Multimodal vortex vibration control method for long-span bridges based on multiple SATMDI

Abstract

This invention relates to a multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI (Self-Active Capacitive Tuned Mass Damper) dampers, belonging to the field of structural vibration control. The invention addresses the problem of insufficient optimization in existing semi-active controlled SATMDI settings. The method comprises the following steps: Step 1: Establishing a finite element model of the bridge and extracting modal information for each order; Step 2: Screening the bridge modes, retaining modes with natural frequencies lower than the highest vortex shedding frequency; Step 3: Determining the vortex-induced vibration amplitude limits [A] for each mode requiring control. n ], and then obtain the allowable value of the peak value of the dynamic amplification coefficient 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; Repeat step 4 P times, regrouping each time, and obtain P results; Step 5, take the grouping strategy with the minimum value among the P results as the optimal grouping strategy, and obtain the minimum additional mass required to control all modes and the corresponding optimal SATMDI deployment position.

Description

Technical Field

[0001] This invention relates to a multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI modes, belonging to the field of structural vibration control. Background Technology

[0002] Vortex-induced vibration (VID) is a self-limiting vibration phenomenon caused by the alternating shedding of vortices as wind flows over a structural surface. While VID generally does not directly lead to bridge structural failure, it is particularly prevalent in long-span bridges with flexible structures, occurring frequently even at low wind speeds. This can severely impact traffic safety and reduce the fatigue life of the bridge structure. As bridge spans increase, they gradually exhibit dense-frequency characteristics, leading to multi-mode VID under design wind speeds, posing new challenges to VID control in long-span bridges.

[0003] One common method for controlling vortex-induced vibration is the use of tuned mass dampers (TMDs). However, as bridge spans increase, the natural frequencies of each mode gradually decrease. When controlling ultra-low frequency vertical vibrations, TMDs are difficult to manufacture due to their excessive static elongation and are also difficult to install within the bridge's main girder. Furthermore, because TMDs operate at a single frequency, controlling multi-mode vortex-induced vibration requires a separate set of TMDs for each mode, increasing control costs and adding mass to the bridge.

[0004] A TMDI (Tunable Mass Inertia Damper) is a novel type of mechanical vibration damper that incorporates an inertial container into the TMD and connects it in parallel with springs and damping units. Compared to traditional TMDs, TMDIs, characterized by high static stiffness and low dynamic stiffness, exhibit significantly better static elongation when controlling vertical vibrations at the same frequency. A semi-active controlled TMDI (Semi-Active Tunable Mass Inertia Damper) incorporates a controller into the inertial container, allowing adjustment of the mass distribution to alter the inertial mass without changing the physical mass, thereby regulating the operating frequency of the SATMDI. Only one SATMDI is needed to control multi-mode vortex-induced vibrations. In reality, the mode shapes at the same location on the main beam are often unequal under different modes. To ensure that the SATMDI meets control requirements in all target modes, its placement must be carefully selected. Inappropriate placement will significantly weaken the vibration damping performance of the SATMDI in a particular mode. Blindly increasing the mass of the SATMDI to meet control requirements in all modes would be an unnecessary waste. There is still room for optimization in the existing semi-active control inertial capacitive tuned mass damper (SATMDI). Summary of the Invention

[0005] To address the issue of insufficient optimization in the existing semi-active control inertial capacitive tuned mass damper (SATMDI), this invention provides a multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDIs, which satisfies the control requirements of each target mode while using the smallest possible additional mass.

[0006] The present invention describes a multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI modes, which includes the following steps:

[0007] Step 1: Establish a finite element model of the bridge and extract modal information for 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 limits for each mode that needs to be controlled [A] n ], thereby obtaining 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 masses m required for each group of modes. total ;

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

[0012] Step 5, using P units of m total The grouping strategy at the minimum value is taken as the optimal grouping strategy. The minimum additional mass required to control all modes under the optimal grouping strategy and the corresponding optimal SATMDI deployment position are obtained.

[0013] Preferably, in step 1, the natural circular frequencies ω of each mode are obtained using a bridge finite element model. n And generalized quality, which is obtained in the following ways:

[0014]

[0015] Among them, M n φ represents the generalized mass of the nth mode, where n = 1, 2, ..., N, and N is the number of modes; n Let m be the mode shape vector of the nth mode of the bridge, m be the nodal mass matrix, and ω be the mode shape vector. n Let be the natural circular frequency of the nth mode;

[0016] Extract modal information for each order, where the natural frequencies of each mode are:

[0017] Preferably, the highest vortex shedding frequency f in step 2 max Obtain by the following formula:

[0018]

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

[0020] Preferably, the allowable peak value of the dynamic amplification factor of each mode of the bridge in step 3 is DMP. n Obtain by the following formula:

[0021]

[0022] Among them, [A] n [A] represents the amplitude limit of the vortex-induced vibration of the nth mode. vu ζ represents the vortex-induced vibration amplitude of the bridge under uncontrolled conditions. n Let be the damping ratio of the nth mode.

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

[0024] Step 41: Sort the remaining modes according to their natural frequencies from smallest to largest, and divide them into G groups according to their natural frequencies. The g-th group includes m modes, where g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group of modes to N. g Each SATMDI has an equal mass, and its initial position is set based on maximizing the mode shape value at its location. The natural circular frequency ω of the highest-order mode within the group is set. gm The initial angular frequency ω0 for all SATMDI signals in this group, i.e., ω0 = ω gm ;

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

[0026] According to the system of equations

[0027]

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

[0029]

[0030] Among them, DMP n ω is the allowable peak value of the dynamic amplification factor for the nth mode of the bridge, where n = 1, 2, ..., m; gn ω is the natural angular frequency of the nth mode in the g-th group;gm Let be the natural circular frequency of the m-th mode in the g-th group;

[0031] Step 43: Based on the mode shape values ​​at each SATMDI location, obtain the minimum additional mass m controlling each mode of the g-th group. g1 ,m g2 ,...,m gm The minimum additional mass m required for the nth mode gn for:

[0032]

[0033] Where, φ gn,i M represents the mode shape value of the nth mode corresponding to the i-th SATMDI position in the g-th group. gn Let g be 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 additional mass of group g has reached the optimal level. If yes, proceed to step 46; otherwise, optimize the position of each SATMDI in the group and return to step 43 to redeploy and optimize.

[0037] Step 46: Determine whether the mass of a single SATMDI in group g exceeds the upper limit allowed by the bridge's load-bearing capacity and manufacturing process. If yes, increase the number of SATMDIs, reset the position of the SATMDIs, and return to step 43 to re-optimize the layout. If no, proceed to step 47.

[0038] Step 47: Determine if 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 redo the layout optimization. If not, output the sum of the minimum additional masses m required for each group of modes. total :

[0039]

[0040] Preferably, in step 45, the criterion for determining whether the minimum additional mass of group g is optimal is: the difference between the minimum additional mass of group g obtained through t iterations of layout optimization and the minimum additional mass obtained through the (t-1)th iteration is less than the tolerance ε, and the current solution is determined to be optimal. The tolerance can be set by the following formula:

[0041]

[0042] In the formula, μ g1 ,μ g2 ,...,μ gm The minimum SATMDI modal mass ratio required for the first to mth modes in group g;

[0043] M g1 M g2 ,...,M gm It represents the generalized mass of the first to mth modes in the g-th group.

[0044] Preferably, the mass of a single SATMDI within group g Calculate using the following formula:

[0045]

[0046] Preferably, the maximum mass of a single SATMDI allowed by the bridge's load-bearing capacity and manufacturing process is 10 tons.

[0047] The beneficial effects of this invention are:

[0048] 1. This invention employs distributed multi-mode SATMDI control of multi-span bridge vortex-induced vibration, avoiding the problem of excessive static elongation that makes manufacturing and installation difficult in traditional vortex-induced vibration control using TMD.

[0049] 2. The SATMDI used in this invention has a frequency regulation function, and only one SATMDI is needed to control multi-mode vortex vibration. Furthermore, this invention requires only a small additional mass to achieve the same control effect as traditional passive TMDs, thus reducing the control cost of vortex vibration. Attached Figure Description

[0050] Figure 1 This is a flowchart of the multi-mode vortex vibration control method for long-span bridges based on multiple SATMDI as described in this invention;

[0051] Figure 2 This is a flowchart of step 4 of the multi-mode vortex vibration control method for long-span bridges based on multiple SATMDI described in this invention. Detailed Implementation

[0052] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0053] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0054] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0055] Specific Implementation Method 1: The following is combined with... Figures 1 to 2 This embodiment describes a multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI (Synchronous Mode Modal Idiometry). The method includes the following steps:

[0056] Step 1: Establish a finite element model of the bridge and extract modal information for 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 limits for each mode that needs to be controlled [A] n ], thereby obtaining 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 masses m required for each group of modes. total ;

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

[0061] Step 5, using P units of m total The grouping strategy at the minimum value is taken as the optimal grouping strategy. The minimum additional mass required to control all modes under the optimal grouping strategy and the corresponding optimal SATMDI deployment position are obtained.

[0062] Step 1: Based on the construction drawings, establish a finite element model of the bridge. Obtain the mode shape vectors and frequencies of each mode through modal analysis. Normalize each mode shape according to the maximum value of the mode shape vector. Combined with the nodal mass, calculate the generalized mass and generalized stiffness of each mode of the bridge, simplifying the bridge into a single-degree-of-freedom main structure.

[0063] The natural circular frequencies ω of each mode were obtained using a finite element model of the bridge. nAnd generalized quality, which is obtained in the following ways:

[0064]

[0065] Among them, M n φ represents the generalized mass of the nth mode, where n = 1, 2, ..., N, and N is the number of modes; n Let m be the mode shape vector of the nth mode of the bridge, m be the nodal mass matrix, and ω be the mode shape vector. n Let be the natural circular frequency of the nth mode;

[0066] Extract modal information for each order, where the natural frequencies of each mode are:

[0067] Step 2: Based on the Storaha number of the bridge's main beam section and the design wind speed, determine the highest possible vortex shedding frequency, screen the various modes of the bridge, and retain the modes whose natural frequencies are lower than the highest frequency.

[0068] The highest vortex shedding frequency f max Obtain by the following formula:

[0069]

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

[0071] Step 3: Determine the vortex-induced vibration amplitude limit for each mode of the bridge, and the allowable peak value (DMP) of the dynamic amplification factor for each mode of the bridge, based on current specifications, traffic safety requirements, and the owner's requirements. n Obtain by the following formula:

[0072]

[0073] Among them, [A] n [A] represents the amplitude limit of the vortex-induced vibration of the nth mode. vu ζ represents the vortex-induced vibration amplitude of the bridge under uncontrolled conditions. n Let be the damping ratio of the nth mode.

[0074] Step 4 includes the following steps:

[0075] Step 41: Sort the remaining modes according to their natural frequencies from smallest to largest, and divide them into G groups according to their natural frequencies. The g-th group includes m modes, where g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group of modes to N. gEach SATMDI has an equal mass, and its initial position is set based on maximizing the mode shape value at its location. The natural circular frequency ω of the highest-order mode within the group is set. gm The initial angular frequency ω0 for all SATMDI signals in this group, i.e., ω0 = ω gm ;

[0076] SATMDI can adjust its operating frequency by changing the inertial mass, but its control performance will significantly decrease when the inertial-to-mass ratio is too high. Therefore, the modes need to be divided into G groups based on the frequencies and allowable peak values ​​of the dynamic amplification coefficients of each level of the bridge's modes. Assuming that each group of modes is independent, the layout optimization design is performed for each group of modes separately. Let the g-th group have m orders of modes, and the number of SATMDIs controlling the g-th group of modes be N. g The initial angular frequency ω0 of SATMDI is taken as the natural angular frequency ω of the highest-order mode in this group. gm .

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

[0078] Assuming the natural frequency of the highest-order mode is less than 0.6 Hz, the modes can be divided into 2 to 3 groups based on their natural frequencies. When determining the initial grouping, for the case of dividing into 2 groups,

[0079]

[0080] For the case of dividing into 3 groups

[0081]

[0082] Among them, f 1n f is the natural frequency of the nth mode in the first group. 2n f is the natural frequency of the nth mode in the second group. 3n is the natural frequency of the nth mode in the third group.

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

[0084] According to the system of equations

[0085]

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

[0087]

[0088] Among them, DMPn ω is the allowable peak value of the dynamic amplification factor for the nth mode of the bridge, where n = 1, 2, ..., m; gn ω is the natural angular frequency of the nth mode in the g-th group; gm Let be the natural circular frequency of the m-th mode in the g-th group;

[0089] To fully utilize the control performance of SATMDI, its operating frequency needs to be optimized. When SATMDI is tuned to the nth mode, the required inertia-to-mass ratio is:

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

[0091]

[0092] Taking the allowable peak value of the dynamic amplification factor of the nth mode as the maximum value of the dynamic amplification factor of this mode under SATMDI control, the system of equations established by the formulas for inertia-mass ratio and peak value of dynamic amplification factor yields the minimum modal mass ratio required by SATMDI to control the nth mode:

[0093]

[0094] Step 43: Based on the mode shape values ​​at each SATMDI location, obtain the minimum additional mass m controlling each mode of the g-th group. g1 ,m g2 ,...,m gm The minimum additional mass m required for the nth mode gn for:

[0095]

[0096] Where, φ gn,i M represents the mode shape value of the nth mode corresponding to the i-th SATMDI position in the g-th group. This mode shape value is obtained based on the mode shape vector mentioned in the bridge finite element model. gn Let g be 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] To ensure that SATMDI can control the amplitude of each mode within this group within its corresponding limit, the minimum additional mass required to control the g-th mode should be taken as the maximum value of the minimum additional mass required to control each mode. By optimizing the placement of this group of SATMDIs 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 additional mass of group g has reached the optimal level. If yes, proceed to step 46; otherwise, optimize the position of each SATMDI in the group and return to step 43 to redeploy and optimize.

[0101] The criterion for determining whether the minimum additional mass of group g is optimal is: the difference between the minimum additional mass of group g obtained through t iterations of layout optimization and the minimum additional mass obtained through the (t-1)th iteration is less than the tolerance ε; the current solution is then considered optimal. Each layout optimization should be performed more than 3 times. Each time, the layout can be changed by adjusting the SATMDI placement position, and the minimum additional mass of this layout scheme can be calculated. The tolerance can be set using the following formula:

[0102]

[0103] In the formula, μ g1 ,μ g2 ,...,μ gm The minimum SATMDI modal mass ratio required for the first to mth modes in group g;

[0104] M g1 M g2 ,...,M gm It represents the generalized mass of the first to mth modes in the g-th group.

[0105] Single SATMDI quality Calculate using the following formula:

[0106]

[0107] Step 46: Determine whether the mass of a single SATMDI in group g exceeds the upper limit allowed by the bridge's load-bearing capacity and manufacturing process. If yes, increase the number of SATMDIs, reset the position of the SATMDIs, and return to step 43 to re-optimize the layout. If no, proceed to step 47.

[0108] The maximum allowable mass of a single SATMDI bridge, based on its load-bearing capacity and manufacturing process, is 10 tons.

[0109] Step 47: Determine if 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 redo the layout optimization. If not, output the sum of the minimum additional masses m required for each group of modes. 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 indicates that the number of SATMDIs is too small and the mode shape value of its placement position in a certain mode is small, which cannot fully exert the control performance. At this time, the minimum additional mass obtained will also be too large, and the number of SATMDIs should be increased. Return to step 43 to re-optimize the placement.

[0112] Repeat step 4 P times, regrouping each time, to obtain P m groups. total Regrouping involves re-dividing the number of groups and the range of natural frequencies within each group based on the natural frequencies.

[0113] Step 5, using P units of m total The grouping strategy at the minimum value is taken as the optimal grouping strategy. The minimum additional mass required to control all modes under the optimal grouping strategy and the corresponding optimal SATMDI deployment position are obtained.

[0114] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI modes, characterized in that, The method includes the following steps: Step 1: Establish a finite element model of the bridge and extract modal information for 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 limits for each mode that needs to be controlled [A] n ], thereby obtaining 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 masses m required for each group of modes. total ; Repeat step 4 P times, regrouping each time, to obtain P m groups. total ; Step 5, using P units of m total The grouping strategy at the minimum value is taken as the optimal grouping strategy. The minimum additional mass required to control all modes under the optimal grouping strategy and the corresponding optimal SATMDI deployment position are obtained. Step 4 includes the following steps: Step 41: Sort the remaining modes according to their natural frequencies from smallest to largest, and divide them into G groups according to their natural frequencies. The g-th group includes m modes, where g = 1, 2, ..., G; set the number of SATMDIs controlling the g-th group of modes to N. g Each SATMDI has an equal mass, and its initial position is set based on maximizing the mode shape value at its location. The natural circular frequency ω of the highest-order mode within the group is set. gm The initial angular frequency ω0 for all SATMDI signals in this group, i.e., ω0 = ω gm ; Step 42: Calculate the vortex vibration amplitude for each mode and control it within [A]. n The required minimum modal mass ratio μ of SATMDI is as follows: gn : According to the system of equations Where, δ gn Inertia-mass ratio; Obtain the minimum modal mass ratio μ of SATMDI gn for: Among them, DMP n ω is the allowable peak value of the dynamic amplification factor for the nth mode of the bridge, where n = 1, 2, ..., m; gn ω is the natural angular frequency of the nth mode in the g-th group; gm Let be the natural circular frequency of the m-th mode in the g-th group; Step 43: Based on the mode shape values ​​at each SATMDI location, obtain the minimum additional mass m controlling each mode of the g-th group. g1 ,m g2 ,...,m gm The minimum additional mass m required for the nth mode gn for: Where, φ gn,i M represents the mode shape value of the nth mode corresponding to the i-th SATMDI position in the g-th group; gn Let g be 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 additional mass of group g has reached the optimal level. If yes, proceed to step 46; otherwise, optimize the position of each SATMDI in the group and return to step 43 to redeploy and optimize. Step 46: Determine whether the mass of a single SATMDI in group g exceeds the upper limit allowed by the bridge's load-bearing capacity and manufacturing process. If yes, increase the number of SATMDIs, reset the position of the SATMDIs, and return to step 43 to re-optimize the layout. If no, proceed to step 47. Step 47: Determine if 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 redo the layout optimization. If not, output the sum of the minimum additional masses m required for each group of modes. total :

2. The method for controlling multi-mode vortex-induced vibration of long-span bridges based on multiple SATMDI as described in claim 1, characterized in that, In step 1, the natural circular frequencies ω of each mode are obtained using a finite element model of the bridge. n And generalized quality, which is obtained in the following ways: M n =φ n τ mφ n Among them, M n φ represents the generalized mass of the nth mode, where n = 1, 2, ..., N, and N is the number of modes; n Let m be the mode shape vector of the nth mode of the bridge, m be the nodal mass matrix, and ω be the mode shape vector. n Let be the natural circular frequency of the nth mode; Extract modal information for each order, where the natural frequencies of each mode are:

3. The multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI as described in claim 2, characterized in that, The highest vortex shedding frequency f in step 2 max Obtain by the following formula: Among them, S t U is the Storaha number of the main girder section of the bridge. d For the design wind speed, D is the characteristic height of the main beam section.

4. The multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI as described in claim 1, characterized in that, The allowable peak value (DMP) of the dynamic amplification factor for each mode of the bridge in step 3. n Obtain by the following formula: Among them, [A] n [A] represents the amplitude limit of the vortex-induced vibration of the nth mode. vu ζ represents the vortex-induced vibration amplitude of the bridge under uncontrolled conditions. n Let be the damping ratio of the nth mode.

5. The multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI as described in claim 1, characterized in that, In step 45, the criterion for determining whether the minimum additional mass of group g is optimal is: if the difference between the minimum additional mass of group g obtained through t iterations of optimization and the minimum additional mass obtained through the (t-1)th iteration is less than the tolerance ε, the current solution is considered optimal; the tolerance can be set by the following formula: In the formula, μ g1 ,μ g2 ,...,μ gm The minimum SATMDI modal mass ratio required for the first to mth modes in group g; M g1 M g2 ,...,M gm It represents the generalized mass of the first to mth modes in the g-th group.

6. The multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI as described in claim 1, characterized in that, Mass of a single SATMDI within group g Calculate using the following formula:

7. The multi-mode vortex-induced vibration control method for long-span bridges based on multiple SATMDI as described in claim 1, characterized in that, The maximum allowable mass of a single SATMDI bridge, based on its load-bearing capacity and manufacturing process, is 10 tons.

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