Cam curve design method for cap screwing machine based on multi-objective method

Abstract

Disclosed is a cam curve design method for a cap screwing machine based on a multi-objective method, belonging to the technical field of cap screwing machine cam design. The present disclosure is to solve the problem that the conventional method for redesigning the cam curve cannot optimize the parameters according to the degree of importance such that the operating stability of a cam mechanism is poor. The method comprises: acquiring key point data determined for realizing the functions of a cam of the cap screwing machine; performing dimensionless processing on the key point data to obtain a dimensionless time T and a dimensionless displacement S; writing an n-order polynomial fitting cam curve displacement function, manually interpolating local control points, and obtaining a polynomial fitting cam curve expression and chart in combination with the constraints of key point displacement by using a least squares method; and selecting cam curve optimization objectives and setting weight coefficients to construct an optimized cam curve. The present disclosure is used for the design of a cam curve.

Claims

1. A method to design a cam curve for a cap screwing machine based on a multi-objective method, comprising the following steps: step 1: acquiring key point data determined for realizing functions of a cam of the cap screwing machine; step 2: performing dimensionless processing on the key point data to obtain dimensionless time T and dimensionless displacement S; step 3: writing an n-order polynomial fitting cam curve displacement function, comparing a fitting error with a set allowable error, and if the comparison result is greater than a set threshold, manually interpolating local control points till the comparison result is within the set threshold; then, obtaining a polynomial fitting cam curve expression and chart in combination with constraints of key point displacement by using a least squares method, n being a positive integer; and step 4: selecting a cam curve optimization objective and setting a weight coefficient to construct an optimized cam curve; step 5: machining the cam with a surface shaped in the optimized cam curve.

2. The method to design a cam curve for a cap screwing machine based on a multi-objective method according to claim 1, wherein in step 1, data that a cam follower of the cap screwing machine arrives at corresponding displacement points at corresponding angles is used as key points, and these key points support cap grasping, bottle entering and cap screwing functions of the cap screwing machine.

3. The method to design a cam curve for a cap screwing machine based on a multi-objective method according to claim 1, wherein in step 3, a polynomial fitting program is written through MATLAB, and n-order polynomial fitting cam curve displacement function is expressed as S=C.sub.0+C.sub.1T+ . . . +C.sub.nT.sup.n, where C.sub.0 is an initial coefficient, C.sub.1 is a 1st coefficient, and C.sub.n is an N.sup.th coefficient.

Description

BRIEF DESCRIPTION OF FIGURES

(1) FIG. 1 is a structure schematic diagram of a cam mechanism of a cap screwing machine according to an embodiment of the present disclosure;

(2) FIG. 2 is a schematic diagram of a cam in FIG. 1;

(3) FIG. 3 is a time-displacement cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(4) FIG. 4 is a time-velocity cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(5) FIG. 5 is a time-acceleration cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(6) FIG. 6 is a time-jerk cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(7) FIG. 7 is a time-jump cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(8) FIG. 8 is a time-dynamic load torque cam curve chart designed by a 7-order polynomial multi-objective optimization method;

(9) FIG. 9 is a time-displacement cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(10) FIG. 10 is a time-velocity cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(11) FIG. 11 is a time-acceleration cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(12) FIG. 12 is a time-jerk cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(13) FIG. 13 is a time-jump cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(14) FIG. 14 is a time-dynamic load torque cam curve chart designed by an improved 2-order harmonic function multi-objective optimization method;

(15) FIG. 15 is a comparison chart of time-displacement curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(16) FIG. 16 is a comparison chart of time-velocity curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(17) FIG. 17 is a comparison chart of time-acceleration curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(18) FIG. 18 is a comparison chart of time-jerk curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(19) FIG. 19 is a comparison chart of time-jump curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(20) FIG. 20 is a comparison chart of time-dynamic load torque curves before and after multi-objective optimization of polynomial and harmonic cam curves according to the present disclosure;

(21) FIG. 21 is a flow chart of a method according to the present disclosure.

DETAILED DESCRIPTION

Example 1

(22) The following describes the present embodiment in combination with FIG. 1 to FIG. 20. A cam curve design method for a cap screwing machine based on a multi-objective method according to the present embodiment includes the following steps:

(23) step 1: acquiring key point data determined for realizing the functions of a cam of the cap screwing machine;

(24) step 2: performing dimensionless processing on the key point data to obtain dimensionless time T and dimensionless displacement S;

(25) step 3: writing an n-order polynomial fitting cam curve displacement function, comparing a fitting error with a set allowable error, and if the comparison result is greater than a set threshold, manually interpolating local control points till the comparison result is within the set threshold; then, obtaining a polynomial fitting cam curve expression and chart in combination with the constraints of key point displacement by using a least squares method; n being a positive integer; and

(26) step 4: selecting cam curve optimization objectives and setting weight coefficients to construct an optimized cam curve.

(27) In step 1, the data that a cam follower of the cap screwing machine arrives at corresponding displacement points at corresponding angles is used as key points, and these key points generally support the cap grasping, bottle entering and cap screwing functions of the cap screwing machine. In step 3, a polynomial fitting program is written through MATLAB, and the n-order polynomial fitting cam curve displacement function can be expressed as: S=C.sub.0+C.sub.1 T+ . . . +C.sub.nT.sup.n (the present disclosure mainly uses 5-order, 6-order or 7-order polynomial). In order to control the fitting error within a certain range, an allowable error of design is preset when programming is conducted. When the fitting error differs from the set allowable error greatly, i.e., the fitting error exceeds the set threshold, local control points can be manually interpolated to reduce the error and effectively control the shape of the cam curve, so that the curve trend is more in accordance with the actual situation. Then, the cam curve expression and chart designed by a polynomial fitting method are obtained in combination with the constraints of key point displacement by using a least squares method, wherein C.sub.0 is an initial coefficient, C.sub.1 is a 1st coefficient, and C.sub.n is an n.sup.th coefficient.

Example 2

(28) The following describes the present embodiment in combination with FIG. 1 to FIG. 20. A cam curve design method for a cap screwing machine based on a multi-objective method according to the present embodiment includes the following steps:

(29) step 1: acquiring key point data determined for realizing the functions of a cam of the cap screwing machine;

(30) step 2: performing dimensionless processing on the key point data to obtain dimensionless time T and dimensionless displacement S;

(31) step 3: writing a cam curve fitting program, determining the order of k based on an improved cam curve displacement function of a k-order harmonic function, adding local control points, fitting key points and the local control points by using a sum formula of a least squares method, and solving a cam curve expression and a cam curve chart; and

(32) step 4: selecting cam curve optimization objectives and setting weight coefficients to construct an optimized cam curve.

(33) The k is more than or equal to 2.

(34) In step 1, the data that a cam follower of the cap screwing machine arrives at corresponding displacement points at corresponding angles is used as key points, and these key points generally support the cap grasping, bottle entering and cap screwing functions of the cap screwing machine. In step 3, a cam curve fitting program is written through MATLAB, and the improved cam curve displacement function of the k-order harmonic function can be expressed as:

(35) $S=\underset{n=1}{\overset{k}{.Math.}}{c}_n\sin \left({\omega }_{n}T+{\phi }_n\right).$
For the selection of k, if the k value is smaller, the possibility of resonance of the mechanism is smaller. When k=1, i.e. first order, the performance of the first-order curve is the same as the motion law of a cycloid curve or a simple harmonic curve, so k is generally rounded from 2. At the same time, in order to control the local shape and well adjust the cam curve, a series of local control points are added, key points and local control points are fitted by using a sum formula of a least squares method to obtain the values of c.sub.n, ω.sub.n and φ.sub.n, and then a cam curve expression and a cam curve chart are obtained. In the formula, c.sub.n is an amplitude, ω.sub.n is an angular frequency, and q is a phase angle.

(36) A multi-objective optimization method is used to optimize the cam curve of polynomial fitting or harmonic function fitting, so that these parameter values can be balanced in advance and the importance thereof can be set according to the actual conditions and requirements, and these parameter values can be optimized to different degrees, and the fluctuation of the optimized characteristic values can be reduced, which is very beneficial to the stability of the cam mechanism during operation and effectively ensures the operating precision of the mechanism.

(37) When multi-objective optimization is used, the optimized objectives usually constrain one another, so that all the objectives cannot be optimized together. Accordingly, when the multi-objective optimization is used to solve practical problems, weights corresponding to the importance of respective objectives are usually assigned according to the different importance of the objectives, and the more important objectives are regarded as main optimization objectives. Thus, in order to achieve relatively excellent kinematic and dynamic characteristics, the optimization objectives need to be balanced, so that the important objectives are optimal, and other objectives are preferable. The multi-objective optimization function is generally written as:

(38) $\{\begin{array}{c}{\min \left(f_1\left(x\right),f_2\left(x\right),\mathrm{.Math.},f_p\left(x\right)\right)}^T,p>1,x>E^P\\ s.t{g}_i\left(x\right)\ge 0,i=1,2,\mathrm{.Math.},m\\ h_i\left(x\right)=0,i=1,2,\mathrm{.Math.},l\end{array},$

(39) In the formula, f.sub.p(x) is a single objective function that needs to be optimized, s.t g.sub.i(x)>0 is a multi-objective optimization inequality constraint, and s.t h.sub.i(x)=0 is an equality constraint. h.sub.i(x) is an equality constraint equation, m is a positive integer, and I is a positive integer.

(40) When a linear weighting method is used to optimize a few objectives, p objective functions f.sub.j(x) are usually endowed with certain weight coefficients λ.sub.1 according to different importance of the objectives and the degree of importance, and then a new objective function is constructed, that is, all the objective functions are weighted and summed to obtain the optimal value of the new objective function. Its mathematical model can be written as:

(41) $\{\begin{array}{c}\min \underset{j=1}{\overset{p}{.Math.}}{\lambda }_j{f}_j\left(x\right),p>1,x>E^P\\ s.t{g}_i\left(x\right)\ge 0,i=1,2,\mathrm{.Math.},m\\ h_i\left(x\right)=0,i=1,2,\mathrm{.Math.},l\end{array}.$

(42) In the above two embodiments, the method of optimizing characteristic parameter values in step 4 is: optimizing the cam curve based on a multi-objective optimization method. First optimization objectives are determined: the characteristic values of maximum velocity V.sub.max, maximum acceleration A.sub.max, maximum jerk J.sub.max, maximum jump Q.sub.max and dynamic load torque characteristic parameter AV.sub.max are compared, the comprehensive performance of the cam curve is distinguished by experience, and the maximum velocity V.sub.max, maximum acceleration A.sub.max, maximum jerk J.sub.max, maximum jump Q.sub.max and dynamic load torque characteristic parameter AV.sub.max are selected as cam curve optimization objectives. Then a weight coefficient is set: a weight coefficient λ.sub.1 is added to the objectives of the maximum velocity V.sub.max, maximum acceleration A.sub.max, maximum jerk J.sub.max, maximum jump Q.sub.max and dynamic load torque characteristic parameter AV.sub.max by using a linear weighting method, and then summing is conducted. In a specific application, the weights are changed according to relevant design requirements.

(43) Finally, the obtained cam curve charts obtained by optimization can be compared to observe the maximum speed value, maximum acceleration value, maximum jerk value, maximum jump value and the like as well as the fluctuation thereof so as to verify the optimization effect.

(44) Aiming at the defects caused by the conventional design method of connecting key points with a straight line and chamfering the junctions when a cam curve of a cap screwing machine is designed, the present disclosure optimizes the cam curve in a multi-objective manner based on polynomial fitting or harmonic function fitting on the premise that the key points determined for realizing the functions of the conventional cam mechanism are reserved, thereby improving the derivability and continuity of the cam curve at the key points, and effectively solving the problems of impact, wear, fatigue damage, noise and the like of the cam mechanism during operation.

(45) The present disclosure adopts a multi-objective optimization method to design a cam curve of a cap screwing machine. First, each key point (generally expressed as certain displacement to be achieved for an angle) determined to complete a cap screwing function of the cap screwing machine is obtained. Second, in order to avoid the influence of dimensions, the expression of each key point is converted into a time-displacement relationship, and then dimensionless processing is performed on the time and the displacement. Finally, local control points are appropriately added on the basis of controlling the shape and error of the curve, key points and local control points are fitted by the multi-objective optimization manner, and a cam curve expression and a cam curve chart are output.

(46) In order to verify the validity of the method according to the present disclosure, the optimized cam curve can be compared with cam curves designed by single polynomial fitting and harmonic function fitting to select an optimal cam curve design method.

(47) The following gives an application example to elaborate the design process of the cam curve:

(48) As shown in FIG. 1, a cam mechanism of a cap screwing machine is mainly composed of a cam center shaft 10, a cam 20, a follower 30, a follower upper roller 40, a follower lower roller 50, etc. During operation, the cam 20 remains fixed, the cam center shaft 10 rotates relative to the cam 20, and the follower upper roller 40 and the follower lower roller 50 roll about the cam center shaft 10 along a cam hub to drive the cam follower 30 to move up and down along the hub.

(49) As shown in FIG. 2, the cam follower of the cap screwing machine needs to arrive at corresponding displacement points at corresponding angles to complete the operations from bottle leaving, cap grasping, bottle entering to cap screwing. These displacement points are referred to as key points, and the design allowable error for the key points is specified to 1 mm. The motion law is that the displacements of the cam mechanism from a bottle leaving point P1) (0°) to a point P2 (230°) till a cap screwing end point (290°) are two predetermined horizontal displacements, and the displacement from a cap screwing start point (170°) to the point P2) (230° is predetermined. The main operating period of the conventional cam is in a return period, and the cap screwing machine needs to complete the main cap grasping, bottle entering and cap screwing operations in the return period, so the design of the curve directly affects the cap screwing speed and quality of the cap screwing machine. Therefore, the cam curve from the point P1 to the cap screwing start point in the return period is mainly optimized.

(50) According to the method described in embodiment 1, as shown in FIG. 3 to FIG. 8, a cam curve chart of the cam curve from the point P1 to the cap screwing start point is designed by using a 7-order polynomial multi-objective optimization method. A 7-order polynomial displacement curve can be written as S=x.sub.1+x.sub.2T.sup.1+x.sub.3T.sup.2+x.sub.4T.sup.3+x.sub.5T.sup.4+x.sub.6T.sup.5+x.sub.7T.sup.6+x.sub.8T.sup.7, and the cam curve can be expressed as:

(51) $\{\begin{array}{c}S=x_1+x_2{T}^1+x_3{T}^2+x_4{T}^3+x_5{T}^4+x_6{T}^5+x_7{T}^6+x_8{T}^7\\ V=x_2+2x_3{T}^1+3x_4{T}^2+4x_5{T}^3+5x_6{T}^4+6x_7{T}^5+7x_8{T}^6\\ A=2x_3+6x_4{T}^1+12x_5{T}^2+20x_6{T}^3+30x_7{T}^4+42x_8{T}^5\\ J=6x_4+24x_5{T}^1+60x_6{T}^2+120x_7{T}^3+210x_8{T}^4\\ Q=24x_5+120x_6{T}^1+360x_7{T}^2+840x_8{T}^3\\ \mathrm{AV}=(x_2+2x_3{T}^1+3x_4{T}^2+4x_5{T}^3+5x_6{T}^4+\\ 6x_7{T}^5+7x_8{T}^6)\times (2x_3+6x_4{T}^1+12x_5{T}^2+\\ 20x_6{T}^3+30x_7{T}^4+42x_8{T}^5)\end{array},$

(52) S is displacement, the interval of T is 0-1, x is a coefficient, x=[x.sub.1, x.sub.2, x.sub.3, x.sub.4, x.sub.5, x.sub.6, x.sub.7, x.sub.8], V is velocity, A is acceleration, J is jerk, Q is jump, and AV is a dynamic load torque characteristic parameter; the cam curve optimization objectives are maximum velocity V.sub.max, maximum acceleration A.sub.max, maximum jerk J.sub.max, maximum jump Q.sub.max and dynamic load torque characteristic parameter AV.sub.max.

(53) Then, f.sub.1(x)=V.sub.max=max|V|, f.sub.2(x)=A.sub.max=max|A|, f.sub.3(x)=J.sub.max=max|J|, f.sub.4(x)=Q.sub.max=max|Q| and f.sub.5(x)=AV.sub.max=max|AV|. The weight coefficient of the maximum velocity V.sub.max and the maximum acceleration A.sub.max is set to 0.35, the weight coefficient of other characteristic parameters is set to 0.1, λ.sub.1=λ.sub.2=0.35, and λ.sub.3=λ.sub.4=λ.sub.5=0.35.

(54) The objective function can be written as:

(55) $f\left(x\right)=\underset{j=1}{\overset{5}{.Math.}}{\lambda }_i{f}_j\left(x\right).fwdarw.\min .$

(56) The cam curve has a lower limit value V.sub.max which is 1, but does not have an upper limit V.sub.max, so too large V.sub.max is not needed during design. The V.sub.max of almost all the common standard cam curves is less than 3, so in order to reduce the time required for calculation, the upper limit value V.sub.max of the optimization objectives can be 3. Therefore, the feasible region of V.sub.max is 1-3. The feasible region of other AV.sub.max is 4-12, the feasible region of J.sub.max is 30-350, the feasible region of Q.sub.max is 40-20,000, and the feasible region of AV.sub.max is 3-21. In addition, the variation range of each key point is set in order to ensure that the error of the displacement curve at each key point is not more than 1 mm. The charts show the optimized cam curve as V.sub.max=1.95, A.sub.max=9.465, J.sub.max=203.2, Q.sub.max=1344, AV.sub.max=14.41.

(57) By using the method described in embodiment 2, as shown in FIG. 9 to FIG. 14, a cam curve chart of the cam curve from the point P1 to the cap screwing start point is designed by using an improved 2-order harmonic optimization method. The displacement function of the harmonic cam curve from the point P1 to the cap screwing start point can be expressed as:
S=x.sub.1 sin(x.sub.2T+x.sub.3)+x.sub.4 sin(x.sub.5T+x.sub.6).

(58) An improved 2-order harmonic optimization cam curve can be obtained according to the 7-order polynomial cam curve optimization method:

(59) $\{\begin{array}{c}S=100\sin \left(0.008791T+3.1314\right)+\\ 0.14096\sin \left(7.2444T+3.3152\right)\\ V=0.87912\cos \left(0.008791T+3.1314\right)+\\ 1.021171\cos \left(7.2444T+3.3152\right)\\ A=-0.007729\sin \left(0.008791T+3.1314\right)-\\ 7.397768\sin \left(7.2444T+3.3152\right)\\ J=-0.000068\cos \left(0.008791T+3.1314\right)-\\ 53.592394\cos \left(7.2444T+3.3152\right)\\ Q=0.0000006\sin \left(0.008791T+3.1314\right)+\\ 388.244738\sin \left(7.2444T+3.3152\right)\end{array}.$

(60) The charts show the optimized cam curve as V.sub.max=1.9, A.sub.max=7.396, J.sub.max=53.59, Q.sub.max=388.2, AV.sub.max=8.932.

(61) As shown in FIG. 15 to FIG. 20, it can be found by comparing the cam curve of 7-order polynomial optimization and the cam curve of improved 2-order harmonic optimization with the curves before optimization that the performance of the 7-order polynomial fitting cam curve before optimization is worst, the fluctuation of various characteristic values is large, but after the overall 7-order polynomial fitting cam curve is optimized, the maximum velocity V.sub.max is reduced from 2.34 to 1.95 by 16.67%; the maximum acceleration A.sub.max is reduced from 69.91 to 9.465 by 86.5%; the maximum jerk J.sub.max is reduced from 1292 to 203.2 by 84.27%; the maximum jump Q.sub.max is reduced from 13450 to 1344 by 90%; and the dynamic load torque characteristic parameter AV.sub.max is reduced from 52.93 to 14.41 by 72.78%. All the characteristic parameters are reduced greatly, where the maximum jump Q.sub.max is even reduced by 90%, so the kinematic and dynamic performance is improved obviously. After the improved 2-order harmonic cam curve is optimized, the maximum velocity V.sub.max is reduced from 2.002 to 1.9 by 5.09%; the maximum acceleration A.sub.max is reduced from 8.806 to 7.398 by 16.0%; the maximum jerk J.sub.max is reduced from 68.34 to 53.59 by 21.58%; the maximum jump Q.sub.max is reduced from 533 to 388.3 by 27.15%; and the dynamic load torque characteristic parameter AV.sub.max is reduced from 10.98 to 8.911 by 18.84%. All the characteristic parameters are reduced less than those of the polynomial cam curve but still reduced a lot, where the maximum jump Q.sub.max is even reduced by 27.15%, so the kinematic and dynamic performance is improved relatively obviously. Finally, the optimized improved 2-order harmonic cam curve is regarded as an optimal cam curve, a three-dimensional model thereof can be imported into a UG machining module to generate a machining locus, and the machining of the cam of the cap screwing machine is completed on a machining center.

(62) Therefore, the multi-objective optimization method of the present disclosure is very effective for the design of the cam curve. In the actual engineering design, the objectives can be endowed with different weight optimization cam curves according to different importance of the optimization objectives and different engineering conditions.

(63) Since the cam curve continuity of the cam designed by the method of the present disclosure is improved, when the rotation speed is improved, the mechanism will not vibrate violently, and the production efficiency of the cap screwing machine is greatly improved. Taking the cam mechanism of the cap screwing machine in the specific embodiment as an example, after the rotation speed is improved, the cap screwing ability of a single cap screwing head of the cap screwing machine is improved from original 1,800 to 2,600 per hour, and the cap screwing efficiency is improved by 30.8%. Since the cam mechanism is designed with 18 cap screwing heads, 46,800 beverage bottle caps can be screwed per hour on a production line, which will bring considerable benefits to the beverage filling enterprises.

(64) The method of the present disclosure is not only limited to the cam of the cap screwing machine, and is applicable to cams in packaging machines such as filling machines and bottle blowing machines. The cam curve design method of the present disclosure can greatly improve the operating performance of a mechanical system, effectively control the unfavorable factors such as noise and vibration during the operation of the machine and improve the operating efficiency under the premise of high-speed driving of the machine.