**3. Simulations**

#### *3.1. Numerical Simulations Results*

In this paper, numerical simulations are carried out using Matlab, and the programs are available in supplementary materials online. In order to ensure the physical significance of the dynamic equation, the damping is significant. Generally, the damping matrix can be regarded as a linear combination of the mass matrix (Equation (20)) and the stiffness matrix (Equation (21)) in a mechanical system dynamics equation, and can be written as:

$$\mathbf{C} = 0.02\mathbf{M} + 0.02\mathbf{K}.\tag{26}$$

After inserting the damping matrix (Equation (26)) into the dynamic Equation (17), the system dynamic equation can be written as:

$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{F}.\tag{27}$$

In (27), **.x** is the velocity vector:

$$
\dot{\mathbf{x}} = \begin{bmatrix}
\dot{\mathbf{x}} \ \dot{\mathbf{y}} \ \dot{\mathbf{z}} \ \dot{\mathbf{y}} \ \dot{\boldsymbol{\theta}}
\end{bmatrix}^{\mathrm{T}}.\tag{28}
$$

This paper intends to use the SLK3661W double-deck linear mining vibrating screen as an exploration object, which has certain parameters, such as screen box mass (18,944 kg), spring stiffness of each unloading side (1,242,400 N/m), spring stiffness of each loading side (931,800 N/m), screen deck dimension (3.6 × 6.1 m), processing capacity (350–400 t/h), motor speed (1480 r/min), and electric power (55 kW). Numerical simulations were carried out, based on the Newmark-β algorithm, and the main parameters used in the simulation are shown in Table 1. Additionally, the coordinate of force action point was (−0.2, 0, 0) in the body frame. The total time of the simulation was tm, while the time step was dt.


**Table 1.** Simulation parameters table.

Under normal conditions, the system vibrations included *x*, *y*, and *θ*, while *z* = *ϕ* = *γ* = 0. The displacement curves of the mass center are shown in Figure 5.

**Figure 5.** Displacement curves of the mass center under normal conditions, including two translational displacements (*<sup>x</sup>*, *y*) and one angular displacement (*θ*). Additionally, *z* = *ϕ* = *γ* = 0.

As shown in Figure 5, the displacements are large initially, then gradually decrease to a stable range. The stable state amplitudes (peak to peak values of displacement) are as follows:

> *x* = 7.82 mm, *y* = 7.80 mm, *θ* = 4.79 × 10−<sup>4</sup> rad.

Under spring failure conditions, the value of *<sup>k</sup>*1*y* was decreased to 652,260 N/m and the simulation was run again. The system vibrations include *x*, *y*, and *z*, as well as *γ*, *ϕ*, and *θ*. The displacement curves of the mass center are shown in Figure 6.

**Figure 6.** Displacement curves of the rigid body mass center under spring failure conditions, including three translational displacements (*<sup>x</sup>*, *y*, *z*) and three angular displacements (*γ*, *ϕ*, *θ*).

As shown in Figure 6, the displacements are large initially, then gradually decrease to a stable range. The stable state amplitudes (peak to peak values of displacement) are as follows:

*x* = 7.82 mm, *y* = 7.79 mm, *z* = 0.96 mm, *γ* = 0.2 × 10−<sup>4</sup> rad, *ϕ* = 0.75 × 10−<sup>4</sup> rad, *θ* = 9.40 × 10−<sup>4</sup> rad.

According to the analysis of the simulations above, the results showed that the four elastic supports of the whole system were symmetric on the *<sup>x</sup>*-*y* plane under normal conditions. The system vibrations included two translations and one rotation; namely, the rigid body only moved in the *<sup>x</sup>*–*y* plane. In addition, the system vibrations changed into a very complicated spatial motion with spring stiffness decrease, which included three translations and three rotations. Meanwhile, the amplitudes changed at the same time.

Therefore, the proposed six-degree-of-freedom model is feasible for exploring the mining vibrating screen dynamic characteristics with spring stiffness decrease caused by spring failures, and vice versa.

#### *3.2. Spring Failure Simulations Results*

Under normal conditions, the four elastic support points were symmetrical (point 1 = point 3, point 2 = point 4) in the proposed model. However, this symmetry broke under spring failure conditions, and hence six types of failure were selected for the simulation analysis, as shown in Table 2. Aimed at obtaining the influence rule of the spring failures, only the spring stiffness in the *y* direction was changed in the simulations.


**Table 2.** Types of spring failure.

Notes: 1 failure; 2 normal.

Due to the difference of each spring's stiffness and stiffness change, the stiffness variation coefficient (SVC) for normalization was defined as:

$$
\Delta k\_i = \frac{k\_{i\bar{j}0} - k\_{i\bar{j}}}{k\_{i\bar{j}0}} \times 100\%, \ (i = 1, 2, 3, 4; \ j = 1, 2, \dots, n). \tag{29}
$$

In (29), *i* is the elastic support point sequence number, *j* is the stiffness sequence number, *kij*0 is the normal spring stiffness in the *y* direction (as shown in Table 2), and *kij* is the various spring stiffness in the *y* direction.

Setting the value of *βz* as 89◦ in Table 2, the amplitudes of the four elastic support points in all directions were selected to be normal amplitudes. Due to the difference of each amplitude and amplitude change, the amplitude variation coefficient (AVC) for normalization was defined as:

$$
\Delta\lambda\_{id} = \frac{\lambda\_{id0} - \lambda\_{id}}{\lambda\_{id0}} \times 100\%, \ (i = 1, 2, 3, 4; d = x, y, z). \tag{30}
$$

In (30), *i* is the elastic support point sequence number, *d* is one of the three directions, *λid*0 is the normal amplitude of one elastic support point, and *λid* is the various amplitudes of the same elastic support point.

#### 3.2.1. Single Spring Failure Simulations Results

In the case of *k*1 failures, the spring stiffness variation coefficient (Δ*k*1) was changed from 0 to 30%, and hence the amplitude variation coefficients of the four elastic support points in all directions changed together.

As shown in Figure 7, if the spring stiffness variation coefficient (Δ*k*1) increased, the amplitude variation coefficients of all elastic support points in the *x* direction decreased, while all amplitude variation coefficients in the *z* direction increased. In the *y* direction, the amplitude variation coefficients of points 2 and 4 increased, while the amplitude variation coefficients of points 1 and 3 decreased.

**Figure 7.** The amplitude variation coefficient curves of four elastic support points, including the amplitude variation coefficients in the *x*, *y,* and *z* directions.

#### 3.2.2. Double Spring Failures Simulations Results

In the case of *k*1 and *k*2 failures, the spring stiffness variation coefficient (Δ*k*1 and Δ*k*2) was changed from 0% to 30%, and hence the amplitude variation coefficients of four elastic support points in all directions changed together.

As shown in Figure 8, if the spring stiffness variation coefficient increased, the amplitude variation coefficients of the four elastic support points in the *x* direction decreased, increased, or stayed the same (i.e., indeterminate) under the coupling action of *k*1 and *k*2 failures. The amplitude variation coefficients of all elastic support points in the *x* direction decreased, increased, or stayed the same (i.e., indeterminate) under the coupling action of *k*1 and *k*2 failures as well. Meanwhile, the amplitudes of variation coefficients in the *z* direction always increased, as well as Δ*λ*1*z* = Δ*λ*3*z* and Δ*λ*2*z* = Δ*λ*4*z*.

**Figure 8.** The amplitude variation coefficient surfaces of four elastic support points, including the amplitude variation coefficients in the *x*, *y,* and *z* directions.
