Dynamic characteristic and reliability of the vibrating screen are important indicators of large vibrating screen. Considering the influence of coupling motion of each degree of freedom, the dynamic model with six degrees of freedom (6 DOFs) of the vibrating screen is established based on the Lagrange method, and modal parameters (natural frequencies and modes of vibration) of the rigid body are obtained. The finite element modal analysis and harmonic response analysis are carried out to analyze the elastic deformation of the structure. By using the parametric modeling method, beam position is defined as a variable, and an orthogonal experiment on design is performed. The BP neural network is used to model the relationship between beam position and maximal elastic deformation of the lateral plate. Further, the genetic algorithm is used to optimize the established neural network model, and the optimal design parameters are obtained.
The vibrating screen is one of the key equipment for coal processing, which is widely used in grading, desliming, sculpting, and dewatering of coal [
At present, ANNs have become a preferred alternative way to solve any of complex, highly coupled, and nonlinear problems. Rad et al. constructed an expert system used Bayesian regulation back-propagation neural network and vibration monitoring data for electric motor status diagnosis [
The article is structured as follows. In Section
ZS2560 is a linear vibrating screen widely used in coal separation. As shown in Figure
Schematic diagram of the ZS2560 linear vibrating screen.
At present, the simplified 2-degree or 3-degree freedom mass-spring vibration model is often used in dynamic analysis of the vibrating screen [
Sketch map of 6 degrees of freedom vibration model.
As shown in Figure Kinetic energy of the system can be expressed as Potential energy of the system can be expressed as
The Lagrange equation with 6 degrees of freedom is shown as follows:
Substituting Equations (
Accordingly, the mass matrix and stiffness matrix of the system are constructed as follows:
The vibration equation of the linear vibration screen is established based on the Lagrange method and it is shown that
Modal analysis is a common method for studying dynamic characteristics of mechanical structures and design optimization. Modal analysis of the vibrating screen can be used to obtain modal parameters, such as natural frequency and natural vibration mode, as well as to provide a reference for the revision and structure design optimization of the subsequent simulation calculation model [
Assuming that the excitation force is equal to zero, the free vibration equation of vibrating screen is defined by the following equation:
According to the vibration theory, we can assume that the solution of Equation (
Substituting Equation (
In order to obtain the generic natural frequencies of the dynamic system, the eigenvalues problem has to be solved by the following equation:
The above equation is the algebraic equation of the 6 power real coefficient of
In this paper, we use a large linear vibrating screen with the size of 2500 mm × 6000 mm as a research object. The kinetic parameters are shown in Table
The kinetic parameters of the ZS2560 vibrating screen.
Parameter | Numerical value |
---|---|
|
7290 kg |
|
11875 kg·m2 |
|
19632 kg·m2 |
|
27233 kg·m2 |
|
1.2 × 106 N/m,4 × 106 N/m, 1.2 × 106 N/m |
|
−0.543 m |
|
2.52 m rear front |
2.33 m rear back | |
|
1.54 m |
The calculated natural frequency and corresponding system vibration are shown in Table
Modal calculation results of the vibrating screen (1∼6 order).
Order | Natural frequency (Hz) | Natural modes of vibration |
---|---|---|
1 | 1.9947 | [0, 0, 1, 0.0834, 0.0157, 0] |
2 | 2.0346 | [−1, −0.002, 0, 0, 0, 0.0151] |
3 | 3.5803 | [0, 0, 0.0383, 0.0262, −1, 0] |
4 | 3.7300 | [0, 1, 0, 0, 0.0, 0] |
5 | 4.6042 | [0, 0, −0.1491, 1, 0.0152, 0] |
6 | 6.6498 | [0.0566, 0.0429, 0, 0, 0, 1] |
The first six vibration modes of the system are presented graphically in Figures
Translational vibration mode of 1–6 orders. 1:
Torsional vibration mode of 1–6 orders. 4:
To verify the results obtained by theoretical analysis and to optimize the structure dynamics, the finite element model of vibrating screen was set up using the finite element software ANSYS (Figure
Grid partition of vibrating screen and boundary condition setting.
Modal picture of vibrating screen (1∼6 orders). (a) 1st order. (b) 2nd order. (c) 3rd order (d) 4th order. (e) 5th order (f) 6th order.
The comparison of results obtained by theoretical analysis and finite element analysis for the first six order modes is shown in Table
Comparison of FEM results with theoretical values.
Order | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
FEM results (Hz) | 1.9774 | 2.0166 | 3.6141 | 3.7023 | 4. 8486 | 6.2717 |
Theoretical values (Hz) | 1.9947 | 2.0346 | 3.5803 | 3.7300 | 4.6042 | 6.6498 |
Error (%) | −0.87 | −0.88 | 0.94 | −0.74 | 5.31 | −5.69 |
In particular, the error of the torsional mode (3rd, 5th, and 6th orders) is greater than that of the translational mode (1st, 2nd, and 4th orders), which is due to the linearization simplification of torsion in the theoretical model. The motion trend of the sieve, as shown in Figure
For analysis of the elastic deformation of the vibration screen in the movement process, the 7th, 8th, 9th, and 10th order modes were extracted, which are deformable modes, and modal frequency lies on structural stiffness of the vibrating screen. In Figure
Modal picture of vibrating screen (7∼10 order). (a) 7 order. (b) 8 order. (c) 9 order. (d) 10 order.
Modal calculation results of vibrating screen (7∼10 order).
Order | Natural frequency |
---|---|
7 | 11.028 |
8 | 13.372 |
9 | 14.697 |
10 | 21.968 |
On the basis of modal analysis, the harmonic response analysis was carried out based on the modal superposition method, and the exciting force
The relationship between structure design variables of vibration system and its dynamic characteristics is highly nonlinear. For complex systems, the relation between these parameters and performance cannot be expressed explicitly by a linear function, and direct optimization of these parameters is costly to calculate. However, the artificial neural networks (ANNs) have a very strong nonlinear mapping ability [
BP neural network prediction model for structural elastic deformation. (a) Input and output parameters of BP neural network model. (b) BP neural network model structure.
The sigmoidal function (
The input parameters of the network are the horizontal and vertical installation position of beams which is shown in Figure
Schematic diagram of design variable for the vibrating screen.
Design of the neural network model requires ANNs training with a certain number of samples with a proper distribution to make the neural network learn and express the relation accurately. In order to meet the requirements of both test workload and design accuracy, the orthogonal test design method is used to select a few representative schemes from a large number of test schemes.
In design optimization, first the design variables were defined, and then positions of 6 strengthening beams in
Through the orthogonal design with 12 factors and 3 levels, 27 sets of parameter combinations were obtained. The finite element model was established, and the corresponding elastic deformation was calculated.
In order to more intuitively reveal the trend of the test results changing with the level of each factor, the range analysis method was used to analyze the results of the orthogonal test directly. The trend of the 12 factors in this experiment is shown in Figure
Trend diagram of relationship between beam positions and elastic deformation.
On the basis of the orthogonal test, the qualitative law of structural deformation was obtained through the range analysis method. In order to more accurately design the structure, neural network modeling and parameter optimization are required. According to the structure shown in Figure
Comparison between the training results and the simulation results.
Comparison between the testing results and the simulation results.
The GA simulates the Darwinian evolution genetic selection, i.e., the evolution process of survival of the fittest rules with the same group of chromosome, by a random search algorithm combining the information transformation mechanism, initialization parameter coding, and initial population, and then by using the crossover operation and mutation, natural selection operator, parallel iteration, and optimization solutions [
Optimization flow chart of the large vibrating screen.
In order to reduce the elastic deformation by searching the optimal beams position, the genetic algorithm is selected to optimize the BP network model established, and it is defined by
The upper and lower bounds of design variables (mm).
|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
800 | 1200 | 1722.4 | 1596 | 4020 | 1705 | 5107 | 1219 | 900 | 500 | 1100 | 120 |
|
1200 | 1600 | 2122.4 | 1996 | 4620 | 2005 | 5607 | 1619 | 1400 | 800 | 1500 | 240 |
Sensitivity analysis can determine the gradient relationship between the design variable and the objective function and reflect the contribution of the design variable to the change of the objective function.
The differential form sensitivity of the design variable is defined by the following equation:
Based on the BP neural network model established in Section
The sensitivity of design variables.
As shown in Figure
The genetic algorithm parameter settings are shown in Table
The genetic algorithm parameter settings.
GA parameters | Explain |
---|---|
Coding method | Binary coding |
Ordering method | Ranking compositor |
Selection method | Roulette |
Cross method | Random multipoint intersection |
Variation method | Disperse multipoint variation |
Population size | 200 |
Crossover probability | 0.8 |
Mutation probability | 0.1 |
Evolutionary algebra | 200 |
The iterative histories for objective function are shown in Figure
The iterative histories for objective function.
The optimal design variables obtained by the genetic algorithm are compared with the original design variables in Table
Optimized design variables.
Parameters | Original value (mm) | Optimal value (mm) | Round (mm) |
---|---|---|---|
|
1126 | 814.4550945 | 814.5 |
|
1400 | 1274.839705 | 1275 |
|
1922.4 | 1751.020621 | 1751 |
|
1696 | 1688.835332 | 1689 |
|
4120 | 4587.176421 | 4587 |
|
1805 | 1924.636191 | 1925 |
|
5307 | 5571.593556 | 5572 |
|
1419 | 1432.600318 | 1433 |
|
939.6 | 1319.516474 | 1320 |
|
570 | 786.8099837 | 787 |
|
1300 | 1355.805261 | 1356 |
|
170 | 219.2089448 | 219 |
Maximal elastic deformation | 5.68 | — | 5.283 |
From Table
A new structural solution for high loaded vibrating screens was proposed in this paper, and the following conclusions are obtained: The dynamic model of the vibrating screen with 6 degrees of freedom is established based on the Lagrange equation. By numerical calculation, the first six orders natural frequency and the corresponding vibration mode of rigid body as well as the distribution law of rigid body modes are obtained by numerical calculation. Among them, the 1st mode (1.9947 Hz), 2nd mode (2.0346 Hz), and 4th mode (3.5803 Hz) are mainly translational motion, and the 3rd mode (3.5803 Hz), 5th mode (4.6042 Hz), and 6th mode (6.6498 Hz) are mainly translational motion. By means of the vibration mode, the complex space motion of the vibrating screen is dynamically decoupled and the dynamic decoupling condition of the vibrating screen is given. The excitation frequency range that influences the steady state operation of the vibrating screen is analyzed, and the maximum error between the finite element model and the theoretical value is 5.69%, the feasibility and accuracy of the finite element model in describing the vibration characteristics of the system are verified. Based on the orthogonal test and finite element analysis, the BP neural network is used to model the relationship between design variables and elastic deformation of the vibrating screen. And the sensitivity of each variable to structural elastic deformation is analyzed; the elastic deformation is more sensitive to the Genetic algorithm is used for optimization. According to the obtained results, the maximum elastic deformation was reduced by 6.99% compared to the original design.
This optimization design is simple and flexible; it shortens the design period and applies to other similar products.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work has been supported in part by the National Natural Science Foundation of China (Project nos. 51775544 and U1508210).
The amplitude vector of the system in the case of free vibration
Threshold values for network connections
The excitation matrix of the system
The moments of inertia of the vibrating screen with respect to
The moments of inertia of the vibrating screen with respect to
The moments of inertia of the vibrating screen with respect to
The product of inertia in the
The product of inertia in the
The product of inertia in the
The stiffness of springs in the
The stiffness of springs in the
The stiffness of springs in the
The spatial position of the spring (m)
The mass of the vibrating screen (kg)
The product of eccentric mass diameter of the exciter (kg·m)
The number of input layer nodes
The number of hidden layer nodes
The number of output elements
The translational degrees of freedom in the
The rotation degrees of freedom in the
The rotation speed (rad/s)
The vibrating direction angle (rad)
The normal distance between the center of mass and the excitation force direction (m)
The modal frequency (Hz)
The network connection weight value
The implicit excitation function.