Keywords

1 Introduction

The movable pulley block is a vital component of the crane lifting mechanism. The study of a defective movable pulley block system’s dynamic performance has significant theoretical and practical importance. Many researchers have conducted dynamic analysis on different types of cranes, among which Adamiec-Wójcik [1] has conducted dynamic theoretical derivation and simulation research on offshore multi-boom cranes by establishing static and dynamic models. Through the use of ADAMS, Yin Xiao-lei [2] developed a rigid-flexible coupling simulation model for cranes and optimized the vibration of the crane. Cibicik and Egeland [3] took the steering arm of a floating crane as the research object, extended the Kane motion equation to modeling, and studied the pitch motion dynamics of the flexible arm driven by a hydraulic cylinder. With the FZQ40 tower crane, Lu Yan [4] investigated the impact coefficient of a jib tower crane’s sudden unloading and concluded that several parameter changes, including amplitude, balance weight, and a ratio of unloading mass to lifting mass, affect the impact coefficient. Based on the stiffness equivalence of truss structures, mass equivalence, and rotational inertia equivalence of mechanism dynamic analysis, Zong Hao [5] constructed an equivalent model for the rotary operation of a 2800 t. m tower crane, then dynamic analysis was then applied to the simplified tower crane structure. Using ADAMS software, Wang Xin et al. [6] dynamically analyzed the sudden unloading of the crane boom, whose research showed that under the same lifting capacity conditions, the larger the elevation angle of the crane’s boom, the greater the inclination angle of the boom after sudden unloading, meanwhile, the shorter the unloading time, the greater the caster angle of the jib.

There are also many studies on the working state of steel wire ropes. Li Bo-lun [7] and Jiao Qian-qian [8] odelled and simulated the fracture dynamics of a pulley block system with two wire ropes, respectively. Additionally, when a wire rope suddenly fails, the stress on the remaining wire rope doubles. Qing Huang et al. [9] studied the multi-body dynamics of lifting steel wire ropes. In their respective simulation analyses of the winding and swaying of crane steel wire rope, Yan Jing-Feng [10] and Cao Xu-yang [11] found that the lifting weight has no effect on deviation but that the fixed pulley’s position does, with the deviation decreasing as it gets closer to the fixed pulley’s distance from the drum. Wang Yong-long [12] built the vertical vibration dynamics model of crane hoisting weight when the hoisting weight is displaced horizontally and the wire rope amplitude is changed through the rope compensation method and proposed a theoretical method to reduce the vibration by the speed regulation of the reel. Although there are many simulations and analyses of crane wire ropes, but dynamic simulation of crane movable pulley systems with defects needs more in-depth and comprehensive research.

This study aims to establish a pulley block simulation platform based on ANSYS contact analysis using the APDL command. Through simulation and comparison, the relationship between the impact of the pulley block system and the input load is obtained, and factors affecting the recoil force will be analyzed, which provides theoretical guidance for optimizing the dynamic performance of the defective movable pulley block system. This paper is organized as follows: Sect. 2 builds and verifies parametric models of the BEAM and PLANE element rope pulley block through static simulation. In Sect. 3, dynamic simulations are performed on an intact pulley block model under two dangerous working conditions: lifting off the ground and sudden unloading. The relationship between the impact force of the pulley block and the input load is investigated, and the factors influencing the impact force are analyzed. Section 4 establishes models of defective wire rope and defective hook, and dynamic analyses are implemented on the hook-defective model for hook rupture and the wire rope-defective model for double-sided wire rope rupture., then the factors influencing the recoil force are analyzed. Finally, the research conclusion is summarized in Sect. 5.

2 Establishment and Verification of Parameterized Model for Pulley Block

2.1 Establishment of Parameterized Model for Pulley Block

To reduce the number of elements, simulation time, and modeling workload, the pulley block model needs to be simplified by building a 2D plane pulley block model. The pulley is simulated using the plane element PLANE182, while the wire rope is simulated using both the beam element BEAM3 and the plane element PLANE182. The pulley block model is illustrated in Fig. 1.

Fig. 1
2 diagrams of a pulley block in two ways, using the plane elements for ropes and using the beam elements for ropes.

Pulley block model

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The established pulley block model is parameterized using a number of adjustable parameters, including pulley diameter, wire rope diameter, pulley thickness, material elastic modulus, density, and Poisson's ratio. These parameters are defined using the APDL command and are associated with the relevant dimensions as variables. The key points in the model are defined by these parameters, allowing for the size of the pulley block to be changed as a whole by altering the values of the parameters. This parameterization enables the model to accommodate different sizes and materials for the pulley block.

2.2 Verification of Parameterized Model for Pulley Block

First, a contact model based on contact elements is established for the pulley block. The CONTA171 element is selected as the contact element, and the TARGE169 element as the target element. Contact pairs and models are formed based on the previously established beam element rope pulley block model. The pulley block's static structure is modeled by constraining all degrees of freedom at the rope ends. Then, a concentrated force of 10000N is applied downward at the center of the movable pulley, and static simulation is performed. As illustrated in Fig. 2, the contour displacement plot in the X and Y directions, the equivalent stress diagram of the wire rope, and the support reaction force of the pulley block are all recorded. The theoretical reaction force is 5000N, and the combined force with the maximum deviation from the theoretical value is 4960.5N. The maximum relative error of the beam element rope model based on contact elements is 0.79%.

Fig. 2
A. A screenshot of the P R R S O L command window displays F X and F Y values. B, C, and D. 3 simulation diagrams of the pulley system plot the distribution of equivalent stress of steel wire rope, contour displacement in the X direction, and contour displacement in the Y direction.

Static simulation result

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3 Dynamic Simulation of Intact Pulley Block System

3.1 Dynamic Analysis Under Lifting from Ground Conditions

A condition with large fluctuations in the lifting load is selected from the observed lifting load data because it has a more noticeable effect on the crane and its operation. The selected relationship curve between lifting load and time is illustrated in Fig. 3.

Fig. 3
A line graph of tension versus time. It plots a trend that moves horizontally at 0 till 40 seconds, rises to 2 times 10 power 4 at 50 seconds, exhibits a fluctuating pattern at 50 seconds, and becomes almost flat after 60 seconds.

The relationship curve between lifting load and time

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After applying the lifting load, a transient dynamic analysis is carried out, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 4.

Fig. 4
Four line graphs. A. U Y versus time graph presents the declining displacement of the moving pulley center. B. Stress versus time graph displays sudden boost and zig-zag pattern in steel wire rope stress. C and D. Force versus time exhibits a sudden rise and an erratic pattern in the X-direction.

Dynamic simulation results under lifting load

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Data analysis shows that the maximum load applied in lifting off the ground condition is 27.77 KN, and the maximum vertical recoil force is 27.998 KN, with an impact amplification factor of 1.0082. The force of the horizontal impact is insignificant and can be disregarded. The maximum load during lifting off the ground occurs at 51.498 s, and the maximum vertical recoil force occurs at 51.560 s. Due to the damping in the pulley block system, the response of the pulley block lags behind the lifting load.

3.2 Dynamic Analysis Under Sudden Unloading Conditions

The sudden unloading condition is simulated by setting multiple load steps. The initial load P0 is set to 10000N and at 0.1s the initial load was unloaded within 1e-6s. After setting the load steps, transient dynamic analysis is implemented, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 5.

Fig. 5
Four line graphs plot wavering trends for variation with time. A. Displacement curve of the center of the moving pulley. B. Stress curve of the steel wire rope. C. Support reaction curve in the Y direction of the fixed pulley. D. Support reaction curve in the X direction of the fixed pulley.

Dynamic simulation results under sudden unloading conditions

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It is clear from examining the force–time curve of the support reaction at the fixed pulley's center that abruptly unloading the pulley block causes a quick drop in the force of the vertical reaction, which ultimately causes unfavorable vibrations that have a significant negative impact on the crane's overall functionality. The graph shows a maximum vertical recoil force of 7438.55N and an impact coefficient of 0.7438. The horizontal recoil force is negligible and can be ignored.

3.3 Analysis of Influencing Factors

  1. (1)

    Unloading Time: The unloading time of the sudden unloading condition is varied to explore its influence on the impact reaction force of the pulley block. The maximum recoil force is recorded, and a relationship curve between the maximum recoil force and the unloading time is plotted, as shown in Fig. 6. The relationship chart demonstrates that the maximum recoil force produced by the pulley block system diminishes as the unloading time increases.

    Fig. 6
    A line graph of m recoil force in Newton versus time in seconds plots a descending trend from (0.001, 8000) to (0.01, 0).

    The relationship curve between the maximum recoil force and the unloading time

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  2. (2)

    Wire Rope Length: To explore the influence of wire rope length on the impact force, the distance between the movable pulley and the fixed pulley in the model varies, and dynamic simulations are performed under sudden unloading conditions. As shown in Fig. 7, a relationship curve between the maximum recoil force and the pulley distance (wire rope length) is plotted after the maximum recoil force in the vertical direction is recorded. From the curve, it can be observed that there is generally a positive correlation between rope length and maximum recoil force. The longer the rope length, the greater its maximum recoil force generated in the pulley block.

    Fig. 7
    A line graph of maximum recoil force in Newton versus multiple pulley radius. It exhibits elevations with slight curves, indicating varied force levels with changing pulley distance.

    The relationship curve between the maximum recoil force and the pulley distance (wire rope length)

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4 Dynamic Simulation of Defective Pulley Block System

4.1 Dynamic Simulation of Hook System with Defects

In order to simulate the defect, the COMBIN37 element's switch function is employed to turn off the spring element at a predetermined time point, simulating the rupture or disengagement of the crane hook. The hook rupture or disengagement is achieved by setting multiple load steps, with different steps such as applying the initial load P0 = 10000N, breaking the hook at 0.1 s, and applying subsequent analysis in different load steps. A transient dynamic analysis is carried out, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 8.

Fig. 8
Four graphs plot wavering trends for variation with time. A. Displacement curve of the center of the moving pulley. B Stress curve of the steel wire rope. C. The support reaction curve in the Y direction of the fixed pulley. D. Support reaction curve in the X direction of the fixed pulley.

Dynamic simulation results of hook fracture

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By comparing the dynamic simulation results of the pulley block hook rupture impact and the sudden unloading condition, it is found that the pulley block responses are consistent as long as the hook rupture time and sudden unloading time are the same. Figure 9 shows the comparison diagrams of the displacement–time curve of the center of the movable pulley and the vertical support reaction force of the fixed pulley under the two conditions.

Fig. 9
A graph of reaction force versus time in seconds plots a fit trend that moves horizontally at 1 times 10 power 4 till 0.1 seconds and fluctuates between minus 8 times 10 power 3 and 8 times 10 power 3.

Comparison diagram of vertical support reaction force between hook fracture and sudden unloading of the fixed pulley

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4.2 Dynamic Simulation of Steel Wire Rope System with Defects

The spring element COMBIN37 in ANSYS simulates the defect and establishes a model with breakpoints on the wire ropes on both sides of the moving pulley. The model contains two COMBIN37 elements, the first COMBIN37 element has one node connected to the right end of the rope, and the other node uses a fixed constraint as support, while the other COMBIN37 unit is connected to the left wire rope and is symmetrical with the right one. The defective model of the wire rope is shown in Fig. 10.

Fig. 10
A diagram illustrates the defective wire rope model with major labels COMBIN 37 on both sides. It highlights the identification and significance of the combined factor in the rope's structural integrity.

The defective model of the wire rope

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Considering self-weight, an external load of P0 = 10000N is applied as a concentrated force at the node below the movable pulley of the COMBIN37 element and makes the wire rope break at 0.1s. A transient dynamic analysis is conducted under the condition of wire rope rupture unloading, and the displacement–time curve of the center of the movable pulley, the stress-time curve on the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 11.

Fig. 11
Four line graphs. A. Displacement U Y versus time plots a concave downward descending curve. B. Stress versus time. It plots a pinching waveform of descending amplitude. C and D. Force versus time for Y and X directions. Both plot a pinching waveform of descending amplitude.

Simulation analysis curve of broken rope in systems with defects

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The vertical support reaction force rapidly decreases and produces vibrations based on the pulley's self-weight when the wire rope breaks at 0.1 s, which can have a significant impact on the crane behavior, according to the force–time curve of the vertical support reaction at the fixed pulley's center. The maximum impact force recorded is 7421.3N, with an impact coefficient of 0.74213. In contrast to sudden unloading and hook rupture conditions, where the horizontal impact force is negligible and can be ignored, when the steel wire ropes on both sides of the moving pulley break simultaneously in a defective system will cause a more significant horizontal impact force, resulting in vibrations and oscillations in the horizontal support reaction force.

5 Conclusion

This paper establishes a parametric model of the crane pulley block using APDL commands of ANSYS. Based on contact analysis, static simulation verification is carried out on the contact model of the beam element rope pulley block. Dynamic simulations are conducted on the intact pulley block model under two dangerous working conditions: lifting off the ground and sudden unloading. Dynamic simulations are also performed on the hook-defective and the wire rope-defective pulley block models for hook and double-sided wire rope rupture, respectively. Through simulation comparison, it is concluded that the longer the unloading time, the smaller the impact generated in the pulley block under sudden unloading conditions. The longer the wire rope length, the greater the impact force generated in the pulley block under sudden unloading conditions. When the unloading time is consistent, the dynamic analysis of the intact pulley block under sudden unloading and the defective pulley block with hook rupture are equivalent. Unlike the sudden unloading and hook rupture conditions, where the horizontal impact force can be ignored, the steel wire ropes on both sides of the moving pulley break simultaneously in a defective system will cause a more significant horizontal impact force. This study demonstrates that ANSYS contact analysis with COMBIN37 as the defective element can accurately and efficiently apply dynamic simulation to a crane movable pulley system with defects.