CN106528952A - Pseudorandom vector-based assumed mode set construction method - Google Patents

Abstract

The invention discloses a pseudorandom vector-based assumed mode set construction method. The method comprises the following steps of constructing a dynamics equation of an N-freedom system by adopting a finite unit method; performing eigenvalue analysis on the system to obtain a plurality of order modes of the system; generating a group of pseudorandom vectors, wherein the number of the pseudorandom vectors is determined according to the number of the obtained modes; constructing an assumed mode set by adopting the obtained system modes and the pseudorandom vector group; and forming a new mode set by the obtained assumed mode set and the system modes to construct a residual flexibility matrix of the system. According to the method, the assumed mode set can be constructed according to any known mode vector group, and the process for calculating the residual flexibility matrix by a sub-structure mode synthesis technology is greatly simplified.

Description

Pseudo-random vector-based assumed modal set construction method

Technical Field

The invention belongs to the technical field of structural dynamics analysis, and particularly relates to a pseudo-random vector-based construction method for an assumed modal set.

Background

With the development of science and technology, more and more large-scale complex structures need to be faced in the engineering structure design process. When dynamic analysis is performed on such large and complex structures, the problem that the finite element model is too large and the calculation cost is too high is often faced, so that the design period of the whole structure is invisibly prolonged.

Currently, a substructure mode synthesis method is an effective technical means for reducing the scale of a model. The free interface substructure modal synthesis method can be more conveniently butted with experimental data, so that the method is widely applied to engineering structure design. In the process of reducing the model, the method mainly adopts a mode truncation method, and only retains low-order modes which obviously influence the structure dynamics characteristics so as to achieve the purpose of reducing the model. However, the method of directly eliminating the free interface of the high-order mode is often low in calculation precision and cannot meet the structural design requirement at the present stage. Therefore, the method becomes a main idea for improving the calculation accuracy of the free interface substructure modal synthesis method by considering the residual flexibility, namely, the influence of the reserved high-order mode on the structural dynamic characteristics.

The main process of calculating the residual compliance matrix by the existing free interface modal synthesis method is shown in fig. 1, and mainly has the following two problems:

1) generally, there is no unified calculation mode to process the two cases of the non-rigid-body mode and the rigid-body mode separately.

2) When solving the problem of the remaining flexibility matrix of the substructure with the rigid body mode, it is usually necessary to assume that a group of constraints are applied to the substructure to obtain an additional statically determinate constraint matrix, convert the original motion equation into a motion equation with constrained displacement through the matrix, and then solve the remaining flexibility matrix. This approach is cumbersome to implement.

Disclosure of Invention

The invention aims to provide a novel construction method of an assumed modal set based on a pseudo-random vector, which can construct the assumed modal set according to any known modal vector group and simplify the process of calculating a residual flexibility matrix by adopting a substructure modal synthesis technology.

The technical solution for realizing the purpose of the invention is as follows: a pseudo-random vector-based construction method for an assumed modal set comprises the following steps:

step 1, constructing an N-degree-of-freedom system kinetic equation by adopting a finite element method;

step 2, analyzing the characteristic value of the system in the step 1 to obtain a plurality of orders of modes of the system;

step 3, generating a group of pseudo-random vectors, wherein the number of the pseudo-random vectors is determined according to the mode number obtained in the step 2;

step 4, constructing a hypothesis mode set by adopting the system mode obtained in the step 2 and the pseudo-random vector group obtained in the step 3;

and 5, forming a new mode set by the assumed mode set obtained in the step 4 and the system mode obtained in the step 2, and constructing a system residual flexibility matrix.

Compared with the prior art, the invention has the following remarkable advantages: (1) a set of hypothetical modalities can be constructed from any set of known modality vectors; (2) the two situations of the structure non-rigid body mode and the structure rigid body mode are not required to be processed respectively, a unified calculation mode is provided, and the process of calculating the residual flexibility matrix by the substructure mode comprehensive technology is greatly simplified.

The present invention is described in further detail below with reference to the attached drawing figures.

Drawings

FIG. 1 is a flow chart of a conventional method for calculating a residual compliance matrix by a free interface modal synthesis method.

FIG. 2 is a flow chart of a pseudo-random vector-based hypothetical modality set construction method of the present invention.

Detailed Description

The method is based on the pseudo-random vector set to construct a vector set which is orthogonal to all known modal vector weights, and can be used for constructing equivalent high-order modes, constructing residual flexibility matrixes and the like, and the problems of high-frequency component loss and insufficient analysis precision caused by modal truncation are solved.

With reference to fig. 2, the method for constructing a pseudo-random vector-based hypothetical modality set according to the present invention includes the following steps:

step 1, constructing an N-degree-of-freedom system kinetic equation by adopting a finite element method, which comprises the following specific steps:

wherein M is a system mass matrix, K is a system stiffness matrix, u is a generalized displacement vector, and f is a generalized load vector.

Step 2, analyzing the characteristic value of the system in the step 1 to obtain a plurality of orders of modes of the system;

the eigenvalue analysis is performed on the formula (1) to obtain a plurality of orders of modal vectors, wherein the vectors formed by the plurality of orders of modal are recorded as

Step 3, generating a group of pseudo-random vectors, wherein the number of the pseudo-random vectors is determined according to the modal number obtained in the step 2, and the method specifically comprises the following steps:

given a set of pseudorandom vectors x, x ═ x₁x₂… x_n]And the number N of the pseudorandom vectors is the order of the mode obtained by subtracting the step 2 from the total degree of freedom N of the system.

And 4, constructing a hypothetical mode set by using the system mode obtained in the step 2 and the pseudo-random vector group obtained in the step 3, wherein the hypothetical mode set comprises the following specific steps:

suppose that

Wherein,andrespectively refers to a vector group composed of several orders of modesThe j-th column and the r-th column of (1),for the vector to be constructed, a_jsTo find the coefficient, a is found according to the formula (2) and the formula (3)_js：

A to be obtained_jsSubstituting formula (2) with:

as above, a set of vectors weighted orthogonal to the set of known vectors is constructed by the set of pseudo-random vectors, i.e. the set of hypothetical modes to be constructed.

And 5, forming a new mode set by the assumed mode set obtained in the step 4 and the system mode obtained in the step 2, and constructing a system residual flexibility matrix, wherein the method specifically comprises the following steps:

the constructed vector groupAnd set of known vectorsCombining into a new vector group phi, wherein s is 1,2, …, n, j is 1,2, …, m, phi is marked as

And (3) converting a substructure kinetic equation from a physical coordinate system to a modal coordinate system through phi aiming at the N-degree-of-freedom substructure model, and solving a corresponding residual flexibility matrix.

The residual flexibility matrix of the substructure obtained by the method can meet the problem of substructure synthesis under any boundary conditions, and the problems of the conventional substructure synthesis technology are well solved.

Example 1

With reference to the flow shown in fig. 2, taking the construction of the equivalent high-order mode as an example, the specific implementation steps are as follows:

step 1: for a certain N-degree-of-freedom dynamic system, the motion equation can be expressed as

Wherein M is a system mass matrix, K is a system stiffness matrix, u is a generalized displacement vector, and f is a generalized load vector.

Step 2: through characteristic value analysis, the first-order low-order mode of the structure is obtainedIs marked as

And step 3: a set of pseudo-random vectors is given, and x is recorded as x ═ x₁x₂... x_N-l]。

And 4, step 4: assuming any first order equivalent higher order mode asAnd satisfy

Wherein,andrespectively referring to the low-order modal vector set obtained in step 2J-th and r-th columns of (a)_jsThe coefficients are to be found.

By substituting formula (8) for formula (9)

Can be obtained by finishing

As can be seen from the weighted orthogonality of the modes with respect to the quality matrix, if and only if j r,then there are

Can be obtained from the equation (11)

Thus, the undetermined coefficient a is obtained_rsr＝1,2,…,l；s＝1,2,…,n-l

A to be obtained_rsThe substitution formula (8) is

Equation (14) can also be written as

This results in the desired assumed higher order mode. As can be seen from the above process, this set of assumed higher-order modes is orthogonal to the known modal weights.

And 5: combining the assumed high-order mode obtained in the step 4 with the vector group obtained by the eigenvalue analysis to form a new vector group phi

Let u be phi p, expanded to obtain

Wherein p is a modal coordinate, formula (17) is substituted for formula (7), and phi is used^TBy left-multiplying the obtained equation, the obtained equation can be obtained

The equation (18) is expanded with a second set of equations

Wherein,

laplace transformation is performed on the formula (19) to obtain

Taylor expansion is carried out on the above formula, only the first item is reserved, and then inverse Laplace transformation is carried out to obtain the final product

Thus, the

Order toI.e. the remaining compliance matrix.

The physical coordinates of the kinetic equation can be converted into modal coordinates through the formula, and meanwhile, the degree of freedom is reduced. And when the residual flexibility matrix is solved, whether the structure contains rigid body modes or not does not need to be considered, so that the model reduction process is greatly simplified, and the problem of the existing substructure mode synthesis technology in calculating the residual flexibility matrix is solved.

Claims (6)

1. A pseudo-random vector-based construction method for an assumed modal set is characterized by comprising the following steps:

step 1, constructing an N-degree-of-freedom system kinetic equation by adopting a finite element method;

step 2, analyzing the characteristic value of the system in the step 1 to obtain a plurality of orders of modes of the system;

step 3, generating a group of pseudo-random vectors, wherein the number of the pseudo-random vectors is determined according to the mode number obtained in the step 2;

step 4, constructing a hypothesis mode set by adopting the system mode obtained in the step 2 and the pseudo-random vector group obtained in the step 3;

and 5, forming a new mode set by the assumed mode set obtained in the step 4 and the system mode obtained in the step 2, and constructing a system residual flexibility matrix.

2. The pseudo-random vector-based hypothetical modality set construction method according to claim 1, wherein the N-degree-of-freedom system kinetic equation is constructed by using a finite element method in step 1, and specifically comprises the following steps:

$M\stackrel{\·\·}{u}+Ku=f---\left(1\right)$

wherein M is a system mass matrix, K is a system stiffness matrix, u is a generalized displacement vector, and f is a generalized load vector.

3. The pseudo-random vector-based hypothesis mode set construction method according to claim 1, wherein step 2 is to perform eigenvalue analysis on the system in step 1 to obtain a plurality of orders of modes of the system, wherein the plurality of orders of modes form a vector group

4. The pseudo-random vector-based assumed modality set construction method according to claim 1, wherein the step 3 generates a set of pseudo-random vectors, and the number of the pseudo-random vectors is determined according to the modality number obtained in the step 2, specifically as follows:

given a set of pseudorandom vectors x, x ═ x₁x₂… x_n]Wherein the number n of the pseudo-random vectors is the total degree of freedom of the systemThe number N minus the order of the modality obtained in step 2.

5. The pseudo-random vector-based assumed modality set construction method according to claim 1, wherein the assumed modality set is constructed in step 4 by using the system modality obtained in step 2 and the pseudo-random vector group obtained in step 3, and specifically as follows:

suppose that

Wherein,andrespectively refers to a vector group composed of several orders of modesThe j-th column and the r-th column of (1),for the vector to be constructed, a_jsTo find the coefficient, a is found according to the formula (2) and the formula (3)_js

A to be obtained_jsSubstituting formula (2) with:

as above, a set of vectors weighted orthogonal to the set of known vectors is constructed by the set of pseudo-random vectors, i.e. the set of hypothetical modes to be constructed.

6. The pseudo-random vector-based assumed modality set construction method according to claim 1, wherein step 5 is to combine the assumed modality set obtained in step 4 and the system modality obtained in step 2 into a new modality set for constructing a system residual compliance matrix, and specifically includes the following steps:

the constructed vector groupAnd set of known vectorsCombining into a new vector group phi, wherein s is 1,2, …, n, j is 1,2, …, m, phi is marked as

And (3) converting a substructure kinetic equation from a physical coordinate system to a modal coordinate system through phi aiming at the N-degree-of-freedom substructure model, and solving a corresponding residual flexibility matrix.