and performing a polycondensation, the following reduced model is obtained:

[0028]

[0029] in,

[0030]

[0031]

[0032] Furthermore, in the substructure method described in step three, the wheel disc is used as the main substructure, and the free interface method is selected for its interface treatment; each blade is used as a slave substructure, and the fixed interface method is selected for interface treatment, thereby obtaining a secondary reduced model.

[0033] Furthermore, the secondary reduction comprises the following steps:

[0034] ① For the blade substructure, solve the characteristic equation

[0035]

[0036] Get the main mode matrix Φ of the blade substructure b ,

[0037]

[0038] Where n is the main mode order retained;

[0039] And constrain the modal matrix Ψ as follows b :

[0040]

[0041] Then we get the quadratic reduction matrix

[0042]

[0043] Then we can get the mass matrix and stiffness matrix of the blade substructure after secondary reduction.

[0044]

[0045] ② For the roulette plate structure, the coordinate transformation relationship before and after reduction is established as follows:

[0046]

[0047] in, Reduce the coordinates of the roulette plate structure once; is the quadratic reduced coordinate; The first n main modes are retained for the roulette plate structure; Ψ m is the remaining m-order high-order mode set;

[0048] Ignore the influence of inertia force, simplify the free vibration equation into a static equation, and substitute the coordinate transformation relationship of the wheel plate structure into it,

[0049]

[0050] in,

[0051]

[0052] Residual Flexibility Matrix From the formula

[0053]

[0054] The quadratic reduction matrix of the roulette wheel is derived as

[0055]

[0056] Where D is the internal degree of freedom of the wheel, Γ is the interface degree of freedom, n is the first n-order mode retained after the wheel is calculated by the characteristic equation, and I is the unit matrix;

[0057] Then the secondary reduced mass matrix and stiffness matrix of the wheel plate structure are obtained as follows:

[0058]

[0059] ③ Assemble the substructures after quadratic reduction to obtain the quadratic reduction matrix of the system for

[0060]

[0061] According to the interface displacement coordination condition, the mass matrix of the integrated blade reduction model is obtained and the stiffness matrix They are

[0062]

[0063] Among them, I is the unit matrix.

[0064] Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0065] The present invention proposes an interface reduction method for high-fidelity dynamic modeling of blade disks. By selecting points on the interface of the divided substructures, a transformation matrix is ​​generated to complete a primary reduction. Then, the substructure method is used to perform a secondary reduction of the reduced model to establish a reduced model of the entire blade disk. This method can significantly improve the calculation efficiency compared to the complete finite element model while ensuring the accuracy of finite element calculations, and has practical engineering significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 It is a schematic diagram of the implementation process of the interface reduction method for high-fidelity dynamic modeling of blade disks in the example of the present invention;

[0067] Figure 2 is a schematic diagram of a blade disk and sector calculation model in an embodiment of the present invention;

[0068] Figure 3 Schematic diagram of different reduction interface point selection results in an embodiment of the present invention;

[0069] Figure 4 is a comparison diagram of relative errors of natural frequencies corresponding to different reduction node sets in an embodiment of the present invention;

[0070] Figure 5 It is a comparison chart of the calculation results and time spent on solving the harmonic response of the complete model, the interface full-node retained substructure model and the interface reduced substructure model in the embodiment of the present invention. DETAILED DESCRIPTION

[0071] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0072] This embodiment uses an integral blade disk as an example. The material of the integral blade disk is titanium alloy, with a Young's modulus of 121 GPa, a Poisson's ratio of 0.3, and a density of 4480 kg / m 3 , the Rayleigh damping coefficients α and β are 153.3547 and 1.6224×10 -7The boundary condition is that the disk is completely fixed at one end face. The finite element model of the whole blade disk, the single sector model part, the blade substructure of the sector model and the sector disk structure are basically as follows. Figure 2 The overall flow chart of the interface reduction method for high-fidelity dynamic modeling of blade disks in this embodiment is basically as follows Figure 1 As shown, this method is first used for a single sector model (such as Figure 1 The finite element model shown in the sector model part of the diagram is reduced and the optimal interface point selection method is selected. The specific process is as follows:

[0073] Step 1: Define the leaf substructures separately ( Figure 1 Substructure A) and sector roulette plate structure ( Figure 1 Substructure B), and generate the coordinate matrix X, mass matrix M and stiffness matrix K of each substructure.

[0074] The blade disk is divided into two types of substructures, and the free vibration equations of each are obtained. Under free vibration, each substructure only interacts at the interface. If the interface force on the substructure is assumed to be F Γ , then

[0075]

[0076] The superscripts b and d represent the blade and wheel plate structures, respectively.

[0077] Step 2: Select nodes on the interface between blades and disks to generate a transformation matrix, and perform a reduction on the blades and disks to obtain a reduced basis.

[0078] Here, you can define a point selection strategy based on different interfaces, such as Figure 3 The complete interface node set and multiple interface reduced node sets shown in FIG. 1 are shown in FIG. 2 ; the interface point selection strategy in this embodiment can be summarized as selecting points at the interface of the blade substructure and the wheel plate structure at a ratio of 50% of the total interface node number based on the premise of uniform selection. Specifically, the following three strategies can be adopted:

[0079] 1) If Figure 3 As shown in the reduced node set 1, the uniformly spaced point selection method is performed both horizontally and vertically, that is, the interface nodes are first selected horizontally and then these nodes are selected vertically.

[0080] 2) If Figure 3 As shown in the reduced node set 2, the longitudinal interval uniform point selection method is to select the interface nodes longitudinally at intervals;

[0081] 3) If Figure 3As shown in the reduced node set 3, the horizontal and vertical staggered interval uniform point selection method is to first perform vertical staggered interval point selection on each row of interface nodes, that is, after performing interval point selection on the kth row, when performing interval point selection on the k+1th row, it is required to be vertically staggered with the selected nodes of the kth row;

[0082] Figure 3 The reduced node sets 4 and 5 shown will be used for comparison with the above-mentioned node reduction strategy in the following.

[0083] Generate a unit matrix of corresponding size according to the degree of freedom of the substructure, and then select node sets and two substructures according to different interfaces to generate different primary reduction matrices. The specific reduction method is:

[0084] Generate a transformation matrix of the blade and wheel plate structure by redefining the rows corresponding to the degrees of freedom and deleting the columns corresponding to the degrees of freedom

[0085] For the i-th interface point selection strategy, the first reduction matrix of the overall structure is:

[0086]

[0087] The following formula:

[0088]

[0089] Substitute it into formula (1) and perform a polycondensation to generate the i-th reduced model of the blade and the impeller plate structure respectively:

[0090]

[0091] in,

[0092]

[0093] Step three, use the substructure method to perform secondary reduction on the primary reduction model to obtain the secondary reduction model used in response analysis calculation.

[0094] For the blade substructure, fixed interface reduction is adopted in this embodiment, and the reduction steps are as follows:

[0095] First, solve the characteristic equation

[0096]

[0097] Get the main mode matrix of the blade substructure

[0098]

[0099] Where n is the order of the retained main mode.

[0100] On the other hand, the constraint mode matrix Ψ b From formula (8), we can get:

[0101]

[0102] Finally, the quadratic reduction matrix is ​​obtained as

[0103]

[0104] At the same time, the mass matrix and stiffness matrix after secondary reduction are obtained as follows:

[0105]

[0106] This embodiment adopts free interface reduction for the wheel plate structure.

[0107] The modes of the roulette plate structure are expressed as the main modes and the remaining high-order modes. The coordinate transformation relationship before and after reduction can be written as follows:

[0108]

[0109] in, Reduce the coordinates of the roulette plate structure once, is the quadratic reduced coordinate, The first n main modes are retained for the roulette plate structure, Ψ m is the remaining m-order high-order mode set.

[0110] The reduction steps are:

[0111] Ignoring the influence of inertia force, the model is simplified to statics equation. Substituting the above coordinate transformation relationship into it, we get:

[0112]

[0113] in,

[0114]

[0115] Residual Flexibility Matrix It can be obtained by the formula

[0116]

[0117] The quadratic reduction matrix of the roulette wheel is derived as follows:

[0118]

[0119] Where I is the unit matrix, D is the internal degree of freedom of the disk, Γ is the interface degree of freedom, and n is the first n-order modes retained after the disk is calculated by the characteristic equation.

[0120] Then the secondary reduced mass matrix and stiffness matrix of the wheel plate structure are obtained as follows:

[0121]

[0122]

[0123] Finally, the mass matrix and stiffness matrix of the substructure after secondary reduction are combined, that is,

[0124]

[0125] Among them, I is the unit matrix, is the quadratic reduction matrix of the system, we have:

[0126]

[0127] In order to verify the influence of different interface reduction node sets (corresponding to different interface point selection strategies) on the calculation accuracy in the above method, modal analysis is also performed in this example, and the relative errors of the first 50 natural frequencies of the reduced model of the substructure method of the complete interface and different interface reduction node sets are compared. The results are shown in Figure 2. Figure 4 shown.

[0128] Among them, the relative average error of the natural frequency of the substructure method of the complete interface is 0.04%;

[0129] The relative average error of the reduced model corresponding to the interface reduced node set 1 is 2.17%;

[0130] The relative average error of the reduced model corresponding to the interface reduced node set 2 is 0.68%;

[0131] The relative average error of the reduced model corresponding to the interface reduced node set 3 is 0.89%;

[0132] The relative average error of the reduced model corresponding to the interface reduced node set 4 is 9.50%;

[0133] The relative average error of the reduced model corresponding to the interface reduced node set 5 is 8.57%.

[0134] Because the calculation accuracy of interface reduction node set 2 is the highest, the reduced model corresponding to interface selection node set 2 is subsequently selected in this embodiment to complete the response calculation analysis. In the practical application of the present invention, the above steps can be used to select a part of the blade disk model for interface reduction node set selection. From the above results, it can be seen that the errors caused by different interface reduction node set selections for the same model are not the same. Although the amount of calculation is paid as a price, in order to improve the accuracy of subsequent calculations, it is necessary to use a partial model to verify the errors of different interface reduction nodes.

[0135] In this section, the whole blade disk model is integrated with the sector group (such as Figure 1 The finite element model shown in the blade disk model part) is used for harmonic response analysis, and the calculation results of the reduced model established by this method (the interface node reduction adopts the optimal interface point selection method in step one) and the complete model are compared to illustrate the advantage of the present invention in calculation efficiency.

[0136] The quadratic reduction method proposed in the present invention is used for the whole blade disk, and each blade and disk are taken as different substructures. The blade adopts the fixed interface substructure method, and the whole disk adopts the free interface substructure method. The interface full node set 1 and the interface reduced node set 2 with the highest calculation accuracy are selected for calculation and comparison with the complete model. The process of establishing the reduced model is the same as the above steps.

[0137] According to the reduction method in this embodiment, the reduced model of the integral blade disk is finally obtained, and its mass matrix and stiffness matrix are respectively

[0138]

[0139] The reduced model is loaded with traveling wave excitation, the frequency range of the harmonic response calculation is set to 1200~3200Hz, and a force of 5N is loaded on the middle node at the top of the blade. The final response solution formula is:

[0140]

[0141] In the formula, is the steady-state solution of the response under traveling wave excitation; is the Rayleigh damping matrix after quadratic reduction, α and β are the Rayleigh damping coefficients. The simple harmonic response of the reduced system can be obtained by direct method.

[0142] Comparison of the harmonic response solutions of the complete model, the interface full node retained substructure model, and the interface reduced substructure model. Figure 5 As shown. Compared with the analysis results of the complete model using the interface node reduction substructure method, the two peak frequency errors between the substructure method after interface reduction and the complete model are 1.28% and 2.78%, and the response amplitude errors are 7.78% and 7.02%, which are within the acceptable response error range. More importantly, in terms of computational efficiency, the interface reduction calculation method is improved by 78.18% compared with the complete model, and the computational efficiency is improved by 69.23% compared with the ordinary interface full-node retained substructure method. That is, the interface reduction method exemplified in this embodiment greatly improves the computational efficiency while ensuring the computational accuracy. Since the computational efficiency has been greatly improved, the computational cost paid for the error verification of different interface reduction node sets in the early stage is acceptable.

[0143] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An interface reduction method for high-fidelity dynamic modeling of blade disks, characterized in that: The following steps are involved: Step 1: Divide the blade disk model into two substructure models: blade and disk, and obtain the corresponding mass and stiffness matrices; Assume that the interface force on the substructure is F Γ , the free vibration equations of the blade and the impeller plate structure are established as follows: Where X is the coordinate matrix, M is the mass matrix, K is the stiffness matrix, and the superscripts b and d represent the blade and impeller structure respectively. are the blade interface force and the disk interface force; Step 2: Select nodes on the interface between blades and disks, generate a transformation matrix, and perform a reduction on the blades and disks to obtain a reduced model, including: Generate a unit matrix of corresponding scale according to the degree of freedom of the substructure, and convert it into a transformation matrix for reduction according to the interface node selection results. When reducing a certain degree of freedom, the transformation matrix of the blade substructure and the wheel plate structure is generated by redefining the row of the corresponding degree of freedom and deleting the column of the corresponding degree of freedom. Then the first reduction matrix of the overall structure is obtained as follows: According to the following coordinate transformation relationship: Will Substituting into the free vibration equation of each substructure and performing a polycondensation, the following reduced model is obtained: in, Step 3: Use the substructure method to perform secondary reduction on the primary reduction model to obtain a secondary reduction model used for response analysis and calculation; wherein the wheel disc is used as the main substructure, and the free interface method is selected for its interface treatment; each blade is used as a slave substructure, and the fixed interface method is selected for interface treatment, thereby obtaining a secondary reduction model, including: ① For the blade substructure, solve the characteristic equation Get the main mode matrix Φ of the blade substructure b , Where n is the main mode order retained; And constrain the modal matrix Ψ as follows b : Then we get the quadratic reduction matrix Then we can get the mass matrix and stiffness matrix of the blade substructure after secondary reduction: ② For the roulette plate structure, the coordinate transformation relationship before and after reduction is established as follows: in, Reduce the coordinates of the roulette plate structure once; is the quadratic reduced coordinate; The first n main modes are retained for the roulette plate structure; Ψ m is the remaining m-order high-order mode set; Ignore the influence of inertia force, simplify the free vibration equation into a static equation, and substitute the coordinate transformation relationship of the wheel plate structure into it, in, Residual Flexibility Matrix Obtained from the following The quadratic reduction matrix of the roulette wheel is derived as Where D is the internal degree of freedom of the wheel, Γ is the interface degree of freedom, n is the first n-order mode retained after the wheel is calculated by the characteristic equation, and I is the unit matrix; Then the secondary reduced mass matrix and stiffness matrix of the wheel plate structure are obtained as follows: ③ Assemble the substructures after quadratic reduction to obtain the quadratic reduction matrix of the system for According to the interface displacement coordination condition, the mass matrix of the integrated blade reduction model is obtained and the stiffness matrix They are Among them, I is the unit matrix.

2. The method according to claim 1, characterized in that The interface point selection strategy in step 2 includes, based on the premise of uniform selection, selecting points at the interface between the blade and the disk in a ratio of 50% of the total number of interface nodes to obtain an interface reduced node set.

3. The method according to claim 2, characterized in that In step 2, select the interface node according to one of the following three strategies: (1) The strategy of uniformly selecting points in both horizontal and vertical directions, that is, first selecting interface nodes in horizontal intervals, and then selecting these nodes in vertical intervals; (2) The longitudinal uniform interval point selection strategy, that is, the longitudinal interval selection of interface nodes; (3) The strategy of selecting points evenly with staggered intervals in the horizontal and vertical directions is to first select points with staggered intervals in the vertical direction for each row of interface nodes.

4. The method according to claim 3, characterized in that In step 2, points are selected according to the three strategies, respectively, to obtain three different interface reduction node sets, and corresponding one-time reduction models are obtained respectively; In step 3, three different first-order reduction models are respectively subjected to second-order reduction to obtain three different second-order reduction models; It also includes using three different quadratic reduction models to perform modal analysis and taking the one with the smallest relative error in natural frequency for response analysis calculation.

5. The method according to claim 4, characterized in that Three different local quadratic reduction models are obtained using a local disk model that includes at least one blade substructure and the partial disk mechanism in which it is located for modal analysis. The interface reduction strategy corresponding to the quadratic reduction model with the smallest relative error of the natural frequency is taken as the interface point selection strategy for completing the model.

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