A method of simulating a rope amplitude of a rope drive system

Abstract

The application discloses a method for simulating the rope vibration amplitude of a rope driving system, which comprises the following steps: based on the rope tension and the axial motion speed of the rope, the rope vibration amplitude is solved by using a rope dynamics equation. In the expression of the rope tension, the initial cable force of the rope, the cable force change caused by the rope damping, the cable force change caused by the time-varying motion of the end effector and the cable force change caused by the lateral vibration of the rope are simultaneously considered, and the expression of the cable force change caused by the time-varying motion of the end effector and the expression of the cable force change caused by the rope damping are adopted, so that the lateral vibration amplitude of the rope midpoint of the rope driving system can be simulated more accurately.

Description

Technical Field

[0001] The present invention relates to the field of engineering measurement and calculation, and in particular to a method for simulating rope vibration amplitude of a rope drive system. Background Art

[0002] A rope-driven system is a system in which a cable connects a driver and an end effector, transmitting the driving force through the cable to control the end effector's motion. It offers advantages such as fast response, high load capacity, flexible controllability, lightweight design, and low cost. Its applications are increasingly widespread in engineering applications, such as tethered satellites, tethered robots, elevator cables, cable-controlled underwater robots, and passenger and freight cableways. Due to the flexibility of the cable, it is easily excited and vibrates. Excessive rope vibration can even cause the cable to break, leading to serious safety accidents. Therefore, rope vibration is a key issue restricting rope-driven systems.

[0003] In the rope-driven system, the flexibility of the rope itself cannot be ignored. The time-varying motion of the end effector will excite the rope, causing the rope to vibrate. Modeling rope vibration is one of the research methods, and one of the key factors in rope vibration modeling is the modeling of rope force. In the existing research on rope-driven systems, there are mainly three methods for modeling rope force: (1) not considering the flexibility of the rope, ignoring the elongation and vibration of the rope, and assuming that the rope force is a constant value (Bu Fannan, Wang Xiaoguang, Chen Hengtong, et al. Analysis of aerodynamic interference characteristics of a rope-driven parallel supported missile model [J]. Journal of Harbin Engineering University, 2022, 43 (3): 407-413.); (2) simplifying the rope into an undamped linear elastic spring, assuming that the change of rope force is linear, and the change of rope force is only related to the rope stiffness and elastic elongation of the rope (Diao X, Ma O. Vibration analysis of cable-driven parallel manipulators [J]. Multibody system Dynamics, 2009, 21(4): 347-360.); (3) The rope is simplified as a damped spring, and the change in rope force is considered to be related to the rope stiffness, elastic elongation, damping, and length change rate, but the influence of rope vibration on the rope force is not considered (Liu Zhihua, Tang Xiaoqiang, Shao Zhufeng, et al. Vibration characteristics of a 6-DOF rope parallel mechanism [J]. Journal of Mechanical Engineering, 2013, 49(03): 49-55.). However, the time-varying motion of the end effector will excite the rope force and then affect the rope force, and the rope force is also affected by the lateral vibration of the rope. Therefore, in the aforementioned rope force modeling method, it is difficult to reflect the actual rope force change, which will lead to a large deviation in the rope vibration modeling, inaccurate simulation of the rope amplitude, and difficult to match the actual working conditions. Summary of the Invention

[0004] The purpose of the present invention is to overcome the above-mentioned defects or problems in the background technology and provide a method for simulating the rope vibration amplitude of a rope drive system, which can more accurately simulate the rope midpoint vibration amplitude of the rope drive system.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] The first technical solution relates to a method for simulating the rope vibration amplitude of a rope drive system, which includes the steps of obtaining the rope vibration amplitude using a rope dynamics equation based on the real-time rope tension and the axial motion speed of the rope. The real-time rope tension T is expressed as:

[0007] T=T0+ΔT l +ΔT c +ΔT δ

[0008] Where T0 represents the initial cable force, ΔT δ represents the change in cable tension caused by the lateral vibration of the rope,

[0009] ΔT l represents the cable force variation caused by the time-varying motion of the end effector, which is expressed as

[0010]

[0011]

[0012] in, S X A Represents the position vector of the rope end support point A in the global coordinate system, Represents the position vector of the center of mass E of the end effector in the global coordinate system at any time, Represents a vector at any time D P is the position vector in the global coordinate system, P represents the pulling point of the rope on the end effector, It represents the position vector of the mass center E of the end effector in the global coordinate system at time t = 0, Represents the vector at time t=0 D P is the position vector in the global coordinate system, the global coordinate system is OXYZ, k l represents the stiffness coefficient; represents the fitting function, which is expressed as:

[0013]

[0014] Where f represents the motion frequency of the end effector; n, a n 、b n represents the fitting parameters;

[0015] ΔT c represents the change in cable force due to rope damping, which is expressed as:

[0016]

[0017] Where c represents the damping coefficient of the rope, L represents the length vector of the rope that changes in real time in the global coordinate system, |L| represents the modulus of the length vector of the rope that changes in real time, Represents the velocity vector of the end effector's center of mass E in the global coordinate system, It represents the angular velocity vector of the end effector around the center of mass E in the global coordinate system, X E ,Y E ,Z E is the translation of the mass center E of the end effector along the three coordinate axes OXYZ, θ r ,θ p ,θ y It is the rotation of the center of mass E of the end effector around the three coordinate axes OX, OY, and OZ.

[0018] The second technical solution is based on the first technical solution, wherein the stiffness coefficient k l Expressed as:

[0019]

[0020] Among them, T a represents the amplitude of cable force change, χ max represents the maximum value of χ, min represents the minimum value of χ.

[0021] The third technical solution is based on the first technical solution, wherein the rope force change ΔT caused by the lateral vibration of the rope δ Expressed as:

[0022]

[0023] where X w Indicates the amplitude of the rope midpoint, L indicates the length of the rope that changes in real time, m e represents the elastic modulus of the rope, C * Indicates the cross-sectional area of ​​the rope.

[0024] The fourth technical solution is based on the third technical solution, wherein the axial movement speed of the rope is expressed as:

[0025]

[0026] The fifth technical solution is based on the fourth technical solution, wherein the rope dynamics equation is expressed as:

[0027]

[0028] in represents the second derivative of the amplitude at the midpoint of the rope, and ρ represents the density of the rope.

[0029] From the above description of the present invention, it can be seen that compared with the prior art, the present invention has the following beneficial effects:

[0030] This application comprehensively considers the effects of initial rope force, rope lateral vibration, time-varying motion of the end effector, and rope damping on the real-time tension of the rope, and innovatively proposes an expression model for the rope force changes caused by the time-varying motion of the rope end effector and an expression model for the rope force changes caused by rope damping, thereby establishing a nonlinear real-time rope tension model. Experiments have shown that the rope midpoint vibration amplitude of the rope drive system can be simulated more accurately.

[0031] Among them, according to the applicant's research experience, since the movement of the end effector is controlled by controlling the change of the rope length in the rope drive system, the change frequency of the rope force and rope length is the same as the movement frequency of the end effector. Therefore, in the fitting function proposed in this application, (1) based on the Fourier series and the kinematic principle of the rope drive system, the relationship between the movement frequency of the end effector and the change of the rope force is established, so that the simulation of the change of the rope force caused by the time-varying movement of the end effector is more accurate; (2) the fitting function adopts the form of sine and cosine, which is simple in form and can fit any form of function by increasing the order, which is powerful; (3) the solution of the fitting parameters can be quickly solved by using the fitting function library of general software such as MATLAB or Origin, which helps to improve the calculation efficiency.

[0032] In addition, the stiffness coefficient k is obtained by combining the single-degree-of-freedom low-frequency oscillation test and the numerical calculation of the rope length variation. l , which describes the dynamic stiffness relationship between the end effector motion and the cable force change, and is consistent with the static stiffness expression in the prior art ( Where k represents the static stiffness, m e represents the elastic modulus of the rope, C * represents the cross-sectional area of ​​the rope, and L represents the length of the rope. ) are essentially different, which can make the simulation of the cable force change caused by the time-varying motion of the end effector more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0034] Figure 1 Schematic diagram of the rope drive system. DETAILED DESCRIPTION

[0035] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are preferred embodiments of the present invention and should not be regarded as excluding other embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0036] In the claims, description and drawings of the present invention, unless otherwise clearly defined, the use of terms such as "first", "second" or "third" is for the purpose of distinguishing different objects rather than for describing a specific order.

[0037] In the claims, specification and the above-mentioned drawings of the present invention, unless otherwise expressly defined, directional words such as the terms "center", "transverse", "longitudinal", "horizontal", "vertical", "top", "bottom", "inside", "outside", "up", "down", "front", "back", "left", "right", "clockwise", "counterclockwise" and the like indicating directions or positional relationships are based on the directions and positional relationships shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific direction or be constructed and operated in a specific direction, and therefore cannot be understood as limiting the specific scope of protection of the present invention.

[0038] In the claims, description and above-mentioned drawings of the present invention, unless otherwise expressly defined, the terms "fixed connection" or "fixed connection" should be understood in a broad sense, that is, any connection method without displacement relationship and relative rotation relationship between the two parties, that is, including non-detachable fixed connection, detachable fixed connection, integral connection and fixed connection through other devices or elements.

[0039] In the claims, description and drawings of the present invention, if the terms "include", "have" and their variations are used, they are intended to mean "including but not limited to".

[0040] This embodiment provides a method for simulating rope vibration amplitude of a rope drive system, which includes the steps of obtaining the rope vibration amplitude by solving a rope dynamics equation based on the real-time tension of the rope and the axial motion speed of the rope.

[0041] like Figure 1 It should be noted that two coordinate systems are used in this paper when conducting the following analysis, namely the global coordinate system OXYZ and the local coordinate system Ax w yw , point A is used to represent the end point of the rope in the rope drive system, point O is any point in space, and Ay w This is the axial direction of the rope. In the calculation, the positions of point O and point A are fixed, and the position of point O determines the initial position vector of the end effector.

[0042] For the rope dynamics equation, we have:

[0043] Based on the Euler motion equation of continuous medium, the force analysis of the rope element can be obtained:

[0044]

[0045] Where ρ represents the density of the rope, C * represents the cross-sectional area of ​​the rope, ds is the arc length of the rope element, Denotes the rope element in Ax w The rate of change of velocity in the direction, T represents the real-time tension of the rope, x w represents the lateral vibration displacement of the rope, y w Represents the axial coordinate of the rope.

[0046] It is known that in the rope drive system, the rope moves axially and the rope element is at Ax w Speed ​​in direction v x is the displacement y w and time t, that is, v x =v x (y w , t), so

[0047]

[0048] in, Indicates that the rope element is in Ay w Velocity change rate in the direction (rope axial direction), v y is the axial speed of the rope.

[0049] For the rope element, ds≈dy w , we can get

[0050]

[0051] According to the Galerkin method,

[0052]

[0053] Among them, X w represents the amplitude of the rope midpoint as a function of time t, y w ∈[0, L], L represents the length of the rope that changes in real time, and in order to study the vibration response of the rope midpoint, there is

[0054]

[0055] Available

[0056]

[0057] in, Represents the second derivative of the amplitude at the midpoint of the rope.

[0058] The position of the end effector in the global coordinate system is [X E ,T E ,Z E ,θ r ,θ p ,θ y ] T , where X E ,Y E ,Z E is the translation of the mass center E of the end effector along the three coordinate axes OXYZ, in mm, θ r ,θ p ,θ y It is the rotation of the center of mass E of the end effector around the three coordinate axes OX, OY, and OZ, in degrees.

[0059] For the real-time tension T of the rope, we have

[0060] T=T0+ΔT l +ΔT c +ΔT δ

[0061] Among them, T0 represents the initial cable force, which can be measured by the sensor.

[0062] ΔT l represents the cable force variation caused by the time-varying motion of the end effector, which is expressed as

[0063]

[0064]

[0065] in, S X A Represents the position vector of the rope end support point A in the global coordinate system, Represents the position vector of the center of mass E of the end effector in the global coordinate system at any time, Represents a vector at any time D P is the position vector in the global coordinate system, P represents the pulling point of the rope on the end effector, It represents the position vector of the mass center E of the end effector in the global coordinate system at time t = 0, Represents the vector at time t=0 D P is the position vector in the global coordinate system, and the global coordinate system is OXYZ; k l represents the stiffness coefficient, which can be obtained by the following formula or by other methods.

[0066]

[0067] Among them, T a The amplitude representing the cable force change can be obtained from a single degree of freedom oscillation test (given the motion posture of the end effector, [0,0,0,0,θ p =A test *sin(2π*f test *t),0] T ,Pick f min Represents the minimum value of the end effector motion frequency. test Represents the end effector motion amplitude in the single degree of freedom oscillation test and can be set to the actual motion amplitude of the end effector. During the end effector motion, the rope tension is measured by the tension sensor, that is, the rope force change amplitude T is obtained. a . );χ max represents the maximum value of χ, min represents the minimum value of χ.

[0068] represents the fitting function, which is expressed as:

[0069]

[0070] Where f represents the motion frequency of the end effector; n, a n 、b n Represents the fitting parameters; the known motion pose of the end effector [X E ,Y e ,Z e ,θ r ,θ p ,θ y ] T , we can find the curve of χ changing with time; fitting parameters n, a n 、b n The solution can be obtained by using the fitting function library of general software such as MATLAB or Origin, or by using other function fitting methods.

[0071] ΔT c represents the change in cable force caused by the rope damping, which can be expressed as:

[0072]

[0073] Where c represents the damping coefficient of the rope, () T represents the transpose of the matrix, L represents the length vector of the rope that changes in real time in the global coordinate system, |L| represents the modulus of the length vector of the rope, which changes in real time. Represents the velocity vector of the end effector's center of mass E in the global coordinate system, Represents the angular velocity vector of the end effector around the center of mass E in the global coordinate system.

[0074] In this embodiment, the calculation formula of the rope damping coefficient c is:

[0075]

[0076] Where, ζ represents the rope damping ratio, ρ represents the density of the rope, and C * represents the cross-sectional area of ​​the rope, L0 represents the initial rope length, m e represents the elastic modulus of the rope.

[0077] ΔT δ represents the change in cable force caused by the lateral vibration of the cable, which can be first expressed as

[0078]

[0079] Among them, dx w ,dy w They represent the rope element along the coordinate axis x in the local coordinate system. w 、y w Since the slant-to-span ratio of the rope is very small, that is, And in the aforementioned Therefore, according to Taylor expansion, we can get:

[0080]

[0081] Therefore, the real-time tension T of the rope can be expressed as

[0082]

[0083] For the axial motion speed v of the rope y , first of all there is

[0084]

[0085] Taking its derivative, we can know that:

[0086]

[0087] Therefore, the dynamic equations of the rope are combined The expression of the real-time tension T of the rope and the axial speed v of the rope y The expression of is solved by using the Runge-Kutta numerical integration method to obtain the amplitude-frequency response of the rope vibration.

[0088] The method in this embodiment is verified below by taking a single-rope drive system as an example.

[0089] Set the initial position vector of the end effector Position vector of the rope end support point A in the global coordinate system S X A =[165,415,-1280] T mm, the coordinates of the traction point P of the rope on the end effector are (-207, 80, 0) mm, the end effector performs pitch motion (the pitch motion amplitude is 2°, the motion frequency is 60 Hz), the rope damping ratio is 0.06, and the rope density ρ = 1400 kg / m 3 The rope elastic modulus is 11.9 GPa, the initial rope force T0 = 36.98 N, and the initial rope length L0 = 757.9 mm. Under the same end effector motion frequency and rope diameter, the amplitude of the rope midpoint is obtained. The results are shown in the following table:

[0090]

[0091] The experimental measurement involved exciting one end of the rope with an exciter and measuring the rope amplitude with a high-speed camera. D1 disregarded rope flexibility, neglected rope elongation, and assumed a constant rope force. D2 simplified the rope into an undamped linear elastic spring. This indicates that the model provided in this embodiment simulates the rope vibration amplitude of the rope drive system with minimal error, more closely resembling actual conditions.

[0092] In this embodiment, by comprehensively considering the effects of the initial cable force, the lateral vibration of the rope, the time-varying motion of the end effector and the rope damping on the real-time tension of the rope, and innovatively proposing an expression model for the cable force changes caused by the time-varying motion of the end effector of the rope and an expression model for the cable force changes caused by the rope damping, a nonlinear real-time tension model of the rope is established. Experiments have shown that the vibration amplitude of the rope midpoint of the rope drive system can be simulated more accurately.

[0093] In the fitting function of this embodiment, (1) based on the Fourier series and the kinematic principles of the rope drive system, a correlation is established between the motion frequency of the end effector and the cable force change, so that the simulation of the cable force change caused by the time-varying motion of the end effector is more accurate; (2) the fitting function adopts the sine and cosine forms, which are simple in form and can fit functions of any form by increasing the order, and has powerful functions; (3) the solution of the fitting parameters can be quickly solved by using the fitting function library of general software such as MATLAB or Origin, which helps to improve the calculation efficiency.

[0094] In this embodiment, the stiffness coefficient k is obtained by combining the single-degree-of-freedom low-frequency oscillation test and the numerical calculation of the rope length change. l , which describes the dynamic stiffness relationship between the end effector motion and the cable force change, and is consistent with the static stiffness expression in the prior art ( Where k represents the static stiffness, m e represents the elastic modulus of the rope, C * represents the cross-sectional area of ​​the rope, and L represents the length of the rope. ) are essentially different, which can make the simulation of the cable force change caused by the time-varying motion of the end effector more accurate.

[0095] The method provided in this embodiment can also be applied to a multi-rope drive system, as follows:

[0096] Establish the matrix equation to simulate the rope vibration amplitude of the multi-rope drive system:

[0097]

[0098] Among them, vector X w =[X w1 ,X w2 ,…,X wn ] T ,X wi (i=1,2,…,n) represents the amplitude of the midpoint of the i-th rope, and the matrix K w =diag(K w1 ,K w2 ,…,K wn ), ρ represents the density of the rope, C * represents the cross-sectional area of ​​the rope, T i (i=1,2,…,n) represents the real-time tension of the i-th rope,

[0099]

[0100] v yi represents the axial motion speed of the i-th rope,

[0101] The real-time tension T of the i-th rope i and axial velocity v yi The definitions of variables in the expressions are the same as those for the single-rope drive system.

[0102] Write the above equation in the form of state equation:

[0103]

[0104] in,

[0105] The Runge-Kutta numerical integration method is used to solve the problem and the amplitude-frequency response of each rope vibration can be obtained.

[0106] The above description and embodiments are intended to explain the scope of protection of the present invention, but do not constitute a limitation thereto. Modifications, equivalent substitutions, or other improvements to the embodiments of the present invention or portions thereof that can be obtained by a person of ordinary skill in the art through logical analysis, reasoning, or limited experimentation based on the teachings of the present invention or the above embodiments, combined with common knowledge, ordinary technical knowledge in the field, and / or prior art, should all be included within the scope of protection of the present invention.

Claims

1. A method for simulating the rope vibration amplitude of a rope drive system, characterized in that: The method includes the steps of obtaining the rope vibration amplitude using the rope dynamics equation based on the rope real-time tension and the rope axial motion speed. The rope real-time tension T is expressed as: T=T0+ΔT l +ΔT c +ΔT δ Where T0 represents the initial cable force, ΔT δ represents the change in cable force caused by the lateral vibration of the cable, the ΔT δ Expressed as: where X w Indicates the amplitude of the rope midpoint, L indicates the length of the rope that changes in real time, m e represents the elastic modulus of the rope, C * Indicates the cross-sectional area of ​​the rope; ΔT l represents the cable force variation caused by the time-varying motion of the end effector, which is expressed as in, S X A Represents the position vector of the rope end support point A in the global coordinate system, Represents the position vector of the center of mass E of the end effector in the global coordinate system at any time, Represents a vector at any time D P is the position vector in the global coordinate system, P represents the pulling point of the rope on the end effector, It represents the position vector of the mass center E of the end effector in the global coordinate system at time t = 0, Represents the vector at time t=0 D P is the position vector in the global coordinate system, the global coordinate system is OXYZ, k l represents the stiffness coefficient; represents the fitting function, which is expressed as: Where f represents the motion frequency of the end effector; n, a n 、b n represents the fitting parameters; ΔT c represents the change in cable force due to rope damping, which is expressed as: Where c represents the damping coefficient of the rope, L represents the length vector of the rope that changes in real time in the global coordinate system, |L| represents the modulus of the length vector of the rope that changes in real time, Represents the velocity vector of the end effector's center of mass E in the global coordinate system, It represents the angular velocity vector of the end effector around the center of mass E in the global coordinate system, X E ,Y E ,Z E is the translation of the mass center E of the end effector along the three coordinate axes OXYZ, θ r ,θ p ,θ y is the rotation of the mass center E of the end effector around the three coordinate axes OX, OY, and OZ; The axial motion speed of the rope is expressed as: Where ΔT C and v y ( ) in the expression T Represents the transpose of a matrix; The rope dynamics equation is expressed as: in represents the second derivative of the amplitude at the midpoint of the rope, and ρ represents the density of the rope.

2. The method for simulating rope vibration amplitude of a rope drive system according to claim 1, characterized in that: The stiffness coefficient k l Expressed as: Among them, T a represents the amplitude of cable force change, χ max represents the maximum value of χ, min represents the minimum value of χ.

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