Dry friction damper multi-target robust optimization method considering multi-source uncertainty

Abstract

The invention relates to the technical field of aero-engines, and discloses a dry friction damper multi-target robust optimization method considering multi-source uncertainty, and the method specifically comprises the following steps: 1, quantifying the uncertainty parameters of a dry friction damper; step 2, performing hyper-Latin cube (LHS) sampling on value ranges of geometric parameters and contact parameters caused by abrasion to obtain parameter combination sampling capable of expressing global characteristics in a global sample space; 3, calculating a corresponding nonlinear dynamic response by using a multi-dimensional harmonic balance method (MHBM) by using the nonlinear parameters obtained by sampling; by means of the mode of synchronously optimizing key design parameters of the friction-damping system, the optimal robustness design of the friction-damping system is allowed to be carried out under parameter uncertainty with the purpose of vibration reduction, the damping effect of the dry friction damper is improved, the Pareto optimal solution of the neural network is solved through the genetic algorithm, and the optimal stability of the friction-damping system is improved. Therefore, the multi-objective optimization is realized and the calculation efficiency is improved.

Description

Technical Field

[0001] The invention relates to the technical field of aeroengines, and in particular to a multi-objective robust optimization method for a dry friction damper considering multi-source uncertainties. Background Art

[0002] In order to achieve vibration reduction of aircraft engine turbine blades, dry friction dampers are an important vibration reduction component. In actual manufacturing, dry friction dampers are affected by processing tolerances and assembly tolerances. The size of these tolerances varies randomly, resulting in the actual contact state of the dry friction damper being inconsistent with the ideal state, which ultimately affects the vibration reduction effect. Secondly, the working state of aircraft engines is changeable, and the working speed of turbine blades changes dramatically and unpredictably, which makes it difficult for dry friction dampers to achieve good vibration reduction effects under various working conditions, bringing great challenges to the design of dry friction dampers. Finally, long-term operation of turbine blades under high temperature conditions will cause wear on the contact surface between the blades and the dry friction damper. This wear will change the state of the contact surface, thereby affecting the contact between the blades and the dry friction damper, thereby affecting the vibration reduction effect. Therefore, in order to ensure the accuracy of the prediction of the nonlinear response of the system, the uncertainty of the parameters must be quantified and propagated into the numerical simulation.

[0003] In the past, the robustness optimization of dry friction dampers in the design process widely adopted parameter sensitivity analysis of forced vibration response. This method can only optimize a single objective at a time, and is inefficient and time-consuming during the optimization process. It cannot ensure the stability of the structure under the influence of uncertain factors.

[0004] To this end, we propose a multi-objective robust optimization method for dry friction dampers considering multi-source uncertainties. Summary of the invention

[0005] The present invention mainly solves the technical problems existing in the above-mentioned prior art and provides a multi-objective robust optimization method for a dry friction damper taking into account multi-source uncertainties.

[0006] In order to achieve the above object, the present invention adopts the following technical solution, a multi-objective robust optimization method for a dry friction damper considering multi-source uncertainty, which specifically includes the following steps:

[0007] Step 1: Quantify the uncertainty parameters of the dry friction damper;

[0008] Step 2: Perform super Latin cube (LHS) sampling on the range of geometric parameters and contact parameters caused by wear to obtain parameter combination sampling that can describe global characteristics in the global sample space;

[0009] Step 3: Using the nonlinear parameters obtained by sampling, the corresponding nonlinear dynamic response is calculated using the multidimensional harmonic balance method (MHBM);

[0010] Step 4: Calculate the mean and variance of the frequency response function obtained in step 3, use the multi-source uncertainty parameters as the input set of training samples, the mean and variance of the frequency response function as the output set of training samples, and use the BP neural network for training;

[0011] Step 5: Perform robustness design with the optimization goal of reducing the dispersion of vibration amplitude and frequency response function, and use multi-objective genetic algorithm (NSGA-II) for robustness optimization to obtain the optimized Pareto optimal solution.

[0012] Preferably, the specific method of step one is: establishing a finite element model of a dry friction damper, characterizing the uncertainty of the geometric shape of the damper, quantifying the uncertainty of geometric parameters, processing the micro-wear experimental data of the dry friction damper, obtaining the uncertainty quantification of contact parameters caused by wear, and quantifying them using the mean and standard deviation of the uncertainty parameters.

[0013] Preferably, the specific method of step 2 is: the LHS test design of n points in M ​​dimensions can be constructed by an n×m matrix M, and the sampling points are defined as follows:

[0014]

[0015] Where uij is a uniform random variable on [0,1], and Mij is the i-th element arranged in the j-dimensional space. Preferably, the specific method of step three is: the vibration differential equation of the dry friction system can be expressed in the following form:

[0016]

[0017] Among them, M, C, and K are the system mass matrix, damping matrix, and stiffness matrix, respectively, with dimensions of Nd×Nd, Nd is the number of degrees of freedom, u is the system displacement, which is the unknown quantity to be solved, fnl represents the nonlinear force, and fsin(wt) is the external excitation force at the excitation frequency. The steady-state response displacement solution of the nonlinear system can be expressed as the following Fourier series:

[0018]

[0019] Where Nh is the truncated harmonic order, w is the external force excitation frequency, U0 is the Fourier series corresponding to the "0" order harmonic, Uck and Usk are the Fourier series corresponding to the k-order harmonic, assuming that the system degree of freedom is Nd, then in the calculation process, the Nd (2Nh + 1)-dimensional Fourier series vector is arranged as follows:

[0020]

[0021] At the same time, a displacement-coefficient conversion matrix is ​​defined as follows:

[0022]

[0023] Where I is the identity matrix of Nd×Nd dimensions, represents the Kronecker product, the displacement formula can be rewritten as follows:

[0024] u(t)=H(ωt i )

[0025] Introduce a frequency derivative operator:

[0026]

[0027] Then the velocity and acceleration can be expressed using the frequency derivative operator:

[0028]

[0029] Similarly, nonlinear forces and external excitation forces can also be expressed as the product of the transformation matrix and the corresponding Fourier coefficients:

[0030] f(t i )=H(ωt i )×[F 0 ,F c1 ,F s1 ,...F ck ,F sk ,...] T

[0031] =H(ωt i )F

[0032]

[0033] From this we can get:

[0034] G(ω,U)=P(ω)U+F nl (U)-F=0

[0035] P(ω)=ω 2 N M ▽ 2 +ωN C ▽+N K

[0036] Among them, P is the dynamic stiffness matrix, NM, NC, NK are block diagonal matrices composed of mass matrix M, mass matrix C, and stiffness matrix K, which are solved by the Newton-Raphson iterative method.

[0037] Preferably, the specific method of step 4 is: when the BP neural network is forward propagated, the activation values ​​from the input layer to the hidden layer and from the hidden layer to the output layer are:

[0038]

[0039] During back propagation, the error between the output layer and the hidden layer is:

[0040]

[0041] Then, the chain rule is used through the BP neural network to calculate the contribution of the error in each group of neurons, and the weights and thresholds are adjusted iteratively to minimize the error to obtain an accurate proxy model.

[0042] Preferably, the specific method of step five is: Pareto optimal solution refers to a special solution in a multi-objective optimization problem that satisfies the following conditions: one objective cannot be improved without compromising any other objective, which means that in a given problem, no other solution can achieve better results on all objectives. The NSGA-Ⅱ algorithm includes individual coding, fitness, selection operator, crossover operator, mutation operator and crowding degree. The individual coding is specifically: using a finite length binary string composed of 1 and 0 (called a chromosome) to encode the individual. It is assumed that each variable uses a chromosome of length n, and each position in the chromosome is defined by qij, qij∈[0,1]. In order to decode the chromosome and map it to a specific interval in the solution space, the design variable xi is defined as follows

[0043]

[0044] The NSGA-Ⅱ algorithm obtains multi-objective optimization solutions through several generations of "genetic evolution". In the evolution process, all individuals in a certain generation constitute a population, and each individual in the initial population is generated by random coding. The initial population size is generally selected between 20 and 200; the fitness is specifically: the evolution process of the genetic algorithm is driven by fitness. The fitness function is obtained by making some deformations of the objective function. Each chromosome corresponds to a fitness value. Different individuals are compared with each other through fitness. The size of the fitness value directly determines the individual's genetic opportunity, that is, the probability of survival; the selection operator is specifically: selection The operator performs genetic operations by giving individuals with higher fitness values ​​a higher probability. Each individual is assigned a probability of being selected, allowing individuals with poor fitness to be selected by chance. This mechanism ensures that individuals with poor fitness will not be lost from the group too early, avoiding premature convergence. Common selection operators include roulette strategy and tournament strategy. In the roulette strategy, the chance of an individual being selected is proportional to its fitness. For a relatively small initial group, there is a risk of premature convergence. The tournament strategy selects the best individual through random selection and competition. The tournament strategy does not need to sort the group. The algorithm More efficient, and more importantly, it can ensure the diversity of the population. The present invention adopts the selection operator of the tournament strategy; the crossover operator is specifically: the crossover operator is responsible for combining the chromosome information from the two parents to generate new offspring individuals. Common crossover operators for binary coding include single-point crossover, two-point crossover and uniform crossover. The present invention selects uniform crossover. The genes in the offspring chromosomes (bits in binary) are all generated by copying the corresponding genes from the parent chromosomes with equal probability; the mutation operator is specifically: the mutation operator simulates the gene mutation in the evolutionary process, which can make the genetic algorithm have a strong local search ability, thereby maintaining the genetic diversity of the population. The mutation operator is achieved by randomly selecting a gene position in the individual and changing its value from 0 (or 1) to 1 (or 0); the crowding degree is specifically: in the target space, NSGA-II compares the candidate individuals in pairs according to the objective function, and all individuals can form multiple frontier layers in sequence. Individuals in the same Pareto layer have a greater crowding degree and are better. In different Pareto layers, if an individual is better than the parent in the sense of Pareto optimization, the entire generation population approaches the Pareto frontier and is evenly distributed.

[0045] The present invention provides a multi-objective robust optimization method for a dry friction damper taking multi-source uncertainties into consideration.

[0046] It has the following beneficial effects:

[0047] 1. This multi-objective robust optimization method for dry friction dampers considering multi-source uncertainties, by synchronously optimizing the key design parameters of the friction-damping system, allows for the optimal robustness design of the friction-damping system under parameter uncertainty for the purpose of vibration reduction, improves the damping effect of the dry friction damper, and solves the Pareto optimal solution of the neural network through a genetic algorithm, thus achieving multi-objective optimization and improving computational efficiency.

[0048] 2. This multi-objective robust optimization method for dry friction dampers that considers multi-source uncertainties has effectively solved various problems in practical applications of dry friction dampers by comprehensively optimizing the vibration reduction performance of aircraft engine turbine blades. It not only considers uncertainty factors in the manufacturing process such as processing tolerances and assembly tolerances, but also fully considers actual operating conditions such as the variable working state of aircraft engines and wear caused by long-term high-temperature operation, thereby ensuring the accuracy and practicality of the optimization results.

[0049] 3. The multi-objective robust optimization method of the dry friction damper considering multi-source uncertainties adopts the multi-objective genetic algorithm (NSGA-II) with the advantages of high efficiency and stability. It can quickly converge to the optimal solution while ensuring the diversity of the population, further improving the optimization efficiency. Therefore, the multi-objective robust optimization method of the dry friction damper considering multi-source uncertainties has broad application prospects and important practical value. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is a flow chart of the method of the present invention;

[0051] Figure 2 It is a schematic diagram of the damping structure of the present invention;

[0052] Figure 3 A schematic diagram of the Pareto frontier solution set of the present invention;

[0053] Figure 4 It is a schematic diagram of the frequency response function before and after the optimization of the present invention. DETAILED DESCRIPTION

[0054] Embodiment 1: A multi-objective robust optimization method for a dry friction damper considering multi-source uncertainties, such as Figure 1As shown, the specific steps include: Step 1: quantify the uncertainty parameters of the dry friction damper; Step 2: perform super Latin cube (LHS) sampling on the value range of the geometric parameters and the contact parameters caused by wear to obtain the parameter combination sampling that can describe the global characteristics in the global sample space; Step 3: use the nonlinear parameters obtained by sampling to calculate the corresponding nonlinear dynamic response using the multidimensional harmonic balance method (MHBM); Step 4: calculate the mean and variance of the frequency response function obtained in step 3, use the multi-source uncertainty parameters as the input set of training samples, and the mean and variance of the frequency response function as the output set of training samples, and use the BP neural network for training; Step 5: perform robust design with the optimization goal of reducing the vibration amplitude and the dispersion of the frequency response function, and use the multi-objective genetic algorithm (NSGA-II) for robust optimization to obtain the optimized Pareto optimal solution. By simultaneously optimizing the key design parameters of the friction-damping system for the purpose of vibration reduction, the optimal robustness design of the friction-damping system can be carried out under parameter uncertainty, the damping effect of the dry friction damper can be improved, and the Pareto optimal solution of the neural network is solved by a genetic algorithm, which achieves multi-objective optimization and improves computational efficiency.

[0055] Embodiment 2: Based on Embodiment 1, Figure 2 As shown, the specific method of step one is: establish a finite element model of the dry friction damper, characterize the uncertainty of the geometric shape of the damper, quantify the uncertainty of the geometric parameters, process the experimental data of the micro-motion wear of the dry friction damper, obtain the uncertainty quantification of the contact parameters caused by wear, and quantify it using the mean and standard deviation of the uncertainty parameters. The specific method of step two is: the LHS test design of n points in M ​​dimensions can be constructed by the n×m matrix M, and the sampling points are defined as follows:

[0056]

[0057] Where uij is a uniform random variable on [0,1], and Mij is the i-th element arranged in the j dimension. By comprehensively optimizing the vibration reduction performance of aircraft engine turbine blades, various problems of dry friction dampers in practical applications are effectively solved. Not only the uncertainty factors in the manufacturing process such as processing tolerance and assembly tolerance are considered, but also the actual operating conditions such as the variable working state of aircraft engines and the wear caused by long-term high-temperature operation are fully considered, thereby ensuring the accuracy and practicality of the optimization results.

[0058] Embodiment 3: Based on Embodiment 1 and Embodiment 2, Figure 3 As shown, the specific method of step three is: the vibration differential equation of the dry friction system can be expressed as follows:

[0059]

[0060] Among them, M, C, and K are the system mass matrix, damping matrix, and stiffness matrix, respectively, with dimensions of Nd×Nd, Nd is the number of degrees of freedom, u is the system displacement, which is the unknown quantity to be solved, fnl represents the nonlinear force, and fsin(wt) is the external excitation force at the excitation frequency. The steady-state response displacement solution of the nonlinear system can be expressed as the following Fourier series:

[0061]

[0062] Where Nh is the truncated harmonic order, w is the external force excitation frequency, U0 is the Fourier series corresponding to the "0" order harmonic, Uck and Usk are the Fourier series corresponding to the k-order harmonic, assuming that the system degree of freedom is Nd, then in the calculation process, the Nd (2Nh + 1)-dimensional Fourier series vector is arranged as follows:

[0063]

[0064] At the same time, a displacement-coefficient conversion matrix is ​​defined as follows:

[0065]

[0066] Where I is the identity matrix of Nd×Nd dimensions, represents the Kronecker product, the displacement formula can be rewritten as follows:

[0067] u(t)=H(ωt i )

[0068] Introduce a frequency derivative operator:

[0069]

[0070] Then the velocity and acceleration can be expressed using the frequency derivative operator:

[0071]

[0072] Similarly, nonlinear forces and external excitation forces can also be expressed as the product of the transformation matrix and the corresponding Fourier coefficients:

[0073] f(t i )=H(ωt i )×[F 0 ,F c1 ,F s1 ,...F ck ,F sk ,...] T

[0074] =H(ωt i)F

[0075]

[0076] From this we can get:

[0077] G(ω,U)=P(ω)U+F nl (U)-F=0

[0078] P(ω)=ω 2 N M ▽ 2 +ωN C ▽+N K

[0079] Among them, P is the dynamic stiffness matrix, NM, NC, NK are block diagonal matrices composed of mass matrix M, mass matrix C, and stiffness matrix K, which are solved by the Newton-Raphson method iteration method. The multi-objective genetic algorithm (NSGA-II) adopted has the advantages of high efficiency and stability. It can quickly converge to the optimal solution while ensuring the diversity of the population, further improving the optimization efficiency. Therefore, the multi-objective robust optimization method of dry friction damper considering multi-source uncertainty has broad application prospects and important practical value.

[0080] Embodiment 4: Based on Embodiment 1, Embodiment 2 and Embodiment 3, Figure 3 As shown, the specific method of step 4 is: when the BP neural network is forward propagated, the activation values ​​from the input layer to the hidden layer and from the hidden layer to the output layer are:

[0081]

[0082] During back propagation, the error between the output layer and the hidden layer is:

[0083]

[0084] Then, the chain rule is used through the BP neural network to calculate the contribution of the error in each group of neurons, and the weights and thresholds are adjusted iteratively to minimize the error to obtain an accurate proxy model.

[0085] Embodiment 5: Based on Embodiment 1, Embodiment 2, Embodiment 3 and Embodiment 4, Figure 3As shown, the specific method of step five is: Pareto optimal solution refers to a special solution in multi-objective optimization problems that satisfies the following conditions: one objective cannot be improved without compromising any other objective, which means that in a given problem, no other solution can achieve better results on all objectives. The NSGA-Ⅱ algorithm includes individual coding, fitness, selection operator, crossover operator, mutation operator and crowding degree. Individual coding is specifically: using a finite length binary string composed of 1 and 0 (called chromosome) to encode individuals. Assume that each variable uses a chromosome of length n, and each position in the chromosome is defined by qij, qij∈[0,1]. In order to decode the chromosome and map it to a specific interval in the solution space, the design variable xi is defined as follows

[0086]

[0087] The NSGA-Ⅱ algorithm obtains multi-objective optimization solutions through several generations of "genetic evolution". In the evolution process, all individuals in a certain generation constitute a population, and each individual in the initial population is generated by random coding. The initial population size is generally selected between 20 and 200; the fitness is specifically: the evolution process of the genetic algorithm is driven by fitness. The fitness function is obtained by making some deformations of the objective function. Each chromosome corresponds to a fitness value. Different individuals are compared with each other through fitness. The size of the fitness value directly determines the individual's genetic opportunity, that is, the probability of survival; the selection operator is specifically: selection The operator performs genetic operations by giving individuals with higher fitness values ​​a higher probability. Each individual is assigned a probability of being selected, allowing individuals with poor fitness to be selected by chance. This mechanism ensures that individuals with poor fitness will not be lost from the group too early, avoiding premature convergence. Common selection operators include roulette strategy and tournament strategy. In the roulette strategy, the chance of an individual being selected is proportional to its fitness. For a relatively small initial group, there is a risk of premature convergence. The tournament strategy selects the best individual through random selection and competition. The tournament strategy does not need to sort the group. The algorithm More efficient, and more importantly, it can ensure the diversity of the population. The present invention adopts the selection operator of the tournament strategy; the crossover operator is specifically: the crossover operator is responsible for combining the chromosome information from the two parents to generate new offspring individuals. Common crossover operators for binary coding include single-point crossover, two-point crossover and uniform crossover. The present invention selects uniform crossover. The genes in the offspring chromosomes (bits in binary) are all generated by copying the corresponding genes from the parent chromosomes with equal probability; the mutation operator is specifically: the mutation operator simulates the gene mutation in the evolutionary process, which can make the genetic algorithm have a strong local search ability, thereby maintaining the genetic diversity of the population. The mutation operator is achieved by randomly selecting a gene position in the individual and changing its value from 0 (or 1) to 1 (or 0); the crowding degree is specifically: in the target space, NSGA-II compares the candidate individuals in pairs according to the objective function, and all individuals can form multiple frontier layers in sequence. Individuals in the same Pareto layer have a greater crowding degree and are better. In different Pareto layers, if an individual is better than the parent in the sense of Pareto optimization, the entire generation population approaches the Pareto frontier and is evenly distributed.

[0088] Working principle of the present invention: The structural design variables of the dry friction damper considered in the present invention are as follows: Figure 2As shown in Figure 1, the geometry of the damper can be characterized by the following parameters: w1 and w2 are the widths of the upper and lower dampers; b is the height of the base; h2 is the height of the trapezoid; θL, θR are the angles of the damper bottom; ld is the length of the damper; hL, hR are the lengths of the left and right hypotenuses of the damper. The reference values ​​w1 = 3mm, w2 = 33mm, ld = 70mm, b = 2mm and θL = θR = 30° represent the original design. The lengths of other geometric parameters can be obtained based on geometric knowledge, and the uncertainty of the geometric parameters is quantified, as shown in Table 1.

[0089] Table 1 Uncertainty of geometric features

[0090]

[0091] The uncertainty caused by contact surface wear is regarded as a noise parameter, and the above parameters are sampled by LHS. The response functions of the sampled parameters after considering the uncertainty are calculated respectively. The mean amplitude of each group of uncertainty response and the amplitude attenuation rate are used as the sample output of the BP neural network to obtain the relevant proxy model. The NSGA-Ⅱ algorithm is used for multi-objective optimization. The initial population size is set to 500, the number of iterations is 400, and the S1 and S2 of the individual objective functions in the population are sampled by 100,000 Monte Carlo sampling to ensure the convergence of the calculation results. The final Pareto solution set is shown in Figure 2 , RD1, RD2, and RD3 are selected, the amplitude means are 0.423736mm, 0.49018mm, and 0.5975324mm, respectively, and the amplitude attenuation rates are 5.66994%, 1.47331%, and 0.3062%, respectively. The frequency response function of the third group that has undergone robustness optimization is as follows Figure 4 shown.

[0092] The above shows and describes the basic principles and main features of the present invention and the advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A multi-objective robust optimization method for dry friction dampers considering multi-source uncertainties is characterized by: The specific steps include: Step 1: Quantify the uncertainty parameters of the dry friction damper; Step 2: Perform super Latin cube (LHS) sampling on the range of geometric parameters and contact parameters caused by wear to obtain parameter combination sampling that can describe global characteristics in the global sample space; Step 3: Using the nonlinear parameters obtained by sampling, the corresponding nonlinear dynamic response is calculated using the multidimensional harmonic balance method (MHBM); Step 4: Calculate the mean and variance of the frequency response function obtained in step 3, use the multi-source uncertainty parameters as the input set of training samples, the mean and variance of the frequency response function as the output set of training samples, and use the BP neural network for training; Step 5: Perform robustness design with the optimization goal of reducing the dispersion of vibration amplitude and frequency response function, and use multi-objective genetic algorithm (NSGA-II) for robustness optimization to obtain the optimized Pareto optimal solution.

2. The multi-objective robust optimization method for dry friction damper considering multi-source uncertainty according to claim 1 is characterized in that: The specific method of step one is: establishing a finite element model of the dry friction damper, characterizing the uncertainty of the geometric shape of the damper, quantifying the uncertainty of the geometric parameters, processing the micro-wear experimental data of the dry friction damper, obtaining the uncertainty quantification of the contact parameters caused by wear, and quantifying it using the mean and standard deviation of the uncertainty parameters.

3. The multi-objective robust optimization method for dry friction damper considering multi-source uncertainty according to claim 1 is characterized in that: The specific method of step 2 is: the LHS test design of n points in M ​​dimensions can be constructed by an n×m matrix M, and the sampling points are defined as follows: where uij is a uniform random variable on [0,1] and Mij is the i-th element of the permutation in the j dimension.

4. The multi-objective robust optimization method for dry friction damper considering multi-source uncertainty according to claim 1 is characterized in that: The specific method of step three is: the vibration differential equation of the dry friction system can be expressed as follows: Among them, M, C, and K are the system mass matrix, damping matrix, and stiffness matrix, respectively, with dimensions of Nd×Nd, Nd is the number of degrees of freedom, u is the system displacement, which is the unknown quantity to be solved, fnl represents the nonlinear force, and fsin(wt) is the external excitation force at the excitation frequency. The steady-state response displacement solution of the nonlinear system can be expressed as the following Fourier series: Where Nh is the truncated harmonic order, w is the external excitation frequency, U0 is the Fourier series corresponding to the "0" order harmonic, Uck and Usk are the Fourier series corresponding to the k-order harmonic, assuming that the system degree of freedom is Nd, then in the calculation process, the Nd (2Nh + 1)-dimensional Fourier series vector is arranged as follows: At the same time, a displacement-coefficient conversion matrix is ​​defined as follows: Where I is the identity matrix of Nd×Nd dimensions, represents the Kronecker product, the displacement formula can be rewritten as follows: u(t)=H(ωt i )U Introduce a frequency derivative operator: Then the velocity and acceleration can be expressed using the frequency derivative operator: Similarly, nonlinear forces and external excitation forces can also be expressed as the product of the transformation matrix and the corresponding Fourier coefficients: f(t i )=H(ωt i )×[F 0 ,F c1 ,F s1 ,...F ck ,F sk ,...] T =H(ωt i )F From this we can get: G(ω,U)=P(ω)U+F nl (U)-F=0 Among them, P is the dynamic stiffness matrix, NM, NC, NK are block diagonal matrices composed of mass matrix M, mass matrix C, and stiffness matrix K, which are solved by the Newton-Raphson iterative method.

5. The multi-objective robust optimization method for dry friction damper considering multi-source uncertainty according to claim 1, characterized in that: The specific method of step 4 is: when the BP neural network is forward propagated, the activation values ​​from the input layer to the hidden layer and from the hidden layer to the output layer are: During back propagation, the error between the output layer and the hidden layer is: Then, the chain rule is used through the BP neural network to calculate the contribution of the error in each group of neurons, and the weights and thresholds are adjusted iteratively to minimize the error to obtain an accurate proxy model.

6. The multi-objective robust optimization method for dry friction damper considering multi-source uncertainty according to claim 1 is characterized in that: The specific method of step five is: Pareto optimal solution refers to a special solution in a multi-objective optimization problem that satisfies the following conditions: one objective cannot be improved without compromising any other objective, which means that in a given problem, no other solution can achieve better results on all objectives. The NSGA-Ⅱ algorithm includes individual coding, fitness, selection operator, crossover operator, mutation operator and crowding degree. Individual coding is specifically: using a finite length binary string composed of 1 and 0 (called a chromosome) to encode the individual. Assume that each variable uses a chromosome of length n, and each position in the chromosome is defined by qij, qij∈[0,1]. In order to decode the chromosome and map it to a specific interval in the solution space, the design variable xi is defined as follows The NSGA-Ⅱ algorithm obtains multi-objective optimization solutions through several generations of "genetic evolution". In the evolution process, all individuals in a certain generation constitute a population, and each individual in the initial population is generated by random coding. The initial population size is generally selected between 20 and 200; the fitness is specifically: the evolution process of the genetic algorithm is driven by fitness. The fitness function is obtained by making some deformations of the objective function. Each chromosome corresponds to a fitness value. Different individuals are compared with each other through fitness. The size of the fitness value directly determines the individual's genetic opportunity, that is, the probability of survival; the selection operator is specifically: selection The operator performs genetic operations by giving individuals with higher fitness values ​​a higher probability. Each individual is assigned a probability of being selected, allowing individuals with poor fitness to be selected by chance. This mechanism ensures that individuals with poor fitness will not be lost from the group too early, avoiding premature convergence. Common selection operators include roulette strategy and tournament strategy. In the roulette strategy, the chance of an individual being selected is proportional to its fitness. For a relatively small initial group, there is a risk of premature convergence. The tournament strategy selects the best individual through random selection and competition. The tournament strategy does not need to sort the group. The algorithm More efficient, and more importantly, it can ensure the diversity of the population. The present invention adopts the selection operator of the tournament strategy; the crossover operator is specifically: the crossover operator is responsible for combining the chromosome information from the two parents to generate new offspring individuals. Common crossover operators for binary coding include single-point crossover, two-point crossover and uniform crossover. The present invention selects uniform crossover. The genes in the offspring chromosomes (bits in binary) are all generated by copying the corresponding genes from the parent chromosomes with equal probability; the mutation operator is specifically: the mutation operator simulates the gene mutation in the evolutionary process, which can make the genetic algorithm have a strong local search ability, thereby maintaining the genetic diversity of the population. The mutation operator is achieved by randomly selecting a gene position in the individual and changing its value from 0 (or 1) to 1 (or 0); the crowding degree is specifically: in the target space, NSGA-II compares the candidate individuals in pairs according to the objective function, and all individuals can form multiple frontier layers in sequence. Individuals in the same Pareto layer have a greater crowding degree and are better. In different Pareto layers, if an individual is better than the parent in the sense of Pareto optimization, the entire generation population approaches the Pareto frontier and is evenly distributed.

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