A multi-objective optimization method for tuned mass damper based on bridge vortex-induced amplitude

Abstract

The application provides a multi-objective optimization method of a tuned mass damper based on bridge vortex-induced amplitude. The main technical scheme is as follows: step one: a dynamic equation of a bridge-tuned mass damper system under the action of vortex-induced force is established; step two: an analytical solution of the bridge vortex-induced amplitude is derived; step three: the bridge vortex-induced amplitude is taken as a constraint condition, and a damper stroke index and a system robustness index are defined as two optimization objectives; step four: a multi-objective optimization process containing the constraint is carried out, and a multi-objective optimization algorithm is used for solving; and step five: the optimization result is presented in the form of a Pareto front, and a suitable damper and design parameters are selected. The method is suitable for parameter design of all tuned mass dampers used for controlling bridge vortex-induced vibration, can not only effectively control the vortex-induced vibration of the bridge by the tuned mass damper, but also simultaneously reduce the damper stroke and increase the robustness of the system to the change of the inherent frequency of the bridge, and realizes optimization of the comprehensive performance of the system.

Description

Technical Field

[0001] This invention relates to the field of multi-objective optimization of tuned mass dampers, and more particularly to a multi-objective optimization method for tuned mass dampers based on bridge vortex-induced amplitude. Background Technology

[0002] As bridge spans increase, the natural frequencies of the structure become increasingly close to the dominant frequencies of wind loads, leading to a greater sensitivity of the bridge structure to wind loads. Vortex-induced vibration is a common vibration mode in long-span bridges under wind loads. When the eddy current shedding frequency approaches the bridge's natural frequency, the bridge will generate excessively large vortex-induced amplitudes, which will not only reduce the bridge's service life but may even lead to structural failure. Therefore, the problem of vortex-induced vibration in bridges has become a key issue in the design of long-span bridges.

[0003] To mitigate adverse vibrations in bridges, researchers have designed and developed various types of tuned mass dampers to control vortex-induced vibrations. Tuned mass dampers have proven to be one of the simplest and most effective auxiliary devices for controlling structural vibrations. Their principle is to induce resonance between the substructure and the main structure to dissipate external excitation energy. Furthermore, the design parameters of the tuned mass damper determine its performance; therefore, optimizing these design parameters is essential.

[0004] Currently, the determination of damper parameters mainly employs empirical methods or single-objective optimization methods. Single-objective optimization considers only one objective function, sacrificing other system performance aspects during the optimization process. For tuned mass dampers, the dynamic amplification factor is often the top priority among single-objective functions, but this means that the optimization process cannot account for the damper's control effect's sensitivity to the main structure's frequency and its large space requirement. Multi-objective optimization methods, on the other hand, can consider multiple optimization objectives during the optimization process, achieving optimal overall system performance. Therefore, for tuned mass dampers used to control vortex-induced vibrations in bridges, a multi-objective optimization method is needed to optimize their design parameters. Summary of the Invention

[0005] Technical problem: The technical problem to be solved by the present invention is to provide a multi-objective optimization method for tuned mass dampers based on bridge vortex-induced amplitude, which addresses the deficiencies in the prior art.

[0006] Technical solution:

[0007] A multi-objective optimization method for tuned mass dampers based on bridge vortex-induced amplitude is proposed to determine the design parameters of the damper, including the following steps:

[0008] Step 1: Select different tuned mass dampers to control the vortex-induced vibration of the bridge and establish the dynamic equations of the bridge-tuned mass damper system under the action of vortex-induced force.

[0009] Step 2: Solve the dynamic equations using the Krylov-Bogoliubov method to obtain the analytical solution for the vortex-induced amplitude of the bridge;

[0010] Step 3: Using the bridge vortex-induced vibration amplitude as a constraint, define two optimization objectives: the damper stroke index and the system robustness index.

[0011] Step 4: Using the damper design parameters as research parameters, conduct a constrained multi-objective optimization process and solve it using a multi-objective optimization algorithm;

[0012] Step 5: The optimization results are presented in the form of a Pareto front, which is used to select a tuned mass damper and design parameters to control the vortex-induced vibration of the bridge.

[0013] Furthermore, in step one, the main girder of the bridge is treated as a single-degree-of-freedom mass point, and its damping and stiffness are determined based on its material characteristics; a tuned mass damper (TMD), a series double-tuned mass damper (DTMD), and a tuned mass damped inertial container (TMDI) are selected to control the vortex-induced vibration of the main girder; the vortex-induced force model adopts the Scanlan semi-empirical nonlinear model; the dynamic equations of the bridge-tuned mass damper system are established and unified into the following form:

[0014]

[0015] In the formula: m1, k1, and c1 are the equivalent mass, stiffness, and damping of the main beam; m2, k2, and c2, and m3, k3, and c3 represent the mass, stiffness, and damping of the two tuned mass dampers TMD1 and TMD2 connected in series in the DTMD; y1 is the displacement of the main beam; y2 is the relative displacement between TMD1 and the main beam; y3 is the relative displacement between TMD1 and TMD2; b is the inertia coefficient; F VIV The vortex-induced force acting on the main beam is simulated by a semi-empirical nonlinear model, as shown below:

[0016]

[0017] In the formula: ρ is the air density; U is the average wind speed; B is the crosswind characteristic dimension of the main beam; Y1 and ε are the linear aerodynamic damping coefficient and the nonlinear aerodynamic damping coefficient, respectively.

[0018] Furthermore, in step two, the Krylov-Bogoliubov method is used to solve the dynamic equations, and the expression for the vortex-induced amplitude A1 of the main beam is obtained as follows:

[0019]

[0020] In the formula: α2, α4, Q 21 Q 31 Q 32 P 32 P 21 All are user-defined simplified parameters, defined as follows:

[0021] In the above expressions, μ2, μ3, β, M i (i = 1, 2, 3) are all user-defined simplified parameters, defined as follows:

[0022] μ2=M2 / M1

[0023] μ3=M3 / M1

[0024] β=b / M1

[0025]

[0026] In the above expressions, φ is the basic vertical vibration mode of the bridge; ξ i (i = 1, 2, 3) and ω i (i = 1, 2, 3) represents the modal damping ratio and modal frequency of the bridge, TMD1, and TMD2; ω is the excitation frequency of vortex-induced vibration; and L is the bridge span.

[0027] Furthermore, in step three, the damper stroke index f is defined. S and the system robustness index f R The two optimization objectives are expressed as follows:

[0028]

[0029] In the formula:

[0030]

[0031] A1 is the vortex-induced amplitude of the main beam, a e P(e) represents the vortex-induced vibration amplitude of the main beam when a relative parameter error *e* exists, where *e* represents the error in the bridge's natural frequency; P(e) represents the probability distribution density of the relative parameter error; l and e r Represents the lower and upper limits of error;

[0032] Damper stroke index f S The smaller the value, the smaller the damper stroke; the system robustness index f R The smaller the value, the less sensitive the system is to changes in the bridge's natural frequency.

[0033] Furthermore, in step four, the constrained multi-objective optimization process is represented by the following model:

[0034] min F(x)={f S (x),f R (x)}

[0035] stA1≤A limit

[0036] x=(μ k ,ω2,ω3,ξ2,ξ3,β)∈X

[0037] In the formula:

[0038] μ k To simplify the parameters, they are defined as μ. k =M3 / M2

[0039] F represents the optimization objective function; A limit The upper limit of the actual required amplitude of the vortex-induced vibration of the main beam; x and X represent the optimization variables and the variable space;

[0040] A multi-objective optimization algorithm is used to solve the problem, and the optimization results are presented in the form of Pareto fronts.

[0041] Beneficial effects:

[0042] Compared with the prior art, the present invention has the following outstanding substantive features and significant advantages:

[0043] First, the method of this invention is applicable to the parameter design of tuned mass dampers used to control vortex-induced vibration of bridges. In this invention, the dynamic equations of the bridge-tuned mass damper system under vortex-induced force are established and solved to obtain analytical solutions for the vortex-induced amplitude of the bridge.

[0044] Secondly, the method of this invention uses the vortex-induced vibration amplitude of the bridge as a constraint condition, and simultaneously defines two optimization objectives: the damper stroke index and the system stability index, establishing a constrained multi-objective optimization model. This method not only enables the tuned mass damper to effectively control the vortex-induced vibration of the bridge, but also simultaneously reduces the damper stroke and increases the system's robustness to changes in the bridge's natural frequency, achieving optimal overall system performance. Attached Figure Description

[0045] Figure 1 This is a flowchart of the optimized design method of the present invention;

[0046] Figure 2 This is a schematic diagram of a bridge-tuned mass damper system model according to an embodiment of the present invention;

[0047] Figure 3 These are the displacement-time history curves of the main beam model without tuned mass dampers in this embodiment of the invention.

[0048] Figure 4 These are the Pareto fronts of the three types of tuned mass dampers in the embodiments of the present invention;

[0049] Figure 5 These are the displacement time history curves of three types of tuned mass dampers and corresponding main beams according to embodiments of the present invention.

[0050] Figure 6 These are curves showing the relationship between the main beam displacement and frequency ratio under the control of three types of tuned mass dampers according to embodiments of the present invention. Detailed Implementation

[0051] To more clearly illustrate the objectives, technical solutions, and advantages of the present invention, the present invention will be further explained in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0052] A multi-objective optimization method for tuned mass dampers based on bridge vortex-induced amplitude, such as... Figure 1 As shown, it includes the following steps:

[0053] Step 1: Select different tuned mass dampers to control the vortex-induced vibration of the bridge and establish the dynamic equations of the bridge-tuned mass damper system under the action of vortex-induced force.

[0054] Step 2: Solve the dynamic equations using the Krylov-Bogoliubov method to obtain the analytical solution for the vortex-induced amplitude of the bridge;

[0055] Step 3: Using the bridge vortex-induced vibration amplitude as a constraint, define two optimization objectives: the damper stroke index and the system robustness index.

[0056] Step 4: Using the damper design parameters as research parameters, conduct a constrained multi-objective optimization process and solve it using a multi-objective optimization algorithm;

[0057] Step 5: The optimization results are presented in the form of Pareto front. Based on the actual engineering requirements, a suitable tuned mass damper and design parameters are selected to control the vortex-induced vibration of the bridge.

[0058] The following embodiments use three different tuned mass dampers to control vortex-induced vibration of bridges, and establish the dynamic equations of the three bridge-tuned mass damper systems under vortex-induced force. However, this is not the only embodiment; in other embodiments, the tuned mass damper can be selected as needed.

[0059] In step one, the main girder of the bridge is treated as a single-degree-of-freedom mass point, and its damping and stiffness are determined based on its material characteristics. A tuned mass damper (TMD), a series double tuned mass damper (DTMD), and a tuned mass damped inertial container (TMDI) are selected to control the vortex-induced vibration of the main girder. The vortex-induced force model adopts the Scanlan semi-empirical nonlinear model. For ease of demonstration and calculation, the three types of tuned mass dampers are combined, such as... Figure 2 As shown, the dynamic equations of the bridge-tuned mass damper system can be unified into the following form:

[0060]

[0061] In the formula: m1, k1, and c1 are the equivalent mass, stiffness, and damping of the main beam; m2, k2, and c2, and m3, k3, and c3 represent the mass, stiffness, and damping of the two tuned mass dampers TMD1 and TMD2 connected in series in the DTMD; y1 is the displacement of the main beam; y2 is the relative displacement between TMD1 and the main beam; y3 is the relative displacement between TMD1 and TMD2; b is the inertia coefficient; F VIV The vortex-induced force acting on the main beam is simulated by a semi-empirical nonlinear model, as shown below:

[0062]

[0063] In the formula: ρ is the air density; U is the average wind speed; B is the crosswind characteristic dimension of the main beam; Y1 and ε are the linear aerodynamic damping coefficient and the nonlinear aerodynamic damping coefficient, respectively.

[0064] In step two, the Krylov-Bogoliubov method is used to solve the dynamic equations, and the expression for the vortex-induced amplitude A1 of the main beam is obtained as follows:

[0065]

[0066] In the formula: α2, α4, Q 21 Q 31 Q 32 P 32 P 21 All are user-defined simplified parameters, defined as follows:

[0067]

[0068] In the above expressions, μ2, μ3, β, M i (i = 1, 2, 3) are all user-defined simplified parameters, defined as follows:

[0069] μ2=m2 / M1

[0070] μ3=m3 / M1

[0071] β=b / M1

[0072]

[0073] In the above expressions, φ is the basic vertical vibration mode of the bridge; ξ i (i = 1, 2, 3) and ω i (i = 1, 2, 3) represents the modal damping ratio and modal frequency of the bridge, TMD1, and TMD2; ω is the excitation frequency of vortex-induced vibration; and L is the bridge span.

[0074] In step three, the damper stroke index f is defined. S and the system robustness index f R The two optimization objectives are expressed as follows:

[0075]

[0076] In the formula:

[0077]

[0078] A1 is the vortex-induced amplitude of the main beam, a e P(e) represents the vortex-induced vibration amplitude of the main beam when a relative parameter error *e* exists, where *e* represents the error in the bridge's natural frequency; P(e) represents the probability distribution density of the relative parameter error; l ,e r These represent the lower and upper limits of the error.

[0079] Damper stroke index f S The smaller the value, the smaller the damper stroke; the system robustness index f R The smaller the value, the less sensitive the system is to changes in the bridge's natural frequency.

[0080] In step four, the constrained multi-objective optimization process can be represented by the following model:

[0081] min F(x)={f S (x),f R (x)}

[0082] stA1≤A limit

[0083] x=(μ k ,ω2,ω3,ξ2,ξ3,β)∈X

[0084] In the formula:

[0085] μ k To simplify the parameters, they are defined as μ. k=M3 / M2

[0086] F represents the optimization objective function; A limit The upper limit of the actual required vortex-induced vibration amplitude of the main beam is given by x and X, which represent the optimization variables and the variable space, respectively.

[0087] The bridge model parameters are selected as follows: M1 = 8888900 kg, M2 + M3 = 88889 kg, A limit =50mm, ρ=1.293kg / m3, B=4.5m, U=30m / s, Y1=179.108, ε=2561.84, ω1=5.06425rad / s. Figure 3 The displacement-time history curves of the main beam of a bridge model without tuned mass dampers under vortex-induced force are shown.

[0088] The solution is obtained using a multi-objective particle swarm optimization algorithm, and the optimization results are presented in the form of a Pareto front. Figure 4 The optimization results for TMD, TMDI, and DTMD in two objective function spaces are presented. For ease of presentation, each set of fronts consists of 30 Pareto optimal solutions, which are distributed as evenly as possible in the space. Clearly, the damper stroke index f... S With system robustness index f R Inversely proportional.

[0089] For TMD and TMDI, the Pareto front is located in the middle region on the left side of the figure, indicating that these solutions have a small damper stroke index f. S S and system robustness index f R In other words, both the optimized TMD and TMDI can control the amplitude of the main girder with a smaller stroke. It is noteworthy that the Pareto front of the TMDI is almost directly above that of the TMD. This means that the optimized TMD and TMDI have essentially the same stroke behavior, but the TMD is more robust to changes in the bridge's natural frequency.

[0090] For DTMD, the Pareto front is located at the bottom of the graph, i.e., the system robustness index f. R The smallest value indicates that the optimized DTMD exhibits the most stable performance in response to changes in the bridge's natural frequency. Furthermore, the projection of the DTMD's Pareto front on the x-axis is significantly wider than that of the TMD and TMDI, because the DTMD contains two mass blocks, resulting in a larger stroke.

[0091] For fair comparison and intuitive observation, a representative point was selected from the three Pareto fronts, where the main beam amplitude was 10 mm. The parameters of the representative point were substituted into the dynamic equations, and the displacement-time history curves of the damper and the main beam were obtained, as shown below. Figure 5 As shown. It can be seen that, with Figure 3Compared to the uncontrolled bridge shown in the image, all three types of mass dampers can achieve good control of the bridge.

[0092] Introducing frequency ratio λ=ω e / ω1. Figure 6 Furthermore, the displacement-frequency ratio curves of the main girder under the control of three tuned mass dampers are presented. It can be clearly seen that the curve shapes of the three mass dampers are similar. The minimum amplitude of all three mass dampers occurs at λ=1, at which point the main girder amplitude is 10mm, and the amplitude increases as λ moves further away from 1. Obviously, the curve of DTMD is wider and flatter than that of TMD and TMDI, indicating that within the amplitude range required by the standard (50mm), the λ range of DTMD is much larger than the corresponding ranges of TMD and TMDI. In other words, the bridge-DTMD system has the best robustness to changes in the bridge's natural frequency.

[0093] The above descriptions are merely embodiments of the present invention. Commonly known structures and characteristics are not described in detail here. Those skilled in the art are aware of all common technical knowledge in the field prior to the application date or priority date, are aware of all existing technologies in that field, and have the ability to apply conventional experimental methods prior to that date. Those skilled in the art can, under the guidance of this application, improve and implement this solution in combination with their own capabilities. Some typical known structures or methods should not be obstacles to the implementation of this application. It should be noted that those skilled in the art can make several modifications and improvements without departing from the structure of the present invention. These should also be considered within the scope of protection of the present invention, and will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A method for multi-objective optimization of a tuned mass damper based on bridge vortex-induced amplitude, for determining design parameters of a damper, characterized in that, The method comprises the following steps: Step 1: selecting different tuned mass dampers to control the vortex-induced vibration of a bridge, and establishing the dynamic equation of the bridge-tuned mass damper system under the action of vortex-induced force; Step 2: solving the dynamic equation by using the Krylov-Bogoliubov method to obtain an analytical solution of the vortex-induced amplitude of the bridge; Step 3: taking the vortex-induced amplitude of the bridge as a constraint condition, defining two optimization objectives of a damper stroke index and system robustness index; Step 4: taking the damper design parameters as research parameters, carrying out a multi-objective optimization process with constraints, and solving by using a multi-objective optimization algorithm; Step 5: the optimization result is presented in the form of a Pareto front, and the tuned mass damper and the design parameters are selected to control the vortex-induced vibration of the bridge; In the step 1, the main girder of the bridge is taken as a single-degree-of-freedom particle, the damping and stiffness thereof are determined according to the material characteristics, the tuned mass damper TMD, the series double tuned mass damper and the tuned mass damper inertial container TMDI are selected to control the vortex-induced vibration of the main girder, the vortex-induced force model adopts the Scanlan semi-empirical nonlinear model, and the dynamic equation of the bridge-tuned mass damper system is established and unified in the following form: where: , , is the equivalent mass, stiffness, damping of the main girder, , , and , , denote the mass, stiffness, damping of TMD1, TMD2; is the displacement of the main girder; is the relative displacement between TMD1 and the main girder; denotes the relative displacement between TMD1 and TMD2; is the mass ratio; is the vortex-induced force acting on the main girder, which is simulated by a semi-empirical nonlinear model as follows: where: is the air density; is the average wind speed; is the characteristic crosswind dimension of the girder; and are the linear and nonlinear aerodynamic damping coefficients, respectively; In the second step, the Krylov-Bogoliubov method is used to solve the dynamic equation to obtain the expression of the amplitude of the vortex-induced vibration of the main beam The expression is shown as follows: In the formula: is the fundamental vertical mode of the bridge; and denote the modal damping ratio and modal frequency of the bridge, TMD1 and TMD2; is the excitation frequency of vortex-induced vibration; L is the bridge span, ; In step three, define the damper travel index and the system robustness index Two optimization objectives, expressed as follows: In the formula: the vortex-induced amplitude of the girder, denotes the vortex-induced amplitude of the girder in the presence of a relative parametric error denotes the vortex-induced amplitude of the girder in the presence of a relative parametric error denotes the error in the natural frequency of the bridge; denotes the probability density of the relative parametric error; and denotes the lower and upper error limits; Damper stroke indicator The smaller the value of the damper stroke indicator, the smaller the damper stroke represents; System robustness indicator The smaller the value of the system robustness indicator, the smaller the sensitivity of the system to the change of the bridge natural frequency represents.

2. The bridge vortex-induced vibration amplitude-based tuned mass damper multi-objective optimization method of claim 1, wherein, In the step 4, the multi-objective optimization process with constraints is represented by the following model: In the formula: , represents an optimization objective function; is an upper limit value of the actual demand of the vortex-induced amplitude of the main girder; and represents an optimization variable and a variable space; The multi-objective optimization algorithm is used for solving, and the optimization result is presented in the form of a Pareto front.

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