Robust optimization design method and system for multiple dynamic vibration absorbers

Abstract

The invention provides a multi-dynamic vibration absorber robust optimization design method and system, and relates to the technical field of dynamic vibration absorbers, the design method is based on a scene theory, and comprises the following steps: S1, establishing a mechanical model of a deterministic multi-dynamic vibration absorber; s2, determining a probability distribution form and a sampling range of each uncertain parameter variable, abstracting a robustness optimization design problem of the uncertain multi-dynamic vibration absorber into a minimum-maximum optimization problem, and establishing a mathematical model of the minimum-maximum optimization problem; s3, converting an infinite constraint optimization problem into a finite constraint optimization problem by utilizing a scene theory, and establishing an uncertainty multi-dynamic vibration absorber robustness optimization model based on the scene theory; and S4, solving the uncertainty multi-dynamic vibration absorber robustness optimization model based on the scene theory by using a self-adaptive constraint processing double-population evolutionary algorithm as an optimization algorithm, and searching optimal design parameters of the dynamic vibration absorber. According to the invention, the reliability of the design of the multi-dynamic vibration absorber can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic vibration absorbers, and in particular to a robust optimization design method and system for multiple dynamic vibration absorbers. Background Art

[0002] The main principle of the dynamic vibration absorber is to transfer most of the vibration energy from the main vibration system to the dynamic vibration absorber. At this time, the amplitude of the main vibration system will be greatly reduced, avoiding the impact of vibration on the main vibration system. The traditional passive dynamic vibration absorber consists of three parts: mass, spring and damping. By designing these parameters, a good vibration reduction effect can be achieved. However, the current design and optimization of the passive dynamic vibration absorber structure are mostly based on deterministic models, without considering the influence of uncertain parameters in the main vibration system and the dynamic vibration absorber. In actual engineering, factors such as manufacturing accuracy, measurement errors and changes in the external environment will affect the vibration reduction performance of the dynamic vibration absorber. If the influence of various uncertain parameters is ignored and the dynamic vibration absorber is directly analyzed and designed, the natural frequency of the designed dynamic vibration absorber is inconsistent with the vibration reduction frequency of the main vibration system, which does not meet the vibration reduction requirements. At the same time, its vibration reduction performance decreases with the increase of uncertainty.

[0003] A common method for designing dynamic vibration absorbers considering the influence of uncertainty factors is to model such situations as minimum-maximum optimization problems for robust optimization design. This problem aims to minimize an objective function while satisfying a set of convex uncertainty set constraints. However, such problems with finite optimization variables and infinite optimization constraints are usually difficult to solve and are often NP-hard problems. In addition, the robust optimization design of most dynamic vibration absorbers only considers the situation where the main vibration system has uncertainty. In actual working conditions, both the dynamic vibration absorber and the main vibration system are subject to uncertainty factors, and there may be more than one arrangement of the dynamic vibration absorber. When considering the uncertainty of multiple dynamic vibration absorbers, the constraints of its robust optimization design problem also become complicated, and traditional numerical methods may not be able to solve it due to the high computational cost.

[0004] In view of this, the inventor of the present application has designed a robust optimization design method and system for multiple dynamic vibration absorbers in order to overcome the above-mentioned technical problems. Summary of the invention

[0005] The technical problem to be solved by the present invention is to overcome the defect that the current design and optimization of passive dynamic vibration absorber structures in the prior art are mostly based on deterministic models, without considering the influence of uncertain parameters in the main vibration system and the dynamic vibration absorber, and to provide a robust optimization design method and system for multiple dynamic vibration absorbers.

[0006] The present invention solves the above technical problems through the following technical solutions:

[0007] The present invention provides a method for robust optimization design of multiple dynamic vibration absorbers, which is characterized in that the design method is based on scenario theory and includes the following steps: S1, establishing a mechanical model of a deterministic multiple dynamic vibration absorber; S2, determining the probability distribution form and sampling range of each uncertain parameter variable, abstracting the robust optimization design problem of the uncertain multiple dynamic vibration absorber into a minimum-maximum optimization problem, and establishing its mathematical model; S3, using scenario theory to convert the infinite constraint optimization problem into a finite constraint optimization problem, and establishing a robust optimization model of the uncertain multiple dynamic vibration absorber based on scenario theory; S4, using a dual-population evolutionary algorithm for adaptive constraint processing as an optimization algorithm, solving the robust optimization model of the uncertain multiple dynamic vibration absorber based on scenario theory, and finding the optimal design parameters of the dynamic vibration absorber.

[0008] According to one or more embodiments of the present invention, in step S1, the motion equation of the main vibration system is expressed as: Where M represents the mass of the main vibration system; K represents the stiffness of the main vibration system; n is the number of dynamic vibration absorbers installed on the main vibration system; k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber, x i represents the displacement of the ith dynamic vibration absorber (i = 1, 2, ..., n); x is the displacement of the main vibration system; f is the external harmonic excitation, f = F0e jωt ; The motion equation of the dynamic vibration absorber is expressed as: Among them, m i represents the mass of the ith dynamic vibration absorber, k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber (i=1,2,…,n); x i is the displacement of the dynamic vibration absorber.

[0009] According to one or more embodiments of the present invention, in step S1, the motion equation of the dynamic vibration absorber is rewritten into a complex form as follows: x = Xe jωt ,x i =X i e jωt ,f=Fe jωt ; Solve the above equations to obtain the displacement amplitude ratio expression of the main vibration system:

[0010] Where h1(ω) represents the displacement amplitude ratio of the main vibration system; X represents the displacement amplitude of the main vibration system; X st is the static deformation of the main vibration system; is the excitation frequency ratio; is the mass ratio of the ith n-fold dynamic vibration absorber; is the natural angular frequency ratio of the ith dynamic absorber, is the natural angular frequency of the multi-dynamic vibration absorber; is the damping ratio of the ith dynamic vibration absorber; B =ω / Ω n is the forced vibration frequency ratio.

[0011] According to one or more embodiments of the present invention, in step S2, the mass M and stiffness K of the main vibration system and the mass m of the ith dynamic vibration absorber among the n dynamic vibration absorbers are i , stiffness, c i and damping k i Set as interval variables, and establish the mathematical model of robustness optimization of the uncertain multiple dynamic vibration absorber. The expression of the mathematical model is as follows: min[maxh1(ω|m i ,c i ,k i )]; stm i ,c i ,k i ∈

[0012] Δ; where Δ is a continuous uncertainty set, the mass M and stiffness K of the main vibration system are uniformly distributed in the interval [(1-α)×M, (1+α)×M] and the interval [(1-α)×K, (1+α)×K]; the optimization design variable is the stiffness k of the i-th dynamic vibration absorber among the n dynamic vibration absorbers i and damping c I , which is in the interval [(1-α)×k I ,(1+α)×k I ] and the interval [(1-α)×c i ,(1+α)×c i Evenly distributed inside.

[0013] According to one or more embodiments of the present invention, step S3 introduces scenario theory and considers [m I ,c I ,k I ]∈Δ, Δ is a continuous uncertainty set, according to the probability distribution P of Δ r Through Monte Carlo random sampling, N scenarios are obtained as optimization constraints. When solving the subsequent optimization problem, only the sampled N scenarios are considered, and the original infinite constraint set is no longer considered. The semi-infinite optimization problem is further converted into a finite constraint problem, and the slack variable h is introduced.

[0014] According to one or more embodiments of the present invention, step S3 comprises the following steps: 31 , given the number of optimization variables d, risk parameter ∈ and confidence parameter β, the number of samples N required for the optimization constraint satisfies the following formula: Where d is the number of optimization variables; N is the number of samples of optimization constraints; ∈∈(0,1) is the given risk parameter; β∈(0,1) is the given confidence parameter; The probability not less than (1-β) satisfies most of the constraints in the uncertainty set Δ, and the unsatisfied part does not exceed ∈, that is: in, is the optimal solution to the scene convex optimization problem; S 32 , N optimization constraints are obtained through Monte Carlo random sampling, and the original semi-infinite optimization problem is transformed into a standard finite optimization problem. The robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory is expressed as: Among them, h is the slack variable.

[0015] According to one or more embodiments of the present invention, in step S4, the dual-population evolutionary algorithm with adaptive constraint processing solves complex multi-constraint problems through a dual-population collaboration mechanism; the main population dynamically explores feasible solutions to the optimization problem in the search space, and the auxiliary population retains valuable information of the unconstrained optimization problem to help the main population overcome infeasible obstacles; and the search state is dynamically changed through an adaptive constraint processing strategy.

[0016] According to one or more embodiments of the present invention, step S4 comprises the following steps: 41 , initialize the basic parameters of the dual population evolutionary algorithm, and the optimization variable is the damping c of the dynamic vibration absorber I and stiffness k i , that is, to find the optimal solution (c i ,k i ) opt ; S 42 , randomly generate two populations, namely the main population and the auxiliary population, each population contains N1 individuals, and calculate the fitness value; where the fitness value is calculated according to the cost function h1(ω|m i ,c i ,k i ) is calculated; S 43 , determine whether the main population and auxiliary population meet the iteration end conditions; if they do, output the optimal design parameters (c I ,k i ) opt ; If not satisfied, continue to iterate; S 44, using binary tournament selection to select mating parents, and performing update crossover and mutation operations to generate offspring; S 45 , combine the offspring with their respective populations to form a mixed population, and calculate the fitness value of each individual in the mixed population; S 46 , the main population adopts an adaptive constraint processing strategy for independent update; the auxiliary population adopts the traditional auxiliary population selection mechanism to determine the dominance relationship between individuals, ignore the existence of constraints, and give priority to selecting individuals with the best performance on a single objective function;

[0017] S 47 , execute the environmental selection strategy, the main population adopts the EFb environmental selection strategy, and selects the best N1 individuals as the next generation population through the EFb environmental selection strategy; the auxiliary population adopts the PIb environmental selection strategy, and selects the most adaptable N1 individuals as the next generation population through the PIb environmental selection strategy, and then proceeds to step S 43 .

[0018] According to one or more embodiments of the present invention, step S 46 The following steps are included: 461 , calculate the fitness F(x) of individual x in population P, the calculation formula is: G(y)=|{z∈P|y<z}; where F(x) represents the fitness of individual x; y is an individual in population P; D x is x and The distance between the nearest neighbors; S 462 , calculate and determine the proportion of feasible individuals in the main population, the calculation formula is: Among them, r k represents the proportion of feasible individuals in the kth generation population; z f and z represent the number of feasible individuals and individuals in the main population respectively; k Is it less than or equal to the critical value γ? If so, go to step S 463 If not, go to step S 464 ; S 463 , the evolution of the main population is set to the exploration state, using the infeasible individuals with lower constraint violations and better target values ​​to explore the feasible area more deeply, and then proceed to step S 465 ; S 464 , the evolution of the main population is set to a diffusion state, the population diffuses forward along the edge of the feasible region and obtains a good distribution, and the feasible individuals with the optimal target value are preferentially selected; S 465 , output the mixed population of the main population and the offspring.

[0019] According to one or more embodiments of the present invention, step S 47In it, the algorithm flow of the EFb environment selection strategy is as follows: S 471a Input the mixed population M1 of the main population and offspring, the population size N1, and the fitness value F(x); S 472a Judge whether the fitness of the individuals in M1 is less than 1; if so, store the individuals in the archive A1; if not, store the individuals in the archive A2; S 473a Select the next-generation main population from A1 and A2, including the following three cases: (1) If |A1| = N1, directly select A1 as the next-generation population; (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; among them, A2 is sorted in ascending order according to the fitness value; (3) If |A1| > N1, then select the first N1 individuals in A1; among them, A1 is sorted in ascending order according to the individual density.

[0020] According to one or more embodiments of the present invention, the step S 47 In it, the algorithm flow of the PIb environment selection strategy is as follows: S 471b Input the mixed population M2 of the auxiliary population and offspring, the population size N1, and the fitness value F(x); S 472b Judge whether the fitness of the individuals in M2 is less than 1; if so, store the individuals in the archive A1; if not, store the individuals in the archive A2; S 473b Select the next-generation auxiliary population from A1 and A2, including the following three cases: (1) If |A1| = N1, directly select A1 as the next-generation population; (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; among them, A2 is sorted in ascending order according to the fitness value; (3) If |A1| > N1, then select the first N1 individuals in A1; among them, A1 is sorted in ascending order according to the individual density.

[0021] The present invention also provides a robust optimization design system for a multi-power absorber, characterized in that the design system uses the above-mentioned robust optimization design method for a multi-power absorber, and the design system includes: a data processing module for establishing a mechanical model of a deterministic multi-power absorber; and determining the probability distribution form and sampling range of each uncertain parameter variable, abstracting the robust optimization design problem of the uncertain multi-power absorber into a min-max optimization problem, and establishing its mathematical model; an optimization model modeling module for converting the infinite-constraint optimization problem into a finite-constraint optimization problem by using the scenario theory, and establishing a robust optimization model for the uncertain multi-power absorber based on the scenario theory; a solving module for using the dual-population evolutionary algorithm with adaptive constraint handling as the optimization algorithm to solve the robust optimization model for the uncertain multi-power absorber based on the scenario theory, and finding the optimal design parameters of the power absorber.

[0022] The positive and progressive effects of the present invention are as follows:

[0023] The robust optimization design method and system of multiple dynamic vibration absorbers of the present invention have at least the following advantages:

[0024] The robust optimization design method and system for multiple dynamic vibration absorbers of the present invention are designed for the situation where both the main vibration system and the dynamic vibration absorber have uncertain parameters. By introducing scenario theory, the NP problem is converted into a finite constraint convex optimization problem. At the same time, in view of the situation where a large number of constraint samples are required for multiple dynamic vibration absorbers, a dual-population evolutionary algorithm for adaptive constraint processing is used in combination with a variety of constraint processing techniques to balance goals and constraints to adapt to diverse feasible areas, effectively solving the complex multi-constraint problem introduced by scenario theory. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The above and other features, properties and advantages of the present invention will become more apparent through the following description in conjunction with the accompanying drawings and embodiments, in which the same reference numerals always represent the same features, wherein:

[0026] Figure 1 It is a structural schematic diagram of a mechanical model of a multiple dynamic vibration absorber in one embodiment of a robust optimization design method for a multiple dynamic vibration absorber of the present invention.

[0027] Figure 2 It is a schematic diagram of the scenario theory in one embodiment of the robust optimization design method for multiple dynamic vibration absorbers of the present invention.

[0028] Figure 3 It is a flow chart of an embodiment of the robust optimization design method of multiple dynamic vibration absorbers of the present invention. DETAILED DESCRIPTION

[0029] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0030] Embodiments of the present invention will now be described in detail with reference to the accompanying drawings. Reference will now be made in detail to preferred embodiments of the present invention, examples of which are shown in the accompanying drawings. Wherever possible, the same reference numerals will be used in all drawings to represent the same or similar parts. In addition, although the terms used in the present invention are selected from well-known and commonly used terms, some of the terms mentioned in the specification of the present invention may be selected by the applicant at his or her discretion, and their detailed meanings are explained in the relevant parts of the description herein. In addition, it is required to understand the present invention not only by the actual terms used, but also by the meaning implied by each term.

[0031] The present invention provides a robust optimization design method for multiple dynamic vibration absorbers, which is based on scenario theory and includes the following steps:

[0032] Step S1, establishing a mechanical model of a deterministic multi-dynamic vibration absorber;

[0033] Step S2, determining the probability distribution form and sampling range of each uncertain parameter variable, abstracting the robustness optimization design problem of uncertain multiple dynamic vibration absorbers into a minimum-maximum optimization problem, and establishing its mathematical model;

[0034] Step S3, using scenario theory to convert the infinite constraint optimization problem into a finite constraint optimization problem, and establishing an uncertainty multiple dynamic vibration absorber robustness optimization model based on scenario theory;

[0035] Step S4, using the dual population evolutionary algorithm with adaptive constraint processing as an optimization algorithm, solving the robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory, and finding the optimal design parameters of the dynamic vibration absorber.

[0036] The robust optimization design method for multiple dynamic vibration absorbers of the present invention is based on scenario theory. Through this optimization design method, the reliability of the design of multiple dynamic vibration absorbers can be improved when there are uncertain parameters in the main vibration system and the dynamic vibration absorbers.

[0037] like Figure 1 As shown, as a preferred implementation of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, in step S1, the motion equation of the main vibration system is expressed as:

[0038]

[0039] Where M represents the mass of the main vibration system; K represents the stiffness of the main vibration system; n is the number of dynamic vibration absorbers installed on the main vibration system; k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber, x i represents the displacement of the ith dynamic vibration absorber (i = 1, 2, ..., n); x is the displacement of the main vibration system; f is the external harmonic excitation, f = F0e jωt ;

[0040] The motion equation of the dynamic vibration absorber is expressed as:

[0041]

[0042] Among them, m i represents the mass of the ith dynamic vibration absorber, k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber (i=1,2,…,n); x iis the displacement of the dynamic vibration absorber.

[0043] like Figure 1 , which is a schematic diagram of the structure of the mechanical model of the multiple dynamic vibration absorber in one embodiment of the robust optimization design method of the multiple dynamic vibration absorber of the present invention.

[0044] As a preferred implementation of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, in step S1, the motion equation of the dynamic vibration absorber is rewritten into a complex form as follows:

[0045] x=Xe jωt ,x i =X i e jωt ,f=Fe jωt

[0046] Solving the above equations, we can obtain the displacement amplitude ratio expression of the main vibration system:

[0047]

[0048] Where h1(ω) represents the displacement amplitude ratio of the main vibration system; X represents the displacement amplitude of the main vibration system; X st is the static deformation of the main vibration system; is the excitation frequency ratio; is the mass ratio of the ith n-fold dynamic vibration absorber; is the natural angular frequency ratio of the ith dynamic absorber, is the natural angular frequency of the multi-dynamic vibration absorber; is the damping ratio of the ith dynamic vibration absorber; B =ω / Ω n is the forced vibration frequency ratio.

[0049] As a preferred embodiment of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, in step S2, the mass M and stiffness K of the main vibration system and the mass m of the i-th dynamic vibration absorber among the n-weight dynamic vibration absorbers are calculated. i , stiffness, c i and damping k i Set as interval variables, establish its uncertainty multiple dynamic vibration absorber robustness optimization mathematical model, in order to achieve the optimal design of multiple dynamic vibration absorber under uncertainty, the mathematical model is an infinite constraint optimization problem, its expression is as follows:

[0050] min[maxh1(ω|m i ,c i ,k i )]

[0051] stm i,c i ,k i ∈Δ

[0052] Where Δ is a continuous uncertainty set, the mass M and stiffness K of the main vibration system are uniformly distributed in the interval [(1-α)×M, (1+α)×M] and the interval [(1-α)×K, (1+α)×K]; the optimization design variable is the stiffness k of the i-th dynamic vibration absorber among the n dynamic vibration absorbers. i and damping c i , which is in the interval [(1-α)×k i ,(1+α)×k i ] and the interval [(1-α)×c i ,(1+α)×c i Evenly distributed inside.

[0053] As a preferred embodiment of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, step S3 introduces scenario theory and takes into account [m i ,c i ,k i ]∈Δ, Δ is a continuous uncertainty set, according to the probability distribution P of Δ r Through Monte Carlo random sampling, N "scenarios" are obtained as optimization constraints. When solving the subsequent optimization problem, only the N "scenarios" obtained by sampling are considered, and the original infinite constraint set is no longer considered. The semi-infinite optimization problem is further converted into a finite constraint problem, and the slack variable h is introduced to reduce the difficulty of solving the problem.

[0054] As a preferred implementation of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, step S3 includes the following steps:

[0055] Step S 31 , given the number of optimization variables d, risk parameter ∈ and confidence parameter β, the number of samples N required for the optimization constraint satisfies the following formula:

[0056]

[0057] Where d is the number of optimization variables; N is the number of samples of optimization constraints; ∈∈(0,1) is the given risk parameter; β∈(0,1) is the given confidence parameter;

[0058] The probability not less than (1-β) satisfies most of the constraints in the uncertainty set Δ, and the unsatisfied part does not exceed ∈, that is:

[0059]

[0060] in, It is the optimal solution to the scene convex optimization problem;

[0061] Step S 32 , N optimization constraints are obtained through Monte Carlo random sampling, and the original semi-infinite optimization problem is transformed into a standard finite optimization problem. The robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory is expressed as:

[0062]

[0063] Among them, h is the slack variable.

[0064] like Figure 2 , which is a diagrammatic illustration of the scenario theory in one embodiment of the robust optimization design method for multiple dynamic vibration absorbers of the present invention.

[0065] As a preferred implementation of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, in step S4, the dual-population evolutionary algorithm for adaptive constraint processing solves the complex multi-constraint problem through a dual-population collaboration mechanism;

[0066] The main population dynamically explores feasible solutions to the optimization problem in the search space, and the auxiliary population retains valuable information of the unconstrained optimization problem to help the main population overcome infeasible obstacles. Through adaptive constraint processing strategies, the search state (exploration state and diffusion state) is dynamically changed to improve the effective use of search resources and help overcome the obstacles brought by complex constraints.

[0067] As a preferred implementation of the robust optimization design method for multiple dynamic vibration absorbers of the present invention, step S4 includes the following steps:

[0068] Step S 41 , initialize the basic parameters of the dual population evolutionary algorithm, and the optimization variable is the damping c of the dynamic vibration absorber i and stiffness k i , that is, to find the optimal solution (c i ,k i ) opt ;

[0069] Step S 42 , randomly generate two populations, namely the main population (majPop) and the auxiliary population (supPop), each population contains N1 individuals, and calculate the fitness value; where the fitness value is calculated according to the cost function h1(ω|m i ,c i ,k i ) to perform calculations;

[0070] Step S 43, determine whether the main population (majPop) and the auxiliary population (supPop) meet the iteration end conditions; if they do, output the optimal design parameters (c i ,k i ) opt ; If not satisfied, continue to iterate;

[0071] Step S 44 , using binary tournament selection to select mating parents, and performing update crossover and mutation operations to generate offspring;

[0072] Step S 45 , combine the offspring with their respective populations to form a mixed population, and calculate the fitness value of each individual in the mixed population;

[0073] Step S 46 , the main population (majPop) adopts an adaptive constraint processing strategy for independent update; the auxiliary population (supPop) adopts the selection mechanism of the traditional auxiliary population (supPop) to determine the dominance relationship between individuals, ignore the existence of constraints, and give priority to selecting individuals with the best performance on a single objective function;

[0074] Step S 47 , execute the environment selection strategy, the main population (majPop) adopts the EFb environment selection strategy, and selects the best N1 individuals as the next generation population through the EFb environment selection strategy; the auxiliary population (supPop) adopts the PIb environment selection strategy, and selects the most adaptable N1 individuals as the next generation population through the PIb environment selection strategy, and then proceeds to step S 43 .

[0075] Preferably, step S 46 In the paper, the adaptive constraint processing strategy divides the evolution of the main population (majPop) into two states, namely the exploration state and the diffusion state. These two states use different constraint processing methods to dynamically deploy search resources.

[0076] It should be noted that step S 47 In the paper, the process of environmental selection strategy is the key step of the dual population evolutionary algorithm, which provides continuous motivation for the efficient evolution of the population. In order to cope with complex infeasible obstacles and strengthen the selection pressure, the EFb environmental selection strategy and the PIb environmental selection strategy are designed for the main population (majPop) and the auxiliary population (supPop), respectively.

[0077] As a preferred embodiment of the robust optimization design method of multiple dynamic vibration absorbers of the present invention, the step S 46 The steps include:

[0078] Step S 461, calculate the fitness F(x) of individual x in population P, the calculation formula is:

[0079]

[0080] G(y)=|{z∈P|y<z}

[0081] Among them, F(x) represents the fitness of individual x; y is the individual in population P; D x is the distance between x and its √zth nearest neighbor;

[0082] Step S 462 , Calculate the proportion of feasible individuals in the main population (majPop), the calculation formula is:

[0083]

[0084] Among them, r k represents the proportion of feasible individuals in the kth generation population; z f and z represent the number of feasible individuals and individuals in the main population (majPop), respectively;

[0085] Judgement k Is it less than or equal to the critical value γ? If so, go to step S 463 If not, go to step S 464 ;

[0086] Step S 463 , the evolution of the main population (majPop) is set to the exploration state, using the infeasible individuals with lower constraint violations and better target values ​​to explore the feasible area more deeply, and then go to step S 465 ;

[0087] Step S 464 , the evolution of the main population (majPop) is set to a diffusion state, the population diffuses forward along the edge of the feasible region, and obtains a good distribution, giving priority to feasible individuals with the optimal target value;

[0088] Step S 465 , output the main population (majPop) and the mixed population of offspring.

[0089] As a preferred embodiment of the robust optimization design method of multiple dynamic vibration absorbers of the present invention, the step S 47 In the EFb environment selection strategy, the algorithm flow is as follows:

[0090] Step S 471a , input the main population (majPop) and the mixed population M1 of the offspring, the population size N1, and the fitness value F(x);

[0091] Step S472a 1. Determine whether the fitness of the individuals in M1 is less than 1; if so, store the individuals in archive A1; if not, store the individuals in archive A2;

[0092] Step S 473a 2. Select the next-generation main population (majPop) from A1 and A2, including the following three cases:

[0093] (1) If |A1| = N1, directly select A1 as the next-generation population;

[0094] (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; where A2 is sorted in ascending order of fitness value;

[0095] (3) If |A1| > N1, then select the first N1 individuals in A1; where A1 is sorted in ascending order of individual density (i.e., the distance between each individual and its √z nearest neighbor individual).

[0096] As a preferred embodiment of the robust optimization design method of the multi-power absorber of the present invention, in the step S 47 , the algorithm flow of the PIb environmental selection strategy is as follows:

[0097] Step S 471b 3. Input the auxiliary population (supPop) and the mixed population M2 of the offspring, the population size N1, and the fitness value F(x);

[0098] Step S 472b 4. Determine whether the fitness of the individuals in M2 is less than 1; if so, store the individuals in archive A1; if not, store the individuals in archive A2;

[0099] Step S 473b 5. Select the next-generation auxiliary population (supPop) from A1 and A2, including the following three cases:

[0100] (1) If |A1| = N1, directly select A1 as the next-generation population;

[0101] (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; where A2 is sorted in ascending order of fitness value;

[0102] (3) If |A1| > N1, then select the first N1 individuals in A1; where A1 is sorted in ascending order of individual density (i.e., the distance between each individual and its √z nearest neighbor individual).

[0103] As Figure 3 shown, it is a schematic flow of an embodiment of the robust optimization design method of the multi-power absorber of the present invention.​

[0104] As described above, the robust optimization design method for multiple dynamic vibration absorbers of the present invention uses scenario theory to convert the infinite constraint optimization problem into a finite constraint optimization problem, establishes a robust optimization model for uncertain multiple dynamic vibration absorbers based on scenario theory, uses the dual-population Jinhua algorithm for adaptive constraint processing as an optimization algorithm, solves the robust optimization model for uncertain multiple dynamic vibration absorbers based on scenario theory, and finds the optimal design parameters of the dynamic vibration absorber.

[0105] The robust optimization design method for multiple dynamic vibration absorbers of the present invention is designed for the case where both the main vibration system and the dynamic vibration absorber have uncertain parameters. By introducing scenario theory, the NP problem is converted into a finite constraint convex optimization problem. At the same time, in view of the fact that a large number of constraint samples are required for multiple dynamic vibration absorbers, a dual-population evolutionary algorithm for adaptive constraint processing is used in combination with a variety of constraint processing techniques to balance objectives and constraints to adapt to diverse feasible areas, thereby effectively solving the complex multi-constraint problem introduced by scenario theory.

[0106] The present invention also provides a robust optimization design system for multiple dynamic vibration absorbers, wherein the design system uses the robust optimization design method for multiple dynamic vibration absorbers as described above, and the design system comprises:

[0107] The data processing module is used to establish the mechanical model of the deterministic multiple dynamic vibration absorber; determine the probability distribution form and sampling range of each uncertain parameter variable, abstract the robustness optimization design problem of the uncertain multiple dynamic vibration absorber into a minimum-maximum optimization problem, and establish its mathematical model;

[0108] The optimization model modeling module uses scenario theory to transform the infinite constraint optimization problem into a finite constraint optimization problem, and establishes an uncertainty multi-dynamic vibration absorber robustness optimization model based on scenario theory;

[0109] The solution module uses the dual-population evolutionary algorithm with adaptive constraint processing as the optimization algorithm to solve the robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory and find the optimal design parameters of the dynamic vibration absorbers.

[0110] The robust optimization design system for multiple dynamic vibration absorbers of the present invention has the advantages of the robust optimization design method for multiple dynamic vibration absorbers of the present invention. By introducing scenario theory, the NP problem is converted into a finite constraint convex optimization problem. At the same time, in view of the fact that multiple dynamic vibration absorbers require a large number of constraint samples, a dual-population evolutionary algorithm for adaptive constraint processing is used in combination with a variety of constraint processing techniques to balance goals and constraints to adapt to diverse feasible areas, thereby effectively solving the complex multi-constraint problem introduced by scenario theory.

[0111] Although the specific embodiments of the present invention are described above, it should be understood by those skilled in the art that these are only examples, and the protection scope of the present invention is defined by the appended claims. Those skilled in the art may make various changes or modifications to these embodiments without departing from the principles and essence of the present invention, but these changes and modifications all fall within the protection scope of the present invention.

Claims

1. A robust optimization design method for multiple dynamic vibration absorbers, characterized in that: The described design method is based on the scenario theory and includes the following steps: S1. Establish the mechanical model of the deterministic multiple dynamic vibration absorber; S2. Determine the probability distribution form and sampling range of each uncertain parameter variable, abstract the robust optimization design problem of the uncertain multiple dynamic vibration absorber as a min-max optimization problem, and establish its mathematical model; S3. Use the scenario theory to convert the infinite constraint optimization problem into a finite constraint optimization problem, and establish a robust optimization model of the uncertain multiple dynamic vibration absorber based on the scenario theory; S4. Use the dual-population evolutionary algorithm with adaptive constraint handling as the optimization algorithm to solve the robust optimization model of the uncertain multiple dynamic vibration absorber based on the scenario theory, and find the optimal design parameters of the dynamic vibration absorber.

2. The robust optimization design method for multiple dynamic vibration absorbers according to claim 1, characterized in that: In the step S1, the motion equation of the main vibration system is expressed as: Where M represents the mass of the main vibration system; K represents the stiffness of the main vibration system; n is the number of dynamic vibration absorbers installed on the main vibration system; k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber, x i represents the displacement of the ith dynamic vibration absorber (i = 1, 2, ..., n); x is the displacement of the main vibration system; f is the external harmonic excitation, f = F0e jωt ; The motion equation of the dynamic vibration absorber is expressed as: Among them, m i represents the mass of the ith dynamic vibration absorber, k i represents the stiffness of the ith dynamic vibration absorber, c i represents the damping of the ith dynamic vibration absorber (i=1,2,…,n); x i is the displacement of the dynamic vibration absorber.

3. The robust optimization design method for multiple dynamic vibration absorbers according to claim 2, characterized in that: In the step S1, the motion equation of the dynamic vibration absorber is rewritten in the complex form as follows: x=Car jωt ,x i =X i e jωt ,f=Fe jωt Solve the above equations to obtain the displacement amplitude ratio expression of the main vibration system: Where h1(ω) represents the displacement amplitude ratio of the main vibration system; X represents the displacement amplitude of the main vibration system; X st is the static deformation of the main vibration system; is the excitation frequency ratio; is the mass ratio of the ith n-fold dynamic vibration absorber; is the natural angular frequency ratio of the ith dynamic absorber, is the natural angular frequency of the multi-dynamic vibration absorber; is the damping ratio of the ith dynamic absorber; B =ω / Ω n is the forced vibration frequency ratio.

4. The robust optimization design method for multiple dynamic vibration absorbers according to claim 2, characterized in that: In step S2, the mass M and stiffness K of the main vibration system and the mass m of the ith dynamic vibration absorber among the n dynamic vibration absorbers are calculated. i , stiffness, c i and damping k i It is set as an interval variable, and a mathematical model for robustness optimization of its uncertain multiple dynamic vibration absorber is established. The expression of the mathematical model is as follows: min[maxh1(ω|m i ,c i ,k i )] s.t.m i ,c i ,k i ∈Δ Where Δ is a continuous uncertainty set, the mass M and stiffness K of the main vibration system are uniformly distributed in the interval [(1-α)×M, (1+α)×M] and the interval [(1-α)×K, (1+α)×K]; the optimization design variable is the stiffness k of the i-th dynamic vibration absorber among the n dynamic vibration absorbers. i and damping c i , which is in the interval [(1-α)×k i ,(1+α)×k i ] and the interval [(1-α)×c i ,(1+α)×c i Evenly distributed inside.

5. The robust optimization design method for multiple dynamic vibration absorbers according to claim 4, characterized in that: The step S3 introduces the scenario theory and considers [m i ,c i ,k i ]∈Δ, Δ is a continuous uncertainty set, according to the probability distribution P of Δ r Through Monte Carlo random sampling, N scenarios are obtained as optimization constraints. When solving the subsequent optimization problem, only the sampled N scenarios are considered, and the original infinite constraint set is no longer considered. The semi-infinite optimization problem is further converted into a finite constraint problem, and the slack variable h is introduced.

6. The robust optimization design method for multiple dynamic vibration absorbers according to claim 5, characterized in that: The step S3 includes the following steps: S 31 , given the number of optimization variables d, risk parameter ∈ and confidence parameter β, the number of samples N required for the optimization constraint satisfies the following formula: Where d is the number of optimization variables; N is the sampling number of optimization constraints; v∈(0,1) is the given risk parameter; β∈(0,1) is the given confidence parameter; The probability not less than (1-β) satisfies most of the constraints in the uncertainty set Δ, and the unsatisfied part does not exceed v, that is: in, It is the optimal solution to the scene convex optimization problem; S 32 , N optimization constraints are obtained through Monte Carlo random sampling, and the original semi-infinite optimization problem is transformed into a standard finite optimization problem. The robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory is expressed as: st|h1(ω|m i ,c i ,k i )| ∞ ≤h,i=1,……n Where h is the slack variable.

7. The robust optimization design method for multiple dynamic vibration absorbers according to claim 5, characterized in that: In the step S4, the dual-population evolutionary algorithm with adaptive constraint handling solves complex multi-constraint problems through the dual-population cooperation mechanism; The main population dynamically explores feasible solutions to the optimization problem in the search space, and the auxiliary population retains valuable information of the unconstrained optimization problem to help the main population cross the infeasible obstacles; through the adaptive constraint handling strategy, the search state is dynamically changed.

8. The robust optimization design method for multiple dynamic vibration absorbers according to claim 7, characterized in that: The step S4 includes the following steps: S 41 , initialize the basic parameters of the dual population evolutionary algorithm, and the optimization variable is the damping c of the dynamic vibration absorber i and stiffness k i , that is, to find the optimal solution (c i ,k i ) opt ; S 42 , randomly generate two populations, namely the main population and the auxiliary population, each population contains N1 individuals, and calculate the fitness value; where the fitness value is calculated according to the cost function h1(ω|m i ,c i ,k i ) to perform calculations; S 43 , determine whether the main population and auxiliary population meet the iteration end conditions; if they do, output the optimal design parameters (c i ,k i ) opt ; If not satisfied, continue to iterate; S 44 , using binary tournament selection method to select mating parents, and performing update crossover and mutation operations to generate offspring; S 45 , combine the offspring with their respective populations to form a mixed population, and calculate the fitness value of each individual in the mixed population; S 46 , the main population adopts an adaptive constraint processing strategy for independent update; the auxiliary population adopts the traditional auxiliary population selection mechanism to determine the dominance relationship between individuals, ignore the existence of constraints, and give priority to selecting individuals with the best performance on a single objective function; S 47 , execute the environmental selection strategy, the main population adopts the EFb environmental selection strategy, and selects the best N1 individuals as the next generation population through the EFb environmental selection strategy; the auxiliary population adopts the PIb environmental selection strategy, and selects the most adaptable N1 individuals as the next generation population through the PIb environmental selection strategy, and then proceeds to step S 43 .

9. The robust optimization design method for multiple dynamic vibration absorbers according to claim 8, characterized in that: The step S 46 The steps include: S 461 , calculate the fitness F(x) of individual x in population P, the calculation formula is: G(y)=|{z∈P|y<z} Among them, F(x) represents the fitness of individual x; y is the individual in population P; D x is x and The distance between nearest neighbors; S 462 , calculate and determine the proportion of feasible individuals in the main population, the calculation formula is: Among them, r k represents the proportion of feasible individuals in the kth generation population; z f and z represent the number of feasible individuals and individuals in the main population, respectively; Judgement k Is it less than or equal to the critical value γ? If so, go to step S 463 If not, go to step S 464 ; S 463 , the evolution of the main population is set to the exploration state, using the infeasible individuals with lower constraint violations and better target values ​​to explore the feasible area more deeply, and then proceed to step S 465 ; S 464 , the evolution of the main population is set to a diffusion state, the population diffuses forward along the edge of the feasible region and obtains a good distribution, giving priority to feasible individuals with the optimal target value; S 465 , output the mixed population of the main population and the offspring.

10. The robust optimization design method for multiple dynamic vibration absorbers according to claim 8, characterized in that: The step S 47 In the EFb environment selection strategy, the algorithm flow is as follows: S 471a , input the mixed population M1 of the main population and offspring, the population size N1, and the fitness value F(x); S 472a , determine whether the fitness of the individual in M1 is less than 1; if so, store the individual in file A1; if not, store the individual in file A2; S 473a , select the next generation main population from A1 and A2, including the following three cases: (1) If |A1| = N1, directly select A1 as the next-generation population; (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; where A2 is sorted in ascending order of fitness value; (3) If |A1| > N1, then select the first N1 individuals in A1; where A1 is sorted in ascending order of individual density.

11. The robust optimization design method for multiple dynamic vibration absorbers according to claim 8, characterized in that: The step S 47 In , the PIb environment selection strategy algorithm flow is as follows: S 471b , input the mixed population M2 of auxiliary population and offspring, population size N1, fitness value F(x); S 472b , determine whether the fitness of the individual in M2 is less than 1; if so, store the individual in file A1; if not, store the individual in file A2; S 473b , select the next generation of auxiliary population from A1 and A2, including the following three cases: (1) If |A1| = N1, directly select A1 as the next-generation population; (2) If |A1| < N1, select all of A1 and the first N1 individuals of A2; where A2 is sorted in ascending order of fitness value; (3) If |A1| > N1, then select the first N1 individuals in A1; where A1 is sorted in ascending order of individual density.

12. A robust optimization design system for multiple dynamic vibration absorbers, characterized in that: The described design system uses the robust optimization design method of the multiple dynamic vibration absorber as described in any one of claims 1-11. The design system includes: A data processing module, which is used to establish the mechanical model of the deterministic multiple dynamic vibration absorber; and determine the probability distribution form and sampling range of each uncertain parameter variable, abstract the robust optimization design problem of the uncertain multiple dynamic vibration absorber as a min-max optimization problem, and establish its mathematical model; An optimization model modeling module, which uses the scenario theory to convert the infinite constraint optimization problem into a finite constraint optimization problem, and establish a robust optimization model of the uncertain multiple dynamic vibration absorber based on the scenario theory; The solution module uses the dual-population evolutionary algorithm with adaptive constraint processing as the optimization algorithm to solve the robustness optimization model of uncertain multiple dynamic vibration absorbers based on scenario theory and find the optimal design parameters of the dynamic vibration absorbers.

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