Calculation method of frequency response function of concentric shaft structure

Abstract

The invention particularly relates to a method for calculating frequency response functions of concentric shaft structures. The method comprises the following steps of: firstly, respectively dividing two concentric shaft structures into n-1 sub-junctions along the axial direction; wherein the frequency response functions of each pair of concentric substructures are coupled according to the balance compatibility condition between the two end points of the substructures. And finally, according to the obtained coupling frequency response function of the concentric substructure pair, obtaining a coupling frequency response function of two concentric structures according to a substructure coupling method. The calculated concentric-axis structure frequency response function is used for extracting modal parameters of the concentric-axis structure application system, and stability analysis of the concentric-axis structure application system is carried out based on the modal parameters. According to the concentric shaft structure frequency response function calculation method, the dimension of the matrix in the calculation process can be kept unchanged all the time, and repeated derivation is not needed in the calculation process.

Description

Technical Field

[0001] The present invention relates to the field of concentric shaft structure performance analysis, and relates to a method for calculating the frequency response function of a concentric shaft structure, specifically to a method for calculating the frequency response function of a tool holder - chuck and tool coupling structure, which can be applied to predicting the tip frequency response function of the tool shaft system in milling. Background Art

[0002] The frequency response function of the concentric shaft structure has various important applications in the fields of engineering and science, mainly used for analyzing and optimizing the dynamic characteristics of the structure, especially crucial in the analysis of cutting stability. The modal parameters of the system can be calculated from the frequency response function, which, as an important input parameter for cutting stability prediction, plays a vital role in predicting cutting stability. The tool holder - chuck and the tool in the milling system are typical coaxial structures. The frequency response function of the concentric shaft structure involved in the present invention can calculate the frequency response function after the coupling of the tool holder - chuck and the tool, and then be used for predicting the tip frequency response function in the tool shaft system and the cutting stability prediction of key components in aerospace.

[0003] The existing methods for obtaining the frequency response function of the concentric shaft structure mainly calculate the frequency response function of the concentric shaft structure based on the balance and compatibility conditions of n points on two concentric shaft structures. However, its solution process depends on the number of points, and the dimension of the intermediate process matrix is strongly related to the number of points. Therefore, finding a calculation method for the response function of the concentric shaft structure coupling that does not depend on the number of coupling points is of great significance for predicting the stability of the milling process of key components in the aerospace field. It has important theoretical and practical application values.

[0004] Reference 1 "T.L. Schmitz, D. Powell, Kevin Won, W.G. Duncan, G. Scott Sawyer, J.C. Ziegert, Shrink fit tool holder connection stiffness / damping modeling for frequency response prediction in milling, International Journal of Machine Tools and Manufacture 47(2007)1368–1380." discloses a method for calculating the frequency response function of a concentric shaft substructure that depends on the number of points. First, n points are evenly distributed on two coaxial structures with a length of L, and the distance between each point is L / n. According to the balance and compatibility conditions between the direct substructures and the compatibility conditions between the substructures and the assembled body, the frequency response function at the end of the assembled body is obtained. This frequency response function consists of two vectors related to the number of points in two dimensions and a matrix related to the number of points in one dimension.

[0005] Reference 2 "T. Schmitz, A. Honeycutt, M. Gomez, M. Stokes, E. Betters, Multi-point coupling for tool point receptance prediction, Journal of Manufacturing Processes 43(2019)2–11." discloses a method for calculating the frequency response function of the tool tip after the coupling of the tool holder - tool concentric shaft. First, the spindle - tool holder - tool system is divided into a spindle - flange substructure, an overhanging tool holder substructure, and a tool substructure. The spindle - flange substructure is obtained through the inverse substructure coupling method based on experimental results. The tool substructure is calculated by the Timoshenko beam theory. The tool holder substructure consists of two parts: a tool holder with a concentric shaft and a tool holder. Its frequency response function is calculated through the balance and compatibility conditions of 7 points on the tool holder and the tool. The calculation process is similar to that in Reference 1 and is strongly related to the number of points.

[0006] The typical characteristics of the above references are: the derivation process of the proposed method depends on the number of points, and the dimension of the intermediate matrix in the calculation process is strongly related to the number of points. Therefore, the calculation process of the frequency response function of the concentric shaft structure should not depend on the number of points.

[0007] It should be noted that the information disclosed in the above background art section is only used to enhance the understanding of the background of the present invention, and thus may include information that does not constitute the prior art known to those of ordinary skill in the art. Summary of the Invention

[0008] The present invention provides a method for calculating the frequency response function of a concentric axis structure, which can always keep the dimension of the calculation process matrix unchanged and does not require repeated derivation of the calculation process, thereby overcoming the defects existing in the prior art to a certain extent.

[0009] Other features and advantages of the present invention will become apparent from the following detailed description, or may be learned in part by practice of the present invention.

[0010] According to a first aspect of the present invention, there is provided a method for calculating a frequency response function of a concentric axis structure, the method comprising:

[0011] The concentric axis structures A1 and B1 are evenly divided into n-1 substructures along the axial direction;

[0012] For each pair of corresponding substructures A1_η and B1_η, where η = 1 to n-1; based on the equilibrium conditions and compatibility conditions at the two end points of the substructure, calculate the frequency response functions of the excitation and response of the subassembly C1_η composed of the substructure pair A1_η and B1_η at the two end points respectively;

[0013] By iteratively coupling all sub-assemblies C1_η, the frequency response function of the overall assembly is finally obtained;

[0014] The calculated frequency response function of the concentric axis structure is used to extract modal parameters of the concentric axis structure application system, and stability analysis of the concentric axis structure application system is performed based on the modal parameters.

[0015] In some exemplary embodiments,

[0016] The frequency response functions of the excitation and response of the computing sub-assembly C1_η at two endpoints include:

[0017] Establish the response equations of substructures A1_η and B1_η at both end points respectively;

[0018] According to the compatibility condition and equilibrium condition of the rigid connection, the frequency response functions of the excitation and response of C1_η at the two endpoints are obtained by simultaneous solution.

[0019] In some exemplary embodiments, the frequency response functions of the excitation and response of C1_η at two endpoints include:

[0020] The frequency response function of C1_η where both the excitation and response are at point η;

[0021] Frequency response function of C1_η with excitation at η+1 and response at point η;

[0022] The frequency response function with the excitation of C1_η at point η and the response at η + 1;

[0023] The frequency response function with both the excitation and response of C1_η at point η + 1.

[0024] In some exemplary embodiments, the solution of the frequency response function with both the excitation and response of C1_η at point η includes:

[0025] Substitute the response equations of substructures A1_η and B1_η at the two endpoints into the compatibility condition formula that satisfies rigid connection to obtain the first formula;

[0026] Substitute the condition formula that satisfies balance when the excitation is applied at the first port and between substructures A1_η and B1_η into the first formula to obtain the second formula;

[0027] Establish the frequency response function of sub - assembly C1_η when both the excitation and response are at point η;

[0028] Substitute the response equations of the two endpoints of substructure A1_η and the compatibility condition formula between substructure A1_η and sub - assembly C1_η into the frequency response function of sub - assembly C1_η when both the excitation and response are at point η to obtain the third formula;

[0029] Simultaneously solve the second formula and the third formula to obtain the final frequency response function with both the excitation and response of C1_η at point η.

[0030] In some exemplary embodiments, the solution of the frequency response function with the excitation of C1_η at η + 1 and the response at point η includes:

[0031] Establish the initial frequency response function of sub - assembly C1_η when the excitation is at point η + 1 and the response is at point η to obtain the fourth formula;

[0032] Substitute the condition formula that satisfies balance when the excitation is applied at the second port and between substructures A1_η and B1_η into the second formula to obtain the fifth formula;

[0033] Simultaneously solve the fourth formula and the fifth formula to obtain the final frequency response function with the excitation of C1_η at point η + 1 and the response at point η.

[0034] In some exemplary embodiments, the solution of the frequency response function with the excitation of C1_η at point η and the response at η + 1 includes:

[0035] Establish the initial frequency response function of sub - assembly C1_η when the excitation is at point η1 and the response is at point η + 1 to obtain the sixth formula;

[0036] Simultaneously solve the sixth formula and the third formula to obtain the final frequency response function with the excitation of C1_η at point η and the response at η + 1.

[0037] In some exemplary embodiments, the solution of the frequency response function of C1_η for both excitation and response at the η+1 point includes:

[0038] Establish an initial frequency response function of the sub-assembly C1_η when both excitation and response are at the η+1 point, obtaining the seventh equation;

[0039] Simultaneously solve the seventh equation and the fifth equation to obtain the final frequency response function of C1_η when both excitation and response are at the η+1 point.

[0040] In some exemplary embodiments, the iterative coupling process includes:

[0041] Couple the adjacent sub-assemblies C1_η and C1_η+1 axially through the equilibrium and compatibility conditions at the connection point to generate a new sub-assembly C1_η-(η+1);

[0042] Couple them in sequence until all sub-assemblies are coupled into an integral structure.

[0043] According to a second aspect of the present invention, there is provided an application of a method for calculating the frequency response function of a coaxial structure, wherein the coaxial structure is a coupling structure of a tool holder-chuck and a tool in a milling system. Based on the calculated frequency response functions of the tool holder-chuck and the tool, analyze the modal parameters of the coupling structure of the tool holder-chuck and the tool, and predict the cutting stability of key components in aerospace based on the modal parameters of the coupling structure of the tool holder-chuck and the tool.

[0044] According to a third aspect of the present invention, there is provided a computer program product having a computer program stored thereon, and when the computer program is executed by a processor, it implements the method for calculating the frequency response function of the coaxial structure described in the first aspect above.

[0045] For a combined body composed of two coaxial structures, the method for calculating the frequency response function of the coaxial structure provided by the embodiments of the present invention first divides the two coaxial structures along their axes into n-1 sub-structures respectively. For each corresponding pair of sub-structures, calculate the frequency response function of the sub-assembly composed of the two sub-structures according to the equilibrium and compatibility conditions at their two end points. Then couple each sub-assembly in sequence according to the equilibrium and compatibility conditions at the connection point to obtain the frequency response function of the final combined body. Compared with Document 1, there is no need for complex and repeated experimental steps; compared with Document 2, it can make the dimension of the intermediate process matrix always 4×4 without depending on the number of points used during the calculation process.

[0046] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The accompanying drawings here are incorporated into the specification and constitute a part of this specification, showing embodiments in accordance with the present invention, and are used together with the specification to explain the principles of the present invention. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can obtain other drawings based on these drawings without creative efforts.

[0048] Figure 1 Schematic diagram of the calculation method flow of the concentric shaft structure frequency response function for an exemplary embodiment of the present invention;

[0049] Figure 2 Schematic diagram of the calculation principle of the concentric shaft structure frequency response function for an exemplary embodiment of the present invention;

[0050] Figure 3 Schematic diagram of the substructure coupling for an exemplary embodiment of the present invention;

[0051] Figure 4 Amplitude of the frequency response function of the HSK-E32-ER20M-050S type tool holder - ER20 chuck of the present invention when the clamping diameter is 4 mm and the clamping length is 20 mm;

[0052] Figure 5 Real part of the frequency response function of the HSK-E32-ER20M-050S type tool holder - ER20 chuck of the present invention when the clamping diameter is 4 mm and the clamping length is 20 mm;

[0053] Figure 6 Imaginary part of the frequency response function of the HSK-E32-ER20M-050S type tool holder - ER20 chuck of the present invention when the clamping diameter is 4 mm and the clamping length is 20 mm. Detailed implementation manners

[0054] Example embodiments will now be described more fully with reference to the accompanying drawings. However, the example embodiments can be implemented in various forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this invention will be more complete and comprehensive, and the concept of the example embodiments will be fully conveyed to those skilled in the art. The features, structures, or characteristics described can be combined in any suitable manner in one or more embodiments.

[0055] In addition, the accompanying drawings are only schematic illustrations of the present invention and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and thus repeated descriptions thereof will be omitted. Some of the block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities may be implemented in software form, or implemented in one or more hardware modules or integrated circuits, or implemented in different networks and / or processor devices and / or microcontroller devices.

[0056] Aiming at the defect that the calculation process of the frequency response function of the existing coaxial structure depends on the selected number of points n, resulting in the intermediate process matrix being 2n×2n and different numbers of points corresponding to different derivation processes, the present invention proposes a calculation method for the frequency response function of the coaxial structure in which the dimension of the intermediate process matrix is always 4×4 and different numbers of points do not need to be repeatedly derived. The method first divides two coaxial structures into n - 1 substructures along the axial direction respectively. Among them, the frequency response functions of each pair of substructures of the coaxial structure are coupled according to the balance compatibility condition between their two endpoints. Finally, the coupled frequency response functions of the coaxial substructure pairs obtained are used to obtain the coupled frequency response functions of the two coaxial structures according to the classical substructure coupling method.

[0057] Expected technical effect: The calculation method for the frequency response function of the coaxial structure provided by the present invention can always keep the dimension of the calculation process matrix unchanged, and the calculation process does not need to be repeatedly derived.

[0058] Aiming at the disadvantages and deficiencies of the existing technology, in this exemplary embodiment, a calculation method for the frequency response function of a coaxial structure that does not depend on the number of points to solve the dimension of the process matrix is provided. Refer to Figure 1 As shown, the calculation method for the frequency response function of the coaxial structure may specifically include the following steps:

[0059] Step S1: Divide the coaxial structures A1 and B1 into n - 1 substructures along the axial direction evenly;

[0060] Step S11: The responses of the two endpoints on the η-th substructure divided from the A1 structure can be expressed as:

[0061]

[0062] In the formula, and are the responses of the A1_η substructure at the η-th point and the η + 1-th point respectively. and are the excitations of the A1_η substructure at the η-th point and the η + 1-th point respectively. is the frequency response function of substructure A1_η when the excitation is at point q and the response is at point p.

[0063] Step S12. The responses of the two end points on the η-th substructure where the B1 structure is divided can be expressed as:

[0064]

[0065] where and are the responses of substructure B1_η at points η and η + 1 respectively. and are the excitations of substructure B1_η at points η and η + 1 respectively. is the frequency response function of substructure B1_η when the excitation is at point q and the response is at point p.

[0066] Step S2. For each pair of corresponding substructures A1_η and B1_η, where η = 1 to n - 1; according to the equilibrium condition and compatibility condition of the two end points of the substructure, calculate the 4×4 frequency response function matrix of the sub-combination C1_η composed of substructures A1_η and B1_η;

[0067] Step S2 specifically includes the following steps:

[0068] Step S21. If the connection between substructure B1_η and A1_η is rigid, their compatibility condition can be expressed as:

[0069]

[0070] Step S22. Substitute the responses of substructures B1_η and A1_η in Steps S11 and S12 into the compatibility condition in Step S21, and we have:

[0071]

[0072] Step S23. When the excitation is applied at point η, the equilibrium condition between substructures B1_η and A1_η can be expressed as:

[0073]

[0074] where is the excitation of structure C1_η at point η.

[0075] Step S24. Substitute the equilibrium condition in Step S23 into Step S22, and we have:

[0076]

[0077] Step S25. The frequency response function of the sub-assembly C1_η when both the excitation and the response are at the η point can be expressed as:

[0078]

[0079] Step S26. The compatibility condition between the sub-structure A1_η and the sub-assembly C1_η can be expressed as:

[0080]

[0081] In the formula, and are the responses of the C1_η sub-structure at the η point and the η + 1 point respectively.

[0082] Step S27. Substituting Step S11 and Step S26 into Step S25, the frequency response function of the sub-assembly C1_η when both the excitation and the response are at the η point can be further expressed as:

[0083]

[0084] Step S28. The two ratios and in Step S27 can be obtained through Step S24 and can be expressed as:

[0085]

[0086] Step S29. Substituting Step S28 into Step S27, the frequency response function of the sub-assembly C1_η when both the excitation and the response are at the η point can be obtained:

[0087]

[0088] Step S210. The frequency response function of the sub-assembly C1_η when the excitation is at η + 1 and the response is at the η point can be expressed as:

[0089]

[0090] Step S211. When the excitation is applied at the η + 1 point, the balance condition between the sub-structures B1_η and A1_η can be expressed as:

[0091]

[0092] Step S212. The two ratios and in Step S211 can be obtained by substituting Step S211 into Step S24 and can be expressed as:

[0093]

[0094] Step S213: Substitute Step S211 into Step S210. When the excitation is at η + 1 and the response is at η, the frequency response function of the sub-assembly C1_η can be further expressed as:

[0095]

[0096] Step S214: The frequency response function of the sub-assembly C1_η with the excitation at η and the response at η + 1 can be expressed as:

[0097]

[0098] Step S215: Substitute Step S28 into Step S214. When the excitation is at η and the response is at η + 1, the frequency response function of the sub-assembly C1_η can be further expressed as:

[0099]

[0100] Step S216: The frequency response function of the sub-assembly C1_η with the excitation at η + 1 and the response at η + 1 can be expressed as:

[0101]

[0102] Step S217: Substitute Step S212 into Step S216. When the excitation is at η + 1 and the response is at η + 1, the frequency response function of the sub-assembly C1_η can be further expressed as:

[0103]

[0104] Step S218: Repeating Steps S11 to S217 for η = 1 to n - 1 can obtain the frequency response functions of each sub-assembly C1_1 to C1_n - 1 with the excitation and response at the two endpoints respectively.

[0105] Step S3: By iteratively coupling all the sub-assemblies C1_η, the frequency response function of the overall assembly is finally obtained;

[0106] Exemplarily, coupling the first sub-assembly C1_1 and the second sub-assembly C1_2 axially according to the balance and compatibility conditions at 2 points can obtain a new sub-assembly C1_1 - 2, as Figure 3 shown in a.

[0107]

[0108] In the formula, represents the frequency response function of the response at point p and the excitation at point q after coupling from the sub-assembly C1_1 to C1_2.

[0109] Continue to couple in the above manner and loop the above process until the last substructure C1_n-1 is coupled, and the frequency response function of the structure C1 can be obtained. The last step of this loop is:

[0110]

[0111] In the formula, represents the frequency response function of the response at point p and the excitation at point q after coupling from the sub-assembly C1_1 to C1_n-2. represents the frequency response function of the response of the assembly C1 at point p and the excitation at point q.

[0112] The frequency response function of the coaxial structure calculated above is used to extract the modal parameters of the coaxial structure application system, and the stability analysis of the coaxial structure application system is carried out based on the modal parameters.

[0113] Next, each step and its application of the calculation method of the frequency response function of the coaxial structure in this exemplary embodiment will be described in more detail with reference to the accompanying drawings and specific embodiments.

[0114] Example 1:

[0115] In the milling tool shaft system, the tool holder-chuck and the tool are typical coaxial structures, and the prediction of the frequency response function of their coupling results is crucial for the prediction of the tool tip frequency response function.

[0116] Select a cylindrical rod with a length of 20 mm and a diameter of 4 mm, that is, tool A2 in the milling tool shaft system; and a tool holder-ER20 chuck of type HSK-E32-ER20M-050S, that is, B2. The coupling result can be obtained by following the steps described in the embodiment for A2 and B2, which can be used for predicting the frequency response function of the milling tool shaft system.

[0117] A method for calculating the frequency response function of a tool holder-chuck and tool coupling structure includes the following steps:

[0118] Step 1: Divide structures A2 and B2 along their axial directions into 4 substructures evenly, and the length of each substructure is 25 mm.

[0119] Step 2: The responses of the two endpoints on the ηth (η = 1 to 4) substructure divided from the A2 structure can be expressed as:

[0120]

[0121] In the formula, and are the responses of the A2_η substructure at points η and η + 1 respectively. and They are the excitations of the A2_η sub-structure at the η point and the η+1 point respectively. It is the frequency response function of the A2_η sub-structure when the excitation is at the q point and the response is at the p point.

[0122] Step 3: The responses at the two end points of the η-th sub-structure into which the B2 structure is divided can be expressed as:

[0123]

[0124] In the formula, and They are the responses of the B2_η sub-structure at the η point and the η+1 point respectively. and They are the excitations of the B2_η sub-structure at the η point and the η+1 point respectively. It is the frequency response function of the B2_η sub-structure when the excitation is at the q point and the response is at the p point.

[0125] Step 4: There is a rigid connection between the B2_η sub-structure and the A2_η sub-structure, and their compatibility condition can be expressed as:

[0126]

[0127] Step 5: Substituting the responses of the A2_η sub-structure and the B2_η sub-structure in Step 2 and Step 3 into the compatibility condition in Step 4, we get:

[0128]

[0129] Step 6: When the excitation is applied at the η point, the balance condition between the B2_η sub-structure and the A2_η sub-structure can be expressed as:

[0130]

[0131] Step 7: Substituting the balance condition in Step 6 into Step 5, we get:

[0132]

[0133] Step 8: The frequency response function of the C2_η sub-assembly with the excitation and response both at the η point can be expressed as:

[0134]

[0135] Step 9: According to the compatibility condition between the A2_η sub-structure and the C2_η sub-assembly, it can be expressed as:

[0136]

[0137] In the formula, and They are the responses of the C2_η sub-structure at the η point and the η + 1 point respectively.

[0138] Step Ten: Substitute Steps Two and Nine into Step Eight. The frequency response function of the sub-assembly C2_η with both the excitation and response at the η point can be further expressed as:

[0139]

[0140] Step Eleven: The two ratios and in Step Ten can be obtained through Step Seven and can be expressed as:

[0141]

[0142] Step Twelve: Substitute Step Eleven into Step Ten to obtain the frequency response function of the sub-assembly C2_η with both the excitation and response at the η point:

[0143]

[0144] Step Thirteen: The frequency response function of the sub-assembly C2_η with the excitation at the η + 1 point and the response at the η point can be expressed as:

[0145]

[0146] Step Fourteen: When the excitation is applied at the η + 1 point, the balance condition between the sub-structures B2_η and A2_η can be expressed as:

[0147]

[0148] Step Fifteen: The two ratios and in Step Fourteen can be obtained by substituting Step Fourteen into Step Seven and can be expressed as:

[0149]

[0150] Step Sixteen: Substitute Step Fifteen into Step Thirteen. The frequency response function of the sub-assembly C2_η with the excitation at the η + 1 point and the response at the η point can be further expressed as:

[0151]

[0152] Step Seventeen: The frequency response function of the sub-assembly C2_η with the excitation at the η point and the response at the η + 1 point can be expressed as:

[0153]

[0154] Step XVIII: Substitute Step XI into Step XVII, and the frequency response function of the combined body C2_η and the excitation at point η and the response at point η + 1 can be further expressed as:

[0155]

[0156] Step XIX: The frequency response function of the sub-combined body C2_η and the excitation at η + 1 and the response at point η + 1 can be expressed as:

[0157]

[0158] Step XX: Substitute Step XV into Step XIX, and the frequency response function of the sub-combined body C2_η and the excitation at η + 1 and the response at point η + 1 can be further expressed as:

[0159]

[0160] Step XXI: Repeating Steps II to XX for η = 1 to 4 can obtain the frequency response functions of each sub-structure C2_1 to C2_4 with the excitation and response at the two endpoints respectively.

[0161] Step XXII: Couple the first sub-combined body C2_1 and the second sub-combined body C2_2 axially according to the balance and compatibility conditions at 2 points to obtain a new sub-combined body C,

[0162]

[0163] In the formula, represents the frequency response function of the response at point p (p = 1, 3) and the excitation at point q (q = 1, 3) after coupling from the sub-combined body C2_1 to C2_2.

[0164] Step XXIII: Take the new sub-combined body C2_1-2 obtained in Step XXII as the structure C2_1 in Step XXII, and take the next sub-combined body C2_3 as C2_2 in Step XXII and continue to couple in the manner of Step XXII. Repeat the above process until the frequency response function of the structure C2 can be obtained. The last step of this cycle is:

[0165]

[0166] In the formula, represents the frequency response function of the response at point p (p = 1, 4) and the excitation at point q (q = 1, 4) after coupling from the sub-combined body C2_1 to C2_3. represents the frequency response function of the response at point p (p = 1, 5) and the excitation at point q (q = 1, 5) of the combined body C2.

[0167] Through the above coupling of the tool shank - chuck and the tool substructure, the frequency response function of their combined body C2 can be calculated. The frequency response function of the tool shank - chuck and the tool after coupling can be further used to calculate the frequency response function of the tool tip of the milling system (refer to the reference A new receptance coupling substructure analysismethodology to predict tool tip dynamics). Based on the frequency response function of the tool tip, the stability of the milling system can be analyzed and applied to the cutting stability prediction of key components in aerospace.

[0168] As Figure 4 shown are the amplitudes of the frequency response functions of the tool shank - chuck and the tool substructure for the HSK - E32 - ER20M - 050S tool shank - ER20 chuck in Example 1 when the clamping diameter is 4 mm and the clamping length is 20 mm. As Figure 5 shown are the real parts of the frequency response functions of the tool shank - chuck and the tool substructure for the HSK - E32 - ER20M - 050S tool shank - ER20 chuck in Example 1 when the clamping diameter is 4 mm and the clamping length is 20 mm. As Figure 6 shown are the imaginary parts of the frequency response functions of the tool shank - chuck and the tool substructure for the HSK - E32 - ER20M - 050S tool shank - ER20 chuck in Example 1 when the clamping diameter is 4 mm and the clamping length is 20 mm.

[0169] Those skilled in the art will readily think of other embodiments of the present invention after considering the specification and practicing the invention herein. This application is intended to cover any variations, uses, or adaptations of the present invention, which follow the general principles of the present invention and include known common knowledge or conventional technical means in the technical field not disclosed by the present invention. The specification and examples are only regarded as exemplary, and the true scope and spirit of the present invention are pointed out by the claims.

[0170] It should be understood that the present invention is not limited to the exact structures described above and shown in the drawings, and various modifications and changes can be made without departing from its scope. The scope of the present invention is only defined by the appended claims.

Claims

1. A calculation method for the frequency response function of a concentric axis structure, characterized in that, The method comprises: The concentric axis structures A1 and B1 are evenly divided into n-1 substructures along the axial direction; For each pair of corresponding substructures A1_η and B1_η, where η = 1 to n-1; based on the equilibrium conditions and compatibility conditions at the two end points of the substructure, calculate the frequency response functions of the excitation and response of the subassembly C1_η composed of the substructure pair A1_η and B1_η at the two end points respectively; By iteratively coupling all sub-assemblies C1_η, the frequency response function of the overall assembly is finally obtained; The calculated frequency response function of the concentric axis structure is used to extract modal parameters of the concentric axis structure application system, and stability analysis of the concentric axis structure application system is performed based on the modal parameters.

2. The method according to claim 1, wherein The frequency response functions of the excitation and response of the computing sub-assembly C1_η at two endpoints include: Establish the response equations of substructures A1_η and B1_η at both end points respectively; According to the compatibility condition and equilibrium condition of the rigid connection, the frequency response functions of the excitation and response of C1_η at the two endpoints are obtained by simultaneous solution.

3. The method according to claim 2, wherein The frequency response functions of the excitation and response of C1_η at the two endpoints include: The frequency response function of C1_η where both the excitation and response are at point η; Frequency response function of C1_η with excitation at η+1 and response at point η; Frequency response function of C1_η with excitation at point η and response at η+1; The excitation and response of C1_η are both at the frequency response function of η+1.

4. The method according to claim 3, wherein The solution of the frequency response function of the excitation and response of C1_η at point η includes: Substituting the response equations of substructures A1_η and B1_η at the two endpoints into the compatibility condition for the rigid connection, we obtain the first equation; Substituting the equilibrium condition between the first port and the substructures A1_η and B1_η into the first equation, we obtain the second equation: Establish the frequency response function of the subassembly C1_η when both the excitation and the response are at point η; Substituting the response equations of the two endpoints of substructure A1_η and the compatibility condition between substructure A1_η and subassembly C1_η into the frequency response function of subassembly C1_η when both the excitation and response are at point η, we obtain the third equation: Solving the second and third equations together yields the final frequency response function of C1_η, where both the excitation and response are at point η.

5. The method according to claim 4, characterized in that, The solution of the frequency response function of the excitation of C1_η at η+1 and the response at point η includes: Establish the initial frequency response function of the subassembly C1_η when the excitation is at point η+1 and the response is at point η, and the fourth equation is obtained; Substituting the equilibrium condition between the excitation applied to the second port and the substructures A1_η and B1_η into the second equation, we obtain the fifth equation: Solving the fourth and fifth equations together yields the final frequency response function of C1_η, where the excitation is at point η+1 and the response is at point η.

6. The method according to claim 5, wherein The solution of the frequency response function of the C1_η excitation at point η and the response at η+1 includes: Establish the initial frequency response function of the subassembly C1_η when the excitation is at point η1 and the response is at point η+1, and the sixth equation is obtained; Solving the sixth and third equations together yields the final frequency response function of C1_η, where the excitation is at point η and the response is at point η+1.

7. The method according to claim 6, characterized in that, The solution of the frequency response function of the excitation and response of C1_η at point η+1 includes: Establish the initial frequency response function of the subassembly C1_η when both the excitation and response are at point η+1, and obtain the seventh equation; Solving the seventh and fifth equations together yields the final frequency response function C1_η when both the excitation and response are at point η+1.

8. The method according to claim 1, wherein The iterative coupling process includes: Couple the adjacent subassemblies C1_η and C1_η+1 along the axial direction through the balance and compatibility conditions of the connection point to generate a new subassembly C1_η-(η+1); The subassemblies are coupled in sequence until all subassemblies are coupled into an integral structure.

9. Application of a calculation method for the frequency response function of a concentric axis structure, characterized in that, The coaxial structure is a coupling structure of the tool holder-chuck and the tool in the milling system. The modal parameters of the coupling structure of the tool holder-chuck and the tool are analyzed based on the calculated frequency response function of the tool holder-chuck and the tool. The cutting stability of key aerospace components is predicted based on the modal parameters of the coupling structure of the tool holder-chuck and the tool.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by a processor, the method for calculating the frequency response function of the concentric axis structure according to any one of claims 1 to 8 is implemented.

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