Effect of Hoisting Load on Transverse Vibrations of Hoisting Catenaries in Floor Type Multirope Friction Mine Hoists

The effect of hoisting load on transverse vibrations of hoisting catenaries during lifting in coal mines was investigated in this study. Firstly, dynamic analyses of the vertical hoisting rope were performed. The results show that transverse vibration plays the dominating role in the coupled dynamics of the vertical rope subjected to external excitation induced by axial fluctuations of head sheave, and the rope tension can be approximated by the quasistatic tension. Secondly, employing dynamic simulations, the effect of hoisting load on the transverse vibrations of hoisting catenaries was discussed. The results show that, under the second-order excitation frequency, a discrepant large transverse amplitudewill be excitedwhen the hoisting load ranges from0 to 5000 kg, leading to collision between the two adjacent catenaries and accelerating the rupture of rope. To solve this problem, according to simulation curves, the self-weight of conveyance (preload) can be optimized from 39500 kg to 49500 kg. Eventually, on-site measurements were performed on the studied machine, validating the accuracy of the effect of hoisting load on transverse vibrations of hoisting catenaries.This investigationwill greatly support facilitymaintenance,machine design, and even engineering optimization for floor type multirope friction mine hoists.


Introduction
Steel wire ropes are widely used in many industrial applications due to their high axial strength and bending flexibility [1].In this latter application, cables connecting the mine hoist and the hoisting conveyance have played vital roles in mine hoisting systems because their endurance strength and fatigue life have great effect on the mine production and safety of miners [2].A mine multirope hoisting system is shown in Figure 1.It comprises a driving friction pulley, four hoisting ropes, two sets of head sheaves, two skips, and three tail ropes.The four hoisting ropes pass from the same friction pulley over the four singular head sheaves to the skip constrained to move in a vertical shaft, forming four upper and lower catenaries and four vertical ropes.
Due to axial tensile loads and axial fluctuations of the head heaves, large amplitude transverse vibrations are usually associated with the hoisting catenaries as shown in Figure 2. The resonance may make two adjacent catenaries collide with each other and thereby accelerate the rupture of the rope.
In the dynamics of hoisting rope in mine hoists, some researchers have done some significant work in recent years.Cao et al. [3,4] established the coupled extensional-torsional model for a frictional hoisting system with an oriented pulley and the extensional-torsional-lateral model for a winding hoisting system with a head heave.Wang et al. [5] investigated the lateral response of the moving hoisting conveyance in cable-guided hoisting system and revealed that the maximum lateral displacement is linearly proportional to the excitation amplitude.Goroshko [6] described the longitudinal-torsional vibrations of ropes, the elongation of which is obtained during untwisting and the twisting moment occurring under tension.Gong [7] proposed that resonance will be excited when an external excitation frequency approaches one mode of natural frequencies of the hoisting rope in a single-rope mine drum hoist; however, he did not study the influence factors.Kaczmarczyk [8] formulated an integral longitudinal model of the catenary-vertical hoisting cable system with a periodic excitation to analyze the passage through resonance.To investigate the dynamic behavior of a hoisting cable, Kaczmarczyk and Ostachowicz [9,10] derived a distributedparameter mathematical model by employing the classical moving coordinate frame approach and Hamilton's principle with no regard to the transverse motions of head sheave in a single-rope mine drum hoist.Mankowski and Cox [11] studied the longitudinal response of mine hoisting cable to a kinetic shock load.Wang et al. [12] investigated the effect of terminal mass on fretting and fatigue parameters of the hoisting rope during a lifting cycle in coal mine.
From the literature studies mentioned above, few literatures have been focused on transverse vibrations of the hoisting catenaries in a multirope friction mine hoist with varying hoisting load.Therefore, the objective of the present study is to investigate the effect of hoisting load on the transverse vibrations of hoisting catenaries in a floor type multirope friction mine hoist.Firstly, the dynamics of the vertical translating hoisting rope, which can help explore the actual rope tension under different hoisting loads, was investigated.Secondly, theoretical correlation model between hoisting load and transverse vibrations of hoisting catenaries was established in order to study the influence of hoisting load on the transverse displacements of hoisting catenaries.Eventually, on-site measurements were carried out to validate the accuracy of the effect of hoisting load on the transverse displacements of hoisting catenary, and thereby the optimization of the hoisting load can be performed to reduce large amplitude transverse vibrations of the hoisting catenary.

Dynamics of Hoisting Rope Tension
Due to vibrations of the flexible vertically translating hoisting rope during lifting [13], dynamic analyses of the vertical translating hoisting rope can help to understand the actual rope tension and provide basic data for the analyses of transverse vibrations of hoisting catenaries.

Dynamic Model.
To describe the dynamic behaviors of vertical hoisting ropes in colliery, assuming that the four vertical translating ropes have the same dynamic properties, only one rope was adopted as the study object and thereby a sliding coordinate system [14], oxy, was constructed as shown in Figure 3. Due to axial fluctuations of head sheaves, the transverse vibrations of vertical ropes are predominantly along the  direction in plane as shown in Figure 3.The hoisting rope has a time-varying length () at time instant , a vertically translating velocity V() = l (), acceleration () = l (), and a total length , where the overdot denotes time differentiation.The conveyance attached to the tip of the hoisting cable was modeled as a mass point with mass   ().The mass of the tail rope was integrated into mass of the conveyance.The displacement of the upper end of the cable, specified by (), represents external excitation that is induced by the axial fluctuations of the head sheave.The transverse and longitudinal displacements of the rope particle instantaneously located in spatial position  at time  are represented by (, ) and (, ), respectively.() denotes the actual position of the sliding coordinate origin which represents the hoisting conveyance.
The equations governing the dynamic behavior of the translating cable [15] in Figure 3 in the - plane are where the subscripts  and  denote partial differentiation; , , and  are the linear density, Young's modulus, and crosssectional area of the hoisting rope, respectively; and (, ) is the quasistatic tension in spatial position  at time  given by where  is the acceleration of gravity.Because the mass of the tail rope was integrated into the mass of the conveyance, hence the time-varying mass of the mass point in Figure 3 can be given by where the parameters   and   are relative to the mass of the conveyance and the actual hoisting load, respectively, and   denotes the linear density of the tail rope.The initial conditions of the cable are given by Normal Collision Upper catenaries Lower catenaries   The governing equation in (2) with the time-dependent boundary conditions in (6) can be transformed to one with homogeneous boundary conditions.The transverse displacement is expressed in the form [16]  (, ) =  (, ) + ℎ () , (7) where (, ) and ℎ(, ) are the parts that satisfy the corresponding homogeneous and nonhomogeneous boundary conditions, respectively.It is reasonable to assume that the lateral displacement from excitation varies from zero at the conveyance to () at the head sheave uniformly [15,16].Thus, the absolute displacement ℎ(, ) can be defined to be a first-order polynomial in  and expressed as Substituting ( 6) into (2) yields where is the additional forcing term induced by transforming the governing equation with time dependent nonhomogeneous boundary conditions to one with homogeneous boundary conditions.According to ( 6)-( 7), the boundary conditions of the vertical rope in Figure 3 where the subscript  denotes partial differentiation.Similarly, the partial derivatives of ℎ(, ) versus  and  are expressed as where the terms about ℎ * are given as follows: Substituting ( 3), (13), and ( 14) into ( 9)and omitting the highorder terms [16] yield where Substituting ( 12) into ( 1) and omitting the high-order terms yield where Galerkin method could be used to transform ( 16) and ( 18) into a set of ordinary differential equations by separating the transverse displacement as where  represents the order of the mode,   () and   () are the generalized coordinates, and the mode shape is Substituting ( 20) into ( 16) and ( 18), then multiplying ( 16) and ( 18) by   () −0.5 (), ( = 1, 2, . . ., ), and integrating ( 16) and ( 18) over the interval of 0 and 1, ordinary differential equations can be obtained as where  = [ 1 ,  2 , . . .,   ]  and  = [ 1 ,  2 , . . .,   ]  are the vectors of generalized coordinates and  ( lon ,  tr ),  ( lon ,  tr ),  ( lon ,  tr ), and  ( lon ,  tr ) are the mass, damping, stiffness matrices, and the force vector, respectively, and  ( lon ,  tr ) is the coupled term.Entries of these matrices are formulated as where   is the Kronecker delta defined by   = 1 if  =  and   = 0 if  ̸ = .

Hoisting Rope Tension.
The dynamic tension at the particle located in spatial position  at time  is [17]  (, ) =  (, ) +  (, ) , where (, ) and (, ) are the quasistatic and vibratory axial forces at the particle with the coordinate  at time , respectively, and (, ) is the strain in spatial position  at time  which can be approximated as [18]  (, ) =   (, ) + 0.5 Dynamic responses were calculated for a vertical hoisting rope in a multirope friction mine hoist.The parameters for the model shown in Figure 3 were  = 7.86 m/s,   = 10.48 m/s,  = 1.5246 × 10 8 N/m, and   = 39500 kg.The rope is assumed to be at rest initially; hence (, 0) =   (, 0) = (, 0) =   (, 0) = 0. Consider the upward movement profile in Figure 4, which is divided into three stages.The velocity function V() is given by the following formula: where the initial velocities at the 1st to 3rd stage, V 01 to V 03 , are 0, 9.31 m/s, and 9.31 m/s, respectively; the accelerations at the 1st to 3rd stage are 0.7 m/s 2 , 0, and −0.7 m/s  The boundary excitations are induced by axial fluctuations of head sheaves specified by () =  sin(), where  = 0.005 m and  =  × 2V()/., the diameter of the head sheave, has a value of 4.5 m.Applying the conclusion in reference [19] that the first three-order frequency components play the dominating roles in the axial fluctuations of head sheaves in mine hoists, hence,  = 1, 2, 3.The transverse displacements at the sheave end, which also act as the external displacement excitations, are demonstrated in Figure 5.To obtain the rope tension at the sheave end, according to (24)-(25), the dynamic strains at the sheave end should be calculated firstly and can be expressed as Under the above external displacement excitations, the longitudinal and transverse strains at the top of the vertical rope (sheave end,  = ()) were calculated as shown in Figures 6 and 7.The convergence of ( 22) can be examined by varying the number of included modes.In this case, the number of included modes can be determined as 10.
It can be inferred from Figures 6 and 7 that, under the 1stto 3rd-order excitations, the longitudinal strains at the sheave end are far less than the transverse no matter with full load or no load.Hence, it can be concluded that the transverse strain plays the dominating role in the coupled dynamics of the vertical hoisting rope.According to (27), the length of the vertical rope, (), and its exponent were used as the main divisor which can make the dynamic strain of the rope at the sheave end so small.For example, according to (27), the transverse strain at the sheave end can be expressed as 0.5(()/()) 2 and the range of () is −0.005 m to 0.005 m which is far less than the range of the length of the vertical rope (22 m to 699 m); therefore, it accounts for such small dynamic strain at the sheave end.Additionally, there is no obvious difference between the responses of the dynamic strain under the 1st-to 3rd-order excitations.Therefore, there is no dominant frequency of the oscillation.Applying (24), the quasistatic and vibratory axial force at the sheave end during the hoisting process can be obtained as shown in Figure 8.It can be noted that the vibratory axial force at the sheave end is much smaller than the quasistatic, no matter whether under the condition of no or full hoisting load.Hence, the tension in the vertical rope can be approximated by the quasistatic tension.

Effect of Hoisting Load on Transverse Vibrations of Hoisting Catenaries
The hoisting catenaries in a multirope friction mine hoist are typically moving cables with constant length and usually suffer intense transverse vibrations induced by axial fluctuations of head sheaves.As shown in Figures 2 and 9, large amplitude transverse vibrations will contribute to collision between two adjacent catenaries, accelerating the rupture of the rope.In the present work, the effect of hoisting load on transverse vibrations of catenaries was investigated.

Dynamic Model.
To describe the transverse vibrations of hoisting catenaries, a fixed coordinate system, oxy, is established.Due to axial fluctuations of head sheaves, the transverse vibrations of catenaries are predominantly along the  direction in plane as shown in Figure 9.The four catenaries wrap around the same friction pulley and attach to the four singular sheaves, respectively.In the present study, only one catenary was adopted to conduct the research for simplicity.The friction pulley is modeled as a fixed-center pulley, and the singular head sheave is modeled as a point subjected to axial fluctuating displacement specified by (), where  is the hoisting time.
The transverse displacement of the catenary in the position  at time ,   (, ), has been derived in our previous work [19]; its governing equation can be expressed as in which () and V() are the hoisting acceleration and velocity, respectively;  is the linear density of the hoisting rope; and   (, ) is the axial tension of the catenary located in position  at time .Due to the continuity of deflection across the head sheave, it is required that the dynamic tension in the catenary equals that at the top of the vertical rope [9] as shown in Figure 8.What is more, the effect of gravity due to a catenary inclination is small compared to the total quasistatic tension; hence, the effect of gravity can be omitted and thereby the axial tension in each point of the catenary can be approximated identically.Consequently, the axial tension in the catenary can be expressed as Applying the conclusion in Section 2.3 that the tension in the vertical rope can be approximated by the quasistatic tension, hence, according to (3)-( 4), the axial tension in the catenary can be derived as Assume that the transverse displacement of a catenary can be expressed as where   () is a set of generalized displacements of the catenary and sin() is a set of trial functions;   is the constant length of the catenary.Substituting (31) into (28) yields

Shock and Vibration
The 3rd-order excitation According to Galerkin method, the requirement is Inserting (32) into (33) and integrating the resulting equation yield the simplified equations of motion: (34)

Transverse Vibrations of the Catenary with Varying
Hoisting Load.In this section, the maximum transverse amplitudes at the center of the lower catenary with varying hoisting load are simulated.The fluctuating displacement of the head sheave can be specified by () =  sin(), where  = 2V()/ is the rotational frequency of the head sheave; V() is the hoisting speed in Figure 4(c); , the diameter of the head sheave, has a value of 4.5 m; and  is the number of the included order.According to the conclusion mentioned in [19] that the first three-order frequency components play the dominating roles in the axial fluctuations of head sheaves, therefore,  = 3.The constant length of the lower catenary, Lc, is 43.88 m, and the other parameters are the same as those introduced in Section 2.3.Finally, according to (31), the effects of hoisting load on transverse vibrations can be investigated as shown in Figures 10,11,and 12, respectively.It can be seen from Figures 10, 11, and 12 that the maximum amplitude at the center of the catenary increases with the increasing excitation amplitude.When the excitation amplitude is constant, with varying hoisting load under the first-and third-order excitation frequency, the maximum amplitude changes smoothly and has relatively small value that is less than 35 mm.However, as shown in Figure 11, under the second-order excitation frequency, a discrepant large transverse amplitude emerges when the hoisting load is within the range from 0 to 5000 kg.In this study, the rope spacing between two adjacent catenaries in the multirope friction mine hoist has a constant value of 350 mm.Therefore, if the maximum transverse amplitudes of two adjacent catenaries both exceed the dangerous threshold of 175 mm, collision between the two adjacent catenaries will result as shown in Figure 2, accelerating the rupture of the rope.Especially in the case that the hoisting load is about 2500 kg, even the excitation amplitude is as small as 1 mm and the maximum transverse amplitude under the second-order excitation frequency has exceeded the dangerous threshold of 175 mm, which is strictly prohibited in the industrial field of coal mines.
To guarantee the safety production in coal mines, large amplitude transverse vibrations must be avoided.The mine hoist mainly operates under two working conditions, that is, no hoisting load with   = 0 and full hoisting load with   = 20000 kg in this case.The self-weight of the conveyance (preload) in this case is   = 39500 kg.Hence, to avoid the discrepant large transverse amplitude, the self-weight of the conveyance (preload) can be optimized from 39500 kg to 49500 kg by adding 10000 kg, equal to transforming the original hoisting load range [0, 20000 kg] to a new range [10000 kg, 30000 kg] as shown in Figure 11.Eventually, the discrepant large transverse amplitude and the collision between two adjacent catenaries can be thoroughly avoided, making the mine hoist operate in stability.However, at the same time, it should be noted that increasing preload leads to higher mean stress which certainly in turn affects fatigue life of the rope.The JKMD-4.5 × 4(III) E floor type multirope friction mine hoist is the machine that the present work is focusing on, and the permitted maximum tension of a rope in this machine is 900 KN.Before optimization, according to (30), the maximum tension of a rope during the hoisting process with full load is 213.88KN, and the corresponding rope tension is 240.13KN after increasing the preload by 10000 kg.The maximum optimized rope tension, 240.13 KN, is still far less than the permitted, 900 KN.There is no doubt that increasing the rope tension will shorten the fatigue life of a rope; however, if collision between two adjacent catenaries occurs because of resonance as shown in Figure 11, though the fatigue life of ropes may be shortened, increasing preload can rapidly avoid resonance and still make the machine operate in safety for a long time.Therefore, increasing preload is still an acceptable method to rapidly remove the collision between catenaries.Furthermore, the proposed technique can also help the engineers to choose reasonable hoisting parameters in the design phase of the machine to avoid potential failures.

Measurement and Validation
4.1.On-Site Measurements.During the hoisting process as shown in Figure 4, real object tests were performed on the studied multirope friction hoist in the Yaoqiao Coal Mine of Shanghai Datun Joint Co. Ltd.
In the present work, lower catenary 4 was adopted as the study object for simplicity.Hence, to obtain the transverse displacements of the center point of lower catenary 4, a high-speed camera, which was placed beneath the moving catenaries at an arbitrary appropriate distance, was applied to record the sequential images during the hoisting process with no and full hoisting load, respectively.The processed image was shown in Figure 13.A guide line, which was perpendicular to the hoisting catenaries, was first inserted in the sequential images.Then, at the start frame in which the static equilibrium catenary was recorded, a tracking window was defined in the intersection area of lower catenary 4 and the guide line.To calculate the actual vibration displacement,  it is imperative to obtain the scaling factor .The length of line  in Figure 13(b) is relative to the rope spacing which has a constant value of 350 mm.The coordinates of points  and  are (544, 747) and (650, 718), respectively.And the pixel distance of line  can be calculated as 109.9.Thus, the scaling factor  can be calculated as 3.18 mm/pixel.Eventually, through tracking the dynamic intersection area by employing mean shift tracking algorithm [20][21][22] and then recording the location of the tracker, the transverse vibration displacements of the center point of the lower hoisting catenary 4 can be obtained as demonstrated in Figure 14.
During the measurement of transverse displacements of the lower catenary 4, the axial fluctuating displacements of lower head sheave 4 were recorded by using inductive displacement transducer as shown in Figure 15.The inductive displacement transducers fixed in the retainer were adjusted to face the outer rim of the sheaves.Employing fast Fourier transform, the amplitude spectrum of the axial fluctuating displacements of lower head sheave 4 can be obtained as shown in Figure 16.

Validation.
It can be seen from Figure 16 that the first three-order frequency components play the dominating roles in the axial fluctuating displacements of the head sheave.Thus, the external displacement excitations at the sheave end of the lower hoisting catenary 4 can be specified by 1.46 sin() mm, 6.1 sin(2) mm, and 2.1 sin(3) mm, respectively, where  is the fundamental angular frequency of the head sheave.Under the above external excitations, according to (31) and employing the corresponding parameters introduced in Section 3.2, the time histories of transverse displacements at the center of lower catenary 4 under no and full hoisting load were given in Figures 17 and 18, respectively.
Under the circumstance of full hoisting load with   = 20000 kg, it can be seen from Figure 17 that the firstorder response amplitude is much smaller than the secondand third-order amplitudes and thereby can be negligible.The maximum values of the second-and third-order response amplitude in Figure 17 are 28.96mm and 10.1 mm, respectively.According to superposition principle, the sum of the second-and third-order maximum amplitudes is 39.06 mm, which is much close to the measured 39.1 mm shown in Figure 14.Furthermore, the shapes of the vibrating waveforms under the second-and third-order excitations in Figure 17, especially the second-order excitation, are similar to the measured shapes as shown in Figure 14.
Under the circumstance of no hoisting load, it can be seen from Figure 18 that the first-and third-order response amplitudes are much smaller than the second-order response amplitude and thereby can be negligible.The maximum value of the second-order response amplitude in Figure 18 is 155.6 mm, which is much close to the measured 161.2 mm shown in Figure 14.Additionally, the shape of the vibrating waveform under the second-order excitation in Figure 18 is similar to the measured shape as shown in Figure 14.Figures 14 and 18 are interesting since experimental and numerical time histories are similar and present "beat."These seem associated mainly with the second-order excitation.To account for this issue, under the condition of no load, the corresponding amplitude spectrum of Figure 14(a) was given in Figure 19(a).It can be seen that the main vibration frequency is 1.36 Hz relating to the lower catenary 4. The hoisting velocity during the constant speed stage is 9.31 m/s; therefore, the fundamental excitation frequency can be obtained as 0.66 Hz by calculating the rotational frequency.It can be noted that the second-order excitation frequency of 1.32 Hz is much close to 1.36 Hz, the measured vibration frequency.Therefore, it can be concluded that the second-order excitation frequency is the main excitation frequency.To more clearly and accurately account for this issue, by varying the excitation frequency under the excitation 0.005 sin(), the maximum transverse amplitudes at the center of a catenary are depicted as shown in Figure 19   closer to the resonance frequency than the first-and thirdorder and the "beat" thereby resulted.
Some differences may result from model errors, which are acceptable when the primary purpose is fault diagnosis.In conclusion, the validity and applicability of the dynamic model and governing equation of the transverse vibrations of hoisting catenaries can be confirmed.Most important of all, the accuracy of the effect of hoisting load on transverse vibrations of hoisting catenaries can be validated, providing reasonable basis for the optimization of the hoisting load and even great help during the design phrase of the machine.

Conclusions
The dynamics of the vertical translating hoisting rope is particularly focused on in the present study.The theoretical model and numerical solution scheme provide an efficient approach to analyzing the transverse and longitudinal coupled vibrations of the vertical translating hoisting rope.The simulation results indicate that the transverse vibration plays the dominating role in the coupled dynamics of the vertical hoisting rope subjected to external excitation induced by axial Shock and Vibration fluctuations of the head sheave, and the rope tension can be approximated by the quasistatic tension.
According to the quasistatic tension of the hoisting rope, the effects of hoisting load on the transverse vibrations of the hoisting catenary were discussed employing dynamic simulations during lifting.Under the second-order excitation frequency, a discrepant large transverse amplitude will be excited when the hoisting load ranges from 0 to 5000 kg.Therefore, collision between two adjacent catenaries will result and the rupture of the rope will be thereby accelerated.To solve this problem, the self-weight of the conveyance (preload) can be optimized from 39500 kg to 49500 kg by adding 10000 kg according to the simulation curves.
On-site measurements were performed on the studied multirope friction hoist in coal mines.The simulation results were well consistent with the practically measured results, confirming the validity and applicability of the dynamic model and governing equation of the transverse vibrations of hoisting catenaries and validating the accuracy of the effect of hoisting load on transverse vibrations of hoisting catenaries.
Eventually, this investigation will provide great theoretical basis to realize resonance avoidance of the hoisting catenaries by optimizing the hoisting load in a multirope friction mine hoist.Furthermore, the research results will also be a great help during the design phrase of the machine.

Figure 1 :
Figure 1: Schematic of a floor type multirope friction hoist in coal mine.

Figure 3 :
Figure 3: Diagram of hoisting rope in coal mine.

(
) and 4(b), the initial and final lengths of the vertical rope are 699 m and 22 m, respectively, and the initial and final lengths of the tail rope are 14 m and 691 m, respectively.

Figure 4 :
Figure 4: Upward movement profile of the conveyance: (a) time-varying length of the tail rope; (b) time-varying length of the vertical rope; (c) hosting velocity; (d) hoisting acceleration.

Figure 5 :
Figure 5: The transverse displacements at the sheave end with different orders.

Figure 6 :
Figure 6: The longitudinal and transverse strains at the sheave end under the condition of no hoisting load,   = 0.

Figure 7 :
Figure 7: The longitudinal and transverse strains at the sheave end under the condition of full hoisting load,   = 20000 kg.

Figure 8 :Figure 9 :
Figure 8: The quasistatic and vibratory axial force at the sheave end during the hoisting process.

Figure 10 :
Figure 10: Maximum amplitude at the center of the catenary under the first-order excitation angular frequency with varying hoisting load.

Figure 11 :
Figure 11: Maximum amplitude at the center of the catenary under the second-order excitation angular frequency with varying hoisting load.

Figure 12 :
Figure 12: Maximum amplitude at the center of the catenary under the third-order excitation angular frequency with varying hoisting load.

Figure 13 :Figure 14 :
Figure 13: The processed image: (a) the global image and (b) the local image.

Figure 15 :
Figure 15: Measurement of the axial fluctuating displacements of head sheaves.
(b).It can be got that the second-order excitation angular frequency of 8.28 rad/s is

Figure 16 :
Figure 16: Amplitude spectrum of axial fluctuations of lower head sheave 4.

Figure 17 :
Figure 17: Simulation of transverse vibrations at the center of lower catenary 4 under the condition of full hoisting load.

Figure 18 :Figure 19 :
Figure 18: Simulation of transverse vibrations at the center of lower catenary 4 under the condition of no hoisting load.
The new variables are  * (, ) = (, ) and  * (, ) = (, ), and the new variable for ℎ(, ) is ℎ * (, ).The partial derivatives of (, ) and (, ) with respect to  and  relative to those of  * (, ) and  * (, ) with respect to  and  are 2, respectively; the hoisting time at the 1st stage is 13.3 s, and 59.4 s and 13.3 s correspond to the 2nd and 3rd stages.The maximum velocity and acceleration are 9.31 m/s and 0.7 m/s 2 , respectively, and the total travel time is 86 s.According to the time-varying lengths of the tail rope and the vertical rope shown in Figures4(a