CN114912324A - Distributed MTMDI-based multi-order vortex-induced vibration control method for large-span bridge - Google Patents

Distributed MTMDI-based multi-order vortex-induced vibration control method for large-span bridge Download PDF

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CN114912324A
CN114912324A CN202210554369.1A CN202210554369A CN114912324A CN 114912324 A CN114912324 A CN 114912324A CN 202210554369 A CN202210554369 A CN 202210554369A CN 114912324 A CN114912324 A CN 114912324A
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mtmdi
order
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张连振
彭思杰
孙勇
卫学亮
吴红林
孙永明
刘宇
张大光
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Harbin Institute of Technology
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • E—FIXED CONSTRUCTIONS
    • E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
    • E01D—CONSTRUCTION OF BRIDGES, ELEVATED ROADWAYS OR VIADUCTS; ASSEMBLY OF BRIDGES
    • E01D22/00—Methods or apparatus for repairing or strengthening existing bridges ; Methods or apparatus for dismantling bridges
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
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Abstract

A distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge belongs to the field of structural vibration control. The method aims to solve the problem that the conventional TMD cannot be applied due to overlarge static extension when the vertical vibration of a low-frequency structure is controlled. The invention comprises the following steps: establishing a finite element model of a target bridge, obtaining modal information of a structure through modal analysis, wherein the modal information comprises frequency, vibration mode and modal quality, determining the order of a target modal according to the design passing wind speed and the Stero-Roh number of a main beam section, determining the arrangement position of each sub TMDI of MTMDI according to a vibration mode vector, determining the modal quality ratio of each order, calculating the physical quality of each sub TMDI according to the vibration mode value of the arrangement position, determining the inertia quality of each TMDI according to the installation space in a beam, and determining the optimal parameter of each MTMDI group by adopting an iterative MTMDI parameter optimization method. The invention is used for large-span bridges.

Description

Distributed MTMDI-based multi-order vortex-induced vibration control method for large-span bridge
Technical Field
The invention belongs to the field of structural vibration control, and particularly relates to a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge.
Background
When the near-earth wind in the boundary layer bypasses the bridge, flow separation is generated on the windward side of the main beam, and vortex shedding which changes alternately is generated on the upper surface and the lower surface of the main beam, and vortex-induced resonance is generated when the vortex shedding frequency is close to or equal to a certain self-vibration frequency of the structure. The vortex vibration is wind-induced amplitude-limiting vibration with self-excitation property, which can not cause catastrophic damage to the bridge like flutter and gallop, but because the frequency of the large-span bridge is lower and is a dense-frequency structure, the vortex-induced vibration is easy to frequently appear even at low wind speed, thereby seriously affecting the driving comfort and safety, causing long-term fatigue problem and reducing the service life of the bridge. Therefore, the method has very important significance for eliminating or inhibiting the vortex-induced vibration of the large-span bridge.
Since vortex-induced vibration generally represents single-mode vibration, Tuned Mass Damper (TMD) has outstanding control performance on single-mode vibration, and is one of the common methods for controlling vortex-induced vibration of large-span bridges. However, due to the constructive characteristics of TMD, its static elongation is only related to its own frequency, i.e. δ is g/ω 2 When the vibration control is performed on a large-span bridge, the frequency is low, the static extension of the TMD is too large, the TMD is not suitable under the condition that the installation space inside the main beam is limited, and extra measures such as a lever and a prestressed spring are often needed to limit the static extension of the main beam. According to the existing research, under the condition that the mass ratio of TMD is not changed, a plurality of TMD with different frequencies are adoptedMTMD composed of TMD sub-ratio of rate and damping ratio has better control effect and robustness. In addition, the frequency distribution of the large-span bridge is dense, the possibility of vortex vibration of multi-order modes exists, the multi-order modes need to be controlled, and as the total mass of the TMD is large and the transverse partition plates in the large-span bridge beam are distributed densely, the large-mass TMD needs to be dispersed into a plurality of sub TMDs with relatively small mass and arranged along the bridge span direction inevitably. However, TMD cannot be applied to control vertical vibration of a low-frequency structure due to too large static elongation.
Disclosure of Invention
The invention provides a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge, aiming at solving the problem that the conventional TMD cannot be applied due to overlarge static elongation when controlling the vertical vibration of a low-frequency structure, and considering the possibility of multi-order vortex vibration existing in the large-span bridge due to dense mode, and the method is mainly characterized by a distributed MTMDI arrangement position selection method and an iterative MTMDI parameter optimization method.
In order to achieve the purpose, the invention is realized by the following technical scheme:
a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge comprises the following steps:
s1, establishing a finite element model of the target bridge, and obtaining structural modal information including frequency, vibration mode and modal quality through modal analysis;
s2, determining the order of the target mode according to the designed passing wind speed and the Stero-Ha number of the main beam section;
calculating vortex shedding frequency by a Strouhal formula, obtaining a target mode by the vortex shedding frequency corresponding to the maximum passing wind speed,
determining the order N of the target mode, wherein the Strolouha formula is as follows:
Figure BDA0003654308640000021
f is vortex shedding frequency, St is the Strouhal number of the section of the main beam, D is the height of the main beam, and U is the designed passing wind speed;
s3, determining the arrangement position of each sub TMDI of the MTMDI according to the vibration mode vector;
s4, determining modal mass ratio of each order, calculating physical mass of each sub TMDI according to the vibration type value of the arrangement position, and determining inertia mass of each TMDI according to the installation space in the beam;
and S5, determining the optimal parameters of each set of MTMDI by adopting an iterative MTMDI parameter optimization method.
The specific implementation of step S5 is:
s5.1, independently optimizing each group of MTMDI without considering the synergistic effect among MTMDI in different modes to obtain the initial value of each group of MTMDI parameter;
s5.2, optimizing the MTMDI corresponding to each order of target mode, wherein when the ith order mode is optimized, the frequency and the damping ratio of the ith MTMDI group are optimization variables, the other MTMDI groups are regarded as non-resonance MTMDI, the parameters are kept unchanged, and the effective mass m passes through enri The influence of the parameters on parameter optimization is taken into account, and each updated set of MTMDI parameters is obtained;
s5.3, calculating the offset of the updated frequency and damping ratio of the MTMDI and the change of the amplitude of each-order modal power amplification coefficient of the main beam under the combined action of all the MTMDI to obtain three offset vectors of iteration results, and measuring the size of the offset vectors by adopting a two-norm method;
and S5.4, judging whether the sizes of the three iteration result offset vectors are smaller than a preset allowable value or not, if not, repeating the step S5.2-5.3, and if the two norms of the three iteration result offset vectors are smaller than the preset allowable value, outputting an optimal MTMDI parameter.
Further, the step S3 is implemented by the following steps:
s3.1, determining a candidate position for installing the MTMDI according to the internal structure of the main beam and the practical installation space limit, and determining each order modal vibration type value corresponding to the candidate position;
and S3.2, each order mode is controlled by adopting a group of MTMDI, each candidate position is only provided with 1 TMDI, and the TMDI is placed from large to small one by one from low mode to high mode according to the vibration type value of each candidate position.
Further, the method comprisesStep S3.1, according to the coordinate of the candidate position of MTMDI installation, the corresponding sub-matrix phi is provided from the vibration type matrix c ＝[φ c1 …φ ci …φ cN ]Wherein phi ci The mode shape value of the mode shape of the ith order of the candidate position is taken as the mode shape value of the candidate position;
s3.2, obtaining the arrangement position of each group of MTMDI, wherein the corresponding vibration mode matrix is phi c Is sub-matrix of
Figure BDA0003654308640000031
Wherein psi si The matrix is a q multiplied by N matrix, q is the number of sub TMDI of the MTMDI, N is the order of the target mode, and the superscript T is the transposition.
Further, step S3.1 the interior of the main beam is configured as a diaphragm distribution.
Further, the step S4 is implemented by the following steps:
s4.1, determining the mass ratio of MTMDI (maximum Transmission mdi) of each order according to the vibration reduction requirement;
s4.2, mass ratio mu of MTMDI according to ith order mode i And calculating the physical quality of the sub TMDI, wherein the calculation formula is as follows:
Figure BDA0003654308640000032
wherein M is i The ith order modal quality of the bridge; m is i The physical mass of the neutron TMDI in the ith group of MTMDI; q is the sub TMDI number of the MTMDI group;
Figure BDA0003654308640000033
the model value of the ith order mode corresponding to the position of the h sub TMDI;
s4.3, determining the maximum static elongation of the TMDI according to the mounting space in the beam, so as to determine the inertial mass of the TMDI; actual frequency ω of h-th sub TMDI h And damping ratio xi h Are respectively as
Figure BDA0003654308640000034
Figure BDA0003654308640000035
Wherein m is h ,c h ,k h And b h The physical mass, damping coefficient, rigidity and inertial mass of the h-th sub TMDI respectively;
the static elongation of the h-th sub TMDI is:
Figure BDA0003654308640000036
wherein g is the acceleration of gravity.
Further, step S5.1 obtains initial values X of parameters of MTMDI of each group 0 ＝[P 10 …P i0 …P N0 ]In which P is i0 The initial parameters of the ith set of MTMDI are a 2 xq matrix, the first action is the actual frequency, and the second action is the actual damping ratio;
step S5.4 iteratively sets the termination condition as:
max(Norm1,Norm2,Norm3)＜ε
wherein ε is a predetermined tolerance.
Further, when step S5.2 optimizes the ith order MTMDI, the frequency response function of the single degree of freedom system under the action of the MTMDI is taken as the objective function, the optimization variables are the frequency and the damping ratio of the ith order MTMDI, and the initial value is P i(k-1) The parameter of the rest of the non-resonant MTMDI is unchanged and is X k-1 Through effective mass m enri Taking into account the influence on parameter optimization, obtaining the absolute value H through an optimization algorithm i Frequency and damping ratio P of i-th group MTMDI with minimum maximum value of (j omega) | ik ；
The ith order MTMDI optimized objective function | H i (j ω) | is:
Figure BDA0003654308640000041
wherein M is i ,C i And K i The ith order modal mass, damping and rigidity of the main structure are respectively; omega is vortex shedding frequency; j is an imaginary unit; m is ei The effective mass of the MTMDI represents the dynamic action of the MTMDI on the main structure;
Figure BDA0003654308640000042
m is to be ei Divided into two parts, the first part being the effective mass of the resonant MTMDI, i.e. the ith MTMDI, denoted m eri (ii) a The second part is the effective mass of the non-resonant MTMDI, namely the effective mass of the rest modal MTMDI, and is marked as m enri ；
When the ith order vortex vibration is taken as a target, the optimized variables are only the frequency and the damping ratio of the ith order MTMDI, the MTMDI parameter of the last iteration is taken as an initial value, and all the MTMDI parameters are contained in m eri The parameters of the rest non-resonant MTMDI are the MTMDI parameters obtained by the last iteration, and are kept unchanged and pass through the effective mass m enri Accounting for its effect on current MTMDI tuning;
through the optimization of the step, parameters of each group of MTMDI after the current iteration order, namely X, are obtained k ＝[P 1k …P ik …P Nk ]In which P is ik The initial parameters of the ith set of MTMDI after the kth iteration are a 2 xq matrix, the first action is the actual frequency, and the second action is the actual damping ratio.
Further, step S5.3 calculates the updated offset of the frequency and damping ratio of the MTMDI and the change in the amplitude of the modal power amplification coefficient of each step of the main beam under the combined action of all MTMDI, and calculates three offset vectors of the iteration result:
ω offset ＝X k (1,:)-X k-1 (1,:)
ξ offset ＝X k (2,:)-X k-1 (2,:)
DMF offset ＝DMF k -DMF k-1
wherein, ω is offset And xi offset For MTMDI actual frequency and damping before and after iterationA specific offset; DMF (dimethyl formamide) offset The amplitude of modal power amplification coefficients of each order of the main structure is deviated before and after iteration; DMF (dimethyl formamide) k After the kth iteration, the amplitude of each order modal power amplification coefficient of the main structure under the action of all MTMDI is 1 xN row vectors, wherein the ith element is defined as:
DMF k (1,i)＝max(|H i (jω)|)/(1/K i )
the size of the offset vector is measured by using the two-norm of the vector, which is defined as follows:
Figure BDA0003654308640000051
where x is a row vector with n elements, the two-norm of the three resulting offset vectors is:
Norm1＝||ω offset || 2
Norm2＝||ξ offset || 2
Norm3＝||DMF offset || 2
the invention has the beneficial effects that:
the invention relates to a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge, which aims at the problem that the conventional TMD cannot be applied due to overlarge static elongation when controlling the vertical vibration of a low-frequency structure,
a double-end Mass element (Inerter) is used for improving TMD, a novel Inerter Damper, namely a Tuned Mass Damper Inerter (TMDI) is obtained, because the Inerter Damper has a dynamic negative stiffness effect, compared with the traditional TMD, the static elongation of the TMDI is obviously reduced, the practical problem that a large-span bridge has dense modal and multi-order modes which can generate vortex vibration is considered, a multi-order vortex-induced vibration control method of the large-span bridge based on distributed MTMDI (multiple Tuned Mass Damper) is provided, the characteristic that the vortex-induced vibration is single-modal vibration is combined, a selection method of the arrangement position of the MTMDI is provided, the synergistic effect between MTMDIs of different modes is considered, an iterative MTMDI parameter optimization method is provided, the frequency response functions of the modes obtained by the optimization method are uniformly distributed, and the power amplification coefficients of the modes are smaller than the large power amplification coefficient when the power amplification coefficients of the modes are not considered, the MTMDI optimized by the optimization method provided by the invention has better control effect.
Drawings
FIG. 1 is a flow chart of a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge according to the invention;
FIG. 2 is a flowchart of step S5 of the method for controlling multi-order vortex-induced vibration of a large-span bridge based on a distributed MTMDI according to the present invention;
FIG. 3 is a schematic structural diagram of the MTMDI controlling the single degree of freedom system according to the present invention;
FIG. 4 is a comparison graph of the effects of the distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in detail with reference to the accompanying drawings and the detailed description. It is to be understood that the embodiments described herein are illustrative of the present invention and are not to be construed as limiting thereof, i.e., the described embodiments are merely a subset of the embodiments of the invention and are not all embodiments. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations, and the present invention may have other embodiments.
Thus, the following detailed description of specific embodiments of the present invention presented in the accompanying drawings is not intended to limit the scope of the invention as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the detailed description of the invention without inventive step, are within the scope of protection of the invention.
For further understanding of the contents, features and effects of the present invention, the following embodiments are exemplified in conjunction with the accompanying drawings and the following detailed description:
a distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge comprises the following steps:
s1, establishing a finite element model of the target bridge, and obtaining modal information of the structure through modal analysis, wherein the modal information comprises frequency, vibration mode and modal quality;
further, step S1 is to establish a finite element model of the target bridge using finite element software according to the bridge design drawing, and obtain modal information of the structure including frequency, mode shape and modal mass through modal analysis, wherein the mode shape is normalized according to the maximum amplitude.
Fig. 1 is a flowchart of a multi-order vortex-induced vibration control method for a large-span bridge based on a distributed MTMDI in the embodiment.
S2, determining the order of the target mode according to the designed passing wind speed and the Stero-Ha number of the main beam section;
in step S2, the vortex shedding frequency is calculated by the strouhal formula, the target mode is obtained by the vortex shedding frequency corresponding to the maximum passing wind speed, and the order N of the target mode is determined, where the strouhal formula is:
Figure BDA0003654308640000061
f is vortex shedding frequency, St is girder section Strouhal number, D is girder height, and U is designed passing wind speed.
Furthermore, because the vortex-induced vibration is amplitude-limiting vibration, the structural damage of the bridge cannot be directly caused, but the driving safety of the bridge can be affected, so that when the target mode of the bridge vortex vibration control is selected, only the mode below the vortex shedding frequency corresponding to the maximum passing wind speed needs to be considered. And determining the maximum passing wind speed according to the specification, and determining the target modal order N by combining the Strouhal number of the section of the main beam.
S3, determining the arrangement position of each sub TMDI of the MTMDI according to the vibration mode vector;
the specific implementation of step S3 includes the following sub-steps:
s3.1, determining a candidate position for installing the MTMDI according to the internal structure of the main beam and the actual installation space limit, and determining modal vibration type values of each order corresponding to the candidate position;
and S3.2, each order mode is controlled by adopting a group of MTMDI, each candidate position is only provided with 1 TMDI, and the TMDI is placed from large to small one by one from low mode to high mode according to the vibration type value of each candidate position.
Further, in step S3.1, according to the coordinates of the candidate positions for MTMDI installation, the corresponding sub-matrix Φ is extracted from the vibration mode matrix c ＝[φ c1 …φ ci …φ cN ]Wherein phi ci The mode shape value of the mode shape of the ith order at the candidate position is set;
further, step S3.1, the inner structure of the main beam is distributed by diaphragm plates;
further, the arrangement position of each set of MTMDI is obtained in step S3.2, and the corresponding vibration mode matrix is phi c Is sub-matrix of
Figure BDA0003654308640000071
Wherein psi si The matrix is a q multiplied by N matrix, q is the number of sub TMDI of the MTMDI, N is the order of the target mode, and the superscript T is the transposition.
Further, in the conventional multi-modal vibration control of the structure by the TMDs, the TMDs for each order of modes are generally required to be arranged at the peak point of the target mode and at the nodes of other modes as much as possible so as to reduce the influence between adjacent modes, and research shows that in the multi-modal vibration of the structure, the mutual influence between the TMDs for different modes is unfavorable for the overall vibration reduction effect. The vortex-induced vibration of the bridge is single-mode vibration, and the problem of coupling between different modes does not exist, so that the damper only needs to be arranged at the peak point of the target mode.
Furthermore, each order of target modes is controlled by adopting a group of MTMDI, each group of MTMDI comprises q sub TMDI with the same mass and different frequencies and damping ratios, and because the number of peak points of the vibration mode is increased along with the increase of the frequency, compared with a low-order mode, the high-order mode has more peak points to arrange the MTMDI, when the MTMDI is arranged, the TMDI is arranged from the low mode to the high mode one by one according to the size of the vibration mode value of each candidate position.
S4, determining modal mass ratio of each order, calculating physical mass of each sub TMDI according to the vibration type value of the arrangement position, and determining inertia mass of each TMDI according to the installation space in the beam;
the specific implementation of step S4 includes the following sub-steps:
s4.1, determining the mass ratio of MTMDI (maximum Transmission mdi) of each order according to the vibration reduction requirement;
s4.2, mass ratio mu of MTMDI according to ith order mode i And calculating the physical quality of the sub TMDI, wherein the calculation formula is as follows:
Figure BDA0003654308640000081
wherein M is i The ith order modal quality of the bridge; m is i The physical mass of the neutron TMDI in the ith group of MTMDI; q is the sub TMDI number of the MTMDI group;
Figure BDA0003654308640000082
the model value of the ith order mode corresponding to the position of the h sub TMDI;
s4.3, determining the maximum static elongation of the TMDI according to the mounting space in the beam, so as to determine the inertial mass of the TMDI;
actual frequency ω of h-th sub TMDI h And damping ratio xi h Are respectively as
Figure BDA0003654308640000083
Figure BDA0003654308640000084
Wherein m is h ,c h ,k h And b h The physical mass, damping coefficient, rigidity and inertial mass of the h-th sub TMDI respectively;
the static elongation of the h-th sub TMDI is:
Figure BDA0003654308640000085
wherein g is the acceleration of gravity.
Further, according to the existing research, it is shown that increasing the inertial mass of the TMDI reduces the control effect of the TMDI, so that it is necessary to select an appropriate inertial mass to ensure that the TMDI has a sufficient control effect and to enable the mounting of the TMDI in the beam with a small static elongation.
And S5, determining the optimal parameters of each group of MTMDI by adopting an iterative MTMDI parameter optimization method.
The specific implementation of step S5 includes the following sub-steps:
s5.1, independently optimizing each group of MTMDI without considering the synergistic effect among MTMDI in different modes to obtain the initial value of each group of MTMDI parameter;
s5.2, optimizing the MTMDI corresponding to each order of target mode, wherein when the ith order mode is optimized, the frequency and the damping ratio of the ith MTMDI group are optimization variables, the other MTMDI groups are regarded as non-resonance MTMDI, the parameters are kept unchanged, and the effective mass m passes through enri The influence of the parameters on parameter optimization is taken into account, and each set of updated MTMDI parameters is obtained;
s5.3, calculating the offset of the updated frequency and damping ratio of the MTMDI and the change of the amplitude of each-order modal power amplification coefficient of the main beam under the combined action of all the MTMDI to obtain three offset vectors of iteration results, and measuring the size of the offset vectors by adopting a two-norm method;
and S5.4, judging whether the sizes of the three iteration result offset vectors are smaller than a preset allowable value or not, if not, repeating the step 5.2-5.3, and if the two norms of the three iteration result offset vectors are smaller than the preset allowable value, outputting an optimal MTMDI parameter.
Fig. 2 is a flowchart of step S5 of the method for controlling multi-step vortex-induced vibration of a large-span bridge based on a distributed MTMDI in this embodiment.
Further, step S5.1 obtains initial values X of parameters of MTMDI of each group 0 ＝[P 10 …P i0 …P N0 ]In which P is i0 The initial parameters of the ith set of MTMDI are a 2 xq matrix, the first action is the actual frequency, and the second action is the actual damping ratio;
further, when step S5.2 optimizes the ith order MTMDI, the frequency response function of the single degree of freedom system under the action of the MTMDI is taken as the objective function, the optimized variables are the frequency and the damping ratio of the ith order MTMDI, and the initial value is P i(k-1) The parameter of the rest of the non-resonant MTMDI is unchanged and is X k-1 To pass through the effective mass m enri Taking into account the influence on parameter optimization, obtaining the absolute value H through an optimization algorithm i Frequency and damping ratio P of i-th group MTMDI with minimum maximum value of (j omega) | ik ；
The ith order MTMDI optimized objective function | H i (j ω) | is:
Figure BDA0003654308640000091
wherein, M i ,C i And K i The ith order modal mass, damping and rigidity of the main structure are respectively; omega is vortex shedding frequency; j is an imaginary unit; m is ei The effective mass of the MTMDI represents the dynamic action of the MTMDI on the main structure;
Figure BDA0003654308640000092
m is to be ei Divided into two parts, the first part being the effective mass of the resonant MTMDI, i.e. the ith MTMDI, denoted m eri (ii) a The second part is the effective mass of the non-resonant MTMDI, namely the effective mass of the rest modal MTMDI, which is marked as m enri ；
When the ith order vortex vibration is taken as a target, the optimized variables are only the frequency and the damping ratio of the ith order MTMDI, the MTMDI parameter of the last iteration is taken as an initial value, and all the MTMDI parameters are contained in m eri Of the restThe parameters of the non-resonance MTMDI are the parameters of the MTMDI obtained by the last iteration, and are kept unchanged and pass through the effective mass m enri Accounting for its effect on current MTMDI tuning;
through the optimization of the step, parameters of each group of MTMDI after the current iteration order, namely X, are obtained k ＝[P 1k …P ik …P Nk ]In which P is ik The initial parameters of the ith set of MTMDI after the kth iteration are a 2 xq matrix, the first action is the actual frequency, and the second action is the actual damping ratio.
Further, step S5.3 calculates the updated offset of the frequency and damping ratio of the MTMDI and the change in the amplitude of the modal power amplification coefficient of each step of the main beam under the combined action of all MTMDI, and calculates three offset vectors of the iteration result:
ω offset ＝X k (1,:)-X k-1 (1,:)
ξ offset ＝X k (2,:)-X k-1 (2,:)
DMF offset ＝DMF k -DMF k-1
wherein, ω is offset And xi offset The actual frequency and damping ratio of the MTMDI before and after iteration are deflected; DMF (dimethyl formamide) offset The amplitude of modal power amplification coefficients of each order of the main structure is deviated before and after iteration; DMF (dimethyl formamide) k After the kth iteration, the amplitude of each order modal power amplification coefficient of the main structure under the action of all MTMDI is 1 xN row vectors, wherein the ith element is defined as:
DMF k (1,i)＝max(|H i (jω)|)/(1/K i )
the size of the offset vector is measured by using the two-norm of the vector, which is defined as follows:
Figure BDA0003654308640000101
where x is a row vector of n elements, the two-norm of the three resulting offset vectors is:
Norm1＝||ω offset || 2
Norm2＝||ξ offset || 2
Norm3＝||DMF offset || 2
further, step S5.4 iteratively sets the termination condition as:
max(Norm1,Norm2,Norm3)＜ε
wherein ε is a predetermined tolerance.
Further, step S5.1 is to obtain an initial value of iteration, first, without considering the synergistic effect between MTMDI of different modes, and considering only the effect of MTMDI of the mode when controlling a certain order of vortex-induced vibration, individually optimizing each set of MTMDI to obtain an initial value of a parameter of each set of MTMDI;
further, MTMDI corresponding to each order of target mode is optimized, when the ith order mode is optimized, the parameters of the ith group of MTMDI are optimization variables, the rest groups of MTMDI are regarded as non-resonance MTMDI, and the influence of the non-resonance MTMDI on the current mode is counted in the form of effective mass. Obtaining updated MTMDI parameters of each group;
since the vortex-induced vibration is single-mode vibration, when the vortex amplitude value is calculated, the main structure can be converted into a single-degree-of-freedom system in a mode space, the MTMDI is adopted to control the vortex-induced vibration, and the MTMDI can be regarded as controlling the single-degree-of-freedom system in the mode space, as shown in fig. 3. The vortex-induced forces can be approximated as simple harmonic forces in general design.
Furthermore, because the large-span bridge has dense modes, the frequencies of the MTMDI groups are relatively close, and the mutual influence among the MTMDI aiming at different modes cannot be ignored. While the objective functions for the different modalities are different, the optimization problem is effectively a multi-objective optimization problem, with each objective function being a function of all MTMDI parameters. Aiming at the multi-objective optimization problem, the embodiment adopts an iterative MTMDI parameter optimization method to determine the optimal parameters of each group of MTMDI.
Fig. 4 is a comparison graph of the control effect of the MTMDI obtained by the optimization method of the present invention and the control effect of the MTMDI without considering the synergistic effect during the optimization under the condition that the mass ratio of the MTMDI is the same, and it can be found that the frequency response functions of the modes obtained by the optimization method of the present invention are uniformly distributed, and the power amplification coefficients of the modes of the orders are smaller than the power amplification coefficients when the synergistic effect is not considered, which indicates that the MTMDI optimized by the optimization method of the present invention has a better control effect.
It is noted that relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in a process, method, article, or apparatus that comprises the element.
While the application has been described above with reference to specific embodiments, various modifications may be made and equivalents may be substituted for elements thereof without departing from the scope of the application. In particular, the various features of the embodiments disclosed herein may be used in any combination that is not inconsistent with the structure, and the failure to exhaustively describe such combinations in this specification is merely for brevity and resource conservation. Therefore, it is intended that the application not be limited to the particular embodiments disclosed, but that the application will include all embodiments falling within the scope of the appended claims.

Claims (8)

1. A distributed MTMDI-based multi-order vortex-induced vibration control method for a large-span bridge is characterized by comprising the following steps: the method comprises the following steps:
s1, establishing a finite element model of the target bridge, and obtaining modal information of the structure through modal analysis, wherein the modal information comprises frequency, vibration mode and modal quality;
s2, determining the order of the target mode according to the designed passing wind speed and the Stero-Ha number of the main beam section;
calculating vortex shedding frequency by a Strouhal formula, obtaining a target mode by the vortex shedding frequency corresponding to the maximum passing wind speed,
determining the order N of the target mode, wherein the Strolouha formula is as follows:
Figure FDA0003654308630000011
f is vortex shedding frequency, St is the Stero-Ha number of the section of the main beam, D is the height of the main beam, and U is the designed passing wind speed;
s3, determining the arrangement position of each sub TMDI of the MTMDI according to the vibration mode vector;
s4, determining modal mass ratio of each order, calculating physical mass of each sub TMDI according to the vibration type value of the arrangement position, and determining inertia mass of each TMDI according to the installation space in the beam;
s5, determining the optimal parameters of each group of MTMDI by adopting an MTMDI parameter optimization method based on iteration;
the specific implementation of step S5 is:
s5.1, independently optimizing each group of MTMDI without considering the synergistic effect among MTMDI in different modes to obtain the initial value of each group of MTMDI parameter;
s5.2, optimizing the MTMDI corresponding to each order of target mode, wherein when the ith order mode is optimized, the frequency and the damping ratio of the ith MTMDI group are optimization variables, the other MTMDI groups are regarded as non-resonance MTMDI, the parameters are kept unchanged, and the effective mass m passes through enri The influence of the parameters on parameter optimization is taken into account, and each updated set of MTMDI parameters is obtained;
s5.3, calculating the offset of the updated frequency and damping ratio of the MTMDI and the change of the amplitude of each-order modal power amplification coefficient of the main beam under the combined action of all the MTMDI to obtain three offset vectors of iteration results, and measuring the size of the offset vectors by adopting a two-norm method;
and S5.4, judging whether the sizes of the three iteration result offset vectors are smaller than a preset allowable value or not, if not, repeating the step S5.2-5.3, and if the two norms of the three iteration result offset vectors are smaller than the preset allowable value, outputting an optimal MTMDI parameter.
2. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 1, which is characterized in that: the specific implementation of step S3 includes the following sub-steps:
s3.1, determining a candidate position for installing the MTMDI according to the internal structure of the main beam and the actual installation space limit, and determining modal vibration type values of each order corresponding to the candidate position;
and S3.2, each order mode is controlled by adopting a group of MTMDI, each candidate position is only provided with 1 TMDI, and the TMDI is placed from large to small one by one from low mode to high mode according to the vibration type value of each candidate position.
3. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 2, which is characterized in that: step S3.1, according to the coordinate of the candidate position of MTMDI installation, the corresponding sub-matrix phi is provided from the vibration mode matrix c ＝[φ c1 …φ ci …φ cN ]Wherein phi ci The mode shape value of the mode shape of the ith order of the candidate position is taken as the mode shape value of the candidate position;
s3.2, obtaining the arrangement position of each group of MTMDI, wherein the corresponding vibration mode matrix is phi c Is sub-matrix of
Figure FDA0003654308630000021
Wherein psi si The matrix is a q multiplied by N matrix, q is the number of sub TMDI of the MTMDI, N is the order of the target mode, and the superscript T is the transposition.
4. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 3, which is characterized in that: and S3.1, constructing the inner part of the main beam into a diaphragm plate distribution.
5. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 4, characterized in that: the specific implementation of step S4 includes the following sub-steps:
s4.1, determining the mass ratio of MTMDI (maximum Transmission mdi) of each order according to the vibration reduction requirement;
s4.2, mass ratio mu of MTMDI according to ith order mode i And calculating the physical quality of the sub TMDI, wherein the calculation formula is as follows:
Figure FDA0003654308630000022
wherein M is i The ith order modal quality of the bridge; m is i The physical mass of the neutron TMDI in the ith group of MTMDI; q is the sub TMDI number of the MTMDI group;
Figure FDA0003654308630000023
the model value of the ith order mode corresponding to the position of the h sub TMDI;
s4.3, determining the maximum static elongation of the TMDI according to the mounting space in the beam, so as to determine the inertial mass of the TMDI;
actual frequency ω of h-th sub TMDI h And damping ratio xi h Are respectively as
Figure FDA0003654308630000024
Figure FDA0003654308630000025
Wherein m is h ,c h ,k h And b h The physical mass, damping coefficient, rigidity and inertial mass of the h-th sub TMDI respectively;
the static elongation of the h-th sub TMDI is:
Figure FDA0003654308630000031
wherein g is the acceleration of gravity.
6. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 5, wherein: s5.1 obtaining each set of MTMDI parameter initial value X 0 ＝[P 10 …P i0 …P N0 ]In which P is i0 The initial parameters of the ith set of MTMDI are a 2 xq matrix, the first action is actual frequency, and the second action is actual damping ratio;
step S5.4 iteratively sets the termination condition as:
max(Norm1,Norm2,Norm3)＜ε
wherein ε is a predetermined tolerance.
7. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 6, characterized in that: s5.2, when the ith-order MTMDI is optimized, the frequency response function of the single-degree-of-freedom system under the action of the MTMDI is taken as an objective function, the optimized variables are the frequency and the damping ratio of the ith-order MTMDI, and the initial value is P i(k-1) The parameter of the rest of the non-resonant MTMDI is unchanged and is X k-1 Through effective mass m enri Taking into account the influence on parameter optimization, obtaining the absolute value H through an optimization algorithm i Frequency and damping ratio P of ith MTMDI group with minimum maximum value of (j omega) | ik ；
The ith order MTMDI optimized objective function | H i (j ω) | is:
Figure FDA0003654308630000032
wherein M is i ,C i And K i The ith order modal mass, damping and rigidity of the main structure are respectively; omega is vortex shedding frequency; j is an imaginary unit; m is ei The effective quality of MTMDI represents the dynamic action of MTMDI on the main structureUsing;
Figure FDA0003654308630000033
m is to ei Divided into two parts, the first part being the effective mass of the resonant MTMDI, i.e. the ith MTMDI, denoted m eri (ii) a The second part is the effective mass of the non-resonant MTMDI, namely the effective mass of the rest modal MTMDI, which is marked as m enri ；
When the ith order vortex vibration is taken as a target, the optimized variables are only the frequency and the damping ratio of the ith order MTMDI, the MTMDI parameter of the last iteration is taken as an initial value, and all the MTMDI parameters are contained in m eri The parameters of the rest non-resonant MTMDI are the MTMDI parameters obtained by the last iteration, and are kept unchanged and pass through the effective mass m enri Accounting for its effect on current MTMDI tuning;
through the optimization of the step, parameters of each group of MTMDI after the current iteration order, namely X, are obtained k ＝[P 1k …P ik …P Nk ]In which P is ik The initial parameters of the ith set of MTMDI after the kth iteration are a 2 xq matrix, the first action is the actual frequency, and the second action is the actual damping ratio.
8. The distributed MTMDI-based multi-order vortex-induced vibration control method for the large-span bridge according to claim 7, which is characterized in that: step S5.3, calculating the updated offset of the frequency and the damping ratio of the MTMDI and the change of the amplitude value of each-order modal power amplification coefficient of the main beam under the combined action of all the MTMDI, and calculating three offset vectors of iteration results:
ω offset ＝X k (1,:)-X k-1 (1,:)
ξ offset ＝X k (2,:)-X k-1 (2,:)
DMF offset ＝DMF k -DMF k-1
wherein, ω is offset And xi offset The actual frequency and damping ratio deviation of the MTMDI before and after iteration; DMF (dimethyl formamide) offset For each order mode of main structure before and after iterationDeviation of amplitude of power amplification factor; DMF (dimethyl formamide) k After the kth iteration, the amplitude of each order modal power amplification coefficient of the main structure under the action of all MTMDI is 1 xN row vectors, wherein the ith element is defined as:
DMF k (1,i)＝max(|H i (jω)|)/(1/K i )
the size of the offset vector is measured by using the two-norm of the vector, which is defined as follows:
Figure FDA0003654308630000041
where x is a row vector of n elements, the two-norm of the three resulting offset vectors is:
Norm1＝||ω offset || 2
Norm2＝||ξ offset || 2
Norm3＝||DMF offset || 2 。
CN202210554369.1A 2022-05-20 2022-05-20 Distributed MTMDI-based multi-order vortex-induced vibration control method for large-span bridge Pending CN114912324A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115637638A (en) * 2022-10-26 2023-01-24 哈尔滨工业大学 Variable inertial mass semi-active tuned mass damper inertial container and frequency tuning method
CN115637638B (en) * 2022-10-26 2024-01-12 哈尔滨工业大学 Inertial container of variable inertial semi-active tuning mass damper and frequency tuning method

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