Kinetic parameter prediction method for cutting process of thin-wall part

Abstract

The invention provides a thin-wall part cutting process kinetic parameter prediction method, which comprises the following steps: dispersing a middle surface of a side wall of a thin-wall part into N plate units, obtaining initial modal parameters in the process through a modal hammering experiment, solving an inherent frequency vector lambda m and a characteristic value matrix phi m of the part, and calculating the kinetic parameters of the thin-wall part in the process. And a motion equation of the part in a physical space is converted into a modal space for further analysis, a structural dynamic modification method is established, and modified part modal parameters can be quickly obtained according to the actually measured modal parameters of the part and the stiffness matrix and mass matrix variation at the mth step. The initial kinetic parameter prediction of the workpiece is realized, and the calculation efficiency is greatly improved.

Description

Technical Field

[0001] The present invention relates to the milling machining technology of thin-walled parts, and particularly to a method for predicting the dynamic parameters during the cutting process of thin-walled parts. Background Art

[0002] In order to meet the requirements of high performance, high strength, and reduced structural weight, thin-walled integral structural parts are widely used in the aerospace manufacturing industry, such as integral blisks, integral panels, and stringers of aircraft engines. Generally, such structural parts have problems such as thin walls and poor rigidity, and are prone to chatter during milling machining, which may lead to non-compliance of the part surface with the machining quality requirements or even scrapping of the part.

[0003] A key step in the milling chatter analysis of thin-walled parts is to predict the dynamic parameters during the milling process of thin-walled parts. The most accurate means of prediction is to conduct a series of modal test experiments. However, due to the need for sensor installation and downtime measurement, it is very difficult to implement modal experiments on a large scale in industrial applications. Especially in a complex machining environment, there will be many difficulties in excitation operations and sensor installation. Another means is to achieve it through finite element simulation. However, finite element simulation involves problems such as repeated modeling and difficult mesh generation, with very low computational efficiency and high technical difficulty, and is not suitable for industrial applications either. Summary of the Invention

[0004] In order to improve the prediction efficiency of the dynamic parameters during the machining process of thin-walled parts, the present invention proposes a comprehensive and efficient method for predicting the dynamic parameters during the machining process of thin-walled parts.

[0005] The present invention adopts the following technical solutions to achieve the above object. A method for predicting the dynamic parameters during the cutting process of thin-walled parts, the steps are as follows:

[0006] Step 1: Discretize the mid-plane of the side wall of the thin-walled part into N plate elements, and assemble the overall stiffness matrix K and mass matrix M of the part according to the corresponding positions from the divided element stiffness matrices and mass matrices, expressed as follows:

[0007]

[0008] Wherein, i represents the i-th element, and respectively represent the stiffness matrix and mass matrix of the i-th element, Gi is the element degree-of-freedom transformation matrix, α R and β R are respectively called the mass damping coefficient and the stiffness damping coefficient, and the multi-order modal damping ratios and natural frequencies can be experimentally determined and determined by the least squares method;

[0009] Step 2: Obtain the initial modal parameters in the above process through a modal hammering experiment;

[0010] Step 3: The units divided for the initial state of the part are exactly the same. Therefore, the initial stiffness matrix and mass matrix of all units are and During the cutting process, the element stiffness matrix and mass matrix of the area where the material removed in the m-th step are changed to and At the m-th step, the overall stiffness matrix and overall mass matrix of the part can be expressed as:

[0011]

[0012] In the formula, is the number of elements corresponding to the area where the material is removed in each step, S is the total number of steps, represents rounding up;

[0013] Step 4: Solve the natural frequency vector Λ m and the eigenvalue matrix Φ m , and the detailed steps are as follows:

[0014] SO1: After obtaining the overall stiffness matrix, mass matrix and damping matrix of the discrete structure of the entire thin-walled part, the motion equation in the physical space of the part can be expressed in the following matrix form:

[0015]

[0016] Among them, and Q represent the acceleration, velocity and displacement vectors of the part respectively;

[0017] S02: If the vibration mode of the part conforms to the modal superposition theory and considering the undamped free vibration form of the part, the equation represented by S01 degenerates to:

[0018]

[0019] Its characteristic equation in the physical space is:

[0020] (K m -λ m M m )Φ m = 0

[0021] In the formula, λ m is the eigenvalue vector. Further solution can obtain an infinite set of eigenvalues and eigenvectors, which are expressed in matrix form as follows:

[0022] Λ m = diag(λ m，1 ,λ m，2 ,…,λ m,n )

[0023] where λ m，1 <λ m，2 <…<λ m,n Let ω m,i be the natural frequency of the part. Correspondingly, the eigenvectors can be arranged in order as follows:

[0024]

[0025] In the matrix, are the modal vibration modes of each order of the part, and the corresponding matrix Φ m is called the modal matrix;

[0026] Step 5: Transform the motion equation of the part in the physical space into the modal space for further analysis. The specific steps are as follows:

[0027] SO1: First, regularize the modal matrix. After mass normalization, the stiffness matrix K m and the mass matrix M m satisfy the following conditions:

[0028]

[0029] where I is the identity matrix;

[0030] S02: Then, introduce the coordinate transformation from the physical space to the modal space:

[0031] Q(t) = Φ m Γ(t) (4)

[0032] where Γ(t) is the vibration displacement of the part in the modal coordinates;

[0033] S03: Substitute equation (4) into equation (2) and multiply on the left by to obtain:

[0034]

[0035] Considering the damping matrix and the cutting force matrix in equation (1), the motion equation of the part in the modal space can be finally expressed as:

[0036]

[0037] where ξ m and ω m are the damping ratio matrix and the natural frequency matrix respectively, both of which are diagonal matrices. The damping ratio matrix can be obtained from the modal hammering experiment;

[0038] Step 6: Next, based on the perturbation principle, assuming that the modification amount of the part structure is very small, a structural dynamic modification method is established. For ease of distinction, the subscript "0" is used to represent the initial state before part processing. The following modification relationships of the stiffness and mass matrices can be established:

[0039]

[0040] Equation (2) can be modified to:

[0041]

[0042] Step 7: Represent Equation (6) in the modal space and left-multiply by the initial mode shape matrix to obtain:

[0043]

[0044] Combined with the orthogonality condition expressed by Equation (3), Equation (7) can be further written as:

[0045]

[0046] Its characteristic equation is expressed as:

[0047]

[0048] Step 8: Considering that actual modal analysis can often only determine a set of truncated natural frequencies and mode shapes, to address this issue, the structural dynamic modification scheme is processed in the modal stage. The modal matrix Φ0 and eigenvalue matrix Λ0 of the initial structure are further expressed in a block form:

[0049]

[0050] where, and respectively represent the known first p-order modal mode shapes and the corresponding natural frequency matrix; and represent the unknown last q-order modal mode shapes and the corresponding natural frequencies;

[0051] Step 9: Substitute Equation (9) into Equation (8) for congruence transformation to obtain:

[0052]

[0053] where,

[0054]

[0055] where,

[0056] After Equation (10), Multiply the Obtained:

[0057]

[0058] In the formula,

[0059]

[0060] When λ q → ∞, obtained:

[0061]

[0062] In the formula, ||E|| → 0, Equation (11) can be simplified. Therefore, its n lowest eigenvalues are determined by the following characteristic equation:

[0063]

[0064] After modal truncation through the above steps, the measured modal parameters of the part and the changes in the stiffness matrix and mass matrix at the m-th step can be used to quickly obtain the modified modal parameters of the part.

[0065] The present invention predicts the changes in the dynamic characteristics during the machining process of thin-walled parts based on the method of elastic thin plate theory - finite element discretization - structural dynamic modification. Based on the finite element solution method for elastic thin plate vibration, the vibration of the part during the machining process is simplified to the vibration solution problem of a thin plate under a vertical mid-plane excitation load, realizing the prediction of the initial dynamic parameters of the workpiece. Then, considering the changes in the dynamic characteristics of the workpiece caused by material removal, the time-varying dynamic parameters of the workpiece system are quickly obtained according to the theory of structural dynamic modification. The prediction results are highly consistent with the finite element simulation and experimental measurement results. Compared with the finite element simulation, the calculation efficiency is greatly improved. Description of the Drawings

[0066] Figure 1 is a milling schematic diagram of the thin-walled part established by the present invention;

[0067] In the figure: 1 - cutter, 2 - material layer to be removed, 3 - mid-plane, 4 - thin-walled part. Specific Embodiments

[0068] The following further describes the present invention with reference to the drawings. Refer to Figure 1 , a method for predicting the dynamic parameters during the cutting process of thin-walled parts, which takes the following specific steps:

[0069] Step 1: For a thin-walled cuboid part 4, its bottom surface is fully constrained in all degrees of freedom by clamping. The material layer 2 to be removed is milled by the cutter 1. The middle surface 3 of the side wall of the thin-walled part 4 (length L = 160 mm, width W = 45 mm, thickness h = 6 mm) is discretized into N plate elements. The overall stiffness matrix K and mass matrix M of the part are assembled according to the corresponding positions from the divided element stiffness matrices and mass matrices, and are expressed as follows:

[0070]

[0071] where i represents the i-th element, and represent the stiffness matrix and mass matrix of the i-th element respectively. G i is the element degree-of-freedom transformation matrix. α R and β R are respectively called the mass damping coefficient and the stiffness damping coefficient. The multi-order modal damping ratios and natural frequencies can be experimentally measured and determined by the least squares method.

[0072] Step 2: The initial first-order natural frequency of the part in Step 1 is obtained as 2954.1 Hz through the modal hammering experiment.

[0073] Step 3: It is assumed that the elements divided in the initial state of the part are exactly the same. Therefore, the initial stiffness matrices and mass matrices of all elements are and During the cutting process, the element stiffness matrix and mass matrix of the area where the material removed in the m-th step are changed to and At the m-th step, the overall stiffness matrix and overall mass matrix of the part can be expressed as:

[0074]

[0075] In the formula, is the number of elements corresponding to the area where the material is removed in each step, S is the total number of steps, represents rounding up.

[0076] Step 4: Solve the natural frequency vector Λ m and the eigenvalue matrix Φ m , and the detailed steps are as follows:

[0077] SO1: After obtaining the overall stiffness matrix, mass matrix, and damping matrix of the discrete structure of the entire thin-walled part, the motion equation in the physical space of the part can be expressed in the following matrix form:

[0078]

[0079] where, Acceleration, velocity, and displacement vectors of the part are represented by \(a\) and \(Q\) respectively.

[0080] S02: If the vibration mode of the part conforms to the modal superposition theory, considering the undamped free vibration mode of the part, the equation represented by S01 degenerates to:

[0081]

[0082] The characteristic equation in the physical space is:

[0083] (K m -λ m M m )Φ m =0

[0084] where λ m is the eigenvalue vector. Further solving can obtain an infinite set of eigenvalues and eigenvectors, which are expressed in matrix form as follows:

[0085] Λ m =diag(λ m，1 ,λ m，2 ,…,λ m,n )

[0086] where λ m，1 <λ m，2 <…<λ m,n . Let ω m,i be the natural frequency of the part. Correspondingly, the eigenvectors can be arranged in order as:

[0087]

[0088] In the matrix, is the modal vibration mode of each order of the part, and the corresponding matrix Φ m is called the mode shape matrix.

[0089] Step Five: Transform the motion equation of the part in the physical space into the modal space for further analysis. The specific steps are as follows:

[0090] SO1: First, perform regularization on the mode shape matrix. After mass normalization, the stiffness matrix K m and the mass matrix M m satisfy the following conditions:

[0091]

[0092] where I is the identity matrix.

[0093] S02: Then, introduce the coordinate transformation from the physical space to the modal space:

[0094] Q(t) = Φ m Γ(t) (4)

[0095] Where Γ(t) is the vibration displacement of the part in the modal coordinate system.

[0096] S03: Substitute Equation (4) into Equation (2) and multiply left by We can obtain:

[0097]

[0098] Considering the damping matrix and cutting force matrix in Equation (1), the motion equation of the part in the modal space can be finally expressed as:

[0099]

[0100] Where ξ m and ω m are the damping ratio matrix and natural frequency matrix respectively, both of which are diagonal matrices. The damping ratio matrix can be obtained from the modal hammer test. The first-order damping ratio of the part in this embodiment is 0.021.

[0101] Step Six: Next, based on the perturbation principle, assuming that the structural modification of the part is very small, a structural dynamic modification method is established. For easy distinction, the subscript "0" is used to represent the initial state of the part before machining. The following modification relationships of the stiffness and mass matrices can be established:

[0102]

[0103] Equation (2) can be modified as:

[0104]

[0105] Step Seven: Represent Equation (6) in the modal space and multiply left by the initial mode shape matrix We get:

[0106]

[0107] Combined with the orthogonality condition expressed by Equation (3), Equation (7) can be further written as:

[0108]

[0109] Its characteristic equation is expressed as:

[0110]

[0111] Step 8: Considering that the actual modal analysis can often only determine a set of truncated natural frequencies and mode shapes, to address this issue, the structural dynamic modification scheme is processed in the modal stage, and the modal matrix Φ0 and eigenvalue matrix Λ0 of the initial structure are further expressed in a block form

[0112]

[0113] where and respectively represent the known first p-order modal vibration modes and the corresponding natural frequency matrix and represent the unknown last q-order modal vibration modes and the corresponding natural frequencies

[0114] Step 9: Substitute Equation (9) into Equation (8) to perform a congruence transformation, and obtain:

[0115]

[0116] where

[0117]

[0118] where

[0119] Multiply the last n×n rows of Equation (10) by to obtain:

[0120]

[0121] where

[0122]

[0123] When λ q →∞, we get:

[0124]

[0125] where ||E||→0, Equation (11) can be simplified. Therefore, its n lowest eigenvalues are determined by the following characteristic equation:

[0126]

[0127] After modal truncation by the above method, only the measured modal parameters of the part and the changes in the stiffness matrix and mass matrix at the m-th step are required to quickly obtain the modified modal parameters of the part. The entire material removal is divided into 8 times. The comparison of the natural frequencies obtained in all 8 times by this method with the experimental values and finite element simulation values is shown in Table 1 in the appendix:

[0128] Number Experimental value / Hz This method / Hz Finite element simulation value / Hz 1 2954.10 2963.9 3038.8 2 3161.62 3210.3 3282.6 3 3039.55 3050.0 3130.3 4 2683.10 2691.9 2427.4 5 2805.17 2936.0 2722.1 6 2731.93 2824.6 2623.0 7 2390.13 2328.8 2211.1 8 2221.99 2117.0 2031.7

[0129] As can be seen from Table 1, compared with the simulation values, the results obtained by this method are closer to the experimental values, and the computational time-consuming of this method is only about 10% of that of the finite element simulation method.

Claims

1. A method for predicting dynamic parameters of thin-walled parts cutting process, characterized in that: Here are the steps: Step 1: Discretize the mid-surface of the side wall of the thin-walled part into N plate elements, and integrate the overall stiffness matrix K and mass matrix M of the part according to the corresponding position groups of the divided element stiffness matrix and mass matrix, which are expressed as follows: Among them, i represents the i-th unit, and Represent the stiffness matrix and mass matrix of the i-th unit, G i is the unit degree of freedom conversion matrix, α R and β R They are called mass damping coefficient and stiffness damping coefficient respectively, and can be experimentally determined to obtain multi-order modal damping ratios and natural frequencies by the least squares method; Step 2: Obtain the initial modal parameters in the above process through modal hammer test; Step 3: Set the units divided by the initial state of the part to be exactly the same, so the initial stiffness matrix and mass matrix of all units are and The element stiffness matrix and mass matrix of the area where the material removed in the mth step during cutting is located become and At the mth step, the overall stiffness matrix and overall mass matrix of the part can be expressed as: In the formula, is the number of cells corresponding to the area where the material is removed in each step, S is the total number of steps, Indicates rounding up; Step 4: Solve the natural frequency vector Λ of the part m and the eigenvalue matrix Φ m , the detailed steps are as follows: SO1: After obtaining the overall stiffness matrix, mass matrix and damping matrix of the discrete structure of the entire thin-walled part, the motion equation of the physical space of the part can be expressed in the following matrix form: in, and Q represent the acceleration, velocity, and displacement vector of the part, respectively; S02: If the vibration mode of the part conforms to the modal superposition theory, considering the undamped free vibration form of the part, the equation represented by S01 degenerates into: The characteristic equation of its physical space is: (K m -l m M m )F m =0 In the formula, λ m is the eigenvalue vector. Further solving can obtain an infinite set of eigenvalues ​​and eigenvectors, which can be expressed in matrix form as follows: L m =diag(λ m，1 ,l m，2 ,…,l m,n ) In the formula, λ m，1 <λ m，2 <…<λ m,n ,make ω m,i That is the natural frequency of the part, and correspondingly, the eigenvectors can be arranged in order as follows: In the matrix, That is, the modal vibration shapes of each order of the part, and the corresponding matrix Φ m It is called the mode matrix; Step 5: Convert the motion equation of the part in physical space into modal space for further analysis. The specific steps are as follows: SO1: First, the vibration matrix is ​​regularized. After mass normalization, the stiffness matrix K m and the mass matrix M m The following conditions are met: Where I is the unit matrix; S02: Then, introduce the coordinate transformation from physical space to modal space: Q(t)=Φ m C(t) (4) In the formula, Γ(t) is the vibration displacement of the part in the modal coordinates; S03: Substitute equation (4) into equation (2) and multiply on the left You can get: Considering the damping matrix and cutting force matrix in equation (1), the motion equation of the part modal space can be finally expressed as: In the formula, ξ m and ω m The ratios are the damping ratio matrix and the natural frequency matrix, both of which are diagonal matrices. The damping ratio matrix can be obtained from the modal hammer test. Step 6: Next, based on the perturbation principle, assuming that the part structure modification amount is small, a structural dynamic modification method is established; for easy distinction, the subscript "0" is used to represent the initial state of the part before processing, and the following modification relationship between the stiffness and mass matrix can be established: Formula (2) can be modified as follows: Step 7: Express equation (6) in the modal space and multiply it by the initial vibration matrix get: Combined with the orthogonalization condition expressed in formula (3), formula (7) can be further written as: Its characteristic equation is expressed as: Step 8: Considering that the actual modal analysis can often only determine a set of truncated natural frequencies and vibration modes, in order to solve this problem, the structural dynamic modification scheme is processed in the modal stage, and the modal matrix Φ0 and eigenvalue matrix Λ0 of the initial structure are further expressed in block form: In the formula, and Respectively represent the known first p-order mode shapes and the corresponding natural frequency matrices; and Represents the unknown post-q-order mode shape and the corresponding natural frequency; Step 9: Substitute equation (9) into equation (8) and perform congruence transformation to obtain: In the formula, In the formula, After formula (10) Multiplication get: In the formula, When q →∞, we get: Where ||E||→0, equation (11) is simplifiable, so its n lowest eigenvalues ​​are determined by the following characteristic equation: After the modal truncation is performed in the above steps, the modified modal parameters of the part can be quickly obtained by the measured modal parameters of the part and the changes in the stiffness matrix and mass matrix at the mth step.

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