A Simulation Method for the Dynamic Characteristics of Turning Sections of a Track-Type Belt Conveyor

Abstract

This invention relates to the field of dynamic characteristics research on belt conveyors, and in particular to a simulation method for the dynamic characteristics of a track-type belt conveyor during turning sections. The method includes: performing force analysis on the trailer unit and organizing its degrees of freedom; establishing the overall coordinate system, vehicle coordinate system, and contact coordinate system, and their transformation relationships; establishing the dynamic equations of the trailer unit; determining the dynamic parameters of the trailer unit; establishing the motion equations of the left and right wheel pairs of the trailer unit based on Hamilton's principle, thereby obtaining the linear second-order differential equations of the trailer unit; and using numerical calculation software to compile a calculation program to simulate the wheel-rail vibration characteristics of the trailer unit during turning sections. This invention is based on multi-degree-of-freedom multibody dynamics theory, constructing nonlinear dynamic equations for the belt conveyor system, and analyzing the six-degree-of-freedom motion equations of the trailer unit.

Description

Technical Field

[0001] This invention relates to the field of research on the dynamic characteristics of belt conveyors, and in particular to a simulation method for the dynamic characteristics of turning sections of track-type belt conveyors. Background Technology

[0002] Belt conveyors are widely used in the coal industry's transportation processes. For long-distance cross-country belt conveyors, track-mounted belt conveyors can effectively reduce operating resistance. Both track-mounted and ordinary belt conveyors use flexible conveyor belts for material transport, but during operation, the acceleration and tension of the conveyor belt change differently over time at different locations. Furthermore, the trailer is subjected not only to the interaction force between the wheel and rail, but also to the dynamic tension of the conveyor belt on the trailer and the force exerted by the wire rope. These combined forces directly affect the stability and safety of the trailer's operation, and consequently, the interaction between the wheel and rail. As the conveyor's operating speed increases, the interaction force between the wheel and rail also increases, leading to increased wheel-rail contact fatigue, accelerated material wear, and consequently, wheel wear problems and abnormal vehicle vibration caused by wheel wear. The causes of wheel-rail wear are complex, mainly involving wheel-rail coupled vibration, wheel-rail material wear and plastic deformation, and wheel-rail rolling contact fatigue, but existing technologies lack methods for accurately studying its dynamic characteristics. Summary of the Invention

[0003] This invention aims to solve the above problems and provides a simulation method for the dynamic characteristics of the turning section of a track-type belt conveyor. The technical solution adopted is as follows:

[0004] A simulation method for the dynamic characteristics of a turning section of a track-type belt conveyor is provided. The method defines a trolley unit as comprising 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 in the turning section of the trolley track, and the inner rail is raised. The direction perpendicular to the conveyor belt surface 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 in space perpendicular to both the vertical and longitudinal directions is defined as the transverse direction. 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.

[0005] The simulation method for the dynamic characteristics of the turning section of a belt conveyor includes the following steps:

[0006] S1. Perform force analysis on the trailer frame, each wheel and the conveyor belt unit in the trailer unit and sort out the degrees of freedom, including longitudinal translation, lateral translation, vertical translation, lateral rolling, nodding and yaw rotation;

[0007] S2. Establish a coordinate system, which includes a global coordinate system, a vehicle body coordinate system, and a contact coordinate system. The global coordinate system is fixed to the ground, the vehicle body coordinate system is fixed to the trailer unit, and the origin of the contact coordinate system is defined at the contact center between each wheel and the trailer track. Establish the attitude angle of the wheel in the vehicle body coordinate system and establish the relative transformation relationship between the vehicle body coordinate system and the global coordinate system.

[0008] S3. Based on d'Alembert's principle and spatial force analysis, establish the dynamic equations of the motorcycle unit;

[0009] S4. Determine the dynamic parameters of the trolley unit: Determine the solution parameters for the conveyor belt, including the linear mass of the material, the linear mass of the trolley, the relationship between the wheel angular velocity and the trolley frame velocity, the moment of inertia of the wheels, and the wheel inertial torque; calculate the force distribution between the conveyor belt and the upper surface arc of the trolley frame when the conveyor belt is carrying material, and calculate the location of the point of application of the force exerted by the conveyor belt on the surface arc of the trolley frame according to the material quantity; perform force analysis and verification for the trolley unit in the turning section; calculate the conveyor belt tension and the resistance of the belt conveyor; and verify the conveyor belt sag condition.

[0010] S5. Treating the wheelset as an elastic body, the trailer unit is symmetrically divided in the vertical direction along the transverse direction. The dynamic characteristics of the left and right wheel rails are analyzed for the divided half of the trailer unit. The motion equations of the left and right wheelsets of the trailer unit are established according to Hamilton's principle, and then the linear second-order differential equation of the trailer unit is obtained.

[0011] S6. Compile a calculation program using numerical calculation software to simulate the wheel-rail vibration characteristics of the trailer unit during its travel on a turning section.

[0012] Based on the above scheme, step S2 includes:

[0013] Define a global coordinate system O-XYZ, with its origin O coinciding with the center of mass of the trailer unit at its initial position; the X-axis is along the initial longitudinal direction of travel of the trailer frame, with forward being positive; the Z-axis is in the same direction as gravity, with downward being positive; the Y-axis is perpendicular to the XOZ plane and points to the right along the X direction.

[0014] Define the vehicle coordinate system O C -X C Y C Z C Its origin O C At the center of gravity of the trailer frame, X C Pointing to the front of the trailer frame, Y C Pointing to the right side of the trailer frame, Z C Pointing towards the bottom of the trailer frame;

[0015] Define contact coordinate system O W -X W Y W ZW Its origin O W At the contact center of each wheel, X W The axis points in front of the contact surface between the wheel and the trailer track, Y W Pointing to the right side of the contact surface, Z W Pointing downwards from the contact surface;

[0016] If the wheel's yaw angle and roll angle are in the same direction and equal in magnitude to the trailer's yaw angle and roll angle, then the wheel attitude angles include: wheel yaw angle ψ w Side roll angle φ w And the rotation angle σ of the wheel about its center of mass w ;

[0017] 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. C 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 ,

[0018] The direction cosine matrices are as follows:

[0019] (2-1)

[0020] (2-2)

[0021] (2-3)

[0022] Based on the global coordinate system O and the vehicle body coordinate system O T The transformation relationship between them, where N represents the global coordinate system O and the vehicle body coordinate system O C The direction cosine matrix, i.e.:

[0023] (2-4).

[0024] Based on the above scheme, in step S3, when establishing the dynamic equation, structural simplification assumptions, load and mechanical assumptions, motion assumptions, friction and resistance assumptions, and environmental factor assumptions are made.

[0025] The structural simplification assumptions are as follows: the conveyor belt is regarded as a uniform continuous medium, and the trailer frame, wheels and trailer rails are regarded as rigid bodies; the conveyor belt is defined as an ideal elastic body, and the conveyor belt unit is simplified as an ideal linear spring and an ideal viscous damping unit;

[0026] The load and mechanical assumptions are as follows: the load distribution on the conveyor belt is static and uniform, and the direction of the tension of the steel wire ropes at the front and rear of the trolley frame is always longitudinal.

[0027] The motion assumption is that the motion of the belt conveyor is stable and there are no transient effects during start-up and shutdown.

[0028] The friction and resistance assumptions are as follows: the friction coefficient between the trailer track, the trailer frame, and the wheels is assumed to be uniform; air resistance is ignored.

[0029] The environmental factor assumption is that ambient temperature has no effect on the mechanical properties of the material.

[0030] Based on the above scheme, the dynamic equations of the motorcycle unit in step S3 include:

[0031] The longitudinal motion equation is

[0032] (3-1)

[0033] The equation of lateral motion is

[0034] (3-2)

[0035] The equation of vertical motion is

[0036] (3-3)

[0037] The equation of motion for rolling is

[0038] (3-4)

[0039] The equation of motion for nodding is

[0040] (3-5)

[0041] The equation for head-shaking motion is:

[0042] (3-6)

[0043] in, F q1 The traction force of the conveyor belt on the trolley, in N; Fe Total resistance of the trolley unit, N; F s1 , F s2 These are the tensions of the front and rear steel wire ropes of the motorcycle, in N; F xhl2 , F xhr1 , F xhl4 , F xhr3 These are the longitudinal forces exerted by the left and right rails on the left and right wheels, respectively, in N; F zhl2 , F zhr1 , F zhl4 , F zhr3 These are the vertical forces exerted by the left and right rails on the left and right wheels, respectively, in N; F yhl2 , F yhr1 , F yhl4 , F yhr3 These are the lateral forces exerted by the left and right rails on the left and right wheels, respectively, in N; d 2 is half the lateral distance between the left and right application points of the type ② force, in meters; the type ① force is defined as the force exerted by the conveyor belt on the arc surface of the trailer, and the type ② force is defined as the force exerted by the trailer frame on the left and right wheel axles, with application points of type ① and type ② respectively; F x1L1 , F y1L1 , F z1L1 , F x1R1 , F y1R1 , F z1R1 The force is classified as type ①, and it is assumed to act on the same vertical plane as type ② forces. Therefore, the transverse distances to the center of mass are equal, i.e., the lever arms are equal. ; Rc Let the radius of curvature, in meters, be the radius of curvature corresponding to the center of gravity of the trolley on the track. r 0 represents the nominal rolling radius, in meters. F zb The vertical force exerted by the conveyor belt on the middle of the trailer is expressed in N. φ sec The superelevation angle corresponding to the center of the trolley on the track is given in rad. F x1 , F x3 For a type ① action acting on the wheel, N; h 01 , h 03 These are the vertical distances, in meters, from the left and right sides, type ①, to the center of gravity of the motorcycle. h 2 represents the vertical distance from the point of action (type ②) to the center of mass of the vehicle, in meters (m). h 1 is the vertical distance from the point of action of type ① to the center of mass of the vehicle, m; a0 is half the distance between the contact points of the left and right wheels, m; d L The lateral distance, in meters, is from the point of contact between the left wheel and the rail to the center of gravity of the trolley. d R The lateral distance, in meters, is from the point of contact between the right wheel and the rail to the center of gravity of the trolley. r L The actual contact radius of the left wheel is in meters (m). r R The actual contact radius of the right wheel is in meters (m). I cx Vertical moment of inertia of the motorcycle unit, kg·m 2 ; X c Let be the longitudinal displacement of the trailer unit, in meters (m). M c The mass of the trailer unit is expressed in kg. Y c Let m be the lateral displacement of the trailer unit; Z c Let be the vertical displacement of the trailer unit, in meters (m). The roll angle of the trailer unit is expressed in rad. v The speed of the trolley unit is expressed in m / s. M ywl2 , M ywl4 The lateral component of the spin torque of the left wheel-rail is given in N·m. M ywr1 , M ywr3 The lateral component of the right wheel-rail spin torque is given by N·m. I cy Let be the lateral rotational inertia of the trailer unit, in kg·m2 ; β c The nodal angle of the trailer unit is measured in rad. M zwl2 , M zwl4 Let N be the vertical component of the spin torque of the left wheel-rail system; M zwr1 , M zwr3 For the vertical component of the right wheel-rail spin torque, N·m; I cz Let be the vertical moment of inertia of the trailer unit, in kg·m 2 ; ψ c The yaw angle of the trailer unit is expressed in rad.

[0044] Based on the above scheme, step S4 includes:

[0045] S4-2. Force distribution between the conveyor belt and the curved surface of the trolley frame

[0046] Approximating the curved surface of the trailer as a straight surface intersecting the bottom surface, and establishing geometric relationships, the areas are calculated as follows:

[0047] (4-7)

[0048] In the formula, S The total area of ​​the material is m. 2 B is a constant that is related to the radius of curvature of the wheel-rail contact point along the x and y directions. S 1 represents the area of ​​the material above the left arc surface of the trailer, in meters. 2 ; S 2 represents the area of ​​the material directly above the horizontal section, in meters. 2 ; S 3 represents the area of ​​the material above the right arc surface of the trailer, in meters. 2 ; S a The total area of ​​the material above the left and right curved surfaces of the trailer is in meters. 2 ; l 1 、l 2 、l 3 represents the length of the material on the left trailer's curved surface, the length of the middle horizontal section, and the length of the material on the right trailer's curved surface, respectively, in meters; B 0 represents the width of the conveyor belt carrying the material, in meters (m). λ The inclination angle of the trailer's curved surface is expressed in rad. θ Angle of repose, in rad;

[0049] Based on actual working conditions, the tilt angle of the trailer's arc surface... The conveyor belt is 1.2 m wide, and the material stacking angle is taken as follows. ;

[0050] Given the above material area distribution, and defining type ③ force as the force exerted by the trolley frame on the left and right wheel axles, the magnitude of the vertical component of type ③ force is:

[0051] (4-8)

[0052] In the formula q B For the mass of the conveyor belt line, kg / m; q W Material mass, kg / m;

[0053] Furthermore, the location of the point of application of type ③ forces can be divided into the following three cases:

[0054] a: Unloaded, meaning the only weight on the trailer's curved surface is the conveyor belt. The point of contact between the conveyor belt and the trailer's curved surface is taken as half the vertical height of the conveyor belt, approximately half the vertical height of the trailer's curved surface.

[0055] b: Low load, that is, a small amount of material is only distributed in the horizontal section, and the weight of the conveyor belt is only on the arc surface of the trailer. Then the position of the point of action is the same as that of type a.

[0056] c: Multiple loads, meaning the material has already covered the upper part of the trailer's arc surface, then the point of contact between the conveyor belt and the trailer's arc surface is taken as half the vertical height of the trailer's arc surface covering the material;

[0057] For case c, the height of the stress points on the left and right arc surfaces of the trailer frame is defined as follows: h 01 , h 03 Then it means the following:

[0058] (4-9)

[0059] Under full load conditions, the height of the force-bearing point can be approximated using an equation; based on geometric relationships, the magnitude of the lever arm of type ③ force can be derived; the lateral distance from the point of application to the center of mass of the trolley frame... d 01 , d 03 They are respectively:

[0060] (4-10)

[0061] The lever arm of the other three types of forces is a fixed value, determined by the structure of the trailer frame;

[0062] S4-4. Perform stress analysis and verification on the motorcycle unit in the turning section.

[0063] Analysis of the centripetal force on the turning balance of the trailer unit:

[0064] The horizontal centripetal force of the conveyor belt is

[0065] (4-15)

[0066] In the formula, F zL The tension of the conveyor belt at the trailer unit, in N; l c The distance between the trailers is in meters (m). R c Let m be the corresponding radius of curvature. α c The turning angle corresponding to one trailer unit is expressed in rad.

[0067] The centripetal force of the conveyor belt on the curved surface of the trolley frame is

[0068] (4-16)

[0069] Based on geometric relationships, the normal force on the curved surface of the trailer frame...

[0070] (4-17)

[0071] Tangential force on the curved surface of the trailer frame

[0072] (4-18)

[0073] The total tangential force on the arc surface of the motorcycle

[0074] (4-19)

[0075] In the formula, F y The total centripetal force caused by the conveyor belt in the turning section, in N; F y1 , F y2 , F y3 The centripetal forces, in N, are generated by the conveyor belt on the left, middle, and right curved surfaces of the trolley, respectively. F n1 , F n2 , F n3 These are the normal forces exerted by the conveyor belt on the left, middle, and right curved surfaces of the trailer, respectively, in N; F m The total tangential force on the trailer's curved surface is expressed in N. F m1 , F m2 , F m3 The values ​​are the tangential forces (N) on the left, middle, and right arc surfaces of the trailer, respectively.

[0076] Due to the raised inner rail curve, under the weight of the conveyor belt and the material, as well as the centripetal force of the conveyor belt, the tangential friction between the conveyor belt and the trailer's curved surface is:

[0077] (4-20)

[0078] In the formula, F ch The frictional force, in N, is the tangential force between the conveyor belt and the trailer's curved surface. µ 1. µ 2. µ 3 represents the coefficient of friction between the conveyor belt and the left, middle, and right arc surfaces of the trailer, respectively. ρ The density of the material is kg / m³. 3 ;

[0079] The downward force of the conveyor belt along the tangential direction of the trailer's arc surface is

[0080] (4-21)

[0081] In the formula, F mB The downward force of the conveyor belt along the tangential direction of the trailer's arc surface, in N;

[0082] The downward force of the material along the tangential direction of the trailer's arc surface is

[0083] (4-22)

[0084] In the formula, F mw The sliding force of the material along the tangential direction of the trailer's arc surface, in N;

[0085] Therefore, the sum of the tangential frictional force and the downward force between the conveyor belt and the trailer's curved surface should not be less than the total axial force exerted on the trailer's curved surface due to the conveyor belt tension, i.e.

[0086] (4-23).

[0087] Preferably, step S5 includes:

[0088] Based on Hamilton's principle, the vertical motion equation of the trailer unit can be analyzed to determine...

[0089] (5-4)

[0090] (5-5)

[0091] (5-6)

[0092] Analysis of the rolling motion equations of the motorcycle unit shows that

[0093] (5-7)

[0094] In the formula, m w3 , m w4 , m c2 The masses of the left wheel, right wheel, and vehicle body are respectively, in kg; y w3 , y w4 , y c2 These are the lateral displacements of the front wheels, rear wheels, and vehicle body, respectively, in meters. z w3 , z w4 , z c2 These represent the vertical displacements of the front wheels, rear wheels, and vehicle body, respectively, in meters (m). l 03 The distance between the centers of gravity of the left and right wheels is in meters (m).

[0095] Define the mass of the i-th motor vehicle unit 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 Treating the conveyor belt as a linear system, the linear second-order differential equation in matrix form is as follows:

[0096] (5-8)

[0097] In the formula, M is the mass matrix; C is the damping matrix; K is the stiffness matrix; and all of them are... Array; Let F be the system's acceleration, velocity, and displacement matrices; and let F be the force matrix acting on each element. Column vectors;

[0098] Rearranging the above equations into matrix form of linear second-order differential equations yields the following result:

[0099] (5-9)

[0100] (5-10)

[0101] (5-11)

[0102] (5-12)

[0103] (5-13).

[0104] Preferably, step S6 uses MATLAB software to compile the calculation process, including the following steps:

[0105] S6-1. Input the dynamic equations of the trailer unit;

[0106] S6-2. Input the dynamic parameters of the trailer unit determined in step S4;

[0107] S6-3. Establish the equations for the lateral, vertical, and rolling motion of the trailer unit;

[0108] S6-4. Set up the trolley track excitation model;

[0109] S6-5. Introducing the trace method;

[0110] S6-6. Determine the wheel-rail rigid contact point;

[0111] S6-7. Elastically corrected contact points;

[0112] S6-8. Output relevant contact parameters;

[0113] S6-9. Generate the trailer unit matrix and external force matrix;

[0114] S6-10. Introduce a numerical integration algorithm to solve the equations of motion;

[0115] S6-11. Select the time step and determine the convergence of the algorithm;

[0116] S6-12. Calculate the vibration characteristics of the left and right wheelsets;

[0117] S6-13. Use the result of the previous time step as the initial iteration value to calculate the vibration state;

[0118] S6-14. Determine whether the actual calculation time t is greater than the planned calculation time T. If t>T is satisfied, continue to step S6-15. If t>T is not satisfied, return to step S6-12 to continue the calculation.

[0119] S6-15. Solve the dynamic equations of the wheelset and the chassis and output the dynamic response results.

[0120] Preferably, it also includes three-dimensional dynamic simulation verification, which includes the following steps:

[0121] S7-1. Create a 3D model of the track-type belt conveyor and import it into the simulation software;

[0122] S7-2. Modal analysis of the trailer frame and wheels;

[0123] S7-3. Output the three-dimensional vibration acceleration changes of the front and rear wheels on one side when running at different belt speeds on the turning section of the trailer track;

[0124] S7-4. Output the triaxial vibration acceleration changes of the front and rear wheels on one side when the spacing between the trailer units is different on the trailer track in the turning section;

[0125] S7-5. Output the three-dimensional vibration acceleration changes of the front and rear wheels on one side when the inner rail of the trailer is raised at different angles during the turning section.

[0126] The beneficial effects of this invention are as follows: Based on the multi-degree-of-freedom multibody dynamics theory, a nonlinear dynamic equation for the track-type belt conveyor system is constructed, focusing on the operating characteristics of the conveyor. The six-degree-of-freedom motion equations of the trailer unit are analyzed to obtain the motion and orientation relationships of key structures. For the vertical motion of the left and right wheel sets of the trailer, the vibration equations of a half-trailer model are constructed using Hamilton's principle. Using MATLAB software, the Newmark-β integral method is employed to solve the global domain system motion equations under different influencing factors (belt speed, trailer spacing, and inner rail elevation angle) during transportation in the spatial turning section, thereby analyzing the coupled vibration characteristics between the trailer and the track. The MATLAB numerical simulation results are verified using three-dimensional simulation to ensure the accuracy of the simulation results. Attached Figure Description

[0127] Figure 1 : A schematic diagram of the track-type belt conveyor structure to which this invention applies;

[0128] Figure 2 : Schematic diagram of the trailer unit structure of the present invention;

[0129] Figure 3 The present invention includes a front view of the dynamic model and degree-of-freedom distribution of the motorcycle unit.

[0130] Figure 4 Side view of the dynamic model and degree-of-freedom distribution of the vehicle unit of this invention;

[0131] Figure 5 Top view of the dynamic model and degree-of-freedom distribution of the vehicle unit in this invention;

[0132] Figure 6 : Spatial coupling force analysis diagram of the trailer unit of this invention;

[0133] Figure 7 : A schematic cross-sectional view of the track-type belt conveyor in its carrying state according to the present invention;

[0134] Figure 8 : Schematic diagram of the centripetal force balancing during turning of the trailer unit of this invention;

[0135] Figure 9 The present invention provides a dynamic model of the lateral segmentation of the trailer unit.

[0136] Figure 10 : MATLAB compilation flowchart of this invention;

[0137] Figure 11 : Normal force distribution diagram within the contact patch of this invention;

[0138] Figure 12 : Plastic deformation diagram of the elastic-plastic contact region of this invention;

[0139] Figure 13 : Relationship between maximum contact pressure and elastic deformation in this invention. Detailed Implementation

[0140] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0141] A simulation method for the dynamic characteristics of the turning section of a track-type belt conveyor, such as... Figure 1 As shown, the belt conveyor mainly consists of a conveyor belt 1, rotating wheels at the head and tail, trolleys, and trolley tracks 2. The trolleys are connected in series by steel wire ropes, and the tracks employ a double-layer design with reversing devices at the head and tail. When a trolley reaches the rotating wheel at the head or tail, the conveyor belt 1 separates from the trolley, and the trolley enters the rotating wheel and changes tracks. The trolley uses a U-shaped belt 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 trolley tracks result in low frictional resistance during trolley operation. Furthermore, the smooth U-shaped structure of the trolley body not only increases the contact area between the conveyor belt and the trolley but also effectively reduces the local pressure of the material on the conveyor belt. During operation, the trolleys and the conveyor belt above them remain relatively stationary. The numerous trolleys arranged in the conveyor's running section to assist in supporting the conveyor belt help eliminate indentation resistance and reduce the stringent requirements for conveyor belt sag.

[0142] like Figure 2 As shown, a trolley unit is defined as including 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 in the turning section of the trolley track, and the inner rail is raised. The 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 in space perpendicular to both the vertical and longitudinal directions is defined as the transverse direction. 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.

[0143] The simulation method for the dynamic characteristics of the turning section of a belt conveyor includes the following steps:

[0144] 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.

[0145] S2. Establish a coordinate system, which includes a global coordinate system, a vehicle body coordinate system, and a contact coordinate system. The global coordinate system is fixed to the ground, the vehicle body coordinate system is fixed to the trailer unit, and the origin of the contact coordinate system is defined at the contact center between each wheel and the trailer track. Establish the attitude angle of the wheel in the vehicle body coordinate system and establish the relative transformation relationship between the vehicle body coordinate system and the global coordinate system.

[0146] Specifically, a global coordinate system O-XYZ is defined, with its origin O coinciding with the center of mass of the trailer unit at its initial position; the X-axis is along the initial longitudinal direction of travel of the trailer frame, with forward being positive; the Z-axis is in the same direction as gravity, with downward being positive; and the Y-axis is perpendicular to the XOZ plane and points to the right along the X direction.

[0147] Define the vehicle coordinate system O C -X C Y C Z C Its origin O C At the center of gravity of the trailer frame, X C Pointing to the front of the trailer frame, Y C Pointing to the right side of the trailer frame, Z C Pointing towards the bottom of the trailer frame;

[0148] Define contact coordinate system O W -X W Y W Z W Its origin O W At the contact center of each wheel, X W The axis points in front of the contact surface between the wheel and the trailer track, Y W Pointing to the right side of the contact surface, Z W Pointing downwards from the contact surface;

[0149] If the wheel's yaw angle and roll angle are in the same direction and equal in magnitude to the trailer's yaw angle and roll angle, then the wheel attitude angles include: wheel yaw angle ψ w Side roll angle φ w And the rotation angle σ of the wheel about its center of mass w ;

[0150] 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. C 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 ,

[0151] The direction cosine matrices are as follows:

[0152] (2-1)

[0153] (2-2)

[0154] (2-3)

[0155] Based on the global coordinate system O and the vehicle body coordinate system O T The transformation relationship between them, where N represents the global coordinate system O and the vehicle body coordinate system O C The direction cosine matrix, i.e.:

[0156] (2-4).

[0157] S3. Based on d'Alembert's principle and spatial force analysis, such as Figure 6 As shown, the dynamic equations of the motorcycle unit are established;

[0158] When establishing the dynamic equations, assumptions are made regarding structural simplification, load and mechanical properties, motion, friction and resistance, and environmental factors.

[0159] The structural simplification assumptions are as follows: the conveyor belt is regarded as a uniform continuous medium, and the trailer frame, wheels and trailer rails are regarded as rigid bodies; the conveyor belt is defined as an ideal elastic body, and the conveyor belt unit is simplified as an ideal linear spring and an ideal viscous damping unit;

[0160] The load and mechanical assumptions are as follows: the load distribution on the conveyor belt is static and uniform, and the direction of the tension of the steel wire ropes at the front and rear of the trolley frame is always longitudinal.

[0161] The motion assumption is that the motion of the belt conveyor is stable and there are no transient effects during start-up and shutdown.

[0162] The friction and resistance assumptions are as follows: the friction coefficient between the trailer track, the trailer frame, and the wheels is assumed to be uniform; air resistance is ignored.

[0163] The environmental factor assumption is that ambient temperature has no effect on the mechanical properties of the material.

[0164] The dynamic equations of the motorcycle unit include:

[0165] The longitudinal motion equation is

[0166] (3-1)

[0167] The equation of lateral motion is

[0168] (3-2)

[0169] The equation of vertical motion is

[0170] (3-3)

[0171] The equation of motion for rolling is

[0172] (3-4)

[0173] The equation of motion for nodding is

[0174] (3-5)

[0175] The equation for head-shaking motion is:

[0176] (3-6)

[0177] in, F q1 The traction force of the conveyor belt on the trolley, in N; FeTotal resistance of the trolley unit, N; F s1 , F s2 These are the tensions of the front and rear steel wire ropes of the motorcycle, in N; F xhl2 , F xhr1 , F xhl4 , F xhr3 These are the longitudinal forces exerted by the left and right rails on the left and right wheels, respectively, in N; F zhl2 , F zhr1 , F zhl4 , F zhr3 These are the vertical forces exerted by the left and right rails on the left and right wheels, respectively, in N; F yhl2 , F yhr1 , F yhl4 , F yhr3 These are the lateral forces exerted by the left and right rails on the left and right wheels, respectively, in N; d 2 is half the lateral distance between the left and right application points of the type ② force, in meters; the type ① force is defined as the force exerted by the conveyor belt on the arc surface of the trailer, and the type ② force is defined as the force exerted by the trailer frame on the left and right wheel axles, with application points of type ① and type ② respectively; F x1L1 , F y1L1 , F z1L1 , F x1R1 , F y1R1 , F z1R1 The force is classified as type ①, and it is assumed to act on the same vertical plane as type ② forces. Therefore, the transverse distances to the center of mass are equal, i.e., the lever arms are equal. ; R c Let the radius of curvature, in meters, be the radius of curvature corresponding to the center of gravity of the trolley on the track. r 0 represents the nominal rolling radius, in meters. F zb The vertical force exerted by the conveyor belt on the middle of the trailer is expressed in N. φ sec The superelevation angle corresponding to the center of the trolley on the track is given in rad. F x1 , F x3 For a type ① action acting on the wheel, N; h01 , h 03 These are the vertical distances, in meters, from the left and right sides, type ①, to the center of gravity of the motorcycle. h 2 represents the vertical distance from the point of action (type ②) to the center of mass of the vehicle, in meters (m). h 1 is the vertical distance from the point of action of type ① to the center of mass of the vehicle, m; a0 is half the distance between the contact points of the left and right wheels, m; d L The lateral distance, in meters, is from the point of contact between the left wheel and the rail to the center of gravity of the trolley. d R The lateral distance, in meters, is from the point of contact between the right wheel and the rail to the center of gravity of the trolley. r L The actual contact radius of the left wheel is in meters (m). r R The actual contact radius of the right wheel is in meters (m). I cx Vertical moment of inertia of the motorcycle unit, kg·m 2 ; X c Let be the longitudinal displacement of the trailer unit, in meters (m). M c The mass of the trailer unit is expressed in kg. Y c Let m be the lateral displacement of the trailer unit; Z c Let be the vertical displacement of the trailer unit, in meters (m). The roll angle of the trailer unit is expressed in rad. v The speed of the trolley unit is expressed in m / s. M ywl2 , M ywl4 The lateral component of the spin torque of the left wheel-rail is given in N·m. M ywr1 , M ywr3 The lateral component of the right wheel-rail spin torque is given by N·m. I cy Let be the lateral rotational inertia of the trailer unit, in kg·m 2 ; β c The nodal angle of the trailer unit is measured in rad. M zwl2 , M zwl4 Let N be the vertical component of the spin torque of the left wheel-rail system; M zwr1 , M zwr3 For the vertical component of the right wheel-rail spin torque, N·m; I czLet be the vertical moment of inertia of the trailer unit, in kg·m 2 ; ψ c The yaw angle of the trailer unit is expressed in rad.

[0178] S4. Determine the dynamic parameters of the trolley unit: Determine the solution parameters for the conveyor belt, including the linear mass of the material, the linear mass of the trolley, the relationship between the wheel angular velocity and the trolley frame velocity, the moment of inertia of the wheels, and the wheel inertial torque; calculate the force distribution between the conveyor belt and the upper surface arc of the trolley frame when the conveyor belt is carrying material, and calculate the location of the point of application of the force exerted by the conveyor belt on the surface arc of the trolley frame according to the material quantity; perform force analysis and verification for the trolley unit in the turning section; calculate the conveyor belt tension and the resistance of the belt conveyor; and verify the conveyor belt sag condition.

[0179] S4-1. Parameters for solving the conveyor belt problem:

[0180] Based on actual working conditions and referring to the conveyor belt standard GB / T9770, the linear mass of the ST1600 conveyor belt is as follows:

[0181] (4-1)

[0182] Material line quality:

[0183] (4-2)

[0184] Cart cable quality:

[0185] (4-3)

[0186] The relationship between wheel angular velocity and vehicle speed is as follows:

[0187] (4-4)

[0188] Moment of inertia of a wheel:

[0189] (4-5)

[0190] The inertial torque of the wheel:

[0191] (4-6)

[0192] In the formula, Q The design transport capacity of the conveyor is t / h; q B The mass of the conveyor belt is expressed in kg / m. q w Material mass, kg / m; q c The mass of the trailer line is expressed in kg / m. vb The conveyor belt speed is in m / s. ω l2 , ω l4 , ω r1 , ω r3 The angular velocities of wheels 2 (left), 4 (left), 1 (right), and 3 (right) are respectively, in rad / s; v c The speed of the motorcycle is expressed in m / s. I wy2 , I wy4 , I wy1 , I wy3 The lateral moments of inertia of wheels 2 (left), 4 (left), 1 (right), and 3 (right), respectively, in kg·m. 2 ; m w The mass of a single wheel is expressed in kg.

[0193] S4-2. Force distribution between the conveyor belt and the curved surface of the trolley frame

[0194] The curved surface of the trailer is approximated as a straight surface intersecting the bottom surface, and a geometric relationship is established, such as... Figure 7 As shown, the areas are calculated as follows:

[0195] (4-7)

[0196] In the formula, S The total area of ​​the material is m. 2 B is a constant that is related to the radius of curvature of the wheel-rail contact point along the x and y directions. S 1 represents the area of ​​the material above the left arc surface of the trailer, in meters. 2 ; S 2 represents the area of ​​the material directly above the horizontal section, in meters. 2 ; S 3 represents the area of ​​the material above the right arc surface of the trailer, in meters. 2 ; S a The total area of ​​the material above the left and right curved surfaces of the trailer is in meters. 2 ; l 1 、l 2 、l 3 represents the length of the material on the left trailer's curved surface, the length of the middle horizontal section, and the length of the material on the right trailer's curved surface, respectively, in meters; B 0 represents the width of the conveyor belt carrying the material, in meters (m). λ The inclination angle of the trailer's curved surface is expressed in rad. θ Angle of repose, in rad;

[0197] Based on actual working conditions, the tilt angle of the trailer's arc surface... The conveyor belt is 1.2 m wide, and the material stacking angle is taken as follows. ;

[0198] Given the above material area distribution, and defining type ③ force as the force exerted by the three pairs of left and right wheel axles on the trolley frame, then the magnitude of the vertical component of type ③ force is:

[0199] (4-8)

[0200] In the formula q B For the mass of the conveyor belt line, kg / m; q W Material mass, kg / m;

[0201] Furthermore, the location of the point of application of type ③ forces can be divided into the following three cases:

[0202] a: Unloaded, meaning the only weight on the trailer's curved surface is the conveyor belt. The point of contact between the conveyor belt and the trailer's curved surface is taken as half the vertical height of the conveyor belt, approximately half the vertical height of the trailer's curved surface.

[0203] b: Low load, that is, a small amount of material is only distributed in the horizontal section, and the weight of the conveyor belt is only on the arc surface of the trailer. Then the position of the point of action is the same as that of type a.

[0204] c: Multiple loads, meaning the material has already covered the upper part of the trailer's arc surface, then the point of contact between the conveyor belt and the trailer's arc surface is taken as half the vertical height of the trailer's arc surface covering the material;

[0205] For case c, the height of the stress points on the left and right arc surfaces of the trailer frame is defined as follows: h 01 , h 03 Then it means the following:

[0206] (4-9)

[0207] Under full load conditions, the height of the point of force application can be approximated using an equation. Based on geometric relationships, the magnitude of the lever arm of type ③ force can be derived. The lateral distance from the point of application to the center of mass of the trailer frame is also considered. d 01 , d 03 They are respectively:

[0208] (4-10)

[0209] The lever arm of the other three types of forces is a fixed value, determined by the structure of the trailer frame;

[0210] S4-3. Belt Tension Calculation

[0211] Given the tension F of the conveyor belt just entering the characteristic track. zL1 The minimum sag of the conveyor belt is used to check whether the characteristic tension meets the requirements.

[0212] (4-11)

[0213] If the initial belt tension is unknown, then the relationship between the tension just entering the drive roller and the tension just exiting the drive roller is used, i.e.

[0214] (4-12)

[0215] In the formula, F zLr The initial tension upon entering the drum is N; F zLc The tension at the moment of exiting the roller is N; e is the base of the natural logarithm; µ bq The coefficient of friction between the conveyor belt and the drive drum surface is 0.25, given the wet rubber covering with diamond grooves. β q The wrap angle of the conveyor belt on the drive roller, expressed in rad;

[0216] The driving force of the drive unit can be determined from the power of the drive roller.

[0217] (4-13)

[0218] Based on the condition that the conveyor belt does not slip, it can be known that...

[0219] (4-14)

[0220] In the formula, K A This is a spare coefficient for friction. Here, we take 1.5;

[0221] S4-4. Perform stress analysis and verification on the motorcycle unit in the turning section.

[0222] Analysis of the centripetal force on the turning balance of the trailer unit, such as Figure 8 As shown, then

[0223] The horizontal centripetal force of the conveyor belt is

[0224] (4-15)

[0225] In the formula, F zL The tension of the conveyor belt at the trailer unit, in N; l c The distance between the trailers is in meters (m). Rc Let m be the corresponding radius of curvature. α c The turning angle corresponding to one trailer unit is expressed in rad.

[0226] The centripetal force of the conveyor belt on the curved surface of the trolley frame is

[0227] (4-16)

[0228] Based on geometric relationships, the normal force on the curved surface of the trailer frame...

[0229] (4-17)

[0230] Tangential force on the curved surface of the trailer frame

[0231] (4-18)

[0232] The total tangential force on the arc surface of the motorcycle

[0233] (4-19)

[0234] In the formula, F y The total centripetal force caused by the conveyor belt in the turning section, in N; F y1 , F y2 , F y3 The centripetal forces, in N, are generated by the conveyor belt on the left, middle, and right curved surfaces of the trolley, respectively. F n1 , F n2 , F n3 These are the normal forces exerted by the conveyor belt on the left, middle, and right curved surfaces of the trailer, respectively, in N; F m The total tangential force on the trailer's curved surface is expressed in N. F m1 , F m2 , F m3 The values ​​are the tangential forces (N) on the left, middle, and right arc surfaces of the trailer, respectively.

[0235] Due to the raised inner rail curve, under the weight of the conveyor belt and the material, as well as the centripetal force of the conveyor belt, the tangential friction between the conveyor belt and the trailer's curved surface is:

[0236] (4-20)

[0237] In the formula, F chThe frictional force, in N, is the tangential force between the conveyor belt and the trailer's curved surface. µ 1. µ 2. µ 3 represents the coefficient of friction between the conveyor belt and the left, middle, and right arc surfaces of the trailer, respectively. ρ The density of the material is kg / m³. 3 ;

[0238] The downward force of the conveyor belt along the tangential direction of the trailer's arc surface is

[0239] (4-21)

[0240] In the formula, F mB The downward force of the conveyor belt along the tangential direction of the trailer's arc surface, in N;

[0241] The downward force of the material along the tangential direction of the trailer's arc surface is

[0242] (4-22)

[0243] In the formula, F mw The sliding force of the material along the tangential direction of the trailer's arc surface, in N;

[0244] Therefore, the sum of the tangential frictional force and the downward force between the conveyor belt and the trailer's curved surface should not be less than the total axial force exerted on the trailer's curved surface due to the conveyor belt tension, i.e.

[0245] (4-23)

[0246] Then, through calculation, the reasonable range of inner rail elevation angles can be deduced.

[0247] S4-5. Resistance Calculation

[0248] Units are divided according to the distance between the motorcycles. l c With arc length U The relationship between characteristic orbits is used to divide them into U / l c Let there be n feature points, denoted as 1, 2, ..., i. U / l c If it is not an integer, then take another digit; to reduce the workload of calculation, select 3 feature points and study the pose of the vehicle unit at the feature points to obtain the triaxial force of wheel-rail interaction; the 3 feature points are feature point 1, feature point (1+i) / 2, and feature point i; that is, entering the feature track, just exiting the feature track, and the middle position of the feature track;

[0249] The mass of a trailer unit consists of the mass of the conveyor belt line, the mass of the material line, and the mass of the trailer frame and wheels line, namely:

[0250] (4-24)

[0251] In the formula, M G To simplify the total unit mass, use kg;

[0252] The motion resistance of a track-type belt conveyor includes the main resistance along the conveyor belt path. F fz Additional resistance on each component F ff Special resistance F ft And increase resistance F fs The main resistances include wheel-rail rolling friction resistance, conveyor belt bending resistance, and material internal friction resistance, for simplified calculation:

[0253] (4-25)

[0254] In the formula: F fz The main resistance is N; δ q The conveyor belt inclination angle is expressed in degrees (°). f The rolling friction coefficient is 0.03 between the wheel and the rail.

[0255] Additional resistance includes the inertial resistance of the material in the receiving section and the frictional resistance between the material and the conveyor belt; the frictional resistance between the material and the sidewall of the guide chute in the acceleration zone of the receiving section; the bending resistance of the conveyor belt around the rollers; the bearing resistance of the non-drive rollers; and the frictional resistance between the conveyor belt in the receiving section and the sealing skirt of the guide chute. When calculating additional resistance, an additional resistance coefficient is usually multiplied by the main resistance, i.e.:

[0256] (4-26)

[0257] The special resistance is basically the same as that of traditional belt conveyors. It is worth noting that additional bending resistance will be generated in the convex arc section, namely:

[0258] (4-27)

[0259] Increase resistance:

[0260] (4-28)

[0261] Therefore, the total resistance of a single motor unit is:

[0262] (4-29)

[0263] Therefore, the system linear resistance is:

[0264] (4-30)

[0265] The total resistance along the entire turning section is:

[0266] (4-31)

[0267] In the formula, F ff For additional resistance, N; C 1 represents the additional drag coefficient; F fw Add bending resistance, N, to the convex arc segment; F zlq For the tension at the starting point of the convex arc segment, F zlz For the tension at the end of the convex arc segment, α t The central angle corresponding to the convex arc segment; F fw To increase resistance, N; h q To increase the height, m; F e The total resistance of the trailer unit is N; F e0 The system linear resistance is N; F eq Total resistance during the turning segment, in N; L q Total length of the turning section, in meters;

[0268] S4-6. Conveyor Belt Tension Calculation

[0269] According to the "Design and Selection Manual for DTII(A) Type Fixed Belt Conveyor," the conveyor belt sag condition was checked, and the specified relative sag should not exceed 1%. The minimum tension required for the load-bearing section is:

[0270] (4-32)

[0271] The minimum tension on the return segment is:

[0272] (4-33)

[0273] In the formula, F zLcmin , F zLhmin These are the minimum tensions in the carrying section and the return section, respectively, in N; h maxThe maximum allowable sag of the conveyor belt is set to 0.02.

[0274] Based on the actual working conditions on site, the track-type belt conveyor adopts a counterweight tensioning method, then the characteristic point i The conveyor belt tension at this location is:

[0275] (4-34)

[0276] In the formula, F zLi For the first i Conveyor belt tension at each trailer unit, N; F zL（i-1） For the first i-1 Conveyor belt tension at each trailer unit, N; F e（i-1~i） For the first i-1 Unit to i Inter-unit running resistance, N; m （i-1~i） The equivalent mass of the vehicle with the track system is kg; α The average acceleration / deceleration of the conveyor belt, in m / s² 2 .

[0277] S5. Treating the wheelset as an elastic body, the motorcycle unit is symmetrically divided laterally in the vertical direction. The dynamic characteristics of the left and right wheel rails are analyzed for each of the divided halves of the motorcycle unit. Figure 9 As shown; based on Hamilton's principle, the motion equations of the left and right wheel pairs are established, and then the linear second-order differential equations of the motorcycle unit are obtained;

[0278] Based on Hamilton's principle, the analysis of the lateral motion equation of the motorcycle unit shows that...

[0279] (5-1)

[0280] (5-2)

[0281] (5-3)

[0282] Analysis of the vertical motion equations of the trailer unit shows that

[0283] (5-4)

[0284] (5-5)

[0285] (5-6)

[0286] Analysis of the rolling motion equations of the motorcycle unit shows that

[0287] (5-7)

[0288] In the formula, m w3 , m w4 , m c2 The masses of the left wheel, right wheel, and vehicle body are respectively, in kg; y w3 , y w4 , y c2 These are the lateral displacements of the front wheels, rear wheels, and vehicle body, respectively, in meters. z w3 , z w4 , z c2 These represent the vertical displacements of the front wheels, rear wheels, and vehicle body, respectively, in meters (m). l 03 The distance between the centers of gravity of the left and right wheels is in meters (m).

[0289] Define the mass of the i-th motor vehicle unit 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 Treating the conveyor belt as a linear system, the linear second-order differential equation in matrix form is as follows:

[0290] (5-8)

[0291] In the formula, M is the mass matrix; C is the damping matrix; K is the stiffness matrix; and all of them are... Array; Let F be the system's acceleration, velocity, and displacement matrices; and let F be the force matrix acting on each element. Column vectors;

[0292] Rearranging the above equations into matrix form of linear second-order differential equations yields the following result:

[0293] (5-9)

[0294] (5-10)

[0295] (5-11)

[0296] (5-12)

[0297] (5-13).

[0298] S6. Compile a calculation program using numerical calculation software to simulate the wheel-rail vibration characteristics of the trailer unit during its journey on a turning section. Specifically, use MATLAB software to compile the calculation process, such as... Figure 10 As shown, it includes the following steps:

[0299] S6-1. Input the dynamic equations of the trailer unit;

[0300] S6-2. Input the dynamic parameters of the trailer unit determined in step S4;

[0301] S6-3. Establish the equations for the lateral, vertical, and rolling motion of the trailer unit;

[0302] S6-4. Set up the trolley track excitation model;

[0303] S6-5. Introducing the trace method;

[0304] S6-6. Determine the wheel-rail rigid contact point;

[0305] S6-7. Elastically corrected contact points;

[0306] S6-8. Output relevant contact parameters;

[0307] S6-9. Generate the trailer unit matrix and external force matrix, as shown in Equations 5-8 to 5-13;

[0308] S6-10. Introducing numerical integration algorithms to solve the equations of motion; the most efficient solution method for solving linear differential equations with time-varying coefficients, such as dynamic equations, is the step-by-step integration method;

[0309] S6-11. Select the time step and determine the algorithm's convergence; to maintain the algorithm's stability and prevent the high-frequency components of structural vibration from increasing unchecked, which would render the entire integral meaningless, combine Newmark- The characteristics of the integral method, after verification, are generally taken as follows: The algorithm is unconditionally stable. Then, based on the convergence rate requirement, a selection is made. This ensures that the algorithm meets the convergence limit requirements in the table. Based on the definition and conditions of the convergence region, when the time step is sufficiently small, the convergence rate generally meets the requirements. Therefore, the algorithm selected in this paper... , The Newmark algorithm is constructed by dividing the time step into sufficiently small steps;

[0310] S6-12. Calculate the vibration characteristics of the left and right wheelsets;

[0311] S6-13. Use the result of the previous time step as the initial iteration value to calculate the vibration state;

[0312] S6-14. Determine whether the actual calculation time t is greater than the planned calculation time T. If t>T is satisfied, continue to step S6-15. If t>T is not satisfied, return to step S6-12 to continue the calculation.

[0313] S6-15. Solve the dynamic equations of the wheelset and the chassis and output the dynamic response results.

[0314] It also includes three-dimensional dynamic simulation verification, which includes the following steps:

[0315] S7-1. Create a 3D model of the track-type belt conveyor and import it into the simulation software;

[0316] S7-2. Modal analysis of the trailer frame and wheels;

[0317] S7-3. Output the three-dimensional vibration acceleration changes of the front and rear wheels on one side when running at different belt speeds on the turning section of the trailer track;

[0318] S7-4. Output the triaxial vibration acceleration changes of the front and rear wheels on one side when the spacing between the trailer units is different on the trailer track in the turning section;

[0319] S7-5. Output the three-dimensional vibration acceleration changes of the front and rear wheels on one side when the inner rail of the trailer is raised at different angles during the turning section.

[0320] It also includes the theoretical elastoplastic analysis of wheel-rail rolling contact, including the following steps:

[0321] A1. Based on the assumption of an elastic half-space, it is assumed that the surfaces of the two contacting objects are smooth and the curvature within the elliptical contact region is constant. The normal gap between the two contacting bodies can be expressed by a Taylor polynomial. By neglecting higher-order terms, the normal gap within the contact patch is obtained. h for:

[0322] (a-1)

[0323] In the formula, A and B The constant is along the contact point with the trailer rail. x Xianghe y It is related to the radius of curvature in the direction;

[0324] A and B It can be represented as:

[0325] (a-2)

[0326] (a-3)

[0327] 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, °;

[0328] A2. Analyze the normal force distribution in the contact area between the wheel and the trailer track. The normal force distribution in the contact area is as follows: Figure 11 As shown, according to the Boussinesq-Cerruti formula in elasticity, the semi-major axis of the elliptical contact patch can be obtained. a and short half shaft b They are respectively:

[0329] (a-4)

[0330] In the formula, F p The normal contact load is N; m、n It is a constant, and BA / A+B The value is related to the value of , which can be obtained by looking up a table using interpolation.

[0331] Equivalent elastic modulus E for:

[0332] (a-5)

[0333] The normal pressure inside the elliptical contact patch is:

[0334] (a-6)

[0335] The relative deformation of the wheel-rail contact is:

[0336] (a-7)

[0337] 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;p z The normal pressure within the elliptical contact patch is N; ε Let m be the relative deformation of the wheel-rail contact. r f Is with BA / A+B A constant related to the value of ;

[0338] A3. When two ellipsoids are in contact, if the maximum pressure in the contact area does not exceed the material's yield pressure, the contact area undergoes elastic deformation, still satisfying the Hertz contact pressure distribution; if the maximum pressure in the contact area exceeds the yield pressure, plastic deformation will occur in the central region of the contact patch, such as... Figure 12 As shown, the normal pressure in the elastic deformation region and the plastic deformation region is:

[0339] (a-8)

[0340] Assuming the material satisfies the Mises yield criterion, the normal contact force at the initial yield of the two contacting bodies is:

[0341] (a-9)

[0342] In the formula, p y The initial yield pressure is given in MPa. p 0 * The maximum contact pressure is assumed to occur when the entire contact area undergoes only elastic deformation, satisfying the following conditions. N; S t , S s These represent the areas of the elastic deformation region and the plastic deformation region, respectively, in meters. 2 ; Y The yield strength of the material with lower strength in the two contacting bodies, in MPa; ζ The Poisson's ratio of the material;

[0343] The contact pressure is greatest at the center point of the elliptical contact patch, and its maximum contact pressure can be expressed as:

[0344] (a-10)

[0345] when E t =E When this value is 0, it indicates a linear elastic contact condition. According to Hertz contact theory, extending this to the case of an elliptical contact patch, the yield pressure... p y Major axis corresponding to initial yield a y With short axis by The relationship is:

[0346] (a-11)

[0347] Then yield pressure p y and initial yield load F y0 satisfy:

[0348] (a-12)

[0349] Considering the continuity of contact pressure, the following exists in the elastoplastic transition region:

[0350] (a-13)

[0351] From the above formula, the semi-major axis of the plastic zone can be obtained. a p and short half shaft b p They are respectively:

[0352] (a-14)

[0353] Integrating over the contact area yields the contact force as follows:

[0354] (a-15);

[0355] A4. Simplifying the above equation yields...

[0356] (a-16)

[0357] In the formula, a y , b y , respectively, represent the major and minor semi-axis corresponding to the yield pressure and the initial yield, in meters; F y0 The initial yield load is N; a p , b p These are the major and minor semi-axises of the plastic region, respectively, in meters;

[0358] A5. According to Hertz contact theory, the relationship between the maximum contact pressure and elastic deformation in an elastic contact model is as follows:

[0359] (a-17)

[0360] like Figure 13 As shown, normal contact pressure p and They are directly proportional. Assuming the contact between objects follows an elastoplastic linear strengthening relationship, the pressure... p It can be represented as:

[0361] (a-18)

[0362] In the formula, δ Let m be the elastic deformation. δ y Let m be the elastic critical deformation. k The enhancement coefficient;

[0363] When the strengthening coefficient When , it represents an ideal elastic-plastic material model, that is, when the maximum deformation in the contact area is greater than the critical deformation. δ y At that time, the maximum pressure in the contact area remained at [time value]. p y When the strengthening coefficient When , it represents a bilinear strengthening material model;

[0364] Then elastic-plastic displacement δ for

[0365] (a-19).

[0366] It also includes wheel-rail contact dynamics simulation analysis, including the following steps:

[0367] 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 rail support structure, and establish a Cartesian coordinate system O-XYZ, where the origin O is the initial wheel-rail contact point, and X, Y and Z represent the longitudinal direction (wheel-rail rolling direction), the transverse direction and the vertical direction, respectively.

[0368] B2. Set the model size parameters and material parameters, and set the simulation experience stages, including the initial stress stage and the steady-state rolling contact stage, wherein the steady-state rolling contact stage is the contact stage under stress equilibrium;

[0369] B3. Output the transient stress and deformation of the trailer under different load conditions, and analyze the location of stress concentration;

[0370] B4. Output the outline diagram and stress distribution diagram of the wheel-rail contact area when moving at different speeds under different load weights.

[0371] The present invention has been described above by way of example, but the present invention is not limited to the specific embodiments described above. Any modifications or variations made based on the present invention shall fall within the scope of protection claimed by the present invention.

Claims

1. A method of simulating the dynamic behaviour of a curved section of an orbital belt conveyor, characterized in that, The definition of the trolley unit comprises a trolley frame, four wheels rotationally connected with the trolley frame, and a conveying belt unit supported on the trolley frame, the trolley unit moves on a trolley track turn section, and the inner track is raised; the direction perpendicular to the conveying belt surface 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 the 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 belt conveyor turn section dynamics characteristic simulation method comprises the following steps: S1. Force analysis is performed on the trolley frame, each wheel and the conveying 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. The dynamics equation of the trolley unit is established according to the D'Alembert principle and spatial force analysis; S4. The dynamics parameters of the trolley unit are determined: the conveying belt solving parameters 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 conveying belt and the surface arc of the trolley frame is calculated when the conveying belt carries the material, the position of the acting point of the force of the conveying belt on the surface arc of the trolley frame is calculated according to the material quantity classification; the force analysis and checking of the trolley unit on the turn section are performed; the conveying belt tension and the belt conveyor resistance are calculated; and the sag condition of the conveying belt is checked; S5. The wheel pair is regarded as an elastic body, the trolley unit is symmetrically divided along the transverse direction in the vertical direction, the dynamic characteristics of the left and right wheel tracks of the divided half trolley unit are analyzed; the motion equation of the left and right wheel pairs is 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 running process in the turn section are simulated.

2. The belt conveyor turn section dynamics characteristic simulation method according to claim 1, wherein S2. comprises: defining the general coordinate system O-XYZ, the origin O of which 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 positive in the forward direction; the Z-axis is the same as the direction of gravity, and positive in the downward direction; 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 dolly unit has six degrees of freedom, three of which are used to describe the position of the dolly frame's center of mass relative to the global coordinate system, i.e. X, Y, Z coordinates, and the other three are used to describe the attitude of the dolly frame, i.e. yaw angle ψ, pitch angle β and roll angle φ, also known as Euler angle coordinates; any attitude of the dolly coordinate system can be obtained by three sequential rotations, i.e. rotating the dolly coordinate system O C from the global coordinate system O around its Z C axis by an angle ψ to obtain the reference system O C 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 dolly coordinate system O C , O C X C Y C indicates the reference system O C 2 , O C X C Y C indicates the reference system O C 1 , O C X C Y C indicates the final dolly coordinate system O 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, characterized in that, in step S3, the dynamics equation is established by performing structure simplification assumption, load and mechanics assumption, motion assumption, friction and resistance assumption and environmental factor assumption; the structure simplification assumption is that the conveying belt is regarded as a uniform continuous medium, the trolley frame, the wheel and the trolley track are regarded as rigid bodies; the conveying belt is defined as an ideal elastic body, and the conveying belt unit is simplified as an ideal linear spring and an ideal viscous damping unit. The load and mechanical assumption is that the load distribution on the conveyor belt is static and uniform, and the direction of the steel wire rope tension on the front and rear of the trolley frame is always along the longitudinal direction; The motion assumption is that the motion of the belt conveyor is stable, and there is no transient effect during starting and stopping; The friction and resistance assumption is that the uniform friction coefficient is assumed between the trolley track and the trolley frame and the wheel; and the air resistance is ignored; The environmental factor assumption is that the environmental temperature has no effect on the mechanical properties of the material.

4. The method according to claim 3, characterized in that, The trolley unit dynamics equation in step S3 includes: The longitudinal motion equation is （3-1） The lateral 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） in, F q1 The traction force of the conveyor belt on the trolley, in N; Fe Total resistance of the trolley unit, N; F s1 , F s2 These are the tensions of the front and rear steel wire ropes of the motorcycle, in N; F xhl2 , F xhr1 , F xhl4 , F xhr3 These are the longitudinal forces exerted by the left and right rails on the left and right wheels, respectively, in N; F zhl2 , F zhr1 , F zhl4 , F zhr3 These are the vertical forces exerted by the left and right rails on the left and right wheels, respectively, in N; F yhl2 , F yhr1 , F yhl4 , F yhr3 These are the lateral forces exerted by the left and right rails on the left and right wheels, respectively, in N; d 2 is half the lateral distance between the left and right application points of the type ② force, in meters; the type ① force is defined as the force exerted by the conveyor belt on the arc surface of the trailer, and the type ② force is defined as the force exerted by the trailer frame on the left and right wheel axles, with application points of type ① and type ② respectively; F x1L1 , F y1L1 , F z1L1 , F x1R1 , F y1R1 , F z1R1 The force is classified as type ①, and it is assumed to act on the same vertical plane as type ② forces. Therefore, the transverse distances to the center of mass are equal, i.e., the lever arms are equal. ; R c Let the radius of curvature, in meters, be the radius of curvature corresponding to the center of gravity of the trolley on the track. r 0 represents the nominal rolling radius, in meters. F zb The vertical force exerted by the conveyor belt on the middle of the trailer is expressed in N. φ sec The superelevation angle corresponding to the center of the trolley on the track is given in rad. F x1 、 F x3 N is the ① type action on the wheel; h 01 、 h 03 L1 and L2 are the vertical distances from the left and right ① type action points to the center of mass of the trolley, respectively, m; h 2 is the vertical distance from the ② type action point to the center of mass of the trolley, m; h 1 is the vertical distance from the ① type action point to the center of mass of the trolley, m; a0 is half of the distance between the left and right wheel contact points, m; d L L is the lateral distance from the left wheel and track contact action point to the center of mass of the trolley, m; d R R is the lateral distance from the right wheel and track contact action point to the center of mass of the trolley, m; r L R is the actual contact radius of the left wheel, m; r R R is the actual contact radius of the right wheel, m; I cx Izz is the vertical rotational inertia of the trolley unit, kg·m 2 ; X c X is the longitudinal displacement of the trolley unit, m; M c M is the mass of the trolley unit, kg; Y c Y is the lateral displacement of the trolley unit, m; Z c Z is the vertical displacement of the trolley unit, m; is the roll angle of the trolley unit, rad; v V is the running speed of the trolley unit, m / s; M ywl2 、 M ywl4 Mx is the lateral component of the left wheel track spin moment, N·m; M ywr1 、 M ywr3 Mx is the lateral component of the right wheel track spin moment, N·m; I cy Iyy is the lateral rotational inertia of the trolley unit, kg·m 2 ; β c θ is the nodding angle of the trolley unit, rad; M zwl2 、 M zwl4 My is the vertical component of the left wheel track spin moment, N·m; M zwr1 、 M zwr3 is the vertical component of the right wheel track spin moment, N-m; I cz is the vertical rotational inertia of the bogie unit, kg-m 2 ; ψ c is the swing angle of the bogie unit, rad.

5. The method according to claim 4, characterized in that, The step S4 includes: S4-2. Force system distribution between the conveyor belt and the trolley frame camber The trolley camber is approximated as a straight surface intersecting with the bottom surface, the geometric relationship is established, and each area is calculated as follows: (4-7) In the formula, S A is the total area of the material, m 2 ; B is a constant, related to the radius of curvature of the wheel-rail contact point in the x and y directions; S 1 is the material area above the left arc surface of the trolley, m 2 ; S 2 is the material area directly above the horizontal section, m 2 ; S 3 is the material area above the right arc surface of the trolley, m 2 ; S a The total material area above the left and right arc surfaces of the trolley is m 2 ; l 1 、l 2 、l 3 are the lengths of the left trolley arc surface material, the middle horizontal section, and the right trolley arc surface material, respectively, m; B 0 is the width of the conveyor belt carrying the material, m; λ is the inclination angle of the trolley arc surface, rad; θ is the material accumulation angle, rad; According to the actual working condition, the arc surface inclination angle of the trolley , the conveying belt width is 1.2 m, and the material accumulation angle is ; Given the above material area distribution, define the ③ type force as the force of the trolley frame 3 on the left and right wheel shafts, then the vertical component of the ③ type force is: (4-8) where q B is the mass of the conveyor belt, kg / m; q W is the mass of the material, kg / m; And the position of the ③ type force point is divided into the following three cases: a: empty load, that is, 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 one half of the vertical height of the conveyor belt, which is approximately one half of the vertical height of the trolley camber; b: less load, that is, a small amount of material is only distributed on the horizontal section, and only the weight of the conveyor belt on the trolley camber, then the action point position is the same as the a type case; c: more load, that is, the material has covered the trolley camber, then the action point of the conveyor belt and the trolley camber is taken from one half of the vertical height of the trolley camber covered by the material; For the case of c, the height of the force point of the left and right camber of the dolly frame is defined as h 01 、 h 03 , which is represented as follows: (4-9) The height of the stress point under the full load condition of the conveying belt can adopt an approximate equation. According to the geometric relationship, the size of the force arm of the ③ type force can be derived; the transverse distance from the action point to the center of mass of the trolley frame d 01 、 d 03 are respectively: (4-10) Other ③ type force arm sizes are fixed values determined by the trolley frame structure; S4-4. Force analysis and checking of the trolley unit on the turning section The turning balance centripetal force of the trolley unit is analyzed, then The horizontal centripetal force of the conveyor belt is （4-15） wherein F zL N is the belt tension at the bogie unit, N; l c L is the distance between the bogie units, m; R c R is the corresponding radius of curvature, m; α c is the corresponding turning angle of a bogie unit, rad; Then the centripetal force of the conveyor belt on the trolley frame camber is （4-16） According to the geometric relationship, the normal force of the trolley frame camber is （4-17） The tangential force of the trolley frame camber is （4-18） Then the total tangential force of the trolley camber is （4-19） In the formula, F y Total centripetal force caused by the turning section conveyor belt, N; F y1 , F y2 , F y3 Centripetal force caused by the conveyor belt on the left side arc surface, the middle section, and the right side arc surface of the trolley, respectively, N; F n1 , F n2 , F n3 Normal pressure of the conveyor belt on the left side arc surface, the middle section, and the right side arc surface of the trolley, respectively, N; F m Total tangential force of the trolley arc surface, N; F m1 , F m2 , F m3 Tangential force on the left side arc surface, the middle section, and the right side arc surface of the trolley, respectively, N; Due to the curve of the inner rail being raised, under the gravity of the conveyor belt and the material and the centripetal force of the conveyor belt, the tangential friction force between the conveyor belt and the trolley camber is （4-20） In the formula, F ch N is the friction force between the conveyor belt and the arc tangent of the trolley; µ 1, µ 2, µ 3 are the friction coefficients of the left side arc, the middle section, and the right side arc of the conveyor belt and the trolley, respectively. The sliding force of the conveyor belt along the tangential direction of the trolley camber is （4-21） In the formula, F mB N is the tangential downforce of the conveyor belt along the arc surface of the bogie. The sliding force of the material along the tangential direction of the trolley camber is （4-22） In the formula, F mw Fg is the tangential force of the material along the arc surface of the bogie, N. Therefore, the sum of the tangential friction force and the sliding force between the conveyor belt and the trolley camber should not be less than the total axial force of the conveyor belt acting on the trolley camber due to the tension, that is, （4-23）。 6. The method according to claim 4, characterized in that, The step S5 includes: According to the Hamilton principle, the vertical motion equation of the trolley unit is analyzed （5-4） （5-5） （5-6） The roll motion equation of the trolley unit is analyzed （5-7） wherein, m w3 , m w4 , m c2 are the mass of the left wheel, right wheel and vehicle body, kg, respectively; y w3 , y w4 , y c2 are the lateral displacement of the front wheel, rear wheel and vehicle body, m, respectively; z w3 , z w4 , z c2 are the vertical displacement of the front wheel, rear wheel and vehicle body, m, respectively; l 03 is the distance between the mass centers of the left and right wheels, m. 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 （5-8） 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, which is （5-9） （5-10） （5-11） （5-12） （5-13）。 7. The method according to claim 1, characterized in that, The step S6 uses MATLAB software to compile the calculation process, including the following steps, S6-1. Input the trolley unit dynamics equation; S6-2. Input the trolley unit dynamics parameters determined in step S4; S6-3. Establish the lateral, vertical and roll motion equations of the trolley unit; S6-4. Set the trolley track excitation model; S6-5. Introduce the trace method; S6-6. Determine the rigid contact point of the wheel and rail; S6-7. Elastic correction of the contact point; S6-8. Output the related contact parameters; S6-9. Generating the bogie unit matrix and the external force matrix; S6-10. Introducing the numerical integration algorithm to solve the motion equation; S6-11. Selecting the time step and judging the convergence of the algorithm; S6-12. Calculating the vibration characteristics of the left and right wheelsets; S6-13. Taking the results of the previous time step as the initial iteration value to calculate the vibration state; S6-14. Judging whether the actual calculation time t is greater than the planned calculation time T, if t > T, continue to step S6-15, if t < T, return to step S6-12 for calculation; S6-15. Solving the wheelset dynamics equation and the bogie frame dynamics equation and outputting the dynamic response results.

8. The method according to claim 1, characterized in that, It also includes three-dimensional dynamic simulation verification, which includes the following steps: S7-1. Establishing a three-dimensional model of the rail-mounted belt conveyor and importing it into the simulation software; S7-2. Modal analysis of the bogie frame and wheels; S7-3. Outputting the three-directional vibration acceleration changes of the single-side front and rear wheels when running at different belt speeds on the turning section bogie track; S7-4. Outputting the three-directional vibration acceleration changes of the single-side front and rear wheels when the spacing between the bogie units is different on the turning section bogie track; S7-5. Outputting the three-directional vibration acceleration changes of the single-side front and rear wheels when the track lifting angle is different in the turning section bogie track.

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