Steel wire rope dynamics modeling method based on multi-body dynamics

Abstract

The invention provides a steel wire rope dynamics modeling method based on multi-body dynamics in order to solve the problems that an existing steel wire rope dynamics modeling method is time-consuming in calculation, difficult to simulate the bending wave propagation effect of a steel wire rope in a high-energy impact scene and difficult to meet the requirement for high-precision and high-efficiency solving of stress characteristics of the steel wire rope under complex working conditions. Firstly, a global coordinate system is established, and enough cylindrical rigid body units and local coordinate systems fixedly connected to the centroids of the cylindrical rigid body units are established to describe the whole steel wire rope; secondly, a motion equation of a steel wire rope rigid body unit multi-body system is given; then establishing a flexible connection equation between adjacent rigid body units; and finally, establishing a three-dimensional contact force equation of the rigid body unit and other objects so as to establish a dynamic model of the steel wire rope under the high-energy impact. The dynamic model can effectively simulate the propagation process of bending waves in the steel wire rope under high-energy impact, and the dynamic response of the steel wire rope is accurately solved.

Description

Technical Field

[0001] The invention belongs to the field related to steel wire rope dynamics modeling and relates to a steel wire rope dynamics modeling method. Background Art

[0002] Wire ropes are ropes made of multiple twisted steel wires. They possess high axial strength and stiffness, as well as bending flexibility. They are widely used in lifting, rockfall protection, and aircraft arrestment. Unlike typical wire rope applications, wire ropes can experience bending wave propagation under high-energy impact. At the moment of high-energy impact, the wire rope undergoes significant local deformation. This deformation propagates through the rope as a stress wave, causing the rope to exhibit a wave-like bending pattern, starting from the impact point and propagating toward both ends. As the bending wave propagates along the rope, it encounters impedance changes in the rope's internal structure and external contacts, generating new stress waves that propagate along the rope axis in the opposite direction toward the impact point. The superposition of these reflected stress waves can lead to significant local stress in the rope. Therefore, establishing an accurate dynamic model of wire ropes under high-energy impact is crucial for studying their dynamic characteristics and optimizing their structure.

[0003] The paper "Li L, Peng Y, Wang Y, et al. Modeling the Transient Dynamics of Arresting Hooks and Cables through the Parameter Inversion Method[J]. Aerospace, 2023, 11(1): 20." uses the finite element method to establish a wire rope simulation model. The experimental data are used to perform parameter inversion to obtain the external elastic parameters under different tensions. Substituting these parameters back into the wire rope simulation model can obtain a relatively accurate wire rope dynamic model. However, this method requires fine mesh division, is time-consuming to calculate, and has difficulty simulating the bending wave propagation effect of the wire rope under high-energy impact. As a result, it is difficult to meet the requirements of high-precision and efficient solution of the force characteristics of the wire rope under complex working conditions, and cannot provide a scientific basis for the design, optimization and safe use of wire ropes under complex working conditions.

[0004] Existing wire rope modeling methods struggle to balance efficiency and accuracy when dealing with high-energy impact scenarios. Therefore, it is necessary to propose a wire rope dynamics modeling method under high-energy impact to efficiently and accurately solve the dynamic response of wire ropes under different high-energy impact conditions, providing a scientific basis for the design, optimization, and safe use of wire ropes. Summary of the Invention

[0005] In order to overcome the technical problems that the existing wire rope dynamics modeling methods are time-consuming to calculate and difficult to simulate the bending wave propagation effect of wire ropes in high-energy impact scenarios, resulting in their inability to meet the requirements of high-precision and efficient solution of wire rope force characteristics under complex working conditions, the present invention proposes a wire rope dynamics modeling method based on multi-body dynamics.

[0006] The technical solution adopted by the present invention to solve the technical problem is:

[0007] A wire rope dynamics modeling method based on multi-body dynamics is special in that the wire rope is discretized into a multi-body system composed of multiple cylindrical rigid body units, and the motion equation of the multi-body system, the flexible connection equation between two adjacent rigid body units in the multi-body system, and the three-dimensional contact force equation between the rigid body unit and other objects are established respectively. The motion equation, flexible connection equation and three-dimensional contact force equation constitute the wire rope dynamics equation.

[0008] Furthermore, the method further includes the steps of compiling the established wire rope dynamics equation into a script that can be run by dynamics analysis software and importing the script into the script to solve the dynamic response of the wire rope under high-energy impact.

[0009] Furthermore, the wire rope is discretized into a multi-body system consisting of n cylindrical rigid body units with a length of ΔL and a diameter of D, where n=L / ΔL, and L is the total length of the wire rope; each rigid body unit has six degrees of freedom.

[0010] Furthermore, the established motion equation of the multi-body system is:

[0011]

[0012]

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019]

[0020] Among them, Q i represents the generalized force vector acting on the mass center of the rigid body element i in the multi-body system when a high-speed object collides with the wire rope; Q xi, Q yi , Q zi is the force component of the center of mass of rigid body element i in the x, y, and z directions in the global coordinate system CSO, Q θxi , Q θyi , Q θzi are the moment components of the rigid body element i about the x, y and z axes in the global coordinate system CSO; q, Q and denote the generalized coordinates, generalized forces and generalized velocities of the multi-body system respectively; q i The generalized coordinate vector representing the center of mass position of rigid body element i; and are the position vector and rotation vector of the center of mass of rigid body unit i respectively; q xi , q yi , q zi is the position coordinate of the center of mass of rigid body unit i in the global coordinate system CSO, x, y, z axis, q ψi , q θi , q φi is the rotation angle of the rigid body element i around the x-axis, y-axis and z-axis in the global coordinate system CSO; and They are point P in the local coordinate system CSO i and the position in the global coordinate system CSO; A=A(θ i ) is from the local coordinate system CSO i The three-dimensional orthogonal rotation transformation matrix transformed to the global coordinate system CSO; E k is the kinetic energy of the multi-body system; M is the mass matrix composed of the masses of all rigid body units; is the constraint force of the equation corresponding to the Lagrange multiplier method; the origin O of the global coordinate system is the center of the circle of any end face of the wire rope, the x-axis is perpendicular to the paper and outward, the y-axis is horizontal and to the right, and the z-axis is determined according to the right-hand rule; the local coordinate system CSO i The coordinate origin O i Coincident with the center of mass of rigid body unit i, x i The axis is perpendicular to the paper and points outward, coincides with the axis of rigid body unit i and points to the direction of rigid body unit i+1. i The positive direction of the y axis i The directions of the axes are determined according to the right-hand rule.

[0021] Furthermore, the flexible connection equation between two adjacent rigid body units in the established multi-body system is:

[0022]

[0023] K = [ k 11 0 0 0 0 0 0 k 22 0 0 0 0 0 0 k 33 0 0 0 0 0 0 k 44 0 0 0 0 0 0 k 55 0 0 0 0 0 0 k 66 ]

[0024]

[0025] C = [ c 11 0 0 0 0 0 0 c 22 0 0 0 0 0 0 c 33 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 55 0 0 0 0 0 0 c 66 ]

[0026] is the generalized force acting on rigid body element i; Q0 is the generalized preload applied to rigid body element i; K and C are the stiffness matrix and damping matrix of the rigid body element respectively; and are the mass center positions of rigid body unit i and rigid body unit i+1 respectively; G and E are the shear modulus and elastic modulus of the wire rope respectively; c aa , a=1,2,…,6 is obtained by experiments or finite element simulation.

[0027] Furthermore, the three-dimensional contact force equation between the rigid body unit and other objects is established as:

[0028]

[0029] F c =F cn n

[0030]

[0031] F cf = F cf τ

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040] Among them, F c F is the contact force on a rigid body unit when other objects collide with it; cn and F cf are the normal impact force and tangential friction force on the rigid body unit respectively; n and τ are the unit normal vector and unit tangent vector on the rigid body unit at the contact detection point with other objects respectively; K cis the contact stiffness coefficient; e is the collision nonlinearity index; δ is the penetration depth at the contact detection point; step(δ) is the contact damping coefficient; R1 and R2 represent the curvature radius of a pair of contact surfaces at the contact point; μ h and E h h are the Poisson's ratio and elastic modulus of the object material; c are m is the maximum contact damping coefficient; m is the maximum penetration depth; μ f is the friction coefficient; μ d and μ s are the dynamic and static friction coefficients, v d and v s are the dynamic and static critical speeds respectively; v τ is the relative sliding velocity of the two contacting objects; for Calculate the step function in the formula; .

[0041] The beneficial effects of the present invention are:

[0042] The present invention first establishes a global coordinate system, and establishes a sufficient number of cylindrical rigid units and local coordinate systems fixed to their center of mass to describe the entire wire rope; secondly, the motion equations describing the multi-body system of the wire rope rigid unit are given; then, the flexible connection equations between adjacent rigid units are established; finally, the three-dimensional contact force equations between the rigid unit and other objects are established, thereby establishing a dynamic model of the wire rope under high-energy impact. This dynamic model can effectively simulate the propagation process of the bending wave in the wire rope under high-energy impact, and can accurately solve the dynamic response of the wire rope. The modeling method of the present invention avoids the problem that the finite element method requires fine grid division and low solution efficiency. The method of the present invention effectively improves the accuracy and solution efficiency of the wire rope dynamic model, helps to study the dynamic characteristics of the wire rope under complex working conditions, and provides support for the design and optimization of the wire rope. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a schematic diagram of the rigid body unit describing the wire rope.

[0044] Figure 2 It is a schematic diagram of the relationship between generalized forces between rigid body elements in the local coordinate system.

[0045] Figure 3 It is a schematic diagram of the contact relationship between rigid body units and other objects.

[0046] Figure 4 This is a schematic diagram of the wire rope being impacted.

[0047] Figure 5It is the geometric model of the wire rope under high energy impact.

[0048] Figure 6 It is the propagation process of wire rope bending wave under high energy impact.

[0049] Figure 7 This is a diagram showing the change in tension in the wire rope under high-energy impact.

[0050] Figure 8 This is a diagram showing the change in contact force between the wire rope rigid body unit and the high-energy object under high-energy impact. DETAILED DESCRIPTION

[0051] The present invention will be further described below in conjunction with the accompanying drawings.

[0052] The wire rope dynamics modeling method based on multi-body dynamics proposed in the present invention specifically includes the following steps:

[0053] Step 1: Establish a global coordinate system;

[0054] In order to describe the relative position of the wire rope in the three-dimensional model space, a global coordinate system CSO is established in the three-dimensional model space. The origin O of the global coordinate system CSO is the center of the circle of any end face of the wire rope, the x-axis is perpendicular to the paper and outward, the y-axis is horizontal and to the right, and the position of the z-axis is determined according to the right-hand rule, as follows: Figure 1 shown.

[0055] Step 2: Discretize the wire rope into a multi-body system consisting of multiple cylindrical rigid body elements and establish the local coordinate system of the rigid body elements;

[0056] The wire rope is considered a multibody system consisting of n cylindrical rigid elements of length ΔL and diameter D, where n = L / ΔL, and L is the total length of the wire rope. Each rigid element has six degrees of freedom: translation along the x, y, and z directions and rotation about the x, y, and z axes.

[0057] According to the relative position of the wire rope in space, the rigid body unit i and the local coordinate system CSO fixed to its center of mass are established in turn. i (i=1,2,…,n), i is the local coordinate system of the i-th rigid body element, local coordinate system CSO i The coordinate origin O i Coincident with the center of mass of rigid body unit i, x i The axis is perpendicular to the paper and points outward, coincides with the axis of rigid body unit i and points to the direction of rigid body unit i+1. i The positive direction of the y axis is determined by the right-hand rule. i The direction of the axis, such as Figure 1 shown.

[0058] Step 3: Establish the motion equation of the wire rope multi-body system;

[0059] In order to describe the spatial motion of the wire rope, the position vector and rotation vector of the center of mass of the rigid body unit i are defined as follows:

[0060] r i = [ q xi , q yi , q zi ] T (3-1)

[0061] θ i = [ q θ xi , q θ yi , q θ zi ] T (3-2)

[0062] Where q xi , q yi , q zi is the position coordinate of the center of mass of rigid body unit i in the global coordinate system CSO, x, y, z axis, q ψi , q θi , q φi is the rotation angle of rigid body element i around the x-axis, y-axis and z-axis in the global coordinate system CSO.

[0063] The center of mass position of rigid body element i is expressed by the generalized coordinate vector q i Expressed as:

[0064] q i = [ r i T , θ i T ] T (3-3)

[0065] The global position of any point P in rigid element i is:

[0066] (3-4)

[0067] in, and They are point P in the local coordinate system CSO i and the position in the global coordinate system CSO; A=A(θ i ) is from the local coordinate system CSO i The three-dimensional orthogonal rotation transformation matrix transformed to the global coordinate system CSO.

[0068] When a high-energy impact object collides with the wire rope, the generalized force vector Q acting on the center of mass of the rigid body element i in the wire rope multi-body system is i Expressed as:

[0069] Q i = [ Q xi , Q yi , Q zi , Q θ xi , Q θ yi , Q θ zi ] T (3-5)

[0070] Where Q xi , Q yi , Q zi is the force component of the center of mass of rigid body element i in the x, y, and z directions in the global coordinate system CSO, Q θxi , Q θyi, Q θzi are the moment components of the rigid body element i about the x, y and z axes in the global coordinate system CSO.

[0071] Taking all rigid body elements as a whole, the wire rope multi-body system is obtained. The generalized coordinates q and generalized forces Q of the wire rope multi-body system are expressed as:

[0072] q = [ q 1 T , q 2 T , ⋅⋅⋅ , q n T ] T (3-6)

[0073] Q = [ Q 1 T , Q 2 T , ⋅⋅⋅ , Q n T ] T (3-7)

[0074] The kinetic energy of the wire rope multi-body system is:

[0075] (3-8)

[0076] Where M is the mass matrix composed of the masses of all rigid body elements, is the generalized velocity of the wire rope multi-body system.

[0077] Considering the constraint equation, the motion equation of the wire rope multi-body system is established using the Lagrangian method with multipliers:

[0078] (3-9)

[0079] Where, is the constraint force of the equation corresponding to the Lagrange multiplier method.

[0080] Step 4: Establish the flexible connection equation between two adjacent rigid body elements in the wire rope;

[0081] The wire rope is a flexible body. In order to accurately simulate the characteristics of the flexible body, it is necessary to establish a flexible connection between two adjacent rigid body units. Figure 2 The relationship between the generalized forces between rigid body elements in the local coordinate system is shown. The generalized force on rigid body element i is It depends on its generalized displacement and generalized velocity relative to the rigid body element i+1, and its expression is:

[0082] (4-1)

[0083] Where Q0 is the generalized preload applied to the rigid element i; K and C are the stiffness matrix and damping matrix of the rigid element, respectively. K and C depend on the structure and material properties of the rigid element. and are the center of mass positions of rigid body unit i and rigid body unit i+1 respectively;

[0084] The expression of the stiffness matrix K is:

[0085] K = [ k 11 0 0 0 0 0 0 k 22 0 0 0 0 0 0 k 33 0 0 0 0 0 0 k 44 0 0 0 0 0 0 k 55 0 0 0 0 0 0 k 66 ] (4-2)

[0086] The deformation of the wire rope is regarded as linear elastic deformation, and the diagonal element value k of the stiffness matrix K is calculated using the superposition principle. jj (j=1,2,…,6) is:

[0087] (4-3)

[0088] Where G is the shear modulus of the wire rope, and E is the elastic modulus of the wire rope.

[0089] The expression of the damping matrix C is:

[0090] C = [ c 11 0 0 0 0 0 0 c 22 0 0 0 0 0 0 c 33 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 55 0 0 0 0 0 0 c 66 ] (4-4)

[0091] The diagonal element values ​​c of the damping matrix C aa (a=1,2,…,6) is generally obtained through experiments or finite element simulation, and its value is usually between 100 and 1000.

[0092] The above equations (4-1) to (4-4) constitute the flexible connection equations between rigid body element i and rigid body element i + 1. For a wire rope multibody system containing n rigid body elements, there are a total of n-1 flexible connection equations, which are established using the same method as above.

[0093] Step 5: Establish the three-dimensional contact force equation between the rigid body unit and other objects;

[0094] When a high-speed object (a high-energy impact object) collides with a wire rope, the object may slide along the rope due to eccentric or non-perpendicular contact. Simultaneously, the rope may also slip against objects such as pulleys. Modeling this contact process involves contact detection and contact force calculation.

[0095] Place a series of contact detection points P inside the wire rope j , assuming that the object being touched is c with a radius of R c Cylinders, such as Figure 3 As shown. Contact detection is performed in the local coordinate system CSO of object c. n+1 The local coordinate system CSO n+1 The origin O is the center of mass of object c, z n+1 The axis coincides with the cylinder axis and is perpendicular to the paper and outward, n+1 The axis is horizontal to the right, and y is determined according to the right-hand rule. n+1 The position of the axis, such as Figure 3The contact force calculation is performed in the global coordinate system CSO to facilitate the subsequent assembly of the wire rope system equations.

[0096] Set any contact detection point P j The position vector of is transformed from the global coordinate system CSO to the local coordinate system CSO of the object c n+1 In:

[0097] (5-1)

[0098] Among them, r c is the position vector of object c in the global coordinate system CSO, A c is the local coordinate system CSO of object c n+1 The three-dimensional orthogonal rotation transformation matrix transformed to the global coordinate system CSO; and They are contact detection points P j In the global coordinate system CSO and the local coordinate system CSO n+1 The position vector in .

[0099] Contact detection point P j The penetration depth δ is expressed as:

[0100] (5-2)

[0101] When a high-speed object collides with a rigid body unit of the wire rope, the rigid body unit is subjected to a contact force F c By the normal impact force F cn and tangential friction F cf Composition, that is

[0102] (5-3)

[0103] The impact force is equivalent to a nonlinear spring damper, and the normal impact force F cn =F cn n, where n is the unit normal vector of the contacting rigid body element.

[0104] Normal impact force F cn The expression is:

[0105] (5-4)

[0106] Where, e is the collision nonlinearity index, which reflects the nonlinearity of the force deformation of the material. Its value can be obtained by consulting the published professional manuals, where the value for steel materials is 1.5; K c is the contact stiffness coefficient, which can be calculated as follows:

[0107] (5-5)

[0108] Where R1 and R2 represent the respective curvature radii of a pair of contact surfaces at the contact point. In this invention, they refer to the respective curvature radii of the high-speed object c and the rigid body unit at the contact surface. σ1 and σ2 are related to the material parameters of the two contacting objects and are expressed as:

[0109] (5-6)

[0110] Where μ h and E h are the Poisson's ratio and elastic modulus of the material of object h; step(δ) is the contact damping coefficient, whose value is given by the following formula:

[0111] (5-7)

[0112] Where c m is the maximum contact damping coefficient, which is usually set to the contact stiffness coefficient K c 0.1-1%; is the maximum penetration depth, and its suitable value is 0.1mm.

[0113] So far, the normal impact force F during the contact process has been obtained cn .

[0114] Tangential friction force F cf = F cf τ, where τ is the unit tangent vector of the rigid body element in contact;

[0115] The tangential friction force F is calculated using the velocity-based friction model. cf , whose values ​​are:

[0116] (5-8)

[0117] Where μ f is the friction coefficient, and its expression is:

[0118] (5-9)

[0119] Where μ d and μ s are the dynamic and static friction coefficients, v d and v s They are dynamic and static critical speeds respectively. These coefficients can be obtained by consulting the manual based on the material properties; v τ is the relative sliding velocity of the two contacting objects;

[0120] The step function f in formula (5-9) is defined as:

[0121] (5-10)

[0122] in, .

[0123] So far, the tangential friction force F during the contact process has been obtained. cf .

[0124] Equations (5-1) to (5-10) constitute the three-dimensional contact force F between the rigid body element i and other objects. c According to the possible contact situations in the actual wire rope system, for example, if the high-speed object c contacts m rigid body units in the wire rope multi-body system, the same method can be used to establish m three-dimensional contact force equations in sequence.

[0125] Step 6: Assemble the dynamic equations of the wire rope system;

[0126] The motion equations (3-1) to (3-9) of the wire rope multi-body system established in step 3, the flexible connection equations (4-1) to (4-4) between two adjacent rigid body units established in step 4, and the three-dimensional contact force equations (5-1) to (5-10) between the rigid body unit and other objects established in step 5 together constitute the wire rope system equation. The wire rope system equation is compiled into a script that can be run by the dynamic analysis software, and then the script is imported into the dynamic analysis software to solve the dynamic response of the wire rope under high-energy impact, that is, the generalized force Q and contact force F of the wire rope under high-energy impact. c and the propagation phenomenon of bending wave propagation.

[0127] Example:

[0128] This embodiment takes the impact of the wire rope perpendicular to the central axis of the wire rope as an example. Figure 4 、 5 As shown in Figure 1, the wire rope is wound around two fixed pulleys and is divided into three sections: the left and right sections are both 10 m long and connected to spring dampers at their ends; the middle section is 20 m long and directly contacts the high-energy impact object c. The wire rope has a diameter of D = 40 mm and an elastic modulus of E = 1.95 × 10 5 MPa, shear modulus G = 0.75 × 10 5 MPa, Poisson's ratio μ=0.3.

[0129] The method for performing dynamic modeling on the steel wire rope in this embodiment is as follows:

[0130] Step 1: Establish a global coordinate system;

[0131] In order to describe the relative position of the wire rope in the three-dimensional space of the model, a global coordinate system CSO is established in the three-dimensional space. The origin O of the global coordinate system CSO is the center of the left wire rope end face, the x-axis is perpendicular to the paper and outward, the y-axis is horizontal and to the right, and the position of the z-axis is determined according to the right-hand rule, as shown in the following example: Figure 1 shown.

[0132] Step 2: Consider the wire rope as a multi-body system consisting of multiple cylindrical rigid body elements and establish the local coordinate system of the rigid body elements;

[0133] Will Figure 4 The steel wire rope shown is regarded as a multi-body system composed of n cylindrical rigid units. Each rigid unit has six degrees of freedom and can perform translational motion along the x, y, and z directions and rotational motion relative to the x, y, and z axes. The length of a single rigid unit ΔL = 50 mm and the diameter D = 40 mm, so the number of rigid units in the steel wire rope multi-body system n = 800.

[0134] According to the relative position of the wire rope in space, the rigid body unit i and the local coordinate system CSO fixed to its center of mass are established in turn. i (i=1,2,…,800). Local coordinate system CSO i The coordinate origin O i Coincident with the center of mass of rigid body unit i, x i The axis is perpendicular to the paper and outward, z i The axis coincides with the axis of rigid body unit i and points to the direction of rigid body unit i+1 as z i The positive direction of the y axis is determined by the right-hand rule. i The position of the axis, such as Figure 1 shown.

[0135] Step 3: Establish the motion equation of the wire rope multi-body system;

[0136] In order to describe the spatial motion of the wire rope, the position vector and rotation vector of the center of mass of the rigid body element i are:

[0137] r i = [ q xi , q yi , q zi ] T (3-1)

[0138] θ i = [ q θ xi , q θ yi , q θ zi ] T (3-2)

[0139] Among them, q xi , q yi , q zi is the position coordinate of the center of mass of rigid body unit i in the global coordinate system CSO, x, y, z axis, q ψi , q θi , q φiis the rotation angle of rigid body element i around the x-axis, y-axis and z-axis in the global coordinate system CSO.

[0140] The position of the center of mass of the rigid body element i is expressed by the generalized coordinate vector q i Expressed as:

[0141] q i = [ r i T , θ i T ] T (3-3)

[0142] The global position of any point P in rigid element i is:

[0143] (3-4)

[0144] in, and They are point P in the local coordinate system CSO i and the position in the global coordinate system CSO; A=A(θ i ) is from the local coordinate system CSO i The three-dimensional orthogonal rotation transformation matrix transformed to the global coordinate system CSO.

[0145] When a high-energy impact object collides with the wire rope, the generalized force vector Q acting on the center of mass of the rigid body element i in the wire rope multi-body system is i Expressed as

[0146] Q i = [ Q xi , Q yi , Q zi , Q ψ i , Q θ i , Q φ i ] T (3-5)

[0147] Where Q xi , Q yi , Q zi is the force component of the center of mass of rigid body element i in the x, y, and z directions in the global coordinate system CSO, Q θxi , Q θyi , Q θzi are the moment components of the rigid body element i about the x, y and z axes in the global coordinate system CSO.

[0148] Taking all rigid body elements as a whole, the generalized coordinates q and generalized forces Q of the wire rope multi-body system are expressed as:

[0149] q = [ q 1 T , q 2 T , ⋅⋅⋅ , q n T ] T (3-6)

[0150] Q = [ Q 1 T , Q 2 T , ⋅⋅⋅ , Q n T ] T (3-7)

[0151] The kinetic energy of the wire rope multi-body system is:

[0152] (3-8)

[0153] Considering the constraint equation, the Lagrangian method with multipliers is used to establish the system motion equation of the wire rope multi-body system:

[0154] (3-9)

[0155] Where, is the constraint force of the equation corresponding to the Lagrange multiplier method.

[0156] Step 4: Establish the flexible connection equation between rigid body elements;

[0157] In order to construct the dynamic model of the wire rope, it is necessary to define the flexible connection equations between the rigid body units. Figure 2 The relationship between the generalized forces between rigid body elements in the local coordinate system is shown. The generalized force on rigid body element i is:

[0158] (4-1)

[0159] Where Q0 is the generalized preload; K and C are the stiffness matrix and damping matrix of the rigid body element, respectively. K and C depend on the structure and material properties of the rigid body element. and are the center of mass positions of rigid body unit i and rigid body unit i+1 respectively;

[0160] The expression of the stiffness matrix K is:

[0161] K = [ k 11 0 0 0 0 0 0 k 22 0 0 0 0 0 0 k 33 0 0 0 0 0 0 k 44 0 0 0 0 0 0 k 55 0 0 0 0 0 0 k 66 ] (4-2)

[0162] Calculation can be obtained in formula (9-2) k jj (j=1,2,…,6) is:

[0163] (4-3)

[0164] Using a vibration table to apply excitation of different frequencies and amplitudes to the wire rope and measure the force-displacement / velocity response, the damping matrix C can be solved as follows:

[0165] C = [ 110 N ⋅ s / m 0 0 0 0 0 0 110 N ⋅ s / m 0 0 0 0 0 0 110 N ⋅ s / m 0 0 0 0 0 0 900 N ⋅ s ⋅ m / rad 0 0 0 0 0 0 900 N ⋅ s ⋅ m / rad 0 0 0 0 0 0 900 N ⋅ s ⋅ m / rad ] (4-4)

[0166] The above equations (4-1) to (4-4) constitute the flexible connection equations (also called constraint equations) between rigid body element i and rigid body element i+1. For the cable multi-body system of this embodiment, there are a total of 799 such independent complete flexible connection equations.

[0167] Step 5: Establish the three-dimensional contact force equation between the rigid body unit and other objects;

[0168] Assume that the high energy impact object c in this embodiment has a radius of R c =50mm cylinder, Poisson's ratio μ c =0.3, elastic modulus E c = 2×10 5 MPa.

[0169] Place a series of contact detection points P inside the wire rope j ,like Figure 3 As shown. Contact detection is performed in the local coordinate system CSO of object c. n+1 , while the contact force calculation is performed in the global coordinate system CSO in order to assemble the wire rope system equations.

[0170] Set any contact detection point P j The position vector of is transformed from the global coordinate system CSO to the local coordinate system CSO of the object c n+1 In:

[0171] (5-1)

[0172] Among them, r c is the position vector of object c in the global coordinate system CSO, A c is the local coordinate system CSO of object c n+1 The 3×3 orthogonal rotation matrix of and They are contact detection points P j In the global coordinate system CSO and the local coordinate system CSO n+1 The position vector in .

[0173] Contact detection point P j The penetration depth δ is:

[0174] (5-2)

[0175] Contact force F c By the normal impact force F cn and tangential friction F cf Composition, that is

[0176] (5-3)

[0177] The impact force is equivalent to a nonlinear spring damper, and the normal impact force F c =F cn n, n is the unit normal vector of the contacted rigid body element;

[0178] F cn The expression is:

[0179] (5-4)

[0180] Since it is a metal-to-metal collision, the collision nonlinear index e=1.5; K c is the contact stiffness coefficient, which can be calculated as follows:

[0181] (5-5)

[0182] (5-6)

[0183] For the contact between high-energy impact object c and wire rope, the contact stiffness coefficient K c =1.73×10 10 N / m. step(δ) is the contact damping coefficient. In this embodiment, the maximum penetration depth d in the contact damping coefficient is m =1×10 -4 m, maximum contact damping coefficient c m Let contact stiffness coefficient K be c 0.5%, then c m =8.65×10 7 The calculated contact damping coefficient is:

[0184] (5-7)

[0185] So far, the normal impact force F during the contact process has been obtained cn .

[0186] Tangential friction force F cf =F cf τ, where τ is the unit tangent vector of the rigid body element in contact;

[0187] The tangential friction force F is calculated using the velocity-based friction model. cf , whose expression is:

[0188] (5-8)

[0189] Where μ f is the friction coefficient, and its expression is:

[0190] (5-9)

[0191] Where μ d and μ s are the dynamic and static friction coefficients, v d and v s They are dynamic and static critical speeds respectively. These coefficients can be obtained by consulting the manual based on the material properties; v τis the relative sliding speed of the two contacting objects; in this embodiment, the contact of steel materials, the dynamic friction coefficient μ d =0.25, static friction coefficient μ s =0.3, dynamic critical speed v d =1×10 - 4 m / s and static critical speed v s =1×10 -2 m / s.

[0192] The step function f in formula (5-9) is defined as:

[0193] (5-10)

[0194] in, .

[0195] So far, the tangential friction force F during the contact process has been obtained. cf .

[0196] The above equations (5-1) to (5-10) constitute the three-dimensional contact force F between the rigid body unit i and the high-energy impact object c. c According to the possible contact situations in the actual system, for example, if the high-energy impact object c contacts m rigid body units in the wire rope multi-body system, then m such three-dimensional contact force equations are established in sequence.

[0197] Step 6: Assemble the dynamic equations of the wire rope system;

[0198] The motion equations (3-1) to (3-9) of the wire rope multi-body system established in step 3, the flexible connection equations (4-1) to (4-4) between two adjacent rigid body units established in step 4, and the three-dimensional contact force equations (5-1) to (5-10) between the rigid body unit and other objects established in step 5 together constitute the wire rope system equation. The wire rope system equation is compiled into a script that can be run by the dynamic analysis software and imported into the dynamic analysis software to obtain the propagation of the wire rope bending wave. Figure 6 As shown in the figure, after the high-energy impact object c collides with the wire rope, the generated bending wave propagates from the impact point along both ends of the wire rope. When it encounters the fixed pulley and collides, the bending wave propagates back along the wire rope axis toward the impact point. The left end of the middle section of the wire rope is 5m away from the impact point as the detection point. The resultant force of the generalized force Q in the x, y, and z directions in the global coordinate system CSO is the tension on the wire rope at this detection point. The tension changes as shown in the figure. Figure 7As shown in . It can be seen that the first peak of the tensile force is the stress propagated here after the collision, at which time the bending wave propagates in the forward direction; the second peak is the stress superposition propagated here due to the reverse propagation of the bending wave. The high-energy impact object c may contact multiple rigid body units in the wire rope model. Taking the contact force between the 200th rigid body unit from left to right in the middle section of the wire rope and the high-energy impact object c as an example, the contact force change curve is shown as follows Figure 8 As shown. Figure 6 Similarly, due to the propagation of bending waves, the contact force also fluctuates. Figure 6 、 Figure 7 and Figure 8 It can be proved that the wire rope dynamic model constructed in this embodiment effectively simulates the bending wave propagation process of the wire rope under high-energy impact and can solve the dynamic response of the wire rope under high-energy impact.

Claims

1. A wire rope dynamics modeling method based on multi-body dynamics, characterized by: The wire rope is discretized into a multi-body system consisting of multiple cylindrical rigid units, and the motion equation of the multi-body system, the flexible connection equation between two adjacent rigid units in the multi-body system, and the three-dimensional contact force equation between the rigid unit and other objects are established respectively. The motion equation, flexible connection equation and three-dimensional contact force equation constitute the dynamic equation of the wire rope.

2. The wire rope dynamics modeling method based on multi-body dynamics according to claim 1 is characterized in that: The method also includes the steps of compiling the established wire rope dynamic equation into a script that can be run by dynamic analysis software and importing the script into the script to solve the dynamic response of the wire rope under high-energy impact.

3. The wire rope dynamics modeling method based on multi-body dynamics according to claim 1 or 2, characterized in that: The wire rope is discretized into a multi-body system consisting of n cylindrical rigid body units with a length of ΔL and a diameter of D, where n=L / ΔL, and L is the total length of the wire rope; each rigid body unit has six degrees of freedom.

4. The wire rope dynamics modeling method based on multi-body dynamics according to claim 3 is characterized in that: The established motion equation of the multi-body system is: Among them, Q i represents the generalized force vector acting on the mass center of the rigid body element i in the multi-body system when a high-speed object collides with the wire rope; Q xi , Q yi , Q zi is the force component of the center of mass of rigid body element i in the x, y, and z directions in the global coordinate system CSO, Q θxi ,Q θyi , Q θzi are the moment components of the rigid body element i about the x, y and z axes in the global coordinate system CSO; q, Q and denote the generalized coordinates, generalized forces and generalized velocities of the multi-body system respectively; q i The generalized coordinate vector representing the center of mass position of rigid body element i; and are the position vector and rotation vector of the center of mass of rigid body unit i respectively; q xi , q yi , q zi is the position coordinate of the center of mass of rigid body unit i in the global coordinate system CSO, x, y, z axis, q ψi , q θi , q φi is the rotation angle of the rigid body element i around the x-axis, y-axis and z-axis in the global coordinate system CSO; and They are point P in the local coordinate system CSO i and the position in the global coordinate system CSO; A=A(θ i ) is from the local coordinate system CSO i The three-dimensional orthogonal rotation transformation matrix transformed to the global coordinate system CSO; E k is the kinetic energy of the multi-body system; M is the mass matrix composed of the masses of all rigid body units; is the constraint force of the equation corresponding to the Lagrange multiplier method; the origin O of the global coordinate system is the center of the circle of any end face of the wire rope, the x-axis is perpendicular to the paper and outward, the y-axis is horizontal and to the right, and the z-axis is determined according to the right-hand rule; the local coordinate system CSO i The coordinate origin O i Coincident with the center of mass of rigid body unit i, x i The axis is perpendicular to the paper and points outward, coincides with the axis of rigid body unit i and points to the direction of rigid body unit i+1. i The positive direction of the y axis i The directions of the axes are determined according to the right-hand rule.

5. The wire rope dynamics modeling method based on multi-body dynamics according to claim 4 is characterized in that: The flexible connection equation between two adjacent rigid body units in the established multi-body system is: is the generalized force acting on rigid body element i; Q0 is the generalized preload applied to rigid body element i; K and C are the stiffness matrix and damping matrix of the rigid body element respectively; and are the mass center positions of rigid body unit i and rigid body unit i+1 respectively; G and E are the shear modulus and elastic modulus of the wire rope respectively; c aa , a=1,2,…,6 is obtained by experiments or finite element simulation.

6. The wire rope dynamics modeling method based on multi-body dynamics according to claim 5, characterized in that: The established three-dimensional contact force equation between the rigid body unit and other objects is: F c =F cn n F cf = F cf t Among them, F c F is the contact force on a rigid body unit when other objects collide with it; cn and F cf are the normal impact force and tangential friction force on the rigid body unit respectively; n and τ are the unit normal vector and unit tangent vector on the rigid body unit at the contact detection point with other objects respectively; K c is the contact stiffness coefficient; e is the collision nonlinearity index; δ is the penetration depth at the contact detection point; step(δ) is the contact damping coefficient; R1 and R2 represent the curvature radius of a pair of contact surfaces at the contact point; μ h and E h h are the Poisson's ratio and elastic modulus of the object material; c are m is the maximum contact damping coefficient; m is the maximum penetration depth; μ f is the friction coefficient; μ d and μ s are the dynamic and static friction coefficients, v d and v s are the dynamic and static critical speeds respectively; v τ is the relative sliding velocity of the two contacting objects; for Calculate the step function in the formula; .

Images