Nonlinear Friction Dynamics Identification Method for the Secondary Relief Face of Shoulder Milling Cutters under Vibration

Abstract

Nonlinear friction dynamics identification method for the secondary flank of a shoulder milling cutter under vibration, belonging to the field of milling cutter processing technology. It includes the identification of the contact between the secondary flank of the milling cutter and the machining transition surface under vibration; the calculation method of the instantaneous friction characteristic variables of the secondary flank of the milling cutter; the phase space reconstruction method of the tribological behavior at the tool-work interface; the calculation method of the correlation dimension of the instantaneous friction characteristic variables of the secondary flank of the milling cutter; the calculation method of the K-entropy of the instantaneous friction characteristic variables of the secondary flank of the milling cutter; the dynamic evaluation method of the correlation dimension and K-entropy of the instantaneous friction characteristic variables of the secondary flank of the milling cutter. It reveals the dynamic characteristics and nonlinear friction dynamics characteristics of the instantaneous friction characteristic variables of the secondary flank of the milling cutter, avoiding ignoring the differences in the influence characteristics of the milling cutter structure, milling vibration and tooth error on the instantaneous cutting behavior of each tooth and the influence on the nonlinear dynamics of the instantaneous friction characteristic variables at different characteristic points of the tooth.

Description

Technical Field

[0001] The present invention relates to a method for identifying the non - linear friction dynamics of the flank face of a square - shoulder milling cutter under vibration, and belongs to the technical field of milling cutter machining. Background Art

[0002] The identification of the non - linear friction dynamics behavior of the flank face of a square - shoulder milling cutter is to reveal the dynamic change process experienced by the flank face of the milling cutter tooth during the milling process, which is an important index for evaluating the stability of the milling process. Affected by cutting vibration, tooth error and transient cutting thermo - mechanical coupling field, the friction process of the flank face of the cutter tooth shows variability, and it is difficult to identify the dynamic characteristics of the flank face of the cutter tooth.

[0003] Existing research on the method for solving the non - linear friction dynamics of the flank face of a milling cutter uses single - factor aspects such as acoustic emission signals or friction temperature to reveal the non - linear friction dynamics characteristics of the flank face of the milling cutter. This method ignores the influence of vibration and tooth error on the cutting process and cannot reveal the non - linear friction dynamics characteristics of the flank face of the milling cutter tooth during the entire milling stroke.

[0004] Therefore, it is urgent to propose a new type to solve the above - mentioned technical problems. Summary of the Invention

[0005] The research and development purpose of the present invention is to solve the problems of the existing method for solving the non - linear friction dynamics of the flank face of a milling cutter, which ignores the influence of vibration and tooth error on the cutting process and cannot reveal the non - linear friction dynamics characteristics of the flank face of the milling cutter tooth during the entire milling stroke. The present invention proposes a method for identifying the non - linear friction dynamics of the flank face of a square - shoulder milling cutter under vibration. This method uses the influence characteristics of vibration and tooth error on the instantaneous cutting behavior of the milling cutter and the cutter tooth to construct an instantaneous contact pose model of the flank face of the milling cutter, identify the contact relationship between the flank face of the milling cutter and the machining transition surface. Based on the atomic interface theory, obtain the instantaneous friction velocity, instantaneous friction energy consumption, and instantaneous friction coefficient of different points on the flank face of the milling cutter. Through the phase - space solution model of the tribological behavior at the tool - work interface, obtain the phase trajectory of the instantaneous friction characteristic variables of different points on the flank face of the milling cutter. Use the correlation dimension solution method to obtain the correlation dimension of the phase trajectory of the friction characteristic variables of different points on the flank face of the milling cutter, describe the high - density nature of its phase trajectory. Use the K - entropy solution method to obtain the K - entropy of the phase trajectory of the friction parameters of different points on the flank face of the milling cutter, describe the internal randomness of its phase trajectory. Use the Euclidean distance to distinguish the dynamic characteristics of the correlation dimension and K - entropy of the instantaneous friction characteristic variables at different positions on the flank face of the milling cutter. A brief overview of the present invention is given below in order to provide a basic understanding of certain aspects of the present invention. It should be understood that this overview is not an exhaustive overview of the present invention. It is not intended to identify the key or important parts of the present invention, nor is it intended to limit the scope of the present invention.

[0006] Technical solution of the present invention:

[0007] Nonlinear friction dynamics identification method for the secondary flank of a vibration - acting shoulder milling cutter, comprising:

[0008] S1, identification of the contact between the secondary flank of the milling cutter and the machining transition surface under vibration;

[0009] S2, calculation method for instantaneous friction characteristic variables of the secondary flank of the milling cutter;

[0010] S3, phase - space reconstruction method for the tribological behavior at the tool - workpiece interface;

[0011] S4, calculation method for the correlation dimension of instantaneous friction characteristic variables of the secondary flank of the milling cutter;

[0012] S5, calculation method for the K - entropy of instantaneous friction characteristic variables of the secondary flank of the milling cutter;

[0013] S6, dynamics evaluation method for the correlation dimension and K - entropy of instantaneous friction characteristic variables of the secondary flank of the milling cutter.

[0014] Further, the S1 includes:

[0015] Utilize the workpiece structure parameters, cutting parameters, milling cutter structure parameters, tooth structure parameters, tooth errors, and the influence characteristics of milling vibration on the instantaneous cutting behavior of the milling cutter and teeth to construct an instantaneous pose model of the milling cutter and teeth, and then identify the contact relationship between the secondary flank of the milling cutter and the machining transition surface.

[0016] Further, the S2 includes:

[0017] Obtain the instantaneous friction characteristic variables at different points on the secondary flank of the milling cutter, namely the instantaneous friction velocity, instantaneous friction energy consumption, and instantaneous friction coefficient, through the common tangent plane equation of the secondary flank of the tooth and the transition surface, the instantaneous pose of the tooth, the instantaneous relative motion velocity of points on the secondary flank of the tooth, the instantaneous contact temperature, and the instantaneous contact stress.

[0018] Further, the S3 includes:

[0019] Obtain the phase - space reconstruction parameters by solving the instantaneous friction characteristic variables, delay coordinate components, and embedding dimension, thereby obtaining the phase trajectories of the instantaneous friction characteristic variables, i.e., the phase trajectories of the instantaneous friction velocity, instantaneous friction energy consumption, and instantaneous friction coefficient.

[0020] Further, the S4 includes:

[0021] Obtain the correlation dimension of the phase trajectories of the friction characteristic variables at different points on the secondary flank of the milling cutter through the phase trajectories of the instantaneous friction characteristic variables by using the correlation dimension calculation method.

[0022] Further, the S5 includes:

[0023] Through the phase trajectory of the instantaneous friction characteristic variables, using the K-entropy calculation method, the K-entropy of the friction parameter phase trajectories at different points on the flank face of the milling cutter pair is obtained.

[0024] Furthermore, the S6 includes:

[0025] Through the correlation dimension and K-entropy of the phase trajectory of the instantaneous friction characteristic variables, the dynamic characteristics of the correlation dimension and K-entropy of the instantaneous friction characteristic variables at different positions on the flank face of the milling cutter pair are discriminated using the Euclidean distance.

[0026] The present invention has the following beneficial effects:

[0027] 1. The present invention considers the influence of milling vibration and tooth errors on the tooth structure and instantaneous cutting behavior, establishes an instantaneous pose model of the flank face of the milling cutter pair, identifies the contact between the flank face of the milling cutter pair and the transition surface, adopts a phase space reconstruction model, reveals the phase trajectory characteristics of the friction characteristic variables at different points on the flank face of the milling cutter pair in a high-dimensional space, and obtains the high density and internal randomness of the friction characteristic variable phase trajectories at different points on the flank face of the milling cutter pair through the correlation dimension calculation model and the K-entropy calculation model, and uses the Euclidean distance to obtain the dynamic evaluation method of the correlation dimension and K-entropy of the instantaneous friction characteristic variables at different positions on the flank face of the milling cutter pair;

[0028] 2. The present invention proposes a calculation method for the instantaneous friction characteristic variables of the flank face of the milling cutter pair. Using the cutting pose and the atomic interface theory, it reveals the dynamic characteristics of the instantaneous friction characteristic variables of the flank face of the milling cutter pair, and solves the problem that the existing methods ignore the differences in the influence characteristics of the milling cutter structure, milling vibration and tooth errors on the instantaneous cutting behavior of each tooth;

[0029] 3. The present invention proposes a phase space reconstruction method for the flank face of the milling cutter pair. Combining the false nearest neighbor method and the autocorrelation function method, it reveals the phase trajectory evolution characteristics of the instantaneous friction speed, instantaneous friction energy consumption, and instantaneous friction coefficient at different points on the flank face of the milling cutter pair; and proposes a correlation dimension calculation method and a K-entropy calculation method for the instantaneous friction parameters of the flank face of the milling cutter pair. Combining the instantaneous friction characteristic variables of different characteristic points of the tooth, it reveals the non-linear friction dynamics characteristics of different points of the tooth, and solves the problem that the existing solutions ignore the influence of the milling cutter structure, milling vibration and tooth errors on the non-linear dynamics of the instantaneous friction characteristic variables of different characteristic points of the tooth. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 is a flowchart of a method for identifying the non-linear friction dynamics of the flank face of a square shoulder milling cutter pair under the action of vibration;

[0031] Figure 2It is the milling cutter structure of the present invention and its instantaneous cutting position and attitude diagram, where (a) is the structure diagram of the square shoulder milling cutter, (b) is the motion state diagram of the milling cutter and the cutting teeth under vibration, and (c) is the instantaneous cutting attitude diagram of the square shoulder milling cutter;

[0032] Figure 3 It is the cutting tooth locus diagram of the milling cutter of the present invention;

[0033] Figure 4 It is the change diagram of the instantaneous offset angle of the milling cutter of the present invention;

[0034] Figure 5 It is the schematic diagram for judging the contact between the flank of the cutting tooth and the machining transition surface of the present invention;

[0035] Figure 6 It is the schematic diagram of the cutting tooth and the extraction and analysis of the vibration signal of the present invention, where (a) is the fluctuation diagram of the milling vibration time domain signal, and (b) is the schematic diagram for selecting the characteristic points on the flank of the cutting tooth;

[0036] Figure 7 It is the friction speed distribution diagram at the cutting-in stage of the three-tooth characteristic points of the present invention;

[0037] Figure 8 It is the friction energy consumption distribution diagram at the cutting-in stage of the three-tooth characteristic points of the present invention;

[0038] Figure 9 It is the friction coefficient distribution diagram at the cutting-in stage of the three-tooth characteristic points of the present invention;

[0039] Figure 10 It is the change diagram of the friction variable delay coordinate component of the present invention, where (a) is the friction speed delay coordinate component, (b) is the friction energy consumption delay coordinate component, and (c) is the friction coefficient delay coordinate component;

[0040] Figure 11 It is the state distribution diagram of the friction speed single variable in 2D and 3D spaces of the present invention, where (a) is the state of S p,m and S q,m in the 2D space state, and (b) is the state of S p,m and S q,m in the 3D space state;

[0041] Figure 12 It is the embedding dimension schematic diagram of the friction variable of the present invention, where (a) is the embedding dimension of the friction speed, (b) is the embedding dimension of the friction energy consumption, and (c) is the embedding dimension of the friction coefficient;

[0042] Figure 13 It is the distribution diagram of the friction variable phase space coordinate determination method of the present invention, where (a) is the distribution diagram of the friction speed phase space coordinate determination method, (b) is the distribution diagram of the friction energy consumption phase space coordinate determination method, and (c) is the distribution diagram of the friction coefficient phase space coordinate determination method;

[0043] Figure 14 It is a schematic diagram of the phase trajectory evolution process of the friction characteristic variables at point K1 on the flank face of the tool tooth pair of the present invention. Among them, (a) is a schematic diagram of the phase trajectory evolution of the friction speed, (b) is a schematic diagram of the phase trajectory evolution of the friction energy consumption, and (c) is a schematic diagram of the phase trajectory evolution of the tool tooth friction coefficient;

[0044] Figure 15 It is a schematic diagram of the phase trajectory evolution process of the friction characteristic variables at point K2 on the flank face of the tool tooth pair of the present invention. Among them, (a) is a schematic diagram of the phase trajectory of the friction speed at point K2, (b) is a schematic diagram of the phase trajectory of the friction energy consumption at point K2, and (c) is a schematic diagram of the phase trajectory of the friction coefficient at point K2;

[0045] Figure 16 It is a schematic diagram of the correlation dimension of the friction characteristic variables at point K1 of the present invention;

[0046] Figure 17 It is a schematic diagram for comparing the correlation dimensions of the friction parameters at points K1 and K2 of the present invention. Among them, (a) is a schematic diagram of the correlation dimension of the friction speed at points K1 and K2, (b) is a schematic diagram of the correlation dimension of the friction energy consumption at points K1 and K2, and (c) is a schematic diagram of the correlation dimension of the friction coefficient at points K1 and K2;

[0047] Figure 18 It is a schematic diagram of the change of K entropy of the friction characteristic variables at point K1 of the present invention;

[0048] Figure 19 It is a comparison distribution diagram of the K entropy of the friction characteristic variables at points K1 and K2 of the present invention. Among them, (a) is a schematic diagram of the K entropy of the friction speed at points K1 and K2, (b) is a schematic diagram of the K entropy of the friction energy consumption at points K1 and K2, and (c) is a schematic diagram of the K entropy of the friction coefficient at points K1 and K2; Detailed implementation manners

[0049] To make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be described below through specific embodiments shown in the accompanying drawings. However, it should be understood that these descriptions are only exemplary and do not intend to limit the scope of the present invention. In addition, in the following description, the descriptions of well-known structures and technologies are omitted to avoid unnecessarily confusing the concepts of the present invention.

[0050] The connections mentioned in the present invention are divided into fixed connections and detachable connections. The fixed connections (i.e., non-detachable connections) include, but are not limited to, conventional fixed connection methods such as hemming connection, rivet connection, bonding connection, and welding connection. The detachable connections include, but are not limited to, conventional detachable methods such as screw connection, snap connection, pin connection, and hinge connection. When the specific connection method is not clearly defined, it is defaulted that at least one connection method can always be found among the existing connection methods to achieve this function, and those skilled in the art can select it according to their needs. For example: welding connection is selected for fixed connection, and hinge connection is selected for detachable connection.

[0051] Specific Embodiment 1: In combination with Figures 1-19 To illustrate this embodiment, the method for identifying the non-linear friction dynamics of the flank face of the milling cutter pair under vibration in this embodiment includes:

[0052] S1. Identification of the contact between the flank face of the milling cutter pair and the machining transition surface under vibration;

[0053] Utilize the workpiece structure parameters, cutting parameters, milling cutter structure parameters, cutter tooth structure parameters, cutter tooth errors, and the influence characteristics of milling vibration on the instantaneous cutting behavior of the milling cutter and cutter teeth to construct an instantaneous pose model of the milling cutter and cutter teeth, and then identify the contact relationship between the flank face of the milling cutter pair and the machining transition surface.

[0054] The milling cutter structure, cutter tooth errors, milling method, cutting parameters, and milling vibration directly affect the instantaneous contact relationship between the flank face of the cutter tooth pair and the machining transition surface. Among them, the instantaneous cutting pose of the indexable square shoulder milling cutter and its cutter teeth under vibration is as Figure 2 shown.

[0055] Figure 2 In d -X d Y d Z d is the milling cutter structure coordinate system, O i -a i b i c i is the cutter tooth coordinate system, Δz i is the axial error of the milling cutter tooth; Δr i is the radial error of the milling cutter tooth; r i is the radius of the circle where the i-th cutter tooth is located; r max is the radius of the circle where the maximum outer diameter is located; is the tooth space angle; η ii+1 is the angle between the i-th cutter tooth and the Y d axis; l i is the distance from the lowest point of the i-th cutter tooth to the end face of the milling cutter; l1 is the distance from the lowest point of the milling cutter to the end face of the milling cutter; D is the diameter of the milling cutter body.

[0056] The o-xyz is the workpiece coordinate system, and L0, W0, and H0 are the workpiece length, width, and height directions respectively; the o0-uvw is the cutting motion coordinate system of the milling cutter without vibration, and the u, v, and w axes are parallel to the workpiece length, width, and height directions respectively; O c -UVW is the cutting motion coordinate system under the vibration of the milling cutter, and a e is the cutting width of the workpiece; n is the rotational speed of the milling cutter; v f is the feed speed of the milling cutter; θ(t) is the instantaneous attitude angle of the milling cutter under vibration; is the initial cutting-in moment of milling, and the position where the tip of the i-th tooth of the milling cutter is located; is the instantaneous initial cutting-in angle of the milling cutter tooth; is the angle between the milling cutter structure coordinate system and the cutting motion coordinate system under vibration at the moment of t0; O c -a i ’b i ’c i ’ is the cutting motion coordinate system under the vibration of the tooth, and A x (t), A y (t), A z (t) are the vibration displacements of the tooth in the x, y, and z directions respectively; l is the overhang length of the milling cutter; e is the starting point of the overhang amount of the square shoulder milling cutter; θ'(t) is the instantaneous direction angle of the tooth; θ1(t) is the projection of the instantaneous attitude angle of the tooth on the b i o i c i plane; θ2(t) is the projection of the instantaneous attitude angle of the tooth on the a i o i c i plane.

[0057] From Figure 2 , under the influence of vibration and tooth error, the trajectory equation of any point on the flank of the tooth pair in the workpiece coordinate system is:

[0058]

[0059] Among them, is the conversion relationship between the tooth coordinate system and the workpiece coordinate system.

[0060] The instantaneous angle between the V axis and the Y axis in the UVW plane is:

[0061]

[0062] Among them, is the initial cutting-in moment of the milling cutter, that is, the angle between the V axis and the Y axis in the UVW plane at t = 0 is:

[0063]

[0064] Among them, x o0 (t), y o0 (t), z o0 (t) is the instantaneous position coordinate of the origin o of the milling cutter cutting coordinate system without vibration action at o in the workpiece coordinate system o-xyz:

[0065]

[0066] Among them, θ1(t) is the i projection of the 'axis on the i o i c i plane and the instantaneous included angle with the i axis, and θ2(t) is the i projection of the 'axis on the i o i c i plane and the instantaneous included angle with the i axis is:

[0067]

[0068] θ(t) is the instantaneous offset angle of the milling cutter cutting coordinate system under vibration action:

[0069]

[0070] Define one revolution of the cutter tooth as a cutting cycle. Considering the influence of cutter tooth errors, according to the vibration characteristics reflected by the vibration signal obtained from experiments, solve the cutting motion trajectory and instantaneous offset angle of the cutter tooth according to Equation (1) and Equation (6), as Figure 3 , 4 shown.

[0071] According to the milling cutter and cutter tooth pose, solve the instantaneous contact relationship between the flank of the cutter tooth pair and the machining transition surface as Figure 5 shown.

[0072] As Figure 5 shown, select point e i (x1, y1, z1) on the flank of the cutter tooth pair, and make a normal vector i from the cutting plane where e is located towards the machining transition surface, which intersects the machining transition surface at point p(x2, y2, z2). Let Take the direction from the characteristic point of the cutter tooth flank to the outside of the cutter tooth as the positive direction, and its magnitude is Then the determination of its instantaneous contact relationship:

[0073] (1) When , Taking the positive direction, e is tangent to p, belonging to the contact critical point;

[0074] (2) When At this time, is in the negative direction, and there is no contact between e and p;

[0075] (3) When At this time, is in the positive direction, and there is friction between e and p.

[0076] S2, the method for calculating the instantaneous friction characteristic variables of the flank face of the milling cutter sub;

[0077] The instantaneous friction characteristic variables of different points on the flank face of the milling cutter sub, namely the instantaneous friction velocity, the instantaneous friction energy consumption, and the instantaneous friction coefficient, are obtained through the common tangent plane equation of the flank face of the cutter tooth sub and the transition surface, the instantaneous pose of the cutter tooth, the instantaneous relative motion velocity of the points on the flank face of the cutter tooth sub, the instantaneous contact temperature, and the instantaneous contact stress.

[0078] In order to characterize the dynamic characteristics of the tribological behavior at the tool-workpiece interface, the friction velocity of the characteristic points in the contact area between the flank face of the cutter tooth sub and the machining transition surface is calculated. Since a large amount of energy is consumed during the instantaneous contact between the tool and the workpiece, in order to analyze the non-linear dynamic characteristics of the instantaneous friction energy consumption on the flank face of the cutter tooth sub, a friction energy consumption calculation model is constructed. To more fully illustrate the dynamic characteristics of the tribological behavior, a friction coefficient calculation model is built. Using the above calculation models, two experimental schemes of titanium alloy milling experiments are carried out with a three-tooth square shoulder milling cutter with a diameter of 25 mm provided by Walter Company and its titanium alloy milling process plan. The milling method is dry up milling. The cutter tooth that first cuts into the workpiece is defined as cutter tooth 1, and the milling parameters are shown in Table 1.

[0079] Table 1 Cutter tooth error of the milling cutter and milling parameters

[0080]

[0081] In the experimental scheme, the cutter teeth and the extraction of vibration signals are as Figure 6 shown. T1 is the cutting-in stage with 35 cycles, T2 is the cutting stage with 460 cycles, and T3 is the cutting-out stage with 35 cycles. 5 cycles are equally divided in each stage.

[0082] Select the point with the highest wear depth and define the coordinates as point K1, and the coordinates are (1171, 228).

[0083] The trajectory parameter equation of any point on the cutter tooth is obtained by solving formula (1) as shown in the formula. Taking the partial derivative of the trajectory of any point on the cutter tooth with respect to time, the component velocities in the x, y, and z axis directions of the workpiece coordinate system are obtained as shown in v nx 、v ny 、v nz .

[0084] Relative motion speed v of any point n The solution method is as shown in Equation (7).

[0085]

[0086] Relative motion speed v n Unit vector in the workpiece coordinate system is (v nx , v ny , v nz ). Pass through the intersection point o of the flank face of the tool tooth pair and the transition surface of the workpiece r , pass through o r and the common tangent plane P of the flank face of the tool tooth pair and the transition surface of the workpiece being machined i is:

[0087] Pass through o r and the common tangent plane P of the flank face of the tool tooth pair and the transition surface of the workpiece being machined i is as shown in Equation (8).

[0088]

[0089] where A i ' is the flank face of the tool tooth pair in the workpiece coordinate system.

[0090] Make the projection of v n on the common tangent plane P i and pass through point o r to make the unit vector in the projection direction is (l px , l py , l pz ).

[0091]

[0092] In the workpiece coordinate system, the friction speed v of any point on the flank face is m as shown in Equation (10).

[0093] v m = v n ·cosθ m (10)

[0094] where θ m is the angle between the relative motion speed v n and the unit vector , and θ m is complementary to θ c . The calculation method of θ c is:

[0095]

[0096] Among them, θ c is the included angle between the relative motion speed v n and the friction speed v m .

[0097] In the workpiece coordinate system, the friction speed direction vector of the point on the secondary flank is as shown in Equation (12).

[0098]

[0099] Using the instantaneous transformation relationship matrix between the cutter tooth coordinate system and the workpiece coordinate system, the friction speed components v mXi , v mYi , v mZi of the point on the secondary flank of the cutter tooth in the cutter tooth coordinate system are obtained, as shown in Equation (13).

[0100]

[0101] Among them, v mXi is the friction speed of the point on the secondary flank of the cutter tooth along the X i axis direction in the cutter tooth coordinate system; v mYi is the friction speed of the point on the secondary flank of the cutter tooth along the Y i axis direction in the cutter tooth coordinate system; v mZi is the friction speed of the point on the secondary flank of the cutter tooth along the Z i axis direction in the cutter tooth coordinate system.

[0102] The friction speed of any point in the cutter tooth coordinate system is

[0103]

[0104] By solving Equation (14), the friction speed at the position of point K1 on the secondary flank of the three teeth of the milling cutter in the cutting-in stage is obtained, as Figure 7 shown as:

[0105] The change rate E v of the absorbed energy is obtained from the interface atom theory as:

[0106]

[0107] Among them, E is the absorbed energy at time t, and υ is the forced vibration frequency of the atom as shown in Equation (16):

[0108]

[0109] The instantaneous energy distribution function of the absorption is as shown in Equation (17).

[0110]

[0111] where: a is the lattice constant a = 2.9506×10 -10 m; h is the Planck constant h = 6.62607015×10 -34 J·s; k is the Boltzmann constant k = 1.380649×10 -23 J / K; t es1 and t es2 are respectively the initial time and the termination time of the instantaneous absorbed energy; T is the calculation method of the temperature rise at the atomic interface as shown in Equation (31).

[0112]

[0113] where: η is the relative atomic mass of the atom, which is 4.34×10 -26 kg, ω n is the natural frequency of the atom, which is 4.39×10 11 rad / s, H is the amplitude of the excitation force of the interface potential field, which is 1×10 -9 N.

[0114] The normal pressure at the frictional contact point is F p , and the friction coefficient is μ. Then, the work done by the frictional force at the contact point within the dt time is:

[0115] dW = μ(a i , b i , c i , t)·F p (a i , b i , c i , t)·v m (a i , b i , c i , t)·dt (19)

[0116] Assume that during the frictional movement of the cutting interface, all the frictional work is converted into the heat energy of the system;

[0117] dW = dE (20)

[0118] The distribution function of the friction coefficient is obtained from Equation (20) as shown in Equation (21).

[0119]

[0120] S3, the phase space reconstruction method for the tribological behavior of the cutting interface;

[0121] The phase space reconstruction parameters are obtained by solving the instantaneous friction characteristic variables, the delayed coordinate components, and the embedding dimension, thereby obtaining the phase trajectories of the instantaneous friction characteristic variables, i.e., the phase trajectories of the instantaneous friction velocity, the instantaneous friction energy consumption, and the instantaneous friction coefficient.

[0122] S3.1, Method for resolving delay coordinate components of friction characteristic variable time series

[0123] The delay coordinate component τ is a key parameter for transforming the friction characteristic variable time series from one-dimensional space into equivalent coordinates in high-dimensional space. The friction characteristic variables include friction velocity, friction energy consumption, and friction coefficient. The autocorrelation function is used to determine the delay coordinate component.

[0124] According to Figures 7-9 , for the friction characteristic variable time series autocorrelation function of point K1 on the secondary flank of cutter tooth 1 in the first cycle is:[[]]

[0125]

[0126] where s α is the friction characteristic variable time series, g is the total number of friction characteristic variables, β is the mean value of the friction characteristic variable time series, σ 2 (s) is the standard deviation of the friction characteristic variable time series, τ0 is the friction characteristic variable delay coordinate component, A(τ0) is the autocorrelation function of the friction characteristic variable, which can represent the degree of mutual correlation between two moments α and α + τ0. When A(τ0) first reaches or approaches (1 - 1 / e) times the initial value A(0), the corresponding τ0 is the appropriate delay coordinate component value of the friction characteristic variable. Using the above method, according to Figures 7-9 obtain the delay coordinate component of the friction characteristic variable of K1 on the secondary flank of cutter tooth 1 in the first cycle as shown in Figure 10 .

[0127] From Figure 10 , it can be known that the delay coordinate component of the friction velocity is 7, the delay coordinate component of the friction energy consumption is 3, and the delay coordinate component of the friction coefficient is 2.

[0128] S3.2, Method for resolving embedding dimension of friction variable time series

[0129] Selecting an appropriate embedding dimension m can fully represent the information of the original dynamic system. The false nearest neighbor method is used to determine the embedding dimension. According to Figures 7-9 , for the state sequence of the friction characteristic variable time series of point K1 on the secondary flank of cutter tooth 1 in the first cycle in the m-dimensional space is:

[0130] S α =(s α , s α+τ ..., s α+(m-1)τ ), 1 ≤ i ≤ g - (m - 1)τ (23)

[0131] where τ is the friction variable delay coordinate component.

[0132] The friction variable has a nearest neighbor point S at each point in the m-dimensional space α For each point S ε According to Figure 7 , take a point s in the friction velocity p , which is S in the two-dimensional space p,m =(s p , s p+τ ), find its nearest neighbor point S in the two-dimensional space q,m =(s q , s q+τ ). When the dimension increases to three, S p,m is S in the three-dimensional space p,m+1 =(s p , s p+τ , s p+2τ ), and S q,m is S in the three-dimensional space q,m+1 =(s q , s q+τ , s q+2τ ), as shown in Figure 11 .

[0133] The distance between two points in the m-dimensional space is R p (m), and the distance in the (m + 1)-dimensional space is R p (m + 1). The distance between R p (m) and R p (m + 1) is a p,m , that is

[0134] a p,m =R p (m + 1)-R p (m) (24)

[0135] When a p,m is very large, it will become a non-neighboring point. That is, set a threshold R T = 10. When a p,m > 10, this point is considered a false neighboring point.

[0136] Gradually increase m from the minimum value of 1, and calculate the proportion of state points with false neighbors in the reconstructed phase space of different dimensions m, which is called the false nearest neighbor rate. When the false nearest neighbor rate is less than 5% or no longer decreases as m increases, it is considered that the phase trajectory has been fully opened, and the appropriate embedding dimension is determined accordingly:

[0137]

[0138] In the above formula, F is the false nearest neighbor rate of the phase space points of the friction variable, and N is the number of phase space points of the friction variable in different dimensions.

[0139] Using the above method according to Figures 7-9Obtain the embedding dimension of the friction characteristic variables of point K1 on the flank face of the cutting edge 1 in the first cycle as follows Figure 12 as shown.

[0140] It can be seen from Figure 11 that the embedding dimensions of the friction velocity time series, the friction energy consumption time series, and the friction coefficient time series are all 3.

[0141] S3.3, Phase space reconstruction method for friction variable time series

[0142] To reveal the non-linear evolution information contained in the time series of the friction behavior of the flank face of the cutting edge pair, the above single-variable time series is extended to a high-dimensional space, and its dynamic characteristics are obtained through phase space reconstruction. Figure 12 In it, the single-variable time series of the friction velocity, friction energy consumption, and friction coefficient of point K1 on the flank face of the milling cutter cutting edge 1 in the first cycle are lifted to dimension m, and s is constructed according to the reasonable delay coordinate component τ α (α = 1, 2,..., N) coordinates of points in the phase space.

[0143]

[0144] Among them, S α is the state sequence of the friction characteristic variables in the high-dimensional space obtained according to Figures 7-9 the sampling interval method, s α is the corresponding friction characteristic variable time series, m is the embedding dimension after the phase space reconstruction of the friction characteristic variables, τ is the equivalent parameter delay coordinate component of the friction characteristic variables from one-dimensional space to high-dimensional space, g is the length of the friction characteristic variable time series, and N is the number of vectors in the reconstructed phase space of the friction characteristic variables.

[0145] To solve the problem that the high-dimensional phase trajectory cannot be observed, using the phase space reconstruction matrix S and the principal component analysis method, the main eigenvalues λ e1 , λ e2 and λ e3 of the friction characteristic variable phase space are obtained, and the corresponding main eigenvectors ξ e1 , ξ e2 , ξ e3 . The above vectors contain the main characteristics of the friction behavior of the flank face of the cutting edge pair, then project the matrix S onto the main vector direction to obtain an N×3 matrix:

[0146] λ e1 ≥λ e2 ≥λ e3 ≥...λ em , α = S[ξ e1 ξ e2 ξ e3 (27)

[0147] The phase trajectory of the friction characteristic variables of the flank face of the cutting tooth can be obtained by the above method. Among them, the evolution processes of the friction speed, friction energy consumption, and friction coefficient of the flank face (1171, 228) of cutting tooth 1 are as follows Figure 13 as shown.

[0148] It can be seen from Figure 14 that during the cutting-in and cutting-out stages of the milling cutter, the phase trajectory of the friction speed of the flank face of the cutting tooth is an irregular circular ring, and the phase trajectories of the friction energy consumption and friction coefficient show an irregular and chaotic motion state in all directions within a limited space; during the middle stage of the milling cutter cutting, the phase trajectory of the friction speed presents an obvious regular circular ring in the restricted area, and the phase trajectories of the friction energy consumption and friction coefficient are irregular circular rings.

[0149] The above analysis results show that during the cutting process of the milling cutter, the phase trajectories of the friction speed, friction energy consumption, and friction coefficient of the flank face of the cutting tooth under the influence of milling vibration and cutting tooth error gradually form a relatively regular state; as the wear of the flank face of the cutting tooth intensifies and the instantaneous contact temperature and contact stress change continuously, the attractor trajectory changes from a relatively regular state to an irregular state, and the phase trajectories of the friction speed, friction energy consumption, and friction coefficient of the flank face of the cutting tooth all show the characteristics of divergence-convergence-divergence, but the phase trajectories of the friction energy consumption and friction coefficient show more complex and chaotic non-linear friction dynamics characteristics than the friction speed.

[0150] To observe the dynamic characteristics of different points on the flank face, the evolution processes of the friction speed, friction energy consumption, and friction coefficient of the flank face (741, 228) of cutting tooth 1 are as follows Figure 14 as shown.

[0151] It can be seen from Figure 15 that the motion trends of the phase trajectories of the friction parameters at point K2 and the phase trajectories of the friction parameters at point K1 are similar. Since points K2 and K1 have experienced similar motion processes, they show an evolution process of divergence-convergence-divergence from cutting-in to cutting-out.

[0152] S4, the method for calculating the correlation dimension of the instantaneous friction characteristic variables of the flank face of the milling cutter;

[0153] Through the phase trajectory of the instantaneous friction characteristic variables and using the correlation dimension calculation method, the correlation dimensions of the phase trajectories of the friction characteristic variables at different points on the flank face of the milling cutter are obtained.

[0154] The correlation dimension is very sensitive to the time process behavior of the dynamic system, can perfectly quantify the properties of the dynamic system, can characterize the high density of its phase trajectory, and reveal the complexity of the friction dynamics parameters of the flank face of the milling cutter. After the phase space reconstruction, the Euclidean distance between the phase points in the friction variable phase space is ||S α -S j||, given a positive number δ, any two phase points with a distance less than δ between them are considered related phase point pairs. After reconstruction, there are a total of N friction variable phase points, corresponding to N 2 pairings, and then calculate the proportion of related phase point pairs among them according to the above principle, which is called the correlation integral (denoted as C m (δ)):

[0155]

[0156] where H(ψ) represents the Heaviside function, that is

[0157]

[0158] Due to the chaotic characteristics of the phase trajectory, different values of δ are selected within a certain range, generally in the interval from 0 to 1, and calculate the corresponding correlation integral C m (δ) and follow the following rules:

[0159]

[0160] D2 represents the correlation dimension and is estimated by the following formula

[0161]

[0162] where is the estimated value of D2. In the specific estimation, generally let the embedding dimension m take values from small to large. For each m, draw the curve formed by the corresponding ln(δ) and ln(C m (δ)), and then remove the intervals at both ends of each curve with slopes of 0 and ∞, and use the least squares method to fit the slope, which is The entire milling process is divided into stages of cutting in, cutting, and cutting out, and calculate the correlation dimensions of the friction speed, friction energy consumption, and friction coefficient at each stage, and obtain the correlation dimension of the friction characteristic variables at point K1 from cutting in to cutting out as shown Figure 15 below.

[0163] From Figure 16 it can be seen that during the process of the milling cutter cutting in to cutting out, the correlation dimensions of the friction speed, friction energy consumption, and friction coefficient change from large to small and then to large, presenting a "bathtub curve" form. This is because the correlation dimension is high and the chaos degree is strong in the cutting-in and cutting-out stages, and the dynamic system presents an unstable state. In the cutting stage, the correlation dimension is low and the chaos degree is low, and the dynamic system presents a stable state.

[0164] To compare the dynamic characteristics at different positions of the flank face, observe the correlation dimensions D 2K1 and D 2K2 of the friction characteristic parameters of K1 and K2 and the correlation dimension D 2K0 under steady state as shownFigure 16 as shown

[0165] From Figure 17 it can be seen that the overall correlation dimension of the friction velocity, friction energy consumption, and friction coefficient at point K1 in the cutting-in stage, cutting stage, and cutting-out stage is larger than that at point K2, indicating that the friction velocity, friction energy consumption, and friction coefficient at point K2 have weaker chaos, lower complexity of the phase trajectory, and more stable dynamic characteristics than point K1 in the dynamic system.

[0166] S5, the method for calculating the K-entropy of the instantaneous friction characteristic variables on the flank face of the milling cutter;

[0167] Through the phase trajectory of the instantaneous friction characteristic variables, the K-entropy of the friction parameter phase trajectories at different points on the flank face of the milling cutter is obtained by using the K-entropy calculation method.

[0168] K-entropy is a quantitative description of the disorder and uncertainty of the system. In the theory of nonlinear dynamics, K-entropy describes the randomness within its phase trajectory, that is, the internally spontaneous generation of randomness, revealing the randomness of the friction dynamic parameters on the flank face of the milling cutter. The K-entropy in chaos theory is introduced into the study of nonlinear dynamics for characterizing the chaotic characteristics of the phase trajectory.

[0169] For a g-dimensional dynamic system, the friction characteristic variable phase space is divided into many boxes with side length ε, and the system state is observed at time intervals of τ. Define the joint probability that the phase point is in the α1, α2,...α m th box at times τ, 2τ, 3τ,..., mτ as P(α1, α2,...α m ), and the q-order Renyi entropy is defined as

[0170]

[0171] where, when q = 0, K0 represents the topological entropy, and when q = 1, K1 is the Kolmogoro entropy. Usually, the value of K2 is used to estimate K1. For the theoretical dynamic model, K2 can be calculated from the formula describing the system. For the data reconstructed from the phase space, it is difficult to calculate K2 through the above formula, but K2 can be calculated through the correlation integral algorithm:

[0172]

[0173] where, C m (δ) is the correlation integral with embedding dimension m, calculated through Equation (28), and C m+Δm (δ) is the correlation integral when the embedding dimension increases by Δm. Theoretically, when the embedding dimension increases continuously and approaches infinity, the second-order entropy K2 approaches the K-entropy infinitely. In practical applications, when the embedding dimension m is large enough, the second-order entropy K2 gradually reaches a certain saturation value, and this value can be used as an estimate of the K-entropy.

[0174] The entire milling process is divided into stages of cutting-in, cutting-through, and cutting-out. The friction velocity, friction energy consumption, and K-entropy of the friction coefficient for each stage are calculated by Equation (33), as Figure 18 shown.

[0175] From Figure 18 it can be seen that during the process of the milling cutter cutting in to cutting out, the friction velocity, friction energy consumption, and K-entropy of the friction coefficient decrease first, then increase, showing a "bathtub curve" form. This is because the K-entropy is high, the chaos degree is strong, the randomness within the phase trajectory is strong, and the dynamic system is unstable during the cutting-in and cutting-out stages. While the K-entropy is low, the chaos degree is low, the randomness within the phase trajectory is weak, and the dynamic system is relatively stable during the cutting-through stage.

[0176] To compare the dynamic characteristics at different positions on the flank face and observe the K-entropy K of the friction characteristic parameters K1 and K2 2K1 、K 2K1 and the K-entropy K under steady state 2K0 as Figure 19 shown.

[0177] From Figure 19 it can be known that the friction velocity, friction energy consumption, and friction coefficient at point K2 have a generally smaller K-entropy than those at point K1 during the cutting-in stage, cutting-through stage, and cutting-out stage. It can be shown that the friction velocity, friction energy consumption, and friction coefficient at point K1 have a stronger chaos in the dynamic system, a stronger randomness within the phase trajectory, and more unstable dynamic characteristics than those at point K2.

[0178] S6, the correlation dimension of the instantaneous friction characteristic variables of the flank face of the milling cutter and the dynamic evaluation method of K-entropy.

[0179] Through the correlation dimension and K-entropy of the phase trajectory of the instantaneous friction characteristic variables, the dynamic characteristics of the correlation dimension and K-entropy of the instantaneous friction characteristic variables at different positions on the flank face of the milling cutter are discriminated using the Euclidean distance.

[0180] The Euclidean distance of the correlation dimension of the friction characteristic variables at point K1 is:[[]]

[0181]

[0182] where γ is the number of periods selected from cutting-in to cutting-out,

[0183] The Euclidean distance of the correlation dimension of the friction characteristic variables at point K2 is:[[]]

[0184]

[0185] Table 2 is obtained from Equation (34) and Equation (35):

[0186] Table 2 Euclidean distances of the correlation dimensions of the friction characteristic variables at points K1 and K2

[0187] Cut-in stage Cutting stage Cut-out stage Overall stage <![CDATA[K1 friction velocity]]> 4.94 2.91 4.94 7.61 <![CDATA[K2 friction velocity]]> 4.85 2.28 4.24 6.86 <![CDATA[K1 Friction Energy Consumption]]> 8.24 4.05 6.98 11.37 <![CDATA[K2 Frictional Energy Consumption]]> 6.84 3.32 6.50 10.01 <![CDATA[Coefficient of friction K1]]> 8.26 5.53 8.27 12.52 <![CDATA[Coefficient of friction of K2]]> 6.86 3.53 6.99 10.37

[0188] It can be seen from Table 2 that the Euclidean distance from the correlation dimension of the friction characteristic variables at point K1 to the correlation dimension of the steady-state variables is mostly larger than that from the correlation dimension of the friction characteristic variables at point K2 to the correlation dimension of the steady-state variables, indicating that the friction velocity, friction energy consumption, and friction coefficient at point K2 have weak chaos in the dynamic system, low complexity of the phase trajectory, and more stable dynamic characteristics than point K1.

[0189] The Euclidean distance of the K-entropy of the friction characteristic variables at point K1 is as follows:

[0190]

[0191] The Euclidean distance of the K-entropy of the friction characteristic variables at point K2 is as follows:

[0192]

[0193] Table 3 is obtained through Equations (36) and (37):

[0194] Table 3 Euclidean distances of the K-entropy of the friction characteristic variables at points K1 and K2

[0195] Cut-in stage Cutting stage Cut-out stage Overall stage <![CDATA[K1 friction velocity]]> 0.73 0.52 0.7 1.16 <![CDATA[K2 friction velocity]]> 0.69 0.43 0.75 1.14 <![CDATA[K1 Friction Energy Consumption]]> 2.78 0.87 2.66 3.95 <![CDATA[K2 Frictional Energy Consumption]]> 2.71 0.65 2.43 3.71 <![CDATA[Coefficient of friction K1]]> 2.80 0.90 2.89 4.13 <![CDATA[Coefficient of friction K2]]> 2.62 0.79 2.57 3.76

[0196] It can be seen from Table 3 that the Euclidean distance from the K-entropy of the friction characteristic variables at point K1 to the K-entropy of the steady-state variables is mostly larger than that from the K-entropy of the friction characteristic variables at point K2 to the K-entropy of the steady-state variables, indicating that the friction velocity, friction energy consumption, and friction coefficient at point K2 have weak chaos in the dynamic system, low randomness within the phase trajectory, and more stable dynamic characteristics than point K1.

[0197] It should be noted that in the above embodiments, as long as the technical solutions do not conflict, they can be arranged and combined. Those skilled in the art can exhaust all possibilities based on the mathematical knowledge of permutations and combinations. Therefore, the present invention will no longer explain the technical solutions after permutation and combination one by one, but it should be understood that the technical solutions after permutation and combination have been disclosed by the present invention.

[0198] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for identifying the non-linear friction dynamics of the secondary flank face of a shoulder milling cutter under vibration action, characterized in that Including: S1, identification of the contact between the flank face of the milling cutter sub and the machining transition surface under vibration; S2, calculation method for the instantaneous friction characteristic variables of the flank face of the milling cutter sub; S3, phase space reconstruction method for the tribological behavior at the tool-workpiece interface; S4, calculation method for the correlation dimension of the instantaneous friction characteristic variables of the flank face of the milling cutter sub; S5, calculation method for the K-entropy of the instantaneous friction characteristic variables of the flank face of the milling cutter sub; S6, dynamic evaluation method for the correlation dimension and K-entropy of the instantaneous friction characteristic variables of the flank face of the milling cutter sub; The said S1 includes: Using the workpiece structure parameters, cutting parameters, milling cutter structure parameters, cutter tooth structure parameters, cutter tooth errors, and the influence characteristics of milling vibration on the instantaneous cutting behavior of the milling cutter and cutter teeth, constructing the instantaneous pose model of the milling cutter and cutter teeth, and then identifying the contact relationship between the flank face of the milling cutter sub and the machining transition surface; The said S2 includes: Obtaining the instantaneous friction characteristic variables of different points on the flank face of the milling cutter sub, namely the instantaneous friction velocity, instantaneous friction energy consumption, and instantaneous friction coefficient, through the common tangent plane equation of the flank face of the cutter tooth sub and the transition surface, the instantaneous pose of the cutter tooth, the instantaneous relative motion velocity of the points on the flank face of the cutter tooth sub, the instantaneous contact temperature, and the instantaneous contact stress; The said S3 includes: Obtaining the phase space reconstruction parameters by solving the instantaneous friction characteristic variables, delay coordinate components, and embedding dimension, and thus obtaining the phase trajectories of the instantaneous friction characteristic variables, namely the phase trajectories of the instantaneous friction velocity, instantaneous friction energy consumption, and instantaneous friction coefficient; The said S4 includes: Obtaining the correlation dimension of the phase trajectories of the friction characteristic variables of different points on the flank face of the milling cutter sub through the phase trajectories of the instantaneous friction characteristic variables and using the correlation dimension calculation method; The said S5 includes: Obtaining the K-entropy of the phase trajectories of the friction parameters of different points on the flank face of the milling cutter sub through the phase trajectories of the instantaneous friction characteristic variables and using the K-entropy calculation method; The said S6 includes: Using the Euclidean distance to judge the dynamic characteristics of the correlation dimension and K-entropy of the instantaneous friction characteristic variables at different positions on the flank face of the milling cutter sub through the correlation dimension and K-entropy of the phase trajectories of the instantaneous friction characteristic variables.

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