WO2018223774A1 - Method for indirectly acquiring continuous distribution mechanical parameter field of non-homogeneous materials - Google Patents

Method for indirectly acquiring continuous distribution mechanical parameter field of non-homogeneous materials Download PDF

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WO2018223774A1
WO2018223774A1 PCT/CN2018/083274 CN2018083274W WO2018223774A1 WO 2018223774 A1 WO2018223774 A1 WO 2018223774A1 CN 2018083274 W CN2018083274 W CN 2018083274W WO 2018223774 A1 WO2018223774 A1 WO 2018223774A1
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orthogonal polynomial
frequency response
field
mechanical
response function
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吴邵庆
范刚
费庆国
李彦斌
姜东�
韩晓林
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东南大学
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  • the invention relates to a method for indirect acquisition of a mechanical parameter field of continuous distribution of non-uniform materials, and belongs to the technical field of structural dynamics inverse problems.
  • Composite materials have the advantages of small specific gravity, large specific strength and specific modulus, and are widely used in aerospace, machinery and other fields.
  • the molding process of such composite materials is mostly complicated, and holes, defects and internal stresses are often present in the materials, and the macroscopic mechanical properties are characterized by non-uniformity. If uniformized parameters are used to represent the mechanical properties of non-uniform materials, the results of the mechanical analysis will be inaccurate and may cause catastrophic consequences.
  • the theoretical analysis method uses the theoretical analysis model of the material to predict the mechanical parameters, and can obtain relatively coarse results.
  • the finite element calculation method obtains the unit cell model of the material, and generally uses the stiffness average method to obtain the mechanical parameters of the material, which exists due to the microscopic modeling process. Uncertainty, the material equivalent mechanical parameters predicted by the unit cell model are deviated from the actual values; the experimental measurement method can obtain more reliable material parameters, but due to the limitations of the experimental conditions, only part of the material mechanical parameters can be obtained. For mechanical parameters that are difficult to obtain by experimental measurements, methods can be indirectly identified by a combination of theoretical/numerical analysis and testing.
  • the indirect identification is a method of measuring the mechanical parameters of the composite material under the force of the force or responding to the inversion of the mechanical parameters of the material.
  • the model of the material parameters obtained by the indirect identification method can accurately reflect the mechanical characteristics of the structure.
  • Most of the existing methods for obtaining mechanical material parameters identify the mechanical parameters of material homogenization, and cannot consider the non-uniformity of the mechanical parameters with spatial distribution. It is difficult to guarantee the accuracy of structural mechanics modeling and analysis results. Therefore, it is necessary to propose a method to obtain the distribution field of mechanical parameters of inhomogeneous materials.
  • the object of the present invention is to provide an indirect acquisition method for mechanical parameters of continuous distribution of non-uniform materials, to solve the problem of obtaining mechanical parameters of non-uniform materials, and to provide accurate parameters for structural mechanics modeling and analysis using such non-uniform materials.
  • a method for indirectly obtaining a mechanical field of continuous distribution of non-uniform materials characterized in that the method comprises the following steps:
  • step S1 The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of the modal test and the test frequency response function of the non-uniform material beam in step S1 include:
  • step S2 The indirect acquisition method for continuous distributed mechanical parameter field of non-uniform material according to claim 1 , wherein the mechanical parameter field simulation and structural dynamics model of continuous material distribution based on orthogonal polynomial expansion in step S2 are established.
  • the specific steps include:
  • P k (x) is the k-th order generalized orthogonal polynomial basis function
  • b k is the k-th order generalized orthogonal polynomial coefficient
  • l is the beam length
  • step S4 The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of identifying the orthogonal polynomial coefficients in the material parameter model based on the optimization algorithm in step S4 include:
  • S j is the sensitivity matrix for calculating the frequency response function to estimate the orthogonal polynomial coefficient vector, namely:
  • step S5 The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific step of reconstructing the mechanical field of the non-uniform material mechanical parameter in step S5 comprises: obtaining orthogonality by convergence Polynomial coefficient vector b and equation (1) are reconstructed to obtain the distribution field of mechanical parameters of inhomogeneous materials.
  • the invention has the following advantages:
  • the existing material mechanical parameter identification technology generally only recognizes the uniform distribution of material mechanical parameters or through the division of finite elements, can identify the mechanical parameters of the material distributed with space, but can not identify the mechanical parameter field of the material continuously distributed with space.
  • the technology provided by the invention can identify the mechanical parameter field of the non-uniform material continuously distributed with space by using the measured frequency response function at the limited measuring point, and can better reflect the non-uniformity of the material with the spatial distribution than the prior art identification result. Conducive to improve the accuracy of subsequent mechanical modeling and analysis;
  • Figure 1 is a logic flow diagram of the method of the present invention.
  • FIG. 2 is a schematic view of a composite material beam in an embodiment
  • FIG. 3 is a schematic diagram of a finite element model and a measurement point number in the embodiment.
  • Figure 4 is a frequency response function curve at a typical measuring point in the embodiment.
  • FIG. 5 is an elastic modulus distribution field of the composite beam obtained by the identification in the embodiment.
  • the elastic modulus distribution field E(x) of the material in a certain direction is identified by the technique of the present invention, and specifically includes the following steps:
  • the composite material beam shown in Fig. 2 is produced, and the two ends of the beam are clamped by rigid constraints; the modal test system under hammer excitation is built, and the beam is evenly divided into 11 segments, and the measurement point number is shown in Fig. 3, and the mode is developed.
  • the experimental acceleration frequency response function matrix H B ( ⁇ ) at each measuring point is obtained, and the partial frequency response function curve is as shown in FIG. 4 .
  • the initial value of the coefficient b k in the orthogonal polynomial model in (1) is assumed.
  • the finite element model of the composite beam with two ends is established.
  • the unit division is based on the marking in the modal test.
  • the number of elements is 11, the stiffness matrix K e of the beam element is:
  • N represents the second derivative of the beam element shape function matrix pair x
  • I represents the beam moment of inertia of the beam
  • l e is the unit length
  • x e is the left node coordinate of the element
  • T represents the matrix transpose
  • the total stiffness matrix of the beam of inhomogeneous material can be obtained by superposition of its element stiffness matrix:
  • the total mass matrix M of the beam can also be represented by a similar step as a form in which the generalized orthogonal polynomial coefficients are multiplied by the corresponding matrix:
  • the total damping matrix C of the beam can be calculated from the total mass matrix M and the total stiffness matrix K.
  • the calculated acceleration frequency response function matrix H A ( ⁇ ) can be further solved as shown in the following equation:
  • S j is the sensitivity matrix for calculating the frequency response function to estimate the orthogonal polynomial coefficient vector, namely:
  • the elastic modulus distribution field and the linear density distribution field to be identified are respectively obtained by the equations (1) and (5).
  • the elastic modulus distribution field E(x) of the obtained non-uniform composite beam continuously distributed in the axial direction is shown in Fig. 5.

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Abstract

A method for indirectly acquiring a continuous distribution mechanical parameter field of non-homogeneous materials, comprising the following steps: S1, carrying out a non-homogeneous material beam modal test and acquiring a test frequency response function; S2, simulating a continuous distribution mechanical parameter field of materials and establishing a structural dynamics model based on orthogonal polynomial expansion; S3, solving a sensitivity matrix of an acceleration frequency response function relative to an orthogonal polynomial coefficient; S4, identifying an orthogonal polynomial coefficient in a material parameter model based on an optimization algorithm; and S5, reconstructing a mechanical parameter distribution field of the non-homogeneous materials. The method can solve the problem of acquiring mechanical parameters of non-homogeneous materials, and provide accurate parameters for structural mechanics modeling and analysis using the non-homogeneous materials.

Description

一种非均匀材料连续分布力学参数场间接获取方法Indirect acquisition method for mechanical parameter field of continuous distribution of non-uniform materials 技术领域：Technical field:
本发明涉及一种非均匀材料连续分布力学参数场间接获取方法，属于结构动力学反问题技术领域。The invention relates to a method for indirect acquisition of a mechanical parameter field of continuous distribution of non-uniform materials, and belongs to the technical field of structural dynamics inverse problems.
背景技术：Background technique:
复合材料具有比重小，比强度和比模量大等优点，被广泛用于航空航天，机械等领域。然而，该类复合材料成型工艺大多较复杂，材料中常存在孔洞、缺陷以及内应力等，其宏观力学性能表现出非均匀性等特点。如采用均一化的参数去表示非均匀材料的力学特征，会导致力学分析结果的不准确，可能引起灾难性后果。Composite materials have the advantages of small specific gravity, large specific strength and specific modulus, and are widely used in aerospace, machinery and other fields. However, the molding process of such composite materials is mostly complicated, and holes, defects and internal stresses are often present in the materials, and the macroscopic mechanical properties are characterized by non-uniformity. If uniformized parameters are used to represent the mechanical properties of non-uniform materials, the results of the mechanical analysis will be inaccurate and may cause catastrophic consequences.
常用获取非均匀复合材料力学参数的手段有理论分析，有限元计算和试验测量等。理论分析方法利用材料的理论分析模型预测力学参数，能够获取较为粗略地结果；有限元计算方法通过建立材料的单胞模型，一般利用刚度平均法获取材料的力学参数，由于微观建模过程中存在不确定性，利用单胞模型预测获得的材料等效力学参数与实际值存在偏差；试验测量法能够获取较为可信的材料参数，但是由于试验条件的限制，只能获取部分的材料力学参数。对于试验测量难以获取的力学参数，可以通过理论/数值分析与试验相结合的手段来间接识别方法。相比前三种途径，间接识别是通过实测复合材料结构受力情况下的变形或响应反演材料力学参数的方法，利用间接识别方法获取的材料参数建立模型，能够较为准确反映结构的力学特征。现有的材料力学参数获取方法大多识别的是材料均一化的力学参数，并无法考虑其力学参数随空间分布的非均匀性，难以保证结构力学建模和分析结果的精度。因此，提出一种能够获取非均匀材料力学参数分布场的方法非常必要。Common methods for obtaining mechanical parameters of non-uniform composite materials include theoretical analysis, finite element calculation and experimental measurement. The theoretical analysis method uses the theoretical analysis model of the material to predict the mechanical parameters, and can obtain relatively coarse results. The finite element calculation method obtains the unit cell model of the material, and generally uses the stiffness average method to obtain the mechanical parameters of the material, which exists due to the microscopic modeling process. Uncertainty, the material equivalent mechanical parameters predicted by the unit cell model are deviated from the actual values; the experimental measurement method can obtain more reliable material parameters, but due to the limitations of the experimental conditions, only part of the material mechanical parameters can be obtained. For mechanical parameters that are difficult to obtain by experimental measurements, methods can be indirectly identified by a combination of theoretical/numerical analysis and testing. Compared with the first three methods, the indirect identification is a method of measuring the mechanical parameters of the composite material under the force of the force or responding to the inversion of the mechanical parameters of the material. The model of the material parameters obtained by the indirect identification method can accurately reflect the mechanical characteristics of the structure. . Most of the existing methods for obtaining mechanical material parameters identify the mechanical parameters of material homogenization, and cannot consider the non-uniformity of the mechanical parameters with spatial distribution. It is difficult to guarantee the accuracy of structural mechanics modeling and analysis results. Therefore, it is necessary to propose a method to obtain the distribution field of mechanical parameters of inhomogeneous materials.
发明内容Summary of the invention
本发明的目的是提供一种非均匀材料连续分布力学参数场间接获取方法，解决非均匀材料的力学参数获取问题，为使用该类非均匀材料的结构力学建模与分析提供准确的参数。The object of the present invention is to provide an indirect acquisition method for mechanical parameters of continuous distribution of non-uniform materials, to solve the problem of obtaining mechanical parameters of non-uniform materials, and to provide accurate parameters for structural mechanics modeling and analysis using such non-uniform materials.
上述的目的通过以下技术方案实现：The above objectives are achieved by the following technical solutions:
1.一种非均匀材料连续分布力学参数场间接获取方法，其特征在于，该方法包括以下步骤：A method for indirectly obtaining a mechanical field of continuous distribution of non-uniform materials, characterized in that the method comprises the following steps:
S1.非均匀材料梁模态试验与试验频响函数获取；S1. Modal test of the beam of non-uniform material and acquisition of the frequency response function of the test;
S2.基于正交多项式展开的材料连续分布力学参数场模拟与结构动力学模型建立；S2. Mechanical parameter field simulation and structural dynamics model establishment of material continuous distribution based on orthogonal polynomial expansion;
S3.加速度频响函数相对正交多项式系数的灵敏度矩阵求解；S3. Solving the sensitivity matrix of the acceleration frequency response function relative to the orthogonal polynomial coefficient;
S4.基于优化算法的材料参数模型中正交多项式系数识别；S4. Identification of orthogonal polynomial coefficients in a material parameter model based on an optimization algorithm;
S5.非均匀材料力学参数分布场重构。S5. Reconstruction of the distribution field of mechanical parameters of inhomogeneous materials.
2.根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S1中所述的非均匀材料梁模态试验与试验频响函数获取的具体步骤包括：The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of the modal test and the test frequency response function of the non-uniform material beam in step S1 include:
S11：制作该非均匀材料梁，使材料的非均匀特性沿梁轴向分布，并对梁的一端或两端施加约束，使梁不能自由移动。S11: making the non-uniform material beam, so that the non-uniform property of the material is distributed along the axial direction of the beam, and restraining one end or both ends of the beam, so that the beam cannot move freely.
S12：利用标记将两端固支梁分成m等份，采用力锤敲击梁，测量在第i点锤击激励下第j点处的动响应R j(t)，并记录力锤激励信号f i(t)，其中动响应可以是结构位移，速度、加速度等响应，t表示时间； S12: Using the mark to divide the fixed beam at both ends into m equal parts, tapping the beam with a hammer, measuring the dynamic response R j (t) at the jth point under the hammering excitation at the i-th point, and recording the hammer excitation signal f i (t), wherein the dynamic response may be a structural displacement, a velocity, an acceleration, etc., and t represents a time;
S13：分别对激励信号f i(t)和动响应信号R j(t)进行傅立叶变换，获得频域内的激励信号f i(ω)和动响应信号R j(ω)，ω表示频率； S13: Perform Fourier transform on the excitation signal f i (t) and the dynamic response signal R j (t) respectively to obtain an excitation signal f i (ω) and a dynamic response signal R j (ω) in the frequency domain, where ω represents a frequency;
S14：计算第i点锤击激励下第j点处的位移频率响应函数(简称频响函数)H ji(ω)，组成实测频响函数矩阵H B(ω)。 S14: Calculate a displacement frequency response function (abbreviated as a frequency response function) H ji (ω) at the jth point under the hammering excitation of the i-th point, and form a matrix of the measured frequency response function H B (ω).
3.根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S2中所述的基于正交多项式展开的材料连续分布力学参数场模拟与结构动力学模型建立的具体步骤包括：3 . The indirect acquisition method for continuous distributed mechanical parameter field of non-uniform material according to claim 1 , wherein the mechanical parameter field simulation and structural dynamics model of continuous material distribution based on orthogonal polynomial expansion in step S2 are established. The specific steps include:
S21：假设非均匀材料沿梁轴向随空间连续分布的力学参数场函数为Q(x)，x表示沿梁轴向的位置坐标，建立基于正交多项式展开的材料参数分布场模型：S21: Assuming that the mechanical parameter field function of the non-uniform material continuously distributed along the axial direction of the beam is Q(x), x represents the position coordinate along the axial direction of the beam, and a material parameter distribution field model based on orthogonal polynomial expansion is established:
Figure PCTCN2018083274-appb-000001
Figure PCTCN2018083274-appb-000001
其中P k(x)是第k阶广义正交多项式基函数；b k为第k阶广义正交多项式系数，l为梁长； Where P k (x) is the k-th order generalized orthogonal polynomial basis function; b k is the k-th order generalized orthogonal polynomial coefficient, and l is the beam length;
S22：以模态试验中的等分方式划分有限单元，以广义正交多项式P k(x)为力学参数分布场函数，求解复合材料梁的总体质量矩阵M和总体刚度矩阵K； S22: dividing the finite element by an aliquot in the modal test, and using the generalized orthogonal polynomial P k (x) as a mechanical parameter distribution field function to solve the overall mass matrix M and the overall stiffness matrix K of the composite beam;
S23：基于瑞利阻尼假设，由梁的总质量矩阵M和总刚度矩阵K计算得到总阻尼矩阵C，建立梁结构的动力学模型；S23: Based on the Rayleigh damping hypothesis, the total damping matrix C is calculated from the total mass matrix M of the beam and the total stiffness matrix K, and a dynamic model of the beam structure is established;
S24：求解梁的计算频响函数矩阵H A(ω)： S24: Solving the calculated frequency response function matrix H A (ω) of the beam:
H A(ω)＝(-ω 2M+iωC+K) -1  (2)。 H A (ω)=(−ω 2 M+iωC+K) -1 (2).
4.根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S4中所述的基于优化算法的材料参数模型中正交多项式系数识别的具体步骤包括：The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of identifying the orthogonal polynomial coefficients in the material parameter model based on the optimization algorithm in step S4 include:
S41：以基于模态试验的实测频响函数H B(ω)与基于有限元模型的计算频响函数H A(ω)之差最小值为优化目标，构造如下优化问题： S41: the test based on the measured modal frequency response function H B (ω) based on the difference H A (ω) of the finite element model optimization target minimum frequency response function, the optimization problem is constructed as follows:
MinJ(b)＝||H B(ω)-H A(ω,b)||  (3)， MinJ(b)=||H B (ω)-H A (ω,b)|| (3),
其中b为待估计的正交多项式系数向量，b＝[b 1 … b n]，||□||表示矩阵范数； Where b is the orthogonal polynomial coefficient vector to be estimated, b=[b 1 ... b n ], ||□|| represents the matrix norm;
S42：基于灵敏度分析方法迭代求解优化问题，即第j个迭代步求解下式：S42: Iteratively solves the optimization problem based on the sensitivity analysis method, that is, the jth iteration step solves the following formula:
H B(ω)-H j A(ω,y j)＝S j(b j+1-b j)  (4)， H B (ω)-H j A (ω, y j )=S j (b j+1 -b j ) (4),
其中S j为计算频响函数对待估计正交多项式系数向量的灵敏度矩阵，即： Where S j is the sensitivity matrix for calculating the frequency response function to estimate the orthogonal polynomial coefficient vector, namely:
Figure PCTCN2018083274-appb-000002
Figure PCTCN2018083274-appb-000002
当
Figure PCTCN2018083274-appb-000003
时(ε为某个小数，如10 -3)，迭代收敛得到正交多项式系数向量b。 when
Figure PCTCN2018083274-appb-000003
When ε is a fractional number, such as 10 -3 , the iterative convergence yields an orthogonal polynomial coefficient vector b.
5.根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S5中所述的非均匀材料力学参数分布场重构的具体步骤包括：由收敛得到正交多项式系数向量b和式(1)重构获取非均匀材料力学参数分布场。The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific step of reconstructing the mechanical field of the non-uniform material mechanical parameter in step S5 comprises: obtaining orthogonality by convergence Polynomial coefficient vector b and equation (1) are reconstructed to obtain the distribution field of mechanical parameters of inhomogeneous materials.
有益效果：Beneficial effects:
本发明与现有技术相比，具有以下优点：Compared with the prior art, the invention has the following advantages:
1、现有的材料力学参数识别技术一般只能识别均匀分布的材料力学参数或者通过有限单元的划分，能够识别随空间分布的材料力学参数，但是无法识别随空间连续分布的材料力学参数场。本发明中提供的技术能够利用有限测点处的实测频响函数识别非均匀材料随空间连续分布的力学参数场，比现有技术的识别结果能更好反映材料随空间分布的非均匀性，有利于提高后续力学建模和分析的精度；1. The existing material mechanical parameter identification technology generally only recognizes the uniform distribution of material mechanical parameters or through the division of finite elements, can identify the mechanical parameters of the material distributed with space, but can not identify the mechanical parameter field of the material continuously distributed with space. The technology provided by the invention can identify the mechanical parameter field of the non-uniform material continuously distributed with space by using the measured frequency response function at the limited measuring point, and can better reflect the non-uniformity of the material with the spatial distribution than the prior art identification result. Conducive to improve the accuracy of subsequent mechanical modeling and analysis;
2、将随空间连续分布的材料力学参数场函数采用正交多项式展开模型模拟，能够将力学参数分布场函数的估计问题转换为正交多项式系数的估计问题，大大降低了参数识别问题的维数和难度，具有易操作和计算效率高的特点。2. The field function of material mechanics parameters continuously distributed with space is simulated by orthogonal polynomial expansion model, which can convert the estimation problem of mechanical parameter distribution field function into the estimation problem of orthogonal polynomial coefficients, which greatly reduces the dimension of parameter identification problem. And difficulty, with easy operation and high computational efficiency.
附图说明DRAWINGS
图1为本发明方法的逻辑流程框图。Figure 1 is a logic flow diagram of the method of the present invention.
图2为实施例中复合材料梁示意图2 is a schematic view of a composite material beam in an embodiment
图3为实施例中有限元模型和测点编号示意图3 is a schematic diagram of a finite element model and a measurement point number in the embodiment.
图4为实施例中典型测点处的频响函数曲线Figure 4 is a frequency response function curve at a typical measuring point in the embodiment.
图5为实施例中识别获得的复合材料梁弹性模量分布场FIG. 5 is an elastic modulus distribution field of the composite beam obtained by the identification in the embodiment.
具体实施方式detailed description
下面通过实施例的方式，对本发明技术方案进行详细说明，但实施例仅是本发明的其中一种实施方式，应当指出：对于本技术领域的技术人员来说，在不脱离本发明原理的前提下，还可以以更换固定方式，识别获取其他力学参数等方式做出若干改进和等同替换，这些对本发明权利要求进行改进和等同替换后的技术方案，均落入本发明的保护范围。The technical solutions of the present invention are described in detail below by way of the embodiments, but the embodiments are only one of the embodiments of the present invention, and it should be noted that those skilled in the art can not deviate from the principles of the present invention. In the following, it is also possible to make a number of improvements and equivalent substitutions in a manner of replacing and fixing, identifying other mechanical parameters, and the like, and the technical solutions of the present invention are improved and equivalently substituted.
实施例：对一非均匀复合材料，利用本发明的技术识别该材料沿某一方向的弹性模量分布场E(x)，具体包括以下步骤：Embodiment: For a non-uniform composite material, the elastic modulus distribution field E(x) of the material in a certain direction is identified by the technique of the present invention, and specifically includes the following steps:
1、非均匀材料梁模态试验与试验频响函数获取。1. Non-uniform material beam modal test and experimental frequency response function acquisition.
制作如图2所示复合材料梁，将梁两端利用刚性约束夹持；搭建锤击激励下模态试验系统，将梁均匀划分为11段，测点编号如图3所示，开展模态试验，获得各测点处的试验加速度频响函数矩阵H B(ω)，部分频响函数曲线如图4中所示。 The composite material beam shown in Fig. 2 is produced, and the two ends of the beam are clamped by rigid constraints; the modal test system under hammer excitation is built, and the beam is evenly divided into 11 segments, and the measurement point number is shown in Fig. 3, and the mode is developed. In the experiment, the experimental acceleration frequency response function matrix H B (ω) at each measuring point is obtained, and the partial frequency response function curve is as shown in FIG. 4 .
2、基于正交多项式展开的材料连续分布力学参数场模拟与结构动力学模型建立。2. Dynamic parameter field simulation and structural dynamics model establishment of material continuous distribution based on orthogonal polynomial expansion.
假定复合材料梁沿轴向随空间连续分布的弹性模量场函数E(x)具有如下Legendre正交多项式展开的形式：It is assumed that the elastic modulus field function E(x) of the composite beam continuously distributed along the axial direction with space has the following form of Legendre orthogonal polynomial expansion:
Figure PCTCN2018083274-appb-000004
Figure PCTCN2018083274-appb-000004
先假设(1)式中正交多项式模型中系数b k的初值，依据图4建立两端固支复合材料梁有限元模型，单元划分依据模态试验中的标记等分，单元个数为11，则梁单元的刚度矩阵K e为： Firstly, the initial value of the coefficient b k in the orthogonal polynomial model in (1) is assumed. According to Fig. 4, the finite element model of the composite beam with two ends is established. The unit division is based on the marking in the modal test. The number of elements is 11, the stiffness matrix K e of the beam element is:
Figure PCTCN2018083274-appb-000005
Figure PCTCN2018083274-appb-000005
其中N”(x)表示梁单元形函数矩阵对x的两阶导数，I表示梁的截面惯性矩，l e为单元长度，x e为单元左节点坐标，上标T表示矩阵转置，。将式(1)中的正交多项式展开模型代入式(2)中，可得： Where N"(x) represents the second derivative of the beam element shape function matrix pair x, I represents the beam moment of inertia of the beam, l e is the unit length, x e is the left node coordinate of the element, and the superscript T represents the matrix transpose. Substituting the orthogonal polynomial expansion model in equation (1) into equation (2), we can obtain:
Figure PCTCN2018083274-appb-000006
Figure PCTCN2018083274-appb-000006
则非均匀材料梁的总刚度矩阵可以由其单元刚度矩阵叠加得到：The total stiffness matrix of the beam of inhomogeneous material can be obtained by superposition of its element stiffness matrix:
Figure PCTCN2018083274-appb-000007
Figure PCTCN2018083274-appb-000007
其中
Figure PCTCN2018083274-appb-000008
among them
Figure PCTCN2018083274-appb-000008
如果考虑非均匀材料的线密度随空间分布ρA(x)，将复合材料梁沿轴向的线密度分布场利用Legendre广义正交多项式展开，如下式所示：If we consider the linear density of the inhomogeneous material with the spatial distribution ρA(x), the linear density distribution field of the composite beam along the axial direction is expanded by the Legendre generalized orthogonal polynomial, as shown in the following equation:
Figure PCTCN2018083274-appb-000009
Figure PCTCN2018083274-appb-000009
则梁的总质量矩阵M也可以通过类似步骤表示为广义正交多项式系数与对应矩阵相乘叠加的形式：Then the total mass matrix M of the beam can also be represented by a similar step as a form in which the generalized orthogonal polynomial coefficients are multiplied by the corresponding matrix:
Figure PCTCN2018083274-appb-000010
Figure PCTCN2018083274-appb-000010
其中
Figure PCTCN2018083274-appb-000011
基于瑞利阻尼假设，梁的总阻尼矩阵C可以由总质量矩阵M和总刚度矩阵K计算得到。由此，可以进一步求解计算加速度频响函数矩阵H A(ω)，如下式所示： among them
Figure PCTCN2018083274-appb-000011
Based on the Rayleigh damping assumption, the total damping matrix C of the beam can be calculated from the total mass matrix M and the total stiffness matrix K. Thus, the calculated acceleration frequency response function matrix H A (ω) can be further solved as shown in the following equation:
H A(ω)＝(-ω 2M+iωC+K)- 1  (7) H A (ω)=(-ω 2 M+iωC+K) -1 (7)
3、加速度频响函数相对正交多项式系数的灵敏度矩阵求解。3. The sensitivity matrix of the acceleration frequency response function is solved relative to the orthogonal polynomial coefficient.
加速度频响函数H A(ω)对待识别参数向量b的灵敏度为： The sensitivity of the acceleration frequency response function H A (ω) to the identification parameter vector b is:
Figure PCTCN2018083274-appb-000012
Figure PCTCN2018083274-appb-000012
其中，下标j表示第j个迭代步，
Figure PCTCN2018083274-appb-000013
表示位移频响函数，Z为动刚度矩阵，Z关于待识别参数b的一阶导数为： Where subscript j represents the jth iteration step,
Figure PCTCN2018083274-appb-000013
Represents the displacement frequency response function, Z is the dynamic stiffness matrix, and the first derivative of Z with respect to the parameter b to be identified is:
Figure PCTCN2018083274-appb-000014
Figure PCTCN2018083274-appb-000014
仅考虑杨氏模量E的非均匀性，则：Considering only the non-uniformity of Young's modulus E, then:
Figure PCTCN2018083274-appb-000015
Figure PCTCN2018083274-appb-000015
则：then:
Figure PCTCN2018083274-appb-000016
Figure PCTCN2018083274-appb-000016
4、基于优化算法的材料参数模型中正交多项式系数识别。4. Orthogonal polynomial coefficient identification in material parameter model based on optimization algorithm.
以基于模态试验的实测频响函数H B(ω)与基于有限元模型的计算频响函数H A(ω)之差最小值为优化目标，构造如下优化问题： Based on the measured modal test frequency response function H B (ω) based on ([omega]) of the difference between H A finite element model optimization target minimum frequency response function, the optimization problem is constructed as follows:
MinJ(b)＝||H B(ω)-H A(ω,b)||  (12) MinJ(b)=||H B (ω)-H A (ω,b)|| (12)
其中b为待估计的正交多项式系数向量，b＝[b 1 … b n]，||□||表示矩阵范数； Where b is the orthogonal polynomial coefficient vector to be estimated, b=[b 1 ... b n ], ||□|| represents the matrix norm;
S42：基于灵敏度分析方法迭代求解优化问题，即第j个迭代步求解下式：S42: Iteratively solves the optimization problem based on the sensitivity analysis method, that is, the jth iteration step solves the following formula:
H B(ω)-H j A(ω,y j)＝S j(b j+1-b j)  (13) H B (ω)-H j A (ω, y j )=S j (b j+1 -b j ) (13)
其中S j为计算频响函数对待估计正交多项式系数向量的灵敏度矩阵，即： Where S j is the sensitivity matrix for calculating the frequency response function to estimate the orthogonal polynomial coefficient vector, namely:
Figure PCTCN2018083274-appb-000017
Figure PCTCN2018083274-appb-000017
当
Figure PCTCN2018083274-appb-000018
时(ε为某个小数，如10 -3)，迭代收敛得到正交多项式系数向量b。 when
Figure PCTCN2018083274-appb-000018
When ε is a fractional number, such as 10 -3 , the iterative convergence yields an orthogonal polynomial coefficient vector b.
5、非均匀材料力学参数分布场重构。5. Reconstruction of the distribution field of mechanical parameters of inhomogeneous materials.
由式(1)和式(5)分别重构获取待识别的弹性模量分布场和线密度分布场。图5中给出了获取的非均匀复合材料梁沿轴向连续分布的弹性模量分布场E(x)。The elastic modulus distribution field and the linear density distribution field to be identified are respectively obtained by the equations (1) and (5). The elastic modulus distribution field E(x) of the obtained non-uniform composite beam continuously distributed in the axial direction is shown in Fig. 5.

Claims (5)

  1. 一种非均匀材料连续分布力学参数场间接获取方法，其特征在于，该方法包括以下步骤：An indirect acquisition method for mechanical parameter field of continuous distribution of non-uniform materials, characterized in that the method comprises the following steps:
    S1.非均匀材料梁模态试验与试验频响函数获取；S1. Modal test of the beam of non-uniform material and acquisition of the frequency response function of the test;
    S2.基于正交多项式展开的材料连续分布力学参数场模拟与结构动力学模型建立；S2. Mechanical parameter field simulation and structural dynamics model establishment of material continuous distribution based on orthogonal polynomial expansion;
    S3.加速度频响函数相对正交多项式系数的灵敏度矩阵求解；S3. Solving the sensitivity matrix of the acceleration frequency response function relative to the orthogonal polynomial coefficient;
    S4.基于优化算法的材料参数模型中正交多项式系数识别；S4. Identification of orthogonal polynomial coefficients in a material parameter model based on an optimization algorithm;
    S5.非均匀材料力学参数分布场重构。S5. Reconstruction of the distribution field of mechanical parameters of inhomogeneous materials.
  2. 根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S1中所述的非均匀材料梁模态试验与试验频响函数获取的具体步骤包括：The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of the modal test and the test frequency response function of the non-uniform material beam in step S1 include:
    S11：制作该非均匀材料梁，使材料的非均匀特性沿梁轴向分布，并对梁的一端或两端施加约束，使梁不能自由移动。S11: making the non-uniform material beam, so that the non-uniform property of the material is distributed along the axial direction of the beam, and restraining one end or both ends of the beam, so that the beam cannot move freely.
    S12：利用标记将两端固支梁分成m等份，采用力锤敲击梁，测量在第i点锤击激励下第j点处的动响应R j(t)，并记录力锤激励信号f i(t)，其中动响应可以是结构位移，速度、加速度等响应，t表示时间； S12: Using the mark to divide the fixed beam at both ends into m equal parts, tapping the beam with a hammer, measuring the dynamic response R j (t) at the jth point under the hammering excitation at the i-th point, and recording the hammer excitation signal f i (t), wherein the dynamic response may be a structural displacement, a velocity, an acceleration, etc., and t represents a time;
    S13：分别对激励信号f i(t)和动响应信号R j(t)进行傅立叶变换，获得频域内的激励信号f i(ω)和动响应信号R j(ω)，ω表示频率； S13: Perform Fourier transform on the excitation signal f i (t) and the dynamic response signal R j (t) respectively to obtain an excitation signal f i (ω) and a dynamic response signal R j (ω) in the frequency domain, where ω represents a frequency;
    S14：计算第i点锤击激励下第j点处的位移频率响应函数(简称频响函数)H ji(ω)，组成实测频响函数矩阵H B(ω)。 S14: Calculate a displacement frequency response function (abbreviated as a frequency response function) H ji (ω) at the jth point under the hammering excitation of the i-th point, and form a matrix of the measured frequency response function H B (ω).
  3. 根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S2中所述的基于正交多项式展开的材料连续分布力学参数场模拟与结构动力学模型建立的具体步骤包括：The method for indirectly obtaining a mechanical field of continuous distribution of non-uniform materials according to claim 1, wherein the method for constructing a mechanical parameter field of the continuous distribution of the material based on the orthogonal polynomial expansion and the structural dynamics model are established in step S2. The steps include:
    S21：假设非均匀材料沿梁轴向随空间连续分布的力学参数场函数为Q(x)，x表示沿梁轴向的位置坐标，建立基于正交多项式展开的材料参数分布场模型：S21: Assuming that the mechanical parameter field function of the non-uniform material continuously distributed along the axial direction of the beam is Q(x), x represents the position coordinate along the axial direction of the beam, and a material parameter distribution field model based on orthogonal polynomial expansion is established:
    Figure PCTCN2018083274-appb-100001
    Figure PCTCN2018083274-appb-100001
    其中P k(x)是第k阶广义正交多项式基函数；b k为第k阶广义正交多项式系数，l为梁长； Where P k (x) is the k-th order generalized orthogonal polynomial basis function; b k is the k-th order generalized orthogonal polynomial coefficient, and l is the beam length;
    S22：以模态试验中的等分方式划分有限单元，以广义正交多项式P k(x)为力学参数分布场函数，求解复合材料梁的总体质量矩阵M和总体刚度矩阵K； S22: dividing the finite element by an aliquot in the modal test, and using the generalized orthogonal polynomial P k (x) as a mechanical parameter distribution field function to solve the overall mass matrix M and the overall stiffness matrix K of the composite beam;
    S23：基于瑞利阻尼假设，由梁的总质量矩阵M和总刚度矩阵K计算得到总阻尼矩阵C，建立梁结构的动力学模型；S23: Based on the Rayleigh damping hypothesis, the total damping matrix C is calculated from the total mass matrix M of the beam and the total stiffness matrix K, and a dynamic model of the beam structure is established;
    S24：求解梁的计算频响函数矩阵H A(ω)： S24: Solving the calculated frequency response function matrix H A (ω) of the beam:
    H A(ω)＝(-ω 2M+iωC+K) -1   (2)。 H A (ω)=(−ω 2 M+iωC+K) -1 (2).
  4. 根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S4中所述的基于优化算法的材料参数模型中正交多项式系数识别的具体步骤包括：The method for indirectly obtaining a mechanical parameter field of a non-uniform material continuous distribution according to claim 1, wherein the specific steps of identifying the orthogonal polynomial coefficients in the material parameter model based on the optimization algorithm in step S4 include:
    S41：以基于模态试验的实测频响函数H B(ω)与基于有限元模型的计算频响函数H A(ω)之差最小值为优化目标，构造如下优化问题： S41: the test based on the measured modal frequency response function H B (ω) based on the difference H A (ω) of the finite element model optimization target minimum frequency response function, the optimization problem is constructed as follows:
    MinJ(b)＝||H B(ω)-H A(ω,b)||   (3)， MinJ(b)=||H B (ω)-H A (ω,b)|| (3),
    其中b为待估计的正交多项式系数向量，b＝[b 1 … b n]，||□||表示矩阵范数； Where b is the orthogonal polynomial coefficient vector to be estimated, b=[b 1 ... b n ], ||□|| represents the matrix norm;
    S42：基于灵敏度分析方法迭代求解优化问题，即第j个迭代步求解下式：S42: Iteratively solves the optimization problem based on the sensitivity analysis method, that is, the jth iteration step solves the following formula:
    H B(ω)-H j A(ω,y j)＝S j(b j+1-b j)   (4)， H B (ω)-H j A (ω, y j )=S j (b j+1 -b j ) (4),
    其中S j为计算频响函数对待估计正交多项式系数向量的灵敏度矩阵，即： Where S j is the sensitivity matrix for calculating the frequency response function to estimate the orthogonal polynomial coefficient vector, namely:
    Figure PCTCN2018083274-appb-100002
    Figure PCTCN2018083274-appb-100002
    当
    Figure PCTCN2018083274-appb-100003
    时(ε为某个小数，如10 -3)，迭代收敛得到正交多项式系数向量b。     when
    Figure PCTCN2018083274-appb-100003
    When ε is a fractional number, such as 10 -3 , the iterative convergence yields an orthogonal polynomial coefficient vector b.
  5. 根据权利要求1所述的非均匀材料连续分布力学参数场间接获取方法，其特征在于，步骤S5中所述的非均匀材料力学参数分布场重构的具体步骤包括：由收敛得到正交多项式系数向量b和式(1)重构获取非均匀材料力学参数分布场。The method for indirectly obtaining a mechanical field of continuous distribution of non-uniform material according to claim 1, wherein the specific step of reconstructing the field of the mechanical parameter of the non-uniform material in step S5 comprises: obtaining an orthogonal polynomial coefficient by convergence Vector b and equation (1) reconstruct to obtain the distribution field of mechanical parameters of inhomogeneous materials.
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