CN110170886A - A kind of Camshaft Grinding processing method based on T-S fuzzy control - Google Patents

Abstract

The invention discloses a camshaft grinding method based on T-S fuzzy control, which is realized by the following steps: 1. While the grinding wheel rotates at high speed, use the computer's numerical control program to control the grinding wheel feed system as the X axis and the cam rotation system as the C axis in the CNC camshaft grinder; 2. According to the cam lift value, grinding wheel radius, probe radius, cam base circle radius and formulas (1), (2), (4) and (5) provided by the user, use MATLAB software tools to formulate formulas (1), (2) ) to fit the displacement curve of the grinding wheel feed X ( θ ) and the cam angle α ( θ ), and the value of X ( θ ) ‑α ( θ ) formed by the displacement curve is automatically generated by the programming software for the CNC machining subroutine of the grinding wheel feed , so as to realize the lateral reciprocating motion of the grinding wheel following the rotation of the cam, and at the same time use the MATLAB software tool to fit the speed curve of the cam rotation speed F ( θ ) and the cam rotation angle α ( θ ) from equations (2) and (5), the speed curve The formed F ( θ ) - α ( θ ) value is automatically generated by the programming software for the cam rotation NC machining subroutine to realize the cam rotation motion. Ideal grinding accuracy, processing efficiency and surface quality can be obtained.

Description

A Camshaft Grinding Method Based on T-S Fuzzy Control

technical field

The invention belongs to a grinding process method, in particular to a camshaft grinding process method based on T-S fuzzy control.

Background technique

The camshaft is one of the key parts of the automobile engine. Due to its special shape, the camshaft is ground by non-circular grinding. The machining accuracy and efficiency not only determine the quality and production cost of the product, but also affect the engine work performance. At present, the traditional camshaft grinding method is that the grinding wheel feed (X axis) adopts the ideal grinding wheel feed displacement motion equation, while the cam rotation (C axis) speed adopts an empirical model and the cam rotation speed is modified by experienced engineers and technicians. After repeated trial grinding processing to determine. This camshaft grinding method is difficult to meet the high precision, high efficiency and high flexibility requirements of modern automobile parts processing.

Scholars at home and abroad have been committed to the research of precision and high-efficiency non-circular grinding technology. Xu Dihong of Hunan University et al. The surface of the tangent point moves at a constant linear velocity and is corrected according to a constant removal rate. The camshaft grinding mathematical model based on this theory has achieved practical application value, but its machining accuracy and efficiency still need to be further improved. Chinese patent (ZL201010278922.0) discloses "a camshaft numerical control grinding method", which predicts the cam rotation (C axis) speed by limiting the non-circular segment grinding wheel feed speed, acceleration, and jerk The shaft is ground with high precision, but for the oil pump cam and the cam with a large lift value, the part with a large lift curvature at the waist will produce a large cam lift error. The document "Optimization method for the rotation speed of the rotating shaft of the CNC camshaft grinder" ("Journal of Mechanical Engineering", 2014.15) proposes to use forward and reverse synchronous acceleration by limiting the rotation speed of the non-circular segment cam, the feed speed of the grinding wheel, the acceleration, and the jerk. The control method dynamically solves the meeting point of the maximum feed speed of forward and reverse interpolation to realize the optimal interpolation of the grinding wheel feed, so as to reduce the impact of acceleration on the machine tool and improve the grinding accuracy and efficiency of the machine tool. But the surface quality still needs to be further improved.

Contents of the invention

The purpose of the invention is to provide a camshaft grinding method based on T-S fuzzy control which can improve the camshaft grinding precision and efficiency.

Realize that the technical scheme that the present invention adopts is as follows:

The camshaft grinding method based on T-S fuzzy control provided by the invention is realized by the following steps:

Step 1. While the grinding wheel rotates at high speed, use the numerical control program of the computer to control the grinding wheel feed system as the X axis and the cam rotation system as the C axis in the CNC camshaft grinder;

Step 2. According to the cam lift value, grinding wheel radius, probe radius, cam base circle radius and formulas (1), (2), (4) and (5) provided by the user, use the MATLAB software tool to formulate the formula (1) , (2) Fit the displacement curve of the grinding wheel feed X(θ) and the cam angle α(θ), and the X(θ)-α(θ) value formed by the displacement curve is automatically generated by the programming software for the numerical control of the grinding wheel feed The subroutine is processed to realize the lateral reciprocating motion of the grinding wheel following the rotation of the cam. At the same time, the speed curve of the cam rotation speed F(θ) and the cam rotation angle α(θ) is fitted by the formula (2) and (5) using MATLAB software tools. The F(θ)-α(θ) value formed by the speed curve is automatically generated by the programming software for the cam rotation numerical control machining subroutine to realize the cam rotation movement;

X(θ)=OO ₂ -r ₂ -r (1)

In the formula: X(θ) is the displacement of the grinding wheel feed, r is the radius of the base circle of the cam, O is the center of the base circle of the cam, O ₁ is the center of the measuring head of the roller follower, O ₂ is the center of the grinding wheel, φ is the angle between the line connecting the cam base circle center O to the probe roller center O ₁ and the cam instant center M to the probe roller center O ₁ , O ₁ O ₂ =r ₂ -r ₁ , r ₂ is the radius of the grinding wheel, r ₁ is the radius of the roller follower probe, OO ₁ = r+H(θ)+r ₁ , H(θ) is the lift value of the cam, θ is the cam base circle profile The angle between the line connecting point A to the cam base circle center O and the line connecting the cam base circle center O to the probe roller center O ₁ , α(θ) is the midpoint A of the cam base circle profile and the cam base circle The angle between the line connecting the center O and the center O ₂ of the grinding wheel to the center O of the base circle of the cam, ρ(θ) is the polar radius from the tangent point P between the cam profile and the grinding wheel to the center O of the base circle of the cam, β(θ ) is the angle between the line connecting the midpoint A of the cam base circle profile to the center O of the cam base circle and the tangent point P between the cam profile and the grinding wheel to the line O of the cam base circle center O, ω(θ) is the cam rotation Rotation angular velocity of the tangent point P between the cam profile and the grinding wheel when the cam is lifted, _ω0 is the angular velocity of the tangent point P between the cam profile and the grinding wheel when the cam rotates to the base circle, F(θ) is the cam rotation to the roller from The rotation speed at the side head rotation angle θ of the movable member, n(θ) is the rotation speed from the cam rotation to the side head rotation angle θ of the roller follower;

Step 3. Calculate the first-order derivative, the second-order derivative, and the third-order derivative respectively to the grinding wheel feed displacement formula (1) to obtain formulas (6), (7), and (8);

In the formula, v(θ) is the feed speed of the grinding wheel, a(θ) is the feed acceleration of the grinding wheel, and j(θ) is the feed jerk of the grinding wheel;

Use MATLAB software tools to fit equations (2) and (6) to get the speed curve of the grinding wheel feed speed v(θ) and cam angle α(θ), and use MATLAB software tools to fit equations (2) and (7) Get the acceleration curve of the grinding wheel feed acceleration a(θ) and the cam rotation angle α(θ); the maximum rotation speed of the camshaft exceeds the rotation speed of the base circle (36000deg/min) by more than 1.2 times, according to formulas (9) and (10) for the grinding wheel Feedrate, acceleration, jerk, cam rotation speed are limited:

In the formula, v(θ _i ) is the feed speed of the grinding wheel in the i-th interpolation period, a(θ _i ) is the feed acceleration of the grinding wheel in the i-th interpolation period, and j(θ _i ) is the grinding wheel speed in the i-th interpolation period Feed jerk, k is the speed limit ratio, v _max is the allowable maximum speed of grinding wheel feed, a _max is the allowable maximum acceleration of grinding wheel feed, j _max is the allowable maximum jerk of grinding wheel feed, ω _max is the maximum rotational angular velocity;

Step 4. According to the S-type acceleration and deceleration control method, the cam lift value, the formula (16) of acceleration mode 1 and the formulas (18), (19) and (20) of acceleration mode 2, solve the grinding wheel feed interpolation period T _si and the interpolation cycle of the cam return to predict the time it takes for the cam to rotate 1 degree for the grinding wheel feed;

Calculate the interpolation period T _si of acceleration mode 1 by formula (16):

ΔX(T _si ) is the displacement of the grinding wheel feed per rotation of 1 degree, t _i1 is the acceleration phase time, t _i2 is the deceleration phase time, v _i-1 is the initial feed speed, j _max is the allowable grinding wheel feed The maximum jerk, s(t _i1 ) is the feed displacement of the grinding wheel at stage t _i1 , s(t _i2 ) is the feed displacement of the grinding wheel at stage t _i2 , and T _si is an interpolation cycle when the grinding wheel feeds ΔX(T _si );

Calculate the interpolation period T _si of acceleration mode 2 by formulas (18), (19) and (20):

a _max t _i2 ² +(j _max t _i1 ² +2a _max t _i1 +2v _i-1 )t _i2 +4v _i-1 t _i1 +j _max t _i1 ³ +a _max t _i1 ² -2ΔX(T _si ) =0 (18)

t _i1 =t _i3 =a _max /j _max (19)

T _si =t _i1 +t _i2 +t _i3 =2t _i1 +t _i2 (20)

Among them: ΔX(T _si ) is the displacement of the grinding wheel per rotation of 1 degree, t _i1 is the time of acceleration and acceleration phase, t _i2 is the time of uniform acceleration phase, t _i3 is the time of deceleration and acceleration phase, v _i-1 is the initial progress Giving speed, j _max is the allowable maximum jerk of the grinding wheel feed, a _max is the allowable maximum acceleration of the grinding wheel feed, T _si is an interpolation cycle when the grinding wheel feed ΔX(T _si );

Step 5. obtain the cam rotation speed value by formula (21), (22);

T _w = T _si (21)

Among them, T ₀ is the base circle interpolation period of the cam rotation, n ₀ is the base circle speed, and the value is 100rmp, T _w is the interpolation period from the cam rotation to the lift section, and F″”(θ) is the cam rotation to the lift section predicted speed;

Step 6. Establish a one-dimensional T-S fuzzy controller model with a single-input roller follower measuring head rotation angle θ and a single-output cam rotation speed F″(θ), and the T-S fuzzy controller simplified model is as follows:

R _i : if θ is A _i then F _i ′(θ)＝a _i0 +a _i1 θ (25)

Among them, R _i (i=1, 2, ... 12) represents the i-th fuzzy rule, θ is the roller follower probe rotation angle as the input variable of the fuzzy controller, and A _i is the input variable θ on the domain of discourse Fuzzy subset, F _i ′(θ) represents the cam rotation speed output of the i-th fuzzy rule, a _i0 and a _i1 are the subsequent parameters of the i-th fuzzy rule, F″(θ) is the cam of the entire fuzzy controller Rotation speed output, since there is only one input quantity θ, μ _i (θ) is A _i (θ), A _i (θ) is the satisfaction degree of the i-th fuzzy rule θ to the fuzzy subset A _i , h _i (θ ) is the membership function on the domain of input variables, that is, the antecedent parameter;

Step 7. By analyzing the curves of grinding wheel feed displacement X(θ) and acceleration a(θ), obtain the range of rotation angle θ of each roller follower probe, grinding wheel feed displacement X(θ), and grinding wheel feed Acceleration a(θ), cam rotation speed initial value b _i , cam rotation speed slope k _i basic parameter fuzzy rule table of the camshaft segment, and draw a schematic diagram of the cam rotation speed linear segment prediction model at the same time;

Step 8. Calculation of TS fuzzy controller consequent parameters: TS fuzzy controller consequent parameters a _i0 and a _i1 are determined by formula (28);

Among them, c _i is the starting value of the roller follower probe rotation angle of the ith segment in the range of each segment of the roller follower probe rotation angle in the basic parameters of the camshaft segment; k _i is the speed slope of each segment, b _i is the initial value of the cam rotation speed of each segment in ;

Step 9. Calculate T-S fuzzy controller antecedent parameter: T-S fuzzy controller antecedent parameter is determined by formula (31):

Substituting (31) into (30), and then substituting (30) into (29), the expression of the membership function h _i (θ) of the TS fuzzy controller is as follows:

h _i (θ) = f _tri (θ, c _i-1 , (c _i +c _i+1 )/2, c _i+2 ) (32)

In formula (32), θ is the rotation angle of the roller follower probe, c _i is the initial value of the rotation angle of the roller follower probe in the i-th segment, h _i (θ) is the membership function of the input variable universe , represents the membership degree of the fuzzy subset A _i when the roller follower probe turns to θ;

Step 10. Substitute equations (32) and (25) into equation (26), and use MATLAB software tool to fit the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ) after TS fuzzy control, and L ₁ The cam rotation speed F′(θ) and the cam rotation angle α(θ) predict the linear segment approximation curve and L ₂ The cam rotation speed F″(θ) after the TS fuzzy control is compared with the cam rotation angle α(θ) prediction curve, If the prediction curve of L ₂ cam rotation speed F″(θ) and cam rotation angle α(θ) is smooth and close to the curve of L ₁ cam rotation speed F′(θ) and cam rotation angle α(θ) prediction straight line segment curve, then the desired effect is achieved; Otherwise, it is necessary to optimize the TS fuzzy controller to make the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ) smoother and at the same time be as close as possible to L ₁ cam rotation speed F′(θ) Predict the straight line segment approaching the curve with the cam rotation angle α(θ);

Step 11. The optimization of the predicted value of the cam rotation speed is to use the adjacent fuzzy subset membership function to adjust the influence law of the output data in order to obtain a more ideal curve. There are mainly two situations:

Situation 1: Suppose only two adjacent fuzzy rules R ₁ and R ₂ are activated for a certain rotation angle θ ₁ of the roller follower probe, that is, the membership degree of θ ₁ only to fuzzy subsets A ₁ and A ₂ is not zero , get formula (33) from formula (26), when F ₂ ′(θ ₁ )>F ₁ ′(θ ₁ ), F″(θ ₁ )>F ₁ ′(θ ₁ ), by adjusting the adjacent membership The initial value c _i-1 of the rotation angle of degree function h ₂ (θ), see formula (31), if c _i-1 increases to (c _{i -1} +ci )/2, h ₂ (θ ₁ ) decreases , increase, decreases, thus F″(θ ₁ ) will decrease and approach F ₁ ′(θ ₁ ), so that the predicted speed approaches the ideal straight line segment, and the curve is optimized;

F ₁ ′(θ ₁ ), F ₂ ′(θ ₁ ) are the outputs of the cam rotation speed of the roller follower measuring head rotation angle θ ₁ on the fuzzy rules R ₁ and R ₂ respectively, and F″(θ ₁ ) is the input is the total output of the fuzzy controller when θ ₁ , h ₁ (θ ₁ ), h ₂ (θ ₁ ) are the membership degrees of the roller follower probe rotation angle θ ₁ to the fuzzy subsets A ₁ and A ₂ ;

Case 2: Assume that a certain rotation angle θ ₂ of the roller follower measuring head is used as the input of the fuzzy controller, and the cam rotation speed F″(θ ₂ ) is used as the output of the fuzzy controller, and the equation (34) is obtained from the equation (26), when When F ₁ ′(θ ₂ )<F ₂ ′(θ ₂ ), F″(θ ₂ )<F ₂ ′(θ ₂ ), it can be measured by adjusting the roller follower of the membership function h ₁ (θ) If the end value of head rotation angle c _i+2 is reduced to (c _i+1 +c _i+2 )/2, h ₁ (θ ₂ ) decreases, increase, Increase (F ₁ ′(θ ₂ )-F ₂ ′(θ ₂ ) is a negative number), so F″(θ ₂ ) will increase and tend to F ₂ ′(θ ₂ ), making the predicted speed approach the ideal Straight line segment, so that the curve is optimized.

F ₁ ′(θ ₂ ), F ₂ ′(θ ₂ ) are the outputs of the cam rotation speed of the roller follower measuring head rotation angle θ ₂ in the fuzzy rules R ₁ and R ₂ respectively, and F″(θ ₂ ) is the input is the total output of the fuzzy controller when θ ₂ , h ₁ (θ ₂ ), h ₂ (θ ₂ ) are the membership degrees of the roller follower probe rotation angle θ ₂ to the fuzzy subsets A ₁ and A ₂ ;

According to the above principles, the optimized formula from formula (32) after repeated tests is as follows:

h _i (θ)=f _tri (θ, ( _ci-1 + _ci )/s, ( _ci + _ci+1 )/s, ( _ci+1 + _ci+2 )/s) ( 35)

Among them, the value of s is between 1.90 and 2.10. According to the curve, it can be increased or decreased by 0.01 each time, and the L ₂ cam rotation speed F″(θ) and the cam rotation angle α(θ) can be adjusted to predict that the curve is smooth and close to the L ₁ cam rotation speed F '(θ) and the cam angle α(θ) predict the curve of the straight line segment.

Based on the above documents, the present invention analyzes and perfects the camshaft grinding mathematical model, and uses the T-S fuzzy controller to predict the cam rotation speed under the condition of providing specific grinding parameters and lift values. The ideal grinding accuracy, processing efficiency and surface quality are obtained.

As one of the important methods of intelligent control, fuzzy control imitates human decision-making process to a certain extent, among which T-S fuzzy control adopts piecewise linear model to approximate a nonlinear cam speed global fuzzy model. The introduction of fuzzy control in the grinding process of camshaft can maintain the stability of grinding and make the grinding surface quality achieve a good effect.

The technical solution of the present invention will be further described below in conjunction with the accompanying drawings.

Description of drawings

Figure 1 is the mathematical model of CNC camshaft grinding.

Fig. 2 is the simulation curve of grinding wheel feed and cam rotation motion.

Fig. 3 is a schematic diagram of the S-type acceleration and deceleration mode of the grinding wheel feed.

Fig. 4 is a schematic diagram of a linear segment model for predicting cam rotation speed.

Fig. 5 is a triangular membership function graph.

Fig. 6 is a prediction curve diagram of cam rotation speed F"(θ) and cam rotation angle a(θ).

Figure 7 shows the optimal front roller follower probe rotation angle θ and cam rotation speed F″(θ) in case 1, the adjacent fuzzy rules R ₁ , R ₂ cam rotation speeds F ₁ ′(θ) and F ₂ ′(θ ) graph.

Fig. 8 is a comparison diagram before and after optimization of the membership function h ₂ (θ) in case 1.

Figure 9 shows the optimized roller follower probe rotation angle θ and cam rotation speed F″'(θ), adjacent fuzzy rules R ₁ , R ₂ cam rotation speeds F ₁ ′(θ) and F ₂ ′( θ) graph.

Figure 10 shows the optimal front roller follower probe rotation angle θ and cam rotation speed F″(θ) in case 2, the adjacent fuzzy rules R ₁ , R ₂ cam rotation speeds F ₁ ′(θ) and F ₂ ′(θ ) graph.

Fig. 11 is a comparison diagram before and after optimization of the membership function h ₁ (θ) in case 2.

Figure 12 shows the optimized roller follower probe rotation angle θ and cam rotation speed F″′(θ), adjacent fuzzy rules R ₁ , R ₂ cam rotation speeds F ₁ ′(θ) and F ₂ ′( θ) graph.

Fig. 13 is a flow chart of the cam rotation speed prediction curve algorithm.

Fig. 14 is the linear segment approximation curve predicted by the cam rotation speed F'(θ) and the cam rotation angle a(θ).

Fig. 15 is a prediction curve of cam rotation speed F"(θ) and cam rotation angle a(θ).

Fig. 16 is a graph showing the distribution curve of the membership degree function for the prediction of the rotation speed of the cam.

Fig. 17 is a prediction curve of cam rotation speed F"'(θ) and cam rotation angle a(θ) after optimization.

Fig. 18 is a graph showing the distribution curve of the membership degree function for the prediction of the rotation speed of the cam after optimization.

Fig. 19 is a comparison chart of the predicted curve of cam rotation speed and cam angle a(θ) after repeated optimization and the original machinable curve.

Detailed ways

The steps and principle of the inventive method are as follows:

1). While the grinding wheel rotates at high speed, use the computer's numerical control program to control the grinding wheel feed system as the X axis and the cam rotation system as the C axis in the CNC camshaft grinder.

The mathematical model of CNC camshaft grinding is shown in Fig. 1. The grinding mathematical model of the camshaft is established according to the grinding point tangent point tracking, the principle of constant grinding removal rate and the shape of the cam profile, that is, while the grinding wheel rotates at high speed, the grinding wheel (X axis) of the camshaft grinding machine is controlled by the numerical control command to advance Grinding the surface of the cam profile is realized by the two-axis linkage with the rotation of the cam (C axis).

In Fig. 1, 1 is the cam, 2 is the grinding wheel, 3 is the measuring head of the roller follower, O is the center of the base circle of the cam, O ₁ is the center of the measuring head of the roller follower, O ₂ is the center of the grinding wheel, and A is the cam The midpoint of the base circle profile, which is the starting point of grinding wheel grinding. The measuring head of the roller follower rolls on the surface of the cam, and the running track of the roller center O ₁ is K _t . Draw a vertical line between OM and O ₁ O, extend O ₂ O ₁ , and intersect the cam and the grinding wheel at point P, and intersect with OM at point M, the tangential velocity at point P of the grinding wheel is v _s , and the tangential velocity at point P of the cam is v _j , the rotation speed of the cam is n ₁ , the rotation speed of the grinding wheel is n ₂ , the angular velocity at point P where the cam is tangent to the grinding wheel is ω(θ), and the angular velocity at which the cam rotates to the base circle is ω ₀ . The radius of the base circle of the cam is OA=r, the radius of the measuring head of the roller follower is PO ₁ =r ₁ , and the radius of the grinding wheel is O ₂ P=r ₂ . θ is the angle between the line connecting the midpoint A of the cam base circle profile to the cam base circle center O and the line connecting the probe roller center O ₁ to the cam base circle center O (that is, the roller follower probe rotation angle ), α(θ) is the angle between the line connecting the midpoint A of the cam base circle profile and the cam base circle center O and the line connecting the grinding wheel center O ₂ to the cam base circle center O, β(θ) is the angle in the cam base circle profile The angle between point A and the line connecting the cam base circle center O and the tangent point P between the cam profile and the grinding wheel to the line connecting the cam base circle center O, φ is the cam base circle center O to the probe roller center O ₁ The angle between the connection line of the cam and the line from the cam instant center M to the probe roller center O ₁ .

The follower (lifter) of the cam mechanism comes in three different forms: knife-edge lifter, roller lifter, and flat lifter. The knife-edge lifter can be regarded as the roller lifter when the roller radius is 0; the plane lifter can be regarded as the roller lifter when the roller radius is infinite. Therefore, only the mathematical model of the feed displacement of the grinding wheel in the form of the roller tappet is required to solve the problem of the feed displacement of the grinding wheel in the other two tappet forms. According to the several servo relations in the above figure, formulas (1) and (2) can be deduced, where formula (1) is the feed motion equation of the grinding wheel (X axis) (see "Optimization method for the rotational speed of the workpiece rotating shaft of CNC camshaft grinder", "Chinese Journal of Mechanical Engineering", 2014.15).

2). According to the principle of constant grinding removal rate and the relationship between arc micro-displacement and angular velocity, polar diameter and angle, formulas (3), (4) and (5) can be derived (MALKIN S. "Grinding technology Theory and applications of machining with abrasives ", Industrial Press Inc., 2008).

X(θ)=OO ₂ -r ₂ -r (1)

In the formula: r is the radius of the base circle of the cam, r ₁ is the radius of the probe of the roller follower, r ₂ is the radius of the grinding wheel, O is the center of the base circle of the cam, O ₁ is the center of the probe of the roller follower, and O ₂ is The center of the grinding wheel, φ is the angle between the line connecting the cam base circle center O to the probe roller center O ₁ and the cam instant center M to the probe roller center O ₁ , H(θ) is the cam Lift value, θ is the angle between the line connecting the cam base circle center point A to the cam base circle center O and the line connecting the cam base circle center O to the probe roller center O ₁ (that is, the roller driven ρ(θ) is the polar radius from the tangent point P between the cam profile and the grinding wheel to the cam base circle center O, β(θ) is the distance from the midpoint A of the cam base circle profile to the cam base circle center O The angle between the connection line and the tangent point P between the cam profile and the grinding wheel to the cam base circle center O, ω(θ) is the tangent point P between the cam profile and the grinding wheel when the cam rotates to the cam lift Angular velocity of rotation, _ω0 is the angular velocity of rotation of the tangent point P where the cam profile is tangent to the grinding wheel when the cam turns to the base circle of the cam, X(θ) is the displacement of the grinding wheel feed, O ₁ O ₂ =r ₂ -r ₁ , OO ₁ =r+H(θ)+r ₁ , a _p is the cut-in depth at the point where the cam rotates to α(θ), v _j is the linear velocity at point P of the cam at the point where the cam rotates to α(θ), v _s is the linear velocity at point P of the grinding wheel at the point where the cam rotates to α(θ), Q _W' is the removal rate per unit width. If the cam turns to a certain angle, the cam P point line speed It can guarantee that a _ep is a constant, which means that the metal removal rate per unit width is equal, that is, the grinding force is constant. Among them, the linear velocity v ₀ of a certain tangent point of the base circle, v ₀ =ω ₀ r, F(θ) is the rotational speed at which the cam rotates to the side head rotation angle θ of the roller follower, and n(θ) is the rotation speed at which the cam rotates to the side head rotation angle θ of the roller follower.

According to the cam lift value provided by the user (see appendix 1), the radius of the grinding wheel, the radius of the measuring head, the radius of the base circle of the cam and the formulas (1), (2), (4) and (5), using the MATLAB software tool, the formula (1), (2) Fit the displacement curve of grinding wheel feed X(θ) and cam angle α(θ) (see Figure 2(a)), the displacement curve formed by X(θ)-α(θ ) value is automatically generated by the programming software for the CNC machining subroutine of the grinding wheel feed, so as to realize the lateral reciprocating motion of the grinding wheel following the rotation of the cam. At the same time, using the MATLAB software tool, the cam rotation speed F( θ) and the cam rotation angle α(θ) speed curve (see Figure 2(d)), the F(θ)-α(θ) value formed by the speed curve is automatically generated by the programming software for the cam rotation NC machining subroutine to Realize cam rotation motion.

3). Calculate the first-order derivative, second-order derivative, and third-order derivative of formula (1), respectively, and obtain formulas (6), (7), and (8):

In the formula, v(θ) is the feed speed of the grinding wheel, a(θ) is the feed acceleration of the grinding wheel, and j(θ) is the feed jerk of the grinding wheel.

According to the cam lift value provided by the user (see Appendix 1), use MATLAB software tools to fit the formulas (1), (5), (6), (7) and formula (2) respectively, and the results are shown in Figure 2 The simulation curves of grinding wheel feed and cam rotation motion shown in Fig. 2(a) are the curves of grinding wheel feed displacement X(θ) versus cam rotation angle α(θ), and Fig. 2(b) is the curve of grinding wheel feed speed v(θ ) to the cam rotation angle α(θ) curve, Figure 2(c) is the curve of the grinding wheel feed acceleration a(θ) to the cam rotation angle α(θ), and Figure 2(d) is the cam rotation angular velocity ω(θ) to the cam rotation angle α (θ) curve. For 2) and (6), the speed curve of the grinding wheel feed speed v(θ) and the cam angle α(θ) can be fitted (see Figure 2(b)), and the equations (2), (7 ) can fit the acceleration curve of the grinding wheel feed acceleration a(θ) and the cam angle α(θ) (see Figure 2(c)); from Figure 2(d), it can be seen that: when the maximum rotation speed of the camshaft reaches 197991deg/min, when it exceeds the rotation speed of the base circle (36000deg/min) by about 5 times, that is, it exceeds the driving capacity of the camshaft rotation system, and the feed speed, acceleration, jerk, cam The rotation speed is limited as follows:

In the formula, v(θ _i ) is the feed speed of the grinding wheel in the i-th interpolation period, a(θ _i ) is the feed acceleration of the grinding wheel in the i-th interpolation period, and j(θ _i ) is the feed rate of the grinding wheel in the i-th interpolation period. Given the jerk, v _max is the allowable maximum speed of the grinding wheel feed, a _max is the allowable maximum acceleration of the grinding wheel feed, j _max is the allowable maximum jerk of the grinding wheel feed, ω _max is the maximum rotational angular velocity of the cam. In the S-type acceleration and deceleration control method, the displacement is used to calculate the interpolation period. Therefore, it is necessary to convert each rotation of the cam into a rotation period T _w , and then calculate the grinding wheel feed according to the relationship between the grinding wheel feed and the cam rotation. Period T _si . The conversion method is as follows:

In the formula, T _w is the interpolation period from the cam rotation to the lift section, T _si is the interpolation period of the grinding wheel feed, n(θ) is the rotation speed of the cam rotation to θ, ΔX(θ) is the T _si The feed displacement of the interpolation period grinding wheel, Δv(θ) is the feed speed of the grinding wheel at the T _si interpolation period, Δa(θ) is the feed acceleration of the T _si interpolation period, and Δj(θ) is the T _si Grinding wheel feed jerk for an interpolation period, F(θ) is the predicted speed from the cam rotation to the lift section.

4). From the feed displacement curve of the grinding wheel in Figure 2(a), it can be seen that the curve can be divided into 6 segments: AB, BC, CD, DE, EF, and FG, where AB and FG are base circle segments, and the grinding wheel does not feed. The feed of the grinding wheel in the BC section is from the base circle of the cam to the middle of the cam lift (that is, the acceleration section), and the grinding wheel in the CD section is from the middle of the cam lift to the cam lobe (that is, the deceleration section); the DE section is from the cam lobe to the middle of the cam lift ( That is, the acceleration section), and return from the middle of the EF cam lift to the cam base circle (that is, the deceleration section). Then, according to the S-type acceleration and deceleration control method, it is decomposed into five acceleration and deceleration modes. Figure 3 shows four acceleration and deceleration control modes, of which Figure 3(a) is the acceleration mode 1, which only has two stages of acceleration and deceleration; Figure 3 (b) is acceleration mode 2, after three stages of acceleration, uniform acceleration, and acceleration and deceleration; Figure 3(c) is deceleration mode 1, only deceleration and deceleration, deceleration and acceleration two stages; Figure 3(d) is deceleration mode 2 , after deceleration and deceleration, uniform deceleration and deceleration and acceleration three stages, the fifth is the constant speed mode. The curves of feed speed, acceleration and jerk are drawn respectively in the figure. The advantage of this method is that when the grinding wheel is fed to a certain target position, the acceleration is zero, which can reduce mechanical impact, (see "Numerical Control Camshaft Grinding Machine Workpiece Rotational Axis Speed Optimization method", "Chinese Journal of Mechanical Engineering", 2014.15). We calculate the equations (15), (16) and (17), (18), (19), and (20) respectively to obtain the interpolation period T _si of acceleration mode 1 and acceleration mode 2. Because of the reversibility of acceleration and deceleration, it is only necessary to discuss the interpolation of acceleration mode 1 and acceleration mode 2, and the entire interpolation period can be obtained, that is, the rotation speed of the cam (C axis) can be predicted.

Acceleration mode 1: see Fig. 3(a), a _i ≤ a _max , divide each interpolation cycle into two stages, according to Fig. 3(a), the following formula can be obtained:

ΔX(T _si ) is the displacement of the grinding wheel per rotation of 1 degree, t _i1 is the jerk phase time, t _i2 is the deceleration phase time, v _i-1 is the initial feed speed, j _max is the maximum jerk, s _i1 (t _i1 ) is the feed displacement of the grinding wheel at stage t _i1 , s _i2 (t _i2 ) is the feed displacement of the grinding wheel at stage t _i2 , a _i is the final acceleration of t _i1 , v _i1 is the final velocity of t _i1 , and v _i2 is t The end speed of _i2 , T _si is an interpolation cycle.

Acceleration mode 2: see Figure 3(b), a _i ≥ a _max , each interpolation cycle is divided into three stages, according to Figure 3(b), the following formula can be expressed:

a _max t _i2 ² +(j _max t _i1 ² +2a _max t _i1 +2v _i-1 )t _i2 +4v _i-1 t _i1 +j _max t _i1 ³ +a _max t _i1 ² -2ΔX(T _si ) =0 (18)

t _i1 =t _i3 =a _max /j _max (19)

T _si =t _i1 +t _i2 +t _i3 =2t _i1 +t _i2 (20)

Among them: ΔX(T _si ) is the displacement of the grinding wheel per rotation of 1 degree, t _i1 is the time of jerk and acceleration phase, t _i2 is the time of uniform acceleration phase, t _i3 is the time of deceleration and acceleration phase, v _i-1 is the initial progress Giving speed, j _max is the maximum jerk, s _i1 (t _i1 ) is the feed displacement of the t _i1 grinding wheel, s _i2 (t _i2 ) is the feed displacement of the t _i2 grinding wheel, s _i3 (t _i3 ) is the t _i3 segment Grinding wheel feed displacement, a _i is the end acceleration of t _i1 , a _max is the maximum acceleration, v _i1 is the end speed of t _i1 , v _i2 is the end speed of t _i2 , v _i3 is the end speed of t _i3 , T _si is an interpolation cycle .

5). Since the grinding wheel feed and the cam rotation are two-axis linkage, that is, the interpolation period T _si of the grinding wheel feed is the cam rotation interpolation period T _w . By formula (21), (22) can obtain the cam rotation speed value.

T _w = T _si (21)

Among them, T ₀ is the base circle interpolation period of the cam rotation, n ₀ is the base circle speed, the value is 100rmp, T _w is the interpolation period from the cam rotation to the lift section, and F′(θ) is the prediction of the cam rotation to the lift section speed.

(6). According to the basic principle of TS fuzzy control, the discourse domain of each input variable is divided into several fuzzy subsets, and the fuzzy subsets of each input variable are combined with each other according to certain experience to establish an input-output linear relational expression, which is determined The fuzzy implication relation R _i "If the input...then the output..." (If...then...), is a fuzzy rule; use a local segmental linear model to approximate any global non-linear function. Its expression is as follows (see the literature "TS Fuzzy Controller Design and Optimization Method Research", Yan Xiaoxi, 2007.05):

R _i : if z ₁ is A _i1 and z ₂ is A _i2 ......z _j is A _ij ......and z _n is A _in , then F _i (z)＝a _i0 +a _i1 z ₁ +a _i2 z ₂ +...+a _ij z _j +...+a _in z _n (23)

Among them, i(i=1,2,...,n) represents the number of fuzzy rules, j(j=1,2,...,n) represents the number of input variables, R _i represents the i-th fuzzy rule , z=[z ₁ , z ₂ ,..., z _j ,..., z _n ] ^T represents the input vector of the fuzzy controller. A _ij (z _j ) is a fuzzy subset, F _i (z) represents the output of the i-th fuzzy rule, a _i0 , a _i1 ,..., a _in are the subsequent parameters of the i-th fuzzy rule, F( z) is the output of the fuzzy controller, μ _i (z) is the degree of satisfaction of the i-th fuzzy rule defined as the product form, A _ij (z _j ) is the degree of satisfaction of z _j to A _ij , h _ij (z _j ) is the membership function on the domain of input variables.

According to equations (23) and (24), a one-dimensional fuzzy controller model with single input (roller follower probe rotation angle θ) and single output (cam rotation speed F″(θ)) can be established, that is, the T-S fuzzy controller can be The simplification is as follows.

R _i : if θ is A _i then F _i ′(θ)＝a _i0 +a _i1 θ (25)

Among them, R _i (i=1, 2, ... 12) (see Table 1) represents the i-th fuzzy rule, and θ is the rotation angle of the roller follower probe as the input variable of the fuzzy controller. A _i is a fuzzy subset, F _i ′(θ) is the cam rotation speed output of the i-th fuzzy rule, a _i0 and a _i1 are the subsequent parameters of the i-th fuzzy rule, F″(θ) is the entire fuzzy control The cam rotation speed output of the device, since there is only one input variable, μ _i (θ) is directly expressed as the satisfaction degree of the i-th fuzzy rule θ to A _i , A _i (θ) is the satisfaction degree of θ to A _i , h _i ( θ) is the membership function on the domain of input variables.

7). Through the curve of grinding wheel feed displacement X(θ) and cam rotation angle α(θ) (see Figure 2(a)) and the curve of grinding wheel feed acceleration a(θ) and cam rotation angle α(θ) (see Figure 2 (c)), the range of rotation angle θ of the measuring head of each section of roller follower, the feed displacement X(θ) of the grinding wheel, the feed acceleration a(θ) of the grinding wheel, the initial value of the cam rotation speed b _i , Camshaft segmentation basic parameter fuzzy rule table of cam rotation speed slope _ki (see Table 1). According to Table 1, the schematic diagram of cam rotation speed prediction linear segment model shown in Fig. 4 can be drawn.

From the simulation curve of the grinding wheel feed acceleration a(θ) and the cam rotation angle α(θ) (see Figure 2(c)), it can be seen that the grinding wheel feed acceleration curve can be divided into A ₁ , A ₂ ,..., A ₁₂ segment (see column 1 in Table 1), A ₁ , A ₂ , A ₁₁ , and A ₁₂ are cam base circle segments, A ₁ and A ₁₂ segment cam rotation speeds are generally 36000deg/min, A ₂ segment is the base circle to cam In the lifting section, the speed should drop from b ₂ (36000deg/min) to b ₃ rapidly. Section A ₃ is the slow acceleration section of the cam lift (see Figure 2(c)), and the speed should drop slowly from b ₃ to b ₄ . Section A ₄ is the fast acceleration section of the cam lift, and the speed should drop from b ₄ to b ₅ . Section A ₅ is the rapid deceleration section of the cam lift, and the speed increases from b ₅ to b ₆ . Section A ₆ is the slow-speed deceleration section of the cam lift, and the speed increases from b ₆ to b ₇ . Section A ₇ is the slow acceleration section of the cam lift, and the speed decreases from b ₇ to b ₈ . Section A ₈ is the rapid acceleration section of the cam lift, and the speed decreases from b ₈ to b ₉ . Section A ₉ is the rapid deceleration section of the cam lift, and the speed increases from b ₉ to b ₁₀ . Section A ₁₀ is the slow deceleration section of the cam lift, and the speed increases from b ₁₀ to b ₁₁ . Section A ₁₁ is the section from the end of the cam lift to the base circle, and the speed increases rapidly from b ₁₁ to b ₁₂ (36000deg/min). Finally, the ₁₂ sections of the base circle A are completed, and according to the roller follower measuring head rotation angle θ, X(θ) displacement curve, and a(θ) acceleration curve, the rules in columns 2, 3, and 4 of Table 1 are obtained, that is, the cam cycle is completed. processing process. In column 5 of Table 1, the initial value b _i of the rotation speed of each segment is obtained by the acceleration and deceleration control method, and the slope of the rotation speed in column 6 of Table 1 is shown in the notes below Table 1.

Table 1 Fuzzy rule table of basic parameters of camshaft segmentation

Note: A _i is the fuzzy subset of the i-th segment; c _i is the initial value of the roller follower probe rotation angle of the i-th segment; i=1, 2, ..., 12; among them,

k _i =(b _i+1 -b _i )/(c _i+1 -c _i ); k ₁ =0; k ₁₂ =0; b ₁ =b ₂ =b ₁₂ =36000, b ₄ =b ₄ = b ₈ =b ₁₀ , b ₁₁ =b ₃ , b ₅ =b ₉ .

8). Calculation of TS fuzzy controller subsequent parameters a _i0 , a _i1 : Substitute the parameters in columns 2, 5, and 6 of Table 1 into formula (25);

F _i ′(θ)=b _i +k _i (θ-c _i )=k _i θ+b _i -k _i c _i (27)

From formulas (25) and (27), the following formula can be obtained:

Where i=1, 2, ..., 12; c _i is the initial roller follower probe rotation angle in the _i -th column in column 2 of Table 1, and ki is the slope of the corresponding segment in column 6 of Table 1 , b _i is the initial value of the cam rotation speed corresponding to the fifth column of Table 1.

9). Calculate the antecedent parameters of the TS fuzzy controller: the membership function h _i (θ) of the TS fuzzy controller is:

h _i (θ) = f _tri (θ, x ₁ , x ₂ , x ₃ ) (29)

Where θ is the rotation angle of the roller follower probe, f _tri (θ, x ₁ , x ₂ , x ₃ ) is the triangular membership function, h _i (θ) is the membership function, which means that the roller follower probe has a certain The degree of membership (satisfaction degree) of each rotation angle θ to the fuzzy subset A _i , x ₁ , x ₂ , x ₃ are the antecedent parameters of the fuzzy controller, and x ₁ ≤ x ₂ ≤ x ₃ ;

The present invention adopts the triangular membership function, and the triangular membership function expression is (seeing document " fuzzy control and MATLAB simulation thereof ", Shi Xinmin etc., Tsinghua University Press, 2008.03):

Among them, θ is the input variable, x ₁ , x ₂ , and x ₃ are the parameters of the triangular membership function, x ₁ ≤ x ₂ ≤ x ₃ ; the curve of the triangular membership function is shown in Figure 5. It can be seen from the figure that x ₃ -x The smaller the value of ₁ , the sharper the function shape;

The antecedent parameter of T-S fuzzy controller of the present invention is determined by formula (31):

Substituting (31) into (30), and then substituting (30) into (29), the membership function h _i (θ) of the TS fuzzy controller can be written as:

h _i (θ) = f _tri (θ, c _i-1 , (c _i +c _i+1 )/2, c _i+2 ) (32)

In formula (32), θ is the rotation angle of the roller follower probe, c _i is the initial value of the rotation angle of the roller follower probe in the second column of Table 1, h _i (θ) is the input variable theory The membership function on the domain indicates the membership degree (satisfaction degree) of a certain rotation angle θ of the roller follower probe to the fuzzy rule i.

10). Substitute equations (25) and (32) into equation (26), and use the MATLAB software tool to fit the curve of the cam rotation speed F″(θ) and the roller follower measuring head rotation angle θ. Then rotate the cam The speed F″(θ) and the cam rotation angle α(θ) are fitted to obtain the prediction curve of the cam rotation speed F″(θ) and the cam rotation angle α(θ) after the T-S fuzzy controller. Figure 6 shows the cam rotation speed F″(θ ) and cam angle α(θ) prediction curve.

In Fig. 6, L ₁ represents the cam rotation speed F′(θ) and the cam rotation angle a(θ) predicted linear segment approximation curve, and L ₂ represents the cam rotation speed F″(θ) and the cam rotation angle a(θ) after TS fuzzy control ) forecast curve).

It can be seen from Figure 6 that the predicted curves of cam rotation speed F″(θ) and cam rotation angle α(θ) after the output of the TS fuzzy controller are not close to the cam rotation speed F′(θ) and cam rotation angle α(θ) in sections A ₆ and A ₇ The rotation angle α(θ) predicts the straight line segment and the curve is not smooth enough. In order to make the cam rotation speed F″(θ) close to the predicted straight line segment (Figure 6 curve L ₁ ) while maintaining the cam rotation speed F″(θ) and the cam rotation angle α( To smooth the prediction curve of θ), it is necessary to optimize the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ) after TS fuzzy control (curve L ₂ in Figure 6). 11). The optimization of the predicted value of the cam rotation speed is to adjust the value of the cam rotation speed F″(θ) by adjusting the antecedent parameters (membership function h _i (θ)) of the adjacent fuzzy rules, so as to optimize the output of the fuzzy controller The curve makes it smooth while being closer to the ideal state, mainly in the following two situations:

Situation 1: Fig. 7 shows the optimized front roller follower probe rotation angle θ and cam rotation speed F″(θ), adjacent fuzzy rules R ₁ , R ₂ cam rotation speed F ₁ ′(θ) and F ₂ ′(θ ) graph.

Assume that a certain rotation angle θ ₁ of the roller follower probe is used as the input of the fuzzy controller, and the cam rotation speed F′(θ ₁ ) is used as the output of the fuzzy controller. F ₁ ′(θ ₁ ), F ₂ ′(θ ₁ ) are the cam rotation speeds of a certain rotation angle θ ₁ of the roller follower measuring head on the fuzzy rules R ₁ and R ₂ respectively. From formula (26) to formula (33), that is to say, the relational formula of the cam rotation speed F″(θ ₁ ) output by the fuzzy controller is as follows:

Among them: h ₁ (θ ₁ ), h ₂ (θ ₁ ) represent the degree of membership (satisfaction) of a certain rotation angle θ ₁ of the roller follower probe to the fuzzy subsets A ₁ and A ₂ , by adjusting h ₂ ( θ ₁ ) to adjust the value of F″(θ ₁ ), so that the output curve of the fuzzy controller is closer to the ideal state (that is, the predicted linear segment model F′(θ)), the specific adjustment steps are as follows:

If the intersection of the straight line segments adjacent to the fuzzy rules R ₁ and R ₂ in Fig. 7 of the cam rotation speeds F ₁ ′(θ) and F ₂ ′(θ) is taken as the boundary, it can be known from the left half of Fig. 7 that F″(θ ₁ )>F ₁ ′(θ ₁ ), and the cam rotation speed F ₂ ′(θ ₁ ) on the adjacent fuzzy rule R ₂ is greater than the cam rotation speed F ₁ ′(θ ₁ ) on the fuzzy rule R ₁ , that is, F ₂ ′ (θ ₁ )＞F ₁ ′(θ ₁ ), you can adjust the initial value c _i-1 of the roller follower probe rotation angle of the membership function h ₂ (θ), if it is increased to ( _ci-1 +c _i )/2 (see Figure 8), h ₂ (θ ₁ ) becomes smaller to h ₂ ′(θ ₁ ), substituting into formula (33), increase, As a result, F″(θ ₁ ) will decrease and become closer to F ₁ ′(θ ₁ ). Similarly, the right half of Figure 7 can also reduce F by adjusting the membership function h ₁ (θ). "(θ) brings the right half closer to F ₂ '(θ).

Combining with the above optimization rules, optimize the left half of Fig. 7 to obtain the F″′(θ) curve output by the fuzzy controller for the optimized roller follower measuring head rotation angle θ in case 1. Fig. 9 is the optimized roller follower measured The head rotation angle θ fuzzy controller outputs F″'(θ), and the adjacent fuzzy rules R ₁ and R ₂ output F ₁ ′(θ) and F ₂ ′(θ) curves.

In Fig. 7, straight line A represents the straight line segment of F ₁ ′(θ) of fuzzy rule R ₁ , straight line B represents the straight line segment of F ₂ ′(θ) adjacent to fuzzy rule R ₂ , and curve C represents the output F” of the fuzzy controller (θ)).

The straight line A in Fig. 8 is the membership degree function h ₂ (θ) before optimization, and the straight line B is the membership degree function h ₂ '(θ) after optimization).

In Fig. 9, straight line A represents the straight line segment of F ₁ ′(θ) of fuzzy rule R ₁ , straight line B represents the straight line segment of F ₂ ′(θ) adjacent to fuzzy rule R ₂ , and curve C represents the output of fuzzy controller after optimization F"'(θ)).

Situation 2: As shown in Figure 10, another type of roller follower probe rotation angle θ and cam rotation speed F″(θ), adjacent fuzzy rules R ₁ , R ₂ cam rotation speed F ₁ ′(θ) and Graph of F ₂ '(θ).

Also assume that a certain rotation angle θ ₂ of the roller follower measuring head in Fig. 10 is used as the input of the fuzzy controller, and the cam rotation speed F″(θ ₂ ) is used as the output of the fuzzy controller. F ₁ ′(θ ₂ ), F ₂ ′ (θ ₂ ) is the rotation speed of the cam at a certain rotation angle θ ₂ of the roller follower measuring head on the fuzzy rules R ₁ and R _2. From the formula (26), the formula (34) is obtained, that is, the output cam of the fuzzy controller The relational formula of rotation speed F″(θ ₂ ) is as follows:

Among them: h ₁ (θ ₂ ), h ₂ (θ ₂ ) represent the degree of membership (satisfaction) of a certain rotation angle θ ₂ of the roller follower probe to the fuzzy subsets A ₁ and A ₂ , by adjusting h ₁ ( θ ₂ ) to adjust the value of F″(θ ₂ ), so that the output curve of the fuzzy controller is closer to the ideal state (that is, the predicted linear segment model F′(θ)), the specific adjustment is:

If the intersection point of the straight line segment of the cam rotation speed F ₁ ′(θ) and F ₂ ′(θ) of the fuzzy rules R ₁ and R ₂ is used as the boundary, a certain rotation angle θ of the roller follower measuring head in the right half of Fig. 10 ₂ ’s output F″(θ ₂ )<F ₂ ′(θ ₂ ), and the output F ₁ ′(θ ₂ ) on the adjacent fuzzy rule R ₁ is smaller than the output F ₂ ′(θ ₂ ) on the fuzzy rule R ₁ ), that is, F ₁ ′(θ ₂ )<F ₂ ′(θ ₂ ), which can be adjusted by adjusting the end value c _i+2 of the roller follower probe rotation angle of the membership function h ₁ (θ), if it is reduced to ( c _i+1 +c _i+2 )/2 (as shown in Figure 11), then h ₁ (θ ₂ ) decreases to h ₁ ′(θ ₂ ), increase, increase (F ₁ ′(θ ₂ )-F ₂ ′(θ ₂ ) is a negative number), thus F″(θ ₂ ) will increase and tend to F ₂ ′(θ ₂ ). Similarly, for The left half of 10 can also adjust the membership function h ₂ (θ), so that the increase of F″(θ) makes the left half closer to F ₁ ′(θ).

Combined with the above rules, the left half and right half of Figure 10 can be optimized to obtain the F"'(θ) curve output by the fuzzy controller for the optimized roller follower probe angle θ in case 2. Figure 12 is the optimized case 2 Curves of roller follower probe rotation angle θ versus cam rotation speed F″′(θ), adjacent fuzzy rules R ₁ , R ₂ cam rotation speeds F ₁ ′(θ) and F ₂ ′(θ).

In Fig. 10, straight line A represents the straight line segment of F ₁ ′(θ) of fuzzy rule R ₁ , straight line B represents the straight line segment of F ₂ ′(θ) adjacent to fuzzy rule R ₂ , and curve C represents the output F” of the fuzzy controller (θ)).

The straight line A in Fig. 11 is the membership degree function h ₁ (θ) before optimization, and the straight line B is the membership degree function h ₁ '(θ) after optimization).

In Fig. 12, straight line A represents the straight line segment of F ₁ ′(θ) of fuzzy rule R ₁ , straight line B represents the straight line segment of F ₂ ′(θ) adjacent to fuzzy rule R ₂ , and curve C represents the output of fuzzy controller after optimization F"'(θ)).

According to the above principles, the optimized formula from formula (32) after repeated tests is as follows:

h _i (θ)=f _tri (θ, ( _ci-1 + _ci )/s, ( _ci + _ci+1 )/s, ( _ci+1 + _ci+2 )/s) ( 35)

Among them, s is the optimization adjustment coefficient, the value is between 1.90-2.10, generally ₂ , and can be increased or decreased by 0.01 each time according to the curve, and the L2 cam rotation speed F″(θ) and cam rotation angle α(θ) in Figure 6 can be adjusted ) prediction curve is smooth and close to L ₁ cam rotation speed F'(θ) and cam angle α(θ) prediction line section curve.

The algorithm flowchart of the present invention is shown in FIG. 13 .

12). Take the machining and grinding of a camshaft provided by a company as an example to calculate. It is known that the camshaft adopts flat probe roller r ₁ =999999; cam base circle radius r=17.5; grinding wheel radius r ₂ =175, cam rotation speed n ₁ =100rmp, grinding wheel speed n ₂ =4367rmp. The cam lift value is shown in Appendix 1.

① Speed limit calculation

From formula (10):

Angular velocity at lift point:

From formula (22):

Before entering the starting point of the lift, the cam speed should be reduced in advance to ensure the smooth operation of the cam.

② According to Appendix 1 and the S-type acceleration and deceleration method, the segmented parameter table of the camshaft is shown in Table 2.

Before entering the starting point of the cam lift, the cam speed should be reduced to 8737deg/min in advance to ensure the smooth operation of the cam. Substituting the above conditions into formula (1), calculating the first derivative, second derivative and third derivative of formula (1), the maximum feeding speed of the grinding wheel v _max = 0.252mm/deg; the maximum acceleration a _max = 0.024mm can be calculated /deg ² ; maximum jerk j _max =0.0085mm/deg ³ . Then according to formulas (4), (11), (12), (13), (14), the maximum feeding speed of the grinding wheel v _max = 87mm/s; the maximum acceleration a _max = 1795mm/s ² ; the maximum jerk j _max =150791 mm/s ³ . According to the cam lift value and the acceleration and deceleration control method, the speed value in column 7 of Table 2 can be calculated.

Table 2 Camshaft Segment Parameter Table

According to the camshaft segmented parameter table in Table 2, combined with the schematic diagram of the cam rotation speed prediction linear segment model (see Figure 4), the cam rotation speed F′(θ) and the cam rotation angle α(θ) can be fitted by MATLAB software tools The predicted straight line segment approximation curve is shown in Figure 14.

③ Substituting the parameters in column 2 of Table 1 into formula (32) to get formula (36), since A ₁ and A ₁₂ are horizontal straight line segments, and the slopes of A ₂ and A ₁₁ are quite different from those of other segments , in order to avoid the slopes of A ₂ and A ₁₁ from affecting other segments, the membership degree of other segments in the fuzzy subsets A ₂ and A ₁₂ is 0, so h ₂ (θ) and h ₁₂ (θ) are determined as right triangle membership degrees function.

Substituting the parameter starting angle c _i in the third column of Table 2 into formula (36), the following formula is obtained:

④ Substitute the parameters in columns 3, 7, and 8 in Table 2 into formula (28), then substitute formula (28) into formula (25), then substitute formulas (25) and (37) into formula (26), use MATLAB Fitting equations (26) and (2), the prediction curve of cam rotation speed F"(θ) and cam rotation angle α(θ) can be obtained as shown in Figure 15.

In Fig. 15, L ₁ represents the cam rotation speed F′(θ) and the cam rotation angle a(θ) predicted linear segment approximation curve, and L ₂ represents the cam rotation speed F″(θ) and the cam rotation angle a(θ) after TS fuzzy control ) forecast curve).

In addition, substituting formula (37) into formula (30) and using MATLAB to simulate, the distribution curve of membership function can be obtained as shown in Figure 16.

It can be seen from Fig. 15 that after TS fuzzy control, the prediction curve of cam rotation speed F"(θ) and cam rotation angle α(θ) (curve L ₂ in Fig. 15) is not close to the cam rotation speed prediction line segment in A ₆ and A ₇ curve, and the curve is not smooth enough, it is necessary to optimize the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ), so that the optimized cam rotation speed F″’(θ) and cam rotation angle α(θ) prediction The curve is as close as possible to the cam rotation speed prediction line segment curve (curve L ₁ in FIG. 15 ) while ensuring smoothness.

⑤ According to the optimization method in step 11, formula (38) can be obtained according to formula (35), so as to optimize the cam rotation speed prediction curve in Fig. 15 .

⑥

Substituting the parameter starting angle c _i in the third column of Table 2 into formula (38) can be obtained:

Similarly, substitute the parameters in columns 3, 7, and 8 of Table 2 into formula (28), then substitute formula (28) into formula (25), then substitute formulas (25) and (39) into formula (26), use MATLAB Fitting equations (26) and (2) to obtain the prediction curve of cam rotation speed F″'(θ) versus cam rotation angle α(θ) after optimization is shown in Figure 17.

Substitute Equation (38) into Equation (30), and perform MATLAB simulation on Equation (30) to obtain the optimized membership function h _i (θ) distribution curve as shown in Figure 18.

It can be seen from the above figure that the cam rotation speed prediction curve obtained by the optimized membership function through fuzzy control is smoother and closer to the linear segment approximation curve of the cam rotation speed. On the basis of the actual effect, the membership function is continuously tested and optimized to obtain a more reasonable curve.

⑥Result Analysis

According to the above method of optimizing the cam rotation speed prediction curve (step 11), through continuous optimization, use MATLAB to fit the cam rotation speed F"'(θ) and the cam rotation angle α(θ) to obtain the cam rotation speed as shown in Figure 19 F″'(θ) vs. cam rotation angle α(θ) prediction curve (curve B in Fig. 19), compare the simulation result diagram obtained by the final optimization with the curve diagram of part of the original method that can realize processing. Curve A is the camshaft rotation speed curve that can be processed, and curve B is the cam rotation speed F"'(θ) and cam rotation angle α(θ) prediction curve finally optimized in the present invention.

The efficiency of the cam rotation speed processing curve of the original method has been significantly improved, and the stability is better.

Applications

Now take the Toyota Koki CNC camshaft grinder modified by a company as the test equipment, and the camshaft profile detector as the testing equipment, and process and grind the camshaft of a certain company as an example. It is known that the camshaft adopts flat probe roller r ₁ =999999; cam base circle radius r=17.5; grinding wheel radius r ₂ =175, cam speed n ₁ =100rmp, and grinding wheel speed n ₂ =3000rmp. The cam lift value is shown in Appendix 1.

Through the C# language design in Visual Studio 2008, design the user parameter setting interface, displacement and speed subroutine generation software, and transplant the software to the Siemens 840D SL CNC system of the Operate Programming Package, just input the cam lift value, grinding wheel radius, roller The parameters of the radius of the follower probe and the radius of the base circle can automatically generate the subroutine of the grinding wheel displacement NC machining and the cam rotation NC machining subroutine of the NC machining through the software. The simulation curve of the cam rotation NC machining subroutine is shown in Figure 19. A is the subroutine simulation curve of cam rotation NC machining generated by the original method, and B is the subroutine simulation curve of cam rotation NC machining generated by this method. After grinding and testing, the test results are shown in Appendix 3. It can be seen from the fourth column of appendix 3 that the cam lift error is less than 0.02mm, which is better than the national standard (see appendix 4). Appendix 4 is a comparison table of technical performance. From the table, it can be seen that the performance index of the CNC camshaft grinder modified by Jinan Maiteli Company is equivalent to that of the CNC camshaft grinder of JUNKER Company in Germany. The specific method is as follows:

1. Input the data in the cam lift value (seeing appendix 1), the radius of the grinding wheel, the radius of the roller follower measuring head, and the radius of the base circle to the user interface of the present invention, and obtain the displacement control of the grinding wheel by the above-mentioned step 2) The NC machining subroutine of the NC machining subroutine can obtain the X-axis displacement value curve of the grinding wheel holder (see Figure 2 (a, b, c)) and the NC machining subroutine of the grinding wheel displacement control.

2. The data in the camshaft lift value (seeing appendix 1), the grinding wheel radius, the roller follower measuring head radius, and the base circle radius value are input to the user interface of the present invention, and the correction obtained by the above step 7) The NC machining subroutine for cam rotation speed control can obtain the cam rotation speed NC subroutine graph (see Figure 19) and the C-axis rotation speed subroutine of the cam, and can also display the minimum value of the cam rotation speed at the same time.

Appendix 1 The cam lift value provided by the manufacturer

Appendix 2 Acceleration and deceleration algorithm to calculate the predicted value of cam rotation speed

Appendix 3 Cam machining error detection table

Appendix 4 Technical Performance Comparison Table

Claims (1)

1. The camshaft grinding method based on T-S fuzzy control is realized by the following steps: Step 1. While the grinding wheel rotates at high speed, use the numerical control program of the computer to control the grinding wheel feed system as the X axis and the cam rotation system as the C axis in the CNC camshaft grinder; Step 2. According to the cam lift value, grinding wheel radius, probe radius, cam base circle radius and formulas (1), (2), (4) and (5) provided by the user, use the MATLAB software tool to formulate the formula (1) , (2) Fit the displacement curve of the grinding wheel feed X(θ) and the cam angle α(θ), and the X(θ)-α(θ) value formed by the displacement curve is automatically generated by the programming software for the numerical control of the grinding wheel feed The subroutine is processed to realize the lateral reciprocating motion of the grinding wheel following the rotation of the cam. At the same time, the speed curve of the cam rotation speed F(θ) and the cam rotation angle α(θ) is fitted by the formula (2) and (5) using MATLAB software tools. The F(θ)-α(θ) value formed by the speed curve is automatically generated by the programming software for the cam rotation numerical control machining subroutine to realize the cam rotation movement; X(θ)=OO ₂ -r ₂ -r (1) In the formula: X(θ) is the displacement of the grinding wheel feed, r is the radius of the base circle of the cam, O is the center of the base circle of the cam, O ₁ is the center of the measuring head of the roller follower, O ₂ is the center of the grinding wheel, φ is the angle between the line connecting the cam base circle center O to the probe roller center O ₁ and the cam instant center M to the probe roller center O ₁ , O ₁ O ₂ =r ₂ -r ₁ , r ₂ is the radius of the grinding wheel, r ₁ is the radius of the roller follower probe, OO ₁ = r+H(θ)+r ₁ , H(θ) is the lift value of the cam, θ is the cam base circle profile The angle between the line connecting point A to the cam base circle center O and the line connecting the cam base circle center O to the probe roller center O ₁ , α(θ) is the midpoint A of the cam base circle profile and the cam base circle The angle between the line connecting the center O and the center O ₂ of the grinding wheel to the center O of the base circle of the cam, ρ(θ) is the polar radius from the tangent point P between the cam profile and the grinding wheel to the center O of the base circle of the cam, β(θ ) is the angle between the line connecting the midpoint A of the cam base circle profile to the center O of the cam base circle and the tangent point P between the cam profile and the grinding wheel to the line O of the cam base circle center O, ω(θ) is the cam rotation Rotation angular velocity of the tangent point P between the cam profile and the grinding wheel when the cam is lifted, _ω0 is the angular velocity of the tangent point P between the cam profile and the grinding wheel when the cam rotates to the base circle, F(θ) is the cam rotation to the roller from The rotation speed at the side head rotation angle θ of the movable member, n(θ) is the rotation speed from the rotation of the cam to the side head rotation angle θ of the roller follower; Step 3. Calculate the first-order derivative, the second-order derivative, and the third-order derivative respectively to the grinding wheel feed displacement formula (1) to obtain formulas (6), (7), and (8); In the formula, v(θ) is the feed speed of the grinding wheel, a(θ) is the feed acceleration of the grinding wheel, and j(θ) is the feed jerk of the grinding wheel; Use MATLAB software tools to fit equations (2) and (6) to get the speed curve of the grinding wheel feed speed v(θ) and cam angle α(θ), and use MATLAB software tools to fit equations (2) and (7) Get the acceleration curve of the grinding wheel feed acceleration a(θ) and the cam rotation angle α(θ); the maximum rotation speed of the camshaft exceeds the rotation speed of the base circle (36000deg/min) by more than 1.2 times, according to formulas (9) and (10) for the grinding wheel Feedrate, acceleration, jerk, cam rotation speed are limited: In the formula, v(θ _i ) is the feed speed of the grinding wheel in the i-th interpolation period, a(θ _i ) is the feed acceleration of the grinding wheel in the i-th interpolation period, and j(θ _i ) is the grinding wheel speed in the i-th interpolation period Feed jerk, k is the speed limit ratio, v _max is the allowable maximum speed of grinding wheel feed, a _max is the allowable maximum acceleration of grinding wheel feed, j _max is the allowable maximum jerk of grinding wheel feed, ω _max is the maximum rotational angular velocity; Step 4. According to the S-type acceleration and deceleration control method, the cam lift value, the formula (16) of acceleration mode 1 and the formulas (18), (19) and (20) of acceleration mode 2, solve the grinding wheel feed interpolation period T _si and the interpolation cycle of the cam return to predict the time it takes for the cam to rotate 1 degree for the grinding wheel feed; Calculate the interpolation period T _si of acceleration mode 1 by formula (16): ΔX(T _si ) is the displacement of the grinding wheel feed per rotation of 1 degree, t _i1 is the acceleration phase time, t _i2 is the deceleration phase time, v _i-1 is the initial feed speed, j _max is the allowable grinding wheel feed The maximum jerk, s(t _i1 ) is the feed displacement of the grinding wheel at stage t _i1 , s(t _i2 ) is the feed displacement of the grinding wheel at stage t _i2 , and T _si is an interpolation cycle when the grinding wheel feeds ΔX(T _si ); Calculate the interpolation period T _si of acceleration mode 2 by formulas (18), (19) and (20): a _max t _i2 ² +(j _max t _i1 ² +2a _max t _i1 +2v _i-1 )t _i2 +4v _i-1 t _i1 +j _max t _i1 ³ +a _max t _i1 ² -2ΔX(T _si ) =0 (18) t _i1 =t _i3 =a _max /j _max (19) T _si =t _i1 +t _i2 +t _i3 =2t _i1 +t _i2 (20) Among them: ΔX(T _si ) is the displacement of the grinding wheel per rotation of 1 degree, t _i1 is the time of acceleration and acceleration phase, t _i2 is the time of uniform acceleration phase, t _i3 is the time of deceleration and acceleration phase, v _i-1 is the initial progress Giving speed, j _max is the allowable maximum jerk of the grinding wheel feed, a _max is the allowable maximum acceleration of the grinding wheel feed, T _si is an interpolation cycle when the grinding wheel feed ΔX(T _si ); Step 5. obtain the cam rotation speed value by formula (21), (22); T _w = T _si (21) Among them, T ₀ is the base circle interpolation period of the cam rotation, n ₀ is the base circle speed, and the value is 100rmp, T _w is the interpolation period from the cam rotation to the lift section, and F″”(θ) is the cam rotation to the lift section predicted speed; Step 6. Establish a one-dimensional T-S fuzzy controller model with a single-input roller follower measuring head rotation angle θ and a single-output cam rotation speed F″(θ), and the T-S fuzzy controller simplified model is as follows: R _i : if θ is A _i then F _i ′(θ)＝a _i0 +a _i1 θ (25) Among them, R _i (i=1, 2, ... 12) represents the i-th fuzzy rule, θ is the roller follower probe rotation angle as the input variable of the fuzzy controller, and A _i is the input variable θ on the domain of discourse Fuzzy subset, F _i ′(θ) represents the cam rotation speed output of the i-th fuzzy rule, a _i0 and a _i1 are the subsequent parameters of the i-th fuzzy rule, F″(θ) is the cam of the entire fuzzy controller Rotation speed output, since there is only one input quantity θ, μ _i (θ) is A _i (θ), A _i (θ) is the satisfaction degree of the i-th fuzzy rule θ to the fuzzy subset A _i , h _i (θ ) is the membership function on the domain of input variables, that is, the antecedent parameters; Step 7. By analyzing the curves of grinding wheel feed displacement X(θ) and acceleration a(θ), obtain the range of rotation angle θ of each roller follower probe, grinding wheel feed displacement X(θ), and grinding wheel feed Acceleration a(θ), cam rotation speed initial value b _i , cam rotation speed slope k _i basic parameter fuzzy rule table of the camshaft segment, and draw a schematic diagram of the cam rotation speed linear segment prediction model at the same time; Step 8. Calculation of TS fuzzy controller consequent parameters: TS fuzzy controller consequent parameters a _i0 and a _i1 are determined by formula (28); Among them, c _i is the starting value of the roller follower probe rotation angle of the ith segment in the range of each segment of the roller follower probe rotation angle in the basic parameters of the camshaft segment; k _i is the speed slope of each segment, b _i is the initial value of the cam rotation speed of each segment in ; Step 9. Calculate T-S fuzzy controller antecedent parameter: T-S fuzzy controller antecedent parameter is determined by formula (31): Substituting (31) into (30), and then substituting (30) into (29), the expression of the membership function h _i (θ) of the TS fuzzy controller is as follows: h _i (θ) = f _tri (θ, c _i-1 , (c _i +c _i+1 )/2, c _i+2 ) (32) In formula (32), θ is the rotation angle of the roller follower probe, c _i is the initial value of the rotation angle of the roller follower probe in the i-th segment, h _i (θ) is the membership function of the input variable universe , represents the membership degree of the fuzzy subset A _i when the roller follower probe turns to θ; Step 10. Substitute equations (32) and (25) into equation (26), and use MATLAB software tool to fit the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ) after TS fuzzy control, and L ₁ The cam rotation speed F′(θ) and the cam rotation angle α(θ) predict the linear segment approximation curve and L ₂ The cam rotation speed F″(θ) after the TS fuzzy control is compared with the cam rotation angle α(θ) prediction curve, If the prediction curve of L ₂ cam rotation speed F″(θ) and cam rotation angle α(θ) is smooth and close to the curve of L ₁ cam rotation speed F′(θ) and cam rotation angle α(θ) prediction straight line segment curve, then the desired effect is achieved; Otherwise, it is necessary to optimize the TS fuzzy controller to make the prediction curve of cam rotation speed F″(θ) and cam rotation angle α(θ) smoother and at the same time be as close as possible to L ₁ cam rotation speed F′(θ) Predict the straight line segment approaching the curve with the cam rotation angle α(θ); Step 11. The optimization of the predicted value of the cam rotation speed is to use the adjacent fuzzy subset membership function to adjust the influence law of the output data in order to obtain a more ideal curve. There are mainly two situations: Situation 1: Suppose only two adjacent fuzzy rules R ₁ and R ₂ are activated for a certain rotation angle θ ₁ of the roller follower probe, that is, the membership degree of θ ₁ only to fuzzy subsets A ₁ and A ₂ is not zero , get formula (33) from formula (26), when F ₂ ′(θ ₁ )>F ₁ ′(θ ₁ ), F″(θ ₁ )>F ₁ ′(θ ₁ ), by adjusting the adjacent membership The initial value c _i-1 of the rotation angle of the degree function h ₂ (θ), (see formula (31)), if c _i-1 increases to ( _ci-1 + _ci )/2, h ₂ (θ ₁ ) decrease, increase, Decrease, so F″(θ ₁ ) will decrease and approach F ₁ ′(θ ₁ ), so that the predicted speed approaches the ideal straight line segment, and the curve is optimized; F ₁ ′(θ ₁ ), F ₂ ′(θ ₁ ) are the outputs of the cam rotation speed of the roller follower measuring head rotation angle θ ₁ on the fuzzy rules R ₁ and R ₂ respectively, and F″(θ ₁ ) is the input is the total output of the fuzzy controller when θ ₁ , h ₁ (θ ₁ ), h ₂ (θ ₁ ) are the membership degrees of the roller follower probe rotation angle θ ₁ to the fuzzy subsets A ₁ and A ₂ ; Case 2: Assume that a certain rotation angle θ ₂ of the roller follower measuring head is used as the input of the fuzzy controller, and the cam rotation speed F″(θ ₂ ) is used as the output of the fuzzy controller, and the equation (34) is obtained from the equation (26), when When F ₁ ′(θ ₂ )<F ₂ ′(θ ₂ ), F″(θ ₂ )<F ₂ ′(θ ₂ ), it can be measured by adjusting the roller follower of the membership function h ₁ (θ) If the end value of head rotation angle c _i+2 is reduced to (c _i+1 +c _i+2 )/2, h ₁ (θ ₂ ) decreases, increase, Increase (F ₁ ′(θ ₂ )-F ₂ ′(θ ₂ ) is a negative number), so F″(θ ₂ ) will increase and tend to F ₂ ′(θ ₂ ), making the predicted speed approach the ideal Straight line segment, so that the curve is optimized. F ₁ ′(θ ₂ ), F ₂ ′(θ ₂ ) are the outputs of the cam rotation speed of the roller follower measuring head rotation angle θ ₂ in the fuzzy rules R ₁ and R ₂ respectively, and F″(θ ₂ ) is the input is the total output of the fuzzy controller when θ ₂ , h ₁ (θ ₂ ), h ₂ (θ ₂ ) are the membership degrees of the roller follower probe rotation angle θ ₂ to the fuzzy subsets A ₁ and A ₂ ; According to the above principles, the optimized formula from formula (32) after repeated tests is as follows: h _i (θ)=f _tri (θ, ( _ci-1 + _ci )/s, ( _ci + _ci+1 )/s, ( _ci+1 + _ci+2 )/s) ( 35) Among them, the value of s is between 1.90 and 2.10. According to the curve, it can be increased or decreased by 0.01 each time, and the L ₂ cam rotation speed F″(θ) and the cam rotation angle α(θ) can be adjusted to predict that the curve is smooth and close to the L ₁ cam rotation speed F '(θ) and the cam angle α(θ) predict the curve of the straight line segment.