CN110032794B

CN110032794B - Method for constructing and verifying dynamic cutting force model of milling cutter under vibration action

Info

Publication number: CN110032794B
Application number: CN201910284912.9A
Authority: CN
Inventors: 姜彬; 张庭政; 赵培轶; 赵俊峰
Original assignee: Harbin University of Science and Technology
Current assignee: Harbin University of Science and Technology
Priority date: 2019-04-10
Filing date: 2019-04-10
Publication date: 2022-06-24
Anticipated expiration: 2039-04-10
Also published as: CN110032794A

Abstract

The invention discloses a method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration, belongs to the technical field of milling cutters, and aims to solve the problems that the distribution and the change characteristics of instantaneous cutting force of cutter tooth milling infinitesimal cannot be disclosed, the instantaneous cutting force relation between cutter teeth cannot be disclosed, and the dynamic change process of the cutting force cannot be accurately reflected in the conventional research on cutting force modeling. Step a, solving instantaneous cutting behavior of the milling cutter; step b, solving the boundary condition of the single-tooth cutting under the vibration action; step c, establishing a parameter model of the instantaneous cutting layer of the cutter teeth of the milling cutter and calculating; d, establishing an instantaneous cutting force model of the milling cutter teeth and solving; and e, completing the construction and verification of the dynamic cutting force model of the milling cutter. The dynamic cutting force model construction and verification method for the milling cutter under the vibration action can accurately reflect the dynamic change of the cutting force in the cutting process, and carries out triple verification on the dynamic cutting force prediction model.

Description

Method for constructing and verifying dynamic cutting force model of milling cutter under vibration action

Technical Field

The invention relates to a method for constructing and verifying a dynamic cutting force model of a milling cutter, in particular to a method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration, and belongs to the technical field of milling cutters.

Background

Vibration in the cutting process can cause the cutting action to be changed continuously, the parameters of a cutting layer to be changed continuously, and cutting force to change frequently, so that the cutting load in the milling process is changed, and finally cutter abrasion and the quality of a processed surface are reduced. At the same time, vibration interacts with the cutting forces and as cutting progresses, this interaction is increasingly intensified, eventually leading to cutting instability. In order to deeply reveal the interaction relationship between vibration and cutting force, control milling stability, and improve the processing quality and the processing efficiency of a workpiece, a dynamic cutting force model of the milling cutter needs to be constructed. The existing milling force modeling methods mainly comprise three methods, namely an empirical formula model, a theoretical formula model and a mechanical model.

The empirical formula model describes the relationship between the milling force and the milling parameter through a set of milling force coefficients. The method comprises the following steps of milling workpieces made of different materials under different cutting conditions through cutters made of different materials and geometric parameters of the same type to obtain a large amount of cutting force experimental data, and determining undetermined coefficients through curve fitting. The empirical formula model is only suitable for predicting the milling force under the fixed conditions of a cutter material, an angle, a workpiece material and the like, has poor general performance and cannot reflect the dynamic change process of the cutting force.

The theoretical formula model is a cutting force theoretical formula deduced through material mechanics, can reflect the internal relation among all influencing factors of the cutting force, is beneficial to analyzing problems, simplifies a plurality of conditions in the deduction process, has a large difference with the real situation, is complex in calculation, and is not generally used for solving the transient cutting force.

The mechanical model is a cutting thickness calculation model established through geometrical parameters and cutting parameters of the cutter, and the cutting force is regarded as the product of the cutting area and the unit cutting force. The mechanical model of the cutting force can reveal the dynamic change process of the cutting force in the cutting process, the prediction precision is high, and the simulation and the reproduction by utilizing a program are simple, so that the mechanical model is relatively widely applied. The existing mechanical model mainly adopts instantaneous undeformed cutting thickness to calculate cutting force, cannot reveal the distribution and the change characteristics of the instantaneous cutting force of the cutter teeth of the milling cutter, cannot reveal the instantaneous cutting force relationship between the cutter teeth, and cannot accurately reflect the dynamic change process of the cutting force.

Disclosure of Invention

The invention aims to provide a dynamic cutting force model construction and verification method for a milling cutter under the action of vibration, which aims to solve the problem that the influence of cutting force on milling vibration is only considered in the existing research on cutting force modeling.

A method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration comprises the following steps:

step a, solving instantaneous cutting behavior of the milling cutter;

step b, solving the boundary condition of the single-tooth cutting under the vibration action;

c, establishing a parameter model of the instantaneous cutting layer of the cutter teeth of the milling cutter and calculating;

d, establishing an instantaneous cutting force model of the milling cutter teeth and solving;

and e, completing the construction and verification of the dynamic cutting force model of the milling cutter.

Preferably: the method for solving the instantaneous cutting behavior of the milling cutter in the step a comprises the following steps:

step a1, constructing a cutter tooth instantaneous contact relation model under the influence of milling vibration and cutter tooth errors;

step a2, establishing a tool nose cutting motion track model according to the step a1, wherein the model reflects the influence of milling vibration on the tool nose in the titanium alloy milling process to generate displacement increment, so that the actual cutting motion track of the tool nose is changed;

step a3, milling vibration causes the whole milling cutter to deflect, so that the milling cutter forms an attitude included angle increment relative to an initial state, and an instantaneous cutting attitude model of the milling cutter is established according to the step a1, and reflects the attitude change before and after the milling cutter deflects;

and a4, establishing a function equation according to the model relations in the step a3, and solving the instantaneous attitude included angle of the milling cutter.

Preferably: the solving of the single-tooth cutting boundary condition under the vibration action in the step b comprises the following steps:

b1, the spiral edge is influenced by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth errors, instantaneous cutting areas of all points on a cutting edge can be changed continuously. Based on a differential principle, dividing a milling cutter tooth i into k infinitesimal parts from a reference bottom surface, and establishing a milling cutter structure, milling infinitesimal part and cutting edge space position model which can represent cutter tooth errors and the spatial positions of milling infinitesimal parts;

b2, establishing a cutter tooth instantaneous contact angle model which can solve the axial boundary condition of the cutting edge instantaneously participating in cutting.

Preferably, the following components: the step c of establishing a parameter model of the instantaneous cutting layer of the milling cutter teeth and calculating comprises the following steps:

c1, the milling vibration and the cutter tooth error change the posture and the structure of the milling cutter, thereby influencing the parameters of the cutting layer. An instantaneous cutting layer parameter model considering milling vibration and cutter tooth errors is established, and the model can more accurately calculate the instantaneous cutting area of the cutter tooth in the cutting process;

and c2, deducing an instantaneous cutting thickness mathematical model of the cutter tooth i, and calculating the instantaneous cutting area of the milling infinitesimal.

Preferably: the step d of establishing an instantaneous cutting force model of the milling cutter teeth and solving the model comprises the following steps:

d1, constructing a cutting edge instantaneous cutting force system, and revealing dynamic changes of the size, direction and boundary conditions of the cutter tooth instantaneous cutting force distribution under the influence of milling vibration and cutter tooth errors;

in step d2, as the cutting process proceeds, the direction of the infinitesimal cutting force on the cutting edge is also changing. Establishing a force decomposition model in a infinitesimal reference plane and a milling cutter coordinate system, and decomposing infinitesimal cutting force into the milling cutter coordinate system, namely resolving the cutting resultant force borne by the cutter teeth;

d3, decomposing the principal cutting force of the infinitesimal into three directions of a milling cutter coordinate system, decomposing the infinitesimal forces in the three directions of the milling cutter coordinate system into three directions of a workpiece coordinate system through a coordinate change matrix, and finally integrating the infinitesimal forces along the axial direction according to the boundary condition of single-tooth cutting to solve the resultant cutting force on the cutting edge which instantaneously participates in cutting.

Preferably: the construction and verification of the dynamic cutting force model of the milling cutter in the step e comprise the following steps:

step e1, the instantaneous cutting force relationship between the cutter teeth and the cutter teeth is also constantly changing under the influence of milling vibrations and cutter teeth errors. Meanwhile, the number of teeth which instantaneously participate in cutting directly determines the magnitude of instantaneous cutting force of the milling cutter. Establishing a spatial position relation and an instantaneous cutting contact angle model of adjacent cutter teeth, representing the spatial position relation and the instantaneous cutting contact angle of the adjacent cutter teeth, and then giving an instantaneous multi-tooth cutting criterion of the milling cutter;

e2, establishing an instantaneous cutting force system of the milling cutter under the action of vibration in order to reveal the relation of instantaneous cutting force between cutter teeth;

and e3, accumulating and summing the instantaneous cutting force of each cutter tooth which instantaneously participates in cutting, and solving the dynamic cutting force of the whole milling cutter. Constructing a dynamic cutting force mathematical model of the milling cutter;

and e4, calculating the correlation degree of the actually measured dynamic cutting force and the simulated dynamic cutting force by adopting an improved correlation analysis algorithm, namely judging the approximation degree of the dynamic cutting force model for predicting the change characteristic of the actual dynamic cutting force. Constructing an incidence matrix of the actually measured dynamic cutting force and the simulated dynamic cutting force;

step e5, judging the resolving accuracy of the maximum value and the minimum value of the dynamic cutting force model calculation cutting force through the relative error;

and e6, verifying the coincidence degree of the dynamic change characteristic predicted by the dynamic cutting force model and the dynamic cutting force change characteristic measured by experiments along the three directions of the milling cutter feed speed, the cutting width and the cutting depth through correlation analysis. Through the calculation of the relative error of the maximum value and the minimum value of the cutting force, the calculation accuracy of the maximum value and the minimum value of the cutting force of the dynamic cutting force model along the three directions of the feed speed, the cutting width and the cutting depth of the milling cutter can be verified.

Compared with the existing product, the invention has the following effects:

the method establishes a model of the instantaneous contact relation between the milling cutter and the workpiece, solves the milling cutter track and posture, and reveals the instantaneous cutting behavior of the cutter teeth according to the milling cutter track and posture. And (3) constructing a cutter tooth instantaneous cutting force system, and disclosing the distribution and the change characteristics of the cutter tooth instantaneous cutting force of the milling cutter. And constructing an instantaneous cutting force system of the milling cutter, and disclosing the instantaneous cutting force relation between the cutter teeth. Finally, a dynamic cutting force model capable of accurately reflecting the dynamic change of the cutting force in the cutting process is established, and the dynamic cutting force prediction model is subjected to triple verification through comparison of simulation and experimental dynamic cutting force curves, improved grey correlation analysis and calculation of the cutting force maximum value relative error and the cutting force minimum value relative error, so that the reliability of the dynamic cutting force model is ensured;

by establishing an instantaneous contact relation model of the milling cutter and a workpiece, establishing a cutting motion track model of a tool tip and an instantaneous cutting attitude model of the milling cutter under the action of vibration, disclosing an influence mechanism of milling vibration on the cutting motion track and the instantaneous cutting attitude of the milling cutter, and providing a method for quantitatively describing a dynamic forming process of a machined surface; establishing an instantaneous cutting layer parameter model under the influence of vibration through a milling cutter cutting motion track model and a milling cutter instantaneous cutting attitude model, and truly reflecting the dynamic change process of cutting layer parameters in the cutting process; solving the instantaneous cutting force of the cutting edge infinitesimal of the milling cutter relative to the workpiece through an instantaneous cutting layer parameter model, establishing an instantaneous cutting force system on the cutting edge and revealing the relation between the infinitesimal cutting forces on the cutting edge; the method comprises the steps of providing a cutting edge length boundary condition which instantaneously participates in cutting under the influence of vibration, establishing a cutter tooth instantaneous cutting force model, establishing an instantaneous multi-tooth cutting criterion, establishing a milling cutter dynamic cutting force model, revealing the dynamic change characteristic of the cutting force in the machining process, and providing a basis for milling process evaluation and design.

Drawings

FIG. 1 is a flow chart of a method for constructing and verifying a dynamic cutting force model of a milling cutter;

FIG. 2 is a schematic view of a model of the instantaneous cutting contact relationship of a milling cutter;

FIG. 3 is a schematic diagram of the instantaneous cutting motion trail of the tool tip under the action of vibration;

FIG. 4 is a schematic illustration of a vibration acceleration signal;

FIG. 5 is a diagram illustrating the fitting result of vibration acceleration signals at a cutting stroke of 1 m;

FIG. 6 is a diagram showing the fitting result of vibration acceleration signal at the cutting stroke of 6 m;

FIG. 7 is a simulation diagram of the tooth path of the milling cutter under the action of vibration;

fig. 8 is a schematic view of the instantaneous cutting attitude of the milling cutter;

FIG. 9 is a graph of attitude angle versus time;

FIG. 10 is a schematic diagram of milling cutter micro-element division;

FIG. 11 is a schematic view of the spatial position of the cutting edge of the milling cutter;

FIG. 12 is a schematic view of the instantaneous contact angle of the cutter teeth;

FIG. 13 is a schematic view of a simulation of a vibration-free cutting layer forming process;

FIG. 14 is a schematic view showing a simulation of a formation process of a vibration-cut layer;

FIG. 15 is a schematic view of a cutting edge infinitesimal instantaneous cutting thickness model;

FIG. 16 shows a cut thickness model at X_g-O_g-Y_gA projection schematic on a plane;

FIG. 17 is a graph showing simulation results of instantaneous cutting area;

FIG. 18 is a schematic view of the instantaneous cutting force system of the cutting edge;

FIG. 19 is a schematic view of the resolution of forces in a milling infinitesimal reference plane and a milling cutter coordinate system;

FIG. 20 is a schematic view showing the spatial relationship between adjacent teeth;

FIG. 21 is a schematic view of the instantaneous cutting contact angle between adjacent teeth;

FIG. 22 is a schematic view of the instantaneous cutting force system of the milling cutter;

FIG. 23 is a feed speed direction cutting force graph;

FIG. 24 is a cutting force profile in the cutting width direction;

fig. 25 is a cutting force graph in the cutting depth direction.

Detailed Description

Preferred embodiments of the present invention are explained in detail below with reference to the accompanying drawings.

In a first specific embodiment, a method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration is shown in fig. 1, wherein N represents the number of cutter teeth of the milling cutter, the process of milling titanium alloy by the milling cutter is analyzed, and an instantaneous contact relation model of the milling cutter under the action of vibration is established; calculating the actual cutting motion track and attitude of the milling cutter according to the contact relation of the milling titanium alloy cutter worker; constructing a milling cutter structure model, carrying out cutting edge infinitesimal division, representing the spatial position of a cutting edge, and determining the cutting boundary condition of a single tooth; according to the track and the posture, establishing a parameter model of the instantaneous cutting layer of the cutter teeth under the vibration action, and establishing an instantaneous cutting force system of the cutting edge; giving a multi-tooth cutting criterion of the milling cutter under the vibration action, constructing an instantaneous cutting force system of the milling cutter, and finally establishing a dynamic cutting force model of milling titanium alloy, wherein the method comprises the following steps:

step a, solving instantaneous cutting behavior of the milling cutter;

Preferably, the following components: the method for solving the instantaneous cutting behavior of the milling cutter in the step a comprises the following steps:

step a1, constructing a cutter tooth instantaneous contact relation model under the influence of milling vibration and cutter tooth errors; as shown in FIG. 2, in the figure, O_g-X_gY_gZ_gAs a coordinate system of the workpiece, o_d1-a₁b₁c₁、o_d2-a₂b₂c₂、o_d3-a₃b₃c₃Respectively being milling cutter coordinate systems at different moments o_d(t₀)-a(t₀)b(t₀)c(t₀) Is the milling cutter coordinate system at the initial feed position. o_dv1-a_v1b_v1c_v1、o_dv2-a_v2b_v2c_v2、o_dv3-a_v3b_v3c_v3And respectively milling cutter coordinate systems with changed postures caused by milling vibration at different moments. s₀ ⁱ¹-a_i1b_i1c_i1、s₀ ⁱ²-a_i2b_i2c_i2、s₀ ⁱ³-a_i3b_i3c_i3Respectively being milling cutter tooth coordinate systems, x, at different moments_d、y_d、z_dRespectively the position, x, of the milling cutter in the coordinate system of the workpiece_g1、y_g1、z_g1The positions of the milling cutters in the workpiece coordinate system at the initial moment are respectively, and m is the number of cutting strokes of the milling cutters. o_d1(t₀) As initial position of the milling cutter on the first cutting pass, o_d1(t_e) For the end position of the milling cutter on the first cutting stroke, o_dm(t₀) The initial position of the milling cutter on the mth cutting stroke;

step a2, establishing a tool nose cutting motion track model which reflects the influence of milling vibration on a tool nose in the titanium alloy milling process to generate displacement increment, so that the actual cutting motion track of the tool nose is changed; as shown in FIG. 3, point a is the theoretical tool tip position point, point a' is the real position point of the tool tip point under the influence of milling vibration, and A_x(t)、A_y(t)、A_zAnd (t) is the vibration displacement of the tool nose in the x, y and z directions respectively.

An equation of an actual cutting motion track of the tool tip under the milling vibration action is shown as a formula (1).

In the formula, delta₁For milling by vibrationThe included angle of the posture of the milling cutter is a-o_dAngle of projection on the c plane, δ₂The included angle of the milling cutter posture under the milling vibration action is b-o_d-projection angle on the c plane;

step a3, milling vibration causes the whole milling cutter to deflect, so that the milling cutter forms an attitude included angle increment relative to an initial state, and an instantaneous cutting attitude model of the milling cutter is established, wherein the model reflects the attitude change before and after the milling cutter deflects; as shown in fig. 8, point e is a starting point for calculating the overhang of the milling cutter, point l is the overhang of the milling cutter, and point δ is the included angle of the posture of the milling cutter under the action of milling vibration.

Step a4, establishing a function equation according to each model relation in the step a3, solving the milling cutter instantaneous attitude included angle, and calculating the milling cutter instantaneous attitude included angle according to the formula (2) shown in fig. 8.

b1, the spiral edge is influenced by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth errors, instantaneous cutting areas of all points on a cutting edge can be changed continuously. Based on a differential principle, dividing a milling cutter tooth i into k infinitesimal parts from a reference bottom surface, and establishing a milling cutter structure, milling infinitesimal part and cutting edge space position model which can represent cutter tooth errors and the spatial positions of milling infinitesimal parts; as shown in FIGS. 10 and 11, s in the figure_o ⁱ-a_ib_ic_iIs the coordinate system of the milling cutter tooth i. s is_o ⁱFor tooth i point, s_cminMeasuring the reference cutter tooth tip point, i.e. the axially lowest tip point, s, for the axial error of the milling cutter_rmaxMeasuring the reference cutter tooth point, i.e. the point of maximum radius of gyration, for the radial error of milling cutter_cThe distance between the point of the lowest cutter tooth tip of the milling cutter and the end face of the milling cutter, l_iThe distance r from the tip point of the ith cutter tooth to the end face of the milling cutter_0maxAs milling cuttersMaximum radius of gyration, r₀ ⁱIs the radius of gyration, θ, of tooth i_iThe included angle between the ith cutter tooth and the next cutter tooth is shown, and beta is the helical angle of the milling cutter. Δ c₁First infinitesimal length, Δ c, measured from the milling cutter datum plane for tooth i_mIs the infinitesimal length, Δ c, of the point m of the cutting edge of tooth i_kThe length of the infinitesimal at the highest point of the cutting tooth i participating in cutting. c. C_zAxial height, m, of the micro-element division for tooth i_iIs a knife tooth i edge line infinitesimal delta c_mLower boundary point, using the negative a-axis direction as reference, defining theta_i(t) is the instantaneous position angle of the tooth tip point of tooth i, θ_mFor milling cutters having an axial height c above the cutting edge_mThe instantaneous position angle of time. And (4) representing the axial error and the radial error of the cutter teeth of the milling cutter as shown in a formula (3).

Δc_i＝l_c-l_i；Δr_i＝r_0max-r₀ ⁱ (3)

And (3) calculating the instantaneous position angles of the tool nose point of the tool tooth i and any point on the cutting edge, as shown in formulas (4) and (5).

In the formula, t₀At an initial 0 time, θ_i(t₀) Is t₀The position angle of the cutter tooth i at the moment;

b2, establishing a cutter tooth instantaneous contact angle model which can solve the axial boundary condition of a cutting edge instantaneously participating in cutting, as shown in figure 12, wherein theta is_stThe initial cutting angle of the cutter tooth i is the point A of the nicking cutter during initial cutting, and theta_wtThe initial cutting angle of the point of the knife edge of the knife tooth i is B, theta_pThe axial height of the cutting edge of the cutter tooth i is a_pThe axial height of the position angle of the tool nose point reaches a_pThe time point of the tool is C, theta_etThe position angle when the cutter tooth i completely cuts the workpiece is shown, the point of the cutter point when completely cutting is shown as D,

the maximum contact angle of cutter tooth i.

The axial boundary of the cutting edge where the tooth i instantaneously participates in cutting is discussed with reference to fig. 12, as shown in equation (6).

In the formula, c_iThe axial boundary of the cutting edge where the cutter tooth instantaneously participates in cutting.

Preferably: the step c of establishing a parameter model of the instantaneous cutting layer of the milling cutter teeth and calculating comprises the following steps:

c1, the milling vibration and the cutter tooth error change the posture and the structure of the milling cutter, thereby influencing the parameters of the cutting layer. An instantaneous cutting layer parameter model considering milling vibration and cutter tooth errors is established, and the model can more accurately calculate the instantaneous cutting area of the cutter tooth in the cutting process; as shown in fig. 15 and 16, in the figure, o_di-a_ib_ic_iIs t₂Time milling cutter coordinate system o_di-1-a_i-1b_i-1c_i-1Is t₁Time milling cutter coordinate system, delta_iIs t₂Angle of attitude delta of instant milling cutter_i-1Is t₁Angle of attitude theta of milling cutter at any moment_miIs axial height c_mInstantaneous position angle of cutting tooth i, theta_mi-1Is axial height c_mThe instantaneous position angle of the cutter tooth i-1,

is the deflection angle theta of the radius of gyration when the cutter tooth i and the cutter tooth i-1 rotate through the same point_vIs o_mi′o_mi-1' and X_gAngle of positive axis, r₀ ^i-1Is the radius of gyration, o, of the cutter tooth i-1_mi′、θ_mi-1′、m_i′、m_i-1' are each o_mi、θ_mi-1、m_i、m_i-1Projection point in the cutting depth direction of the workpiece coordinate system, r₀ ⁱ′、r₀ ^i-1′、h_D(t)' are each r₀ ⁱ、r₀ ^i-1、h_D(t) a projection length in a cutting depth direction of the workpiece coordinate system;

in step c2, a mathematical model of the instantaneous cutting thickness of tooth i is derived according to fig. 15 and 16, as shown in equation (7).

The instantaneous cutting area of the milling infinitesimal is calculated and shown in the formula (8).

d1, constructing a cutting edge instantaneous cutting force system, and revealing dynamic changes of the size, direction and boundary conditions of the cutter tooth instantaneous cutting force distribution under the influence of milling vibration and cutter tooth errors; as shown in FIG. 18, in the figure, d_Ftm、d_Frm、d_FamIs any point m on the cutter tooth i_iTangential, radial and axial forces of minor elements of the cutting edge relative to the workpiece, h_DmIs m on the cutting edge_iInstantaneous cutting thickness at point, h_Dm+1Is m on the cutting edge_iInstantaneous cut thickness at +1 point;

in step d2, as the cutting process proceeds, the direction of the infinitesimal cutting force on the cutting edge is also changing. Establishing a force decomposition model in a infinitesimal reference plane and a milling cutter coordinate system, and decomposing infinitesimal cutting force into the milling cutter coordinate system, namely resolving the cutting resultant force borne by the cutter teeth; as shown in FIG. 19, in the figure, d_Ftm、d_Fam、d_FrmRespectively is the knife tooth i is m_iTangential force of infinitesimal cutting edge and axial direction of infinitesimal cutting edgeForce, infinitesimal radial force, d_Fami、d_FbmiAre respectively m_iAt the tangential force of the cutting edge infinitesimal along a_vAxial direction and b_vAxial force component. To point m_iMain cutting force d of micro element_FtmThe solution is performed as shown in equation (9).

d_Ftm＝pA_D(t) (9)

Wherein p is the unit cutting force;

step e1, the instantaneous cutting force relationship between the cutter teeth and the cutter teeth is also constantly changing under the influence of milling vibrations and cutter teeth errors. Meanwhile, the number of teeth which instantaneously participate in cutting directly determines the magnitude of instantaneous cutting force of the milling cutter. Establishing a spatial position relation and an instantaneous cutting contact angle model of adjacent cutter teeth, representing the spatial position relation and the instantaneous cutting contact angle of the adjacent cutter teeth, and then giving an instantaneous multi-tooth cutting criterion of the milling cutter; as shown in FIGS. 20 and 21, s in the figure₀ ⁱThe point of the cutter tooth i, s₀ ⁱ ^-1Point of knife point, s, of knife tooth i-1₀ ^i-2Is the point of the knife point, r, of the knife tooth i-2₀ ⁱIs the radius of gyration, r, of tooth i₀ ^i-1Is the radius of gyration of cutter tooth i-1.

Referring to fig. 20 and 21, the milling cutter participating in the cutting tooth number criterion is shown as equation (10).

In the formula, N_tTo participate in cutting the tooth number.

Step e2, in order to reveal the relation of instantaneous cutting force between the cutter teeth, establish the instantaneous cutting force system of the milling cutter under the vibration; as shown in fig. 22, where is δ_tiIs t_iMilling cutter attitude angle under influence of moment milling vibration, F_t1Xi、F_t1Yi、F_t1ZiAre each t₁The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t2Xi、F_t2Yi、F_t2ZiAre each t₂The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t3Xi、F_t3Yi、F_t3ZiAre each t₃The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t1Xi-1、F_t1Yi-1、F_t1Zi-1Are each t₁The cutting force of the cutter tooth i-1 in the feed speed direction, the cutting width direction and the cutting depth direction at the moment F_t2Xi-1、F_t2Yi-1、F_t2Zi-1Are each t₂The cutting force of the cutter tooth i-1 in the feed speed direction, the cutting width direction and the cutting depth direction at the moment F_t3Xi-1、F_t3Yi-1、F_t3Zi-1Are each t₃Cutting force of the cutter tooth i-1 in the feeding speed direction, the cutting width direction and the cutting depth direction at the moment;

and e3, accumulating and summing the instantaneous cutting force of each cutter tooth which instantaneously participates in cutting, and solving the dynamic cutting force of the whole milling cutter. Constructing a dynamic cutting force mathematical model of the milling cutter; as shown in equation (11).

In the formula, F_XCutting force of the milling cutter in the direction of feed speed, F_YCutting force of the milling cutter in the cutting width direction, F_ZCutting force of the milling cutter along the cutting depth direction;

and e4, calculating the correlation degree of the actually measured dynamic cutting force and the simulated dynamic cutting force by adopting an improved correlation analysis algorithm, namely judging the approximation degree of the dynamic cutting force model for predicting the change characteristic of the actual dynamic cutting force. Constructing an incidence matrix of the actually measured dynamic cutting force and the simulated dynamic cutting force; as shown in equation (12).

γ1＝(γ(A_m1X,B_m1X)γ(A_m1Y,B_m1Y)γ(A_m1Z,B_m1Z)) (12)

Wherein gamma 1 is a correlation matrix of the simulated dynamic cutting force and the measured dynamic cutting force, and gamma (A)_m1X，B_m1X)、γ(A_m1Y，B_m1Y)、γ(A_m1Z，B_m1Z) The correlation degrees of the simulated dynamic cutting force and the measured dynamic cutting force of the milling cutter in the three directions of the feeding speed direction, the cutting width direction and the cutting depth direction at the initial cutting-in stage of the milling cutter are respectively.

And giving a criterion of the strength of the correlation degree, as shown in a formula (13).

γ(A_m1,B_m1)≥0.7 (13)

In the formula, A_m1And B_m1Respectively are the behavior sequences of the actual measurement and the simulation of the dynamic cutting force.

When the correlation degree is more than or equal to 0.7, the method belongs to strong correlation, and the approximation degree of the change characteristics of the actual measurement and simulation dynamic cutting force is considered to be high;

step e5, judging the resolving accuracy of the maximum value and the minimum value of the dynamic cutting force model calculation cutting force through the relative error; the relative error can be defined as:

W＝|(F_imitation-F_True)/F_True|×100％ (14)

Wherein W is a relative error, F_ImitationTo simulate cutting forces, F_TrueIs the actual cutting force.

And (3) providing a criterion for resolving the maximum and minimum values of the dynamic cutting force, wherein the criterion is shown as a formula (15).

W≤20％ (15)

When the relative error is less than or equal to 20%, the maximum cutting force and the minimum value resolving precision are considered to meet the requirement;

In the second embodiment, the first embodiment of the invention,

vibration change characteristic of milling titanium alloy

(1) The experiment is carried out on a three-axis milling center (VDL-1000E), a forward milling mode is adopted, the rotating speed n of a main shaft is 1290r/min, and the feeding speed v is_fAt 573 mm/min. The DHDAS5922 dynamic signal test system was used to test for vibration.

The vibration acceleration signal is collected as shown in fig. 4.

In the figure, T₁、T₂、T₃And T₄Respectively idle running, initial cutting-in stage, smooth cutting stage and idle cutting-out stage.

(2) Taking the vibration signal in the cutting speed direction during cutting of the cutter obtained in the experiment as an example, the matlab is used to perform noise reduction on the vibration acceleration signal, and the sine function fitting is performed on the vibration acceleration signal after noise reduction, and the specific fitting result is shown in fig. 5 and 6.

And (3) performing equation fitting on the vibration acceleration signal curve, and performing secondary integration to obtain a tool vibration displacement equation ax (t) as shown in the formula (16).

In the formula, a_x(t) is the equation of the transient acceleration of the tool in the x-direction, a₁～a₈、b_x1～b_x8、c_x1～c_x8The coefficients are respectively, and the values of the coefficients during cutting are shown in table 1:

TABLE 1 fitting Curve equation coefficients at cutting strokes of 1m and 6m

The two stroke coefficients in the comparison table show that under different cutting strokes of the same milling cutter, the coefficients of corresponding parameter items in the vibration displacement equation of the cutter are obviously different, and 16.7 percent of the coefficients show opposite properties. Therefore, in the whole cutting process, the posture of the milling cutter is changed continuously due to the influence of milling vibration, and the posture is changed differently under different strokes. The dynamic change process of the milling cutter vibration in the cutting process can be quantitatively expressed by adopting the formula (16).

Cutting motion track of tool tip under vibration action

And (3) simulating the tool nose track by using matlab to obtain the real cutting motion track of the tool nose points of the five tool teeth of the milling cutter under the milling vibration action, as shown in fig. 7.

As can be seen from fig. 7, the cutting motion trajectory of each cutter tooth exhibits a transient change under the action of the milling vibration.

Instantaneous cutting attitude of milling cutter under vibration action

(1) The milling vibration causes the whole milling cutter to deflect, so that the milling cutter forms an attitude included angle increment relative to the initial state. And establishing an instantaneous cutting attitude model of the milling cutter, wherein the model reflects the attitude change before and after the milling cutter deflects. As shown in fig. 8.

In the figure, point e is the starting point for calculating the overhang of the milling cutter, point l is the overhang of the milling cutter, point delta is the included angle of the posture of the milling cutter under the vibration action, and point delta is₁Is delta at a-o_dAngle of projection on the c plane, δ₂Is delta at b-o_d-projection angle on the c-plane. From fig. 8, the instantaneous attitude angle of the milling cutter and the instantaneous position angle of any point on the cutting edge of the tooth i can be calculated, as shown in equation (17).

From the formula (17), the instantaneous attitude angle of the milling cutter is influenced by the vibration displacement and the overhang amount of the milling cutter.

(2) The change curve of the milling cutter rotating for one circle of attitude included angle along with time under the vibration action is obtained by applying matlab simulation to the milling cutter spatial attitude angle, and is shown in fig. 9.

The model analysis shows that the milling cutter attitude included angle is a space change angle in the milling process, the change of the space included angle can lead the projection angles of the milling cutter attitude included angle in two directions to change, and the dynamic change of the space included angle is directly related to the vibration displacement generated in the milling process.

Boundary condition of single-tooth cutting under vibration

(1) The helical edge is affected by the structure and participates in the cutting point by point. Under the action of milling vibration and cutter tooth errors, the instantaneous cutting area of each point on the cutting edge can be changed continuously. Based on the differential principle, the milling cutter tooth i is divided into k infinitesimal parts from the reference bottom surface, a milling cutter structure, milling infinitesimal part and cutting edge space position model is established, and the model can represent cutter tooth errors and the space positions of the milling infinitesimal parts. As shown in fig. 10 and 11.

In the figure, s_o ⁱ-a_ib_ic_iAs a coordinate system of the milling cutter tooth i, a_iIn the direction of the cutting speed of the cutter tooth i, b_iTo the direction of the circle center of the milling cutter pointing to the tool nose point, c_iThe direction is the axial direction of the milling cutter. s is_o ⁱFor milling cutter teeth i point, s_cminMeasuring the reference cutter tooth tip point, i.e. the axially lowest tip point, s, for the axial error of the milling cutter_rmaxMeasuring the reference cutter tooth point, i.e. the point of maximum radius of gyration, for the radial error of the milling cutter_cThe distance between the point of the lowest cutter tooth tip of the milling cutter and the end face of the milling cutter, l_iThe distance r from the tip point of the ith cutter tooth to the end face of the milling cutter_0maxIs the maximum radius of gyration, r, of the milling cutter₀ ⁱRadius of gyration, theta, of the ith tooth_iThe included angle between the ith cutter tooth and the next cutter tooth is shown, and beta is the helical angle of the milling cutter. Δ c₁First infinitesimal length, Δ c, measured from the milling cutter datum plane for tooth i_mIs a cutting tooth i cutting edge m_iInfinitesimal length at a point, Δ c_kThe length of the infinitesimal at the highest point where the cutter tooth i participates in cutting. c. C_zFor the cutter tooth i advancesAxial height of line infinitesimal division, m_iIs a knife tooth i edge line infinitesimal delta c_mThe coordinate point of the lower boundary is defined by theta with the negative direction of the a axis as a reference_i(t) is the instantaneous position angle of the tooth tip point of tooth i, θ_mFor milling cutters having an axial height c above the cutting edge_mThe instantaneous position angle of time.

The structural parameters of a solid carbide end mill with a diameter of 20mm are shown in table 2.

TABLE 2 Overall cemented carbide end mill construction parameters with a diameter of 20mm

And (4) representing the axial error and the radial error of the milling cutter tooth, as shown in a formula (18).

Δc_i＝l_c-l_i；Δr_i＝r_0max-r₀ ⁱ (18)

The radial and axial errors of the milling cutter were measured by a tool setting gauge, and the measurement results are shown in table 3.

TABLE 3 cutter tooth error

And (3) calculating the instantaneous position angles of the tool nose point of the tool tooth i and any point on the cutting edge, as shown in formulas (19) and (20).

In the formula, t₀At an initial 0 time, θ_i(t₀) Is t₀The position angle of the cutter tooth i at the moment.

(2) And establishing a cutter tooth instantaneous contact angle model which can solve the axial boundary condition of the cutting edge instantaneously participating in cutting. As shown in fig. 12.

In the figure, theta_stIs the initial cutting angle of the cutter tooth i, and the corresponding cutter point is A, theta_wtThe initial cutting angle of the point of the knife tooth i is B, theta corresponding to the point of the knife point_pThe axial height of the cutting edge of the cutter tooth i is a_pThe position angle of the time tool point is C, theta corresponding to the tool point_etThe position angle when the cutter tooth i completely cuts the workpiece is D corresponding to the cutter point,

the maximum contact angle of cutter tooth i.

Referring to fig. 12, the instant axial boundary of the cutting edge where the tooth i participates in cutting is discussed, as shown in equation (21).

In the formula, c_iThe instantaneous cutting edge axial boundary where the tooth participates in cutting.

As can be seen from equation (21), the length of the cutting edge that instantaneously participates in cutting dynamically changes with time. As the cutting process progresses, the edge length participating in cutting undergoes a process of gradually increasing, then keeping the same, and finally gradually decreasing.

Instantaneous cutting layer parameter of milling cutter under vibration action

(1) The cut layer formation process was simulated using matlab as shown in fig. 13 and 14.

As can be seen from fig. 13 and 14, the cutting edges of the milling cutter teeth are offset under the action of vibration, so that the shapes of the cutting layers formed by adjacent teeth are changed, and the instantaneous cutting thickness and cutting width of the teeth are changed.

(2) Milling vibration and cutter tooth errors change the posture and the structure of the milling cutter, thereby influencing parameters of a cutting layer. An instantaneous cutting layer parameter model considering milling vibration and cutter tooth errors is established, and the model can more accurately calculate the instantaneous cutting area of the cutter tooth in the cutting process. As shown in fig. 15 and 16.

In the figure, o_di-a_ib_ic_iIs t₂Time milling cutter coordinate system o_di-1-a_i-1b_i-1c_i-1Is t₁Time milling cutter coordinate system, delta_iIs t₂Angle of inclination delta of milling cutter_i-1Is t₁Angle of attitude theta of milling cutter at any moment_miIs axial height c_mInstantaneous position angle of cutting tooth i, theta_mi-1Is axial height c_mThe instantaneous position angle of the cutter tooth i-1,

is the position angle theta when the cutter tooth i and the cutter tooth i-1 rotate through the same point_vIs o_mi′o_mi-1' and X_gAngle of positive axis, r₀ ^i-1Is the radius of gyration of cutter tooth i-1, A_xIs o_miPoint to o_mi-1 distance of point in the direction of feed speed, A_yIs o_miPoint to o_mi-1The distance of the point in the cutting width direction. o_mi′、θ_mi-1′、m_i′、m_i-1' are each o_mi、θ_mi-1、m_i、m_i-1Projection point in the cutting depth direction of the workpiece coordinate system, r₀ ⁱ′、r₀ ^i-1′、h_D(t)' is each r₀ ⁱ、r₀ ⁱ ^-1、h_D(t) the projection length in the cutting depth direction of the workpiece coordinate system, and the instantaneous cutting thickness of the cutter tooth i are as shown in the formula (22).

The instantaneous cutting area of the milling infinitesimal is calculated as shown in the formula (23).

(3) And (3) simulating an ideal state, only considering errors, only considering milling vibration and the instantaneous cutting area of the errors and the milling vibration in the cutting process by using the matlab. The simulation results are shown in fig. 17.

As can be seen from fig. 17, both static errors and milling vibrations of the milling cutter during the milling process cause the instantaneous cutting thickness of the milling cutter to change, and exhibit a dynamic change.

Instantaneous cutting force of milling cutter teeth

(1) And (3) constructing a cutting edge instantaneous cutting force system, and revealing dynamic changes of the size, direction and boundary conditions of the cutter tooth instantaneous cutting force distribution under the influence of milling vibration and cutter tooth errors. As shown in fig. 18.

In FIG. 18, d_Ftm、d_Frm、d_FamIs any point m on the cutter tooth i_iTangential, radial and axial forces of the cutting edge micro-elements relative to the workpiece.

Each infinitesimal point on the spiral edge has hysteresis when cutting into a workpiece, and the instantaneous cutting thicknesses of the infinitesimals on the cutting edge participating in cutting are different, so that the instantaneous cutting forces of the infinitesimals on the cutting edge are influenced to be different.

(2) Along with the cutting process, the direction of the infinitesimal cutting force on the cutting edge is also changed continuously, a force decomposition model in the infinitesimal reference plane and the milling cutter coordinate system is established, the infinitesimal cutting force is decomposed into the milling cutter coordinate system, and the cutting resultant force borne by the cutter teeth can be solved. As shown in fig. 19.

In the figure, d_Ftm、d_Fam、d_FrmRespectively is an arbitrary point m on the cutter tooth i_iTangential force, axial force, radial force, d_Fami、d_FbmiAre respectively m_iAt the tangential force of the cutting edge infinitesimal along a_vAxial direction and b_vAxial force component. To point m_iMain cutting force d of micro element_FtmThe solution is performed as shown in equation (24).

d_Ftm＝pA_D(t) (24)

Wherein p is a unit cutting force.

Decomposing the principal cutting force of the micro-element into three directions of a milling cutter coordinate system, decomposing the micro-element forces in the three directions of the milling cutter coordinate system into three directions of a workpiece coordinate system through a coordinate change matrix, and finally integrating the micro-element forces along the axial direction according to the boundary conditions of the single-tooth cutting in the implementation example 4 to solve the resultant cutting force on the cutting edge which instantaneously participates in the cutting.

Multi-tooth cutting criterion for milling cutter under vibration action

The instantaneous cutting force relationship between the cutter teeth and the cutter teeth is also constantly changing under the influence of milling vibrations and cutter tooth errors. Meanwhile, the number of teeth which instantaneously participate in cutting directly determines the magnitude of instantaneous cutting force of the milling cutter. And establishing a spatial position relation and an instantaneous cutting contact angle model of adjacent cutter teeth, representing the spatial position relation and the instantaneous cutting contact angle of the adjacent cutter teeth, and giving an instantaneous multi-tooth cutting criterion of the milling cutter. As shown in fig. 20 and 21.

In the figure, s₀ ⁱThe point of the cutter tooth i, s₀ ^i-1Is the point of the knife tip, s, of the knife tooth i-1₀ ^i-2Is the point of the knife point, r, of the knife tooth i-2₀ ⁱIs the radius of gyration, r, of tooth i₀ ^i-1Is the radius of gyration of cutter tooth i-1.

As can be seen from fig. 20 and 21, the instantaneous cutting-participating tooth number is related to the initial entry angle, the complete cutting angle and the included angle between the cutter teeth, and the criterion of the cutting-participating tooth number of the milling cutter is given as shown in equation (25).

In the formula, N_tTo participate in cutting the tooth number.

The instantaneous cutting tooth number of the milling cutter is 2 by solving the milling parameter in the embodiment 1 through the formula (10).

Method for constructing and verifying milling cutter transient cutting force model under vibration action

(1) To reveal the relationship of the instantaneous cutting force between the cutter teeth, the instantaneous cutting force system of the milling cutter under the action of vibration is established, as shown in fig. 22.

In the figure, δ_tiIs t_iMilling cutter attitude angle under influence of moment milling vibration, F_t1Xi、F_t1Yi、F_t1ZiAre each t₁The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t2Xi、F_t2Yi、F_t2ZiAre each t₂The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction at the moment F_t3Xi、F_t3Yi、F_t3ZiAre each t₃The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t1Xi、F_t1Yi、F_t1ZiAre each t₁The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t2Xi、F_t2Yi、F_t2ZiAre each t₂The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction, F_t3Xi、F_t3Yi、F_t3ZiAre each t₃The cutting force of the cutter tooth i in the feed speed direction, the cutting width direction and the cutting depth direction is timed.

As can be seen from fig. 22, during the milling process, as the posture of the milling cutter is changed, the relationship between the cutting forces of the cutter teeth that are instantaneously involved in the cutting is also changed.

(2) And accumulating and summing the instantaneous cutting force of each cutter tooth instantaneously participating in cutting to solve the dynamic cutting force of the whole milling cutter. And (3) constructing a dynamic cutting force mathematical model of the milling cutter, as shown in a formula (26).

In the formula, F_XCutting force of the milling cutter in the direction of feed speed, F_YCutting force of the milling cutter in the cutting width direction, F_ZThe cutting force of the milling cutter in the cutting depth direction.

(3) And (3) carrying out noise reduction treatment on the cutting force obtained in the experiment by using matlab, and simulating the milling force in three directions of a workpiece coordinate system. The obtained simulation results are shown in fig. 23 to 25.

In the figure, t_tFor the initial cutting-in time, t, of the milling cutter teeth_eFor the final cutting-out moment of the teeth of the milling cutter, T is the period of one turn of the milling cutter, JT_iThe milling cutter is rotated through a cycle of a single tooth.

As can be seen from fig. 23 to 25, the simulated curve is matched with the variation trend of the actual curve, but the waveform has a deviation, and the analysis reason is that the vibration of the workpiece is measured and has a deviation from the vibration of the milling cutter at the instantaneous cutting position, so that the deviation of the simulated curve from the actual curve waveform is caused.

(4) And extracting data of inflection points of the actually measured dynamic cutting force and the simulated dynamic cutting force in three directions of the feeding speed direction, the cutting width direction and the cutting depth direction. And constructing a simulation and actual measurement dynamic cutting force behavior sequence. And taking the actually measured dynamic cutting force behavior sequence as a reference sequence, and taking the simulated dynamic cutting force behavior sequence as a comparison sequence. And calculating by adopting an improved grey correlation analysis algorithm, and establishing a correlation matrix of the actually measured dynamic cutting force and the simulated dynamic cutting force, as shown in a formula (27).

Wherein gamma 1 is a correlation matrix of simulated dynamic cutting force and measured dynamic cutting force, and gamma (A)_m1X，B_m1X)、γ(A_m1Y，B_m1Y)、γ(A_m1Z，B_m1Z) Respectively simulating the relevance of dynamic cutting force and actual dynamic cutting force of the milling cutter along the feed speed direction, the cutting width direction and the cutting depth direction at the initial cutting stage of the milling cutter into the workpiece, wherein gamma (A)_m2X，B_m2X)、γ(A_m2Y，B_m2Y)、γ(A_m2Z，B_m2Z) The relevance of simulated dynamic cutting force and actually measured dynamic cutting force of the milling cutter along the feed speed direction, the cutting width direction and the cutting depth direction at the middle stage of cutting a workpiece by the milling cutter is gamma (A)_m3X，B_m3X)、γ(A_m3Y，B_m3Y)、γ(A_m3Z，B_m3Z) And respectively simulating the relevance between the dynamic cutting force and the actually measured dynamic cutting force of the milling cutter along the feeding speed direction, the cutting width direction and the cutting depth direction at the stage of cutting the workpiece by the milling cutter.

And giving a correlation strength criterion, as shown in a formula (28).

γ(A_m1,B_m1)≥0.7 (28)

When the degree of correlation is 0.7 or more, the degree of approximation of the change characteristics of the actual measurement and simulated dynamic cutting forces is considered to be high, because the degree of correlation is strong.

And observing the calculation result of the correlation degree, wherein the correlation degree of the simulated cutting force and the measured cutting force of the milling cutter in the feeding speed direction and the cutting width direction at the initial stage of cutting into the workpiece is smaller than that of the other two stages, the correlation degree in the cutting width direction is the minimum, the correlation degree in the feeding speed direction is the second order, but the correlation degrees are all between 0.7 and 0.9, and the correlation degree belongs to strong correlation. The reason why the above phenomenon occurs is analyzed that the initial cutting position of the cutter tooth is deviated, and a cutter back-off occurs due to a sudden increase in load when the cutter tooth initially cuts into the workpiece. The intermediate stage and the cut-out stage do not have the above two problems, and therefore the degree of association is high. The reason why the correlation between the simulated dynamic cutting force in the cutting direction and the actually measured dynamic cutting force is high in the initial cutting stage is that the cutting position deviation in the cutting direction is not large.

(5) The calculation accuracy of the dynamic cutting force model is judged by the relative error. The relative error can be defined as:

W＝|(F_imitation-F_True)/F_True|×100％ (29)

The relative error between the simulation of the three stages of the cutting process and the maximum value and the minimum value of the actual dynamic cutting force is calculated by the equation (29), and the calculation result of the relative error is shown in table 4.

TABLE 4 simulation and actual cutting force relative error

In the table, W_maxIs the relative error between the measured maximum value and the simulated maximum value of the dynamic cutting force, W_minThe relative error between the measured minimum value and the simulated minimum value of the dynamic cutting force is obtained.

And (3) providing a criterion for resolving the maximum and minimum values of the dynamic cutting force, wherein the criterion is shown as a formula (30).

W≤20％ (30)

When the relative error is less than or equal to 20%, the maximum cutting force and the minimum value resolving accuracy are considered to meet the requirement.

As can be seen from table 4, the relative error in the feed speed direction and the cutting width direction is larger in the initial cutting stage, and the relative error in the cutting width direction is larger than the relative error in the feed speed direction, but the differences are not large, and the relative errors in the cutting depth direction are both smaller than 5.4% and smaller than those in the other two directions. The above phenomenon is also caused by the deviation of the initial cutting position of the cutter teeth and the occurrence of a cutter back-off due to a sudden increase in load at the time of initial cutting into the workpiece. These two causes cause a large relative error in the direction of the feed speed at the initial cutting-in stage, but since the positional deviation in the cutting depth direction during cutting is small relative to the positional deviations in the other two directions, the relative error in the cutting depth direction is significantly smaller than the relative errors in the other two directions.

The present embodiment verifies the degree of coincidence between the dynamic change characteristic predicted by the dynamic cutting force model and the dynamic cutting force change characteristic measured by an experiment through correlation analysis in three directions of the milling cutter feed speed, the cutting width, and the cutting depth. 78% of the calculation results of the correlation degree are greater than 0.9, which shows that the change characteristic of the dynamic cutting force curve predicted by the instantaneous cutting force model constructed by the invention is very similar to the change characteristic of the dynamic cutting force curve obtained by experiments. And the relative error between the simulation of the dynamic cutting force and the actual cutting force is less than 16.3 percent through the calculation result of the maximum value relative error and the minimum value relative error of the cutting force. The calculation accuracy of the maximum value and the minimum value of the cutting force of the dynamic cutting force model along the three directions of the milling cutter feeding speed, the cutting width and the cutting depth is verified to be better. By adopting the two verification methods, the dynamic cutting force model can correctly reflect the change characteristics and the level of the dynamic cutting force in the actual machining process

During milling, there is an interaction between the cutting forces and the vibrations, i.e. the cutting forces cause a change in the vibrations. Meanwhile, the vibration can react to the cutting force, and the contact relation of the tool bit and the parameters of the cutting layer are changed by influencing the cutting motion track of the tool bit and the cutting posture of the tool bit, so that the cutting force is changed finally. Therefore, in order to completely reveal the interaction relationship between the vibration and the cutting force in the cutting process, the influence mechanism of the cutting force on the vibration and the influence mechanism of the vibration on the cutting force need to be simultaneously discovered.

The existing empirical formula model is required to depend on a large number of milling tests, is only suitable for predicting the milling force under the fixed conditions of cutter materials, angles, workpiece materials and the like, has poor general performance and cannot reflect the dynamic change process of the cutting force.

The existing theoretical formula model simplifies a plurality of conditions in the derivation process, has a large difference from the real situation, is complex in calculation and is not generally used for solving the transient cutting force.

The existing mechanical model does not establish a cutting force system on the cutting edge and cannot reflect the relation between infinitesimal cutting forces on the cutting edge. And the average cutting layer area is adopted when the instantaneous cutting force is solved, so that the stress condition on the cutting edge cannot be truly reflected. And the relation of the force between the adjacent cutter teeth is not given, so that the cutting force generated by the milling cutter instantly cannot be predicted more accurately.

Aiming at the problem that the milling vibration has serious influence on the cutting stability and the quality of a processed surface in the milling process, the invention researches the influence mechanism of the vibration on the cutting force on the basis of the existing influence mechanism of the cutting force on the vibration. An instantaneous cutting force system of the cutter teeth is constructed, the dynamic change characteristic of infinitesimal cutting force distribution on the cutting edge is disclosed, the boundary condition of the cutting edge which instantaneously participates in cutting is solved, the instantaneous cutting force system of the milling cutter is established, and the change relation of the instantaneous cutting force system between the cutter teeth is disclosed. And constructing a dynamic cutting force model of the milling cutter, predicting the dynamic cutting force in the cutting process, and providing an effective basis for realizing the control of milling stability.

This embodiment is only illustrative of the patent and does not limit the scope of protection thereof, and those skilled in the art can make modifications to its part without departing from the spirit of the patent.

Claims

1. A method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration is characterized by comprising the following steps:

step a, solving instantaneous cutting behavior of the milling cutter;

the method for solving the instantaneous cutting behavior of the milling cutter in the step a comprises the following steps:

step a2, establishing a tool nose cutting motion track model according to the step a1, wherein the model reflects the influence of milling vibration on a tool nose in the titanium alloy milling process to generate displacement increment, so that the actual cutting motion track of the tool nose is changed;

step a4, establishing a function equation according to the model relations in the step a3, and solving the instantaneous attitude included angle of the milling cutter;

the solving of the single-tooth cutting boundary condition under the vibration action in the step b comprises the following steps:

b1, the spiral edge is influenced by the structure, participates in cutting point by point, and cutting is carried out under the action of milling vibration and cutter tooth errorThe instantaneous cutting area of each point on the edge can be changed continuously, and the milling cutter teeth are formed on the basis of the differential principleiDivision from the base of referencekEstablishing a milling cutter structure, milling infinitesimal division and cutting edge space position model, wherein the model represents the cutter tooth error and the spatial position of the milling infinitesimal;

b2, establishing a cutter tooth instantaneous contact angle model, and solving the axial boundary condition of the cutting edge which instantaneously participates in cutting;

the step c of establishing a parameter model of the instantaneous cutting layer of the milling cutter teeth and calculating comprises the following steps:

c1, the milling vibration and the cutter tooth error change the posture and the structure of the milling cutter, so that the milling cutter has influence on the parameters of the cutting layer, an instantaneous cutting layer parameter model considering the milling vibration and the cutter tooth error is established, and the model can more accurately calculate the instantaneous cutting area of the cutter tooth in the cutting process;

step c2, deriving cutter teethiCalculating the instantaneous cutting area of the milling infinitesimal by using the instantaneous cutting thickness mathematical model;

the step d of establishing an instantaneous cutting force model of the milling cutter teeth and solving the model comprises the following steps:

d2, along with the cutting process, the direction of the infinitesimal cutting force on the cutting edge is continuously changed, a force decomposition model in the infinitesimal reference plane and the milling cutter coordinate system is established, the infinitesimal cutting force is decomposed into the milling cutter coordinate system, namely, the cutting resultant force borne by the cutter teeth is solved;

d3, decomposing the principal cutting force of the infinitesimal into three directions of a milling cutter coordinate system, decomposing the infinitesimal forces in the three directions of the milling cutter coordinate system into three directions of a workpiece coordinate system through a coordinate change matrix, and finally integrating the infinitesimal forces along the axial direction according to the boundary condition of single-tooth cutting to solve the resultant cutting force on the cutting edge which instantaneously participates in cutting;

e, completing the construction and verification of the dynamic cutting force model of the milling cutter;

the step e of completing the construction and verification of the dynamic cutting force model of the milling cutter comprises the following steps:

step e1, under the action of milling vibration and cutter tooth errors, the instantaneous cutting force relationship between cutter teeth and cutter teeth is constantly changed, meanwhile, the number of teeth instantaneously participating in cutting directly determines the magnitude of the instantaneous cutting force of the milling cutter, a space position relationship and instantaneous cutting contact angle model of adjacent cutter teeth are established, the space position relationship and instantaneous cutting contact angle of the adjacent cutter teeth are represented, and the instantaneous multi-tooth cutting criterion of the milling cutter is given;

step e3, accumulating and summing the instantaneous cutting force of each cutter tooth which instantaneously participates in cutting, namely solving the dynamic cutting force of the whole milling cutter, and constructing a mathematical model of the dynamic cutting force of the milling cutter;

step e4, calculating the correlation degree of the actual measurement dynamic cutting force and the simulation dynamic cutting force by adopting an improved correlation analysis algorithm, namely judging the approximation degree of the dynamic cutting force model for predicting the change characteristic of the actual dynamic cutting force, and constructing a correlation matrix of the actual measurement dynamic cutting force and the simulation dynamic cutting force;

and e6, verifying the matching degree of the dynamic change characteristic predicted by the dynamic cutting force model and the dynamic cutting force change characteristic measured by the experiment in the three directions of the milling cutter feeding speed, the cutting width and the cutting depth through correlation analysis, and verifying the resolving accuracy of the maximum value and the minimum value of the cutting force of the dynamic cutting force model in the three directions of the milling cutter feeding speed, the cutting width and the cutting depth through the calculation of the relative error of the maximum value and the minimum value of the cutting force.