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

Abstract

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. In order to solve the existing research on cutting force modeling, the present invention cannot reveal the distribution of the instantaneous cutting force and the Its changing characteristics cannot reveal the instantaneous cutting force relationship between the cutter teeth and the cutter teeth, and cannot accurately reflect the dynamic change process of the cutting force. Step a, solve the instantaneous cutting behavior of the milling cutter; step b, solve the boundary condition of single-tooth cutting under the action of vibration; step c, establish and calculate the instantaneous cutting layer parameter model of the milling cutter tooth; step d, establish the milling cutter tooth Instantaneous cutting force model and solution; step e, complete the construction and verification of the milling cutter dynamic cutting force model. The method for constructing and verifying the dynamic cutting force model of a milling cutter under the action of vibration of the present invention can accurately reflect the dynamic cutting force model of the dynamic change of the cutting force in the cutting process, and perform triple verification on the dynamic cutting force prediction model.

Description

A method for constructing and verifying dynamic cutting force model of milling cutter under the action of vibration

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 technique

Vibration during the cutting process will lead to continuous changes in cutting behavior, resulting in constant changes in cutting layer parameters, resulting in frequent changes in cutting force, resulting in changes in cutting load during milling, and ultimately resulting in tool wear and reduced surface quality. At the same time, the vibration interacts with the cutting force, and as the cutting progresses, this interaction continues to strengthen, eventually leading to cutting instability. In order to deeply reveal the interaction between vibration and cutting force, control the stability of milling, and improve the machining quality and efficiency of workpiece, it is necessary to build a dynamic cutting force model of milling cutter. There are three main methods for modeling milling force, which are empirical formula model, theoretical formula model and mechanical model.

The empirical formula model describes the relationship between milling force and milling parameters through a set of milling force coefficients. Through the same type of tools with different materials and geometric parameters, the workpieces of different materials are milled under different cutting conditions, and a large number of experimental data of cutting force are obtained, and the undetermined coefficients are determined by curve fitting. The empirical formula model must rely on a large number of milling experiments, and is only suitable for the prediction of milling force under fixed conditions such as tool material, angle, and workpiece material. It has poor general performance and cannot reflect the dynamic change process of cutting force.

The theoretical formula model is a theoretical formula of cutting force derived from material mechanics, which can reflect the internal relationship between various influencing factors of cutting force and is beneficial to analyze problems, but it simplifies many conditions in the derivation process, which is different from the real situation. It is large and complicated to calculate, so it is generally not used to solve the transient cutting force.

The mechanical model establishes a cutting thickness calculation model through the tool geometric parameters and cutting parameters, and regards the cutting force 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, with high prediction accuracy and simple reproduction by program simulation, so the mechanical model is relatively widely used. The existing mechanical model mainly uses the instantaneous undeformed cutting thickness to calculate the cutting force, which cannot reveal the distribution and variation characteristics of the instantaneous cutting force of the cutter teeth of the milling cutter, and cannot reveal the instantaneous cutting force relationship between the cutter teeth and the cutter teeth. The dynamic change process of cutting force cannot be accurately reflected.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration, so as to solve the problem that the existing research on cutting force modeling only considers the influence of cutting force on milling vibration.

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

Step a, solve the instantaneous cutting behavior of the milling cutter;

Step b, solve the boundary condition of single-tooth cutting under the action of vibration;

Step c, establish and calculate the instantaneous cutting layer parameter model of milling cutter teeth;

Step d, establish and solve the instantaneous cutting force model of milling cutter teeth;

In step e, the construction and verification of the dynamic cutting force model of the milling cutter is completed.

Preferably: the solution to the instantaneous cutting behavior of the milling cutter in step a includes the following steps:

Step a1, constructing a model of the instantaneous contact relationship between the cutter and teeth under the influence of milling vibration and cutter-tooth error;

Step a2, according to step a1, establish a tool tip cutting motion trajectory model, which reflects that the tool tip is affected by the milling vibration in the process of milling titanium alloys, resulting in a displacement increment, so that the actual cutting motion trajectory of the tool tip changes state;

In step a3, the milling vibration causes the milling cutter to deflect as a whole, so that the milling cutter forms an attitude angle increment relative to the initial state. According to step a1, the instantaneous cutting attitude model of the milling cutter is established, and the model reflects the attitude change of the milling cutter before and after the deflection. ;

In step a4, a function equation is established according to the relationship of each model in step a3, and the angle of the instantaneous attitude of the milling cutter is solved.

Preferably: in step b, the solution of the boundary condition of single-tooth cutting under the action of vibration includes the following steps:

b1, the helical edge is affected by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth error, the instantaneous cutting area of each point on the cutting edge will change continuously. Based on the differential principle, the cutter tooth i of the milling cutter is divided into k micro-elements from the base bottom surface, and the milling cutter structure, milling micro-element division and cutting edge spatial position model are established, which can characterize the cutter tooth error and the space of milling micro-elements. Location;

b2, establish a model of the instantaneous contact angle of the cutter teeth, which can solve the axial boundary condition of the cutting edge that instantaneously participates in cutting.

Preferably: in step c, establishing and calculating the instantaneous cutting layer parameter model of the milling cutter tooth includes the following steps:

c1, milling vibration and cutter tooth error change the attitude and structure of the milling cutter, thus affecting the parameters of the cutting layer. A parameter model of instantaneous cutting layer considering milling vibration and cutter tooth error is established, which can more accurately calculate the instantaneous cutting area of cutter teeth during the cutting process;

In step c2, the mathematical model of the instantaneous cutting thickness of the cutter tooth i is derived, and the instantaneous cutting area of the milling element is calculated.

Preferably: in step d, establishing and solving the instantaneous cutting force model of milling cutter teeth includes the following steps:

In step d1, the instantaneous cutting force system of the cutting edge is constructed, which can reveal the dynamic changes of the magnitude, direction and boundary conditions of the instantaneous cutting force distribution of the cutter teeth under the influence of milling vibration and cutter tooth error;

In step d2, as the cutting process proceeds, the direction of the micro-element cutting force on the cutting edge is also constantly changing. The decomposition model of the force in the micro-element reference plane and the milling cutter coordinate system is established, and the micro-element cutting force is decomposed into the milling cutter coordinate system, and the resultant cutting force on the cutter teeth can be calculated;

Step d3, decompose the main cutting force of the micro-element into the three directions of the milling cutter coordinate system, and then decompose the micro-element force in the three directions of the milling cutter coordinate system into the three directions of the workpiece coordinate system through the coordinate change matrix. The boundary condition of single-tooth cutting is to integrate the micro-element force along the axial direction to solve the resultant cutting force on the cutting edge that participates in cutting instantaneously.

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

In step e1, under the action of milling vibration and cutter tooth error, the instantaneous cutting force relationship between cutter teeth and cutter teeth is also constantly changing. At the same time, the instantaneous number of teeth involved in cutting directly determines the instantaneous cutting force of the milling cutter. The spatial position relationship and instantaneous cutting contact angle model of adjacent cutter teeth is established to characterize the spatial position relationship and instantaneous cutting contact angle of adjacent cutter teeth, and then the instantaneous multi-tooth cutting criterion of milling cutter can be given;

Step e2, in order to reveal the relationship of the instantaneous cutting force between the cutter teeth and the cutter teeth, establish the instantaneous cutting force system of the milling cutter under the action of vibration;

In step e3, the instantaneous cutting force of each tooth that instantaneously participates in cutting is accumulated and summed, and the dynamic cutting force of the entire milling cutter can be calculated. Build a mathematical model of the dynamic cutting force of the milling cutter;

In step e4, the improved correlation analysis algorithm is used to calculate the correlation degree between the measured dynamic cutting force and the simulated dynamic cutting force, so as to judge the approximate degree of the dynamic cutting force model predicting the variation characteristics of the actual dynamic cutting force. Construct the correlation matrix between the measured dynamic cutting force and the simulated dynamic cutting force;

In step e5, the relative error is used to judge the calculation accuracy of the dynamic cutting force model for calculating the maximum value and the minimum value of the cutting force;

Step e6, through the correlation analysis, the degree of agreement between the dynamic change characteristics predicted by the dynamic cutting force model and the dynamic cutting force change characteristics measured by the experiment can be verified along the three directions of the milling cutter feed rate, cutting width and cutting depth. Through the calculation of the relative error of the maximum and minimum cutting force, the calculation accuracy of the maximum and minimum cutting force along the three directions of milling cutter feed rate, cutting width and cutting depth can be verified by the dynamic cutting force model.

Compared with existing products, the present invention has the following effects:

The invention establishes an instantaneous contact relationship model between the milling cutter and the workpiece, calculates the trajectory and posture of the milling cutter, and reveals the instantaneous cutting behavior of the cutter teeth according to the trajectory and posture of the milling cutter. The instantaneous cutting force system of the cutter teeth is constructed, and the distribution and variation characteristics of the instantaneous cutting force of the milling cutter teeth are revealed. The instantaneous cutting force system of the milling cutter is constructed, and the instantaneous cutting force relationship between the cutter teeth is revealed. Finally, a dynamic cutting force model that can accurately reflect the dynamic changes of cutting force in the cutting process is established, and through the comparison of simulation and experimental dynamic cutting force curves, improved grey correlation analysis, and the relative error of the maximum cutting force and the relative error of the minimum cutting force The calculation of the dynamic cutting force prediction model is triple-verified, which ensures the reliability of the dynamic cutting force model;

By establishing the instantaneous contact relationship model between the milling cutter and the workpiece, establishing the cutting motion trajectory model of the tool nose and the instantaneous cutting attitude model of the milling cutter under the action of vibration, the influence mechanism of the milling vibration on the cutting motion trajectory and the instantaneous cutting attitude of the milling cutter is revealed. A method is provided to describe the dynamic formation process of the machined surface; the instantaneous cutting layer parameter model under the influence of vibration is established through the cutting motion trajectory model of the milling cutter and the instantaneous cutting attitude model of the milling cutter, which more realistically reflects the dynamics of the cutting layer parameters in the cutting process. Change process; through the instantaneous cutting layer parameter model, the instantaneous micro-element cutting force of the milling cutter cutting edge relative to the workpiece is solved, the instantaneous cutting force system on the cutting edge is established, and the relationship between the micro-element cutting forces on the cutting edge is revealed; The boundary condition of the cutting edge length that instantaneously participates in cutting under the influence of vibration is given, the instantaneous cutting force model of cutter teeth is established, the instantaneous multi-tooth cutting criterion is established, the dynamic cutting force model of milling cutter is established, and the dynamic change of cutting force in the machining process is revealed. The characteristics provide a basis for the evaluation and design of the milling process.

Description of drawings

Figure 1 is a flow chart of the construction and verification method of the dynamic cutting force model of the milling cutter;

Fig. 2 is the schematic diagram of the instantaneous cutting contact relationship model of milling cutter;

Figure 3 is a schematic diagram of the instantaneous cutting motion trajectory of the tool tip under the action of vibration;

Fig. 4 is the schematic diagram of vibration acceleration signal;

Figure 5 is a schematic diagram of the fitting result of the vibration acceleration signal when the cutting stroke is 1m;

Figure 6 is a schematic diagram of the fitting result of the vibration acceleration signal when the cutting stroke is 6m;

Fig. 7 is the simulation diagram of milling cutter tooth trajectory under the action of vibration;

8 is a schematic diagram of the instantaneous cutting attitude of a milling cutter;

Fig. 9 is the change curve diagram of attitude angle with time;

Figure 10 is a schematic diagram of the micro-element division of a milling cutter;

Figure 11 is a schematic diagram of the spatial position of the cutting edge of a milling cutter;

Figure 12 is a schematic diagram of the instantaneous contact angle of the cutter teeth;

Fig. 13 is a simulation schematic diagram of a vibration-free cutting layer formation process;

Fig. 14 is the simulation schematic diagram of the vibration cutting layer formation process;

Figure 15 is a schematic diagram of the instantaneous cutting thickness model of the cutting edge micro-element;

Figure 16 is a schematic diagram of the projection of the cutting thickness model on the _Xg - _Og - _Yg plane;

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

Figure 18 is a schematic diagram of the instantaneous cutting force system of the cutting edge;

Fig. 19 is the exploded schematic diagram of the force in the milling micro-element reference plane and the milling cutter coordinate system;

Figure 20 is a schematic diagram of the spatial positional relationship between adjacent cutter teeth;

Figure 21 is a schematic diagram of the instantaneous cutting contact angle between adjacent cutter teeth;

Figure 22 is a schematic diagram of the instantaneous cutting force system of a milling cutter;

Fig. 23 is a graph of cutting force in the direction of feed rate;

Figure 24 is a graph of cutting force in the cutting width direction;

Fig. 25 is a graph of cutting force in the depth of cut direction.

Detailed ways

Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

Embodiment 1, a method for constructing and verifying a dynamic cutting force model of a milling cutter under the action of vibration, as shown in Figure 1, where N represents the number of teeth of the milling cutter, analyze the process of milling titanium alloy by the end mill, and establish the vibration The instantaneous contact relationship model of the milling cutter under the action; according to the contact relationship between the cutter and the milling cutter, the actual cutting motion trajectory and attitude of the milling cutter are calculated; the structure model of the milling cutter is constructed, the cutting edge is divided into micro-elements, the spatial position of the cutting edge is characterized, and the single tooth is determined. Cutting boundary conditions; according to the trajectory and attitude, establish the instantaneous cutting layer parameter model of the cutter teeth under the action of vibration, and construct the instantaneous cutting force system of the cutting edge; give the multi-tooth cutting criterion of the milling cutter under the action of vibration, and construct the instantaneous cutting force system of the milling cutter, Finally, the dynamic cutting force model of milling titanium alloy is established, including the following steps:

Step a, solve the instantaneous cutting behavior of the milling cutter;

Step b, solve the boundary condition of single-tooth cutting under the action of vibration;

Step c, establish and calculate the instantaneous cutting layer parameter model of milling cutter teeth;

Step d, establish and solve the instantaneous cutting force model of milling cutter teeth;

In step e, the construction and verification of the dynamic cutting force model of the milling cutter is completed.

Preferably: the solution to the instantaneous cutting behavior of the milling cutter in step a includes the following steps:

Step a1, construct the instantaneous contact relationship model of cutter and tooth under the influence of milling vibration and cutter tooth error; as shown in Figure 2, in the figure, O _g -X _g Y _g Z _g is the workpiece coordinate system, o _d1 -a ₁ b ₁ c ₁ , o _d2 -a ₂ b ₂ c ₂ , o _d3 -a ₃ b ₃ c ₃ are the milling cutter coordinate systems at different times, respectively, o _d (t ₀ )-a(t ₀ )b(t ₀ )c( t ₀ ) is the coordinate system of the milling cutter at the initial feed position. o _dv1 -a _v1 b _v1 c _v1 , o _dv2 -a _v2 b _v2 c _v2 , and o _dv3 -a _v3 b _v3 c _v3 are the coordinate systems of the milling cutter after the attitude changes caused by the milling vibration at different times. s ₀ ⁱ¹ -a _i1 b _i1 c _i1 , s ₀ ⁱ² -a _i2 b _i2 c _i2 , s ₀ ⁱ³ -a _i3 b _i3 c _i3 are the milling cutter tooth coordinate systems at different times, x _d , y _d , z _d is the position of the milling cutter in the workpiece coordinate system, x _g1 , y _g1 , and z _g1 are the position of the milling cutter in the workpiece coordinate system at the initial moment, respectively, and m is the number of cutting strokes of the milling cutter. o _d1 (t ₀ ) is the initial position of the milling cutter on the first cutting stroke, o _d1 (t _e ) is the end position of the milling cutter on the first cutting stroke, and o _dm (t ₀ ) is the mth cutting stroke The initial position of the upper milling cutter;

Step a2, establish the tool nose cutting motion trajectory model, which reflects that the tool nose is affected by the milling vibration in the process of milling titanium alloy, resulting in a displacement increment, so that the actual cutting motion trajectory of the tool nose changes state; as shown in Figure 3 , in the figure, point a is the theoretical tool nose position point, a' is the real position point of the tool nose point under the influence of milling vibration, A _x (t), A _y (t), A _z (t) are the tool nose at Vibration displacement in three directions: x, y, and z.

The actual cutting motion trajectory equation of the tool tip under the action of milling vibration is shown in formula (1).

In the formula, δ ₁ is the projection angle of the milling cutter attitude angle on the ao _d -c plane under the action of milling vibration, and δ ₂ is the projection angle of the milling cutter attitude angle under the action of milling vibration on the bo _d -c plane. ;

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

In step a4, a function equation is established according to each model relationship in step a3, and the instantaneous attitude angle of the milling cutter is solved, and the instantaneous attitude angle of the milling cutter can be calculated from FIG. 8, as shown in formula (2).

Preferably: in step b, the solution of the boundary condition of single-tooth cutting under the action of vibration includes the following steps:

b1, the helical edge is affected by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth error, the instantaneous cutting area of each point on the cutting edge will change continuously. Based on the differential principle, the cutter tooth i of the milling cutter is divided into k micro-elements from the base bottom surface, and the milling cutter structure, milling micro-element division and cutting edge spatial position model are established, which can characterize the cutter tooth error and the space of milling micro-elements. position; as shown in Figure 10 and Figure 11, in the figure, s _o ⁱ -a _i b _i c _i is the coordinate system of the milling cutter tooth i. s _o ⁱ is the tool nose point of the cutter tooth i, s _cmin is the reference tool tooth tool nose point for the measurement of the axial error of the milling cutter, that is, the lowest axial tool nose point, and s _rmax is the reference tool tooth cutter for the measurement of the radial error of the milling cutter. The cusp, that is, the tool tip with the largest turning radius, l _c is the distance from the tip of the lowest tooth of the milling cutter to the end face of the milling cutter, li is the distance from the tip of the _ith tooth to the end face of the milling cutter, r _0max is the maximum turning radius of the milling cutter, r ₀ ⁱ is the turning radius of the cutter tooth i, θ _i is the angle between the i-th cutter tooth and the next cutter tooth, and β is the milling cutter helix angle. Δc ₁ is the length of the first micro-element measured from the base plane of the milling cutter for tooth i, Δc _m is the micro-element length at point m of the cutting edge of tooth i, and Δc _k is the micro-element length at the highest point where tooth i participates in cutting. Meta length. c _z is the axial height of the micro-element division of the tooth i, m _i is the lower boundary point of the micro-element Δc _m of the blade line of the tooth i, with the negative direction of the a-axis as the benchmark, define θ _i (t) as the knife tooth i The instantaneous position angle of the cusp, θ _m is the instantaneous position angle when the axial height of the milling cutter cutting edge is _cm . The axial error and radial error of the milling cutter teeth are characterized, as shown in formula (3).

Δc _i =lc -l _i ; Δr _i = _{r 0max} -r ₀ ⁱ (3)

Calculate the instantaneous position angle of the tip point of the cutter tooth i and any point on the cutting edge, as shown in equations (4) and (5).

In the formula, t ₀ is the initial time 0, θ _i (t ₀ ) is the position angle of the tooth i at the time t ₀ ;

为刀齿i的最大接触角。b2, establish a model of the instantaneous contact angle of the cutter teeth, which can solve the axial boundary conditions of the cutting edge that participates in cutting instantaneously, as shown in Figure 12, in the figure, θ _st is the initial penetration angle of the cutter tooth i, and the cutter at the initial penetration moment The sharp point is A, θ _wt is the initial cutting angle of the tool nose point of the tooth i, the tool nose point is B at the initial cutting time, and θ _p is the position of the tool nose point when the axial height on the cutting edge of the tooth i is a _p Angle, when the axial height reaches a _p , the tool nose point is C, θ _et is the position angle when the tool tooth i completely cuts out the workpiece, and the tool nose point is D when the workpiece is completely cut out,

is the maximum contact angle of tooth i.

According to Fig. 12, the axial boundary of the cutting edge in which the cutter tooth i instantaneously participates in cutting is discussed, as shown in Equation (6).

In the formula, c _i is the axial boundary of the cutting edge where the cutter teeth participate in cutting instantaneously.

Preferably: in step c, establishing and calculating the instantaneous cutting layer parameter model of the milling cutter tooth includes the following steps:

为刀齿i与刀齿i-1转过同一个点时回转半径的偏角，θ_v为o_mi′o_mi-1′与X_g轴正向的夹角，r₀ ^i-1为刀齿i-1的回转半径，o_mi′、θ_mi-1′、m_i′、m_i-1′分别为o_mi、θ_mi-1、m_i、m_i-1在工件坐标系切削深度方向上的投影点，r₀ ⁱ′、r₀ ^i-1′、h_D(t)′分别为r₀ ⁱ、r₀ ^i-1、h_D(t)在工件坐标系切削深度方向上的投影长度；c1, milling vibration and cutter tooth error change the attitude and structure of the milling cutter, thus affecting the parameters of the cutting layer. A parameter model of instantaneous cutting layer considering milling vibration and cutter tooth error is established, which can more accurately calculate the instantaneous cutting area of cutter teeth during the cutting process; as shown in Figure 15 and Figure 16, in the figures, o _di -a _i b _i c _i is the milling cutter coordinate system at time t ₂ , o _di-1 -a _i-1 b _i-1 c _i-1 is the milling cutter coordinate system at time t ₁ , δ _i is the milling cutter attitude angle at time t ₂ , δ _i-1 is the attitude angle of the milling cutter at t ₁ , _θmi is the instantaneous position angle of the cutter tooth _i at the axial height cm, and _{θmi -} 1 is the instantaneous position angle of the cutter tooth i-1 at the axial height cm ,

is the declination angle of the radius of gyration when the tooth i and the tooth i-1 rotate through the same point, θ _v is the angle between o _mi 'o _mi-1 ' and the positive direction of the X _g axis, r ₀ ^i-1 is the knife The radius of gyration of tooth _i -1, o _mi ′, θ _mi-1 ′, mi ′, mi- ₁ ′ are the cutting depths of _{o mi} , θ _mi-1 , mi , and mi- ₁ in the workpiece coordinate system respectively The projection points in the direction, r ₀ ⁱ ′, r ₀ ^i-1 ′, h _D (t)′ are the projection points of r ₀ ⁱ , r ₀ ^i-1 , h _D (t) in the depth of cut direction of the workpiece coordinate system, respectively. projection length;

Step c2, according to Fig. 15 and Fig. 16, the mathematical model of the instantaneous cutting thickness of the cutter tooth i is derived as shown in formula (7).

The calculation of the instantaneous cutting area of the milling element is shown in formula (8).

Preferably: in step d, establishing and solving the instantaneous cutting force model of milling cutter teeth includes the following steps:

Step d1, construct the instantaneous cutting force system of the cutting edge, which can reveal the dynamic changes of the magnitude, direction and boundary conditions of the instantaneous cutting force distribution of the cutter teeth under the influence of milling vibration and cutter tooth error; as shown in Figure 18, in the figure, d _Ftm , d _Frm , d _Fam are the tangential force, radial force and axial force of the cutting edge element relative to the workpiece at any point m _i on the tooth i, h _Dm is the instantaneous cutting at the point m _i on the cutting edge Thickness, h _Dm+1 is the instantaneous cutting thickness at point m _i +1 on the cutting edge;

In step d2, as the cutting process proceeds, the direction of the micro-element cutting force on the cutting edge is also constantly changing. Establish the decomposition model of the micro-element reference plane and the force in the milling cutter coordinate system, and decompose the micro-element cutting force into the milling cutter coordinate system, and then the resultant cutting force on the cutter teeth can be calculated; as shown in Figure 19, in the figure, d _Ftm , d _Fam , and d _Frm are the micro-element tangential force, micro-element axial force, and micro-element radial force of the cutting edge at m _i on tooth i, respectively, d _Fami , d _Fbmi are the micro-element tangential force of the cutting edge at m _i The component force of the element tangential force along the a _v -axis direction and the b _v -axis direction. The calculation of the main cutting force d _Ftm at the point m _i is shown in formula (9).

d _Ftm = pA _D (t) (9)

In the formula, p is the unit cutting force;

Step d3, decompose the main cutting force of the micro-element into the three directions of the milling cutter coordinate system, and then decompose the micro-element force in the three directions of the milling cutter coordinate system into the three directions of the workpiece coordinate system through the coordinate change matrix. The boundary condition of single-tooth cutting is to integrate the micro-element force along the axial direction to solve the resultant cutting force on the cutting edge that participates in cutting instantaneously.

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

In step e1, under the action of milling vibration and cutter tooth error, the instantaneous cutting force relationship between cutter teeth and cutter teeth is also constantly changing. At the same time, the instantaneous number of teeth involved in cutting directly determines the instantaneous cutting force of the milling cutter. The spatial position relationship and instantaneous cutting contact angle model of adjacent cutter teeth is established to characterize the spatial position relationship and instantaneous cutting contact angle of adjacent cutter teeth, and the instantaneous multi-tooth cutting criterion of milling cutter can be given; as shown in Figure 20 and Figure 21 In the figure, s ₀ ⁱ is the tool nose point of tooth i, s ₀ ^{i -1} is the tool nose point of tooth i-1, s ₀ ^i-2 is the tool nose point of tooth i-2, r ₀ ⁱ is the radius of gyration of the tooth i, and r ₀ ^i-1 is the radius of gyration of the tooth i-1.

According to Fig. 20 and Fig. 21, the criterion for the number of teeth that the milling cutter participates in is given, as shown in formula (10).

In the formula, N _t is the number of teeth involved in cutting.

Step e2, in order to reveal the relationship between the instantaneous cutting force between the cutter teeth and the cutter teeth, establish the instantaneous cutting force system of the milling cutter under the action of vibration; as shown in Figure 22, in the figure, δ _ti is the moment t _i under the influence of milling vibration. The attitude angle of the milling cutter, F _t1Xi , F _t1Yi , and F _t1Zi are the cutting forces of the cutter i in the feed speed direction, the cutting width direction and the cutting depth direction at t ₁ , respectively, and F _t2Xi , F _t2Yi , F _t2Zi are respectively The cutting force of tooth i in the direction of feed speed, cutting width and depth of cut at time _t2 , F _t3Xi , F _t3Yi , F t3Zi are the direction of feed speed, cutting width and F _t3Zi respectively at time _t3 . The cutting force in the direction of the depth of cut, F _t1Xi-1 , F _t1Yi-1 , and F _t1Zi-1 are the cutting forces of the tooth i-1 in the direction of feed speed, width of cut and depth of cut at time t ₁ , respectively, F _t2Xi-1 , F _t2Yi-1 , F _t2Zi-1 are the cutting forces of cutter i-1 in the direction of feed speed, width of cut and depth of cut at time t ₂ , respectively, F _t3Xi-1 , F _{t3Yi- 1.} F _t3Zi-1 is the cutting force of cutter i-1 in the direction of feed speed, cutting width and cutting depth at time _t3 , respectively;

In step e3, the instantaneous cutting force of each tooth that instantaneously participates in cutting is accumulated and summed, and the dynamic cutting force of the entire milling cutter can be calculated. A mathematical model of the dynamic cutting force of the milling cutter is constructed; as shown in formula (11).

In the formula, F _X is the cutting force of the milling cutter along the feed speed direction, F _Y is the cutting force of the milling cutter along the cutting width direction, and F _Z is the cutting force of the milling cutter along the cutting depth direction;

In step e4, the improved correlation analysis algorithm is used to calculate the correlation degree between the measured dynamic cutting force and the simulated dynamic cutting force, so as to judge the approximate degree of the dynamic cutting force model predicting the variation characteristics of the actual dynamic cutting force. The correlation matrix between the measured dynamic cutting force and the simulated dynamic cutting force is constructed; as shown in formula (12).

γ1=(γ(A _m1X ,B _m1X )γ(A _m1Y ,B _m1Y )γ(A _m1Z ,B _m1Z )) (12)

In the formula, γ1 is the correlation matrix between the simulated dynamic cutting force and the measured dynamic cutting force, γ(A _m1X , B _m1X ), γ(A _m1Y , B _m1Y ), γ(A _m1Z , B _m1Z ) are the initial cutting of the milling cutter, respectively. The stage milling cutter simulates the correlation between the dynamic cutting force and the measured dynamic cutting force along the feed speed direction, the cutting width direction, and the cutting depth direction.

The criterion for the strength of correlation is given, as shown in Equation (13).

γ(A _m1 ,B _m1 )≥0.7 (13)

where A _m1 and B _m1 are the behavior sequences of the measured and simulated dynamic cutting forces, respectively.

When the correlation degree is greater than or equal to 0.7, it belongs to a strong correlation. At this time, it is considered that the variation characteristics of the measured and simulated dynamic cutting forces have a high degree of approximation;

In step e5, the relative error is used to judge the calculation accuracy of the dynamic cutting force model to calculate the maximum value and the minimum value of the cutting force; the relative error can be defined as:

W=|( _Fsimulation - _Ftrue )/ _Ftrue |×100% (14)

In the formula, W is the relative error, F is the _simulated cutting force, and F _is the actual cutting force.

The maximum and minimum calculation accuracy criteria of dynamic cutting force are given, as shown in formula (15).

W≤20% (15)

When the relative error is less than or equal to 20%, it is considered that the calculation accuracy of the maximum and minimum cutting force meets the requirements;

Step e6, through the correlation analysis, the degree of agreement between the dynamic change characteristics predicted by the dynamic cutting force model and the dynamic cutting force change characteristics measured by the experiment can be verified along the three directions of the milling cutter feed rate, cutting width and cutting depth. Through the calculation of the relative error of the maximum and minimum cutting force, the calculation accuracy of the maximum and minimum cutting force along the three directions of milling cutter feed rate, cutting width and cutting depth can be verified by the dynamic cutting force model.

Specific implementation two,

Vibration Variation Characteristics of Milling Titanium Alloys

(1) The experiment was carried out on a three-axis milling machining center (VDL-1000E), using the milling method of climb milling, the spindle speed n was 1290r/min, and the feed rate v _f was 573mm/min. Vibration was tested with DHDAS5922 dynamic signal test system.

The collected vibration acceleration signal is shown in Figure 4.

In the figure, T ₁ , T ₂ , T ₃ and T ₄ are idling, initial cut-in, smooth cutting and cut-out idling stages, respectively.

(2) Take the vibration signal in the cutting speed direction obtained in the experiment as an example, use matlab to perform noise reduction processing on the vibration acceleration signal, and perform sine function fitting on the vibration acceleration signal after noise reduction, and the specific fitting results As shown in Figure 5 and Figure 6.

Equation fitting is performed on the vibration acceleration signal curve, and the tool vibration displacement equation Ax(t) is obtained by quadratic integration, as shown in equation (16).

In the formula, a _x (t) is the transient acceleration equation in the x-direction of the tool, a ₁ ～a ₈ , b _x1 ～b _x8 , and c _x1 ～c _x8 are coefficients, respectively, and their values during cutting are shown in Table 1:

Table 1 Coefficients of fitting curve equation when cutting stroke is 1m and 6m

Comparing the two stroke coefficients in the table, it can be seen that the coefficients of the corresponding parameter terms in the tool vibration displacement equation are obviously different for the same milling cutter under different cutting strokes, and 16.7% of the coefficients show opposite properties. Therefore, during the whole cutting process, due to the influence of milling vibration, the attitude of the milling cutter changes continuously, and the attitude changes under different strokes. Equation (16) can be used to quantitatively express the dynamic change process of milling cutter vibration during cutting.

Tool nose cutting motion trajectory under the action of vibration

Using matlab to simulate the tool nose trajectory, the real cutting motion trajectory of the five tooth tool nose points of the milling cutter under the action of milling vibration is obtained, as shown in Figure 7.

It can be seen from Figure 7 that under the action of milling vibration, the cutting motion trajectory of each cutter tooth presents a transient change process.

Instantaneous cutting attitude of milling cutter under vibration

(1) The milling vibration causes the overall deflection of the milling cutter, so that the milling cutter forms an attitude angle increment relative to the initial state. The instantaneous cutting attitude model of the milling cutter is established, which reflects the attitude changes before and after the milling cutter is deflected. As shown in Figure 8.

In the figure, point e is the starting point for calculating the overhang of the milling cutter, l is the overhang of the milling cutter, δ is the attitude angle of the milling cutter under the action of vibration, δ ₁ is the projection angle of δ on the ao _d -c plane, δ ₂ is the projection angle of δ on the bo _d -c plane. From Figure 8, the instantaneous attitude angle of the milling cutter and the instantaneous position angle of any point on the cutting edge of the cutter tooth i can be calculated, as shown in formula (17).

It can be seen from equation (17) that the vibration displacement and the overhang of the milling cutter jointly affect the instantaneous attitude angle of the milling cutter.

(2) Using matlab to simulate the space attitude angle of the milling cutter, the variation curve of the attitude angle with time under the action of vibration of the milling cutter rotating one circle is shown in Figure 9.

From the above model analysis, it can be seen that the attitude angle of the milling cutter is a spatial change angle during the milling process, and the change of the space angle will cause its projection angle in both directions to change. The dynamic change of the space angle is different from the milling process. The resulting vibration displacement is directly related.

Boundary condition of single tooth cutting under vibration

(1) The helical edge is affected by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth error, the instantaneous cutting area of each point on the cutting edge will change continuously. Based on the differential principle, the cutter tooth i of the milling cutter is divided into k micro-elements from the base bottom surface, and the milling cutter structure, milling micro-element division and cutting edge spatial position model are established, which can characterize the cutter tooth error and the space of milling micro-elements. Location. As shown in Figure 10 and Figure 11.

In the figure, s _o ⁱ -a _i b _i c _i is the coordinate system of the cutter tooth i of the milling cutter, the a _i direction is the cutting speed direction of the cutter tooth i, the b _i direction is the direction that the milling cutter circle center points to the tool nose point, and c _i The direction is the direction of the milling cutter axis. s _o ⁱ is the tool nose point of the milling cutter tooth i, s _cmin is the reference tool tooth tool nose point of the milling cutter axial error measurement, that is, the axial lowest tool nose point, s _rmax is the milling cutter radial error measurement reference tool tooth cutter The cusp, that is, the tool tip with the largest turning radius, l _c is the distance from the tip of the lowest tooth of the milling cutter to the end face of the milling cutter, li is the distance from the tip of the _ith tooth to the end face of the milling cutter, r _0max is the maximum turning radius of the milling cutter, r ₀ ⁱ is the turning radius of the ith tooth, θ _i is the angle between the ith tooth and the next tooth, and β is the helix angle of the milling cutter. Δc ₁ is the length of the first micro-element measured by the cutter tooth i from the base plane of the milling cutter, Δc _m is the micro-element length at the point m _i of the cutting edge of the tooth i, and Δc _k is the maximum point where the tooth i participates in cutting. Element length. c _z is the axial height of the micro-element division of the tooth i, m _i is the coordinate point of the lower boundary of the micro-element Δc _m of the tooth i edge line, taking the negative direction of the a-axis as the benchmark, define θ _i (t) as the tooth i The instantaneous position angle of the tool tip, θ _m is the instantaneous position angle when the axial height on the cutting edge of the milling cutter is _cm .

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

Table 2 Structural parameters of solid carbide end mills with a diameter of 20mm

The axial error and radial error of the milling cutter teeth are characterized, as shown in equation (18).

Δc _i =lc -l _i ; Δr _i = _{r 0max} -r ₀ ⁱ (18)

The radial and axial errors of the milling cutter are measured by the tool setter, and the measurement results are shown in Table 3.

Table 3 Tooth error

Calculate the instantaneous position angle of the tip point of the cutter tooth i and any point on the cutting edge, as shown in equations (19) and (20).

In the formula, t ₀ is the initial time 0, and θ _i (t ₀ ) is the position angle of the tooth i at the time t ₀ .

(2) Establish a model of the instantaneous contact angle of the cutter teeth, which can solve the axial boundary conditions of the cutting edge that instantaneously participate in cutting. As shown in Figure 12.

为刀齿i的最大接触角。In the figure, θ _st is the initial cutting angle of the tool tooth i, the corresponding tool nose point is A, θ _wt is the initial cutting angle of the tool tooth i tool nose point, the corresponding tool nose point is B, and θ _p is the cutting edge of the tool tooth i The position angle of the tool nose point when the upper axial height is a _p , the corresponding tool nose point is C, θ _et is the position angle when the tool tooth i completely cuts out the workpiece, and the corresponding tool nose point is D,

is the maximum contact angle of tooth i.

According to Fig. 12, discuss the instantaneous cutting edge axial boundary of cutter i participating in cutting, as shown in equation (21).

In the formula, c _i is the instantaneous cutting edge axial boundary where the cutter teeth participate in cutting.

It can be seen from equation (21) that the length of the cutting edge that participates in cutting instantaneously changes dynamically with time. As the cutting process progresses, the length of the cutting edge involved in the cutting first gradually increases, then remains unchanged, and finally decreases gradually.

Instantaneous cutting layer parameters of milling cutter under vibration

(1) Use matlab to simulate the formation process of the cutting layer, as shown in Figure 13 and Figure 14.

It can be seen from Figure 13 and Figure 14 that under the action of vibration, the cutting edge of the milling cutter teeth is offset, which causes the shape of the cutting layer formed by the adjacent teeth to change, thereby changing the instantaneous cutting thickness and cutting width of the teeth.

(2) Milling vibration and cutter tooth error change the attitude and structure of the milling cutter, thus affecting the parameters of the cutting layer. A parameter model of instantaneous cutting layer considering milling vibration and cutter tooth error is established, which can more accurately calculate the instantaneous cutting area of cutter teeth in the cutting process. As shown in Figure 15 and Figure 16.

为刀齿i与刀齿i-1转过同一个点时位置角，θ_v为o_mi′o_mi-1′与X_g轴正向的夹角，r₀ ^i-1为刀齿i-1的回转半径，A_x为o_mi点到o_mi-1点在进给速度方向上的距离，A_y为o_mi点到o_mi-1点在切削宽度方向上的距离。o_mi′、θ_mi-1′、m_i′、m_i-1′分别为o_mi、θ_mi-1、m_i、m_i-1在工件坐标系切削深度方向上的投影点，r₀ ⁱ′、r₀ ^i-1′、h_D(t)′分别为r₀ ⁱ、r₀ ^{i -1}、h_D(t)在工件坐标系切削深度方向上的投影长度，刀齿i的瞬时切削厚度如式(22)所示。In the figure, o _di -a _i b _i c _i is the coordinate system of the milling cutter at time t ₂ , o _di-1 -a _i-1 b _i-1 c _i-1 is the coordinate system of the milling cutter at time t ₁ , and δ _i is The attitude angle of the milling cutter at t ₂ , δ _i-1 is the attitude angle of the milling cutter at t ₁ , θ _mi is the instantaneous position angle of the cutter tooth i at the axial height c _m , and θ _mi-1 is the axial height c _m The instantaneous position angle of the cutter tooth i-1,

is the position angle when the tooth i and the tooth i-1 rotate through the same point, θ _v is the angle between o _mi 'o _mi-1 ' and the positive direction of the X _g axis, r ₀ ^i-1 is the tooth i- 1 radius of gyration, A _x is the distance from o _mi point to o _mi -1 point in the direction of feed rate, A _y is the distance from o _mi point to o _mi-1 point in the cutting width direction. _{o mi} ′, θ _mi-1 ′, mi ′, mi- ₁ ′ are the projection points of _{o mi} , θ _mi-1 , mi , and mi- ₁ on the workpiece coordinate system cutting depth direction, respectively, r ₀ ⁱ ′, r ₀ ^i-1 ′, h _D (t)′ are the projection lengths of r ₀ ⁱ , r ₀ ^{i -1} , h _D (t) in the workpiece coordinate system cutting depth direction, respectively, the instantaneous The cutting thickness is shown in formula (22).

The calculation of the instantaneous cutting area of the milling element is shown in formula (23).

(3) Using matlab to simulate the ideal state in the cutting process, only considering the error, only considering the milling vibration and the instantaneous cutting area considering the error and milling vibration. The simulation results are shown in Figure 17.

It can be seen from Fig. 17 that both the static error and the milling vibration of the milling cutter during the milling process will cause the instantaneous cutting thickness of the milling cutter to change and present a dynamic change.

Instantaneous cutting force of milling cutter teeth

(1) The instantaneous cutting force system of the cutting edge is constructed, which can reveal the dynamic changes of the magnitude, direction and boundary conditions of the instantaneous cutting force distribution of the cutter teeth under the influence of milling vibration and cutter tooth error. As shown in Figure 18.

In Fig. 18, d _Ftm , d _Frm , and d _Fam are the tangential force, radial force and axial force of the cutting edge element relative to the workpiece at any point m _i on the tooth i.

Each micro-element on the helical edge has a hysteresis when cutting into the workpiece, and the instantaneous cutting thickness of each micro-element on the cutting edge involved in cutting is different, which affects the instantaneous cutting force of each micro-element on the cutting edge.

(2) With the progress of the cutting process, the direction of the micro-element cutting force on the cutting edge is also constantly changing, and the decomposition model of the force in the micro-element reference plane and the milling cutter coordinate system is established, and the micro-element cutting force is decomposed into the milling cutter coordinates. In the system, the resultant cutting force on the cutter teeth can be calculated. As shown in Figure 19.

In the figure, d _Ftm , d _Fam , d _Frm are the micro-element tangential force, micro-element axial force, and micro-element radial force of the cutting edge at any point m _i on the tooth i, respectively, d _Fami , d _Fbmi are m respectively The component forces of the tangential force of the cutting edge element at _i along the directions of the a _v axis and the b _v axis. The calculation of the main cutting force d _Ftm of the micro element at the point m _i is shown in formula (24).

d _Ftm = pA _D (t) (24)

where p is the unit cutting force.

The main cutting force of the micro-element is decomposed into the three directions of the milling cutter coordinate system, and then the micro-element force in the three directions of the milling cutter coordinate system is decomposed into the three directions of the workpiece coordinate system through the coordinate change matrix. Finally, according to the implementation example 4 In the single-tooth cutting boundary condition, the micro-element force is integrated along the axial direction, and the resultant cutting force on the cutting edge that participates in cutting instantaneously is solved.

Criterion for multi-tooth cutting of milling cutter under vibration

Under the action of milling vibration and cutter tooth error, the instantaneous cutting force relationship between cutter teeth and cutter teeth is also constantly changing. At the same time, the instantaneous number of teeth involved in cutting directly determines the instantaneous cutting force of the milling cutter. The spatial position relationship and instantaneous cutting contact angle model of adjacent cutter teeth is established to characterize the spatial position relationship and instantaneous cutting contact angle of adjacent cutter teeth, and then the instantaneous multi-tooth cutting criterion of milling cutter can be given. As shown in Figure 20 and Figure 21.

In the figure, s ₀ ⁱ is the tool nose point of tooth i, s ₀ ^i-1 is the tool nose point of tooth i-1, s ₀ ^i-2 is the tool nose point of tooth i-2, r ₀ ⁱ is the radius of gyration of the tooth i, and r ₀ ^i-1 is the radius of gyration of the tooth i-1.

It can be seen from Figure 20 and Figure 21 that the instantaneous number of teeth involved in cutting is related to the initial penetration angle, complete cutting angle and the angle between the teeth of the cutter teeth.

In the formula, N _t is the number of teeth involved in cutting.

From the milling parameters in Example 1, it is calculated by formula (10) that the number of instantaneously involved cutting teeth of the milling cutter is 2.

Construction and verification method of transient cutting force model of milling cutter under vibration

(1) In order to reveal the relationship between the instantaneous cutting force between the cutter teeth and the cutter teeth, the instantaneous cutting force system of the milling cutter under the action of vibration is established, as shown in Figure 22.

In the figure, δ _ti is the attitude angle of the milling cutter under the influence of milling vibration at time t _i , F _t1Xi , F _t1Yi , and F _t1Zi are the direction of feed speed, cutting width and depth of cut of cutter i at time _t1 , respectively F _t2Xi , F _t2Yi , F _t2Zi are the cutting forces of the cutter i in the feed speed direction, cutting width direction and cutting depth direction at t ₂ , respectively, F _t3Xi , F _t3Yi , F _t3Zi are respectively t ₃ The cutting force of tooth i in the direction of feed speed, cutting width and depth of cut at time, F _t1Xi , F _t1Yi , F _t1Zi are the direction of feed speed, cutting width and cutting depth of tooth i at time _t1 respectively The cutting force in the direction, F _t2Xi , F _t2Yi , F _t2Zi are the cutting forces of the cutter i in the feed speed direction, the cutting width direction and the cutting depth direction at t ₂ , respectively, F _t3Xi , F _t3Yi , F _t3Zi are respectively The cutting force of the cutter i in the direction of feed speed, width of cut and depth of cut at time _t3 .

It can be seen from Fig. 22 that during the milling process, with the continuous change of the attitude of the milling cutter, the cutting force relationship between the cutter teeth participating in the cutting instantaneously is also constantly changing.

(2) Accumulate and sum the instantaneous cutting force of each tooth that participates in cutting instantaneously, and then the dynamic cutting force of the whole milling cutter can be calculated. A mathematical model of the dynamic cutting force of the milling cutter is constructed, as shown in equation (26).

In the formula, F _X is the cutting force of the milling cutter along the feed rate direction, F _Y is the cutting force of the milling cutter along the cutting width direction, and F _Z is the cutting force of the milling cutter along the cutting depth direction.

(3) Use matlab to denoise the cutting force obtained in the experiment, and simulate the milling force in the three directions of the workpiece coordinate system. The simulation results are shown in Figure 23 to Figure 25.

In the figure, t _t is the initial cut-in time of the milling cutter tooth, t _e is the final cut-out time of the milling cutter tooth, T is the period that the milling cutter makes one revolution, and JT _i is the period that the milling cutter makes one revolution of a single tooth .

It can be seen from Figure 23 to Figure 25 that the simulation curve is consistent with the change trend of the actual curve, but there is a deviation in the waveform. The reason is that the vibration of the workpiece is measured, which deviates from the vibration of the milling cutter at the instantaneous cutting position, resulting in The deviation between the simulation curve and the actual curve waveform is calculated.

(4) Extract the data of the measured dynamic cutting force and the inflection point data of the simulated dynamic cutting force along the feed speed direction, the cutting width direction, and the cutting depth direction. Build simulated and measured dynamic cutting force behavior sequences. The measured dynamic cutting force behavior sequence is used as the reference sequence, and the simulated dynamic cutting force behavior sequence is used as the comparison sequence. The improved grey relational analysis algorithm is used for calculation, and the relational matrix between the measured dynamic cutting force and the simulated dynamic cutting force is established, as shown in formula (27).

In the formula, γ1 is the correlation matrix between the simulated dynamic cutting force and the measured dynamic cutting force, γ(A _m1X , B _m1X ), γ(A _m1Y , B _m1Y ), γ(A _m1Z , B _m1Z ) are the initial cutting of the milling cutter, respectively. The correlation between the simulated dynamic cutting force and the measured dynamic cutting force of the milling cutter along the feed rate direction, the cutting width direction, and the cutting depth direction at the workpiece stage, γ(A _m2X , B _m2X ), γ(A _m2Y , B _m2Y ), γ( A _m2Z , B _m2Z ) are the correlation between the simulated dynamic cutting force and the measured dynamic cutting force along the feed rate direction, cutting width direction, and cutting depth direction of the milling cutter in the intermediate stage of the milling cutter cutting the workpiece, respectively, γ(A _m3X , B _m3X ) , γ(A _m3Y , B _m3Y ), γ(A _m3Z , B _m3Z ) are the simulated dynamic cutting force and the measured dynamic cutting force of the milling cutter along the feed speed direction, the cutting width direction, and the cutting depth direction at the stage of the milling cutter cutting out the workpiece, respectively. degree of relevance.

The strength criterion of correlation is given, as shown in Eq. (28).

γ(A _m1 ,B _m1 )≥0.7 (28)

where A _m1 and B _m1 are the behavior sequences of the measured and simulated dynamic cutting forces, respectively.

When the correlation degree is greater than or equal to 0.7, it belongs to a strong correlation. At this time, it is considered that the variation characteristics of the measured and simulated dynamic cutting forces have a high degree of approximation.

Observing the calculation results of the correlation degree, the correlation degree between the simulated cutting force and the measured cutting force in the feed speed direction and the cutting width direction of the milling cutter at the initial stage of the milling cutter cutting into the workpiece is smaller than that in the other two stages, and the correlation degree in the cutting width direction is smaller than that in the other two stages. The correlation degree is the smallest, followed by the correlation degree in the feed speed direction, but both are between 0.7 and 0.9, which are strong correlations. The reason for analyzing the above phenomenon is that the initial cutting position of the cutter teeth is deviated, and when the initial cutting into the workpiece, due to the sudden increase of the load, the cutter will yield. The above two problems do not exist in the intermediate stage and the cut-out stage, so the correlation degree is high. The high correlation between the simulated dynamic cutting force in the depth-of-cut direction and the measured dynamic cutting force in the initial cut-in stage is due to the small deviation of the cutting position in the depth-of-cut direction.

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

W=|( _Fsimulation - _Ftrue )/ _Ftrue |×100% (29)

In the formula, W is the relative error, F is the _simulated cutting force, and F _is the actual cutting force.

The relative error between the simulation and the actual dynamic cutting force maximum and minimum values of the three stages in the cutting process is calculated by formula (29). The calculation results of the relative error are shown in Table 4.

Table 4 Relative error of simulation and actual cutting force

In the table, W _max is the relative error between the measured maximum value of dynamic cutting force and the simulated maximum value, and W _min is the relative error between the measured minimum value of dynamic cutting force and the simulated minimum value.

The maximum and minimum calculation accuracy criteria of dynamic cutting force are given, as shown in formula (30).

W≤20% (30)

When the relative error is less than or equal to 20%, the calculation accuracy of the maximum and minimum cutting force is considered to meet the requirements.

It can be seen from Table 4 that the relative error between the feed speed direction and the cutting width direction in the initial cutting stage is larger, and the relative error in the cutting width direction is larger than the relative error in the feed speed direction, but the difference is not large. The errors are all less than 5.4%, smaller than the other two directions. The reason for the above phenomenon is also the deviation of the initial cutting position of the cutter teeth and the sudden increase of the load when the initial cutting into the workpiece will cause the knife to yield. These two reasons lead to a relatively large relative error in the direction of the feed speed in the initial cutting stage, but since the cutting position deviation in the cutting depth direction is small relative to the position deviation in the other two directions during the cutting process, the relative error in the cutting depth direction is significantly smaller than relative error in the other two directions.

This embodiment verifies the degree of agreement between the dynamic change characteristics predicted by the dynamic cutting force model and the experimentally measured dynamic cutting force change characteristics along the three directions of milling cutter feed rate, cutting width and cutting depth through correlation analysis. 78% of the calculation results of the correlation degree are greater than 0.9, indicating that the variation characteristics of the dynamic cutting force curve predicted by the instantaneous cutting force model constructed by the present invention are very similar to the variation characteristics of the dynamic cutting force curve obtained by the experiment. And through the calculation results of the relative error of the maximum value of cutting force and the relative error of the minimum value, it is found that the relative error of dynamic cutting force simulation and actual cutting force is less than 16.3%. It is verified that the dynamic cutting force model has a good solution accuracy of the maximum and minimum cutting force along the three directions of milling cutter feed rate, cutting width and cutting depth. Using the above two verification methods, it is confirmed that the dynamic cutting force model can correctly reflect the changing characteristics and level of the dynamic cutting force in the actual machining process

In the milling process, there is an interaction between the cutting force and the vibration, that is, the cutting force will cause the change of the vibration. At the same time, the vibration will react to the cutting force. By affecting the cutting motion trajectory of the tool tip and the cutting attitude of the tool, the contact relationship between the tool and the cutting layer and the parameters of the cutting layer are changed, and finally the cutting force changes. Therefore, in order to fully reveal the interaction between vibration and cutting force in the cutting process, it is necessary to investigate the influence mechanism of cutting force on vibration and the influence mechanism of vibration on cutting force at the same time.

The existing empirical formula models must rely on a large number of milling experiments, and are only suitable for the prediction of milling force under fixed conditions such as tool material, angle and workpiece material.

The existing theoretical formula model simplifies many conditions in the derivation process, which is quite different from the real situation, and the calculation is complicated, so it is generally not used to solve the transient cutting force.

The existing mechanical model does not establish the cutting force system on the cutting edge, and cannot reflect the relationship between the micro-element cutting forces on the cutting edge. In addition, the average cutting layer area is used to solve the instantaneous cutting force, which cannot truly reflect the force on the cutting edge. Moreover, the force relationship between adjacent cutter teeth is not given, so the instantaneous cutting force generated by the milling cutter cannot be predicted more accurately.

Aiming at the problem that the milling vibration has a serious impact on the cutting stability and the quality of the machined surface during the milling process, the present invention studies the impact mechanism of the vibration on the cutting force based on the existing impact mechanism of the cutting force on the vibration. The instantaneous cutting force system of the cutter teeth is constructed, the dynamic variation characteristics of the micro-element cutting force distribution on the cutting edge are revealed, the boundary conditions of the cutting edge participating in cutting instantaneously are solved, the instantaneous cutting force system of the milling cutter is established, and the cutting edge is revealed. The changing relationship of the instantaneous cutting force system between the teeth and the cutter teeth. The dynamic cutting force model of the milling cutter is constructed to predict the dynamic cutting force in the cutting process, which provides an effective basis for the control of milling stability.

This embodiment is only an exemplary description of this patent, and does not limit its protection scope. Those skilled in the art can also make partial changes to it, as long as it does not exceed the spirit of this patent, it is within the protection scope of this patent.

Claims (1)

1. a milling cutter dynamic cutting force model construction and verification method under the action of vibration, is characterized in that, comprises the following steps: Step a, solve the instantaneous cutting behavior of the milling cutter; In the step a, the solution to the instantaneous cutting behavior of the milling cutter includes the following steps: Step a1, constructing a model of the instantaneous contact relationship between the cutter and teeth under the influence of milling vibration and cutter-tooth error; Step a2, according to step a1, establish a tool tip cutting motion trajectory model, which reflects that the tool tip is affected by the milling vibration in the process of milling titanium alloys, resulting in a displacement increment, so that the actual cutting motion trajectory of the tool tip changes state; In step a3, the milling vibration causes the milling cutter to deflect as a whole, so that the milling cutter forms an attitude angle increment relative to the initial state. According to step a1, the instantaneous cutting attitude model of the milling cutter is established, and the model reflects the attitude change of the milling cutter before and after the deflection. ; Step a4, establish a functional equation according to each model relationship in step a3, and solve the instantaneous attitude angle of the milling cutter; Step b, solve the boundary condition of single-tooth cutting under the action of vibration; In the step b, the solution of the boundary condition of single-tooth cutting under the action of vibration includes the following steps: b1, the helical edge is affected by the structure and participates in cutting point by point. Under the action of milling vibration and cutter tooth error, the instantaneous cutting area of each point on the cutting edge will change continuously. Based on the differential principle, the milling cutter tooth i is changed from The base bottom surface is divided into k micro-elements, and the milling cutter structure, milling micro-element division and cutting edge spatial position model are established. The model represents the cutter tooth error and the spatial position of the milling micro-element; b2, establish a model of the instantaneous contact angle of the cutter teeth, which solves the axial boundary conditions of the cutting edge that instantaneously participates in cutting; Step c, establish and calculate the instantaneous cutting layer parameter model of milling cutter teeth; In the step c, the parameter model of the instantaneous cutting layer of the milling cutter tooth is established and the calculation includes the following steps: c1, milling vibration and cutter tooth error change the attitude and structure of the milling cutter, thus affecting the cutting layer parameters. A model of instantaneous cutting layer parameters considering milling vibration and cutter tooth error is established, which can more accurately calculate the cutting layer parameters during the cutting process. Instantaneous cutting area of cutter teeth; Step c2, derive the mathematical model of the instantaneous cutting thickness of the cutter tooth i , and calculate the instantaneous cutting area of the milling element; Step d, establish and solve the instantaneous cutting force model of milling cutter teeth; In the step d, establishing and solving the instantaneous cutting force model of milling cutter teeth includes the following steps: Step d1, construct the instantaneous cutting force system of the cutting edge to reveal the dynamic changes of the magnitude, direction and boundary conditions of the instantaneous cutting force distribution of the cutter teeth under the influence of milling vibration and cutter tooth error; Step d2, with the progress of the cutting process, the direction of the micro-element cutting force on the cutting edge is also constantly changing, and the decomposition model of the force in the micro-element reference plane and the milling cutter coordinate system is established, and the micro-element cutting force is decomposed into the milling cutter coordinates. In the system, that is, the resultant cutting force on the cutter teeth is calculated; Step d3, decompose the main cutting force of the micro-element into the three directions of the milling cutter coordinate system, and then decompose the micro-element force in the three directions of the milling cutter coordinate system into the three directions of the workpiece coordinate system through the coordinate change matrix. The boundary condition of single tooth cutting is to integrate the micro-element force along the axial direction to solve the resultant cutting force on the cutting edge that participates in cutting instantaneously; Step e, complete the construction and verification of the dynamic cutting force model of the milling cutter; The construction and verification of the dynamic cutting force model of the milling cutter in the step e includes the following steps: In step e1, under the action of milling vibration and cutter tooth error, the instantaneous cutting force relationship between cutter teeth and cutter teeth is also constantly changing. At the same time, the instantaneous number of teeth involved in cutting directly determines the instantaneous cutting force of the milling cutter. The spatial position relationship and instantaneous cutting contact angle model of adjacent cutter teeth is established to characterize the spatial position relationship and instantaneous cutting contact angle of adjacent cutter teeth, that is, the instantaneous multi-tooth cutting criterion of milling cutter is given; Step e2, in order to reveal the relationship of the instantaneous cutting force between the cutter teeth and the cutter teeth, establish the instantaneous cutting force system of the milling cutter under the action of vibration; Step e3: Accumulate and sum the instantaneous cutting forces of each tooth that participates in cutting instantaneously, that is, to calculate the dynamic cutting force of the entire milling cutter, and construct a mathematical model of the dynamic cutting force of the milling cutter; In step e4, the improved correlation analysis algorithm is used to calculate the correlation degree between the measured dynamic cutting force and the simulated dynamic cutting force, that is, to determine the approximate degree of the dynamic cutting force model predicting the variation characteristics of the actual dynamic cutting force, and to construct the measured dynamic cutting force and the simulated dynamic cutting force. force correlation matrix; In step e5, the relative error is used to judge the calculation accuracy of the dynamic cutting force model for calculating the maximum value and the minimum value of the cutting force; Step e6, through correlation analysis, verify the degree of agreement between the dynamic change characteristics predicted by the dynamic cutting force model and the dynamic cutting force change characteristics measured by the experiment along the three directions of milling cutter feed rate, cutting width and cutting depth. The relative error calculation of the maximum value and the minimum value is used to verify the calculation accuracy of the maximum and minimum value of the cutting force along the three directions of the milling cutter feed rate, cutting width and cutting depth of the dynamic cutting force model.

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