CN110162815B - Multi-objective optimization method of eight-connecting-rod mechanical press based on NSGA-II algorithm - Google Patents

Multi-objective optimization method of eight-connecting-rod mechanical press based on NSGA-II algorithm Download PDF

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CN110162815B
CN110162815B CN201810965045.0A CN201810965045A CN110162815B CN 110162815 B CN110162815 B CN 110162815B CN 201810965045 A CN201810965045 A CN 201810965045A CN 110162815 B CN110162815 B CN 110162815B
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杨将新
陈安杰
曹衍龙
曹彦鹏
任立飞
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Zhejiang University ZJU
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Abstract

The invention provides an NSGA-II algorithm-based eight-connecting-rod mechanical press multi-objective optimization method, which comprises the steps of determining an optimization objective, design variables and constraint conditions, and establishing a mathematical model; simplifying an eight-link mechanism model; setting an initial design variable and an initial eccentric body rotating speed, and performing kinematic analysis on the eight-bar linkage model to obtain kinematic parameters including a slide block stroke, a slide block speed and a slide block acceleration; performing dynamic analysis on the eight-connecting-rod mechanism model according to the kinematic parameters and the tonnage information of the press to obtain the crank torque; optimizing the speed fluctuation of the sliding block and the maximum crank torque by using an NSGA-II algorithm, solving a multi-objective optimization mathematical model, and obtaining a Pareto optimal solution set of design variables. The design variable obtained by the method has high accuracy and quick optimization, and the design and development time of an enterprise is reduced.

Description

Multi-objective optimization method of eight-connecting-rod mechanical press based on NSGA-II algorithm
Technical Field
The invention relates to the field of mechanical transmission, in particular to an eight-connecting-rod mechanical press multi-objective optimization method based on NSGA-II algorithm.
Background
The mechanical press is a typical multi-variety and small-batch mechanical product, which is generally designed by taking a complete machine as a unit, and with the development of the automobile industry and the update of old stamping equipment, the current product structure gradually develops from the original standard machine accounting for 60 percent and the customer customizing machine accounting for 40 percent to the customer customizing machine accounting for 60 to 70 percent, which is undoubtedly great for the technical challenge. And mechanical presses are produced in an order mode, each press needs to be redesigned according to the order requirements of customers, 3-5 machines need to be designed simultaneously on one production line sometimes, and the conventional design of each machine generally needs to be subjected to a series of repeated processes of data retrieval, scheme conception, calculation analysis, drawing, documentation and the like, so that the related contents and steps are more, the formulas, the diagrams and the data volume are large, the mutual restriction factors are many, and a large amount of manpower is consumed, so the design period is long, and the task is heavy.
However, some orders can be realized only by finely adjusting design parameters on the basis of a mature model, and conventional manual modification of the model is not only prone to errors, but also consumes a large amount of time, so that the design and development period of an enterprise is greatly increased.
Disclosure of Invention
The invention aims to provide an NSGA-II algorithm-based eight-connecting-rod mechanical press multi-objective optimization method, which is high in accuracy of obtained design variables and fast in optimization and reduces design and development time of enterprises.
The technical scheme adopted by the invention for solving the technical problems is as follows: the multi-objective optimization method of the eight-connecting-rod mechanical press based on the NSGA-II algorithm comprises the following steps:
(1) determining an optimization target, design variables and constraint conditions, and establishing a multi-objective optimization mathematical model taking the minimum fluctuation value of the running speed of the sliding block in a specified area and the minimum maximum crank torque value in the specified area as objective functions;
(2) establishing an eight-connecting-rod mechanism three-dimensional model of the eight-connecting-rod mechanical press, and obtaining a simplified eight-connecting-rod mechanism model;
(3) setting an initial design variable and an initial eccentric body rotating speed, and performing kinematic analysis on the eight-bar linkage model to obtain kinematic parameters including a slide block stroke, a slide block speed and a slide block acceleration;
(4) performing dynamic analysis on the eight-connecting-rod mechanism model according to the kinematic parameters and the tonnage information of the press to obtain the crank torque;
(5) and optimizing the speed fluctuation of the sliding block and the maximum crank torque by using an NSGA-II algorithm, and solving a multi-objective optimization mathematical model to obtain a Pareto optimal solution set of the design variables.
Preferably, in step (1), the multi-objective optimization mathematical model is
Figure BDA0001774755340000021
Wherein the content of the first and second substances,
Figure BDA0001774755340000022
minf2(X)＝Tmax，
X＝[l1，l2，l3，l4，l5，l6，l7，l8，a，b，α，β]，
S，tXmin≤X≤Xmax
gu(X)≤0(u＝1，2，…，)，
in the formula, minf1(X) is an objective function, minf, with the minimum fluctuation value of the running speed of the sliding block in the designated area2(X) is an objective function for which the maximum crank torque value in the specified region is minimal; t ismaxIs the maximum crank torque; x is a design variable,/1，l2，l3，l4，l5，l6，l7，l8The length of each rod is 8, a and b are the positions of two hinges, and alpha is the upper and lower rocking bars3、l4An included angle between the two legs, beta is a tripod6、l7The included angle of the two sides; gu(X) is a constraint.
Preferably, in step (3), the method for obtaining kinematic parameters comprises the following steps:
(3.1) establishing a kinematic equation set of the eight-bar mechanism model by utilizing a vector closure rule according to the simplified eight-bar mechanism model;
(3.2) solving a kinematic equation set of the eight-bar linkage model by using a New-Raphson algorithm to obtain a slider stroke solving curve; solving a first derivative of the slide stroke solving curve to obtain a slide speed solving curve; solving a second derivative of the slide stroke solving curve to obtain a slide acceleration solving curve;
(3.3) establishing a three-dimensional model of the eight-bar linkage mechanism, and obtaining a slide block stroke simulation curve, a slide block speed simulation curve and a slide block acceleration simulation curve according to the three-dimensional model simulation processing of the eight-bar linkage mechanism;
(3.4) verifying whether the solving curve is consistent with the simulation curve, and if so, respectively outputting solving parameters of the slide stroke, the slide speed and the slide acceleration; otherwise, returning to the step (3.1).
Preferably, in step (3.1), the expression of the kinematic equation set of the eight-bar linkage model is as follows:
Figure BDA0001774755340000031
wherein: l1,l2,l3,l4,l5,l6,l7,l8Respectively, the length of each rod is m; a and b are respectively a hinge center O and a hinge center O1Distances in the X-axis and Y-axis directions in units of m; alpha, beta are respectively a rod l3And l4，l6And l7The unit of the included angle of (A) is rad; theta1,θ2,θ3,θ4,θ5,θ6,θ7,θ8The angular displacement corresponding to each rod is respectively, the unit is rad, and all the angles rotate anticlockwise to the angles of each component by taking the positive direction of an X axis as a starting point; y is the displacement of the slider in m.
Preferably, in the step (4), the method for obtaining the crank torque includes the steps of:
(4.1) carrying out dynamic static analysis on each rod of the eight-bar linkage according to the simplified eight-bar linkage model to obtain the stress condition of each rod;
(4.2) listing a dynamic static balance equation set of the eight-link mechanism according to theoretical mechanics and Newton's law;
(4.3) substituting the kinematic parameters into an equation set and solving to obtain a crank torque solving curve;
(4.4) carrying out simulation processing according to the eight-connecting-rod mechanism three-dimensional model to obtain a crank torque simulation curve;
(4.5) verifying whether the solving curve of the crank torque is consistent with the simulation curve or not, and if so, respectively outputting crank torque solving parameters; otherwise, returning to the step (4.1).
Preferably, in the step (4), the method for obtaining the crank torque includes the steps of:
(4.11) according to the simplified eight-bar linkage model, carrying out stress analysis on each bar of the eight-bar linkage to obtain the stress condition of each bar;
(4.12) integrating the stress condition of each rod, and listing an imaginary work equation by using an imaginary work principle;
(4.13) substituting the kinematic parameters into an imaginary work equation and solving to obtain a crank torque solving curve;
(4.14) establishing a three-dimensional model of the eight-bar linkage mechanism, and carrying out simulation processing according to the three-dimensional model of the eight-bar linkage mechanism to obtain a crank torque simulation curve;
(4.15) verifying whether the solving curve of the crank torque is consistent with the simulation curve or not, and if so, respectively outputting crank torque solving parameters; otherwise, returning to the step (4.11).
As a preferable scheme, in the step (4.4), a three-dimensional model of the eight-bar linkage is established by using solid works software, and the three-dimensional model of the eight-bar linkage is subjected to simulation processing by using software Adams.
Preferably, in the step (4.14), an eight-link mechanism three-dimensional model is established by using Solidworks software, and the eight-link mechanism three-dimensional model is subjected to simulation processing by using software Adams.
The invention has the advantages that: 1. by providing the multi-objective optimization method of the eight-connecting-rod mechanical press based on the NSGA-II algorithm, the optimal solution set of the design variables is obtained, and the redesign of the press can be met only by finely adjusting the design variables on the basis of the mature press model.
2. The accuracy of the rod system parameters obtained through optimization is high, the time spent in the optimization process is short, and the design and development period of an enterprise is greatly shortened.
3. In the process of kinematics and dynamics analysis, an Adams simulation method is adopted for verification, so that the accuracy of the kinematics and dynamics analysis is ensured.
4. In the dynamic analysis process, each rod in the mechanism is analyzed by a dynamic static analysis method, a column equation of the Dalberger principle is utilized, a matrix is combined, solution is carried out, all supporting reaction forces of the mechanism are obtained, crank driving torque is obtained through the obtained supporting reaction forces and kinematic parameters, the solved T error is smaller, and the accuracy is higher.
5. The crank driving torque is directly solved by utilizing the virtual work principle, the intermediate link of solving the thrust reaction is not needed, the crank driving torque T can be directly solved by utilizing the virtual work principle, the calculation amount can be greatly reduced, and the calculation efficiency is improved.
Drawings
Fig. 1 is a schematic structural view of an eight-link mechanism.
Fig. 2 is a schematic diagram of motion analysis of an eight-bar linkage.
FIG. 3 is a graph of slider travel, velocity, acceleration.
FIG. 4 is a simulation graph of the stroke, velocity and acceleration of the slider obtained from Adams.
FIG. 5 is a graph of the solution for crank torque T for both methods.
Fig. 6 is a diagram of a slider F.
FIG. 7 shows the main rod l8The force is tested.
Figure 8 is a tripod diagram.
FIG. 9 shows a lower link l5The force is tested.
Fig. 10 is a rocker force diagram.
FIG. 11 shows the upper rod2The force is tested.
FIG. 12 is a force analysis diagram of the eccentric body.
Fig. 13 is a force analysis diagram of the system.
FIG. 14 is a plot of crank torque T from Adams simulations.
FIG. 15 is a flow chart of the present invention.
The labels in the figure are: eccentric body 1, go up pull rod 2, go up rocker 3, lower rocker 4, lower link 5, tripod 6, main pull rod 7, slider 8, hinge 9, hinge 10.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
The invention provides an NSGA-II algorithm-based multi-objective optimization method for an eight-connecting-rod mechanical press, which aims at a certain type of press with a certain stroke and tonnage, needs to adjust design variables (rod system parameters) on the basis of the existing mature press, and establishes a multi-objective optimization mathematical model by taking the minimum fluctuation value of the running speed of a sliding block in a designated area and the minimum maximum crank torque value in the designated area as optimization targets. In actual operation, the design variables (rod system parameters) of the mature press are used as a median value, float a certain proportion up and down and used as an upper limit and a lower limit, constraint conditions are adjusted simultaneously, then a multi-objective optimization mathematical model is established, a plurality of groups of solutions (rod system parameters) are found by using an NSGA-II algorithm to enable two objective functions to reach the optimal of the parato, a parato optimal solution set is obtained, and basically all the design variables (rod system parameters) need to be finely adjusted in the parato optimal solution set.
NSGA-II algorithm
NSGA-II is one of the most popular multi-target genetic algorithms at present, reduces the complexity of the non-inferior ranking genetic algorithm, has the advantages of high running speed and good convergence of solution sets, and becomes the basis of the performance of other multi-target optimization algorithms.
The present embodiment was optimized based on a press model T4L 8-1200. The eight-connecting-rod mechanism comprises an eccentric body 1, an upper pull rod 2, an upper rocker 3, a lower rocker 4, a lower pull rod 5, a tripod 6, a main pull rod 7 and a sliding block 8, and all the rods are connected through hinges. Go up rocker 3 and lower rocker 4 and pass through the hinge 10 and articulate, go up pull rod 2 and last rocker 3 and articulate, the tripod 6 is located the eccentric body 1 outside with articulated connection form cover, and the center of rotation of eccentric body 1 is hinge 9, and 5 one ends of lower pull rod are articulated with lower rocker 4, and the other end of lower pull rod 5 is articulated with tripod 6. One end of the main pull rod 7 is hinged with the tripod 6, and the other end of the main pull rod 7 is hinged with the sliding block 8.
As shown in fig. 15, the multi-objective optimization method for an eight-bar mechanical press based on the NSGA-II algorithm provided by the embodiment of the present invention includes the following steps:
(1) and determining an optimization target, design variables and constraint conditions, and establishing a multi-objective optimization mathematical model taking the minimum fluctuation value of the running speed of the sliding block in the designated area and the minimum value of the maximum crank torque value in the designated area as objective functions.
The optimization target is based on the operation stability of the sliding block and the power of the motor, the minimum fluctuation value of the operation speed of the sliding block in a designated area and the minimum value of the maximum crank torque value in the designated area are taken as the optimization targets, and the smaller the fluctuation of the operation speed of the sliding block is, the more stable the sliding block is; the smaller the maximum crank torque, the less motor power is required.
Specifically, the expression of the multi-objective optimization mathematical model is as follows:
Figure BDA0001774755340000061
wherein, minf1(X) is an objective function, minf, with the minimum fluctuation value of the running speed of the sliding block in the designated area2(X) is an objective function for which the maximum crank torque value in the specified region is the minimum. The objective function is expressed as follows:
Figure BDA0001774755340000071
minf2(X)＝Tmax，
in the formula, V (X, theta)1) For a crank angle of theta1When the speed of the sliding block is actually increased,
Figure BDA0001774755340000073
is theta1Equal to the average speed of the inner slide block of 90-150 DEG, and n is theta1The number of control points is between 90 and 150 degrees and depends on the segmentation precision.
Wherein, X is a design variable,
X＝[l1，l2，l3，l4，l5，l6，l7，l8，a，b，α，β]，
in the formula I1，l2，l3，l4，l5，l6，l7，l8Each having a rod length of 8 rods, i.e. /)1Is an eccentric body 1, l2Is an upper pull rod 2, l3Is an upper rocker 3, l4Is a lower rocker 4, l5Is a lower pull rod 5; l6And l7Are respectively the length of two side rods of the tripod 68The main pull rod 7, a and b are respectively the center O of the hinge 9 and the center O of the hinge 101Distances in the X-axis and Y-axis directions; alpha is an up-down rocker L3、L4Beta is a tripod L6Main pull rod L7The angle between the two sides, see fig. 1 and 2.
Wherein the constraint is gu(X) ≦ 0(u ═ 1, 2, …, m), including the conditions that the crank present is required to satisfy, the motion noninterference condition, and the conditions that the maximum pressure angle is less than a critical value, the slide travel is between the maximum and minimum allowed values.
The constraint conditions can be divided into boundary constraints and behavior constraints, and the boundary constraints are used for limiting the variation range X of a certain design variablemin≤X≤Xmax，XminIs the lower limit of X, XmaxThe upper limit of X.
The behavioral constraints are based on certain properties of the structure, including:
a. conditions of existence of crank
Figure BDA0001774755340000072
Figure BDA0001774755340000081
Figure BDA0001774755340000082
In which a and b are eachIs the center O of the hinge 9 and the center O of the hinge 101Distances in the X-axis and Y-axis directions.
b. The stroke of the slide block meets the tolerance requirement
g4(X)＝Smin-S≤0
g5(X)＝S-Smax≤0
SminMinimum allowed slide travel, take Smin＝S-0.1(mm)；
SmaxMaximum allowed slide travel, take Smax＝S+0.5(mm)。
c. Meet the assembly conditions
g6(X)＝|l5-l6|-min(lAC)≤0，
g7(X)＝max(lAC)-|l5-l6|≤0，
In the formula
Figure BDA0001774755340000083
K＝l4cos(θ3+α)-l1cos(θ1)+a，
M＝l4sin(θ3+α)-l1sin(θ1)+b。
d. Satisfy the maximum pressure angle constraint condition
g8(X)＝γ8-40°≤0，
In the formula, gamma8Is the pressure angle of the slider.
(2) And establishing an eight-connecting-rod mechanism three-dimensional model of the eight-connecting-rod mechanical press, simplifying the eight-connecting-rod mechanism three-dimensional model, and obtaining the simplified eight-connecting-rod mechanism model.
In the embodiment, the solid works software is firstly adopted to establish the eight-link mechanism three-dimensional model of the eight-link mechanical press, and of course, in other embodiments, other software with modeling function can be also used to establish the eight-link mechanism three-dimensional model. And then simplifying the eight-link mechanism three-dimensional model to obtain a simplified eight-link mechanism model.
In this embodiment, a method for simplifying a three-dimensional model of an eight-bar linkage mechanism is disclosed in reference "eight-bar linkage mechanical press dynamics analysis", which was published in 8 months of 2012 and was written by charpy chain, tension, etc.
(3) Setting an initial design variable and an initial eccentric body rotating speed, and carrying out kinematic analysis on the eight-connecting-rod mechanism model to obtain kinematic parameters including the slide block stroke, the slide block speed and the slide block acceleration.
In the present embodiment, the initial design variable and the eccentric body rotation speed are set. Design variables, i.e. the parameters of the bar system, including the bar length l of 8 bars1、l2、l3、l4、l5、l6、l7、l8Center O of hinge 9 and center O of hinge 101Distances a, b in X-and Y-axis directions, up-and-down rocking levers l3、l4Included angle alpha between, tripod l6、l7The included angle beta of the two sides.
In this embodiment, kinematic analysis is performed on the eight-bar linkage of the eight-bar mechanical press to obtain kinematic parameters including the slide stroke, the slide velocity, and the slide acceleration. Since each kinematic parameter is a function of time t, it is possible to plot each kinematic parameter as a curve (the abscissa is time t and the ordinate is each kinematic parameter), which is the final representation.
The method comprises the following specific steps of carrying out kinematic analysis on the eight-bar linkage model to obtain kinematic parameters:
and (3.1) establishing a kinematic equation set of the eight-link mechanism model according to a vector closure rule.
As can be seen from FIG. 2, the product is OABO1In the formed vector closed quadrangle, the vector closed quadrangle can be obtained according to the vector closed rule
Figure BDA0001774755340000091
Writing equation (2-1) in complex form:
Figure BDA0001774755340000092
the equation (2-2) is developed by an Euler formula to respectively obtain a real part equation and an imaginary part equation:
Figure BDA0001774755340000093
by analogy, from OADCO1The formed vector encloses a pentagon, and according to the vector rule, the following can be obtained:
Figure BDA0001774755340000094
writing equations (2-4) in complex form:
Figure BDA0001774755340000095
the equation (2-5) is developed by an Euler formula to respectively obtain a real part equation and an imaginary part equation:
Figure BDA0001774755340000101
similarly, a vector enclosed quadrilateral composed of OAEFs can be obtained according to the vector rule:
Figure BDA0001774755340000102
equations (2-7) are written in plural form:
Figure BDA0001774755340000103
the real part equation and the imaginary part equation are respectively obtained by expanding the formula (2-8) by an Euler formula:
Figure BDA0001774755340000104
the united vertical type (2-3), the formula (2-6) and the formula (2-9) form an equation set
Figure BDA0001774755340000105
Knowing theta4＝θ3+α (2-11)
θ7＝θ6-β (2-12)
The formula (2-11) and the formula (2-12) are respectively substituted into the formula equation set (2-10) to obtain
Figure BDA0001774755340000106
The system of equations (2-13) is a non-linear system of equations, wherein:
l1,l2,l3,l4,l5,l6,l7,l8the respective rod lengths (unit: m) are known quantities. a and b are respectively the center O of the hinge 9 and the center O of the hinge 101The distances in the X-axis and Y-axis directions (unit: m) are known quantities. Alpha, beta are respectively a rod l3And l4，l6And l7The angle (unit: rad) of (d) is a known quantity. Theta1,θ2,θ3,θ5,θ6,θ8Respectively, corresponding to the angular displacement (unit: rad) of each rod, and all the angles are rotated counterclockwise to the angle of each member with the X-axis positive direction as a starting point, see fig. 2. Wherein theta is1ω t is the eccentric 1 (i.e. l)1) The rotation angle is a known quantity, since ω ═ 2 × pi × n, n is the crank speed (unit: rps) is known, θ2,θ3,θ5,θ6,θ8Is an unknown quantity, y is the displacement of the slider (unit: m), is an unknown quantity.
Then, executing the step (3.2), solving a kinematic equation set of the eight-bar linkage model by using a New-Raphson algorithm to obtain a slide block stroke curve; solving a first derivative of the slide stroke curve to obtain a slide speed curve; the second derivative of the slider travel curve is calculated to obtain the slider acceleration curve, see fig. 3.
In this embodiment, a solution method of a nonlinear equation set, a newton-raphson method, is used to solve the kinematic equation set of the eight-bar linkage model.
Newton-Raphson solution principle
The newton-raphson method, newton's iteration, is an iterative method of solving nonlinear equations, starting with some given initial vector and continuing incrementally until all results are "close enough" to be an accurate solution.
Without loss of generality, assume a 2-equation simultaneous solution problem containing 2 unknowns:
Figure BDA0001774755340000111
in the formula (2-14), q1And q is2When the unknown quantity is required to be obtained,
order to
Figure BDA0001774755340000112
In the formula (2-15), the metal salt,
Figure BDA0001774755340000113
for predictive value of solution, Δ qiA small correction factor for the difference between the estimated value and the solution to the equation.
Using Taylor series to convert f in formula (2-14)i(q1,q2) In the estimation
Figure BDA0001774755340000114
Is unfolded, and then the back part of the bag is unfolded,
Figure BDA0001774755340000115
in the formula (2-16), - (. DELTA.q)1),ο(Δq2) Is a high-order term, and in order to make the formula (2-16) only contain linear form and omit the high-order term, the high-order term is substituted in the formula (2-14) and written into a matrix formThe formula is as follows:
Figure BDA0001774755340000121
from the formulae (2-17):
Figure BDA0001774755340000122
equations (2-18) are mathematical models of newton-raphson that solve two non-linear equations simultaneously.
Wherein:
Figure BDA0001774755340000123
a Jacobian matrix which is a non-linear system of equations.
Generalizing equation (2-18) to n variables, n equations, is as follows
Figure BDA0001774755340000124
Wherein the Jacobian matrix
Figure BDA0001774755340000125
Figure BDA0001774755340000131
And solving the motion equation of the eight-bar linkage model by utilizing Newton-Raphson.
Obtaining a correction factor according to the Newton-Raphson algorithm in the previous section by using an equation set (2-13)
Figure BDA0001774755340000132
Jacobian matrix of it
Figure BDA0001774755340000133
Adjusting theta according to the following formula (2-23)2,θ3,θ5,θ6,θ8And y is substituted into the system of equations (2-13) until its second order norm is less than a small positive number (e.g., 10)-6) Obtaining angular displacement theta of component2,θ3,θ5,θ6,θ8And the displacement y of the slider.
Figure BDA0001774755340000134
In the formula (I), the compound is shown in the specification,
Figure BDA0001774755340000135
respectively representing initial values of the corresponding angular displacements; delta theta2,Δθ3,Δθ5,Δθ6,Δθ8Respectively, indicate increments.
The first derivative of the equation (2-13) is obtained with respect to the time t to obtain the angular velocity of the member and the velocity of the slider, and J is the same as the equation (2-22).
Figure BDA0001774755340000141
And (3) calculating a second derivative of the time t by the formula (2-13) to obtain the angular acceleration of the component and the acceleration of the slide block, wherein J is the same as the formula (2-22).
Figure BDA0001774755340000142
Wherein
Figure BDA0001774755340000143
And (3.3) establishing an eight-link mechanism three-dimensional model, and performing simulation processing on the eight-link mechanism three-dimensional model to obtain a slide block stroke simulation curve, a slide block speed simulation curve and a slide block acceleration simulation curve, which are shown in the figure 4.
In the embodiment, a Solidworks software is adopted to establish an eight-bar linkage three-dimensional model. Of course, in other embodiments, other software with modeling functions may be used to create a three-dimensional model of the eight-bar linkage.
In this embodiment, the eight-bar linkage three-dimensional model is imported into Adams software, and a slider stroke simulation curve, a slider speed simulation curve, and a slider acceleration simulation curve are obtained through simulation processing. During simulation processing, the properties of the added material comprise density, elastic modulus and Poisson ratio; adding constraints, wherein the constraints comprise gravity, a revolute pair and a revolute pair; a drive is added, the drive comprising a crank speed.
Finally, the step (3.4) is executed, whether the solving curves of the slide stroke, the slide speed and the slide acceleration are consistent with the simulation curve or not is verified respectively, and if so, the slide stroke solving parameter, the slide speed solving parameter and the slide acceleration solving parameter are output respectively; otherwise, returning to the step (3.1) to the step (3.4). The crank torque obtained by dynamics solving is verified through Adams software simulation, and the accuracy of the solving process is guaranteed.
(4) And performing dynamic analysis on the eight-connecting-rod mechanism model according to the kinematic parameters and the tonnage information of the press to obtain the crank torque T.
As an embodiment, the method for obtaining the crank torque T by performing dynamic analysis on the eight-bar linkage model comprises the following steps:
firstly, executing the step (4.1), and carrying out dynamic static analysis on each rod of the eight-bar linkage mechanism according to the simplified eight-bar linkage mechanism model to obtain the stress condition of each rod.
Then, step (4.2) is executed, and a dynamic static balance equation set of the eight-link mechanism is listed according to theoretical mechanics and Newton's law through the stress condition of each rod.
And (4.3) executing the step, substituting the kinematic parameters obtained in the step (3) into an equation set and solving to obtain a solution curve of the crank torque, and referring to fig. 5.
The process of carrying out dynamic static force analysis on each rod of the eight-bar linkage mechanism comprises the following steps:
(4.111) force analysis of slider
Fig. 6 is a force diagram of the slider F. As shown in FIG. 6, the mass of the slider is mFThe constraint reaction force of the revolute pair F is RxFAnd RyFThe positive pressure of the guide rail is N, and the load is P. Centroid s is obtained from theoretical mechanicsFThe equilibrium equations for the forces on the real and imaginary axes, respectively, are shown in (3-1) and (3-2)
RxF-N＝0 (3-1)
Figure BDA0001774755340000151
(4.112) Main Pull rod l8Analysis of force
FIG. 7 shows the main rod l8The force is tested. As shown in FIG. 7, the main link l8Mass m8The constraint reaction force of the revolute pair E is RxEAnd RyEThe constraint reaction force of the revolute pair F is RxFAnd RyFCenter of mass s8Distance r from revolute pair Ec8Around the center of mass s8Moment of inertia J8Derived from theoretical mechanics
Figure BDA0001774755340000161
Figure BDA0001774755340000162
Figure BDA0001774755340000163
Knowledge of kinematics can be used to derive the main link l8Center of mass s8The components of the acceleration in the real axis and the imaginary axis are as follows
Figure BDA0001774755340000164
Figure BDA0001774755340000165
Combining the formulae (3-1) to (3-5) in a matrix form
Figure BDA0001774755340000166
The reaction force R can be obtained by substituting the formula (3-6) and the formula (3-7) into the formula (3-8)xE,RyE,RxF,RyF,N。
(4.113) force analysis of tripod
Figure 8 is a tripod diagram. As shown in FIG. 8, the tripod has a mass m67The constraint reaction force of the revolute pair E is RxEAnd RyEThe constraint counter force of the revolute pair D is RxDAnd RyDThe constraint reaction force of the revolute pair A is RxA2And RyA2Center of mass s67Distance r from revolute pair Ac67And has an angle theta with the positive direction of the X-axis67Around the center of mass s67Moment of inertia J67Derived from theoretical mechanics
Figure BDA0001774755340000167
Figure BDA0001774755340000168
Figure BDA0001774755340000171
The mass center s of the tripod can be deduced from the kinematic knowledge67The components of the acceleration in the real axis and the imaginary axis are as follows:
Figure BDA0001774755340000172
Figure BDA0001774755340000173
(4.114) lower drawbar l5Analysis of force
FIG. 9 shows a lower link l5The force is tested. As shown in FIG. 9, the lower lever l5Mass m5The constraint counter force of the revolute pair D is RxDAnd RyDThe constraint counter force of the revolute pair C is RxCAnd RyCCenter of mass s5Distance r from revolute pair Cc5Around the center of mass s5Moment of inertia J5Derived from theoretical mechanics
Figure BDA0001774755340000174
Figure BDA0001774755340000175
Figure BDA0001774755340000176
The lower pull rod l can be derived from the knowledge of kinematics5Center of mass s5The components of the acceleration in the real axis and the imaginary axis are as follows
Figure BDA0001774755340000177
Figure BDA0001774755340000178
Combining the formulae (3-9) to (3-11), the formulae (3-14) to (3-16) in matrix form can be written as formula (3-19), where RxE,RyEAs already found in the previous step. The reaction force R can be solved by substituting the formulas (3-12), (3-13), (3-17) and (3-18) into the formula (3-19)xA2,RyA2,RxC,RyC,RxD,RyD。
Figure BDA0001774755340000181
(4.115) force analysis of Rocker
Fig. 10 is a rocker force diagram. As shown in FIG. 10, the rocker has a mass m34The constraint reaction force of the revolute pair B is RxBAnd RyBThe constraint counter force of the revolute pair C is RxCAnd RyCRevolute pair O1With a constraint reaction force of Rx1And Ry1Center of mass s34To revolute pair O1A distance of rc34And has an angle theta with the positive direction of the X-axis34Around the center of mass s34Moment of inertia J34Derived from theoretical mechanics
Figure BDA0001774755340000182
Figure BDA0001774755340000183
Figure BDA0001774755340000184
The mass center s of the rocker can be deduced from the knowledge of kinematics34The components of the acceleration in the real axis and the imaginary axis are as follows
Figure BDA0001774755340000185
Figure BDA0001774755340000186
(4.116) Upper brace l2Analysis of force
FIG. 11 shows the upper rod2The force is tested. As shown in fig. 11, the upper tie bar l2Mass m2The constraint reaction force of the revolute pair B is RxBAnd RyBThe constraint reaction force of the revolute pair A is RxA1And RyA1Center of mass s2Distance r from revolute pair Ac2Around the center of mass s2Moment of inertia J2Derived from theoretical mechanics
Figure BDA0001774755340000191
Figure BDA0001774755340000192
Figure BDA0001774755340000193
The upper pull rod l can be derived from the knowledge of kinematics2Center of mass s2The components of the acceleration in the real axis and the imaginary axis are as follows:
Figure BDA0001774755340000194
Figure BDA0001774755340000195
combining the formulae (3-20) to (3-22), the formulae (3-25) to (3-27) in matrix form can be written as formula (3-30), wherein RxC,RyCAs already found in the previous step. The reaction force R can be solved by substituting the formulas (3-23), (3-24), (3-28) and (3-29) into the formula (3-30)xA1,RyA1,RxB,RyB,Rx1,Ry1。
Figure BDA0001774755340000196
(4.117) stress analysis of eccentric body
FIG. 12 is a force analysis diagram of the eccentric body. As shown in FIG. 12, the eccentric mass is m1The constraint reaction force of the revolute pair A is RxAAnd RyAThe constraint counter force of the revolute pair O is RxOAnd RyOCenter of mass s1Distance r from revolute pair Oc1Around the center of mass s1Moment of inertia J1The driving torque of the crank is T, and can be obtained by theoretical mechanics
Figure BDA0001774755340000201
Figure BDA0001774755340000202
Figure BDA0001774755340000203
The knowledge of kinematics can deduce the mass center s of the eccentric body1The components of the acceleration in the real axis and the imaginary axis are as follows
Figure BDA0001774755340000204
Figure BDA0001774755340000205
Combining the formulae (3-31) to (3-33) in matrix form, can be written as formula (I), wherein RxA,RyAR has been determined separately in the first two stepsxA2,RxA1And RyA2,RyA1I.e. RxA2+RxA1＝RxA，RyA2+RyA1＝RyA. The reaction force R can be obtained by substituting the formulas (3-34) and (3-35)xO,RyOAnd a crank drive torque T,
Figure BDA0001774755340000206
as another example, the crank torque T is solved directly using the virtual work principle. The principle of virtual work refers to that for a particle system with ideal constraints, the essential balance is: the sum of the virtual work done by the primary forces acting on the particle system at any virtual displacement equals zero.
In this embodiment, first, step (4.11) is executed, and according to the simplified eight-bar linkage model, stress analysis is performed on each bar of the eight-bar linkage to obtain stress conditions of each bar;
(4.12) listing an imaginary work equation by using an imaginary work principle according to the stress condition of each rod;
(4.13) substituting the kinematic parameters obtained in the step (3) into an imaginary work equation and solving to obtain a crank torque solution curve, which is shown in FIG. 5.
Fig. 13 is a force analysis diagram of the system. According to the force analysis of fig. 13, the system is given a virtual displacement, the crank is rotated through a very small angle δ θ, the slider gets a downward displacement δ s, and the virtual work equation is listed:
T*δθ+m1g*δy1+m2g*δy2+m34g*δy34+m5g*δy5+m67g*δy67+m8g*δy8+mFg*δs-P*δs＝0
(3-37)
δ y in the formula (3-37)iProjection of the virtual displacement at the centroid of each bar onto the y-axis, deltay1＝rc1cosθ1ω，
Figure BDA0001774755340000207
Figure BDA0001774755340000211
Figure BDA0001774755340000212
The crank rotating through a very small angle
Figure BDA0001774755340000213
Displacement of
Figure BDA0001774755340000214
T is crank torque; l1Is an eccentric body,/2Is a pull rod,/3Is a rocker arm l4Is a lower rocker arm l5Is a lower link,/6And l7Length of the two side rods of the tripod8Is a main pull rod; theta1,θ2,θ3,θ4,θ5,θ6,θ7,θ8Respectively, corresponding to the angular displacement (unit: rad), theta, of each rod34,θ67Respectively represent a rocker (said rocker is an upper rocker l3And a lower rocker l4Linked together) and angular displacements of the tripod;
Figure BDA0001774755340000215
are respectively a rod l2,l4,l5,l7,l8The angular velocity of (a) of (b),
Figure BDA0001774755340000216
the angular velocity of the rocker and the angular velocity of the tripod are respectively; delta theta is a minimum angle rotated by the crank, delta s is displacement rotated by the crank, and P is working pressure loaded on the sliding block; and omega is 2 pi n, and n is the crank speed.
In this embodiment, the crank speed n is 18rpm, the nominal pressure is 1200 tons, P is 3000KN, P is the working pressure loaded on the slide block, and the nominal pressure 1200 tons is 12000KN, because the press in this embodiment is a four-point eight-link mechanism, and shares 300 tons, i.e. 3000KN, on average per point. The crank torque T can be obtained by substituting the kinematic parameters obtained by the solution into the equations (3-37).
And (4.4) establishing an eight-connecting-rod mechanism three-dimensional model, and performing simulation processing on the eight-connecting-rod mechanism three-dimensional model to obtain a crank torque simulation curve, which is shown in fig. 14.
In the embodiment, a Solidworks software is adopted to establish an eight-bar linkage three-dimensional model. Of course, in other embodiments, other software with modeling functions may be used to create a three-dimensional model of the eight-bar linkage.
In this embodiment, the three-dimensional model of the eight-bar linkage is imported into Adams software, and a crank torque simulation curve is obtained through simulation processing. During simulation processing, the properties of the added material comprise density, elastic modulus and Poisson ratio; adding constraints, wherein the constraints comprise a revolute pair and a revolute pair; adding a drive comprising a crank rotation speed; nominal force was added.
Finally, executing the step (4.5), checking whether the solved curve is consistent with the simulation curve, and if so, respectively outputting torque parameters of the crank; otherwise, returning to the step (4.1) or the step (4.11).
The crank torque obtained by dynamics solving is verified through Adams software simulation, and the accuracy of the solving process is guaranteed.
(5) And optimizing the speed fluctuation of the sliding block and the maximum crank torque by using an NSGA-II algorithm, and solving a multi-objective optimization mathematical model to obtain a Pareto optimal solution set of the design variables.
In this step, the speed of the slide is one of the kinematic parameters, which is a function of the time t, and a curve of which can be plotted (the abscissa is the time t and the ordinate is the magnitude v of the speed of the slide), and which is periodic, i.e. repeated at intervals. This time is the period T, and when analyzing the speed fluctuation of the slide block, only a small time [ T ] in T is taken1,t2]To calculate its speed standard deviation, see the objective function f1(x) The formula, i.e., the standard deviation of speed, represents the speed fluctuation. Similarly, the crank torque is also a function of time T, and is also periodically varied, but here we only take a small time T in T3,t4]Find the maximum value of this time period as TmaxThe maximum crank torque is obtained.
The embodiments described in this specification are merely illustrative of implementations of the inventive concept and the scope of the present invention should not be considered limited to the specific forms set forth in the embodiments but rather by the equivalents thereof as may occur to those skilled in the art upon consideration of the present inventive concept.

Claims (5)

1. The multi-objective optimization method of the eight-connecting-rod mechanical press based on the NSGA-II algorithm is characterized by comprising the following steps of:
(1) determining an optimization target, design variables and constraint conditions, and establishing a multi-objective optimization mathematical model taking the minimum fluctuation value of the running speed of the sliding block in a specified area and the minimum maximum crank torque value in the specified area as objective functions; the multi-objective optimization mathematical model is
Figure FDA0003118338420000011
Wherein the content of the first and second substances,
Figure FDA0003118338420000012
minf2(X)＝Tmax，
in the formula, V (X, theta)1) For a crank angle of theta1When the speed of the sliding block is actually increased,
Figure FDA0003118338420000013
is theta1Equal to the average speed of the inner slide block of 90-150 DEG, and n is theta1The number of control points is between 90 and 150 degrees and depends on the segmentation precision;
X＝[l1，l2，l3，l4，l5，l6，l7，l8，a，b，α，β]，
S.t Xmin≤X≤Xmax
gu(X)≤0(u＝1，2，…，8)，
in the formula, minf1(X) is an objective function, minf, with the minimum fluctuation value of the running speed of the sliding block in the designated area2(X) is an objective function for which the maximum crank torque value in the specified region is minimal; x is a design variable,/1，l2，l3，l4，l5，l6，l7，l8Respectively are eccentric body, upper pull rod, upper rocker, lower pull rod, two side rod lengths of tripod and rod length of main pull rod, a, b are respectively the positions of two hinges, alpha is upper and lower rocker l3、l4An included angle between the two legs, beta is a tripod6、l7The included angle of the two sides; gu(X) is a constraint condition;
(2) establishing an eight-connecting-rod mechanism three-dimensional model of the eight-connecting-rod mechanical press, and obtaining a simplified eight-connecting-rod mechanism model;
(3) setting an initial design variable and an initial eccentric body rotating speed, and performing kinematic analysis on the eight-bar linkage model to obtain kinematic parameters including a slide block stroke, a slide block speed and a slide block acceleration;
(4) performing dynamic analysis on the eight-connecting-rod mechanism model according to the kinematic parameters and the tonnage information of the press to obtain the crank torque;
(5) optimizing the speed fluctuation of the sliding block and the maximum crank torque by using an NSGA-II algorithm, and solving a multi-objective optimization mathematical model to obtain a Pareto optimal solution set of design variables;
in the step (3), the method for obtaining the kinematic parameters comprises the following steps:
(3.1) establishing a kinematic equation set of the eight-bar mechanism model by using the simplified eight-bar mechanism model and a vector closure rule;
(3.2) solving a kinematic equation set of the eight-bar linkage model by using a New-Raphson algorithm to obtain a slider stroke solving curve; solving a first derivative of the slide stroke solving curve to obtain a slide speed solving curve; solving a second derivative of the slide stroke solving curve to obtain a slide acceleration solving curve;
(3.3) establishing a three-dimensional model of the eight-bar linkage mechanism, and obtaining a slide block stroke simulation curve, a slide block speed simulation curve and a slide block acceleration simulation curve according to the three-dimensional model simulation processing of the eight-bar linkage mechanism;
(3.4) verifying whether the solving curve is consistent with the simulation curve, and if so, respectively outputting solving parameters of the slide stroke, the slide speed and the slide acceleration; otherwise, returning to the step (3.1);
in the step (3.1), the kinematic equation set expression of the eight-bar linkage model is as follows:
Figure FDA0003118338420000031
wherein: l1,l2,l3,l4,l5,l6,l7,l8Respectively, the length of each rod is m; a and b are respectively a hinge center O and a hinge center O1Distances in the X-axis and Y-axis directions in units of m; alpha, beta are respectively a rod l3And l4，l6And l7The unit of the included angle of (A) is rad; theta1,θ2,θ3,θ5,θ6,θ8The angular displacement corresponding to each rod is respectively, the unit is rad, and all the angles rotate anticlockwise to the angles of each component by taking the positive direction of an X axis as a starting point; y is the displacement of the slider in m;
the solution method of the kinematic equation set of the eight-bar linkage model comprises the following steps:
Figure FDA0003118338420000032
in the formula (2-14), q1And q is2When the unknown quantity is required to be obtained,
order to
Figure FDA0003118338420000033
In the formula (2-15), the metal salt,
Figure FDA0003118338420000034
for the estimated value of the solution, Δ qiA small correction factor for the difference between the estimated value and the equation solution;
using Taylor series to convert f in formula (2-14)i(q1,q2) In the estimation
Figure FDA0003118338420000035
Is unfolded, and then the back part of the bag is unfolded,
Figure FDA0003118338420000036
in the formula (2-16)
Figure FDA0003118338420000037
Is a high-order term, in order to make the formula (2-16) only contain linear form and omit the high-order term, the high-order term is substituted in the formula (2-14) and written into a matrix form, and the following steps are provided:
Figure FDA0003118338420000041
from the formulae (2-17):
Figure FDA0003118338420000042
the formula (2-18) is a Newton-Raphson mathematical model for simultaneous solution of two nonlinear equations;
wherein:
Figure FDA0003118338420000043
a Jacobian matrix which is a nonlinear system of equations;
generalizing equation (2-18) to n variables, n equations, is as follows
Figure FDA0003118338420000044
Wherein the Jacobian matrix
Figure FDA0003118338420000051
Solving a motion equation of the eight-bar linkage model by utilizing Newton-Raphson;
obtaining a correction factor according to the Newton-Raphson algorithm in the previous section by using an equation set (2-13)
Figure FDA0003118338420000052
Jacobian matrix of it
Figure FDA0003118338420000053
Adjusting theta according to the following formula (2-23)2,θ3,θ5,θ6,θ8Substituting y into equation set (2-13) until the second-order norm is less than a small positive number to obtain angular displacement theta of the component2,θ3,θ5,θ6,θ8And the displacement y of the slide;
Figure FDA0003118338420000061
in the formula (I), the compound is shown in the specification,
Figure FDA0003118338420000062
respectively representing initial values of the corresponding angular displacements; delta theta2,△θ3,△θ5,△θ6,△θ8Respectively represent increments;
the first derivative of the formula (2-13) on the time t is obtained to obtain the angular speed of the component and the speed of the slide block, and J is the same as the formula (2-22);
Figure FDA0003118338420000063
the second derivative is obtained for time t by the formula (2-13), the angular acceleration of the component and the acceleration of the slide block are obtained, and J is the same as the formula (2-22);
Figure FDA0003118338420000064
wherein
Figure FDA0003118338420000071
2. The NSGA-II algorithm-based eight-bar mechanical press multi-objective optimization method of claim 1, wherein in the step (4), the method for obtaining the crank torque comprises the steps of:
(4.1) carrying out dynamic static analysis on each rod of the eight-bar linkage according to the simplified eight-bar linkage model to obtain the stress condition of each rod;
(4.2) listing a dynamic static balance equation set of the eight-link mechanism according to theoretical mechanics and Newton's law;
(4.3) substituting the kinematic parameters into an equation set and solving to obtain a crank torque solving curve;
(4.4) carrying out simulation processing according to the eight-connecting-rod mechanism three-dimensional model to obtain a crank torque simulation curve;
(4.5) verifying whether the solving curve of the crank torque is consistent with the simulation curve or not, and if so, respectively outputting crank torque solving parameters; otherwise, returning to the step (4.1).
3. The NSGA-II algorithm-based eight-bar mechanical press multi-objective optimization method of claim 2, wherein in the step (4), the method for obtaining the crank torque comprises the steps of:
(4.11) according to the simplified eight-bar linkage model, carrying out stress analysis on each bar of the eight-bar linkage to obtain the stress condition of each bar;
(4.12) according to the stress condition of each rod, using the virtual work principle to list equations;
(4.13) substituting the kinematic parameters into an equation and solving to obtain a crank torque solving curve;
(4.14) establishing a three-dimensional model of the eight-bar linkage mechanism, and carrying out simulation processing according to the three-dimensional model of the eight-bar linkage mechanism to obtain a crank torque simulation curve;
(4.15) verifying whether the solving curve of the crank torque is consistent with the simulation curve or not, and if so, respectively outputting crank torque solving parameters; otherwise, returning to the step (4.11).
4. The NSGA-II algorithm-based multi-objective optimization method for the eight-bar mechanical press as claimed in claim 3, wherein in the step (4.4), Solidworks software is adopted to establish a three-dimensional model of the eight-bar mechanism, and software Adams is used to perform simulation processing on the three-dimensional model of the eight-bar mechanism.
5. The NSGA-II algorithm-based multi-objective optimization method for the eight-bar mechanical press as claimed in claim 4, wherein in the step (4.14), Solidworks software is adopted to establish an eight-bar mechanism three-dimensional model, and software Adams is used to perform simulation processing on the eight-bar mechanism three-dimensional model.
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