Experimental research on effect of wire rope transverse vibration on friction transmission stability in a friction hoisting system

Tribology International 115 (2017) 233–245

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Experimental research on effect of wire rope transverse vibration on friction transmission stability in a friction hoisting system Yongbo Guo a, Dekun Zhang b, *, Xuehui Yang b, Cunao Feng b, Shirong Ge a a b

School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, 221116, China School of Materials Science and Engineering, China University of Mining and Technology, Xuzhou, 221116, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Transverse vibration Wire rope Friction transmission Transmission stability

The transverse vibration of a wire rope and the effect on friction transmission stability in a friction hoisting system are investigated. A simulation model is established by Adams/cable and then it is validated by an experimental device using the time-frequency technology. The influence of different factors (including speed, acceleration, load, rope length) on transverse vibration is discussed. A method is employed to weaken the transverse vibration by adjusting the initial acceleration time. The results show that, the transverse vibration reduces the actual contact area of the wrap angle. The intensification of transverse displacement at the tangent point of the pulley reduces the friction force of the following rope in the wrap angle and increases the instability of the friction force.

1. Introduction

investigate the dynamic behavior of deep mine hoisting cables. The discrete model describes the modal interactions between lateral oscillations of the catenary cable. Based on the Hamilton principle, Zhu et al. [10] established a new wave method to get the exact response of an elevator string with constant tension and arbitrarily varying length for general initial conditions and external excitation. And in his recent study [11], a spatial discretization and substructure method is developed to accurately calculate dynamic responses of one-dimensional structural systems. The methodology is applied to several moving elevator cable-car systems in Part II [12]. In terms of vibration control, Yao et al. [5] established assessment criteria for axial fluctuating displacements of head sheaves to avoid large transverse displacements of inclined ropes in a multi-rope friction hoist. Chen et al. [13] found that, when the lateral and rotational damper coefficients are properly balanced, the damping enhancement of the cable can reach up to 30 percent compared to the case with only the lateral damper. The friction characteristics between the steel wire rope and friction lining are also widely studied by many scholars in the field of friction. Ge et al. [14] investigated the coefficient of friction (COF) between a wire rope and polyvinylchloride (PVC) lining at low sliding speeds as a function of velocity and load. They found that the COF decreased with an increase in velocity or pressure. Peng et al. [15] found that an increase in contact stress of lining within a certain range will increase the friction coefficient. Zhang et al. [16] studied the bending fatigue behavior and failure mechanisms of wire ropes. When working around nylon pulleys, wire ropes exhibit a slowly increasing of fracture rate and total damage in

Multi-rope friction hoist has wide applications in mine hoists [1] and elevators [2]. An assembly of a multi-rope friction hoist system is shown in Fig. 1, which comprises a friction pulley, guide pulley, several steel wire ropes, several tail ropes and two conveyances. The device, driven by friction between ropes and friction lining fixed at the surface of a pulley, has the advantages of small power, small size and large driving force, etc. In recent years, as the building height and mining depth increase, the rope length and hoist speed also increase a lot, which leads to the increase of transverse vibration of steel wire rope [3,4]. Severe transverse vibration will cause rope collisions and affect the stationarity of the lifting conveyance [5]. Especially in a deep heavy or high speed hoist, the rope vibration will cause dynamic contact characteristic between the rope and lining (dynamic stress distribution, local slip etc.) [6], thereby generating a power vibration source in the key drive module of the system. The vibration of the power source reacts with the wire rope. It induces a coupling effect which is detrimental to the stability of the entire hoisting system. Rope sliding will occur if the coupling effect is serious due to the insufficient friction force, which might cause over-winding, over-falling and even cage falling [7] Therefore, the transverse vibration of rope and the effect on the friction transmission stability need to be combined for discussion. Many researches were focused on steel wire rope vibration, including transverse, longitudinal and coupling characteristics. Kaczmarczyk [8,9] established a classical distributed-parameter mathematical model to * Corresponding author. Tel.: þ86 13952207958. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.triboint.2017.05.033 Received 29 March 2017; Received in revised form 16 May 2017; Accepted 22 May 2017 Available online 23 May 2017 0301-679X/© 2017 Elsevier Ltd. All rights reserved.

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(mass, inertia, beam-formulation-based longitudinal, bending and torsional stiffnesses) in Cable module. There is also an inline joint primitive between each cable part in order to provide control over the bending angle and transverse displacement at the ends of each beam. The rope segments are represented graphically as spheres so as to exactly reflect the geometry used in the rope-to-pulley contact detection process. Contact with pulleys is applied with forces using an optimized analytical formulation in the plane together with an appropriate lateral guidance approximation. The discretized rope will compute precise rope vibrations and forces on pulleys in scenarios where the mass and inertia effects of the rope are important. Pulleys can be offset from base plane and rotated in and out of plane during design time and disengage from the cable during the course of a simulation. The joints between rope elements are defined by the BEAM [19] in the Cable module. The BEAM statement defines a massless elastic beam with a uniform cross section. The beam transmits forces and torques between two markers in accordance with either linear Timoshenko beam theory or the non-linear Euler-Bernoulli theory. Euler-Bernoulli theory is evolved on the basis of Timoshenko beam theory. The theory has more precise characteristics in the case of high axial forces in the beam. Thus it is suitable for rope hoisting conditions, which the axial load can often reach several tons or even tens of tons. Fig. 2 shows two markers (I and J) that define the beam extremities and indicates at twelve forces (s1 to s12) they produce. The x-axis of the J marker defines the centroidal axis of the beam. The y-axis and z-axis of the J marker are the principal axes of the cross section. They are perpendicular to the x-axis and to each other. When the beam is in an undeflected position, the I marker has the same angular orientation as the J marker, and the I marker lies on the x-axis of the J marker a distance LENGTH (L) away. The beam statement applies the following forces to the marker I in response to the relative motion of the marker I with respect to the marker J. Where,

Friction pulley

Hoisting rope

Guide pulley

Conveyance

Tail rope

Fig. 1. Diagram of multi-rope friction hoist system.

one lay length. The bending fatigue life of wire ropes is twice longer than that of ropes working around steel pulleys. Zhu et al. [17] found that both the whole temperature and the equivalent stresses of the brake shoe increased and then decreased during mine hoist emergency braking; region of 1–3 mm below the friction surface was suffered to larger temperature gradients and stresses. Zhang et al. [1] studied the effects of the microstructure and basic properties of different friction linings on friction. Three commercial grade linings with similar compositions were investigated (trade names: K25, G30, and GM-3). Reciprocating sliding friction and wear tests indicated that higher contents of methylene and filler improve the friction coefficient. From the literature studies mentioned above, previous efforts mainly focuses on the rope vibration itself or friction characteristics themselves between the friction lining and rope. Most of researches of rope vibration are based on the Hamilton principle. The boundary conditions at the rope meeting or separation point on the pulley are all zero [8,10]. Thus, the junction of hanging segment and contact segment of the wire rope is truncated by the boundary condition. Hence, these numerical methods do not establish a relation between the rope dynamics and the contact within the wrap angle. Wang et al. [6,18] investigated the roles of hoisting parameters on contact states, slip amplitude and stress distributions along the contact path using finite element analyses. They found that the dynamic contact status during hoisting consisted of a slip regime and mixed regime. But the article studied the effect of the longitudinal tension of the wire rope on the contact, and did not involve the transverse vibration. The researches of friction characteristics between the rope and friction lining are mostly focused on the mechanism of friction and material properties. Most of their experimental conditions are static. But the actual contact and slip conditions between the rope and pulley within the wrap angle are changing dynamically. Few studies of rope transverse vibration and the effect on the friction transmission stability during friction hoisting were reported. The objective of the present study is to explore rope transverse vibration and effects on dynamic friction transmission stability in a friction hoist system. The results are conducive to improve the friction transmission stability in a friction hoist system. Section 2 presents a simulation model and experimental device. In Section 3, the transverse vibration characteristics of rope and effect of different parameters on vibration are discussed; a method of transverse vibration suppression is employed. The transverse vibration of rope close to the friction pulley and the friction characteristics of rope within the wrap angle are obtained. The relation between transverse vibration of rope and friction behaviors is revealed.

   

s1 and s7 are axial forces. s5, s6, s11, and s12 are bending moments about the y-axis and z-axis. s4 and s10 are twisting moments about the x-axis. s2, s3, s8, and s9 are shear forces.

The Timoshenko beam theory uses a linear translational and linear rotational action-reaction force between the two markers I and J. The forces the beam produces are linearly dependent on the displacements, rotations, and corresponding velocities between the markers at its endpoints. The following constitutive equations define how the beam applies a force and a torque to the I marker depending on the displacement rotation and velocity of the I marker relative to the J marker.

y s8

y

s11 s2

s10

I

s5 s4 s1

2. Research methods

x

J s6

s12 z

s3

z

L

2.1. Cable module support theory The rope is discretized with appropriate parts, joints and forces

Fig. 2. Definition of the beam. 234

s7

x

s9

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2

3

K11 0 6 0 K22 6 7 6 7 0 7 ¼ 6 0 6 0 7 0 6 5 4 0 0 0 K62 2 c11 c21 6 c21 c22 6 6 c31 c32 6 6 c41 c42 6 4 c51 c52 c61 c62

Fx Fy Fz Tx Ty Tz

0 0 K33 0 K53 0

0 0 0 K44 0 0

c31 c32 c33 c43 c53 c63

c41 c42 c43 c44 c54 c64

0 0 K35 0 K55 0 c51 c52 c53 c54 c55 c65

3 32 xL 0 7 6 K26 7 76 y 7 6 z 7 0 7 7 76 7 6 0 7 76 a 7 4 5 b 5 0 c K66 32 3 Vx c61 6 7 c62 7 76 Vy 7 6 7 c63 7 76 Vz 7 6 7 c64 7 7 6 ωx 7 c65 54 ωy 5 c66 ωz

L0 is the instantaneous vector from the marker J to the marker I. The Euler-Bernoulli theory, which the Cable module employed, uses a non-linear translational and non-linear rotational action-reaction force between two markers I and J. The following constitutive equations define how the beam applies a force and a torque to the marker I depending on the displacement rotation and velocity of the marker I relative to the marker J.

2 6 6 6 6 4

(1)

where,

2

N ¼ K11 ðx  LÞ þ d

Fig. 3 shows the hoist system model built by the parametric integration module Cable in Adams. Wherein, the total length, diameter, density, Young's modulus and divided element number of steel wire rope are 8m, 8 mm, 4.78  105 kg/mm3 (0.24 kg/m), 105 N/mm2 and 250, respectively. The Hertz contact formulation stiffness coefficient, exponent, maximal damping and friction coefficient between rope and friction lining are 104 N/mm, 2, 0.1 and 0.27, respectively. The friction pulley sizes are shown in the figure. The values are: depth ¼ 4 mm, angle ¼ 0.01 rad, radius ¼ 4 mm, diameter ¼ 640 mm and width ¼ 50 mm. The wrap angle is 180 . Sliding guide is used as the weight guide system. There are rectangular grooves on both sides of the block. Each gap between the block and guide is 1 mm. The Hertz contact formulation stiffness coefficient, exponent, maximal damping and friction coefficient between block and guide are 105 N/mm, 2.2, 10 and 0.01, respectively. The initial abscissas of the lifting side and lowing side are 320 mm and 320 mm, respectively. The model simulation time is the lifting time. The step size and error value of the solution are 0.01 s and 0.001, respectively.

(2)

(3)

where, E is the Young's modulus of elasticity, A is the cross-sectional area, L is the length of the beam, Iyy and Izz are principal area moments of inertia about the y-axis and z-axis, respectively. G is shear modulus. ASY and ASZ specifies the correction factor (that is, the shear area ratio) for shear deflection in the y direction and z direction, respectively. For Timoshenko beams [20](1.11 for the solid circle):

ASY ¼

A ∫ Iy2 A



Qy Iz

2 dA

(4)

where, Qy is the first moment of cross-sectional area to be sheared by a force in the z direction, and Iz is the cross section dimension in the z direction. The equilibrating force and torque at the marker J are:

Fj ¼ Fi Tj ¼ Ti  L0 xFi

(7)

2.2. Simulation model

where,

  Py ¼ 12E⋅Izz⋅ASY GAL2  2 Pz ¼ 12E⋅Iyy⋅ASZ GAL

(6)

The term d corresponds to the axial viscous forces. For the wire rope, the model is not refined to the inner layer structure. So there is no viscous force. The default value is 0.

The damping matrix Cij and stiffness matrix Kij, are symmetric, that is, Cij ¼ Cji and Kij ¼ Kji. You define twenty one unique damping coefficients when the BEAM statement is written. The theory defines each Kij as follows:

9 ¼ EA=L   >  > > 3 > ¼ 12E⋅Izz L 1 þ Py  > > > 2 > ¼ 6E⋅Izz L 1 þ Py  > > = 3 ¼ 12E⋅Iyy L ð1 þ P Þ z  2 ¼ 6E⋅Izz L ð1 þ Pz Þ > > > > ¼ G⋅Ixx=L > > > > ¼ ð4 þ P ÞE⋅Iyy=½Lð1 þ P Þ z z >      > ; ¼ 4 þ Py E⋅Izz L 1 þ Py

3 32 xL 0 0 0 0 0 0 7 6 6 0 6=5L 0 0 0 1=10 7 76 y 7 6 7 76 z 7 60 7 0 6=5L 0 1=10 0 7 76 7 ¼ ½F0   N 6 7 6 60 7 0 0 0 0 0 7 76 a 7 6 5 40 0 1=10 0 2L=15 0 54 b 5 c 0 1=10 0 0 0 2L=15 3

where, [F0] corresponds to the Timoshenko's constitutive equations shown above. N is the axial force on the beam computed as:

 Fx, Fy and Fz are the translational force components in the coordinate system of the J marker.  x, y and z are the translational displacements of the I marker with respect to the J marker  Vx, Vy and Vz are the time derivatives of x, y, and z, respectively.  Tx, Tz and Tz are the rotational force components in the coordinate system of the J marker.  a, b and c are the relative rotational displacements of the I marker with respect to the J marker.  ωx, ωy and ωz are the angular velocity components of the I marker with respect to the J marker.  Cij are the entries of the damping matrix either specified by the CMATRIX option or computed by using the CRATIO option. All Cij entries default to zero.

8 K11 > > > > K22 > > > > K26 > > > > > > K44 > > > K > > : 55 K66

Fx Fy Fz Tx Ty Tz

(5) Fig. 3. Simulation model of the friction transmission. 235

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2.3. Experimental device An experimental device which can simulate the dynamic friction transmission of friction hoist is designed. The principle is shown in Fig. 4. The photos of the experiment device are shown in Fig. 5. A friction pulley is installed on a top platform and driven by a traction machine. A number of friction linings are affixed at the friction pulley. A steel wire rope wrapped around the pulley with one end hanging loaded conveyance and the either end hanging empty conveyance. Tension sensors are installed between wire rope and the conveyances to measure the rope tension. The rope tension difference is considered to be the friction force between the friction pulley and rope. The conveyances move along the vertical direction on the guide rail through the guide shoes. Encoder 1 is installed at the loaded conveyance to measure the vertical displacement of rope. Triaxis accelerometer is installed at the wire rope with an elastic bracket to measure the rope vibration in three directions. 3. Results and discussion Fig. 6 shows lifting speed and acceleration of the hoist system. Table 1 lists the parameters of the experimental system. 3.1. Transverse rope vibration 3.1.1. Transverse vibration characteristics and model validation The rope end vibration (rope and conveyance joint) is directly related to the operational stability of the hoisting conveyance. The three-axis acceleration sensor is used to measure the transverse vibration acceleration of the rope end. As shown in Fig. 7, the transverse vibration presents a three-stage change. The vibration is small at initial lifting stage, then the vibration amplitude gradually augments with the increase of lifting speed. The vibration amplitude increases from 0 to 4m/s2 at the acceleration stage. The vibration becomes stronger at the constant speed stage, and the maximum amplitude reaches -8m/s2. The vibration amplitude decreases slowly at the deceleration stage. But the vibration at the beginning of the deceleration stage is significantly stronger than that at the ending of the acceleration stage. Therefore, the transverse vibration of rope is both related to the lifting speed and the rope length. In the lifting ending, the rope length is getting shorter, which leads to the vibration acceleration intensification [21]. In addition, the rope vibration remains to exist when the system stops [22,23]. It indicates that the transverse vibration has a delay characteristic. It is easy to induce the rope dynamic instability when the vibration is strong, which will affect the operational stability of the hoisting conveyance [10]. Fig. 8 shows the simulation and experimental results of the rope transverse vibration under different lifting speeds. The faster the rope

Fig. 5. Photos of the dynamic friction transmission experiment device.

Tension sensor 2

Maximum lifting height 6m

0.4

0.8 -1

1.0

0.6

0.0

t3

-0.4

t4

t7

Acceleration Velocity

0.4 0.2

t5

0.0

0.5

1.0

1.5 2.0 Time/s

2.5

t6 0.0

3.0

3.5

Fig. 6. Hoisting speed and acceleration curve.

Friction lining(K25)

Tri-axis accelerometer

0.8

-1.2

Pulley

Wire rope

1.2

t2

-0.8

¦ Ø Encoder 2

t1

Velocity/m·s

Acceleration/m·s-2

1.2

runs, the greater the amplitude of the transverse vibration acceleration is. When the maximum lifting speed is changed from 0.7 to 1.5m/s, the maximum transverse acceleration obtained from the simulation increases from 6.9 to 11.7 m/s2, and the amplitude of the acceleration increases from 3.7–6.9m/s2 to 13.2–11.7m/s2. The maximum transverse acceleration obtained from the experiment increases from 5.6 to 12 m/s2, and the amplitude of the acceleration increases from 3~5.6 m/s2 to 11.4–12 m/s2. The amplitude of the vibration acceleration increases first and then decreases. The simulation results are similar to the experimental results, which reflects the adaptability of the simulation model to the variable speed parameters. The transverse vibration signal of rope is non-stationary. Its discrete statistics are time-varying. Hence, the time-frequency analysis of vibration signal is carried out. The Adaptive Optimal Kernel Time-Frequency Representation Technique is used to obtain the time-frequency

Tension sensor 2 Empty container guide rail

Encoder 1 Loaded container Fig. 4. Schematic diagram of the dynamic friction transmission experiment device of friction hoist. 236

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corresponds to the signal characteristics of the acceleration itself. When the lifting speed increases from 0.7m/s to 1.5m/s, the peak of the transverse vibration power spectrum increases obviously. The maximum increases from 133.9 to 309 m2 s. This indicates that the vibration energy increases significantly. It can be found from the frequency characteristics that the vibration frequency increases with the increase of maximum lifting speed. The basic frequency increases from approximately 15Hz–25Hz. This is due to the shortening of the wire rope [26]. The agglomeration of each frequency component is enhanced, indicating that the trend of resonance is stronger. There are four large power signals at the low speed of 0.7m/s, approximately concentrated at 14Hz, 20Hz, 25Hz and 30Hz, respectively. The basic frequency is approximately 14Hz. The frequency multi-polarization is caused by the traction system, wire rope, load and other factors in the process of lifting. When the maximum speed increases to 1.1m/s, the number of high-power frequency components, overall frequency and intensity all increase. The basic frequency increases to approximately 23Hz. When the maximum lifting speed is 1.5m/s, there is only one obvious frequency component and drops from 33Hz to 25Hz. The basic frequency is approximately 24Hz. A rising frequency component can be found in all the experiment time-frequency figures. This is due to the shortening of rope length [27,28]. Therefore, a sufficient rope length can be reserved to reduce the transverse vibration frequency. Fig. 9(b) shows the time-frequency analysis of the simulation signals (Fig. 8) under different maximum speeds. It can be seen that the distribution of the vibration frequency in the time domain is concentrated at the constant speed stage. It can be seen that the distribution of the vibration frequency in the time domain is similar to the experimental results, i.e., is concentrated in the constant speed stage. Three vibration basic frequencies under three maximum speeds are 12Hz, 23Hz and 24Hz, respectively, which is very close to the experimental results. The simulation results are similar to the experimental results in terms of vibration amplitude and frequency characteristics. The validity of the Cable model is verified.

Table 1 Parameters of the experimental system. Shaft parameters

Load mass (kg) Conveyance mass (kg) Maximum lifting height (m)

37.75 32.66 6

Multi-rope friction hoist Friction pulley diameter (m) Wrap angle ( ) Friction lining

0.64 180 K25

Lifting Rope

Type Diameter (mm) Mass per meter of lifting rope (kg) Elastic modulus (MPa)

6  19WS þ FC 8 0.24 105

Kinematic parameters

Initial static tension of lifting side (N) Initial static tension of lowering side (N) Acceleration (m/s2) Maximum speed (m/s) Acceleration time (s) Constant speed time (s)

691.22 369.95 0.92 1.1 1.2 1.2

Acceleration/m·s

-2

10 5 0 -5 Constant speed

Acceleration

Deceleration

-10 0.0

0.5

1.0

1.5

2.0 Time/s

2.5

3.0

3.5

Fig. 7. Transverse vibration of the lifting rope end.

characteristic and power spectrum of the signal [24,25]. The rope vibration intensity at different times and frequencies can be demonstrated. Fig. 9(a) shows the time-frequency analysis of the experiment signals (Fig. 8) under different maximum speeds. It can be seen that the transverse vibration signal of the rope has a plurality of components. The warm zone is concentrated in the middle of the time domain, which

0

1

2

3

3.1.2. Effect of different parameters on transverse vibration Fig. 10 shows the transverse vibrations of the lifting rope end under different maximum lifting speeds, accelerating and lifting side loads. The increase of maximum lifting speed enlarges the transverse vibration obviously. As it is shown in Fig. 10(a), when the maximum lifting speed

4

0

0.7m/s

10

3

4 0.7m/s

5

0

0

-5

-5

-10

-10 1.1m/s

5 0 -5

-10

1.1m/s

10

-2

10

Acceleration/m·s

-2

2

10

5

Acceleration/m·s

1

5 0 -5

-10

1.5m/s

10 5

1.5m/s

10 5

0

0

-5

-5

-10

-10 0

1

2 Time/s

3

4

0

(a) Simulation results

1

2 Time/s

3

(b) Experimental results

Fig. 8. Transverse vibration of the lifting rope end of different maximum speed. 237

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Max speed

(a) experiment

(b) simulation Spectrum

Time Frequency Representation 35

35

30

30

30

20 15

25

Frequency(Hz)

25

20 15

Frequency(Hz)

35

30

25 20 15 10

10

5

5

5

0

0

0

0

0.5

1

1.5

2

0

50

100

0

0.5

1

Spectrum 35

30

30

20 15

20 15 10

5

5 1.5 2 Time(s)

2.5

3

0

3.5

Time Frequency Representation

30

20

10

0

100

0

200

20

10

0

Spectrum

0.5

1

1.5 2 Time(s)

2.5

3

0

3.5

25

25

15

20 15

Frequency(Hz)

30

25

Frequency(Hz)

35

30

Frequency(Hz)

35

20

20 15

15

10

10

10

5

5

5

5

0

0

4

0

100

200

300

0

0

1

2 Time(s)

3

200

20

10

3

100

Spectrum

30

25

0

Time Frequency Representation

35

2 Time(s)

60

30

30

1

40

40

35

0

20

Spectrum

40

25

10

1

0

Time Frequency Representation

Frequency(Hz)

25

0.5

0

2

Frequency(Hz)

35

0

1.5 Time(s)

Frequency(Hz)

Frequency(Hz)

15

5

0

Frequency(Hz)

20

10

Time Frequency Representation

1.5m/s

25

10

Time(s)

1.1m/s

Spectrum

35

Frequency(Hz)

0.7m/s

Frequency(Hz)

Time Frequency Representation

4

0

0

200

400

600

Fig. 9. Time frequency representation of transverse vibration of the lifting rope end.

of acceleration, the rope is tightened due to the increase of inertial force, and the transverse vibration decreases. Even though the rope becomes tight as the load increases, the maximum vibration amplitude is increased. This indicates that the effect of large loads on system dynamics is greater than that of the rope tension. The transverse vibration makes the rope easily produce greater transverse force. It may cause a collision between ropes and then induce the impact of rope threads [5].

changes from 0.7 to 1.5m/s, the maximum transverse acceleration increases from 5.6 to 12 m/s2 and the span of the amplitude increases from 3~5.6 m/s2 to 11.4–12 m/s2. Fig. 10(b) presents that the transverse vibration decreases with the increase of acceleration. As the acceleration increases from 0.84 to 1 m/s2, the maximum transverse acceleration decreases from 11.2 to 7.8 m/s2 and the span of amplitude increases from 4~5.6 m/s2 to 11.4–12 m/s2. It is shown in Fig. 10(c) that the transverse vibration amplitude slightly increased when the lifting load changes from 450 to 690N. The standard deviations are 1.48, 1.57 and 1.6 m/s2, respectively. Besides, when the load increases, the moment which the vibration began to intensify is delayed. The duration of strong vibration is narrowed throughout the whole lifting period. This indicates that high load has a higher sensitivity to acceleration and deceleration. Under the experimental conditions in this paper, it can be seen that the transverse rope vibration has the highest sensitivity to the speed. Fig. 11 shows the maximum vibration amplitude under different parameters, such as speeds, accelerations and loads. The maximum vibration amplitude shows trends to increase, decrease and increase, respectively, with the increase of the parameters. According to 3.1.2, the increase of speed leads to the concentration of each frequency components and thus induces the intensification of vibration. With the increase

3.1.3. Transverse vibration suppression It can be seen from the above analysis that the load and speed have significant effects on the transverse vibration. But the reduction of load and speed will inevitably reduce the hoisting efficiency. Literature [29] shows that the trapezoidal acceleration curve has an optimal dynamic response as compared to parabolic, sinusoidal and triangular acceleration curves. Therefore, based on the trapezoidal acceleration curve, an attempt is made to control the rope end vibration by modifying the initial acceleration time of the trapezoidal acceleration curve (As shown in Fig. 6, t1 is initial acceleration time). In order not to affect the efficiency of the upgrade, we adjusted the maximum acceleration, that is, with the same time to reach the maximum speed. To guarantee the same hoist efficiency, the maximum acceleration is set to obtain the same maximum

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0

1

2

3

10

4 0.7m/s

0

1

2

3

10

0

4

0.84m/s

2

0

-5

-5

-5

-10

-10

Acceleration/m·s

-2

0

0.92m/s

10

5

5 0

0

-5

-5

-5

-10

-10

-10

10

1m/s

2

5

0

0

0

-5

-5

-10

-10

-5

0

1

2 Time/s

3

4

(a). Different maximum speeds(m/s)

690N

10

5

5

570N

10

0

10

450N

-10 2

5

1.5m/s

3

5

5

0

1.1m/s

2

10

5

10

1

-10

0

1

2 Time/s

3

4

Vibration began to intensify

0 2

1

2 Time/s

3

(b). Different accelerations(m/s ) (c). Different loads (N)

Fig. 10. Transverse vibration of the lifting rope end under different parameters.

vibration is obviously weakened when the acceleration time is chosen the vibration period of LFW. The standard deviation analysis shows that the average amplitude of the vibration fluctuation is reduced by 13% and 18%, respectively, compared with 0.75 times and 1.25 times of the LFW vibration period. The average amplitude of the vibration fluctuation is reduced by 61% compared with the rectangle acceleration curve. The Time-Frequency Representation shows that, when the initial acceleration time is 0.14 s, the frequency distribution shows a low aggregation. Hence, the power spectral value is lower (maximum 112.47 m2 s). When the initial acceleration time is 0.175 s, the frequency components have high aggregation and the power spectral value is the highest (maximum 178.92 m2 s). Therefore, when the LFW vibration period is selected as the initial acceleration time, the frequency components will be far away from each other. Then the resonant frequency band is dispersed, and the vibration of the steel wire rope is weakened.

speed within the same acceleration time. Related researches showed that the transverse and longitudinal vibration of rope had a high coupling effect [8,9,30]. Set the vibration period of longitudinal fundamental wave (LFW) of rope as the initial acceleration time. The vibration frequency of rope LFW is [31]:

λ ωm ¼ L

sffiffiffiffiffiffi EF ρ

(8)

where, E is the elastic modulus, F is the cross-sectional area, ρ is the mass per unit length, L is the length between the friction pulley tangent point and lifting conveyance. λ is the solution of transcendental equation λ tan λ ¼ nρL Q [32], wherein, Q is the rope end quality, n is the number

Maximum vibration amplitude/m·s

-2

of rope. The vibration period of rope LFW is obtained, i.e. Tj ¼ 2π/ ωm ¼ 0.14 s. Fig. 12 shows the effects of different initial acceleration times (0.75 times, 1 times and 1.25 times of the LFW vibration period, respectively) on the transverse vibration of the rope ends. The maximum acceleration is 3.56 m/s2 when the initial acceleration time is 0.14 s. This value increase to 4.05 m/s2 and 5.28 m/s2, respectively, when the initial acceleration time are 0.105 s and 0.175 s. The transverse rope

3.2. Effect of transverse vibration on friction transmission stability The contact region of rope and friction pulley is the dual friction region of rope and friction lining, and it is the direct source of friction force. The rope dynamics produce the dynamic contact, slipping and jumping from slot on the friction interface. These phenomena lead to the friction instability in local contact area between the rope and friction lining. It can direct influence the stability and reliability of friction transmission in hoisting system.

14 12

3.2.1. Transverse rope vibration close to friction pulley Fig. 13 shows the transverse vibration acceleration of the 150th, 160th, 170th and 180th rope elements in a hosting cycle. The distances from the original position of each element to the meeting point of rope and friction pulley are 1.197m, 0.877m, 0.557m and 0.237m, respectively. The maximum lifting speed is 1.1 m/s and maximum acceleration is 0.92 m/s2. The acceleration signal figure presents that there is a smoothing region in the curve of transverse rope vibration. It represents that the rope goes into the wrap angle. The acceleration amplitude of two sides of smoothing region increases obviously. It demonstrates that the transverse vibration is much stronger when the rope gets close to the separating point or the meeting point. But The transverse vibration of the 150th rope element reaches the maximum value 127 m/s2 at 1.46 s. The severe

10 8 6 4 2 0

450 570 690 0.84 0.92 1 0.7 1.1 1.5 -1 -2 Speed/m·s Acceleration/m·s Load/N Fig. 11. Statistical results of transverse vibration. 239

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1.0

1.5

2.0

2.5

3.0

3.5 0.105

3 0 -3

40

30 20

0

0.140

30 20

10

0

0.5

1

1.5 2 Time(s)

2.5

3

0

3.5

0

50

50

40

40

50

100

0

-3 -6 6

0.175

Frequency(Hz)

3 Frequency(Hz)

-2

40

10

-6 6

Acceleration/m·s

50

Frequency(Hz)

0.5

Frequency(Hz)

0.0 6

Spectrum

50

30

20 10

30 20 10

0

0

50

50

-3

40

40

0.0

0.5

1.0

1.5

2.0 Time/s

2.5

3.0

3.5

0

0.5

1

1.5 2 Time(s)

2.5

3

30

20 10 0

0

3.5

Frequency(Hz)

-6

Frequency(Hz)

3

0

50

100

30 20

10

0

0.5

1

1.5 2 Time(s)

2.5

3

3.5

0

0

50

100

150

Fig. 12. Transverse vibration of the lifting rope end of different initial acceleration time.

vibration region transforms from the lifting side to the lowering side with the winding of the rope. It is because the rope length shortens gradually 0.0

0.5

1.0

1.5

2.0

2.5

at the lifting side and extends at the lowering side during the process of friction transmission. The transverse stiffness of extending rope 3.0

40

3.5 150

th

0 -40 -80 -120 40

160

th

180th

Acceleration/m·s

-2

20 0

170th

-20 170

20

th

160th

0 -20

150th

-40 180

60

th

30 0 -30 0.0

0.5

1.0

1.5 2.0 Time/s

2.5

3.0

3.5

Fig. 13. Transverse vibration acceleration of the lifting rope of different locations. 240

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decreases, which causes the increase of vibration acceleration amplitude. The transverse rope displacement is obtained by the quadratic time integration of the acceleration in Fig. 13. Before the integration, the vibration equilibrium positions of both sides should be found out first. Because of the experimental tower height restrictions, the transverse displacements of simulation under two kinds of lifting height, i.e., 2.62m and 4.81m, are presented in Fig. 14. The vibration cycle characteristics can be expressed as complete as possible because they cover the 1 and 1.5 cycles of the lifting and lowering side, respectively. It shows that the vibration equilibrium positions are not at the initial positions, i.e., ±320 mm, but 319.8 mm on the lifting side and 319.7 mm on the lowering side. This is due to the deformation of both the wire rope and friction lining, which are caused by the tensile load and radial load, respectively. Fig. 15 shows the transverse vibration displacements of selected rope elements. The transverse displacement of each element varies randomly. The general vibration range of each point is within ±1 mm at lifting side and ±2 mm at lowering side. Besides, the displacement curve distributes more serried when the rope winds close to the meeting point or separating point. It illustrates that the vibrational frequency increases. The maximum values of vibration displacement amplitude shows the opposite changing rule at the lifting side and lowering side. The maximum amplitudes are 0.99, 0.46, 0.3 and 0.23 mm at the lifting side, respectively, showing a decreasing trend. The maximum amplitudes are 0.93, 1.41, 1.62 and 1.63 mm at the lowering side, respectively, showing an increasing trend. The vibration displacement amplitude of the lowering side is larger than that of the lifting side. This is due to the larger load of lifting side, and thus the rope tension is larger, and the rope vibration displacement is smaller. At the lifting side, the vibration displacement amplitude of each rope element expands first and then narrows. The reason for the expansion is the increase of rope speed. The reason for the narrowing is that the wire rope approaches the friction pulley tangent point. The vibration displacement amplitude is weakened due to the limitation of the rope groove. At the lowering side, the vibration displacement amplitude of each element shows an overall expanding trend (as showed in Fig. 14(b)), and the expanding amplitude is larger than that at the lifting side. The phenomenon is dominated by the extension of rope. Besides, there is a slight narrowing in the process of expansion. This is because this moment is the transition period between constant speed and deceleration. The lower side rope has an upward inertial force to tighten the rope. Then the displacement amplitude narrows. Fig. 16 shows that the transverse displacements of the selected rope elements shift from the equilibrium position to the direction away from the center of friction pulley. The effect of shifting leads to that the actual contact area of rope and friction pulley is less than the designed wrap angle. Therefore, when the transverse vibration is large, the rope jumps 0

1

2

3

4

400

0.5

1.0

1.5

2.0

2.5

3.0

0.99

-320.5 0.0

0.93

0.5

1.0

320.0

1.5

319.5 3.0

th

-319.5

Displacement/mm

0.46

1.41

321

-320.0

0

-320.5 0.0

-200

320 0.5

1.0

1.5

319 2.5

3.0

th

170 -319.5

200

1.62

0.3

321

-320.0

0

0.0

320 0.5

1.0

319 2.5

180

200

321

-200

1.63

320

-320.0 0.0

319

0.5

2.0

-0.5

0.0

0.5

1.0

1.5

2.5

2.0 2.5 Time/s

3.0

3.0

3.5

3.5

4.0

4.5

Fig. 15. Transverse vibration of the lifting rope of different locations.

from slot easily. More seriously, it would cause insufficient friction force and rope skid accidents. Table 2 shows the transverse vibration acceleration and transverse displacement of four selected rope elements when they rotate to the meeting point and separating point of friction pulley. The displacement values of all selected rope elements are negative at the meeting point, and the accelerations are positive. The results show oppositely at the separating point. This indicates that the rope which is located away from friction pulley has the movement trend of getting close to the friction pulley. Furthermore, the transverse rope displacement is larger at the separating point than that at the meeting point, because the rope tension is small at the lowering side.

6

0

1

2

3

4

6

4.81m

320.0

Displacement/mm

5

319.6

319.2 2.62m

320.0

319.6

319.2 5

3.5

0.23

-319.5

-320.0 4

3.0

th

400

-319.8

3 Time/s

3.5

400

-200

2.62m

2

3.5

160

200

0

4.5

320.5

-320.0

400

-320.0

1

4.0

th

-200

-319.8

0

3.5

150

-319.5

0

5

-319.6

0.0 -319.0

200

4.81m

-319.6

Displacement/mm

-0.5

6

(a). Lifting side

0

1

2

3 Time/s

4

(b). Lowering side

Fig. 14. Transverse vibration displacement of the lifting rope end of different lifting height. 241

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Normal force Friction force

Reduce the actual contact area Fig. 16. Diagram of effect of transverse vibration on contact between the rope and pulley.

Fig. 17. Normal force and Friction force of rope element of different locations.

3.2.3. Correlation of transverse vibration and friction transmission Fig. 20 shows the experiment value of total friction force between friction pulley and rope under different maximum speeds. The friction force is largest in the acceleration stage. The friction force fluctuates wildly at the beginning of this stage and then becomes stable. The fluctuation amplitude can reach 24% of stable value, and the relative large fluctuation is related to the rectangular acceleration curve. A short acceleration time leads to a large inertial impact, which is the reason of large fluctuation generation. During the acceleration stage, the three friction force curves are close to each other. Meanwhile, the transverse rope vibration is also small at the acceleration stage according to section 3.1. The fluctuation amplitude of friction force increases significantly with the maximum speed. The static stable value of friction force is 100N, and the fluctuation amplitude is 7%, 20% and 47% of static stable value, respectively. Especially, the phenomenon like beat wave is generated at the beginning of constant speed stage under the maximum speed of 1.5m/s [5]. During deceleration stage, the fluctuation of friction force is larger than that in the stable region of acceleration stage, and the fluctuation increases with the maximum lifting speed. The trend is similar to the transverse vibration in section 3.1.3. Therefore, it can be inferred that the rope transverse vibration is related to the friction stability. The friction force becomes unstable when the transverse vibration intensifies. Through model simulation, the transverse displacements at the meeting points between rope elements and friction pulley are obtained. The 150th rope element in the constant speed stage is chosen as the research object. The absolute value of the transverse displacement from the equilibrium position is 0.14 mm when it reaches the meeting point on the friction pulley. And Fig. 21 presents the friction forces of the following rope elements, i.e., 150th, 151th and 152th elements. The values are 12.2N, 12.75N and 12.67N, respectively. This is due to the fact that the state between rope element and friction pulley changes from separation to full contact. But then the load decreases, and the friction force decreases gradually which is shown in Fig. 18(b). The 135th and 155th rope elements with different vibration displacement are selected from a number of rope elements that go through the tangent point. The transverse displacements are 0.15 mm and 0.145 mm, respectively. At the time they go through the tangent point, the friction forces of the following elements are shown in Fig. 21. The larger the rope transverse displacement is at tangent point, the less the friction force of the following rope elements is. When the transverse displacement increase from 0.14 mm to 0.15 mm, the falling value of the three groups of friction force is close to 1N. It is 8% of total friction force of rope element. The simulation result demonstrates that the transverse rope displacement, which is on the direction away from the friction pulley at the tangent point, has an adverse influence on the contact between the following rope and friction lining. Therefore, through controlling transverse vibration, the friction stability of the rope and friction lining can be guaranteed to a certain degree. The safety and reliability of friction hoist are improved.

3.2.2. Friction characteristics of rope within wrap angle Fig. 17 shows an instantaneous state during the simulation. Red arrows represent the pulley forces towards the rope elements. It can be found that the rope force is decreasing from the meeting point to the separation point. This is closely related to the wire rope being subjected to heavy loads and to light loads. In addition, the force direction is deflected radially to the right. Therefore this force can be decomposed into normal force and friction force. Fig. 18 shows the normal force and tangential friction force between the friction lining and rope during the process that four selected rope elements rotate through the wrap angle. The normal force and friction force show the decline trend and accompany with fluctuation. The overall ranges of normal force of selected rope elements are 69–43N, 70–41N, 76–41N and 75–41N, respectively. The fluctuation range, which is not caused by acceleration and deceleration, is within 4.3%. Related research has shown that the local maximum pressure is about 45 times higher than the average one from the “plane method” formula [33]. The normal force of the local rope obtained in this paper can provide more accurate parameter support for the friction linning force analysis and fretting fatigue behavior of hoisting rope wires [34]. The overall ranges of the friction force are 4–15N, 7–15N, 7–18N and 7–17N, respectively. The friction force produces relatively large declining amplitude at the moments from acceleration to constant speed and from constant speed to deceleration. The declining rates shown in the figure are 103N/s, 161N/s and 207N/s, respectively. Whereas the declining amplitude of the normal force is relative small. The fluctuation range of the friction force, which is not caused by acceleration and deceleration, is within 15.6%. According to 2.1, the simulation model uses coulomb friction theory to calculate friction. Therefore, the dynamic change of friction coefficient can be obtained. Fig. 19 presents that the friction coefficient is maximum at the acceleration stage, followed by the friction coefficient at the constant speed stage, and the friction coefficient gets minimum at the deceleration stage. This is because the contact force of friction pairs is large at the acceleration stage and small at the deceleration stage. The friction coefficient shows the increasing trend during the acceleration stage (increased from 0.2 to 0.26 according to the 180th rope element). It indicates that the contact force decrease leads to friction coefficient increase [35]. During the constant speed, the friction coefficient fluctuates on the basis of 0.2, and the maximum fluctuation range is 25%. At the end of constant speed, the fluctuation amplitude of the friction coefficient increases, and the contact force is low right now, which means the stability of friction pairs decreases when the contact force decreases to a certain value.

Table 2 Transverse vibration of the rope. Rope elements

150 160 170 180

(1.197m) (0.877m) (0.557m) (0.237m)

Meeting point

Separation point

Acceleration (m/s2)

Displacement (mm)

Acceleration (m/s2)

Displacement (mm)

3.4 3.72 2.98 1.51

0.14 0.137 0.142 0.135

1.89 3.43 4.01 3.58

0.258 0.254 0.264 0.259

4. Conclusions A simulation model and experimental device of the friction hoisting system are established. The simulation results are similar to the 242

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60

Acceleration Constant speed Deceleration 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 150th

15

40

10

20

5

Acceleration Constant speed Deceleration 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 150th

0

0

160

60 40

10

20

5

0 th

170

60

160th

15

40

Friction force/N

Normal force of the element/N

th

0

170th

15 10 5

20

0

0

180th

60

180th

15 10

30

5 0

0 0.0

0.5

1.0

1.5 2.0 Time/s

2.5

3.0

3.5

0.0

0.5

(a). Normal force

1.0

1.5 2.0 Time/s

2.5

3.0

3.5

(b). Friction force

Fig. 18. Normal force and Friction force of rope element of different locations.

Acceleration Constant speed Deceleration 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 150th

Mearsured friction force/N

400

0.2 0.1 0.0 160th

Friction coefficient

0.2 0.1

300 200 100 0

-100

0.0 170th

The maximum speed(m/s) 0.7 1.1 1.5

beat frequency 0.0

0.5

1.0

1.5 Time/s

2.0

2.5

Fig. 20. Measured friction force of different maximum speed.

0.2 experimental results in terms of vibration amplitude and frequency characteristics. The transverse rope vibration is small at the initial stage of the lifting. The amplitude of the vibration acceleration increases with the speed. The vibration at the initial stage of the deceleration phase is stronger than that at the end of the acceleration phase. The increase of the lifting speed can increase the vibration acceleration and vibration displacement, and the shortening of the rope will increase the vibration acceleration and reduce the vibration displacement. A sufficient rope length can be reserved to reduce the transverse vibration frequency. Influences of different hoisting parameters on the transverse rope end vibration are obtained. The vibration intensified with the maximum lifting speed increase, and weakened with the maximum acceleration increase, and intensified with the lifting load increase. When the initial acceleration time is equal to the vibration period of longitudinal fundamental rope wave of the rope, the agglomeration of the frequency

0.1 0.0 180th

0.2 0.1 0.0 0.0

0.5

1.0

1.5 2.0 Time/s

2.5

3.0

3.5

Fig. 19. Friction coefficient between rope element and friction lining of different locations. 243

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153th

+10.0

152th

Friction force/N

3.0 2.5

158th

157th

151th

2.0

138th 156th

1.5

137th

1.0 136th

0.5 0.0

Dis

pla

ce

0.1

4

0.1 4 0.1 5 5

me

nt

3 ts en

/m

m

1 ng wi llo

2 elem e rop

Fo

153/158/138 152/157/137 151/156/136

Tangent point

150/155/135

Fig. 21. Friction force of following rope elements of different transverse displacement at the meeting points.

components can be reduced, and the resonant frequency band is dispersed. The average amplitude of the vibration fluctuation is reduced by 61% as compared with the rectangle acceleration curve. Rope vibration characteristics near the tangent point of the friction pulley are obtained. The closer the rope is to the tangent point, the greater the transverse vibration frequency is. The severe vibration region transforms from the lifting side to the lowering side with the rope winding. The maximum vibration value reaches 127m/s2. The transverse displacements shift from the equilibrium position to the direction away from the center of friction pulley. The effect of the shifting leads to that the actual contact area of rope and friction pulley is less than the designed wrap angle. Correlation between transverse vibration and friction stability is established. The friction characteristics between the rope and friction pulley are obtained at any time and any position. The normal and friction forces show the decline trend and accompany with fluctuation, and the friction force fluctuates within 15.6%. The fluctuation amplitude of the friction coefficient increases when the contact force decreases to a certain value, which affects the friction stability. Measured friction force fluctuates greater when the transverse vibration is intense. The simulation results indicate that the stronger the transverse vibration of the rope is at the tangent point, the smaller the friction force of the following rope is, i.e., drop is by 8%. Therefore, through controlling transverse vibration, the friction stability of the rope and friction lining can be guaranteed to a certain degree. The safety and reliability of friction hoist are improved.

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Acknowledgements The research reported here is supported by the National Natural Science Foundation of China (Grant No. 51375479), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). References [1] Zhang DK, Chen K, Zhang XJ. Comparison of the thermophysical properties, microstructures, and frictional behavior of lining materials used in mine hoists. Wear 2016;354–355:1–9. [2] Herrera I, Romero E. Software design to calculate and simulate the mechanical response of electromechanical lifts. J Phys Conf Ser IOP Publ 2016;721(1), 012009. [3] Sandilo SH, van Horssen WT. On a cascade of autoresonances in an elevator cable system. Nonlinear Dynam 2015;80(3):1613–30. [4] Kou BF, Liu QZ, Liu CY, Liang Q. Characteristic research on the transverse vibrations of wire rope during the operation of mine flexible hoisting system. J China Coal Soc 2015;40(5):1194–8. 244

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