CN111414714B - Thin-wall section characteristic deformation identification method - Google Patents
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Abstract
The invention discloses a thin-wall section characteristic deformation identification method, which comprises the following steps: step 1) constructing a thin-wall structure high-order model considering section deformation, and deducing a control differential equation of the thin-wall structure by adopting a Hamilton principle; step 2) solving generalized characteristic values and corresponding characteristic vectors of a control differential equation by using a finite element method; step 3) performing reduced order approximation processing on the characteristic matrix to identify an axial variation mode of the basis function; and 4) orthogonally decomposing the characteristic vector into components which are collinear with the axial change mode of the basis function to obtain the proportional relation between the components and the axial change mode, and multiplying the original deformation mode by the corresponding proportional coefficient to generate a new deformation mode. The method can realize numerical value in a simple and visual mode, and the derived deformation mode has clear hierarchy and physical interpretability, thereby being beneficial to truly reflecting the dynamic behavior of the thin-wall structure, greatly improving the calculation efficiency and reducing the calculation cost.
Description
Technical Field
The invention relates to a thin-wall section characteristic deformation identification method, and belongs to the field of thin-wall structure dynamics analysis.
Background
The thin-wall structure is a structure consisting of thin plates, thin shells and slender rod pieces, can bear larger load by using smaller weight and less materials, and is widely applied to various engineering machines. In the working process, the thin-wall structure is easy to stretch, bend and deform in cross section due to external force, the working performance of engineering equipment is directly influenced, even the life and property safety of workers is threatened, and the research on the deformation of the thin-wall structure cross section has important practical significance. At present, a method for establishing a one-dimensional model is commonly adopted for analyzing a thin-wall structure, but the one-dimensional model considering section deformation still has some defects in the aspect of mode identification: too many deformation modes are adopted, so that the model is more complex and the calculation cost is high; deformation modes are difficult to adapt to various geometric and boundary conditions.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a thin-wall section characteristic deformation identification method which can be numerically realized in a simple and visual mode, can provide a physical explanation of a deformation mode generated by the method, considers the structure geometry, material parameters and boundary conditions when determining the deformation mode, and is beneficial to truly reflecting the dynamic behavior of a thin-wall structure.
The invention mainly adopts the technical scheme that:
a thin-wall section characteristic deformation identification method comprises the following steps:
step 1: constructing a high-order model of the thin-wall structure considering section deformation, approximating the section deformation of the thin-wall structure through a group of basis functions, defining a three-dimensional displacement field of the thin-wall structure, obtaining a strain field and a stress field under the assumption of small displacement, and deducing a control differential equation of the thin-wall structure by adopting a Hamilton principle;
and 2, step: solving generalized characteristic values and corresponding characteristic vectors of a control differential equation by using a finite element method, wherein in each characteristic vector, an axial change mode of an amplitude function corresponds to a structural behavior;
and step 3: decomposing the feature vector by adopting a principal component analysis method, and performing reduced order approximation processing on the feature matrix to identify an axial variation mode of the basis function;
and 4, step 4: and orthogonally decomposing the characteristic vector into components which are collinear with the axial change mode of the basis function to obtain the proportional relation between the components and the axial change mode, and multiplying the original deformation mode by the corresponding proportional coefficient to generate a new deformation mode.
Preferably, in the step 1, the deformation of the thin-wall structure section includes out-of-plane axial deformation and in-plane tangential, normal and rotational deformation; the basis functions are derived from node displacement interpolation applied on a thin-wall cross section; the thin-wall structure has four degrees of freedom, including three translation and one rotation; the displacement field is obtained according to Kirchhoff sheet assumption and thin-wall bending characteristics, and comprises displacement components in the axial direction, the tangential direction and the normal direction, wherein the axial direction and the tangential direction displacement components are interpolated by adopting a linear Lagrange function, and the normal direction displacement components are interpolated by adopting a cubic Hermite function; the governing differential equation is established according to the Hamiltonian principle.
Preferably, in step 2, the governing differential equation is a typical second-order differential equation, the governing differential equation is solved by a finite element dispersion method, and the set of the obtained feature vectors is shown in formula (1):
Φ＝[χ (1) χ (2) … χ (k) … χ (N) ] (1)；
wherein, χ (k) Is a k-order feature vector, each feature vector is formed by combining amplitude functions, as shown in formula (2):
χ (k) ＝[D 1 D 2 … D N ] (2)。
preferably, in the step (3), the feature matrix may be decomposed into two sub-matrices, corresponding to the out-of-plane deformation mode and the in-plane deformation mode, respectively, as shown in equation (3):
wherein the content of the first and second substances,
respectively corresponding to the number of out-of-plane and in-plane deformation modes;
decomposing the feature vector by adopting a principal component analysis method, and specifically expressing the feature vector as shown in a formula (4):
χ (k) ＝U (k) ∑ (k) V (k) (4)；
wherein, the correlation matrix U (k) 、∑ (k) Respectively, as follows:
in the formulae (5) and (6),
basis vector, σ, representing axial deformation mode 1 (k) The representation matrix sigma 1 (k) On the main diagonalElements, i.e. singular values, and σ 1 (k) To σ r (k) The characteristic vectors are arranged from large to small, and r represents the rank of the characteristic vector;
the number of deformation patterns generated by the combination depends on the largest integer p that satisfies the following relationship:
in the formula (7), μ represents a contribution value of the basis vector, μ 0 For the threshold, r represents the rank of the feature vector.
Preferably, in the step (4), the eigenvectors corresponding to the out-of-plane and in-plane deformation modes are orthogonally decomposed into components collinear with the corresponding basis vectors, and the proportionality coefficients λ of the eigenvectors and the components collinear with the corresponding basis vectors are obtained i,j (i is not less than 1 and not more than N, j is not less than 1 and not more than p), which is specifically expressed as:
combining the scaling coefficients into a weight matrix lambda (k) Specifically, the following is shown in (9):
λ (k) ＝[λ 1 (k) λ 2 (k) … λ p (k) ] (9)。
has the beneficial effects that: the invention provides a thin-wall section characteristic deformation identification method, which has the following advantages compared with the prior art:
(1) the section deformation mode identification method provided by the invention can realize numerical value in a simple and visual mode, and the derived deformation mode has clear hierarchy and physical interpretability;
(2) the deformation mode determined by the invention considers the influence of factors such as structure geometry, material parameters, boundary conditions and the like, and is beneficial to truly reflecting the dynamic behavior of the thin-wall structure;
(3) the invention utilizes the weight matrix to combine the basis functions to generate a new deformation mode, thereby greatly reducing the number of the deformation modes, greatly improving the calculation efficiency and reducing the calculation cost on the premise of reproducing the structural deformation as accurately as possible;
(4) the characteristic deformation identification method provided by the invention has generality and is suitable for thin-wall structures with any prism sections.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic view of a prism section thin-wall structure;
FIG. 3 is a thin-wall structure cross-sectional node diagram;
FIG. 4 is a schematic diagram of one of three out-of-plane deformation modes identified by first-order eigenvectors;
FIG. 5 is a schematic diagram of one of three out-of-plane deformation modes identified by first-order eigenvectors;
FIG. 6 is a schematic diagram of one of three out-of-plane deformation modes identified by first-order eigenvectors.
Detailed Description
In order to make those skilled in the art better understand the technical solutions in the present application, the technical solutions in the embodiments of the present application are clearly and completely described below, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
A thin-wall section characteristic deformation identification method comprises the following steps:
step 1: constructing a thin-wall structure high-order model considering section deformation, approximating the section deformation of the thin-wall structure through a group of basis functions, defining a three-dimensional displacement field of the thin-wall structure, obtaining a strain field and a stress field under the assumption of small displacement, and deducing a control differential equation of the thin-wall structure by adopting a Hamilton principle;
step 2: solving generalized characteristic values and corresponding characteristic vectors of a control differential equation by using a finite element method, wherein in each characteristic vector, an axial change mode of an amplitude function corresponds to a structural behavior;
and 3, step 3: decomposing the feature vector by adopting a principal component analysis method, and performing reduced order approximation processing on the feature matrix to identify an axial variation mode of the basis function;
and 4, step 4: and orthogonally decomposing the characteristic vector into components collinear with the axial change mode of the basis function to obtain the proportional relation between the components and the axial change mode, and multiplying the original deformation mode by the corresponding proportional coefficient to generate a new deformation mode.
Preferably, in the step 1, the deformation of the thin-wall structure section includes out-of-plane axial deformation and in-plane tangential, normal and rotational deformation; the basis functions are derived from node displacement interpolation applied to the thin-wall cross section; the thin-wall structure has four degrees of freedom, including three translation and one rotation; the displacement field is obtained according to Kirchhoff sheet assumption and thin-wall bending characteristics, and comprises displacement components in the axial direction, the tangential direction and the normal direction, wherein the axial direction and the tangential direction displacement components are interpolated by adopting a linear Lagrange function, and the normal direction displacement components are interpolated by adopting a cubic Hermite function; the governing differential equation is established according to the Hamiltonian principle.
Preferably, in step 2, the governing differential equation is a typical second-order differential equation, the governing differential equation is solved by using a finite element dispersion method, and the set of the obtained feature vectors is shown in formula (1):
Φ＝[χ (1) χ (2) … χ (k) … χ (N) ] (1)；
wherein, χ (k) Is a k-order feature vector, each feature vector is formed by combining amplitude functions, as shown in formula (2):
χ (k) ＝[D 1 D 2 … D N ] (2)。
preferably, in the step (3), the feature matrix may be decomposed into two sub-matrices, corresponding to the out-of-plane deformation mode and the in-plane deformation mode, respectively, as shown in equation (3):
wherein the content of the first and second substances,
respectively corresponding to the number of out-of-plane and in-plane deformation modes;
decomposing the feature vector by adopting a principal component analysis method, and specifically expressing the feature vector as shown in a formula (4):
χ (k) ＝U (k) ∑ (k) V (k) (4)；(V (k) is a square matrix generated by the decomposition of the eigenvector, and only two other matrixes are used in the application, so the detailed description is omitted
Wherein, the correlation matrix U (k) 、∑ 1 (k) Respectively, as follows:
in the formulae (5) and (6),
basis vector, σ, representing axial deformation mode 1 (k) The representation matrix sigma 1 (k) Elements on the principal diagonal, i.e. singular values, and a 1 (k) To sigma r (k) The sequences are arranged from large to small, and r is the rank of the feature vector;
the number of deformation patterns generated by the combination depends on the largest integer p that satisfies the following relationship:
in the formula (7), μ represents a contribution value of the basis vector, μ 0 For the threshold, r represents the rank of the feature vector.
Preferably, in the step (4), the step (4) is to be respectively carried outOrthogonal decomposition of the eigenvectors corresponding to the out-of-plane and in-plane deformation modes into components collinear with the corresponding basis vectors, to obtain their proportionality coefficients λ i,j (i is more than or equal to 1 and less than or equal to N, j is more than or equal to 1 and less than or equal to p), which is specifically expressed as follows:
combining the scaling coefficients into a weight matrix lambda (k) Specifically, the following means (9):
λ (k) ＝[λ 1 (k) λ 2 (k) … λ p (k) ] (9)。
example 1:
the first-order characteristic vector out-of-plane deformation mode of the prism section thin-wall structure shown in fig. 2 is taken as an identification object, and the specific implementation steps are as follows:
step 1, a thin-wall section node diagram is shown in a graph 3, the deformation of a thin-wall section is captured by using 6 primary nodes and 4 secondary nodes, the out-of-plane deformation corresponds to the axial deformation of the section, 10 original out-of-plane deformation modes are adopted, a high-order model of the thin-wall structure considering the section deformation is established, a basis function is defined by applying node displacement interpolation on the thin-wall section, the section deformation of the thin-wall structure is captured by the basis function in an accurate manner as far as possible, the bending characteristic of the thin wall is considered, a three-dimensional displacement field is constructed according to the Kirchoff sheet assumption, a strain field and a stress field are obtained under the small displacement assumption, and a dynamic control differential equation of the thin-wall structure is deduced by combining the Hamilton principle.
Step 2, solving a control differential equation of the thin-wall structure by using a finite element dispersion method, solving a quality matrix and a rigidity matrix through a difference format and numerical integration, further solving a generalized characteristic value and a corresponding characteristic vector of the control differential equation, wherein in each characteristic vector, an axial change mode of an amplitude function corresponds to a structure behavior,
the first order feature vector is represented as: phi ═ x [, ] (1) ]And the characteristic vector χ (1) Is formed by combining amplitude functions, specifically expressed as：χ (1) ＝[χ 1 χ 2 … χ N ]。
And 3, decomposing a first-order eigenvector by adopting a principal component analysis method because the eigenvector is not a square matrix, and performing reduced order approximation processing on the eigenvector to identify the axial variation mode of the basis function, wherein the first-order eigenvector is decomposed as follows:
χ (1) ＝U (1) ∑ (1) V (1) ；
∑ 1 (1) ＝diag(σ 1 (1) σ 2 (1) … σ r (1) )；
in the above formula, the first and second carbon atoms are,
the basis vectors representing the axial deformation patterns, not all the variation patterns are useful for pattern recognition, and only valid variation patterns are meaningful for the recognition of the deformation patterns. The number of deformation patterns generated by the combination depends on the largest integer p that satisfies the following relationship:
μ denotes the contribution of the basis vector, μ 0 When the threshold value is set to be 0.000001, r represents the rank of the feature vector, and the effective rank p is determined to be 3, 3 out-of-plane deformation modes are identified by the first-order feature vector.
Step 4, taking out the submatrix of the characteristic matrix corresponding to the out-of-plane deformation mode, orthogonally decomposing the characteristic vector into components collinear with the identified change mode to obtain the proportional relation of the components and the proportional coefficient lambda i,j (1. ltoreq. i.ltoreq.N, 1. ltoreq. j.ltoreq.p) is expressed as:
the scaling factors are used to combine the basis functions of the deformation modes, but not all weight vectors are valid, since some deformation modes may recur in different modes. To ensure independence, a newly identified weight vector λ j (1) Should preserve lambda (1) Otherwise, new weight vectors should be deleted from the final weight matrix, and the scaling factors are combined into a weight matrix λ (1) Specifically, the following are shown:
weighting matrix lambda (1) The 3 line vectors are respectively multiplied by the original 10 out-of-plane deformation modes to generate 3 new out-of-plane deformation modes in a combined mode, as shown in figures 4-6, the identified section deformation mode has structural explanatory property, the classic bending of the thin-wall structure is reflected, the dynamic behavior of the thin-wall structure is truly reflected, the number of the deformation modes is reduced, and the calculation efficiency of the model is improved.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and amendments can be made without departing from the principle of the present invention, and these modifications and amendments should also be considered as the protection scope of the present invention.
Claims (3)
1. A thin-wall section characteristic deformation identification method is characterized by comprising the following steps:
step 1: constructing a thin-wall structure high-order model considering section deformation, approximating the section deformation of a thin-wall structure through a group of basis functions, defining a three-dimensional displacement field of the thin-wall structure, obtaining a strain field and a stress field under the assumption of small displacement, and deducing a control differential equation of the thin-wall structure by adopting a Hamilton principle, wherein the section deformation of the thin-wall structure comprises out-of-plane axial deformation and in-plane tangential, normal and rotational deformation; the basis functions are derived from node displacement interpolation applied to the thin-wall cross section; the thin-wall structure has four degrees of freedom, including three translation and one rotation; the displacement field is obtained according to Kirchhoff thin plate hypothesis and thin wall bending characteristics, and comprises displacement components in the axial direction, the tangential direction and the normal direction, wherein the axial displacement component and the tangential displacement component are interpolated by adopting a linear Lagrange function, and the normal displacement component is interpolated by adopting a cubic Hermite function; the control differential equation is established according to the Hamilton principle;
step 2: solving generalized characteristic values and corresponding characteristic vectors of a control differential equation by using a finite element method, wherein in each characteristic vector, an axial change mode of an amplitude function corresponds to a structural behavior;
and 3, step 3: decomposing the feature vector by adopting a principal component analysis method, and performing order-reduction approximate processing on the feature matrix to identify an axial change mode of the basis function, wherein the feature matrix is decomposed into two sub-matrices which respectively correspond to an out-of-plane deformation mode and an in-plane deformation mode, and the expression (3) is as follows:
wherein the content of the first and second substances,
respectively corresponding to the number of out-of-plane and in-plane deformation modes;
decomposing the feature vector by adopting a principal component analysis method, and specifically expressing the feature vector as shown in a formula (4):
χ (k) ＝U (k) ∑ (k) V (k) (4)；
wherein, the correlation matrix U (k) 、∑ (k) Respectively, as follows:
in the formulae (5) and (6),
basis vector, σ, representing axial deformation mode 1 (k) The representation matrix sigma 1 (k) Elements on the main diagonal, i.e. singular values, and σ 1 (k) To σ r (k) The sequences are arranged from large to small, and r represents the rank of the feature vector;
the number of deformation patterns generated by the combination depends on the largest integer p that satisfies the following relationship:
in the formula (7), μ represents a contribution value of the basis vector, μ 0 R represents the rank of the feature vector as a threshold;
and 4, step 4: and orthogonally decomposing the characteristic vector into components which are collinear with the axial change mode of the basis function to obtain the proportional relation between the components and the axial change mode, and multiplying the original deformation mode by the corresponding proportional coefficient to generate a new deformation mode.
2. The method for identifying the characteristic deformation of the thin-wall section according to claim 1, wherein in the step 2, the governing differential equation is a typical second-order differential equation, the governing differential equation is solved by adopting a finite element dispersion method, and the set of the obtained characteristic vectors is shown as a formula (1):
Φ＝[χ (1) χ (2) …χ (k) …χ (N) ] (1)；
wherein, χ (k) Is a k-order feature vector, each feature vector is formed by combining amplitude functions, as shown in formula (2):
χ (k) ＝[D 1 D 2 …D N ] (2)。
3. the method for identifying the characteristic deformation of the thin-wall section according to claim 1, wherein in the step 4, the characteristic vectors corresponding to the out-of-plane and in-plane deformation modes are orthogonally decomposed into components which are collinear with the corresponding basis vectors respectively, and the proportionality coefficients λ of the components are obtained i,j Wherein i is more than or equal to 1 and less than or equal to N, j is more than or equal to 1 and less than or equal to p, and the concrete expression is as follows:
combining the scaling coefficients into a weight matrix lambda (k) Specifically, the following is shown in (9):
λ (k) ＝[λ 1 (k) λ 2 (k) …λ p (k) ] (9)。
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