CN112329332B - Intrinsic frequency and vibration mode configuration method based on added rigidity - Google Patents

Intrinsic frequency and vibration mode configuration method based on added rigidity Download PDF

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CN112329332B
CN112329332B CN202011100229.4A CN202011100229A CN112329332B CN 112329332 B CN112329332 B CN 112329332B CN 202011100229 A CN202011100229 A CN 202011100229A CN 112329332 B CN112329332 B CN 112329332B
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任军
曹秋玉
吴瀚海
李其良
张强豪
何文浩
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Hubei University of Technology
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Abstract

The invention discloses a method for configuring natural frequency and vibration mode based on added rigidity, which comprises the steps of firstly determining the position of the added rigidity of a structure according to actual engineering (the added rigidity of the structure refers to the added supporting rigidity on the degree of freedom or the added connecting rigidity between two degrees of freedom), then expressing the matrix increment of the structure after the added rigidity in a vector form, finally converting the problem of changing the structural quantity required by solving and obtaining the ideal natural frequency and vibration mode into a numerical optimization problem containing the ideal natural frequency, vibration mode and rigidity added quantity, and solving the size of the required added rigidity 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

Intrinsic frequency and vibration mode configuration method based on added rigidity
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 rigidity, and particularly relates to a method for adding the natural frequency and the vibration mode required by rigidity configuration on an original structure based on an original structure frequency response function model. The mode of adding rigidity to the original structure here means that support rigidity is added to each degree of freedom of the original structure or connection rigidity is added between each degree of freedom.
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. Generally, the mode of adding mass is convenient to implement. In some practical projects, however, adding stiffness to a particular structural system is the only allowable and effective method. For example, some barrel-shaped structures cannot improve their dynamic characteristics by adding mass to the surface of the structure because the functional requirements do not allow for changing their appearance, and structural modifications that add connecting springs internally can be chosen. The process of adding the structural configuration natural frequency and the mode shape needs a relevant dynamic model based on the original structure, such as a modal model (composed of modal frequency and modal vector), a space 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 natural frequency and mode configuration method based on the added rigidity has important engineering significance.
Disclosure of Invention
In order to solve the technical problems, the invention provides a method for adding rigidity to configure 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 method for configuring natural frequency and vibration mode based on added rigidity is characterized in that: assuming that the differential equation of vibration for a general linear n-degree-of-freedom undamped vibration system is expressed as:
Figure GDA0003480011030000011
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 GDA0003480011030000012
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula (1) to obtain:
Figure GDA0003480011030000021
in the formula, Zn×nThe dynamic stiffness matrix is of an original structure; 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;
the method comprises the following steps:
step 1: suppose that a size dk is added to the original structure systemijThe stiffness, after addition of the stiffness, equation (2) translates to:
Figure GDA0003480011030000022
the matrix Δ K of equation (3) is:
Figure GDA0003480011030000023
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure GDA0003480011030000024
when i is j, it means that the added stiffness is the support stiffness in the degree of freedom i, and
Figure GDA0003480011030000025
if stiffness is added simultaneously in the first t degrees of freedom, equation (3) translates to:
Figure GDA0003480011030000026
step 2: converting the variables of formula (5) into the form of formula (6);
Figure GDA0003480011030000027
the differential equation of motion of the structure after adding the rigidity is as follows:
Figure GDA0003480011030000028
and step 3: will configure the natural frequency omega as desireddAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure GDA0003480011030000031
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 GDA0003480011030000032
wherein, γdIs a weight coefficient;
and 4, step 4: 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 stiffnesses that enable equation (8) to hold, within a given stiffness modification. However, in practical engineering, the condition of equation (8) is often not present, so that the optimal solution in the stiffness modification range is found by using the genetic algorithm selection and inheritance mechanism, and the calculated stiffness can obtain the minimum value of equation (9) as much as possible. After a group of required rigidity is solved through a genetic algorithm, the rigidity is added to an 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 rigidity required to be added (connected between the degrees of freedom) on each degree of freedom of the original structure, and the ideal natural frequency and vibration mode can be obtained after the rigidity is added (connected between the degrees of freedom) on each degree of freedom of the original structure, 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 five-DOF vibration system model with added support stiffness according to an embodiment of the present invention;
FIG. 3 shows the five-DOF vibration system with added connection stiffness dk according to the embodiment of the invention13、dk14、dk34、dk25A schematic diagram of a post-model;
FIG. 4 shows the five-DOF vibration system with added connection stiffness dk according to the embodiment of the invention13、dk23、dk35、dk45、dk15A schematic diagram of a post-model;
FIG. 5 is a frequency response function H of a first order natural frequency and mode shape configured with added support stiffness according to an embodiment of the present invention14，H15Original value, modified value compare schematic diagram;
FIG. 6 is a schematic diagram of an ideal mode shape with a first order natural frequency and mode shape configured with added support stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
FIG. 7 is a frequency response function H of a first-order natural frequency and mode of a configuration with added connection stiffness according to an embodiment of the present invention11，H15Original value, modified value compare schematic diagram;
FIG. 8 is a schematic diagram of an embodiment of the present invention in which a first order natural frequency and mode shape are configured for ideal mode shape with the addition of connection stiffness to obtain a mode shape comparison;
FIG. 9 is a frequency response function H of a second order natural frequency and mode shape configured with added support stiffness according to an embodiment of the present invention14,H15Original value, modified value compare schematic diagram;
FIG. 10 is a schematic diagram of a first-order ideal mode shape with a second-order natural frequency and mode shape configured by adding support stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
FIG. 11 is a schematic diagram of a second-order ideal mode shape with the second-order natural frequency and mode shape configured by adding the supporting stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
FIG. 12 is a frequency response function H of a second order natural frequency and mode shape configured by adding connection stiffness according to an embodiment of the present invention14,H15Original value, modified value pairA schematic diagram;
FIG. 13 is a schematic diagram of a first-order ideal mode shape with second-order natural frequency and mode shape configured by adding connection stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
FIG. 14 is a diagram illustrating a second-order ideal mode shape with second-order natural frequency and mode shape configured by adding connection stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
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 method for configuring natural frequency and vibration mode by adding rigidity, which is characterized by comprising the following steps: assuming that the differential equation of vibration for a general linear n-degree-of-freedom undamped vibration system is expressed as:
Figure GDA0003480011030000041
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 GDA0003480011030000042
representing an acceleration vector.
Let the vibration system respond by x ueiωtAnd substituting it into the formula (1) to obtain
Figure GDA0003480011030000043
In the formula, Zn×nThe dynamic stiffness matrix is of an original structure; 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.
The differential equation of vibration of the structure after adding stiffness becomes:
Figure GDA0003480011030000051
wherein, Δ K is the corresponding stiffness matrix increment; wherein the stiffness is a support stiffness or a connection stiffness;
(1) assuming that the original structure frequency response function matrix is H, adding dk to the 1 st and 2 … i degree of freedom of the original structure11,dk22,…,dkiiSupporting rigidity of i<N; the structural stiffness matrix increment is Δ K after the addition of the bearing stiffness, Δ K being expressed as:
Figure GDA0003480011030000052
then equation (3) translates to equation (4-1) after adding the bearing stiffness:
Figure GDA0003480011030000053
if the n-degree-of-freedom undamped vibration system only adds the rigidity dk at the ith positioniiThe incremental stiffness matrix of the structure after adding the supporting stiffness is delta K, and the delta K is expressed as follows:
Figure GDA0003480011030000054
(2) assuming that the frequency response function matrix of the original structure is H, adding the rigidity dk between the 1 st, 2 nd, 1 st, 3 rd 3 … 1 th, i th degrees of freedom of the original structure12,dk13,…,dk1iThe increment of the structural rigidity matrix after the connection rigidity is added is delta K, and the delta K is expressed as the following form:
Figure GDA0003480011030000061
then equation (3) translates to equation (4-2) after adding the stiffness of the connection:
Figure GDA0003480011030000062
if the n-degree-of-freedom undamped vibration system only adds the rigidity dk at the ith and the jth positionsijThe increment of the rigidity matrix of the structure after the connection rigidity is added is delta K, and the delta K is expressed as the following form:
Figure GDA0003480011030000063
the specific implementation of the embodiment includes the following steps:
step 1: suppose that a size dk is added to the original structure systemijThe stiffness, after addition of the stiffness, equation (2) translates to:
Figure GDA0003480011030000064
the matrix Δ K of equation (3) is:
Figure GDA0003480011030000065
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure GDA0003480011030000066
when i is j, it means that the added stiffness is the support stiffness in the degree of freedom i, and
Figure GDA0003480011030000071
if stiffness is added simultaneously in the first t degrees of freedom, equation (3) translates to:
Figure GDA0003480011030000072
step 2: when the variables in equation (7) are expressed by equation (8), then:
Figure GDA0003480011030000073
then equation (8) can be collated as:
Figure GDA0003480011030000074
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]Column i.
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 t degrees of freedom are involved in the addition of stiffness, the available vector y is addedTExpressed, as follows:
Figure GDA0003480011030000075
and step 3: converting equation (9) to the form of equation (4), equation (9) is described as:
Figure GDA0003480011030000076
Figure GDA0003480011030000077
and 4, step 4: the problem of configuring the natural frequency and the mode shape is expressed as an optimization problem, and an optimization formula is shown as a formula (13):
Figure GDA0003480011030000078
wherein, γdAre weight coefficients.
And 5: selecting the desired natural frequency omegadAnd the mode of vibration udAnd sets the corresponding weight coefficient. The purpose of the optimization of equation (13) is to find a set of required stiffnesses that enable equation (12) to hold, within a given stiffness modification. However, in practical engineering, the condition of equation (12) is often not present, so that the optimal solution in the stiffness modification range is found by using the genetic algorithm selection and inheritance mechanism, and the calculated stiffness can obtain the minimum value of equation (13) as much as possible. After a group of required rigidity is solved through a genetic algorithm, the rigidity is added to an 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 stiffness range of the spring added to the respective degrees of freedom of the original structure is 0-300 kN/m when the natural frequency and the mode shape are configured, the model schematic diagrams after the addition are shown in FIG. 2, FIG. 3 and FIG. 4:
TABLE 1 cantilever beam physics parameter table
Figure GDA0003480011030000081
TABLE 2 natural frequencies and corresponding modes of vibration of the original structure
Figure GDA0003480011030000082
A first example is now given: configuring a first-order natural frequency and a vibration mode based on the added supporting rigidity, wherein the required configuration natural frequency and the required vibration mode are respectively omegad＝55.00Hz，ud＝[0；0.01；-0.1；0.8；1.0]TThe schematic structure after the addition is shown in FIG. 2. The results of the optimization of the added support stiffness obtained by selecting a genetic algorithm for optimization are shown in table 3.
TABLE 3 original system structure with added supporting rigidity to configure first-order natural frequency and vibration mode
Figure GDA0003480011030000083
Figure GDA0003480011030000091
As shown in fig. 5 and fig. 6, the frequency response function comparison graph of the structure after adding the support stiffness and the original structure, and the comparison graph of the vibration mode obtained by the required configuration and adding the support stiffness are respectively shown, and the analysis of the comparison results is shown in table 4:
TABLE 4 addition of support stiffness to obtain a comparison with desired configuration natural frequency and mode shape
Figure GDA0003480011030000092
Second example of an embodiment: configuring a first-order natural frequency and a vibration mode based on adding connection rigidity, wherein the natural frequency and the vibration mode required to be configured are omega respectivelyd＝55.00Hz，ud＝[0；0.01；-0.1；0.8；1.0]T. The structural schematic diagram after addition is shown in fig. 3, the genetic algorithm is selected for optimization, and the result of optimizing the added connection stiffness is shown in table 5.
TABLE 5 original system structure with added connection stiffness to configure first-order natural frequency and vibration mode
Figure GDA0003480011030000093
As shown in fig. 7 and 8, the graphs are respectively a comparison graph of the frequency response function graph of the structure after adding the connection stiffness and the original structure, and a comparison graph of the vibration modes obtained by adding the connection stiffness and the required connection stiffness, and the analysis of the comparison results is shown in table 6:
TABLE 6 comparison of natural frequency and mode for desired configuration with addition of connection stiffness
Figure GDA0003480011030000094
Figure GDA0003480011030000101
A third example is now given: configuring a second-order natural frequency and a vibration mode based on the added supporting rigidity, wherein the required configuration natural frequency and the required vibration mode are respectively omegad1＝35.00Hz，ud1＝[1.0；1.5；-0.5；-0.2；0]T，ωd2＝55.00Hz，ud2＝[0；0.1；-0.1；0.7；1.0]TThe schematic structure after the addition is shown in FIG. 2. The result of the support stiffness addition obtained by optimizing the selection of the genetic algorithm is shown in table 7.
TABLE 7 original system structure with added supporting rigidity configuration second-order natural frequency and vibration mode
Figure GDA0003480011030000102
As shown in fig. 9, 10 and 11, the graphs are respectively a graph comparing the frequency response function of the structure after adding the support stiffness with the original structure, a graph comparing the first-order required configuration with the mode shape obtained by adding the support stiffness, and a graph comparing the second-order required configuration with the mode shape obtained by adding the support stiffness, and the analysis of the comparison results is shown in table 8:
TABLE 8 addition of support stiffness to obtain a comparison with desired configuration natural frequency and mode shape
Figure GDA0003480011030000103
Figure GDA0003480011030000111
A fourth example is now given: configuring second-order natural frequency and vibration mode based on adding connection rigidity, and configuring natural frequency sumVibration modes are respectively omegad1＝37.00Hz，ud1＝[0.2；1.0；-0.5；-0.4；-0.2]T，ωd2＝51.00Hz，ud2＝[0.4；-0.05；-0.2；1.0；0.4]TThe schematic structure after the addition is shown in FIG. 4. The result of the support stiffness addition obtained by optimizing the selection of the genetic algorithm is shown in table 9.
TABLE 9 original system structure with added connection stiffness configuration second order natural frequency and vibration mode
Figure GDA0003480011030000112
As shown in fig. 12, 13, and 14, the graphs are respectively a graph comparing the frequency response function of the structure after adding the connection stiffness with the original structure, a graph comparing the first-order required configuration with the mode shape obtained by adding the support stiffness, and a graph comparing the second-order required configuration with the mode shape obtained by adding the support stiffness, and the analysis of the comparison results is shown in table 8:
TABLE 10 comparison of natural frequency and mode for desired configuration with addition of connection stiffness
Figure GDA0003480011030000113
Figure GDA0003480011030000121
As shown in fig. 5, fig. 6, fig. 7, fig. 8, tables 3, table 4, table 5, and table 6, in the process of configuring the first-order natural frequency and the mode shape, the respective degrees of freedom of the original structure are added with the supporting stiffness or the connecting stiffness, a group of stiffness parameters are obtained through an optimization algorithm, the obtained results are substituted into the original structure to calculate the added structural frequency response function and the added mode shape, and the results show 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. 9, fig. 10, fig. 11, fig. 12, fig. 13, fig. 14, table 7, table 8, table 9, and table 10, in the process of configuring the second-order natural frequency and the mode shape, the respective degrees of freedom of the original structure are added with the supporting stiffness or the connecting stiffness, a set of stiffness 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 calculation example provided by the invention is used for the situation that the five-degree-of-freedom system adds rigidity at all five mass points. And as can be seen from the given theoretical description, the added rigidity scheme applicable to the method is flexible and changeable, and can be added at one or more points except all the mass points. However, reducing the number of points to be added reduces the accuracy of the result of the frequency and mode arrangement to some extent, particularly in the case of a multi-step frequency and mode arrangement. In engineering, the specific adding rigidity number and the corresponding adding position can be determined according to actual working conditions and 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 (3)

1. A method for configuring natural frequency and vibration mode based on added rigidity is characterized in that: assuming that the differential equation of vibration for a general linear n-degree-of-freedom undamped vibration system is expressed as:
Figure FDA0003480011020000011
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 FDA0003480011020000012
represents an acceleration vector;
let x ueiωtSubstituting the formula (1) to obtain:
Figure FDA0003480011020000013
in the formula, Zn×nThe dynamic stiffness matrix is of an original structure; 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;
the method comprises the following steps:
step 1: suppose that a size dk is added to the original structure systemijThe stiffness, after addition of the stiffness, equation (2) translates to:
Figure FDA0003480011020000014
the matrix Δ K of equation (3) is:
Figure FDA0003480011020000015
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure FDA0003480011020000016
when i ═j is the added stiffness is the support stiffness in the degree of freedom i, and
Figure FDA0003480011020000017
if stiffness is added simultaneously in the first t degrees of freedom, equation (3) translates to:
Figure FDA0003480011020000018
step 2: converting the variables of formula (5) into the form of formula (6);
Figure FDA0003480011020000021
the differential equation of motion of the structure after adding the rigidity is as follows:
Figure FDA0003480011020000022
and step 3: will configure the natural frequency omega as desireddAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure FDA0003480011020000023
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 FDA0003480011020000024
wherein, γdIs a weight coefficient;
and 4, step 4: selecting the desired natural frequency omegadAnd the mode of vibration udSetting corresponding weight coefficients;
searching an optimal solution in a rigidity modification range by utilizing a genetic algorithm selection and genetic mechanism, so that the calculated rigidity can obtain a minimum value in a formula (9) as much as possible; after a group of required rigidity is solved through a genetic algorithm, the rigidity is added to an original structure, and the configuration of the natural frequency and the vibration mode can be completed.
2. The method for configuring natural frequency and mode shape based on added stiffness of claim 1, wherein: in step 1, assuming that the original structure frequency response function matrix is H, adding dk to the 1 st and 2 … i degree of freedom of the original structure11,dk22,…,dkiiSupporting rigidity of i<N; the structural stiffness matrix increment is Δ K after the addition of the bearing stiffness, Δ K being expressed as:
Figure FDA0003480011020000025
then equation (3) translates to equation (4-1) after adding the bearing stiffness:
Figure FDA0003480011020000031
if the n-degree-of-freedom undamped vibration system only adds the rigidity dk at the ith positioniiThe incremental stiffness matrix of the structure after adding the supporting stiffness is delta K, and the delta K is expressed as follows:
Figure FDA0003480011020000032
3. the method for configuring natural frequency and mode shape based on added stiffness of claim 1, wherein: in step 1, assuming that the original structure frequency response function matrix is H, adding the stiffness dk between the 1 st, 2 nd, 1 st, 3 rd 3 … 1 th, i th degrees of freedom of the original structure12,dk13,…,dk1iThe increment of the structural rigidity matrix after the connection rigidity is added is delta K, and the delta K is expressed as the following form:
Figure FDA0003480011020000033
then equation (3) translates to equation (4-2) after adding the stiffness of the connection:
Figure FDA0003480011020000034
if the n-degree-of-freedom undamped vibration system only adds the rigidity dk at the ith and the jth positionsijThe increment of the rigidity matrix of the structure after the connection rigidity is added is delta K, and the delta K is expressed as the following form:
Figure FDA0003480011020000041
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