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Abstract

This article demonstrates the tension distribution estimation method for the ring chain transmission system of scraper conveyer, which is based on multi-body dynamics theory and linear state observer. As designed, a multi-body dynamic model of ring chain transmission system is constructed using ADAMS dynamic design software, and a mathematical model solved by MATLAB functions is established and adopted to verify system performance of the dynamic model by comparing the simulation results. In practical terms, tension monitoring of ring chains is virtually unreasonable due to difficulties of direct measurement and restrictions of high-cost tension sensors during the process of mechanized coal mining. Besides, the change law of tension distribution of the ring chain transmission system has mainly been obtained via heuristic experience, coupled with simple calculations. The proposed estimation method makes it feasible to obtain all the contact forces of ring chains based on the measurable state variables. For research purpose, the state-space modeling allows the reconstruction of the tension observer and flexible application for numerical solutions based on Runge–Kutta equations. However, the high-dimensional tension observer has inherent drawbacks of heavy computational loads. In such cases, a novel dimension-minimized algorithm is proposed for high-order matrices solution, which makes the observer very attractive for applications in dimensionality reduction. Furthermore, the practicability of the proposed estimation method is verified with simulation evaluation and deviation analysis.

Keywords

Tension monitoring multi-body dynamics contact pairs tension distribution dimension-minimized algorithm

Introduction

The ring chain transmission system (RCTS), which connects the driving devices and conveying system, has been extensively applied in metallurgy, aerospace, and other industries,^1,2 especially in mechanical mining. The scraper conveyer (Figure 1) is a set of pivotal transport equipment whose vital subsystem is the RCTS and has been widely employed to transport coals and gangues³ with the characteristics of long distance and heavy duty in the fully mechanized coal mining face. Accordingly, the coal production performance can be directly attributed to the reliability and efficiency of the RCTS.

Figure 1.

Schematic of scraper conveyor in coal mine.

Mainly consisted of the driving system, sprockets, scrapers, ring chains, and middle troughs, the RCTS is characterized by complex dynamic behavior and discrete nature dominated by a large number of contacting chains, which attracted a sprawl of attention to investigate the complex interactions between different components. In recent scientific literature, scholars have carried out relatively thorough research on contact force models of the RCTS.^4–8 In addition, researches on dynamic modeling have been widely carried out to learn about characteristics of diverse transmission systems. Saghafi and Farshidianfar⁹ designed an optimal gear transmission system to control and eliminate the chaotic behaviors using a nonlinear dynamic model. By discretizing the sprocket tooth profile into seven possible contact regions, Pereira et al.¹⁰ presented a novel approach to deal with the modeling and dynamic analysis of planar chain drives. Zhang et al.¹¹ studied the 6-degree-of-freedom (DOF) dynamic modeling method of geared rotor system characterized by its complexity which comes mainly from the shafts by the mesh stiffness due to gear pairs and variable bearing coefficients of load power changing. Fuglede and Thomsen¹² proposed the kinematic and dynamic model with an approach to calculate the chain wrapping length, aiming at providing a framework for conducting and understanding both numerical and experimental investigations of roller chain drive dynamics. Xu et al.¹³ constructed the virtual prototype model of the intermittent roller chain to simulate the dynamic response of the chain system; in addition, different motion laws and the influence from tension force were also considered in the simulation investigation.

Generally, the tension variation of ring chains can effectively reflect the working state of scraper conveyor. Applied in long distance transportation, the RCTS is composed of a large number of ring chains, which brings about great restriction for direct measurements of chain tension. As a result, tension monitoring of ring chains has not been considered in the latest research works, and most of the studies stop proceeding at the level of theoretical calculation.^14–16 In recent years, with the increasing in concerns about parameter prognostic, the theory of state observer has been widely applied and studied in various fields. Ali et al.¹⁷ proposed a novel hybrid fuzzy-sliding mode observer to estimate various parameters with a high rate of accuracy even with disturbances and noise in the model. Roux et al.¹⁸ developed a nonlinear grinding mill observer model for estimation of the states and parameters. Xu and Chen¹⁹ analyzed the degradation of lithium-ion batteries with the sequentially observed discharging profiles using the observation model based on a general state-space model. Liang et al.²⁰ described a disturbance observer-based force estimation method and predicted the contact force between the subject body and the environment in robotic and mechatronic systems.

In this article, a virtual prototype model of the RCTS is presented by multi-body dynamics simulation software ADAMS based on kinematics and dynamics, which serves as a dynamic model to study the incidence relations between the dynamic characteristics and the running states of the RCTS under normal working condition. For further study, a mathematical model of the RCTS is established based on mass-spring-damping modules and simulated with MATLAB package. Sequentially, a combination of virtual prototyping model and mathematical model is implemented to check the correctness of dynamic modeling method. To date, few researches related to tension distribution estimation of the RCTS have been undertaken in the literature. With available states derived from tension sensors as the inputs of the state observer, we propose the observer-based tension distribution estimation method, which could estimate the unmeasurable contact forces for the first time. To minimize the computational cost, the novel dimension-minimized algorithm shows great advantages in dealing with high-dimensional solution.

The article is organized as follows: section “Virtual prototype modeling of RCTS” presents the virtual prototype model of the actual RCTS. Section “Tension distribution estimation method” details the design of the tension distribution estimation method based on a linear state observer. Section “Estimation process and simulation results” describes the calculation process of the novel dimension-minimized algorithm and illustrates simulation results to evaluate the performance of the proposed tension distribution estimation method. Finally, followed by acknowledgments, we offer conclusions in the “Conclusion” section.

Virtual prototype modeling of RCTS

Basic theory of contact pairs

The RCTS works as a representative multi-body system (MBS) whose representation of contact mechanics is an eminently challenging task to model the complex contacting bodies. To perform our study, the polygonal contact model (PCM)⁶ is adopted to deal with complex contact surfaces with satisfactory results and is equally applicable to the contact pairs of chains.

As a contact algorithm to solve contact problems, the polygonal contact model has two prominent features: on one hand, the surfaces of arbitrary body in a MBS can be described in the polygonal form, and the commonly used method for construction of polygonal surfaces is non-uniform rational B-splines (NURBS); on the other hand, the contact forces determination between any two bodies in a MBS is based on the elastic foundation model.

The research on classical multi-body dynamics has practical significance to raise efficiency of MBS simulations by the hypothesis that the elastic foundation model considered in the RCTS is treated as rigid bodies. Assuming that the equivalent rigid bodies are covered with thin elastic layers, as follows

P_{N} = \frac{K}{B} \cdot S_{N}

(1)

where $P_{N}$ is the normal pressure and $S_{N}$ is the normal pressure displacement. Equation (1) shows the relation between $P_{N}$ and $S_{N}$ , neglecting the tangential share stress in a layer whose thickness is B. The elastic modulus K of thin and linear–elastic layers is given by equation (2), where E is Young’s modulus and μ is Poisson’s ratio (μ < 0.45)

K = \frac{1 - μ}{(1 + μ) (1 - 2 μ)} \cdot E

(2)

Most applications cannot satisfy the assumption of thin layers, we can derive the layer stiffness from experiments or qualified estimating, and we can also apply the elastic foundation model

C = \frac{K}{B}

(3)

Besides, the elastic foundation model is built on the basis of contact path discretization. For contact element N of area $A_{N}$ , through the integration of equations (1) and (3), the normal force is obtained through

F_{N} = C \cdot A_{N} \cdot S_{N}

(4)

Generally, the establishment of contact pairs plays a crucial role in the process of dynamics simulation. As for the RCTS, a greater number of contacting bodies are included, such as sprockets, ring chains, scrapers, and middle troughs. Contact pairs consist of two bodies moving correspondingly to each other and are specially adopted to simulate the interactions among main contacting bodies. In this article, contact pairs are discussed between sprocket and plat chain, and between vertical chain and plat chain. Referring to the PCM theory, Figure 2 illustrates the surfaces discretization of the contact pairs using the NURBS.

Figure 2.

Discretization of contacting surfaces: (a) contact pair between sprocket and plat chain and (b) contact pair between vertical chain and plat chain.

With efforts of discretization, quadrangular and pentagonal surfaces of the contacting bodies can be feasibly converted into two and three triangles, respectively.⁶ Taking the inertial coordinate system I as the reference, local coordinate systems S, P, V, and q and point coordinate systems $S'$ , $P'$ , $V'$ , and $q'$ are defined for convenience of descriptions. Normally, the inertial coordinate I is given to determine the position of local coordinate systems, based on which we can further obtain the location of any point on the surfaces of the rigid body. For contact pairs consist of bodies A and B, there is a minimum distance between point $K_{1}$ on the surface of A and point $K_{2}$ on the surface of B; similarly, we get points $O'_{s}$ , $O'_{q}$ and $O'_{v}$ , $O'_{p}$ as shown in Figure 2(a) and (b). In addition, any point on the surface of sprockets can be represented by the unit normal vector $S'_{1}$ and the unit tangential vector $S'_{2}$ ; correspondingly, the unit normal vector $V'_{1}$ and the unit tangential vector $V'_{2}$ are adopted to represent any point on the surface of chains. The vector $R_{m}$ and $R_{n}$ are written as

R_{m} = r_{p} + \vec{O_{q} {O'}_{q}} - r_{s} + \vec{O_{s} {O'}_{s}}

(5)

R_{n} = r_{t} + \vec{O_{p} {O'}_{p}} - r_{v} + \vec{O_{v} {O'}_{v}}

(6)

The minimum distance between $O'_{s}$ and $O'_{q}$ , and $O'_{v}$ and $O'_{p}$ , i.e. $S_{m} and S_{n}$ are written by:

S_{m} = R_{m} \cdot S'_{1}

(7)

S_{n} = R_{n} \cdot V'_{1}

(8)

where if $S_{m} or S_{n} > 0$ , the two bodies are in separation; if $S_{m} or S_{n} = 0$ , the two bodies are in contact; and if $S_{m} or S_{n} < 0$ , the two bodies are in penetration.

Dynamic modeling process

In this article, we establish the three-dimensional (3D) physical model of scraper conveyor using the creo2.0 software as shown in Figure 1, and the dimension of the chain is ϕ48 × 152. To study the interactions between main components of actual RCTS, a virtual prototyping model has been built. In addition, contact pairs are effectively used for dynamic modeling and computing, and polygonal contact theory is directly employed to detect contact pairs of the ring chain system via collision detection.⁶

It could be obtained through model simplification that the dynamic model of the RCTS consists of one double-chain system, one transmission trough system, and one double-drive sprocket system, as shown in Figure 3. The double-chain system works as the left single chain subsystem and right single chain subsystem, both of which are composed of lots of plat chains, scrapers, and vertical chains. The transmission trough is treated as an independent entity through Boolean operations of transitional troughs and middle troughs. The kinematic restriction of two rigid bodies in contact includes two main aspects: kinematic constraints and contact force relations; the restriction of the system is shown in Figure 4.

Figure 3.

Dynamic model of the actual RCTS.

Figure 4.

Restriction of the system.

Results analysis

In practical engineering, the double-drive transmission system is driven by electromotor in actual transmission system. Thus, we apply rotation motion on the revolute joints, which are built between drive sprocket and system frame, and the sprocket rotates in the counterclockwise direction. In order to ensure compliance with actual working conditions, we take the rotation angular velocities of the sprockets as the inputs of the RCTS. The left single chain subsystem and right single chain subsystem are similar in kinematic and dynamic behaviors, either of them works as a closed transmission chain system formed by linking the plate chains and the vertical chains together. In our work, we just consider contact forces between two adjacent chains of the left single chain subsystem under the effect of uniform loads.

Figure 5 shows the variation of the tension between plate chain and vertical chain running on middle troughs. After a smooth and steady startup process, drive sprockets rotate at a speed of 2.56 rad/s. Overall, the value of the contact force reaches a steady level from 1 to 4.1 s and presents irregular cyclical changes.

Figure 5.

Tension of chain running on middle troughs.

As shown in Figure 6, considering the situation of chains running from transitional troughs to drive sprocket 1, the operation process is divided into three stages: $S_{1}$ , $S_{2}$ , and $S_{3}$ . For stage $S_{1}$ , the chains run on transitional troughs; tension of the contact pair is much larger than chains moving on middle troughs. Then, the tension between plate chain and vertical chain has decreased greatly with quite small numerical value of the whole engagement process, instantaneous impact force occurs when the plate chain engages with the sprocket, and the transient peaks of the impact force are presented as in stage $S_{2}$ . Stage $S_{3}$ describes the tension variation of the contact pair when the plate chain starts to separate from the sprocket.

Figure 6.

Tension between plate and vertical in pre- and post-engaging process.

Accordingly, the variation of driving torque of sprockets is exhibited in Figure 7. After starting up, drive sprockets 1 and 2 rotate at the same angular velocity accompanied by erratic numerical fluctuation of torque value at stages S01 and S02, respectively. Then, either of the sprockets reaches stable state like stages S12, S23, and S34 shown in Figure 7(a) and stages S56, S67, and S78 shown in Figure 7(b). For the dynamic model, arbitrary two adjacent scrapers are equidistantly arranged. It is evident that the driving torque exhibits higher value with a sudden increase when the scraper begins to engage with the sprocket. Particularly, a disordered variation of the torque value occurs during the engagement of chains and sprockets, due to transversal and longitudinal vibration, load impact and uneven chain velocity of the transmission mechanism.

Figure 7.

Driving torque of sprockets: (a) sprocket 1 and (b) sprocket 2.

Tension distribution estimation method

Discrete modeling process

Due to the viscoelastic properties of the ring chains, the article utilizes MATLAB package and discretizes the left single chain subsystem into a system of Kelvin models based on the finite element method, further establishes the mathematical model, and studies the longitudinal dynamic characteristics of the discrete system. The assumptions are given as follows:

To ensure the reliability of the discrete system, the simulation process is described by stiffness and damping and so on. Aiming to conform to the actual RCTS, all parameters required are set in accordance with the virtual prototype model, whose driving torques serve as the inputs of the mathematical model.

As the inclination angle of transitional trough has little effect on the longitudinal force of ring chains due to long distance transportation of the RCTS, we establish the mathematical model which neglects the effect of the inclined plane relative to dynamic model of the actual RCTS, as shown in Figure 3. In this article, all contact pairs on the inclined plane are treated without taking the inclination angle into consideration.

According to the one-dimensional (1D) longitudinal discrete system (Figure 8) built in this article, any one of the chains works as a discrete element. For simplicity, equivalent mass of scrapers, chains, and external loads is evenly distributed to the elements along the transmission trough. For the head and tail of the discrete system, when the chain stakes seat in the teeth of a sprocket, the sprocket and chains wrapping around are integrated as a single discrete element.

Moreover, the dynamic behaviors of the discrete system are discussed and analyzed neglecting polygonal action and meshing impact during the engagement process. In our work, we just take the left single chain subsystem as the research object.

Figure 8.

One-dimensional longitudinal discrete system.

As shown in Figure 8, the elements are assumed to be elastically connected to each other and modeled as Kelvin modules, which allows force transfer of the left single chain subsystem. The viscoelastic model is dominated by elasticity and damping, and contact force between contiguous elements is given by

F_{j} = k_{j} (x_{j} - x_{j + 1}) + c_{j} (x'_{j} - x'_{j + 1})

(9)

where $k_{j}$ and $c_{j}$ are, respectively, the stiffness coefficient and damping coefficient of element j; $x_{j}$ and $x'_{j}$ are the displacement and velocity of element j. The discrete system is simplified as n elements, and the sprocket moves counterclockwise as the arrows. The Newton–Euler approach serves to construct mathematical equations to study the relations between the kinematic characteristics and dynamic behaviors, as follows

\begin{array}{l} m_{1} {x^{″}}_{1} + (c_{1} + c_{n} + f_{1}) {x^{'}}_{1} - c_{1} {x^{'}}_{2} - c_{n} {x^{'}}_{n} + (k_{1} + k_{n}) x_{1} \\ - k_{1} x_{2} - k_{n} x_{n} = \frac{T_{t}}{R_{t}} \end{array}

(10)

\begin{matrix} {m_{n}}_{/ 2 + 1} {x ″}_{n / 2 + 1} + (c_{n / 2} + c_{n / 2 + 1} + f_{n / 2 + 1}) {x'}_{n / 2 + 1} \\ - c_{n / 2} {x'}_{n / 2} - c_{n / 2 + 1} {x'}_{n / 2 + 2} \\ + (k_{n / 2} + k_{n / 2 + 1}) x_{n / 2 + 1} - k_{n / 2} x_{n / 2} \\ - k_{n / 2 + 1} x_{n / 2 + 2} = \frac{T_{h}}{R_{h}} \end{matrix}

(11)

\begin{array}{l} m_{j} {x^{″}}_{j} + (c_{j - 1} + c_{j} + f_{j}) {x^{'}}_{j} - c_{j - 1} {x^{'}}_{j - 1} - c_{j} {x^{'}}_{j + 1} \\ + (k_{j - 1} + k_{j}) x_{j} - k_{j - 1} x_{j - 1} - k_{j} x_{j + 1} = 0 \end{array}

(12)

\begin{array}{l} m_{n} {x^{″}}_{n} + (c_{n - 1} + c_{n} + f_{n}) {x^{'}}_{n} - c_{n - 1} {x^{'}}_{n - 1} - c_{n} {x^{'}}_{1} \\ + (k_{n - 1} + k_{n}) x_{n} - k_{n - 1} x_{n - 1} - k_{n} x_{1} = 0 \end{array}

(13)

where $m_{j}$ , $k_{j}$ , $c_{j}$ , $f_{j}$ , $x_{j}$ , $x'_{j}$ , and $x_{j}^{″}$ represent the equivalent mass, stiffness coefficient, damping coefficient, resistance coefficient, displacement, velocity, and acceleration of element j, respectively. $T_{t}$ and $T_{h}$ represent the driving torques of the sprockets, respectively. Mathematical equations of the elements at the head and tail are written by equations (10) and (11), that is, $j = n / 2 + 1$ and $j = 1$ . Through equation (11), dynamic behavior of element j ( $2 \leq j \leq n / 2$ and $n / 2 + 1 \leq j \leq n - 1$ ) is described, and equation (12) is appropriate for research of element n.

Numerical solutions

State-space equation

Based on equations (10)–(12), the differential equations of the time-invariant linear system are reflected by

MX ″ + CX' + KX = F

(14)

where M ∈ R^n × n, C ∈ R^n × n, and K ∈ R^n × n are the mass matrix, the damping matrix, and the stiff matrix of the elements, respectively, $M = diag (m_{1} m_{2} \dots m_{n / 2 + 1} \dots m_{n - 1} m_{n})$ . Represented by the n × 1 matrix, $X$ , $X'$ , and $X ″$ are described as the displacement vector, the velocity vector, and the acceleration vector, respectively. F ∈ Rⁿ^× 1 donates as the external force vector, $F = [T_{t} / R_{t}; \dots; T_{h} / R_{h}; F_{n / 2 + 2}; \dots F_{n - 1}; F_{n}]_{n \times 1}$ . According to the state-space method, we can obtain that

Z' = AZ + BU

(15)

Y = HZ

(16)

Matrices and vectors are established as

A = {[\begin{matrix} 0_{n \times n} & I_{n \times n} \\ - M^{- 1} K & - M^{- 1} C \end{matrix}]}_{2 n \times 2 n} B = {[\begin{matrix} 0_{n \times 1} \\ M^{- 1} F \end{matrix}]}_{2 n \times 1}

K = {[\begin{matrix} k_{1} + k_{n} & - k_{1} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 & - k_{n} \\ - k_{1} & k_{1} + k_{2} & - k_{2} & 0 & . . . & 0 & 0 & 0 \\ 0 & - k_{2} & k_{2} + k_{3} & - k_{3} & \cdot \cdot \cdot & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - k_{n - 2} & k_{n - 2} + k_{n - 1} & - k_{n - 1} \\ - k_{n} & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & - k_{n - 1} & k_{n - 1} + k_{n} \end{matrix}]}_{n \times n}

C = {[\begin{matrix} c_{n} + c_{1} + f_{1} & - c_{1} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 & - c_{n} \\ - c_{1} & c_{1} + c_{2} + f_{2} & - c_{2} & 0 & . . . & 0 & 0 & 0 \\ 0 & - c_{2} & c_{2} + c_{3} + f_{3} & - c_{3} & \cdot \cdot \cdot & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - c_{n - 2} & c_{n - 2} + c_{n - 1} + f_{n - 1} & - c_{n - 1} \\ - c_{n} & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & - c_{n - 1} & c_{n - 1} + c_{n} + f_{n} \end{matrix}]}_{n \times n}

where A, B, Y, and H are constant system matrix, control matrix, output matrix, and transformation matrix, respectively. Y is defined as the output of the system, Z is the state vector and U is the input vector, $Z = [x_{1}; x_{2}; \dots x_{n / 2 + 1}; \dots x_{n - 1}; \begin{matrix} x_{n}; & {x'}_{1}; & {x'}_{2}; & \begin{matrix} \cdot \cdot \cdot & {x'}_{n / 2 + 1}; & \cdot \cdot \cdot & \begin{matrix} {x'}_{n - 1}; & {x'}_{n} \end{matrix} \end{matrix} \end{matrix}]_{2 n \times 1}$ and $U = [T_{1}; T_{h}]_{2 \times 1}$ .

Solutions of differential equation

With respect to the state-space equations, we develop the general scheme as

Z' = f (Z, U)

(17)

Y = g (Z, U)

(18)

Functions $f (\cdot)$ and $g (\cdot)$ are employed to solve differential equations in continuous time T based on the initial values of U, Y, and Z. The runtime T of the RCTS is discretized into N calculation steps with h sampling period, that is, $T = N \cdot h$ , for time index $n_{0} (1 \leq n_{0} \leq N) : x_{1} (n_{0}), x_{2} (n_{0}), \dots, x_{n - 1} (n_{0}), x_{n} (n_{0}), x'_{1} (n_{0}), x'_{2} (n_{0}), \dots, x'_{n - 1} (n_{0}) and x'_{n} (n_{0})$ are the state parameters of the system, $T_{t} (n_{0})$ and $T_{h} (n_{0})$ are the input parameters, and $Y (n_{0})$ works as the output. For state parameters $x$ and $x'$

x' = φ (t, x)

(19)

Since the mathematical equations are continuously differentiable, in this section, the solution method is derived using classical fourth-order Runge–Kutta formula,²¹ which can be given by

x (n_{0} + 1) = x (n_{0}) + \frac{1}{6} (K_{1} + 2 K_{2} + 2 K_{3} + K_{4})

(20)

Y (n_{0} + 1) = g (x (n_{0} + 1), u (n_{0}))

(21)

where $K_{1}$ , $K_{2}$ , $K_{3}$ , and $K_{4}$ are defined as

{\begin{matrix} K_{1} = h φ (t_{n_{0}}, x (n_{0})) \\ K_{2} = h φ (t_{n_{0}} + h / 2, x (n_{0}) + K_{1} / 2) \\ K_{3} = h φ (t_{n_{0}} + h / 2, x (n_{0}) + K_{2} / 2) \\ K_{4} = h φ (t_{n_{0}} + h, x (n_{0}) + K_{3}) \end{matrix}

(22)

Tension distribution method

The actual RCTS (Figure 3) is characterized by contact pairs between vertical chains and plate chains, and plate chains and sprockets. To date, a contradiction is formed due to practical engineering requirements and the measurement restrictions. The observer-based tension distribution method is proposed to monitor tension of the RCTS by estimating contact forces of ring chains and can solve the above contradiction.

Equations (17) and (18) are assembled to reconstruct the linear observer²² and provide sufficient guidance for the design of the proposed estimation method, and we have

\bar{Z}' = (A - GH) \bar{Z} + BU + GY

(23)

\bar{Y} = \tilde{H} \bar{Z}

(24)

where $\bar{Z}$ serves as the reconstructed state vector with dimensions of 2n × 1, $\bar{Y}$ works as the reconstructed output matrix, $\tilde{H}$ is a n-order identity matrix, $G$ is the feedback gain matrix with dimension of $q \times n$ , and q is defined as the number of available states corresponding to tension sensors. Derived from direct measurement, the available states are expressed by Y as in equation (18) and serve as the state feedback of the designed state observer (Figure 9), whose inputs are derived from vectors U and Y.

Figure 9.

Reconstruction process of the state observer.

Based on full-order design, the state observer can estimate all state variables of the state-space model and get the change rules of displacement, velocity, acceleration, and contact force of all discrete elements (Figure 8) at any time set by sampling period h. With high relevance in applications, the observer-based tension distribution estimation method can get tension variation of actual RCTS by replacing the unmeasurable contact forces with observed results.

Estimation process and simulation results

Construction of the spatial tension distribution model

As mentioned in section “Tension distribution estimation method,” the proposed estimation method is employed to estimate unmeasurable contact forces of the actual RCTS reflected by the reconstructed state variables. The RCTS is modeled as n ( $n = 146$ ) discrete elements by Kelvin models in a series connections. In order to analyze the transient dynamic characteristics, surface-to-surface contact is defined between two elements. As Figure 10 describes, the plate chain and vertical chain are connected by a contact pair, and their surfaces are complicated and constantly changing. For being convenient to study, contact surface is represented as the contact point.

Figure 10.

Simplified contact points.

According to the positions of the contact pairs between two adjacent elements, the closed RCTS can be transformed into a spatial tension distribution model. Furthermore, the model is beneficially established to research the longitudinal tension characteristics of the RCTS, where each contact pair is replaced by a specific numbered spatial contact point with one translational DOF, that is, point $i (1 \leq i \leq n)$ . The driving sprocket center of rotation is fixed in the space and the simplified spatial contact points move relatively to the system frame while the tension holds them in finite position. With reference to Figure 11, the distribution trajectory is defined with a certain shape, based on which to confirm the position of each spatial contact point and monitor the tension variation when contact pairs passing through certain spatial points.

Figure 11.

Spatial tension distribution model.

For the left single chain subsystem, the mass of main components and bulk materials being carried vary with time and space and are evenly distributed according to the distribution of spatial contact points. Numerical simulation is carried out under uniform working loads and pretension force, and both of the drive sprockets run at a constant speed of 2.56 rad/s during the steady operation stage. The positive direction of the arrow as shown in Figure 11 is chosen, and the speed curve is shown in Figure 12.

Figure 12.

Angular velocity of the drive sprocket.

Novel dimension-minimized algorithm

In this article, the observer-based tension distribution estimation method has a low computational efficiency due to high dimensionality of the system, and the order of the reconstructed observer is inherently high. As a consequence, a novel dimension-minimized algorithm suitable for the large-scale system is introduced. The resolution procedure is summarized as follows:

1. Tension variation of spatial contact points yields directly to the state variables of the observer; we first confirm that the total number of state variables is 2n determined by 2 discrete sprocket elements and $n - 2$ chain elements as mentioned in section “Virtual prototype modeling of RCTS.” The left single chain subsystem is depicted as the upper and lower chain subsystem represented as points $I_{1} (1 \leq I_{1} \leq n / 2)$ and $I_{2} (n / 2 + 1 \leq I_{2} \leq n)$ , respectively.

2. Then, we integrate $I_{0}$ consecutive chain elements as a whole research object along the running direction, and the dimension of observer reconstructed can be greatly reduced, we can availably get a M₀-order observer, where

M_{0} = 4 N_{0} + 4

(25)

3. On combining equations (14)–(16), the new constructed observer is reflected by $T_{0}$ elements, whose equivalent mass M and displacement X are, respectively, established as

M = {\begin{matrix} M_{1} = m_{1} \\ M_{2} = \sum_{i = 2}^{I_{0} + 1} m_{i} \\ ⋮ \\ M_{j} = \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} m_{i} \\ ⋮ \\ M_{N_{0}} = \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} m_{i} \\ M_{N_{0} + 1} = \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} m_{i} \\ M_{N_{0} + 2} = m_{74} \\ M_{N_{0} + 3} = \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} m_{i} \\ ⋮ \\ M_{j} = \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} m_{i} \\ ⋮ \\ M_{2 N_{0} + 1} = \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} m_{i} \\ M_{2 N_{0} + 2} = \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} m_{i} \end{matrix} X = {\begin{matrix} X_{1} = x_{1} \\ X_{2} = \sum_{i = 2}^{I_{0} + 1} x_{i} / I_{0} \\ ⋮ \\ X_{j} = \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} x_{i} / I_{0} \\ ⋮ \\ X_{N_{0}} = \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} x_{i} / I_{0} \\ X_{N_{0} + 1} = \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} x_{i} / I_{0} \\ X_{N_{0} + 2} = x_{74} \\ X_{N_{0} + 3} = \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} x_{i} / I_{0} \\ ⋮ \\ X_{j} = \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} x_{i} / I_{0} \\ ⋮ \\ X_{2 N_{0} + 1} = \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} x_{i} / I_{0} \\ X_{2 N_{0} + 2} = \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} x_{i} / I_{0} \end{matrix}

The stiffness coefficient K and damping coefficient C are described as

K = {\begin{matrix} K_{1} = k_{1} \\ K_{2} = 1 / 1 \sum_{i = 2}^{I_{0} + 1} 1 / k_{i} \sum_{i = 2}^{I_{0} + 1} 1 / k_{i} \\ ⋮ \\ K_{j} = 1 / 1 \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} 1 / k_{i} \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} 1 / k_{i} \\ ⋮ \\ K_{N_{0}} = 1 / 1 \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} 1 / k_{i} \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} 1 / k_{i} \\ K_{N_{0} + 1} = 1 / 1 \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} 1 / k_{i} \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} 1 / k_{i} \\ K_{N_{0} + 2} = k_{74} \\ K_{N_{0} + 3} = 1 / 1 \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} 1 / k_{i} \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} 1 / k_{i} \\ ⋮ \\ K_{j} = 1 / 1 \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} 1 / k_{i} \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} 1 / k_{i} \\ ⋮ \\ K_{2 N_{0} + 1} = 1 / 1 \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} 1 / k_{i} \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} 1 / k_{i} \\ K_{2 N_{0} + 2} = 1 / 1 \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} 1 / k_{i} \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} 1 / k_{i} \end{matrix} C = {\begin{matrix} C_{1} = c_{1} \\ C_{2} = 1 / 1 \sum_{i = 2}^{I_{0} + 1} 1 / c_{i} \sum_{i = 2}^{I_{0} + 1} 1 / c_{i} \\ ⋮ \\ C_{j} = 1 / 1 \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} 1 / c_{i} \sum_{i = (j - 2) I_{0} + 2}^{(j - 1) I_{0} + 1} 1 / c_{i} \\ ⋮ \\ C_{N_{0}} = 1 / 1 \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} 1 / c_{i} \sum_{i = (N_{0} - 2) I_{0} + 2}^{(N_{0} - 1) I_{0} + 1} 1 / c_{i} \\ C_{N_{0} + 1} = 1 / 1 \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} 1 / c_{i} \sum_{i = (N_{0} - 1) I_{0} + 2}^{N_{0} I_{0} + 1} 1 / c_{i} \\ C_{N_{0} + 2} = c_{74} \\ C_{N_{0} + 3} = 1 / 1 \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} 1 / c_{i} \sum_{i = 75}^{(N_{0} + 1) I_{0} + 2} 1 / c_{i} \\ ⋮ \\ C_{j} = 1 / 1 \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} 1 / c_{i} \sum_{i = (j - N_{0} - 3) I_{0} + 75}^{(j - 2) I_{0} + 2} 1 / c_{i} \\ ⋮ \\ C_{2 N_{0} + 1} = 1 / 1 \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} 1 / c_{i} \sum_{i = (N_{0} - 2) I_{0} + 75}^{(2 N_{0} - 1) I_{0} + 2} 1 / c_{i} \\ C_{2 N_{0} + 2} = 1 / 1 \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} 1 / c_{i} \sum_{i = (N_{0} - 1) I_{0} + 75}^{2 N_{0} I_{0} + 2} 1 / c_{i} \end{matrix}

Also, the parameters used are defined as

N_{0} = \frac{(n - 2)}{2 I_{0}}

(26)

T_{0} = \frac{(n - 2)}{I_{0} + 2}

(27)

As described in aforementioned method, the dimension of the new reconstructed observer is variable and determined by parameter I₀. For this study, we take q available state variables as the inputs of the observer by equal interval sampling, which is corresponding to contact forces of q spatial contact points measured by tension sensors; besides, the transformation matrix H is performed to define the minimum value of parameter q. In this section, the estimated tension vector of $T_{0}$ reconstructed spatial contact points is defined as $\tilde{f} = [\begin{matrix} {\tilde{f}}_{1} \begin{matrix} ; \end{matrix} & {\tilde{f}}_{2} \begin{matrix} ; \end{matrix} & \cdot \cdot \cdot & \begin{matrix} {\tilde{f}}_{T_{0} - 1} \begin{matrix} ; \end{matrix} & {\tilde{f}}_{T_{0}} \end{matrix} \end{matrix}]$ . For the purpose of tension estimation, we can finally estimate contact forces of all the n spatial contact points of the actual RCTS as shown in Figure 11, through contact forces measured by q tension sensors, and the tension vector is defined as $\bar{F} = [\begin{matrix} {\bar{F}}_{1} & ; {\bar{F}}_{2}; & \cdot \cdot \cdot & \begin{matrix} {\bar{F}}_{n - 1}; & {\bar{F}}_{n} \end{matrix} \end{matrix}]$ , that is, $\bar{F} = (\begin{matrix} {\bar{F}}_{1} & \begin{matrix} ; \end{matrix} {\bar{F}}_{2} \begin{matrix} ; \end{matrix} & \begin{matrix} \cdot \cdot \cdot & {\bar{F}}_{1 + N_{0} I_{0}}; \end{matrix} {\bar{F}}_{74} \begin{matrix} ; \end{matrix} & \begin{matrix} \begin{matrix} \cdot \cdot \cdot & {\bar{F}}_{73 + N_{0} I_{0}} \end{matrix}; & {\bar{F}}_{74 + N_{0} I_{0}} \end{matrix} \end{matrix}]$ . To illustrate the algorithm in a more detailed way, equations (28) and (29) are, respectively, employed to calculate the contact forces of the upper and lower chain subsystem

{\bar{F}}_{I 1} = {\begin{matrix} {\bar{F}}_{1} = {\tilde{f}}_{1} \\ {\bar{F}}_{2} = χ_{1} \\ {\bar{F}}_{3} = χ_{1} + 1 \cdot σ_{1} \\ ⋮ \\ {\bar{F}}_{1 + I_{0}} = χ_{1} + (I_{0} - 1) \cdot σ_{1} \\ {\bar{F}}_{2 + I_{0}} = χ_{2} \\ {\bar{F}}_{3 + I_{0}} = χ_{2} + 1 \cdot σ_{2} \\ ⋮ \\ {\bar{F}}_{1 + 2 I_{0}} = χ_{2} + (I_{0} - 1) \cdot σ_{2} \\ ⋮ \\ {\bar{F}}_{2 + (N_{0} - 1) I_{0}} = χ_{N_{0}} \\ {\bar{F}}_{3 + (N_{0} - 1) I_{0}} = χ_{N_{0}} + 1 \cdot σ_{N_{0}} \\ ⋮ \\ {\bar{F}}_{1 + N_{0} I_{0}} = χ_{N_{0}} + (I_{0} - 1) \cdot σ_{N_{0}} \end{matrix}

(28)

{\bar{F}}_{I 2} = {\begin{matrix} {\bar{F}}_{74} = {\tilde{f}}_{N_{0} + 2} \\ {\bar{F}}_{75} = χ_{N_{0} + 1} \\ {\bar{F}}_{76} = χ_{N_{0} + 1} + 1 \cdot σ_{N_{0} + 1} \\ ⋮ \\ {\bar{F}}_{74 + I_{0}} = χ_{N_{0} + 1} + (I_{0} - 1) \cdot σ_{N_{0} + 1} \\ {\bar{F}}_{75 + I_{0}} = χ_{N_{0} + 2} \\ {\bar{F}}_{76 + I_{0}} = χ_{N_{0} + 2} + 1 \cdot σ_{N_{0} + 2} \\ ⋮ \\ {\bar{F}}_{74 + 2 I_{0}} = χ_{N_{0} + 2} + (I_{0} - 1) \cdot σ_{N_{0} + 2} \\ ⋮ \\ {\bar{F}}_{75 + (N_{0} - 1) I_{0}} = χ_{2 N_{0}} \\ {\bar{F}}_{76 + (N_{0} - 1) I_{0}} = χ_{2 N_{0}} + 1 \cdot σ_{2 N_{0}} \\ ⋮ \\ {\bar{F}}_{74 + N_{0} I_{0}} = χ_{2 N_{0}} + (I_{0} - 1) \cdot σ_{2 N_{0}} \end{matrix}

(29)

where $χ$ and $σ$ , are respectively, described as

\begin{array}{l} χ = {\begin{array}{l} χ_{1} = ({\tilde{f}}_{1} + {\tilde{f}}_{2}) / 2 \\ \begin{array}{l} χ_{2} = ({\tilde{f}}_{2} + {\tilde{f}}_{3}) / 2 \\ ⋮ \end{array} \\ χ_{N_{0} - 1} = ({\tilde{f}}_{N_{0} - 1} + {\tilde{f}}_{N_{0}}) / 2 \\ χ_{N_{0}} = ({\tilde{f}}_{N_{0}} + {\tilde{f}}_{N_{0} + 1}) / 2 \\ χ_{N_{0} + 1} = ({\tilde{f}}_{N_{0} + 2} + {\tilde{f}}_{N_{0} + 3}) / 2 \\ \begin{array}{l} χ_{N_{0} + 2} = ({\tilde{f}}_{N_{0} + 3} + {\tilde{f}}_{N_{0} + 4}) / 2 \\ ⋮ \end{array} \\ χ_{2 N_{0} - 1} = ({\tilde{f}}_{2 N_{0}} + {\tilde{f}}_{2 N_{0} + 1}) / 2 \\ χ_{2 N_{0}} = ({\tilde{f}}_{2 N_{0} + 1} + {\tilde{f}}_{2 N_{0} + 2}) / 2 \end{array} \\ σ = {\begin{array}{l} σ_{1} = ({\tilde{f}}_{2} + {\tilde{f}}_{1}) / (I_{0} - 1) \\ \begin{array}{l} σ_{2} = ({\tilde{f}}_{3} + {\tilde{f}}_{2}) / (I_{0} - 1) \\ ⋮ \end{array} \\ σ_{N_{0} - 1} = ({\tilde{f}}_{N_{0}} + {\tilde{f}}_{N_{0} - 1}) / (I_{0} - 1) \\ σ_{N_{0}} = ({\tilde{f}}_{N_{0} + 1} + {\tilde{f}}_{N_{0}}) / (I_{0} - 1) \\ σ_{N_{0} + 1} = ({\tilde{f}}_{N_{0} + 3} + {\tilde{f}}_{N_{0} + 2}) / (I_{0} - 1) \\ \begin{array}{l} σ_{N_{0} + 2} = ({\tilde{f}}_{N_{0} + 4} + {\tilde{f}}_{N_{0} + 3}) / (I_{0} - 1) \\ ⋮ \end{array} \\ σ_{2 N_{0} - 1} = ({\tilde{f}}_{2 N_{0} + 1} + {\tilde{f}}_{2 N_{0}}) / (I_{0} - 1) \\ σ_{2 N_{0}} = ({\tilde{f}}_{2 N_{0} + 2} + {\tilde{f}}_{2 N_{0} + 1}) / (I_{0} - 1) \end{array} \end{array}

Simulation results and discussion

Through a proper set of initial condition for the normal starting up of the RCTS, the longitudinal tension distribution rules are discussed during stage of constant speed ( $1 \leq t \leq 3$ ) as described in Figure 12. Considering the variation of position for each contact point $i (1 \leq i \leq 146)$ , the estimation results at different levels of $I_{0}$ are presented in Figure 13.

Figure 13.

Tension distribution estimation results: (a) I₀= 2, (b) I₀ = 3, (c) I₀ = 4, (d) I₀ = 6, (e) I₀ = 8, and (f) I₀ = 9.

As shown in Figure 13(a)–(f), tension differences are presented as the 3D curved surfaces. Meanwhile, the estimation result reveals that tension of the upper chain and lower chain subsystem exhibits an overall increase in the ascending order of contact points $i : (1 \leq i \leq 73) and 74 \leq i \leq 146$ . Moreover, for each $I_{0}$ , the tension of spatial contact points during the whole operating process presents a consistent and nonlinear trend at different times. It is clearly seen that the maximum and minimum values occur, respectively, at point 1 and point 73, point 74 and point 146 for the upper chain and lower chain subsystem, respectively, which indicates that all the contact pairs move toward the higher-tension positions. Besides, contact points adjacent to the head and tail of RCTS exhibit higher stress concentration. To conclude, the tension distribution rules of the system are obviously influenced by where contact points are located.

To validate the consistency of estimation performance, comparisons between actual tension $f$ derived from the virtual prototype model and estimation tension $\bar{F}$ obtained from the observer are given and discussed, where $f = [\begin{matrix} f_{1} \begin{matrix} ; \end{matrix} & f_{2} \begin{matrix} ; \end{matrix} & \cdot \cdot \cdot & \begin{matrix} f_{n - 1}; & f_{n} \end{matrix} \end{matrix}]$ and $F = [\begin{matrix} F_{1} \begin{matrix} ; \end{matrix} & F_{2} \begin{matrix} ; \end{matrix} & \cdot \cdot \cdot & \begin{matrix} F_{n - 1} \begin{matrix} ; \end{matrix} & F_{n} \end{matrix} \end{matrix}]$ . In addition, F is defined as the theoretical tension of the RCTS based on the mathematical model. In practical applications, the proposed estimation method is capable of obtaining the unknown contact forces of all contact points of the system at any time. The stage of constant speed ( $t = 2.5 s$ ) is selected as the research object to illustrate estimation accuracy corresponding to the simulated trajectories displayed in Figure 14.

Figure 14.

Tension variation trajectories: (a) I₀ = 2, (b) I₀ = 3, (c) I₀ = 4, (d) I₀ = 6, (e) I₀ = 8, and (f) I₀ = 9.

It could be clearly seen from the simulation results that estimation errors of the tension occur at different levels of $I_{0}$ and the root mean squared (RMS) of estimation errors can provide practical solutions to quantify the deviations. For quantitative evaluation, the RMS errors are calculated, respectively, between values of estimation and actual tension, theoretical and actual tension, the estimation and theoretical tension, that is, $ζ_{F}$ , $δ_{F}$ , and $ξ_{F}$

RMS ζ_{F} = \frac{\sqrt{(1 / n) {\sum_{i = 1}^{n} ({\bar{F}}_{i} - f_{i})}^{2}}}{\sqrt{(1 / n) \sum_{i = 1}^{n} f_{i}^{2}}}

(30)

RMS δ_{F} = \frac{\sqrt{(1 / n) {\sum_{i = 1}^{n} (F_{i} - f_{i})}^{2}}}{\sqrt{(1 / n) \sum_{i = 1}^{n} f_{i}^{2}}}

(31)

RMS ξ_{F} = \frac{\sqrt{(1 / n) {\sum_{i = 1}^{n} ({\bar{F}}_{i} - F_{i})}^{2}}}{\sqrt{(1 / n) \sum_{i = 1}^{n} F_{i}^{2}}}

(32)

The RMS values for different levels of $I_{0}$ are explained as shown in Table 1, where evaluation results corresponding to Figure 14(a)–(f) are shown, and the approximation errors are compared in the following.

Table 1.

RMS values.

I ₀	q	M₀	ξ_F	ζ_F	δ_F
2	15	148	0.0146	0.0593	0.0558
3	10	100	0.0232	0.0617
4	7	76	0.0333	0.0674
6	5	52	0.0534	0.0785
8	4	40	0.0749	0.0955
9	3	36	0.0856	0.1040

RMS: root mean squared.

We can confirm that the approximation value of $δ_{F}$ is 0.0558, which provides a reliable basis to verify the correctness of the virtual prototype model. As shown in Figure 15, it is obtained that $ξ_{F}$ shows low values and as I₀ increases from 2 to 9, the approximation errors are prone to higher levels, which is the same for each time step of the simulation process. When the value of I₀ is 9, q reaches 0.1040, correspondingly, the best set of q is 3, which can completely satisfy the engineering demand by effectively reducing the number of the tension sensors used in actual measurement system and shows practical potentials for computational efficiency. The proposed estimation method reveals reliable estimating ability and provides a useful tool to monitor the tension of the RCTS.

Figure 15.

Histogram of calculation results.

Conclusion

This article investigates the tension distribution rules of the RCTS, which can provide reference for transient dynamic analysis to monitor tension variation of the ring chains. Due to the complexity of the system, a virtual prototyping model has been presented based on multi-body dynamics theory and polygonal contact mode, for determination of matching the actual system and investigating dynamic contact properties under uniformly distributed loads during the practical transmission process. To this end, the system is discretized into n elements reflected by Kelvin models and expressed as a mathematical model. For research purpose, the demonstrated tension distribution estimation method is designed based on linear observer and the state-space formulation.

For the high requirements of computational efficiency of the high-dimensional reconstructed observer, we raise the novel dimension-minimized algorithm to deal with the numerical simulations based on the choice of I₀. More precisely, the smaller the set of I₀, the higher the order of the observer M₀, and the minimum q measurable states are applied to estimate contact forces of the contact points in the spatial tension distribution model. Subsequently, comparisons are made through RMS errors within a reasonable range, which have validated the correctness of the virtual prototyping model and satisfactory estimation results of the proposed tension distribution estimation method.

The tension variation reflects the working state of the actual RCTS. For this work, we first combine dynamic modeling and state estimation method to predict tension of the ring chains, which indicates excellent estimation ability. There still remains extensive research space for our future works, as the scraper conveyor works in odious conditions. In addition, we further take the speed difference between driving sprockets and the non-uniform distributed load between the ring chains into consideration in our later research. The ongoing project aims to adapt the estimation method to variable working conditions with respect to abnormal loads and rotational speeds, which serves as a theoretical basis for real-time tension monitoring and fault detecting of the RCTS, and allows for system maintenance and improvement of coal safety and high-efficient production in the future work.

Footnotes

Academic Editor: Yangmin Li

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the Key Project of National Natural Science Foundation of China (U1510205), the Natural Science Foundation of Jiangsu Province (No. BK 20160251), Xuzhou Research program (KC14H0138), the Fundamental Research Funds for the Central Universities (2014Y05), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

References

Childs

PRN

. Chapter 12: belt and chain drives. In: Childs

PRN

(ed.) Mechanical design engineering handbook. Amsterdam: Elsevier, 2014, pp.459–512.

Smith

Mobley

RK.

Chapter 8: chain drives. In: Smith

Mobley

(eds) Rules of thumb for maintenance and reliability engineers. Amsterdam: Elsevier, 2008, pp.123–127.

Meng

Cheng

XH.

Analysis on current status and development of scraper conveyer. Coal Eng 2014; 46: 58–60 (in Chinese).

Pfeiffer

Wolfsteiner

Mechanik

LBF

. Relative kinematics of multibody contacts. In: Proceedings of the international mechanical engineering congress & exposition, Nashville, TN, 14–19 November 1999, pp.16–21.

Song

Kraus

Kumar

et al . Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J Appl Mech: T ASME 2001; 68: 118–128.

Hippmann

. An algorithm for compliant contact between complexly shaped surfaces in multibody dynamics. Multibody Dynam 2003.

Rodriguez

. Dynamic simulation of multibody systems in simultaneous, indeterminate contact and impact with friction. Dissertations and Theses—Gradworks, 2014, https://uta-ir.tdl.org/uta-ir/handle/10106/24399

Zhang

Meng

. Dynamic modeling of scraper conveyor sprocket transmission system and simulation analysis. In: Proceedings of the 2011 international conference on mechatronics and automation (ICMA), Beijing, China, 7–10 August 2011, vol. 103, pp.1390–1394. New York: IEEE.

Saghafi

Farshidianfar

. An analytical study of controlling chaotic dynamics in a spur gear system. Mech Mach Theory 2016; 96: 179–191.

10.

Pereira

Ambrósio

Ramalho

. Dynamics of chain drives using a generalized revolute clearance joint formulation. Mech Mach Theory 2015; 92: 64–85.

11.

Zhang

Zhai

Han

et al . Dynamics of a geared parallel-rotor system subjected to changing oil-bearing stiffness due to external loads. Finite Elem Anal Des 2015; 106: 32–40.

12.

Fuglede

Thomsen

. Kinematic and dynamic modeling and approximate analysis of a roller chain drive. J Sound Vib 2016; 366: 447–470.

13.

Liu

Yang

et al . Simulation on the dynamics of intermittent roller chain drive based on virtual prototyping technology. China Mech Eng 2009; 20: 15–19 (in Chinese).

14.

G-M

Mao

. Research of tension analyzing and calculating of chains for scraper conveyor. Coal Mine Mach 2007; 7: 1–2 (in Chinese).

15.

Wang

Zhang

. Dynamic simulation analysis of the ring chain transmission system of the armoured face conveyor. J Tianjin Univ Sci Technol 2013; 28: 60–64 (in Chinese).

16.

Baiyan

. Dynamic behavior analysis on the ring chain transmission system of an armoured face conveyor. J Mech Eng 2012; 48: 50–56 (in Chinese).

17.

Ali

Hoang

Hussain

et al . Hybrid observer for parameters estimation in ethylene polymerization reactor: a simulation study. Appl Soft Comput 2016; 49: 687–698.

18.

Roux

JDL

Steinboeck

Kugi

et al . An EKF observer to estimate semi-autogenous grinding mill hold-ups. J Process Contr 2017; 51: 27–41.

19.

Chen

. A state-space-based prognostics model for lithium-ion battery degradation. Reliab Eng Syst Safe 2017; 159: 47–57.

20.

Liang

Huang

Chen

et al . Force estimation and failure detection based on disturbance observer for an ear surgical device. ISA T 2017; 66: 476–484.

21.

Beyhan

. Runge–Kutta model-based nonlinear observer for synchronization and control of chaotic systems. ISA T 2013; 52: 501–509.

22.

Ding

. Combined state and least squares parameter estimation algorithms for dynamic systems. Appl Math Model 2014; 38: 403–412.

Tension monitoring for the ring chain transmission system using an observer-based tension distribution estimation method

Abstract

Keywords

Introduction

Virtual prototype modeling of RCTS

Basic theory of contact pairs

Dynamic modeling process

Results analysis

Tension distribution estimation method

Discrete modeling process

Numerical solutions

State-space equation

Solutions of differential equation

Tension distribution method

Estimation process and simulation results

Construction of the spatial tension distribution model

Novel dimension-minimized algorithm

Simulation results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References