Large deformation of longitudinal-torsional-lateral coupled effect of round balance rope in mine friction hoisting system

Abstract

Torsion deformable spatial beam and Costello theory are used to establish longitudinal-torsional-lateral coupled model of round balance rope. Based on Coulomb’s friction law and longitudinal-torsional-lateral coupled model, the nonlinear coupled dynamic model with friction constraint of round balance rope is established. Meanwhile, time-varying multi-segments non–equal-length element transformation method (TMN-ETM) is proposed to save computation time. Then, natural frequencies, lateral responses are calculated when the coupled stiffness coefficient is zero. And the calculations are compared with traditional solution method. The results show that only about few areas in the round balance rope loop have large stress norm value, while the rest parts have small stress values. Besides, dynamic responses of the balance rope with balance rope suspension rotor releasing are conducted. When the static friction is converted to dynamic friction, the friction torque and angle acceleration will be mutated at the switching instants, and slip-stick transition in angle acceleration occurs.

Introduction

Flexible cables are light in weight and easy to fold which have been widely employed in diverse engineering applications, such as suspension bridges, ¹ cable-pulley systems, ² aerospace tether transportation system, ³ elevators,^4,5 overhead crane system,^6,7 cable-suspended swinging system, ⁸ and mine hoisting systems. ⁹ Hoisting ropes, conveyances, guide sheave, traction sheave, and balance ropes constitute multi-rope friction hoisting system in mine shaft. The upper ends of the balance rope are suspended at the bottom of two conveyances via balance rope suspension rotors (BRSRs). It is set up to balance the gravity of hoisting rope and obtain equal moments in mine friction hoisting system. ¹⁰ The BRSR is used to release the torsional force caused by the self-weight of round balance rope during operation. Therefore, the working state and mechanical properties of the balance rope directly affect the safe operation of friction hoisting system. ¹¹ The round balance rope is only subjected to its own gravity during its movement, and it’s space oscillation displacement is large. Due to the inertia of round balance rope and its complex helical structure, it will cause lateral, longitudinal, and twisting or torsional vibration when round balance rope moves up and down. ¹²

The dynamic model of round balance rope established by traditional coordinate system is relatively complex. Thus, Shabana et al. ¹³ proposed absolute nodal coordinate formulation (ANCF) and applied it to the model and analyzed large deformation of cantilever beam. Lan et al. ¹⁴ proposed a new high-voltage electricity wire model to simulate the dynamic behavior of the wire after failed tensioning and during the nonlinear sliding between the wire and the iron tower. Wang et al. ¹⁵ presented the dynamic model of flexible filament bundle with viscoelasticity based on ANCF and analyzed its dynamic behavior. Zhang et al. ¹⁶ established a dynamic model of flat balance rope under multiple constraints with friction via using ANCF with non–equal-length element division method. These scholars focused on studying the translational motion of slender beam in plane, but ignored spatial torsion effect of slender beam.

Vetyukov et al. ¹⁷ proposed a rod model for large bending and torsion of an elastic strip, aiming at studying the bends and twists under the action of gravity force and varying span length between two clamped ends. Dmitrochenko et al. ¹⁸ studied the dynamical behavior of a uniform-rotating helicoseir with one fixed using ANCF. Yoo et al. ¹⁹ proposed a new finite element of a thin spatial beam using ANCF and Frenet frame, and implemented the model of a thin strip rotating around a vertical axis. However, Frenet frame is undefined at the inflection points and straight segments of the beam where its curvature is zero, leading to singularities and errors in their numerical analysis. ²⁰

Shanana et al. ²¹ proposed using ANCF to model three-dimensional beam elements with the slope continuity, thereby analyzed its large rotation, torsion, shear, deformation, and the rotation of the beam cross section at the nodal points. Lots of scholars use 3D ANCF beam to analyze the dynamics of rotating shaft.^22,23 However, 3D ANCF beam elements yielded too large torsional and flexural rigidities so that shear locking effectively suppressed the asymmetric bending mode for a slender flexible spatial beam. Schwab et al. ²⁴ proposed new absolute nodal coordinate formulation with the elastic line approach to calculate a slender flexible spatial beam. Dombrowski ²⁵ developed a large deformation beam element to describe slender beams with non-ignorable torsion effects. The number of degrees of freedom of this element is reduced from 24 to 14 compared to the three-dimensional flexible beam element proposed in reference. ²⁶ This will effectively alleviate the problem that the computational efficiency is too slow due to too many degrees of freedom (DOFs). Yang et al. ²⁷ proposed an improved variable-length beam element based on ANCF and arbitrary Lagrangian–Eulerian description to build dynamic model of a one-dimensional medium with mass transportation and a non-ignorable torsion effect, so as to research the dynamic behavior of drill pipe. However, this dynamic model does not take into account the coupled relationship between longitudinal stiffness and torsional stiffness.

Therefore, the torsion deformable spatial beam element and Costello wire rope theory are used to establish longitudinal-torsional-lateral coupled vibration of round balance rope with BRSR in friction hoisting system. Time-varying multi-segments non–equal-length element transformation method (TMN-ETM) is used to reduce the dimension of the dynamic equation matrix and save dynamics simulation time of longitudinal-torsional-lateral coupled balance rope vibration. Hence, large deformation of longitudinal-torsional-lateral coupled effect of round balance rope will be studied.

Dynamic equation of round balance rope

Physical description

Mine friction hoisting system is mainly composed of traction sheave, conveyance, hoisting rope, BRSR, and balance rope. The simplified model of friction hoisting system is shown in Figure 1(a). The BRSR is used to release the torsional force caused by the self-weight of round balance rope during operation. The outermost strand twist of multi-layer strand wire rope is formulated as outer strand twist, and the inner strand twist is formulated as inner strand twist. Due to the opposite twist direction between the inner rope and the outer rope, the torsional moment generated by the inner rope and the outer rope of the multi-layer strand wire rope can offset each other, but there is still some torsional moment in the round wire rope as shown in Figure 1(b).

Figure 1.

(a) Simplified model of friction hoisting system. (b) Schematic of the round balance rope.

Costello wire rope theory, shown in Figure 1(b), is used to describe the coupling relationship between longitudinal and torsional stiffness of round balance rope. When the tensile-torsional deformation of the wire rope is considered and its bending deformation is ignored, the constitutive relation of the wire rope can be expressed as

$T^d=Q_a\epsilon +Q_b\tau ,M^d=Q_c\epsilon +Q_d\tau$

(1)

ε and τ represent axial strain and torsional curvature. T ^d and M ^d represent dynamic tension and dynamic torque in the wire rope, respectively. ¹² Q _a and Q _d are longitudinal and torsional stiffness coefficients of the round balance rope, and Q _b and Q _c are coupled stiffness coefficients of the round balance rope. ²⁸ The wire rope coupling coefficient can be obtained by importing the 3D model of the wire rope into the finite element software and then applying different boundary conditions.

However, the above constitutive relation ignores the bending deformation of the wire rope, and cannot describe the large spatial deformation behavior of the round balance rope. Therefore, it is necessary to establish a global coordinate system to describe the spatial coordinate relationship and large spatial deformation behavior of the round balance rope.

Description of coordinate systems for spatial round wire rope

In the slack low tension rope, spatial positional posture of the round balance rope can be described from two aspects: (i) the translation of the local coordinate system of the round balance rope and (ii) the rotation of the local coordinate system on the round balance rope. The former can be represented by the position vector of the origin of the local coordinate system, namely, the position vector of the point on the center line of the round balance rope. The latter uses the direction cosine matrix Θ to realize the mapping from the local coordinate system to the global inertial coordinate system.

As shown in Figure 2(a), O-XYZ is the global inertial coordinate system, and the global position vector of the beam centerline is r _c (x,t), where x is the arc length coordinate of the round balance rope centerline, and t represents time.

Figure 2.

(a) Round wire rope in the inertial coordinate system. (b) Schematic of κ ₁ , κ _2, and τ, (c) Schematic of spatial longitudinal-torsional-lateral coupled model of round balance rope.

Let us define the local coordinate system on the round balance rope section as t(x), n(x) and b(x), where t(x) is the tangent axis of the round balance rope centerline, n(x) and b(x) represent the normal axis and sub normal axis, respectively. The tangent basis vector can be described as

$\mathbf{t}=\frac{{\mathbf{r}}_c^{\prime }}{\sqrt{{{\mathbf{r}}_c^{\prime }}^{\mathrm{T}}{\mathbf{r}}_c^{\prime }}}$

(2)

${\mathbf{r}}_c^{\prime }$ represents the first-order spatial slope vector at any point r _c on the axis of rope.

The directional cosine matrix from the local coordinate system to the global coordinate system is $\left[\begin{array}{ccc}\mathbf{t}& \mathbf{n}& \mathbf{b}\end{array}\right]$ , and the position vector of any point p in the corresponding section of the beam is

${\mathbf{r}}_p={\mathbf{r}}_c+\left[\begin{array}{ccc}\mathbf{t}& \mathbf{n}& \mathbf{b}\end{array}\right]{\left[\begin{array}{ccc}0& y& z\end{array}\right]}^{\mathrm{T}}$

(3)

where [0, y, z]^T is the coordinate vector of point p in the local coordinate system.

Assuming that the inertial coordinate system is rotated by Z-Y-X parallel to the local coordinate system and the corresponding Euler angle is Tait–Bryan angle, ²⁵ the rotation matrix Θ can be expressed as

$\mathrm{\Theta }=\left[\begin{array}{ccc}{c}_2{c}_3& c_2{s}_3& -s_2\\ -c_1{s}_3+s_1{s}_2{c}_3& c_1{c}_3+s_1{s}_2{s}_3& s_1{c}_2\\ s_1{s}_3+{\mathrm{c}}_1{\mathrm{s}}_2{\mathrm{c}}_3& -s_1{c}_3+c_1{s}_2{s}_3& c_1{c}_2\end{array}\right]$

(4)

where $c_{\alpha }=\cos \left({\vartheta }_{\alpha }\right),s_{\alpha }=\sin \left({\vartheta }_{\alpha }\right),\alpha =\mathrm{1,2,3}$ . ${\vartheta }_3$ , ${\vartheta }_2$ , and ${\vartheta }_1$ represent the angle of rotation around the Z-axis, Y-axis, and X-axis, respectively. Since it is assumed that the beam axis is always perpendicular to the cross section, the rotation matrix Θ can be expressed as the directional cosine matrix ${\left[\begin{array}{ccc}\mathbf{t}& \mathbf{n}& \mathbf{b}\end{array}\right]}^{\mathrm{T}}$

$\mathbf{t}=\left[\begin{array}{c}{c}_2{c}_3\\ c_2{s}_3\\ -s_2\end{array}\right],\mathbf{n}=\left[\begin{array}{c}-c_1{s}_3+s_1{s}_2{c}_3\\ c_1{c}_3+s_1{s}_2{s}_3\\ s_1{c}_2\end{array}\right],\mathbf{b}=\left[\begin{array}{c}{s}_1{s}_3+{\mathrm{c}}_1{\mathrm{s}}_2{\mathrm{c}}_3\\ -s_1{c}_3+c_1{s}_2{s}_3\\ c_1{c}_2\end{array}\right]$

(5)

Considering the orthogonality of tangent axis t(x), normal axis n(x), and bi-normal axis b(x), there exist scalar functions κ ₁ (x), κ ₂ (x), and τ(x) ²⁰ shown in Figure 2(b)

$\begin{array}{l}{\mathbf{t}}^{\prime }\left(x\right)={\kappa }_1\left(x\right)\mathbf{n}\left(x\right)+{\kappa }_2\left(x\right)\mathbf{b}\left(x\right)\\ {\mathbf{n}}^{\prime }\left(x\right)=-{\kappa }_1\left(x\right)\mathbf{t}\left(x\right)+\tau \left(x\right)\mathbf{b}\left(x\right)\\ {\mathbf{b}}^{\prime }\left(x\right)=-{\kappa }_2\left(x\right)\mathbf{t}\left(x\right)-\tau \left(x\right)\mathbf{n}\left(x\right)\end{array}$

(6)

where a prime denotes differentiation with respect to x and

${\kappa }_1\left(x\right)={\left({\mathbf{t}}^{\prime }\left(x\right)\right)}^{\mathrm{T}}\mathbf{n}\left(x\right),{\kappa }_2\left(x\right)={\left({\mathbf{t}}^{\prime }\left(x\right)\right)}^{\mathrm{T}}\mathbf{b}\left(x\right),\tau \left(x\right)={\left({\mathbf{n}}^{\prime }\left(x\right)\right)}^{\mathrm{T}}\mathbf{b}\left(x\right)$

(7)

where κ₁(x) and κ₂(x) represent bending curvatures and τ(x) indicates the torsional curvature of the local coordinate system about the tangent vector. The torsional curvature can be simplified by substituting equation (5) into equation (7)

$\tau ={{\vartheta }_1}^{\prime }-\sin \left({\vartheta }_2\right){{\vartheta }_3}^{\prime }$

(8)

In summary, in order to reduce the number of degrees of freedom in the element, the positional posture of the round balance rope can be described by the position vector r_c(x,t) on the axis, the rotation angle ${\vartheta }_1$ of local coordinate system around the round balance rope axis. Schematic of spatial longitudinal-torsional-lateral coupled model of round balance rope is shown in Figure 2(c). Angular coordinates ø is redefined to replace ${\vartheta }_1$ . They can be uniformly expressed as generalized position vectors at any point p on the axis of the element, namely

${\mathbf{r}}_p={\left[\begin{array}{cc}{\left({\mathbf{r}}_c\right)}^{\mathrm{T}}& {\varphi }^p\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{cccc}{r}_X^p& r_Y^p& r_Z^p& {\varphi }^p\end{array}\right]}^{\mathrm{T}}=\mathbf{S}{\left[\begin{array}{cc}{\left({\mathbf{e}}_{k-1}\right)}^{\mathrm{T}}& {\left({\mathbf{e}}_k\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$

(9)

$\begin{array}{l}{\mathbf{r}}_c={\mathbf{r}}_p\left(1:3,:\right)={\left[\begin{array}{ccc}{r}_X^p& r_Y^p& r_Z^p\end{array}\right]}^{\mathrm{T}}={\mathbf{S}}_t{\left[\begin{array}{cc}{\left({\mathbf{e}}_{k-1}\right)}^{\mathrm{T}}& {\left({\mathbf{e}}_k\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}\\ {\mathbf{r}}_c^{\prime }={\mathbf{r}}_p^{\prime }\left(1:3,:\right)={\left[\begin{array}{ccc}{r^{\prime }}_X^p& {r^{\prime }}_Y^p& {r^{\prime }}_Z^p\end{array}\right]}^{\mathrm{T}}={\mathbf{S}}_t^{\prime }{\left[\begin{array}{cc}{\left({\mathbf{e}}_{k-1}\right)}^{\mathrm{T}}& {\left({\mathbf{e}}_k\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}\\ {\varphi }^p={\mathbf{r}}_p\left(4,:\right)={\mathbf{S}}_{\varphi }{\left[\begin{array}{cc}{\left({\mathbf{e}}_{k-1}\right)}^{\mathrm{T}}& {\left({\mathbf{e}}_k\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}\\ {{\varphi }^{\prime }}^p={\mathbf{r}}_p^{\prime }\left(4,:\right)={\mathbf{S}}_{\varphi }^{\prime }{\left[\begin{array}{cc}{\left({\mathbf{e}}_{k-1}\right)}^{\mathrm{T}}& {\left({\mathbf{e}}_k\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}\\ {\mathbf{S}}_t=\mathbf{S}\left(1:3,:\right),{\mathbf{S}}_{\varphi }=\mathbf{S}\left(4,:\right),{\mathbf{S}}_t^{\prime }={\mathbf{S}}^{\prime }\left(1:3,:\right),{\mathbf{S}}_{\varphi }^{\prime }={\mathbf{S}}^{\prime }\left(4,:\right)\end{array}$

(10)

The matrix S and ${\mathbf{S}}^{\prime }$ are shape function, and its corresponding first-order spatial slope in Appendix A.

The round balance rope is divided into seven segments in the axis direction as shown in Figure 3, and they are named part I, part II, part III, part IV, part V, part VI, and part VII, respectively. It is divided into ( $N_{t1}$ + $N_{t2}$ + $N_{t3}$ + $N_{t4}$ + $N_{t5}$ + $N_{t6}$ + $N_{t7}$ ) elements in the axis direction by TMN-ETM, separately. Using TMN-ETM, the length of any element is $l_{t1}$ , $l_{t2}$ , $l_{t3}$ , $l_{t4}$ , $l_{t5}$ , $l_{t6}$ , and $l_{t7}$ in part I, part II, part III, part IV, part V, part VI, and part VII, respectively.

Figure 3.

Multi-segments non–equal-length element division method.

Equation derivation

Dynamic model of round balance without BRSR

The element kinetic energy, strain energy, dissipated energy, virtual work done by the external load, gravity, concentrated force, and concentrated torque of round balance rope can be expressed as

$T_k=\frac{1}{2}\rho {\int }_{V_t}{\left(\frac{D}{Dt}\left({\mathbf{r}}_c+\mathbf{n}y+\mathbf{b}z\right)\right)}^{\mathrm{T}}\frac{D}{Dt}\left({\mathbf{r}}_c+\mathbf{n}y+\mathbf{b}z\right)\mathrm{d}V$

(11)

$U_k=\frac{1}{2}{\int }_{l_t}\left(T^d\epsilon +M^d\tau \right)\mathrm{d}x+\frac{1}{2}{\int }_{l_t}\left(E{I}_y{\kappa }_1^2+E{I}_z{\kappa }_2^2\right)\mathrm{d}x$

(12)

$\delta W_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p},\mathrm{k}}={\beta }_{ck}\left(\begin{array}{c}{Q}_a{\int }_{l_t}\dot{\epsilon }\delta \epsilon \mathrm{d}x+Q_d{\int }_{l_t}\dot{\tau }\delta \tau \mathrm{d}x+\left(Q_b+Q_c\right){\int }_{l_t}\left(\dot{\tau }\delta \epsilon +\dot{\epsilon }\delta \tau \right)\mathrm{d}x\\ +{\int }_{l_t}\left(E{I}_y{\dot{\kappa }}_1\delta {\kappa }_1+E{I}_z{\dot{\kappa }}_2\delta {\kappa }_2\right)\mathrm{d}x\end{array}\right)$

(13)

$\delta W_{\mathrm{e}\mathrm{x}\mathrm{t},\mathrm{k}}={\int }_{l_t}\delta {\left({\mathbf{r}}_p\right)}^{\mathrm{T}}\left(f_{ke}+f_{kg}\right)\mathrm{d}x+{\int }_{l_t}\delta {\left({\mathbf{r}}_p\overline{\delta }\left(x-\overline{x}\right)\right)}^{\mathrm{T}}\left(f_c+T_c\right)\mathrm{d}x$

(14)

where ${\dot{\mathbf{r}}}_p$ is the first derivative of the global position vector ${\mathbf{r}}_p$ at any point p on the central axis of rope. EI _y and EI _z represent bending stiffness coefficient. ε represents axial strain

$\epsilon =\frac{1}{2}\left({{\mathbf{r}}^{\prime }}_c^{\mathrm{T}}{\mathbf{r}}_c^{\prime }-1\right)$

(15)

$\overline{\delta }$ is the Dirac delta function in equation (14). ${\beta }_{ck}$ is the damping coefficient

$\dot{\epsilon }={\left(\frac{\partial \epsilon }{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}{\dot{\mathbf{e}}}_k,\dot{\tau }={\left(\frac{\partial \tau }{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}{\dot{\mathbf{e}}}_k,{\dot{\kappa }}_1={\left(\frac{\partial {\kappa }_1}{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}{\dot{\mathbf{e}}}_k,{\dot{\kappa }}_2={\left(\frac{\partial {\kappa }_2}{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}{\dot{\mathbf{e}}}_k$

(16)

Based on element energy expressions represented by equations (11)–(14), the dynamic equation of round balance rope can be yielded from Lagrange equation

$\{\begin{array}{c}\mathbf{M}\left(\mathbf{e}\right)\ddot{\mathbf{e}}+\mathbf{C}\left(\mathbf{e},\dot{\mathbf{e}}\right)+{\mathbf{F}}^{\mathrm{e}\mathrm{x}}+\mathbf{Q}\left(\mathbf{e}\right)+{\mathbf{W}}_N^{\mathrm{T}}{\mathbf{\lambda }}_N=0\\ {\mathbf{\Phi }}_N=0\end{array}$

(17)

where M(e) is the mass matrix, C (e,ė), Q(e), and F^ex are damping, elastic force, and generalized external force vectors of the round balance rope

$\mathbf{M}\left(\mathbf{e}\right)={\int }_{l_t}\left(\rho A{\mathbf{S}}_t^{\mathrm{T}}{\mathbf{S}}_t+\rho I_y{\left(\frac{\partial \mathbf{n}}{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}\frac{\partial \mathbf{n}}{\partial {\mathbf{e}}_k}+\rho I_z{\left(\frac{\partial \mathbf{b}}{\partial {\mathbf{e}}_k}\right)}^{\mathrm{T}}\frac{\partial \mathbf{b}}{\partial {\mathbf{e}}_k}\right)\mathrm{d}x$

(18)

$\mathbf{C}\left(\mathbf{e},\dot{\mathbf{e}}\right)={\beta }_{ck}{\left(\begin{array}{c}{Q}_a{\int }_{l_t}\dot{\epsilon }\frac{\partial \epsilon }{\partial {\mathbf{e}}_k}\mathrm{d}x+\left(Q_b+Q_c\right){\int }_{l_t}\left(\dot{\tau }\frac{\partial \epsilon }{\partial {\mathbf{e}}_k}+\dot{\epsilon }\frac{\partial \tau }{\partial {\mathbf{e}}_k}\right)\mathrm{d}x\\ +Q_d{\int }_{l_t}\dot{\tau }\frac{\partial \tau }{\partial {\mathbf{e}}_k}\mathrm{d}x+{\int }_{l_t}\left(E{I}_y{\dot{\kappa }}_1\frac{\partial {\kappa }_1}{\partial {\mathbf{e}}_k}+E{I}_z{\dot{\kappa }}_2\frac{\partial {\kappa }_2}{\partial {\mathbf{e}}_k}\right)\mathrm{d}x\end{array}\right)}^{\mathrm{T}}$

(19)

$\mathbf{Q}\left(\mathbf{e}\right)={\left(\begin{array}{c}{Q}_a{\int }_{l_t}\epsilon \frac{\partial \epsilon }{\partial {\mathbf{e}}_k}\mathrm{d}x+\left(Q_b+Q_c\right){\int }_{l_t}\left(\tau \frac{\partial \epsilon }{\partial {\mathbf{e}}_k}+\epsilon \frac{\partial \tau }{\partial {\mathbf{e}}_k}\right)\mathrm{d}x\\ +Q_d{\int }_{l_t}\tau \frac{\partial \tau }{\partial {\mathbf{e}}_k}\mathrm{d}x+{\int }_{l_t}\left(E{I}_y{\kappa }_1\frac{\partial {\kappa }_1}{\partial {\mathbf{e}}_k}+E{I}_z{\kappa }_2\frac{\partial {\kappa }_2}{\partial {\mathbf{e}}_k}\right)\mathrm{d}x\end{array}\right)}^{\mathrm{T}}$

(20)

${\mathbf{\Phi }}_N$ denotes the constraints equation of the first and the last nodes

$\{\begin{array}{cccc}\begin{array}{l}{g}_1=e_{\mathrm{1,1}}-x_L=0,\\ g_4=e_{N_t+\mathrm{1,1}}-x_R=0,\\ g_7=e_{\mathrm{1,4}}=0,\\ g_{11}=e_{N_t+\mathrm{1,4}}=0,\end{array}& \begin{array}{l}{g}_2=e_{\mathrm{1,2}}-y_L=0,\\ g_5=e_{N_t+\mathrm{1,2}}-y_R=0,\\ g_8=e_{\mathrm{1,5}}=0,\\ g_{12}=e_{N_t+\mathrm{1,5}}=0,\end{array}& \begin{array}{l}{g}_3=e_{\mathrm{1,3}}-z_L=0,\\ g_6=e_{N_t+\mathrm{1,3}}-z_R=0,\\ g_9=e_{\mathrm{1,6}}+1=0,\\ g_{13}=e_{N_t+\mathrm{1,6}}-1=0,\end{array}& \begin{array}{l}\\ \\ g_{10}=e_{\mathrm{1,7}}=0,\\ g_{14}=e_{N_t+\mathrm{1,7}}=0\end{array}\end{array}$

(21)

$x_L$ , $y_L$ , $z_L$ , $x_R$ , $y_R$ , $z_R$ are the horizontal, lateral, and vertical coordinates of the left and right ends. $e_{i,j}$ represents the absolute coordinates of the i-th node on the round balance rope.

The round balance rope is fixedly connected to conveyance without BRSR

${\mathbf{\Phi }}_N=\left[\begin{array}{cccccccccccccc}{g}_1& g_2& g_3& g_4& g_5& g_6& g_7& g_8& g_9& g_{10}& g_{11}& g_{12}& g_{13}& g_{14}\end{array}\right]$

(22)

${\mathbf{\lambda }}_N$ is Lagrange multiplier corresponding to constraints of the first and last nodes. ${\mathbf{W}}_N$ is the first-order partial derivative of the constraints coordinate of the first and last nodes.

Dynamic model of round balance with BRSR

If the round balance rope is connected to the conveyance through BRSR, the constraint equation ${\mathbf{\Phi }}_N$

${\mathbf{\Phi }}_N=\left[\begin{array}{cccccc}{g}_1& g_2& g_3& g_4& g_5& g_6\end{array}\right]$

(23)

when both BRSRs are stuck, constraint equation ${\mathbf{\Phi }}_N$ can be written as equation (22).

${\mathrm{\Phi }}_T$ denotes the relative tangential displacement of torsion constraint joint

$g_1^{\prime }=e_4-{\varphi }_L,g_2^{\prime }=e_{dof-4}-{\varphi }_R$

(24)

where ${\varphi }_L$ and ${\varphi }_R$ are torsional constraint coordinates of left end and right end.

${\mathbf{\lambda }}_T$ is the friction force of torsion constraint joint. ${\mathbf{W}}_T$ is the vector of tangential force direction

${\mathbf{W}}_T=\partial {\mathrm{\Phi }}_T/\partial \mathbf{e}$

(25)

Coulomb’s dry friction law ²⁹ can be used to describe the contact friction force at the BRSR

${\lambda }_{T,i}=\{\begin{array}{c}\left[\begin{array}{cc}-1,& 1\end{array}\right]{\mu }_{s,i}{T}_{m,i}\begin{array}{cc},& \end{array}\begin{array}{cc}{\dot{\mathrm{g}}}_{T,i}=0,& \end{array}\\ -\frac{{\dot{\mathrm{g}}}_{T,i}}{\Vert {\dot{\mathrm{g}}}_{T,i}\Vert }{\mu }_{d,i}{T}_{m,i}\begin{array}{cc},& \end{array}\begin{array}{cc}{\dot{\mathrm{g}}}_{T,i}\ne 0,& \end{array}\end{array}i=\mathrm{1,2}$

(26)

${\mu }_{s,i}$ and ${\mu }_{d,i}$ are static friction coefficient and dynamic friction coefficient, and assumed to be constant. ³⁰ Using this projection function ²⁹

${\lambda }_T={\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{C_T\left(T_m\right)}\left({\lambda }_T-{\dot{\mathrm{g}}}_T\right)$

(27)

where the convex set $C_T$ covers the range of admissible friction forces as follows

$C_T\left(T_m\right)=\left\{x\in \mathrm{R}|x\le \mu \left|T_m\right|\right\}$

(28)

$T_m$ represents the torque force generated by the pressure at the BRSR between the round balance rope and BRSR.

Combining equations (17) and (27) yields the dynamic equations with friction, which are expressed as

$\{\begin{array}{c}\mathbf{M}\left(\mathbf{e}\right)\ddot{\mathbf{e}}+\mathbf{C}\left(\mathbf{e},\dot{\mathbf{e}}\right)+{\mathbf{F}}^{\mathrm{e}\mathrm{x}}+\mathbf{Q}\left(\mathbf{e}\right)+{\mathbf{W}}_N^{\mathrm{T}}{\mathbf{\lambda }}_N+{\mathbf{W}}_T^{\mathrm{T}}{\mathbf{\lambda }}_T=0\\ {\mathbf{\Phi }}_N=0\\ {\mathbf{\lambda }}_T-{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{C_T\left(T_m\right)}\left({\lambda }_T-{\mathbf{W}}_T\dot{\mathbf{e}}\right)=0\end{array}$

(29)

Equations (17) and (29) can be solved by the extended generalized-α algorithm. ¹⁶

Element transformation

As shown in Figure 4, the node position of the balance rope is transformed from Figure 4(a)to Figure 4(b). After the fourth segment is transformed, the number of the element in the fourth segment is evenly distributed on the left and right sides. The number of elements in the first segment is increased by one, the number of elements in the seventh segment is decreased by one, and the number of elements in other segments remains unchanged.

Figure 4.

TMN-ETM node transformation of the round balance rope.

When three nodes need to be merged into two nodes at the critical point of the second segment and the first segment, it can be achieved by equation (30)

$\left[\begin{array}{c}{\mathbf{e}}^{l-1}\\ {\mathbf{e}}^{l+1}\end{array}\right]={\mathbf{A}}_l{\left[\begin{array}{ccc}{\mathbf{e}}^{l-1,\mathrm{T}}& {{\mathbf{e}}^{l,}}^{\mathrm{T}}& {\mathbf{e}}^{l+1,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$

(30)

${\mathbf{A}}_l=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]\otimes {\mathbf{I}}_8$

(31)

where ${\mathbf{I}}_8$ is the eighth-order identity matrix. When five nodes need to be merged into three nodes at the critical point of the third segment and the second segment, it can be achieved by equation (32)

$\left[\begin{array}{c}{\mathbf{e}}^{m-1}\\ {\mathbf{e}}^{m+1}\\ {\mathbf{e}}^{m+3}\end{array}\right]={\mathbf{A}}_m{\left[\begin{array}{ccccc}{\mathbf{e}}^{m-1,\mathrm{T}}& {\mathbf{e}}^{m,\mathrm{T}}& {\mathbf{e}}^{m+1,\mathrm{T}}& {\mathbf{e}}^{m+2,\mathrm{T}}& {\mathbf{e}}^{m+3,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$

(32)

${\mathbf{A}}_m=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]\otimes {\mathbf{I}}_8$

(33)

when nine nodes need to be merged into five nodes at the critical point of the fourth segment and the third segment, it can be achieved by equation (34)

$\left[\begin{array}{c}{\mathbf{e}}^{n-1}\\ {\mathbf{e}}^{n+1}\\ {\mathbf{e}}^{n+3}\\ {\mathbf{e}}^{n+5}\\ {\mathbf{e}}^{n+7}\end{array}\right]={\mathbf{A}}_n{\left[\begin{array}{ccccccccc}{\mathbf{e}}^{n-1,\mathrm{T}}& {\mathbf{e}}^{n,\mathrm{T}}& {\mathbf{e}}^{n+1,\mathrm{T}}& {{\mathbf{e}}^{n+2,}}^{\mathrm{T}}& {\mathbf{e}}^{n+2,\mathrm{T}}& {\mathbf{e}}^{n+4,\mathrm{T}}& {\mathbf{e}}^{n+5,\mathrm{T}}& {\mathbf{e}}^{n+6,\mathrm{T}}& {\mathbf{e}}^{n+7,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$

(34)

${\mathbf{A}}_n=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\otimes {\mathbf{I}}_8$

(35)

when two nodes need to interpolate three nodes at the critical point of the sixth segment and seventh segment, it can be achieved by the formula

${\left[\begin{array}{ccc}{\mathbf{e}}^{p-1,\mathrm{T}}& {\mathbf{e}}^{{\left(p-1\right)}^{\prime },\mathrm{T}}& {\mathbf{e}}^{p,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}={\mathbf{B}}_p\left[\begin{array}{c}{\mathbf{e}}^{p-1}\\ {\mathbf{e}}^p\end{array}\right]$

(36)

${\mathbf{B}}_p={\left[\begin{array}{cc}{\mathbf{B}}_p^{\mathrm{F}}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}\end{array}\right]}^{\mathrm{T}}$

(37)

${\mathbf{B}}_p^{\mathrm{F}}=\left[\begin{array}{c}{\mathbf{I}}_8\\ {\mathbf{0}}_8\end{array}\right],{\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}=\left[\begin{array}{ccc}{\mathbf{S}}^{\mathrm{T}}\left(\frac{1}{2}\right)& {{\mathbf{S}}^{\prime }}^{\mathrm{T}}\left(\frac{1}{2}\right)& {\mathbf{B}}_p^{\mathrm{L}}\end{array}\right],{\mathbf{B}}_p^{\mathrm{L}}=\left[\begin{array}{c}{\mathbf{0}}_{8\times 8}\\ {\mathbf{I}}_8\end{array}\right]$

(38)

where 0_8×8is the 8-order zero matrix. When three nodes need to interpolate five nodes at the critical point of the fifth segment and sixth segment, it can be achieved by the formula

${\left[\begin{array}{ccccc}{\mathbf{e}}^{q-1,\mathrm{T}}& {\mathbf{e}}^{{\left(q-1\right)}^{\prime },\mathrm{T}}& {\mathbf{e}}^{q,\mathrm{T}}& {\mathbf{e}}^{q^{\prime },\mathrm{T}}& {\mathbf{e}}^{q+1,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}={\mathbf{B}}_q\left[\begin{array}{c}{\mathbf{e}}^{q-1}\\ {\mathbf{e}}^q\\ {\mathbf{e}}^{q+1}\end{array}\right]$

(39)

${\mathbf{B}}_q={\left[\begin{array}{ccc}{\mathbf{B}}_p^{\mathrm{F}}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}& {\mathbf{0}}_{8\times 16}\\ {\mathbf{0}}_{8\times 8}& {\mathbf{0}}_{8\times 16}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}\end{array}\right]}^{\mathrm{T}}$

(40)

where 0_8×16is the zero matrix of eight rows and sixteen columns. When five nodes need to interpolate nine nodes at the critical point of the fourth segment and fifth segment, it can be achieved by the formula

${\left[\begin{array}{ccccccccc}{\mathbf{e}}^{w-1,\mathrm{T}}& {\mathbf{e}}^{{\left(w-1\right)}^{\prime },\mathrm{T}}& {\mathbf{e}}^{w,\mathrm{T}}& {\mathbf{e}}^{w^{\prime },\mathrm{T}}& {\mathbf{e}}^{w+1,\mathrm{T}}& {\mathbf{e}}^{{\left(w+1\right)}^{\prime },\mathrm{T}}& {\mathbf{e}}^{w+2,\mathrm{T}}& {\mathbf{e}}^{{\left(w+2\right)}^{\prime },\mathrm{T}}& {\mathbf{e}}^{w+3,\mathrm{T}}\end{array}\right]}^{\mathrm{T}}{\mathbf{e}}^{w^{\prime },\mathrm{T}}={\mathbf{B}}_w\left[\begin{array}{c}{\mathbf{e}}^{w-1}\\ {\mathbf{e}}^w\\ {\mathbf{e}}^{w+1}\\ {\mathbf{e}}^{w+2}\\ {\mathbf{e}}^{w+3}\end{array}\right]$

(41)

${\mathbf{B}}_w={\left[\begin{array}{ccccc}{\mathbf{B}}_p^{\mathrm{F}}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}\\ {\mathbf{0}}_{8\times 8}& {\mathbf{0}}_{8\times 16}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}\\ {\mathbf{0}}_{8\times 8}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}& {\mathbf{0}}_{8\times 16}\\ {\mathbf{0}}_{8\times 8}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}& {\mathbf{0}}_{8\times 16}& {\mathbf{B}}_p^{\mathrm{s}\mathrm{p}}\end{array}\right]}^{\mathrm{T}}$

(42)

Among them, ${\mathbf{e}}^{\mathrm{()},\mathrm{T}}$ is the element node before transformation, ${\mathbf{e}}^{\mathrm{()}\prime ,\mathrm{T}}$ is the newly increased element node after transformation.

At the same time, velocity coordinate ė and acceleration coordinate $\ddot{\mathbf{e}}$ are multiplied by equations (31), (33), (35), (37), (40), and (42) to get new generalized velocity coordinate ${\dot{\mathbf{e}}}_{new}$ , and acceleration coordinate ${\ddot{\mathbf{e}}}_{new}$ .

Flow chart of the TMN-ETM round balance rope is shown in Figure 5, the normal time step h is divided into a slightly smaller time step ${\beta }_{s}h$ and a slightly larger time step ${\beta }_{l}h$ , namely, ${\beta }_s<0.5<{\beta }_l,$ and ${\beta }_s+{\beta }_l=1$ . A slightly smaller time step Newton–Raphson iteration will be used to eliminate the disturbance caused in the element transformation process.

Figure 5.

Flow chart of the TMN-ETM round balance rope.

Application in round balance rope

Validation of the TMN-ETM

Numerical simulations are applied to a 120 m round balance rope, and the physical parameters and simulation parameters are listed in Tables 1 and 2.

Table 1.

120 m round balance rope physical parameters.

Parameter	Value	Description
L	164 m	Total length of the balance rope
H	120 m	Vertical distance between two ends of the balance rope
D	2.5 m	Horizontal distance between two ends of the balance rope
a max	0.75 m/s2	Maximum operating acceleration
v max	6 m/s	Maximum operating speed
ρA	6.42 kg/m	Linear density of the balance rope
EIy, EIz	68.78 N·m2	Bending stiffness coefficient
Q a	1.097·108 N	Longitudinal stiffness coefficient
Q d	2.751·103 N·m2	Torsional stiffness coefficient
Q b ,Q c	/,/N·m2	Coupled stiffness coefficients
ρIy, ρIz	0.00142 kg·m2	Moment of inertia

Table 2.

Simulation parameters and computation time of the 120 m round balance rope.

Q b = Q c = 0	Element	DOF	Time, h	Q b ≠ 0, Q c ≠ 0	Element	DOF	Time, h
TMN-ETM	154	1240	1.04	TMN-ETM	154	1240	1.04
EEM ANCF	656	3942	7.73	EEM	656	5256	14.25

In order to verify the correctness of nonlinear coupled dynamic model with TMN-ETM, the coupled stiffness coefficients of the round balance rope in the calculation process are reduced to 0, namely, Q _b = Q _c = 0 N·m. Under the same physical parameters, the natural frequencies, the vibration displacement response are compared with ANSYS. The left side and right side observation positions are 3 m above the lowest point of balance rope in initial condition, respectively.

The first four out-plane and in-plane natural frequencies of the round balance rope obtained from ANSYS and TMN-ETM are depicted in Figures 6(a) and (b), separately. H is the height difference between the left and right conveyance. The solid line and dashed line represent the frequencies obtained from ANSYS and TMN-ETM, respectively. From Figure 6, it can be seen that the solid line and dashed line are completely coincident. That is to say, the frequencies obtained by the TMN-ETM and ANSYS are in excellent agreement. It is found that a curve veering phenomenon occurs in out-plane natural frequencies from Figure 6(a). However, neither frequency veering nor mode shift exists in in-plane natural frequencies of the round balance rope. In-plane natural frequencies do not change too much with the position of conveyances.

Figure 6.

First four out-plane and in-plane natural frequencies of the 120 m round balance rope.

For the round balance rope under the non-excitation condition, the dynamic analysis is carried out by MATLAB. The lateral displacements of the selected space positions on the round balance rope are shown in Figure 7. The solid line and dashed line represent the displacements obtained from ANCF with equal-length element method (EEM), and nonlinear coupled dynamic model with TMN-ETM, respectively. From Figure 7, it can be seen that the lateral displacements of the selected space positions are in good agreement by two methods. That is to say, the lateral displacements of the round balance rope obtained by the TMN-ETM meet the error requirements when the coupling stiffness coefficient is zero.

Figure 7.

Lateral displacements of the selected positions.

When the coupled stiffness coefficients of the round balance rope are not zero, set Q _b = 2.371 × 10⁵ N·m, Q _c = 2.301 × 10⁵ N·m, the correctness of nonlinear coupled dynamic model with TMN-ETM is verified by comparing dynamic response obtained by the nonlinear coupled dynamic model with EEM and nonlinear coupled dynamic model with TMN-ETM. For the round balance rope under the non-excitation condition, the dynamic analysis is carried out by MATLAB. The out-plane displacements and in-plane displacements of the selected space positions on round balance rope are shown in Figure 8. The solid line and dashed line represent the displacements obtained from the EEM and TMN-ETM, respectively. The arrow direction indicates the movement direction of round balance rope. As shown in Figure 8, out-plane lateral displacement at left observation point is negative, while out-plane lateral displacement at right observation point is positive. The left side out-plane lateral displacement and the right side out-plane lateral displacements do not jump between positive and negative values. The torsional displacements of the selected space positions on round balance rope are displayed in Figure 9. The solid line and dashed line represent the displacements obtained from the EEM and TMN-ETM, respectively. Since the coupled stiffness coefficients of the round balance rope is not zero, the longitudinal vibration of the round balance rope will cause torsional vibration of the round balance rope shown in Figure 9, which will drive the torsion of the round balance rope, and then generate out-plane lateral vibration.

Figure 8.

Lateral displacements of the selected positions of the 120 m round balance rope.

Figure 9.

Torsional displacements of the selected positions of the 120 m round balance rope.

It can be noticed from Figures 8and 9that out-plane, in-plane, and torsional displacements of the selected space positions are in good agreement by two methods. That is, the lateral displacements and torsional displacements of the round balance rope obtained by the TMN-ETM meet the error requirements when the coupling stiffness coefficient is not zero.

Figures 10 and 11 show the element stress norm diagram of the round balance rope removing first and last element nodes. Figures 10(a)and 11(a)demonstrate the element bending stress and element torsional stress norm diagram by EEM, respectively. The highlighted part in the element bending stress norm diagram described in Figure 10(a)represents the element stress norm value of the round balance rope bending part, and the curve formed by the highlighted part corresponds to the displacement curve of conveyance. It can be seen from Figure 10(a)that only about few areas in the round balance rope loop have large stress norm value and are in bending state, while the rest parts have a small stress value and are in approximately straight state. Similarly, the same pattern appears in the element torsional stress diagram from Figure 11(a). Therefore, it is only necessary to perform finer element division in the places where the bending stress value and the torsional stress value are large, while the elements in other areas can be slightly rougher, thereby reducing the number of elements. Figures 10(b)and 11(b)illustrate the element bending stress and element torsional stress norm diagram by TMN-ETM. It can be seen from Figures 10and 11that the maximum values of element bending stress and the element torsional stress norm value obtained by the two methods in the calculation process are the same, and the amplitudes are relatively close. The element bending stress and element torsional stress norm value are stable over the time domain, and there is no sudden change in stress due to node transformation. This shows that the proposed method does not produce spurious stress caused by the longer element length of vertical segment. It can be proved that the proposed method is stable and convergent over the entire motion time domain.

Figure 10.

Element bending stress norm value of the 120 m round balance rope.

Figure 11.

Element torsional stress norm value of the 120 m round balance rope.

Figure 12(a) shows kinetic energy of the round balance rope. The blue solid line represents the kinetic energy obtained by EEM. The red dashed line and magenta dash-dotted line represent the kinetic energy obtained before and after the TMN-ETM node transformation, respectively. It can be seen that kinetic energy of the balance rope is in good agreement by EEM and TMN-ETM. The kinetic energy error between EEM and TMN-ETM is shown in Figure 12(b). It can be seen from the diagram that the relative kinetic energy error generated through the TMN-ETM is less than 1 J, which is much lower than the kinetic energy generated by the balance rope during the movement (difference of four orders of magnitude). The relative error of kinetic energy before and after TMN-ETM node transformation is shown in Figure 12(c). It can be seen from the figure that the relative error of kinetic energy generated by velocity change after TMN-ETM node transformation is less than 10⁻⁴ J, so the kinetic energy error generated by TMN-ETM node transformation can be ignored.

Figure 12.

(a) Balance rope kinetic energy. (b) Kinetic energy error between EEM and TMN-ETM. (c) Kinetic energy error before and after TMN-ETM node transformation.

Computation time of the 120 m round balance rope is shown in Table 2. Compared with the ANCF with EEM, the element number of nonlinear coupled dynamic model with TMN-ETM is decreased by 502, and the computation time of simulation when Q _b = Q _c = 0 N·m under non-excitation is decreased by 86.5%. Compared with the nonlinear coupled dynamic model with EEM, the element number of TMN-ETM is decreased by 502, and the computation time of simulation when Q _b ≠ 0, Q _c ≠ 0 N·m under non-excitation is decreased by 92.7%. That is to say, nonlinear coupled dynamic model with TMN-ETM is stable and efficient.

By means of the same physical parameters, the natural frequencies and vibration displacements under non-excitation condition are compared with ANSYS and ANCF with EEM. We can detect that TMN-ETM is reliable and trustworthy, and TMN-ETM can be used to establish the dynamic model and analyze dynamic behaviors of the round balance rope.

Dynamic responses of balance rope with BRSR releasing

The physical parameters of the 240 m round balance rope are used in Table 3. T _F is the friction torque. μ _s,i and μ _d,i are static and dynamic friction coefficient. μ _s,i = 0.2 and μ _d,i = 0.16. T _m represents the torque force, set T _m = 50 N⋅m.

Table 3.

240 m round balance rope physical parameters.

Parameter	Value	Parameter	Value	Parameter	Value
L	284 m	v max	6 m/s	Q a	$2.46\cdot {10}^8\mathrm{N}$
H	240 m	ρA	17.5 kg/m	Q b	1.46·106 N·m
D	2.5 m	EIy, EIz	$3.21\cdot {10}^2\mathrm{N}\cdot {\mathrm{m}}^2$	Q c	1.52·106 N·m
a max	0.75 m/s2	ρIy, ρIz	0.01252 kg·m2	Q d	2.38·104 N·m2

For the round balance rope under torsional friction constraint, that is equations (43),

$T_F=\{\begin{array}{cc}\begin{array}{ccc}\begin{array}{c}{\lambda }_{T,i},\\ {\lambda }_{T,i},\\ -{\dot{\mathrm{g}}}_{T,i}/\Vert {\dot{\mathrm{g}}}_{T,i}\Vert {\mu }_{T,i}{T}_m,\end{array}& & \begin{array}{cc}\Vert {\lambda }_{T,i}\Vert \le {\mu }_{\mathrm{s}}{T}_m& \& {\dot{\mathrm{g}}}_{T,i}=0,\\ \Vert {\lambda }_{T,i}\Vert \ge {\mu }_d{T}_m& \& {\dot{\mathrm{g}}}_{T,i}=0,\\ \Vert {\lambda }_{T,i}\Vert \ge {\mu }_d{T}_m& \& {\dot{\mathrm{g}}}_{T,i}\ne 0,\end{array}\end{array}& \end{array}i=\mathrm{1,2}$

(43)

Figures 13(a)–(d) show the three-dimensional configurations of the round balance rope, its projection on XY plane and ZY plane, and torsional displacement of the round balance rope with BRSR releasing at different times, respectively. It can be seen from Figure 13that in the process of balance rope releasing torsional force, in-plane lateral displacements of the balance rope change little (Figure 13(b)), and out-plane lateral displacements and torsional displacements change violently (Figures 13(a), (c) and (d)), resulting in large out-plane lateral and torsional displacements. The out-plane lateral balance rope loop swings back and forth from negative region to positive region.

Figure 13.

(a) Three-dimensional configuration of the round balance rope at different times, and (b) its projection on the XY plane, and (c) its projection on the ZY plane, and (d) torsional displacement of the round balance rope.

Angle acceleration, angle velocity, torsional displacement, and friction torque at right BRSR are displayed in Figures 14(a)–(d). It can be observed from Figure 14that at the moment of balance rope releasing torsional force, the angle acceleration, angle velocity, torsional displacement, and friction torque at the right end of the balance rope change dramatically and oscillate back and forth. When the torsional force of the balance rope is less than the static friction torque to be overcome, the friction between the balance rope and BRSR is static friction. Otherwise, the friction is converted to dynamic friction. At the switching instants, the friction torque and angle acceleration will also be mutated as depicted in Figures 14(a) and (d), and accompanied by slip-stick transition in angle acceleration, for instance, around t = 20 s and t = 20.8 s.

Figure 14.

Angle acceleration, angle velocity, torsional displacement, and friction torque at right BRSR.

Conclusion

The torsion deformable spatial beam element and Costello wire rope theory are used to establish the longitudinal-torsional-lateral coupled dynamic model of the round balance rope in mine friction hoisting system. The nonlinear coupled dynamic model with friction constraint of the round balance rope under BRSR is proposed based on longitudinal-torsional-lateral coupled model and Coulomb’s friction law. Based on the spatial configuration characteristics of the balance rope, TMN-ETM is proposed to reduce the dimension of the dynamic equation matrix. By means of the same parameters, natural frequencies, lateral responses are compared with ANSYS and ANCF with EEM when the coupled stiffness coefficient is zero. Thereby, it can be concluded that TMN-ETM is stable and efficient.

Motion characteristics of the round balance rope without BRSR are analyzed when the coupled stiffness coefficient is not zero. The maximum torsional displacement and out-plane lateral displacement of the balance rope, caused by longitudinal-torsional coupling effect, appears at the right side balance rope loop below the balance rope. Both left and right side out-plane lateral displacements do not jump between positive and negative values during the motion of conveyance. Only about few areas in the round balance rope loop have large stress norm value and are in bending state, while the rest parts have a small stress value.

Dynamic responses of the balance rope with BRSR releasing are conducted. In the process of releasing the torsional force of the balance rope, in-plane lateral displacements of the balance rope change little, and out-plane lateral displacements and torsional displacements change violently. The out-plane balance rope loop swings back and forth from negative region to positive region. When the static friction is converted to dynamic friction, the friction torque and angle acceleration will be mutated at the switching instants, and accompanied by slip-stick transition in angle acceleration.