Method for calculating natural vibration frequency and damping ratio of crowd-structure coupling system

Abstract

The invention provides a method for calculating natural vibration frequency and damping ratio of a crowd-structure coupling system, and relates to the field of structural dynamics, each pedestrian on the structure is regarded as an independent spring-mass-damper system, the structure is regarded as a multi-degree-of-freedom system, a motion equation of the walking crowd-structure coupling system and a frequency response function thereof are deduced, and the natural vibration frequency and damping ratio of the crowd-structure coupling system are calculated. Carrying out simulation and parameterization analysis on the dynamic characteristics of the coupling system, and further proposing a closed expression of the natural vibration frequency and the damping ratio of the coupling system; according to the method, through parametric modeling and Monte Carlo simulation, the natural vibration frequency and the damping ratio of the coupling system are converted into a closed expression which only depends on key dimensionless variables such as the mass ratio and the normalized frequency, and the problems that an existing method is complex in theory and difficult to apply to actual engineering design are solved; the dynamic characteristics of the coupling system can be quickly and accurately calculated, and a reliable basis is provided for human-induced structural vibration design of actual engineering.

Description

Technical Field

[0001] This invention belongs to the field of structural dynamics, specifically relating to a method for calculating the natural frequency and damping ratio of a system coupled with a human body and a structure. Background Technology

[0002] Previous studies have largely investigated the impact of crowds on the dynamic characteristics and vibration response of structures by establishing motion equations for coupled crowd-structure systems. However, this method is difficult to apply to practical engineering design due to the complexity of the equations, requiring the calculation of complex eigenvalues ​​of matrices to obtain the natural frequencies and damping ratios of the coupled system, or the use of time history analysis to solve high-degree-of-freedom motion equations. Regarding the influence of crowds on structural dynamic characteristics, while related studies have qualitatively indicated that the presence of crowds alters the dynamic characteristics of structures (e.g., increasing the damping ratio, decreasing or increasing the natural frequency), quantitative studies are lacking to describe the impact. This makes it impossible to directly obtain the damping ratio and natural frequency of a coupled system composed of a specific crowd and a specific structure, and thus, to provide key design parameters for the structure.

[0003] For example, in the patent application CN20181172441.4 entitled "A Method for Structural Vibration Response Analysis Based on Random Crowd Walking Model", both the patent and this patent treat pedestrians as a spring-mass-damper system to establish the motion equations of the coupled system and use matrix form to describe the interaction between the structure and the crowd. However, this patent predicts the vibration response of the structure under random crowd walking by dynamically simulating the pedestrian movement process and using time history analysis to iteratively solve the equations of the high-degree-of-freedom coupled system. Its calculation process is complex and computationally intensive, making it difficult to directly apply to the design of human-induced structural vibration.

[0004] In view of this, the present invention proposes a method for calculating the natural frequency and damping ratio of a crowd-structure coupled system, which is used to solve the problem that the existing technology has high computational complexity and is difficult to apply directly to engineering design. Summary of the Invention

[0005] To address the above issues, this invention proposes a method for calculating the natural frequency and damping ratio of a pedestrian-structure coupled system. Each pedestrian on the structure is considered an individual spring-mass-damper system, and the structure is viewed as a multi-degree-of-freedom system. The motion equations and frequency response functions of the pedestrian-structure coupled system are derived, and the dynamic characteristics of the coupled system are simulated and parameterized. Finally, closed-form expressions for the natural frequency and damping ratio of the coupled system are proposed.

[0006] A method for calculating the natural frequency and damping ratio of a crowd-structure coupled system, characterized by comprising the following steps:

[0007] S1: Discretize the structure using the lumped mass method and consider only the vertical degree of freedom of the structure. Treat each pedestrian on the structure as a separate spring-mass-damper system, and use the center of mass of each pedestrian as the dynamic degree of freedom to construct a coupled system model of the pedestrian crowd-structure.

[0008] S2: Based on the definition of the influence coefficient method, the stiffness matrix, damping matrix and mass matrix of the coupled system model are derived respectively, and the motion equation of the pedestrian crowd-structure coupled system is expressed based on this.

[0009] According to the modal decomposition method, the displacements of the structure are expanded through its natural modes, and the matrix in the equation of motion is then derived accordingly. Transpose the matrix by left-multiplying the structure matrix;

[0010] The contact force between the pedestrian and the structure and the displacement of the crowd are decomposed, and the walking load is normalized by body weight. The decomposed equations are substituted into the equations of motion to further solve for the overall equations of motion of the coupled system model.

[0011] S3: The state-space method is used to transform the overall motion equations to obtain the frequency response function matrices corresponding to the input and output. The first row of the frequency response function matrix represents the frequency response function of the structural acceleration response. For general lightweight simply supported structures, the dynamic characteristics of the crowd-structure coupled system can be directly obtained from the frequency response function matrix. The natural frequencies of the coupled system model are then obtained. With damping ratio ;

[0012] S4: Fully traverse important variables such as structural mass, structural damping, structural stiffness, population size, population mass, human body damping, and human body stiffness;

[0013] For natural frequency Damping ratio Monte Carlo simulations were performed for each working condition to obtain the mean curves of the natural frequency and damping ratio changes from the Monte Carlo simulations. Based on these curves, the natural frequency and damping ratio were calculated separately to obtain their closed-form expressions.

[0014] The beneficial effects of the above technical solution are as follows:

[0015] (1) Traditional methods require establishing high-order coupled motion equations and solving complex domain eigenvalue problems. The computational complexity increases exponentially with the increase of degrees of freedom. However, this invention transforms the natural frequency and damping ratio of the coupled system into closed expressions that depend only on key dimensionless variables such as mass ratio and normalized frequency through parametric modeling and Monte Carlo simulation. This simplifies the calculation process into explicit algebraic operations, eliminating the need for iterative solving of matrix eigenvalues, making the calculation more convenient.

[0016] (2) The closed expression proposed in this invention characterizes the frequency shift effect in the form of a piecewise function and describes the damping ratio gain in a linear relationship. Engineers can directly input the structural fundamental frequency, the proportion of mass of the population, and the statistical characteristic values ​​of human dynamic parameters to quickly obtain the dynamic characteristics of the coupled system. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of the overall framework of the present invention;

[0018] Figure 2 This is a schematic diagram of a pedestrian crowd coupling system model in a specific embodiment of the present invention;

[0019] Figure 3 This is a schematic diagram of the half-power bandwidth method in a specific embodiment of the present invention;

[0020] Figure 4 This is a schematic diagram illustrating frequency variations under different operating conditions in a specific embodiment of the present invention;

[0021] Figure 5 This is a schematic diagram of frequency changes in a specific embodiment of the present invention;

[0022] Figure 6 This is a fitting graph of the normalized frequency values ​​corresponding to the extreme values ​​in a specific embodiment of the present invention;

[0023] Figure 7 This is a frequency extremum fitting graph in a specific embodiment of the present invention;

[0024] Figure 8 This is a schematic diagram illustrating the damping ratio variation under different operating conditions in a specific embodiment of the present invention;

[0025] Figure 9 This is a schematic diagram illustrating the change in damping ratio in a specific embodiment of the present invention;

[0026] Figure 10 This is a fitting graph of the maximum and minimum damping ratio in a specific embodiment of the present invention;

[0027] Figure 11 This is a schematic diagram of the mid-span acceleration time history of a pedestrian bridge under walking conditions of 96 and 108 bpm in a specific embodiment of the present invention.

[0028] Figure 12 This is a schematic diagram comparing the measured values ​​and predicted values ​​under weak constraints in a specific embodiment of the present invention. Detailed Implementation

[0029] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments. The following specific embodiments will help those skilled in the art to further understand the present invention, but do not limit the present invention in any way. It should also be noted that, for ease of description, only the parts related to the present invention are shown in the drawings, and not all of them.

[0030] Example 1, such as Figure 1 As shown, a method for calculating the natural frequency and damping ratio of a crowd-structure coupled system is implemented in the following steps:

[0031] S1: The structure is discretized using the lumped mass method, and only the vertical degrees of freedom are considered. Each pedestrian on the structure is treated as a separate spring-mass-damper system, with the center of mass of each pedestrian as the dynamic degree of freedom. Figure 2 As shown, a walking crowd-structure coupled system model is constructed.

[0032] S2: Based on the definition of the influence coefficient method, derive the stiffness matrix, damping matrix, and mass matrix of the coupled system model; based on the modal decomposition method, expand the displacement of the structure through its natural modes, and extract the first matrix from the equation of motion. Multiply the transpose of the structural matrix by left-hand side. Decompose the contact forces between the pedestrian and the structure, as well as the displacements of the crowd, and normalize the pedestrian load to body weight. Substitute the decomposed equations into the equations of motion to further solve for the equations of motion of the crowd-structure coupled system. Specifically, the equations of motion are expressed as:

[0033]

[0034] In the formula, , , These are the mass, damping, and stiffness matrices of the structure, respectively. , , These represent the mass, damping, and stiffness matrices of the crowd, respectively. This is a positioning matrix used to determine the location of pedestrians on a structure. Unlike fixed-point loads such as jumping loads, pedestrian loads have both temporal and spatial characteristics. The pedestrian position matrix... The location will change over time, and its position can be calculated by walking speed (see equation (5)). If the pedestrian group can form a stable pedestrian flow, it can be assumed that the pedestrian's position in the structure remains unchanged, and the positioning matrix will be used. It can be considered a constant. and These are represented as displacement vectors for the structure and the crowd, respectively.

[0035]

[0036]

[0037] In the formula, m p,ii c p,ii and k p,ii These are the mass, damping, and stiffness of the i-th walker, respectively.

[0038] To simplify the model, it is assumed that the pedestrian's position in the structure remains unchanged, and the pedestrian's localization matrix is... Treat it as a constant:

[0039]

[0040]

[0041] In the formula, Let be the displacement of the i-th degree of freedom of the structure; Let be the displacement of the i-th walker.

[0042] According to the modal decomposition method, the displacement of a structure can be expanded through its natural modes of vibration, taking into account the preceding... Mode shape:

[0043]

[0044]

[0045] In the formula, It is the array coordinate vector of the structure; It is the matrix of the structure, and the j-th column of the matrix is ​​the j-th order mode shape function. The first column of the matrix in equation (1) is... Left-hand multiplication We can obtain:

[0046]

[0047] In the formula, , and These are the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix of the structure, respectively, calculated according to equation (9).

[0048]

[0049] Furthermore, the contact force between pedestrians and the structure can be decomposed into self-driving force and crowd-structure coupling force; the displacement of the crowd Divided into self-driving items and population-structure coupling term :

[0050]

[0051]

[0052] In the formula, x self (t) represents the force exerted on the structure by the up-and-down movement of the human body, x inter (t) represents the force exerted by a pedestrian walking on a flexible structure due to structural vibrations and the fact that both the human body and the structure have mass, stiffness, and damping. and Related to structural vibration, and It is unrelated to structural vibration.

[0053] Will When acting on a rigid structural surface, the self-driving term of the crowd is solved using Newton's second law, yielding:

[0054]

[0055]

[0056] Assuming that the crowd will not change their walking pattern due to minor vibrations in the structure, the self-driving force of each pedestrian is equal to the ground reaction force generated when they move on a rigid surface:

[0057]

[0058] To eliminate differences caused by pedestrian weight, the load is normalized to the weight:

[0059]

[0060]

[0061] In the formula, It is a weight matrix of the population. This represents the weight of the i-th walker; It is the weight-normalized pedestrian load.

[0062] Substituting equations (11)-(15) into equation (8), we obtain the motion equations of the pedestrian-structure coupled system:

[0063] .

[0064] S3: The state-space method is used to transform the motion equations of the coupled system to obtain the input. and output Corresponding frequency response function Frequency response function matrix The first row of elements The frequency response function, expressed as the structural acceleration response, is used to directly obtain the dynamic characteristics of a crowd-structure coupled system for general lightweight simply supported structures, as shown in the attached figure. Figure 3 As shown, the peak value method is used to obtain the natural frequency of the coupled system. The damping ratio of the coupled system is obtained using the half-power bandwidth method. Specifically, equation (17) is transformed using the state-space method:

[0065]

[0066] In the formula, , and Calculated using the following formulas:

[0067]

[0068]

[0069]

[0070] In the formula, , , and The overall mass, damping, and stiffness matrices in equation (18) are:

[0071]

[0072] The input can be obtained from equation (18). and output Corresponding frequency response function :

[0073]

[0074] Frequency response function matrix The first row of elements The frequency response function, expressed as the structural acceleration response, is used to directly obtain the dynamic characteristics of crowd-structure coupled systems for general lightweight simply supported structures, such as... Figure 3 As shown, the peak value method is used to obtain the natural frequency of the coupled system, and the half-power bandwidth method is used to calculate the damping ratio of the coupled system.

[0075]

[0076]

[0077] In the formula, The coupling frequency corresponding to the maximum value of the frequency response function amplitude. and These are the maximum amplitude values. The frequency corresponding to a multiple of 1.

[0078] S4: Comprehensively iterates through important variables such as structural mass, structural damping, structural stiffness, population size, population mass, human damping, and human stiffness. It also considers the coupled natural frequencies. and coupling damping ratio Monte Carlo simulations were performed for each operating condition to obtain the mean curves of the changes in the natural frequency and damping ratio of the coupled system. Based on these curves, the coupled natural frequency and coupled damping ratio were calculated separately, yielding their closed-form expressions. Specifically, the modal damping ratio of the structure... The natural frequency of the empty structure is calculated in increments of 0.01 to 0.04. Taking a frequency variation of 1~10 Hz with an interval of 0.01, the first-order mode shape can be taken as a half-sine type. Among them, the damping of the structure and stiffness It can be calculated using the following formula:

[0079]

[0080]

[0081] The relationship between changes in population size, population quality, and structural quality is transformed into a quality ratio. Changes, take The variation range is 0.01~0.30, with intervals of 0.01. Human body natural frequency. It follows a normal distribution N(3.25, 0.32). 2 [Hz]; The human body damping ratio follows a normal distribution N(0.30, 0.352) [-]. Damping and stiffness The calculation is consistent with equations (26) and (27).

[0082] To better study the relationship between the parameters, the following variables are introduced:

[0083]

[0084]

[0085]

[0086] In the formula, This represents the frequency change value. This represents the change in damping ratio. The ratio of the natural frequency of an empty structure to the mean natural frequency of the human body (hereinafter referred to as the normalized frequency).

[0087] Substituting the parameters mentioned above into equations (24) and (25), and considering that the crowd is evenly distributed on the pedestrian bridge, the coupling frequency is adjusted for each working condition. and coupling damping ratio Perform simulations, such as Figure 4 As shown, different mass ratios and empty structure fundamental frequency The trend of natural frequency variation in the lower coupled system.

[0088] The average value of the data under the four damping ratio conditions was used to calculate... The expression. The value in There are noticeable changes in the vicinity, so a piecewise function approach is used for calculation. Based on... The trend of the curve, such as Figure 5 As shown, the curve is divided into three parts at its minimum and maximum values. The calculation process consists of three steps:

[0089] The first step, due to different mass ratios Below The normalized frequencies corresponding to the minimum and maximum values Non-fixed values, the minimum value corresponding to and the maximum value corresponding The relationship between the mass ratio and the graph is plotted as follows: Figure 6 As shown, linear functions are used for calculation respectively:

[0090]

[0091]

[0092] The second step is to determine the maximum and minimum values ​​corresponding to different mass ratios. Plot the minimum and maximum values ​​corresponding to different mass ratios on a graph, as shown below. Figure 7 As shown, for the minimum value The maximum value is calculated using a linear function. Calculations are performed using power functions:

[0093]

[0094]

[0095] The third step is to represent the curve segments of Part I and Part III using exponential functions, namely:

[0096]

[0097] In the formula, The decay exponent for the descending segment of the curve is calculated as follows: , Part II can be approximated as a straight line segment, and can be calculated by interpolation from the minimum and maximum values.

[0098] like Figure 8 The figure shows different mass ratios. and empty structure fundamental frequency The trend of damping ratio variation in the coupled system. The average value of data under four damping ratio conditions is used to calculate... The expression for the change in coupling damping ratio and the change in normalized frequency is as follows: Figure 9 As shown, The relationship between the maximum value and the mass ratio is as follows: Figure 10 As shown, it changes approximately linearly. Combining this with the frequency response function formula, the following formula is used to calculate the change in damping ratio:

[0099]

[0100] In the formula, , , , The parameters were calculated with values ​​of 0.01, -0.494, 0.081, and 0.005, respectively, and a closed-form expression for the damping ratio of the coupled system was finally obtained.

[0101] Example 2: This invention patent employs a structural acceleration response spectrum model that considers the coupling effect between crowds and structures. By comparing the measured values ​​of the structural response with the predicted values ​​calculated using a closed-form expression based on pedestrian walking data from a steel pedestrian bridge at a civil engineering laboratory of a university in Denmark, the applicability and accuracy of the closed-form expression for the natural frequency and damping ratio of the crowd-structure coupled system proposed in this paper are verified.

[0102] In this embodiment, the pedestrian bridge has a span of 15 m and a width of 1.84 m. The main beams are made of two UNP350 channel steels, and the secondary beams are made of UNP200 channel steels. The main and secondary beams are connected by angle steel and bolts. To adjust the structural modal mass and make its fundamental frequency close to the walking frequency range of pedestrians, counterweight lead blocks are added to the secondary beams. The structural modal parameters are obtained by using a vibrator to excite the structure with a sinusoidal wave in the open span. The mode shape analysis results show that the fundamental frequency of the structure is 1.78 Hz, the first mode shape is half-sine, and the damping ratio can be determined to be 0.69% based on the free decay curve. The static load test at mid-span measured the stiffness of the structure under concentrated force to be approximately 507940 N / m, according to the formula... The calculated modal mass of the structure is 4060 kg.

[0103] In this embodiment, 12 participants were involved, with their average weight taken as W=700 N according to specifications. Since the first mode shape of the structure is approximately sinusoidal, the modal mass of the human body can be taken as half the total mass of the group. Under weak constraints, the mass ratio of the group to the structure is... Under weak constraints, the mass ratio of the crowd to the structure is such that the test subject walks on the bridge following the frequency of a metronome, proceeding in a straight line along one side, turning around at the bridgehead and returning along the other side, thus forming a stable density of crowd flow, such as... Figure 12 As shown. The metronome-guided step frequency conditions include six conditions: 96, 100, 104, 108, 112, and 120 bpm (60 bpm = 1 Hz). Each condition is repeated twice. The collected displacement data is differentiated twice to obtain the corresponding acceleration data, as shown. Figure 11 The figure shows the measured acceleration time history at the mid-span under the conditions of 96 and 108 bpm.

[0104] Since this structure is a lightweight pedestrian bridge, it is significantly affected by the coupling effect between the pedestrian and the structure. Therefore, the dynamic characteristics of the structure are calculated using closed-form expressions for the coupled system frequency and damping ratio. The calculation yields the frequency of the coupled system under weak constraints. Damping ratio is Normalized frequency The predicted structural response was obtained by using a structural acceleration response spectrum model that considers the crowd-structure coupling effect. Simultaneously, the structural response under weakly constrained conditions without considering the crowd-structure coupling effect was also calculated and compared with the measured maximum structural response. The results are as follows: Figure 12 As shown in the figure, it can be concluded that the predicted structural acceleration response calculated using the closed-form expression proposed in this patent, considering the crowd-structure coupling effect, is very close to the measured maximum acceleration value. Ignoring the influence of the coupling effect would lead to an overestimation of the predicted value. By comparing and analyzing the measured and predicted values ​​of the structural response, it can be concluded that the closed-form expression for the dynamic characteristics of the coupled system proposed in this patent can be well used for predicting human-induced structural vibration response.

Claims

1. A method for calculating the natural frequency and damping ratio of a crowd-structure coupled system, characterized in that, The method comprises the following steps: S1: discretize the structure by using the lumped mass method, and only consider the vertical degree of freedom of the structure, regard each pedestrian on the structure as a single spring-mass-damper system, take the center of mass of each pedestrian as the dynamic degree of freedom, and construct a pedestrian group-structure coupling system model; S2: according to the definition of the influence coefficient method, derive the stiffness matrix, the damping matrix and the mass matrix of the coupling system model respectively, and based on this, express the motion equation of the pedestrian group-structure coupling system; According to the mode decomposition method, the displacement of the structure is expanded by its inherent mode, and the matrix in the motion equation is transposed by multiplying the left row of the structure matrix pattern matrix. row left multiply structure matrix pattern matrix to transpose; S3: decompose the contact force between the pedestrian and the structure and the displacement of the pedestrian group, normalize the pedestrian load by weight, and substitute the decomposed formulas into the motion equation to further solve the overall motion equation of the coupling system model; S3: The state space method is used to transform the overall motion equation to obtain the frequency response function matrix corresponding to the input and output. The first row element of the frequency response function matrix represents the frequency response function of the structure acceleration response. For a general lightweight simply supported structure, the dynamic characteristics of the crowd-structure coupled system can be directly obtained from the frequency response function matrix to obtain the natural frequency of the coupled system model and damping ratio ; S4: comprehensively traverse important variables such as structure mass, structure damping, structure stiffness, pedestrian group size, pedestrian group mass, human body damping and human body stiffness. natural frequency and damping ratio Monte Carlo simulation is performed for each working condition to obtain the mean curve of the Monte Carlo simulation of the natural frequency and damping ratio variation value. Based on this, the natural frequency and damping ratio are calculated respectively to obtain their closed expressions.

2. The method of claim 1, wherein, The coupling system model in the step S1 comprises: regarding each pedestrian on the structure as a single single-degree-of-freedom spring-mass-damper sub-model, taking the center of mass of each pedestrian as the dynamic degree of freedom, discretizing the structure by using the lumped mass method, and only considering the vertical degree of freedom of the structure.

3. The method of claim 1, wherein, The motion equation in the step S2 is shown in formula (1): wherein , , are the mass, damping, stiffness matrices of the structure, respectively, , , are the mass, damping, stiffness matrices of the crowd, respectively; To locate the matrix, to determine the position of the crowd in the structure; and are represented as displacement vectors for the structure and the crowd, respectively.

4. The method of claim 3, wherein, To simplify the model, it is assumed that the position of the pedestrian on the structure remains constant, the positioning matrix of the pedestrian is considered constant.

5. The method of claim 3, wherein, The structure and the displacement vector of the crowd and Specifically, as shown in equation (2): wherein is the displacement of the structure for the i-th degree of freedom; is the displacement of the i-th walker; According to the modal decomposition method, the displacement of the structure can be expanded by its natural modes, and the contribution of each mode is considered in the order of the mode number mode number: wherein is the array coordinate vector of the structure; is the array matrix of the structure.

6. The method of claim 5, wherein, The decomposition of the contact force between the pedestrian and the structure and the displacement of the pedestrian group in the step S2 comprises the process of weight normalization of the pedestrian load, which comprises: decomposing the contact force between the pedestrian and the structure into self-driving force and pedestrian-structure coupling force; Displacement of the crowd is divided into a self-driving term and a crowd-structure coupling term : where x self (t) is the force on the structure due to the up-and-down motion of the human body, x inter (t) is the force on the structure due to the walking of a person on the flexible structure, due to the vibration of the structure and the fact that both the person and the structure have mass, stiffness and damping; and and are related to the vibration of the structure, and are not related to the vibration of the structure. The When acting on the rigid structure surface, the self-driven term of the crowd is solved by using Newton's second law, and the following equation is obtained: where, , , are the mass, damping and stiffness of the crowd respectively, assuming that the crowd does not change its walking pattern due to the small vibration of the structure, and the self-driving force of the pedestrian is equal to the ground reaction force when it is moving on a rigid ground: In order to eliminate the difference caused by the weight of pedestrians, the load is normalized by weight: wherein is a matrix of body weights of the population, whose elements represent the body weight of a specific pedestrian; is the body weight normalized pedestrian population load.

7. The method of claim 3, wherein, The whole motion equation construction process in the step S2 includes multiplying the left of the matrix in the formula (1) by the first column of the matrix in the formula (2) It can be obtained that​ wherein , and are the generalized mass matrix, the generalized damping matrix and the generalized stiffness matrix of the structure, respectively; The overall motion equation of the coupling system model is calculated by substituting formula (4) to formula (9) into formula (10):

8. The method of claim 7, wherein, The step S3 comprises: converting formula (11) by using the state space method: wherein x represents the state vector of the coupled system, containing the displacements and velocities of all degrees of freedom in the structure and the crowd; A represents the state matrix, containing the inherent dynamic characteristics determined by the mass, damping and stiffness matrices of the system; B represents the input matrix, representing the distribution mechanism of external excitation in the generalized coordinates; The input and output corresponding frequency response functions can be obtained from equation (12) Frequency response function matrix The first row element of The frequency response function is expressed as the structural acceleration response, the coupled system model is a general light simply supported structure, so its dynamic characteristics can be directly obtained from the frequency response function, the natural frequency of the coupled system model is obtained by peak value method, and the damping ratio of the coupled system model is calculated by half power bandwidth method: In the formula, The coupling frequency corresponding to the maximum value of the frequency response function amplitude. and These are the maximum amplitude values. The frequency corresponding to the multiple, is the coupling damping ratio.

9. The method of claim 8, further characterized by, The Monte Carlo simulation in the step S4 includes selecting modal damping ratio of the structure The natural frequency of the empty structure is calculated at intervals of 0.01 from 0.01 to 0.04 The first order modal shape can be taken as a half-sine type at intervals of 0.01 from 1 to 10 Hz The damping ratio of the structure and the stiffness of the structure can be calculated according to the following formula In the formula, denotes the structure quality, and the variable denotes the change relationship between the population size, the population quality and the structure quality, and takes The change range of the variable is 0.01-0.30, and the interval is 0.

01. Considering the natural frequency of the human body It follows a normal distribution N(3.25, 0.32). 2 The human body damping ratio follows a normal distribution N(0.30, 0.352) [-]. Therefore, the human body damping... and human body stiffness The calculation is consistent with equations (16) and (17); In order to better study the relationship between parameters, the following variables are introduced: wherein is the frequency change value, is the damping ratio change value, is the normalized frequency, is the mean value of the natural frequency of the human body, is the ratio of the normalized frequency to the mean value of the natural frequency of the human body. The above variables are brought into equations (14) and (15), and the crowd is evenly distributed on the footbridge, the coupling frequency and coupling damping ratio are simulated for each working condition, and the change trend of the natural frequency of the coupling system under different mass ratios and natural frequency of the empty structure is obtained. The data for each damping ratio condition is averaged to calculate the expression The value of the function has a relatively sharp change in the vicinity of which is calculated using a piecewise function. According to The curve is divided into three parts, the first, second and third, at the minimum and maximum values of the curve, according to the trend of the curve.

10. The method of claim 9, further characterized by, The construction process of the closed expression in the step S4 comprises: S4-1: due to different mass ratios of minimum and maximum values corresponding to normalized frequency non-fixed values, the minimum value corresponding to and the maximum value corresponding to the relationship with the mass ratio is plotted in the normalized frequency value fitting binary orthogonal diagram, and a linear function is used for calculation respectively:​ S4-2: Determine the extreme values corresponding to different quality ratios, plot the minimum and maximum values corresponding to different quality ratios in the frequency extreme value fitting binary orthogonal graph, and for the minimum value Calculate the maximum value using a linear function Calculate using a power function: S4-3: the curve segments of the first part and the third part are expressed in the form of exponential function, that is: In the formula, is the decay exponent of the curved descending segment, which is calculated by , ; The second part can be approximately equal to a straight line segment, which can be calculated by interpolation from the minimum value and the maximum value; S4-4: According to different mass ratios and the empty structure natural frequency The trend of the change of the under-coupling system damping ratio, the data of each damping ratio condition is averaged to calculate expression; S4-5: Based on the trend of the change in coupling damping ratio and normalized frequency, and referring to... From the relationship between the maximum value and the mass ratio, we can see that it changes approximately linearly. Combining this with the frequency response function formula, the following formula is used to calculate the change in damping ratio: wherein , , , are the calculation parameters, and the parameter values are 0.01, -0.494, 0.081, 0.005, respectively, and finally the closed expression of the damping ratio of the coupled system is obtained.

Images