A method for simulating dynamic characteristics of a trailer-type belt conveyor

Abstract

The present application relates to the technical field of belt conveyor dynamics characteristic research, especially to a kind of dynamics characteristic simulation method of bogie type belt conveyor, which includes force analysis and collation degree of freedom to bogie frame, each wheel and conveyor belt unit in bogie unit respectively;Overall coordinate system, vehicle body coordinate system and contact coordinate system and conversion relationship are established;Dynamics equation of wheelset and bogie frame is respectively established;Dynamics parameters of bogie unit are determined;According to Hamilton principle, bogie frame vertical motion equation and bogie frame nodding motion equation are established, and then linear second-order differential equation of bogie unit is obtained;Numerical calculation software is used to compile calculation program, and wheel-rail vibration characteristics of bogie unit in straight line segment advancing process are simulated.The present application is based on multi-degree-of-freedom multibody dynamics theory, constructs nonlinear dynamics equation of bogie type belt conveyor system, and analyzes six-degree-of-freedom motion equation of wheelset, bogie frame and coupling thereof.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of belt conveyor dynamics characteristic research, and particularly relates to a bogie type belt conveyor dynamics characteristic simulation method. BACKGROUND

[0002] The belt conveyor is widely applied to the transportation link of the coal industry. For the long-distance cross-country belt conveyor, the bogie type belt conveyor can effectively reduce the running resistance. The bogie type belt conveyor and the ordinary belt conveyor both adopt the flexible conveying belt to transport the materials, but the acceleration and the tension of the conveying belt at different positions change with time in different rules during the running. In addition, the bogie is not only subjected to the interaction force between the wheel and the rail, but also subjected to the dynamic tension of the conveying belt and the force of the steel wire rope. The resultant force directly affects the stability and the safety of the bogie running, and further affects the interaction between the wheel and the rail. With the increase of the running speed of the conveyor, the interaction force between the wheel and the rail increases, which causes the aggravation of the wheel-rail contact fatigue and the acceleration of the material wear, and further causes the wheel wear problem and the abnormal vibration of the vehicle caused by the wheel wear. The causes of the wheel-rail wear are relatively complex, mainly involve the wheel-rail coupling vibration, the wheel-rail material wear and plastic deformation, the wheel-rail rolling contact fatigue and other factors, but the existing technical solutions still lack the method for accurately researching the dynamic characteristics. SUMMARY

[0003] The present application aims at solving the above problems, and provides a bogie type belt conveyor dynamics characteristic simulation method, which adopts the technical scheme as follows.

[0004] A bogie type belt conveyor dynamics characteristic simulation method, defines that the bogie unit comprises one bogie frame, four wheels rotationally connected with the bogie frame and one conveying belt unit supported on the bogie frame, the bogie unit moves along a straight line on the bogie rail, defines that the direction perpendicular to the conveying belt surface is the vertical direction, the direction of the bogie frame moving along the bogie rail is the longitudinal direction, and the direction perpendicular to the vertical direction and the longitudinal direction in the space is the transverse direction, defines that the side roll is the rotation around the longitudinal direction, the nodding is the rotation around the transverse direction, and the shaking head is the rotation around the vertical direction.

[0005] The bogie type belt conveyor dynamics characteristic simulation method comprises the following steps.

[0006] S1. The force analysis of the bogie frame, each wheel and the conveying belt unit in the bogie unit is carried out, and the degrees of freedom are sorted, wherein the degrees of freedom include the longitudinal translation, the transverse translation, the vertical translation, the roll, the nodding and the shaking head rotation.

[0007] S2. Establish a coordinate system, which includes a general coordinate system, a vehicle body coordinate system and a contact coordinate system, the general coordinate system is fixed to the earth, the vehicle body coordinate system is fixed to the bogie unit, the origin of the contact coordinate system is defined at the contact center of each wheel and the bogie track, the attitude angle of the wheel in the vehicle body coordinate system is established, and the relative conversion relationship between the vehicle body coordinate system and the general coordinate system is established;

[0008] S3. According to the d'Alembert principle and spatial force analysis, the dynamic equations of the wheel set and the bogie frame are established respectively;

[0009] S4. Determine the dynamics parameters of the bogie unit: determine the solution parameters of the conveyor belt, including the mass of the material line, the mass of the bogie line, the relationship between the angular velocity of the wheel and the speed of the bogie frame, the moment of inertia of the wheel and the inertial moment of the wheel; calculate the force system distribution between the conveyor belt and the upper surface arc surface of the bogie frame when the conveyor belt carries the material, calculate the action point position of the force of the conveyor belt on the surface arc surface of the bogie frame according to the material quantity classification; calculate the tension of the conveyor belt and the resistance of the belt conveyor; check the sag condition of the conveyor belt;

[0010] S5. The wheel set is regarded as an elastic body, the bogie unit is vertically divided along the longitudinal direction, and the dynamic characteristics of the front and rear wheel tracks of the divided half bogie unit are analyzed; the vertical motion equation and the nodding motion equation of the bogie frame are established according to the Hamilton principle, and then the linear second-order differential equation of the bogie unit is obtained;

[0011] S6. Use numerical calculation software to compile the calculation program, and simulate the vibration characteristics of the bogie unit in the straight line section.

[0012] On the basis of the above scheme, the step S2 comprises:

[0013] A general coordinate system O-XYZ is defined, the origin O of which coincides with the mass center of the bogie unit at the initial position of the bogie unit; the X axis is along the initial longitudinal driving direction of the bogie frame, and positive forward; the Z axis is the same as the direction of gravity, and positive downward; the Y axis is perpendicular to the XOZ plane and points to the right along the X direction;

[0014] A vehicle body coordinate system O C -X C Y C Z C is defined, the origin O C of which is at the mass center of the bogie frame, the X C axis points to the front of the bogie frame, the Y C axis points to the right of the bogie frame, and the Z C axis points downward;

[0015] A contact coordinate system O W -X W Y W Z Wwhose origin O W At the contact center of each wheel, X W The axis points forward of the contact surface between the wheel and the track, Y W Points to the right side of the contact surface, Z W Points downward of the contact surface;

[0016] The wheel attitude angle includes the wheel yaw angle ψ w , roll angle φ w and the rotation angle σ w of the wheel around the wheel mass center, when the yaw angle and roll angle of the wheel are the same as those of the bogie frame;

[0017] The bogie unit has six degrees of freedom, three of which are used to describe the position of the mass center of the bogie frame relative to the global coordinate system, i.e. X, Y, Z coordinates, and the other three are used to describe the attitude of the bogie frame, i.e. yaw angle ψ, pitch angle β and roll angle φ, also known as Euler angle coordinates; any attitude of the bogie coordinate system can be obtained by three successive rotations, i.e. the vehicle body coordinate system O C Starting from the same as the global coordinate system O, rotating around its Z C axis by an angle ψ to obtain the reference system O 2 , then rotating the reference system O C 2 around its Y C ' axis by an angle β to obtain the reference system O C 1 , and finally rotating the reference system O C 1 around its X C " axis by an angle φ to obtain the vehicle body coordinate system O C , O C X C 'Y C ' represents the reference system O C 2 , O C X C "Y C " represents the reference system O C 1 , O C X C Y C represents the final vehicle body coordinate system O T ,

[0018] The direction cosine matrix is respectively:

[0019] (2-1)

[0020] (2-2)

[0021] (2-3)

[0022] According to the conversion relationship between the global coordinate system O and the vehicle body coordinate system O T , N is the direction cosine matrix of the global coordinate system O and the vehicle body coordinate system O C , that is:

[0023] (2-4).

[0024] On the basis of the above scheme, the wheelset dynamics equation in step S3 includes:

[0025] The wheelset longitudinal motion equation is

[0026] (3-1)

[0027] The lateral motion equation is

[0028] (3-2)

[0029] The vertical motion equation is

[0030] (3-3)

[0031] The roll motion equation is

[0032] (3-4)

[0033] The nodding motion equation is

[0034] (3-5)

[0035] The shaking motion equation is

[0036] (3-6)

[0037] Wherein, F xhl4 , F xhr3 Fxl, Fxr are the longitudinal forces of the left and right wheelsets on the left and right wheels, respectively, N; F yhl4 , F yhr3 Fyl, Fyr are the lateral forces of the left and right wheelsets on the left and right wheels, respectively, N; F zhl4 , F zhr3 Fzl, Fzr are the vertical forces of the left and right wheelsets on the left and right wheels, respectively, N; M xwl4 , M ywl4 , Mzwl4 , respectively, are the longitudinal, lateral and vertical components of the spin moment of the left wheel rail, N·m; M xwr3 , M ywr3 , M zwr3 , respectively, are the longitudinal, lateral and vertical components of the spin moment of the right wheel rail, N·m; F x2L1 , F x2R1 , respectively, are the longitudinal forces of the bogie frame on the left and right wheel shafts, N; F y2L1 , F y2R1 , respectively, are the lateral forces of the bogie frame on the left and right wheel shafts, N; F z2L1 , F z2R1 , respectively, are the vertical forces of the bogie frame on the left and right wheel shafts, N; φ sew is the rail cant angle corresponding to the center of the wheel pair on the bogie track, rad; R w is the radius of curvature corresponding to the center of the wheel pair on the bogie track, m; a0 is half of the distance between the contact points of the left and right wheels, m; r 0 is the nominal rolling radius of the wheel, m; d 2 is half of the lateral distance between the action points of the bogie frame on the left and right wheel shafts, m; d L is the lateral distance from the contact action point of the left wheel and the rail to the mass center of the bogie frame, m; d R is the lateral distance from the contact action point of the right wheel and the bogie track to the mass center of the bogie frame, m; r L is the actual contact radius of the left wheel, m; r R is the actual contact radius of the right wheel, m;

[0038] The dynamics equation of the bogie frame in the step S4 comprises:

[0039] The longitudinal motion equation of the bogie frame is

[0040] (3-7)

[0041] The lateral motion equation of the bogie frame is

[0042] (3-8)

[0043] The vertical motion equation of the bogie frame is

[0044] (3-9)

[0045] The rolling motion equation of the trolley frame is

[0046] (3-10)

[0047] The nodding motion equation of the trolley frame is

[0048] (3-11)

[0049] The shaking motion equation of the trolley frame is

[0050] (3-12)

[0051] wherein, F xb is the traction force of the conveying belt on the trolley frame, N; F zb is the vertical force of the conveying belt on the middle of the trolley frame, N; F s1 , F s2 are the front and rear steel wire rope tensions of the trolley frame, N; F x1L1 , F x1R1 are the longitudinal forces of the conveying belt on the left and right side arc surfaces of the trolley frame, N; F y1L1 , F y1R1 are the transverse forces of the conveying belt on the left and right side arc surfaces of the trolley frame, N; F z1L1 , F z1R1 are the vertical forces of the conveying belt on the left and right side arc surfaces of the trolley frame, N; h 01 , h 03 are the vertical distances from the left and right force points of the trolley frame to the center of mass of the trolley frame, m; h 2is the vertical distance from the wheel shaft force point to the center of mass of the trolley frame, m; φ sec is the rail superelevation angle corresponding to the center of the trolley frame on the trolley rail, rad; d 01 , d 03 are the horizontal distances from the left and right force points of the trolley frame to the center of mass of the trolley frame, m; l 02 is half of the longitudinal distance from the front and rear action points of the trolley frame on the left and right wheel shafts, m; R cR is the curvature radius corresponding to the gravity center of the trolley frame on the trolley track, m;

[0052] F x1L1 F y1L1 F z1L1 F x1R1 F y1R1 F z1R1 is defined as the force of the conveying belt on the trolley frame arc surface, and is assumed to act on the same vertical plane as the force of the trolley frame on the wheel shaft, so the distance to the centroid is equal, i.e. the force arm

[0053] On the basis of the above scheme, the step S5 comprises:

[0054] According to the Hamilton principle, the vertical motion equation of the trolley frame is analyzed to obtain

[0055] (5-1)

[0056] (5-2)

[0057] (5-3)

[0058] The nodding motion equation of the trolley frame is analyzed to obtain

[0059] (5-4)

[0060] wherein, m w1 m w2 m c1 m is the mass of the front wheel, rear wheel and trolley frame, respectively, kg; z w1 z w2 z c1 is the vertical displacement of the front wheel, rear wheel and trolley frame, respectively, m; F zb1 is the pressure of the conveying belt on the trolley frame, N;

[0061] the mass of the i-th trolley unit is defined as m i the stiffness coefficient is k i the damping coefficient is c i and the running resistance is​​​​​​​​​​w i , displacement is x i ; the conveyor belt is regarded as a linear system, and a linear second-order differential equation matrix is adopted

[0062] (5-5)

[0063] In the formula, M is the mass matrix; C is the damping matrix; K is the stiffness matrix; all of which are square matrix; is the system acceleration, velocity and displacement matrix; F is the force matrix acting on each unit, which is column vector;

[0064] The above equation is arranged into the matrix form of the linear second-order differential equation to obtain

[0065] (5-6)

[0066] (5-7)

[0067] (5-8)

[0068] (5-9)

[0069] (5-10).

[0070] Preferably, the step S6 uses MATLAB software to compile the calculation process, including the following steps,

[0071] S6-1. Input the wheelset dynamics equation and the bogie frame dynamics equation;

[0072] S6-2. Input the bogie unit dynamics parameters determined in step S4;

[0073] S6-3. Bogie frame vertical motion equation and nodding motion equation;

[0074] S6-4. Set the bogie track excitation model;

[0075] S6-5. Generate the bogie unit matrix and external force matrix;

[0076] S6-6. Introduce a numerical integration algorithm to solve the motion equation;

[0077] S6-7. Select the time step and judge the convergence of the algorithm;

[0078] S6-8. Calculate the vibration characteristics of the front and rear wheelsets;

[0079] S6-9. Calculate the vibration state with the last time step result as the initial iteration value;

[0080] S6-10. Determine whether the actual calculation time t is greater than the planned calculation time T, if t > T is satisfied, continue to step S6-11, if t > T is not satisfied, return to step S6-8 to continue calculation;

[0081] S6-11. Solve the wheelset dynamics equation and the bogie frame dynamics equation and output the dynamic response result.

[0082] Preferably, it further comprises a three-dimensional dynamic simulation verification, which comprises the following steps:

[0083] S7-1. Establish a three-dimensional model of the bogie type belt conveyor and import it into the simulation software;

[0084] S7-2. Perform modal analysis on the bogie frame and wheels;

[0085] S7-3. Output the three-direction vibration acceleration changes of the single-side front wheel and rear wheel under different bogie spacings when running at different belt speeds on the straight section bogie track;

[0086] S7-4. Output the three-direction vibration acceleration changes of the single-side front wheel and rear wheel when subjected to different load impacts on the straight section bogie track;

[0087] S7-5. Output the three-direction vibration acceleration changes of the single-side front wheel and rear wheel when the bogie spacing is different on the straight section bogie track.

[0088] Preferably, it further comprises a wheel-rail rolling contact theory elastic-plastic analysis, comprising the following steps:

[0089] A1. Based on the elastic half-space assumption, it is considered that the surfaces of two contact objects are smooth and the curvature in the elliptical contact area is constant, the normal gap between the two contact objects can be represented by a Taylor polynomial, and by ignoring the high-order terms, the normal gap in the contact patch is obtained h :

[0090] (a-1)

[0091] In the formula, A and B are constants related to the curvature radii of the contact points of the bogie track in the x direction and the y direction;

[0092] A and B can be represented as:

[0093] (a-2)

[0094] (a-3)

[0095] In the formula, R rx 、R ry Here, represents the radius of curvature of the trailer track along the direction of wheel rolling and laterally, respectively, in meters (m). R wx 、R wy Here, denoted as the radius of curvature of the wheel's rolling circle and the radius of curvature along the transverse direction, respectively, in meters (m). θ 1 is included in the wheel and carriage rails R rx and R wx The angle between the two principal curvature planes, °;

[0096] A2. Analyzing the normal force distribution in the contact area between the wheel and the trailer track, the semi-major axis of the elliptical contact patch can be obtained using the Boussinesq-Cerruti formula in elasticity. a and short half shaft b They are respectively:

[0097] (a-4)

[0098] In the formula, F p The normal contact load is N; m, n It is a constant, and B - A / A + B The value is related to the value of , which can be obtained by looking up a table using interpolation.

[0099] Equivalent elastic modulus E for:

[0100] (a-5)

[0101] The normal pressure inside the elliptical contact patch is:

[0102] (a-6)

[0103] The relative deformation of the wheel-rail contact is:

[0104] (a-7)

[0105] In the formula, E The equivalent elastic modulus of the wheel and rail is expressed in MPa. ζ 1. ζ 2 represents the Poisson's ratio of the wheel and rail materials, respectively; E 1. E 2 represents the elastic modulus of the wheel and rail, respectively, in MPa; pz is the normal pressure in the elliptical contact patch, N; ε is the relative deformation of wheel-rail contact, m; r f is a constant related to the value of B - A / A + B ;

[0106] A3. When the maximum pressure in the contact area does not exceed the yield pressure of the material, the contact area deforms elastically, still satisfying the Hertz contact pressure distribution; when the maximum pressure in the contact area exceeds the yield pressure, the central region of the contact patch deforms plastically, and the normal pressure of the elastic deformation region and the plastic deformation region is:

[0107] (a-8)

[0108] Assuming that the material satisfies the Mises yield criterion condition, the normal contact force when the two contact bodies initially yield is:

[0109] (a-9)

[0110] where, p y is the initial yield pressure, MPa; p 0 * is the maximum contact pressure when it is assumed that only elastic deformation occurs in the entire contact area, satisfying , N; S t , S s are the areas of the elastic deformation region and the plastic deformation region, respectively, m 2 ; Y is the yield strength of the material with lower strength in the two contact bodies, MPa; ζ is the Poisson's ratio of the material;

[0111] The maximum contact pressure is at the center point of the elliptical contact patch, which can be expressed as:

[0112] (a-10)

[0113] When E t =E , it indicates a linear elastic contact situation. According to the Hertz contact theory formula, it is extended to the case of elliptical contact patch, then the yield pressure p y The long axis a y and the short axis b y corresponding to the initial yield are related as:

[0114] (a-11)

[0115] The yield pressure p y And the initial yield load F y0 Satisfies:

[0116] (a-12)

[0117] Considering the continuity of the contact pressure, in the elastic-plastic transition region, there is:

[0118] (a-13)

[0119] The plastic zone long semi-axis a p And the short semi-axis b p Respectively:

[0120] (a-14)

[0121] The contact force can be obtained by integrating in the contact area:

[0122] (a-15);

[0123] A4. Simplifying the above formula, we get

[0124] (a-16)

[0125] In the formula, a y , b y The yield pressure and the initial yield corresponding long and short semi-axes, m, respectively; F y0 The initial yield load, N; a p , b p The long and short semi-axes of the plastic region, m, respectively;

[0126] A5. According to the Hertz contact theory, the relationship between the maximum contact pressure of the elastic contact model and the elastic deformation is:

[0127] (a-17)

[0128] Assuming that the object contact satisfies the elastic-plastic linear hardening relationship, its pressure p Can be expressed as:

[0129] (a-18)

[0130] wherein, δ is the elastic deformation, m; δ y is the elastic critical deformation, m; k is the hardening coefficient;

[0131] When the hardening coefficient , then it represents the ideal elastic-plastic material model, i.e. when the maximum deformation of the contact area is greater than the critical deformation δ y , the maximum pressure of the contact area is still p y ; when the hardening coefficient , then it represents the bilinear hardening material model;

[0132] The elastic-plastic displacement δ is

[0133] (a-19).

[0134] On the basis of the above scheme, it further includes wheel-rail contact display dynamic simulation analysis, including the following steps:

[0135] B1. Establish a three-dimensional wheel-rail rolling contact finite element theoretical model, including the mass on the wheel, spring-damping unit, wheel-rail system and track support structure, and establish a Cartesian coordinate system O-XYZ, wherein the origin O is the initial contact point of the wheel-rail, X, Y and Z represent the longitudinal direction (wheel-rail rolling direction), transverse direction and vertical direction respectively;

[0136] B2. Set the model size parameters and material parameters, and set the simulation experience stage, including the stress initial stage and the steady-state rolling contact stage, and the steady-state rolling contact stage is the contact stage under the stress balance state;

[0137] B3. Output the transient stress and deformation of the trolley under different load weight conditions, and analyze the stress concentration position;

[0138] B4. Output the wheel-rail contact area contour graph and stress distribution graph when moving at different speeds under different load weight conditions.

[0139] The beneficial effects of the present application are: around the operation characteristics of the bogie type belt conveyor, based on the multi-degree-of-freedom multi-body dynamics theory, the nonlinear dynamics equation of the bogie type belt conveyor system is constructed, the six-degree-of-freedom motion equation of the wheel set and the bogie frame and the coupling thereof is analyzed, so that the motion posture relationship of the key structure is obtained, and the internal relationship of the motion of each component is clarified; for the vertical motion of the front and rear wheel sets of the bogie, the vibration equation of the half bogie model is constructed by using the Hamilton principle, the Newmark-beta integral method is used to solve the global system motion equation under different influence factors (belt speed, load impact, bogie spacing) in the straight line transportation process by using MATLAB software programming, and then the coupling vibration characteristics of the bogie-rail are analyzed; the MATLAB numerical simulation results are verified by combining the three-dimensional simulation mode, and the accuracy of the simulation results is ensured. BRIEF DESCRIPTION OF DRAWINGS

[0140] Figure 1 : The bogie type belt conveyor structure diagram suitable for the present application;

[0141] Figure 2 : The bogie unit structure diagram of the present application;

[0142] Figure 3 : The dynamics model and the front view of the degree of freedom distribution of the bogie unit of the present application;

[0143] Figure 4 : The dynamics model and the side view of the degree of freedom distribution of the bogie unit of the present application;

[0144] Figure 5 : The dynamics model and the top view of the degree of freedom distribution of the bogie unit of the present application;

[0145] Figure 6 : The wheel set space stress analysis diagram of the present application;

[0146] Figure 7 : The bogie frame space stress analysis diagram of the present application;

[0147] Figure 8 : The carrying state section diagram of the bogie type belt conveyor of the present application;

[0148] Figure 9 : The dynamics model of the bogie unit after longitudinal segmentation of the present application;

[0149] Figure 10 : The MATLAB compilation flowchart of the present application;

[0150] Figure 11 : The normal force distribution diagram in the contact spot of the present application;

[0151] Figure 12 : The plastic deformation diagram of the elastic-plastic contact area of the present application;

[0152] Figure 13 : Relationship between maximum contact pressure and elastic deformation in this invention. Detailed Implementation

[0153] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0154] A simulation method for the dynamic characteristics of a trailer-type belt conveyor, such as Figure 1 As shown, the trailer-type belt conveyor mainly consists of a conveyor belt 1, rotating wheels at the head and tail, trailers, and trailer tracks 2. The trailers are connected in series by wire ropes, and the tracks employ a double-layer design with reversing devices at the head and tail. When a trailer reaches the rotating wheel at the head or tail, the conveyor belt 1 separates from the trailer, and the trailer enters the rotating wheel and changes tracks. The trailer uses a U-shaped support structure, allowing it to pass smoothly through the rotating wheel without interference, while the conveyor belt, like traditional general-purpose belt conveyors, is deflected by rollers. The smooth characteristics of the trailer tracks result in low frictional resistance during trailer operation. Furthermore, the U-shaped smooth structure of the trailer body not only increases the contact area between the conveyor belt and the trailer but also effectively reduces the local pressure of the material on the conveyor belt. During operation, the trailers and the conveyor belt above them remain relatively stationary. The numerous trailers arranged in the conveyor's running section to assist in supporting the conveyor belt help eliminate indentation resistance and reduce the stringent requirements on conveyor belt sag.

[0155] like Figure 2 As shown, a trolley unit is defined as including a trolley frame 3, four wheels 4 rotatably connected to the trolley frame 3, and a conveyor belt 1 unit supported on the trolley frame 3. The trolley unit moves in a straight line on the trolley track 2. The direction perpendicular to the surface of the conveyor belt is defined as vertical, the direction in which the trolley frame moves along the trolley track is defined as longitudinal, and the direction in space that is perpendicular to both the vertical and longitudinal directions is defined as transverse. Side rolling is defined as rotation around the longitudinal direction, nodding is defined as rotation around the transverse direction, and head shaking is defined as rotation around the vertical direction.

[0156] The simulation method for the dynamic characteristics of a trailer-type belt conveyor includes the following steps:

[0157] S1. As Figures 3 to 5 As shown, the force analysis and degrees of freedom of the trailer frame 3, each wheel 4 and the conveyor belt unit 1 in the trailer unit are respectively performed. The degrees of freedom include longitudinal translation, lateral translation, vertical translation, lateral rolling, nodding and yaw rotation.

[0158] S2. Establish a coordinate system, which includes a general coordinate system, a vehicle body coordinate system and a contact coordinate system, the general coordinate system is fixed to the ground, the vehicle body coordinate system is fixed to the bogie unit, the origin of the contact coordinate system is defined at the contact center of each wheel and the bogie track, the attitude angle of the wheel in the vehicle body coordinate system is established, and the relative conversion relationship between the vehicle body coordinate system and the general coordinate system is established;

[0159] Specifically, a general coordinate system O-XYZ is defined, the origin O of which coincides with the mass center of the bogie unit at the initial position of the bogie unit; the X axis is along the initial longitudinal driving direction of the bogie frame, positive forward; the Z axis is the same as the direction of gravity, positive downward; and the Y axis is perpendicular to the XOZ plane and points to the right along the X direction;

[0160] A vehicle body coordinate system O C -X C Y C Z C is defined, the origin O C of which is at the mass center of the bogie frame, the X C axis points forward of the bogie frame, the Y C axis points to the right of the bogie frame, and the Z C axis points downward of the bogie frame;

[0161] A contact coordinate system O W -X W Y W Z W is defined, the origin O W of which is at the contact center of each wheel, the X W axis points forward of the contact surface of the wheel and the bogie track, the Y W axis points to the right of the contact surface, and the Z W axis points downward of the contact surface;

[0162] The wheel attitude angle includes the wheel yaw angle ψ w , the wheel roll angle φ w and the rotation angle σ w of the wheel around the mass center of the wheel, provided that the yaw angle and the roll angle of the wheel are the same as the yaw angle and the roll angle of the bogie frame;

[0163] The bogie unit has six degrees of freedom, of which three degrees of freedom are used to describe the position of the mass center of the bogie frame relative to the general coordinate system, i.e. the X, Y and Z coordinates, and the other three degrees of freedom are used to describe the attitude of the bogie frame, i.e. the yaw angle ψ, the pitch angle β and the roll angle φ, also known as Euler angle coordinates; any attitude of the bogie coordinate system can be obtained through three successive rotations, i.e. the vehicle body coordinate system O C is first made consistent with the general coordinate system O, then the reference system O C is rotated by an angle ψ around the Z 2 axis thereof, and then the reference system O C 2around its Y C axis by an angle β to get the reference frame O C 1 Finally, the reference frame O C 1 around its X C axis by an angle φ to get the body coordinate frame O C , O C X C 'Y C ' denotes the reference frame O C 2 , O C X C "Y C " denotes the reference frame O C 1 , O C X C Y C denotes the final body coordinate frame O T ,

[0164] The direction cosine matrixes are:

[0165] (2-1)

[0166] (2-2)

[0167] (2-3)

[0168] According to the transformation relationship between the global coordinate frame O and the body coordinate frame O T , N is the direction cosine matrix of the global coordinate frame O and the body coordinate frame O C , that is:

[0169] (2-4).

[0170] S3. According to the D'Alembert principle and spatial force analysis, the dynamic equations of the wheelset and the bogie frame are established respectively as shown in Figure 6

[0171] The longitudinal motion equation of the wheelset is

[0172] (3-1)

[0173] The lateral motion equation is

[0174] (3-2)

[0175] The vertical motion equation is

[0176] (3-3) ​

[0177] The equation of motion for rolling is

[0178] (3-4)

[0179] The equation of motion for nodding is

[0180] (3-5)

[0181] The equation for head-shaking motion is:

[0182] (3-6)

[0183] in, F xhl4 , F xhr3 These are the longitudinal forces exerted by the left and right trailer tracks on the left and right wheels, respectively, in N; F yhl4 , F yhr3 These are the lateral forces exerted by the left and right trailer tracks on the left and right wheels, respectively, in N; F zhl4 , F zhr3 These are the vertical forces exerted by the left and right trailer tracks on the left and right wheels, respectively, in N; M xwl4 , M ywl4 , M zwl4 These are the longitudinal, lateral, and vertical components of the left-side wheel-rail spin torque, respectively, in N·m; M xwr3 , M ywr3 , M zwr3 These are the longitudinal, lateral, and vertical components of the right-side wheel-rail spin torque, respectively, in N·m; F x2L1 , F x2R1 These are the longitudinal forces exerted by the trailer frame on the left and right wheel axles, respectively, in N; F y2L1 , F y2R1 These are the lateral forces exerted by the trailer frame on the left and right wheel axles, respectively, in N; F z2L1 , F z2R1 The vertical force exerted by the trailer frame on the left and right wheel axles, in N; φ sew The track superelevation angle corresponding to the center of the wheelset on the trailer track, in rad; R wR is the curvature radius corresponding to the center of the wheelset on the trolley track, m; a0is the half of the distance between the contact points of the left and right wheels, m; r 0 is the nominal rolling radius of the wheel, m; d 2 is the half of the horizontal distance between the action points of the trolley frame on the left and right wheel shafts, m; d L is the horizontal distance from the contact action point of the left wheel to the mass center of the trolley frame, m; d R is the horizontal distance from the contact action point of the right wheel to the mass center of the trolley frame, m; r L is the actual contact radius of the left wheel, m; r R is the actual contact radius of the right wheel, m;

[0184] As shown in Figure 7 , the dynamic equation of the trolley frame includes:

[0185] The longitudinal motion equation of the trolley frame is

[0186] (3-7)

[0187] The lateral motion equation of the trolley frame is

[0188] (3-8)

[0189] The vertical motion equation of the trolley frame is

[0190] (3-9)

[0191] The roll motion equation of the trolley frame is

[0192] (3-10)

[0193] The pitch motion equation of the trolley frame is

[0194] (3-11)

[0195] The yaw motion equation of the trolley frame is

[0196] (3-12)

[0197] wherein, F xb is the traction force of the conveyor belt on the trolley frame, N; F zb is the vertical action force of the conveyor belt on the middle of the trolley frame, N; F s1 , F s2These represent the tension of the front and rear steel wire ropes of the trailer frame, in N; F x1L1 , F x1R1 These are the longitudinal forces exerted by the conveyor belt on the left and right curved surfaces of the trolley frame, respectively, in N; F y1L1 , F y1R1 These represent the lateral forces exerted by the conveyor belt on the left and right curved surfaces of the trailer frame, respectively, in N; F z1L1 , F z1R1 These are the vertical forces exerted by the conveyor belt on the left and right curved surfaces of the trailer frame, respectively, in N; h 01 , h 03 The vertical distance from the stress points on the left and right sides of the trailer frame to the center of mass of the trailer frame, respectively, in meters; h 2 is the vertical distance from the point of force application on the wheel axle to the center of mass of the trailer frame, in meters; φ sec The superelevation angle, in rad, is the angle of elevation corresponding to the center of the trailer frame on the trailer track. d 01 , d 03 These are the lateral distances, in meters, from the left and right stress points of the trailer frame to the center of gravity of the trailer frame. l 02 The longitudinal distance between the front and rear points of action of the trailer frame on the left and right wheel axles is half of the distance in meters. R c Let the radius of curvature (m) be the radius of curvature corresponding to the center of gravity of the trolley frame on the trolley track.

[0198] F x1L1 , F y1L1 , F z1L1 , F x1R1 , F y1R1 , F z1R1 Defined as the force exerted by the conveyor belt on the arc surface of the trailer frame, and assumed to act on the same vertical plane as the force exerted by the trailer frame on the wheel axle, therefore the lateral distance to the center of mass is equal, i.e., the lever arm. .

[0199] S4. Determine the dynamics parameters of the trolley unit: determine the parameters of the conveyor belt, including the mass of the material line, the mass of the trolley line, the relationship between the angular velocity of the wheel and the speed of the trolley frame, the moment of inertia of the wheel, and the inertial moment of the wheel; calculate the force system distribution between the conveyor belt and the upper surface arc of the trolley frame when the conveyor belt carries material, calculate the position of the action point of the force of the conveyor belt on the surface arc of the trolley frame according to the classification of the amount of material; calculate the tension of the conveyor belt and the resistance of the belt conveyor; check the sag condition of the conveyor belt;

[0200] S4-1. Parameters of the conveyor belt:

[0201] According to the actual working condition and combined with the reference of the conveyor belt standard GB / T9770, it is known that the line mass of the ST1600 conveyor belt is:

[0202] (4-1)

[0203] The mass of the material line is:

[0204] (4-2)

[0205] The mass of the trolley line is:

[0206] (4-3)

[0207] The relationship between the angular velocity of the wheel and the speed of the trolley frame is:

[0208] (4-3)

[0209] The moment of inertia of the wheel is:

[0210] (4-4)

[0211] The inertial moment of the wheel is:

[0212] (4-5)

[0213] In the formula, Q is the design transportation capacity of the conveyor, t / h; q B is the mass of the conveyor belt line, kg / m; q w is the mass of the material line, kg / m; q c is the mass of the trolley line, kg / m; v b is the speed of the conveyor belt, m / s; ω l2 , ω l4 , ω r1 ,ω r3 ωL2, ωL4, ωR1, ωR3 are the angular velocities of the left 2, left 4, right 1, right 3 wheels, respectively, rad / s; v c v is the trolley running speed, m / s; I wy2 、 I wy4 、 I wy1 、 I wy3 IxxL2, IxxL4, IxxR1, IxxR3 are the transverse moments of inertia of the left 2, left 4, right 1, right 3 wheels, respectively, kg·m 2 ; m w m is the mass of a single wheel, kg.

[0214] S4-2. Force system distribution between the conveyor belt and the trolley frame camber

[0215] The trolley camber is approximated as a straight surface intersecting with a bottom surface, and a geometric relationship is established as shown in Figure 8 , and each area is calculated as follows:

[0216] (4-5)

[0217] In the formula, S A is the total area of the material, m 2 ; S 1 is the material area above the left trolley camber, m 2 ; S 2 is the material area directly above the horizontal section, m 2 ; S 3 is the material area above the right trolley camber, m 2 ; S a A is the total material area above the left and right trolley cambers, m 2 ; l 1 、 l 2 、l 3 are the lengths of the material on the left trolley camber, the horizontal section, and the right trolley camber, respectively, m; B 0 is the width of the conveyor belt carrying the material, m; λ is the inclination angle of the trolley camber, rad; θ is the material accumulation angle, rad.

[0218] According to the actual working conditions, the inclination angle of the trolley camber , the width of the conveyor belt is 1.2 m, and the material accumulation angle is taken as .

[0219] Given the above material area distribution, define the ① type force as the force of the trolley frame 3 on the left and right wheel shafts, then the size of the vertical component of the ① type force is:

[0220] (4-6)

[0221] And the position of the ① type force point is divided into the following three cases:

[0222] a: empty load (only the weight of the conveyor belt on the trolley camber), then the action point of the conveyor belt and the trolley camber is taken from the vertical height of the conveyor belt, which is approximately half the vertical height of the trolley camber.

[0223] b: light load (a small amount of material is only distributed on the horizontal section, and only the weight of the conveyor belt is on the trolley camber), then the action point position is the same as the a type case.

[0224] c: heavy load (the material has covered above the trolley camber), then the action point of the conveyor belt and the trolley camber is taken from the vertical height of the covered material, which is half the vertical height of the trolley camber.

[0225] For the c type case, the trolley frame left and right camber force point height is defined as h 01 、 h 03 which is represented as follows:

[0226] (4-7)

[0227] Under the full load condition of the conveyor belt, the force point height can be approximated by the equation. According to the geometric relationship, the size of the ① type force arm can be derived. The horizontal distance from the action point to the center of mass of the trolley frame d 01 、 d 03 are respectively:

[0228] (4-8)

[0229] The size of the other ① type force arm is a fixed value, determined by the structure of the trolley frame.

[0230] S4-3. Belt tension calculation

[0231] Given the conveyor belt tension F zL1 just entering the characteristic track, according to the minimum sag test of the conveyor belt, whether the characteristic tension meets the requirements, that is,

[0232] (4-9)

[0233] The initial belt tension is unknown, then according to the relationship between the driving drum tension just entering the drum and the tension just leaving the drum, that is,

[0234] (4-10)

[0235] where, F zLr is the tension of the belt just entering the drum, N; F zLc is the tension of the belt just leaving the drum, N; e is the base of the natural logarithm; µ bq is the coefficient of friction between the belt and the surface of the drive drum, 0.25 for a rubber-covered belt with diamond-shaped grooves and wet; β q is the wrap angle of the belt on the drive drum, rad;

[0236] is the driving force of the drive device, which can be determined from the power of the drive drum

[0237] (4-11)

[0238] is the driving force of the drive device, which can be determined from the power of the drive drum

[0239] (4-12)

[0240] where, K A is the coefficient of friction between the belt and the surface of the drive drum, 0.25 for a rubber-covered belt with diamond-shaped grooves and wet; , which is taken as 1.5 here;

[0241] S4-4. Resistance calculation

[0242] According to the division of the unit, according to the distance between the trolleys l c and the arc length U , the characteristic track is divided into U / l c characteristic points, denoted as 1, 2, …, i characteristic points, if U / l c is not an integer, then take one more bit; in order to reduce the calculation workload, 3 characteristic points are selected, and the pose of the trolley unit at the characteristic points is studied to obtain the three-direction force of the wheel-rail interaction; the 3 characteristic points are characteristic point 1, characteristic point (1+i) / 2, and characteristic point i; that is, the entry of the characteristic track, the just exit of the characteristic track, and the middle position of the characteristic track;

[0243] The mass of the trolley unit is composed of the mass of the belt line, the mass of the material line, the mass of the trolley frame and the wheel line, that is:

[0244] (4-13)

[0245] where,M G M = mass of unit, kg

[0246] The motion resistance of the trolley type belt conveyor includes the main resistance on the belt line F fz , the additional resistance on each component F ff , the special resistance F ft and the lifting resistance F fs ; the main resistance includes the wheel-rail rolling friction resistance, the belt bending resistance, the material internal friction resistance, the simplified calculation is:

[0247] (4-14)

[0248] wherein: F fz is the main resistance, N δ q is the belt inclination, ° f is the rolling friction coefficient, the rolling friction coefficient between the wheel and the rail is taken as 0.03

[0249] The additional resistance includes the inertia resistance of the material in the receiving section and the friction resistance between the material and the belt; the friction resistance between the material and the side wall of the material guide groove in the acceleration area of the receiving section; the bending resistance of the belt around the roller, the bearing resistance of the non-driving roller, the friction resistance between the belt in the receiving section and the sealing apron of the material guide groove. In the calculation of the additional resistance, an additional resistance coefficient is usually multiplied to the main resistance, that is:

[0250] (4-15)

[0251] The special resistance is basically the same as that of the conventional belt conveyor, and it is worth noting that additional bending resistance will be generated in the convex arc section, that is:

[0252] (4-16)

[0253] The lifting resistance:

[0254] (4-17)

[0255] Therefore, the total resistance of a trolley unit is:

[0256] (4-18)

[0257] Therefore, the system line resistance is:

[0258] (4-18)

[0259] Total resistance of the turning section:

[0260] (4-20)

[0261] Where, F ff N is the additional resistance; C 1 is the additional resistance coefficient; F fw N is the additional bending resistance of the convex arc section; F zlq is the initial tension of the convex arc section, F zlz is the terminal tension of the convex arc section, α t is the corresponding central angle of the convex arc section; F fw N is the lifting resistance; h q is the lifting height, m; F e N is the total resistance of the trolley unit; F e0 N is the system line resistance; F eq Total resistance of the turning section, N; L q Total length of the turning section, m;

[0262] S4-5. Conveyor belt tension calculation

[0263] According to the "DTII (A) type fixed belt conveyor design selection manual", the relative sag condition is checked, which stipulates that the relative sag shall not exceed 1%. The minimum tension required for the load carrying section is:

[0264] (4-21)

[0265] The minimum tension of the return section is:

[0266] (4-22)

[0267] Where, F zLcmin , F zLhmin N is the minimum tension of the load carrying section and the return section, respectively; h max is the maximum allowable sag of the conveyor belt, which is taken as 0.02.

[0268] According to the actual working condition of the site, the trolley type belt conveyor adopts heavy hammer tensioning form, so the conveyor belt tension at the characteristic point i is:

[0269] (4-23)

[0270] wherein, F zLi is the belt tension at the nth bogie unit, N; i zL（i-1） is the belt tension at the nth bogie unit, N; F e（i-1~i） is the running resistance between the mth and the nth bogie unit, N; i-1 （i-1~i） is the equivalent mass of the belt track system, kg; F 2 is the average acceleration / deceleration of the belt, m / s i-1 i m w1 α w2

[0271] S5. The wheel pair is regarded as an elastic body, and the bogie unit is divided into two halves in the longitudinal direction in the vertical direction. The dynamic characteristics of the front and rear wheel tracks of the divided half bogie unit are analyzed, as shown in FIG. 5; the vertical motion equation and the nodding motion equation of the bogie frame are established according to the Hamilton principle, and then the linear second-order differential equation of the bogie unit is obtained; Figure 9

[0272] Specifically, the vertical motion equation of the bogie frame is analyzed according to the Hamilton principle, and the following equation is obtained

[0273] (5-1)

[0274] (5-2)

[0275] (5-3)

[0276] The nodding motion equation of the bogie frame is analyzed, and the following equation is obtained

[0277] (5-4)

[0278] wherein, m w1 , m w2 , m c1 respectively, are the masses of the front wheel, the rear wheel and the bogie frame, kg; z w1 , z w2 , z c1 respectively, are the vertical displacements of the front wheel, the rear wheel and the bogie frame, m; F zb1 ​​​​N for the pressure of the conveyor belt on the bogie frame;

[0279] Let the mass of the ith bogie unit be m i Let the stiffness coefficient be k i Let the damping coefficient be c i Let the running resistance be w i Let the displacement be x i ; the conveyor belt is regarded as a linear system, and the matrix form of the linear second-order differential equation is

[0280] (5-5)

[0281] where M is the mass matrix; C is the damping matrix; K is the stiffness matrix; all of which are square matrices; is the system acceleration, velocity, and displacement matrix; F is the force matrix acting on each unit, which is a column vector;

[0282] The above equation is arranged into the matrix form of the linear second-order differential equation to obtain

[0283] (5-6)

[0284] (5-7)

[0285] (5-8)

[0286] (5-9)

[0287] (5-10).

[0288] S6. Use numerical calculation software to compile a calculation program to simulate the wheel-rail vibration characteristics of the bogie unit during straight-line travel, specifically, as shown in Figure 10 the calculation flow is compiled using MATLAB software, including the following steps,

[0289] S6-1. input the wheelset dynamics equation and the bogie frame dynamics equation;

[0290] S6-2. input the bogie unit dynamics parameters determined in step S4;

[0291] S6-3. establish the bogie frame vertical motion equation and nodding motion equation;

[0292] S6-4. set the bogie track excitation model;

[0293] S6-5. Generating the bogie unit matrix and the external force matrix, as shown in formulas 5-5 to 5-10;

[0294] S6-6. Introducing a numerical integration algorithm to solve the motion equation; the most effective solution for solving the linear differential equation group with time-varying coefficients, i.e. the motion equation, is the step-by-step integration method;

[0295] S6-7. Selecting a time step and judging the convergence of the algorithm; in order to first maintain the stability of the algorithm and avoid the high-frequency components of the structural vibration from being unrestrictedly increased, making the entire integration meaningless. Combined with the characteristics of the Newmark-integration method, it is verified that the time step is generally taken as , , the algorithm is unconditionally stable. Then, according to the requirement of the convergence rate, the time step is selected as , so as to meet the requirement of the convergence limit in the table. From the definition and condition of the convergence domain, when the division of the time step is small enough, the convergence rate can generally meet the requirement. Therefore, in this paper, the Newmark algorithm with unconditional stability is constructed by selecting , a small enough time step;

[0296] S6-8. Calculating the vibration characteristics of the front and rear wheel pairs;

[0297] S6-9. Taking the result of the previous time step as the initial iterative value to calculate the vibration state;

[0298] S6-10. Judging whether the actual calculation time t is greater than the planned calculation time T, if t > T is met, step S6-11 is continued, if t > T is not met, step S6-8 is returned to continue the calculation;

[0299] S6-11. Solving the wheel pair dynamics equation and the bogie frame dynamics equation and outputting the dynamic response result.

[0300] It also includes three-dimensional dynamic simulation verification, which includes the following steps:

[0301] S7-1. Establishing a three-dimensional model of the bogie type belt conveyor and importing it into the simulation software;

[0302] S7-2. Performing modal analysis on the bogie frame and the wheels;

[0303] S7-3. Outputting the three-direction vibration acceleration changes of the single-side front and rear wheels under different belt speeds and different bogie spacings when running on the straight section bogie track;

[0304] S7-4. Outputting the three-direction vibration acceleration changes of the single-side front and rear wheels when subjected to different load impacts on the straight section bogie track;

[0305] S7-5. Output the three-directional vibration acceleration changes of the single-side front wheel and the rear wheel when the trolley spacing on the straight section trolley track is different.

[0306] Further comprising the elastoplastic analysis of wheel-rail rolling contact, including the following steps:

[0307] A1. Based on the elastic half-space assumption, it is considered that the surfaces of two contact objects are smooth and the curvature in the elliptical contact area is constant, the normal gap between the two contact objects can be expressed by Taylor polynomial, and the normal gap in the contact spot is obtained by ignoring the high-order terms h :

[0308] (a-1)

[0309] In the formula, A and B are constants, related to the curvature radius of the contact point of the trolley track along x and y ;

[0310] A and B can be expressed as:

[0311] (a-2)

[0312] (a-3)

[0313] In the formula, R rx 、R ry are the curvature radii of the trolley track along the rolling direction and the transverse direction of the wheel, respectively, m; R wx 、R wy are the curvature radii of the rolling circle of the wheel and the transverse direction, respectively, m; θ 1 is the angle between the two principal curvature planes containing R rx and R wx , °;

[0314] A2. Analyze the normal force distribution of the contact area between the wheel and the trolley track, the normal force distribution of the contact area is shown in Figure 11 , according to the Boussinesq-Cerruti formula in elasticity mechanics, the long semi-axis a and the short semi-axis b of the elliptical contact spot are:

[0315] (a-4)

[0316] In the formula, F p N is the wheel-rail normal contact load; m, n is a constant related to the value of B - A / A + B , which can be obtained by interpolation from a table, where

[0317] Equivalent elastic modulus E is:

[0318] (a-5)

[0319] The normal pressure in the elliptical contact patch is:

[0320] (a-6)

[0321] The relative deformation of wheel-rail contact is:

[0322] (a-7)

[0323] In the formula, E Equivalent elastic modulus of wheel-rail, MPa; ζ 1, ζ 2 are the Poisson's ratios of wheel and rail materials, respectively; E 1, E 2 are the elastic moduli of wheel and rail, MPa, respectively; p z Normal pressure in the elliptical contact patch, N; ε Relative deformation of wheel-rail contact, m; r f is a constant related to the value of B - A / A + B ;

[0324] A3. When the maximum pressure in the contact area does not exceed the material yield pressure, the contact area deforms elastically and still satisfies the Hertz contact pressure distribution. When the maximum pressure in the contact area exceeds the yield pressure, the central region of the contact patch will deform plastically, as shown in Figure 12 , and the normal pressure in the elastic and plastic deformation regions is:

[0325] (a-8)

[0326] Assuming that the material satisfies the Mises yield criterion condition, the initial yield normal contact force of the two contact bodies is:

[0327] (a-9)

[0328] In the formula, p y Initial yield pressure, MPa; p 0* For the maximum contact pressure when the whole contact area is only elastically deformed, it satisfies , N; S t , S s are the areas of the elastic deformation area and the plastic deformation area, respectively, m 2 ; Y is the yield strength of the material with lower strength in the two contact bodies, MPa; ζ is the Poisson's ratio of the material;

[0329] The maximum contact pressure is at the center point of the elliptical contact patch, and the maximum contact pressure can be expressed as:

[0330] (a-10)

[0331] When E t =E , it means the linear elastic contact condition. According to the Hertz contact theory formula, it is extended to the case of elliptical contact patch, and the yield pressure p y The long axis a y and the short axis b y corresponding to the initial yield are related as:

[0332] (a-11)

[0333] The yield pressure p y and the initial yield load F y0 satisfy:

[0334] (a-12)

[0335] Considering the continuity of the contact pressure, there is:

[0336] (a-13)

[0337] From the above formula, the long semi-axis a p and the short semi-axis b p of the plastic zone are:

[0338] (a-14)

[0339] The contact force can be obtained by integrating in the contact area:

[0340] (a-15);

[0341] A4. Simplifying the above equation, we have

[0342] (a-16)

[0343] where, a y , b y are the long and short semi-axes of the plastic region, m; F y0 is the initial yield load, N; a p , b p are the long and short semi-axes of the plastic region, m;

[0344] A5. From the Hertz contact theory, the relationship between the maximum contact pressure and the elastic deformation of the elastic contact model is:

[0345] (a-17)

[0346] As shown in Figure 13 , the normal contact pressure p is proportional to . Assuming that the object contact satisfies the elastic-plastic linear hardening relationship, the pressure p can be expressed as:

[0347] (a-18)

[0348] where, δ is the elastic deformation, m; δ y is the elastic critical deformation, m; k is the hardening coefficient;

[0349] When the hardening coefficient , it represents an ideal elastic-plastic material model, i.e., when the maximum deformation of the contact area is greater than the critical deformation δ y . The maximum pressure of the contact area is still p y ; when the hardening coefficient , it represents a bilinear hardening material model;

[0350] then the elastic-plastic displacement δ is

[0351] (a-19).

[0352] Also included is a wheel-rail contact display dynamic simulation analysis, including the following steps:

[0353] B1. Establish a three-dimensional wheel-rail rolling contact finite element theoretical model, including the wheel mass, spring-damper unit, wheel-rail system and track support structure, and establish a Cartesian coordinate system O-XYZ, wherein the origin O is the initial contact point of the wheel-rail, X, Y, Z respectively represent the longitudinal direction (wheel-rail rolling direction), lateral direction and vertical direction;

[0354] B2. Set the model size parameters and material parameters, and set the simulation experience stage, including the stress initial stage and the steady-state rolling contact stage, the steady-state rolling contact stage is the contact stage under the stress balance state;

[0355] B3. Output the transient stress and deformation of the trolley under different load weight conditions, and analyze the stress concentration position;

[0356] B4. Output the wheel-rail contact area contour map and stress distribution map when moving at different speeds under different load weight conditions.

[0357] The above has been described by way of example, but the present application is not limited to the above specific embodiments, and any modification or change made on the basis of the present application falls within the scope of the present application.

Claims

1. A method of simulating the dynamic characteristics of a towed belt conveyor, characterized in that The trolley unit comprises a trolley frame, four wheels rotatably connected to the trolley frame, and a conveyor belt unit supported on the trolley frame, the trolley unit moves along a straight line on a trolley track; the vertical direction perpendicular to the surface of the conveyor belt is defined as the vertical direction, the direction in which the trolley frame moves along the trolley track is defined as the longitudinal direction, and the direction perpendicular to the vertical direction and the longitudinal direction in space is defined as the transverse direction; the side roll is defined as the rotation around the longitudinal direction, the nodding is defined as the rotation around the transverse direction, and the shaking is defined as the rotation around the vertical direction; The trolley type belt conveyor dynamics characteristic simulation method comprises the following steps: S1. Force analysis is performed on the trolley frame, each wheel and the conveyor belt unit in the trolley unit, and the degrees of freedom are sorted, wherein the degrees of freedom include longitudinal translation, transverse translation, vertical translation, roll, nodding and shaking rotation; S2. A coordinate system is established, the coordinate system comprises a general coordinate system, a vehicle body coordinate system and a contact coordinate system, the general coordinate system is fixed to the ground, the vehicle body coordinate system is fixed to the trolley unit, and the origin of the contact coordinate system is defined at the contact center of each wheel and the trolley track, the attitude angle of the wheel in the vehicle body coordinate system is established, and the relative conversion relationship between the vehicle body coordinate system and the general coordinate system is established; S3. According to the D'Alembert principle and spatial force analysis, the dynamics equations of the wheelset and the trolley frame are respectively established; S4. The dynamics parameters of the trolley unit are determined: the solving parameters of the conveyor belt are determined, including the material line mass, the trolley line mass, the relationship between the wheel angular velocity and the trolley frame speed, the rotational inertia of the wheel and the inertia moment of the wheel; the force system distribution between the conveyor belt and the surface arc of the trolley frame is calculated when the conveyor belt is loaded with materials, the position of the acting point of the force of the conveyor belt on the surface arc of the trolley frame is calculated according to the material quantity classification; the conveyor belt tension and the belt conveyor resistance are calculated; and the sag of the conveyor belt is checked; S5. The wheelset is regarded as an elastic body, the trolley unit is symmetrically divided along the longitudinal direction in the vertical direction, and the dynamic characteristics of the front and rear wheel tracks of the half trolley unit after the division are analyzed; the vertical motion equation and the nodding motion equation of the trolley frame are established according to the Hamilton principle, and then the linear second-order differential equation of the trolley unit is obtained; S6. A numerical calculation software is used to compile a calculation program, and the vibration characteristics of the wheel track of the trolley unit during the straight line running process are simulated.

2. The method according to claim 1, wherein, The step S2 comprises: a general coordinate system O-XYZ is defined, the origin O of the general coordinate system O-XYZ coincides with the mass center of the trolley unit at the initial position of the trolley unit; the X-axis is along the initial longitudinal driving direction of the trolley frame, and the forward direction is positive; the Z-axis is the same as the direction of gravity, and the downward direction is positive; and the Y-axis is perpendicular to the XOZ plane and points to the right along the X direction; The body coordinate system O is defined as follows: C - X C - Y C - Z C with its origin O C at the center of mass of the bogie frame, X C pointing forward, Y C pointing to the right, Z C pointing downward; The contact coordinate system O is defined W - X W Y W Z W with its origin O W At the contact center of each wheel, X W points in the forward direction of the wheel with respect to the contact surface of the track, Y W points to the right side of the contact surface, Z W points downward with respect to the contact surface; The wheel attitude angle includes a wheel yaw angle ψ w , a wheel roll angle φ w , and a wheel rotation angle σ w about the wheel center of mass. The trailer unit has six degrees of freedom. Three degrees of freedom describe the position of the trailer frame's center of mass relative to the global coordinate system, namely the X, Y, and Z coordinates. The other three degrees of freedom describe the trailer frame's attitude, namely the yaw angle ψ, the pitch angle β, and the roll angle φ, also known as Euler angle coordinates. Any attitude of the trailer coordinate system can be obtained through three sequential rotations, namely the vehicle coordinate system O. C Starting from the coordinate system O, and revolving around its Z... C Rotating the axis by an angle ψ yields the reference frame O. 2 Then the reference frame O C 2 around its Y C Rotating the axis by an angle β yields the reference frame O. C 1 Finally, the reference frame O C 1 Around its X C Rotating the axis by an angle φ yields the vehicle coordinate system O. C O C X C 'Y C 'Indicates reference frame O C 2 O C X C Y C "Indicates reference frame O" C 1 O C X C Y C Represents the final coordinate system O of the vehicle body T , then the direction cosine matrixes are respectively: （2-1） （2-2） （2-3） According to the conversion relationship between the global coordinate system O and the vehicle body coordinate system O T , N is the direction cosine matrix of the global coordinate system O and the vehicle body coordinate system O C , that is: （2-4）。 3. The method according to claim 2, wherein, the wheelset dynamics equation in the step S3 comprises: the longitudinal motion equation of the wheelset is （3-1） the transverse motion equation is （3-2） the vertical motion equation is （3-3） the roll motion equation is （3-4） the nodding motion equation is （3-5） the shaking motion equation is （3-6） wherein, F xhl4 , F xhr3 Fy, Fy are the longitudinal forces of the left and right wheel-rail pairs on the left and right wheels, respectively; F yhl4 , F yhr3 Fz, Fz are the lateral forces of the left and right wheel-rail pairs on the left and right wheels, respectively; F zhl4 , F zhr3 Fz, Fz are the vertical forces of the left and right wheel-rail pairs on the left and right wheels, respectively; M xwl4 , M ywl4 , M zwl4 Mx, My, Mz are the longitudinal, lateral, and vertical components of the self-rotation moment of the left wheel-rail, respectively; M xwr3 , M ywr3 , M zwr3 Mx, My, Mz are the longitudinal, lateral, and vertical components of the self-rotation moment of the right wheel-rail, respectively; F x2L1 , F x2R1 Fy, Fy are the longitudinal forces of the left and right wheel-axle pairs on the left and right wheel-axles, respectively; F y2L1 , F y2R1 Fz, Fz are the lateral forces of the left and right wheel-axle pairs on the left and right wheel-axles, respectively; F z2L1 , F z2R1 Fz, Fz are the vertical forces of the left and right wheel-axle pairs on the left and right wheel-axles, respectively; the step S4 comprises: sew is the rail superelevation angle corresponding to the wheel-pair center on the wheel-rail; R w is the curvature radius corresponding to the wheel-pair center on the wheel-rail; a0 is half of the distance between the contact points of the left and right wheels; r 0 is the nominal rolling radius of the wheel; d 2 is half of the lateral distance between the action points of the left and right wheel-axles on the wheel-rail; d L is the lateral distance from the contact action point of the left wheel and the wheel-rail to the mass center of the wheel-rail; d R is the lateral distance from the contact action point of the right wheel and the wheel-rail to the mass center of the wheel-rail; r L is the actual contact radius of the left wheel; r R is the actual contact radius of the right wheel; the longitudinal motion equation of the trolley frame is the transverse motion equation of the trolley frame is （3-7） the vertical motion equation of the trolley frame is （3-8） the roll motion equation of the trolley frame is （3-9） the nodding motion equation of the trolley frame is （3-10） the shaking motion equation of the trolley frame is （3-11） the step S5 comprises: （3-12） Wherein, F xb is the traction force of the conveyor belt on the trolley frame; F zb is the vertical force of the conveyor belt on the middle of the trolley frame; F s1 , F s2 are the front and rear steel wire rope tension of the trolley frame, respectively; F x1L1 , F x1R1 are the longitudinal force of the conveyor belt on the left and right side of the trolley frame, respectively; F y1L1 , F y1R1 are the transverse force of the conveyor belt on the left and right side of the trolley frame, respectively; F z1L1 , F z1R1 are the vertical force of the conveyor belt on the left and right side of the trolley frame, respectively; h 01 , h 03 are the vertical distance from the left and right force points of the trolley frame to the center of mass of the trolley frame, respectively; h 2 is the vertical distance from the wheel shaft force point to the center of mass of the trolley frame; the vertical motion equation of the trolley frame is analyzed according to the Hamilton principle, and the following equation is obtained sec is the track superelevation angle corresponding to the center of the trolley frame on the trolley track; d 01 , d 03 are the horizontal distance from the left and right force points of the trolley frame to the center of mass of the trolley frame, respectively; l 02 is half of the longitudinal distance from the front and rear action points of the trolley frame to the left and right wheel shafts; R c is the radius of curvature corresponding to the center of gravity of the trolley frame on the trolley track; F x1L1 、 F y1L1 、 F z1L1 、 F x1R1 、 F y1R1 、 F z1R1 The force defined as the force of the conveyor belt to the arc surface of the trolley frame, and assuming the same vertical plane with the force of the trolley frame to the wheel shaft, so the horizontal distance to the mass center is equal, that is, the force arm .

4. The method according to claim 3, wherein, ​ ​ （5-1） （5-2） （5-3） The nodding motion equation of the bogie frame can be obtained by analyzing （5-4） wherein m w1 , m w2 , m c1 respectively the mass of the front wheel, the rear wheel, the bogie; z w1 , z w2 , z c1 respectively the vertical displacement of the front wheel, the rear wheel, the bogie; F zb1 the pressure of the conveyor belt on the bogie; The mass of the ith dolly unit is defined as m i The stiffness coefficient is k i The damping coefficient is c i The running resistance is w i The displacement is x i The conveying belt is regarded as a linear system, and a linear second-order differential equation matrix is adopted （5-5） where M is the mass matrix; C is the damping matrix; K is the stiffness matrix; all of which are square matrices; are the system acceleration, velocity, and displacement matrices; F is the force matrix acting on each element, which is a column vector. The above equation is arranged into a matrix form of a linear second-order differential equation to obtain （5-6） （5-7） （5-8） （5-9） （5-10）。 5. The method according to claim 1, wherein, The step S6 uses MATLAB software to compile a calculation process, including the following steps, S6-1. input the wheelset dynamics equation and the bogie frame dynamics equation; S6-2. input the bogie unit dynamics parameters determined in step S4; S6-3. establish the vertical motion equation and the nodding motion equation of the bogie frame; S6-4. set the bogie track excitation model; S6-5. generate the bogie unit matrix and the external force matrix; S6-6. introduce a numerical integration algorithm to solve the motion equation; S6-7. select a time step, and judge the convergence of the algorithm; S6-8. calculate the vibration characteristics of the front and rear wheelsets; S6-9. take the result of the previous time step as an initial iteration value to calculate the vibration state; S6-10. judge whether the actual calculation time t is greater than the planned calculation time T, if t > T, continue to step S6-11, if t < T, return to step S6-8 to continue the calculation; S6-11. solve the wheelset dynamics equation and the bogie frame dynamics equation and output the dynamic response result.

6. The method according to claim 1, wherein, It also includes three-dimensional dynamic simulation verification, including the following steps: S7-1. establish a three-dimensional model of the bogie type belt conveyor and import it into the simulation software; S7-2. perform modal analysis on the bogie frame and the wheels; S7-3. output the three-directional vibration acceleration changes of the single-side front and rear wheels under different belt speeds and different bogie spacings on the straight section bogie track; S7-4. output the three-directional vibration acceleration changes of the single-side front and rear wheels under different load impacts on the straight section bogie track; S7-5. output the three-directional vibration acceleration changes of the single-side front and rear wheels under different bogie spacings on the straight section bogie track.

7. The method according to claim 1, wherein, It also includes elastic-plastic analysis of wheel-rail rolling contact theory, including the following steps: A1. Based on the elastic half-space assumption, it is considered that the surface of the two contact objects is smooth and the curvature in the elliptical contact area is constant, the normal gap between the two contact objects can be expressed by Taylor polynomial, and the normal gap in the contact spot is obtained by ignoring the high order term h is: （a-1） wherein A and B are constants related to the radius of curvature of the contact point of the dolly track along x and y and A and B may be represented as: （a-2） （a-3） wherein R rx 、R ry Rtand Rlare the radii of curvature of the trolley rail in the rolling direction and in the transverse direction, respectively; R wx 、R wy Rr and Rlare the radii of curvature of the rolling circle and in the transverse direction, respectively; θ 1 is the angle between the two principal planes of curvature of the wheel and the trolley rail R rx and R wx the angle between the two principal planes of curvature of the wheel and the trolley rail A2. Analyzing the normal force distribution of the contact area between the wheel and the track, according to the Boussinesq-Cerruti formula in the elastic mechanics, the long semi-axis and the short semi-axis of the elliptical contact patch are obtained as follows: a and b respectively. （a-4） wherein F p is the wheel-rail normal contact force; m, n is a constant, related to the value of B-A / A+B , which can be obtained by interpolation from a table, wherein equivalent elastic modulus E is: （a-5） The normal pressure in the elliptical contact patch is: （a-6） The relative deformation of wheel-rail contact is: （a-7） wherein E E is the equivalent elastic modulus of wheel and rail; ζ 1, ζ 2 are the Poisson's ratios of the wheel and rail materials, respectively; E 1, E 2 are the elastic moduli of the wheel and rail, respectively; p z P is the normal pressure in the elliptical contact patch; ε δ is the relative deformation of wheel and rail contact; r f is a constant related to the value of B-A / A+B is a constant related to the value of A3. When the maximum pressure in the contact area does not exceed the material yield pressure, the contact area is elastically deformed, still satisfying the Hertz contact pressure distribution; when the maximum pressure in the contact area exceeds the yield pressure, the central area of the contact patch will be plastically deformed, and the normal pressure of the elastic deformation area and the plastic deformation area is: （a-8） Assuming that the material satisfies the Mises yield criterion condition, the normal contact force when the two contact bodies initially yield is: （a-9） wherein p y is the initial yield pressure; p 0 * is the maximum contact pressure assuming only elastic deformation in the entire contact area, satisfying ; S t , S s are the areas of the elastic and plastic deformation regions, respectively; Y is the yield strength of the material with lower strength in the two contact bodies; ζ is the Poisson's ratio of the material; The maximum contact pressure at the center point of the elliptical contact patch can be expressed as: (a-10) When E t =E Then, it means the linear elastic contact case, according to the Hertz contact theory formula, extended to the case of elliptical contact spot, then the yield pressure p y The long axis corresponding to the initial yield a y The short axis b y The relationship is: (a-11) then the yield pressure p y and the initial yield load F y0 satisfies: (a-12) Considering the continuity of the contact pressure, there is: (a-13) From the above equation, the plastic zone long semi-axis a p and the short semi-axis b p are respectively: (a-14) Integrating in the contact area, the contact force is: (a-15)； A4. Simplifying the above equation, we obtain （a-16） wherein a y , b y are the long and short semi-axes corresponding to the yield pressure and to the initial yield, respectively; F y0 is the initial yield load; a p , b p are the long and short semi-axes of the plastic region, respectively; A5. According to the Hertz contact theory, the relationship between the maximum contact pressure of the elastic contact model and the elastic deformation is: （a-17） Assuming that the contact of the object satisfies the elastic-plastic linear hardening relationship, the pressure p can be expressed as: （a-18） In the formula, δ is the elastic deformation amount; δ y is the elastic critical deformation amount; k is the reinforcement coefficient; When the hardening coefficient is equal to 0, then the ideal elastic-plastic material model is represented, i.e. when the maximum deformation of the contact area is greater than the critical deformation δ y , the maximum pressure of the contact area is still p y ; when the hardening coefficient is equal to 1, then the bilinear hardening material model is represented; then the elastic-plastic displacement δ is （a-19）。 8. The method according to claim 7, wherein, It also includes wheel-rail contact display dynamic simulation analysis, including the following steps: B1. Establish a three-dimensional wheel-rail rolling contact finite element theoretical model, including the mass on the wheel, spring-damper unit, wheel-rail system and track support structure, and establish a Cartesian coordinate system O-XYZ, wherein the origin O is the initial contact point of wheel-rail, X, Y and Z represent the longitudinal direction (wheel-rail rolling direction), lateral direction and vertical direction respectively; B2. Set the model size parameters and material parameters, and set the simulation experience stage, including the stress initial stage and the steady-state rolling contact stage, the steady-state rolling contact stage is the contact stage under the stress balance state; B3. Output the transient stress and deformation of the trolley under different load weight conditions, and analyze the stress concentration position; B4. Output the wheel-rail contact area contour map and stress distribution map when moving at different speeds under different load weight conditions.

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