CN110170886B - Camshaft grinding machining method based on T-S fuzzy control - Google Patents
Camshaft grinding machining method based on T-S fuzzy control Download PDFInfo
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- CN110170886B CN110170886B CN201910315529.5A CN201910315529A CN110170886B CN 110170886 B CN110170886 B CN 110170886B CN 201910315529 A CN201910315529 A CN 201910315529A CN 110170886 B CN110170886 B CN 110170886B
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B24—GRINDING; POLISHING
- B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
- B24B1/00—Processes of grinding or polishing; Use of auxiliary equipment in connection with such processes
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B24—GRINDING; POLISHING
- B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
- B24B19/00—Single-purpose machines or devices for particular grinding operations not covered by any other main group
- B24B19/08—Single-purpose machines or devices for particular grinding operations not covered by any other main group for grinding non-circular cross-sections, e.g. shafts of elliptical or polygonal cross-section
- B24B19/12—Single-purpose machines or devices for particular grinding operations not covered by any other main group for grinding non-circular cross-sections, e.g. shafts of elliptical or polygonal cross-section for grinding cams or camshafts
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Abstract
The invention discloses a camshaft grinding processing method based on T-S fuzzy control, which is realized by the following steps: 1. when the grinding wheel rotates at a high speed, a numerical control program of a computer is utilized to control 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; 2. according to the cam lift value, the grinding wheel radius, the measuring head radius, the cam base circle radius and the expressions (1), (2), (4) and (5) provided by a user, fitting the grinding wheel feeding according to the expressions (1) and (2) by using an MATLAB software toolX(θ) Angle of rotation with camα(θ) The displacement curve formedX(θ)‑α(θ) The value is automatically generated into a grinding wheel feeding numerical control processing subprogram by programming software so as to realize that the grinding wheel rotates along with the cam to do transverse reciprocating motion, and meanwhile, an MATLAB software tool is used for fitting the rotating speed of the cam according to the formulas (2) and (5)F(θ) Angle of rotation with camα(θ) The velocity profile ofF(θ)‑α(θ) The value is automatically generated into a cam rotation numerical control processing subprogram by programming software so as to realize the cam rotation motion. The ideal grinding precision, machining efficiency and surface quality can be obtained.
Description
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:
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 the base circle of the camFrom the center O to 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 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 roller1Alpha (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, beta (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);
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 the grinding wheel feed speed v (theta) and the cam rotation angle alpha (theta) to the formulas (2) and (6) by using an MATLAB software tool, and fitting acceleration curves of the grinding wheel feed acceleration a (theta) and the cam rotation angle alpha (theta) to the formulas (2) and (7) by using the MATLAB software tool; the maximum rotation speed of the camshaft exceeds the rotation speed of the base circle by more than 1.2 times, the rotation speed of the base circle is 36000deg/min, and the feed speed, the acceleration, the jerk and the rotation speed of the cam of the grinding wheel are limited 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 ismaxMaximum speed allowed for wheel feed, amaxMaximum acceleration allowed for wheel feed, jmaxMaximum allowable jerk, ω, for wheel feedmaxIs the maximum rotational angular velocity;
and 4, solving a grinding wheel feed interpolation period T according to an S-shaped acceleration and deceleration control method and an expression (16) of an acceleration mode 1 with two stages of acceleration and deceleration and expressions (18), (19) and (20) of an acceleration mode 2 with three stages of acceleration, uniform acceleration and deceleration, wherein the values of the cam lift are obtainedsiAnd 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(Tsi) For wheel feed of 1 degree per revolutionAmount of displacement of, ti1To add acceleration phase time, ti2To reduce the acceleration phase time, vi-1At an initial feed rate, jmaxMaximum jerk allowed for wheel feed, s (t)i1) Is ti1Segment grinding wheel feed displacement, s (t)i2) Is ti2Segment grinding wheel feed displacement, TsiFeeding 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：
amaxti2 2+(jmaxti1 2+2amaxti1+2vi-1)ti2+4vi-1ti1+jmaxti1 3+amaxti1 2-2ΔX(Tsi)＝0 (18)
ti1＝ti3＝amax/jmax (19)
Tsi＝ti1+ti2+ti3＝2ti1+ti2 (20)
Wherein: Δ X (T)si) For a displacement of the grinding wheel feed per 1 degree of rotation, ti1To add acceleration phase time, ti2For a period of uniform acceleration, ti3To reduce the acceleration phase time, vi-1At an initial feed rate, jmaxMaximum allowable jerk for wheel feed, amaxMaximum acceleration allowed for wheel feed, TsiFeeding 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);
Tw＝Tsi (21)
wherein, T0Interpolating the period, n, for the base circle of the cam rotation0The base circle speed is taken as 100rmp, TwFor cam rotating to lift sectionInterpolation period, F' (theta) is the predicted speed of the cam rotating to the lift section;
and 6, establishing a one-dimensional T-S fuzzy controller model of a single-input roller follower measuring head rotation angle theta and a single-output cam rotation speed F' (theta), wherein the simplified model of the T-S fuzzy controller is as follows:
Ri:ifθ is Ai then Fi'(θ)＝ai0+ai1θ (25)
wherein R isi(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, aiFor fuzzy subsets on the input variable theta discourse, Fi' (θ) shows the cam rotation speed output of the ith fuzzy rule, ai0、ai1For 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, mui(theta) is Ai(θ)，Ai(theta) is the ith fuzzy rule theta to fuzzy subset AiDegree of satisfaction of hi(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 speediCam rotational speed slope kiThe 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;
wherein c isiThe 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 ofiFor the slope of the speed of rotation of the segments, biA 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 parameters are determined by equations (29), (30), (31):
hi(θ)＝ftri(θ,x1,x2,x3) (29)
where θ is the roller follower feeler rotation angle, ftri(θ,x1,x2,x3) Is a triangular membership function, hi(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 AiDegree of membership (degree of satisfaction), x1、x2、x3Are the front-part parameters of the fuzzy controller, and x1≤x2≤x3(ii) a Theta is an input variable, x1、x2、x3Respectively parameters, x, of triangular membership functions1≤x2≤x3；
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 controlleri(θ) the expression is as follows:
hi(θ)＝ftri(θ,ci-1,(ci+ci+1)/2,ci+2) (32)
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 ci-1Is increased by (c)i-1+ci) 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' (theta)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 anglei+2If the decrease is (c)i+1+ci+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) Is negative), thus F ″ (θ)2) Will increase and tend to F2'(θ2) The predicted speed is close to an ideal straight line segment, and a 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' (theta)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:
hi(θ)＝ftri(θ,(ci-1+ci)/s,(ci+ci+1)/s,(ci+1+ci+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 the cam rotation speed F' (theta) and the cam rotation angle alpha (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 α (θ) predicted.
FIG. 7 shows optimization of the front roller follower stylus 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 optimized probe rotation angle θ and cam rotation speed F' (θ) of the roller follower in case 1, and 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, and the 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 probe rotation angle θ and cam rotation speed F' (θ) of the roller follower 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 α (θ).
Fig. 15 is a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle α (θ).
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 α (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 rotational speed and the cam rotational angle α (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. The measuring head of the roller driven part rolls on the surface of the cam, and the center O of the roller1The running track is Kt. 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 vsThe tangential velocity of the P point of the cam is vjThe 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 between the connecting lines of the center O of the cam base circle (namely the rotation angle of the measuring head of the roller follower), and alpha (theta) is the connecting line of the midpoint A of the profile of the cam base circle and the center O of the grinding wheel2The included angle between the connecting line of the center O of the cam base circle, beta (theta) is the included angle between the connecting line of the center A of the cam base circle and the center O of the cam base circle and the tangent point P between the cam profile and the grinding wheel and the connecting line of the center O of the cam base circle, phi is the 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 geometrical relationships in the above figures, equations (1) and (2) can be derived, where equation (1) is the feed motion equation of the grinding wheel (X-axis) (see "method for optimizing the rotating speed of workpiece rotating shaft in numerically controlled camshaft grinder", proceedings 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 roller1The angle between the connecting lines (namely the rotation angle of the measuring head of the roller follower), rho (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, beta (theta) is the angle between the connecting line from the midpoint A of the cam base circle profile to the center O of the cam base circle and 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 tangent point0The 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,
OO1＝r+H(θ)+r1，
apfor the depth of cut, v, at which the cam rotates to alpha (theta)jFor the cam P to point line speed, v, at a position where the cam rotates to alpha (theta)sFor the point-line speed, Q, of the grinding wheel P at which the cam rotates to alpha (theta)W’Is the unit width removal rate. If the cam rotates to a certain angle, the cam P point line speed
Can ensure aepConstant 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 (theta) is the rotation speed of the cam to the roller follower side head rotation angle theta, and n (theta) is the rotation speed of the cam to the rollerAnd the rotation speed of the sub-driven member at the side head rotation angle theta.
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), an MATLAB software tool can be used for fitting a displacement curve of the grinding wheel feed X (theta) and the cam rotation angle alpha (theta) (see figure 2(a)) through the equations (1) and (2), a grinding wheel feed numerical control machining subprogram is automatically generated through programming software according to the X (theta) -alpha (theta) value formed by the displacement curve so as to realize that the grinding wheel rotates along with the cam to do transverse reciprocating motion, meanwhile, a speed curve of the cam rotation speed F (theta) and the cam rotation angle alpha (theta) (see figure 2(d)) can be fitted through the equations (2) and (5) through the MATLAB software tool, the F (theta) -alpha (theta) value formed by the speed curve is automatically generated through the programming software, to effect the cam rotational movement.
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 a grinding wheel feed and cam rotation motion simulation curve shown in FIG. 2, wherein FIG. 2(a) is a curve of grinding wheel feed displacement X (theta) to a cam rotation angle alpha (theta), FIG. 2(b) is a curve of grinding wheel feed speed v (theta) to a cam rotation angle alpha (theta), FIG. 2(c) is a curve of grinding wheel feed acceleration a (theta) to a cam rotation angle alpha (theta), and FIG. 2(d) is a curve of cam rotation angle speed omega (theta) to a cam rotation angle alpha (theta). Speed curves of the grinding wheel feed speed v (theta) and the cam rotation angle alpha (theta) can be fitted for the pairs 2 and 6 (see fig. 2(b)), and acceleration curves of the grinding wheel feed acceleration a (theta) and the cam rotation angle alpha (theta) can be fitted for the pairs 2 and 7 (see fig. 2(c)) by using an MATLAB software tool; as can be seen from fig. 2 (d): when the maximum rotation speed of the camshaft reaches 197991deg/min and exceeds the base circle rotation speed (36000deg/min) by about 5 times, namely exceeds the driving capability of the camshaft rotation system, the feed speed, the acceleration, the jerk and the cam rotation speed of the grinding wheel need to be limited according to the following 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 v to the grinding wheel feed for the ith interpolation periodmaxMaximum speed allowed for wheel feed, amaxMaximum acceleration allowed for wheel feed, jmaxMaximum allowable jerk, ω, for wheel feedmaxThe 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 camwThen, according to the relationship between the grinding wheel feed and cam rotation communication, calculating grinding wheel feed period Tsi. The conversion method comprises the following steps:
in the formula, TwInterpolating periods, T, for cam rotation to lift rangesiFor 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 TthsiThe feed displacement of the grinding wheel in each interpolation period, where Deltav (theta) is TthsiThe feed speed of the grinding wheel in each interpolation period, Δ a (θ) being TthsiThe feed acceleration of the grinding wheel is delta j (theta) in each interpolation periodsiAnd 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). We have shown the formulae (15), (16) and (17)) The interpolation periods T of the acceleration mode 1 and the acceleration mode 2 can be obtained by performing the calculations of (18), (19), and (20), respectivelysi. 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), ai≤amaxEach interpolation period is divided into two stages, and the following formula can be obtained according to fig. 3 (a):
ΔX(Tsi) For a displacement of the grinding wheel feed per 1 degree of rotation, ti1To add acceleration phase time, ti2To reduce the acceleration phase time, vi-1At an initial feed rate, jmaxAt maximum jerk, si1(ti1) Is ti1Segment grinding wheel feed displacement, si2(ti2) Is ti2Segment grinding wheel feed displacement, aiIs ti1End acceleration, vi1Is ti1Terminal velocity, vi2Is ti2End velocity, TsiIs an interpolation period.
The acceleration mode 2: see FIG. 3(b), ai≥amaxEach interpolation period is divided into three stages, and the following formula can be obtained according to fig. 3 (b):
amaxti2 2+(jmaxti1 2+2amaxti1+2vi-1)ti2+4vi-1ti1+jmaxti1 3+amaxti1 2-2ΔX(Tsi)＝0 (18)
ti1＝ti3＝amax/jmax (19)
Tsi＝ti1+ti2+ti3＝2ti1+ti2 (20)
wherein: Δ X (T)si) For a displacement of the grinding wheel feed per 1 degree of rotation, ti1To add acceleration phase time, ti2To even the acceleration stage time, ti3To reduce the acceleration period, vi-1At an initial feed rate, jmaxAt maximum jerk, si1(ti1) Is ti1Segment grinding wheel feed displacement, si2(ti2) Is ti2Segment grinding wheel feed displacement, si3(ti3) Is ti3Segment grinding wheel feed displacement, aiIs ti1End acceleration, amaxIs the maximum acceleration, vi1Is ti1Terminal velocity, vi2Is ti2Terminal velocity, vi3Is ti3End velocity, TsiIs 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 wheelsiIs the interpolation period T of the cam rotationw. The cam rotation speed values can be obtained from equations (21) and (22).
Tw＝Tsi (21)
Wherein, T0Interpolating the period, n, for the base circle of the cam rotation0The base circle speed is taken as 100rmp, TwF' (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, establishing an input-output linear relation, and determining a fuzzy implication relationRiIf 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):
Ri:if z1 is Ai1 and z2 is Ai2……zj is Aij……and zn is Ain,then
Fi(z)＝ai0+ai1z1+ai2z2+......+aijzj+......+ainzn (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 variablesiDenotes the ith fuzzy rule, z ═ z1，z2，...,zj，...，zn]TRepresenting the input vector of the fuzzy controller. A. theij(zj) To blur the subset, Fi(z) output of the ith fuzzy rule, ai0、ai1、...、ainIs a parameter of the i-th fuzzy rule, F (z) is the output of the fuzzy controller, mui(z) is the satisfaction degree of the ith fuzzy rule defined as the product form, Aij(zj) Is zjTo AijDegree of satisfaction of hij(zj) 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.
Ri:if θ is Ai then Fi'(θ)＝ai0+ai1θ (25)
Wherein R isi(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. theiTo blur the subset, Fi' (θ) is the cam rotation speed output of the ith fuzzy rule, ai0、ai1For the latter part of the ith fuzzy rule, F "(θ) is the cam rotation speed output of the entire fuzzy controller, since there is only one input variable, μi(θ) is directly expressed as the ith fuzzy rule θ to AiDegree of satisfaction, Ai(theta) is theta to AiDegree of satisfaction of hiAnd (theta) is a membership function on the input variable domain.
7) Analyzing a curve (see fig. 2(a)) of the grinding wheel feed displacement X (theta) and the cam angle alpha (theta) and a curve (see fig. 2(c)) of the grinding wheel feed acceleration a (theta) and the cam angle alpha (theta), obtaining 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 speediCam rotational speed slope kiThe 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 alpha (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。A5With a segment of cam liftA rapid deceleration section, the speed of rotation from5Rise 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 methodiTable 1 column 6 rotational speed slope is noted below table 1.
TABLE 1 fuzzy rule table of basic parameters of camshaft subsection
Note: a. theiFuzzy subsets for the ith segment; c. CiThe initial value of the measuring head rotation angle of the ith section of roller driven part is obtained; 1,2, 12; wherein k isi＝(bi+1-bi)/(ci+1-ci)；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 controlleri0、ai1: substituting the parameters of the 2 nd, 5 th and 6 th columns of the table 1 into the formula (25);
Fi'(θ)＝bi+ki(θ-ci)＝kiθ+bi-kici (27)
the following equations (25) and (27) can be obtained:
wherein i 1, 2.., 12; c. CiStarting roller follower stylus angle, k, for section i in column 2 of Table 1iIs the slope of the corresponding segment in column 6 of Table 1, biThe 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 controlleri(θ) is:
hi(θ)＝ftri(θ,x1,x2,x3) (29)
where θ is the roller follower feeler rotation angle, ftri(θ,x1,x2,x3) Is a triangular membership function, hi(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 AiDegree 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 controlleri(θ) can be written as:
hi(θ)＝ftri(θ,ci-1,(ci+ci+1)/2,ci+2) (32)
in the formula (32), theta is the angle of the probe of the roller follower, ciIs the initial value of the roller follower feeler rotation angle, h, of the i-th section in the 2 nd row of table 1iAnd (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) Substituting the equations (25) and (32) into the equation (26), and fitting a curve of the cam rotation speed F' (theta) and the roller follower feeler rotation angle theta by using an MATLAB software tool. And then fitting the cam rotation speed F ' (theta) and the cam rotation angle alpha (theta) to obtain a predicted curve of the cam rotation speed F ' (theta) and the cam rotation angle alpha (theta) after the T-S fuzzy controller, wherein FIG. 6 is the predicted curve of the cam rotation speed F ' (theta) and the cam rotation angle alpha (theta).
L in FIG. 61An approximation curve L representing a predicted straight line segment of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta)2Shows a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle α (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 alpha (theta) after the output of the T-S fuzzy controller is in A6、A7The section is not close to the predicted straight line section of the cam rotation speed F '(theta) and the cam rotation angle alpha (theta) and the curve is not smooth enough, in order to make the cam rotation speed F' (theta) close to the predicted straight line 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 alpha (theta), the predicted curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) after T-S fuzzy control needs to be carried out (the curve L in figure 6)2) And (6) optimizing. 11) Cam rotation speed preThe measured value optimization is realized by adjusting the parameters (membership function h) of the front part of the adjacent fuzzy rulesi(θ)) to adjust the value of the cam rotation speed F "(θ) to optimize the fuzzy controller output curve to be smooth and at the same time closer to the ideal state, there are two main cases:
case 1: FIG. 7 shows optimization of the front roller follower feeler rotation angle θ and the cam rotation speed F ″ (θ), 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 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 in the vicinity of the fuzzy rule 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 probei-1If the increase is (c)i-1+ci) /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 part of fig. 7 is optimized by combining the optimization rules to obtain an optimized roller follower feeler rotation angle theta fuzzy controller output F '(theta) curve of case 1, and fig. 9 is an optimized roller follower feeler rotation angle theta fuzzy controller output F' (theta) 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 with a probe rotation angle θ, a cam rotation speed F ″ (θ), and an adjacent fuzzy rule R1、R2Cam wheelRotation 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 a formula (34) from the formula (26), namely obtaining the cam rotation speed F' (theta) output by the fuzzy controller2) 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 ″ (θ)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 anglei+2If the decrease is (c)i+1+ci+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) Is negative), thus 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(theta) whereby F "(theta) increases to bring the left half closer to F1'(θ)。
By combining the above rules, the left half part and the right half part of fig. 10 can be optimized to obtain a curve of the probe rotation angle θ fuzzy controller output F '″ (θ) of the roller driven member after optimization under the condition 2, and fig. 12 shows the probe rotation angle θ of the roller driven member after optimization under the condition 2, the cam rotation speed F' ″ (θ) 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.
Based on the above principle, the formula (32) is optimized through repeated experiments as follows:
hi(θ)＝ftri(θ,(ci-1+ci)/s,(ci+ci+1)/s,(ci+1+ci+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 curve2Cam rotation speed F ″ (θ) andthe predicted curve of the cam rotation angle alpha (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.
Rotation 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.
And secondly, a camshaft section parameter table obtained by the appendix 1 and the S-shaped acceleration and deceleration method is shown in a 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 wheelmax0.252 mm/deg; maximum acceleration amax＝0.024mm/deg2(ii) a Maximum jerk jmax＝0.0085mm/deg3. And according to the formulae (4), (11), (1)2) (13), (14) can calculate the maximum grinding wheel feed speed vmax87 mm/s; maximum acceleration amax＝1795mm/s2(ii) a Maximum jerk jmax＝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 cam rotation speed F' (θ) and the cam rotation angle α (θ) prediction straight line segment approximation as shown in fig. 14.
③ substituting the 2 nd row parameters of Table 1 into formula (32) to obtain formula (36), because of 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(θ) is determined as a right triangle membership function.
The 3 rd column parameter of Table 2 is set to the starting angle ciSubstituting formula (36) to give the following formula:
(iv) substituting the parameters listed in the 3 rd, 7 th and 8 th rows in 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 to obtain a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle α (θ), as shown in fig. 15.
L in FIG. 151An approximation curve L representing a predicted straight line segment of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta)2Shows a predicted curve of the cam rotation speed F ″ (θ) and the cam rotation angle α (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 predicted linear piecewise curve of the cam rotation speed, the curve is not smooth enough, and the predicted curve of the cam rotation speed F '(theta) and the cam rotation angle alpha (theta) needs to be optimized, so that the optimized predicted curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) is close to the predicted linear piecewise curve of the cam rotation speed as much as possible while smoothness is ensured (the curve L in figure 15)1)。
The optimization method according to step 11 may obtain equation (38) according to equation (35) to optimize the cam rotation speed prediction curve of fig. 15.
⑥
The 3 rd column parameter of Table 2 is set to the starting angle ciSubstitution of formula (38) can be:
similarly, the parameters listed in the 3 rd, 7 th and 8 th rows of table 2 are substituted into formula (28), formula (28) is substituted into formula (25), formulas (25) and (39) are substituted into formula (26), and MATLAB is used to fit formulas (26) and (2) to obtain the optimized predicted curve of the cam rotation speed F' (theta) to the cam rotation angle alpha (theta) as shown in FIG. 17.
Substituting the formula (38) into the formula (30), and carrying out MATLAB simulation on the formula (30) to obtain the optimized membership function hi(theta) distribution curveAs 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.
Analysis of results
According to the method for optimizing the cam rotation speed prediction curve (step 11), a prediction curve (curve B in fig. 19) of the cam rotation speed F '(theta) to the cam rotation angle alpha (theta) can be obtained by continuously optimizing and fitting the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) by using MATLAB, and a simulation result graph obtained by final optimization is compared with a curve graph which can be processed by a part of original methods. Curve a is a camshaft rotation speed curve which can realize processing, and curve B is a predicted curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) which are finally obtained through optimization.
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 4 technical Properties Table
Claims (1)
1. The camshaft grinding processing method based on T-S fuzzy control 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 displacement curves of a grinding wheel feed X (theta) and a cam corner alpha (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 by using an MATLAB software tool, automatically generating a grinding wheel feed numerical control machining subprogram by using programming software according to the X (theta) -alpha (theta) values formed by the displacement curves to realize that the grinding wheel rotates along with the cam to do transverse reciprocating motion, and fitting speed curves of a cam rotation speed F (theta) and the cam corner alpha (theta) by the formulas (2) and (5) by using the MATLAB software tool to realize that the cam rotates by using the programming software to automatically generate the cam rotation numerical control machining subprogram;
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 roller1Alpha (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, beta (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 the grinding wheel feed speed v (theta) and the cam rotation angle alpha (theta) to the formulas (2) and (6) by using an MATLAB software tool, and fitting acceleration curves of the grinding wheel feed acceleration a (theta) and the cam rotation angle alpha (theta) to the formulas (2) and (7) by using the MATLAB software tool; the maximum rotation speed of the camshaft exceeds the rotation speed of the base circle by more than 1.2 times, the rotation speed of the base circle is 36000deg/min, and the feed speed, the acceleration, the jerk and the rotation speed of the cam of the grinding wheel are limited 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 ismaxMaximum speed allowed for wheel feed, amaxMaximum acceleration allowed for wheel feed, jmaxMaximum allowable jerk, ω, for wheel feedmaxIs the maximum rotational angular velocity;
step 4, adding and subtracting according to the S typeSpeed control method, formula (16) of acceleration mode 1 with two stages of acceleration and deceleration and formula (18), (19) and (20) of acceleration mode 2 with three stages of acceleration, uniform acceleration and deceleration are obtained by cam lift, and grinding wheel feed interpolation period T is solvedsiAnd 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(Tsi) For a displacement of the grinding wheel feed per 1 degree of rotation, ti1To add acceleration phase time, ti2To reduce the acceleration phase time, vi-1At an initial feed rate, jmaxMaximum jerk allowed for wheel feed, s (t)i1) Is ti1Segment grinding wheel feed displacement, s (t)i2) Is ti2Segment grinding wheel feed displacement, TsiFeeding 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：
amaxti2 2+(jmaxti1 2+2amaxti1+2vi-1)ti2+4vi-1ti1+jmaxti1 3+amaxti1 2-2ΔX(Tsi)＝0 (18)
ti1＝ti3＝amax/jmax (19)
Tsi＝ti1+ti2+ti3＝2ti1+ti2 (20)
Wherein: Δ X (T)si) For a displacement of the grinding wheel feed per 1 degree of rotation, ti1To add acceleration phase time, ti2For a period of uniform acceleration, ti3To reduce the acceleration phase time, vi-1At an initial feed rate, jmaxMaximum allowable jerk for wheel feed, amaxMaximum acceleration allowed for wheel feed, TsiFeeding 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);
Tw＝Tsi (21)
wherein, T0Interpolating the period, n, for the base circle of the cam rotation0The base circle speed is taken as 100rmp, TwF '(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 a single-input roller follower measuring head rotation angle theta and a single-output cam rotation speed F' (theta), wherein the simplified model of the T-S fuzzy controller is as follows:
Ri:if θ is Ai then Fi'(θ)＝ai0+ai1θ (25)
wherein R isi(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, aiFor fuzzy subsets on the input variable theta discourse, Fi' (θ) shows the cam rotation speed output of the ith fuzzy rule, ai0、ai1For 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, mui(theta) is Ai(θ)，Ai(theta) is the ith fuzzy rule theta to fuzzy subset AiDegree of satisfaction of hi(theta) is a membership function on the input variable domain, namely a front-part parameter;
step 7, openAnalyzing 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 follower, the grinding wheel feed displacement X (theta), the grinding wheel feed acceleration a (theta), and the initial value b of the cam rotation speediCam rotational speed slope kiThe 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 ai0、ai1Determined by equation (28);
wherein c isiThe 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 ofiFor the slope of the speed of rotation of the segments, biA cam rotation speed start value for each segment;
step 9, calculating the front part parameters of the T-S fuzzy controller: the T-S fuzzy controller antecedent parameters are determined by equations (29), (30), (31):
hi(θ)＝ftri(θ,x1,x2,x3) (29)
where θ is the roller follower feeler rotation angle, ftri(θ,x1,x2,x3) Is a triangular membership function, hi(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 AiDegree of membership, x1、x2、x3Respectively fuzzy controllersFront part parameters, and x1≤x2≤x3(ii) a Theta is an input variable, x1、x2、x3Respectively parameters, x, of triangular membership functions1≤x2≤x3；
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 controlleri(θ) the expression is as follows:
hi(θ)＝ftri(θ,ci-1,(ci+ci+1)/2,ci+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 alpha (theta) after T-S fuzzy control by an MATLAB software tool, and converting L into L1An approximation curve of a straight line segment predicted by the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) and L2The rotational speed F' (theta) of the cam after the fuzzy control of T-S is compared with the predicted curve of the rotational angle alpha (theta) of the cam if L2The predicted curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) is smooth and close to L1The linear section curve is predicted by the cam rotation speed F' (theta) and the cam rotation angle alpha (theta), so that an ideal effect is achieved; otherwise, the T-S fuzzy controller needs to be optimized, so that the predicted curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (theta) is more smooth and is close to L as much as possible1Predicting a straight-line segment approximation curve of the cam rotation speed F' (theta) and the cam rotation angle alpha (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-1(see formula (31)) if ci-1Is increased by (c)i-1+ci) 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' (theta)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 anglei+2If the decrease is (c)i+1+ci+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) Is negative), thus F ″ (θ)2) Will increase and tend to F2'(θ2) The predicted speed is close to an ideal straight line segment, and a 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' (theta)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:
hi(θ)＝ftri(θ,(ci-1+ci)/s,(ci+ci+1)/s,(ci+1+ci+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 the cam rotation speed F' (theta) and the cam rotation angle alpha (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.
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| CN107520752A (en) * | 2016-06-22 | 2017-12-29 | 许亚夫 | A kind of high precision numerical control camshaft grinding machine control system |
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| EP0079750A1 (en) * | 1981-11-12 | 1983-05-25 | WALLER, Francis E. | Parallel cylinder internal combustion engine |
| CN101486162A (en) * | 2009-01-16 | 2009-07-22 | 北京诺思泰格精密技术有限公司 | X-Y-C three axis linkage high tracking counter shaft-crank shaft grinding machine |
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