Semi-Active Suppression of Longitudinal Vibration in Mine Hoisting Ropes Using Magnetorheological Damper and Output-Feedback Adaptive Sliding-Mode Control

Abstract

Severe longitudinal vibrations and abnormal tension fluctuations in hoisting ropes pose significant threats to the safe and stable operation of mine hoisting systems. To address these issues, this paper proposes a semi-active vibration-suppression strategy combining a magnetorheological damper (MRD) with output-feedback adaptive sliding-mode control (ASMC). A dynamic model of the MRD-equipped hoisting system is developed using Hamilton’s principle. The nonlinear hysteresis of the MRD is described by a simplified extended hyperbolic tangent function model (SEHTFM), and an inverse model converts the desired control force into a feasible real-time current command. Using only displacement and velocity measurements at the conveyance–rope connection, the ASMC compensates for matched uncertainties, including boundary excitation, modeling and truncation errors, and force-realization errors. Numerical simulations compare an optimized passive viscous damper benchmark, SMC–MRD, and ASMC–MRD responses under varying payloads, accelerations, and hoisting speeds. During constant-speed operation, ASMC–MRD achieves peak reduction rates of 82.8% in dynamic displacement and 77.6% in dynamic tension relative to the optimized passive benchmark. The results demonstrate accurate force realization with small bounded tracking errors and improved robustness under variable operating conditions.

1. Introduction

Mine hoists are essential components of underground mining systems and play a critical role in the vertical transportation of materials. To meet the growing demand for high-output and high-efficiency deep mining, modern hoisting systems are being developed with higher payload capacities, increased hoisting speeds, and longer lifting distances. However, the combination of long suspended ropes, high operating speeds, and heavy terminal loads makes flexible wire ropes particularly prone to pronounced longitudinal vibrations. Excessive longitudinal vibration can cause rope slackening, dynamic instability, severe impact loading, and structural damage to the hoisting system, thereby posing a serious threat to the safe and stable operation of deep mine hoists [1,2,3,4].

To mitigate rope vibration and tension fluctuations in mine hoisting systems, extensive research has focused on active hydraulic control at the floating sheave. In this approach, the actuator adjusts the sheave motion to regulate rope dynamics and axial tension. Wang et al. [5,6] designed a full-dimensional state observer to compensate for time-varying sensor delays and changing rope lengths and further developed an observer-based output-feedback controller to suppress axial vibrations in time-varying hoisting ropes. For axially excited ropes, a state feedback controller without higher-order differential terms was developed using the backstepping method for coupled partial differential equation–ordinary differential equation (PDE–ODE) systems [7]. Under uncertain axial disturbances, an output-feedback controller combining backstepping with active disturbance rejection control was proposed to attenuate rope vibrations [8]. An infinite-dimensional observer was also designed to enable input-feedback control of the two moving sheaves, thereby reducing tension differences between the ropes [9]. Li et al. [10,11,12] developed a fuzzy adaptive sliding-mode controller for rope-tension coordination under uncertainties and disturbances. Subsequently, nonlinear adaptive backstepping control with a disturbance observer was applied to two-rope hoisting systems. In addition, rope tension has been regulated using a control strategy that combines a backstepping controller with a barrier Lyapunov function. Ding et al. [13,14] developed Hamilton-based distributed-parameter models and adaptive robust boundary controllers to suppress axial vibrations and tension imbalances in ropes and containers.

High-speed elevator systems exhibit dynamic characteristics similar to those of mine hoisting systems, including long flexible ropes, moving terminal masses, and time-varying rope lengths. Therefore, vibration-control methods developed for elevator ropes can provide useful references for suppressing vibrations in mine hoisting ropes. In recent years, increasing attention has been devoted to the vibration control of elevator ropes. Zhu and Chen [15] proposed an electromagnetic control device to dissipate vibration energy in elevator ropes and investigated the lateral responses of moving ropes in a high-rise elevator under both uncontrolled and controlled conditions. Benosman and Fukui [16,17,18] examined the effects of upward and downward compensation-sheave motions on the lateral vibration of elevator ropes and developed an active variable-stiffness control method based on the Lyapunov functional approach. Knezevic et al. [19] proposed a cooperative speed-control enhancement method incorporating a stopper filter, the Goertzel algorithm, and the Kiefer search algorithm to mitigate vibration-induced degradation in elevator ride comfort and speed regulation. Santo et al. [20] investigated the nonlinear horizontal response of a three-degree-of-freedom elevator model considering guide-rail deformation and proposed a state-dependent Riccati-equation-based controller to improve passenger comfort. Nguyen et al. [21,22] developed nonlinear active control strategies for high-rise elevators to mitigate seismic vibrations of hoisting and compensation ropes while maintaining nominal tension; these strategies were verified through stability analysis and numerical simulations. Tian et al. [23] studied the nonlinear longitudinal vibrations of high-speed elevators using a multi-degree-of-freedom model with time-varying rope properties, which was solved using a variable-step Runge–Kutta method and validated experimentally.

Despite these advances, most active vibration-control methods for mine hoist and elevator ropes still rely on hydraulic or electromagnetic actuators, which are typically associated with high energy consumption, structural complexity, and demanding maintenance requirements. These limitations reduce their suitability for deep mining applications, where reliability, compactness, and energy efficiency are critical. Studies on time-varying hoisting ropes and elevator-rope dynamics have further shown that variations in rope length, terminal mass, operating speed, and boundary excitation can significantly affect longitudinal vibration and tension fluctuations [1,2,3,4,23].

In contrast, MRDs offer fast response, adjustable damping, low power consumption, and inherent semi-active stability [24]. Existing studies have extended MRD-based cable control from parameter optimization to integrated damper–sensor–controller design, including adaptive feedback control using limited sensing information [25], target-force-based semi-active control for coupled and parametric cable vibrations [26], and data-driven current regulation and hysteresis modeling using deep reinforcement learning and long short-term memory neural networks [27,28]. In recent years, semi-active vibration-control methods based on MRDs have been further developed. Yang et al. investigated phase deviation in semi-active suspension control and proposed an inertial-suspension-based compensation strategy, highlighting the effects of phase deviation and realizable-force constraints in semi-active systems [29]. Du et al. proposed a semi-active control strategy for nonlinear parametric vibration suppression of super-long stay cables equipped with a magnetorheological fluid damper, showing the applicability of MRD-based semi-active control to cable vibration mitigation [26]. Rodríguez-Torres et al. combined an MRD with active disturbance rejection control to compensate for unmodeled dynamics and parametric uncertainties in semi-active structural vibration suppression [30]. Gonçalves et al. investigated MRD-based semi-active vibration control for high-speed elevator systems, demonstrating the potential of MRDs in rope-related vertical transportation systems [31]. Although these studies confirm the effectiveness of MRD-based semi-active control, the use of this approach for suppressing longitudinal vibration in mine hoisting ropes with time-varying rope lengths and variable operating conditions remains insufficiently investigated.

Although MRD-based semi-active control has been widely studied, its application to mine hoisting ropes remains limited. Mine hoisting ropes are moving flexible structures with time-varying rope length, moving terminal mass, and stage-dependent excitation. In addition, an MRD can provide controllable damping force but cannot independently carry the mean static tensile load. Therefore, a physically consistent MRD-equipped hoisting model, an appropriate passive benchmark, and force-realization analysis are needed to evaluate the actual benefit of semi-active MRD control in mine hoisting systems.

To address these issues, this study develops an MRD-based semi-active longitudinal vibration-suppression strategy for mine hoisting ropes by combining an inverse MRD model with output-feedback adaptive sliding-mode control. The main contributions are as follows:

(1)

A dynamic model of an MRD-equipped mine hoisting system is established by considering time-varying rope length, terminal mass, upper-boundary excitation, and a parallel load-bearing spring element.

(2)

An inverse SEHTFM-based MRD model is employed to convert the desired control force into feasible real-time current commands.

(3)

An optimized passive viscous damper is introduced as the benchmark, and the reduction rates of SMC–MRD and ASMC–MRD are evaluated relative to this benchmark.

(4)

The force–displacement and force–tracking characteristics of the MRD are analyzed to verify the semi-active force-realization capability under different operating conditions.

The remainder of this paper is organized as follows. Section 2 establishes the mathematical model of the MRD-equipped mine hoisting system and formulates the inverse MRD model for real-time current-command generation. Section 3 develops the output-feedback ASMC and presents the corresponding stability analysis. Section 4 evaluates the vibration-suppression performance, force-realization capability, and robustness of the proposed ASMC–MRD strategy through numerical simulations. Finally, Section 5 summarizes the main conclusions and outlines directions for future research.

2. Mathematical Model

2.1. Dynamic Model of the MRD

Nonlinear MRD modeling methods based on hyperbolic-tangent-type descriptions and enhanced hysteretic formulations have been reported in previous studies [32,33], providing useful references for describing the current-dependent hysteretic behavior of MR dampers. In particular, enhanced hysteretic models, such as the enhanced Bouc–Wen model for rotational MR dampers, have demonstrated the importance of accurately representing nonlinear MRD force–displacement and force–velocity behavior [34]. To describe the nonlinear hysteretic behavior of the MRD and facilitate the generation of real-time current commands through inverse-model-based control, the SEHTFM is adopted in this paper. Owing to its compact mathematical form and ease of parameter identification, this model is suitable for implementation in the proposed semi-active vibration-suppression system.

For a prescribed control current, the damping force generated by the MRD is expressed as

$$F_m\left(t\right)=f_{my}\tanh \left({\alpha }_m\left({\dot{x}}_m+v_{mh}\mathrm{sgn}\left(x_m\right)\right)\right)+c_{mpo}{\dot{x}}_m+f_{m0},$$

(1)

where x_m and ${\dot{x}}_m$ denote the relative displacement and relative velocity across the MRD, respectively, f_my is the hysteresis scale factor, α_m is the hysteresis slope factor, v_mh is the hysteresis half-width, c_mpo is the post-yield damping coefficient, and f_m0 is the bias force.

Because the mechanical response of the MRD depends strongly on both the applied current and the damper motion state, the parameters in Equation (1) are expressed as functions of the control current and the motion-dependent variable as follows:

$$\left\{\begin{array}{l}{f}_{my}=d_{m0}+d_{m1}{i}_m+d_{m2}{i}_m^2\\ {\alpha }_m=a_{m1}{\left(1+v_{me}\right)}^{a_{m0}}\\ v_{mh}=v_{me}{b}_{me}\\ c_{mpo}=c_{m0}+\left(c_{m1}{i}_m+c_{m2}{i}_m^2\right)\times c_{m4}{\left(1+v_{me}\right)}^{c_{m3}}\\ v_{me}=\sqrt{{\dot{x}}_m^2-x_m{\ddot{x}}_m}\end{array}\right.,$$

(2)

where i_m is the control current.

The parameter vector of the SEHTFM is defined as

$${\Phi }_m=\left[\begin{array}{cccccccccccc}{d}_{m0}& d_{m1}& d_{m2}& a_{m0}& a_{m1}& b_{me}& c_{m0}& c_{m1}& c_{m2}& c_{m3}& c_{m4}& f_{m0}\end{array}\right].$$

The forward MRD model provides the basis for deriving the inverse model used to generate real-time current commands. In practical implementation, the desired control force specified by the controller must be mapped to a feasible current command for the MRD. Accordingly, at each sampling instant, the required current is determined by solving the following bounded scalar optimization problem:

$$i_m^{*}=\arg {\min }_{0\le i_m\le i_{\max }}\left|F_m\left(i_m\right)-F_m^{*}\right|,$$

(3)

where $F_m^{\ast }$ denotes the desired control force specified by the controller, F_m(i_m) is the damping force generated by the MRD under the control current i_m, and i_max denotes the maximum allowable control current of the MRD.

To enable real-time current-command generation, the forward MRD model based on the SEHTFM is reformulated as a quadratic function of the control current as follows:

$$F_m\left(i_m\right)=A_m{i}_m^2+B_m{i}_m+C_m,$$

(4)

where A_m, B_m, and C_m are motion-state-dependent coefficients defined as

$$\left\{\begin{array}{l}{A}_m=d_{m2}\tanh \left({\alpha }_m\left({\dot{x}}_m+v_{mh}\mathrm{sgn}{x}_m\right)\right)+c_{m2}{c}_{m4}{\left(1+v_{me}\right)}^{c_{m3}}{\dot{x}}_m\\ B_m=d_{m1}\tanh \left({\alpha }_m\left({\dot{x}}_m+v_{mh}\mathrm{sgn}{x}_m\right)\right)+c_{m1}{c}_{m4}{\left(1+v_{me}\right)}^{c_{m3}}{\dot{x}}_m\\ C_m=d_{m0}\tanh \left({\alpha }_m\left({\dot{x}}_m+v_{mh}\mathrm{sgn}{x}_m\right)\right)+c_{m0}{\dot{x}}_m+f_{m0}\end{array}\right.,$$

This quadratic representation retains the nonlinear current dependence incorporated in Equation (2) while allowing the inverse model to be evaluated efficiently for real-time implementation. Under the assumption that F_m(i_m) is monotonic over the admissible current range, the current command that best realizes the desired control force can be obtained from the piecewise closed-form solution to the bounded force-tracking problem in Equation (3) as follows:

$$i_m^{\ast }=\left\{\begin{array}{ll}0,\hfill & A_m=0 , B_m=0\hfill \\ {\displaystyle \frac{F_m^{\ast }-C_m}{B_m}},\hfill & A_m=0, B_m\ne 0 , 0\le {\displaystyle \frac{F_m^{\ast }-C_m}{B_m}}\le i_{\max }\hfill \\ -{\displaystyle \frac{B_m}{2A_m}},\hfill & A_m\ne 0, \Delta =0 , 0\le -{\displaystyle \frac{B_m}{2A_m}}\le i_{\max }\hfill \\ \min \left\{i_{mk}\left|i_{mk}\in \left[0,i_{\max }\right]\right.,k=1,2\right\},\hfill & A_m\ne 0, \Delta >0 , \exists i_{mk}\in [0,i_{\max }]\hfill \\ \left\{\begin{array}{ll}0,\hfill & J_0\le J_{\max }\hfill \\ i_{\max },\hfill & J_0>J_{\max }\hfill \end{array}\right.,\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

(5)

where

$\Delta =B_m^2-4A_m(C_m-F_m^{\ast })$ is the discriminant of the quadratic force-matching equation,

$i_{m1,2}={\displaystyle \frac{-B_m\pm \sqrt{\Delta }}{2A_m}}$ are its real roots when Δ > 0, and

$J_0=\left|C_m-F_m^{\ast }\right|$ and $J_{\max }=\left|A_m{i}_{\max }^2+B_m{i}_{\max }+C_m-F_m^{\ast }\right|$ denote the force-tracking errors at the lower and upper bounds of the admissible current range, respectively. When two admissible real roots exist, the smaller root is selected to reduce the required control current.

The remaining cases correspond to situations in which the linear solution is outside the admissible current range, no real roots exist, the repeated root is inadmissible, or neither real root satisfies the current constraint. In these cases, the optimal current command is selected by comparing the force-tracking errors at the two endpoints of the admissible interval. This closed-form inverse-model solution enables efficient real-time calculation of the required current command, which is essential for high-bandwidth semi-active vibration suppression.

2.2. Dynamic Model of the MRD-Equipped Hoisting System

A schematic configuration of the MRD-equipped mine hoisting system is shown in Figure 1. One end of the hoisting rope is wound around the drum, whereas the other end passes over the head sheave and is connected to the conveyance through a load-bearing spring element and an MRD arranged in parallel. The load-bearing spring element represents the suspension path that carries the mean static tensile force of the hoisting rope. The MRD is installed across the same relative displacement path and provides a controllable dissipative force for suppressing the dynamic vibration component. As the drum rotates, the rope is wound or unwound, thereby driving the upward or downward motion of the conveyance. The MRD is modeled as a lumped semi-active damping element. The load-bearing spring element mainly contributes to the quasi-static load path, whereas the MRD force enters the dynamic vibration equation as a controllable damping force. Two local coordinate systems are introduced. The contact point between the drum and the catenary rope segment is selected as the origin of the coordinate system O₁X₁Y_1, whereas the contact point between the head sheave and the vertical rope segment is selected as the origin of the coordinate system O₂X₂Y₂. l_v(t) is the length of the vertical rope, and l_c is the length of the catenary rope. x_c(t) is the catenary rope arc length coordinate, x_c(t) ∈ [0, l_c]. x_v(t) is the time-varying arc-length coordinate of the vertical rope, x_v(t) ∈ [0, l_v(t)]. The longitudinal displacements of the catenary and vertical rope segments are denoted by u_c(x_c,t) and u_v(x_v,t), respectively. An overdot and a prime denote partial differentiation with respect to time t and the corresponding arc-length coordinate, respectively.

The kinetic energy of the MRD-equipped hoisting system is expressed as

$$\begin{array}{cc}{T}_d=& {\displaystyle \frac{1}{2}}\rho {{\displaystyle {\int }_0^{l_c}\left[V_d\left(t\right)+{\dot{u}}_c\left(x_c,t\right)+V_d\left(t\right){u^{\prime }}_c\left(x_c,t\right)\right]}}^{2}d{x}_c\hfill \\ & +{\displaystyle \frac{1}{2}}{\displaystyle \frac{I_h}{R_h^2}}{\left[V_d\left(t\right)+{\dot{u}}_c\left(l_c,t\right)\right]}^2\hfill \\ & +{\displaystyle \frac{1}{2}}\rho {{\displaystyle {\int }_0^{l_v\left(t\right)}\left[V_d\left(t\right)+{\dot{u}}_v\left(x_v,t\right)+V_d\left(t\right){u^{\prime }}_v\left(x_v,t\right)\right]}}^{2}d{x}_v\hfill \\ & +{\displaystyle \frac{1}{2}}(m+m_d){\left[V_d\left(t\right)+{\dot{u}}_v\left(l_v\left(t\right),t\right)+V_d\left(t\right){u^{\prime }}_v\left(l_v\left(t\right),t\right)\right]}^2\hfill \end{array},$$

(6)

where ρ is the linear density of the rope; m denotes the equivalent terminal mass, including the conveyance and its payload; m_d is the mass of the MRD; I_h and R_h are the moment of inertia and radius of the head sheave; V_d(t) denotes the hoisting speed; and α stands for the inclination angle of the catenary rope with respect to the horizontal direction. Hereafter, the subscripts c and v correspond to the catenary rope segment and vertical rope segment, respectively.

The potential energy of the MRD-equipped hoisting system is expressed as

$$\begin{array}{cc}{P}_d=& {\displaystyle {\int }_0^{l_c}\left((m+m_d)g+\rho g{l}_v\left(t\right)-\rho g\left(l_c-x_c\right)\sin \alpha \right){u^{\prime }}_c}d{x}_c\hfill \\ & +{\displaystyle {\int }_0^{l_v\left(t\right)}\left((m+m_d)g+\rho g{l}_v\left(t\right)-\rho g{x}_v\right){u^{\prime }}_v}d{x}_v\hfill \\ & +{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{l_c}EA{u^{\prime }}_c^2}d{x}_c+{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{l_v\left(t\right)}EA{u^{\prime }}_v^2}d{x}_v\hfill \\ & -(m+m_d)g{u}_v\left(l_v\left(t\right),t\right)\hfill \end{array},$$

(7)

where EA is the axial stiffness of the rope and g is the gravitational acceleration.

The virtual work done by non-conservative forces is expressed as

$$\delta W=-c_h\left(V_d\left(t\right)+{\dot{u}}_c\left(l_c,t\right)\right)\delta u_c\left(l_c,t\right)-F_m\left(t\right)\delta u_v\left(l_v\left(t\right),t\right),$$

(8)

where c_h denotes the damping coefficient at the contact interface between the rope and the head sheave.

By applying Hamilton’s principle, we obtain

$${\displaystyle {\int }_{t_1}^{t_2}\delta \left(T_d-P_d\right)}dt+{\displaystyle {\int }_{t_1}^{t_2}\delta W}dt=0.$$

(9)

Applying Leibniz’s rule and integration by parts to Equation (9), the coupled governing equations and associated boundary conditions of the MRD-equipped hoisting system are obtained as follows:

$$\rho \left({\ddot{u}}_c+{\dot{V}}_d\left(t\right){\dot{u}}_c+2V_d\left(t\right){{\dot{u}}^{\prime }}_c+{\left(V_d\left(t\right)\right)}^2{u^{″}}_c+{\dot{V}}_d\left(t\right)\right)-EA{u^{″}}_c=0, x_c\in \left(0,l_c\right),$$

(10)

$$\rho \left({\ddot{u}}_v+2V_d\left(t\right){{\dot{u}}^{\prime }}_v+{\dot{V}}_d{\dot{u}}_v+{\left(V_d\left(t\right)\right)}^2{u^{″}}_v+{\dot{V}}_d\left(t\right)\right)-EA{u^{″}}_v=0, x_v\in \left(0,l_v\left(t\right)\right),$$

(11)

$$u_c\left(0,t\right)=0,$$

(12)

$$EA{u^{\prime }}_c\left(l_c,t\right)-c_h\left(V_d\left(t\right)+{\dot{u}}_c\left(l_c,t\right)\right)-EA{u^{\prime }}_v\left(0,t\right)={\displaystyle \frac{I_h}{R_h^2}}{\ddot{u}}_c\left(l_c,t\right),$$

(13)

$$\begin{array}{l}\left(m+m_d\right)\left[\left({\ddot{u}}_v+2V_d\left(t\right){{\dot{u}}^{\prime }}_v+{\dot{V}}_d\left(t\right){\dot{u}}_v+{\left(V_d\left(t\right)\right)}^2{u^{″}}_v+{\dot{V}}_d\left(t\right)\right)\right]\\ +EA{u^{\prime }}_v\left(l_v\left(t\right),t\right)+F_b\left(t\right)=0\end{array}.$$

(14)

At the lower boundary of the vertical rope segment, the total force transmitted through the parallel connection can be expressed as

$$F_b(t)=F_p(t)+F_m(t),$$

(15)

where F_p(t) denotes the static or quasi-static force provided by the load-bearing spring element. The spring force balances the mean tensile load associated with the conveyance, payload, MRD mass, and rope self-weight.

Equations (10) and (11) govern the longitudinal vibrations of the catenary and vertical rope segments, respectively, whereas Equations (12)–(14) specify the associated boundary conditions.

In an actual mine hoisting system, the catenary rope segment, the head sheave, and the vertical rope segment are dynamically coupled. The motion of the catenary rope segment and the rotation of the head sheave influence the response of the vertical rope segment through the displacement and tension variations transmitted at its upper boundary. Although a fully coupled model retaining both rope segments can be established using Hamilton’s principle, explicitly discretizing both segments would increase the model order and complicate the subsequent controller design. Because the catenary rope segment is considerably shorter than the vertical rope segment and the present study focuses on suppressing the dominant longitudinal vibration at the conveyance–rope connection, a control-oriented model of the vertical rope segment is adopted. In this reduced model, the effects of the catenary rope–head sheave subsystem and the residual coupling dynamics are represented by an equivalent bounded displacement excitation applied at the upper boundary of the vertical rope segment. Accordingly, the upper boundary condition of the reduced vertical rope model is prescribed as

$$u_v\left(0,t\right)=w_h\left(t\right),$$

(16)

where w_h(t) denotes the equivalent displacement excitation transmitted from the catenary rope–head sheave subsystem to the upper end of the vertical rope segment. When w_h(t) = 0, the reduced model corresponds to the fixed-upper-boundary case.

The governing equation in Equation (11), together with the boundary conditions in Equations (14) and (16), constitutes an infinite-dimensional PDE model of the vertical rope segment. To facilitate the design of the ASMC, this PDE model is approximated by a finite-dimensional ODE model through the Galerkin method. For this purpose, a normalized spatial coordinate is introduced for the vertical rope segment as

$${\xi }_v={\displaystyle \frac{x_v}{l_v\left(t\right)}}\in \left[0,1\right].$$

(17)

Accordingly, the displacement field of the vertical rope segment is rewritten as $u_v\left(x_v,t\right)={\overline{u}}_v\left({\xi }_v,t\right)$. Because the upper boundary condition in Equation (16) is nonhomogeneous, the following transformation is introduced as

$${\overline{u}}_v({\xi }_v,t)=(1-{\xi }_v)w_h(t)+{\tilde{u}}_v({\xi }_v,t), {\tilde{u}}_v(0,t)=0.$$

(18)

This transformation converts the nonhomogeneous upper boundary condition into a homogeneous one. Since the lifting term $(1-{\xi }_v)w_h(t)$ vanishes at the lower boundary, i.e., at ξ_v = 1, it follows that

$${\overline{u}}_v(1,t)={\tilde{u}}_v(1,t).$$

(19)

Therefore, the displacement at the lower end of the vertical rope segment, which is selected as the controlled output, remains unaffected by the transformation.

Substitution of Equation (18) into the governing equation in Equation (11) and the lower boundary condition in Equation (14) shows that the terms involving w_h(t) act as an equivalent external disturbance on the vertical rope subsystem. To streamline the subsequent controller design, these terms are incorporated into the generalized disturbance vector rather than being written explicitly.

To achieve a suitable compromise between approximation accuracy and computational efficiency, a four-mode Galerkin approximation is adopted [34]. Accordingly, the transformed displacement field is approximated as

$${\tilde{u}}_v\left({\xi }_v,t\right)={\displaystyle \sum _{i=1}^4{\varphi }_{v,i}\left({\xi }_v\right)}{q}_i\left(t\right),$$

(20)

where ϕ_v,i(ξ_v) denotes the i-th mode shape function of the vertical rope segment and q_i(t) denotes the corresponding generalized coordinate.

The mode shape functions adopted in the Galerkin approximation are expressed as

$${\varphi }_{v,i}\left({\xi }_v\right)=\sin \left({\displaystyle \frac{2i-1}{2}}\pi {\xi }_v\right), i=1,2,3,4.$$

(21)

Substituting Equations (18) and (20) into the governing equation in Equation (11), enforcing the lower boundary condition in Equation (14), projecting the resulting residual onto each mode shape function ϕ_v,i(ξ_v), and integrating over the normalized spatial domain ([0, 1]) yield the finite-dimensional equations of motion. The terms induced by the nonhomogeneous upper-boundary excitation w_h(t) are represented as an equivalent disturbance input to the vertical rope subsystem. To obtain a computationally tractable model for controller design, the residual effects of modal truncation, parameter perturbations, modeling inaccuracies, and unmodeled dynamics are treated as lumped bounded uncertainties. Accordingly, the finite-dimensional second-order model used for ASMC design is expressed as

$$M\left(t\right)\ddot{q}\left(t\right)+C\left(t\right)\dot{q}\left(t\right)+K\left(t\right)q\left(t\right)=B{F}_m\left(t\right)+D_h\left(t\right)+D_r\left(t\right),$$

(22)

where M(t), C(t), and K(t) are the generalized mass, damping, and stiffness matrices, respectively; B denotes the MRD input vector; D_h(t) represents the equivalent disturbance induced by the upper-boundary excitation transmitted from the catenary rope–head sheave subsystem; D_r(t) denotes the lumped bounded uncertainty associated with modal truncation, parameter perturbations, modeling inaccuracies, and unmodeled dynamics; and q(t) = [q₁(t), q₂(t), q₃(t), q₄(t)]^T is the generalized coordinate vector.

Once Equation (22) is solved for q(t), the longitudinal displacement field of the vertical rope segment can be reconstructed using Equations (18) and (20).

3. Design of an Output-Feedback Adaptive Sliding-Mode Controller

In this paper, an output-feedback ASMC is developed to suppress the longitudinal vibration of the MRD-equipped mine hoisting system. The proposed controller uses the measured displacement and velocity at the conveyance–rope connection to determine the desired control force. This desired force is subsequently converted into a feasible MRD current command through the inverse MRD model subject to the prescribed current constraint. The closed-loop vibration-suppression scheme is shown in Figure 2.

In this paper, compared with the model-based feedback control methods reported in [35,36], the proposed ASMC accounts for the boundary excitation transmitted from the catenary rope–head sheave subsystem, time-varying rope-length effects, modal truncation errors, parameter perturbations, output-feedback approximation errors, residual unmodeled dynamics, and bounded MRD force-realization errors through an equivalent matched disturbance.

3.1. Control-Oriented Dynamic Model and Sliding Surface

Based on the finite-dimensional model obtained in Equation (22), the longitudinal vibration dynamics of the vertical rope segment can be expressed in the control-oriented form as follows:

$$M\left(t\right)\ddot{q}+C\left(t\right)\dot{q}+K\left(t\right)q=B{u}_a\left(t\right)+D_0\left(t\right),$$

(23)

where

$$u_a\left(t\right)=F_m\left(t\right), D_0\left(t\right)=D_h\left(t\right)+D_r\left(t\right), B\left(t\right)=-\mathit{\varphi }\left(1\right).$$

Here, u_a(t) denotes the actual control force generated by the MRD. Premultiplying Equation (23) by M⁻¹(t) gives

$$\ddot{q}=-M^{-1}\left(t\right)\left[C\left(t\right)\dot{q}+K\left(t\right)q\right]+M^{-1}\left(t\right)B{u}_a\left(t\right)+M^{-1}\left(t\right)D_0\left(t\right).$$

(24)

The controlled output is selected as the longitudinal displacement at the conveyance–rope connection as

$$y\left(t\right)={\mathit{\varphi }}^T\left(1\right)q\left(t\right), \dot{y}\left(t\right)={\mathit{\varphi }}^T\left(1\right)\dot{q}\left(t\right),$$

(25)

where ϕ (1) is the vector of mode shape functions evaluated at the lower end of the vertical rope segment, corresponding to the conveyance–rope connection.

For controller design and stability analysis, the following assumptions are introduced.

Assumption 1.

The matrix M(t) is symmetric and positive definite for all t. In addition, M(t), C(t), K(t), and their relevant time-varying coefficients are bounded over the operating range of the hoisting system.

Assumption 2.

The output-channel input gain is defined as

$$g_{\mathit{\varphi }}\left(t\right)={\mathit{\varphi }}^T\left(1\right)M^{-1}\left(t\right)B=-{\mathit{\varphi }}^T\left(1\right)M^{-1}\left(t\right)\mathit{\varphi }\left(1\right).$$

(26)

Since M(t) is uniformly symmetric positive definite and ϕ(1) ≠ 0, g_ϕ(t) maintains a constant negative sign and is bounded away from zero within the operating range of the hoisting system. There exists a positive constant g_min satisfying

$$\left|g_{\varphi }\left(t\right)\right|\ge g_{\min }>0.$$

The linear sliding surface is defined as

$$s\left(t\right)=\dot{y}\left(t\right)+\lambda y\left(t\right), \lambda >0,$$

(27)

where λ determines the desired convergence rate of the output dynamics on the sliding surface.

Differentiating Equation (27) and using Equations (24) and (25), we obtain

$$\dot{s}\left(t\right)=F_{nom}\left(q,\dot{q},t\right)+g_{\varphi }\left(t\right)u_a\left(t\right)+d_{\varphi }\left(t\right),$$

(28)

where

$$F_{nom}\left(q,\dot{q},t\right)\triangleq -{\mathit{\varphi }}^T\left(1\right)M^{-1}\left(t\right)\left[C\left(t\right)\dot{q}\left(t\right)+K\left(t\right)q\left(t\right)\right]+\lambda {\mathit{\varphi }}^T\left(1\right)\dot{q}\left(t\right),$$

(29)

and

$$d_{\varphi }\left(t\right)={\mathit{\varphi }}^T\left(1\right)M^{-1}\left(t\right)D_0\left(t\right).$$

(30)

g_ϕ(t) is nonzero. According to Assumption 2, the disturbance term d_ϕ(t) can be represented as an equivalent disturbance acting through the control-input channel as

$${\delta }_l\left(t\right)={\displaystyle \frac{d_{\varphi }\left(t\right)}{g_{\varphi }\left(t\right)}}.$$

(31)

Accordingly, the sliding-surface dynamics associated with the actual MRD force can be rewritten as

$$\dot{s}\left(t\right)=F_{nom}\left(q,\dot{q},t\right)+g_{\varphi }\left(t\right)\left[u_a\left(t\right)+{\delta }_0\left(t\right)\right].$$

(32)

Equation (32) shows that the upper-boundary excitation and the residual model uncertainties enter the sliding-surface dynamics through the same channel as the actual MRD control force.

3.2. Output-Feedback Approximation and Lumped Uncertainty

In practical mine hoist systems, the complete modal state vectors q and $\dot{q}$ are not directly measurable. Therefore, an output-feedback approximation is introduced using only the measured displacement y(t) and velocity $\dot{y}\left(t\right)$ at the conveyance–rope connection.

For long cable and hoisting-rope systems, the low-frequency modal components usually dominate the longitudinal vibration response [37,38]. Based on this modal-dominance property, the modal response used in the nominal feedback term is approximated by its first-mode component, whereas the residual influence of the neglected modes is incorporated into the lumped uncertainty. Let $v_1={\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]}^T$. Under the first-mode approximation, the modal coordinates can be expressed as

$$q\left(t\right)\approx v_1{\displaystyle \frac{y\left(t\right)}{{\varphi }_1}}, \dot{q}\approx v_1{\displaystyle \frac{\dot{y}\left(t\right)}{{\varphi }_1}}, {\varphi }_1\ne 0,$$

(33)

where ϕ₁ is the first component of the modal-shape vector at the controlled boundary. Substituting Equation (33) into Equation (29), the output-based approximation of the nominal function is as follows:

$$\widehat{F}(y,\dot{y},t)=\left[\alpha (t)+\lambda \right]\dot{y}(t)+\beta (t)y(t),$$

(34)

where

$$\alpha (t)=-{\displaystyle \frac{{\mathit{\varphi }}^{\mathrm{T}}\left(1\right)M^{-1}(t)C(t)v_1}{{\varphi }_1}}, \beta (t)=-{\displaystyle \frac{{\mathit{\varphi }}^{\mathrm{T}}(1)M^{-1}(t)K(t)v_1}{{\varphi }_1}}$$

(35)

The approximation error between the full-state nominal function and its output-based approximation is defined as

$$\Delta F\left(t\right)=F_{\mathrm{nom}}(q,\dot{q},t)-\widehat{F}(y,\dot{y},t).$$

(36)

The term ΔF(t) accounts for the effects of neglected higher-mode contributions and output-feedback approximation errors. These effects are not ignored; rather, they are incorporated into the lumped uncertainty and compensated online by the adaptive sliding-mode term.

The desired control force generated by the ASMC is denoted as $u\left(t\right)=F_m^{\ast }\left(t\right)$. Using the inverse MRD model derived in Section 2.1, the corresponding real-time current command is obtained as

$$i_m(t)={\mathrm{sat}}_{[0,i_{\max }]}\left\{f_{\mathrm{MRD}}^{-1}\left[u_d(t),y(t),\dot{y}(t)\right]\right\},$$

(37)

where ${\mathrm{sat}}_{[0,i_{\max }]}\left\{\cdot \right\}$ denotes the saturation operator within the admissible current range [0, i_max] and $f_{\mathrm{MRD}}^{-1}(\cdot )$ represents the bounded inverse mapping of the MRD, which is realized via the closed-form solution of the force-tracking problem in Equation (5).

Due to current constraints, hysteresis residuals of the MRD, parameter variations and implementation errors, the actual output force of the MRD deviates from the desired control force. Thus,

$$u_a\left(t\right)=u\left(t\right)+e_F\left(t\right), \left|e_F\left(t\right)\right|\le {\overline{e}}_F,$$

(38)

where $e_F\left(t\right)=F_m\left(t\right)-F_m^{\ast }\left(t\right)$ is the bounded force realization error of the MRD. The desired control force may not be realized exactly when it lies outside the force range achievable by the MRD under the instantaneous motion state and admissible current constraint. The resulting bounded discrepancy is represented by e_F(t) and incorporated into the equivalent matched lumped disturbance.

Combining Equations (32), (36) and (38) gives the implementable sliding-surface dynamics based on output feedback as

$$\dot{s}\left(t\right)=\widehat{F}\left(y,\dot{y},t\right)+g_{\varphi }\left(t\right)\left[u\left(t\right)+{\delta }_l\left(t\right)\right],$$

(39)

where the matched lumped uncertainty is defined as

$${\delta }_l(t)={\delta }_0(t)+{\displaystyle \frac{\Delta F(t)}{g_{\varphi }(t)}}+e_F\left(t\right).$$

(40)

Accordingly, δ_l(t) includes the influence of the upper-boundary excitation, time-varying rope-length effects contained in the reduced model, modal truncation errors, parameter perturbations, output-feedback approximation errors, residual unmodeled dynamics, and bounded MRD force-realization errors.

Assumption 3.

The equivalent matched disturbance δ_l(t) is bounded and varies sufficiently slowly with time. Specifically, there exist unknown positive constants δ and D_l such that

$$\left|{\delta }_l\left(t\right)\right|\le \overline{\delta }, \left|{\dot{\delta }}_l\left(t\right)\right|\le D_l.$$

(41)

Assumption 4.

The force-realization error induced by the bounded inverse MRD mapping and residual modeling inaccuracies is bounded and satisfies Equation (38). Consequently, its effect is included in the equivalent matched lumped disturbance δ_l(t).

All components of δ_l(t) enter Equation (39) through the same input channel g_ϕ(t) as the desired control force, and the matching condition required for SMC is satisfied.

3.3. Output-Feedback ASMC Law

To compensate online for the unknown and time-varying lumped disturbance δ_l(t) without requiring its explicit upper bound in the control law, the desired control force is designed as

$$u\left(t\right)={\displaystyle \frac{1}{g_{\varphi }\left(t\right)}}\left[\widehat{F}\left(y,\dot{y},t\right)+k_{1}s\left(t\right)+k_2\Phi (s)\right]-{\widehat{\delta }}_l\left(t\right),$$

(42)

where k₁ > 0 and k₂ > 0 are control gains and ${\widehat{\delta }}_l\left(t\right)$ is the online estimate of the lumped uncertainty δ_l(t).

The continuous switching function is defined as

$$\Phi (s\left(t\right))={\displaystyle \frac{s\left(t\right)}{\left|s\left(t\right)\right|+\epsilon }} , \epsilon >0,$$

(43)

where ε denotes the boundary layer parameter. This continuous approximation reduces chattering by smoothing the switching action in the vicinity of the sliding surface, thereby improving practical implementability in the presence of measurement noise and MRD current constraints.

The disturbance estimate is updated by the σ-modified adaptive law as

$$\dot{\widehat{\delta }}\left(t\right)=\gamma g_{\varphi }\left(t\right)s\left(t\right)-\sigma \widehat{\delta }\left(t\right), \gamma >0, \sigma >0,$$

(44)

where γ is the adaptation gain and σ is a leakage coefficient.

The first term in Equation (44) adjusts the estimated disturbance according to the sliding variable and the input-channel gain, whereas the leakage term prevents unbounded drift of the disturbance estimate in the presence of persistent disturbances, measurement noise, limited excitation, or actuator constraints.

The desired control force calculated by Equation (42) is converted into a feasible current command by Equation (37). Therefore, the proposed controller retains the low-power and inherently stable characteristics of semi-active MRD actuation while providing online compensation for the principal uncertainties affecting longitudinal vibration suppression.

3.4. Controller Implementation and Parameter Tuning

Substituting the ASMC law in Equation (42) into the sliding dynamics in Equation (39) yields the closed-loop sliding-surface dynamics as

$$\dot{s}\left(t\right)=-k_{1}s\left(t\right)-k_2\Phi \left(s(t)\right)+g_{\varphi }\left(t\right)\tilde{\delta }\left(t\right).$$

(45)

The disturbance estimation error is defined as

$${\tilde{\delta }}_l\left(t\right)={\delta }_l\left(t\right)-{\widehat{\delta }}_l\left(t\right).$$

(46)

Combined with Equation (44), the dynamics of the estimation error are derived as

$${\dot{\tilde{\delta }}}_l\left(t\right)={\dot{\delta }}_l\left(t\right)-\gamma g_{\varphi }\left(t\right)s\left(t\right)+\sigma {\widehat{\delta }}_l\left(t\right).$$

(47)

Equivalently, Equation (47) can be rewritten as

$${\dot{\tilde{\delta }}}_l\left(t\right)={\dot{\delta }}_l\left(t\right)-\gamma g_{\varphi }\left(t\right)s\left(t\right)+\sigma {\delta }_l\left(t\right)-\sigma {\tilde{\delta }}_l\left(t\right).$$

(48)

The controller parameters are selected by considering both vibration-suppression performance and physical implementation constraints. In all cases, the resulting desired control force is converted into a bounded current command through the inverse MRD model, ensuring compatibility with the semi-active actuation constraint.

4. Results and Discussion

4.1. Simulation Parameters and Operating Conditions

To evaluate the vibration-suppression performance of the proposed ASMC–MRD strategy, numerical simulations are conducted based on the dynamic model of the MRD-equipped mine hoisting system developed in Section 2 and the output-feedback ASMC designed in Section 3. The principal parameters of the mine hoisting system used in the simulations are listed in

Table 1. Principal parameters of the mine hoisting system used in the simulations.

Parameters	Value	Parameters	Value
Rh (m)	2.45	m (kg)	14,000
Ih (kg/m2)	1.5 × 104	ρ (kg/m)	8.6
EA (N)	2.96 × 108	ch	0.5
md (kg)	200	g (m/s2)	9.8

.

The operating profile of the mine hoist is shown in Figure 3. The hoisting cycle follows a typical trapezoidal velocity profile consisting of three stages: acceleration from 0 to 20 s with a = 0.65 m/s², constant-speed operation from 20 to 64 s with V_d = 13 m/s and deceleration from 64 to 84 s with a = −0.65 m/s². As shown in Figure 3a, the hoisting displacement increases throughout the upward hoisting process. Consequently, the length of the vertical rope segment becomes shorter as the conveyance ascends.

4.2. Mechanical Characteristics of the MRD

Before evaluating the vibration-suppression performance of the proposed ASMC–MRD strategy, the mechanical response of the MRD model is examined under different control currents. The experimentally identified parameters reported in Ref. [33] are adopted to implement the SEHTFM described in Section 2.1. The parameter values used in the subsequent numerical simulations are summarized in

Table 2. Parameters of the SEHTFM used for the MRD simulations.

Parameters	Value	Parameters	Value
dm0 (N)	221.0	cm0 (N·s/mm)	7.285
dm1 (N/A)	7512.1	cm1 (A−1)	1.279
dm2 (N/A2)	−2223	cm2 (A−2)	−0.4
am0	−0.940	cm3	−0.977
am1 (s/mm)	5.296	cm4 (N·s/mm)	911.6
bme	0.2362	fm0 (N)	110

.

Since the MRD parameters are identified using displacement and velocity expressed in millimeters and millimeters per second, respectively, the relative displacement and velocity inputs to the SEHTFM are converted from SI units to the corresponding millimeter-based units before evaluating the MRD force.

Based on the parameters listed in Table 2, the force–velocity and force–displacement relationships of the MRD under different control currents are obtained using the SEHTFM, as shown in Figure 4.

As shown in Figure 4a, the SEHTFM can reproduce the typical hysteresis loops of the MRD. The area enclosed by the hysteresis loop, which reflects the energy dissipation capacity, increases significantly as the control current increases. Figure 4b further shows that the damping force increases nonlinearly with both the control current and the relative velocity, demonstrating the nonlinear damping characteristics of the MRD. These numerical responses are consistent with the expected mechanical behavior of an MRD based on the experimentally identified parameters reported in Ref. [33]. The experimental identification and validation of the MRD model were conducted in Ref. [33] and are therefore not repeated in the present study. Here, the SEHTFM-based MRD model is used as the semi-active actuator model for evaluating the proposed ASMC–MRD vibration-suppression strategy.

4.3. Vibration-Suppression Performance of the ASMC–MRD Strategy

Following the examination of the mechanical response of the adopted MRD model, the proposed ASMC–MRD strategy is applied to suppress the longitudinal vibration of the vertical rope segment in the mine hoisting system. The controller parameters used in the following simulations are selected in accordance with the stability analysis and parameter-tuning considerations presented in Section 3.4. Specifically, the sliding-surface coefficient, linear feedback gain, switching gain, adaptation gain, leakage coefficient, and boundary-layer parameter are set to λ = 1.20, k₁ = 1.00, k₂ = 0.030, γ = 1.0 × 10⁷, σ = 0.08, and ε = 0.002, respectively. Here, λ specifies the desired convergence rate of the controlled output on the sliding surface, k₁ determines the linear convergence contribution, k₂ enhances robustness against the equivalent matched lumped disturbance, γ and σ are the adaptation gain and leakage coefficient, respectively, and ε defines the boundary layer used to smooth the switching function and reduce chattering.

4.3.1. Static Equilibrium Components of the Hoisting Rope

Before analyzing the dynamic responses, the static equilibrium components of the vertical rope segment are determined. The static tension and the corresponding equilibrium displacement of the hoisting rope are calculated as follows:

$$F_s\left(t\right)=\left[\left(m+m_d\right)+\rho l_v\left(t\right)\right]g,$$

(49)

$$u_s\left(t\right)={\displaystyle \frac{\left(m+m_d\right)gl\left(t\right)}{EA}}+{\displaystyle \frac{\rho g{l}^2\left(t\right)}{2EA}},$$

(50)

where F_s(t) represents the static tension at the load end of the hoisting rope and u_s(t) denotes the static equilibrium displacement at the conveyance–rope connection induced by the terminal mass, the MRD mass, and the self-weight of the vertical rope segment.

As shown in Figure 5, both the static tensile force and the equilibrium displacement decrease continuously during upward hoisting because the length of the vertical rope segment decreases as the conveyance moves upward. The uncontrolled and controlled static components are almost identical because they are determined by the quasi-static load path and operating configuration rather than by the semi-active damping action. In the parallel configuration shown in Figure 1, the load-bearing spring element carries the mean tensile load, whereas the MRD regulates only the residual dynamic vibration component.

Therefore, the vibration-suppression performance is evaluated using the dynamic displacement and dynamic tension fluctuations after removing the static equilibrium components. This treatment explains why the static tension and static equilibrium displacement are almost unchanged by MRD control and ensures that the following comparisons focus on the vibration-suppression capability of the passive benchmark, SMC–MRD, and ASMC–MRD.

4.3.2. Optimized Passive Viscous Damper Benchmark

To provide a rigorous benchmark for evaluating the vibration-suppression performance of the proposed semi-active MRD control strategies, an optimized passive viscous damper was introduced for comparison. The purpose of this benchmark is to distinguish the performance improvement caused by the presence of damping from the additional improvement achieved by semi-active force regulation. Therefore, the passive viscous damper was installed at the same position as the MRD, while the load-bearing spring element and the hoisting-system parameters were kept unchanged.

For the passive benchmark case, the controllable MRD force was replaced by the force generated by a conventional viscous damper with a constant damping coefficient. The passive damping force was defined as

$$F_p(t)=-c_p{\dot{x}}_m(t)$$

(51)

where F_p(t) is the passive damping force, c_p is the constant viscous damping coefficient, and ${\dot{x}}_m(t)$ is the relative velocity across the damper. The negative sign indicates that the passive damping force is opposite to the relative motion and therefore dissipates vibration energy. Unlike the MRD, the passive viscous damper does not require current input or a control algorithm. Its mechanical behavior is completely determined by the fixed damping coefficient c_p.

The value of c_p has a direct influence on the vibration response of the hoisting rope. An excessively small damping coefficient cannot provide sufficient energy dissipation, whereas an excessively large damping coefficient may increase force transmission and deteriorate the dynamic tension response. Therefore, the passive damping coefficient was optimized before it was used as the comparison benchmark. The optimization problem was formulated as

$$c_p^{opt}=\arg {\min }_{c_p\in [c_{\min },c_{\max }]}J(c_p)$$

(52)

where $c_p^{opt}$ is the optimal passive damping coefficient and $[c_{\min },c_{\max }]$ is the prescribed search range of the damping coefficient.

Because the vibration-suppression performance of the hoisting rope is evaluated using dynamic displacement and dynamic tension, the objective function was defined by combining the normalized RMS values of these two responses:

$$J(c_p)=w_1{\displaystyle \frac{\mathrm{RMS}[y_d(t;c_p)]}{\mathrm{RMS}[y_d(t;0)]}}+w_2{\displaystyle \frac{\mathrm{RMS}[T_d(t;c_p)]}{\mathrm{RMS}[T_d(t;0)]}}$$

(53)

where y_d(t; c_p) and T_d(t; c_p) denote the dynamic displacement and dynamic tension under the passive damping coefficient c_p, respectively. y_d(t; 0) and T_d(t; 0) denote the corresponding uncontrolled responses. The weighting factors w₁ and w₂ were used to balance the contributions of displacement suppression and tension suppression. In this study, equal weights were adopted, namely w₁ = w₂ = 0.5, because both dynamic displacement and dynamic tension are important indicators of the operating stability of the system.

The optimal damping coefficient was obtained through a parameter-search procedure. Specifically, the damping coefficient was varied within the range [c_min, c_max], and the hoisting-system simulation was repeated for each candidate value of cp. For each simulation, the RMS values of the dynamic displacement and dynamic tension were calculated, and the corresponding objective-function value J(c_p) was obtained. The value of c_p that minimized J(c_p) was selected as the optimal passive damping coefficient.

Figure 6 shows the variation in the objective function J(c_p) with the passive damping coefficient c_p. The value of J(c_p) decreases as c_p increases, indicating that a larger passive damping coefficient improves the RMS-based vibration-suppression performance within the prescribed range. To ensure a fair comparison with the MRD-based semi-active control strategies, the passive damping coefficient was constrained according to the attainable damping-force level of the MRD and practical implementation considerations. Within the admissible range of c_p ∈ [0, 1.0 × 10⁴] N⋅s/m, the minimum value of J(c_p) was obtained at $c_p^{opt}$ = 1.0 × 10⁴ N⋅s/m. Therefore, this value was selected as the optimized passive viscous damper benchmark.

After the optimal damping coefficient was determined, it was kept unchanged in all subsequent simulations. This treatment is consistent with the physical nature of a passive viscous damper, whose damping coefficient cannot be adjusted online during operation. The optimized passive viscous damper was then used as the benchmark for comparison with the SMC–MRD and ASMC–MRD strategies. Therefore, the following numerical comparisons include three cases: the optimized passive viscous damper, the SMC-controlled MRD, and the ASMC-controlled MRD.

4.3.3. Dynamic Vibration-Suppression Performance Relative to the Passive Benchmark

After the optimized passive benchmark is determined, the dynamic responses of the vertical rope segment are compared among the passive, SMC–MRD, and ASMC–MRD cases. The static equilibrium components are removed before calculating the dynamic displacement and dynamic tension so that the evaluation reflects the vibration-suppression performance rather than the quasi-static loading state.

As shown in Figure 7, the time-domain responses of the hoisting rope are compared under the optimized passive damper, SMC–MRD, and ASMC–MRD cases. The optimized passive damper provides a fixed damping action and therefore serves as a strong benchmark. After semi-active MRD control is introduced, both SMC–MRD and ASMC–MRD reduce the dynamic displacement and tension fluctuations relative to this passive benchmark in most operating stages. Figure 8 provides a more detailed comparison between SMC–MRD and ASMC–MRD. The differences between the two controlled responses become particularly evident near the operating-stage transitions, where transient excitations are strongest. Compared with SMC–MRD, ASMC–MRD generally produces smaller peak displacement and peak tension responses, indicating that the adaptive disturbance-compensation mechanism improves the regulation of the MRD force under time-varying operating conditions.

To further quantify the vibration-suppression performance observed in Figure 7 and Figure 8, the reduction rate relative to the optimized passive viscous damper is introduced as the evaluation index and is defined as.

$$X_{\mathrm{red}}(\%)=\left(1-{\displaystyle \frac{X_{\mathrm{con}}}{X_{\mathrm{pas}}}}\right)\times 100\%,$$

(54)

where X_con and X_pas denote the response indices of the MRD-controlled case and the optimized passive viscous damper benchmark, respectively. The index X may represent either the peak value or the RMS value of the dynamic displacement or dynamic tension.

For the RMS-based evaluation, the mean component of each signal within the corresponding operating stage is removed to exclude the influence of the slowly varying equilibrium component. The RMS value is calculated as

$$X_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(x_i-\overline{x})}^2}},$$

(55)

where x_i denotes the displacement or tension response at the i-th sampling instant. $\overline{x}$ denotes its arithmetic mean over the corresponding operating stage, and N is the number of sampling points in that stage.

Based on the quantitative results in

Table 3. Comparison of vibration-suppression performance of SMC–MRD and ASMC–MRD relative to the optimized passive damper in different operating phases.

Operating Phase	Control Strategy	Peak Disp. (mm)	RMS Disp. (mm)	Disp. Reduction, Peak/RMS (%)	Peak Tension (kN)	RMS Tension (kN)	Tension Reduction, Peak/RMS (%)
Acceleration	Passive	61.07	6.79	—	23.45	2.34	—
	SMC–MRD	58.18	5.25	4.74/22.75	22.42	1.71	4.38/26.93
	ASMC–MRD	51.22	4.60	16.12/32.25	19.97	1.51	14.85/35.37
Constant speed	Passive	22.56	3.63	—	9.85	1.53	—
	SMC–MRD	12.33	2.23	45.33/38.62	5.25	0.92	46.70/39.82
	ASMC–MRD	4.04	1.96	82.08/45.92	2.23	0.78	77.36/49.02
Deceleration	Passive	11.61	1.64	—	18.11	1.78	—
	SMC–MRD	11.59	1.46	0.13/11.21	17.89	1.15	1.21/35.58
	ASMC–MRD	10.56	1.41	8.97/14.5	16.42	1.06	9.33/40.45

, Figure 9 compares the reduction rates of SMC–MRD and ASMC–MRD relative to the optimized passive viscous damper in different operating phases. Overall, ASMC–MRD provides stronger peak-displacement suppression than SMC–MRD, especially during constant-speed operation, where the peak dynamic displacement is reduced by 82.08% relative to the passive benchmark. In contrast, the peak dynamic-tension reduction during constant-speed operation is limited because the optimized passive damper already provides comparable tension-peak attenuation in this interval. The RMS reduction rates remain positive for both displacement and tension, indicating that the MRD-based controllers further reduce the overall vibration level relative to the passive damper. During acceleration and deceleration, the peak reductions are smaller because the rope response is strongly affected by transient inertia-induced excitation and the MRD can only provide semi-active damping within its attainable force range.

In summary, Figure 7, Figure 8 and Figure 9 and Table 3 demonstrate that the proposed semi-active MRD strategies can improve the dynamic response relative to the optimized passive viscous damper benchmark. The advantage of ASMC–MRD is most evident in the suppression of transient peak displacement and in RMS vibration attenuation. Although the optimized passive damper can provide comparable tension-peak suppression in some intervals, its fixed damping coefficient cannot adapt online to variations in vibration state, whereas the MRD-based strategies can regulate the damping force through the current command. This adaptive force-regulation capability is beneficial for reducing dynamic impact loading and improving the operational smoothness of the mine hoisting system.

4.3.4. Real-Time Current Regulation and Force-Tracking Performance of the MRD

After evaluating the vibration-suppression performance of the vertical rope segment, the control current and force-realization performance of the MRD are examined to clarify the implementation behavior of the proposed ASMC–MRD strategy.

As shown in Figure 10, the inverse MRD model generates the corresponding bounded current command in real time, whose time-domain response is depicted. The current is not continuously maintained at a high level throughout the hoisting process. Instead, significant current commands are concentrated primarily in the transient stages associated with changes in the operating condition. Pronounced current peaks, with a maximum value of approximately 1.6 A, occur during initial startup, near the transition from acceleration to constant-speed operation at approximately 20 s, and near the transition from constant-speed operation to deceleration at approximately 64 s. These current peaks indicate that the ASMC increases the MRD control action when the hoisting system is subjected to strong transient excitation and intensified rope vibration. In contrast, during most of the constant-speed stage, the current command remains close to zero, except for short-duration decaying responses following the stage transitions. This behavior demonstrates that the controller reduces the required MRD current once the vibration response has been attenuated. Consequently, the proposed strategy provides effective vibration suppression without requiring a persistently large control current, which is consistent with the low-power operating principle of semi-active MRD control.

To evaluate whether the current command generated by the inverse MRD model can realize the target damping force specified by the ASMC, Figure 11 compares the target damping force with the actual MRD damping force. As shown in Figure 11a, the two curves are nearly coincident throughout the hoisting process, indicating that the inverse MRD model effectively converts the target damping force into a realizable current command. Small deviations occur primarily during startup and near the transitions between operating stages, where the target force changes rapidly. The force peaks observed during initial startup and near the transitions at approximately 20 s and 64 s are consistent with the current-command response shown in Figure 10. Figure 12 further presents force–displacement tracking curves over representative intervals. The actual MRD damping force remains within the force boundaries corresponding to 0 A and the maximum current, and it follows the target damping force closely over most of the displacement range. These results verify the feasibility of the semi-active force-realization process.

To quantitatively evaluate the damping-force tracking performance shown in Figure 11, the force-tracking error is defined as follows:

$$e_F(t)=F_m(t)-F_m^{*}(t).$$

(56)

For each operating stage, the maximum absolute tracking error and the RMS tracking error are calculated as

$$e_{F,\max }=\max |e_F(t)|,$$

(57)

$$e_{F,\mathrm{RMS}}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{e}_F^2}(t_i)},$$

(58)

where N is the number of sampling points within the considered operating stage. The force consistency index is further defined as follows:

$$\mathrm{FCI}=\left(1-{\displaystyle \frac{e_{F,\mathrm{RMS}}}{\mathrm{RMS}(F_m^{*})}}\right)\times 100\%$$

(59)

A larger FCI indicates better agreement between the target damping force and the actual MRD damping force. As shown in Figure 11b, the force-tracking error remains small during all three operating stages, and the stage-wise FCI values remain close to 100%. These results indicate that the actual MRD force closely follows the target force generated by the ASMC. Therefore, the inverse MRD model provides accurate real-time force realization, and the remaining force-realization error is small and bounded, consistent with the disturbance assumption introduced in Section 3.

4.4. Robustness Evaluation Under Variable Operating Conditions

To evaluate the robustness of the proposed ASMC–MRD strategy under variable mine hoisting conditions, numerical simulations are conducted for different payloads, acceleration levels, and hoisting speeds. In all robustness cases, the optimized passive viscous damper is used as the benchmark, and the reduction rates of SMC–MRD and ASMC–MRD are calculated relative to the passive benchmark. This allows the additional benefit of semi-active force regulation to be distinguished from the vibration attenuation already provided by a fixed passive damper.

4.4.1. Robustness to Payload Variations

The dynamic responses under payloads of 10,000, 14,000, and 20,000 kg are compared, as shown in Figure 13, Figure 14 and Figure 15.

Figure 13 shows the time-domain responses of the hoisting rope under payloads of 10,000, 14,000, and 20,000 kg. As the payload increases, the dynamic displacement and tension responses become more pronounced, especially during startup and at the speed-transition instants around 20 s and 64 s. Compared with the optimized passive damper, both SMC–MRD and ASMC–MRD reduce the vibration amplitudes and accelerate the decay of residual oscillations. ASMC–MRD produces smaller peak responses than SMC–MRD, indicating that the adaptive term improves the regulation of the MRD force when the payload changes.

Figure 14 further compares the corresponding vibration reduction rates relative to the optimized passive damper. ASMC–MRD generally achieves higher peak reduction rates than SMC–MRD under the investigated payload conditions, particularly during the constant-speed phase, where the response is mainly dominated by residual vibration. The RMS reduction rates show a similar trend, although the differences between SMC–MRD and ASMC–MRD are less pronounced than those of the peak reduction rates. These results indicate that the main advantage of ASMC–MRD lies in transient peak suppression and adaptive force regulation under varying payloads.

Figure 15 presents the force–displacement tracking characteristics of the MRD under different payloads. For all payload cases, the actual MRD damping force follows the target damping force closely over most of the displacement range, indicating that the inverse MRD model can effectively transform the target control force into a feasible current command. It can be observed that the actual MRD force remains within the admissible semi-active force range. As the payload increases from 10,000 kg to 20,000 kg, the MRD displacement range becomes wider, and the required damping force increases, but the target and actual force curves still show good agreement. These results demonstrate the effectiveness and robustness of the proposed force-realization process under different payload conditions.

4.4.2. Robustness to Acceleration Variations

The dynamic responses under acceleration levels of 0.5, 0.65, and 0.8 m/s² are compared, as shown in Figure 16, Figure 17 and Figure 18.

Figure 16 presents the dynamic displacement and tension responses under acceleration levels of 0.50, 0.65, and 0.80 m/s². With increasing acceleration, the operating-stage transitions occur earlier, and the transient excitation becomes stronger. Compared with the optimized passive damper, both SMC–MRD and ASMC–MRD suppress the vibration responses effectively, while ASMC–MRD yields smaller transient peaks and faster attenuation after stage transitions. This indicates that the adaptive disturbance-compensation mechanism can improve the use of the MRD force when the acceleration level changes.

As quantified in Figure 17, the reduction rates are calculated relative to the optimized passive damper. ASMC–MRD provides higher peak reduction rates than SMC–MRD at all acceleration levels for most operating stages. This advantage is most evident in the constant-speed phase, where ASMC–MRD maintains a relatively high peak-suppression capability even when the acceleration increases. The RMS reduction rates confirm the effectiveness of both controllers; however, the difference between SMC–MRD and ASMC–MRD is generally smaller for RMS indices than for peak indices. This again shows that the adaptive term mainly enhances transient peak suppression rather than only reducing the average vibration energy.

Figure 18 shows the force–displacement tracking characteristics of the MRD under different acceleration levels. For all acceleration cases, the actual MRD damping force follows the target damping force closely over most of the displacement range, demonstrating that the inverse MRD model can effectively realize the desired semi-active control force. As the acceleration increases from 0.50 m/s² to 0.80 m/s², the displacement range and the required damping force increase because stronger inertial excitation is introduced during the acceleration stage. Nevertheless, the actual MRD force remains within the admissible force range. These results indicate that the proposed force-realization method maintains good tracking performance under different acceleration conditions.

4.4.3. Robustness to Hoisting Speed Variations

The dynamic responses at hoisting speeds of 7, 10, and 13 m/s are compared, as shown in Figure 19, Figure 20 and Figure 21.

Figure 19 compares the dynamic responses of the hoisting rope at hoisting speeds of 7, 10, and 13 m/s, arranged in ascending order. As the hoisting speed increases, the acceleration and deceleration intervals become shorter, and the stage-transition excitation becomes stronger. Compared with the optimized passive damper, both SMC–MRD and ASMC–MRD reduce the response amplitudes, with ASMC–MRD showing better suppression of transient peaks. The response differences among the three speed cases indicate that speed variation mainly affects the duration of each operating stage and the intensity of transition-induced excitation.

Figure 20 shows that ASMC–MRD generally achieves higher peak reduction rates than SMC–MRD relative to the optimized passive damper under different hoisting speeds, especially in the constant-speed phase. The RMS reduction rates remain positive for both controllers, confirming that semi-active MRD control can further reduce vibration after the passive benchmark has been optimized. Because the speed axis is arranged from 7 to 13 m/s, the figure more clearly shows how the reduction rates vary with increasing maximum hoisting speed.

Figure 21 presents the force–displacement tracking characteristics of the MRD under different maximum hoisting speeds. Compared with the acceleration-variation cases, the force–displacement loops under different speeds exhibit more similar shapes because the payload, acceleration, deceleration, MRD parameters, and controller parameters are kept unchanged. Changing the maximum hoisting speed mainly affects the duration of the constant-speed stage and the time-varying rope length, rather than the basic semi-active force-realization mechanism. In all speed cases, the actual MRD damping force agrees well with the target damping force over most of the displacement range and remains bounded by the attainable force limits at 0 A and the maximum current. This indicates that the inverse MRD model can generate feasible current commands consistently under different hoisting speeds. Overall, the results verify that the proposed ASMC–MRD strategy has stable force-tracking capability and robust semi-active force-realization performance under different maximum hoisting speeds.

4.4.4. Overall Robustness Under Varying Operating Conditions

Taken together, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 demonstrate that variations in payload, acceleration, and hoisting speed substantially influence the longitudinal vibration response of the vertical rope segment, particularly during startup and transitions between operating stages. Under all investigated operating conditions, both SMC–MRD and ASMC–MRD provide additional vibration suppression relative to the optimized passive damper. Compared with SMC–MRD, the proposed ASMC–MRD strategy consistently provides improved suppression of transient peak responses. Although its improvement in RMS reduction is comparatively moderate, ASMC–MRD exhibits a clear advantage in reducing peak displacement and peak tension responses, which are particularly relevant to transient dynamic loading and operational safety in mine hoisting systems. These results demonstrate that the proposed ASMC–MRD strategy provides effective vibration attenuation and improved robustness under variable operating conditions.

5. Conclusions

This paper investigated the semi-active suppression of longitudinal vibration in mine hoisting ropes using an MRD in combination with output-feedback ASMC. A physically consistent MRD-equipped hoisting-system model was established by introducing a load-bearing spring element in parallel with the MRD. The spring element carries the mean static tensile load, whereas the MRD provides adjustable dissipative force for suppressing the dynamic vibration component. The nonlinear hysteretic behavior of the MRD was described using the SEHTFM, and an inverse MRD model was established to convert the desired control force into a feasible real-time current command. Based on the theoretical development and numerical simulation results, the following conclusions are drawn:

• An output-feedback ASMC was developed using only the measured displacement and velocity at the conveyance–rope connection. The proposed controller determines the desired control force, while the inverse MRD model generates the corresponding feasible current command. The σ-modified adaptive law compensates online for the equivalent matched lumped disturbance, and the continuous boundary-layer function reduces chattering. The force–tracking and force–displacement results show that the target damping force can be realized accurately within the attainable semi-active force boundaries of the MRD.
• An optimized passive viscous damper was introduced as the benchmark. Its damping coefficient was selected by minimizing a normalized RMS-based objective function within the prescribed admissible range, and the obtained passive damping coefficient was kept fixed in all subsequent simulations. Relative to this passive benchmark, both SMC–MRD and ASMC–MRD further reduce the dynamic displacement and dynamic tension responses. The advantage of ASMC–MRD is most evident in transient peak suppression and in RMS vibration attenuation, whereas the optimized passive damper can provide comparable tension-peak reduction in some intervals because its damping coefficient is already optimized for the nominal condition.
• Robustness simulations under variations in payload, acceleration, and hoisting speed demonstrate that the proposed ASMC–MRD strategy maintains effective vibration attenuation under changing operating conditions. Compared with SMC–MRD, ASMC–MRD provides improved reduction in peak displacement and peak tension responses, indicating stronger adaptability to transient excitations and variable operating parameters. These findings support the potential of MRD-based semi-active control as a compact and low-power approach for improving the operational smoothness and safety of mine hoisting systems.

The present study is limited to theoretical modeling and numerical simulation. Future work will focus on experimental validation using a hoisting-rope test platform or a field-scale system, optimization of the MRD installation configuration, and further investigation of the effects of sensor noise, time delay, and real-time hardware implementation.