CN112329331B - Natural frequency and vibration mode configuration method based on mixed addition of mass and rigidity - Google Patents

Natural frequency and vibration mode configuration method based on mixed addition of mass and rigidity Download PDF

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CN112329331B
CN112329331B CN202011100220.3A CN202011100220A CN112329331B CN 112329331 B CN112329331 B CN 112329331B CN 202011100220 A CN202011100220 A CN 202011100220A CN 112329331 B CN112329331 B CN 112329331B
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任军
张强豪
吴瀚海
李其良
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 mixed addition of mass and rigidity, which comprises the steps of firstly determining the position of mixed addition of the mass and the rigidity of a structure according to actual engineering (the mixed addition of the mass and the rigidity of the structure refers to adding mass and supporting rigidity on a degree of freedom or connecting rigidity between two degrees of freedom), then expressing the matrix increment of the structure after the addition (the matrix increment refers to the matrix increment after the mixed addition of the mass and the rigidity) in a vector mode, finally converting the problem of changing the structural quantity required by solving and obtaining the ideal natural frequency and the vibration mode into a numerical optimization problem containing the ideal natural frequency, the vibration mode and the rigidity addition quantity, and solving the size of the mass and the rigidity required to be added through a genetic algorithm. The method improves the design efficiency, avoids the design blindness, reduces the design cost and has practical engineering application value.

Description

Natural frequency and vibration mode configuration method based on mixed addition of mass and 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 mixed addition of mass and rigidity, and particularly relates to a method for mixed addition of first-order or multi-order natural frequency and vibration mode required by mass and rigidity configuration on an original structure based on an original structure frequency response function model. The mode of adding mass and rigidity to the original structure in a mixed mode refers to adding mass and supporting rigidity or connection rigidity between the mass and the supporting rigidity at 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. Some methods for configuring the natural frequency and the mode shape by a structural modification mode of locally adding mass or rigidity exist, but the method is only suitable for the case of simply adding mass or rigidity. In some cases, the structure is limited by the dynamic characteristics of the structure and the special requirements of natural frequency and mode configuration, and a better configuration effect cannot be obtained by a structural modification mode of simply adding mass or rigidity. The invention provides a natural frequency and vibration mode configuration method based on mass-rigidity mixed addition. 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 natural frequency and mode configuration method based on mass-rigidity mixed addition in a frequency response function model, which belongs to the research category of 'inverse problem' in structural 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 vibration mode configuration method based on mixed addition of mass and rigidity is characterized in that: assuming that the differential equation of vibration for a general linear n-degree-of-freedom undamped vibratory vibration system is expressed as:
Figure BDA0002725067360000011
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 BDA0002725067360000012
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure BDA0002725067360000021
wherein Z isn×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 mass and stiffness structure is added in the ith degree of freedom, and the mass and the stiffness are respectively dmi,dkijB, carrying out the following steps of; then equation (2) translates after the mass-stiffness mixture addition:
Figure BDA0002725067360000022
the added structure has a stiffness matrix increment of Δ M and a mass matrix increment of Δ K, and the matrix Δ M and Δ K of formula (3) are:
Figure BDA0002725067360000023
Figure BDA0002725067360000024
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure BDA0002725067360000025
when i is j, it means that the added stiffness is the support stiffness in the degree of freedom i, and
Figure BDA0002725067360000026
assuming simultaneous mass-stiffness hybrid addition in t degrees of freedom, equation (3) translates to:
Figure BDA0002725067360000027
step 2: converting the variables of equation (5) to the form of equation (6):
Figure BDA0002725067360000028
therefore, the differential equation of motion of the structure after the mass-stiffness addition can be expressed as:
Figure BDA0002725067360000031
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
and step 3: will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure BDA0002725067360000032
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 BDA0002725067360000033
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 mass-rigidity modification range by utilizing a genetic algorithm selection and genetic mechanism, so that the calculated mass and rigidity can obtain a minimum value in a formula (8) as much as possible; after a group of required mass-rigidity is solved through a genetic algorithm, the mass and the rigidity are 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 added rigidity and added mass 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 and the mass are 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.
Drawings
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 mass-support stiffness hybrid additive model (model 1) according to an embodiment of the invention;
FIG. 3 is a frequency response function H of the first order natural frequency and mode shape of an additive mass-to-support stiffness configuration according to an embodiment of the present invention11，H15Original value, modified value compare schematic diagram;
FIG. 4 is a schematic diagram of an ideal mode shape for a first order natural frequency and mode shape for an additive mass-to-support stiffness configuration according to an embodiment of the present invention, comparing the mode shapes;
FIG. 5 is a frequency response function H of the second order natural frequency and mode shape of an additive mass-support stiffness configuration according to an embodiment of the present invention11，H15Original value, modified value compare schematic diagram;
FIG. 6 is a schematic diagram of a first-order ideal mode shape of a second-order natural frequency and mode shape configured with added mass-to-support stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
FIG. 7 is a schematic diagram of a second-order ideal mode shape with the second-order natural frequency and mode shape configured with added mass-to-support stiffness to obtain a mode shape comparison according to an embodiment of the present invention;
fig. 8 is a schematic diagram of a five-degree-of-freedom vibration system mass-connection stiffness hybrid addition model (model 2) 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 adding and configuring natural frequency and vibration mode based on the mass and rigidity of an original structure frequency response function model, which assumes that the vibration differential equation of a general linear n-degree-of-freedom undamped vibration system is expressed as follows:
Figure BDA0002725067360000041
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 BDA0002725067360000042
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure BDA0002725067360000043
wherein Z isn×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;
(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 … i (i) th of the original structure<N) degrees of freedom with addition of dk11,dk22,…,dkiiHas a bearing stiffness dm1,dm2,…,dmiThe vibration differential equation of the structure after the mass-rigidity mixture is added becomes:
Figure BDA0002725067360000044
wherein, the corresponding matrix increment delta Z is as follows:
Figure BDA0002725067360000051
thus, when added as a mass-bearing stiffness mix, equation (3) can be converted to equation (4-1):
Figure BDA0002725067360000052
(2) 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 … i (i) th of the original structure<N) degrees of freedom with addition of size dm1,dm2,…,dmiIn the 1 st, 2 nd, 1 st, 3 rd 3 … 1 th, i (i) of the original structure<N) degrees of freedom with an added stiffness dk12,dk13,…,dk1iThe connection rigidity of (2). The structural matrix increment Δ Z after the mass-attachment stiffness addition becomes:
Figure BDA0002725067360000053
thus, the mass-stiffness-to-connection hybrid addition converts equation (3) to equation (4-2):
Figure BDA0002725067360000054
Figure BDA0002725067360000061
the method of this embodiment includes the following steps:
step 1: suppose that a mass and stiffness structure is added in the ith degree of freedom, and the mass and the stiffness are respectively dmi,dkij(ii) a Then equation (2) translates after the mass-stiffness mixture addition:
Figure BDA0002725067360000062
the added structure has the rigidity matrix increment of delta M and the mass matrix increment of delta K, and the matrix delta M and the matrix delta K of the formula (5) are as follows:
Figure BDA0002725067360000063
Figure BDA0002725067360000064
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure BDA0002725067360000065
when i is j, it means that the added stiffness is the support stiffness in the degree of freedom i, and
Figure BDA0002725067360000066
assuming simultaneous mass-stiffness hybrid addition in t degrees of freedom, equation (5) translates to:
Figure BDA0002725067360000067
step 2: converting the variables of equation (7) to the form of equation (8):
Figure BDA0002725067360000068
therefore, the differential equation of motion of the structure after the mass-stiffness addition can be expressed as:
Figure BDA0002725067360000069
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
during the configuration process, it is assumed that it is idealNatural frequency and vibration mode are respectively omegad，udThen the frequency response function corresponding to the natural frequency is H (ω)d) According to different engineering conditions, assuming t degrees of freedom for adding rigidity are needed, the available vector y is addedTExpressed, as follows:
Figure BDA0002725067360000071
and step 3: will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (9), after transformation:
Figure BDA0002725067360000072
Figure BDA0002725067360000073
therefore, the problem of the configuration of the natural frequency and the mode shape is converted into an optimization problem as shown in equation (13):
Figure BDA0002725067360000074
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 (13) is to find a set of required masses and stiffnesses that enable equation (12) to hold, within a given mass-stiffness modification. However, in practical engineering, the condition of equation (12) is often not present, so that the optimal solution in the mass-stiffness modification range is found by using the genetic algorithm selection and inheritance mechanism, and the calculated mass and stiffness can make equation (13) obtain a minimum value as much as possible. After solving a group of required mass-rigidity by genetic algorithm, the natural frequency and vibration mode configuration can be completed by adding the mass and rigidity to the original structure。
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, the physical parameters of which are 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 and the stiffness range of the added mass are 0-2 kg and 0-300 kN/m respectively when the natural frequency and the mode shape are configured, the schematic diagram of the model (model 1) after the addition is shown in FIG. 2:
TABLE 1 cantilever beam physics parameter table
Figure BDA0002725067360000075
TABLE 2 natural frequencies and corresponding modes of vibration of the original structure
Figure BDA0002725067360000081
A first example is now given: configuring a first-order natural frequency and a vibration mode based on mass-support stiffness mixed addition, wherein the required configuration natural frequency and the required vibration mode are omega respectivelyd＝39.00Hz，ud＝[1.00；-0.55；0.20；0.00；0.05]TThe schematic structure after the addition is shown in FIG. 2. The results of the addition of mass-support stiffness obtained by optimization by selecting a genetic algorithm are shown in table 3.
TABLE 3 original system structure quality-supporting rigidity mixed addition configuration first-order natural frequency and vibration mode
Figure BDA0002725067360000082
Fig. 3 and 4 show a frequency response function comparison graph of the structure after the mass-support stiffness mixture is added and the original structure, and a comparison graph of the required configuration mode and the mode obtained after the mass-support stiffness mixture is added, respectively. Comparative analysis is shown in table 4:
TABLE 4 comparison of natural frequency and mode shape to desired configuration obtained after mixed addition of mass-bearing stiffness
Figure BDA0002725067360000083
Figure BDA0002725067360000091
Second example of an embodiment: configuring second-order natural frequency and vibration mode based on mass-support stiffness mixed addition, wherein the required configuration natural frequency and vibration mode are omega respectivelyd1＝39.00Hz，ud1＝[1.00；-0.55；0.20；0.00；0.05]T；ωd2＝55.00Hz，ud2＝[0.00；0.01；-0.10；0.80；1.00]TAnd the structural schematic diagram after the addition is shown in fig. 2, the genetic algorithm is selected for optimization, and the addition result of the optimized connection stiffness is shown in table 5.
TABLE 5 original system structure quality-supporting rigidity mixed addition configuration second-order natural frequency and vibration mode
Figure BDA0002725067360000092
As shown in fig. 5, 6, and 7, a comparison graph of a frequency response function of a structure after the mass-support stiffness mixture is added and an original structure, a comparison graph of a mode shape obtained after the first-order required configuration and the mass-support stiffness mixture are added, and a comparison graph of a mode shape obtained after the second-order required configuration and the mass-support stiffness mixture are added are shown, respectively. Comparative analysis is shown in table 6:
TABLE 6 comparison of natural frequency and mode shape to desired configuration obtained after mixed addition of mass-bearing stiffness
Figure BDA0002725067360000093
Figure BDA0002725067360000101
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 respective degrees of freedom of the original structure are added with mass-support stiffness, a set of mass-stiffness parameters is obtained through an optimization algorithm, the obtained results are substituted into the original structure to calculate the frequency response function and the mode shape of the added structure, 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. 5, 6, 7, 5, and 6, 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 mass-support stiffness, a set of mass-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 mass and rigidity at all five mass points. And as can be seen from the theoretical description, the mass-rigidity mixed addition scheme applicable to the method is flexible and changeable, besides the mixed addition at all the mass points, the mass can be added at one or more points, as shown in fig. 8, the mass is added at 1,3 and 5 points, and the connection rigidity is added between 1 and 3 and between 3 and 5 points. In engineering, the specific mass-rigidity mixed adding quantity 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 foregoing description of the preferred embodiments is in some detail and should not 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 meets and bounds of the claims of the invention may be made by those skilled in the art without departing from the scope of the invention.

Claims (3)

1. A natural frequency and vibration mode configuration method based on mixed addition of mass and rigidity is characterized in that: assuming that the differential equation of vibration for a general linear n-degree-of-freedom undamped vibratory vibration system is expressed as:
Figure FDA0003547885910000011
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 FDA0003547885910000012
represents an acceleration vector;
let the vibration system respond by x ueiωtSubstituting the formula into the formula (1) to obtain:
Figure FDA0003547885910000013
wherein Z isn×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 mass and stiffness structure is added in the ith degree of freedom, and the mass and the stiffness are respectively dmi,dkijThen equation (2) translates to after the mass-stiffness mixture is added:
Figure FDA0003547885910000014
the added structure has a stiffness matrix increment of Δ M and a mass matrix increment of Δ K, and the matrix Δ M and Δ K of formula (3) are:
Figure FDA0003547885910000015
Figure FDA0003547885910000016
wherein when i ≠ j, it means that the added stiffness is the connection stiffness between the degrees of freedom i, j, and
Figure FDA0003547885910000017
when i is j, it means that the added stiffness is the support stiffness in the degree of freedom i, and
Figure FDA0003547885910000018
assuming simultaneous mass-stiffness hybrid addition in t degrees of freedom, equation (3) translates to:
Figure FDA0003547885910000019
step 2: converting the variables of equation (5) to the form of equation (6):
Figure FDA0003547885910000021
therefore, the differential equation of motion of the structure after the mass-stiffness addition can be expressed as:
Figure FDA0003547885910000022
in the formula, alphaiElement of line i, { V } of { alpha }iIs a matrix [ V ]]The ith column;
and step 3: will ideally have a natural frequency omegadAnd the mode of vibration udSubstituting into formula (7), after transformation:
Figure FDA0003547885910000023
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 FDA0003547885910000024
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 mass-rigidity modification range by utilizing a genetic algorithm selection and genetic mechanism, so that the calculated mass and rigidity can obtain a minimum value in a formula (8) as much as possible; after a group of required mass-rigidity is solved through a genetic algorithm, the mass and the rigidity are added to an original structure, and the configuration of the natural frequency and the vibration mode can be completed.
2. The mass and stiffness hybrid addition based natural frequency and mode shape configuration method 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 structure respectively11,dk22,…,dkiiHas a bearing stiffness dm1,dm2,…,dmiMass of (a), (b), (c), (d) and (d)<N; mass-rigidity mixed addition rear structureBecomes:
Figure FDA0003547885910000025
wherein, the corresponding matrix increment delta Z is as follows:
Figure FDA0003547885910000031
thus, equation (3-1) is converted to equation (4-1) as the mass-support stiffness mix is added:
Figure FDA0003547885910000032
3. the mass and stiffness hybrid addition based natural frequency and mode shape configuration method of claim 1, wherein: in step 1, assuming that the original structure frequency response function matrix is H, adding the size dm to the 1 st and 2 … i degrees of freedom of the original structure respectively1,dm2,…,dmiThe added stiffness of dk between the 1 st, 2 nd, 1 st, 3 rd 3 … 1 th, i th freedom degree of the original structure12,dk13,…,dk1iConnection stiffness of i<N; the structural matrix increment Δ Z after the mass-attachment stiffness addition becomes:
Figure FDA0003547885910000033
thus, equation (3) translates to equation (4-2) as the mass-stiffness mixture is added:
Figure FDA0003547885910000041
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