A Real-Time Average Dynamic Gradient Calculation Method for Trucks Crossing Bridges Based on Vehicle-Bridge Coupling Theory

Abstract

The application discloses a truck bridge real-time average dynamic slope calculation method based on a vehicle-bridge coupling theory and belongs to the field of vehicle-bridge coupling vibration. The method comprises the following steps: establishing a vehicle dynamics model; solving the roll angle displacement, the swing angle displacement and the lateral displacement of a wheel set; modeling the main girder and the pier by using a space beam element, and simulating the main cable, the sling and the stay cable by using a space rod element; calculating the bridge additional deformation by using a finite element software, and solving the bridge dynamic response according to the bridge additional deformation, the roll angle displacement, the swing angle displacement and the lateral displacement of the wheel set to obtain the bridge node displacement under each wheel set of the truck; differentiating the bridge node displacement under each wheel set of each truck to obtain the real-time dynamic slope when the truck passes through the bridge; and weighting the real-time dynamic slope when the truck passes through the bridge according to the weight of each section of the truck to obtain the overall average equivalent slope when the truck passes through the bridge. The application improves the calculation accuracy of the dynamic slope when the truck passes through the bridge.

Description

Technical Field

[0001] This invention belongs to the field of vehicle-bridge coupled vibration, and particularly relates to a method for calculating the real-time average dynamic slope of trucks crossing bridges based on vehicle-bridge coupling theory. Background Technology

[0002] As railway bridges develop towards longer spans and higher flexibility, significant deformation occurs under train loads, altering the bridge track alignment and consequently the wheel-rail contact state, impacting train stability. For freight trains, high axle loads and long formations exacerbate this deformation, directly affecting the gradient at which freight trains actually cross. Especially after superimposing the bridge's longitudinal profile design, excessive gradients pose a greater challenge to freight train traction performance, potentially leading to insufficient traction capacity and even coupler breakage in heavily loaded trains. Therefore, dynamic gradient calculations for vehicle crossings are necessary during the bridge design phase.

[0003] Traditional slope calculation methods typically treat the train as a rigid body and use a relatively simple static moving load method to calculate bridge deformation and slope. However, in actual freight train bridge crossings, complex dynamic coupling problems are involved, especially for freight trains with high axle loads. Ignoring these dynamic coupling problems will lead to a decrease in the accuracy of the results. Therefore, it is necessary to establish refined dynamic models for each subsystem and conduct vehicle-bridge coupled solutions. Based on the vehicle-bridge coupled solution results, dynamic slope evaluation is carried out.

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

[0005] To address the aforementioned shortcomings in the existing technology, this invention provides a method for calculating the real-time average dynamic slope of trucks crossing bridges based on vehicle-bridge coupling theory, which improves the accuracy of dynamic slope calculation when trucks cross bridges.

[0006] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: a method for calculating the real-time average dynamic gradient of trucks crossing bridges based on vehicle-bridge coupling theory, comprising the following steps:

[0007] S1. Establish a vehicle dynamics model that takes into account truck type, weight and formation;

[0008] S2. Obtain the roll angle, yaw angle, and lateral displacement of the wheelset based on the vehicle dynamics model;

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

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

[0011] S5. Differentiate the bridge node displacements under each wheel pair of the truck to obtain the real-time dynamic slope when the truck crosses the bridge.

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

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

[0014] Furthermore, the motion equations of the wheelset, the frame, and the vehicle body all include lateral motion, vertical motion, roll motion, yaw motion, and rotational motion:

[0015]

[0016]

[0017]

[0018] Among them, M c For vehicle body mass; M t For structural quality; M w For wheelset mass; I wx I is the moment of inertia of the wheelset about the X-axis; wy I is the moment of inertia of the wheelset about the Y-axis; wz I is the moment of inertia of the wheelset about the Z-axis; tx I is the moment of inertia of the frame about the X-axis; ty Let I be the moment of inertia of the frame about the Y-axis; tz I is the moment of inertia of the frame about the Z-axis. cx Let I be the moment of inertia of the vehicle body about the X-axis; cy Let I be the moment of inertia of the vehicle body about the Y-axis; cz K represents the moment of inertia of the vehicle body about the Z-axis. tx For the longitudinal stiffness of the secondary suspension; K ty For the lateral stiffness of the secondary suspension; K tz C represents the vertical stiffness of the second-order suspension. tx For the longitudinal damping of the secondary suspension; C ty For secondary suspension lateral damping; C tz For the second-stage suspension vertical damping; K px K represents the longitudinal stiffness of the primary suspension system. py For primary suspension lateral stiffness; K pz C is the vertical stiffness of the suspension system;px For primary suspension longitudinal damping; C py For primary suspension lateral damping; C pz For primary suspension vertical damping; K my For lateral stop stiffness; K rx For anti-roll stiffness; C sx Damping for anti-hunting shock absorbers; H cb H is the distance between the center of gravity of the vehicle body and the center of gravity 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 center of mass of the frame and the center of mass of the wheelset. c Half the vehicle distance; L t Half of the vehicle's fixed wheelbase; d sk It is half the lateral distance of the central spring; d sc It is half the lateral distance of the second-stage vertical vibration damper; d wk It is half the lateral distance of the axle box spring; d wc The lateral distance of the axle box damper is half; d sx It is half the lateral distance of the anti-hunting shock absorber; δ0 is the lateral stop clearance of the secondary system; Y c Z c φ c ψ c β c These are the vehicle's lateral, vertical, roll, yaw, and nod displacements, respectively. These are the vehicle's lateral, vertical, roll, yaw, and pitching speeds, respectively. These are the vehicle's lateral, vertical, roll, yaw, and pitch accelerations, respectively; Y tj Z tj φ tj ψ tj β tj These represent the lateral, vertical, roll, yaw, and nod displacements of the j-th frame, respectively. These are the lateral, vertical, roll, yaw, and nodding speeds of the j-th frame, respectively. These represent the lateral, vertical, roll, yaw, and nodding accelerations of the j-th frame, respectively; Y wi Z wi φ wi ψ wi β wi These represent the lateral, vertical, side roll, head-shaking, and head-nodding displacements of the i-th pair, respectively. These represent the lateral, vertical, side roll, head-shaking, and head-nodding speeds of the i-th round, respectively. Let ri represent the lateral, vertical, side roll, head-shaking, and head-nodding accelerations of the i-th wheel pair, respectively; r0 is the nominal rolling radius of the wheel; ri Li r RiLet be the rolling radius of the left and right wheels of the i-th wheelset; Ω be the nominal rolling speed of the wheelset; d0 be half the distance between the left and right wheel-rail contact points; F Lxi F Rxi F represents the X-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lyi F Ryi F represents the Y-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lzi F Rzi N represents the Z-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lxi N Rxi N represents the X-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lyi N Ryi N represents the Y-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lzi N Rzi M represents the Z-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lxi M Rxi M represents the X-axis components of the creep torque exerted by the i-th wheel on the left and right wheels; Lyi M Ryi M represents the Y-axis component of the creep torque exerted by the i-th wheel on the left and right wheels; Lzi M Rzi Let be the component of the creep torque on the left and right wheels of the i-th wheel along the Z-axis.

[0019] Further, step S4 specifically includes:

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

[0021] S402. Based on the velocity, displacement and acceleration of the truck in the previous time step, calculate the truck's travel distance based on the kinematic equations, and add the truck's actual position on the bridge to the truck's specific position on the bridge corresponding to the truck's travel distance, thus obtaining the truck's actual position.

[0022] S403. Based on the actual position of the freight car, the side roll angle displacement, yaw angle displacement and lateral displacement of the wheelset, the wheel-rail contact point is found using the trace method.

[0023] S404. Based on the wheel-rail contact point and Hertz's nonlinear elastic contact theory, calculate the wheel-rail contact normal force.

[0024] S405. The wheel-rail creep force is calculated using Kalker's linear creep theory and corrected using Shen's theory to obtain the corrected wheel-rail creep force.

[0025] S406. Based on 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. Based on the wheel-rail interaction force, the bridge-rail interaction force is obtained using the rail equation;

[0027] S408. Treat the bridge-rail interaction force as the external load of the bridge motion equation, and solve the bridge dynamic response using the Newmark-β method to obtain the bridge displacement in each degree of freedom direction of each bridge node.

[0028] S409. Based on the bridge displacements in each degree of freedom direction of each bridge node, the displacements of the bridge nodes under each wheelset of the truck are obtained using cubic spline interpolation.

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

[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) represents the overall average equivalent slope when the truck crosses the bridge; t is time; F s (t) represents the total gradient resistance of the truck; P n Let i be the weight of the nth car; n (t) represents the gradient at the location of the nth car at time t; n is the car number.

[0033] The beneficial effects of this invention are as follows: A refined dynamic model was established for the interaction between freight cars, bridges, and track structures, and a dynamic solution process was constructed. The invention fully considers the dynamic coupling effect between the high axle load and long train formations of freight cars and the bridge during bridge crossings. The cubic spline interpolation method was used to accurately calculate the absolute spatial position of each wheelset of each car during bridge crossings. The average gradient method was used to consider the overall gradient over the entire length of the train, providing an evaluation method for the bridge crossing performance of freight cars and a reference basis for bridge design. Attached Figure Description

[0034] Figure 1 This is a flowchart of the method of the present invention.

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

[0036] Figure 3 This is a schematic diagram of the wheel-rail contact patch coordinate system in an embodiment of the present invention.

[0037] Figure 4 This is a schematic diagram of the dynamic slope results of a truck crossing a bridge under the heating condition in an embodiment of the present invention.

[0038] Figure 5 This is a schematic diagram of the dynamic slope results of a truck crossing a bridge under cooling conditions in an embodiment of the present invention.

[0039] 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 Implementation

[0040] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0041] like Figure 1 As shown,

[0042] In one embodiment of the present invention, a method for calculating the real-time average dynamic gradient of a truck crossing a bridge based on vehicle-bridge coupling theory includes the following steps:

[0043] S1. Establish a vehicle dynamics model that takes into account truck type, weight and formation;

[0044] S2. Obtain the roll angle, yaw angle, and lateral displacement of the wheelset based on the vehicle dynamics model;

[0045] S3. Use spatial beam elements to model the main beam and piers, and use spatial rod elements to simulate the main cable, suspenders and stay cables;

[0046] S4. Calculate the additional deformation of the bridge using finite element software, and solve the dynamic response of the bridge based on the additional deformation of the bridge, the side roll angle displacement, the yaw angle displacement and the lateral displacement of the wheelset, so as to obtain the bridge node displacement under each wheelset of the truck.

[0047] S5. Differentiate the bridge node displacements under each wheel pair of the truck to obtain the real-time dynamic slope when the truck crosses the bridge.

[0048] S6. The real-time dynamic slope of the truck crossing the bridge is weighted according to the weight of each truck section to obtain the overall average equivalent slope of the truck crossing the bridge.

[0049] The vehicle dynamics model includes the motion equations of the wheelsets, the motion equations of the frame, and the motion equations of the vehicle body.

[0050] The motion equations for the wheelset, frame, and vehicle body all include lateral motion, vertical motion, roll motion, yaw motion, and rotational motion.

[0051]

[0052]

[0053] Among them, M c For vehicle body mass; M t For structural quality; M w For wheelset mass; I wx I is the moment of inertia of the wheelset about the X-axis; wy I is the moment of inertia of the wheelset about the Y-axis; wz I is the moment of inertia of the wheelset about the Z-axis; tx I is the moment of inertia of the frame about the X-axis; ty Let I be the moment of inertia of the frame about the Y-axis; tz I is the moment of inertia of the frame about the Z-axis. cx Let I be the moment of inertia of the vehicle body about the X-axis; cy Let I be the moment of inertia of the vehicle body about the Y-axis; cz K represents the moment of inertia of the vehicle body about the Z-axis. tx For the longitudinal stiffness of the secondary suspension; K ty For the lateral stiffness of the secondary suspension; K tz C represents the vertical stiffness of the second-order suspension. tx For the longitudinal damping of the secondary suspension; C ty For secondary suspension lateral damping; C tz For the second-stage suspension vertical damping; K px K represents the longitudinal stiffness of the primary suspension system. py For primary suspension lateral stiffness; K pz C is the vertical stiffness of the suspension system; px For primary suspension longitudinal damping; C py For primary suspension lateral damping; C pz For primary suspension vertical damping; K my For lateral stop stiffness; K rx For anti-roll stiffness; C sx Damping for anti-hunting shock absorbers; H cb H is the distance between the center of gravity of the vehicle body and the center of gravity 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 center of mass of the frame and the center of mass of the wheelset. c Half the vehicle distance; L t Half of the vehicle's fixed wheelbase; d sk It is half the lateral distance of the central spring; d scIt is half the lateral distance of the second-stage vertical vibration damper; d wk It is half the lateral distance of the axle box spring; d wc The lateral distance of the axle box damper is half; d sx It is half the lateral distance of the anti-hunting shock absorber; δ0 is the lateral stop clearance of the secondary system; Y c Z c φ c ψ c β c These are the vehicle's lateral, vertical, roll, yaw, and nod displacements, respectively. These are the vehicle's lateral, vertical, roll, yaw, and pitching speeds, respectively. These are the vehicle's lateral, vertical, roll, yaw, and pitch accelerations, respectively; Y tj Z tj φ tj ψ tj β tj These represent the lateral, vertical, roll, yaw, and nod displacements of the j-th frame, respectively. These are the lateral, vertical, roll, yaw, and nodding speeds of the j-th frame, respectively. These represent the lateral, vertical, roll, yaw, and nodding accelerations of the j-th frame, respectively; Y wi Z wi φ wi ψ wi β wi These represent the lateral, vertical, side roll, head-shaking, and head-nodding displacements of the i-th pair, respectively. These represent the lateral, vertical, side roll, head-shaking, and head-nodding speeds of the i-th round, respectively. Let ri represent the lateral, vertical, side roll, head-shaking, and head-nodding accelerations of the i-th wheel pair, respectively; r0 is the nominal rolling radius of the wheel; ri Li r Ri Let be the rolling radius of the left and right wheels of the i-th wheelset; Ω be the nominal rolling speed of the wheelset; d0 be half the distance between the left and right wheel-rail contact points; F Lxi F Rxi F represents the X-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lyi F Ryi F represents the Y-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lzi F Rzi N represents the Z-axis component of the creep force exerted on the i-th wheel relative to the left and right wheels; Lxi N Rxi N represents the X-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lyi N Ryi N represents the Y-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lzi N RziM represents the Z-axis component of the normal force exerted on the left and right wheels by the i-th wheel; Lxi M Rxi M represents the X-axis components of the creep torque exerted by the i-th wheel on the left and right wheels; Lyi M Ryi M represents the Y-axis component of the creep torque exerted by the i-th wheel on the left and right wheels; Lzi M Rzi Let be the component of the creep torque on the left and right wheels of the i-th wheel along the Z-axis.

[0054] In this embodiment, the vehicle dynamics model considers five degrees of freedom: lateral, vertical, roll, pitch, and yaw, as shown in the table below:

[0055] Table 1 Degrees of freedom of the vehicle dynamics model

[0056]

[0057] 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, and damping parameters are known. The velocities, displacements, and accelerations of the car body, frame, and wheelsets are solved based on the aforementioned vehicle motion equations. Simultaneously, the wheelset displacement φ... w ψ w y w It is also used as an input variable in solving for the wheel-rail contact position. The wheelset displacement at this time step is used as an input variable for the wheel-rail contact position at the next time step.

[0058] Step S4 specifically involves:

[0059] S401. Calculate the additional deformation of the bridge using finite element software;

[0060] In this embodiment, commercial finite element software is used to calculate the bridge alignment under various additional deformation conditions. MIDAS Civil is used for calculating the additional deformation of the bridge. The additional deformation conditions selected are an overall temperature increase of 30°C and an overall temperature decrease of 30°C. The additional deformation X of the bridge at the corresponding abscissa mileage is obtained. Bfi .

[0061] S402. Based on the truck's velocity, displacement, and acceleration from the previous time step, calculate the truck's travel distance using kinematic equations. Then, based on the truck's travel distance and its corresponding position on the bridge, add the truck's position on the bridge to its additional deformation to obtain the truck's actual position.

[0062] X r =X Bfi +X Bi

[0063] Among them, X r X represents the actual location of the truck.Bi This represents the bridge displacement at the corresponding node location.

[0064] S403. Based on the actual position of the freight car, the side roll angle displacement, yaw angle displacement and lateral displacement of the wheelset, the wheel-rail contact point is found using the trace method.

[0065] The expression for the wheel-rail contact point of each carriage of the train is:

[0066]

[0067] l x =-cosφ w sinψ w

[0068] l y =cosφ w cosψ w

[0069] l z =sinφ w

[0070]

[0071] Where, x c The x-coordinate of the wheel-rail contact point; Let O2 be the x-coordinate of the center of the rolling circle of the wheel; x For l x =-cosφ w sinψ w ;R w Let be the radius of the rolling circle of the wheel; tg is the tan function; δ w y is the wheel tread contact angle; c The ordinate of the wheel-rail contact point; Let O2 be the ordinate of the center of the rolling circle of the wheel; y For l y =cosφ w cosψ w ;l z For l z =sinφ w ;y w z represents the lateral displacement of the wheelset; c The vertical coordinate of the wheel-rail contact point; φ is the vertical coordinate of the center O2 of the rolling circle of the wheel; w ψ is the roll angle of the wheelset. w The yaw angle of the wheelset; d w Let be the lateral coordinates of each rolling circle of the wheel tread in the wheelset coordinate system.

[0072] In this embodiment, for train-track interaction, the actual position X of the bridge at the train's travel location is obtained. r Based on the contact geometry between the train wheel tread and the rail head tread, the wheel-rail contact point C is located using the trace method. The geometric relationship of the wheel-rail contact point C is as follows: Figure 2 As shown, the wheel-rail contact point C lies within three planes: the rolling circular plane containing point O2, and the plane with O2 as the center and the wheel rolling circle radius R. w Let the sphere be of radius O1-O1'-C'-C plane; the coordinates (x, y) of the wheel-rail contact point C in the absolute coordinate system can be derived from the equations of the three surfaces. c y c , z c ).

[0073] When the wheelset lateral displacement y w , head-shaking angle ψ w Side roll angle φ w At a certain time, by gradually changing the abscissa d of each rolling circle of the wheel tread in the wheelset coordinate system... w This can form the wheel-rail spatial contact trace at a certain moment; the calculation of the wheel-rail contact point is affected by the displacement of the rail pair and the rail and the change in the wheel pair track irregularity value, and the wheel-rail contact point needs to be recalculated at each moment.

[0074] S404. Based on the wheel-rail contact point and Hertz's nonlinear elastic contact theory, calculate the wheel-rail contact normal force.

[0075] The expression for the wheel-rail contact normal force is:

[0076]

[0077] δN L =δZ L / cos(δ wL +φ w )

[0078] δN R =δZ R / cos(δ wR -φ w )

[0079] Among them, P N (t) represents the wheel-rail contact normal force at the current time step; t is time; G is the wheel-rail contact constant; R is the wheel rolling circle radius; δN(t) is the relative normal compression at the wheel-rail contact point; δN L δZ represents the relative normal compression between the left wheel and rail. L The vertical relative displacement between the wheel and rail at the left contact point; δ wL φ is the contact angle of the left wheel tread. wδN is the roll angle of the wheelset. R δZ represents the relative normal compression between the right-side wheel and rail. R The vertical relative displacement between the wheel and rail at the right contact point; δ wR This is the contact angle of the right wheel tread.

[0080] S405. The wheel-rail creep force is calculated using Kalker's linear creep theory and corrected using Shen's theory to obtain the corrected wheel-rail creep force.

[0081] The expression for the creeping force is:

[0082]

[0083] G(ab)=2G w G r / (G w +G r )

[0084]

[0085] Among them, F x ′ represents the longitudinal creep force corrected using Shen's theory; ε is the correction coefficient; F x F is the initial longitudinal creep force; y ′ represents the lateral creep force modified using Shen's theory; F y M is the initial lateral creep force; z ′ represents the rotational creep torque corrected using Shen's theory; M z F is the initial rotational creep torque; R ′ represents the resultant force of the transverse and longitudinal creep forces after nonlinear correction; F R The sum of the longitudinal and transverse creep forces is given by f; the coefficient of friction between the wheel and rail is given by N; and the total normal force of the wheel-rail contact is given by f. 11 ξ is the longitudinal creep coefficient; x For longitudinal creep rate; f 22 ξ is the lateral creep coefficient; y f is the lateral creep rate. 23 ξ is the rotational lateral creep coefficient; sp f is the rotational creep rate; 33 G(ab) is the rotational creep coefficient; G(ab) is the combined shear modulus of the wheelset and track materials; 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 G is the shear modulus of the wheelset material. r V is the shear modulus of the track material. w1V is the velocity of the contact ellipse on the wheel tread along the e1 axis. r1 V represents the velocity of the contact ellipse on the rail corresponding to axis e1; w2 V is the velocity of the contact ellipse on the wheel tread along the e2 axis. r2 Ω represents the velocity of the contact ellipse along the e2 axis on the rail. w3 Ω represents the velocity of the contact ellipse on the wheel tread around the e3 axis. r3 V is the velocity of the contact ellipse on the rail corresponding to the e3 axis; V is the nominal forward speed of the wheelset on the rail.

[0086] S406. Based on the wheel-rail contact normal force and the corrected wheel-rail creep force, the wheel-rail interaction force F is obtained. T-V ;

[0087] In this embodiment, the wheel-rail contact normal force P of each wheel relative to the left and right wheels is calculated separately. N (t) and creep force can be mapped to F in step S1. Lxi F Rxi —The components of the creep force on the left and right wheels of the i-th wheel along the X-axis (i = 1 to 4); F Lyi F Ryi —The components of the creep force on the left and right wheels of the i-th wheel along the Y-axis (i = 1 to 4); F Lzi F Rzi —The component of the creep force on the left and right wheels of the i-th wheel along the Z-axis (i = 1 to 4); N Lxi N Rxi —The components of the normal force exerted by the i-th wheel on the left and right wheels along the X-axis (i = 1 to 4); N Lyi N Ryi —The components of the normal force exerted on the left and right wheels by the i-th wheel along the Y-axis (i = 1 to 4); N Lzi N Rzi —The Z-axis components of the normal force exerted by the i-th wheel on the left and right wheels (i = 1 to 4); M Lxi M Rxi —The components of the creep torque on the left and right wheels of the i-th wheel on the X-axis (i = 1 to 4); M Lyi M Ryi —The components of the creep torque on the left and right wheels of the i-th wheel on the Y-axis (i = 1 to 4); M Lzi M Rzi —The components of the creep torque on the left and right wheels of the i-th wheel on the Z-axis (i = 1 to 4).

[0088] S407. Based on the wheel-rail interaction force, the bridge-rail interaction force is obtained using the rail equation;

[0089] The vertical, lateral, and torsional vibrations of the rail under the action of wheel-rail forces and bridge rail forces can be expressed as:

[0090]

[0091] The above equation is a fourth-order partial differential equation. For numerical analysis, the Ritz method is used to transform this partial differential equation into a system of second-order ordinary differential equations. To do this, the canonical coordinate q of the rail is introduced. zk (t) (vertical), q yk (t)(horizontal), q tk (t)(torsion), after transformation, yields the second-order ordinary differential equations for rail vibration:

[0092]

[0093]

[0094] in, E represents the second derivative of the vertical rail canonical coordinates; t represents time; E represents the second derivative of the vertical rail canonical coordinates. r I represents the Young's modulus of the rail. ry ρ is the moment of inertia of the rail section about the Y-axis; r A is the mass density of the rail; r q represents the cross-sectional area of ​​the rail; k represents the cutoff mode order number; π represents pi; l represents the length of the rail; q zk (t) represents the vertical canonical coordinates of the rail; N s N represents the number of sleepers within the calculated length range of the rail. w Calculate the number of wheel axles within the length range of the rail; F sVi Z is the vertical reaction force at the i-th support point of the rail; k () represents the vertical vibration mode function of the rail; P V j represents the vertical force exerted by the j-th wheel on the rail; N Z This represents the cutoff mode order of the vertical vibration mode of the rail; I is the second derivative of the canonical coordinates of the transverse rail; z Let q be the moment of inertia of the rail section about the Z-axis; yk (t) represents the normal coordinates of the transverse rail; F sHi Y is the lateral reaction force at the i-th support point of the rail; k () represents the transverse vibration mode function of the rail; P Hj N represents the lateral force exerted by the j-th wheel on the rail. Y This represents the cutoff mode order in the transverse direction of the rail; The second derivative of the canonical coordinates of the torsional rail; G r J is the shear modulus of the rail. rt J is the torsional moment of inertia of the rail section; r0q is the polar moment of inertia of the rail section; tk (t) represents the normal coordinates of the torsional rail; F sTi Φ is the torsional reaction force at the i-th support point of the rail; k () represents the mode shape function of rail torsion; P T j is the torque exerted by the j-th wheel on the rail; x si Let x be the x-coordinate of the i-th support point of the rail; Pj N is the x-coordinate of the j-th wheelset of the rail; T z is the cutoff mode order of the torsional vibration mode of the rail; r (x,t), y r (x,t), φ r (x,t) represent the vertical, lateral, and torsional displacements of the rail at coordinate x at time t, respectively. These represent the vertical, lateral, and torsional velocities of the rail at the x-coordinate at time t, respectively. Let x represent the vertical, lateral, and torsional accelerations of the rail at the x-coordinate at time t.

[0095] S408. Treat the bridge-rail interaction force as the external load of the bridge motion equation, and solve the bridge dynamic response using the Newmark-β method to obtain the bridge displacement in each degree of freedom direction of each bridge node.

[0096] Kinematic equations of the bridge subsystem:

[0097] In the formula, M B C B K B These represent the mass matrix, damping matrix, and stiffness matrix of the bridge subsystem, respectively; X B , These are the displacement, velocity, and acceleration vectors of the bridge subsystem, respectively; F B-T F T-B This refers to the interaction force between the track and the bridge.

[0098] The sleeper forces Fgy, Fgx, and Mgz are located at the sleeper positions. Using cubic spline interpolation, they can be mapped to the bridge node positions, forming the bridge-rail interaction force F. B-T The interpolation method is as follows:

[0099] Since the bridge model is built using the finite element method, the locations of the sleeper application points generally do not coincide with the bridge nodes. Therefore, an interpolation method is used to generate the equivalent loads on the bridge nodes. Considering that the bridge will deform under external loads and be constrained by the supports, the following formula is used for interpolation fitting:

[0100] Y b =N1(x)u yi +N2(x)θzi +N3(x)u yj +N4(x)θ zj

[0101] Z b =N1(x)u zi -N2(x)θ yi +N3(x)u zj -N4(x)θ yj

[0102] φ b =N5(x)θ xi +N6(x)θ xj

[0103]

[0104] In the formula, Y b Z represents the lateral displacement of the centroid of the bridge unit beneath the sleeper. b φ represents the vertical displacement of the centroid of the bridge unit below the sleeper. b denoted as N(x), where N(x) is the torsional displacement of the centroid of the bridge element below the sleeper; N1(x), N2(x), N3(x), N4(x), N5(x), and N6(x) are the shape functions of the spatial beam element; L is the element length; x is the longitudinal coordinate of the sleeper position in the bridge element coordinate system; u yi u yj u zi u zi θ zi θ zj θ yi θ yj θ xi θ xj Let i and j be the lateral, vertical, and torsional displacements of the two nodes on the left and right sides of each element, respectively, where i and j are the components of the two nodes on each element.

[0105] The equivalent bridge nodal load F is obtained by multiplying the spatial beam element shape function with the sleeper load. T-B ;

[0106] F yi =N1(x)F gy

[0107] F zi =N1(x)F gz

[0108] F yj =N3(x)F gy

[0109] F zj =N3(x)F gz

[0110] M xi =N5(x)M gx

[0111] M yi =-N2(x)F gz

[0112] M zi =N2(x)F gy

[0113] M xj =N6(x)M gx

[0114] M yj =-N4(x)F gz

[0115] M zj =N4(x)F gy

[0116] 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 from the track to the bridge are respectively T-B The components in the horizontal, vertical, and torsional directions, as well as the components at nodes i and j of each unit;

[0117] The Newmark-β method integral scheme for solving the equations of motion of the bridge subsystem is as follows:

[0118]

[0119] In the formula, α and β are integration parameters, which are generally taken as 0.25 and 0.5 respectively; Δt is the time integration step size; n represents the nth integration step;

[0120] The displacement X of the bridge subsystem B ,speed acceleration vector As X respectively n , Substituting the values, the motion equations of the bridge subsystem can be solved;

[0121] In the above formula Using X n+1 To represent, and then respectively Substituting the dynamic equations of the bridge subsystem at time n+1, we obtain X. n+1The solution formula is as follows:

[0122]

[0123] In the formula, F n+1 Let be the bridge node load at time n+1; M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively.

[0124] X n+1 The solution formula is as follows:

[0125]

[0126] In the formula, a6=Δ t (1-β), a7=βΔt, solving the above equation yields X n+1 α is the integration parameter, taken as 0.25; β is the integration parameter, taken as 0.5;

[0127] The solution formula is as follows:

[0128]

[0129] S409. Based on the bridge displacements in each degree of freedom direction of each bridge node, the displacements of the bridge nodes under each wheelset of the truck are obtained using cubic spline interpolation.

[0130] The specific interpolation method is the same as the method for obtaining sleeper displacement from bridge displacement interpolation.

[0131] Y wheelset =N1(y)u yi +N2(y)θ zi +N3(y)u yj +N4(y)θ zj

[0132] Z wheelset =N1(y)u zi -N2(y)θ yi +N3(y)u zj -N4(y)θ yj

[0133] φ wheelset =N5(y)θ xi +N6(y)θ xj

[0134]

[0135] In the formula, Y wheelset Z represents the lateral displacement of the centroid of the bridge unit beneath the wheel; wheelset φ represents the vertical displacement of the centroid of the bridge unit beneath the wheel.wheelset y represents the torsional displacement of the centroid of the bridge element below the wheel; N1(y), N2(y), N3(y), N4(y), N5(y), and N6(y) are the shape functions of the spatial beam element; L is the element length; y is the longitudinal coordinate of the wheel position in the bridge element coordinate system; u yi u yj u zi u zi θ zi θ zj θ yi θ yj θ xi θ xj The lateral, vertical, and torsional displacements of the two nodes on the left and right sides of each element are represented by i and j, which are the components of the two nodes of each element. The dynamic slope is mainly determined by Z. wheelset Perform the calculation.

[0136] The vertical displacement Z of the bridge under the wheels is obtained at each moment. wheelset (t), let x = t × v, then Z can be... wheelset (t) is transformed into a function Z of the distance x. wheelset (x), then the gradient of the nth vehicle at its current position after traveling a distance x:

[0137]

[0138] Let t = x / v, then i n (x) is transformed into a function i that corresponds to time t. n (t).

[0139] In this embodiment, the real-time dynamic gradients of trains 1, 20, 40, and 60 are as follows: Figure 2 and Figure 3 As shown.

[0140] The expression for the overall average equivalent slope of the truck crossing the bridge in step S6 is:

[0141] i eq (t)=F s (t) / Σ n P n

[0142] F s (t)=Σ n P n i n (t)

[0143] Among them, i eq (t) represents the overall average equivalent slope when the truck crosses the bridge; t is time; F s (t) represents the total gradient resistance of the truck; P nLet i be the weight of the nth car; n (t) represents the gradient at the location of the nth car at time t; n is the car number.

[0144] In this embodiment, the average equivalent gradient of the train crossing the bridge is as follows: Figure 4 As shown.

[0145] This method, based on vehicle-bridge coupling theory, establishes a system that considers the dynamics of freight cars in various train formations and incorporates the dynamic coupling effect between the train and the bridge. It obtains the dynamic alignment of the bridge under each wheelset when a freight car crosses the bridge, and calculates the equivalent gradient of the freight car crossing based on this dynamic alignment. This provides a precise calculation method for the performance analysis of freight cars crossing bridges and offers a reference for design.

Claims

1. A truck bridge real-time average dynamic slope calculation method based on axle coupling theory, characterized in that, The method comprises the following steps: S1, establishing a vehicle dynamics model considering the truck model, truck weight and marshalling; S2, obtaining the roll angle displacement, yaw angle displacement and lateral displacement of the wheelset according to the vehicle dynamics model; S3, modeling the main beam and the pier using a space beam element, and simulating the main cable, sling and stay cable using a space bar element; S4, calculating the bridge additional deformation using a finite element software, and solving the bridge dynamic response according to the bridge additional deformation, 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; the step S4 is specifically: S401, calculating the bridge additional deformation using a finite element software; S402, calculating the running distance of the truck based on the kinematic equation according to the speed, displacement and acceleration of the truck at the last time step, and adding the specific position on the bridge to the bridge additional deformation to obtain the actual position of the truck based on the specific position on the bridge corresponding to the running distance of the truck; S403, finding the wheel-rail contact point by using the trace method according to the actual position of the truck, the roll angle displacement, the yaw angle displacement and the lateral displacement of the wheelset; S404, calculating the wheel-rail contact normal force based on the Hertz nonlinear elastic contact theory according to the wheel-rail contact point; S405, calculating the wheel-rail creep force by using the linear creep theory of Kalker and correcting the wheel-rail creep force by using the Shen's theory to obtain the 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 is obtained ; S407, obtaining the bridge-rail interaction force by using the rail equation according to the wheel-rail interaction force; S408, taking the bridge-rail interaction force as the external load of the bridge motion equation, solving the bridge dynamic response by the Newmark-β method to obtain the bridge displacement in each degree of freedom direction of each bridge node; S409, obtaining the bridge node displacement under each wheelset of the truck by using the cubic spline interpolation method according to the bridge displacement in each degree of freedom direction of each bridge node; S5, differentiating the bridge node displacement under each wheelset of the truck to obtain the real-time dynamic slope when the truck passes through the bridge; S6, weighting the real-time dynamic slope when the truck passes through the bridge according to the truck weight of each truck to obtain the overall average equivalent slope when the truck passes through the bridge; the expression of the overall average equivalent slope when the truck passes through the bridge in the step S6 is: wherein, is the overall average equivalent slope for the truck crossing the bridge; is time; is the total slope added resistance for the truck; is the first is the weight of the section of the truck; is the first is the time of operation of the section of the truck is the slope at the location where the truck is at time t; is the car number.

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

3. The method for calculating real-time average dynamic slope of a truck crossing a bridge based on the axle coupling theory according to claim 2, characterized in that, The wheelset motion equation, the frame motion equation and the car body motion equation all comprise lateral motion, vertical motion, roll motion, yaw motion and rotation motion: wherein, M c is the car body mass; M t is the frame mass; M w is the wheelset mass; I wx is the wheelset moment of inertia about the X axis; I wy is the wheelset moment of inertia about the Y axis; I wz is the wheelset moment of inertia about the Z axis; I tx is the frame moment of inertia about the X axis; I ty is the frame moment of inertia about the Y axis; I tz is the frame moment of inertia about the Z axis; I cx is the car body moment of inertia about the X axis; I cy is the car body moment of inertia about the Y axis; I cz is the car body moment of inertia about the Z axis; K tx is the secondary suspension longitudinal stiffness; K ty is the secondary suspension lateral stiffness; K tz is the secondary suspension vertical stiffness; C tx is the secondary suspension longitudinal damping; C ty is the secondary suspension lateral damping; C tz is the secondary suspension vertical damping; K px is the primary suspension longitudinal stiffness; K py is the primary suspension lateral stiffness; K pz is the primary suspension vertical stiffness; C px is the primary suspension longitudinal damping; C py is the primary suspension lateral damping; C pz is the primary suspension vertical damping; K my For lateral stop stiffness; K rx To enhance anti-roll stiffness; C sx Damping for anti-hunting vibration dampers; H cb This is the distance between the vehicle's center of gravity and the bolster's center of gravity; H bt This is the distance between the center of mass of the bolster and the center of mass of the frame; H tw This is the distance between the center of mass of the frame and the center of mass of the wheelset; L c Half the vehicle distance; L t It is half of the vehicle's fixed wheelbase; d sk It is half the lateral distance of the central spring; d sc It is half the lateral distance of the second-stage vertical vibration damper; d wk It is half the lateral distance of the axle box spring; d wc It is half the lateral distance of the axle box damper; d sx It is half the lateral distance of the anti-hunting shock absorber; δ 0 represents the lateral stop clearance of the second-stage system; , , , , These are the vehicle's lateral, vertical, roll, yaw, and nod displacements, respectively. , , , , These are the vehicle's lateral, vertical, roll, yaw, and pitching speeds, respectively. , , , , These are the vehicle's lateral, vertical, roll, yaw, and nodding accelerations, respectively. , , , , The first j The frame can be lateral, vertical, roll, yaw, and nod displacements. , , , , The first j Frame lateral, vertical, side roll, yaw, and nodding speeds; , , , , are the lateral, vertical, roll, yaw, pitch accelerations of the frame, respectively; j , , , , are the lateral, vertical, roll, yaw, pitch displacements of the wheelset, respectively; i , , , , are the lateral, vertical, roll, yaw, pitch velocities of the wheelset, respectively; i , , , , are the lateral, vertical, roll, yaw, pitch accelerations of the wheelset, respectively; i r 0 is the nominal rolling radius of the wheel; r Li , r Ri are the rolling radii of the left and right wheels of the wheelset, respectively; Ω is the nominal rolling velocity of the wheelset; i d 0 is half the distance between the left and right wheel-rail contact points; F Lxi , F Rxi are the components of the creep force on the left and right wheels of the wheelset in the x-axis, respectively; i X Lyi , F Ryi are the components of the creep force on the left and right wheels of the wheelset in the y-axis, respectively; F i Lzi , Y Rzi are the components of the creep force on the left and right wheels of the wheelset in the z-axis, respectively; F Lxi , F Rxi are the components of the normal force on the left and right wheels of the wheelset in the x-axis, respectively; i Z Lyi , N Ryi are the components of the normal force on the left and right wheels of the wheelset in the y-axis, respectively; N i <000004A>, X <000004B>are the components of the normal force on the left and right wheels of the wheelset in the z-axis, respectively; N <000004C>, N <000004D>are the components of the lateral, vertical, roll, yaw, pitch accelerations of the frame, respectively; i ​​​​​The normal forces acting on the left and right wheels of the wheelset Y Components on the axis; N Lzi , N Rzi For the first i The normal forces acting on the left and right wheels of the wheelset Z Components on the axis; M Lxi , M Rxi For the first i The creep torque on the left and right wheels of the wheelset is X Components on the axis; M Lyi , M Ryi For the first i The creep torque on the left and right wheels of the wheelset is Y Components on the axis; M Lzi , M Rzi For the first i The creep torque on the left and right wheels of the wheelset is Z Components on the axis.

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