Simulation of the lateral oscillation of rope-guided conveyance based on fluid-structure interaction

2.1. Equations of lateral motion for the conveyance

Generally, the conveyances are made up of structure steel, and the deformation displacement caused by aerodynamic force is much smaller than the lateral motion displacement. So the conveyances are treated as rigid bodies travelling on the guide ropes. The lateral loads on the conveyances vary with the longitudinal velocity and the vertical position.

Equations governing the behaviour of a conveyance travelling on rope guides have been derived previously by a number of authors [1, 2, 11]. However, FSI method hasn’t been reported to simulate the lateral motion of the conveyance.

According to measurements [7, 8], the rotation of the conveyance about the horizon axis is very small, so this degree of freedom is omitted in this study, which is consistent with the reference 2 and reference 3. The equation of lateral motion for the conveyance can be represented as follows:

where $m$ is the mass of conveyance, $x$ is the displacement, $k$ is the lateral equivalent spring stiffness, and $F_{\mathrm{\Sigma }}$ is the disturbing forces which mainly consist of lateral aerodynamic forces and Coriolis force [2, 3].

The lateral equivalent spring stiffness of rope-guided conveyance has nonlinear forms [1, 4, 13] and a linear form [3]. The linear form can be represented as follows [3]:

where $n_R$ is the number of rope guides guiding a single conveyance, $T_L$ is the rope guide tension at conveyance elevation in the shaft, $L$ is the overall length of the rope guides, $L_1$ is the rope guide length between the top of conveyance and the top anchor point, $L_2$ is the rope guide length between the bottom of conveyance and the bottom anchor point, $n_H$ is the number of hoisting ropes for a single conveyance, $T_L$ is the hoisting rope tension at conveyance, $n_T$ is the number of tail ropes for a single conveyance, and $T_T$ is the tail rope tension at conveyance.

2.2. Governing equations of air flow

With respect to dynamic meshes, the integral form of the conservation equation for a general scalar, $\varphi \text{,}$ on an arbitrary control volume, $V\text{,}$ whose boundary is moving can be written as follows [14]:

where $\rho$ is the fluid density, $\overrightarrow{u}$ is the flow velocity vector, ${\overrightarrow{u}}_g$ is the mesh velocity of the moving mesh, $\mathrm{Г}$ is the diffusion coefficient, and $S_{\varphi }$ is the source term of $\varphi$. Here, $\partial V$ is used to represent the boundary of the control volume, $V$.

To construct the mass conservation equation, the momentum equation and the energy equation, the relevant entries for $\varphi$ and their rates of change per unit volume as defined in Eq. (3) are given in Table 1.

2.3. Numeric simulation flowchart

The detailed flowchart of FSI simulation for the rope guide hoisting system is given in Fig. 2.

Fig. 2Flowchart of FSI simulation for the rope guide system

3.1. Main shaft parameters

Shaft plan profiles are provided in Fig. 3, and Table 2 gives the main shaft parameters. The Coriolis force always acts in a westerly direction for an ascending conveyance and in an easterly direction for a descending conveyance, so the Coriolis force couldn’t induce the conveyance to move north or south directly, and then the Coriolis force is neglected in this 2D simulation. Furthermore, it is assumed that the conveyances have no lateral displacement or lateral velocity when they are at the beginning of the simulation.

Fig. 3Skip plan profiles

3.2. User-defined function coupled with ANSYS FLUENT

The transport equations were solved by ANSYS FLUENT with PISO scheme, and user-defined function (UDF) was programmed by C language to define the longitudinal motion and solve the equations of lateral motion for the conveyances with the modified Euler method. The Reynolds number is about 3.4×10⁶, far larger than 2300, so the airflow in shaft is turbulent. The shear-stress transport (SST) $k$-$\omega$ model was adopted to simulate the turbulent airflow, and the detailed theory can be found in reference 15. The displacements of conveyance have been implemented using a dynamic mesh with the remeshing algorithm in ANSYS FLUENT. Fig. 4 illustrates the unstructured mesh and the total number of control volumes is around 4×10⁵. The time step size was set to 0.002 seconds in order to get enough time resolution for the dynamic analysis. The total simulation time is 14 seconds for the travelling conveyances, and a total number of 7000 time steps, with approximately 30 iterations for each time step, were necessary to resolve each case.

Fig. 4Mesh used for calculations

3.3. Results

Before and after two skips pass each other, the series of pressure contours are shown in Fig. 5. Here, $s_{xh}$ denotes the distance between the two centers of gravity of skips.

Fig. 5Pressure contours

a) $s_{xh}=$6.79 m

b) $s_{xh}=$4.85 m

c) $s_{xh}=$2.91 m

d) $s_{xh}=$0.97 m

e) $s_{xh}=$–0.97 m

f) $s_{xh}=$–2.91 m

g) $s_{xh}=$–4.85 m

h) $s_{xh}=$–6.79 m

Fig. 6 shows the lateral aerodynamic force, lateral acceleration, velocity and displacement of the southern full skip during travelling upward, and Fig. 7 shows the lateral aerodynamic force, lateral acceleration, velocity and displacement of the northern empty skip during travelling downward. Here, $y$ denotes the distance between the center of gravity of skip and the cross point.

Fig. 6Simulation results for the southern full skip during travelling upward

a) Lateral aerodynamic force

b) Lateral acceleration

c) Lateral velocity

d) Lateral displacement

Fig. 7Simulation results for the northern empty skip during travelling downward

a) Lateral aerodynamic force

b) Lateral acceleration

c) Lateral velocity

d) Lateral displacement

3.4. Discussion

As Fig. 5 shows, the pressure contours change quickly when two skips pass each other. When two skips are far away, the pressure at upwind surface of skips are larger than other surface. When two skips pass each other, the negative pressure zones grow at the sides of two skips.

As Fig. 6(a) and Fig. 7(a) exhibit, there is a considerable lateral aerodynamic buffeting force when two conveyances pass each other, and the maximum lateral aerodynamic force is about 960 N. As the air flow is turbulent, the lateral aerodynamic force is still oscillating after two conveyances pass each other, but the amplitude becomes much small, about 200 N. And the trend is in accordance with the Hurlin’s measurement on a scale model [10]. In this case, the maximum Coriolis force is about 21.5 N, so the lateral aerodynamic force is mach lager than the Coriolis force.

As Fig. 6 and Fig. 7 illustrate, the lateral aerodynamic buffeting force gives the conveyance a lateral impact acceleration, and then the conveyance begins to oscillate laterally. Moreover the tendency of motion for conveyance is in agreement with the Chen’s measurement on Yaoqiao vertical production shaft [6]. The full skip oscillates from about –34.2 mm to +38.5 mm during travelling upward, and the empty skip oscillates from about +45.1 mm to –65.5 mm during travelling downward. The maximum skip deflection near the wall is less than the opposite direction. The simulation compared with Chen’s measurement is given in Table 3. As this article aims to reveal the details how the lateral aerodynamic force acts on conveyance, some other factors, such as initial rope guide motion and external displacement [3], are ignored, and maximum skip deflection is less than Chen’s measurement [5].