Magnetic levitation line axle coupling system seismic response analysis method with seismic isolation support

Abstract

The invention discloses a seismic response analysis method for a magnetic levitation line axle coupling system with a seismic isolation support. The method comprises the steps that a motion equation of the magnetic levitation line axle coupling system considering nonlinear mechanical behaviors of the seismic isolation support under the seismic action is constructed; and solving the motion equation by considering equivalent linearization and separation iteration to obtain the dynamic response of the maglev line vehicle and the bridge. According to the method, a bridge model considering the nonlinear hysteresis behavior of the seismic isolation support and a magnetic track interaction model based on PID active control are established, and unified seismic analysis is carried out on strong nonlinear energy consumption of the seismic isolation support and specific active control dynamics of a magnetic levitation vehicle; the problem that an existing universal axle coupling analysis method cannot be directly applied to the complex system is solved.

Description

Technical Field

[0001] This invention belongs to the field of maglev rail transit, and particularly relates to a seismic response analysis method for a maglev line vehicle-bridge coupled system with seismic isolation bearings. Background Technology

[0002] As a new type of ground rail transit, medium- and low-speed maglev rail transit operates without any physical contact between the trains and the track. This results in significant advantages such as low track wear, low vibration, and low noise, making it highly competitive and promising in the rail transit system. However, my country is a country prone to earthquakes. With the continuous expansion of maglev lines, the likelihood of maglev trains encountering earthquakes while running on bridges is increasing. Earthquakes not only cause severe vibrations and even damage to the bridges, but also threaten the operational safety of vehicles on the bridge through the vehicle-track interaction, leading to accidents such as rollovers and locking due to insufficient suspension clearance. Seismic isolation technology has been proven to improve the seismic performance of bridge structures, ensuring the structural and operational safety of maglev lines. Therefore, accurately predicting the seismic response of the vehicle-bridge system of medium- and low-speed maglev lines with seismic isolation bearings is crucial for seismic isolation design and ensuring the safety of trains on bridges during earthquakes.

[0003] There are many existing technologies involving response analysis methods for vehicle-bridge coupled systems. For example, prior art with publication number CN113177339A discloses a method for earthquake-wind-wave-vehicle-bridge coupled vibration analysis, relating to vehicle-bridge coupled vibration in the field of highway bridges. This method is used to solve the dynamic response of a long-span bridge structure and the vehicles running on it when simultaneously subjected to loads such as earthquakes, vehicles, wind, and waves. This method provides an effective way to assess the safety of bridge structures and vehicles traveling on them under sudden earthquakes during the operational phase. Prior art with publication number CN114580076A discloses a vibration analysis system for vehicle-bridge coupling based on mechanical effects, relating to vehicle-bridge coupled vibration in the field of railway bridges. This method uses a stability analysis module to perform stability analysis on the running train and then feeds back the train vibration response results. This method solves the problems of complex calculation process, long time consumption and low efficiency of existing vehicle-bridge coupled vibration analysis systems. The prior art with publication number CN115345054A discloses a vehicle-bridge coupled vibration analysis method and system based on time history array interactive iteration, which relates to vehicle-bridge coupled vibration in the field of highway bridges. This method uses data sharing between MATLAB and ANSYS to analyze the vehicle subsystem and bridge subsystem separately, and then performs coupled analysis on the vehicle and bridge subsystems to obtain the vibration response of highway bridges and vehicles. It avoids time step integration iteration and improves computational efficiency.

[0004] In summary, existing technologies primarily address vehicle-bridge coupled vibration in traditional wheel-rail transportation sectors such as highways and railways. The wheel-track interaction mechanism in these systems differs fundamentally from the vehicle-bridge interaction mechanism in maglev trains based on active control. Furthermore, existing technologies rarely consider bridge systems with seismic isolation bearings, failing to account for the strong nonlinear hysteretic behavior of these bearings under seismic conditions and the challenges they pose to system dynamic response analysis. Directly applying existing vehicle-bridge coupled vibration analysis systems would clearly be inaccurate; therefore, a seismic response analysis method is needed for vehicle-bridge coupled systems with seismic isolation bearings in medium- and low-speed maglev lines. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention proposes a seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings.

[0006] The technical solution of the present invention is as follows:

[0007] A seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings, wherein the maglev train-bridge coupled system includes a bridge consisting of an upper track beam and lower piers, seismic isolation bearings connecting the track beam and the piers, and a maglev train suspended above the track beam, comprising:

[0008] Construct the motion equations of a maglev train-bridge coupled system that considers the nonlinear mechanical behavior of seismic isolation bearings under seismic loading;

[0009] The dynamic response of the maglev train and the bridge is obtained by considering equivalent linearization and separation iteration of the equation of motion.

[0010] Furthermore, the specific method for constructing the motion equations of the maglev line vehicle-bridge coupled system considering the nonlinear mechanical behavior of seismic isolation bearings under seismic loading includes:

[0011] A multi-rigid-body dynamics model of the maglev line vehicle is established, and the car and suspension frame in the maglev line vehicle are simplified as rigid bodies. The car is considered to have five degrees of freedom: buoyancy, pitching, yaw, head-swaying, and roll. The suspension frame is considered to have four degrees of freedom: buoyancy, pitching, yaw, and head-swaying. The secondary suspension system in the maglev line vehicle is simplified to a spring-damping system.

[0012] A finite element model of the bridge was established, and the track beams and piers in the bridge were simulated using 2-node, 12-DOF spatial beam elements.

[0013] A nonlinear model of the seismic isolation bearing is established, which is simulated using massless spring elements. The nonlinear behavior of the seismic isolation bearing is simulated using a bilinear hysteresis curve model. The design parameters include the initial elastic stiffness k. d Post-yield stiffness ku Yield displacement S y ;

[0014] Establish a model of seismic motion and track irregularities;

[0015] A magnetic-track interaction model and a seismic action model are established. The magnetic-track interaction model includes the levitation electromagnetic force and the guiding electromagnetic force between the bridge track and the vehicle electromagnet. The active control characteristics of the magnetic levitation electromagnetic force are simulated using a PID controller. The seismic action can be achieved using measured seismic waves or artificially synthesized seismic waves.

[0016] Furthermore, the motion equations of the maglev line vehicle-bridge coupling system include the motion equations of the bridge and the motion equations of the maglev line vehicle, wherein the motion equation of the maglev line vehicle is: The equation of motion for the bridge is In the formula, M V C V K V F represents the mass, damping, and stiffness matrices of the maglev train vehicle; M, C, and K represent the mass, damping, and stiffness matrices of the bridge, respectively. The subscript ss indicates the degrees of freedom of the unsupported nodes, gg indicates the degrees of freedom of the supported nodes, and sg and gl indicate the coupling terms between the supported and unsupported nodes. B,V The electromagnetic force vector between the maglev train vehicle and the track beam; u, and ü are the displacement, velocity, and acceleration vectors of the bridge, respectively.

[0017] Furthermore, the electromagnetic force vector F between the maglev train vehicle and the track beam... B,V Based on the position of the maglev train on the bridge, the calculations are performed under the following four conditions:

[0018] 1) Vehicle entering operating condition:

[0019] ;

[0020] 2) Full load condition:

[0021] ;

[0022] 3) Vehicle leaving the operating condition:

[0023] ;

[0024] 4) No-load condition: ;

[0025] In the formula, [N] represents the shape function of the beam element; L E f is the unit length; my,j and f mz,jThese are the levitation electromagnetic force and guiding electromagnetic force of the j-th electromagnet, respectively; l m x is the length of a single electromagnet; h,j and x t,j Let represent the start and end coordinates of the j-th electromagnet; NE and NL are the electromagnet numbers for the units being entered and exited, respectively.

[0026] Furthermore, the levitation electromagnetic force f of the j-th electromagnet my,j And guiding electromagnetic force f mz,j The calculation formula is as follows:

[0027] ;

[0028] ;

[0029] In the formula, h is the suspension gap; c is the guide gap; μ0 is the air permeability; W m A is the pole width; m N represents the magnetic pole area. c To control the number of current coils; i c To control the current; N const I is the number of constant current coils; const The constant current is used to balance the weight of the vehicles on the maglev line and is calculated by the following formula: In the formula: h0 is the rated levitation clearance of the maglev train when it is in the equilibrium position, which is determined according to the actual design conditions of the maglev train; M c M represents the weight of the carriage. bj denoted as , where is the weight of a single bogie; g is the acceleration due to gravity.

[0030] Furthermore, the calculation formulas for the suspension gap h and the guide gap c are as follows: ;

[0031] In the formula, Y(x, t) and Z(x, t) represent the yaw and vertical displacement of the maglev train, respectively; y(x, t) and z(x, t) represent the vertical and lateral displacements of the bridge, respectively; δ yr (x) indicates that the track is uneven.

[0032] Furthermore, the control current i in the electromagnetic force c Active control and adjustment are performed by a PID controller:

[0033] ;

[0034] In the formula, K P K I K DThese are the proportional, integral, and differential coefficients, respectively; ḣ is the rate of change of the suspension gap, which can be obtained by taking the first derivative of the suspension gap h with respect to time t.

[0035] Furthermore, the equivalent linearization includes:

[0036] The nonlinear restoring force generated by the seismic isolation bearing is used as the right-hand term of the equation of motion for transferring the equivalent external load to the bridge. A virtual linear elastic element is introduced into the equation of motion corresponding to the position of the seismic isolation bearing. The specific method is as follows:

[0037] 1) The stiffness of the seismic isolation bearing is derived from the stiffness matrix K on the left-hand side of the equation of motion. ss The nonlinear restoring force R generated by the seismic isolation bearing is obtained by multiplying the nonlinear stiffness of the seismic isolation bearing by its displacement. NL This is equivalent to external loads placed on the right-hand side of the bridge's equation of motion, i.e. In the formula, K is the elastic stiffness matrix neglecting nonlinear elements; R NL This refers to the nonlinear restoring force generated by the seismic isolation bearing;

[0038] 2) Add a virtual linear elastic element to the equation of motion of the bridge corresponding to the location of the seismic isolation bearing, that is: In the formula, K e This is the stiffness matrix of the added linear elastic element.

[0039] Furthermore, after introducing the virtual linear elastic element, the specific steps for solving the motion equations of the bridge are as follows:

[0040] I) Use the vibration state of the bridge at the previous moment as the initial condition of the maglev line vehicle-bridge coupling system at the current moment;

[0041] II) Based on the bridge displacement response of the previous time step or iteration step, calculate the load term of the virtual linear elastic element, and calculate the relative displacement of the pier and beam to determine the vibration state of the seismic isolation bearing. Determine the nonlinear restoring force of the seismic isolation bearing according to the bilinear hysteresis curve model.

[0042] III) Substitute the difference between the nonlinear restoring force obtained in step II) and the load term of the virtual linear elastic element as the equivalent external load into the right side of the motion equation to solve for the bridge vibration state in the current iteration step.

[0043] IV) Determine whether the dynamic response of the bridge satisfies the convergence condition. If it does, use the response at the current moment as the initial condition for the next moment; if it does not, return to step II) for the next iteration. The convergence condition is determined based on the norm of the bridge's dynamic response vector. ;

[0044] In the formula, D B The dynamic response of the bridge is represented by the superscripts i and i-1, which represent the current iteration step and the previous iteration step, respectively; ε is the tolerance error.

[0045] Furthermore, the specific steps for solving the equations of motion using a separation and iteration method are as follows:

[0046] i) Set the initial motion state of the maglev train and bridge;

[0047] ii) Based on the motion state of the maglev line vehicles and bridges at the previous moment, and superimposed with the track irregularities at the current moment, determine the maglev gap and its rate of change, and determine the control current at the current moment through a PID control algorithm;

[0048] iii) Calculate the electromagnetic levitation force and guiding force based on the current maglev gap status and control current;

[0049] iv) Substitute the electromagnetic force calculated in step iii) into the motion equations of the maglev line vehicle and the bridge as load terms, and solve for the vibration state of the maglev line vehicle and bridge system respectively.

[0050] v) Determine whether the dynamic response of the maglev train and the bridge meets the convergence condition. If it does, save the response at the current time and proceed to the next time step; if it does not, return to step iii) for iteration.

[0051] The criterion for determining convergence is whether the norms of the dynamic response vectors of the maglev line vehicle and the bridge meet the requirements: In the formula, D V and D B These represent the dynamic responses of the maglev train and the bridge, respectively; the superscripts i and i-1 indicate the current iteration step and the previous iteration step, respectively; ε is the tolerance error;

[0052] vi) Repeat step ii) to v) until the maglev train has completely left the bridge.

[0053] Compared with the prior art, the present invention has the following beneficial effects:

[0054] This invention provides a seismic response analysis method for a maglev train vehicle-bridge coupled system with seismic isolation bearings. This method establishes a bridge model considering the nonlinear hysteretic behavior of the seismic isolation bearings and a maglev track interaction model based on PID active control. It unifies the seismic analysis of the strong nonlinear energy dissipation of the seismic isolation bearings with the unique active control dynamics of maglev vehicles, solving the problem that existing general vehicle-bridge coupled analysis methods cannot be directly applied to such complex systems.

[0055] This invention addresses the computational challenges posed by the nonlinearity of seismic isolation bearings by proposing an equivalent linearization strategy. This strategy transforms the complex nonlinear restoring force of the bearings into an equivalent external load, and maintains the numerical stability of the system by introducing virtual linear elastic elements. This design makes it possible to handle the equations of bridge systems containing nonlinear seismic isolation bearings, enabling the application of the efficient modal superposition method for solution, significantly improving solution efficiency and making seismic isolation design optimization based on a large number of parameter comparisons feasible.

[0056] This invention addresses the problem of accurately solving the dynamic coupling of maglev vehicle-bridge systems, employing a separate iterative algorithm. This algorithm decouples the vehicle subsystem from the bridge subsystem, independently solving their motion equations, and then enforces the mechanical coordination and geometric compatibility between the two through an iterative process. This method effectively avoids the enormous computational burden and convergence difficulties associated with establishing and solving ultra-large-scale, strongly nonlinear overall system matrices, significantly reducing computational complexity and memory requirements while maintaining the accuracy of the coupling analysis.

[0057] This invention's method, through systematic solution, can accurately obtain the dynamic response of bridges and maglev vehicles during earthquakes. Based on these results, technicians can scientifically evaluate the comprehensive impact of seismic isolation bearings with different design parameters on the structural safety of bridges and the operational stability and safety of maglev trains, thereby quantitatively determining their seismic isolation effectiveness. This provides a direct and reliable theoretical basis and numerical analysis tool for the seismic design and optimized selection of seismic isolation bearings for maglev transportation lines, and has significant engineering application value. Attached Figure Description

[0058] Figure 1 A schematic diagram of the seismic response analysis of the vehicle-bridge coupling system with seismic isolation bearings for medium-low speed maglev lines provided by the present invention.

[0059] Figure 2 A schematic diagram of a medium- and low-speed maglev train-bridge system encountering an earthquake.

[0060] Figure 3 This is a schematic diagram of the interaction between magnetic tracks;

[0061] Figure 4 This diagram illustrates four states of a maglev vehicle on a bridge unit.

[0062] Figure 5 This is a schematic diagram of the bilinear hysteresis curve model of the seismic isolation bearing. Detailed Implementation

[0063] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0064] Example 1:

[0065] This invention discloses a seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings. The maglev train-bridge coupled system includes a bridge consisting of an upper track beam and lower piers, seismic isolation bearings connecting the track beam and the piers, and a maglev train suspended above the track beam. Figure 1 and Figure 2 As shown, it includes:

[0066] S1. Construct the motion equations of the maglev line vehicle-bridge coupled system considering the nonlinear mechanical behavior of seismic isolation bearings under seismic loading.

[0067] S2. The equations of motion are solved by considering equivalent linearization and separation iteration to obtain the dynamic response of the maglev train and the bridge.

[0068] Furthermore, specific methods for constructing the motion equations of a maglev train-bridge coupled system that considers the nonlinear mechanical behavior of seismic isolation bearings under seismic loading include:

[0069] A multi-rigid-body dynamics model for maglev line vehicles is established, simplifying the car and suspension frame in the maglev line vehicles into rigid bodies. The car is considered to have five degrees of freedom: buoyancy, pitching, yaw, head-swaying, and roll, while the suspension frame is considered to have four degrees of freedom: buoyancy, pitching, yaw, and head-swaying. The secondary suspension system in the maglev line vehicles is simplified to a spring-damping system.

[0070] A finite element model of the bridge was established, and the track beams and piers in the bridge were simulated using 2-node, 12-DOF spatial beam elements.

[0071] A nonlinear model of the seismic isolation bearing is established, which uses massless spring elements for simulation. The nonlinear behavior of the seismic isolation bearing is simulated using a bilinear hysteresis curve model. The design parameters include the initial elastic stiffness k. d Post-yield stiffness k u Yield displacement S y ;

[0072] Establish a ground motion and track irregularity model; furthermore, the ground motion and track irregularity model can be obtained using measured values ​​or artificial simulation methods;

[0073] Establish magnetic-orbit interaction models and seismic action models, such as Figure 3As shown, the magnetic-track interaction model includes the levitation electromagnetic force and the guiding electromagnetic force between the bridge track and the vehicle electromagnet. The active control characteristics of the magnetic levitation electromagnetic force are simulated using a PID controller. The seismic action can be simulated using measured seismic waves or artificially synthesized seismic waves.

[0074] In the finite element model of a bridge, the nonlinear behavior of a bridge with seismic isolation bearings under the coupled action of seismic force and vehicle electromagnetic force is mainly concentrated in the bearing device entering the plastic state, while its piers and superstructure are in the elastic working state.

[0075] Furthermore, the motion equations of the maglev line vehicle-bridge coupled system include the motion equations of the bridge and the maglev line vehicle, wherein the motion equation of the maglev line vehicle is: The equation of motion for the bridge is: In the formula, M V C V K V F represents the mass, damping, and stiffness matrices of the maglev train vehicle; M, C, and K represent the mass, damping, and stiffness matrices of the bridge, respectively. The subscript ss indicates the degrees of freedom of the unsupported nodes, gg indicates the degrees of freedom of the supported nodes, and sg and gl indicate the coupling terms between the supported and unsupported nodes. B,V The electromagnetic force vector between the maglev train vehicle and the track beam; u, and ü are the displacement, velocity, and acceleration vectors of the bridge, respectively.

[0076] Furthermore, such as Figure 4 As shown, the electromagnetic force vector F between the maglev train vehicle and the track beam is... B,V Based on the position of the maglev train on the bridge, the calculations are performed under the following four conditions:

[0077] 1) Vehicle entering the working condition (also known as "vehicle entering the bridge unit"):

[0078] ;

[0079] 2) Full-load condition (also known as "bridge unit full load"):

[0080] ;

[0081] 3) Vehicle departure condition (also known as "vehicle departure from bridge unit"):

[0082] ;

[0083] 4) No-load condition (also known as "bridge unit no-load"): ;

[0084] In the formula, [N] represents the shape function of the beam element; LE f is the unit length; my,j and f mz,j These are the levitation electromagnetic force and guiding electromagnetic force of the j-th electromagnet, respectively; l m x is the length of a single electromagnet; h,j and x t,j Let represent the start and end coordinates of the j-th electromagnet; NE and NL are the electromagnet numbers for the units being entered and exited, respectively.

[0085] Furthermore, the levitation electromagnetic force f of the j-th electromagnet my,j And guiding electromagnetic force f mz,j The calculation formula is as follows:

[0086] ;

[0087] ;

[0088] In the formula, h is the suspension gap; c is the guide gap; μ0 is the air permeability; W m A is the pole width; m N represents the magnetic pole area. c To control the number of current coils; i c To control the current; N const I is the number of constant current coils; const The constant current is used to balance the weight of the vehicles on the maglev line and is calculated by the following formula: In the formula: h0 is the rated levitation clearance of the maglev train when it is in the equilibrium position, which is determined according to the actual design conditions of the maglev train; M c M represents the weight of the carriage. bj denoted as , where is the weight of a single bogie; g is the acceleration due to gravity.

[0089] Furthermore, the formulas for calculating the suspension gap h and the guide gap c are as follows: ;

[0090] In the formula, Y(x, t) and Z(x, t) represent the yaw and vertical displacement of the maglev train, respectively; y(x, t) and z(x, t) represent the vertical and lateral displacements of the bridge, respectively; δ yr (x) indicates that the track is uneven.

[0091] Furthermore, the control current i in the electromagnetic force c Active control and adjustment are performed by a PID controller:

[0092] ;

[0093] In the formula, K P K I KD These are the proportional, integral, and differential coefficients, respectively; ḣ is the rate of change of the suspension gap, which can be obtained by taking the first derivative of the suspension gap h with respect to time t.

[0094] Furthermore, the dynamic responses of the vehicle and the bridge are solved separately using the Runge-Kutta numerical integration method through a separate iteration. The bridge motion equations are equivalently linearized, and the modal superposition method is used to decouple and solve the bridge motion equations. The equivalent linearization includes:

[0095] The nonlinear restoring force generated by the seismic isolation bearing is used as the right-hand term of the equation of motion for transferring the equivalent external load to the bridge. A virtual linear elastic element is introduced into the equation of motion corresponding to the location of the seismic isolation bearing. The specific method is as follows:

[0096] 1) The stiffness of the seismic isolation bearing is derived from the stiffness matrix K on the left-hand side of the equation of motion. ss The nonlinear restoring force R generated by the seismic isolation bearing is obtained by multiplying the nonlinear stiffness of the seismic isolation bearing by its displacement. NL This is equivalent to external loads placed on the right-hand side of the bridge's equation of motion, i.e. In the formula, K is the elastic stiffness matrix neglecting nonlinear elements; R NL The global nodal force vector originating from the nonlinear element is the nonlinear restoring force generated by the seismic isolation bearing;

[0097] 2) The above equations of motion neglect the nonlinear isolation bearings, resulting in an unstable and variable system in the computational model. Therefore, a virtual linear elastic element needs to be added to the bridge's equations of motion corresponding to the location of the isolation bearings (the location of the nonlinear element), i.e.: In the formula, K e This is the stiffness matrix of the added linear elastic element.

[0098] 3) The above equations represent the motion of a linear system, which can be solved using the modal superposition method, i.e.:

[0099] ;

[0100] In the formula, Ф i The i-th mode shape is determined by the initial linear elastic stiffness matrix; Q i denoted as the generalized coordinates corresponding to the i-th mode shape; a0 and a1 are proportionality constants, which can be determined by the first two frequencies and the corresponding damping ratios based on the initial linear elastic stiffness of the structure.

[0101] Furthermore, after introducing virtual linear elastic elements, the process of solving the bridge's motion equations includes iterative calculations based on nonlinear restoring forces, with the specific steps as follows:

[0102] I) The vibration state of the bridge at the previous time t-Δt is used as the initial condition of the maglev line vehicle-bridge coupling system at the current time;

[0103] II) Based on the bridge displacement response u from the previous time step or iteration step, calculate the load term K of the virtual linear elastic element. e u, and calculate the relative displacement of the pier and beam (displacement at the bridge bearing) to determine the vibration state of the seismic isolation bearing, such as Figure 5 As shown, the nonlinear restoring force R of the seismic isolation bearing is determined based on the bilinear hysteresis curve model. NL ;

[0104] III) The difference R between the nonlinear restoring force obtained in step II) and the load term of the virtual linear elastic element. NL -K e u is substituted into the right side of the equation of motion as an equivalent external load, and the vibration state of the bridge in the current iteration step is obtained by solving the equation.

[0105] IV) Determine if the bridge's dynamic response meets the convergence condition. If it does, use the current response as the initial condition for the next step; otherwise, return to step II) for the next iteration. The convergence criterion is based on the norm of the bridge's dynamic response vector. ;

[0106] In the formula, D B The dynamic response of the bridge is represented by the superscripts i and i-1, which indicate the current iteration step and the previous iteration step, respectively. ε is the tolerance error, which is set to 10. -5 .

[0107] Furthermore, the specific steps for solving the equations of motion using a separation and iteration method are as follows:

[0108] i) Set the initial motion state of the maglev train and bridge;

[0109] ii) Based on the motion state of the maglev line vehicles and bridges at the previous time t-Δt, and superimposed with the track irregularity at the current time t, determine the maglev gap and its rate of change, and determine the control current at the current time through the PID control algorithm;

[0110] iii) Calculate the electromagnetic levitation force and guiding force based on the current maglev gap status and control current;

[0111] iv) Substitute the electromagnetic force calculated in step iii) into the vehicle motion equation and bridge motion equation of the maglev line as load terms, respectively, and solve for the vibration state of the maglev line vehicle and bridge system. Specifically, substitute the electromagnetic force in step iii) into the load term on the right side of the vehicle motion equation, and use the Runge-Kutta method to solve the differential equation to obtain the vibration state of the vehicle system. Similarly, substitute the electromagnetic force and the seismic excitation at the current time step into the load term on the right side of the bridge motion equation, and use the Runge-Kutta method to solve the differential equation to obtain the vibration state of the bridge. Update the state of the maglev line vehicle-bridge coupling system as the initial condition for the next iteration step, and keep the control current unchanged during the iteration process.

[0112] v) Determine whether the dynamic response of the maglev train and the bridge meets the convergence condition. If it does, save the response at the current time and proceed to the next time step; if it does not, return to step iii) for iteration.

[0113] The criterion for determining convergence is whether the norms of the dynamic response vectors of the maglev line vehicle and the bridge meet the requirements: In the formula, D V and D B These represent the dynamic responses of the maglev train and the bridge, respectively; the superscripts i and i-1 indicate the current iteration step and the previous iteration step, respectively; ε is the tolerance error, taken as 10. -5 ;

[0114] vi) Repeat step ii) to v) until the maglev train has completely left the bridge.

[0115] Furthermore, based on the seismic response of the vehicle subsystem and bridge subsystem under seismic isolation bearings with different design parameters, the impact of seismic isolation bearings on the safety of bridge structures and vehicles traveling on the bridge is evaluated, and the seismic isolation effect of seismic isolation bearings on the vehicle-bridge coupled system is determined to guide the design of seismic isolation bearing parameters.

[0116] Example 2:

[0117] An electronic device includes a memory and a processor. The memory stores a computer program, and the processor is used to invoke and run the computer program stored in the memory to perform the methods of any of the above embodiments.

[0118] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of any of the above embodiments.

[0119] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for seismic response analysis of a maglev train-bridge coupled system with seismic isolation bearings, wherein the maglev train-bridge coupled system comprises a bridge consisting of an upper track beam and lower piers, seismic isolation bearings connecting the track beam and the piers, and a maglev train suspended above the track beam, characterized in that, include: Construct the motion equations of a maglev train-bridge coupled system that considers the nonlinear mechanical behavior of seismic isolation bearings under seismic loading; The dynamic response of the maglev train and the bridge is obtained by considering equivalent linearization and separation iteration of the equation of motion.

2. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 1, characterized in that, The specific method for constructing the motion equations of the maglev line vehicle-bridge coupled system that considers the nonlinear mechanical behavior of seismic isolation bearings under seismic loading includes: A multi-rigid-body dynamics model of the maglev line vehicle is established, and the car and suspension frame in the maglev line vehicle are simplified as rigid bodies. The car is considered to have five degrees of freedom: heave, pitch, yaw, head-up, and roll. The suspension frame is considered to have four degrees of freedom: heave, pitch, yaw, and head-up. The secondary suspension system in the maglev line vehicle is simplified to a spring-damping system. A finite element model of the bridge was established, and the track beams and piers in the bridge were simulated using 2-node, 12-DOF spatial beam elements. A nonlinear model of the seismic isolation bearing is established, which is simulated using massless spring elements. The nonlinear behavior of the seismic isolation bearing is simulated using a bilinear hysteresis curve model. The design parameters include the initial elastic stiffness k. d Post-yield stiffness k u Yield displacement S y ; Establish a model of seismic motion and track irregularities; A magnetic-track interaction model and a seismic action model are established. The magnetic-track interaction model includes the levitation electromagnetic force and the guiding electromagnetic force between the bridge track and the vehicle electromagnet. The active control characteristics of the magnetic levitation electromagnetic force are simulated using a PID controller. The seismic action can be achieved using measured seismic waves or artificially synthesized seismic waves.

3. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 2, characterized in that, The motion equations of the maglev line vehicle-bridge coupling system include the motion equations of the bridge and the motion equations of the maglev line vehicle, wherein the motion equation of the maglev line vehicle is: The equation of motion for the bridge is In the formula, M V C V K V F represents the mass, damping, and stiffness matrices of the maglev train vehicle; M, C, and K represent the mass, damping, and stiffness matrices of the bridge, respectively. The subscript ss indicates the degrees of freedom of the unsupported nodes, gg indicates the degrees of freedom of the supported nodes, and sg and gl indicate the coupling terms between the supported and unsupported nodes. B,V The electromagnetic force vector between the maglev train vehicle and the track beam; u, and ü are the displacement, velocity, and acceleration vectors of the bridge, respectively.

4. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 3, characterized in that, The electromagnetic force vector F between the maglev train vehicle and the track beam B,V Based on the position of the maglev train on the bridge, the calculations are performed under the following four conditions: 1) Vehicle entering operating condition: ; 2) Full load condition: ; 3) Vehicle leaving the operating condition: ; 4) No-load condition: ; In the formula, [N] represents the shape function of the beam element; L E f is the unit length; my,j and f mz,j These are the levitation electromagnetic force and guiding electromagnetic force of the j-th electromagnet, respectively; l m x is the length of a single electromagnet; h,j and x t,j Let represent the start and end coordinates of the j-th electromagnet; NE and NL are the electromagnet numbers for the units being entered and exited, respectively.

5. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 4, characterized in that, The levitation electromagnetic force f of the j-th electromagnet my,j And guiding electromagnetic force f mz,j The calculation formula is as follows: ； ； In the formula, h is the suspension gap; c is the guide gap; μ0 is the air permeability; W m A is the width of the magnetic poles. m N represents the magnetic pole area. c To control the number of current coils; i c To control the current; N const I is the number of constant current coils; const The constant current is used to balance the weight of the vehicles on the maglev line and is calculated by the following formula: In the formula: h0 is the rated levitation clearance of the maglev train when it is in the equilibrium position, which is determined according to the actual design conditions of the maglev train; M c M represents the weight of the carriage. bj denoted as , where is the weight of a single bogie; g is the acceleration due to gravity.

6. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 5, characterized in that, The formulas for calculating the suspension gap h and the guide gap c are as follows: ; In the formula, Y(x, t) and Z(x, t) represent the yaw and vertical displacement of the maglev train, respectively; y(x, t) and z(x, t) represent the vertical and lateral displacements of the bridge, respectively; δ yr (x) indicates that the track is uneven.

7. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 5, characterized in that, The control current i in the electromagnetic force c Active control and adjustment are performed by a PID controller: ； In the formula, K P K I K D These are the proportional, integral, and differential coefficients, respectively; ḣ is the rate of change of the suspension gap, which can be obtained by taking the first derivative of the suspension gap h with respect to time t.

8. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 5 or 6, characterized in that, The equivalent linearization includes: The nonlinear restoring force generated by the seismic isolation bearing is used as the right-hand term of the equation of motion for transferring the equivalent external load to the bridge. A virtual linear elastic element is introduced into the equation of motion corresponding to the position of the seismic isolation bearing. The specific method is as follows: 1) The stiffness of the seismic isolation bearing is derived from the stiffness matrix K on the left-hand side of the equation of motion. ss The nonlinear restoring force R generated by the seismic isolation bearing is obtained by multiplying the nonlinear stiffness of the seismic isolation bearing by its displacement. NL This is equivalent to external loads placed on the right-hand side of the bridge's equation of motion, i.e. In the formula, K is the elastic stiffness matrix neglecting nonlinear elements; R NL This refers to the nonlinear restoring force generated by the seismic isolation bearing; 2) Add a virtual linear elastic element to the equation of motion of the bridge corresponding to the location of the seismic isolation bearing, that is: In the formula, K e This is the stiffness matrix of the added linear elastic element.

9. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 8, characterized in that, The specific steps for solving the motion equations of the bridge after introducing virtual linear elastic elements are as follows: I) Use the vibration state of the bridge at the previous moment as the initial condition of the maglev line vehicle-bridge coupling system at the current moment; II) Based on the bridge displacement response of the previous time step or iteration step, calculate the load term of the virtual linear elastic element, and calculate the relative displacement of the pier and beam to determine the vibration state of the seismic isolation bearing. Determine the nonlinear restoring force of the seismic isolation bearing according to the bilinear hysteresis curve model. III) Substitute the difference between the nonlinear restoring force obtained in step II) and the load term of the virtual linear elastic element as the equivalent external load into the right side of the motion equation to solve for the bridge vibration state in the current iteration step. IV) Determine whether the dynamic response of the bridge meets the convergence condition. If it does, use the response at the current moment as the initial condition for the next moment. If not satisfied, return to step II) for the next iteration; the convergence condition The criterion for determining convergence is the norm of the bridge's dynamic response vector. ; In the formula, D B The dynamic response of the bridge is represented by the superscripts i and i-1, which represent the current iteration step and the previous iteration step, respectively; ε is the tolerance error.

10. The seismic response analysis method for a maglev train-bridge coupled system with seismic isolation bearings according to claim 9, characterized in that, The specific steps for solving the equations of motion using a separation and iterative approach are as follows: i) Set the initial motion state of the maglev train and bridge; ii) Based on the motion state of the maglev line vehicles and bridges at the previous moment, and superimposed with the track irregularities at the current moment, determine the maglev gap and its rate of change, and determine the control current at the current moment through a PID control algorithm; iii) Calculate the electromagnetic levitation force and guiding force based on the current maglev gap status and control current; iv) Substitute the electromagnetic force calculated in step iii) into the motion equations of the maglev line vehicle and the bridge as load terms, and solve for the vibration state of the maglev line vehicle and bridge system respectively. v) Determine whether the dynamic response of the maglev train and the bridge meets the convergence condition. If it does, save the response at the current time and proceed to the next time step; if it does not, return to step iii) for iteration. The criterion for determining convergence is whether the norms of the dynamic response vectors of the maglev line vehicle and the bridge meet the requirements: In the formula, D V and D B These represent the dynamic responses of the maglev train and the bridge, respectively; the superscripts i and i-1 represent the current iteration step and the previous iteration step, respectively. ε represents the tolerance; vi) Repeat step ii) to v) until the maglev train has completely left the bridge.

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