CN112329333B - Method for configuring natural frequency and vibration mode based on added mass - Google Patents

Method for configuring natural frequency and vibration mode based on added mass Download PDF

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CN112329333B
CN112329333B CN202011100240.0A CN202011100240A CN112329333B CN 112329333 B CN112329333 B CN 112329333B CN 202011100240 A CN202011100240 A CN 202011100240A CN 112329333 B CN112329333 B CN 112329333B
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任军
张强豪
李其良
吴瀚海
曹秋玉
何文浩
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Hubei University of Technology
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Abstract

The invention discloses a natural frequency and vibration mode configuration method based on added mass, which comprises the steps of firstly determining the position of the added mass of a structure according to actual engineering, then expressing the mass matrix increment of the structure after the mass is added in a vector form, finally converting the problem of solving the changed structural quantity needed by obtaining the ideal natural frequency and vibration mode into a numerical optimization problem containing target natural frequency, vibration mode and mass addition quantity, and solving the size of the needed added mass through a genetic algorithm. The method can improve the design efficiency, avoid the blindness of design, reduce the design cost and have practical engineering application value.

Description

Method for configuring natural frequency and vibration mode based on added mass
Technical Field
The invention belongs to the technical field of structural dynamics, relates to a natural frequency and vibration mode configuration method based on added mass, and particularly relates to a method for configuring required natural frequency and vibration mode in a mode of adding mass on an original structure based on an original structure frequency response function model. The manner of adding mass to the original structure herein means adding mass to each degree of freedom of the original structure.
Background
In engineering, in order to make a structure meet specific dynamic property requirements, the designed structure needs to be modified to configure certain natural frequency and mode shape. Structural modifications typically include local mass modifications and stiffness modifications. Wherein the structural modification manner of adding mass to the original structure is the simplest and easiest to implement. The process of configuring the natural frequency and the mode shape by adding the mass requires a relevant dynamic model based on the original structure, such as a modal model (composed of modal frequency and modal vector), a spatial state model (composed of mass matrix, stiffness matrix and damping matrix), a frequency response function model (composed of frequency response function), and the like. The frequency response function can be directly obtained through test measurement, and is easy to obtain and accurate. Therefore, the method for configuring the natural frequency and the mode shape based on the added mass in the frequency response function model has important engineering significance.
Disclosure of Invention
In order to solve the technical problems, the invention provides a method for adding mass configuration inherent frequency and vibration mode in an original structure frequency response function model, which belongs to the research category of 'inverse problem' in structure dynamic modification and aims to improve the design efficiency, avoid the design blindness and reduce the design cost.
The technical scheme adopted by the invention is as follows: a natural frequency and mode configuration method based on added mass is characterized in that a vibration differential equation of a general linear n-degree-of-freedom undamped vibration system is assumed to be expressed as:
Figure GDA0003464783330000011
in the formula, K and M are respectively a rigidity matrix and a quality matrix of an original structure, and x represents a displacement vector;
Figure GDA0003464783330000012
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure GDA0003464783330000013
wherein Hn×nA frequency response function matrix of an original structure; ω represents a frequency variable; t represents a unit of time; u denotes the vibration amplitude, uiRepresenting the amplitude of vibration in the ith degree of freedom;
(1) suppose that mass is added in the ith degree of freedom, with mass size dmiEquation (2) translates to after addition of mass:
Figure GDA0003464783330000021
the matrix Δ M of equation (3) is:
Figure GDA0003464783330000022
wherein the content of the first and second substances,
Figure GDA0003464783330000023
(2) assuming that masses are added simultaneously in the first s degrees of freedom, equation (3) now translates to:
Figure GDA0003464783330000024
converting the variables of equation (5) to the form of equation (6):
Figure GDA0003464783330000025
the differential equation of motion of the structure after adding mass is:
Figure GDA0003464783330000026
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure GDA0003464783330000027
therefore, the problem of the configuration of the natural frequency and the mode shape is converted into an optimization problem as shown in formula (9):
Figure GDA0003464783330000028
wherein, γdIs a weight coefficient;
selecting the desired natural frequency omegadAnd the mode of vibration udAnd sets the corresponding weight coefficient. Optimization of formula (9)The purpose of (c) is to find a set of required qualities that enable equation (8) to be established within a given quality modification. However, in practical engineering, the condition of equation (8) is often not present, so that the optimal solution in the mass modification range is found by using the genetic algorithm selection and inheritance mechanism, and the obtained mass can obtain the minimum value of equation (9) as much as possible. After a group of required masses is solved through a genetic algorithm, the masses are added to the original structure, and the configuration of the natural frequency and the vibration mode can be completed. The method provided by the invention can solve the size of the mass required to be added on each degree of freedom of the original structure, and the original structure can obtain ideal natural frequency and vibration mode after adding the mass on each degree of freedom, thereby achieving the effect of configuring the natural frequency and the vibration mode. The method can improve the design efficiency, avoid the blindness of design, reduce the design cost and have practical engineering application value.
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FIG. 1 is a schematic diagram of a five-degree-of-freedom vibration system model according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a model of a five-degree-of-freedom vibration system after adding mass according to an embodiment of the present invention;
FIG. 3 is a frequency response function H of the first order natural frequency and mode shape of the added mass configuration according to the embodiment of the present invention11，H15Original value, modified value compare schematic diagram;
FIG. 4 is a schematic diagram of an embodiment of the present invention in which an ideal mode shape of a first-order natural frequency and a mode shape is configured by an added mass to obtain a mode shape comparison;
FIG. 5 is a frequency response function H of the second order natural frequency and mode shape of the added mass configuration according to the embodiment of the present invention14,H15Original value, modified value compare schematic diagram;
FIG. 6 is a schematic diagram of the second-order natural frequency and the ideal mode shape of the added mass configuration according to the embodiment of the present invention, and the mode shape comparison is obtained;
Detailed Description
In order to facilitate the understanding and implementation of the present invention for those of ordinary skill in the art, the present invention is further described in detail with reference to the accompanying drawings and examples, it is to be understood that the embodiments described herein are merely illustrative and explanatory of the present invention and are not restrictive thereof.
The invention provides a natural frequency and vibration mode configuration method based on added mass, which assumes that the vibration differential equation of a general linear n-degree-of-freedom undamped vibration system is expressed as follows:
Figure GDA0003464783330000031
in the formula, K and M are respectively a rigidity matrix and a quality matrix of an original structure, and x represents a displacement vector;
Figure GDA0003464783330000032
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure GDA0003464783330000041
in the formula, Hn×nA frequency response function matrix of an original structure; ω represents a frequency variable; t represents a unit of time; u denotes the vibration amplitude, uiRepresenting the amplitude of vibration in the ith degree of freedom;
(1) assuming that the frequency response function matrix of the original structure is H, the frequency response function matrix is respectively at the 1 st and 2 … j (j) of the original structure<N) degrees of freedom with addition of size dm1,dm2,…,dmjThe differential equation of vibration of the structure after adding mass becomes:
Figure GDA0003464783330000042
wherein, the corresponding quality matrix increment delta M is:
Figure GDA0003464783330000043
thus, equation (3) can be converted to equation (4) when mass is added:
Figure GDA0003464783330000044
(2) the frequency response function of the original structure is Hn×nAssuming that the n-degree-of-freedom undamped vibration system adds mass dm only at the ith locationiThe added structure has a mass matrix increment of Δ M, which is expressed as follows:
Figure GDA0003464783330000045
(3) assuming that masses are added simultaneously in the first s degrees of freedom, equation (3) now translates to:
Figure GDA0003464783330000046
converting the variables of equation (6) to the form of equation (7):
Figure GDA0003464783330000051
the differential equation of motion of the structure after adding mass is:
Figure GDA0003464783330000052
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
let the natural frequency and the mode shape of the desired configuration be ωd,udThen the frequency response function corresponding to the required configuration natural frequency is H (ω)d) Assuming that s degrees of freedom are involved in the mass to be added, the available vector y is addedTExpressed, as follows:
yT＝[[dm1][dm2]…[dms]] (9)
will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (8), after transformation:
Figure GDA0003464783330000053
Figure GDA0003464783330000054
therefore, the problem of the configuration of the natural frequency and the mode shape is converted into an optimization problem as shown in equation (12):
Figure GDA0003464783330000055
wherein, γdIs a weight coefficient;
selecting the desired natural frequency omegadAnd the mode of vibration udAnd sets the corresponding weight coefficient. The purpose of the optimization of equation (9) is to find a set of required qualities within a given quality modification range to enable equation (8). However, in practical engineering, the condition of equation (8) is often not present, so that the optimal solution in the mass modification range is found by using the genetic algorithm selection and inheritance mechanism, and the obtained mass can obtain the minimum value of equation (9) as much as possible. After a group of required masses is solved through a genetic algorithm, the masses are added to the original structure, and the configuration of the natural frequency and the vibration mode can be completed.
The invention is described in further detail below with reference to the figures and examples.
Fig. 1 shows a five-degree-of-freedom spring mass vibration system with physical parameters as shown in table 1. The natural frequency and mode shape of the original system are shown in table 2. Assuming that the mass range of the added mass is 0-2 kg in the respective degrees of freedom of the original structure when the natural frequency and the mode shape are configured, the schematic diagram of the model after the addition is shown in FIG. 2:
TABLE 1 cantilever beam physics parameter table
Figure GDA0003464783330000061
TABLE 2 natural frequencies and corresponding modes of vibration of the original structure
Figure GDA0003464783330000062
A first example is now given: configuring a first-order natural frequency and a vibration mode based on the added mass, wherein the natural frequency and the vibration mode required to be configured are omega respectivelyd＝55.00Hz，ud＝[1；-0.55；0.2；0；0.05]TThe added structure of the feature pair is shown in figure 2. The results of the addition mass size obtained by optimizing the selected genetic algorithm are shown in table 3.
TABLE 3 adding mass to the original system structure to configure the first order natural frequency and vibration mode
Figure GDA0003464783330000063
As shown in fig. 3 and fig. 4, the frequency response function comparison graph of the structure after adding mass and the original structure, and the comparison graph of the vibration mode obtained after adding mass and the required configuration are respectively shown, and the analysis of the comparison results is shown in table 4:
TABLE 4 comparison of natural frequency and mode shape with desired configuration obtained after adding mass
Figure GDA0003464783330000064
Figure GDA0003464783330000071
Second example of an embodiment: configuring a second-order natural frequency and a vibration mode based on the added mass, wherein the natural frequency and the vibration mode required to be configured are omega respectivelyd1＝39.00Hz，ωd256.00Hz, and u as vibration moded1＝[1；-0.55；0.1；-0.1；-0.2]T, ud2＝[0；-0.01；0.1；-0.4；1]TThe pair of features of (1). The genetic algorithm was selected for optimization, and the results of the addition quality obtained by the optimization are shown in table 5.
TABLE 5 adding mass configuration second order natural frequency and vibration mode of original system
Figure GDA0003464783330000072
As shown in fig. 5 and fig. 6, the frequency response function comparison graph of the structure after adding mass and the original structure, the comparison between the vibration mode required to be configured and the vibration mode obtained after adding mass are respectively shown, and the analysis of the comparison results is shown in table 6:
TABLE 6 comparison of natural frequency and mode shape with desired configuration obtained after adding mass
Figure GDA0003464783330000073
Figure GDA0003464783330000081
As shown in fig. 3 and 4, and tables 3 and 4, in the process of configuring the first-order natural frequency and the mode shape, the weights are added to the degrees of freedom of the original structure, a set of quality parameters is obtained through an optimization algorithm, the obtained result is substituted into the original structure to calculate the frequency response function and the mode shape of the added structure, and the result shows that: obvious formants appear at the natural frequency required to be configured on the frequency response function contrast diagram, the vibration mode vector obtained by optimizing the added structure is obviously converged towards the required vibration mode vector, experimental data show the reliability of the method, error analysis shows that the method has good precision, and the applicability of the method is demonstrated.
As shown in fig. 5 and 6, and tables 5 and 6, in the process of configuring the second-order natural frequency and the mode shape, the weights are added to the degrees of freedom of the original structure, a set of quality parameters is obtained through an optimization algorithm, the obtained result is substituted into the original structure to calculate the frequency response function and the mode shape of the added structure, and the result shows that: obvious formants appear at the second-order natural frequency required to be configured on the frequency response function contrast diagram, the second-order mode vector obtained by optimizing the added structure is obviously converged towards the required mode vector, experimental data shows the reliability of the method, and error analysis shows that the method has good precision, thereby showing the applicability of the method.
The present invention gives an example for the case where a five degree of freedom system adds mass at all five mass points. And as can be seen from the given theoretical description, the quality adding scheme applicable to the method is flexible and changeable, and can be added at one or more points except all the quality points. However, reducing the added mass point reduces the accuracy of the result of the frequency and mode arrangement to some extent, especially in the case of a multi-step frequency and mode arrangement. In the engineering, the number of the specific adding quality points and the corresponding adding positions can be determined according to the actual working conditions and the structural requirements.
It should be understood that no portion of this specification is explicitly set forth as prior art; the above description of the preferred embodiments is intended to be illustrative, and not to be construed as limiting the scope of the invention, which is defined by the appended claims, and all changes and modifications that fall within the metes and bounds of the claims, or equivalences of such metes and bounds are therefore intended to be embraced by the appended claims.

Claims (2)

1. A natural frequency and mode configuration method based on added mass is characterized in that a vibration differential equation of a general linear n-degree-of-freedom undamped vibration system is assumed to be expressed as:
Figure FDA0003464783320000011
in the formula, K and M are respectively a rigidity matrix and a quality matrix of an original structure, and x represents a displacement vector;
Figure FDA0003464783320000019
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure FDA0003464783320000012
wherein Hn×nA frequency response function matrix of an original structure; ω represents a frequency variable; t represents a unit of time; u denotes the vibration amplitude, uiRepresenting the amplitude of vibration in the ith degree of freedom;
(1) suppose that mass is added in the ith degree of freedom, with mass size dmiEquation (2) translates to after addition of mass:
Figure FDA0003464783320000013
the matrix Δ M of equation (3) is:
Figure FDA0003464783320000014
wherein the content of the first and second substances,
Figure FDA0003464783320000015
(2) assuming that masses are added simultaneously in the first s degrees of freedom, equation (3) now translates to:
Figure FDA0003464783320000016
converting the variables of equation (5) to the form of equation (6):
Figure FDA0003464783320000017
the differential equation of motion of the structure after adding mass is:
Figure FDA0003464783320000018
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure FDA0003464783320000021
therefore, the problem of the configuration of the natural frequency and the mode shape is converted into an optimization problem as shown in formula (9):
Figure FDA0003464783320000022
wherein, γdIs a weight coefficient;
selecting the desired natural frequency omegadAnd the mode of vibration udSetting corresponding weight coefficients; searching an optimal solution in a quality modification range by utilizing a genetic algorithm selection and genetic mechanism, so that the calculated quality can obtain a minimum value in a formula (9) as much as possible; after a group of required masses is solved through a genetic algorithm, the masses are added to the original structure, and the configuration of the natural frequency and the vibration mode can be completed.
2. The method according to claim 1, wherein the original structure is assigned a frequency response function matrix of H, and the added dimension dm is added to the original structure in the 1 st and 2 … j degrees of freedom, respectively1,dm2,…,dmjMass of j<N; the differential equation of vibration of the structure after the addition of mass becomes:
Figure FDA0003464783320000023
wherein, the corresponding quality matrix increment delta M is:
Figure FDA0003464783320000024
thus, equation (3) translates to equation (4) when mass is added:
Figure FDA0003464783320000025
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CN106326501A (en) * 2015-06-15 2017-01-11 上海东浩兰生国际服务贸易（集团）有限公司 Natural frequency and vibration mode calculation method for building construction dynamic analysis

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