Truck bridge-crossing real-time average dynamic gradient calculation method based on axle coupling theory

Abstract

The invention discloses a truck bridge-crossing real-time average dynamic gradient calculation method based on an axle coupling theory, and belongs to the field of vehicle bridge coupling vibration. Solving the side rolling angular displacement, the oscillating angular displacement and the transverse displacement of the wheel set; a space beam unit is used for modeling a main beam and a bridge pier, and a space rod unit is used for simulating a main cable, a sling and a stay cable; finite element software is used for calculating bridge additional deformation, bridge dynamic response is solved according to the bridge additional deformation and the side rolling angular displacement, the oscillating angular displacement and the transverse displacement of the wheel sets, and bridge node displacement under all the wheel sets of the truck is obtained; differentiating the bridge node displacement under each wheel set of each truck to obtain a real-time dynamic gradient when the truck passes through the bridge; and weighting the real-time dynamic gradient when the truck passes through the bridge according to the weight of each section of truck to obtain the overall average equivalent gradient when the truck passes through the bridge. The method improves the calculation accuracy of the dynamic gradient when the truck passes through the bridge.

Description

Technical Field

[0001] The invention belongs to the field of vehicle-bridge coupling vibration, and in particular relates to a method for calculating the real-time average dynamic slope of a truck crossing a bridge based on vehicle-bridge coupling theory. Background Art

[0002] As railway bridges develop towards larger spans and higher flexibility, bridges deform significantly under train loads, causing the bridge track line shape to change at the same time, thereby changing the wheel-rail contact state and affecting the train's running stability. For freight trains, characteristics such as high axle weight and long marshaling make bridge deformation more significant. The deformation of the bridge will have a direct impact on the actual operating slope of the freight train, especially after superimposing the designed longitudinal section of the bridge. Excessive slopes will bring greater challenges to the traction performance of the freight train, which may result in insufficient traction capacity and broken hooks of heavy-loaded trains. Therefore, it is necessary to check and calculate the dynamic slope of the vehicle when crossing the bridge during the bridge design stage.

[0003] Traditional slope calculation methods generally treat trains as rigid bodies and use a relatively simple static moving load method to calculate bridge deformation and slope. However, the actual problem of trucks crossing bridges involves complex dynamic coupling problems, especially for trucks with high axle loads. Ignoring their dynamic coupling problems will lead to a decrease in the accuracy of the results. Therefore, it is necessary to establish a refined dynamic model for each subsystem and carry out vehicle-bridge coupling solution. Based on the vehicle-bridge coupling solution results, dynamic slope evaluation is carried out.

[0004] Therefore, obtaining the dynamic slope of a truck when crossing a bridge based on the vehicle-bridge coupling theory is a technical problem that needs to be solved urgently by those skilled in the art. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory, which improves the calculation accuracy of the dynamic slope when the truck crosses the bridge.

[0006] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory, comprising the following steps:

[0007] S1. Establish a vehicle dynamics model that takes into account truck types, weights, and formations;

[0008] S2. Obtaining the rolling angle displacement, yaw angle displacement and lateral displacement of the wheelset according to the vehicle dynamics model;

[0009] S3, use spatial beam elements to model the main beams and piers, and use spatial rod elements to simulate the main cables, suspenders and stay cables;

[0010] S4. Calculate the additional deformation of the bridge using finite element software, and solve the bridge dynamic response based on the additional deformation of the bridge, the roll angle displacement, the yaw angle displacement and the lateral displacement of the wheelset to obtain the bridge node displacement under each wheelset of the truck;

[0011] S5, differentiating the bridge node displacement under each wheel pair of the truck to obtain the real-time dynamic slope when the truck passes the bridge;

[0012] S6. The real-time dynamic slope of the truck when crossing the bridge is weighted according to the weight of each vehicle section to obtain the overall average equivalent slope of the truck when crossing the bridge.

[0013] Furthermore, the vehicle dynamics model includes wheelset motion equations, frame motion equations and vehicle body motion equations.

[0014] Furthermore, the wheelset motion equation, the frame motion equation and the vehicle body motion equation all include lateral motion, vertical motion, rolling motion, head shaking motion and rotational motion:

[0015]

[0016]

[0017]

[0018] Among them, M c is the vehicle body mass; M t is the frame quality; M w is the wheelset mass; I wx I is the moment of inertia of the wheelset around the X axis; wy I is the moment of inertia of the wheelset around the Y axis; wz I is the moment of inertia of the wheelset around the Z axis; tx I is the moment of inertia of the frame around the X axis; ty I is the moment of inertia of the frame around the Y axis; tz I is the moment of inertia of the frame around the Z axis; cx I is the moment of inertia of the vehicle body around the X axis; cy I is the moment of inertia of the vehicle body around the Y axis; cz K is the moment of inertia of the vehicle body around the Z axis; tx is the longitudinal stiffness of the secondary suspension; K ty is the lateral stiffness of the secondary suspension; K tz is the secondary suspension vertical stiffness; C tx is the longitudinal damping of the secondary suspension; C ty is the lateral damping of the secondary suspension; C tz is the secondary suspension vertical damping; K px is the primary suspension longitudinal stiffness; K py is the lateral stiffness of the primary suspension; K pz is the vertical stiffness of the primary suspension; Cpx is the longitudinal damping of the primary suspension; C py is the lateral damping of the primary suspension; C pz is the primary suspension vertical damping; K my is the lateral stop stiffness; K rx is the anti-roll stiffness; C sx is the anti-snaking shock absorber damping; H cb H is the distance between the center of mass of the vehicle body and the center of mass of the bolster; bt H is the distance between the center of mass of the bolster and the center of mass of the frame; tw L is the distance between the frame mass center and the wheelset mass center; c Half of the vehicle's fixed distance; L t Half of the vehicle's fixed wheelbase; d sk Half of the lateral distance of the central spring; d sc It is half of the lateral distance of the secondary vertical shock absorber; d wk Half of the lateral distance of the axle box spring; d wc Half of the lateral distance of the axle box damper; d sx is half of the lateral distance of the anti-snaking shock absorber; δ0 is the lateral stop clearance of the secondary system; Y c , Z c ,φ c , c , β c They are the lateral, vertical, rolling, head shaking and nodding displacements of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding speeds of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding accelerations of the vehicle body; Y tj , Z tj ,φ tj , tj , β tj are the lateral, vertical, rolling, head shaking and nodding displacements of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding accelerations of the jth frame respectively; Y wi , Z wi ,φ wi , wi , β wi are the lateral, vertical, rolling, head shaking and nodding displacements of the i-th wheel pair respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the i-th wheel pair respectively; are the lateral, vertical, rolling, yaw and nod accelerations of the i-th wheel pair respectively; r0 is the nominal rolling radius of the wheel; r Li 、r Riis the rolling radius of the left and right wheels of the i-th wheelset; Ω is the nominal rolling speed of the wheelset; d0 is half the distance between the left and right wheel-rail contact points; F Lxi 、F Rxi is the component of the creep force on the left and right wheels of the i-th wheel on the X-axis; F Lyi 、F Ryi is the component of the creep force on the left and right wheels of the i-th wheel on the Y axis; F Lzi 、F Rzi is the component of the creep force on the left and right wheels of the i-th wheel on the Z axis; N Lxi 、N Rxi is the component of the normal force on the left and right wheels of the i-th wheel on the X-axis; N Lyi 、N Ryi is the component of the normal force on the left and right wheels on the Y axis of the i-th wheel; N Lzi 、N Rzi is the component of the normal force on the left and right wheels on the Z axis of the i-th wheel; M Lxi 、M Rxi M is the component of the creep torque on the left and right wheels of the i-th wheel on the X-axis; Lyi 、M Ryi M is the component of the creep torque on the Y axis of the i-th wheel pair on the left and right wheels; Lzi 、M Rzi is the component of the creep torque on the Z axis on the left and right wheels of the i-th wheel.

[0019] Furthermore, the step S4 is specifically as follows:

[0020] S401. Calculate the additional deformation of the bridge using finite element software;

[0021] S402, according to the speed, displacement and acceleration of the truck in the previous time step, based on the kinematic equation, the running distance of the truck is calculated, and based on the specific position on the bridge corresponding to the running distance of the truck, the specific position on the bridge and the additional deformation of the bridge are added to obtain the actual position of the truck;

[0022] S403, finding the wheel-rail contact point using a trace method according to the actual position of the truck, the rolling angle displacement, the yaw angle displacement and the lateral displacement of the wheelset;

[0023] S404, according to the wheel-rail contact point, based on Hertz nonlinear elastic contact theory, calculate the wheel-rail contact normal force;

[0024] S405, calculating the wheel-rail creep force using Kalker's linear creep theory, and correcting it using Shen's theory to obtain a corrected wheel-rail creep force;

[0025] S406. According to the wheel-rail contact normal force and the corrected wheel-rail creep force, the wheel-rail interaction force F is obtained.T-V ;

[0026] S407, according to the wheel-rail interaction force, using the rail equation, obtain the bridge-rail interaction force;

[0027] S408, using the bridge-rail interaction force as the external load of the bridge motion equation, solving the bridge dynamic response by the Newmark-β method, and obtaining the bridge displacement in each degree of freedom direction of each bridge node;

[0028] S409. According to the bridge displacements in each degree of freedom direction of each bridge node, the displacements of the bridge nodes under each wheel pair of the truck are obtained using a cubic spline interpolation method.

[0029] Furthermore, in step S6, the overall average equivalent slope of the truck when crossing the bridge is expressed as:

[0030] i eq (t) = F s (t) / ∑ n P n

[0031] F s (t)=∑ n P n i n (t)

[0032] Among them, i eq (t) is the overall average equivalent slope of the truck when it crosses the bridge; t is time; F s (t) is the total slope additional resistance of the truck; P n is the weight of the nth car; i n (t) is the slope of the position of the nth car at running time t; n is the car number.

[0033] The beneficial effects of the present invention are as follows: the invention establishes a refined dynamic model for trucks, bridges and track structures and constructs a dynamic solution process method. The dynamic coupling effect between the high axle weight and long train of trucks and the bridge when crossing the bridge is fully considered. The cubic spline interpolation method is used to more accurately calculate the spatial absolute position of each wheel pair of each vehicle when the truck crosses the bridge, and the average slope method is used to consider the overall slope within the entire vehicle length, providing an evaluation method for the bridge crossing performance of trucks and a reference basis for the design of bridges. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 The present invention is a flow chart of the method.

[0035] Figure 2 Schematic diagram of the geometric relationship of the wheel-rail contact point C in an embodiment of the present invention.

[0036] Figure 3 Schematic diagram of the wheel-rail contact spot coordinate system in an embodiment of the present invention.

[0037] Figure 4 This is a schematic diagram of the dynamic slope results when a truck passes over a bridge under temperature rising conditions in an embodiment of the present invention. Figure 5 This is a schematic diagram of the dynamic slope results when a truck passes a bridge under cooling conditions in an embodiment of the present invention. Figure 6 This is a schematic diagram of the equivalent slope result when a truck crosses a bridge in an embodiment of the present invention. DETAILED DESCRIPTION

[0038] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0039] like Figure 1 As shown,

[0040] In one embodiment of the present invention, a method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory comprises the following steps:

[0041] S1. Establish a vehicle dynamics model that takes into account truck types, weights, and formations;

[0042] S2. Obtaining the rolling angle displacement, yaw angle displacement and lateral displacement of the wheelset according to the vehicle dynamics model;

[0043] S3, use spatial beam elements to model the main beams and piers, and use spatial rod elements to simulate the main cables, suspenders and stay cables;

[0044] S4. Calculate the additional deformation of the bridge using finite element software, and solve the bridge dynamic response based on the additional deformation of the bridge, the roll angle displacement, the yaw angle displacement and the lateral displacement of the wheelset to obtain the bridge node displacement under each wheelset of the truck;

[0045] S5, differentiating the bridge node displacement under each wheel pair of the truck to obtain the real-time dynamic slope when the truck passes the bridge;

[0046] S6. The real-time dynamic slope of the truck when crossing the bridge is weighted according to the weight of each vehicle section to obtain the overall average equivalent slope of the truck when crossing the bridge.

[0047] The vehicle dynamics model includes wheelset motion equations, frame motion equations and vehicle body motion equations.

[0048] The wheelset motion equation, frame motion equation and vehicle body motion equation all include lateral motion, vertical motion, rolling motion, head shaking motion and rotational motion:

[0049]

[0050]

[0051] Among them, M c is the vehicle body mass; M t is the frame quality; M w is the wheelset mass; I wx I is the moment of inertia of the wheelset around the X axis; wy I is the moment of inertia of the wheelset around the Y axis; wz I is the moment of inertia of the wheelset around the Z axis; tx I is the moment of inertia of the frame around the X axis; ty I is the moment of inertia of the frame around the Y axis; tz I is the moment of inertia of the frame around the Z axis; cx I is the moment of inertia of the vehicle body around the X axis; cy I is the moment of inertia of the vehicle body around the Y axis; cz K is the moment of inertia of the vehicle body around the Z axis; tx is the longitudinal stiffness of the secondary suspension; K ty is the lateral stiffness of the secondary suspension; K tz is the secondary suspension vertical stiffness; C tx is the longitudinal damping of the secondary suspension; C ty is the lateral damping of the secondary suspension; C tz is the secondary suspension vertical damping; K px is the primary suspension longitudinal stiffness; K py is the lateral stiffness of the primary suspension; K pz is the vertical stiffness of the primary suspension; C px is the longitudinal damping of the primary suspension; C py is the lateral damping of the primary suspension; C pz is the primary suspension vertical damping; K my is the lateral stop stiffness; K rx is the anti-roll stiffness; C sx is the damping of the anti-snaking shock absorber; H cb H is the distance between the center of mass of the vehicle body and the center of mass of the bolster; bt H is the distance between the center of mass of the bolster and the center of mass of the frame; tw L is the distance between the frame mass center and the wheelset mass center; c Half of the vehicle's fixed distance; L t Half of the vehicle's fixed wheelbase; d sk Half of the lateral distance of the central spring; d sc It is half of the lateral distance of the secondary vertical shock absorber; d wkHalf of the lateral distance of the axle box spring; d wc Half of the lateral distance of the axle box damper; d sx is half of the lateral distance of the anti-snaking shock absorber; δ0 is the lateral stop clearance of the secondary system; Y c , Z c ,φ c , c , β c They are the lateral, vertical, rolling, head shaking and nodding displacements of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding speeds of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding accelerations of the vehicle body; Y tj , Z tj ,φ tj , tj , β tj are the lateral, vertical, rolling, head shaking and nodding displacements of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding accelerations of the jth frame respectively; Y wi , Z wi ,φ wi , wi , β wi are the lateral, vertical, rolling, head shaking and nodding displacements of the i-th wheel pair respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the i-th wheel pair respectively; are the lateral, vertical, rolling, yaw and nod accelerations of the i-th wheel pair respectively; r0 is the nominal rolling radius of the wheel; r Li 、r Ri is the rolling radius of the left and right wheels of the i-th wheelset; Ω is the nominal rolling speed of the wheelset; d0 is half the distance between the left and right wheel-rail contact points; F Lxi 、F Rxi is the component of the creep force on the left and right wheels of the i-th wheel on the X-axis; F Lyi 、F Ryi is the component of the creep force on the left and right wheels of the i-th wheel on the Y axis; F Lzi 、F Rzi is the component of the creep force on the left and right wheels of the i-th wheel on the Z axis; N Lxi 、N Rxi is the component of the normal force on the left and right wheels of the i-th wheel on the X-axis; N Lyi 、N Ryi is the component of the normal force on the left and right wheels on the Y axis of the i-th wheel; N Lzi 、N Rzi is the component of the normal force on the left and right wheels on the Z axis of the i-th wheel; MLxi 、M Rxi M is the component of the creep torque on the left and right wheels of the i-th wheel on the X-axis; Lyi 、M Ryi M is the component of the creep torque on the Y axis of the i-th wheel pair on the left and right wheels; Lzi 、M Rzi is the component of the creep torque on the Z axis on the left and right wheels of the i-th wheel.

[0052] In this embodiment, for the vehicle dynamics model, five degrees of freedom are considered, namely, lateral, vertical, rolling, head shaking, and head nodding, as shown in the following table:

[0053] Table 1 Degrees of freedom of vehicle dynamics model

[0054]

[0055] The normal force and creep force on each wheelset are obtained from the wheel-rail contact relationship. The vehicle's specific mass, moment of inertia, stiffness, damping and other parameters are known parameters. The speed, displacement and acceleration of the vehicle body, frame and wheelset are solved according to the above vehicle motion equations. At the same time, the displacement of the wheelset φ w , w ,y w It is also used as an input variable in the wheel-rail contact position solution. The wheelset displacement at this time step is used as an input variable for the wheel-rail contact position at the next time step.

[0056] The step S4 is specifically as follows:

[0057] S401. Calculate the additional deformation of the bridge using finite element software;

[0058] In this embodiment, commercial finite element software is used to calculate the bridge line shape under various additional deformation conditions. In this embodiment, MIDAS Civil is used to calculate the additional deformation of the bridge. The additional deformation conditions are selected as overall temperature increase of 30°C and overall temperature decrease of 30°C. The additional deformation X of the bridge corresponding to the horizontal coordinate mileage of the bridge is obtained. Bfi .

[0059] S402. According to the speed, displacement and acceleration of the truck in the previous time step, based on the kinematic equation, the running distance of the truck is calculated, and based on the specific position on the bridge corresponding to the running distance of the truck, the specific position on the bridge and the additional deformation of the bridge are added to obtain the actual position of the truck:

[0060] X r =X Bfi +X Bi

[0061] Among them, X r is the actual position of the truck, X Biis the bridge displacement at the corresponding node position of the bridge.

[0062] S403, finding the wheel-rail contact point using a trace method according to the actual position of the truck, the rolling angle displacement, the yaw angle displacement and the lateral displacement of the wheelset;

[0063] The expression of the wheel-rail contact point of each train carriage is:

[0064]

[0065] l x =-cosφ w sinψ w

[0066] l y =cosφ w cosψ w

[0067] l z = sinφ w

[0068]

[0069] Among them, x c is the abscissa of the wheel-rail contact point; is the horizontal coordinate of the center O2 of the wheel rolling circle; l x for l x =-cosφ w sinψ w ; R w is the radius of the wheel rolling circle; tg is the tan function; δ w is the wheel tread contact angle; y c is the ordinate of the wheel-rail contact point; is the ordinate of the center O2 of the wheel rolling circle; l y for l y =cosφ w cosψ w ; l z for l z = sinφ w ;y w is the lateral displacement of the wheelset; z c is the vertical coordinate of the wheel-rail contact point; is the vertical coordinate of the center O2 of the wheel rolling circle; φ w is the rolling angle of the wheelset; ψ w is the wheel set's yaw angle; d w It is the transverse coordinate of each rolling circle of the wheel tread in the wheelset coordinate system.

[0070] In this embodiment, for the train-track interaction, the actual bridge position X at the train travel position is obtained. r According to the contact geometric relationship between the train wheel tread and the rail head tread, the wheel-rail contact point C is found by the trace method. The geometric relationship of the wheel-rail contact point C is as follows: Figure 2 As shown in the figure, the wheel-rail contact point C is within three planes, including the rolling circle plane with O2 as the center and the wheel rolling circle radius R w The spherical surface with a radius of 2, and the plane O1-O1'-C'-C; the coordinates of the wheel-rail contact point C in the absolute coordinate system can be derived through the equations of the three surfaces (x c ,y c , z c ).

[0071] When the wheelset lateral displacement y w 、Head shaking angle ψ w 、Roll angle φ w When a certain time is reached, by gradually changing the horizontal coordinates d of each rolling circle of the wheel tread in the wheelset coordinate system w , the wheel-rail spatial contact trace at a certain moment can be constructed; the calculation of the wheel-rail contact point is affected by the movement displacement of the rail pair and the rail and the change of the wheel-track irregularity value, and the wheel-rail contact point needs to be recalculated at every moment.

[0072] S404, according to the wheel-rail contact point, based on Hertz nonlinear elastic contact theory, calculate the wheel-rail contact normal force;

[0073] The expression of the wheel-rail contact normal force is:

[0074]

[0075] δN L =δZ L / cos(δ wL +φ w )

[0076] δN R =δZ R / cos(δ wR -φ w )

[0077] Among them, P N (t) is the normal force of the wheel-rail contact at the current time step; t is time; G is the wheel-rail contact constant; R is the radius of the wheel rolling circle; δN(t) is the relative normal compression of the wheel-rail contact point; δN L is the relative normal compression between the left wheel and rail; δZ L is the relative vertical displacement of the wheel / rail at the left contact point; δ wL is the contact angle of the left wheel tread; φ wis the rolling angle of the wheelset; δN R is the relative normal compression between the right wheel and rail; δZ R is the relative vertical displacement of the wheel / rail at the right contact point; δ wR is the contact angle of the right wheel tread.

[0078] S405, calculating the wheel-rail creep force using Kalker's linear creep theory, and correcting it using Shen's theory to obtain a corrected wheel-rail creep force;

[0079] The expression of creep force is:

[0080]

[0081] G(ab)=2G w G r / (G w +G r )

[0082]

[0083] Among them, F x ′ is the longitudinal creep force after correction using Shen’s theory; ε is the correction coefficient; F x is the initial longitudinal creep force; F y ′ is the lateral creep force after correction using Shen’s theory; F y is the initial lateral creep force; M z ′ is the rotation creep torque after correction using Shen’s theory; M z is the initial rotation creep torque; F R ′ is the resultant force of the lateral and longitudinal creep forces after nonlinear correction; F R is the resultant force of the lateral and longitudinal creep forces; f is the friction coefficient between the wheel and rail; N is the total normal force of the wheel-rail contact; f 11 is the longitudinal creep coefficient; ξ x is the longitudinal creep rate; f 22 is the lateral creep coefficient; ξ y is the lateral creep rate; f 23 is the rotational lateral creep coefficient; ξ sp is the rotation creep rate; f 33 is the rotation creep coefficient; G(ab) is the composite shear modulus of the wheelset and track material; a is the major semi-axis of the contact ellipse; b is the minor semi-axis of the contact ellipse; C 11 , C 22 , C 23 and C 33 All are Kalker coefficients; G w is the shear modulus of the wheelset material, G r is the shear modulus of the track material; V w1V is the movement speed of the contact ellipse on the wheel tread along the e1 axis; r1 V is the motion speed corresponding to the axis of the contact ellipse e1 on the rail; w2 V is the movement speed of the contact ellipse on the wheel tread along the e2 axis; r2 is the motion speed corresponding to the contact ellipse e2 axis on the rail; Ω w3 is the speed of the contact ellipse on the wheel tread around the e3 axis; Ω r3 is the motion speed corresponding to the axis of the contact ellipse e3 on the rail; V is the nominal forward speed of the wheelset on the rail.

[0084] S406. According to the wheel-rail contact normal force and the corrected wheel-rail creep force, the wheel-rail interaction force F is obtained. T-V ;

[0085] In this embodiment, the wheel-rail contact normal force P of the left and right wheels of each wheel pair is calculated separately. N (t) and creep force, which can be corresponded to F in step S1 Lxi 、F Rxi —The component of the creep force on the left and right wheels on the X axis (i=1-4); F Lyi 、F Ryi —The component of the creep force on the left and right wheels on the Y axis (i=1-4); F Lzi 、F Rzi —The component of the creep force on the left and right wheels on the Z axis of the i-th wheel (i=1-4); N Lxi 、N Rxi —The component of the normal force on the left and right wheels by the i-th wheel on the X-axis (i=1-4); N Lyi 、N Ryi —Component of the normal force on the left and right wheels by the i-th wheel on the Y axis (i=1-4); N Lzi 、N Rzi —The component of the normal force on the left and right wheels on the Z axis of the i-th wheel (i=1-4); M Lxi 、M Rxi —The component of the creep torque on the left and right wheels of the i-th wheel on the X-axis (i=1-4); M Lyi 、M Ryi —The component of the creep torque on the Y axis of the i-th wheel on the left and right wheels (i=1-4); M Lzi 、M Rzi —The component of the creep torque on the left and right wheels of the i-th wheel on the Z axis (i=1~4).

[0086] S407, according to the wheel-rail interaction force, using the rail equation, obtain the bridge-rail interaction force;

[0087] The vertical, lateral and torsional vibrations of the rail under the wheel-rail force and bridge-rail action can be expressed as:

[0088]

[0089] The above equation is a fourth-order partial differential equation. In order to perform numerical analysis, the Ritz method is used to convert the above partial differential equation into a second-order ordinary differential equation. To this end, the canonical coordinates q of the rail are introduced. zk (t) (vertical), q yk (t)(horizontal), q tk (t)(torsion), after transformation, the second-order ordinary differential equations of rail vibration can be obtained:

[0090]

[0091]

[0092] in, is the second-order derivative of the vertical rail canonical coordinate; t is time; E r is the Young's modulus of the rail; I ry is the moment of inertia of the rail section around the Y axis; ρ r is the mass density of the rail; A r is the cross-sectional area of ​​the rail; k is the cut-off modal order number; π is the circumference of a circle; l is the length of the rail; q zk (t) is the vertical rail canonical coordinate; N s N is the number of sleepers within the calculated length of the rail; w The number of axles within the calculated length of the rail; F sVi is the vertical reaction force of the ith support point of the rail; Z k () is the vibration mode function of the rail in the vertical direction; P V j is the vertical force acting on the rail by the jth wheel; N Z is the cutoff mode order of the vertical vibration mode of the rail; is the second-order derivative of the transverse rail canonical coordinate; I z is the moment of inertia of the rail section around the Z axis; q yk (t) is the transverse rail canonical coordinate; F sHi is the lateral reaction force of the ith support point of the rail; Y k () is the vibration mode function of the rail in the transverse direction; P Hj is the lateral force of the jth wheel acting on the rail; N Y is the cutoff mode order in the transverse direction of the rail; is the second-order derivative of the canonical coordinates of the torsion rail; G r is the shear modulus of the rail; J rt is the torsional inertia moment of the rail section; J r0is the polar moment of inertia of the rail section; q tk (t) is the canonical coordinate of the twisted rail; F sTi is the torsional reaction force of the i-th support point of the rail; Φ k () is the vibration mode function of rail torsion; P T j is the torque of the jth wheel acting on the rail; x si is the x coordinate of the ith support point of the rail; Pj is the x-coordinate of the j-th wheelset of the rail; N T is the cutoff mode order of the rail torsional vibration mode; z r (x,t),y r (x,t),φ r (x, t) are respectively the vertical, lateral and torsional displacements of the actual vibration of the rail at the x-coordinate at time t; are respectively the vertical, lateral and torsional velocities of the actual vibration of the rail at the x-coordinate at time t; is the vertical, lateral and torsional acceleration of the actual vibration of the rail at the x-coordinate at time t;

[0093] S408, using the bridge-rail interaction force as the external load of the bridge motion equation, solving the bridge dynamic response by the Newmark-β method, and obtaining the bridge displacement in each degree of freedom direction of each bridge node;

[0094] Kinematic equations of the bridge subsystem:

[0095] Where M B , C B , K B are the mass matrix, damping matrix, and stiffness matrix of the bridge subsystem respectively; X B , are the displacement, velocity and acceleration vectors of the bridge subsystem respectively; F B-T 、F T-B is the interaction force between the track and the bridge.

[0096] The sleeper forces Fgy, Fgx, and Mgz are located at the sleeper positions and can be mapped to the bridge node positions using the cubic spline interpolation method to form the bridge-rail interaction force F B-T The interpolation method is as follows:

[0097] Since the bridge model is established using the finite element method, the position of the sleeper position point generally does not coincide with the bridge node. Therefore, the interpolation method is used to form the equivalent load of the bridge node. Considering that the bridge will deform under external loads and be constrained by the supports, the following formula is used for interpolation fitting:

[0098] Y b =N1(x)u yi +N2(x)θzi +N3(x)u yj +N4(x)θ zj

[0099] Z b =N1(x)u zi -N2(x)θ yi +N3(x)u zj -N4(x)θ yj

[0100] φ b =N5(x)θ xi +N6(x)θ xj

[0101]

[0102] Where Y b is the lateral displacement of the centroid of the bridge unit under the sleeper; Z b is the vertical displacement of the centroid of the bridge unit under the sleeper; φ b is the torsional displacement of the centroid of the bridge unit under the sleeper; N1(x), N2(x), N3(x), N4(x), N5(x), N6(x) are the shape functions of the spatial beam unit; L is the unit length; x is the longitudinal coordinate of the sleeper position in the bridge unit coordinate system; u yi 、u yj 、u zi 、u zi ,θ zi ,θ zj ,θ yi ,θ yj ,θ xi ,θ xj are the lateral, vertical and torsional displacements of the left and right nodes of each unit, and i and j are the components of the two nodes of each unit;

[0103] The equivalent bridge node load F is obtained by multiplying the shape function of the spatial beam element and the sleeper load. T-B ;

[0104] F yi =N1(x)F gy

[0105] F zi =N1(x)F gz

[0106] F yj =N3(x)F gy

[0107] F zj =N3(x)F gz

[0108] M xi =N5(x)M gx

[0109] M yi =-N2(x)F gz

[0110] M zi =N2(x)F gy

[0111] M xj =N6(x)M gx

[0112] M yj =-N4(x)F gz

[0113] M zj =N4(x)F gy

[0114] In the formula, F yi 、F zi 、F yj 、F zj 、M xi 、M yi 、M zi 、M xj 、M yj 、M zj The forces F transmitted to the bridge by the track are T-B Components in the three directions of lateral, vertical and torsional directions and at the two nodes i and j of each element;

[0115] The Newmark-β method integration format for solving the motion equations for the bridge subsystem is:

[0116]

[0117] Where α and β are integration parameters, which can generally be taken as 0.25 and 0.5 respectively; Δt is the time integration step; n represents the nth integration step;

[0118] The displacement of the bridge subsystem X B ,speed Acceleration vector As X n , Substituting this into the equation, we can solve the equation of motion of the bridge subsystem;

[0119] In the above formula Utilizing X n+1 To represent, then respectively Substitute into the bridge subsystem dynamics equation at time n+1 and simplify to obtain X n+1The solution formula is:

[0120]

[0121] In the formula, F n+1 is the bridge node load at time n+1; M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively;

[0122] X n+1 The solution formula is:

[0123]

[0124] In the formula, a6=Δ t (1-β), a7=βΔt, solving the above equation yields X n+1 ; α is the integral parameter, which is 0.25; β is the integral parameter, which is 0.5;

[0125] The solution formula is as follows:

[0126]

[0127] S409. According to the bridge displacements in each degree of freedom direction of each bridge node, the displacements of the bridge nodes under each wheel pair of the truck are obtained using a cubic spline interpolation method.

[0128] The specific interpolation method is consistent with the method of interpolating the bridge displacement to obtain the sleeper displacement.

[0129] Y wheelset =N1(y)u yi +N2(y)θ zi +N3(y)u yj +N4(y)θ zj

[0130] Z wheelset =N1(y)u zi -N2(y)θ yi +N3(y)u zj -N4(y)θ yj

[0131] φ wheelset =N5(y)θ xi +N6(y)θ xj

[0132]

[0133] Where Y wheelset is the lateral displacement of the centroid of the bridge unit under the wheel; Z wheelset is the vertical displacement of the centroid of the bridge unit under the wheel; φwheelset is the torsional displacement of the bridge unit centroid under the wheel; N1(y), N2(y), N3(y), N4(y), N5(y), N6(y) are the shape functions of the spatial beam unit; L is the unit length; y is the longitudinal coordinate of the wheel position in the bridge unit coordinate system; u yi 、u yj 、u zi 、u zi ,θ zi ,θ zj ,θ yi ,θ yj ,θ xi ,θ xj are the lateral, vertical and torsional displacements of the left and right nodes of each unit, i and j are the components of the two nodes of each unit; the dynamic slope is mainly determined by Z wheelset Perform calculations.

[0134] At each moment, the vertical displacement Z of the bridge under the wheel is obtained wheelset (t), let x = t × v, Z wheelset (t) is converted into a function Z about the x-mileage wheelset (x), then the slope of the position where the nth vehicle runs x distance is:

[0135]

[0136] Let t = x / v, we can n (x) is converted into the function i of the time coordinate t n (t).

[0137] In this embodiment, the real-time dynamic slopes of the 1st, 20th, 40th, and 60th trains are as follows: Figure 2 and Figure 3 shown.

[0138] The expression of the overall average equivalent slope when the truck crosses the bridge in step S6 is:

[0139] i eq (t) = F s (t) / Σ n P n

[0140] F s (t)=Σ n P n i n (t)

[0141] Among them, i eq (t) is the overall average equivalent slope of the truck when it crosses the bridge; t is time; F s (t) is the total slope additional resistance of the truck; P nis the weight of the nth car; i n (t) is the slope of the position of the nth car at running time t; n is the car number.

[0142] In this embodiment, the average equivalent slope when the train passes the bridge is as follows: Figure 4 shown.

[0143] This method is based on the vehicle-bridge coupling theory, and establishes a train formation that can take into account the dynamics of the freight train. Taking into account the dynamic coupling effect of the bridge and the train, the dynamic line shape of the bridge under each wheel pair when the freight train passes the bridge is obtained, and the equivalent slope of the freight train when passing the bridge is calculated based on the dynamic line shape of the bridge. It provides an accurate calculation method for the performance analysis of the freight train when passing the bridge, and provides a reference for the design.

Claims

1. A method for calculating the real-time average dynamic slope of a truck crossing a bridge based on vehicle-bridge coupling theory, characterized in that: The following steps are involved: S1. Establish a vehicle dynamics model that takes into account truck types, weights, and formations; S2. Obtaining the rolling angle displacement, yaw angle displacement and lateral displacement of the wheelset according to the vehicle dynamics model; S3, use spatial beam elements to model the main beams and piers, and use spatial rod elements to simulate the main cables, suspenders and stay cables; S4. Calculate the additional deformation of the bridge using finite element software, and solve the bridge dynamic response based on the additional deformation of the bridge, the roll angle displacement, the yaw angle displacement and the lateral displacement of the wheelset to obtain the bridge node displacement under each wheelset of the truck; S5, differentiating the bridge node displacement under each wheel pair of the truck to obtain the real-time dynamic slope when the truck passes the bridge; S6. The real-time dynamic slope of the truck when crossing the bridge is weighted according to the weight of each vehicle section to obtain the overall average equivalent slope of the truck when crossing the bridge.

2. The method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory according to claim 1 is characterized in that: The vehicle dynamics model includes wheelset motion equations, frame motion equations and vehicle body motion equations.

3. The method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory according to claim 2 is characterized in that: The wheelset motion equation, frame motion equation and vehicle body motion equation all include lateral motion, vertical motion, rolling motion, head shaking motion and rotational motion: Among them, M c is the vehicle body mass; M t is the frame quality; M w is the wheelset mass; I wx I is the moment of inertia of the wheelset around the X axis; wy I is the moment of inertia of the wheelset around the Y axis; wz I is the moment of inertia of the wheelset around the Z axis; tx I is the moment of inertia of the frame around the X axis; ty I is the moment of inertia of the frame around the Y axis; tz I is the moment of inertia of the frame around the Z axis; cx I is the moment of inertia of the vehicle body around the X axis; cy I is the moment of inertia of the vehicle body around the Y axis; cz K is the moment of inertia of the vehicle body around the Z axis; tx is the longitudinal stiffness of the secondary suspension; K ty is the lateral stiffness of the secondary suspension; K tz is the secondary suspension vertical stiffness; C tx is the longitudinal damping of the secondary suspension; C ty is the lateral damping of the secondary suspension; C tz is the secondary suspension vertical damping; K px is the primary suspension longitudinal stiffness; K py is the lateral stiffness of the primary suspension; K pz is the vertical stiffness of the primary suspension; C px is the longitudinal damping of the primary suspension; C py is the lateral damping of the primary suspension; C pz is the primary suspension vertical damping; K my is the lateral stop stiffness; K rx is the anti-roll stiffness; C sx is the anti-snaking shock absorber damping; H cb H is the distance between the center of mass of the vehicle body and the center of mass of the bolster; bt H is the distance between the center of mass of the bolster and the center of mass of the frame; tw L is the distance between the frame mass center and the wheelset mass center; c Half of the vehicle's fixed distance; L t Half of the vehicle's fixed wheelbase; d sk Half of the lateral distance of the central spring; d sc It is half of the lateral distance of the secondary vertical shock absorber; d wk Half of the lateral distance of the axle box spring; d wc Half of the lateral distance of the axle box damper; d sx is half of the lateral distance of the anti-snaking shock absorber; δ0 is the lateral stop clearance of the secondary system; Y c , Z c ,φ c , c , β c They are the lateral, vertical, rolling, head shaking and nodding displacements of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding speeds of the vehicle body; They are the lateral, vertical, rolling, shaking and nodding accelerations of the vehicle body; Y tj , Z tj ,φ tj , tj , β tj are the lateral, vertical, rolling, head shaking and nodding displacements of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the jth frame respectively; are the lateral, vertical, rolling, head shaking and nodding accelerations of the jth frame respectively; Y wi , Z wi ,φ wi , wi , β wi are the lateral, vertical, rolling, head shaking and nodding displacements of the i-th wheel pair respectively; are the lateral, vertical, rolling, head shaking and nodding speeds of the i-th wheel pair respectively; are the lateral, vertical, rolling, yaw and nod accelerations of the i-th wheel pair respectively; r0 is the nominal rolling radius of the wheel; r Li 、r Ri is the rolling radius of the left and right wheels of the i-th wheelset; Ω is the nominal rolling speed of the wheelset; d0 is half the distance between the left and right wheel-rail contact points; F Lxi 、F Rxi is the component of the creep force on the left and right wheels of the i-th wheel on the X-axis; F Lyi 、F Ryi is the component of the creep force on the left and right wheels of the i-th wheel on the Y axis; F Lzi 、F Rzi is the component of the creep force on the left and right wheels of the i-th wheel on the Z axis; N Lxi 、N Rxi is the component of the normal force on the left and right wheels of the i-th wheel on the X-axis; N Lyi 、N Ryi is the component of the normal force on the left and right wheels on the Y axis of the i-th wheel; N Lzi 、N Rzi M is the component of the normal force on the left and right wheels on the Z axis; Lxi 、M Rxi M is the component of the creep torque on the left and right wheels of the i-th wheel on the X-axis; Lyi 、M Ryi M is the component of the creep torque on the Y axis of the i-th wheel pair on the left and right wheels; Lzi 、M Rzi is the component of the creep torque on the Z axis on the left and right wheels of the i-th wheel.

4. The method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory according to claim 1 is characterized in that: The step S4 is specifically as follows: S401. Calculate the additional deformation of the bridge using finite element software; S402, according to the speed, displacement and acceleration of the truck in the previous time step, based on the kinematic equation, the running distance of the truck is calculated, and based on the specific position on the bridge corresponding to the running distance of the truck, the specific position on the bridge and the additional deformation of the bridge are added to obtain the actual position of the truck; S403, finding the wheel-rail contact point using a trace method according to the actual position of the truck, the rolling angle displacement, the yaw angle displacement and the lateral displacement of the wheelset; S404, according to the wheel-rail contact point, based on Hertz nonlinear elastic contact theory, calculate the wheel-rail contact normal force; S405, calculating the wheel-rail creep force using Kalker's linear creep theory, and correcting it using Shen's theory to obtain a corrected wheel-rail creep force; S406. According to the wheel-rail contact normal force and the corrected wheel-rail creep force, the wheel-rail interaction force F is obtained. T-V ; S407, according to the wheel-rail interaction force, using the rail equation, obtain the bridge-rail interaction force; S408, using the bridge-rail interaction force as the external load of the bridge motion equation, solving the bridge dynamic response by the Newmark-β method, and obtaining the bridge displacement in each degree of freedom direction of each bridge node; S409. According to the bridge displacements in each degree of freedom direction of each bridge node, the displacements of the bridge nodes under each wheel pair of the truck are obtained using a cubic spline interpolation method.

5. The method for calculating the real-time average dynamic slope of a truck crossing a bridge based on the vehicle-bridge coupling theory according to claim 1 is characterized in that: The expression of the overall average equivalent slope when the truck crosses the bridge in step S6 is: i eq (t)=F s (t) / ∑ n P n F s (t)=Σ n P n i n (t) Among them, i eq (t) is the overall average equivalent slope of the truck when it crosses the bridge; t is time; F s (t) is the total slope additional resistance of the truck; P n is the weight of the nth car; i n (t) is the slope of the position of the nth car at running time t; n is the car number.

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