CN112446087B - Mechanical structure interval multi-objective optimization method based on blind number theory - Google Patents
Mechanical structure interval multi-objective optimization method based on blind number theory Download PDFInfo
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Abstract
The invention discloses a mechanical structure interval multi-objective optimization method based on a blind number theory, and belongs to the field of mechanical structure optimization. The method aims to solve the problem of multi-objective optimization design of a mechanical structure interval under a small amount of sample data. The principle is that a calculation parameter is obtained through numerical calculation, and a mechanical structure performance index target allowable interval is determined based on a blind number theory; defining design variables, uncertainty variables and a multi-objective optimization function, constructing a response surface model and verifying the accuracy of the response surface model; determining a design variable and an optimization target change interval, obtaining weight coefficients of each optimization target based on an analytic hierarchy process, and performing global optimization through a non-dominated sorting genetic algorithm to obtain an optimal result. The invention fully considers the uncertainty factor in the mechanical structure, characterizes the uncertainty of the mechanical structure variable through the blind number, introduces the interval value to replace the fixed value in the traditional mechanical structure design, has simple and reliable calculation, does not depend on the test, saves the time and the resources and better accords with the engineering practice.
Description
Technical Field
The invention discloses a mechanical structure interval multi-objective optimization method based on a blind number theory, and belongs to the field of mechanical structure optimization design.
Background
Mechanical property parameters of materials are always regarded as fixed values in the design process of a traditional empirical mechanical structure, and performance judgment of the mechanical structure is also established on the basis of the fixed values of the mechanical property parameters of the mechanical structure. In fact, the mechanical structure has the influence of uncertain factors such as material properties, machining errors, assembly measurement errors and the like in the actual machining and manufacturing process, and the material parameters and the mechanical property of the structure should be presented as interval values in the actual engineering problem. Describing uncertainty information as an interval value in the background of the existing research is one of the commonly used means in uncertainty research.
With the deepening of uncertainty research, a plurality of expression modes for uncertainty quantity extend, for example, the uncertainty quantity is regarded as a random variable at first, fuzzy mathematics is developed to grey numbers, uncertain mathematics is proposed, and finally a blind number theory is developed, wherein the blind number integrates the advantages of the prior uncertainty information quantity expression, can adapt to a plurality of uncertainty information quantities, and is simple in operation and high in reliability, so that the blind number theory is applied to a plurality of uncertain information-containing fields. The method for designing the axle housing of the all-terrain vehicle is obtained by taking stress deformation of the integral axle housing under the working conditions of full load and uneven road surface driving, the working condition of bearing the maximum tangential force by wheels and the working condition of bearing the maximum lateral force by the wheels as limiting conditions. The guo armor of Beijing university of science and technology expresses the uncertainty of a mechanical mechanism in a complex time-varying process by adopting a blind number, establishes a mechanical structure time-varying reliability calculation model based on the blind number theory, and writes a corresponding calculation program to calculate the time-varying reliability of the mechanical structure. In the research, uncertain information in the system is regarded as a blind number, blind number theory is introduced for calculation, and finally, a corresponding evaluation model is obtained, so that the feasibility of the blind number theory is verified. However, the blind number method relies on test data, and when the number of test sample data is small and the information is incomplete, it is only rarely studied how to perform corresponding design and uncertainty optimization.
In the uncertainty optimization problem, the currently adopted interval analysis method has the greatest advantages of small required sample amount, simple realization of uncertain information expression through the number of intervals, and very high operation efficiency, so the non-probability non-fuzzy interval analysis method becomes an important means for uncertainty optimization. At present, a second university of cannon and military engineering, namely a Fangpeng section model provides an uncertainty analysis method for internal noise of an electric vehicle, and the uncertainty analysis method is used for researching the influence of the uncertainty problem of parameters in an acoustic package on the internal noise of the electric vehicle. The Leventai of the university of Hunan industry regards the fatigue strength of the A-shaped frame of the electric wheel dumper as an interval number, deduces the interval fatigue strength of the A-shaped frame based on a blind number theory and experimental data, and optimizes the fatigue strength of the A-shaped frame by adopting an interval analysis method after evaluation. In the structural uncertainty interval analysis and optimization process, only a single target is considered for optimization, multi-target optimization design is not considered, mutual influence of various factors is ignored, and the application of engineering practice is not facilitated. Therefore, by combining the current research situation, how to obtain the target allowable value of the mechanical structure performance index under the condition of a small amount of sample data, and meanwhile, considering the influence of a plurality of uncertain factors in the optimization design process of the mechanical structure to develop the interval multi-objective optimization design is a problem to be solved urgently.
Disclosure of Invention
In order to solve the problems that the material sample data is less in design of the existing mechanical structure, a target allowable value is difficult to obtain, the section multi-objective optimization design of the mechanical structure under uncertain factors is insufficient and the like, and overcome the defects in the background technology, the invention provides a mechanical structure section multi-objective optimization method based on a blind number theory, and the method comprises the following steps:
(1) obtaining a calculation parameter through numerical calculation, and determining a mechanical structure performance index target allowable interval based on a blind number theory;
(2) defining design variables, uncertainty variables and a multi-objective optimization function, constructing a response surface model and verifying the accuracy of the response surface model;
(3) determining a design variable and an optimization target change interval, obtaining weight coefficients of each optimization target based on an analytic hierarchy process, and performing global optimization through a non-dominated sorting genetic algorithm to obtain an optimal result.
Further, in the step (1), the calculating parameters obtained by the numerical calculation include the following steps:
the method comprises the following steps: establishing a finite element model of a mechanical structure, and verifying the accuracy of the finite element model;
step two: changing the size of the finite element model of the mechanical structure to obtain the calculation parameters of the performance indexes of the mechanical structure;
step three: and combining the calculation parameters of the mechanical structure performance indexes and a blind number theoretical algorithm to obtain the target allowable range of the mechanical structure performance indexes.
Further, the allowable range of the mechanical structure performance index target is shown as the following formula:
[fi]＝f([a1]…[am]…[an]) (1)
in the formula: [ f ] ofi]For the target allowable interval, i is the number of targets, f is an algorithm based on the blind number theory, [ a ]m]For blind number expression of the calculation parameters, n is the number of the calculation parameters.
Further, in the step (2), the design variable is a mechanical structure dimension parameter DjThe uncertainty variable is a mechanical structure material performance parameter EkWherein j and k represent the number of parameters; the method for defining the multi-objective optimization function, constructing the response surface model and verifying the accuracy of the response surface model comprises the following steps:
the method comprises the following steps: the mechanical structure dimension parameter D in the design variablesjAnd saidMechanical structure material performance parameter E in uncertainty variables ofkPerforming Latin hypercube random sampling to obtain sample point data;
step two: according to the sample point data, carrying out finite element calculation to obtain a response value corresponding to the design variable and the uncertainty variable;
step three: determining said design variable DjUncertainty variable E described belowkMaximum response value fimax and minimum response value fimin；
Step four: according to the interval analysis method, the maximum response value fimax and minimum response value fiThe sum of min divided by 2 is taken as the median value ZiMaximum response value fimax and minimum response value fiThe difference between min divided by 2 is taken as the radius value BiThen the median value Z is addediMultiplied by 0.5 plus the radius value BiMultiplying by 0.5 to obtain an equivalent value of the response value;
step five: constructing the response surface model Y between the design variables and the equivalent value based on a second-order response surface methodi(Dj,Ek)；
Step six: randomly selecting 10 groups of sample point data out of the first step, and directly substituting into the data for displaying the mathematical relationship Yi(Dj,Ek) Obtaining an equivalent value; meanwhile, operating according to the second step, the third step and the fourth step through finite element calculation to obtain corresponding equivalent values; comparing the two results, if the relative error is within 10%, considering that the constructed target optimization function is accurate, otherwise, returning to the first step, extracting the sample point data again, and constructing the response surface model;
step seven: defining a multi-objective optimization function as:
max Yi(Dj，Ek) (2)
in the formula, YiThe constraint condition is the target allowable interval of the mechanical structure performance index obtained in the step three; dj、EkRespective constraints need to be satisfied.
Further, the mechanical structure dimension parameter DjMechanical structural Material Performance parameter EkThe constraint conditions are respectively as follows:
in the formula:
for the dimensional parameter D of the mechanical structurejThe average value of (a) of (b),
for the material property parameter E of the mechanical structurekIs measured.
Further, in the step (3), the obtaining of each optimization target weight coefficient based on the analytic hierarchy process includes the following steps:
the method comprises the following steps: defining a target fiHas a weight coefficient of alphaiBy comparing two sub-targets to form alphaiuWhere u is an arbitrary target fiThe weight coefficients of (a) are determined by referring to nine values of 1, 2, 3, 4, 5, 6, 7, 8, and 9, and all the values of (a) are quantizediuForming an initial judgment matrix A;
step two: normalizing the column vectors of the initial judgment matrix A, and normalizing each alpha in the column vectors of the judgment matrix AiuDividing the sum of the column vectors to form a judgment matrix B;
step three: then, the row vectors of the judgment matrix B are normalized, and each beta in the row vectors of the judgment matrix B is processediuDividing by the sum of the row vectors to obtain a ranking vector as:
U＝[λ1，…，λi] (4)
in the formula: lambda [ alpha ]iIn order to optimize the target weight coefficients,
further, in the step (3), the population number adopted by the non-dominated sorting genetic algorithm is 200, the iteration number is 1000, the global optimization is performed, and the optimal result is obtained.
The mechanical structure interval multi-target optimization method based on the blind number theory fully considers uncertainty factors in a mechanical structure, represents uncertainty of mechanical structure variables through the blind numbers, introduces interval values to replace fixed values in the traditional mechanical structure design, develops interval multi-target optimization design based on a hierarchical analysis method, is simple and reliable in calculation, does not depend on tests, saves time and resources, and is more in line with engineering practice.
Drawings
FIG. 1 is a flow chart of a mechanical structure interval multi-objective optimization method based on a blind number theory;
fig. 2 is a finite element model of a frame of a dumper with one electric wheel;
fig. 3 is a bending deformation response diagram of a frame of a dumper with one electric wheel;
fig. 4 is a schematic diagram of a frame strength target allowable interval and a frame working stress allowable interval of a certain electric-wheel dumper;
FIG. 5 is a schematic diagram of the design variables for optimizing the frame structure of a dump truck with certain electric wheels;
fig. 6 is a schematic diagram of response surface model accuracy verification under a certain electric wheel dumper frame interval analysis method.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
The following provides an example of multi-objective optimization design of a frame section of an electric wheel dump truck, but the scope of the invention is not limited to the following implementation examples.
The method comprises the following steps: obtaining a calculation parameter through numerical calculation, and determining a target allowable interval of the mechanical structure based on a blind number theory;
a frame finite element model is established for the mechanical structure of a frame of a certain electric wheel dump truck, see figure 2, and the frame stress is tested for a standard vehicle. After the accuracy of the deformation response value is verified, the radial size and the axial size of the frame are changed to obtain five finite element models of the frame with different sizes, five different loads are input and are brought into the finite element models of the frame for calculation, and the corresponding deformation response value is obtained, as shown in fig. 3.
And calculating the calculation parameters of the frame wheelbase, the wheel base, the load, the deformation response and the like based on a blind number theory, and substituting the calculation parameters into an interval bending rigidity formula:
wherein the content of the first and second substances,
is interval bending rigidity (N.m)2)；
The interval number of the concentrated load (N);
the number of intervals of 3 power of the wheelbase a (m);
the number of intervals of the deformation (deflection) (m) at the point of action of the frame load.
Interval torsional stiffness formula:
wherein the content of the first and second substances,
is interval torsional rigidity (N.m)2/rad)；
The interval number of the concentrated load (N);
the interval number of the wheelbase (m);
the interval number of deflection (m) at the load action point;
the number of intervals is the square of the moment arm (m). Deducing the section bending stiffness of the frame [3.28,20.28]× 109N·m2And interval torsional stiffness [0.838,2.91 ]]×109N·m2(ii)/rad. Similarly, the calculation parameters such as the size parameters of the material test piece and the corresponding tensile limit are combined, the calculation is carried out based on the blind number theory, and the allowable interval value of the strength target of the frame is deduced [439.1,792.5 ]]MPa. The allowable interval value of the strength target and the working stress interval value of the vehicle frame are represented by a numerical axis, as shown in FIG. 4.
Step two: defining design variables, uncertainty variables and a multi-objective optimization function, constructing a response surface model and verifying the accuracy of the response surface model;
for the frame of the electric wheel dumper, the design variables comprise mechanical structure dimension parameters DjThe uncertainty variable includes a mechanical structure material performance parameter Ek. Considering that the frame is formed by welding high-strength low-alloy quenched and tempered steel plates with different thicknesses, and the contribution degrees of the steel plates with different thicknesses to the frame stress are different, the finite element analysis of the frame shows that the frame stress is concentrated at the joint of the top of the frame longitudinal beam and the side plate of the tail longitudinal beam and the joint of the bottom plate of the longitudinal beam and the side plate of the longitudinal beam. Therefore, the plate thicknesses of the longitudinal beam top plate, the rib plate, the longitudinal beam side plate, the frame lifting lug and the like which have large influence on the stress are taken as design variables. As shown in fig. 5. Due to a plurality of uncertain factors in processing and manufacturing of the frame plate, the elastic modulus and the Poisson's ratio of material performance parameters are used as uncertain quantities.
The electric wheel dumper is main carrying equipment in an open mine place, is extremely large in daily transportation bearing, long in working time, rugged in mine road and uneven in road surface, and causes severe working environment. The most typical running conditions comprise a horizontal bending condition, an emergency braking condition, a torsion condition and bending rigidity and torsion rigidity of the frame under the limit condition, so that the dynamic stress, the bending rigidity and the torsion rigidity of the frame are selected as targets. Then, sample values of the design variables are obtained based on the Latin hypercube method, equivalent values are calculated by combining a finite element method, and the design variables and response values thereof are shown in Table 1 (space-limited, only 6 groups are listed).
TABLE 1 design variables and their response value sample points
Determining the maximum response values f of the response values of multiple groups of uncertain variables under a group of design variables based on the design variables and the response value sample points thereof in Table 1imax and minimum response value fimin, according to the interval analysis method, the maximum response value fimax and minimum response value fiThe sum of min divided by 2 is taken as the median value ZiMaximum response value fimax and minimum response value fiThe difference between min divided by 2 is taken as the radius value BiThen the median value Z is addediMultiplied by 0.5 plus the radius value BiMultiplying by 0.5 yields the maximum equivalent of the response value. Response surface model Y for constructing design variables and maximum equivalent stress value based on second-order response surface methodi(Dj,Ek) Wherein, because of more targets, only one item label response surface model is listed:
in order to verify the accuracy of the model (3), 10 groups of design variable sample points and 6 groups of uncertainty variable sample points outside the first step are selected arbitrarily, and a response surface model Y is performed respectivelyi(Dj,Ek) The calculated maximum equivalent response value and the finite element calculation are processed by an interval analysis method, and the difference between the maximum equivalent response value and the finite element calculation is compared, as shown in figure 6, the difference between the calculation result of the response surface model and the simulation result of the finite element is almost the same, which shows that the accuracy of the established response surface model is very high. Therefore, the method can be used as a target optimization function for the dynamic stress and the bending of the frame under the horizontal bending working condition, the emergency braking working condition and the torsion working conditionAnd (3) taking the target optimization functions such as flexural rigidity, torsional rigidity and the like as a multi-target optimization function:
max Yi(Dj，Ek) (4)
in the formula, YiThe constraint condition of (A) is the obtained target allowable interval of the mechanical structure performance index, DjAs a mechanical structural dimensional parameter, EkIs a mechanical structure material performance parameter, wherein j and k represent the number of parameters, i is a plurality of targets, Dj、EkRespective constraints need to be satisfied.
Step three: determining a design variable and an optimization target change interval, obtaining weight coefficients of each optimization target based on an analytic hierarchy process, and performing global optimization through a non-dominated sorting genetic algorithm to obtain an optimal result.
The above-mentioned mechanical structure dimension parameter DjMechanical structural Material Performance parameter EkThe constraint conditions are respectively as follows:
in the formula:
for the dimensional parameter D of the mechanical structurejThe average value of (a) of (b),
for the material property parameter E of the mechanical structurekIs measured.
Further, the sub-target weight coefficients of multiple targets need to be determined, so that the weight coefficients of each optimization target are obtained based on an analytic hierarchy process, and the method comprises the following steps:
the first step is as follows: defining a target fiHas a weight coefficient of alphaiBy comparing two sub-targets to form alphaiuWhere u is an arbitrary target fiThe weight coefficient of (a) is determined by referring to 1, 2, 3, 4, 5, 6, 7, 8, 9 values, and all the values of (a) are quantizediuForming an initial judgment matrix A:
the second step is that: normalizing the column vectors of the initial judgment matrix A, and normalizing each alpha in the column vectors of the initial judgment matrix AiuDividing the sum of the column vectors to form a judgment matrix B:
the third step: then, the row vectors of the judgment matrix B are normalized, and each beta in the row vectors of the judgment matrix B is processediuDividing by the sum of the row vectors to obtain a ranking vector as:
U＝[λ1…λi]＝[0.117 0.178 0.142 0.261 0.302] (8)
in the formula: lambda [ alpha ]iThe target weight coefficients are optimized for each.
Further, the population number of the non-dominated sorting genetic algorithm is 200, the iteration number is 1000, the formula (4) is used as a multi-objective optimization function, global optimization is carried out, an optimal result is obtained, and finally the optimal result is obtained through 51-minute operation and is shown in table 2.
TABLE 2 Multi-objective design variable optimization results
| Design variables | Before optimization | After optimization |
| Longitudinal beam top plate thicknessx1/(mm) | 45.0 | 54.0 |
| Thickness of rib platex2/(mm) | 16.0 | 19.2 |
| Side plate thickness of longitudinal beamx3/(mm) | 25.0 | 30.0 |
| Thickness of lifting lugx4/(mm) | 45.0 | 49.7 |
And further, substituting the design variables before optimization and the design variables after optimization into finite element calculation respectively, and judging the target optimization quantity. And finally, obtaining a target optimization result, wherein the target optimization result of each performance index meets the optimization requirement as shown in the table 3.
TABLE 3 frame target optimization results
Claims (3)
1. A mechanical structure interval multi-objective optimization method based on a blind number theory is characterized by comprising the following steps:
the method comprises the following steps: obtaining a calculation parameter through numerical calculation, and determining a mechanical structure performance index target allowable interval based on a blind number theory; the numerical calculation to obtain the calculation parameters comprises the following steps:
step 1: establishing a finite element model of a mechanical structure, and verifying the accuracy of the finite element model;
step 2: changing the size of the finite element model of the mechanical structure to obtain the calculation parameters of the performance indexes of the mechanical structure;
and step 3: combining the calculation parameters of the mechanical structure performance indexes and a blind number theoretical algorithm to obtain a mechanical structure performance index target allowable interval; the allowable range of the mechanical structure performance index target is shown as the following formula:
[fi]＝f([a1]…[am]…[an]) (1)
in the formula: [ f ] ofi]For the target allowable interval, i is the number of targets, f is an algorithm based on the blind number theory, [ a ]m]Is blind number expression of the calculation parameters, and n is the number of the calculation parameters;
step two: defining design variables, uncertainty variables and a multi-objective optimization function, constructing a response surface model and verifying the accuracy of the response surface model; the design variable is a mechanical structure dimension parameter DjThe uncertainty variable is a mechanical structure material performance parameter EkWherein j and k represent the number of parameters; the method for defining the multi-objective optimization function, constructing the response surface model and verifying the accuracy of the response surface model comprises the following steps:
step 1: the mechanical structure dimension parameter D in the design variablesjAnd the mechanical structure material performance parameter E in the uncertainty variableskPerforming Latin hypercube random sampling to obtain sample point data;
step 2: according to the sample point data, carrying out finite element calculation to obtain a response value corresponding to the design variable and the uncertainty variable;
and step 3: determining said design variable DjUncertainty variable E described belowkMaximum response value fimax and minimum response value fimin；
And 4, step 4: according to the interval analysis method, the maximum response value fimax and minimum response value fiThe sum of min divided by 2 is taken as the median value ZiMaximum response value fimax and minimum response value fiThe difference between min divided by 2 is taken as the radius value BiThen the median value Z is addediMultiplied by 0.5 plus the radius value BiMultiplying by 0.5 to obtainAn equivalent to the response value;
and 5: constructing the response surface model Y between the design variables and the equivalent value based on a second-order response surface methodi(Dj,Ek)；
Step 6: randomly selecting 10 groups of sample point data out of the step 1, and directly substituting into the displayed mathematical relation Yi(Dj,Ek) Obtaining an equivalent value; meanwhile, operating according to the steps 2, 3 and 4 to obtain corresponding equivalent values through finite element calculation; comparing the two results, if the relative error is within 10%, considering that the constructed target optimization function is accurate, otherwise, returning to the step 1, extracting the sample point data again, and constructing the response surface model;
and 7: defining a multi-objective optimization function as:
maxYi(Dj,Ek) (2)
in the formula, YiThe constraint condition of (1) is the target allowable interval of the mechanical structure performance index; dj、EkRespective constraint conditions need to be met; the mechanical structure dimension parameter DjMechanical structural Material Performance parameter EkThe constraint conditions are respectively as follows:
in the formula:
for the dimensional parameter D of the mechanical structurejThe average value of (a) of (b),
for the material property parameter E of the mechanical structurekThe mean value of (a);
step three: determining a design variable and an optimization target change interval, obtaining weight coefficients of each optimization target based on an analytic hierarchy process, and performing global optimization through a non-dominated sorting genetic algorithm to obtain an optimal result.
2. The mechanical structure interval multi-objective optimization method based on the blind number theory as claimed in claim 1, wherein in the third step, the optimization objective weight coefficients obtained based on the analytic hierarchy process comprise the following steps:
the method comprises the following steps: defining a target fiHas a weight coefficient of alphaiBy comparing two sub-targets to form alphaiuWhere u is an arbitrary target fiThe weight coefficients of (a) are determined by referring to nine values of 1, 2, 3, 4, 5, 6, 7, 8, and 9, and all the values of (a) are quantizediuForming an initial judgment matrix A;
step two: normalizing the column vectors of the initial judgment matrix A, and normalizing each alpha in the column vectors of the judgment matrix AiuDividing the sum of the column vectors to form a judgment matrix B;
step three: then, the row vectors of the judgment matrix B are normalized, and each beta in the row vectors of the judgment matrix B is processediuDividing by the sum of the row vectors to obtain a ranking vector as:
U＝[λ1，…，λi] (4)
in the formula: lambda [ alpha ]iTo optimize the target weight coefficients.
3. The method according to claim 1, wherein in step three, the non-dominated sorting genetic algorithm uses a population of 200 and an iteration number of 1000, and the global optimization is performed to obtain the optimal result.
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