Keywords

1 Introduction

Rope-pulley system, which is widely used in the crane industry and plays a significant role, has the advantages of a simple structure, a large bearing capacity, and smooth transmission as a component of the crane hoisting mechanism. The inevitable swaying phenomena of the lifting load has a direct impact on the precision and safety of the operation when the crane performs lifting and lowering operations. With the increasing demand for lifting positioning accuracy, the study of the swing characteristics and influencing factors of lifting weights during the lifting process has become more and more essential.

In the majority of the current studies on lifting load sway and anti-sway control, the weights and dimensions of the ropes and pulleys are typically ignored when the model is established, and the influence of the contact relationship between the rope and pulley is not considered, or the ropes are rigidized. However, the contact relationship between the wire rope and the pulley block in different situations is complex and varied [1], and there must exist a certain relationship with the swing of the lifting load. Therefore, it is necessary to study the influence of the rope-pulley system’s contact relationship in different states on the swing of lifting load.

The contact relationship between wire rope and pulley has been thoroughly studied by a large number of academics both domestically and internationally. Based on Archard's wear rule, Argatove et al. investigated the wear pattern of wire ropes under pressure effect and wear coefficients [2]. Kong et al. proposed a modified simplex method using contact solutions to simulate micromotion wear [3]. Qi et al. established the link between material velocity and spatial velocity for the rope-pulley system, and based on this, constructed a dynamic node rope element [4]. Guo et al. established a discrete wire rope model and a contact force model between pulleys through the macro-command method, which realistically responded to the dynamic characteristics of the mechanism in the operation process [5]. Zhang et al. used ADAMS/Cable module to establish the simulation model of the pulley wire rope system, and derived the dynamic response of the wire rope in the transition stage and stable operation [6].

In this article, the research object is the rope-pulley system considering the mass, winding ratio and matching size of the wire rope and pulley. The dynamics simulation model of the rope-pulley system is established by using the ADAMS/Cable module. By changing the winding ratio and matching size of the rope-pulley system, the influence of the contact relationship on the swing characteristics of the lifting load under different conditions is explored. The research content can provide a certain reference for the study including rope-pulley system.

2 Rope-Pulley System Modeling

To study the influence of different winding ratios on the swing characteristics of lifting load, models of rope-pulley system with four-fold winding ratio, six-fold winding ratio and eight-fold winding ratio are established. By changing the size of the eight-fold winding ratio rope-pulley system model, the influence of the change of the matching size on the swing of the lifting load is studied.

2.1 Modeling Theory

The ADAMS/Cable module creates the rope-pulley system model in two ways, one is a simplified model and the other is a discrete model as shown in Figs. 1 and 2. The simplified model uses a dummy object with pure motion constraints to track the tangent between the pulleys [7]. The simulation speed is fast but the wire rope has no solid model, and the influence of wire rope quality and inertia is not considered. The discrete model is to discretize the wire rope into a finite number of rigid spheres. The contact between the wire rope and the pulley is continuous. Considering the mass and inertia effects of the wire rope, the vibration of the wire rope and the friction between the pulleys can be truly simulated, which is more accurate than the simplified model. Therefore, this paper uses the discrete modeling method of ADAMS/Cable module to establish the pulley wire rope system.

Fig. 1
An illustration of a pulley with a wire rope running through its 3 grooves.

Simplified model

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Fig. 2
An illustration of a pulley with a series of bead like spheres running through its 3 grooves.

Discrete model

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The discrete model uses the finite element idea, and the finite rigid spheres are flexibly connected by the sliding pair, the revolute pair and the one-way force. Compared with the cylindrical discrete model, the spherical discrete model has higher accuracy. The calculation formula of the connection force is shown in Eq. (1).

$$\text{F}=-{\text{K}}{X} - {\text{C}}\dot{{X}}+{\text{F}}_{0}$$
(1)

where \({\text{F}}\) is the force acting on the rope system, \({\text{K}}\) is the stiffness, \({\text{C}}\) is the damping force of the system, \({\text{X}}\) is displacement, \(\dot{\text{X}}\) is the speed and \({\text{F}}_{0}\) is the initial force of the rope system.

The wire rope composed of rigid spheres is wound by contact force with the pulley. The contact force consists of two parts, the elastic force generated by the mutual incision between the rigid sphere and the pulley, and the damping force generated by the relative velocity between the two. The contact force between the wire rope and the pulley is calculated by the impact function method. The generalized form of the impact function is shown in Eq. (2).

$${\text{F}}_{\text{ni}}= \text{K} {\updelta}_{\text{i}}^{\text{e}}+ \text{C} {\text{V}}_{\text{i}}$$
(2)

where \({\text{F}}_{\text{ni}}\) is the normal contact force between two components, \({\text{K}}\) is the stiffness of the contact surface and \({\updelta}_{\text{i}}^{\text{e}}\) is the normal penetration depth of the contact point.

When the wire rope and the pulley contact, the damping value will quickly reach the maximum and remain constant, so the penetration depth is very small. However, considering the convergence of the values in ADAMS, the recommended value of 0.1 mm is used.

2.2 Selection of Main Parameters

According to the specification standard of pulley and wire rope, the pulley of model WJ4202 is selected as an example for modeling, and its size parameters are shown in Table 1.

Table 1 Specification standard of the pulley and wire rope
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Where D is the inner diameter of the pulley, D1 is the outer diameter of the pulley, R is the radius of the pulley groove, b is the width of the pulley and A is the angle of the pulley groove.

The force index E is the coefficient of material stiffness in the instantaneous normal force when the wire rope and the pulley contact. For the metal contact between the pulley and the wire rope, the force index is 1.5. The contact parameters of the pulley surface are shown in Table 2.

Table 2 Contact parameters of the pulley
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Where K is the stiffness coefficient, E is the force index, Cm is the damping coefficient, \(\upmu\) is the friction coefficient, Vt is the friction velocity.

The parameters of the wire rope are shown in Table 3.

Table 3 Parameters of the wire rope
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Where D is the diameter of the wire rope, \(\uprho\) is the density of the wire rope, Cm is the damping coefficient.

The three models of the rope-pulley system are shown in Fig. 3. Where (a) is the four-fold winding ratio model, (b) is the six-fold winding ratio model and (c) is the eight-fold winding ratio model. The system is composed of movable pulley, fixed pulley, wire rope, drum, hook and lifting load. The size of the drum, the mass of the lifting load, the winding way of wire rope and other parameters in the system are all the same except for different winding ratio.

Fig. 3
Three illustrations of a rope around a drum and fixed pulley, which are on the top, and the movable pulley, hook, and load, which are at the bottom. In a, the pulleys have 4 grooves. In b, the pulleys have 6 groves. In c, the pulleys have 8 grooves.

Simulation model of the rope-pulley system

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3 Simulation Analysis of Different Winding Ratio

3.1 Driving Function

STEP function is used as the driving function, and its format is STEP (x, x0, h0, x1, h1). In the function expression, x is the independent variable, x0 is the initial value of the independent variable, h0 is the function value at the initial point, x1 is the end value of the independent variable, and h1 is the function value at the end point.

The driving functions of the three models are shown in Table 4. In a certain period of time, the drum drive wire rope stops after the lifting weight is lifted to the same height at a constant speed.

Table 4 The driving functions of different models
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3.2 Simulation Result

In the established model of the rope-pulley system, the lifting weight load in the xoz plane and rises in the y direction. The simulation time-domain result curves of the lifting load in x direction and z direction are shown in Figs. 4, 5 and 6.

Fig. 4
2 graphs plot C M positions in x and z direction, respectively, versus time and depict oscillatory patterns. In a, oscillations vary between negative 2.5 and 3 m m and between negative 1 and 2 m m. In b, oscillations vary between 147 and 170 m m and between 152 and 166 m m. Values are estimated.

Swing curves of the lifting load centroid of the four-fold winding ratio system

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Fig. 5
2 graphs plot C M positions in x and z direction, respectively, versus time and depict oscillatory patterns. In a, oscillations vary between negative 3 and 3 m m and between negative 2 and 2 m m. In b, oscillations vary between 75 and 135 m m and between 95 and 122 m m. Values are estimated.

Swing curves of the lifting load centroid of the six-fold winding ratio system

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Fig. 6
2 graphs plot C M positions in x and z direction, respectively, versus time and depict oscillatory patterns. In a, oscillations vary between negative 7.5 and 11 m m and between negative 0.2 and 2.5 m m. In b, oscillations vary between 50 and 105 m m and between 65 and 92 m m. Values are estimated.

Swing curves of the lifting load centroid of the eight-fold winding ratio system

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Through the analysis of the curve, it can be seen that in the three cases, the lifting load starts to rise at the seventh second and then swings around the equilibrium position with a regular positive spinning tendency. The amplitude of the oscillation reaches its maximum during the lifting process and begins to decrease at the end of the lifting motion. The vibration decay rate of the eight-fold winding ratio system is faster compared to the other two cases.

The equilibrium position coordinates of the lifting load are calculated by extracting the data in the curve, and then the maximum offset of the center of mass of the lifting load in the x direction and the z direction is calculated according to the equilibrium coordinates. The calculation results are shown in Table 5.

Table 5 The maximum offset of the lifting load centroid with different winding ratio
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According to the data in the table, the maximum offset of the four-fold winding ratio system is the smallest, and the maximum offset of the eight-fold winding ratio system is the largest. Therefore, it can be concluded that with the increase of the winding ratio, the maximum offset of the center of the lifting load increases continuously, but the swing amplitude of the eight-fold winding ratio system attenuates faster and the ability to resist swing is better.

4 Simulation Analysis of Different Matching Parameters

4.1 Matching Parameter Selection

In the actual operation process, it is necessary to choose different types of the rope-pulley mechanism for lifting operation according to the working conditions. Through the matching parameters in Sect. 2.2, it can be seen that the same pulley specification can be matched with different diameters of wire ropes, and different matching parameters will make the contact relationship between the pulley and the wire ropes change, and will also affect the swinging of the lifting weight in the process of operation.

According to the pulley specification standard and the matching size relationship between the pulley and the wire rope, this paper selects nine cases of three groups of matching relationships, as shown in Table 6.

Table 6 Matching size of the pulley and wire rope
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4.2 Simulation Result

In the simulation model of the eight-fold winding ratio rope-pulley system, the parameters are changed according to Table 6. Simulation analysis is carried out under the same other variables such as lift speed and lift height. The simulation time-domain result curves are shown in Table 7.

Table 7 The swing curves of the lifting load centroid with different matching parameters
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By analysing the curve, it can be concluded that when the angle of the pulley groove is unchanged, the model with a wire rope diameter of 25 mm has the fastest attenuation of the swing amplitude in the x direction, and the time to approach a stable state is about 90 s. The models with wire rope diameters of 24 mm and 26 mm reached a nearly stable state in about 350 s and 500 s, respectively. This shows that when the diameter of the wire rope is 25 mm, the wire rope is more suitable for the pulley, and its contact relationship makes the lifting load’s stability better. When the diameter of the wire rope is constant and the angle of the pulley groove is changed, the swing curve of the load is relatively close, and there is no obvious difference.

Similarly, by extracting the data in the curves, the maximum offset of the lifting load centroid in the x direction and z direction is calculated and the results are shown in Table 8.

Table 8 The maximum offset of the lifting load centroid with different matching parameters
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It can be obtained from the data in the table that when the angle of the pulley groove is unchanged, with the increase of the diameter of the wire rope, the maximum offset of the lifting load centroid increases gradually in the x-direction and the z-direction. When the diameter of the wire rope is constant and the angle of the pulley groove is changed, the maximum offset of the lifting load centroid is similar and almost unchanged.

In summary, when other conditions remain unchanged, the diameter of the wire rope has a significant effect on the swing characteristics of the lifting load.

5 Conclusion

In this paper, the discrete simulation model of the rope-pulley system is established by using ADAMS/Cable module. The rope-pulley system model with three kinds of winding ratio and three sets of matching parameters is selected as the research object. Through simulation analysis, the maximum offset and swing curves of the lifting load centroid are discovered. The effect of the contact relationship between the pulley and the wire rope in different states on the swing characteristics of the lifting load is investigated. The conclusions obtained are as follows:

  1. (1)

    The maximum offset of the lifting load centroid increases as the winding ratio increases. The eight-fold winding ratio rope-pulley system has a faster decay rate of swing amplitude and better resistance to sway.

  2. (2)

    The change of contact relationship caused by changing the diameter of wire rope has obvious influence on the swing characteristics of lifting load. When other parameters are constant, the larger the diameter of the wire rope, the greater the maximum offset of the lifting load centroid. The change of the pulley groove angle has no obvious effect on the swing characteristics of the lifting load.