CN110170886A

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

Info

Publication number: CN110170886A
Application number: CN201910315529.5A
Authority: CN
Inventors: 王洪; 刘珏; 段海容; 潘艺萌; 乐书思; 张子军; 李启平; 陈永福
Original assignee: Hunan Normal University
Current assignee: Hunan Normal University
Priority date: 2019-04-19
Filing date: 2019-04-19
Publication date: 2019-08-27
Anticipated expiration: 2039-04-19
Also published as: CN110170886B

Abstract

The invention discloses a kind of Camshaft Grinding processing methods based on T-S fuzzy control, realized by following steps: 1. while grinding wheel high speed rotation, is controlled in digital-control camshaft grinding machine using the numerical control program of computer as the grinding wheel feed system of X-axis and as the cam rotary system of C axis；2. the cam lift value provided according to user, grinding wheel radius, gauge head radius, radius of cam base-circle and formula (1), (2), (4), (5) fit grinding wheel feeding by formula (1), (2) with MATLAB software toolX(θ) and cam angleα(θ) displacement curve, which is formed byX(θ)‑α(θ) value by programming software automatically generates grinding wheel feeding numerical control processing subprogram, to realize that grinding wheel follows cam rotation to make traverse motion, while cam rotation speed is fitted by formula (2), (5) with MATLAB software toolF(θ) and cam angleα(θ) rate curve, which is formed byF(θ)‑α(θ) value by programming software automatically generate cam rotation numerical control processing subprogram, to realize cam rotary motion.Ideal grinding accuracy, processing efficiency and surface quality can be obtained.

Description

Camshaft grinding machining method based on T-S fuzzy control

Technical Field

The invention belongs to a grinding method, and particularly relates to a camshaft grinding method based on T-S fuzzy control.

Background

The camshaft is one of key parts of an automobile engine, and due to the particularity of the shape of the camshaft, a non-circular grinding processing mode is adopted for grinding the camshaft, so that the processing precision and efficiency of the camshaft not only determine the processing quality and production cost of products, but also influence the working performance of the engine. The traditional camshaft grinding method is determined by adopting an ideal grinding wheel feed displacement motion equation for grinding wheel feed (X axis), adopting an empirical model for cam rotation (C axis), modifying the cam rotation speed by an experienced engineer and performing repeated trial grinding. The camshaft grinding method is difficult to meet the requirements of high precision, high efficiency and high flexibility of modern automobile part processing.

Scholars at home and abroad have been devoted to the research on precise and efficient non-circular grinding technology, such as the book of the university of Hunan (research on motion models of tangent point tracking grinding method, journal of mechanical engineering, 2002.6), and consider that non-circular grinding is performed by moving along the surface of a grinding tangent point at a constant linear velocity and correcting according to the constant grinding removal rate. The camshaft grinding mathematical model established according to the theory obtains practical application value, but the processing precision and the efficiency of the camshaft grinding mathematical model still need to be further improved. Chinese patent ZL201010278922.0 discloses a numerical control grinding method for a camshaft, which realizes high-precision grinding of the camshaft by limiting the feed speed, acceleration and jerk of a non-circular section grinding wheel and predicting the rotation (C-axis) speed of the cam, but for an oil pump cam and a cam with a large lift value, a large cam lift error can be generated at a part with large curvature of a waist lifting stroke. The document 'numerical control camshaft grinding machine workpiece rotating shaft rotating speed optimization method' (the journal of mechanical engineering, 2014.15) proposes that a forward and reverse interpolation maximum feed speed meeting point is dynamically solved by limiting the rotating speed of a non-circular section cam, the feed speed, the acceleration and the acceleration of a grinding wheel and adopting a forward and reverse synchronous acceleration control method, so that the optimal interpolation of the feed of the grinding wheel is realized, the impact of the acceleration on a machine tool is reduced, and the grinding machining precision and efficiency of the machine tool are improved. But the surface quality still remains to be further improved.

Disclosure of Invention

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

The technical scheme adopted by the invention is as follows:

the invention provides a camshaft grinding processing method based on T-S fuzzy control, which is realized by the following steps:

step 1, controlling a grinding wheel feeding system serving as an X axis and a cam rotating system serving as a C axis in a numerical control camshaft grinding machine by using a numerical control program of a computer while the grinding wheel rotates at a high speed;

step 2, fitting a displacement curve of a grinding wheel feed X (theta) and a cam corner α (theta) by the formulas (1) and (2) and (4) and (5) according to a cam lift value, a grinding wheel radius, a measuring head radius and a cam base circle radius provided by a user, wherein an MATLAB software tool is used for the displacement curve of the grinding wheel feed X (theta) and the cam corner α (theta), an X (theta) - α (theta) value formed by the displacement curve is automatically generated by programming software to realize that the grinding wheel rotates along with the cam to do transverse reciprocating motion, meanwhile, a MATLAB software tool is used for fitting a speed curve of a cam rotation speed F (theta) and a cam corner α (theta) by the formulas (2) and (5), and an F (theta) - α (theta) value formed by the speed curve is automatically generated by the programming software to realize the cam rotation;

X(θ)＝OO₂-r₂-r (1)

in the formula: x (theta) is the displacement of grinding wheel feed, r is the radius of cam base circle, O is the center of cam base circle₁Is the center of a measuring head of a roller driven part, O₂Is the center of the grinding wheel,phi is from the center of the cam base circle O to the center of the measuring head roller O₁The connecting line and the instantaneous center M of the cam to the circle center O of the measuring head roller₁Angle between the connecting lines of (A), (B), O₁O₂＝r₂-r₁，r₂Is the grinding wheel radius, r₁For roller follower gauge radius, OO₁＝r+H(θ)+r₁H (theta) is the lift value of the cam, theta is the connecting line from the midpoint A of the profile of the cam base circle to the center O of the cam base circle, and the center O of the cam base circle to the center O of the measuring head roller₁α (theta) is the connecting line of the cam base circle contour midpoint A and the cam base circle center O and the grinding wheel center O₂An included angle between a connecting line of the cam profile and the grinding wheel tangent center O, rho (theta) is the polar radius from the tangent point P of the cam profile and the grinding wheel tangent to the cam base circle center O, β (theta) is an included angle between a connecting line from the cam base circle profile midpoint A to the cam base circle center O and a connecting line from the tangent point P of the cam profile and the grinding wheel tangent to the cam base circle center O, omega (theta) is the rotation angular velocity of the tangent point P of the cam profile and the grinding wheel tangent when the cam rotates to the cam lift, and omega (theta) is the rotation angular velocity of the cam profile and the grinding₀The angular speed of a tangent point P of a cam profile and a grinding wheel when the cam rotates to a base circle is F (theta), the rotation speed of the cam to a corner theta of the side head of the roller follower is F (theta), and the rotation speed of the cam to the corner theta of the side head of the roller follower is n (theta);

step 3, respectively solving a first derivative, a second derivative and a third derivative of the grinding wheel feeding displacement formula (1) to respectively obtain formulas (6), (7) and (8);

wherein v (theta) is the feed speed of the grinding wheel, a (theta) is the feed acceleration of the grinding wheel, and j (theta) is the feed acceleration of the grinding wheel;

fitting speed curves of a grinding wheel feed speed v (theta) and a cam corner α (theta) to the formulas (2) and (6) by using an MATLAB software tool, fitting acceleration curves of a grinding wheel feed acceleration a (theta) and a cam corner α (theta) to the formulas (2) and (7) by using the MATLAB software tool, wherein the maximum rotation speed of a camshaft exceeds the rotation speed of a base circle (36000deg/min) by more than 1.2 times, and limiting the grinding wheel feed speed, the acceleration and the cam rotation speed according to the formulas (9) and (10):

wherein v (θ)_i) For the i-th interpolation period, a (theta)_i) For the ith interpolation period, the feed acceleration of the grinding wheel, j (theta)_i) Adding acceleration for the grinding wheel feed in the ith interpolation period, wherein k is the speed limit ratio, and v is_maxMaximum speed allowed for wheel feed, a_maxMaximum acceleration allowed for wheel feed, j_maxMaximum allowable jerk, ω, for wheel feed_maxIs the maximum rotational angular velocity;

and 4, solving a grinding wheel feed interpolation period T according to the S-shaped acceleration and deceleration control method and the expression (16) of the acceleration mode 1 and the expressions (18), (19) and (20) of the acceleration mode 2 of the cam lift value_siAnd an interpolation period of the cam return stroke to predict the time for the grinding wheel to feed when the cam rotates 1 degree;

calculation of interpolation period T of acceleration mode 1 from equation (16)_si：

ΔX(T_si) For a displacement of the grinding wheel feed per 1 degree of rotation, t_i1To add acceleration phase time, t_i2To reduce the acceleration phase time, v_i-1At an initial feed rate, j_maxMaximum jerk allowed for wheel feed, s (t)_i1) Is t_i1Segment grinding wheel feed displacement, s (t)_i2) Is t_i2Segment grinding wheel feed displacement, T_siFeeding the grinding wheel by DeltaX (T)_si) An interpolation period of time;

the interpolation period T of the acceleration system 2 is calculated from the expressions (18), (19) and (20)_si：

a_maxt_i2 ²+(j_maxt_i1 ²+2a_maxt_i1+2v_i-1)t_i2+4v_i-1t_i1+j_maxt_i1 ³+a_maxt_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)

Wherein: Δ X (T)_si) For a displacement of the grinding wheel feed per 1 degree of rotation, t_i1To add acceleration phase time, t_i2For a period of uniform acceleration, t_i3To reduce the acceleration phase time, v_i-1At an initial feed rate, j_maxMaximum allowable jerk for wheel feed, a_maxMaximum acceleration allowed for wheel feed, T_siFeeding the grinding wheel by DeltaX (T)_si) An interpolation period is timed;

step 5, calculating the cam rotation speed value by the formulas (21) and (22);

T_w＝T_si(21)

wherein, T₀Interpolating the period, n, for the base circle of the cam rotation₀The base circle speed is taken as 100rmp, T_wF '(theta) is an interpolation period from cam rotation to a lift section, and F' is the predicted speed from cam rotation to the lift section;

and 6, establishing a one-dimensional T-S fuzzy controller model of the measuring head rotation angle theta of the single-input roller follower and the rotation speed F' (theta) of the single-output cam, wherein the simplified model of the T-S fuzzy controller is as follows:

R_i：if θ is A_ithen F_i′(θ)＝a_i0+a_i1θ (25)

wherein R is_i(i ═ 1, 2.. 12) denotes the ith fuzzy rule, θ is the roller follower stylus rotation angle as the input variable for the fuzzy controller, a_iFor fuzzy subsets on the input variable theta discourse, F_i' (θ) shows the cam rotation speed output of the ith fuzzy rule, a_i0、a_i1For the latter part of the ith fuzzy rule, F' (theta) is the cam rotation speed output of the whole fuzzy controller, since there is only one input quantity theta, mu_i(theta) is A_i(θ)，A_i(theta) is the ith fuzzy rule theta to fuzzy subset A_iDegree of satisfaction of h_i(theta) is a membership function on the input variable domain, namely a front-part parameter;

step 7, analyzing the curves of the grinding wheel feed displacement X (theta) and the acceleration a (theta) to obtain the range of the measuring head rotation angle theta of each section of the roller driven part and the grinding wheel feedGiving displacement X (theta), feed acceleration a (theta) of grinding wheel, and initial value b of cam rotation speed_iCam rotational speed slope k_iThe basic parameter fuzzy rule table of the camshaft subsection draws a schematic diagram of a linear subsection prediction model of the cam rotation speed at the same time;

step 8, calculating the back part parameters of the T-S fuzzy controller: T-S fuzzy controller back part parameter a_i0、a_i1Determined by equation (28);

wherein c is_iThe initial value of the measuring head rotation angle of the roller driven part in the ith section in the measuring head rotation angle range of each section of the roller driven part in the basic parameters of the camshaft sections; k is a radical of_iFor the slope of the speed of rotation of the segments, b_iA cam rotation speed starting value of each segment;

step 9, calculating the front part parameters of the T-S fuzzy controller: the T-S fuzzy controller antecedent parameter is determined by equation (31):

substituting the formula (31) into the formula (30), and then substituting the formula (30) into the formula (29) to obtain the membership function h of the T-S fuzzy controller_i(θ) the expression is as follows:

h_i(θ)＝f_tri(θ，c_i-1，(c_i+c_i+1)/2，c_i+2) (32)

in the formula (32), theta is the angle of the probe of the roller follower, c_iInitial value of measuring head rotation angle h of roller driven part in section i_i(theta) is a membership function on the input variable domain, and represents a fuzzy subset A when the measuring head of the roller driven element rotates to theta_iDegree of membership of;

step 10, substituting the formulas (32) and (25) into the formula (26), and fitting T by an MATLAB software toolPredicted curve of cam rotation speed F' (theta) and cam rotation angle α (theta) after S fuzzy control, and L₁Predicted straight-line segment approximation curve and L of cam rotation speed F' (theta) and cam rotation angle α (theta)₂The cam rotation speed F' (theta) after T-S fuzzy control is compared with the predicted curve of the cam rotation angle α (theta), if L is₂The predicted curve of cam rotational speed F' (theta) versus cam angle α (theta) is smooth and close to L₁Otherwise, optimizing the T-S fuzzy controller to make the predicted curves of the cam rotation speed F' (theta) and the cam rotation angle α (theta) become smoother and approach to L as much as possible₁Predicting a straight-line segment approximation curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta);

step 11, optimizing the predicted value of the cam rotation speed, namely adjusting the influence rule of the output data by using a membership function of an adjacent fuzzy subset to obtain a more ideal curve, wherein the two conditions are mainly as follows:

case 1: assuming a certain angle of rotation theta for the roller follower feeler₁Activating only two adjacent fuzzy rules R₁、R₂I.e. theta₁For only fuzzy subset A₁、A₂Is not zero, formula (33) is obtained from formula (26), when F₂′(θ₁)＞F₁′(θ₁)，F″(θ₁)＞F₁′(θ₁) By adjusting the adjacent membership function h₂Initial value c of rotation angle of (theta)_i-1See formula (31) if c_i-1Is increased by (c)_i-1+c_i) When is/2, h₂(θ₁) The number of the grooves is reduced, and the,the size of the mixture is increased, and the mixture is,decreases, thereby F' (theta)₁) Will be reduced to approach F₁′(θ₁) The predicted speed is close to an ideal straight line segment, and a curve is optimized;

F₁′(θ₁)、F₂′(θ₁) Angle of rotation theta of probe for roller driven member₁In fuzzy rule R₁、R₂Output of the speed of rotation of the upper cam, F' (θ)₁) Is input as theta₁Total output of time-fuzzy controller, h₁(θ₁)、h₂(θ₁) For measuring the angle theta of the roller follower₁For fuzzy subset A₁、A₂Degree of membership of;

case 2: assuming a certain rotation angle theta of the roller follower probe₂As an input to the fuzzy controller, the cam rotational speed F' (θ)₂) As the fuzzy controller output, equation (34) from equation (26), when F₁′(θ₂)＜F₂′(θ₂)，F″(θ₂)＜F₂′(θ₂) By adjusting the membership function h₁(theta) end value c of roller follower feeler rotation angle_i+2If the decrease is (c)_i+1+c_i+2) When is/2, h₁(θ₂) The number of the grooves is reduced, and the,the size of the mixture is increased, and the mixture is,increase (F)₁′(θ₂)-F₂′(θ₂) Negative), thereby F' (θ)₂) Will increase and tend to F₂′(θ₂) And the predicted speed is close to an ideal straight line segment, so that the curve is optimized.

F₁′(θ₂)、F₂′(θ₂) Angle of rotation theta of probe for roller driven member₂In fuzzy rule R₁、R₂Output of the speed of rotation of the upper cam, F' (θ)₂) Is input as theta₂Total output of time-fuzzy controller, h₁(θ₂)、h₂(θ₂) For measuring the angle theta of the roller follower₂For fuzzy subset A₁、A₂Degree of membership of;

based on the above principle, the formula (32) is optimized through repeated experiments as follows:

h_i(θ)＝f_tri(θ，(c_i-1+c_i)/s，(c_i+c_i+1)/s，(c_i+1+c_i+2)/s) (35)

wherein s is 1.90-2.10, and L can be adjusted by increasing or decreasing 0.01 each time according to the curve₂The predicted curve of cam rotational speed F' (theta) versus cam angle α (theta) is smooth and close to L₁The cam rotation speed F' (θ) and the cam rotation angle α (θ) predict a straight line segment curve.

The invention predicts the cam rotation speed by utilizing a T-S fuzzy controller under the condition of providing specific grinding parameters and lift values by analyzing and perfecting a camshaft grinding mathematical model on the basis of the documents. The ideal grinding precision, machining efficiency and surface quality are obtained.

Fuzzy control, which is one of the important methods of intelligent control, simulates a human decision process to a certain extent, wherein T-S fuzzy control adopts a piecewise linear model to approximate a nonlinear cam rotating speed global fuzzy model. Fuzzy control is introduced in the grinding process of the camshaft, so that the grinding stability can be maintained, and the grinding surface quality can achieve a good effect.

The technical scheme of the invention is further explained by combining the attached drawings.

Drawings

FIG. 1 is a numerical control camshaft grinding mathematical model.

FIG. 2 is a simulation curve of grinding wheel feed and cam rotation.

FIG. 3 is a schematic view of the S-shaped acceleration and deceleration method of the grinding wheel feed.

Fig. 4 is a schematic diagram of a cam rotational speed prediction linear segment model.

FIG. 5 is a graph of triangular membership functions.

Fig. 6 is a graph of the cam rotation speed F ″ (θ) versus the cam rotation angle a (θ) predicted.

FIG. 7 is a graph of optimization of the front roller follower feeler rotation angle θ and cam rotation speed F ″ (θ) for case 1, and the proximity fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

FIG. 8 is the membership function h for case 1₂(theta) before and after optimization comparative plot.

FIG. 9 shows the optimized roller follower feeler rotation angle θ and cam rotation speed F' (θ) in case 1, and the proximity fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

FIG. 10 is a graph of optimization of the front roller follower feeler rotation angle θ and cam rotation speed F ″ (θ) for case 2, proximity fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

FIG. 11 is the membership function h for case 2₁(theta) before and after optimization comparative plot.

FIG. 12 shows the optimized roller follower feeler rotation angle θ and cam rotation speed F' (θ) in case 2, and the proximity fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

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

Fig. 14 is a predicted straight-line segment approximation curve of the cam rotation speed F' (θ) and the cam rotation angle a (θ).

Fig. 15 is a curve of the cam rotation speed F ″ (θ) versus the cam rotation angle a (θ) predicted.

FIG. 16 is a graph of a distribution of cam rotational speed prediction membership functions.

Fig. 17 is a predicted curve of the cam rotation speed F' ″ (θ) and the cam rotation angle a (θ) after optimization.

FIG. 18 is a graph of a distribution of predicted membership functions for optimized cam rotational speeds.

Fig. 19 is a graph comparing a predicted curve of the cam rotation speed and the cam rotation angle a (θ) after repeated optimization with an original machinable curve.

Detailed Description

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

1) and controlling a grinding wheel feeding system as an X axis and a cam rotating system as a C axis in the numerical control camshaft grinding machine by using a numerical control program of the computer while the grinding wheel rotates at a high speed.

The numerical control camshaft grinding mathematical model is shown in figure 1. The grinding mathematical model of the camshaft is established according to the grinding point tangent point tracking, the constant grinding removal rate principle and the cam profile shape, namely, the grinding wheel rotates at high speed, and the grinding wheel (X axis) feeding and cam (C axis) rotating two-axis linkage of the camshaft grinding machine is controlled through a numerical control instruction to realize the cam profile surface grinding.

In FIG. 1, 1 is a cam, 2 is a grinding wheel, 3 is a roller follower measuring head, O is the center of a cam base circle₁Is the center of a measuring head of a roller driven part, O₂The center of the grinding wheel is A, and the middle point of the profile of the cam base circle is A, namely the starting point of grinding of the grinding wheel. Roller driven part probeCam surface rolling, roller center O₁The running track is K_t. As OM and O₁Perpendicular to, extend O₂O₁Respectively intersecting a point P tangent to the grinding wheel by the cam and an point M intersected with the OM, wherein the tangential speed of the point P of the grinding wheel is v_sThe tangential velocity of the P point of the cam is v_jThe rotational speed of the cam is n₁The rotational speed of the grinding wheel is n₂The angular speed of the point P tangent to the grinding wheel is omega (theta), and the angular speed of the cam rotating to the base circle is omega₀. The radius of cam base circle is OA ═ r, and the radius of measuring head of roller driven part is PO₁＝r₁The radius of the grinding wheel is O₂P＝r₂. Theta is the connecting line from the center point A of the cam base circle outline to the center O of the cam base circle and the center O of the measuring head roller₁The included angle (namely the measuring head rotation angle of the roller follower) between the connecting line of the center O of the cam base circle is α (theta) which is the connecting line of the center A of the cam base circle contour and the center O of the cam base circle and the center O of the grinding wheel₂An included angle between a connecting line of the center O of the cam base circle, β (theta) is an included angle between a connecting line of the center A of the cam base circle and the center O of the cam base circle and a tangent point P between the cam profile and the grinding wheel and a connecting line of the center O of the cam base circle, phi is an included angle between the center O of the cam base circle and the center O of the measuring head roller₁The connecting line and the instantaneous center M of the cam to the circle center O of the measuring head roller₁The angle between the connecting lines.

The followers (tappets) of the cam mechanism have three different forms: knife-edge tappets, roller tappets, and planar tappets. The knife-edge tappet can be regarded as a roller tappet when the radius of the roller is 0; a planar tappet may be considered as a roller tappet when the roller radius is infinite. Therefore, by only obtaining a mathematical model of the grinding wheel feed displacement in the roller tappet form, the problem of the grinding wheel feed displacement in the other two tappet forms can be solved. From the several servo relationships in the above figure, equations (1) and (2) can be derived, wherein equation (1) is the feed motion equation of the grinding wheel (X axis) (see "method for optimizing the rotating speed of workpiece rotating shaft of numerically controlled camshaft grinder", journal of mechanical engineering, 2014.15).

2) Equations (3), (4) and (5) (MALKIN S. "Grinding technology Theory and applications of machining with aberrations", Industrial Press Inc., 2008) can be derived based on the principle of constant removal rate and the relationship between the arc micro-displacement and the angular velocity, the pole diameter and the angle.

X(θ)＝OO₂-r₂-r (1)

In the formula: r is the cam base radius, r₁Radius of probe of roller follower, r₂Is the radius of grinding wheel, O is the center of cam base circle, O₁Is the center of a measuring head of a roller driven part, O₂Is the center of the grinding wheel, phi is the center of the cam base circle O to the center of the measuring head roller O₁The connecting line and the instantaneous center M of the cam to the circle center O of the measuring head roller₁H (theta) is a lift value of the cam, theta is a connecting line from a center point A of a profile of a base circle of the cam to a center O of the base circle of the cam, and from the center O of the base circle of the cam to a center O of the measuring head roller₁Rho (theta) is the polar radius from the tangent point P of the cam profile tangent to the grinding wheel to the center O of the cam base circle, β (theta) is the angle from the connecting line from the midpoint A of the cam base circle to the center O of the cam base circle to the connecting line from the tangent point P of the cam profile tangent to the grinding wheel to the center O of the cam base circle, omega (theta) is the rotation angular velocity of the tangent point P of the cam profile tangent to the grinding wheel when the cam rotates to the cam lift₀The rotation angular velocity of a tangent point P of a cam profile tangent with a grinding wheel when the cam rotates to a cam base circle, X (theta) is the displacement of the grinding wheel feed,O₁O₂＝r₂-r₁，OO₁＝r+H(θ)+r₁，a_pfor the depth of cut at α (θ) for the cam to turn, v_jFor cam P point line speed, v, at cam rotation α (theta)_sFor the point speed, Q, of the grinding wheel P at the point where the cam rotates to α (theta)_W’Is the unit width removal rate. If the cam rotates to a certain angle, the cam P point line speedCan ensure a_epConstant means that the metal removal rate per unit width is equal, i.e., the grinding force is constant. Wherein, the linear velocity v of a certain tangent point of the base circle₀，v₀＝ω₀r，F (θ) is the rotational speed at which the cam rotates to the roller follower side head angle θ, and n (θ) is the rotational speed at which the cam rotates to the roller follower side head angle θ.

According to the cam lift value provided by a user (see appendix 1), the grinding wheel radius, the measuring head radius, the cam base circle radius and the equations (1), (2), (4) and (5), a MATLAB software tool can be used for fitting a displacement curve of the grinding wheel feed X (theta) and the cam rotation angle α (theta) by the equations (1) and (2) (see figure 2(a)), an X (theta) - α (theta) value formed by the displacement curve can be automatically generated by programming software to realize that the grinding wheel rotates along with the cam to do transverse reciprocating motion, and a speed curve of the cam rotation speed F (theta) and the cam rotation angle α (theta) can be fitted by the equations (2) and (5) (see figure 2(d)) by the MATLAB software tool, and the F (theta) - α (theta) value formed by the speed curve can be automatically generated by the programming software to realize that the cam rotates.

3) Respectively calculating the first derivative, the second derivative and the third derivative of the formula (1) to obtain the formulas (6), (7) and (8):

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

According to a cam lift value provided by a user (see appendix 1), fitting equations (1), (5), (6), (7) and (2) by using an MATLAB software tool to obtain grinding wheel feed and cam rotation motion simulation curves as shown in FIG. 2, wherein FIG. 2(a) is a grinding wheel feed displacement X (theta) versus cam rotation angle α (theta), FIG. 2(b) is a grinding wheel feed speed v (theta) versus cam rotation angle α (theta), FIG. 2(c) is a grinding wheel feed acceleration a (theta) versus cam rotation angle α (theta), FIG. 2(d) is a cam rotation angle speed omega (theta) versus cam rotation angle α (theta) curve, and FIG. 2 and 6) can be fitted to speed curves of grinding wheel feed speed v (theta) and cam rotation angle α (theta) (see FIG. 2(b)), and when the acceleration curves of grinding wheel feed acceleration a (theta) and cam rotation angle α (theta) can be fitted to equations (2) and (7) by using the LAB software tool, the MATLAB software tool can be fitted to acceleration curves (2) and the acceleration curves (c) can be obtained from a cam rotation speed limit of a cam rotation speed, a cam rotation speed of a camshaft, a cam rotation speed of a camshaft, a cam rotation speed limit is about 10, and a rotation speed limit is obtained when the cam rotation speed exceeds a camshaft rotation speed limit is obtained by using a drive force of a:

wherein v (θ)_i) For the i-th interpolation period, a (theta)_i) For the ith interpolation period, the feed acceleration of the grinding wheel, j (theta)_i) Adding acceleration v to the grinding wheel feed for the ith interpolation period_maxMaximum speed allowed for wheel feed, a_maxMaximum acceleration allowed for wheel feed, j_maxAllowing maximum jerk ω for wheel feed_maxThe maximum angular velocity of rotation of the cam. In the S-type acceleration/deceleration control method, the interpolation period is determined by the displacement, and therefore, it is necessary to convert the rotation period T into a rotation period T every 1 degree of rotation of the cam_wThen, according to the relationship between the grinding wheel feed and cam rotation communication, calculating grinding wheel feed period T_si. The conversion method comprises the following steps:

in the formula, T_wInterpolating periods, T, for cam rotation to lift range_siFor the interpolation period of the grinding wheel feed, n (theta) is the rotation speed of the cam rotating to theta, and Delta X (theta) is the Tth_siThe feed displacement of the grinding wheel in each interpolation period, where Deltav (theta) is Tth_siThe feed speed of the grinding wheel in each interpolation period, Δ a (θ) being Tth_siThe feed acceleration of the grinding wheel is delta j (theta) in each interpolation period_siAnd F (theta) is the predicted speed of the cam rotating to the lift section.

4) From the grinding wheel feed displacement graph 2(a), the curve can be divided into 6 segments of AB, BC, CD, DE, EF, FG, wherein AB and FG are base circle segments and the grinding wheel is not fed. The BC section of grinding wheel feeding is from the cam base circle to the middle of the cam lift (namely, an acceleration section), and the CD section of grinding wheel is from the middle of the cam lift to the cam lobe (namely, a deceleration section); the DE range extends from the lobe tip to the middle of the cam lift (i.e., the acceleration range) and the EF cam lift returns to the cam base circle (i.e., the deceleration range). Then, according to the S-type acceleration and deceleration control method, five acceleration and deceleration modes are decomposed, and fig. 3 is 4 acceleration and deceleration control modes, wherein fig. 3(a) is an acceleration mode 1 and only comprises two stages of acceleration and acceleration; FIG. 3(b) is an acceleration mode 2, which is performed by three stages of acceleration, uniform acceleration, and acceleration/deceleration; fig. 3(c) shows a deceleration mode 1, which has only two stages of deceleration and acceleration; fig. 3(d) shows a deceleration mode 2, which goes through three stages of deceleration reduction, uniform deceleration and deceleration acceleration reduction, and the 5 th mode is a uniform speed mode. The curves of feed speed, acceleration and jerk are respectively drawn in the figure, and the method has the advantage that when the grinding wheel is fed to a certain target position, the acceleration is zero, and the mechanical impact can be reduced (see a method for optimizing the rotating speed of a workpiece rotating shaft of a numerical control camshaft grinding machine, the journal of mechanical engineering, 2014.15). By calculating equations (15), (16), (17), (18), (19) and (20), the interpolation periods T of the acceleration pattern 1 and the acceleration pattern 2 can be obtained_si. Since acceleration and deceleration have reversibility, therefore: only by discussing the interpolation of the acceleration mode 1 and the acceleration mode 2, the entire interpolation period can be obtained, and the rotation speed of the cam (C axis) can be predicted.

Acceleration mode 1: see FIG. 3(a), a_i≤a_maxEach interpolation period is divided into two stages, and the following formula can be obtained according to fig. 3 (a):

ΔX(T_si) For a displacement of the grinding wheel feed per 1 degree of rotation, t_i1To add acceleration phase time, t_i2To reduce the acceleration phase time, v_i-1At an initial feed rate, j_maxAt maximum jerk, s_i1(t_i1) Is t_i1Segment grinding wheel feed displacement, s_i2(t_i2) Is t_i2Segment grinding wheel feed displacement, a_iIs t_i1End acceleration, v_i1Is t_i1Terminal velocity, v_i2Is t_i2End velocity, T_siIs an interpolation period.

The acceleration mode 2: see FIG. 3(b), a_i≥a_maxEach interpolation period is divided into three stages, and the following formula can be obtained according to fig. 3 (b):

t_i1＝t_i3＝a_max/j_max(19)

T_si＝t_i1+t_i2+t_i3＝2t_i1+t_i2(20)

wherein: Δ X (T)_si) For a displacement of the grinding wheel feed per 1 degree of rotation, t_i1To add acceleration phase time, t_i2To even the acceleration stage time, t_i3To reduce the acceleration period, v_i-1At an initial feed rate, j_maxAt maximum jerk, s_i1(t_i1) Is composed oft_i1Segment grinding wheel feed displacement, s_i2(t_i2) Is t_i2Segment grinding wheel feed displacement, s_i3(t_i3) Is t_i3Segment grinding wheel feed displacement, a_iIs t_i1End acceleration, a_maxIs the maximum acceleration, v_i1Is t_i1Terminal velocity, v_i2Is t_i2Terminal velocity, v_i3Is t_i3End velocity, T_siIs an interpolation period.

5) As the feed of the grinding wheel is linked to the rotation of the cam, i.e. the interpolation period T of the feed of the grinding wheel_siIs the interpolation period T of the cam rotation_w. The cam rotation speed values can be obtained from equations (21) and (22).

T_w＝T_si(21)

Wherein, T₀Interpolating the period, n, for the base circle of the cam rotation₀The base circle speed is taken as 100rmp, T_wF' (theta) is the predicted speed of cam rotation to lift segment.

(6) Dividing the universe of discourse of each input variable into a plurality of fuzzy subsets according to the basic principle of T-S fuzzy control, combining the fuzzy subsets of each input variable according to certain experience to establish an input-output linear relation, and determining a fuzzy implication relation R_iIf the rule is input, then an output is output, (If... the.) which is a fuzzy rule; and (4) approximating any one global nonlinear function by using a local piecewise linear model. The expression is as follows (see document "study of T-S fuzzy controller design and optimization method", Yan Xiaoxi, 2007.05):

R_i：if z₁is A_i1and z₂is A_i2......z_jis A_ij......and z_nis A_in，then F_i(z)＝a_i0+a_i1z₁+a_i2z₂+......+a_ijz_j+......+a_inz_n(23)

where i (i ═ 1, 2.. times, n) denotes the number of fuzzy rules, j (j ═ 1, 2.. times, n) denotes the number of input variables, and R denotes the number of input variables_iDenotes the ith fuzzy rule, z ═ z₁，z₂，...，z_j，...，z_n]^TRepresenting the input vector of the fuzzy controller. A. the_ij(z_j) To blur the subset, F_i(z) output of the ith fuzzy rule, a_i0、a_i1、...、a_inIs a parameter of the i-th fuzzy rule, F (z) is the output of the fuzzy controller, mu_i(z) is the satisfaction degree of the ith fuzzy rule defined as the product form, A_ij(z_j) Is z_jTo A_ijDegree of satisfaction of h_ij(z_j) Is a membership function on the input variable domain.

According to the equations (23) and (24), a single-input (roller follower measuring head rotation angle theta) and single-output (cam rotation speed F' (theta)) one-dimensional fuzzy controller model can be established, namely, the T-S fuzzy controller can be simplified as follows.

R_i：if θ is A_ithen F_i′(θ)＝a_i0+a_i1θ (25)

Wherein R is_i(i ═ 1, 2.. 12) (see table 1) shows the ith fuzzy rule, and θ is the roller follower stylus rotation angle as the input variable to the fuzzy controller. A. the_iTo blur the subset, F_i' (θ) is the cam rotation speed output of the ith fuzzy rule, a_i0、a_i1For the i-th fuzzy ruleThen the back-piece parameter, F' (theta), is the cam rotational speed output of the entire fuzzy controller since there is only one input variable, mu_i(θ) is directly expressed as the ith fuzzy rule θ to A_iDegree of satisfaction, A_i(theta) is theta to A_iDegree of satisfaction of h_iAnd (theta) is a membership function on the input variable domain.

7) By analyzing the curve of the grinding wheel feed displacement X (theta) and the cam angle α (theta) (see fig. 2(a)) and the curve of the grinding wheel feed acceleration a (theta) and the cam angle α (theta) (see fig. 2(c)), the range of the measuring head angle theta of each segment of the roller follower, the grinding wheel feed displacement X (theta), the grinding wheel feed acceleration a (theta), and the initial value b of the cam rotation speed can be obtained_iCam rotational speed slope k_iThe basic parameter fuzzy rule table (see table 1) of the camshaft segment, according to table 1, a schematic diagram of a 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 (theta) and the cam rotation angle α (theta) (see fig. 2(c)), the grinding wheel feed acceleration curve can be divided into a₁、A₂、......、A₁₂Paragraph (see column 1 of Table 1), A₁、A₂、A₁₁、A₁₂Is a cam base circle segment, A₁、A₁₂The rotating speed of the segment cam is 36000deg/min and A₂The section is a base circle to a cam starting lift section, and the rotating speed is from b₂(36000deg/min) is rapidly reduced to b₃。A₃The section is a cam lift slow speed acceleration section (see fig. 2(c)), and the rotating speed is required to be b₃Slowly decreases to b₄。A₄The section is a cam lift rapid acceleration section, and the rotating speed is from b₄Down to b₅。A₅The section is a cam lift rapid deceleration section, and the rotating speed is changed from b₅Rise to b₆。A₆The section is a cam lift slow speed reduction section, and the rotating speed is changed from b₆Rise to b₇。A₇The section is a cam lift slow speed acceleration section, and the rotating speed is from b₇Down to b₈。A₈The section is a cam lift rapid acceleration section, and the rotating speed is changed from b₈Down to b₉。A₉The section is a cam lift rapid deceleration section, and the rotating speed is changed from b₉Rise to b₁₀。A₁₀The section is a cam lift slow speed reduction section, and the rotating speed is changed from b₁₀Rise to b₁₁。A₁₁The section is from the end of the cam lift to the base circle, and the rotating speed is from b₁₁Quickly rises to b₁₂(36000 deg/min). Finally, the base circle A is completed₁₂And (3) obtaining the rules of the 2 nd, 3 rd and 4 th rows in the table 1 according to the roller follower measuring head rotation angle theta, the X (theta) displacement curve and the a (theta) acceleration curve, namely finishing the machining process of the cam for one circle. In the 5 th row of Table 1, the initial value b of each segment rotational speed is obtained by the acceleration and deceleration control method_iTable 1 column 6 rotational speed slope is noted below table 1.

TABLE 1 fuzzy rule table of basic parameters of camshaft subsection

Note: a. the_iFuzzy subsets for the ith segment; c. C_iThe initial value of the measuring head rotation angle of the ith section of roller driven part is obtained; 1, 2, 12; wherein,

k_i＝(b_i+1-b_i)/(c_i+1-c_i)；k₁＝0；k₁₂＝0；b₁＝b₂＝b₁₂＝36000，b₄＝b₄＝b₈＝b₁₀，b₁₁＝b₃，b₅＝b₉。

8) calculating the back part parameter a of the T-S fuzzy controller_i0、a_i1: substituting the parameters of the 2 nd, 5 th and 6 th columns of the table 1 into the formula (25);

F_i′(θ)＝b_i+k_i(θ-c_i)＝k_iθ+b_i-k_ic_i(27)

the following equations (25) and (27) can be obtained:

wherein i 1, 2.., 12; c. C_iStarting roller follower stylus angle, k, for section i in column 2 of Table 1_iIs the slope of the corresponding segment in column 6 of Table 1, b_iThe cam rotation speed start value corresponding to column 5 of table 1.

9) Calculating the front piece parameters of the T-S fuzzy controller: membership function h of T-S fuzzy controller_i(θ) is:

h_i(θ)＝f_tri(θ，x₁，x₂，x₃) (29)

where θ is the roller follower feeler rotation angle, f_tri(θ，x₁，x₂，x₃) Is a triangular membership function, h_i(theta) is a membership function and represents a certain rotation angle theta of a measuring head of the roller driven part to the fuzzy subset A_iDegree of membership (degree of satisfaction), x₁、x₂、x₃Are the front-part parameters of the fuzzy controller, and x₁≤x₂≤x₃；

The invention adopts a triangle membership function, and the expression of the triangle membership function is as follows (see the literature, "fuzzy control and MATLAB simulation", Shixinmin, etc., Qinghua university Press, 2008.03):

where θ is an input variable, x₁、x₂、x₃Respectively parameters, x, of triangular membership functions₁≤x₂≤x₃(ii) a The triangular membership function curve is shown in FIG. 5, from which x is known₃-x₁The smaller the value of (d), the sharper the function shape;

the front piece parameters of the T-S fuzzy controller are determined by an equation (31):

substituting the formula (31) into the formula (30), and then substituting the formula (30) into the formula (29) to obtain the membership function h of the T-S fuzzy controller_i(θ) can be written as:

h_i(θ)＝f_tri(θ，c_i-1，(c_i+c_i+1)/2，c_i+2) (32)

in the formula (32), theta is the angle of the probe of the roller follower, c_iIs the initial value of the roller follower feeler rotation angle, h, of the i-th section in the 2 nd row of table 1_iAnd (theta) is a membership function on the input variable domain, and represents the membership (satisfaction degree) of a certain rotation angle theta of a measuring head of the roller driven part to the fuzzy rule i.

10) And (3) substituting the equations (25) and (32) into the equation (26), fitting a curve of the cam rotation speed F '(theta) and the roller follower measuring head rotation angle theta by using an MATLAB software tool, fitting the cam rotation speed F' (theta) and the cam rotation angle α (theta) to obtain a predicted curve of the cam rotation speed F '(theta) and the cam rotation angle α (theta) after the T-S fuzzy controller is obtained, and FIG. 6 shows the predicted curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta).

L in FIG. 6₁A predicted straight-line segment approximation curve L representing the cam rotation speed F' (theta) and the cam rotation angle a (theta)₂Shows a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle a (θ) after the T-S fuzzy control).

It can be seen from FIG. 6 that the predicted curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta) after the output of the T-S fuzzy controller is at A₆、A₇The section is not close to the predicted straight section of cam rotation speed F '(theta) and cam rotation angle α (theta) and the curve is not smooth enough to bring the cam rotation speed F' (theta) close to the predicted straight section (curve L of FIG. 6)₁) While keeping smooth the predicted curve of the cam rotation speed F '(theta) and the cam rotation angle α (theta), the predicted curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta) after T-S fuzzy control is needed (the curve of FIG. 6)L₂) And (6) optimizing. 11) The predicted value of the cam rotation speed is optimized by adjusting the parameters (membership function h) of the front parts of the adjacent fuzzy rules_i(θ)) to adjust the value of the cam rotation speed F "(θ) to optimize the fuzzy controller output curve to be smooth while closer to the ideal state, mainly two situations:

case 1: FIG. 7 shows optimization of the front roller follower feeler rotation angle θ and the cam rotation speed F ″ (θ) and the proximity fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

Assuming a certain rotation angle theta of the roller follower probe₁As an input to the fuzzy controller, the cam rotational speed F' (θ)₁) As a fuzzy controller output. F₁′(θ₁)、F₂′(θ₁) Each measuring head of a roller follower at a certain angle of rotation theta₁In fuzzy rule R₁、R₂The rotational speed of the cam. Obtaining a formula (33) from the formula (26), namely obtaining the cam rotation speed F' (theta) output by the fuzzy controller₁) The relationship is as follows:

wherein: h is₁(θ₁)、h₂(θ₁) Indicating a certain angle of rotation theta of the feeler of the roller follower₁For fuzzy subset A₁、A₂By adjusting h₂(θ₁) To adjust the value of F' (theta)₁) To make the output curve of the fuzzy controller closer to the ideal state (i.e. predicting the linear segment model F' (θ)), the specific adjustment steps are as follows:

if the adjacent fuzzy rule R in FIG. 7 is used₁、R₂Cam rotation speed F₁' (theta) and F₂'(θ) is a boundary, and F' (θ) is shown in the left half of FIG. 7₁)＞F₁′(θ₁) And is blurred in the vicinityRule R₂Upper cam rotation speed F₂′(θ₁) Greater than at the fuzzy rule R₁Upper cam rotation speed F₁′(θ₁) I.e. F₂′(θ₁)＞F₁′(θ₁) Can be obtained by adjusting the membership function h₂(theta) initial value c of rotation angle of roller follower probe_i-1If the increase is (c)_i-1+c_i) /2 (see FIG. 8), h₂(θ₁) Is reduced to h₂′(θ₁) A compound of formula (33),the size of the mixture is increased, and the mixture is,decreases, thereby F' (theta)₁) Will be reduced and thus more toward F₁′(θ₁). Similarly, the right half of fig. 7 can also be adjusted by adjusting the membership function h₁(theta), decreasing F' (theta) brings the right half closer to F₂′(θ)。

The left half of fig. 7 is optimized by combining the optimization law to obtain a curve of the probe rotation angle θ fuzzy controller output F '(θ) of the optimized roller follower in case 1, and fig. 9 is a curve of the probe rotation angle θ fuzzy controller output F' (θ) of the optimized roller follower and an adjacent fuzzy rule R₁、R₂Output F₁' (theta) and F₂' (theta) graph.

The straight line A in FIG. 7 represents the fuzzy rule R₁F of (A)₁' (θ) and a straight line B indicates a neighboring blurring rule R₂F of (A)₂'(θ) and curve C represents the output of the fuzzy controller F' (θ)).

In FIG. 8, the straight line A is the membership function h before optimization₂(theta), the straight line B is the optimized membership function h₂’(θ))。

The straight line A in FIG. 9 represents the fuzzy rule R₁F of (A)₁' (θ) and a straight line B indicates a neighboring blurring rule R₂F of (A)₂'(theta) and curve C represents the output F' (theta) of the optimized fuzzy controller.

Case 2: FIG. 10 shows another type of roller follower probe rotation angle θ, cam rotation speed F ″ (θ), and proximity ambiguity rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

Also assume that the roller follower stylus in FIG. 10 has a certain angle of rotation θ₂As an input to the fuzzy controller, the cam rotational speed F' (θ)₂) As a fuzzy controller output. F₁′(θ₂)、F₂′(θ₂) For measuring a certain angle of rotation theta of roller driven member₂In fuzzy rule R₁、R₂The rotational speed of the cam. Obtaining the cam rotation speed F' (theta) output by the fuzzy controller by obtaining an expression (34) from an expression (26)₂) The relationship is as follows:

wherein: h is₁(θ₂)、h₂(θ₂) Indicating a certain angle of rotation theta of the feeler of the roller follower₂For fuzzy subset A₁、A₂By adjusting h₁(θ₂) To adjust F' (theta)₂) The output curve of the fuzzy controller is closer to the ideal state (namely, the prediction straight line segment model F' (theta)) by specifically adjusting the following values:

if the fuzzy rule R is used₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) is a boundary with an intersection of straight line segments, and a certain rotation angle theta of the roller follower probe is shown in the right half of FIG. 10₂Output F' (theta)₂)＜F₂′(θ₂) And in the vicinity of the fuzzy rule R₁Output F of₁′(θ₂) Less than at fuzzy rule R₁Output F of₂′(θ₂) I.e. F₁′(θ₂)＜F₂′(θ₂) Can be obtained by adjusting the membership function h₁(theta) end value c of roller follower feeler rotation angle_i+2If the decrease is (c)_i+1+c_i+2) 2 (see FIG. 11), then h₁(θ₂) Is reduced to h₁′(θ₂)，The size of the mixture is increased, and the mixture is,increase (F)₁′(θ₂)-F₂′(θ₂) Negative), thereby F' (θ)₂) Will increase and tend to F₂′(θ₂). Likewise, the membership function h may also be adjusted for the left half of FIG. 10₂(θ), such that F "(θ) increases to bring the left half closer to F₁′(θ)。

The left half and the right half of fig. 10 can be optimized by combining the above rules to obtain a curve of the output F ' (theta) of the fuzzy controller for the measuring head rotation angle theta of the optimized roller driven member in case 2, and fig. 12 shows the curve of the output F ' (theta) of the fuzzy controller for the measuring head rotation angle theta of the optimized roller driven member in case 2, the rotation speed F ' (theta) of the cam, and the adjacent fuzzy rule R₁、R₂Cam rotation speed F₁' (theta) and F₂' (theta) graph.

The straight line A in FIG. 10 represents the fuzzy rule R₁F of (A)₁' (θ) and a straight line B indicates a neighboring blurring rule R₂F of (A)₂'(θ) and curve C represents the output of the fuzzy controller F' (θ)).

In FIG. 11, the line A is the membership function h before optimization₁(theta), the straight line B is the optimized membership function h₁’(θ))。

The straight line A in FIG. 12 represents the fuzzy rule R₁F of (A)₁' (θ) and a straight line B indicates a neighboring blurring rule R₂F of (A)₂'(theta) and curve C represents the output F' (theta) of the optimized fuzzy controller.

h_i(θ)＝f_tri(θ，(c_i-1+c_i)/s，(c_i+c_i+1)/s，(c_i+1+c_i+2)/s) (35)

wherein s is an optimized adjustment coefficient, the value of s is 1.90-2.10, generally 2, and L in FIG. 6 can be adjusted by increasing or decreasing 0.01 each time according to the curve₂The predicted curve of cam rotational speed F' (theta) versus cam angle α (theta) is smooth and close to L₁The cam rotation speed F' (θ) and the cam rotation angle α (θ) predict a straight line segment curve.

The algorithm flow chart of the present invention is shown in fig. 13.

12) Take machining and grinding a camshaft provided by a company as an example. As is known, the camshaft adopts a flat probe roller r₁999999; the radius r of the cam base circle is 17.5; radius r of grinding wheel₂175, cam rotation speed n₁100rmp, grinding wheel speed n₂4367 rmp. The cam lift values are shown in appendix 1.

① speed limit calculation

Obtained by the formula (10):

angular velocity of start-up lift point:

obtained by the formula (22):

before the lift starting point is entered, the cam rotation speed should be reduced in advance to ensure the smooth operation of the cam.

② the table of parameters for camshaft segmentation obtained by annex 1 and S-type acceleration and deceleration method is shown in Table 2.

Before entering the starting point of the cam lift, the cam rotating speed is reduced to 8737deg/min in advance to ensure that the cam runs smoothly. Substituting the above conditions into formula (1), and calculating the first, second and third derivatives of formula (1) to calculate the maximum feeding speed v of the grinding wheel_max0.252 mm/deg; maximum acceleration a_max＝0.024mm/deg²(ii) a Maximum jerk j_max＝0.0085mm/deg³. The maximum grinding wheel feed speed v can be calculated according to the formulas (4), (11), (12), (13) and (14)_max87 mm/s; maximum acceleration a_max＝1795mm/s²(ii) a Maximum jerk j_max＝150791mm/s³. The speed value in the 7 th column of the table 2 can be calculated according to the cam lift value and the acceleration and deceleration control method.

TABLE 2 camshaft section parameter table

According to the camshaft segmentation parameter table of table 2, in combination with the cam rotation speed prediction straight line segmentation model schematic diagram (see fig. 4), a MATLAB software tool can be used to fit a curve of the predicted straight line segment approximation of the cam rotation speed F' (θ) and the cam rotation angle α (θ) as shown in fig. 14.

③ substitution of the parameters listed in column 2 of Table 1 for equation (36) of equation (32), since A₁、A₁₂Is a horizontal straight line segment, and A₂、A₁₁The slope of (A) is greatly different from the slope of other segments to avoid A₂、A₁₁Influences other segments such that other segments are in the fuzzy subset a₂、A₁₂Is 0, so that h₂(θ)、h₁₂(theta) is determined to be straightThe angular triangle is a membership function.

The 3 rd column parameter of Table 2 is set to the starting angle c_iSubstituting formula (36) to give the following formula:

④ A predicted curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta) is obtained by substituting the parameters listed in columns 3, 7 and 8 of Table 2 into equation (28), then substituting equation (28) into equation (25), then substituting equations (25) and (37) into equation (26), and fitting equations (26) and (2) by MATLAB, as shown in FIG. 15.

L in FIG. 15₁A predicted straight-line segment approximation curve L representing the cam rotation speed F' (theta) and the cam rotation angle a (theta)₂Shows a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle a (θ) after the T-S fuzzy control).

Furthermore, by substituting equation (37) for equation (30) and performing simulation using MATLAB, a distribution curve of the membership function can be obtained as shown in FIG. 16.

As can be seen from fig. 15, the predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle α (θ) after the T-S fuzzy control (curve L in fig. 15)₂) In A₆、A₇The section is not close to the linear piecewise curve of the cam rotation speed prediction, the curve is not smooth enough, and the curve of the cam rotation speed F '(theta) and the prediction curve of the cam rotation angle α (theta) needs to be optimized, so that the optimized curve of the cam rotation speed F' (theta) and the prediction curve of the cam rotation angle α (theta) is close to the linear piecewise curve of the cam rotation speed prediction as much as possible while smoothness is guaranteed (a curve L in figure 15)₁)。

⑤ the optimization method according to step 11 optimizes the cam rotational speed prediction curve of fig. 15 by deriving equation (38) from equation (35).

⑥

The 3 rd column parameter of Table 2 is set to the starting angle c_iSubstitution of formula (38) can be:

similarly, fig. 17 shows a predicted curve of the optimized cam rotation speed F' ″ (θ) versus the cam rotation angle α (θ) obtained by fitting the equations (26) and (2) with MATLAB after substituting the parameters listed in the 3 rd, 7 th and 8 th rows of table 2 for the equation (28), the equation (28) for the equation (25), and the equations (25) and (39) for the equation (26).

Substituting the formula (38) into the formula (30), and carrying out MATLAB simulation on the formula (30) to obtain the optimized membership function h_iThe (θ) distribution curve is shown in fig. 18.

It can be seen from the above graphs that the predicted curve of the cam rotation speed obtained by the optimized membership function through fuzzy control is smoother and is closer to the straight-line segment approximation curve of the cam rotation speed, but in order to achieve a more ideal grinding effect, the membership function is continuously checked and optimized to obtain a more reasonable curve according to the actual effect.

⑥ analysis of results

According to the method for optimizing the cam rotation speed prediction curve (step 11), a prediction curve of the cam rotation speed F '(theta) to the cam rotation angle α (theta) (curve B in FIG. 19) shown in FIG. 19 can be obtained by continuously optimizing and fitting the cam rotation speed F' (theta) to the cam rotation angle α (theta) by using MATLAB, and a simulation result graph obtained by final optimization is compared with a graph of the machining achievable by a part of the original method.

The cam rotation speed processing curve efficiency of the original method is obviously improved, and the stability is good.

Examples of the applications

At present, a numerical control camshaft grinding machine of a Toyota machine modified by a certain company is taken as test equipment, a camshaft contour detector is taken as detection equipment, and a camshaft of the certain company is machined and ground as an example. As is known, the camshaft adopts a flat probe roller r₁999999; the radius r of the cam base circle is 17.5; radius r of grinding wheel₂175 cam speed n₁100rmp, grinding wheel speed n₂3000 rmp. The cam lift values are shown in appendix 1.

Through C # language design in Visual Studio 2008, user parameter setting interface, displacement and speed subprogram generation software is designed, the software is transplanted into an Operate Programming Package Siemens 840D SL numerical control system, only parameters of a cam lift value, a grinding wheel radius, a roller follower measuring head radius and a base circle radius are input, a numerical control machining grinding wheel displacement numerical control machining subprogram and a cam rotation numerical control machining subprogram can be automatically generated through the software, and a simulation curve of the numerical control machining cam rotation subprogram is shown in FIG. 19. A is a subprogram simulation curve of the cam rotation numerical control machining generated by the original method, B is a subprogram simulation curve of the cam rotation numerical control machining generated by the method, and after grinding and detection, the detection result is shown in appendix 3. It can be seen from column 4 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 which it can be seen that the performance index of the yota industrial machine numerically controlled camshaft grinder modified by john mayer is equivalent to that of the numerically controlled camshaft grinder of JUNKER, germany. The method comprises the following steps:

1. and (3) inputting data in the cam lift value (see appendix 1), the radius of the grinding wheel, the radius of a measuring head of the roller follower and the radius of the base circle into a user interface of the numerical control grinding machine, and obtaining a numerical control machining subprogram for controlling the displacement of the grinding wheel in the step 2) to obtain an X-axis displacement value curve (see figures 2(a, b and c)) of the grinding carriage and the numerical control machining subprogram for controlling the displacement of the grinding wheel.

2. The data in the camshaft lift value (see appendix 1), the grinding wheel radius, the roller follower measuring head radius and the base circle radius are input into the user interface of the invention, and the numerical control processing subprogram for controlling the cam rotation speed obtained in the step 7) can obtain a cam rotation speed numerical control subprogram curve (see figure 19) and a cam C-axis rotation speed subprogram, and can also display the minimum value of the cam rotation speed.

Cam Lift values provided by manufacturer appendix 1

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

Appendix 3 cam machining error detects table

Appendix 4 technical Properties Table

Claims

1. The camshaft grinding processing method based on T-S fuzzy control is realized by the following steps:

X(θ)＝OO₂-r₂-r (1)

in the formula: x (theta) is the displacement of grinding wheel feed, r is the radius of cam base circle, O is the center of cam base circle₁Is the center of a measuring head of a roller driven part, O₂Is the center of the grinding wheel,phi is from the center of the cam base circle O to the center of the measuring head roller O₁The connecting line and the instantaneous center M of the cam to the circle center O of the measuring head roller₁Angle between the connecting lines of (A), (B), O₁O₂＝r₂-r₁，r₂Is the grinding wheel radius, r₁For roller follower gauge radius, OO₁＝r+H(θ)+r₁H (theta) is the lift value of the cam, theta is the connecting line from the midpoint A of the profile of the cam base circle to the center O of the cam base circle, and the center O of the cam base circle to the center O of the measuring head roller₁α (theta) is the connecting line of the cam base circle contour midpoint A and the cam base circle center O and the grinding wheel center O₂The included angle between the line and the circle center O of the cam base circle is rho (theta) from a tangent point P of the cam profile tangent with the grinding wheel toThe polar radius of the center O of the cam base circle, β (theta) is the included angle between the connecting line from the center A of the cam base circle profile to the center O of the cam base circle and the connecting line from the tangent point P of the cam profile tangent to the grinding wheel to the center O of the cam base circle, omega (theta) is the rotation angular velocity of the tangent point P of the cam profile tangent to the grinding wheel when the cam rotates to the cam lift, and omega (theta) is the rotation angular velocity of the cam profile tangent to the grinding wheel when the cam₀The angular speed of a tangent point P of a cam profile and a grinding wheel when the cam rotates to a base circle is F (theta), the rotation speed of the cam to a corner theta of the side head of the roller follower is F (theta), and the rotation speed of the cam to the corner theta of the side head of the roller follower is n (theta);

t_i1＝t_i3＝a_max/j_max(19)

T_si＝t_i1+t_i2+t_i3＝2t_i1+t_i2(20)

step 5, calculating the cam rotation speed value by the formulas (21) and (22);

T_w＝T_si(21)

R_i：if θ is A_ithen F_i′(θ)＝a_i0+a_i1θ (25)

wherein R is_i(i ═ 1, 2.. 12) denotes the ith fuzzy rule, θ is the roller follower stylus rotation angle as the input variable for the fuzzy controller, a_iFor fuzzy subsets on the input variable theta discourse, F_i' (θ) shows the cam rotation speed output of the ith fuzzy rule, a_i0、a_i1For the i-th fuzzy ruleThen the back-piece parameter, F' (theta), is the cam rotational speed output of the entire fuzzy controller since there is only one input quantity, theta, mu_i(theta) is A_i(θ)，A_i(theta) is the ith fuzzy rule theta to fuzzy subset A_iDegree of satisfaction of h_i(theta) is a membership function on the input variable domain, namely a front-part parameter;

and 7, analyzing curves of the grinding wheel feed displacement X (theta) and the acceleration a (theta) to obtain the range of the measuring head rotation angle theta of each section of the roller driven piece, the grinding wheel feed displacement X (theta), the grinding wheel feed acceleration a (theta) and the initial value b of the cam rotation speed_iCam rotational speed slope k_iThe basic parameter fuzzy rule table of the camshaft subsection draws a schematic diagram of a linear subsection prediction model of the cam rotation speed at the same time;

h_i(θ)＝f_tri(θ，c_i-1，(c_i+c_i+1)/2，c_i+2) (32)

step 10, substituting the equations (32) and (25) into the equation (26), fitting a predicted curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta) after the T-S fuzzy control through an MATLAB software tool, and converting L into L₁Predicted straight-line segment approximation curve and L of cam rotation speed F' (theta) and cam rotation angle α (theta)₂The cam rotation speed F' (theta) after T-S fuzzy control is compared with the predicted curve of the cam rotation angle α (theta), if L is₂The predicted curve of cam rotational speed F' (theta) versus cam angle α (theta) is smooth and close to L₁Otherwise, optimizing the T-S fuzzy controller to make the predicted curves of the cam rotation speed F' (theta) and the cam rotation angle α (theta) become smoother and approach to L as much as possible₁Predicting a straight-line segment approximation curve of the cam rotation speed F' (theta) and the cam rotation angle α (theta);

case 1: assuming a certain angle of rotation theta for the roller follower feeler₁Activating only two adjacent fuzzy rules R₁、R₂I.e. theta₁For only fuzzy subset A₁、A₂Is not zero, formula (33) is obtained from formula (26), when F₂′(θ₁)＞F₁′(θ₁)，F″(θ₁)＞F₁′(θ₁) By adjusting the adjacent membership function h₂Initial value c of rotation angle of (theta)_i-1(see formula (31)) if c_i-1Is increased by (c)_i-1+c_i) When is/2, h₂(θ₁) The number of the grooves is reduced, and the,the size of the mixture is increased, and the mixture is,decreases, thereby F' (theta)₁) Will be reduced to approach F₁′(θ₁) The predicted speed is close to an ideal straight line segment, and a curve is optimized;

h_i(θ)＝f_tri(θ，(c_i-1+c_i)/s，(c_i+c_i+1)/s，(c_i+1+c_i+2)/s) (35)