Dynamic analyses of hoisting ropes in a multi-rope friction mine hoist and determination of proper hoisting parameters

2.1. Governing equations of motion

In order to describe the transverse vibrations of hoisting ropes in colliery, two fixed coordinate systems $oxy$ and $OXY$ are constructed, as shown in Fig. 2, which correspond to the catenaries and the vertical ropes, respectively. In this model, considering that the effect of gravity due to a catenary inclination is small as comparison to the total quasi-static tension, hoisting ropes are divided into four vertically translating catenaries with constant length $L_c$ and four vertical ropes with time-varying length $l\left(t\right)$ in the same vertical plane as shown in Fig. 2. The skip is simplified as a rigid body of mass $m_e$ attached to the bottom ends of four vertical ropes, $X=l\left(t\right)$, and the suspension of the skip against guide rails is assumed to be rigid. In $oxy$, considering that the axial fluctuations of the friction pulley are feeble and negligible, so the origin $o$ represents the friction pulley which is viewed as a fixed-center pulley without axial displacements. The four singular head sheaves are modeled as four particles subjected to axial displacement excitations induced by the axial fluctuations of the head sheaves, specified by $e_1\left(t\right)$, $e_2\left(t\right)$, $e_3\left(t\right)$ and $e_4\left(t\right)$, where $t$ is hoisting time. During the hoisting process, the skip moves upwards. Then, the vertical hoisting rope has a variable length $l\left(t\right)$, an axial velocity $v\left(t\right)=dl\left(t\right)/dt$ and an axial acceleration $a\left(t\right)=d^{2}l\left(t\right)/d{t}^2$. In order to explicitly illustrate transversely dynamic behaviors of the ropes, only the rope 4 and the head sheave 4 shown in Fig. 2 are adopted to conduct the research. For a vertically translating cable with arbitrarily varying length subjected to an external transverse excitation, the transverse vibration will play a dominating role in the forced longitudinal-transverse vibration, and the longitudinal vibration can be negligible [10]. Therefore, the longitudinal vibration is not considered in the present work.

In this study, the axially moving catenaries and vertical hoisting ropes in the multi-rope friction mine hoist can be modeled as taut strings. The equations governing the transverse vibrations of a string with variable length, velocity and tension can be formulated as [13-14]:

Fig. 2Schematic of the transverse vibration model for hoisting ropes

The boundary conditions are:

and the initial conditions are:

Similarly, the equations governing the transverse vibrations of a string with constant length and constant tension [15-18] can be formulated as:

The boundary conditions are:

and the initial conditions are:

where $Y\left(X,t\right)$ and $y\left(x,t\right)$ are the transverse displacements of the rope particle instantaneously located in spatial position $X$ and $x$ at time $t$ relative to the fixed coordinate system $OXY$ and $oxy$, respectively; and the subscripts $X$, $x$ and $t$ denote the partial differentiation, $\rho$ is the mass of the hoisting rope per unit meter and $\lambda$ is the equivalent damping coefficient which can be determined from the literature by Kaczmarczyk and Ostachowicz [19], the subscript 4 in $e_4\left(t\right)$ will be omitted, $P\left(X,t\right)$ is the axial tensile load located in spatial position $X$ at time $t$ in the vertically translating hoisting rope, arising from its own weight, the weight of the skip, the payload, the longitudinal acceleration and the time-varying mass of three tail ropes. In a multi-rope friction mine hoist, the tensions in the four hoisting ropes are required to be balanced, as a result, $P\left(X,t\right)$, which denotes the axial force of a single vertical rope can be formulated as:

where:

where the parameters $m_e$ and $m_p$ correspond to the mass of the skip and the payload, respectively, ${\rho }_w$ denotes the linear density of a hoisting tail rope, $L$ is the maximum length of vertical hoisting ropes, $l\left(t\right)$ is the instantaneous length of a vertical hoisting rope and $L_m$ represents the minimum length of a tail rope during the ascending cycles, and $g$ is the acceleration of gravity.

Subsequently, according to the continuity of deflection across the head sheave, the axial tensile load in the catenary is required to be $P\left(0,t\right)$ that is the axial force at the top of the vertical rope. There are three tail ropes with time-varying length attached at the bottom end of the skip. In a multi-rope friction mine hoist, the sum of the linear density of the three hoisting tail ropes is required to be equal to that of the four hoisting ropes, therefore, a quasi-static tension $T$ in the catenary at the constant speed stage can be formulated as:

2.2. Solution

It’s difficult to solve partial differential equation with time dependent nonhomogeneous boundary conditions. Especially the equation governing the transverse vibrations of a vertically translating rope is a complex partial differential equation, which has an infinite number of degrees of freedom and many time-varying parameters. As a result, it’s difficult to yield the precise analytical solution. To transform the partial differential equation with an infinite number of degrees of freedom to the ordinary differential equations with a finite number of degrees of freedom, the well-known Galerkin method can be adopted. Subsequently, numerical solution method, such as four-order Runge-Kutta method, can be used to solve ordinary differential equations.

In this section, Eq. (1) governing the transverse vibrations of a vertically translating rope with time-varying length is solved as an example. Firstly, the nonhomogeneous boundary conditions need to be transformed into the homogeneous. By employing the method presented by Zhu and Chen [10], the transverse displacement can be expressed as:

where $U\left(X,t\right)$ is the part that satisfies the corresponding homogeneous boundary conditions and $H\left(X,t\right)$ is the part that satisfies the corresponding nonhomogeneous boundary conditions. Substituting Eq. (10) into Eq. (1) yields:

where $f\left(X,t\right)$ is the additional forcing term induced by transforming the governing equation with time dependent nonhomogeneous boundary conditions to one with homogeneous boundary conditions. The corresponding initial conditions for $U\left(X,t\right)$ can be expressed as:

Substituting Eq. (10) into Eq. (2) and setting:

the homogeneous boundary conditions for $U\left(X,t\right)$ can be obtained. And the function of $H\left(X,t\right)$ can be expressed in a first-order polynomial in $X$:

where $a_0\left(t\right)$ and $a_1\left(t\right)$ are the time-dependent unknown coefficients. Substituting Eq. (13) into Eq. (14) yields:

Subsequently, a new independent variable $\xi =X/l\left(t\right)$ is introduced, and the time-varying spatial domain $\left[0,l\left(t\right)\right]$ for $X$ can be converted to a fixed domain [0,1] for $\xi$. The new dependent variable is $U^{*}\left(\xi ,t\right)=U\left(X,t\right)$ and the new dependent variable for $H\left(X,t\right)$ is $H^{*}\left(\xi ,t\right)$. The partial derivatives of $U\left(X,t\right)$ and $H\left(X,t\right)$ with respect to $X$ and $t$ relative to those of $U^{*}\left(\xi ,t\right)$ and $H^{*}\left(\xi ,t\right)$ with respect to $\xi$ and $t$ are:

where subscript $\xi$ denotes the partial differentiation, and the asterisks in the new variables $U^{*}\left(\xi ,t\right)$, $H^{*}\left(\xi ,t\right)$ and their derivatives are omitted in Eq. (16) and henceforth. Assume that the solution of infinite number of degrees of freedom for Eq. (1) can be formulated in the following form:

where ${\phi }_i\left(\xi \right)$ represents the trial function, $q_i\left(t\right)$ denotes the generalized coordinate, and $n$ is the number of truncation modes. Substituting Eqs. (15), (16) and (17) into Eq. (11), then multiplying Eq. (11) by ${\phi }_j\left(\xi \right)$, ($j=$1, 2,…, $n$) and integrating Eq. (11) over the interval of 0 and 1, ordinary differential equations can be obtained as:

where $\mathbf{Q}={\left[q_1\left(t\right),q_2\left(t\right),\cdots ,q_n\left(t\right)\right]}^T$, the entries of the system matrices and the force vector are:

where ${\delta }_{ij}$ is the Kronecker delta, if $i=j$, then ${\delta }_{ij}=$1; if $i\ne j$, then ${\delta }_{ij}=$0. For the established model in Fig. 2, the solution for $U\left(X,t\right)$ is sought first, and then $Y\left(X,t\right)$ is obtained by applying Eq. (10).

Similarly, this method can also be applied to solve the partial differential Eq. (4) which governs the transverse vibrations of the catenary with constant length. Eventually, the ordinary differential equations transformed from Eq. (4) by using Galerkin method can be formulated as:

where ${\mathbf{Q}}^c={\left[q_1\left(t\right),q_2\left(t\right),\dots ,q_n\left(t\right)\right]}^T$. The entries of the system matrices and the force vector are:

2.3. The convergence of the solution

The convergence of the solution for the governing equations of the established models determines the accuracy of the solution. The convergence of the solution for each model in Fig. 2 can be examined by varying the number of included modes. To determine the number of included modes for Eq. (18) that governs the transverse vibrations of vertically moving hoisting ropes with variable length during the lifting process, a simulation external input, $4\times \mathrm{s}\mathrm{i}\mathrm{n}(2vt/D)$ mm where v is the hoisting velocity and $D$ is the diameter of the head sheave, was used to simulate the axial fluctuations of the head heave. According to the hoisting velocity shown in Fig. 3 and the hoisting parameters given in Table 1, the transverse vibrations of the point at the vertical rope whose location is $x=$20 m, $Y\left(20,t\right)$, were calculated with different numbers of included modes as shown in Fig. 4.

Fig. 3The hoisting velocity and acceleration

a) Hositing velocity

b) Acceleration

From Fig. 4, it can be seen that the convergence of Eq. (18) that governs the transverse vibrations of vertically moving hoisting ropes with variable length is basically achieved with $n=$10. Thus, to guarantee the calculation accuracy, $n=$20, the number of included modes, was selected for the vertical rope with varying length. Similarly, by varying the number of included modes, $n=$4 was selected for the catenary with constant length.

4.1. Dynamic behavior of the vertical rope

Since the validity of the vibration model of the axially moving catenaries has been confirmed, and the modal approach has also been validated as comparison to the exact solution obtained by wave method [10], then the dynamic model can be also applied to the vertically translating ropes with time-varying length.

To predict the transverse vibrations at each point of the vertical hoisting rope 4 during the ascending cycles, the same truly measured displacement excitations shown in Fig. 8(a) are also employed as the inputs of Eq. (18), and the other hoisting parameters are the same as shown in Table 1. The simulation working condition is the same as the measure working condition in Section 3. Then the responses of the transverse vibrations at the vertical rope 4 can be calculated.

In the present study, aiming at preventing rope collision, the largest amplitude at each point of the vertical rope was investigated as shown in Fig. 10. Under the same external excitation, the largest amplitude at lower catenary 4 is 166.8mm as shown in Fig. 6(a) relative to the ascending cycle with no payload, while the largest amplitude at the vertically translating rope 4 is 33.06 mm as shown in Fig. 10. Hence, the transverse vibrations at the inclined catenary with constant length are much more severe than those in the vertical hoisting rope. Furthermore, it can be also obtained that the transverse vibrations at the vertical rope are of small amplitude and of a wide range of frequencies, mainly at the upper level of the shaft, which verifies the conclusion presented by Kaczmarczyk and Ostachowicz [12]. Thus, in the point of avoiding rope collision, the transverse vibrations of the vertical rope can be negligible.

4.2. Dynamic analyses of transverse vibrations of hoisting catenaries

According to the analyses mentioned above, the lateral vibrations are usually associated with the catenaries in a multi-rope friction hoist in coal mines, then it is necessary to explore the effects of the hoisting parameters on the lateral vibrations of catenaries.

The hoisting velocity mainly governs the axial fluctuations of head sheave that act as the external excitations, and the rope tension influenced by the hoisting load mainly governs the intrinsic frequency of the catenary. According to literature [11], intense transverse vibrations of catenaries will be excited if the external excitation frequency approaches the intrinsic frequency of the catenary. Therefore, this section mainly investigates the effects of hoisting velocity and load on the transverse vibrations of hoisting catenaries.

4.2.1. Effect of hoisting velocity on the transverse vibrations of hoisting catenaries

It has obtained from Section 3 that the first three order frequency components play the dominating roles in the periodic axial fluctuations of the head sheaves. The first three order excitation frequencies are 0.66 Hz, 1.32 Hz and 1.98 Hz, respectively, and the corresponding excitation amplitudes are 1.8 mm, 4 mm and 2.2 mm in sequence. Therefore, the corresponding first three order angular frequencies are 4.16 rad/s, 8.32 rad/s and 12.48 rad/s, respectively. The external excitation is specified by $A\mathrm{s}\mathrm{i}\mathrm{n}(2vti/D)$ mm, the excitation amplitude $A$ is assumed to be a constant value of 4 mm which is the maximum excitation amplitude, $i$ represents the excitation order. And the other hoisting parameters are the same as shown in Table 1. The effect of hoisting velocity on the transverse vibrations of catenaries was demonstrated in Fig. 11.

Under the circumstance of full payload, it can be seen from Fig. 11 that the transverse vibrations of hoisting catenaries under the second- and third-order excitations have discrepant larger responses than those under the first-order excitation. Especially when the hoisting velocities are 7.5 m/s and 11.2 m/s, the maximum transverse displacements of the hoisting catenaries are as large as 1092 mm and 946.7 mm which have seriously exceed the rope spacing of 350 mm in the present study, and collision between the two adjacent ropes will certainly resulted in as shown in Fig. 2.

Fig. 10The largest transverse displacements of the vertical rope 4 at each depth by prediction

Fig. 11The maximum transverse displacements at the center of a catenary with varying constant velocities under the circumstance of full payload

4.2.2. Effect of hoisting load on the transverse vibrations of hoisting catenaries

To further investigate the effect of hoisting load on the transverse vibrations of hoisting catenaries, three typical working conditions which are ascending with no-load, full-load and part-load are considered. According to Eq. (9) and Table 1, the tensions in a single catenary under the three typical working conditions are 101000 N, 146000 N and 121000 N, respectively. By varying the excitation frequency under the excitation $4\mathrm{s}\mathrm{i}\mathrm{n}\left(wt\right)$ mm, the largest transverse amplitudes at the center of a lower catenary are depicted as shown in Fig. 12.

From Fig. 12, it can be seen that the resonance frequency curve towards right moves with the increasing rope tension decided by the hoisting load. During the ascending cycle with no-load, it is obvious that the second-order excitation angular frequency of 8.32 rad/s is more close to the resonance frequency range, the second order response amplitude at the center of the lower catenary is 151.9 mm which is much larger than the first-order (2.3 mm) and third-order (11.25 mm) as shown in Fig. 12.

Note that the sum of the first three order response amplitudes is 165.45 mm, which is so close to the practical value 166.8 mm shown in Fig. 6(a), so it has indirectly confirmed the validity of the established model. Additionally, it can be also concluded that, during the dynamic analyses and the latter parameter optimization, it is of no influence to replace the measured first-order excitation amplitude of 1.8mm and the third-order excitation amplitude of 2.2 mm by a larger value of 4 mm. And it is apparent that the second-order response amplitude of 12.69 mm relative to the ascending cycle with full-load is far less than 151.9 mm relative to the ascending cycle with no-load in Fig. 12.

Therefore, it can be concluded that a large response amplitude will be excited when an excitation frequency approaches the dangerous resonance frequency range and the resonance frequency increases with the increasing tension of the rope. If the largest amplitudes are required to be within 50 mm, then a dangerous resonance frequency ranging from 7.7 rad/s to 10.3 rad/s can be determined as shown in Fig. 12.

Fig. 12The largest displacements at the center of a catenary in frequency domain

Fig. 13The numerical relationships between the largest response amplitudes of a catenary and the maximum hoisting speed

5.1. Speed optimization

The rotational frequency of the head sheave, which acts the fundamental excitation frequency, is determined by the hoisting speed of the catenary. Hence, to decrease the large transverse amplitude of 166.8 mm at the lower catenary 4 in this case, the excitation frequency can be changed to stay away from the resonance frequency range by varying the hoisting speed. The first three order frequency components play the dominating roles in the external excitations, considering the conclusion mentioned in section 4.2.2 that it is of no influence to replace the measured first-order excitation amplitude of 1.8 mm and the third-order excitation amplitude of 2.2 mm by a larger value of 4 mm, in this section, they are assumed as $4\mathrm{s}\mathrm{i}\mathrm{n}\left(wt\right)$ mm, $4\mathrm{s}\mathrm{i}\mathrm{n}\left(2wt\right)$ mm and $4\mathrm{s}\mathrm{i}\mathrm{n}\left(3wt\right)$ mm, respectively. Using the superposition principle, during the constant speed stage, the largest response amplitude (RF) at the center of the lower catenary 4 can be formulated as:

where $i=$1, 2, 3. According to the displacement-frequency response curves given in Fig. 12, $F$ is the largest response amplitude under the excitation $4\mathrm{s}\mathrm{i}\mathrm{n}\left(iwt\right)$, $v_{mp}$ is the optimal hoisting speed, $D$ is the diameter of the head sheave and $V_{max}$ is the permitted maximum hoisting speed given in Table 1.

Applying Eq. (30), under the three typical working conditions, the numerical relationships between the largest response amplitudes at a lower catenary and the maximum hoisting speed is given in Fig. 13. If the largest safe transverse amplitude of a catenary should be limited within 50 mm, by employing the analogy method, the maximum hoisting speed can be decreased from 9.36 m/s to 8.13 m/s as shown in Fig. 13. As a result, the largest amplitude of the lower catenary 4 in this case can be decreased from 166.8 mm to 35.56 mm.

5.2. Tension optimization

In colliery, if a smaller response amplitude of a catenary in mine hoists must be realized by decreasing the hoisting speed, then the yearly output of coal will decrease, leading to a lower financial reward. In view of this, speed optimization is not usually an economic way to deal with the large transverse amplitude. Hence, a new method must be developed to deal with this problem without changing speed.

In this section, maintaining the hoisting speed of 9.36 m/s unchanged, the external excitations are $4\mathrm{s}\mathrm{i}\mathrm{n}\left(4.16t\right)$ mm, $4\mathrm{s}\mathrm{i}\mathrm{n}\left(8.32t\right)$ mm and $4\mathrm{s}\mathrm{i}\mathrm{n}\left(12.48t\right)$ mm according to the on-site measurement, which are the same as those in Section 4. To reduce the largest response amplitude of a lower catenary without changing speed, the excitation frequencies should be beyond the resonance frequency range. Based on the conclusion that the resonance frequency increases with the increasing tension, then the tension in a catenary can be varied by adding clump weight to the skip. Thus, the updated tension $T_u$ in the catenary during the constant speed stage can be modified as:

where $m_c$ is the mass of additional clump weight. Under the three typical working conditions, employing the analogy method, the relationships between the largest response amplitudes at the center of the lower catenary and the tension increment are shown in Fig. 14. A smallest response amplitude, 31.42 mm, can be achieved when the tension increment of a rope is 26.7 kN. Then the maximum tension of a catenary during the ascending cycles with full load is 172.7 kN, which is far less than the permitted maximum static tension of the rope ($T_{max}=$900 kN given in Table 1). Therefore, tension optimization is effective and reliable. Substituting tension increment of 26.7 kN and other system parameters into Eq. (31), the mass of the additional clump weight can be calculated as 1.068×10⁴ kg.

Fig. 14The numerical relationships between the largest response amplitudes of a catenary and the tension increment

Fig. 15The non-optimal and optimal displacement-frequency curves

After tension optimization, the modified displacement-frequency response curves are depicted in Fig. 15. It can be seen from Fig. 15 that, under the same external excitation $4\mathrm{s}\mathrm{i}\mathrm{n}\left(8.32t\right)$ mm, the largest response amplitude has explicitly decreased from 151.9 mm (Fig. 12) to 24.39 mm, and the resonance frequency range has been transformed from 7.7 rad/s-10.3 rad/s to 8.7 rad/s-11.3 rad/s. Consequently, the three excitation frequencies, 4.16 rad/s, 8.32 rad/s and 12.48 rad/s, all locate outside the dangerous resonance frequency range, and the rope collision can be thoroughly avoided.