Large-span bridge multi-order modal vortex vibration control method based on multiple SATMDI

Abstract

The invention discloses a multi-stage modal vortex-induced vibration control method for a long-span bridge based on multiple SATMDI, belongs to the field of structural vibration control, and aims at solving the problem that the setting optimization degree of an existing semi-active control inerter tuned mass damper SATMDI is not enough. The method comprises the following steps: step 1, establishing a bridge finite element model, and extracting modal information of each order; 2, screening the modals of the bridge, and retaining the modals of which the inherent frequency is lower than the highest vortex shedding frequency; step 3, determining a vortex vibration amplitude limit value [An] of each order of modality needing to be controlled, and further obtaining an allowable value of a peak value of a dynamic amplification coefficient of each order of modality of the bridge; step 4, dividing the reserved modals into G groups, and obtaining the sum of the minimum additional masses required by each group of modals; repeatedly executing the step 4 for P times, and regrouping each time to obtain P results; and step 5, taking the grouping strategy when the P results are minimum as the optimal grouping strategy, and obtaining the minimum additional mass required for controlling all modals and the corresponding SATMDI optimal layout position.

Description

Technical Field

[0001] The invention relates to a multi-order modal vortex vibration control method for a long-span bridge based on multiple SATMDIs, and belongs to the field of structural vibration control. Background Art

[0002] Vortex-induced vibration is a self-limiting vibration phenomenon caused by the alternating shedding of vortices when wind flows over the surface of a structure. Although vortex-induced vibration generally does not directly lead to damage to the bridge structure, for long-span bridges with flexible structures, vortex-induced vibration is prone to occur at lower wind speeds and occurs frequently, seriously affecting driving safety and reducing the fatigue life of the bridge structure. As the span of a bridge increases, it gradually exhibits dense frequency characteristics, and multi-order modal vortex-induced vibrations will occur at the design wind speed, posing new challenges to the vortex-induced vibration control of long-span bridges.

[0003] At present, one of the commonly used methods for vortex-induced vibration control is to set up a tuned mass damper (TMD). However, as the span of the bridge increases, the natural frequency of each mode gradually decreases. When controlling ultra-low frequency vertical vibration, TMD is difficult to manufacture due to its large static elongation, and it is also difficult to install in the main beam of the bridge. In addition, since the TMD has a single operating frequency, when controlling vortex vibration of multiple modes, a set of TMDs needs to be arranged for each mode, which increases the control cost and the additional mass of the bridge.

[0004] Tuned mass damper (TMDI) is a new type of mechanical shock absorber that introduces an inertia container into TMD and connects it in parallel with spring and damping units. Compared with traditional TMD, TMDI has the characteristics of high static stiffness and low dynamic stiffness. When controlling vertical vibration of the same frequency, its static elongation is significantly better than that of TMD. Semi-active control inertia tuned mass damper (SATMDI) introduces a controller into the inertia container. It can adjust the mass distribution to change the inertia mass without changing the physical mass, thereby adjusting the working frequency of SATMDI. Only one set of SATMDI is needed to achieve the control of multi-order modal vortex vibration. In fact, the vibration mode values ​​at the same position on the main beam are often not equal under different modes. In order to make SATMDI meet the control requirements under each target mode, it is necessary to reasonably select the layout position of SATMDI. If the layout position is not appropriate, the vibration reduction performance of SATMDI under a certain mode will be greatly weakened. Blindly increasing the mass of SATMDI to make each mode meet the control requirements will cause unnecessary waste. There is still room for optimization in the setting of the existing semi-active control inertia tuned mass damper SATMDI. Summary of the invention

[0005] In view of the problem that the setting optimization degree of the existing semi-active control inertia tuned mass damper SATMDI is not enough, the present invention provides a multi-order modal vortex vibration control method for a large-span bridge based on multiple SATMDIs, so as to meet the control requirements of each target mode while using the smallest possible additional mass.

[0006] The multi-order modal vortex vibration control method for a long-span bridge based on multiple SATMDIs of the present invention comprises the following steps:

[0007] Step 1: Establish a finite element model of the bridge and extract modal information of each order;

[0008] Step 2: Screen the modes of the bridge and retain the modes whose natural frequencies are lower than the highest vortex shedding frequency;

[0009] Step 3: Determine the vortex vibration amplitude limit [A n ], and then obtain the allowable value of the peak value of the dynamic amplification factor of each mode of the bridge;

[0010] Step 4: Divide the retained modes into G groups and obtain the sum of the minimum additional mass required for each group of modes m total ;

[0011] Repeat step 4 P times, regrouping each time, and obtain P m total ;

[0012] Step 5: P m total The grouping strategy with the minimum value in the middle is taken as the optimal grouping strategy, and the minimum additional mass required to control all modes and the corresponding optimal SATMDI layout position under the optimal grouping strategy are obtained.

[0013] Preferably, in step 1, the natural circular frequency ω of each mode is obtained through the bridge finite element model. n and generalized mass, which is obtained as follows:

[0014]

[0015] Among them, M n is the generalized mass of the nth mode, n=1,2,…,N, N is the number of modes; φ n is the vibration shape vector of the nth mode of the bridge, m is the node mass matrix, ω n is the natural circular frequency of the nth-order mode;

[0016] Extract the modal information of each order, where the natural frequency of each mode is

[0017] Preferably, the maximum vortex shedding frequency f in step 2 is max Get it by pressing:

[0018]

[0019] Among them, S t is the Storaha number of the bridge main beam section, U d is the design wind speed, and D is the characteristic height of the main beam section.

[0020] Preferably, the allowable value DMP of the peak value of the dynamic amplification factor of each mode of the bridge in step 3 is n Get it by pressing:

[0021]

[0022] Among them, [A n ] is the vortex vibration amplitude limit of the nth order mode, A vu is the vortex vibration amplitude of the bridge without control, ζ n is the damping ratio of the nth-order mode.

[0023] Preferably, step 4 comprises the following steps:

[0024] Step 41, sort the retained modes from small to large according to the natural frequency, and divide them into G groups according to the natural frequency, where the g-th group includes m modes, g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group mode to N g The masses of each SATMDI are equal, and the initial position of each SATMDI is set based on the principle that the vibration mode value at the position is as large as possible; the natural circular frequency ω of the highest-order mode in the group is set gm As the initial circular frequency ω0 of all SATMDI in this group: ω0 = ω gm ;

[0025] Step 42: Calculate the vortex vibration amplitude of each mode and control it within [A n ] is below, the required SATMDI minimum modal mass ratio μ gn :

[0026] According to the system of equations

[0027]

[0028] Get the minimum modal mass ratio μ of SATMDI gn for:

[0029]

[0030] Among them, DMP n is the allowable value of the peak value of the dynamic amplification factor of the nth mode of the bridge, n = 1, 2, ..., m; ω gn is the natural circular frequency of the nth order mode of the gth group; ωgm is the natural circular frequency of the mth-order mode of the gth group;

[0031] Step 43: According to the vibration mode values ​​of each SATMDI location, the minimum additional mass m for controlling each mode of the g-th group is obtained. g1 ,m g2 ,...,m gm , where the minimum additional mass m required for the nth mode is gn for:

[0032]

[0033] Among them, φ gn,i is the vibration shape value of the nth mode corresponding to the i-th SATMDI position of the g-th group. gn is the generalized mass of the nth mode of the gth group;

[0034] Step 44: Take the maximum value of the minimum additional mass required for the m modes in the g-th group as the minimum additional mass m of the g-th group. g :

[0035] m g =max(m g1 ,m g2 ,...,m gm )

[0036] Step 45, determine whether the minimum added mass of the g-th group is optimal, if so, execute step 46; if not, optimize the position of each SATMDI in the group, and return to step 43 to re-optimize the layout;

[0037] Step 46, determine whether the mass of a single SATMDI in the g-th group exceeds the upper limit allowed by the bearing capacity of the bridge and the manufacturing process level. If so, increase the number of SATMDIs, reset the positions of SATMDIs, and return to step 43 to re-optimize the layout; if not, execute step 47;

[0038] Step 47, determine whether the peak value of the dynamic amplification factor of a certain mode is less than 80% of the corresponding allowable value. If so, increase the number of SATMDIs, reset the position of SATMDIs, and return to step 43 to re-optimize the layout; if not, output the sum of the minimum additional mass required for each group of modes m total :

[0039]

[0040] Preferably, in step 45, the criterion for judging whether the minimum added mass of the g-th group reaches the optimum is: if the difference between the minimum added mass of the g-th group obtained by t layout optimization iterations and the minimum added mass obtained by the t-1-th iteration is less than the tolerance ε, the current solution is judged to be the optimum. The tolerance can be set as follows:

[0041]

[0042] In the formula, μ g1 ,μ g2 ,...,μ gm is the minimum modal mass ratio of SATMDI required for the 1st to mth modes of the gth group;

[0043] M g1 ,M g2 ,...,M gm is the generalized mass of the 1st to mth modes of the gth group.

[0044] Preferably, the mass of a single SATMDI in group g is Calculate as follows:

[0045]

[0046] Preferably, the upper limit of the mass of a single SATMDI allowed by the bearing capacity and manufacturing process level of the bridge is 10 tons.

[0047] Beneficial effects of the present invention:

[0048] 1. The present invention adopts distributed multiple SATMDI to control multi-order modal vortex vibration of a long-span bridge, avoiding the problem of excessive static elongation and difficulty in manufacturing and installation in traditional vortex vibration control using TMD.

[0049] 2. The SATMDI used in the present invention has a frequency regulation function, and only one set of SATMDI is required to achieve the control of multi-order modal vortex vibration. At the same time, the present invention only needs a small additional mass to achieve the same control effect as the traditional passive TMD, reducing the control cost of vortex vibration. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 It is a flow chart of the multi-order modal vortex vibration control method of a long-span bridge based on multiple SATMDIs according to the present invention;

[0051] Figure 2 It is a flow chart of step 4 of the multi-order modal vortex-vibration control method for a large-span bridge based on multiple SATMDIs described in the present invention. DETAILED DESCRIPTION

[0052] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0053] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0054] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0055] Specific implementation method 1: The following is combined Figure 1 to Figure 2 This embodiment is described. The multi-order modal vortex vibration control method of a long-span bridge based on multiple SATMDIs described in this embodiment includes the following steps:

[0056] Step 1: Establish a finite element model of the bridge and extract modal information of each order;

[0057] Step 2: Screen the modes of the bridge and retain the modes whose natural frequencies are lower than the highest vortex shedding frequency;

[0058] Step 3: Determine the vortex vibration amplitude limit [A n ], and then obtain the allowable value of the peak value of the dynamic amplification factor of each mode of the bridge;

[0059] Step 4: Divide the retained modes into G groups and obtain the sum of the minimum additional mass required for each group of modes m total ;

[0060] Repeat step 4 P times, regrouping each time, and obtain P m total ;

[0061] Step 5: P m total The grouping strategy with the minimum value in the middle is taken as the optimal grouping strategy, and the minimum additional mass required to control all modes and the corresponding optimal SATMDI layout position under the optimal grouping strategy are obtained.

[0062] Step 1: According to the construction drawings, a finite element model of the bridge is established, and the vibration mode vectors and frequencies of each mode of the bridge are obtained through modal analysis. Each mode mode is normalized according to the maximum value of the vibration mode vector, and the generalized mass and generalized stiffness of each mode of the bridge are calculated in combination with the node mass, simplifying the bridge into a single degree of freedom main structure.

[0063] Obtain the natural circular frequency ω of each mode through the bridge finite element model nand generalized mass, which is obtained as follows:

[0064]

[0065] Among them, M n is the generalized mass of the nth mode, n=1,2,…,N, N is the number of modes; φ n is the vibration shape vector of the nth mode of the bridge, m is the node mass matrix, ω n is the natural circular frequency of the nth-order mode;

[0066] Extract the modal information of each order, where the natural frequency of each mode is

[0067] Step 2: According to the Storahar number of the bridge main beam section and the design wind speed, the highest possible vortex shedding frequency is determined, and the modes of each order of the bridge are screened to retain the modes whose natural frequencies are lower than the highest frequency.

[0068] Maximum vortex shedding frequency f max Get it by pressing:

[0069]

[0070] Among them, S t is the Storaha number of the bridge main beam section, U d is the design wind speed, and D is the characteristic height of the main beam section.

[0071] Step 3: Determine the vortex vibration amplitude limit of each bridge mode and the allowable value DMP of the peak value of the dynamic amplification factor of each bridge mode according to the current specifications, driving safety and owner requirements. n Get it by pressing:

[0072]

[0073] Among them, [A n ] is the vortex vibration amplitude limit of the nth order mode, A vu is the vortex vibration amplitude of the bridge without control, ζ n is the damping ratio of the nth-order mode.

[0074] Step 4 includes the following steps:

[0075] Step 41, sort the retained modes from small to large according to the natural frequency, and divide them into G groups according to the natural frequency, where the g-th group includes m modes, g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group mode to N gThe masses of each SATMDI are equal, and the initial position of each SATMDI is set based on the principle that the vibration mode value at the position is as large as possible; the natural circular frequency ω of the highest-order mode in the group is set gm As the initial circular frequency ω0 of all SATMDI in this group: ω0 = ω gm ;

[0076] SATMDI can adjust its operating frequency by changing the inertial mass, but when the inertia ratio is too large, the control performance of SATMDI will be significantly reduced. Therefore, it is necessary to divide the modes into G groups according to the frequency of each mode of the bridge and the peak allowable value of the dynamic amplification factor. Assuming that each group of modes is independent of each other, the layout optimization design of each group of modes is performed separately. Assuming that the g-th group has m-order modes, the number of SATMDIs controlling the g-th group of modes is N. g , the initial circular frequency ω0 of SATMDI is taken as the natural circular frequency ω of the highest-order mode in this group gm .

[0077] The grouping is described with reference to the embodiments.

[0078] Assuming that the natural frequency of the highest-order mode is less than 0.6 Hz, the modes can be divided into 2 to 3 groups according to the natural frequency. When formulating the initial grouping, for the case of being divided into 2 groups,

[0079]

[0080] For the case of being divided into 3 groups,

[0081]

[0082] Among them, f 1n is the natural frequency of the first group of n-th order modes, f 2n is the natural frequency of the second group of n-th order modes, f 3n is the natural frequency of the third group of n-th order modes.

[0083] Step 42: Calculate the vortex vibration amplitude of each mode and control it within [A n ] is below, the required SATMDI minimum modal mass ratio μ gn :

[0084] According to the system of equations

[0085]

[0086] Get the minimum modal mass ratio μ of SATMDI gn for:

[0087]

[0088] Among them, DMPn is the allowable value of the peak value of the dynamic amplification factor of the nth mode of the bridge, n = 1, 2, ..., m; ω gn is the natural circular frequency of the nth order mode of the gth group; ω gm is the natural circular frequency of the mth-order mode of the gth group;

[0089] In order to give full play to the control performance of SATMDI, it is necessary to adjust the operating frequency of SATMDI to the best. When SATMDI is tuned with the nth order mode, the required inertia ratio is:

[0090] When SATMDI is adjusted to the optimum value, the peak value of the power amplification factor is:

[0091]

[0092] The allowable value of the peak value of the dynamic amplification factor of the n-th mode is taken as the maximum value of the dynamic amplification factor of this mode under SATMDI control. The equation set established by the two formulas of inertia ratio and dynamic amplification factor peak value can obtain the minimum modal mass ratio required by SATMDI when controlling the n-th mode:

[0093]

[0094] Step 43: According to the vibration mode values ​​of each SATMDI location, the minimum additional mass m for controlling each mode of the g-th group is obtained. g1 ,m g2 ,...,m gm , where the minimum additional mass m required for the nth mode is gn for:

[0095]

[0096] Among them, φ gn,i is the mode shape value of the nth mode corresponding to the i-th SATMDI position of the g-th group, which is obtained according to the mode shape vector mentioned in the bridge finite element model. gn is the generalized mass of the nth mode of the gth group;

[0097] Step 44: Take the maximum value of the minimum additional mass required for the m modes in the g-th group as the minimum additional mass m of the g-th group. g :

[0098] m g =max(m g1 ,m g2 ,...,m gm )

[0099] In order to make SATMDI control the amplitude of each mode in this group within the corresponding limit, the minimum additional mass required to control the g-th group of modes should be taken as the maximum value of the minimum additional mass required to control each order of mode. By optimizing the layout position of this group of SATMDI and repeating step 43, the minimum additional mass m required for all g-th modes can be obtained. g1 ,m g2 ,...,m gm .

[0100] Step 45, determine whether the minimum added mass of the g-th group is optimal, if so, execute step 46; if not, optimize the position of each SATMDI in the group, and return to step 43 to re-optimize the layout;

[0101] The criterion for judging whether the minimum added mass of the g-th group is optimal is: the difference between the minimum added mass of the g-th group obtained by t layout optimization iterations and the minimum added mass obtained by the t-1-th iteration is less than the tolerance ε, and the current solution is judged to be optimal. Each group of layout optimization should be more than 3 times, and each time the layout can be changed by adjusting the SATMDI layout position, and the minimum added mass of this layout scheme is calculated. The tolerance can be set as follows:

[0102]

[0103] In the formula, μ g1 ,μ g2 ,...,μ gm is the minimum modal mass ratio of SATMDI required for the 1st to mth modes of the gth group;

[0104] M g1 ,M g2 ,...,M gm is the generalized mass of the 1st to mth modes of the gth group.

[0105] Single SATMDI mass Calculate as follows:

[0106]

[0107] Step 46, determine whether the mass of a single SATMDI in the g-th group exceeds the upper limit allowed by the bearing capacity of the bridge and the manufacturing process level. If so, increase the number of SATMDIs, reset the positions of SATMDIs, and return to step 43 to re-optimize the layout; if not, execute step 47;

[0108] The load-bearing capacity and manufacturing process level of the bridge allow the upper limit of the mass of a single SATMDI to be 10 tons.

[0109] Step 47, determine whether the peak value of the dynamic amplification factor of a certain mode is less than 80% of the corresponding allowable value. If so, increase the number of SATMDIs, reset the position of SATMDIs, and return to step 43 to re-optimize the layout; if not, output the sum of the minimum additional mass required for each group of modes m total :

[0110]

[0111] If the peak value of the dynamic amplification factor of a certain mode is much smaller than the corresponding allowable value, it means that the number of SATMDIs is too small, and the vibration mode value of its layout position in a certain mode is small, and the control performance cannot be fully exerted. At this time, the minimum added mass obtained will also be too large, and the number of SATMDIs should also be increased, and return to step 43 to re-optimize the layout.

[0112] Repeat step 4 P times, regrouping each time, and obtain P m total ; The regrouping is to re-divide the number of groups and the natural frequency range within the group according to the natural frequency.

[0113] Step 5: P m total The grouping strategy with the minimum value in the middle is taken as the optimal grouping strategy, and the minimum additional mass required to control all modes and the corresponding optimal SATMDI layout position under the optimal grouping strategy are obtained.

[0114] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the present invention. It should therefore be understood that many modifications may be made to the exemplary embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in a manner different from that described in the original claims. It should also be understood that the features described in conjunction with a single embodiment may be used in other described embodiments.

Claims

1. A multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDIs, characterized in that: The method comprises the following steps: Step 1: Establish a finite element model of the bridge and extract modal information of each order; Step 2: Screen the modes of the bridge and retain the modes whose natural frequencies are lower than the highest vortex shedding frequency; Step 3: Determine the vortex vibration amplitude limit [A n ], and then obtain the allowable value of the peak value of the dynamic amplification factor of each mode of the bridge; Step 4: Divide the retained modes into G groups and obtain the sum of the minimum additional mass required for each group of modes m total ; Repeat step 4 P times, regrouping each time, and obtain P m total ; Step 5: P m total The grouping strategy with the minimum value in the middle is taken as the optimal grouping strategy, and the minimum additional mass required to control all modes and the corresponding optimal SATMDI layout position under the optimal grouping strategy are obtained.

2. According to claim 1, the multi-order modal vortex vibration control method for a long-span bridge based on multiple SATMDI is characterized in that: In step 1, the natural circular frequency ω of each mode is obtained through the bridge finite element model n and generalized mass, which is obtained as follows: Among them, M n is the generalized mass of the nth mode, n=1,2,…,N, N is the number of modes; φ n is the vibration shape vector of the nth mode of the bridge, m is the node mass matrix, ω n is the natural circular frequency of the nth-order mode; Extract the modal information of each order, where the natural frequency of each mode is 3. According to claim 2, the multi-order modal vortex vibration control method for a long-span bridge based on multiple SATMDI is characterized in that: The highest vortex shedding frequency f in step 2 max Press the formula to obtain: Among them, S t is the Storaha number of the bridge main beam section, U d is the design wind speed, and D is the characteristic height of the main beam section.

4. According to claim 1, the multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDIs is characterized in that: The allowable value DMP of the peak value of the dynamic amplification factor of each mode of the bridge in step 3 n Get it by pressing: Among them, [A n ] is the vortex vibration amplitude limit of the nth order mode, A vu is the vortex vibration amplitude of the bridge without control, ζ n is the damping ratio of the nth-order mode.

5. According to claim 1, the multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDIs is characterized in that: Step 4 includes the following steps: Step 41, sort the retained modes from small to large according to the natural frequency, and divide them into G groups according to the natural frequency, where the g-th group includes m modes, g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group mode to N g The masses of each SATMDI are equal, and the initial position of each SATMDI is set based on the principle that the vibration mode value at the position is as large as possible; the natural circular frequency ω of the highest-order mode in the group is set gm As the initial circular frequency ω0 of all SATMDI in this group: ω0 = ω gm ; Step 42: Calculate the vortex vibration amplitude of each mode and control it in [A n ] is below, the required SATMDI minimum modal mass ratio μ gn : According to the system of equations Get the minimum modal mass ratio μ of SATMDI gn for: Among them, DMP n is the allowable value of the peak value of the dynamic amplification factor of the nth mode of the bridge, n = 1, 2, ..., m; ω gn is the natural circular frequency of the nth order mode of the gth group; ω gm is the natural circular frequency of the mth-order mode of the gth group; Step 43: According to the vibration mode values ​​of each SATMDI location, the minimum additional mass m for controlling each mode of the g-th group is obtained. g1 ,m g2 ,...,m gm , where the minimum additional mass m required for the nth mode is gn for: Among them, φ gn,i is the vibration shape value of the nth mode corresponding to the i-th SATMDI position of the g-th group. gn is the generalized mass of the nth mode of the gth group; Step 44: Take the maximum value of the minimum additional mass required for the m modes in the g-th group as the minimum additional mass m of the g-th group. g : m g =max(m g1 ,m g2 ,...,m gm ) Step 45, determine whether the minimum added mass of the g-th group is optimal, if so, execute step 46; if not, optimize the position of each SATMDI in the group, and return to step 43 to re-optimize the layout; Step 46, determine whether the mass of a single SATMDI in the g-th group exceeds the upper limit allowed by the bearing capacity of the bridge and the manufacturing process level. If so, increase the number of SATMDIs, reset the positions of SATMDIs, and return to step 43 to re-optimize the layout; if not, execute step 47; Step 47, determine whether the peak value of the dynamic amplification factor of a certain mode is less than 80% of the corresponding allowable value. If so, increase the number of SATMDIs, reset the position of SATMDIs, and return to step 43 to re-optimize the layout; if not, output the sum of the minimum additional mass required for each group of modes m total :

6. The multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDI according to claim 5 is characterized in that: In step 45, the criterion for judging whether the minimum added mass of the g-th group is optimal is: if the difference between the minimum added mass of the g-th group obtained by t layout optimization iterations and the minimum added mass obtained by the t-1-th iteration is less than the tolerance ε, the current solution is judged to be optimal. The tolerance can be set as follows: In the formula, μ g1 ,μ g2 ,...,μ gm is the minimum modal mass ratio of SATMDI required for the 1st to mth modes of the gth group; M g1 ,M g2 ,...,M gm is the generalized mass of the 1st to mth modes of the gth group.

7. The multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDI according to claim 5 is characterized in that: The mass of individual SATMDI in group g Calculate as follows:

8. The multi-modal vortex vibration control method for a long-span bridge based on multiple SATMDIs according to claim 5 is characterized in that: The load-bearing capacity and manufacturing process level of the bridge allow the upper limit of the mass of a single SATMDI to be 10 tons.

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