Impact Dynamic Analysis of Suspended Marine Cables

Dapeng Zhang; Jin Yan; Keqiang Zhu

doi:10.53964/mset.2023004

Open Access |

Impact Dynamic Analysis of Suspended Marine Cables

Dapeng Zhang1, Jin Yan1*, Keqiang Zhu2

1Ship and Maritime College, Guangdong Ocean University, Zhanjiang, Guangdong Province, China

2Faculty of Maritime and Transportation, Ningbo University, Ningbo, Zhejiang Province, China

*Correspondence to: Jin Yan, PhD, Professor, Ship and Maritime College, Guangdong Ocean University, Zhanjiang 524005, Guangdong Province, China; E-mail: yanj@gdou.edu.cn

DOI: 10.53964/mset.2023004

Abstract

Objective: It is often impossible to avoid collisions between marine cables, the impact dynamic response of the collisions between marine cable needs to be obtained.

Methods: In this paper, we derive the lumped mass method based on the bending moment and torque; using the lumped parameter method, the suspended cable is discretized into lumped mass models, and the impact force of the suspended cable is considered in the dynamic analysis; the impact dynamics model of the suspension cable is developed by referring to the specific parameters of the suspension cable and combining them with offshore construction processes.

Results: There is a relationship between the impact force and the tension of the cable. The tension and impact force are related in cable collisions. While the maximum effective tension occurs at the upper end of the vertical cable and at the collision point, the horizontal cable's tension at the collision position is minimal. The distribution of curvature mutates at the collision point, causing bending deformation. The horizontal cable experiences greater bending change at the collision position than the vertical cable.

Conclusion: In summary, collision of different parts of vertical cable is not obvious, while horizontal cable is more prone to self-collision and collision with vertical cable. Collision impact on horizontal cable is greater, with minimum tension occurring at collision point. Maximum velocity for horizontal cable appears at collision position, and for vertical cable at bottom. Maximum acceleration for both cables occurs at collision zone.

Keywords: lumped mass method, collision of the marine cable, dynamic analysis, the impact force of the suspended cable, modeling

1 INTRODUCTION

With the increasing emphasis on the development of marine resources, all kinds of offshore structures construction work are becoming more and more frequent[1-3], especially offshore lifting operations[4] and marine cables.

Marine cable systems are widely used in several fields, such as ocean monitoring, military detection, seabed mapping, and naval defense. These systems enable you to acquire various industrial raw materials from the oceans.

Towing instruments can be used to find ocean mineral resources, using sonars to discover offshore seabeds, trawl systems for fishing deep waters, and trawl systems to install underwater cables. All of these methods can enhance sustainability. Underwater towed systems are used for ocean monitoring, ocean research, and military affairs. Generally, the system includes a marine exploration ship, a guided towline, pulled cable, an underwater vehicle, and cable detection and control equipment. Underwater trolling systems are subject to interference from the actual marine environment at all times. In addition, there is a complex interaction between the towing ships, the towed cables, and the hauled bodies[5-10].

Due to various causes, the impact between the marine cables is inevitable during the sea lifting process[11,12]. And the impact between the lifting cables is unfavorable to the construction on the sea. The rapid rebound in the process will cause great damage to the surrounding equipment and personnel[13]. Therefore, it is very necessary to analyze the impact process between the suspended marine cables, so as to ensure the safety of the construction process to the maximum extent.

In sharp contrast to the research of the dynamic analysis on marine lifting, there is less research on the impact between the suspended marine cables[14-16]. It has been found that the lifting suspended marine cables impacts the complex contact setting in comparison to a separate study of marine lifting systems. The twisting of the two cables during the impact process is extremely easy to occur as a result of the complex contact setting. The direct programming workload is too large, resulting in extremely difficult convergence of its calculation process[17]. There are many dynamic modeling methods of cables, such as lumped mass method, finite element method, finite difference method, and so on[18]. Among them, the physical meaning of lumped mass method is clear and its algorithm is simple and easy to understand, with a wide range of applicability and scalability[19-23]. As branched cables with complex topology structures are used extensively in engineering, it is necessary to study their modeling. The modeling of deformations of cable cross-sections with different shapes and complex internal structures would be meaningful and challenging without oversimplification[24].

Gao et al.[25] designed a test setup originally intended to examine structural and functional integrity in order to study impact capacity and structural impact failure mechanisms. The proposed experimental and numerical methods are well correlated and suitable for the assessment of the impact capacity of subsea power cables. They can also assist in the protection design of subsea power cables in the engineering field[25]. Based on ABAQUS, Zhang et al.[26] developed a three-dimensional dynamic response analysis model of the "ship anchor-three-core composite submarine cable-soil" to analyze the effect of anchor impact angle on cable damage. According to the results, anchor speed has a greater impact on cable damage than anchor mass. With an increase in buried depth, the impact angle of the conductor and optical fiber unit decreases. Cables can be protected by burying them at a depth greater than 1.9 times their diameter[26]. In order to assess the safety of a newly designed matrix submarine power cable protector assembled with reinforced concrete blocks, Yoon and Na[27] conducted anchorage tests. As a result of all the tests, it was determined that the mattress was not capable of protecting the power cables from anchor collisions. Pipeline deformation, damage, fracture, and simulation of the safety zone for power cables[27]. The pure mechanical ultimate strength of DC and AC transmission cables under axial loads was determined by Ehlers et al[28].

Combined with the lumped mass method, the hanging cable impact dynamic model simulation was established. And the nonlinear dynamic characteristics of the impact process have been obtained by the time domain coupled dynamic analysis method. In order to maximize the authenticity of the simulation, the time step of the simulation must be shorter than the shortest natural node cycle and should not exceed 1/10 of the shortest natural cycle of the model. Combined with the results of dynamic simulation, some guidance has been given, which is important to ensure safe operation.

2 MATERIALS AND METHODS

2.1 Marine Cable Dynamic Model

The main structure of the cable in this paper is concerned with the marine suspended cables, which can bear large axial tensile force and has greater flexibility, but its bend-resist ability is very weak. In order to reflect the dynamic characteristics of the cable structure accurately, the lumped mass method was used to build cable dynamics model.

The method can be applied to the large deformation dynamic analysis of marine flexible cable. As the effects of bending and torsion are considered, it can also be used in the analysis of the submarine pipelines and risers. As a result of its clear physical meaning and its simple and intuitive algorithm, the advantage of this method is its simplicity and generality[30-32].

The advantage of this method is that it is generally applicable to the coastal cable structure and each node only uses three translational degrees of freedom and a rotational variable. The node is connected by linear elastic element, all the force (damping force, gravity, etc.) are considered to be acting on a node. In the next step, the equations of motion for each node are listed, and numerical integration is performed on the computer to compute the instantaneous position and velocity components, as well as the tension, which changes over time, for each node.

2.2 Establishment of Mathematical Model

In order to simplify the calculation, the cable structure is considered as a smooth, circular cross section and can be extended without any attached components. As the suspended marine cables are above the water surface and the operation is generally in a calm sea, and the main research of this paper is the effect of the impact on the suspended marine cable. As a simplified assumption, we assume there are no wave loads present, so the hydrodynamic load caused by the wave can be ignored. According to the above, the static equilibrium and moment equilibrium relation of the suspension cable are listed:

represents the length direction of the suspended marine cable, is the weight of the unit length of the suspended marine cable.

Torque vector , moment vector and effective tension Te are respectively related to torsional stiffness GIp, bending stiffness EI and axial stiffness EA:

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$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 4.jpg资源 4$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 5.jpg资源 5$

The torque H and the distributed load can be obtained from (2) multiplied by the unit tangent vector to get the scalar formula:

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2.3 The Discrete Numerical Method for Solving the Lumped Mass Method

In general, we assume that the suspended marine cable is slender, flexible, and cylindrical. The numerical solution of the boundary problem for suspended marine cable is the discrete lumped mass method. The basic idea of the lumped mass method is to divide the suspended marine cable into N line segments, and the mass of each segment is lumped to a node, so there are N+1 nodes. The tension T at the end of each section and the shear force V can be seen as a centralized function at a node, any external hydrodynamic loads and other properties are all lumped to the nodes. In the continuity equation, the lumped mass model is replaced by the finite difference method to obtain the numerical solution.

Some mathematical expressions are defined as follows:

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$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 8.jpg资源 8$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 9.jpg资源 9$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 10.jpg资源 10$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 11.jpg资源 11$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 12.jpg资源 12$

Here, ,,, represents the number of nodes, k=0 represents the bottom end node. If the bottom end of the cable is fixed, there is no bending moment and torque at the bottom end node.

Based on the expression form of Equation (11) and Equation (12), Equation (2), Equation (4) and Equation (5) can also be written as:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 13.jpg资源 13$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 14.jpg资源 14$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 15.jpg资源 15$

Combined with Equation (8) and Equation (10), Equation (13) can be written as the following expression:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 16.jpg资源 16$

After combining Equation (14) and Equation (15), we can obtain the expression form of Equation (18):

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Equation (17) can be simplified as Equation (18):

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 18.jpg资源 18$

Then, we get Equation (19) by Multiply to Equation (18):

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As , we can get the following Equation (20):

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 20.jpg资源 20$

After the numerical simulation analysis, the equation of motion of the K-th node is expressed as:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 21.jpg资源 21$

Where

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 22.jpg资源 22$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 23.jpg资源 23$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 24.jpg资源 24$

According to the mathematical relationship, Equation (25) is shown in the following:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 25.jpg资源 25$

As , Equation (25) can be expressed as Equation (26):

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 26.jpg资源 26$

The presented method can also be used for the calculation of the following Equation (27):

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 27.jpg资源 27$

To make it easier to express, we put forward a variable symbol Qk:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 28.jpg资源 28$

Based on Equation (26), Equation (27) and Equation (28), each term of Equation (25) can be expressed as:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 29.jpg资源 29$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 30.jpg资源 30$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 31.jpg资源 31$

The difference of the external force in Equation (21) of motion can be expressed as:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 32.jpg资源 32$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 33.jpg资源 33$

By Equations (1)-(33), Equation (21) can be represented in the form of a matrix:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 34.jpg资源 34$

Where denotes the k-th segmental stiffness matrix, which consists of the following five sub-matrices.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 35.jpg资源 35$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 36.jpg资源 36$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 37.jpg资源 37$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 38.jpg资源 38$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 39.jpg资源 39$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 40.jpg资源 40$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 41.jpg资源 41$

Based on Equation (21), the external forces of Equation (33) are as the following expression:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 42.jpg资源 42$

The velocity and acceleration of the nodes in Equation (42) can be obtained by using the Newmark-β algorithm:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 43.jpg资源 43$

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 44.jpg资源 44$

Therefore, for the time step (n), the equation of motion in the inertial frame is:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 45.jpg资源 45$

[K] is the inertial stiffness matrix of 3N × 3N, {R} and {Fe} are the three-dimensional nodal position vector and the external force vector, respectively. Through the numerical integration of the above-mentioned single-node position vector and velocity vector, we can get the nodal displacement, velocity vector and pipeline tension with time response.

2.4 Calculation of the Impact Force of Suspended Marine Cables

When the collision is considered, two suspended marine cables are respectively equivalent to a flexible body model of several cylinders. When the two cylinders contact each other, the normal contact force is calculated by the model based on the penalty function, the actual objects in the collision process will be equivalent as a nonlinear spring damping model based on penetration depth, which is the most commonly used in collision model. In this collision model, the most direct impact force is the normal contact force. In order to simplify the calculation process, the influence of the normal contact force is mainly considered, and the tangential contact force is not considered. R1 and R2 are the outer radius of the umbilical cable and riser respectively. To make it clear to understand, we name the distance between the two middle axial axes of the umbilical cable and riser δ. When δ<R1 + R2, the two collide and the interference occurs; when δ> R1 + R2, the two don’t collide and the interference doesn’t occur; when δ = R1 + R2, the contact relationship between the two is at the edge where the collision occurs and does not collide; in other words, at this time, the two cables will not collide, The two just began to come into contact with each other, but also did not produce the role of force. As shown in Figure 1. Contact about the relation between these three kinds of cable-to-cable, if placed in the plane, the relationship between the two cables can be described as the contact relationship of the intersection, separation, being externally tangent of two circles.

$C:\Users\Administrator\Desktop\预发布\已预发布\MSET20220253-终审\排版\图片终版\58.jpg58$

Figure 1. Schematic diagram of contact relationship for Cable-cable.

The impact contact force of interference between cable-to-cable in OrcaFlex can be expressed as FI:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 46.jpg资源 46$

Among them, δ is the distance between the two middle axial axes of the cables, Fd is the structural damping force, and K is the coefficient of the contact stiffness and its expression is as the following:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 47.jpg资源 47$

K2, K1, respectively, is the contact coefficient of the two cables, the unit is KN/m.

The expression of structural damping force is:

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\公式\资源 48.jpg资源 48$

Where c is the structural damping coefficient, v is the relative velocity of the two during the collisions, and when the relative velocity is less than zero, it means that there is no damping force between the two cables; when the relative velocity is greater than zero it means that the two cables are getting close with each other, there is the role of damping force. It is important to note that in OrcaFlex the coefficient c is directly assigned a value of 0 in the calculation of the structural damping force if the integration method is chosen for the implicit integration. In this paper, the implicit integration is chosen to simplify the calculation results, so the impact force of this paper is not fully considered the influence of structural damping force. The influence of structural damping force will be further considered in the next study.

2.5 The Establishment of Dynamic Simulation Model in OrcaFlex

In order to simplify the modeling process and enhance the possibility of collision, a suspended marine cable is built in horizontal direction, and a vertical cable is built in vertical direction. Both ends of the horizontal cable are hinged during the whole simulation. The coordinates of the two ends of the horizontal cable are end1 (x1=-33.2m, y1=0m, z1=153.5m), end2 (x2=49.2m, y2=0m, z2=153.5m) respectively. The upper end of the vertical cable is always hinged and suspended 180m above the sea level (specific coordinates: x2=9.8m, y2=8.7m, z2=180m), the lower end is hinged to the sea level above 130.7m during the modeling static equilibrium stage (specific coordinate is x3=7.4m , y3=-64.3m, z3=130.7m).After the static equilibrium stage, the upper end termination constraints is released, as the initial state of the vertical cable two ends are not in a vertical line with a certain angle and its own gravity, The vertical cable starts to rotate around the upper point and eventually collides with the other horizontal cable.

The two cables are the same material and length: the length of the two cables is 100m, the diameter is 0.35m, linear density is 0.12t/m, Poisson's ratio is 0.5, the axial stiffness EA is 6000KN, bending stiffness 50KN/m2, the torsional Stiffness is 0, the normal contact coefficient of 5000KN/m. To ensure effective contact between nodes and nodes during collision, the two cables are divided into 50 segments with a single segment length of 2m. The simulation time is 15s. After the model is finished, it is shown in Figure 2.

$C:\Users\Administrator\Desktop\预发布\已预发布\MSET20220253-终审\排版\图片终版\62.jpg62$

Figure 2. The schematic model of the collision for suspended cables.

3 RESULTS AND DISCUSSIONS

3.1 The Calculation Results

The simulation results show that the 26th node of the vertical cable will collide with the 27th node of the horizontal cable.

3.2 Dynamic Results of the Impact Force of the Cable

With the observation of the results of the impact force in Figure 3, we have found that the more obvious collision is mainly the two short time of collision between the horizontal cable and the vertical cable (at 2s and 5s), after the first collision between the vertical cable and the horizontal cable, the two cables rapidly bounce off each other, and then the second collision occurs again. In addition to these two more obvious collision, the vertical cable of the impact force is maintained at 0KN, which shows that there is no obvious occurrence of collision of different parts of the vertical cable itself in the whole process. For the horizontal cable, the results show that, in addition to the collision with the vertical cable, the collision of the different parts of the horizontal cable itself will occur; that is, a sudden collision of the vertical cable with the horizontal cable will change the spatial configuration of the horizontal cable, resulting in a slight amplitude collision with each other between adjacent nodes adjacent to the 27 node of the horizontal cable, it is not difficult to find this in the time domain image of the impact force between the horizontal cable and the vertical cable. At the same time, it was observed that the collision force (80KN) of the vertical cables is far less than that of the horizontal cables (300KN) when the cables were first collided with each other, indicating that the impact of the cable is far greater than that of the vertical cables , the reason for this phenomenon is: relative to the vertical cable, both ends of the horizontal cable are hinged, and the free end of the vertical cable is free of constraint, when the collision occurs, the impact of the collision is released with the swing of the free end of the vertical cable in time; as the horizontal cable is hinged at both ends, the impact of the collision can not be released in time, therefore, the impact force of the horizontal cable is larger than the vertical cable at the first time, but as its two ends are hinged, so that the different parts cross sections of the horizontal cable itself are more likely to collide with each other.

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Figure 3. Contact impact force curve between two cables.

3.3 Dynamic Analysis Results of the Effective Tension and Bending

The curve of the tension of cables in time domain in Figure 4 shows that: for the lower end of the vertical cable, the lower end of the vertical cable is no longer subjected to the pulling effect after releasing the restriction for the lower end; therefore, after the end of static equilibrium stage, the tension of the lower end of the vertical cable is kept at 0KN during the dynamic stage; the tension of the upper end of the vertical cable and the tension of the 26th node fluctuate continuously with time and the tension of the upper end of the vertical cable is always larger than the tension of the 26th node where the impact has happened, but the tension at the 26 node is larger than that of the lower end of the vertical cable, which is due to severe jitter caused by the collision which makes the nodes below the 26th node have a large traction; for the horizontal cable, the 27 node where the collision happens is not the maximum point of the tension value during the collision, but rather the tension value of the horizontal hinged end1, end2 is relatively large, this is also due to the impact of the collision which is constrained by the two hinged ends is not timely released. At the same time, the maximum value of the tension of the vertical cable (225KN) is larger than the maximum value of the cable tension (150KN). That is to say, for the vertical cable, the maximum value of the effective tension occurs at the upper end, the tension at the collision point is also large, the tension of the free end of is 0; for the horizontal cable, the tension of the collision position is minimal, the tension values of both ends are basically the same. The distribution of the tension along the cable length is shown in the figure below. Figure 5 shows that the distribution of the tension of the horizontal cable is gradually decreasing along the length direction, and the tension of the vertical cable is symmetrical about the collision node.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\64.jpg64$

Figure 4. The tension of cables versus time.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\65.jpg65$

Figure 5. The tension of cables along the length.

It is found that the distributions of the curvature along the length of both the horizontal cable and the vertical cable have a mutation at the collision place, which makes the curvature curve not smooth at the collision position. Meanwhile, it is found that, for the vertical cable, the collision will make the upper 0-10m area and the 10m area around the 26th node(40m-60m) have more obvious bending and the bending of other parts of the vertical cable is not very obvious; for the horizontal cable, the degree of bending from the 27th collision node to both ends of the cable descends in turn. The change of the curvature of the cables at the collision position shows that the impact collision has caused the repeated bending deformation in the contact area of the two cables, and the degree of bending change of the horizontal cable at the collision position is greater than that of the vertical cable at the collision position. Figure 6 and Figure 7 show that the configurations of the curvature and bending moment along the length direction for both the horizontal cable and the vertical cable have a certain degree of similarity, which to a certain extent has verified the correspondence between the curvature and the bending moment.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\66.jpg66$

Figure 6. The distribution curvature of cables.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\67.jpg67$

Figure 7. The bending moment of cables.

3.4 Kinematic Analysis of Suspended Marine Cables

It has been found in Figure 8 that the vertical cable is constantly in the process of increasing the velocity and decreasing the acceleration during the static stage to the stage of the occurrence of the collision, but the horizontal cable is basically in a static state; after the occurrence of the collision, the velocity of the 26th node of the vertical cable is decreased to 0 instantaneously, and then the impact force starts to exert a reverse acceleration on the 26th node of the vertical cable, with time going on, the two cables are out of contact when the 26 node of the vertical cable obtains the maximum velocity and the velocity of the 26th node of the vertical cable goes down . While the horizontal cable is the quiescent state before the collision; at the moment of collision, the speed is accelerated to the maximum value, and then deceleration is started; when the velocity decelerates to 0, the velocity is accelerated in reverse direction, and then decelerates to a maximum value again. It is worth mentioning that, in the whole process, at the moment of the collision, the acceleration of both the 27th node of the horizontal cable and the 26th node of the vertical cable reach the maximum value respectively, and the maximum acceleration of the transverse cable is greater than the maximum acceleration of the vertical cable.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\68.jpg68$

Figure 8. The curves of velocities and accelerations for impact position.

Figure 9 shows that the maximum velocity for the horizontal cable appears at the two cable collision position but for the vertical cable, it is at the bottom where the maximum velocity appears; the areas of maximum acceleration for both the horizontal cable and the vertical cable are all located at the zone of collision: the velocity of the vertical cable is gradually increasing along the length direction, and the velocity of the horizontal cable is decreasing from the collision point to the both ends of the horizontal cable along the length direction, whose distribution is similar to the inverted parabola shape.

$C:\Users\Administrator\Desktop\预发布\MSET20220253\排版\图片终版\69.jpg69$

Figure 9. The curves of velocities and accelerations along the length direction.

4 CONCLUSION

1) There is no obvious occurrence of collision of different parts of the vertical cable itself in the whole process. For the horizontal cable, in addition to the collision with the vertical cable, the collision of the different parts of the horizontal cable itself will occur.

2) The impact of the horizontal cable is far greater than that of the vertical cables, the different parts cross sections of the horizontal cable itself are more likely to collide with each other.

3) For the vertical cable, the maximum value of the effective tension occurs at the upper end, the tension at the collision point is also large; for the horizontal cable, the tension of the collision position is minimal, the tension values of both ends are basically the same.

4) The distributions of the curvature along the length of both the horizontal cable and the vertical cable have a mutation at the collision place; the impact collision has caused the repeated bending deformation in the contact area of the two cables, and the degree of bending change of the horizontal cable at the collision position is greater than that of the vertical cable at the collision position.

5) The maximum velocity for the horizontal cable appears at the two cable collision position but for the vertical cable, it is at the bottom where the maximum velocity appears; the areas of maximum acceleration for both the horizontal cable and the vertical cable are all located at the zone of collision.

Acknowledgements

Not applicable.

Conflicts of Interest

The authors declared that there are no conflict of interest.

Author Contribution

Zhang D, Yan J, and Zhu K wrote and designed the manuscript. All of them contributed to the manuscript and approved the final version.

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