CN115062500A - Structural vibration response analysis method under distributed random excitation - Google Patents

Structural vibration response analysis method under distributed random excitation Download PDF

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CN115062500A
CN115062500A CN202210404217.3A CN202210404217A CN115062500A CN 115062500 A CN115062500 A CN 115062500A CN 202210404217 A CN202210404217 A CN 202210404217A CN 115062500 A CN115062500 A CN 115062500A
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陈义春
何欢
钱瑜婷
何鹏远
王震宇
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a structural vibration response analysis method under distributed random pressure excitation, which relates to the field of random vibration and specifically comprises the following operation steps: firstly, establishing a dynamic finite element discrete model of an engineering structure; secondly, performing modal analysis on the structure; thirdly, considering and determining the number and the positions of the response points; discrete solution is carried out on the distributed random excitation, and the distributed random excitation is converted into multipoint random excitation on a finite element model; fifthly, solving the power spectral density of each concentrated excitation position; exchanging the positions of the excitation points and the response points, exciting the positions of the response points by using finite element software, and solving a transfer function matrix of each response point; and seventhly, solving the response of the response point by combining the harmonic response analysis result and the excitation power spectral density. The invention utilizes the reciprocity of the frequency response function to interchange the positions of the excitation point and the response point, thereby reducing the calculated amount while maintaining the calculation precision and reflecting the actual excitation situation more truly.

Description

Structural vibration response analysis method under distributed random excitation
Technical Field
The invention belongs to the field of multipoint random analysis, and relates to a method for analyzing structural vibration response under distributed random pressure excitation, which is used for converting distributed random excitation into multipoint random excitation and then solving the response of a structure under the random excitation.
Background
Random vibration is a statistically described vibration whose amplitude is not known exactly at any given moment; but its statistical properties of the vibration amplitude, such as the mean, standard deviation and probability of exceeding a certain value, are known exactly, which is usually described in the form of Power Spectral Density (PSD). The response characteristic of the structure is generally described by power spectral density and root mean square, and the methods for calculating the response of the structure include a classical complete quadratic form (CQC) method, a mean square and open root (SRSS) method after transforming the CQC method and a virtual excitation method (PEM).
For wind pressure fluctuation suffered by a spacecraft and a high-rise building, noise excitation caused by rocket and jet engine noise, and distributed pressure random excitation such as pulsating pressure suffered by the aircraft in the process of reentering the atmosphere, the traditional calculation method does not generally consider the transformation of the power spectral density on the space, and generally calculates the response of a structure by utilizing a means of dividing a calculation region when the power spectral density is required to change on the space.
The problems of the existing calculation method mainly occur in the following two aspects: firstly, the algorithm is limited, the CQC method is an accurate method for calculating the structural response, the coupling relation among all vibration modes is completely considered, although the result is accurate and reliable, for a large-scale structure, due to the fact that the freedom degree of a model is too much, a huge calculation amount is caused, and the engineering applicability is not good. The SRSS method ignores the cross terms of modes, but the simplification is only applicable to homogeneous material structures which participate in calculation and have sparse vibration modes and small damping ratios of all orders, however, in practical engineering, most of three-dimensional structures have dense frequencies and vibration modes of all orders, so that the calculation method is very limited. The virtual excitation method is to perform matrix decomposition on excitation power spectral density to obtain a plurality of virtual simple harmonic excitations, and then calculate a response power spectrum through responses under the virtual excitations, and the problem of the method appears in the following aspects, 1: when the matrix is not positively timed, the decomposition of the matrix is difficult, 2: when the excitation power spectrum matrix is large, the decomposed virtual simple harmonic excitation is also large, and the calculated amount is also large when the complex structure and the frequency point number are large, 3: simple harmonic excitation after matrix decomposition is also difficult in practical application. Secondly, for the conversion of the distributed random excitation, it can be known that in practical engineering, the distributed pressure random excitation is certainly not uniform, in other words, the power spectral densities at different positions are also different. In the engineering, a method of dividing a load area is adopted to improve the calculation precision, the finer the divided area is, the more reasonable the load application is, however, no matter how the load area is subdivided, the load working condition is still different from the actual result to a certain extent, and if the divided area is too fine, the problem of overlarge calculation amount is brought, which becomes a problem of difficulty in calculation precision and calculation amount.
Disclosure of Invention
The invention aims to provide a distributed random vibration analysis method, which solves the problems of low calculation precision and large calculation amount of the existing calculation method.
The invention is realized by the following steps:
a method for analyzing structural vibration response under distributed random excitation is characterized by comprising the following steps:
step one, establishing a dynamic finite element discrete model of an engineering structure;
secondly, performing modal analysis on the structure, namely performing modal analysis on the dispersed finite element model under the boundary condition by using finite element software, solving the inherent vibration mode of the discrete finite element model, and considering the position of a response point in the third step;
step three, determining the number and the positions of the response points;
step four, discrete solution is carried out on the distributed random excitation, and the distributed random excitation is converted into multipoint random excitation on a finite element model;
step five, solving the power spectral density of each concentrated excitation position, namely a converted excitation power spectral density matrix;
exchanging the positions of the excitation points and the response points, exciting the positions of the response points by using finite element software, and solving a frequency response function matrix of each response point;
and step seven, solving the response of the response point by combining the harmonic response analysis result and the excitation power spectral density.
Further, in the first step, the engineering structure is discretized into a finite element model, and the vibration equation of the finite element model for the discretized n degrees of freedom can be expressed as:
Figure BDA0003601159240000021
wherein M is an nxn dimensional quality matrix; c is an n multiplied by n dimension orthogonal damping matrix; k is an n x n dimensional stiffness matrix;
Figure BDA0003601159240000022
u is the acceleration, velocity and displacement column vector of the order n respectively; q is a positioning matrix; f equivalent node load vector
Further, the kinetic equation in (1) is modal transformed, that is:
u＝Φq (2)
where Φ is the eigenmode matrix normalized with respect to mass,
Figure BDA0003601159240000031
according to the weighted orthogonality of the natural mode shapes on the mass matrix, the orthogonal damping matrix and the rigidity matrix, the dynamic equations can be decoupled into n mutually independent equations, each equation is identical to a single-degree-of-freedom system in form, wherein the equation of the r element is as follows:
Figure BDA0003601159240000032
finding a frequency response function corresponding to the r-th order mode as:
Figure BDA0003601159240000033
and solving a frequency response function matrix of the system after modal transformation as follows:
Figure BDA0003601159240000034
the frequency response function matrix of the structure can be expressed as:
Figure BDA0003601159240000035
further, the equivalent node load vector f in step four may be obtained by numerically integrating the distributed pressure, and the conversion formula is as follows:
Figure BDA0003601159240000036
Figure BDA0003601159240000037
an equivalent nodal load of the cell that is a cell face load; s e Is the integrated area of the cell; n is a unit shape function moment; t is a surface load vector;
the equivalent node loads of each unit are superposed to form an integral equivalent node load vector, namely f in 4, and the expression is as follows:
Figure BDA0003601159240000038
converting the distributed load into a concentrated load;
assuming that the distributed random excitation is independent of the spatial position, the power spectral density of each point in the solving step five is as follows:
Figure BDA0003601159240000039
wherein S is f (x p ,y p ,ω),S f (x q ,y q ω) is the self-power spectral density at p and q, respectively;
representing the spatial position by using the node number, and obtaining a power spectral density matrix of m node loads after conversion:
Figure BDA0003601159240000041
thus, converted node loads and corresponding power spectral density matrixes are obtained, diagonal elements are self-power spectrums of the excitations, and off-diagonal elements are cross spectrums of the excitations;
the frequency response function matrix obtained by using commercial finite element software is as follows:
Figure BDA0003601159240000042
wherein
Figure BDA0003601159240000043
Calculating the power spectral density of the response point according to a calculation formula of the response power spectral density, wherein the calculation formula is as follows
Figure BDA0003601159240000044
For r degrees of freedom of response, equation (12) can be converted to
Figure BDA0003601159240000045
Generalized frequency response function matrix H required by response solution r (ω) is as follows
Figure BDA0003601159240000046
Obtaining the frequency response function matrix in the formula (13) requires performing frequency response analysis on m random excitations in finite element software respectively.
Further, the sixth step utilizes the reciprocity of the frequency response function of the linear structure, namely
H ij (ω)＝H ji (ω)
The generalized frequency response function matrix in equation (13) can be converted to:
Figure BDA0003601159240000047
namely, it is
Figure BDA0003601159240000051
After the transformation, the original m-time frequency response analysis is changed into r-time frequency response analysis, so that the frequency of the frequency response analysis is reduced, namely, the calculated amount is reduced, and the required generalized frequency response function matrix can be obtained.
Further, the solving method in the seventh step is as follows:
expanding the formula (8) into a m x m-dimensional load change coefficient matrix, namely a load transformation matrix L is
Figure BDA0003601159240000052
At this time, the formula for solving and determining the response result of the response freedom degree is as follows:
Figure BDA0003601159240000053
the invention converts distributed random excitation into multipoint random excitation by establishing a proper engineering finite element discrete model. And determining the position and the number of the response point through modal analysis. On the premise that the position of the response point is known, unit simple harmonic excitation is carried out on the response point. And (4) obtaining all transfer functions of the response points by using frequency response analysis of finite element software. And extracting the calculated transfer function, and carrying out proportional transformation according to the calculated node load size to obtain a harmonic response result of the point under the node load. The harmonic response result after data processing is extracted, the harmonic response result of simple harmonic excitation on the response point and the harmonic response result under each single-point random excitation are completely equivalent, but the calculation amount is greatly reduced. And processing the harmonic response result and the power spectral density of the excitation according to a solving formula of the response power spectral density to obtain the power spectral density of the response point.
The beneficial effects of the invention and the prior art are as follows:
for the random vibration response analysis of distributed random excitation, the traditional calculation method necessarily makes a trade-off in the calculation precision and the calculation efficiency, and has certain defects, the invention firstly determines the position of a response point, avoids a large amount of irrelevant calculation, then converts the distributed random excitation into the centralized excitation according to the conversion relation between the distributed random excitation and the centralized excitation, compared with the calculation of dividing a calculation area, the conversion calculation method has higher precision and can reflect the actual excitation condition more really, and finally, the positions of the excitation point and the response point are exchanged by utilizing the reciprocity of a frequency response function, so that the calculation amount is reduced once more.
Drawings
FIG. 1 is a first order mode-vibration cloud of a simple plate;
FIG. 2 is a diagram of the position of the response point of a determined simple plate;
FIG. 3 is a graph of the load scaling factor of each node after converting distributed random excitation into multipoint random excitation using the method of the present invention;
fig. 4 is a plot of the power spectral density of the response point and the root mean square calculated in accordance with the present invention.
Detailed Description
In order to make the objects, technical solutions and effects of the present invention more clear, the present invention is further described in detail by the following examples. It should be noted that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention discloses a structural vibration response analysis method under distributed random excitation, which comprises the following steps:
step one, establishing a dynamic finite element discrete model of an engineering structure;
step two, performing modal analysis on the simple support plate, as shown in fig. 1;
step three, considering and determining the number and the positions of the response points, and determining the point with the maximum modal displacement as the response point, as shown in fig. 2;
step four, discrete solution is carried out on the distributed random excitation, the distributed random excitation is converted into multi-point random excitation on the finite element model, and the size of each node load is determined, as shown in figure 3;
step five, solving the power spectral density of each concentrated excitation position, namely a converted excitation power spectral density matrix;
exchanging the positions of the excitation points and the response points, exciting the positions of the response points by using finite element software, and solving a frequency response function matrix of each response point;
and step seven, solving the response of the response point by combining the harmonic response analysis result and the excitation power spectral density, as shown in FIG. 4.
The specific process is as follows: in the first step, the engineering structure is discretized into a finite element model, and the vibration equation of the finite element model for the discretized n degrees of freedom can be expressed as:
Figure BDA0003601159240000061
wherein M is an nxn dimensional quality matrix; c is an n multiplied by n dimension orthogonal damping matrix; k is an n x n dimensional stiffness matrix;
Figure BDA0003601159240000062
u is the acceleration, speed and displacement column vector of order n respectively; q is a positioning matrix; f equivalent node loadA charge vector quantity;
and step two, performing modal analysis on the dispersed finite element model under the boundary condition by using finite element software, solving the inherent vibration mode of the discrete finite element model, and determining the position of the response point in the step three.
Performing modal transformation on the kinetic equation in (1), i.e. u ═ Φ q (2)
Where Φ is the eigenmode matrix normalized with respect to mass,
Figure BDA0003601159240000071
according to the weighted orthogonality of the natural mode shapes on the mass matrix, the orthogonal damping matrix and the rigidity matrix, the dynamic equations can be decoupled into n mutually independent equations, each equation is identical to a single-degree-of-freedom system in form, wherein the equation of the r element is as follows:
Figure BDA0003601159240000072
finding a frequency response function corresponding to the r-th order mode as:
Figure BDA0003601159240000073
obtaining a frequency response function matrix of the system after modal transformation as follows:
Figure BDA0003601159240000074
the frequency response function matrix of the structure can be expressed as:
Figure BDA0003601159240000075
the equivalent node load vector f in step four can be obtained by numerically integrating the distributed pressures, and the conversion formula is as follows:
Figure BDA0003601159240000076
Figure BDA0003601159240000077
an equivalent nodal load of the cell that is a cell face load; s e Is the integrated area of the cell; n is a unit shape function moment; t is a surface load vector;
the equivalent node loads of each unit are superposed to form an integral equivalent node load vector, namely f in 4, and the expression is as follows:
Figure BDA0003601159240000078
through the steps, the distributed load is converted into the concentrated load,
in general, we assume that the distributed random excitation is independent of spatial position, and the power spectral density of each point in the solution step five is:
Figure BDA0003601159240000079
wherein S is f (x p ,y p ,ω),S f (x q ,y q ω) is the self-power spectral density at p and q, respectively;
the spatial position is represented by the node number, and the power spectral density matrix of the m node loads after conversion can be obtained:
Figure BDA0003601159240000081
thus, converted node loads and corresponding power spectral density matrixes are obtained, diagonal elements are self-power spectrums of the excitations, and off-diagonal elements are cross spectrums of the excitations;
the generalized frequency response function matrix obtained by using commercial finite element software is as follows
Figure BDA0003601159240000082
Wherein
Figure BDA0003601159240000083
Calculating the power spectral density of the response point according to a calculation formula of the response power spectral density, wherein the calculation formula is as follows
Figure BDA0003601159240000084
For r degrees of freedom of response, equation (12) can be converted to
Figure BDA0003601159240000085
Generalized frequency response function matrix H required by response solution r (ω) is as follows
Figure BDA0003601159240000086
Obtaining the generalized frequency response function matrix in the formula (13) requires performing frequency response analysis on m random excitations in finite element analysis, and the calculated amount is large.
Step six utilizes the reciprocity of the frequency response function of the linear structure, i.e.
H ij (ω)＝H ji (ω)
The generalized frequency response function matrix in equation (13) can be converted to:
Figure BDA0003601159240000091
namely, it is
Figure BDA0003601159240000092
After the transformation, the original one can be convertedThe frequency response analysis of m times is changed into the frequency response analysis of r times, so that the frequency response analysis times are reduced, but a generalized frequency response function matrix required by response point result analysis can be obtained, and the advantage is that the calculation amount is greatly reduced.
The method for calculating the final result in the seventh step comprises the following steps:
expanding the formula (8) into a m x m-dimensional load change coefficient matrix, namely a load transformation matrix L is
Figure BDA0003601159240000093
In this case, the formula for solving and determining the response result of the response freedom degree is
Figure BDA0003601159240000094
The power spectral density of the response point is obtained according to equation (16), as shown in fig. 4.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that modifications can be made by those skilled in the art without departing from the principle of the present invention, and these modifications should also be construed as the protection scope of the present invention.

Claims (6)

1. A method for analyzing structural vibration response under distributed random excitation is characterized by comprising the following steps:
step one, establishing a dynamic finite element discrete model of an engineering structure;
secondly, performing modal analysis on the structure, namely performing modal analysis on the dispersed finite element model by using finite element software under the boundary condition, solving the inherent vibration mode of the discrete finite element model, and considering the position of a response point in the third step;
step three, determining the number and the positions of the response points;
step four, discrete solution is carried out on the distributed random excitation, and the distributed random excitation is converted into multipoint random excitation on a finite element model;
step five, solving the power spectral density of each concentrated excitation position, namely a converted excitation power spectral density matrix;
exchanging positions of the excitation points and the response points, exciting the positions of the response points by using finite element software, and solving a generalized frequency response function matrix of each response point;
and step seven, solving the response of the response point by combining the harmonic response analysis result and the excitation power spectral density.
2. The method of claim 1, wherein in the first step, the engineering structure is discretized into a finite element model, and the equations of the vibration of the finite element model for the discretized n degrees of freedom are expressed as:
Figure FDA0003601159230000011
wherein M is an nxn dimensional quality matrix; c is an n multiplied by n dimension orthogonal damping matrix; k is an n x n dimensional stiffness matrix;
Figure FDA0003601159230000012
u is the acceleration, speed and displacement column vector of order n respectively; q is a positioning matrix; f equivalent node load vector.
3. A method for analyzing the vibration response of a structure under distributed random pressure excitation according to claim 2, wherein the kinetic equation in (1) is modal-transformed, that is:
u＝Φq (2)
where Φ is the eigenmode matrix normalized with respect to mass,
Figure FDA0003601159230000013
according to the weighted orthogonality of the natural mode shapes on the mass matrix, the orthogonal damping matrix and the rigidity matrix, the dynamic equations can be decoupled into n mutually independent equations, each equation is identical to a single-degree-of-freedom system in form, wherein the equation of the r element is as follows:
Figure FDA0003601159230000014
finding a frequency response function corresponding to the r-th order mode as:
Figure FDA0003601159230000021
wherein ζ r And ω r The damping ratio and the natural circular frequency of the r-th order natural mode of the system.
Obtaining a frequency response function matrix of the system after modal transformation as follows:
Figure FDA0003601159230000022
the frequency response function matrix of the structure can be expressed as:
Figure FDA0003601159230000023
4. the method for analyzing structural vibration response under distributed random pressure excitation according to claim 3, wherein the equivalent node load vector f in the fourth step can be obtained by numerically integrating the distributed pressure, and the conversion formula is as follows:
Figure FDA0003601159230000024
Figure FDA0003601159230000025
an equivalent nodal load of the cell that is a cell face load;S e is the integrated area of the cell; n is a unit shape function moment; t is a surface load vector;
the equivalent node loads of each unit are superposed to form an integral equivalent node load vector, namely f in 4, and the expression is as follows:
Figure FDA0003601159230000026
converting the distributed pressure into a concentrated load;
assuming that the distributed random excitation is independent of the spatial position, the power spectral density of each point in the solving step five is as follows:
Figure FDA0003601159230000027
wherein S is f (x p ,y p ,ω),S f (x q ,y q ω) is the self-power spectral density at p and q, respectively;
representing the spatial position by using the node number, and obtaining a power spectral density matrix of m node loads after conversion:
Figure FDA0003601159230000028
thus, converted node loads and corresponding power spectral density matrixes are obtained, diagonal elements are self-power spectrums of the excitations, and off-diagonal elements are cross spectrums of the excitations;
the generalized frequency response function matrix obtained by using commercial finite element software is as follows:
Figure FDA0003601159230000031
wherein
Figure FDA0003601159230000032
Calculating the power spectral density of the response point according to a calculation formula of the response power spectral density, wherein the calculation formula is as follows
Figure FDA0003601159230000033
For r degrees of freedom of response, equation (12) can be converted to
Figure FDA0003601159230000034
Generalized frequency response function matrix H required by response solution r (ω) is as follows
Figure FDA0003601159230000035
Obtaining the generalized frequency response function matrix in formula (13) requires performing frequency response analysis on m random excitations in a finite element respectively.
5. The method according to claim 4, wherein the sixth step utilizes reciprocity of frequency response functions of linear structures, namely
H ij (ω)＝H ji (ω)
The frequency response function matrix in equation (13) can be converted into:
Figure FDA0003601159230000036
namely, it is
Figure FDA0003601159230000037
After the conversion, the original m-time frequency response analysis is changed into r-time frequency response analysis, so that the frequency response analysis times are reduced, namely, the calculation is reducedThe required frequency response function matrix can be obtained at the same time of the quantity.
6. The method for analyzing the structural vibration response under distributed random pressure excitation according to claim 5, wherein the solving method in the seventh step is as follows:
expanding the formula (8) into a m x m-dimensional load change coefficient matrix, namely a load transformation matrix L is
Figure FDA0003601159230000041
At this time, the formula for solving and determining the response result of the response freedom degree is as follows:
Figure FDA0003601159230000042
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116038773A (en) * 2023-03-29 2023-05-02 之江实验室 Vibration characteristic analysis method and device for flexible joint mechanical arm
CN116992269A (en) * 2023-08-02 2023-11-03 上海勘测设计研究院有限公司 Offshore wind power harmonic response extraction method
CN118094784A (en) * 2024-04-28 2024-05-28 东北大学 Determination method of joint response in double-degree-of-freedom accessory system under joint excitation

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116038773A (en) * 2023-03-29 2023-05-02 之江实验室 Vibration characteristic analysis method and device for flexible joint mechanical arm
CN116992269A (en) * 2023-08-02 2023-11-03 上海勘测设计研究院有限公司 Offshore wind power harmonic response extraction method
CN116992269B (en) * 2023-08-02 2024-02-23 上海勘测设计研究院有限公司 Offshore wind power harmonic response extraction method
CN118094784A (en) * 2024-04-28 2024-05-28 东北大学 Determination method of joint response in double-degree-of-freedom accessory system under joint excitation

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