CN113779837A - Structural modal sensitivity analysis method based on novel actual measurement standardization technology - Google Patents
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Abstract
The invention discloses a structural modal sensitivity analysis method based on a novel actual measurement standardization technology, which comprises the following steps of: acquiring a mass and rigidity matrix of a structural system, and calculating the natural frequency and the initial output real mode of the structural system; adjusting the initial output real mode to meet the normalized condition to obtain a normalized real mode; selecting a certain design parameter L, and calculating the sensitivity of the structural system property matrix relative to the design parameter L; first-order sensitivity and second-order sensitivity of the normalized real mode with respect to the design parameter L are calculated, respectively. The method provided by the invention only needs to calculate the order of the mode to be considered for derivation, and is independent of other modes; the frequency and the modal sensitivity can be calculated simultaneously; the problem that diversified actual measurement standardization technologies and sensitivity analysis methods in the fields of model correction, damage identification, structure design, optimization control and the like do not have consistency can be effectively solved.
Description
Technical Field
The invention relates to the technical field of modal standardization, in particular to a structural modal sensitivity analysis method based on a novel actual measurement standardization technology.
Background
The sensitivity of the natural frequency and mode shape of the structure is an important aspect of many engineering applications. The method is widely applied to the aspects of finite element model correction, damage detection, structure optimization, dynamic sensitivity analysis and the like.
Over the past thirty years, many studies have been conducted to correct finite element models using vibration measurements. The existing model correction methods can be roughly divided into two categories, namely a direct method and an iterative method. Direct methods directly modify the elements of the stiffness and mass matrices, which allow the modified analytical model to reproduce the measured vibration mode data, but do not guarantee that it truly represents the physical characteristics of the actual structure. Iterative parameter modification methods, on the other hand, use sensitivity of parameters to modify the finite element analytical model, which take the error between the analyzed data and the measured data as an objective function and attempt to minimize the selected objective function by adjusting a preselected set of physical parameters of the finite element model. Iterative methods such as sensitivity-based parameter modification are more popular than direct methods because they can be implemented in existing finite element codes. Furthermore, there is a ready physical interpretation available for each structural correction parameter, which is related to the unit stiffness and mass of the analytical model. For sensitivity-based correction methods, the performance depends largely on the choice of objective functions and constraints, structural correction parameters, and optimization techniques. The objective function is typically used as a residual between the analytically predicted modal characteristics and the measurement (e.g., natural frequency or modal shape). The selection of structure modification parameters requires considerable physical insight into the actual test structure in order to correctly describe the differences between the analytical model and the test structure.
There are studies that show that the stiffness related parameters contribute relatively much to the target natural frequency, i.e. the natural frequency of the finite element model is more sensitive to these parameters. The sensitivity of the natural frequency corresponding to the correction parameter is an important reference index. Some mass density correction parameters also have a significant effect on the target natural frequency. However, the uncertainty associated with mass density is relatively low, although the natural frequency is sensitive to mass density. Therefore, in many cases, the finite element model modification based on sensitivity is applied to determine the stiffness reduction factor from the actually measured characteristic frequency and the modal shape as an index for identifying the structural damage.
Although many innovative methods have been proposed in recent years, the conventional sensitivity-based method is still widely used for model modification and damage identification, and is compatible with related structure optimization and control technologies, so as to realize important applications in the comprehensive engineering technical field from structure design to in-service health monitoring and the like.
Disclosure of Invention
The invention aims to provide a structural modal sensitivity analysis method based on a novel actual measurement standardization technology, so that the problems in the prior art are solved.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a structural modal sensitivity analysis method based on a novel actual measurement standardization technology comprises the following steps:
s1, acquiring a mass and stiffness matrix of the structural system, and calculating the natural frequency and the initial output real mode of the structural system;
s2, adjusting the initial output real mode to meet the normalization condition to obtain a normalized real mode;
s3, selecting a certain design parameter L, and calculating the sensitivity of the structural system property matrix relative to the design parameter L;
and S4, respectively calculating the first-order sensitivity and the second-order sensitivity of the normalized real mode with respect to the design parameter L.
Preferably, step S1 specifically includes:
s11, obtaining the mass M and rigidity matrix K of the structural system, and calculating the matrix D ═ M-1K；
S12, calculating the characteristic value lambda of D by using Matlab softwareiI.e. the natural frequency of the structural system, and the corresponding feature vector, i.e. the initial output real mode of the structural system
And N is the degree of freedom of the system.
Preferably, in step S2, the real mode is output for the initial output
The process of obtaining the normalized real mode is specifically as follows:
s21, first calculate the normalization constant:
s22, outputting the initial real mode
The normalized real mode is obtained by dividing by the square root of the normalization constant:
preferably, in step S3, the design parameters include geometric parameters, physical parameters, substructure-based parameters, and general elements.
Preferably, in step S3, the formula for calculating the sensitivity of the system property matrix is as follows:
wherein the content of the first and second substances,
refers to the sensitivity of the matrix of properties of the system,
preferably, in step S4, the first-order sensitivity of the normalized real mode is calculated by the process including: by solving the following system of equations
Obtaining the first-order sensitivity of the ith-order normalized real mode to the design parameter L
And first order sensitivity of ith order natural frequency to design parameter L
The meaning of (1) is that the index i may be any number from 1 to N, specifically, if we need to require the sensitivity of the 1 st order modality, i is 1, and if we continue, i may be any number from 1 to N, because this "any" is the user's own requirement, and the user may only require the sensitivity of a certain order modality, or may require the sensitivity of a group of modalities, or may require the sensitivity of all modalities.
Preferably, in step S4, the second-order sensitivity of the normalized real mode is calculated, and the process includes: by solving the following system of equations
Wherein q and r are calculated as
Wherein
Calculated by the following formula
Obtaining the second-order sensitivity of the ith-order normalized real mode to the design parameter L
And second order sensitivity of ith order natural frequency to design parameter L
The invention has the beneficial effects that:
the invention provides a structural modal sensitivity analysis method based on a novel actual measurement standardization technology, and the method provided by the invention only needs to calculate the order of mode to be considered for derivation and has no relation with other modes; the frequency and the modal sensitivity can be calculated simultaneously; the problem that diversified actual measurement standardization technologies and sensitivity analysis methods in the fields of model correction, damage identification, structure design, optimization control and the like do not have consistency can be effectively solved; one of the main obstacles to obtaining a modified model with parameters of physical significance is that the structure of a-priori finite elements may be incorrect. The modified finite element model can be used for response prediction, damage identification and health monitoring of the structure.
Drawings
Fig. 1 is a schematic view of a structural model of a rotating beam provided in example 1.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings. It should be understood that the detailed description and specific examples, while indicating the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention.
Example 1
The embodiment provides a structural modal sensitivity analysis method based on a novel actual measurement normalization technology, which comprises the following steps:
s1, acquiring a mass and stiffness matrix of the structural system, and calculating the natural frequency and the initial output real mode of the structural system;
s2, adjusting the initial output real mode to meet the normalization condition to obtain a normalized real mode;
s3, selecting a certain design parameter L, and calculating the sensitivity of the structural system property matrix;
and S4, respectively calculating the first-order sensitivity and the second-order sensitivity of the normalized real mode.
In step S1, the method specifically includes:
s11, obtaining the mass M and the rigidity matrix K of the structural system, and calculating D ═ M-1K；
S12, calculating the characteristic value lambda of the real mode D by using MATLAB softwareiAnd corresponding initial output real mode
And N is the degree of freedom of the system.
In step S2, adjusting the initial output real mode specifically is adjusting the ith-order initial output mode to obtain a normalized real mode; for initial output real mode
The process of obtaining the normalized real mode specifically comprises the following steps:
s21, first calculate the normalization constant:
s22, outputting the initial real mode
The normalized real mode is obtained by dividing by the square root of the normalization constant:
in step S3, the design parameters include geometric parameters, physical parameters, substructure-based parameters, and general elements.
In step S3, the formula for calculating the sensitivity of the system property matrix is as follows:
wherein the content of the first and second substances,
refers to the sensitivity of the matrix of properties of the system,
in step S4, the first-order sensitivity of the normalized real mode is calculated, and the process includes: by solving the following system of equations
Obtaining the first-order sensitivity of the ith-order normalized real mode to the design parameter L
And first order sensitivity of ith order natural frequency to design parameter L
In step S4, calculating the second-order sensitivity of the normalized real mode, the process includes: by solving the following system of equations
Wherein q and r are calculated as
Wherein
Calculated by the following formula
Obtaining the second-order sensitivity of the ith-order normalized real mode to the design parameter L
And second order sensitivity of ith order natural frequency to design parameter L
By adopting the method, taking the rotating beam structure system in fig. 1 as a specific implementation example, calculation is performed, firstly, the structure system shown in fig. 1 as a high-speed rotating system belongs to an asymmetric system matrix, mass is concentrated at the central position of the beam, and system parameters are as follows: m is0＝10kg/m，M＝10kg，L＝10m， EIy＝9L3/5π2Nm2，EIx＝4L3/5π2Nm2，K1＝K2＝L2/20Nm，
c＝h＝1/4Nsm-1。
Taking the 1 st and 2 nd order normalized real modes as an example, the method specifically comprises the following steps:
1. extracting mass and rigidity matrix of the system, and calculating with software
D＝-M-1K
Calculating the eigenvalue λ of D1,λ2And feature vectors
The eigenvalue is the (undamped) natural frequency of the system, and the eigenvector is the original output real mode.
2. According to the specific steps in S2, taking the 1 st order mode and the 2 nd order mode as examples, the real mode is normalized:
for the order 1 mode, the normalization constant is first calculated:
the normalized real mode is obtained in the 1 st order mode by dividing the initial output real mode by the square root of the normalization constant:
for the order 2 mode, the normalization constant is first calculated:
the normalized real mode of the order 2 mode is obtained by dividing the initial output real mode by the square root of the normalization constant:
the rod length is selected as a design parameter, the rod length L in the embodiment is 10, and the sensitivity of the system property matrix is calculated:
for the 1 st order mode, software is used for calculating to obtain:
for the 2 nd order modality, software is used to calculate:
according to the sensitivity of the system property matrix obtained by the calculation, calculating the first-order sensitivity of the normalized real mode:
for the 1 st order mode, the following system of equations is solved:
a first-order sensitivity of the 1 st-order normalized real mode to the design parameter L is obtained
And first order sensitivity of the 1 st order natural frequency to the design parameter L
For the 2 nd order mode, the following system of equations is solved:
a first-order sensitivity of the 2 nd order normalized real mode to the design parameter L is obtained
And first order sensitivity of the 2 nd order natural frequency to the design parameter L
In this embodiment, only the 1 st order mode and the 2 nd order mode are adopted, and if the above process is repeated, the first-order sensitivity of all modes to the design parameter L can be obtained.
Calculating the second-order sensitivity of the normalized real mode:
for the 1 st order mode, the following system of equations is solved:
a second order sensitivity of the 1 st order mode to the design parameter L is obtained, wherein
Wherein
For a certain order mode, e.g. order 2, the following system of equations is solved:
a second order sensitivity of the 2 nd order mode to the design parameter L is obtained, wherein:
wherein
Similarly, if this process is repeated, the second order sensitivity of all modes to the design parameter L can be found. When the design parameters are such that the perturbation amounts Δ L/L are 0.001 and Δ L/L is 0.0001, the new 1 st and 2 nd order normalized real modes after the perturbation in column 6, and the second order taylor approximation (including the sensitivity calculated herein) in column 7, the error of this approximation is shown in the table below.
Table 1 the sensitivity calculation results for the first-order normalized real mode (L ═ 10).
Table 2 second-order normalized real mode sensitivity calculation results (L ═ 10).
From the above calculation results, it can be known that the method provided by the present invention can effectively solve the problem that the diversified actual measurement normalization technology and the sensitivity analysis method thereof do not have consistency in the fields of model correction, damage identification, structure design, optimization control, etc. One of the main obstacles to obtaining a modified model with parameters of physical significance is that the structure of the a priori finite elements may be incorrect. The modified finite element model can be used for response prediction, damage identification and health monitoring of the structure.
Model modification requires operators to use a more simple and convenient technique for normalizing modal shape (or called normalization technique) to solve the problem of non-uniform order of the measured length, but subsequent analysis work without a sensitivity algorithm corresponding to the technique will cause operation failures, such as identified change of model parameters cannot reflect real damage of the structure, sensitivity analysis of modal vectors cannot cause a component (or substructure) with significantly changed estimated modal parameters to be detected, weighting or selection scheme of the modal vectors cannot measure relative importance of the modal vectors to the damage, damage indexes cannot successfully detect the position of the damage or provide approximate estimation of corresponding damage degree, and the modified model is not a good simulation model of the real structure even if the modified model can reproduce measured data, thereby producing a large distorting effect on the subsequent predicted response.
The robust and efficient sensitivity value is determined by adopting a structural modal sensitivity analysis method based on a novel actual measurement standardization technology, the correct understanding of the sensitivity degree of target natural frequency or modal shape to structural parameters with similar influences can be ensured, the error of engineering operation can be reduced by using a sensitivity analysis method consistent with the actual measurement standardization technology, the effect distortion or the working failure of the engineering operation can be prevented, and even under high-noise corrosion, the damage index based on the calculated sensitivity value can provide good prediction capability for the position and the severity of structural damage.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that it is obvious to those skilled in the art that various modifications and improvements can be made without departing from the principle of the present invention, and the modifications and improvements should be considered within the protection scope of the present invention.
Claims (8)
1. A structural modal sensitivity analysis method based on a novel actual measurement standardization technology is characterized by comprising the following steps:
s1, acquiring a mass and stiffness matrix of the structural system, and calculating the natural frequency and the initial output real mode of the structural system;
s2, adjusting the initial output real mode to meet the normalization condition to obtain a normalized real mode;
s3, selecting a certain design parameter L, and calculating the sensitivity of the structural system property matrix relative to the design parameter L;
and S4, respectively calculating the first-order sensitivity and the second-order sensitivity of the normalized real mode with respect to the design parameter L.
2. The structural modal sensitivity analysis method based on the novel measured normalization technique according to claim 1, wherein the step S1 specifically includes:
s11, obtaining the mass M and rigidity matrix K of the structural system, and calculating the matrix D ═ M-1K；
S12, calculating the characteristic value lambda of D by using Matlab softwareiI.e. the natural frequency of the structural system, and the corresponding feature vector, i.e. the initial output real mode of the structural system
And N is the degree of freedom of the system.
3. The structural modal sensitivity analysis method according to claim 1, wherein in step S2, the normalized real modal is obtained by adjusting the ith-order initial output modality.
4. The structural modal sensitivity analysis method according to claim 3, wherein in step S2, the real mode is selected for the initial output
The process of obtaining the normalized real mode specifically comprises the following steps:
s21, first calculate the normalization constant:
s22, outputting the initial real mode
Dividing by the square root of the normalization constant to obtain the gaugeA normalized real mode:
5. the method for analyzing structural modal sensitivity based on the novel measured normalization technique according to claim 1, wherein in step S3, the design parameters comprise geometric parameters, physical parameters, substructure-based parameters and general elements.
6. The structural modal sensitivity analysis method based on the novel measured normalization technique according to claim 1, wherein in step S3, the formula for calculating the sensitivity of the system property matrix is as follows:
wherein the content of the first and second substances,
refers to the sensitivity of the matrix of properties of the system,
7. the structural modal sensitivity analysis method based on the novel measured normalization technique according to claim 1, wherein in step S4, the first-order sensitivity of the normalized real modal is calculated, and the process includes: by solving the following system of equations
Obtaining the first-order sensitivity of the ith-order normalized real mode to the design parameter L
And first order sensitivity of ith order natural frequency to design parameter L
8. The structural modal sensitivity analysis method based on the novel measured normalization technique according to claim 1, wherein in step S4, the second-order sensitivity of the normalized real modal is calculated, and the process includes: by solving the following system of equations:
wherein q and r are calculated as follows:
wherein
Calculated as follows:
obtaining the second-order sensitivity of the ith-order normalized real mode to the design parameter L
And second order sensitivity of ith order natural frequency to design parameter L
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| CN107357992A (en) * | 2017-07-13 | 2017-11-17 | 东南大学 | Composite structure correction method for finite element model based on cluster analysis |
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| MIAO ZHANG ET AL: "New Sensitivity Analysis Method for Complex Modal Rarameters in Asymmetrical Damped System", 《AIAA JOURNAL》, vol. 59, no. 8, pages 3165 - 3172 * |
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