Function expression method of milling aviation aluminum alloy stress wave in real number field

Abstract

The invention discloses a function expression method for milling an aviation aluminum alloy stress wave in a real number field, relates to the technical field of aviation aluminum alloy milling, and aims to solve the problems that an existing three-dimensional stress wave model is complex in solution, time-consuming and difficult to guarantee in accuracy. The method comprises the following steps: firstly, judging the contact condition of an end mill and a workpiece, and calculating the instantaneous cutting layer thickness of a milling infinitesimal by utilizing a cutter tooth instantaneous contact relationship; on the basis of a one-dimensional stress wave theory, the size of a milling stress wave excitation source is calculated by combining the obtained thickness of the cutting layer; and finally, constructing a longitudinal wave fluctuation equation by using a Lame-Navier equation, and solving a partial differential equation by using a Laplace transformation method to obtain a function expression of stress waves in a real number field. According to the method, the stress wave solving process is simplified, the calculation precision and efficiency are improved, and a basic calculation method is provided for high-stability machining of the milling aviation aluminum alloy and stress wave related research.

Description

Technical Field

[0001] This invention belongs to the field of aerospace aluminum alloy milling technology, specifically relating to a method for expressing stress waves in the real number domain of milled aerospace aluminum alloys. Background Technology

[0002] Aerospace aluminum alloys, with their comprehensive advantages of low density, high specific strength, good corrosion resistance, and excellent machinability, and their ability to meet the manufacturing requirements of complex aerospace parts, have become one of the core structural materials in the aerospace industry. Their cutting stability and machining quality have a significant impact on the application performance of aerospace aluminum alloy workpieces. When milling aerospace aluminum alloy workpieces, the initial equilibrium state of the workpiece's particles is broken when the workpiece is subjected to milling force, causing the particles to deviate from their initial equilibrium positions. This disturbance is transmitted to the particles on the subsurface, generating a wave phenomenon, and ultimately forming a milling stress wave, thereby causing plastic deformation of the workpiece.

[0003] In both brittle and ductile materials, stress waves can cause damage and have a significant impact. Existing research on stress waves generated during milling is limited, and the mathematical models involved in three-dimensional stress wave models are quite complex, introducing numerous mathematical variables and making the solution process overly complicated and time-consuming, thus compromising accuracy. Furthermore, solving the actual wave equation directly in the real number domain is extremely complex, as the wave equation is a partial differential equation. Therefore, it is necessary to solve for the functional expression of stress waves in the real number domain.

[0004] This invention addresses the need for stable cutting of aerospace aluminum alloy workpieces by providing a method for calculating the instantaneous cutting layer thickness of a milling tooth micro-element under the influence of milling vibration, a method for calculating the magnitude of the excitation source of milling stress wave under the action of milling vibration, and a method for expressing the stress wave as a function in the real number domain. Summary of the Invention

[0005] The present invention aims to solve the problems of the complex mathematical model and numerous variables in the existing technology for solving the three-dimensional stress wave of milled aerospace aluminum alloys, which leads to time-consuming solutions and difficulty in ensuring accuracy, as well as the technical problems of directly solving the wave equation (partial differential equation) in the real number domain.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method for expressing stress waves from milled aerospace aluminum alloys as a function in the real number domain includes the following steps:

[0008] S1. Calculation of instantaneous cutting layer thickness of milling micro-element

[0009] Determine whether the end mill comes into contact with the aerospace aluminum alloy workpiece being machined. If contact occurs, analyze and calculate the instantaneous cutting layer thickness h of the milling micro-element using the instantaneous contact relationship between the cutter teeth and the workpiece. D (t,z), h D (t,z) is calculated using the following formula:

[0010] ;

[0011] Where t is time, z is the z-axis direction in the cutting coordinate system, and A x A y They are respectively Compared to The distance traveled along the feed direction and the cutting width direction; , The centers of rotation of the highest axial tip points of cutter teeth i and i-1 are respectively; θ t θ is the angle between the radii of rotation of the two cutting teeth along the projection line of the cutting depth direction; i , The instantaneous position angles of the milling micro-elements, i and i-1, are respectively; r i and Let i and i-1 be the radii of the cutting teeth, respectively. , These are the tilt angles caused by milling vibrations of cutter teeth i and i-1, respectively.

[0012] S2, Calculation of the magnitude of the milling stress wave excitation source

[0013] Based on the instantaneous cutting layer thickness h of the milled micro-element obtained in step S1 D (t,z), under the one-dimensional stress wave theoretical model, the magnitude of the milling stress wave excitation source is calculated based on the workpiece's stress condition. The magnitude of the excitation source is determined by the stress at point i. The calculation formula is as follows:

[0014] ;

[0015] in, For micro-element milling force, h represents the axial milling depth of the micro-element. D (t, dz) represents the instantaneous cutting layer thickness of the milling element;

[0016] S3. Solving the real-domain function expression of stress wave

[0017] Based on the magnitude of the milling stress wave excitation source obtained in step S2, the longitudinal wave equation is constructed using the Lame-Navier equation, and the partial differential equation is solved using the Laplace transform method to obtain the functional expression of the stress wave in the real domain. The functional expression is as follows:

[0018] ;

[0019] in, σ is the symbol for the inverse Laplace transform. i (t) represents a uniformly distributed load on the surface of the workpiece in the x direction, s is a Laplace variable, x is the coordinate of the stress wave propagation path, and c is the wave equation of the longitudinal wave.

[0020] Preferably, in step S1, when the cutting tooth i for the next cut reaches the same position as the cutting tooth i-1 that participated in the previous cut, the angle θ between the radii of rotation of the two cutting teeth along the projection line of the depth of cut is... t Calculated using the following formula:

[0021] ;

[0022] Where, θ i Let r be the instantaneous position angle of the milling micro-element i. i-1 Let be the radius of the cutting tooth i-1; The tilt angle is caused by the milling vibration of the cutter tooth i-1.

[0023] Preferably, in step S1, the instantaneous position angle of the milled micro-element is... Calculated using the following formula:

[0024] ;

[0025] Where β is the helix angle of the milling cutter's cutting edge. Let r be the instantaneous cutting position angle of the cutting tooth i. i Let be the radius of the cutting tooth i, and z be the coordinate of the milling element along the tool axis.

[0026] As a preferred option, in step S1, Compared to The distance A moved along the feed direction x The distance moved in the cutting width direction is calculated using the following formula:

[0027]

[0028] Among them, f z This represents the feed per tooth; t1 and t2 are the times when tooth i and tooth i-1 reach the cutting position, respectively. Let be the vibration displacement of cutter tooth i along the feed direction at time t1. Let be the vibration displacement of cutter tooth i along the feed direction at time t2. Let be the vibration displacement of the cutter tooth i along the cutting width direction at time t1. Let be the vibration displacement of the cutter tooth i along the cutting width direction at time t2.

[0029] Preferably, in step S2, under the one-dimensional stress wave theory model, the angle between the line connecting point i and the center of the milling cutter and the vertical direction is... Calculated using the following formula:

[0030] ;

[0031] Where n is the radial depth of cut, n is the milling cutter speed, and t is time. The radius of rotation of the milling cutter. Angle The maximum angle that can be achieved.

[0032] Preferably, in step S2, the workpiece is subjected to an x-axis force applied by the tool. The net force F acting on point i i The included angle β(t) is calculated using the following formula:

[0033] ;

[0034] in, The y-force applied to the workpiece by the cutting tool.

[0035] Preferably, in step S3, the expression for the Lame-Navier equation is:

[0036] ;

[0037] in For the Laplace operator;

[0038] In the formula:

[0039] ;

[0040] ;

[0041] ;

[0042] Where, θ t Let θ be the angle between the gyration radii, μ and λ be Lamé constants, and E, ν and G be the elastic modulus, Poisson's ratio and shear modulus of the aerospace aluminum alloy material, respectively. For volumetric strain, i represents the three directions in the three-dimensional coordinate system, u i This represents the displacement of a particle in the x, y, and z directions. For u i The second derivative with respect to time t represents acceleration. This refers to the density of aerospace aluminum alloy materials.

[0043] Preferably, in step S3, the wave equation for the longitudinal wave is derived as follows: Assuming the displacement field of the medium is an irrotational field, the wave equation for the longitudinal wave is derived using the Lame-Navier equation:

[0044]

[0045] in: The density of the medium; The stress gradient; It is normal stress; is the coordinate of the propagation path direction; c is the wave equation of the longitudinal wave.

[0046] Preferably, in step S3, the specific process of solving the partial differential equation using the Laplace transform method includes:

[0047] S31. Regarding the longitudinal wave equation Expression for stress σ(x,t) in the x-direction and boundary conditions Taking the Laplace transform over time t, we get:

[0048] ;

[0049] ;

[0050] ;

[0051] in, , , u x (x,t), σ i Laplace transformations of (t) and σ(x,t) with respect to t, where s is a Laplace variable;

[0052] Step S32: Solve the transformed longitudinal wave equation to obtain its general solution:

[0053] ;

[0054] Where A and B are the undetermined coefficients in the general solution;

[0055] Step S33: Determine the undetermined coefficients A and B based on the transformed boundary conditions, and substitute A and B into the transformed stress expression in the x-direction to obtain the transformed stress expression:

[0056] ;

[0057] Step S34: Perform an inverse Laplace transform on the transformed stress expression, based on the retardation properties. Thus, the functional expression of the stress wave in the real number domain is obtained;

[0058] Where u(t) is the unit step function; It is a delay factor; The time lag is the time lag.

[0059] Compared with the prior art, the technical effects and advantages of the present invention are:

[0060] In existing technologies, stress wave research on milled aerospace aluminum alloys largely relies on three-dimensional stress wave models. These models have complex mathematical structures and introduce numerous variables, making the solution process time-consuming and difficult to guarantee accuracy. Furthermore, the wave equation, as a partial differential equation, is extremely difficult to solve directly in the real domain, failing to meet the needs of precise stress wave research. This invention, based on one-dimensional stress wave theory, first calculates the instantaneous cutting layer thickness of the milling micro-element through the instantaneous contact relationship between the cutting teeth and the workpiece. Then, it determines the magnitude of the stress wave excitation source by combining the workpiece's stress conditions. Finally, it constructs the longitudinal wave equation using the Lame-Navier equation and transforms the partial differential equation into an easily solvable form using the Laplace transform method, significantly simplifying the stress wave solution process.

[0061] This method effectively improves the accuracy and efficiency of stress wave calculation, solves the core pain points of existing technologies, provides reliable theoretical support for subsequent research on milling stress waves, and provides a basic calculation method for achieving high stability in the milling of aerospace aluminum alloys, thus helping to ensure the processing quality of aerospace aluminum alloy workpieces. Attached Figure Description

[0062] Figure 1 This is a schematic diagram of the instantaneous contact angle of the blade tooth micro-element of the present invention;

[0063] Figure 2 This is a diagram of a one-dimensional stress wave propagation model of the workpiece in this invention. Detailed Implementation

[0064] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0065] In the milling process, from a dynamic perspective, when the tool comes into contact with the workpiece, the milling force acts on the workpiece surface, causing disturbances in local particles. These disturbances not only propagate to lower-level particles but also propagate from the workpiece surface in the form of waves. The solution process for the three-dimensional stress wave problem involves more complex mathematical models and more variables, making the solution process overly complex and time-consuming.

[0066] This invention proposes a method for expressing the stress wave in the real number domain of milled aerospace aluminum alloys. Based on the principle that the one-dimensional stress wave formed by the milling force applied by the cutting edge to the workpiece surface is sufficient to describe most dynamic behaviors, this method proposes to construct the longitudinal wave equation using the Lame-Navier equation based on the one-dimensional stress wave theory, and to solve the partial differential equation using the Laplace transform method to calculate the functional expression of the stress wave in the real number domain.

[0067] The following combination Figures 1 to 2 This application will be described in further detail;

[0068] To achieve machining stability and ensure machining quality of aerospace aluminum alloy workpieces, this invention provides a method for calculating the instantaneous cutting layer thickness of the milling micro-element under the influence of milling vibration and a method for calculating the magnitude of the excitation source of milling stress wave under the action of milling vibration. The invention also yields a functional expression for the milling stress wave of aerospace aluminum alloy in the real number domain.

[0069] 1. Calculation method for instantaneous cutting layer thickness of milling tooth micro-element under the influence of milling vibration

[0070] After determining whether contact occurs between the end mill and the workpiece, to obtain the magnitude of the instantaneous milling force of the milling micro-element, the instantaneous cutting layer thickness of the milling micro-element is analyzed and calculated using the instantaneous contact relationship between the cutter teeth and the workpiece. Figure 1 As shown.

[0071] Among them, h D (t,z) represents the instantaneous cutting layer thickness of the milling element; , , These are the x, y, and z coordinate reference frames for cutter tooth i; , , These are the coordinate reference systems for the cutter tooth i-1, respectively; , These are the rotation centers of the lowest axial tip points of cutter teeth i and i-1, respectively; , These are the rotation centers of the highest axial tip points of cutter teeth i and i-1, respectively; A x A y They are respectively Compared to The distance moved along the feed direction and the cutting width direction; θ t θ is the angle between the radii of rotation of the two cutting teeth along the projection line of the cutting depth direction; i θ i-1 The instantaneous position angles of the milling micro-elements, i and i-1, are respectively; r i and r i-1 Let i and i-1 be the radii of the cutting teeth, respectively. , The tilt angles caused by milling vibrations of cutter teeth i and i-1 are respectively; m j and n j These are the contact points between the current cutting edge of cutter tooth i and cutter tooth i-1 and the workpiece surface, respectively.

[0072] Depend on Figure 1 The instantaneous cutting layer thickness h of the milling cutter tooth micro-element can be obtained. D (t,z) is:

[0073] (1)

[0074] When the cutting tooth i, which will perform the next cut, reaches the same position as the cutting tooth i-1 that participated in the previous cut, the angle θ between the radii of rotation of the two cutting teeth along the projection line of the depth of cut is... t This can be expressed as:

[0075] (2)

[0076] The instantaneous position angle of the milled micro-element is:

[0077] (3)

[0078] Where β is the helix angle of the milling cutter's cutting edge. Let r be the instantaneous cutting position angle of the cutting tooth i. i Let be the radius of the cutting tooth i, and z be the coordinate of the milling element along the tool axis.

[0079] o j i Compared to o j i-1 The distance traveled along the feed direction and the cutting width direction is:

[0080] (4)

[0081] in, This represents the feed per tooth; t1 and t2 are the times when cutter tooth i and cutter tooth i-1 reach the cutting position, respectively.

[0082] Let be the vibration displacement of cutter tooth i along the feed direction at time t1. Let be the vibration displacement of cutter tooth i along the feed direction at time t2. Let be the vibration displacement of the cutter tooth i along the cutting width direction at time t1. Let i be the vibration displacement of the cutting tooth i along the cutting width direction at time t2;

[0083] From equations (1) to (4), it can be seen that if the orientation angle between the tool and the workpiece axis remains unchanged and the same cutting parameters are used, the inner surfaces of different components can be machined. Although the nominal cutting layer area is the same, the cutting layer area is not equal due to the influence of machining tilt angle, tool tip angle, tool tip radius, tool working main cutting edge angle, and working secondary cutting edge angle.

[0084] By using equations (1) to (11), while keeping the orientation angle between the tool and the workpiece axis unchanged and using the same cutting parameters, different inner surfaces of components can be machined. Although the nominal cutting layer area is the same, the cutting layer area is not equal due to the influence of machining tilt angle, tool tip angle, tool tip radius, tool working main cutting edge angle, and working secondary cutting edge angle.

[0085] 2. Calculation method for the magnitude of the excitation source of milling stress wave under milling vibration.

[0086] In milling, the one-dimensional stress wave formed by the milling force applied by the cutting edge to the workpiece surface is sufficient to describe most of the key dynamic behaviors. Therefore, to improve computational efficiency and facilitate problem handling while ensuring a certain level of accuracy, a milling stress wave solution model is constructed using one-dimensional stress wave theory, such as... Figure 2 As shown.

[0087] Among them, o v The center point of the milling cutter; n is the radial depth of cut; n is the milling cutter speed; v f h is the feed rate. D (t,dz) represents the instantaneous cutting layer thickness of the milling element.

[0088] Under the one-dimensional stress wave theoretical model, according to Figure 2 As shown, and The x- and y-forces applied to the workpiece by the tool; F i The net force acting on point i; Let be the angle between the line connecting point i and the center of the milling cutter and the vertical direction; β(t) is... and F i The included angle, where, for:

[0089] (5)

[0090] in, Angle The maximum angle that can be achieved.

[0091] β(t) is:

[0092] (6)

[0093] The stress at point i is:

[0094] (7)

[0095] in, For micro-element milling force, h represents the axial milling depth of the micro-element. D (t, dz) represents the instantaneous cutting layer thickness of the milling element;

[0096] 3. Functional representation of stress waves in the real number domain

[0097] The magnitude of the milling stress wave excitation source is obtained by solving equation (7). The Lame-Navier equation is a differential equation of motion describing displacement in an elastic body. Using the Lame-Navier equation to analyze the stress wave caused by milling can more conveniently help to understand and describe the displacement, stress distribution, and strain state of each particle during wave propagation, that is:

[0098] (8)

[0099] in For the Laplace operator;

[0100] In the formula:

[0101] (9)

[0102] (10)

[0103] (11)

[0104] Where: θ t The angle between the gyration radii; μ and λ are Lamé constants; E, ν, and G are the elastic modulus, Poisson's ratio, and shear modulus of the material, respectively. For volumetric strain, i represents the three directions in the three-dimensional coordinate system, ρ is the density of the material, and u is the volumetric strain. i This represents the displacement of a particle in the x, y, and z directions. For u i The second derivative with respect to time t represents acceleration.

[0105] Assuming the displacement field of the medium is irrotational, the wave equation for the longitudinal wave is derived as follows:

[0106] (12)

[0107] in, The density of the medium; The stress gradient; It is normal stress; is the coordinate of the propagation path direction; c is the wave equation of the longitudinal wave.

[0108] When stress waves propagate in the cutting teeth, and the direction of stress wave propagation is set to the x-direction, the displacement of the material can be considered as a function of position and time:

[0109] (13)

[0110] Among them, u x u y u z These represent the displacements of the material along the x, y, and z directions, respectively.

[0111] Substituting into equation (12), we can obtain the wave equation for the longitudinal wave as follows:

[0112] (14)

[0113] The stress σ(x,t) in the x-direction is:

[0114] (15)

[0115] During milling, the positional stress on the workpiece surface is equal to the external load it experiences. However, inside the workpiece, away from the surface, the stress wave gradually weakens, eventually approaching zero. Assume that initially, the workpiece's displacement and velocity are both zero. When the workpiece surface is subjected to a uniformly distributed load σ in the x-direction... i When (t), the boundary conditions and initial conditions of the wave equation can be expressed as follows:

[0116] (16)

[0117] In the actual process of solving the wave equation, since the wave equation is a partial differential equation, it is very complicated to solve it directly in the real number domain. To solve this problem, the Laplace transform can be used to simplify the solution. The basic idea is to transform the partial differential equation in the real number domain into an ordinary differential equation in the complex number domain, thereby simplifying the solution process. First, by taking the Laplace transform of the boundary conditions in equations (14), (15) and (16) with respect to time, we get:

[0118] (17)

[0119] (18)

[0120] (19)

[0121] Among them: U x (x,s), , u x (x,t), σ i The Laplace transformation of (t) and σ(x,t) with respect to t, where s is the Laplace variable.

[0122] The general solution to equation (17) is:

[0123] (20)

[0124] Substituting equation (20) into equation (18), the general solution for displacement can be converted into the general solution for stress. Combining this with the stress-based boundary conditions in equation (19), we have:

[0125] (twenty one)

[0126] Substituting A and B into equation (20), and then substituting equation (19) into equation (18), we get:

[0127] (twenty two)

[0128] To obtain the analytical expression for stress in the real domain, an inverse Laplace transform of equation (22) is required. Based on the delay property, we have:

[0129] (twenty three)

[0130] Where u(t) is the unit step function. As a delay factor, The time lag is the time lag.

[0131] Therefore, we can conclude that:

[0132] (twenty four)

[0133] Equation (24) is the analytical expression of the stress wave in the real number domain.

[0134] in, σ is the symbol for the inverse Laplace transform. i (t) represents a uniformly distributed load on the surface of the workpiece in the x direction, s is a Laplace variable, x is the coordinate of the stress wave propagation path, and c is the wave equation of the longitudinal wave.

[0135] Existing three-dimensional stress wave models for solving stress waves involve complex mathematical models and introduce more mathematical variables, resulting in an overly complex and time-consuming solution process that makes it difficult to guarantee the accuracy of the solution. Moreover, in the actual process of solving the wave equation, since the wave equation is a partial differential equation, it is very complex to solve it directly in the real number domain, which cannot meet the requirements for accurate study of stress waves.

[0136] Based on the instantaneous contact relationship between the cutting teeth and the workpiece, this invention analyzes and calculates the instantaneous cutting layer thickness of the milling micro-element. Under the one-dimensional stress wave theory model, the magnitude of the milling stress wave excitation source is calculated according to the stress condition of the workpiece. By setting the displacement field of the medium as an irrotational field, the wave equation of the longitudinal wave is derived. Through Laplace transformation, the solution is simplified, and the analytical expression of the stress wave function in the real number domain is obtained, which effectively simplifies the solution process of the stress wave and improves the calculation accuracy of the stress wave.

[0137] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for expressing stress waves in the real number domain of milled aerospace aluminum alloys, characterized in that, Includes the following steps: S1. Calculation of instantaneous cutting layer thickness of milling micro-element Determine whether the end mill comes into contact with the aerospace aluminum alloy workpiece being machined. If contact occurs, analyze and calculate the instantaneous cutting layer thickness h of the milling micro-element using the instantaneous contact relationship between the cutter teeth and the workpiece. D (t,z), h D (t,z) is calculated using the following formula: Where t is time, z is the z-axis direction in the cutting coordinate system, and A x A y They are respectively Compared to The distance traveled along the feed direction and the cutting width direction; The centers of rotation of the highest axial tip points of cutter teeth i and i-1 are respectively; θ t θ is the angle between the radii of rotation of the two cutting teeth along the projection line of the cutting depth direction; i θ i-1 The instantaneous position angles of the milling micro-elements, i and i-1, are respectively; r i and r i-1 Let i and i-1 be the radii of the cutting teeth, respectively. These are the tilt angles caused by milling vibrations of cutter teeth i and i-1, respectively. S2, Calculation of the magnitude of the milling stress wave excitation source Based on the instantaneous cutting layer thickness h of the milled micro-element obtained in step S1 D (t,z), under the one-dimensional stress wave theoretical model, the magnitude of the milling stress wave excitation source is calculated based on the workpiece stress condition. The magnitude of the excitation source is determined by the stress σ at point i. i (t) represents the value, and the calculation formula is: Wherein dF i (t,dz) represents the milling force of the infinitesimal element, dz represents the axial milling depth of the infinitesimal element, and h D (t,dz) represents the instantaneous cutting layer thickness of the milling element; S3. Solving the real-domain function expression of stress wave Based on the magnitude of the milling stress wave excitation source obtained in step S2, the longitudinal wave equation is constructed using the Lame-Navier equation, and the partial differential equation is solved using the Laplace transform method to obtain the functional expression of the stress wave in the real domain. The functional expression is as follows: Among them, L -1 σ is the symbol for the inverse Laplace transform. i (t) represents a uniformly distributed load on the surface of the workpiece in the x direction, s is a Laplace variable, x is the coordinate of the stress wave propagation path, and c is the wave equation of the longitudinal wave.

2. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 1, characterized in that: In step S1, when the cutting tooth i for the next cut reaches the same position as the cutting tooth i-1 that participated in the previous cut, the angle θ between the radii of rotation of the two cutting teeth along the projection line of the depth of cut is... t Calculated using the following formula: Where, θ i Let r be the instantaneous position angle of the milling micro-element i. i-1 Let δ be the radius of the cutting tooth i-1; i-1 The tilt angle is caused by the milling vibration of the cutter tooth i-1.

3. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 2, characterized in that: In step S1, the instantaneous position angle θ of the milled micro-element i (t,z) is calculated using the following formula: Where β is the helix angle of the milling cutter's cutting edge. Let r be the instantaneous cutting position angle of the cutting tooth i. i Let be the radius of the cutting tooth i, and z be the coordinate of the milling element along the tool axis.

4. The method for expressing the stress wave of milled aerospace aluminum alloy in the real number domain according to claim 3, characterized in that: In step S1, o j i Compared to o j i-1 The distance A moved along the feed direction x The distance moved in the cutting width direction is calculated using the following formula: A x =f z +A x (t1)+A x (t2); A y =A y (t1)+A y (t2); Among them, f z This represents the feed per tooth; t1 and t2 are the times when tooth i and tooth i-1 reach the cutting position, respectively. A x (t1) represents the vibration displacement of the cutter tooth i along the feed direction at time t1, A x (t2) represents the vibration displacement of the cutter tooth i along the feed direction at time t2, A y (t1) represents the vibration displacement of the cutter tooth i along the cutting width direction at time t1, A y (t2) represents the vibration displacement of the cutter tooth i along the cutting width direction at time t2.

5. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 1, characterized in that: In step S2, under the one-dimensional stress wave theoretical model, the angle α(t) between the line connecting point i and the center of the milling cutter and the vertical direction is calculated using the following formula: a e α is the radial depth of cut, n is the milling cutter speed, t is time, r is the milling cutter rotation radius, and α is the radial depth of cut. st Let α(t) be the maximum angle that the included angle can reach.

6. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 5, characterized in that: In step S2, the workpiece is subjected to an x-direction force F′ applied by the tool. x The net force F acting on point i i The included angle β(t) is calculated using the following formula: Among them, F′ y The y-force applied to the workpiece by the cutting tool.

7. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 1, characterized in that: In step S3, the expression for the Lame-Navier equation is: in For the Laplace operator; In the formula: λ = Eν / [(1+ν)(1-2ν)]; μ = G = E / [2(1+ν)]; Where, θ t The angle between the gyration radii is θ, μ and λ are Lamé constants, E, ν and G are the elastic modulus, Poisson's ratio and shear modulus of the aerospace aluminum alloy, respectively; θ is the volumetric strain, i represents the three directions in the three-dimensional coordinate system, and u... i This represents the displacement of a particle in the x, y, and z directions. For u i The second derivative with respect to time t represents the acceleration, and ρ is the density of the aerospace aluminum alloy material.

8. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 7, characterized in that: In step S3, the wave equation for the longitudinal wave is derived as follows: Assuming the displacement field of the medium is irrotational, the wave equation for the longitudinal wave is derived using the Lame-Navier equation as follows: Where: ρ is the density of the medium; σ is the stress gradient; σ is the normal stress; ε is the coordinate of the propagation path direction; c is the wave equation of the longitudinal wave.

9. The method for expressing stress waves in the real number domain of milled aerospace aluminum alloys according to claim 8, characterized in that: In step S3, the specific process of solving the partial differential equation using the Laplace transform method includes: S31. Regarding the longitudinal wave equation Expression for stress σ(x,t) in the x-direction and boundary conditions Taking the Laplace transform over time t, we get: ∑ x (x,s)| x＝0 =∑ i (s),∑ x (x,s)| x＝+∞ =0; Among them, U x (x,s), Σ i (s), Σ x (x,s) represent u x (x,t), σ i Laplace transformations of (t) and σ(x,t) with respect to t, where s is a Laplace variable; Step S32: Solve the transformed longitudinal wave equation to obtain its general solution: Where A and B are the undetermined coefficients in the general solution; Step S33: Determine the undetermined coefficients A and B based on the transformed boundary conditions, and substitute A and B into the transformed stress expression in the x-direction to obtain the transformed stress expression: Step S34: Perform an inverse Laplace transform on the transformed stress expression, based on the retardation property L -1 [∑ i (s)e -sr ]=∑ i (t-τ)u(t-τ) yields the functional expression of the stress wave in the real number domain; Where u(t) is the unit step function; e -sr τ is the delay factor; τ is the lag time.

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